CM Liftings (Book Draft)

CM Liftings Ching-Li Chai Brian Conrad Frans Oort Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 E-mail address: [email protected] Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address: [email protected] Mathematisch Instituut, Budapestlaan 6, Postbus 80.010, Utrecht 3508 TA, The Netherlands E-mail address: [email protected]

1991 Mathematics Subject Classification. Primary 14K10, 14G35; Secondary 14L05, 14F30, 11F32 Key words and phrases. abelian variety, complex multiplication, CM type, isogeny, moduli, Newton polygon, p-divisible group, reflex field, slopes

This book is dedicated to

John Tate for what he taught us, and for his inspiration

v

Preface During the May 2006 Workshop on Abelian Varieties in Amsterdam, the three of us formulated two versions of a refined CM lifting question for abelian varieties over a finite field, abbreviated as (IN) and (I) in 1.7.5 at the end of Chapter 1. The abelian surface counter-example to (IN) in 2.3.1 was found at that time. To our surprise, these two questions were solved within three years and the same example, called the toy model in 4.1.2, plays a crucial role in both. This book is the story of our adventure guided by the CM lifting problem. Ching-Li Chai thanks Hsiao-Ling for her love and support during all these years. He also thanks Utrecht University for hospitality during many visits, including the May 2006 Spring School on Abelian Varieties which concluded with the workshop in Amsterdam; support from NSF grants DMS 0400482 and DMS 0901163 is gratefully acknowledged. Brian Conrad thanks the many participants in the “CM seminar” at the University of Michigan for their enthusiasm on the topic of complex multiplication, as well as Columbia University for its hospitality during a sabbatical visit, and gratefully acknowledges support by NSF grants DMS 0093542 and DMS 0917686. Frans Oort thanks the University of Pennsylvania for hospitality and stimulating environment during several visits. We are also grateful to Burcu Baran, Bas Edixhoven, Ben Moonen, James Parson, and Ren´e Schoof for insightful and memorable discussions.

vii

Contents v Preface

vii

Introduction

1

Chapter 1. Algebraic theory of complex multiplication 1.1. Introduction 1.2. Simplicity, isotypicity, and endomorphism algebras 1.3. Complex multiplication 1.4. CM types 1.5. Abelian varieties over finite fields 1.6. A theorem of Grothendieck and a construction of Serre 1.7. CM lifting questions

7 7 8 15 23 32 36 41

Chapter 2. CM lifting over a discrete valuation ring 2.1. Introduction 2.2. Existence of CM lifting up to isogeny 2.3. CM lifting to a normal domain up to isogeny: counterexamples 2.4. Algebraic Hecke characters 2.5. Theory of complex multiplication 2.6. Local methods

45 45 52 58 64 72 75

Chapter 3. CM lifting of p-divisible groups 3.1. Motivation and background 3.2. Earlier non-liftability results and a new proof 3.3. Finer properties of the a-number 3.4. Complex multiplication for p-divisible groups 3.5. An upper bound for a field of definition 3.6. Appendix: Crystalline representations of CM type

81 81 84 85 87 92 95

Chapter 4. CM lifting of abelian varieties up to isogeny 4.1. Introduction 4.2. Classification and Galois descent by Lie types 4.3. Self-duality and CM lifting 4.4. Complex conjugation and CM lifting

103 103 108 116 126

Appendix A. Some arithmetic results for abelian varieties A.1. The p-part of Tate’s work A.2. The Main Theorem of Complex Multiplication A.3. A converse to the Main Theorem of Complex Multiplication

137 137 145 174

Appendix B. CM lifting via p-adic Hodge theory B.1. A generalization of the toy model B.2. Construct CM lifting by p-adic Hodge theory B.3. Dieudonn´e theories over a perfect field of characteristic p B.4. p-adic Hodge theory and a formula for the closed fiber

179 179 189 199 211

ix

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CONTENTS

Bibliography

221

Notes

225

Glossary of Notations

227

I restricted myself to characeteristic zero: for a short time, the quantum jump to p 6= 0 was beyond the range . . . but it did not take me too long to make this jump. Oscar Zariski

Introduction The arithmetic of abelian varieties with complex multiplication over a number field is fascinating. However this will not be our focus. We study CM theory in mixed characteristics. Abelian varieties over finite fields. In 1940 Deuring showed that an elliptic curve over a finite field can have an endomorphism algebra of rank 4 [23, page 218]. For an elliptic curve in characteristic zero with an endomorphism algebra of rank 2 (rather than rank 1, as in the “generic” case), the j-invariant is called a singular j-invariant. For this reason elliptic curves with even more endomorphisms, in positive characteristic, are called supersingular. Of course, a supersingular elliptic curve isn’t singular. A purist perhaps would like to say “an elliptic curve with supersingular j-value”. However we will adopt the generally used terminology “supersingular elliptic curve” instead. Mumford observed as a consequence of results of Deuring that for any elliptic curves E1 and E2 over a finite field κ of characteristic p > 0 and any prime ` 6= p, the natural map Z` ⊗Z Hom(E1 , E2 )−→ HomZ` [Gal(κ/κ)] (T` (E1 ), T` (E2 )) is an isomorphism [74, page 134] (where on the left side we consider only homomorphisms “defined over κ”). The interested reader might find it an amusing exercise to reconstruct this (unpublished) proof by Mumford. Tate proved that the analogous result holds for abelian varieties over a finite field [74] and he also incorporated the case ` = p by using p-divisible groups. He generalized this result into his influential conjecture [73]:1 an `-adic cohomology class that is fixed under the Galois group should be a Q` -linear combination of fundamental classes of algebraic cycles when the ground field is finitely generated over its prime field. Honda and Tate gave a classification of isogeny classes of simple abelian varieties A over a finite field κ (see [33] and [77]), and Tate refined this by describing the structure of the endomorphism algebra End0 (A) (working in the isogeny category over κ) in terms of the Weil q-number of A, with q = #κ; see [77, Thm. 1]. It follows from Tate’s work (and was originally proved by Honda and Serre) that an abelian variety A over a finite field κ admits sufficiently many complex multiplications in the sense that its endomorphism algebra End0 (A) contains a commutative semi-simple subalgebra of rank 2 dim(A). We will call such an abelian variety (in any characteristic) a CM abelian variety. Grothendieck showed that over any algebraically closed field, an abelian variety that admits sufficiently many complex multiplications is isogenous to an abelian variety defined over a finite extension of the prime field [56]. This was previously known in characteristic zero (by Shimura and Taniyama), and in that case such an abelian variety can be defined over a number field. However in positive characteristic such abelian varieties can fail to be defined over a finite field; examples exist in every dimension > 1. Abelian varieties in mixed characteristics. In characteristic zero, an abelian variety A gives a representation of the endomorphism algebra D = End0 (A) on the Lie algebra Lie(A) of A. If A has complex multiplication by a CM field L of degree 2 dimA then the isomorphism class of the representation of L on Lie(A) is called the CM type of the pair (A, L ,→ End0 (A)). 1Here the prime number ` is assumed to be invertible in the base field. 1

2

INTRODUCTION

As we noted above, every abelian variety over a finite field is a CM abelian variety. Thus, it is natural to ask whether every abelian variety over a finite field can be “CM lifted” to characteristic zero. One of the obstacles2 in this question is that in characteristic zero there is the notion of CM type that is invariant under isogenies, whereas in positive characteristic whatever can be defined in an analogous way is not invariant under isogenies. For this reason we will use the terminology “CM type” only in characteristic zero. For instance, the action of the endomorphism ring R = End(A0 ) of an abelian variety A0 on the Lie algebra of A0 in positive characteristic gives a representation of R/pR on Lie(A0 ). An isogeny f : A0 → B0 may change the endomorphism ring, but even if End(A0 ) = End(B0 ) (under the natural identification between End0 (A0 ) and End0 (B0 )), the representations of this endomorphism ring on Lie(A0 ) and Lie(B0 ) may well be different since Lie(f ) may not be an isomorphism. Moreover, if we have a lifting A of A0 to some characteristic zero local domain, in general the inclusion End(A) ⊂ End(A0 ) is not an equality. If the inclusion End0 (A) ⊂ End0 (A0 ) is an equality then the character of the representation of End(A0 ) on Lie(A0 ) is the reduction of the character of the representation of End(A) on Lie(A). This relation can be viewed as an obstruction to the existence of CM lifting with the full ring of integers of a CM field operating on the lift; see the toy model in 4.1.2 for an illustration. In the case when End(A0 ) contains the ring of integers OL of a commutative semi-simple algebra L ⊂ End0 (A0 ) with [L : Q] = 2 dim(A0 ), the representation of OL /pOL on Lie(A0 ) turns out to be quite useful, despite the fact that it is not an isogeny invariant. Its class in a suitable K-group will be called the Lie type of (A0 , OL ,→ End(A0 )). The above discrepancy between the theories in characteristic zero and characteristic p > 0 is the basic phenomenon underlying this whole book. Before discussing its content, we recall the following theorem of Honda and Tate ([33, Thm. 1 on page 86] and [77, Thm. 2]): For an abelian variety A0 over a finite field κ there is a finite extension κ ⊂ κ0 and an isogeny (A0 )κ0 → B0 such that B0 admits a CM lifting to characteristic zero. Our starting point comes from the following questions. For an abelian variety A0 over a finite field κ, in order to ensure the existence of a CM lifting to a local domain in characteristic zero, (a) is a finite extension κ ⊂ κ0 necessary? (b) is an isogeny (A0 )κ0 → B0 necessary? These questions are formulated in various precise forms in 1.7. An isogeny is necessary. Question (b) was answered in 1992 (see [59]): there exist (many) abelian varieties over an algebraic closure F of the prime field Fp that do not admit any CM lifting to characteristic zero. The main point of [59] is that a CM liftable abelian variety over F can be defined over a small finite field. This idea is further pursued in Chapter 3, where the size of (all possible) fields of definition of the p-divisible group of a given abelian variety over F is turned into an obstruction for the existence of a CM lifting to characteristic 0. We also provide effectively computable examples of abelian varieties over explicit finite fields such that there is no CM lift to characteristic 0. A field extension might be necessary—depending on what you ask. It turns out there are two quite different answers to question (a). The subtle distinction between using normal or general local domains for the lifting went unnoticed for long time. Once this came in focus, various answers became available. If we ask for a CM lifting to a normal domain, in general a field extension is necessary. This is explained in 2.1.3, where we formulate the “residual reflex obstruction”, the idea for 2surely also part of the attraction

INTRODUCTION

3

which goes as follows. Over an algebraically closed field K of characteristic zero, we know that a simple CM abelian variety A with K-valued CM type Φ (for the action of a CM field L) is defined over a number field in K containing the reflex field E(Φ) of the type Φ. Suppose that for every K-valued CM type Φ of L, the residue field of E(Φ) at any prime above p is not contained in the finite field κ with which we began in question (a). In such cases a CM lifting of A0 to a normal local domain of characteristic zero (with residue field κ!) can fail to exist. In 2.3.1 we give such an example, provided by a supersingular abelian surface over a field Fp2 with p2 elements. A much broader class of examples is given in 2.3.5, consisting of absolutely simple abelian varieties (with arbitrarily large dimension) over the prime field Fp . Note that passing to the normalization of a complete local noetherian domain generally enlarges the residue field. Hence, if we drop the condition that the mixed characteristic local domain be normal then the preceding considerations do not give a conclusion. And in fact we were put on the right track by mathematics itself. The phenomenon is best illustrated in the toy model in 4.1.2, which is the same as the example in 2.3.1 already mentioned: an abelian surface over a field with p2 elements which, even up to isogeny, is not liftable to a normal local domain of characteristic zero. On the other hand this abelian surface is CM liftable, after an appropriate isogeny over the same base field Fp2 , to a mixed characteristic non-normal local domain of characteristic zero. This example is easy to construct, and the proof of the existence of a CM lifting, possibly after applying an Fp2 -rational isogeny, is not difficult either. In Chapter 4 we will show that the question of existence of a CM lifting after an appropriate isogeny can be reduced to the same question for this toy model, enabling us to prove: every abelian variety A0 over a finite field κ admits an isogeny A0 → B0 over κ such that B0 admits a CM lifting to a mixed characteristic local domain. There are refined lifting questions that can be asked, such as if we specify at the beginning which CM structure on A0 is to be lifted, or even what its CM type should be. These matters will also be addressed. Our basic method is to “localize” various CM lifting problems to the corresponding problems for p-divisible groups. Although global properties of abelian varieties are often lost in this localization process, the non-rigid nature of p-divisible groups can be an advantage. In Chapter 3 the size of fields of definition of a p-divisible group in characteristic p appears as an obstruction to the existence of CM lifting. The reduction steps in Chapter 4 rely on a classification and descent of CM p-divisible groups in characteristic p with the help of their Lie types. In addition, the “Serre tensor construction” is applied to p-divisible groups, both in characteristic p and in mixed characteristic (0, p); see 1.6.4 and 4.2.10 for this general construction. Survey of the contents of the four chapters. In Chapter 1 we start with a survey of general facts about CM abelian varieties and their endomorphism algebras. In particular, we review results in Honda-Tate theory that describe isogeny classes and endomorphism algebras of abelian varieties over a finite field in terms of Weil numbers. We conclude by formulating various CM lifting problems in 1.7. These are studied in the following chapters. We will see that the questions can be answered with some precision. In Chapter 2 we formulate and study the “residual reflex condition”. Using this condition we construct several examples of abelian varieties over a finite field that do not admit, even after an isogeny over the base field, any CM lifting to a normal local domain (of characteristic zero); see 2.3. It is remarkable that many such examples exist, but we do not know whether we have characterized all possible examples; see 2.3.7. We then study algebraic Hecke characters and review part of the theory of complex multiplication due to Shimura and Taniyama. Using the relationship between algebraic Hecke characters for a CM field L and CM abelian varieties with CM by L (the precise statement of which we review and prove), we use global methods to show that the residual reflex condition is the only obstruction to the existence of CM lifting

4

INTRODUCTION

up to isogeny over a normal local domain of characteristic zero. We also give another proof by local methods (e.g., p-adic Hodge theory). In Chapter 3 we take up methods described in [59]. In that paper classical CM theory in characteristic zero was used. Here we use p-divisible groups instead of abelian varieties and show that the size of fields of definition of a p-divisible group in characteristic p is a nontrivial obstruction to the existence of a CM lifting. We also show, in one case left open in [59, Question C], that an isogeny is necessary. Our methods also provide effectively computed examples. In Chapter 4 we show CM liftability after an isogeny over the finite field (with the lifting over a characteristic zero local domain that need not be normal). That is, an abelian variety A0 over a finite field κ has an isogeny over κ to an abelian variety B0 that admits a CM lift (see 4.1.1). The first key is the easy proof in 4.1.2 that the “toy model” satisfies this property. Then, after an easy reduction to the isotypic case, we apply a theorem of Serre and Tate to localize the problem at p-adic places v of the maximal totally real subfield L0 of a CM field L ⊆ End0 (A0 ) of degree 2 dim(A0 ). This reduces the existence of a CM lifting for the abelian variety A0 to a corresponding problem for the CM p-divisible group A0 [v ∞ ] attached to v. We formulate properties of the CM field L which ensure the existence of a CM lift of A0 [v ∞ ] after an isogeny (over κ) in case the p-adic place v of the totally real field L0 is not “bad” (see 4.1.5, 4.1.6, and 4.3.17). These properties of the ramification, residue field, and Galois group of L pin down precisely that a CM lift up to isogeny can be performed for primes that are not bad in L0 . In the non-bad case we can change the abelian variety by an isogeny (if necessary) so that it acquires a “self-dual” Lie type, and then we produce a CM lifting of the modified p-divisible group; see 4.3.9. If the p-adic place v of the totally real field L0 is “bad” (for the CM lifting problem) then this argument does not work because in that case no modification of the p-divisible group A0 [v ∞ ] by an isogeny (over κ) is self-dual. Instead we change A0 [v ∞ ] by a suitable isogeny so that its Lie type becomes “maximally symmetric”, which we call “striped”. Such a striped CM p-divisible group is too symmetric to be self-dual. 3 However it is isomorphic to the Serre tensor construction applied to the p-divisible group of the toy model; see 4.3.26. After this key step we deduce the existence of a CM lifting of (a modified) A0 [v ∞ ] from the corresponding statements for (the p-divisible group version of) the toy model. In the final stage we lift an abelian variety isogenous to A0 over κ to a characteristic zero local domain that need not be normal. Although OL acts on the closed fiber, we can only assure that a subring of OL of finite index acts on the lifted abelian scheme (see 4.4.7). In Appendix A.1 we provide a self-contained development of the proof of the p-part of Tate’s isogeny theorem for abelian varieties over finite fields of characteristic p, as well as a proof of Tate’s formula for the local invariants at p-adic places for endomorphism algebras of simple abelian varieties over such fields. (An exposition of these results is also given in [51]; our treatment uses less input from non-commutative algebra.) Appendices A.2 and A.3 provide purely algebraic proofs of the Main Theorem of Complex Multiplication for abelian varieties, as well as a converse result, both of which are used in essential ways in Chapter 2. In Appendices B.1 and B.2, we give two versions of a more direct (but more complicated) proof of the existence of CM liftings for a higher-dimensional generalization of the toy model. The first version uses Raynaud’s theory of group schemes of type (p, . . . , p). The second version uses recent developments in p-adic Hodge theory. We hope that material described there will be useful in the future.

3It may seem strange to say that self-duality would be obstructed by having too much symmetry. Is there a clearer way to say this?

INTRODUCTION

5

Notation and terminology. For a field K, we write K to denote an algebraic closure and Ks to denote a separable closure. An extension of fields K 0 /K is primary if K is separably algebraically closed in K 0 (i.e., the algebraic closure of K in K 0 is purely inseparable over K). For a number field L we write OL to denote its ring of integers. If q is a power of a prime, we write Fq to denote a finite field with size q. If T → S is a map of schemes and S 0 is an S-scheme, then TS 0 and T/S 0 denote the S 0 scheme T ×S S 0 if S is understood from context. When S = Spec(R) and S 0 = Spec(R0 ) are affine, we may respectively write TR0 or T/R0 instead when R is understood from context. The dual of an abelian variety A is denoted At . For any abelian varieties A and B over a field K, Hom(A, B) denotes the group of homomorphisms A → B over K, and Hom0 (A, B) denotes Q ⊗Z Hom(A, B). (Since Hom(A, B) → Hom(AK , BK ) is injective, with K/K an algebraic closure, Hom(A, B) is a finite free Z-module since the same holds over K by [53, §19, Thm. 3].) When B = A we write End(A) and End0 (A) respectively, and call End0 (A) the endomorphism algebra of A (over K). The endomorphism algebra End0 (A) is an invariant which only depends on A up to isogeny over K, in contrast with the endomorphism ring End(A). We write A ∼ B to denote that abelian varieties A and B over K are K-isogenous. To avoid any possible confusion with notation found in the literature, we emphasize that what we call Hom(A, B) and Hom0 (A, B) are sometimes denoted by others as HomK (A, B) and Hom0K (A, B) (with the notation Hom(A, B) and Hom0 (A, B) then reserved to mean the analogues for AK and BK over K, or equivalently for AKs and BKs over Ks ). We write AL to denote the adele ring of a number field L, AL,f to denote the factor ring of finite adeles, and A and Af in the case L = Q. The Artin maps of local and global class field theory are taken with the arithmetic normalization, which is to say that local uniformizers are carried to arithmetic Frobenius elements. (We make this choice so that uniformizers correspond to Frobenius endomorphisms in the Main Theorem of Complex Multiplication.) × ab The global reciprocity map for a number field L is denoted recL : A× L /L → Gal(L /L), and × × its composition with the projection AL  AL /L× is denoted rL ; for a local field F we write rF to denote the reciprocity map F × → Gal(F ab /F ). If v is a place of a number field L then Lv denotes the completion of L with respect to v; OL,v denotes the valuation ring OLv of Lv in case v is non-archimedean, Q with residue field κv whose size is denoted qv . For a place w of Q we define Lw := Qw ⊗Q L = v|w Lv , and in case Q w is the `-adic place for a prime ` we define OL,` := Z` ⊗Z OL = v|` OL,v . References. In Mumford’s book [53] the theory of abelian varieties is developed over an algebraically closed base field, and we need the theory over a general field; references addressing this extra generality are Milne’s article [48] (which rests on [53]) and the forthcoming book [29]. Since [29] is not yet in final form we do not refer to it in the main text, but the reader should keep in mind that many results for which we refer to [53] and [48] are also treated in [29]. We refer the reader to [54, Ch. 6] for a self-contained development of the elementary properties of abelian schemes, which we freely use in several places. In addition to familiarity with the basic theory of abelian varieties, we assume familiarity with the classical theory of finite-dimensional semisimple algebras over fields (including the theory of their splitting fields and maximal commutative subfields). A suitable reference for this material is [34, §4.1–4.6], and in 1.2.2–1.2.4 we review some of the facts that we need from that theory. To handle p-torsion phenomena in characteristic p > 0 we use Dieudonn´e theory and pdivisible groups. A brief survey of some basic definitions and properties in this direction is given in 1.4.5. We refer the reader to [75] and [69, §6] for a more systematic discussion of basic facts concerning p-divisible groups, and to [25, Ch. II–III] for a self-contained development of (contravariant) Dieudonn´e theory with applications to p-divisible groups.

The theory of complex multiplication . . . is not only the most beautiful part of mathematics but also of all science. David Hilbert

1

Algebraic theory of complex multiplication 1.1. Introduction

1.1.1. Lifting questions. A natural question early in the theory of abelian varieties is whether every abelian variety in positive characteristic admits a lift to characteristic 0. That is, given an abelian variety A0 over a field κ with char(κ) > 0, does there exist a a local domain R of characteristic zero with residue field κ and an abelian R-scheme A whose special fiber Aκ is isomorphic to A0 ? We may also wish to demand that a specified polarization of A0 or subring of the endomorphism algebra of A0 (or both) also lifts to A. (The functor A Aκ from abelian R-schemes to abelian varieties over κ is faithful, by consideration of finite ´etale torsion levels.) If there is an affirmative solution to such a lifting problem then considerations with direct limits permit one to find a solution for which R is a complete local noetherian domain. Thus, an affirmative solution to a lifting problem as above (for a given A0 ) is often equivalent to an appropriate deformation ring R for A0 admitting a generic point in characteristic 0 (the coordinate ring of the corresponding irreducible component of Spec(R) is such an R). If κ0 /κ is an extension field then there is generally a natural (faithfully flat) local map R → R 0 between the corresponding deformation rings for A0 and A0/κ0 , so if R 0 has a generic point of characteristic 0 then so does R. Hence, to prove an affirmative answer to lifting questions as above it is usually enough to consider the case of algebraically closed κ. For example, the general lifting problem for polarized abelian varieties (allowing polarizations for which the associated symmetric isogeny A0 → At0 is not separable) was solved affirmatively by Norman– Oort [55, Cor. 3.2] when κ is algebraically closed, and from this the general case follows via deformation theory. 1.1.2. Refinements. When a lifting problem as above has an affirmative solution, it is natural to ask if the (complete local noetherian) base ring R for the lifting can be chosen to satisfy nice ring-theoretic properties, such as being normal or a discrete valuation ring. Slicing methods allow one to find an R with dim(R) = 1 (see 2.1.1 for this argument), but normalization generally increases the residue field. Hence, asking that the complete local noetherian domain R be normal or a discrete valuation ring with a specified residue field κ is a non-trivial condition unless κ is algebraically closed. We are interested in versions of the lifting problem for finite κ when we lift not only the abelian variety but also a large commutative subring of its endomorphism algebra. To avoid counterexamples it is sometimes necessary to weaken the lifting problem by permitting the initial abelian variety A0 to be replaced with another in the same isogeny class over κ. In 1.7 we will precisely formulate several such lifting problems involving complex multiplication, and the main result of our work is a rather satisfactory solution to these lifting problems. 1.1.3. Purpose of this chapter. Much of the literature on complex multiplication involves either (i) working over an algebraically closed ground field, (ii) making unspecified finite extensions of the ground field, or (iii) restricting attention to simple abelian varieties. To avoid any uncertainty about the degree of generality in which various foundational results in the theory are valid, as well as to provide a convenient reference for subsequent considerations, 7

8

1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

in this chapter we provide an extensive review of the algebraic theory of complex multiplication over a general base field. This includes special features of the theory over finite fields and over fields of characteristic 0, and for some important proofs we refer to the original literature (e.g., papers of Tate). Some arithmetic aspects (such as reflex fields and the Main Theorem of Complex Multiplication) are discussed in Chapter 2, and Appendix A provides proofs of some results of Tate and of the Main Theorem of Complex Multiplication.

1.2. Simplicity, isotypicity, and endomorphism algebras 1.2.1. Simple abelian varieties. An abelian variety A over a field K is simple (over K) if it is non-zero and contains no non-zero proper abelian subvarieties. Simplicity is not generally preserved under extension of the base field; see Example 1.5.3 for some two-dimensional examples over finite fields and over Q. An abelian variety A over K is absolutely simple (over K) if AK is simple. 1.2.1.1. Lemma. If A is absolutely simple over a field K then for any field extension K 0 /K, the abelian variety AK 0 over K 0 is simple. Proof. This is an application of direct limit and specialization arguments, as we now explain. Assume the assertion fails with some K 0 /K, so by replacing K 0 with an algebraic closure we may arrange that K is algebraically closed. Choose a non-zero proper abelian subvariety B 0 ⊂ AK 0 . By expressing K 0 as a direct limit of finitely generated K-subalgebras, there is a finitely generated K-subalgebra R ⊂ K 0 and an abelian scheme B → Spec(R) that is a closed R-subgroup of AR which descends B 0 ⊂ AK 0 , so the constant positive dimension of the fibers of B → Spec(R) is strictly less than dim(A). Since K is algebraically closed we can choose a K-point x of Spec(R), and the fiber Bx is a non-zero proper abelian subvariety of A, contrary to the simplicity of A over K. For a pair of abelian varieties A and B over a field K, Hom0 (AK 0 , BK 0 ) can be strictly larger than Hom0 (A, B) for some separable algebraic extension K 0 /K. For example, if E is an elliptic curve over Q then considerations with the tangent line over Q force End0 (E) = Q, but it can happen that End0 (EL ) = L for an imaginary quadratic field L (e.g., E : y 2 = x3 − x √ and L = Q( −1)). Scalar extension from number fields to C or from an imperfect field to its perfect closure are useful techniques in the study of abelian varieties, so there is natural interest in considering ground field extensions that are not separable algebraic (i.e., non-algebraic or inseparable). It is an important fact that allowing such general extensions of the base field does not lead to more homomorphisms: 1.2.1.2. Lemma (Chow). Let K 0 /K be an extension of fields that is primary (i.e., K is separably algebraically closed in K 0 ). For abelian varieties A and B over K, the natural map Hom(A, B) → Hom(AK 0 , BK 0 ) is bijective. Proof. See [15, Thm. 3.19] for a proof using faithfully flat descent (which is reviewed at the beginning of [15, §3]). We shall be interested in certain commutative rings acting faithfully on abelian varieties, so we need non-trivial information about the structure of endomorphism algebras of abelian varieties. The study of such rings rests on the following fundamental result. 1.2.1.3. Theorem (Poincar´e reducibility). Let A be an abelian variety over a field K. For any abelian subvariety B ⊂ A, there is is abelian subvariety B 0 ⊂ A such that the multiplication map B × B 0 → A is an isogeny. In particular, if A 6= 0 then there exist pairwise non-isogenous simple abelian varieties Q C1 , . . . , Cs over K such that A is isogenous to Ciei for some ei > 1.

1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS

9

Proof. When K is algebraically closed this result is proved in [53, §19, Thm. 1]. The same method works for perfect K, as explained in [48, Prop. 12.1]. (Perfectness is implicit in the property that the underlying reduced scheme of a finite type K-group is a K-subgroup. For a counterexample over any imperfect field, see [17, Ex. A.3.8].) The general case can be pulled down from the perfect closure via Lemma 1.2.1.2; see the proof of [15, Cor. 3.20] for the argument. 1.2.1.4. Corollary. For a non-zero abelian variety A over a field K and a primary extension of fields K 0 /K, every abelian subvariety B 0 of AK 0 has the form BK 0 for a unique abelian subvariety B ⊂ A. Proof. By the Poincar´e reducibility theorem, abelian subvarieties of A are precisely the images of maps A → A, and similarly for AK 0 . Since scalar extension commutes with the formation of images, the assertion is reduced to the bijectivity of End(A) → End(AK 0 ), which follows from Lemma 1.2.1.2. Since any map between simple abelian varieties over K is either 0 or an isogeny, by general categorical arguments the collection of Ci ’s (up to isogeny) in the Poincar´e reducibility theorem is unique up to rearrangement, and the multiplicities ei are also uniquely determined. 1.2.1.5. Definition. The Ci ’s in the Poincar´e reducibility theorem (considered up to isogeny) are the simple factors of A. By the uniqueness of the simple factors up to isogeny, we deduce: 1.2.1.6. Corollary. Let A be a non-zero abelian variety over a field, with simple factors C1 , . . . , Cs . The non-zero abelian subvarieties of A are generated by the images of maps Ci → A from the simple factors. 1.2.2. Central simple algebras. Using notation from the Poincar´e reducibility theorem, for a non-zero abelian variety A we have Y End0 (A) ' Matei (End0 (Ci )) where {Ci } is the set of simple factors of A and the ei ’s are the corresponding multiplicities. Each End0 (Ci ) is a division algebra, by simplicity of the Ci ’s. Thus, to understand the structure of endomorphism algebras of abelian varieties we need to understand matrix algebras over division algebras, especially those of finite dimension over Q. We therefore next review some general facts about such rings. 1.2.2.1. Definition. A central simple algebra over a field K is an associative K-algebra of finite dimension such that K is the center and the underlying ring is simple (i.e., has no nontrivial two-sided ideals). A central division algebra over K is a central simple algebra over K whose underlying ring is a division algebra. Among the most basic examples of central simple algebras over a field K are the matrix algebras Matn (K) for n > 1. The most general case is given by: 1.2.2.2. Proposition (Wedderburn’s Theorem). Every central simple algebra D over a field K is isomorphic to Matn (∆) = End∆ (∆⊕n ) for some n > 1 and some central division algebra ∆ over K. Moreover, n is uniquely determined by D, and ∆ is uniquely determined up to K-isomorphism. Proof. This is a special case of a general structure theorem for simple rings; see [34, p. 203].

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In addition to matrix algebras, another way to make new central simple algebras from old ones is to use tensor products: 1.2.2.3. Lemma. If D and D0 are central simple algebras over a field K, then so is D ⊗K D0 . For any extension field K 0 /K, DK 0 := K 0 ⊗K D is a central simple K 0 -algebra. Proof. The first part is [34, Cor. 3, p. 219] and the second part is [34, Cor. 2, p. 219]. 1.2.3. Splitting fields. It is a general fact that for any central division algebra ∆ over a field K, ∆Ks is a matrix algebra over Ks . In other words, ∆ is split by a finite separable extension of K. There is a refined structure theory concerning splitting fields and maximal commutative subfields of central simple algebras over fields, and we refer the reader to [34, §4.1–4.6] for a self-contained development of this material. An important result in this direction is: 1.2.3.1. Proposition. Let D be a central simple algebra over a field F , with [D : F ] = n2 . An extension field E/F with degree n embeds as an F -subalgebra of D if and only if E splits D (i.e., DE ' Matn (E)). Moreover, every maximal commutative subfield of D has degree n over F . Proof. See [34, Thm. 4.11, p. 224]. We conclude our general discussion of central simple algebras by reviewing the concept of the K-linear reduced trace map TrdD/K : D → K for a central simple algebra D over a field K. 1.2.3.2. Construction. Let D be a central simple algebra over an arbitrary field K. It splits over a separable closure Ks , which is to say that there is a Ks -algebra isomorphism f : DKs ' Matn (Ks ) onto the n × n matrix algebra for some n > 1. By the Skolem–Noether theorem, all automorphisms of a matrix algebra are given by conjugation by an invertible matrix. Hence, f is well-defined up to composition with an inner automorphism. The matrix trace map Tr : Matn (Ks ) → Ks is invariant under inner automorphisms and is equivariant for the natural action of Gal(Ks /K) , so the composition of the matrix trace with f is a Ks -linear map DKs → Ks that is independent of f and Gal(Ks /K)-equivariant. Thus, this descends to a K-linear map TrdD/K : D → K that is defined to be the reduced trace. In other words, the reduced trace map is a twisted form of the usual matrix trace, just as D is a twisted form of apmatrix algebra. (For d ∈ D, the K-linear left multiplication map x 7→ d · x on D has trace [D : K] TrdD/K (x), as we can see by scalar extension to Ks and p a direct computation for matrix algebras. The elimination of the coefficient [D : K] is the reason for the word “reduced”.) 1.2.4. Brauer groups. For applications to abelian varieties it is important to classify division algebras of finite dimension over Q (such as the endomorphism algebra of a simple abelian variety over a field). If ∆ is such a ring then its center Z is a number field and ∆ is a central division algebra over Z. More generally, the set of isomorphism classes of central division algebras over an arbitrary field has an interesting structure of abelian group. This comes out of the following definition. 1.2.4.1. Definition (Brauer group). Central simple algebras D and D0 over a field K are similar if there exist n, n0 > 1 such that the central simple K-algebras D ⊗K Matn (K) = Matn (D) and D0 ⊗K Matn0 (K) = Matn0 (D0 ) are K-isomorphic. The Brauer group Br(K) is the set of similarity classes of central simple algebras over K, and [D] denotes the similarity class of D. For classes [D] and [D0 ], define [D][D0 ] := [D ⊗K D0 ].

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This composition law on Br(K) is well-defined and makes it into an abelian group with inversion given by [D]−1 = Dopp , where Dopp is the “opposite algebra”. By Proposition 1.2.2.2, each element in Br(K) is represented (up to isomorphism) by a unique central division algebra over K. In this sense, Br(K) is an abelian group structure on the set of isomorphism classes of such division algebras. 1.2.4.2. Example. The computation of the Brauer group of a number field involves computing the Brauer groups of local fields, so we now clear up any possible confusion concerning sign conventions in the description of Brauer groups for non-archimedean local fields. Upon choosing a separable closure Ks of an arbitrary field K, there are two natural procedures to define a functorial group isomorphism Br(K) ' H2 (Ks /K, Ks× ): a conceptual method via non-abelian cohomology as in [66, Ch. X, §5] and an explicit method via crossed-product algebras. By [66, Ch. X, §5, Exer. 2], these procedures are negatives of each other. We use the conceptual method of non-abelian cohomology, but we do not need to make that method explicit here and so we refer the interested reader to [66] for the details. Let K be a non-archimedean local field with residue field κ and let K un denote its maximal unramified subextension in Ks (with κ the residue field of K un ). It is known from local class field theory that the natural map H2 (K un /K, K un× ) → H2 (Ks /K, Ks× ) is an isomorphism, and the normalized valuation mapping K un× → Z induces an isomorphism δ

H2 (K un /K, K un× ) ' H2 (K un /K, Z) ' H1 (Gal(K un /K), Q/Z) = H1 (Gal(κ/κ), Q/Z). There now arises the question of choice of topological generator for Gal(κ/κ): arithmetic or geometric Frobenius? We choose to work with arithmetic Frobenius. (In [63, p. 130] and [66, p. 193] the arithmetic Frobenius generator is also used.) Via evaluation on the chosen topological generator, our conventions lead to a composite isomorphism invK : Br(K) ' Q/Z for non-archimedean local fields K. If one uses the geometric Frobenius convention, then by also adopting the crossed-product algebra method to define the isomorphism Br(K) ' H2 (Ks /K, Ks× ) one would get the same composite isomorphism invK since the two sign differences cancel out in the composite. (Beware that in [63] and [66] the Brauer group of a general field K is defined to be H2 (Ks /K, Ks× ), and so the issue of choosing between non-abelian cohomology or crossed-product algebras does not arise in the foundational aspects of the theory. However, this issue implicitly arises in the relationship of Brauer groups and central simple algebras, such as in [63, pp. 137-8] where the details are omitted.) Since Br(R) is cyclic of order 2 and Br(C) is trivial, for archimedean local fields K there is a unique injective homomorphism invK : Br(K) ,→ Q/Z. By [63, Thm. 3, p. 131], for a finite extension K 0 /K of non-archimedean local fields, composition with the natural map p Br(K) → Br(K 0 ) carries invK 0 to [K 0 : K] · invK . By [66, p. 194, Cor. 3], invK (∆) has order [∆ : K] for any central division algebra ∆ over K. These assertions are trivially verified to also hold for archimedean local fields K. 1.2.4.3. Theorem. Let L be a global field. There is an exact sequence 0

/ Br(L)

/

P

L

v

Br(Lv )

invLv

/ Q/Z

/0

where the direct sum is taken over all places of L and the first map is defined via extension of scalars. Proof. This is [76, §9.7, §11.2].

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For a global field L and central division algebra ∆ over L, invv (∆) denotes invLv (∆Lv ). Theorem 1.2.4.3 says that a central division algebra ∆ over a global field L is uniquely determined up to isomorphism by its invariants invv (∆), and that these mayPbe arbitrarily assigned subject to the conditions invv (∆) = 0 for all but finitely many v and invv (∆) = 0. Moreover, the order of [∆] in Br(L) is the least common “denominator” of the invv (∆) p∈ Q/Z. If K is any field then for a class c ∈ Br(K) its period is its order and its index is [∆ : K] with ∆ the unique central division algebra over K representing the class c. It is a classical fact that the period divides that index and that these integers have the same prime factors (see [66, X.5], especially Lemma 1 and Exercise 3), but in general equality does not hold. (There are function fields of complex 3-folds for which some order-2 elements in the Brauer group cannot be represented by a quaternion algebra, such as in [39, §4], and counterexamples with less interesting fields were first discovered by Brauer.) We have noted above that over local fields there is equality of period and index (the archimedean case being trivial). The following deep result is an analogue over global fields. 1.2.4.4. Theorem. For a central division algebra ∆ over a global field L, the order of [∆] p in Br(L) is [∆ : L]. As a special (and very important) case, elements of order 2 in Br(L) are precisely the Brauer classes of quaternion division algebras for a global field L; as noted above, this fails for more general fields. Since Theorem 1.2.4.4 does not seem to be explicitly stated in any of the standard modern references on class field theory (though there is an allusion to it at the end of [4, Ch. X, §2]), and the structure theory of endomorphism algebras of abelian varieties rests on it, we now give a proof. Proof. Let ∆ have degree n2 over L and let d be the order of [∆] in Br(L), so d|n. Note that d is the least common multiple of the local orders dv of [∆Lv ] ∈ Br(Lv ) for each place v of L, with dv = 1 for complex v, dv |2 for real v, and dv = 1 for all but finitely many v. Using these formal properties of the dv ’s, we may call upon the full power of global class field theory via Theorem 6 in [4, Ch. X] to infer the existence of a cyclic extension L0 /L of degree d such that [L0v0 : Lv ] is a multiple of dv for every place v of L (here, v 0 is any place on L0 over v, and the constraint on the local degree is only non-trivial for dv > 1). In the special case d = 2 (the only case we will require) one only needs weak approximation and Krasner’s Lemma rather than class field theory: take L0 to split a separable quadratic polynomial over L that closely approximates ones that define quadratic separable extensions of Lv for each v such that dv = 2. Since restriction maps on local Brauer groups induce multiplication by the local degree on the local invariants, it follows that ∆L0 is locally split at all places of L0 , whence by the injectivity of the map from the global Brauer group into the direct sum of the local ones (for L0 ) we conclude that the Galois extension L0 /L of degree d splits ∆. (The existence of cyclic splitting fields for all Brauer classes is proved for number fields in [76] and is proved in general in [82], but neither reference seems to exert much control on the degree of the global cyclic extension.) By a general result concerning Brauer groups of arbitrary fields [66, Ch. X, §5, Lemma 1], every Brauer class split by a Galois extension of degree r is represented by a central simple algebra with degree r2 . Applying this fact from algebra in our situation, [∆] = [D] for a central simple algebra D of degree d2 over L. But each Brauer class is represented by a unique central division algebra, and so D must be L-isomorphic to a matrix algebra over ∆. Since [D : L] = d2 and [∆ : L] = n2 with d|n, this forces d = n as desired.

1.2.5. Homomorphisms and isotypicity. The study of maps between abelian varieties over a field rests on the following useful injectivity result.

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13

1.2.5.1. Proposition. Let A and B be abelian varieties over a field K. For any prime ` (allowing ` = char(K)), the natural map Z` ⊗Z Hom(A, B) → Hom(A[`∞ ], B[`∞ ]) is injective, where the target is the Z` -module of maps of `-divisible groups over K (i.e., compatible systems of K-group maps A[`n ] → B[`n ] over K for all n > 1). Proof. We may pass to the case when K is algebraically closed (and hence perfect). When ` 6= char(K) the assertion is a reformulation of the well-known analogous injectivity with `-adic Tate modules (and such injectivity in turn underlies the proof of Z-module finiteness of Hom(A, B)). The proof in terms of Tate modules is given in [53, §19, Thm. 3] for ` 6= char(K), and when phrased in terms of `-divisible groups it works even when ` = p = char(K) > 0. For the convenience of the reader, we now provide the argument for ` = p in such terms. We will use that the torsion-free Z-module Hom(A, B) is finitely generated. Our argument will work for any `. The basic mechanism to be used is the elementary fact that if T : M → N is a Z` -linear map from a finite free Z` -module to a torsion-free Z` -module and T mod ` is injective then T is injective. (Proof: We may replace N with the finite free T (M ), and then describe T in terms of elementary divisors.) We will apply this with N = Hom(A[`∞ ], B[`∞ ]), for which N/`N → Hom(A[`], B[`]) is injective because `

0 → A[`] → A[`∞ ] → A[`∞ ] → 0 is exact for the fppf topology. In terms the problem is to prove that if c1 , . . . , cn ∈ P of a Z-basis {f1 , . . . , fn } of Hom(A, B),P Z` and ci fi kills A[`] then `|ci for all i. The map ci fi : A[`] → B[`] P vanishes, so for ni ∈ Z such that ni ≡ ci (mod `) for all i we see that the homomorphism ni fi : A → B kills A[`]. But ` 0 → A[`] → A → A → 0 P is a short exact sequence of for some h ∈ P finite type K-group schemes, soP ni fi = ` · h P Hom(A, B). Writing h = mi fi for some mi ∈ Z, we get ni ⊗ fi = ` · 1 ⊗ mi fi in Z` ⊗Z Hom(A, B). This implies ni = `mi for all i, so `|ci for all i. A weakening of simplicity that is sometimes convenient is: 1.2.5.2. Definition. An abelian variety A over a field K is isotypic if it is isogenous to C e for a simple abelian variety C over K and some e > 1; that is, up to isogeny, A has a unique simple factor. For a simple factor C of a non-zero abelian variety A over K, the C-isotypic part of A is the isotypic subvariety of A generated by the images of all maps C → A. An isotypic part of A is a C-isotypic part for some such C. Clearly End0 (A) is a semisimple Q-algebra. It is simple if and only if A is isotypic, and it is a division algebra if and only if A is simple. By the Poincar´e reducibility theorem, every non-zero abelian variety A over a field K is naturally isogenous to the product of its distinct isotypic parts, and these distinct parts admit no non-zero maps between them. Hence, if {Bi } is the set of isotypic parts of A then Q End0 (A) = End0 (Bi ) with each End0 (Bi ) a simple algebra of finite dimension over Q. Explicitly, if Ci is the unique simple factor of Bi then a choice of isogeny Bi → Ciei defines an isomorphism from End0 (Bi ) onto the matrix algebra Matei (End0 (Ci )) over the division algebra End0 (Ci ). Beware that the composite ring map End0 (Ci ) → Matei (End0 (Ci )) ' End0 (Bi ) is canonical only when End0 (Ci ) is commutative. In general isotypicity is not preserved by extension of the ground field. To make examples illustrating this possibility, as well as other examples in the theory of abelian varieties, we need to use the operation of Weil restriction of scalars. For a field K and a

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finite K-algebra K 0 , the Weil restriction functor ResK 0 /K from quasi-projective K 0 -schemes to separated (even quasi-projective) K-schemes of finite type is characterized by the functorial property ResK 0 /K (X 0 )(A) = X 0 (K 0 ⊗K A) for K-algebras A. Loosely speaking, Weil restriction is an algebraic analogue of viewing a complex manifold as a real manifold with twice the dimension. In particular, if K 0 /K is an extension of fields then ResK 0 /K (X 0 ) is smooth of pure dimension [K 0 : K]dim(X 0 ) when X 0 is smooth and equidimensional over K. We refer the reader to [8, §7.6] for a self-contained development of the construction and properties of Weil restriction (replacing K with more general rings), and to [17, A.5] for a discussion of further properties (especially of interest for group schemes). In general the formation of Weil restriction naturally commutes with any extension of the base field, `n and for K 0 equal to the product ring K n we have that ResK 0 /K carries a disjoint union i=1 Si Q 0 of quasi-projective K-schemes (viewed as a K -scheme) to the product Si . Thus, scalar extension to Ks shows that if K 0 is a field and is separable over K then ResK 0 /K (A0 ) is an abelian variety over K of dimension [K 0 : K]dim(A0 ) for any abelian variety A0 over K 0 (since [K 0 :K] K 0 ⊗K Ks ' Ks ). If K 0 /K is a field extension of finite degree that is not separable then ResK 0 /K (X 0 ) is never proper when X 0 is smooth and proper of positive dimension [17, Ex. A.5.6]. 1.2.6. Example. Consider a separable quadratic extension of fields K 0 /K and a simple abelian variety A0 over K 0 . Let σ ∈ Gal(K 0 /K) be the non-trivial element, so K 0 ⊗K K 0 ' K 0 × K 0 via x ⊗ y 7→ (xy, σ(x)y). Thus, the Weil restriction A := ResK 0 /K (A0 ) satisfies AK 0 ' A0 × σ ∗ (A0 ), so AK 0 is not isotypic if and only if A0 is not isogenous to its σ-twist. Hence, for K = R examples of non-isotypic AK 0 are obtained by taking A0 to be an elliptic curve over C with analytic model C/(Z ⊕ Zτ ) for τ ∈ C − R such that 1, τ, τ , τ τ are Q-linearly independent. (In Example 1.5.4 we give examples with K = Q.) In cases when AK 0 is non-isotypic, A is necessarily simple. Indeed, if A is not simple then a simple factor of A would be a K-descent of a member of the isogeny class of A0 , contradicting that A0 and σ ∗ (A0 ) are not isogenous. Thus, we have exhibited examples in characteristic 0 for which isotypicity is lost after a ground field extension. The failure of isotypicity to be preserved after a ground field extension does not occur over finite fields: 1.2.6.1. Proposition. If A is an isotypic abelian variety over a finite field K then AK 0 is isotypic for any extension field K 0 /K. Proof. By the Poincar´e reducibility theorem, it is equivalent to show that End0 (AK 0 ) is a simple Q-algebra, so by Lemma 1.2.1.2 we may replace K 0 with the algebraic closure of K in K 0 . That is, we can assume that K 0 /K is algebraic. Writing K 0 = lim Ki0 with {Ki0 } denoting −→ the directed system of subfields of finite degree over K, we have End(AK 0 ) = lim End(AKi0 ). −→ But End(AK 0 ) is finitely generated as a Z-module, so for large enough i we have End0 (AK 0 ) = End0 (AKi0 ). We may therefore replace K 0 with Ki0 for sufficiently large i to reduce to the case when K 0 /K is of finite degree. Let q = #K. The key point is to show that for any abelian variety B 0 over K 0 and any g ∈ Gal(K 0 /K), 0 B and g ∗ (B 0 ) are isogenous. Since Gal(K 0 /K) is generated by the q-Frobenius σq , it suffices to show that B 0 and B 0 (q) := σq∗ (B 0 ) are isogenous. The purely inseparable relative q-Frobenius morphism B 0 → B 0 (q) (arising from the absolute q-Frobenius map B 0 → B 0 over the q-Frobenius of Spec(K 0 )) is such an isogeny. Hence, the Weil restriction ResK 0 /K (B 0 ) Q 0 satisfies ResK 0 /K (B 0 )K 0 ' g g ∗ (B 0 ) ∼ B 0 [K :K] . Take B 0 to be a simple factor of AK 0 (up to isogeny), so ResK 0 /K (B 0 ) is an isogeny factor 0 of ResK 0 /K (AK 0 ) ∼ A[K :K] . By the simplicity of A and the Poincar´e reducibility theorem, it

1.3. COMPLEX MULTIPLICATION

15

follows that ResK 0 /K (B 0 ) is isogenous to a power of A. Extending scalars, ResK 0 /K (B 0 )K 0 is 0

therefore isogenous to a power of AK 0 . But ResK 0 /K (B 0 )K 0 ∼ B 0 [K :K] , so non-trivial powers of AK 0 and B 0 are isogenous. By the simplicity of B 0 and Poincar´e reducibility, this forces B 0 to be the only simple factor of AK 0 (up to isogeny), so AK 0 is isotypic.

1.3. Complex multiplication 1.3.1. Commutative subrings of endomorphism algebras. The following fact motivates the study of complex multiplication in the sense that we shall consider. 1.3.1.1. Theorem. Let A be an abelian variety of dimension g > 0 over a field K and let P ⊂ End0 (A) be a commutative semisimple Q-subalgebra. Then [P : Q] 6 2g, and if equality holds then P is its own centralizer in End0 (A). If equality holds and moreover P is a field of degree 2g over Q then A is isotypic and P is a maximal commutative subfield of End0 (A). Q Proof. Consider the decomposition P = Li into a product of fields. Using the primitive Q idempotents of P , we get a corresponding decomposition Ai of A in the isogeny category of abelian varieties over K, with each Li a commutative subfield of End0 (Ai ) Q Q Ai0 6= 0 and each compatiblyPwith the inclusion End (Ai ) ⊂ End0 (A) and the equality Li = P . Since dim(A) = dim(Ai ), to prove that [P : Q] 6 2g it suffices to treat the Ai ’s separately, which is to say that we may and do assume that P = L is a field. Since D = End0 (A) is of finite rank over Q, clearly [L : Q] is finite. Choose a prime ` different from char(K). The injectivity of the natural map L` := Q` ⊗Q L ,→ EndQ` (V` (A(Ks ))) (see Proposition 1.2.5.1) Q implies that L` acts faithfully on the Q` -vector space V` (A(Ks )) of rank 2g. But L` = w|` Lw , where w runs over all `-adic places of L, so each corresponding factor module V` (A(Ks ))w over Lw is non-zero as a vector space over Lw . Hence, X X 2g = dimQ` V` (A(Ks )) = dimQ` V` (A(Ks ))w > [Lw : Q` ] = [L : Q] w|`

w|`

with equality if and only if V` (A(Ks )) is free of rank 1 over L` . Assume that equality holds, so V` (A(Ks )) is free of rank 1 over L` . If A is not isotypic then by passing to an isogenous abelian variety we may arrange that A = B × B 0 with B and B 0 non-zero abelian varieties such that Hom(B, B 0 ) = 0 = Hom(B 0 , B). Hence, End0 (A) = End0 (B) × End0 (B 0 ) and so L embeds into End0 (B). But 2 dim(B) < 2 dim(A) = [L : Q], so we have a contradiction (since B 6= 0). It remains to prove, without assuming P is a field, that if [P : Q] = 2g then P is its own centralizer in End0 (A). (In case P is a field, so A is isotypic and hence End0 (A) is simple, such a centralizer property would imply that P is a maximal commutative subfield Q of End0 (A), as desired.) Consider once again the field decomposition P = L and the i Q corresponding isogeny decomposition Ai of A as at the beginning of this proof. We have [Li : Q] 6 2 dim(Ai ) for all i, and these inequalities add up to an equality when summed over all i, so in fact [Li : Q] = 2 dim(Ai ) for all i. The preceding analysis shows that each V` (Ai (Ks )) is free of rank 1 over Li,` := Q` ⊗Q Li , and so likewise V` (A(Ks )) is free of rank 1 over P` . Hence, EndP` (V` (A(Ks ))) = P` , so if Z(P ) ⊂ End0 (A) denotes the centralizer of P then the P` -algebra map Z(P )` = Q` ⊗Q Z(P ) → EndQ` (V` (A(Ks ))) is injective (Proposition 1.2.5.1) and lands inside of EndP` (V` (A(Ks ))) = P` . In other words, the inclusion P ⊂ Z(P ) of Q-algebras becomes an equality after scalar extension to Q` , so P = Z(P ) as desired.

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The preceding theorem justifies the interest in the following concept. 1.3.1.2. Definition. Let A be an abelian variety of dimension g > 0 over a field K. We say that A admits sufficiently many complex multiplications (over K) if there exists a commutative semisimple Q-subalgebra P in End0 (A) with rank 2g over Q. The reason for the terminology in Definition 1.3.1.2 is due to certain examples with K = C and P a number field such that the analytic uniformization of A(C) expresses the P -action in terms of multiplication of complex numbers; see Example 1.4.3. The classical theory of complex multiplication focused on the case of Definition 1.3.1.2 in which P is a field, but it is useful to allow P to be a product of several fields (i.e., a commutative semisimple Q-algebra). For example, by Theorem 1.3.1.1 this is necessary if we wish to consider the theory of complex multiplication with A that is not isotypic, or more generally if we want Definition 1.3.1.2 to be preserved under the formation of products. The theory of Shimura varieties provides further reasons to not require P to be a field. Note that we do not consider A to admit sufficiently many complex multiplications merely if it does so after an extension of the base field K. 2 3 1.3.2. Example. many complex multipli√ The elliptic curve y = x − x admits sufficiently cations over Q( −1) but not over Q. More generally, End0 (E) = Q for every elliptic curve E over Q (since the tangent line at the origin is too small to support a Q-linear action by an imaginary quadratic field), so in our terminology an elliptic curve over Q does not admit sufficiently many complex multiplications.

1.3.2.1. Proposition. Let A be a non-zero abelian variety over a field K. The following are equivalent. (1) The abelian variety A admits sufficiently many complex multiplications. (2) Each isotypic part of A admits sufficiently many complex multiplications. (3) Each simple factor of A admits sufficiently many complex multiplications. See Definition 1.2.5.2 for the terminology used in (2). Proof. Let {Bi } be the set of isotypic parts of A, so End0 (Bi ) ' Matei (End0 (Ci )) where C unique simple factor of Bi and ei > 1 is its multiplicity as such. Since End0 (A) = Qi is the 0 End (Bi ), (2) implies (1). It is clear that (3) implies (2). Conversely, assume that End0 (A) Q contains a Q-algebra P satisfying P[P : Q] = 2 dim(A). There is a unique decomposition P = Li with fields L1 , . . . , Ls , and [Li : Q] = 2 dim(A). We saw in the proof of Theorem 1.3.1.1 that by replacing A with an isogenous abelian variety Q 0 we may arrange that A = Ai with Ai a non-zero abelian variety having Q each Q Li ⊂ End (Ai ) 0 0 compatibly with the embedding End (Ai ) ⊂ End (A) and the equality Li = P . Thus, [Li : Q] 6 2 dim(Ai ) for all i, by Theorem 1.3.1.1, yet adding this up over all i yields an equality, so each Ai admits sufficiently many complex multiplications using Li . Since each simple factor of A is a simple factor of some Ai , to prove (3) we are therefore reduced to the case when P = L is a field. Applying Theorem 1.3.1.1 once again, L is its own centralizer in End0 (A) and A is isotypic, say with unique simple factor C appearing with multiplicity e. In particular, End0 (A) = Mate (D) for the division algebra D = End0 (C) of finite rank over Q. If Z denotes the center of D then D is a central division algebra over Z, and L contains Z since L is its own centralizer in D. Letting d = dim(C), Mate (D) contains the maximal commutative subfield L of degree 2g/[Z : Q] = (2d/[Z : Q])e over Z. By Proposition 1.2.3.1, all maximal commutative subfields of a rank-n2 central simple algebra have degree p all maximal commutative subfields of Mate (D) pn over the base field. Hence, have Z-degree e [D : Z], so 2d/[Z : Q] = [D : Z]. In other words, 2d/[Z : Q] is the common Z-degree of all maximal commutative subfields of D = End0 (C), or equivalently 2d is

1.3. COMPLEX MULTIPLICATION

17

the Q-degree of all such fields. But 2d = 2 dim(C), so choosing any maximal commutative subfield of D shows that C admits sufficiently many complex multiplications. 1.3.3. CM algebras and CM abelian varieties. The following three conditions on a number field L are equivalent: (1) L has no real embeddings but is quadratic over a totally real subfield, (2) for every embedding j : L → C, the subfield j(L) ⊂ C is stable under complex conjugation and the involution x 7→ j −1 (j(x)) in Aut(L) is non-trivial and independent of j, (3) there is a non-trivial involution τ ∈ Aut(L) such that for every embedding j : L → C we have j(τ (x)) = j(x) for all x ∈ L. The proof of the equivalence is easy. When these conditions hold, τ in (3) is unique and its fixed field is the maximal totally real subfield L0 ⊂ L (over which L is quadratic). The case L0 = Q corresponds to the case when L is an imaginary quadratic field. 1.3.3.1. Definition. A CM field is a number field L satisfying the equivalent conditions (1), (2), and (3) above. A CM algebra is a product L1 × · · · × Ls of finitely many CM fields (with s > 1). The reason for this terminology is due to the following important result (along with Example 1.4.3). 1.3.3.2. Theorem (Tate). Let A be an abelian variety of dimension g > 0 over a field K. Suppose A admits sufficiently many complex multiplications. Then there exists a CM algebra P ⊂ End0 (A) with [P : Q] = 2 dim(A). In case A is isotypic we can take P to be a CM field. The proof of this theorem will require some effort, especially since we consider an arbitrary base field K. Before we start the proof, it is instructive to consider an example. 1.3.4. Example. Consider A = E 2 with an elliptic curve E over K = C such that L := End0 (E) is an imaginary quadratic field. The endomorphism algebra End0 (A) = Mat2 (L) is simple and contains as its maximal commutative subfields all quadratic extensions of L. Those extensions which are biquadratic over Q are CM fields, and the rest are not CM fields. Hence, in the setup of Theorem 1.3.3.2, even when A is isotypic and char(K) = 0 there can be maximal commutative semisimple subalgebras of End0 (A) that are not CM algebras. However, if char(K) = 0 and A is simple (over K) then End0 (A) is a CM field; see Proposition 1.3.7.3. We will begin the proof of Theorem 1.3.3.2 now, but at a certain point we will need to use deeper input concerning the fine structure of endomorphism algebras of simple abelian varieties over general fields. At that point we will digress to review the required structure theory, and then we will complete the argument. By Proposition 1.3.2.1, to prove the existence of the CM subalgebra P in Theorem 1.3.3.2 it suffices to treat the case when A is simple. Before we turn to the treatment of the simple case, let us first show that the result in the simple case implies that in the isotypic case we can take P to even be a field. For isotypic A, by passing to an isogenous abelian variety we can arrange that A = A0 m for a simple abelian variety A0 over K and some m > 1. Thus, if g 0 = dimA0 then g = mg 0 and End0 (A0 ) contains a CM field P 0 of degree 2g 0 over Q. But End0 (A) ' Matm (End0 (A0 )) contains Matm (P 0 ). To find a CM field P ⊂ End0 (A) of degree 2g = 2g 0 m over Q it therefore suffices to construct a degree-m extension field P of P 0 such that P is a CM field. Let P00 be the maximal totally real subfield of P 0 , so for any totally real field P0 of finite degree over P00 the ring P = P0 ⊗P00 P 0 is a field quadratic over P0 and it is totally complex, so it is a CM field and clearly [P : Q] = [P : P 0 ][P 0 : Q] = 2g 0 [P0 : P00 ]. Hence, to find the required

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CM field P in the isotypic case it suffices to construct a degree-m totally real extension of P00 . To do this we will use the following basic fact from number theory [12, §6]: 1.3.4.1. Theorem (weak approximation). For any number field E and finite set S of places Q of E, the map E → v∈S Ev has dense image. Applying this to P00 , we can construct a monic polynomial f of degree m in P00 [u] that is very close to a totally split monic polynomial of degree m at each real place and is very close to an irreducible (e.g., Eisenstein) polynomial at a single non-archimedean place. It follows that f is totally split at each real place of P00 and is irreducible over P00 , so the ring P0 = P00 [u]/(f ) is a totally real field of degree m over P00 as required. We may and do assume for the remainder of the argument that A is simple. In this case D = End0 (A) is a central division algebra over a number field Z, so the commutative semisimple Q-subalgebra P ⊂ D is a field, and the proof of Proposition 1.3.2.1 shows that the common Q-degree of all maximal commutative subfields of D is 2g. Hence, our problem is to construct a maximal commutative subfield of D that is a CM field. Let TrdD/Q = TrZ/Q ◦ TrdD/Z , where TrdD/Z is the reduced trace. An abelian variety over any field admits a polarization, so choose a polarization of A over K. Let x 7→ x∗ denote the associated Rosati involution on D (so (xy)∗ = y ∗ x∗ and x∗∗ = x). 1.3.5. Lemma. The quadratic form x 7→ TrdD/Q (xx∗ ) on D is positive-definite. p Proof. For any central simple algebra D over any field F whatsoever, let n = [D : F ] and define the variant TrmD/F : D → F on the reduced trace to be the map that sends each y ∈ D to the trace of the F -linear map my : D → D defined by d 7→ yd. We have TrmD/F = n · TrdD/F , as may be checked by extending scalars to Fs and directly computing with elementary matrices (see 1.2.3.2). Hence, in the setting of interest with D = End0 (A) and F = Z we see that it is equivalent to prove positive-definiteness for the quadratic form x 7→ TrmD/Q (xx∗ ), where TrmD/Q = TrZ/Q ◦ TrmD/Z . The positive-definiteness for TrmD/Q 0 (AK ), to which [53, §21, Thm. 1] applies. can be verified by replacing D with EndK We express Lemma 1.3.5 by saying that x 7→ x∗ is a positive involution of D (relative to the linear form TrdD/Q ). The existence of such an involution severely constrains the possibilities for D. First we record the consequences for the center Z. 1.3.6. Lemma. The center Z of D = End0 (A) is either totally real or a CM field, and in the latter case its canonical complex conjugation is induced by the Rosati involution defined by any polarization of A over K. Proof. Fix a polarization and consider the associated Rosati involution x 7→ x∗ on the center Z of D. Clearly Z is stable under this involution. The positive-definite TrdD/Q (xx∗ ) p on D restricts to [D : Z] · TrZ/Q (xx∗ ) on Z, so TrZ/Q (xx∗ ) is positive-definite on Z. If x∗ = x for all x ∈ Z then the rational quadratic form TrZ/Q (x2 ) is positive-definite on Z, so by extending scalars to R we see that Tr(R⊗Q Z)/R (x2 ) is positive-definite. This forces R ⊗Q Z to have no complex factors. Hence, Z is a totally real field in such cases. It remains to show that if the involution x 7→ x∗ is non-trivial on Z for some choice of polarization then Z is a CM field and its intrinsic complex conjugation is equal to this involution on Z. Let Z0 be the subfield of fixed points in Z for this involution, so [Z : Z0 ] = 2 and 2 TrZ0 /Q is the restriction to Z0 of TrZ/Q . Hence, TrZ0 /Q (x2 ) is positive-definite on Z0 , so Z0 is totally real. We aim to prove that Z has no real places, so we assume otherwise and seek a contradiction. Let v be a real place of Z. Since the involution x 7→ x∗ is non-trivial on Z and the field Zv ' R has no non-trivial field automorphisms, the real place v on Z is not fixed

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by the involution x 7→ x∗ . Thus, the real place v ∗ obtained from v under the involution is a real place of Z distinct from v, and so the positive-definiteness of TrZ/Q (xx∗ ) implies (after scalar extension to R) the positive-definiteness of Tr(Zv ×Zv∗ )/R (xx∗ ), where x 7→ x∗ on Zv × Zv∗ = R × R is the involution that swaps the factors. In other words, this is the quadratic form (c, c0 ) 7→ 2cc0 , which by inspection is not positive-definite. 1.3.7. Albert’s classification. To go further with the proof of Theorem 1.3.3.2, we need to review some finer structural properties of endomorphism algebras of simple abelian varieties over arbitrary fields. 1.3.7.1. Definition. An Albert algebra is a pair consisting of a division algebra D of finite dimension over Q and a positive involution x 7→ x∗ on D. For any Albert algebra D and any algebraically closed field K, there exists a simple abelian variety A over K such that End0 (A) is Q-isomorphic to D (with the given involution on D arising from a polarization on A); see [1], [2], [3], [71, Thm. 5 on page 176], and [30, Thm. 13]. For a survey and further references on this topic, see [58]. We will not need this result. Instead, we are interested in the non-trivial constraints on the Albert algebras that arise from polarized simple abelian varieties A over an arbitrary field K when char(K) and dim A are fixed. Before listing these constraints, it is convenient to record Albert’s classification of general Albert algebras (omitting a description of the possibilities for the involution). 1.3.7.2. Theorem (Albert). Let (D, (·)∗ ) be an Albert algebra, and for any place v of the center Z let v ∗ denote the pullback of v along the involution. Exactly one of the following occurs: Type I: D = Z is a totally real field. Type II: D is a central quaternion division algebra over a totally real field Z such that D splits at each real place of Z. Type III: D is a central quaternion division algebra over a totally real field Z such that D is non-split at each real place of Z. Type IV: D is a central division algebra over a CM field Z such that for all finite places v of Z, invv (D) + invv∗ (D) = 0 in Q/Z and moreover D splits at such a v if v = v ∗ . Proof. See [53, §21, Thm. 2] (which also records the possibilities for the involution). Let A be a simple abelian variety over a field K, D = End0 (A), and Z the center of D. Let Z0 be the maximal totally real subfield of Z, so either Z = Z0 or Z is a totally complex p quadratic extension of Z0 . The invariants e = [Z : Q], e0 = [Z0 : Q], d = [D : Z], and g = dim(A) satisfy some divisibility restrictions: • whenever char(K) = 0 we have that ed2 = [D : Q] divides 2g (proof: there is a subfield K0 ⊆ K finitely generated over Q such that A descends to an abelian variety A0 over K0 and the D-action on A in the isogeny category over K descends to an action on A0 in the isogeny category over K0 , so upon choosing an embedding K0 ,→ C we get a Q-linear action by the division algebra D on the 2g-dimensional homology H1 (A0 (C), Q)), • the action of D on V` (A(Ks )) with ` 6= char(K) implies (via [53, §19, Cor. to Thm. 4], whose proof is valid over any base field) that ed|2g, • the structure of symmetric elements in Q ⊗Z Hom(A, At ) ' Q ⊗Z Pic(A)/Pic0 (A) (via [53, §20, Cor. to Thm. 3], whose proof is valid over any base field) yields that [F : Q]|g for every commutative subfield F ⊂ D whose elements are invariant under the involution. • for Type II in any characteristic we have 2e|g (which coincides with the general divisibility ed2 |2g when char(K) = 0 since d = 2 for Type II). To prove it uniformly

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across all characteristics, first note that for Type II we have Y Zv ⊗Z D ' Mat2 (Zv )e . R ⊗Q D = (R ⊗Q Z) ⊗Z D = v|∞

Moreover, by [53, §21, Thm. 2] it can be arranged that under this composite isomorphism the positive involution on D goes over to transpose on each Mat2 (Zv ) = Mat2 (R). Thus, for D of Type II the fixed part of the involution on D has Qdimension 2e and hence Z-degree 2. By centrality of Z in the division algebra D, the condition x∗ = x for x in D of Type II therefore defines a necessarily commutative quadratic extension Z 0 of Z inside of D, so g is divisible by [Z 0 : Q] = 2e as desired. The preceding results are summarized in the following table, taken from the end of [53, §21]. (As we have just seen, the hypothesis there that K is algebraically closed is not necessary.) The invariants of D = End0 (A) are given in the first three columns. In the last two columns we give some necessary divisibility restrictions on these invariants.

Type e I e = e0 II e = e0 III e = e0 IV e = 2e0

d char(K) = 0 char(K) > 0 1 e|g e|g 2 2e | g 2e | g 2 2e | g e|g 2 d e0 d | g e0 d | g

We refer the reader to [53, §21], and to [58] for further information on these invariants. Using the above table, we can prove the following additional facts when the simple A admits sufficiently many complex multiplications. 1.3.7.3. Proposition. Let A be a simple abelian variety of dimension g > 0 over a field K, and assume that A admits sufficiently many complex multiplications. Let D = End0 (A). (1) If char(K) = 0 then D is of Type IV with d = 1 and e = 2g. In particular, D is a CM field. (2) If char(K) > 0 then D is of Type III or Type IV. Proof. By simplicity, D is a division algebra. Let Z be its center, so Z is a commutative field. First suppose char(K) = 0. Let P ⊂ D be a commutative semisimple Q-subalgebra with [P : Q] = 2g. Since D is a division algebra, P is a field. The above table (or the discussion preceding it) says that the degree [D : Q] = ed2 divides [P : Q] = 2g, so the inclusion P ⊂ D is an equality. Thus, D is commutative (i.e., d = 1), so D = Z is a commutative field and hence e := [Z : Q] = 2g by the complex multiplication hypothesis. The table shows that in characteristic 0 we have e|g for Types I, II, and III, so D is of Type IV. Suppose char(K) > 0. In view of the divisibility relations in the table in positive characteristic, D is not of Type I since in such cases D is a commutative field whose Q-degree divides dim(A), contradicting the existence of sufficiently many complex multiplications. Likewise, for Type II we have 2e|g yet 2e = 2[Z : Q] is the Q-degree of a maximal commutative subfield of the central quaternion division algebra D over Z, so there are no such subfields with Q-degree 2g. Since a commutative semisimple Q-subalgebra of D is a field (as D is a division algebra), Type II is therefore not possible if the simple A has sufficiently many complex multiplications. Returning to the proof of Theorem 1.3.3.2, recall that we have reduced the proof to the case when A is simple. Proposition 1.3.7.3(1) settles the case of characteristic 0, and Proposition 1.3.7.3(2) gives that D = End0 (A) is an Albert algebra of Type III or IV when char(K) > 0.

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If D is of Type III then the center Z is totally real and d is even, whereas if D is of Type IV then Z is CM. Thus, we can apply the following general lemma to conclude the proof. 1.3.8. Lemma (Tate). Let D be a central division algebra of degree d2 over a number field Z that is totally real or CM. If Z is totally real then assume that d is even. There exists a maximal commutative subfield L ⊂ D that is a CM field. The parity condition on d is necessary when Z is totally real, since d = [L : Z] by maximality of L in D. Proof. By Proposition 1.2.3.1, any degree-d extension of Z that splits D is a maximal commutative subfield of D. Hence, we just need to find a degree-d extension L of Z that is a CM field and splits D. Let Σ be a finite non-empty set of finite places of Z containing the finite places at which D is non-split. By the structure of Brauer groups of local fields, for any v ∈ Σ the central simple Zv -algebra Dv := Zv ⊗Z D of rank d2 over Zv is split by any extension of Zv of degree d. First assume that Z is totally real, so d is even. By weak approximation (Theorem 1.3.4.1), there is a monic polynomial f over Z of degree d/2 that is close to a monic irreducible polynomial of degree d/2 over Zv for all v ∈ Σ (and in particular f is irreducible over all such Zv , and hence over Z since Σ is non-empty). We can also arrange that for each real place v of Z the polynomial f viewed over Zv ' R is close to a totally split monic polynomial of degree d/2 and hence is totally split over Zv . Thus, Z 0 := Z[u]/(f ) is a totally real extension field of Z with degree d/2. By the same method, we can construct a quadratic extension L/Z 0 that is unramified quadratic over each place v 0 over a place in Σ and is also totally complex (by using approximations to irreducible quadratics over R at the real places of Z 0 ). This L is a CM field and it is designed so that Zv ⊗Z L is a degree-d field extension of Zv for all v ∈ Σ. Hence, DL is split at all places of L (the archimedean ones being obvious), so DL is split. Assume that Z is a CM field. Let Z0 ⊂ Z be the maximal totally real subfield. By the same weak approximation procedure as above (replacing d/2 with d), we can construct a degree d totally real extension Z00 /Z0 such that for each place v0 of Z0 beneath a place v ∈ Σ, the extension Z00 /Z0 has a unique place v00 over v0 and is totally ramified (resp. unramified) at v00 when Z/Z0 is unramified (resp. ramified) at v. Hence, (Z00 )v00 and Zv are linearly disjoint over (Z0 )v0 . We conclude that Z00 and Z are linearly disjoint over Z 0 , so L := Z00 ⊗Z0 Z is a field and each v ∈ Σ has a unique place w over it in L. Clearly [Lw : Zv ] = d for all such w, so L splits D. By construction, L is visibly CM.

1.3.8.1. Corollary. An isotypic abelian variety A with sufficiently many complex multiplications remains isotypic after any extension of the base field. Proof. By Theorem 1.3.3.2, the endomorphism algebra End0 (A) contains a commutative field with Q-degree 2 dim(A). This property is preserved after any ground field extension (even though the endomorphism algebra may get larger), so by the final part of Theorem 1.3.3.2 isotypicity is preserved as well.

1.3.9. CM abelian varieties. It turns out to be convenient to view the CM algebra P in Theorem 1.3.3.2 as an abstract ring in its own right, and to thereby regard the embedding P ,→ End0 (A) as additional structure on A. This is encoded in the following concept. 1.3.9.1. Definition. Let A be an abelian variety over a field K, and assume that A has sufficiently many complex multiplications. Let j : P ,→ End0 (A) be an embedding of a CM algebra P with [P : Q] = 2 dim(A). Such a pair (A, j) is called a CM abelian variety (with complex multiplication by P ).

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Note that in this definition we are requiring P to be embedded in the endomorphism algebra of A over K (and not merely in the endomorphism algebra after an extension of K). For example, according to this definition, no elliptic curve over Q admits a structure of CM elliptic curve (even if such a structure exists after an extension of the base field). As an application of Theorem 1.3.3.2, we establish the following result concerning the possibilities for Z of Type III in Proposition 1.3.7.3(2). This will not be used later. 1.3.9.2. Proposition. Let A, K, and D be aspin Proposition 1.3.7.3(2) with p = char(K), and let Z be the center of D, g = dim(A), d = [D : Z], and e = [Z : Q]. We have ed = 2g, √ and if D is of Type III (so d = 2) then either Z = Q or Z = Q( p). Note that in this proposition, K is an arbitrary field with char(K) > 0; K is not assumed to be finite. p Proof. We always have ed|2g, but ed = [D : Q] and D contains a field P of Q-degree 2g, so 2g|ed. Thus, ed = 2g. Now we can assume A is of Type III, so the field Z is totally real. Since A is of finite type over K and D is finite-dimensional over Q, by direct limit considerations we can descend to the case when K is finitely generated over Fp . Let S be a separated integral Fp -scheme of finite type whose function field is K. Since A is an abelian variety over the direct limit K of the coordinate rings of the non-empty affine open subschemes of S, by replacing S with a sufficiently small non-empty affine open subscheme we can arrange that A is the generic fiber of an abelian scheme A → S. Since S is connected, the fibers of the map A → S all have the same dimension, and this common dimension is g (as we may compute using the generic fiber A). The Z-module End(A) is finitely generated, and each endomorphism of A extends uniquely to a U -endomorphism of AU for some non-empty open U in S (with U perhaps depending on the chosen endomorphism). Using a finite set of endomorphisms that spans End(A) allows us to shrink S so that all elements of End(A) extend to S-endomorphisms of A , or in other words End(A) = End(A ). We therefore have a specialization map D = End0 (A) → End0 (As ) for every point s ∈ S. Fix a prime ` 6= p. Since S is connected and A [`n ] is finite ´etale over S, an S-endomorphism of A [`n ] is uniquely determined by its effect on a single geometric fiber over S. But maps between abelian varieties are uniquely determined by their effect on `-adic Tate modules when ` is a unit in the base field, so we conclude (via consideration of `-power torsion) that the specialization map D → End0 (As ) is injective for all s ∈ S. We can therefore speak of an element of End0 (As ) “lifting” over K in the sense that it is the image of a unique element of D = End0 (A) under the specialization mapping at s. This will be of interest when s is a closed point and we consider Frobenius endomorphisms of As over the finite residue field κ(s) at s. By Theorem 1.3.3.2, we can choose a CM field L ⊂ D with [L : Q] = 2g. In particular, for each s ∈ S the field L embeds into End0 (As ) with [L : Q] = 2g = 2 dim(As ), so each As is isotypic. By Theorem 1.3.1.1, L is its own centralizer in End0 (As ). Take s to be a closed point of S, and let qs denote the size of the finite residue field κ(s) at s. The qs -Frobenius endomorphism ϕs ∈ End0 (As ) is central, so it centralizes L and hence must lie in the image of L. In particular, ϕs lifts to an element of End0 (A) = D that is necessarily central (as we may compute after applying the injective specialization map D ,→ End0 (As )). That is, ϕs ∈ Z ⊂ D for all closed points s ∈ S. Let Z 0 be the subfield of Z generated over Q by the lifts of the endomorphisms ϕs as s varies through all closed points of S. Each Q[ϕs ] is a totally real field since Z is totally real. By Weil’s Riemann Hypothesis for abelian varieties over finite fields (see the discussion following Definition 1.5.1.2), under any embedding ι : Q[ϕs ] ,→ C we have each ι(ϕs )ι(ϕs ) = qs √ for qs = #κ(s) ∈ pZ , so the real number ι(ϕs ) is a power of p. Hence, the subfield Q[ϕs ] ⊂ Z √ √ is either Q or Q( p), so the subfield Z 0 ⊂ Z is either Q or Q( p).

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Choose a prime ` 6= p. Because the A [`n ]’s are finite ´etale over S and K is the function field of S, the representation of Gal(Ks /K) on V` (A(Ks )) factors through the quotient π1 (S, η), where η : Spec(Ks ) → S is the geometric generic point. The Chebotarev Density Theorem for π1 (S, η) [61, App. B.9] says that the Frobenius elements at the closed points of S generate a dense subgroup of the quotient π1 (S, η) of Gal(Ks /K). Thus, the image of Q` [Gal(Ks /K)] in EndQ` (V` (A(Ks ))) is equal to the subalgebra Z`0 := Q` ⊗Q Z 0 generated by the endomorphisms ϕs . We therefore have an injective map Q` ⊗Q D ,→ EndQ` [Gal(Ks /K)] (V` (A(Ks ))) = EndZ`0 (V` (A(Ks ))). By Zarhin’s theorem [87] (see [52, XII, §2] for a proof valid for all p, especially allowing p = 2) this injection is an isomorphism, so we conclude that Z` is central in EndZ`0 (V` (A(Ks ))). But the center of this latter matrix algebra is Z`0 , so the inclusion Z`0 ⊂ Z` is an equality. Hence, √ the inclusion Z 0 ⊂ Z is an equality as well. Since Z 0 is either Q or Q( p), we are done.

1.4. CM types Let A be an isotypic abelian variety of dimension g > 0 over a field K such that A admits sufficiently many complex multiplications. By Theorem 1.3.3.2, we may and do choose a CM field L ⊂ End0 (A) with degree 2g. (Conversely, by Theorem 1.3.1.1, the existence of such an L forces A to be isotypic.) It turns out that the L-linear isogeny class of A is encoded in terms of a rather simple discrete invariant when char(K) = 0 and K is algebraically closed. We wish to review the basic features of this invariant, called the CM type, and to discuss some useful replacements for it in positive characteristic. The order O = L ∩ End(A) in L acts on A over K, so O acts K-linearly on the tangent space T = Lie(A) at the origin. Hence, if char(K) = 0 then L = O ⊗Z Q acts K-linearly on T , whereas if char(K) = p > 0 then O/(p) acts K-linearly on T . In particular, if char(K) = 0 then the isomorphism class of the L ⊗Q K-module T is an invariant of the L-linear isogeny class of A over K, whereas nothing of the sort is true when char(K) = p > 0. 1.4.1. Characteristic 0. We now focus on the Q case char(K) = 0. Let F/K be an algebraically closed extension. Since L ⊗Q K ' KiQfor finite (separable) extensions Ki /K, any L ⊗Q K-module M canonically decomposes as Mi for a Ki -vector space Mi . Thus, if dimK M is finite then the isomorphism class of M is determined by the numbers dimKi Mi , which in turn are determined by the isomorphism class of Q the L ⊗Q F -module M ⊗K F . The F -algebra L ⊗Q F has a very simple form: it is ϕ Fϕ where ϕ ranges through all field embeddings L → F and Fϕ denotes F viewed as an L-algebra via ϕ. Hence, any L ⊗Q F module M 0 decomposes into a corresponding product of eigenspaces Mϕ0 over F on which L acts through ϕ. We conclude that for an L ⊗Q K-module M with finite K-dimension, the isomorphism class of M is determined by the numbers dimF (M ⊗K F )ϕ as ϕ varies through Hom(L, F ). On the set Hom(L, F ) = Hom(L, Q) (with Q the algebraic closure of Q in F ) there is a natural involution defined by precomposition with the intrinsic complex conjugation ι of the CM field L (i.e., the non-trivial automorphism of L over its maximal totally real subfield L0 ). This decomposes the set Hom(L, F ) of size 2g into g “conjugate pairs” of embeddings. In the special case F = C we can also compute the involution on Hom(L, F ) by using composition with complex conjugation on F = C. An especially interesting example is the L ⊗Q F -module M = T ⊗K F with T = Lie(A) for a CM abelian variety A over K with complex multiplication by L. There is a non-trivial constraint on the eigenspaces (T ⊗K F )ϕ for the L-action on T ⊗K F (with ϕ varying through the embeddings ϕ : L → F ):

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1.4.2. Lemma. When char(K) = 0, each eigenspace (T ⊗K F )ϕ is at most 1-dimensional over F . If Φ denotes the set of g distinct embeddings ϕ : L → F for which there is a ϕeigenline in T ⊗K F then Φ contains no “conjugate pairs”. That is, we have a disjoint union ` decomposition Hom(L, F ) = Φ (Φ ◦ ι). Proof. By considerations with direct limits (as in the proof of Proposition 1.2.6.1), we may and do first arrange that K is finitely generated over Q. The choice of algebraically closed extension F/K does not matter, so we can replace F with K. We may then reduce to the case K = F = C, in which case a proof is given via the complex-analytic uniformization in [53, §22]. The preceding considerations lead us to make: 1.4.2.1. Definition. Let L be a CM field of degree 2g over Q and F an algebraically closed field of characteristic 0. An F -valued CM type for L is a subset Φ ⊂ Hom(L, F ) of representatives for the g orbits of the action by the complex conjugation ι of L. That is, Φ consists of ` g distinct elements such that ϕ ◦ ι 6∈ Φ for all ϕ ∈ Φ, or equivalently Hom(L, F ) = Φ (Φ ◦ ι). To emphasize the role of L, we often refer to the pair (L, Φ) as a CM type. The preceding discussion shows that if K is a field of characteristic 0 and F/K is an algebraically closed extension, then the tangent space to a CM abelian variety A over K with complex multiplication by L determines an F -valued CM type Φ for L. This is an invariant of the L-linear isogeny class of A over K. 1.4.2.2. Remark. In general, a CM type takes values in the algebraic closure Q of Q in F , so if we first choose this algebraic closure as an abstract field and then take F to be equipped with a specified embedding of this Q then we can regard the CM type as being independent of F ; this is sometimes useful for passing between different choices of F (such as C and Qp ). 1.4.3. Example. QLet L be a CM field and Φ a C-valued CM type on L. Let (R ⊗Q L)Φ denote R ⊗Q L = v|∞ Lv endowed with the complex structure defined via the isomorphism Lv ' C using the unique element ϕv ∈ Φ pulling back the standard absolute value of C to Q the place v of L for each v|∞. In other words, (R ⊗Q L)Φ = ϕ∈Φ Cϕ where Cϕ denotes C equipped with the L-action via ϕ : L → C. We view the ring of integers OL as a lattice in R ⊗Q L = R ⊗Z OL in the natural way, so the quotient (R ⊗Q L)Φ /OL is a complex torus of dimension [L : Q]/2. In the complex-analytic theory [53, §22] it is proved (using that L is a CM field) that this complex torus admits a Riemann form (with respect to which the action of each c ∈ L has adjoint given by the complex conjugate c ∈ L), and hence is an abelian variety. Let AΦ be the corresponding abelian variety over C. By construction (and GAGA), there is an action by OL on AΦ and hence an embedding L ,→ End0 (AΦ ) as a subfield of Q-degree [L : Q] = 2 dim(AΦ ). This makes AΦ into a CM abelian variety over C with complex multiplication by L. The action by any c ∈ OL ⊂ End(AΦ ) on Y Lie(AΦ ) = Lie(Aan Cϕ Φ ) = (R ⊗Q L)Φ = ϕ∈Φ

is the map (aϕ ) 7→ (ϕ(c)aϕ ) involving multiplication in C. In particular, AΦ equipped with the embedding L ,→ End0 (AΦ ) gives rise to the CM type Φ on L. The CM abelian varieties AΦ are generally not simple, and it is shown in the classical theory [53, §22, First Ex., Thm.] that as we vary Φ through all CM types on L, the AΦ ’s vary (without repetition) through all L-linear isogeny classes of (necessarily isotypic) CM abelian varieties over C with complex multiplication by L.

1.4. CM TYPES

25

1.4.3.1. Remark. Let A be an isotypic abelian variety of dimension g > 0 over a field K of characteristic 0, and let L be a CM field of degree 2g equipped with an embedding j : L ,→ End0 (A). The dual abelian variety At is equipped with a CM structure by L as follows: the embedding L ,→ End0 (At ) is x 7→ j(x)t , where x 7→ x is complex conjugation on L. It is easy to check that this convention is compatible with double duality. The reason for this convention is that it gives At the same CM type as A. To verify this equality of CM types we may reduce to the case when K = C and then use the exhaustive construction in the complex-analytic theory as in Example 1.4.3. Alternatively, still working over C, consider the functorial isomorphism Lie(At ) ' H1 (A, OA )∨ and the functorial Hodge decomposition H1 (A(C), Q) ' H1 (A(C), C)∨ ' Lie(A) ⊕ H1 (A, OA ). Since H1 (A(C), Q) is 1-dimensional as an F -vector space, when At is equipped with the dual action j(x)t (without the intervention of complex conjugation) then its CM type is Hom(L, C) − Φ = Φ. 1.4.4. Descent to a number field. For a CM abelian variety over an algebraically closed field F of characteristic 0, we may make the CM type essentially be independent of F by replacing F with Q (see Remark 1.4.2.2). This enables us to use the complex-analytic theory to prove the following purely algebraic result. 1.4.4.1. Proposition. Let K be an algebraically closed field of characteristic 0. Let L be a CM field, and consider CM abelian varieties A over K with complex multiplication via j : L ,→ End0 (A). The L-linear isogeny class of A is uniquely determined by the K-valued CM type Φ on L associated to (A, j), and every CM type on L arises in this way from some (A, j) over K. Proof. In view of Lemma 1.2.1.2, by expressing K as a direct limit of algebraically closed subfields of finite transcendence degree over Q we can reduce to the case when K has finite transcendence degree over Q. To show that the CM type determines the L-linear isogeny class it suffices (again by Lemma 1.2.1.2) to treat the case K = C. This case was addressed in Example 1.4.3 via the complex-analytic theory, where it was also seen that every CM type Φ on L does arise when K = C. It remains to show that every CM type Φ on L arises when K = Q. Consider the CM abelian variety AΦ over C with complex multiplication by L and CM type Φ as in Example 1.4.3. Recall that OL = L ∩ End(AΦ ) inside of End0 (AΦ ). By expressing C as a direct limit of its finitely generated Q-subalgebras, there is such a subalgebra R for which A with its OL -action over C descends to an abelian scheme A over R equipped with an OL -action. By localization of R, we can arrange that the tangent space Lie(A ) is finite and free as an R-module, and by increasing R to contain the integer ring Q of the Galois closure of L in C we can arrange that the OL -action on Lie(A ) decomposes as ϕ∈Φ Rϕ with Rϕ = R having action by c ∈ OL through multiplication by ϕ(c) ∈ R. For any maximal ideal m of R, the natural map Q → R/m is an isomorphism. Thus, passing to the fiber of A at a closed point of SpecR gives a pair (A, j) over Q with CM type Φ. This proposition has an important consequence for descending the field of definition of a CM abelian variety in characteristic 0, as we will see in Theorem 1.6.2.1. 1.4.4.2. Remark. By Theorem 1.3.1.1, any abelian variety A as in Proposition 1.4.4.1 has a unique simple factor C (in the sense of Definition 1.2.1.5). By Proposition 1.3.2.1, C is a CM abelian variety with complex multiplication by the CM field L0 := End0 (C) (see Proposition 1.3.7.3(1)). Since L0 is canonically identified with the center of End0 (A), it naturally embeds into L. Hence, there is a K-valued CM type Ψ on L0 arising from C, and the pair (L0 , Ψ) is determined by (L, Φ) since A with its complex multiplication by L is determined up to

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L-linear isogeny by Φ (and L0 = L if and only if A = C, which is to say that A is simple). It is therefore natural to seek an intrinsic recipe to directly construct (L0 , Ψ) from (L, Φ), and in particular to characterize in terms of Φ whether or not A is simple. The criterion is this: among the CM fields in L from which Φ is obtained by full preimage under restriction, (L0 , Ψ) is the unique such pair with [L0 : Q] minimal and Φ the full preimage of Ψ. Indeed, since the CM type is Q-valued (Remark 1.4.2.2) and the base field K is algebraically closed, it suffices to treat the case K = C. In this case the desired recipe is established in the complex-analytic theory (see [53, §22, Rem. (1)]). 1.4.5. Dieudonn´ e theory and p-divisible groups. We next wish to discuss the analogue of the preceding considerations when char(K) = p > 0. To give examples and prove theorems, we need a technique for handling p-torsion phenomena in characteristic p > 0. The two main tools for this purpose in what we shall do are Dieudonn´e theory and p-divisible groups. For the convenience of the reader we now briefly summarize some basic facts in this direction, and for additional details we refer to [75] and [69, §6] for p-divisible groups, and [25, Ch. II–III] for (contravariant) Dieudonn´e theory with applications to p-divisible groups. For a perfect field K of characteristic p > 0, let W = W (K) be the ring Witt vectors of K; e.g., if K is finite of size q = pr then W is the ring of integers in an unramified extension of Qp of degree r. Let ϕ be the unique automorphism of W that reduces to the map x 7→ xp on the residue field K. Let DK := W [F, V], where F and V are indeterminates subject to the relations (1) FV = VF = p, (2) Fc = ϕ(c)F and cV = Vϕ(c) for all c ∈ W . Elements of the “Dieudonn´e ring” DK have unique expressions as finite sums X X a0 + aj F j + bj V j j>0

j>0

with coefficients in W , so the center of DK is clearly Zp [F r , V r ] if K has finite size pr and it is Zp otherwise (i.e., if K is infinite). Some of the main conclusions in classical Dieudonn´e theory, as developed from scratch in [25, Ch. I–III], are summarized in the following theorem. 1.4.5.1. Theorem. There is an additive anti-equivalence of categories G D(G) from the category of finite commutative K-group schemes of p-power order to the category of left DK modules of finite W -length. Moreover, the following hold. (1) A group scheme G has order p`W (D(G)) . (2) If K → K 0 is an extension of perfect fields with associated extension W → W 0 of Witt rings (e.g., the absolute Frobenius automorphism of K) then the functor W 0 ⊗W (·) on Dieudonn´e modules is naturally identified with the base-change functor on finite commutative group schemes. In particular, D(G(p) ) ' ϕ∗ (D(G)) as W -modules. (3) Let FG/K : G → G(p) be the relative Frobenius morphism. The ϕ-semilinear action on D(G) induced by D(FG/K ) with respect to the isomorphism D(G(p) ) ' ϕ∗ (D(G)) equals the action of F, and G is connected if and only if F is nilpotent on D(G). Let A be an abelian variety over K with dimension g > 1. Let A[pn ] be the finite commutative pn -torsion subgroup scheme of rank (pn )2g . The directed system A[p∞ ] := (A[pn ])n>1 satisfies the following definition (with h = 2g): 1.4.5.2. Definition. A p-divisible group of height h > 0 over a scheme S is a directed system G = (Gn )n>1 of commutative S-groups Gn such that: Gn is killed by pn , each Gn → S is finite and locally free, G1 → S has constant degree ph , and Gn is identified with Gn+1 [pn ] for all n > 1.

1.4. CM TYPES

27

1.4.5.3. Example. Let R be a complete local noetherian ring with residue characteristic p > 0. For any finite flat commutative R-group scheme H, the connected component H 0 of the identity section is an open and closed R-subgroup scheme and the quotient H ´et := H/H 0 is finite ´etale; these properties can be seen via the special fiber. A p-divisible group G = (Gn ) is connected if each Gn is connected; equivalently, every Gn has infinitesimal special fiber. In [75] it is shown that if G = (Gn ) is a p-divisible group over R then G0 := (G0n ) is a p-divisible group (called the connected component of G) and G´et := (G´ent ) is a p-divisible group (called the ´etale part of G). Somewhat deeper lies the fact (also proved in [75]) that if G is a connected p-divisible group over R then O(G) := lim O(Gn ) is a formal power series ring over ←− R on which the induced formal R-group structure makes [p]∗ : O(G) → O(G) finite flat, and there is an equivalence of categories G

b := Spf(O(G)) G

from the category of connected p-divisible groups over R to the category of commutative formal Lie groups Γ over R for which [p]Γ is an isogeny. The quasi-inverse functor is Γ (Γ[pn ]). This equivalence defines the (relative) dimension and Lie algebra for a connected p-divisible group over R, via the analogous notions for formal Lie groups over R. For later purposes, here is how this construction works in the important example of the b Let CR denote p-divisible group G = A[p∞ ] arising from an abelian R-scheme A. What is G? the category of artin local R-algebras that are module-finite over R (and hence killed by some power of the maximal ideal of R). Every point of A valued in such an algebra and supported at the identity of the special fiber is p-power torsion due to R having residue characteristic b = Spf(O∧ ) has each of its C-points p. Thus, for any C ∈ CR , the formal Lie group A A,0 n 0 b b supported in some A[p ] . It follows that A and G pro-represent the same functor on CR , so b→A b is an isomorphism. In particular, the p-divisible group of A has the the natural map G same (relative) dimension and Lie algebra as A does. For a local ring R with residue field k, the functor A Ak from abelian R-schemes to abelian varieties over k is faithful. This is a consequence of the fact that the subgroup schemes A[N ] for N not divisible by char(k) are finite ´etale and dense in the geometric fibers over Spec(R). 1 The analogous result for p-divisible groups is more subtle when p = char(k), since it is false without a noetherian condition. For example, if R = Zp [ζp∞ ] denotes the (non-noetherian) valuation ring of the p-power cyclotomic extension of Qp and if {ζpn } is a compatible system of primitive p-power roots of unity in R then the R-homomorphism between p-divisible groups Qp /Zp → µp∞ defined by p−n 7→ ζpn is an isomorphism between the generic fibers and induces the zero map between special fibers. Under a noetherian hypothesis, this pathology cannot occur: 1.4.5.4. Proposition. Let (R, m) be a noetherian local ring with residue field k of characteristic p > 0. The functor G Gk from p-divisible groups over R to p-divisible groups over k is faithful. Proof. The problem is to prove that if f : G0 → G is a homomorphism between p-divisible groups over R and fk = 0 then f = 0. For each n > 1, the induced map fn : G0 [pn ] → G[pn ] between finite flat R-group schemes is described via a matrix over R upon choosing R-bases of the coordinate rings (as finite free R-modules). Hence, by the Krull intersection theorem it suffices to prove the vanishing result over R/mN for all N > 1, so we may and do assume that R is an artin local ring. By the functoriality of the connected-´etale sequence, it suffices to treat the following separate cases: G0 and G are both connected, G0 and G are both ´etale, or G0 is ´etale and G is connected. 1Need to insert the precise EGA IV , 11.9 reference for relative schematic density, or give another reference 3 or a proof.

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The case when both are ´etale is obvious. When G0 is ´etale and G is connected then we claim that Hom(G0 , G) = 0. By faithfully flat base change to (the artin local) Rsh , we may assume G0 is constant, so it is a power of Qp /Zp . Hence, we can assume G0 = Qp /Zp , so Hom(G0 , G) = lim G[pn ](R) (inverse limit via p-power maps). Via the equivalence between ←− connected p-divisible groups and formal Lie groups over R on which p is an isogeny (as recorded b in Example 1.4.5.3), this inverse limit is identified with the p-adic Tate module of G(R), where b b G is the formal Lie group associated to G. Hence, the desired vanishing is reduced to proving b b that G(R) has no nonzero infinitely p-divisible elements. In fact, we claim that [pN ] kills G(R) for sufficiently large N . b we may identify the set G(R) b Upon choosing formal parameters for G, with the set of b ordered d-tuples in m, where d = dim G. If g ∈ G(R) has coordinates in an ideal I of R then [p](g) has coordinates in (pI, I 2 ) since [p] has linear part given by p-multiplication on the coordinates. Hence, if we define the sequence of ideals J0 = m and Jn+1 = (pJn , Jn2 ) then we just need JN = 0 for sufficiently large N . More generally, for any ring whatsoever and any ideal J0 , an elementary induction argument shows that Jn ⊂ (p, J0 )n . The nilpotence of m then does the job. Finally, we address the most interesting case, which is connected G0 and G. In this case we switch to the perspective of formal Lie groups and aim to prove that for commutative formal Lie groups Γ and Γ0 over R such that [p]Γ0 is an isogeny, Hom(Γ0 , Γ) → Hom(Γ0k , Γk ) is injective. Consider f ∈ Hom(Γ, Γ0 ) that vanishes modulo an ideal I ⊆ m. Choose formal coordinates {x0i } and {xj } for Γ0 and Γ respectively, so the coefficients of all monomials in f ∗ (xj ) lie in I. Hence, (f ◦ [p]Γ0 )∗ (xj ) = [p]∗Γ0 (f ∗ (xj )) has all coefficients of all monomials lying in (pI, I 2 ). Iterating, if f ∈ Hom(Γ0 , Γ) vanishes over k then f ◦[pn ]Γ0 vanishes modulo the ideal Jn , where J0 = m and Jn+1 = (pJn , Jn2 ). We have already seen that JN = 0 for sufficiently 0 large N , so f ◦ [pN ]Γ0 = 0 for large N . By hypothesis the endomorphism [pN ]Γ0 = [p]N Γ0 of Γ induces an injective endomorphism of the coordinate ring, so f = 0. Now consider p-divisible groups over a field K of characteristic p > 0. Assuming that K is perfect, for any p-divisible group G = (Gn )n>1 over K with height h > 1 we let D(G) denote the DK -module lim D(Gn ). By the same style of arguments used to work out the Z` -module ←− structure of Tate modules of abelian varieties in characteristic 6= ` (resting on knowledge of the orders of the `-power torsion subgroups), we use W -length to replace counting to infer that D(G) is a free W -module of rank h and D(G)/pn D(G) → D(Gn ) is an isomorphism for all n > 1. The p-divisible group G is connected if and only if F is topologically nilpotent on D(G) (since this is equivalent to the nilpotence of F on each D(Gn )). This construction sets up an anti-equivalence between the category of p-divisible groups over K (using the evident notion of morphism) and the category of left DK -modules that are finite and free as W -modules; the W -rank of D(G) is equal to the height of G. The DK -module D(A[p∞ ]) is finite free of rank 2g over W , so it is an analogue of the `-adic Tate module for ` 6= char(K) even though it is contravariant in A. Its DK -module structure is the analogue of the Galois action on `-adic Tate modules, though the action by F and V is highly non-trivial even when K is algebraically closed (whereas the Galois action on Tate modules is trivial for such K). 1.4.5.5. Proposition. Let Z be a p-divisible group of height h > 0 over a field κ of characteristic p, and let κ1 be a perfect extension of κ. (1) If F is a commutative semisimple Qp -subalgebra of End0 (Z) := Qp ⊗Zp End(Z) then [F : Qp ] 6 h with equality if and only if D(Zκ1 )[1/p] is free of rank 1 as a W (κ1 )⊗Zp F module. (2) When equality holds, F is its own centralizer in End0 (Z). If moreover the maximal order OF lies in End(Z) then D(Zκ1 ) is free of rank 1 as a W (κ1 ) ⊗Zp OF -module.

1.4. CM TYPES

29

Proof. We may and do replace κ with κ1 , so κ is perfect. Letting K0 = W (κ)[1/p], we view D(Z)[1/p] as a finite module over the semisimple ring K0 ⊗Qp F . It is harmless to pass to an F -linearly isogenous p-divisible group, so we may decompose Z according to the idempotents of F to reduce to the case when F is a field. Let k be its residue field, κ0 a compositum of k with κ over Fp , and K00 = W (κ0 )[1/p]. Consider the decomposition Y (K00 ⊗j,W (k)[1/p] F ) K0 ⊗Qp F = (K0 ⊗Qp W (k)[1/p]) ⊗W (k)[1/p] F ' j:k→κ0

where j varies through the embeddings over k ∩ κ (⊂ κ0 ). This is a finite product of copies of totally ramified finite extensions of K00 , and the factor fields are permuted simply transitively by the natural F -linear action of Gal(k/(k ∩ κ)). Note that this Galois group is generated by a power of the absolute Frobenius. We conclude Q that any K0 ⊗Qp F -module M canonically decomposes in a compatible F linear way as Mj for vector spaces Mj over the factors fields. Hence, if M is equipped with an injective F -linear endomorphism F that is semilinear over the absolute Frobenius of K0 then F must be an F -linear automorphism and that transitively permutes the Mj ’s via F linear isomorphisms. In particular, if such an M is nonzero then each Mj is a nonzero vector space over the jth factor field, so M as a K0 ⊗Qp F -module would be free of some positive rank ρ and hence of K0 -dimension [F : Qp ]ρ. Now set M = D(Z)[1/p], whose K0 -dimension is h. This is equipped with the required Frobenius-semilinear injective endomorphism (that is moreover F -linear by functoriality), so it is free of some rank ρ > 1 over K0 ⊗Qp F and hence h = [F : Qp ]ρ. It follows that [F : Qp ] 6 h with equality if and only if ρ = 1. This proves (1). Finally, assuming we are in this rank-1 case, it remains to prove that F is its own centralizer in End0 (Z) and that D(Z) is an invertible W (κ) ⊗Zp OF -module when OF ⊆ End(Z). Since W (κ) ⊗Zp OF is a product of finitely many discrete valuation rings that are W (κ)-finite and D(Z) is finite free as a W (κ)-module, the invertibility over W (κ) ⊗Zp OF when OF ⊆ End(Z) follows from the established invertibility after localizing at p. To compute the centralizer of F , first observe that the Dieudonn´e module functor (on the isogeny category) is valued in the category of K0 -vector spaces, so every element f ∈ End0 (Z) that commutes with the F -action induces a K0 ⊗Qp F -linear endomorphism of D(Z)[1/p]. We know that D(Z)[1/p] is free of rank 1 over K0 ⊗Qp F , so D(f ) acts as multiplication by some c ∈ K0 ⊗Qp F . Since D(f ) also commutes with the action of F that is semilinear over the absolute Frobenius σ of K0 , we have (σ ⊗ 1)(c) = c. This forces c ∈ F , as desired. We finish our summary of the theory of p-divisible groups by recording (for later reference) two important results. The first is a deep result of Tate. 1.4.5.6. Theorem (Tate). Let R be a complete discrete valuation ring with perfect residue field of characteristic p > 0 and fraction field M of characteristic 0. For any p-divisible groups G and G0 over R, the natural injective map Hom(G, G0 ) → Hom(GK , G0K ) is bijective. Moreover, if CM denotes the completion of an algebraic closure M of M then there is a canonical CM -linear Gal(M /M )-equivariant isomorphism (1.4.5.1)

CM ⊗Zp Vp (GM ) ' (CM ⊗M Lie(G)M ) ⊕ (CM (−1) ⊗M Lie(Gt )M ).

The full faithfulness of G GK is [75, 4.2], and the construction of the isomorphism (1.4.5.1) occupies most of [75]. Next, we record the Serre–Tate deformation theorem. 1.4.5.7. Theorem (Serre–Tate). Let R be a ring in which p is nilpotent, and let I be an ideal in R such that I n = 0 for some n > 1. Define R0 = R/I, and for an abelian scheme A and p-divisible group G over R let A0 and G0 denote their respective reductions modulo I.

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For any abelian scheme A over R, let A : A[p∞ ]0 ' A0 [p∞ ] denote the canonical isomorphism. The functor A (A0 , A[p∞ ], A ) from the category of abelian schemes over R to the category of triples (A0 , G,  : A[p∞ ]0 ' G0 ) is an equivalence. See [36, 1.2.1] for a proof of this result. The most important application is that for an abelian variety X over a field k of characteristic p > 0, the infinitesimal deformation theory of X coincides with that of its p-divisible group. Likewise, if we fix a subring O ⊂ End(X) or a polarization of X (or both) then via the injection End(X) ⊂ End(X[p∞ ]) (Proposition 1.2.5.1) and the identification of X t [p∞ ] with the Cartier dual of X[p∞ ], 2 the infinitesimal deformation theory of X equipped with this extra structure is the same as that of its p-divisible group equipped with the analogous induced extra structure (e.g., for a p-divisible group G, the analogue of a polarization is an isogeny G → Gt that is skew-symmetric in an evident sense; this is called a quasi-polarization of G). When considering deformation problems that involve endomorphisms or polarizations (viewed as special kinds of isogenies), often Proposition 1.4.5.4 is implicitly used without comment. 1.4.5.8. Example. As a special case of the Serre–Tate deformation theorem, if R is a complete local noetherian ring of residue characteristic p > 0 and A0 is an abelian variety over the residue field, then a deformation of A0 [p∞ ] to a p-divisible group G over R corresponds to a deformation of A0 to a formal abelian scheme A over R. If A0 is equipped with a CM structure and we demand that this structure lifts to A (via the injection End(A) ,→ End(A0 )) then A can fail to be algebraic (i.e., it may not be the formal completion of a proper R-scheme). Explicit CM examples of this type are given in 4.1.2; also see the discussion immediately following the statement of Theorem 2.2.3. To assure algebraicity of A, we need to encode the deformation of a polarization. More specifically, choose a polarization φ0 : A0 → At0 and suppose that the corresponding quasipolarization of A0 [p∞ ] lifts to R (as can happen in at most one way, by Proposition 1.4.5.4). Consider the associated homomorphism φ : A → At that lifts φ0 . If P denotes the formal Poincar´e bundle on A × At (which lifts the Poincar´e bundle P0 on A0 × At0 ) then (1, φ)∗ P is a line bundle on A lifting the line bundle (1, φ0 )∗ P0 on A0 that is ample (due to φ0 being a polarization). Hence, by Grothendieck’s algebraization theorem [22, III1 , 5.1.4], in such cases A is algebraic, so it arises from a unique abelian R-scheme A deforming A0 . There is one special case in which the polarization comes “for free”: the so-called Serre– Tate canonical lifting of an ordinary abelian variety A0 over a perfect field k of characteristic p > 0. By perfectness of k, every p-divisible group over k is uniquely the product of an ´etale p-divisible group and a connected p-divisible group. Assume that A0 has maximal p-rank, namely g = dim A0 ; in other words, assume that A0 [p∞ ]´et has height g and A0 [p∞ ]0 has height g. Since the dual of an ´etale p-divisible group over k is connected and At0 [p∞ ] ' A0 [p∞ ]t = (A0 [p∞ ]0 )t × (A0 [p∞ ]´et )t with At0 isogenous to A0 , for height reasons it follows that the dual of A0 [p∞ ]0 is ´etale. In other words, A0 [p∞ ] ' E 0 t0 × E0 for ´etale p-divisible groups E0 and E00 that are functorial in A0 [p∞ ]. For any complete local noetherian ring R with residue field k, E0 and E00 uniquely lift to respective ´etale p-divisible groups E and E 0 over R, so the deformation E 0 t × E of E 0 t0 × E0 corresponds to a canonical formal deformation A of A0 over R. To prove that this is algebraic, consider a polarization φ0 : A0 → At0 . This symmetric homomorphism induces a skew-symmetric homomorphism t

E 0 0 × E0 = A0 [p∞ ] → At0 [p∞ ] ' A0 [p∞ ]t = E0t × E00 , so this latter map must have the form −f0t × f0 for a homomorphism f0 : E0 → E00 . There is a unique lifting f : E → E 0 , and so −f t × f is a quasi-polarization lifting −f0t × f0 . Thus, A is indeed algebraic. 2We need to give a suitable reference, in Oda’s thesis?

1.4. CM TYPES

31

1.4.6. Positive characteristic. If char(K) > 0 then for an abelian variety A over K of dimension g > 0 admitting an action by an order O in a CM field L of degree 2g over Q there is no corresponding K-linear action by L = Q ⊗Z O on the tangent space T := Lie(A) of A at the origin. Thus, there is not a good notion of CM type on L associated to the embedding of L into End0 (A). More specifically, if char(K) = p > 0 and O := L ∩ End(A) (an order in L) then T has a K-linear action by O/(p) and there is generally no constraint on this action akin to the eigenspace decomposition considered in characteristic 0 (as in Lemma 1.4.2). The lack of such a constraint occurs for a couple of reasons, as we now explain. 1.4.6.1. Example. If p is not totally inert in L or divides the discriminant of O over Z then O/(p) fails to be a field. In such cases, the K-linear O/(p)-action on T admits no notion of eigenspace decomposition that closely resembles the situation in characteristic 0. 1.4.6.2. Example. Suppose that O has discriminant not divisible by p (so O(p) = OL,(p) ) and that p is totally inert in L. In such cases κ := O/(p) is a finite field of degree 2g over Fp and Aut(L/Q) injects into Gal(κ/Fp ), so the canonical complex conjugation on L induces a non-trivial involution on κ. For an algebraically closed Q extension F/K we can consider the eigenspace decomposition of T ⊗K F over κ ⊗Fp F = ϕ Fϕ where ϕ ranges over the 2g distinct embeddings of κ into F . This could fail to resemble the CM types that arise in characteristic 0 because (as we shall see in later examples, such as in Remark 2.3.4) there may be conjugate pairs occurring among the ϕ for which T ⊗K F has a non-zero ϕ-eigenspace with respect to its F -linear κ-action. In such cases, the composite action O → End(A) → End(A)/(p) → EndK (T ) does not “look like the reduction of a CM type”, and so this provides an obstruction for A equipped with its O-action to lift to characteristic 0. There is no dimension obstruction to such lifting, in the sense that each ϕ-eigenspace in T ⊗K F = Lie(AF ) has F -dimension at most 1. To prove this, first note that the Dieudonn´e module D := D(AF [p∞ ]) is free of rank 1 over O ⊗Z W (F ) = O(p) ⊗Z(p) W (F ) = OL ⊗Z W (F ) by Proposition 1.2.5.1 and W (F )-rank considerations, so D/pD is free of rank 1 over κ ⊗Fp F . bF is the identity component of the p-divisible group AF [p∞ ] (Example The formal group A 1.4.5.3), and its tangent space coincides with that of AF . Hence, T ⊗K F ' Lie(AF [p∞ ]) ' D/F(D) by [25, III, Prop. 4.4(ii)], with F : D → D denoting the semilinear Frobenius endomorphism. By naturality this isomorphism is κ ⊗Fp F -linear, so T ⊗K F is monogenic over κ ⊗Fp F . Each ϕ-eigenspace of T ⊗K F is therefore monogenic over F , which is to say is of dimension at most 1 over F . Building on Example 1.4.6.2, an obstruction to the existence of a CM lift in characteristic 0 will be formulated precisely later (see 2.1.6 and 4.1.2). This will be used to exhibit examples (e.g., in 4.1.2) of abelian varieties over finite fields for which there is no lifting to characteristic 0 with sufficiently many complex multiplications. Such examples are interesting due to Corollary 1.5.2.5 below, according to which every abelian variety over a finite field admits sufficiently many complex multiplications. Although the Lie algebra fails to be an isogeny invariant for the study of CM abelian varieties in positive characteristic (and End0 (A) does not act on the tangent space when char(K) > 0), there is an alternative linear object attached to a CM abelian variety A that serves as a good substitute when char(K) = p > 0: the p-divisible group A[p∞ ], or its (contravariant) Dieudonn´e module D(A[p∞ ]) when K is perfect. This rests on Proposition 1.2.5.1, as follows. Taking B = A in Proposition 1.2.5.1, we see that Zp ⊗Z End(A) acts faithfully on A[p∞ ]. Hence, Qp ⊗Q End0 (A) acts faithfully on A[p∞ ] in the isogeny category of p-divisible groups over K. In particular, if K is perfect (e.g., finite) and A is an isotypic CM

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abelian variety over K with complex multiplication by the CM field L (see Theorem 1.3.3.2) then Lp := Qp ⊗Q L acts faithfully and linearly on the vector space D(A[p∞ ])[1/p] of rank 2g over the absolutely unramified p-adic field W (K)[1/p]. This is an analogue for the action by L on the filtered K-vector space H1dR (A/K) of dimension 2g when char(K) = 0, and will be useful in later considerations (e.g., the proof of Theorem 2.2.3) with lifting problems from positive characteristic to characteristic 0. (Note that when char(K) = 0, H1dR (A/K) provides essentially the same information as the CM type arising from the L-action on Lie(A) = H0 (A, Ω1A/K )∨ , in view of the Hodge filtration on H1dR (A/K); cf. Remark 1.4.3.1.)

1.5. Abelian varieties over finite fields In this section we work over a finite field κ with char(κ) = p. 1.5.1. Tate’s theorem and Weil numbers. A fundamental fact in the theory of abelian varieties over finite fields is: 1.5.1.1. Theorem (Tate’s isogeny theorem). Let A be an abelian variety over a finite field κ. The injective map in Proposition 1.2.5.1 is bijective for all primes `. The case ` 6= char(κ) is the main result in [74]; see [53, App. I, Thm. 1] for a proof as well. Unfortunately, Tate did not publish his proof for the case ` = p (though his argument was published in [51]). We provide a proof in Appendix A.1. Tate’s proof of his isogeny theorem is closely tied up with his analysis of the general structure of endomorphism algebras of abelian varieties over finite fields. The essential case, and the one on which we will now focus, is a simple abelian variety A over a finite field κ. In this case D := End0 (A) is a division algebra of finite dimension over Q. If q = #κ then the q-Frobenius endomorphism π = πA : A −→ A is central in D since the q-Frobenius is functorial on the category of κ-schemes. Hence, the number field Q[π] = Q(π) is contained in the center of D. Even without simplicity or isotypicity hypotheses on A, Tate proved (see [53, Thm. 3(a)]) that Q[π] is the center of End0 (A) for any abelian variety A over κ. 1.5.1.2. Definition. In a field of characteristic 0, a Weil q-number is an algebraic integer all of whose Q-conjugates in C have absolute value q 1/2 . The interest in Definition 1.5.1.2 is that Weil proved (see [60, §3]) that for any non-zero abelian variety A over κ and any ` 6= p := char(κ), the Q` -linear q-Frobenius action on V` (A(κs )) has characteristic polynomial fA,q ∈ Z[T ] that is independent of ` and has all roots in C equal to Weil q-numbers. In the special case that A is a simple abelian variety over κ, it follows that under any embedding Z := Q(π) ,→ C, the effect of complex conjugation on Z is characterized by the intrinsic formula π 7→ q/π. Hence, Z is either totally real (with π 2 = q) or a CM field, recovering Lemma 1.3.6 in the special case of finite base fields. In particular, √ when Z is totally real the only possibilities for Z are Q (when q = pn with n even) and Q( p) (when q = pn with n odd). Tate’s isogeny theorem implies some non-trivial results concerning the splitting behavior of the central division algebra D = End0 (A) over Z (with A simple over κ), as follows. Since π encodes the action of the topological q-Frobenius generator of Gal(κs /κ) on the `-adic Tate module of A for ` 6= p, the isogeny theorem gives a Z` -linear isomorphism Z` ⊗Z D ' EndZ` (V` (A(κs ))), Q

where Z` := Q` ⊗Q Z = v|` Zv . Passing to Zv -components for v|`, this implies that D is split at all places of Z away from real and p-adic places. The splitting behavior at the real and (especially) p-adic places is rather more subtle, and this was completely worked out by Tate;

1.5. ABELIAN VARIETIES OVER FINITE FIELDS

33

see Appendix A.1.3 for a proof of the structure of D at the p-adic places of Z and see [74], [77, Thm. 1], and [60, §5] for further discussion about the structure of the central division algebra D over Z. In Tate’s work, he also proved (see [53, App. I, Thm. 3(e)]) that A is isotypic if and only if the common characteristic polynomial fA,q ∈ Z[T ] of the q-Frobenius action on the Tate modules has a single monic irreducible factor over Q, in which case this irreducible factor is obviously the minimal polynomial fπ over Q for the q-Frobenius endomorphism π ∈ End0 (A). (Recall that Q[π] is the center Z of D = End0 (A).) The polynomial fπ only depends on A through its isogeny class (due to the functoriality of q-Frobenius on κ-schemes), and by Weil’s Riemann Hypothesis its Gal(Q/Q)-conjugacy class of roots in C consists of Weil q-numbers, where Q denotes the algebraic closure of Q in C. 1.5.2. The Honda–Tate theorem. Fix an abstract algebraic closure Q of Q and let W (q) denote the set of Weil q-numbers in Q. We consider elements of W (q) to be equivalent when they lie in the same Gal(Q/Q)-orbit, or in other words when they have the same minimal polynomial over Q. The following remarkable result relates Weil q-numbers to isogeny classes of simple abelian varieties over a finite field of size q. 1.5.2.1. Theorem (Honda–Tate). Let κ be a finite field of size q. The assignment A 7→ πA defines a bijection from the set of isogeny classes of simple abelian varieties over κ to the set of Gal(Q/Q)-conjugacy classes of Weil q-numbers. The proof of injectivity in Theorem 1.5.2.1 rests on the work of Tate related to Theorem 1.5.1.1. The proof of surjectivity uses abelian varieties in characteristic 0 (descents to number fields of CM abelian varieties over C; see Theorem 1.6.2.1). We are not aware of a proof which avoids using abelian varieties in characteristic 0. We refer the reader to [33], [77], and [60] for a discussion of the proof of the Honda–Tate theorem. In these references the following consequence of the proof is also addressed, describing the possibilities for the division algebra √ D = End0 (A) in terms of whether the center Z is Q or Q( p) (the totally real cases), or is a CM field. 1.5.2.2. Corollary. Let A be a simple abelian variety over a finite field κ of size q and characteristic p. Let D = End0 (A), π ∈ D the q-Frobenius endomorphism, and Z = Q(π) the center of D. Exactly one of the following occurs. (1) We have π 2 = q = pn with n even. This is precisely the case Z = Q, and occurs exactly when D is a central quaternion division algebra over Q, in which case it is the unique quaternion division algebra over Q ramified at exactly p and at ∞. The corresponding isogeny class of simple abelian varieties consists of supersingular elliptic curves E over κ for which all endomorphisms of Eκ are defined over κ (equivalently, the geometric endomorphism algebra End0 (Eκ ) coincides with End0 (E)). √ (2) We have π 2 = q = pn with n odd. This is precisely the case Z = Q( p), and occurs if and only if D is the unique central quaternion division algebra over Z ramified at exactly the two infinite places of Z. The corresponding isogeny class of simple abelian varieties is represented by the 2-dimensional Weil restriction Resκ0 /κ (E 0 ) where κ0 /κ is a quadratic extension and E 0 is a supersingular elliptic curve over κ0 whose geometric endomorphism algebra is defined over κ0 . (3) The field Z is a CM field. In such cases, D is the central division algebra over Z that is split at all places of Z away from p and for each p-adic place v of Z has local invariant invv (D) = (ordv (π)/ordv (q))[Zv : Qp ] mod Z ∈ Q/Z. The corresponding isogeny class p of simple abelian varieties over κ has members with dimension g = (1/2)[Z : Q] · [D : Z]. The formula for invv (D) in part (3) is proved in A.1.3, resting on preliminary work in A.1.1 and A.1.2.

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1.5.2.3. Remark. In the terminology of Theorem 1.3.7.2, the three cases in Corollary 1.5.2.2 correspond to A that are respectively of Type III with e = 1, Type III with e = 2, and Type IV. By inspection, the formula for invv (D) for p-adic places v of Z in case (3) is also correct in cases (1) and (2). Likewise, the formula for the dimension g in case (3) of Corollary 1.5.2.2 also works in cases (1) and (2). p Since the common Q-degree [Z : Q] [D : Z] of maximal commutative subfields of D is equal to 2g in each case, simple abelian varieties over finite fields always have sufficiently many complex multiplications. By passing to products and using Theorem 1.3.3.2 we obtain the following result (first proved independently by Honda and Serre). 1.5.2.4. Example. Writing q = pr , the elliptic curve cases that occur in part (3) are exactly the following cases, depending on the behavior of p in the imaginary quadratic field Z. Either p splits in Z with π generating the rth power of one of the two primes of Z over p, p is inert in Z with r odd and π = pr/2 ζ for an imaginary quadratic root of unity ζ 6= ±1, or p is ramified in Z and π generates the rth power of the unique prime of Z over p. The latter two are exactly the supersingular cases, and in all of these cases D = Z, so the geometric endomorphism algebra is not entirely defined over κ. Hence, (1) gives all supersingular elliptic curves over κ (up to isogeny) whose geometric endomorphism algebra is defined over κ. 1.5.2.5. Corollary. Every abelian variety A over a finite field admits sufficiently many complex multiplications. If A is isotypic then it admits a structure of CM abelian variety with complex multiplication by a CM field. 1.5.3. Example. As an application of Corollary 1.5.2.2, here are some examples of simple abelian surfaces (over prime fields of any characteristic) that are not absolutely simple. Let κ be a finite field of size p2 , with p a prime. As in Corollary 1.5.2.2(2) applied with q = p, there is a supersingular elliptic curve E over κ whose geometric endomorphism algebra is defined over κ. We can be more explicit when p 6≡ 1 mod 4 (resp. p 6≡ 1 mod 3), as follows. Let π = pζ with ζ 2 = −1 (resp. ζ 2 = −3). This is a p2 -Weil number and Z := Q(π) = Q(ζ) is an imaginary quadratic field in which p is not split. Thus, by Corollary 1.5.2.2, a simple abelian variety over κ with p2 -Frobenius equal to π must have endomorphism algebra Z and dimension 1. This elliptic curve E is supersingular because p is not split in Z. When π 6= ±2i, the isogeny class of E contains no member that is the scalar extension of an elliptic curve over Fp , as otherwise π would have a square root π0 ∈ Z, which is visibly absurd by inspection since π 6= ±2i. (Note that 2i = (1 + i)2 , so the assertion is false for p = 2.) The abelian surface A := Resκ/Fp (E) satisfies Aκ ' E × E 0 where E 0 is the twist of E by the non-trivial automorphism of κ over Fp , so Aκ is not simple. But A is simple if π 6= ±2i, as otherwise there would be a nonzero homomorphism E0 → A from an elliptic curve E0 over Fp , and hence (by the universal property of Weil restriction) a nonzero homomorphism (E0 )κ → E, contrary to what we just saw concerning the isogeny class of E when π 6= ±2i. Note that Aκ is isotypic, since E 0 = E (p) is isogenous to E via the relative Frobenius morphism E → E (p) . Taking K 0 /Q to be a quadratic field in which p is inert, we can lift E over OK 0 ,(p) to get an elliptic curve E over K 0 having good reduction E at pOK 0 . Then A := ResK 0 /Q (E ) is an abelian surface over Q having good reduction Resκ/Fp (E) at p that is simple over Fp , so (via consideration of N´eron models over Z(p) ) A is simple over Q. However, AK 0 ' E × E 0 where E 0 is the twist σ ∗ (E ) by the non-trivial automorphism σ of K 0 over Q, so AK 0 is not simple. 1.5.4. Example. Pushing Example 1.5.3 further over Q, we now prove that if p ≡ 3 (mod 4) (resp. p ≡ 2 mod 3) then E and E 0 are not isogenous, so AK 0 is not isotypic (in contrast with its reduction Aκ ). Suppose that there were an isogeny ψ : E → E 0 , and choose it with minimal degree. In particular, ψ is not divisible by [p]E . We claim that ordp (deg ψ) is odd (and in

1.5. ABELIAN VARIETIES OVER FINITE FIELDS

35

particular, is positive). Suppose otherwise, so deg ψ = mp2n with n > 0 and p - m. Consider the reduction ψ0 : E → E (p) of ψ, so this is an isogeny with degree mp2n as well. In particular, ker ψ0 ⊂ E is a finite subgroup scheme with order mp2n , so its p-part has order p2n . But E is supersingular, so it has a unique subgroup scheme of each p-power order. Hence, the p-part of ker ψ0 is E[pn ], so ψ0 = ψ00 [pn ]E with ψ00 : E → E (p) of degree m. But consider the composite isogeny ψ0

E →0 E (p) →E (p

2)

=E

using the Frobenius isogeny of E (p) . This is an endomorphism of E with degree pm. Since End(E) is an order in Z[ζ] on which the degree is computed as the norm to Z, we get an element of Z[ζ] whose norm in Z is divisible exactly once by p. That is impossible since p ≡ 3 (mod 4), and so completes the verification that deg ψ has p-part pj for some odd j. We conclude that the finite K 0 -subgroup N := ker ψ ⊂ E has non-trivial p-part, and this p-part has cyclic geometric fiber (as otherwise it would contain E [p], contradicting that we arranged ψ to not be divisible by [p]E ). By cyclicity, N [p] is a K 0 -subgroup of E with order p. Consider its scheme-theoretic closure G in the N´eron model of E at pOK 0 ,(p) . This is a finite flat group scheme over R = OK 0 ,(p) of order p, and its special fiber Gκ is an order-p subgroup scheme of the supersingular elliptic curve E, so Gκ ' αp as κ-groups (since Gκ is local-local of order p, so D(Gκ ) = κ with F = V = 0). But R is an absolutely unramified discrete valuation ring, so there are no finite flat group schemes over R with special fiber αp (by the classification results in [78]). This contradiction shows that E and E 0 are not isogenous (so AK 0 is not isotypic), as claimed. In fact, we have proved something stronger: if Kp0 denotes the p-adic completion of K 0 then AKp0 is not isotypic. 1.5.5. CM lifting after a field extension and isogeny. The proof of the surjectivity aspect of the Honda–Tate theorem requires constructing abelian varieties having prescribed properties over finite fields. The idea is to relate simple abelian varieties over finite fields to simple factors of reductions of CM abelian varieties over number fields, at least after some finite extension on the initial finite field. (See [33] or [77, Lemme 3] for details.) One can ask (as Honda did [33, p. 92]) to do better by arranging simplicity to hold for the reduction of a CM abelian variety over a number field (thereby eliminating the need to pass to a simple factor). Tate [77, Thm. 2] proved such an improved lifting theorem (which is really the starting point for the many lifting questions about CM abelian varieties that we will consider in this book): 1.5.5.1. Theorem (Tate). Let A be an isotypic abelian variety over a finite field κ. There exists a finite extension κ0 /κ such that Aκ0 is isogenous to the reduction of a CM abelian variety with good reduction over a p-adic field with residue field κ0 . Proof. By Corollary 1.5.2.5, there is a CM field L ⊂ End0 (A) with [L : Q] = 2 dim(A). The field L is its own centralizer in End0 (A), so it contains an element π which acts by the q-Frobenius endomorphism on A, where q = #κ. Let g = dim(A). Since A is κ-isotypic, Tate’s work on isogenies among abelian varieties over finite fields [74] gives two results for A: (i) the common characteristic polynomial over Q for the action of π on the Tate modules of A is a power of an irreducible polynomial fπ over Q (necessarily the minimal polynomial of π over Q), and (ii) A is κ-isogenous to any g-dimensional isotypic abelian variety over κ whose q-Frobenius is a zero of fπ . Moreover, these properties persist after replacing κ with any finite 0 extension κ0 (and replacing π with π [κ :κ] ), due to Proposition 1.2.6.1. By [77, §3, Thm. 2] (which is stated in the simple case but holds in the isotypic case by the same proof), there exists a number field F ⊂ Qp , a g-dimensional abelian variety A1 over F with good reduction at the induced p-adic place v, an embedding of finite fields κ ,→ κv , and an action of OL on A1 such that the reduction A1 at v has qv -Frobenius in OL given by the action of πv = π [κv :κ] ∈ OL . (Here, qv = #κv .) Since A1 admits a CM structure over κv by a field,

36

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it is κv -isotypic. Thus, since dim(A1 ) = dim(A) and Aκv satisfies FrAκv ,qv = FrA,qv follows from the results in [74] recalled above that A1 is κv -isogenous to Aκv .

= πv , it

1.5.5.2. Remark. A choice of κv -isogeny φ1 : A1 → Aκv may fail to be L-linear but it is certainly Q(πv )-linear since it is compatible with qv -Frobenius endomorphisms. The qv Frobenius generates the center of the endomorphism algebra of any abelian variety over κv , so the Skolem–Noether theorem ensures that any two Q(πv )-embeddings of L into the central simple Q(πv )-algebra End0 (Aκv ) are related through conjugation by a unit. Hence, there is an isogenous endomorphism u ∈ End(Aκv ) such that u ◦ φ1 is L-linear. By renaming this as φ1 , we may assume that φ1 is L-linear. That is, in Theorem 1.5.5.1 we may choose the CM lift so that the action of a specified degree-2g CM field L ⊂ End0 (A) also lifts. It is natural to ask for a strengthening of Theorem 1.5.5.1 in which the isogeny is applied prior to making a residue field extension. As we will record near the end of 1.7, such a stronger form is true and follows from one of the main results proved later in this book.

1.6. A theorem of Grothendieck and a construction of Serre 1.6.1. Isogenies and fields of definition. Let A be an abelian variety over a field K and let K1 ⊂ K be a subfield. We say that A is defined over K1 if there exists an abelian variety A1 over K1 and an isomorphism f : A ' A1/K . We use similarly terminology for a map A → B between abelian varieties over K (i.e., there exists a map h1 : A1 → B1 between abelian varieties over K1 such that (h1 )K is identified with h). For example, suppose K/K1 is a primary extension of fields (i.e., K1 is separably algebraically closed in K) and consider abelian varieties A and B over K such that there are isomorphisms f : A ' A1/K and h : B ' B1/K for abelian varieties A1 and B1 over K1 . By Lemma 1.2.1.2, the pairs (A1 , f ) and (B1 , h) are unique up to unique isomorphism and every map A → B as abelian varieties over K is defined over K1 in the sense that it uniquely descends to a map A1 → B1 as abelian varieties over K1 . Likewise, by Corollary 1.2.1.4, all abelian subvarieties of A are defined over K1 (and even uniquely arise from abelian subvarieties of A1 ). For general extensions K/K1 such K1 -descents may not exist, and when (A1 , f ) does exist it is not necessarily unique (up to isomorphism). 1.6.1.1. Example. Assume char(K) = 0 and let F/K be an algebraically closed extension (a basic example of interest being F = C). We claim that each member of the isogeny class of AF is defined over the algebraic closure K of K in F (and hence over a finite extension of K in F ). To prove this, observe that the kernel of any isogeny ψ : AF → B over F is contained in some torsion subgroup A[n]F , and A[n] becomes constant over K (since A[n] is K-´etale, as char(K) = 0). Hence, we can descend ker ψ to a constant finite subgroup of AK , and the quotient of AK by this gives a descent of (B, ψ) to K ⊂ F . 1.6.1.2. Example. When char(K) = p > 0, the naive analogue of Example 1.6.1.1 fails. An interesting counterexample is A = E 2 for a supersingular elliptic curve E over a field K of characteristic p > 0. The kernel H of the Frobenius isogeny E → E (p) is a local-local K-group of order p, and it is the unique infinitesimal subgroup of E with order p (as any commutative infinitesimal K-group of order p has vanishing Frobenius morphism). The only local-local finite commutative group scheme of order p over a field K of characteristic p is αp . Over perfect fields this is easily proved by a computation with Dieudonn´e modules (as we noted in Example 1.5.4). In Proposition 3.1.3.1 (whose proof is self-contained) we will prove that this property in general descends from the perfect closure (set r = 1 there). Fix a choice of K-subgroup inclusion αp ,→ E over K, so we get a canonical copy of αp2 in A = E 2 (as the kernel of the Frobenius isogeny A → A(p) ). Over an arbitrary field of characteristic p > 0 the Frobenius and Verschiebung morphisms of αp2 vanish, so over any

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37

such field the non-trivial proper subgroups of αp2 are naturally parameterized by lines in the 2-dimensional tangent space Lie(αp2 ); this parameterization is given by the tangent line of the subgroup (see [15, Thm. 3.18] for more details). In particular, the non-trivial proper K-subgroups of αp2 are parameterized by P1 (K), and if K 0 /K is an extension field then the 2 1 0 1 non-trivial proper K 0 -subgroups of αp/K 0 are parameterized by P (K ) (with the subset P (K) 0 consisting of the tangent lines to the K -subgroups defined over K). We conclude that if K 0 /K is a non-trivial extension field then there are K-subgroups 0 2 G ⊂ A0 := AK 0 of order p that are contained in αp/K 0 and do not arise from a K-subgroup of A. In contrast with what we saw in Example 1.6.1.1 for isogeny classes over algebraically closed fields of characteristic 0, we claim that if K is algebraically closed (or more generally if K is separably closed in K 0 , with G0 not defined over K inside of A0 = AK 0 ) then the isogenous quotient A0 /G0 of A0 = AK 0 cannot be defined over K as an abstract abelian variety! Indeed, if there were an isomorphism A0 /G0 ' BK 0 for an abelian variety B over K then the resulting isogeny AK 0 = A0  A0 /G0 ' BK 0 descends to an isogeny A → B over K by Lemma 1.2.1.2 (since K 0 /K is primary). The kernel of this latter isogeny is a K-subgroup of A that descends G0 ⊂ A0 , contrary to how G0 was chosen. Thus, no such B exists. One lesson we learn from Example 1.6.1.1 and Example 1.6.1.2 is that the study of fields of definition for abelian varieties in positive characteristic is rather more subtle than in characteristic 0, even when working over algebraically closed base fields. 1.6.2. Grothendieck’s theorem. To fully appreciate the significance of Example 1.6.1.2, we turn our attention to a striking result of Grothendieck concerning the field of definition of an abelian variety with sufficiently many complex multiplications in positive characteristic. Before stating Grothendieck’s result, we record the analogue in characteristic 0 that is a source of inspiration. 1.6.2.1. Theorem (Shimura–Taniyama). Every abelian variety A with sufficiently many complex multiplications over an algebraically closed field K of characteristic 0 is defined (along with its entire endomorphism algebra) over a number field in K. Proof. By Example 1.6.1.1, without loss of generality we may replace A with an isogenous abelian variety. Thus, by Proposition 1.3.2.1 we can pass to the isotypic (and even simple) case, and so by Theorem 1.3.3.2 the abelian variety A over K admits complex multiplication by a CM field L. Let Φ be the resulting CM type on L. Letting Q denote the algebraic closure of Q in K, we may view Φ as a Q-valued CM type on L. By Proposition 1.4.4.1 (applied over the algebraically closed base field Q) there is a CM abelian variety B over Q with complex multiplication by L and CM type Φ (viewed as valued in Q). The abelian variety BK over K admits complex multiplication by L with associated CM type Φ (viewed as valued in K), so by applying Proposition 1.4.4.1 over K we see that BK is L-linearly isogenous to A over K (as these two abelian varieties over K are endowed with complex multiplication by L yielding the same CM type Φ on L). Fix such an isogeny f : BK → A. As we showed in Example 1.6.1.1, the finite kernel of f descends to a finite subgroup of B. The quotient of B by this descent of ker f is a descent of A = BK /(ker f ) to an abelian variety over Q. By Lemma 1.2.1.2 the endomorphism algebra of A is equal to that of its descent, so we are reduced to the case when the base field K is an algebraic closure Q of Q. The same direct limit argument as used at the start of the proof of Proposition 1.2.6.1 shows that we can descend A to a number field such that all elements of End(A) descend as well.

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Theorem 1.6.2.1 can be formulated with a general ground field K of characteristic 0, but the nature of the descent becomes a bit more subtle. Namely, if A is an abelian variety over a field K of characteristic 0 and if A admits sufficiently many complex multiplications, then there is a finite extension K 0 /K such that AK 0 descends (along with its entire endomorphism algebra) to an abelian variety over a number field contained in K 0 . In this formulation it is crucial to introduce the finite extension K 0 /K, even if we just wish to descend the abelian variety (and not any specific endomorphisms). This is illustrated by quadratic twists of elliptic curves: 1.6.2.2. Example. Consider a CM elliptic curve over C and extend scalars to K = C(t). Let E be the quadratic twist of this scalar extension by a quadratic extension K 0 /K, so E is a CM elliptic curve over K whose `-adic representation for Gal(Ks /K) is non-trivial. No member of the isogeny class of E over K can be defined over C (let alone over Q), as all members of the isogeny class have non-trivial action by Gal(Ks /K) in their `-adic representations. Of course, if we pass up to K then the effect of quadratic twisting goes away and there is no obstruction to descent to Q. 1.6.2.3. Theorem. Every CM abelian variety over a number field has potentially good reduction at all places. This is an application of the semi-stable reduction theorem for abelian varieties; see [68, Thm. 6]. Since every abelian variety over an algebraic closure of Fp descends to a finite field and hence has sufficiently many complex multiplications (by Corollary 1.5.2.5), a naive first guess for an analogue of Theorem 1.6.2.1 is that CM abelian varieties over algebraically closed fields with positive characteristic can be descended to the algebraic closure of the prime field. Example 1.6.1.2 shows that this is false. Allowing isogenies does not eliminate the need for a finite extension: 1.6.2.4. Example. For a finite field κ, Example 1.6.2.2 adapts to work over κ(t) by beginning with an elliptic curve over any κ (for which complex multiplication by an imaginary quadratic field exists in general; see Corollary 1.5.2.5). One can do likewise over κ(t) with κ an algebraic closure of Fp . Motivated by the above counterexamples in positive characteristic, Grothendieck proved a reasonable analogue of Theorem 1.6.2.1: 1.6.2.5. Theorem (Grothendieck). Let A be an abelian variety over a field K with char(K) = p > 0, and assume A admits sufficiently many complex multiplications. Then there exists a finite extension K ⊂ K 0 , a finite subfield κ ⊂ K 0 , and an abelian variety B over κ such that the scalar extensions A ×Spec(K) Spec(K 0 ) and B ×Spec(κ) Spec(K 0 ) over K 0 are isogenous. For an exposition of Grothendieck’s proof, see [56]. The essential difficulty in the proof (in contrast with characteristic 0) is that the isogeny cannot be avoided, as shown by Example 1.6.1.2. The proof of Theorem 1.6.2.5 is immediately reduced to the case when K is finitely generated over the prime field Fp . Grothendieck used the theory of potentially good reduction to find the required K 0 /K and made the descent from K 0 to a finite subfield via a suitable Chow trace (in the sense of [15, §6]). 1.6.3. T. here is a refinement of Grothendieck’s theorem, due to Yu, that clarifies the role of the isogeny and proceeds in a simpler way by using moduli spaces of abelian varieties. To explain the refinement, which we will not need, we require a technique to modify the endomorphism ring; this technique will be very useful later.

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39

As motivation, consider an abelian variety A of dimension g > 0 over a field K such that A admits sufficiently many complex multiplications, and let P ⊂ End0 (A) be a commutative semisimple Q-subalgebra with [P : Q] = 2g. The intersection O := PQ∩ End(A) is Q an order in P but it may not be the maximal order (i.e., it may not be OP := OLi , where Li is the decomposition of P into a finite product of number fields). It is natural to ask if we can pass to an isogenous abelian variety so that the non-maximality problem goes away. The following discussion addresses this issue. 1.6.3.1. Example. Consider the preceding setup with K = C. In this case we have an analytic uniformization Aan = V /Λ in which V is a C-vector space equipped with a C-linear action by P and Λ is a lattice stable under the order O. Then Λ0 := OP · Λ is an OP -stable lattice in V and V /Λ0 is an isogenous quotient of Aan on which OP naturally acts. This algebraizes to an isogenous quotient A0 of A such that under the identification End0 (A0 ) = End0 (A) we have P ∩ End(A0 ) = OP . We need an algebraic variant of the analytic construction in Example 1.6.3.1. Observe that OP · Λ is the image of the natural map OP ⊗O Λ → V . Inspired by this, we are led to ask if these is a way to enlarge an endomorphism ring via a “tensor product” against a finite-index extension of coefficient rings. There is a construction of this sort due to Serre [64], applicable over any base scheme, though it turns out to not be applicable to the above situation because OP is not a projective O-module when O 6= OP . We wish to adapt Serre’s construction to the above situation over a field, so we digress to explain Serre’s procedure. 1.6.4. Serre’s tensor construction. Consider a scheme S, a commutative ring O, and an O-module scheme A over S. Let M be a projective O-module of finite rank. The projectivity ensures that the functor T M ⊗O A(T ) on S-schemes is represented by an S-scheme, denoted M ⊗O A, and that M ⊗O A inherits many nice properties from A such as flatness, smoothness, properness, good behavior with respect to analytification over C, etc. The interested reader can see [16, §7] for details (where non-commutative O are also considered). ϕ The idea of the construction of M ⊗O A is that if Or → Os → M → 0 is a presentation then we want to take M ⊗O A to be the cokernel of the S-group map Ar → As induced by the matrix of ϕ. Over a general base scheme S such a quotient may not exist. However, since M is locally free of finite rank as an O-module (by the projectivity hypothesis) we can instead begin with a presentation of the dual module M ∨ and then dualize to get a left-exact sequence 0 → M → Os → Or with suitable local splitting properties to enable us to construct M ⊗O A as a scheme-theoretic kernel. 1.6.4.1. Example. Let L be a CM field, and let (A, i) and (A0 , i0 ) be CM abelian varieties over an algebraically closed field of characteristic 0, where i and i0 respectively define complex multiplication by L. Assume that via these embeddings, OL lies in the endomorphism rings of the abelian varieties. Finally, assume that there exists an OL -linear isogeny A0 → A. By Proposition 1.4.4.1, it is equivalent to say that the associated CM types Φ0 and Φ on L coincide. We claim that M := Hom((A0 , i0 ), (A, i)) is an invertible OL -module and that the natural map M ⊗OL A0 → A is an isomorphism. (Hence, Pic(OL ) acts simply transitively on the set of isomorphism classes of CM abelian varieties with a fixed CM type (L, Φ) such that the CM order is OL .) By Lemma 1.2.1.2, we may assume that the ground field is C. The OL -modules H1 (A(C), Z) and H1 (A0 (C), Z) are each invertible (due to being Z-flat of rank [L : Q]). By Example 1.4.3, we get OL -linear isomorphisms A(C) = (R ⊗Q L)Φ /a and A0 (C) = (R ⊗Q L)/a0 for nonzero ideals a, a0 ⊂ OL . Hence, elements of M are precisely multiplication on (R ⊗Q L)Φ by those c ∈ L such that ca0 ⊆ a. We conclude that M = HomOL (a0 , a) = aa0 −1 , with M ⊗OL A0 → A given

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by the evident evaluation pairing on C-points. This is an isomorphism because the induced map on homology lattices is the natural pairing M ⊗OL a0 → a that is clearly an isomorphism. The O-linear projectivity hypothesis on M in the construction of M ⊗O A cannot be dropped, even when the base is the spectrum of a discrete valuation ring. For example, if R is a p-adic√discrete valuation ring and E is an elliptic curve over R with endomorphism ring O = Z[p −p] (as can be easily constructed using classical CM theory for elliptic curves), √ then for the non-projective O-module M = Z[ −p] the fppf sheafification of the functor T M ⊗O E(T ) on R-schemes is not representable. (The idea is as follows. First one proves that a representing object, if one exists, must be an elliptic curve E . By presenting M over O using two generators and two relations, we get a quotient map E × E → E whose kernel must be an R-flat divisor in E × E. Studying its defining equation in the formal group of E × E leads to a contradiction.) Here is another interesting kind of counterexample when the base is not normal. 1.6.4.2. Example. We claim that there exists an affine integral scheme S, an elliptic curve A → S, and an imaginary quadratic field L ⊂ End0 (A) := Q ⊗Z End(A) such that there is no isogeny A → A0 to another elliptic curve over S with the equality End0 (A) = End0 (A0 ) carrying OL into End(A0 ). In particular, there is no analogue of Proposition 1.6.4.3 over such a base S. (Note that in any such situation some order of OL lies in End(A0 ) and End(A), as both endomorphism rings are Z-lattices of full rank in the Q-algebra End0 (A) = End0 (A0 ).) For example, let S = Spec(R) where R is the local index-p order Z(p) + p · Z(p) [i] in Z(p) [i] with a prime p ≡ 3 (mod 4) and i2 = −1, and let E be the elliptic curve y 2 = x3 − x viewed over S. The generic fiber has endomorphism ring Z[i], with i acting via [i](x, y) = (−x, −iy), so [i]∗ (dy/x) = i · dy/x. Hence, [i] acts as multiplication by i ∈ Q(i) = Frac(R) on the tangent space, and this property is inherited by the generic fiber of any elliptic curve E 0 over R 0 equipped with an isogeny from E (using the resulting identification End0 (EQ(i) ) = End0 (EQ(i) ) 0 to transfer the Z[i]-action over to EQ(i) in the isogeny category of elliptic curves over Q(i)). 0 But i 6∈ R, so no such E can admit an action by Z[i] over R respecting the action on its generic fiber. Things are better when working over a field. In such cases the initial cokernel idea in the Serre tensor construction is less problematic, so we can avoid dualizing M and hence the method works with weaker hypotheses on M than projectivity. Here is a version for abelian varieties. 1.6.4.3. Proposition. Let A be an abelian variety over a field K. Let O ⊂ End(A) be a commutative subring. For any finitely generated O-module M , the functor T M ⊗O A(T ) on K-schemes has fppf sheafification that is represented by an abelian variety M ⊗O A. For an injective map M → N between torsion-free O-modules with finite cokernel, the induced map M ⊗O A → N ⊗O A is an isogeny. In particular, if O0 is a Z-flat O-algebra that 0 is an isomorphism then the natural map of is finitely generated as an O-module and OQ → OQ abelian varieties A → A0 := O0 ⊗O A is an isogeny and the identification End0 (A) = End0 (A0 ) carries O0 ⊂ End0 (A) into End(A0 ). The notation O0 ⊗O A should not be confused with the standard notation for affine base change of schemes. Proof. Choose a finite presentation of O-modules ϕ

Or −→ Os −→ M → 0. The map ϕ is given by an s×r matrix over O, and so it defines an analogous map [ϕ] : Ar → As between abelian varieties over K. Since we are working over a field, the map [ϕ] is faithfully flat onto its image abelian variety in As . Thus, the abelian variety quotient M ⊗O A := As /[ϕ](Ar )

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represents the cokernel of [ϕ] as fppf abelian sheaves over K. It follows (via the right-exactness of algebraic tensor products) that the abelian variety M ⊗O A represents the fppf sheafification of T M ⊗O A(T ). Let M → N be an injective map between torsion-free O-modules with finite cokernel. There is a map N → M such that both composites M → M and N → N are multiplication by a common non-zero integer n. Hence, we get maps in both directions between M ⊗O A and N ⊗O A whose composites are each equal to multiplication by n, so both maps between M ⊗O A and N ⊗O A are isogenies. The assertions concerning O0 follow by considering the functor represented by O0 ⊗O A. 1.6.5. Variant on Grothendieck’s theorem. Yu’s variant on Theorem 1.6.2.5 (which we will not need) asserts that we can first apply an isogeny and then pass to a finite extension on K (with no further isogeny involved) to get to a situation that descends to a finite field. This goes as follows. Consider the setup in Theorem 1.6.2.5. By Proposition 1.3.2.1, the simple factors all have sufficiently many complex multiplications, so we may and do focus on the case of simple abelian varieties A. Choose a polarization, so D = End0 (A) is endowed with a positive involution. By [86, 2.2], there is a maximal commutative subfield L ⊂ D that is stable under the involution, so L is either totally real or CM. We claim that L is a CM field, or in other words L is not totally real. To prove this property of L, first note that by Proposition 1.3.7.3 (in positive characteristic) the division algebra D is either of Type III or Type IV (in the sense of Theorem 1.3.7.2). Since L contains the center Z of D, for Type IV we get the CM property for L from the fact that Z is CM in such cases. For Type III, the key point is that Z is totally real and D is non-split at all real places of Z. We know that DL is split over L since L is a maximal commutative subfield of D, so L is not totally real. Hence, once again L is a CM field. Applying Proposition 1.6.4.3, we can pass to an isogenous abelian variety to arrange that OL ⊂ End(A). In this special case, it turns out that for some finite extension K 0 /K we can descend AK 0 with its OL -action to a finite field. Note that it actually suffices to just descend the abelian variety AK 0 to a finite field, as then a further finite extension on K 0 will enable us to descend the abelian variety along with its OL -action, by Lemma 1.2.1.2. 1.6.5.1. Theorem (Yu). Let K be a field with positive characteristic, and A an isotypic CM abelian variety over K with CM structure provided by a CM field L ⊂ End0 (A). If OL ⊂ End(A) then there is a finite extension K 0 /K such that AK 0 equipped with its OL -action descends to a finite field contained in K 0 . This result is [86, Thm. 1.3]; it will not be used in what follows.

1.7. CM lifting questions 1.7.1. Basic definitions and examples. Let κ be a field of characteristic p > 0, and consider an abelian variety A0 over κ. By Corollary 1.5.2.5, if κ is finite and A0 is isotypic then we may endow it with a structure of CM abelian variety having complex multiplication by a CM field. Inspired in part by Theorem 1.5.5.1, we wish to pose several questions related to the problem of lifting A0 to characteristic 0 in the presence of CM structures. First we make a general definition unrelated to complex multiplication. 1.7.1.1. Definition. A lifting of A0 to characteristic 0 is a triple (R, A, φ) consisting of a domain R of characteristic 0, an abelian scheme A over R, a surjective map R  κ, and an isomorphism φ : Aκ ' A0 of abelian varieties over κ. Let M = Frac(R), so AM denotes the generic fiber of such an abelian scheme A over R. It is harmless to replace R with its localization at the maximal ideal ker(R  κ), so we generally assume that R is local with residue field κ. If AM admits sufficiently many complex

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multiplications then we say that A is a CM lift of A0 to characteristic 0. The natural map End(A) → End(AM ) is injective, with torsion-free cokernel due to: 1.7.2. Lemma. Let A and B be abelian schemes over an integral scheme S with generic point η. The natural injective map Hom(A, B) → Hom(Aη , Bη ) has torsion-free cokernel. Proof. Consider f : Aη → Bη such that n · f extends to an S-group map h : A → B for a non-zero integer n. The restriction h : A[n] → B[n] between finite flat S-groups vanishes because such vanishing holds on the generic fiber over the integral S. Since [n] : A → A is an fppf covering with kernel A[n], it follows that h factors through this map over S, which is to say h = n · fe for some S-group map fe : A → B. Hence, the map feη − f ∈ Hom(Aη , Bη ) is killed by n, so feη = f . The injective map in Lemma 1.7.2 can fail to be surjective: 1.7.3. Example. Let p be a prime with p ≡ 3 (mod 4), so p is prime in Z[i] (with i2 = −1). Let R be the local order Z(p) + pZ(p) [i] of index p in Z(p) [i], so Frac(R) = Q(i). Let E be the elliptic curve y 2 = x3 − x viewed over R, so the generic fiber EQ(i) has endomorphism ring Z[i] via the action [i](x, y) = (−x, −iy). As we saw in Example 1.6.4.2, Z[i] acts on Lie(EQ(i) ) through scaling via the canonical inclusion Z[i] ,→ Q(i). We claim that End(E) = Z (so End0 (E) := Q ⊗Z End(E) = Q, even though the generic fiber EQ(i) has endomorphism algebra Q(i)). Indeed, if not then End(E) is an order in Z[i] = End(EQ(i) ), so by Lemma 1.7.2 we would have End(E) = Z[i]. In particular, the action by i on EQ(i) would extend to an action on E, and so the resulting multiplier action by i on the tangent line Lie(EQ(i) ) = Lie(E) ⊗R Q(i) would preserve the R-submodule Lie(E). But Lie(E) is a free R-module of rank 1 since R is local and E is R-smooth, so the i-action on this R-module would have to be multiplication by some element r ∈ R. By working over Q(i) we have seen that we get the multiplier i, so necessarily r = i. Since i 6∈ R due to the definition of R, we have a contradiction. In the preceding example, the base ring R is not normal. This is essential, since in the normal case there is no obstruction to extending maps between abelian schemes: 1.7.4. Lemma. For a normal domain R with fraction field M , the functor A abelian schemes over R to abelian varieties over M is fully faithful.

AM from

Proof. This is a special case of a general lemma of Faltings [24, §2, Lemma 1]. For normal R we get a specialization map End0 (AM ) = End0 (A) := Q ⊗Z End(A) → End0 (Aκ ) between endomorphism algebras, and likewise for endomorphism rings. This makes normality a natural property to impose on R when studying questions about CM lifts. In general, if R is not normal then End0 (AM ) may be larger than End0 (A), so it is not evident how to compare the endomorphism algebras on the generic fiber and the κ-fiber. Hence, for general R we just work with the specialization map of endomorphism algebras End0 (A) → End0 (Aκ ). This map can fail to be surjective. An elementary example is an elliptic curve over Z(p) for a prime p (since elliptic curves over finite fields always admit complex multiplication, by Corollary 1.5.2.5, whereas elliptic curves over Q have endomorphism algebra Q). In contrast with Lemma 1.7.2, the specialization map of endomorphism rings End(A) → End(Aκ ) can also have cokernel that is not torsion-free, even when R is normal. (In Chapter 4 we will see many natural examples of this phenomenon in our study of CM lifting problems, when we consider lifting questions for specific orders in CM fields; e.g., see 4.1.2.)

1.7. CM LIFTING QUESTIONS

43

1.7.5. CM lifting problems. To formulate the lifting questions that we shall study in subsequent chapters, let Fq be a finite field of size q and B an abelian variety of dimension g > 0 over Fq . Assume that B is isotypic over Fq (which is necessary and sufficient in order that B admit a structure of CM abelian variety with complex multiplication by a CM field, by Theorem 1.3.1.1 and Corollary 1.5.2.5). Let Bκ denote the scalar extension of B over a finite extension field κ/Fq . Consider the following five assertions concerning the existence of a CM lifting of B or Bκ to characteristic 0. • (CML) CM lifting: there exists a local domain R with characteristic 0 and residue field Fq , an abelian scheme A over R with relative dimension g equipped with a CM field L ⊂ End0 (A) := Q ⊗Z End(A) satisfying [L : Q] = 2g, and an isomorphism φ : AFq ' B as abelian varieties over Fq . • (R) CM lifting after finite residue field extension: there exists a local domain R with characteristic 0 and residue field κ of finite degree over Fq , an abelian scheme A over R with relative dimension g equipped with an action (in the isogeny category over R) by a CM field L with [L : Q] = 2g, and an isomorphism φ : A ×Spec(R) Spec(κ) ' Bκ as abelian varieties over κ. • (I) CM lifting up to isogeny: there exists a local domain R with characteristic 0 and residue field Fq , an abelian scheme A over R with relative dimension g equipped with an action (in the isogeny category over R) by a CM field L with [L : Q] = 2g, and an isogeny AFq → B of abelian varieties over Fq . • (IN) CM lifting to normal domains up to isogeny: there exists a normal local domain R with characteristic 0 and residue field Fq such that (I) is satisfied for B using R. • (RIN) CM lifting to normal domains up to isogeny after finite residue field extension: there exists a normal local domain R with characteristic 0 and residue field κ of finite degree over Fq such that (R) is satisfied for B using R except that φ is only required to be an isogeny over κ rather than an isomorphism. • (sCML) strong CM lifting: (CML) holds for any CM maximal commutative subfield L ⊂ End0 (B), which we may specify a priori. 1.7.5.1. Remark. By expressing a local ring as a direct limit of local subrings essentially of finite type over Z, in the formulation of (R) there is no loss of generality in replacing κ with an algebraic closure of Fq or allowing κ to vary over all extensions of Fq . Likewise, the normality condition in (RIN) is irrelevant because it can always be attained at the cost of a finite residue field extension (by a specialization argument as in 2.1.1). By Remark 1.5.5.2, (RIN) has an affirmative answer for any isotypic B over Fq , and the CM lift can be chosen using any CM maximal commutative subfield L ⊂ End0 (B). There are several questions we wish to answer in the direction of refining this fact: (1) Is a residue field extension necessary? That is, does (IN) hold for every B? (2) If (IN) does not hold for every B, can we characterize when it holds? And how about (I) in general (i.e., drop normality, but permit an isogeny without increasing the residue field)? (3) Is an isogeny necessary? That is, does (R) hold for every B (requiring the local domain R to be normal is not a constraint, since we are allowing a finite extension on κ; cf. Remark 1.7.5.1), or does even (CML) hold for every B? These questions can be made more specific in several respects. For example, since the Q-simple End0 (B) is usually non-commutative, it generally contains more than one CM maximal commutative subfield L (up to conjugacy) and so we can pose the CM lifting questions requiring an order in a particular choice of L to lift to a CM structure over R. We will give examples to show that the choice of such an L can affect the nature of the answer to some of the lifting questions. But even if we know that for a given B and choice of L ⊂ End0 (B) it is possible to construct a CM lift to characteristic 0 on which the action of an order in L

44

1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

also lifts, it could be that the full CM order L ∩ End(B) does not lift. We will give examples where this happens in 4.1.2. The proofs of the following answers form the backbone of subsequent chapters. • By [59, Thm. B], for any g > 2 there exist g-dimensional abelian varieties over an algebraic closure of Fp for which there is no CM lift to characteristic 0. These results are proved in a much stronger form in Chapter 3. Thus, (R) does not hold in general, so in particular (CML) sometimes fails to hold. Hence, an isogeny is necessary; that is, it is better to consider (I) than (CML). • Building on lifting results for p-divisible groups in Chapter 3, in Chapter 4 we prove that (I) holds for every B (so a strengthening of (RIN) holds, applying the isogeny before making a finite extension on the residue field). In fact, for any CM maximal commutative subfield L ⊂ End0 (B) we construct an isogeny B → A0 to an abelian variety over Fq such that A0 has a CM lift to characteristic 0 on which the action of the order Z + pOL in OL also lifts. However, A0 generally depends on L. • In contrast with the success with (I), if we want to impose a normality requirement on R and not increase the residue field (but permit isogenies) then the answer is negative. That is, we will give examples (in Chapter 2) for which (IN) fails. Hence, for the existence of a CM lifting to a normal domain of characteristic 0 we need to allow a finite extension of the initial finite field (and an isogeny), exactly as in Theorem 1.5.5.1. However, there is a good salvage: for each B and choice of L ⊂ End0 (B) we will give (in Chapter 2) concrete necessary and sufficient conditions in terms of a CM type Φ on L for (IN) to have an affirmative answer using a CM lifting to which the action of an order in L (in the isogeny category) also lifts and yields the specified CM type Φ. In Example 2.1.8 we will give examples of B for which this necessary and sufficient condition is satisfied for one choice of L ⊂ End0 (B) (and a suitable Φ) but fails for another choice (and any Φ). • We expect that (sCML) does not hold in general.

2

We trusted in the God who created integers and introduced what we called arithmetic invariants. Jun-Ichi Igusa

CM lifting over a discrete valuation ring 2.1. Introduction In this chapter we consider some obstructions which can be used to study various aspects of CM lifting problems from 1.7.5 over a normal base. The key observation is the residual reflex condition introduced in 2.1.6. In 2.2 we show that CM lifting up to isogeny is possible over Fq if the residual reflex obstruction is avoided in the strongest sense. In 2.3 we give counterexamples to the lifting condition (IN) from 1.7.5. The main result of this chapter (Theorem 2.1.7) is that the residual reflex condition is the only obstruction to the lifting problem (IN). Its proof is given by global methods in 2.5, building on preparations with algebraic Hecke characters in 2.4 (and the Main Theorem of Complex Multiplication, for which we provide a complete proof). An alternative proof via local methods is given in 2.6 by replacing the global arithmetic theory of complex multiplication with local arguments resting on p-divisible groups and p-adic Hodge theory. We prefer the global argument because it is more explicit about the number field over which a given abelian variety over a finite field can be lifted. But the local methods will be used in Chapter 3, which in turn underlies the arguments in Chapter 4. 2.1.1. Some deformation ring arguments. In the formulation of the lifting problem (I) in 1.7.5, there is no loss of generality in requiring R to be a complete local noetherian domain with residue field Fq . For such R, every maximal ideal m of R[1/p] has residue field R[1/p]/m of finite degree over the fraction field of W (Fq ) (by [35, 7.1.9]), and so by replacing R with its image in (the valuation ring of) R[1/p]/m we see that R can even be taken to be an order with residue field Fq in a p-adic integer ring. However, the finite residue field of R may increase after normalization. For this reason, on the one hand (IN) in 1.7.5 is a priori a stronger property than (I), but on the other hand (IN) is a natural condition to consider because abelian schemes over normal base schemes have the pleasant property that homomorphisms between generic fibers extend (uniquely) over the entire base (Lemma 1.7.4). In 2.2 we will give an elementary sufficient criterion for (I) to hold (and in Chapter 4 we will use deeper methods to prove that (I) always holds). In 2.3.5 this is combined with an arithmetic obstruction (explained in 2.1.6) to give absolutely simple examples in which (I) holds but (IN) fails. An interesting special case of (IN) is when the normal base ring R is a p-adic integer ring. We now explain why this is also essentially the general case. Suppose that (IN) holds for some B over Fq using a lifting B 0 of an isogenous B 0 over a characteristic-0 complete local normal noetherian domain R with residue field Fq . There is a polarization of the generic fiber of B 0 over R, and by the normality of R this extends to an isogeny λ : B 0 → B 0 t over R (Lemma 1.7.4) that is necessarily symmetric. This extended map is a polarization due to: 2.1.2. Lemma. Let A → S be an abelian scheme, and L a line bundle on A. The locus U of s ∈ S such that Ls is ample on As is both open and closed in S, and L |AU is U -ample. Proof. By [22, IV3 , 9.6.4], U is open and L |AU is U -ample. To prove that U is closed we can reduce to the case when S is the spectrum of a discrete valuation ring whose generic point η lies in U . We can also assume by passing to a positive power that Lη has a nonzero global section on Aη . The S-group map φL : A → At is an isogeny since it is so on the generic fiber. 45

46

2. CM LIFTING OVER A DISCRETE VALUATION RING

Letting s ∈ S be the closed point, the resulting finiteness of ker φLs implies that Ls is ample provided that it has a nonzero global section [53, §6, App. I]. But the finiteness ensures (by the Vanishing Theorem [53, §16]) that there is exactly one i such that hi (As , Ls ) is non-zero, and the same holds on the generic fiber. On the generic fiber we arranged non-vanishing for i = 0, so by the theorem on cohomology and base change we get the required non-vanishing on the special fiber for i = 0.

The polarization property for λ yields a natural local W (Fq )-algebra map D → R using a deformation ring D for B 0 equipped with its induced polarization and CM structure from B 0 . We claim that the image D1 of D in R is finite flat over W (Fq ). By Theorem 1.6.2.1, CM abelian varieties in characteristic 0 do not vary in positive-dimensional families in the sense that any CM abelian variety over an algebraically closed field k of characteristic zero descends to a number field M ⊂ k. Such a descent has potentially good reduction at all finite places of M (Theorem 1.6.2.3). Taking k to be a geometric generic point over Spec(D[1/p]), we see that the Zariski closure in Spec(D) of every irreducible component of Spec(D[1/p]) (with its reduced structure) has the form Spec(Γ) for a domain Γ that is finite flat over W (Fq ). Thus, the subring D1 ⊂ R is also finite flat over W (Fq ), as desired. e 1 of D1 is a local subring of R. Hence, The normality of R implies that the normalization D e e D1 has residue field Fq , so (IN) holds for B using D1 . In other words, if (IN) holds for B over Fq then up to applying an Fq -isogeny to B the appropriate universal deformation ring has a component of characteristic zero whose normalization is a p-adic integer ring with residue field Fq . 2.1.3. In 2.2 we will use deformation theory to prove that (I) in 1.7.5 is satisfied using any CM subfield L ⊆ End0 (B) such that [L : Q] = 2g for all p-adic places v of L (provided that such a subfield L exists, which we exhibit in some interesting examples in 2.3.5). In Chapter 4 we will prove (I) in general, but the special case in 2.2 is sufficient for some interesting examples in the present chapter. The condition (IN) is subtle because there is an arithmetic obstruction to it coming from classical CM theory. To explain this obstruction, fix an isotypic abelian variety B over Fq with dimension g > 0. Let R be a local normal domain with characteristic 0 and residue field Fq , and let A be an abelian scheme over R with relative dimension g equipped with an action (in the isogeny category over R) by a CM field L with [L : Q] = 2g. Assume that the special fiber AFq is isogenous to B. By direct limit arguments, we can arrange that R is essentially of finite type over Z, so it is excellent. Thus, we can replace R with its completion without losing the normality hypothesis (and this does not change the residue field). Since M := Frac(R) has characteristic 0, the action on A over R by an order in OL uniquely extends to an M -linear action by L on the g-dimensional Lie(AM ). By Lemma 1.4.2, this action viewed over an algebraic closure M decomposes into eigenlines on which L acts through a set Φ of representatives for Homring (L, M ) modulo the action on the CM field L by its intrinsic complex conjugation. We may of course view Φ as a subset of Homring (L, Qp ), where Qp is the algebraic closure of Qp inside of M . (This is called a p-adic CM type to emphasize that its target is Qp , so every φ ∈ Φ induces a p-adic place on L.) 2.1.4. Definition. For a CM field L and a CM type Φ on L valued in an algebraically closed field F of characteristic 0, the reflex field E ⊂ Q ⊆ F is the number field for which Gal(Q/E) is the subgroup of elements of Gal(Q/Q) stabilizing the subset Φ ⊂ Homring (L, Q). Observe that whereas L is an abstract field, by definition the reflex field E associated to (L, Φ) is a subfield of Q.

2.1. INTRODUCTION

47

2.1.4.1. Example. Reflex fields have an interpretation in terms of abelian varieties, as follows. Consider a CM type (L, Φ) arising from an abelian variety A over a field M of characteristic 0 equipped with a choice of algebraic closure M of M . By Galois descent, the reflex field E is the unique minimal subfield of M over which the linear action of L on the M -vector space Lie(A) can be defined. In particular, E ⊂ M . An equivalent description of E is that it is the subfield of M generated over Q by the M -linear traces of the action of every c ∈ L. Proposition 1.4.4.1 suggests that E should be viewed as a “field of definition” for the L-isogeny class of CM abelian varieties over F with CM type (L, Φ). However, it can happen that no member of the isogeny class is actually defined over E. For example, the reflex field of a CM elliptic curve over Q is equal to the associated imaginary quadratic field K (or rather, its canonical image in Q), but it follows from classical CM theory (and Proposition 1.6.4.3) that any field of definition for a member of the isogeny class must contain the Hilbert class field of K. Hence, if K does not have class number 1 then no such elliptic curve can be defined over the reflex field. We will soon see an alternative useful interpretation of E, as a genuine field of definition. 2.1.4.2. Remark. The reflex field E is always a CM field. Indeed, from the description via traces it is clear that E is a subfield of a CM field (such as the Galois closure of L over Q), so it is either CM or totally real. But if it were totally real then a complex conjugation would lie in Gal(Q/E), contradicting the characteristization of this Galois group in terms of the CM type Φ. Since we have introduced the reflex field E, we now briefly digress to introduce the concept of the reflex norm thatQwill play an essential role in the Main Theorem of Complex Multiplication. Consider ϕ∈Φ Q as an L ⊗Q Q-module by making c ∈ L act via c.(xϕ )ϕ∈Φ = (ϕ(c)xϕ )ϕ∈Φ . This is Gal(Q/E)-equivariant when σ ∈ Gal(Q/E) acts in the Q natural way on the right tensor factor of L ⊗Q Q and acts on ϕ∈Φ Q via σ(xϕ ) = (σ(xσ−1 ◦ϕ )). Applying Galois descent, this defines a L ⊗Q E-module tΦ canonically attached to (L, Φ). An alternative characterization of tΦ as an L ⊗Q E-module is that the E-linear action of that P action on it by each c ∈ L has E-linear trace ϕ∈Φ ϕ(c) ∈ E. This determines tΦ up to isomorphism. In the setup of Example 2.1.4.1, tΦ is the L ⊗Q E-linear tangent space of a CM abelian variety over E with CM type (L, Φ) if such an abelian variety exists. However, we saw that usually no such abelian variety exists over E. 2.1.4.3. Definition. Let Φ be a Q-valued CM type on a CM field L, and let E ⊂ Q be the relflex field. The reflex norm NΦ : ResE/Q (Gm ) → ResL/Q (Gm ) is defined via the L-linear determinant of the E-action on tΦ . That is, for a Q-algebra R and × a unit a ∈ ER , NΦ (a) ∈ L× R is the LR -linear determinant of a-multiplication on tΦ ⊗Q R. We now give a useful re-interpretation of the reflex field E and reflex norm NΦ attached to a CM type (L, Φ). For any algebraically closed field F ⊃ Q and any subset Σ ⊂ Hom(L, F ) = Hom(L, Q), consider the F -homomorphism Y hΣ : Gm → Gm = ResL/Q (Gm )F j∈Hom(L,F )

to be the map t 7→ (t, . . . , t, 1, . . . , 1) whose jth component is trivial if j 6∈ Σ and is the identity map if j ∈ Σ. This map clearly descends to the analogous such map defined over Q, and even over the finite Galois splitting field of L over Q. Relative to the evident Q-structures on its source and target, it is clear that the minimal field of definition over Q for hΣ is the number

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2. CM LIFTING OVER A DISCRETE VALUATION RING

field in Q corresponding to the open subgroup of Gal(Q/Q) whose elements are those σ such that σ ◦ Σ = Σ. In particular, hΦ has its minimal field of definition over Q equal to the reflex field E of (L, Φ). Writing hΦ to also denote the descent to E, this E-homomorphism encodes the reflex norm NΦ : 2.1.4.4. Proposition. For any finite separable extension of fields K/k and any commutative affine k-group C of finite type, let NK/k : ResK/k (CK ) → C denote the k-homomorphism that descends the ks -homomorphism Y Ck s → Ck s ResK/k (CK )ks = Res(ks ⊗k K)/ks (Cks ⊗k K ) = j∈Homk (K,ks )

Q

defined by (cj ) 7→ j cj . The composite map ResE/Q (Gm )

ResE/Q (hΦ )

/ ResE/Q (ResL/Q (Gm )E )

NE/Q

/ ResL/Q (Gm )

is NΦ . Proof. Due to the unirationality of tori over fields, the set E × of Q-points in ResE/Q (Gm ) is Zariski-dense. Hence, it suffices to verify the identity on Q-points. That is, for x ∈ E × , we claim that NΦ (x) ∈ L× equals the norm of hΦ (x) ∈ ResL/Q (Gm )(E) = (E⊗Q L)× relative to the finite flat ring homomorphism L → E ⊗Q L. By definition, NΦ (x) is the E-linear determinant of multiplication by x on the E ⊗Q L-module tΦ . But tΦ is locally free over E ⊗Q L with rank 6 1, so it is invertible over a factor ring. By construction, this factor ring as an Ealgebra has its geometric support equal to the product of factors of Q ⊗E (E ⊗Q L) = Q ⊗Q L corresponding to the elements of Φ ⊂ Hom(L, Q). That is, E ⊗Q L = RΦ × RΦ0 where RΦ is the E-algebra whose Q-points (over E) correspond to Φ and we define Φ0 := Hom(L, Q) − Φ. Thus, NΦ (x) = NmRΦ /L (xΦ ) where x ⊗ 1 = (xΦ , xΦ0 ), so NΦ (x) = NmRΦ /L (xΦ )NmRΦ0 /L (1) = N(RΦ ×RΦ0 )/L (xΦ , 1) = Nm(E⊗Q L)/L (xΦ , 1). It now suffices to prove that hΦ (x) = (xΦ , 1) in E ⊗Q L for x ∈ E × . Using extension of × scalars along the canonical inclusion E ,→ Q, for any x ∈ Q the element hΦ (x) ∈ (Q⊗Q L)× = Q × 0 j Q has j-component equal to 1 for j ∈ Φ and equal to x for j ∈ Φ. Since the decomposition Q Q ⊗Q L ' j Q carries the Q-algebra structure on the left over to the diagonal one on the right, the element x ⊗ 1 on the left goes Q over to the diagonal embedding of x on the right. Hence, if x ∈ E then xΦ ∈ Q ⊗E RΦ = Φ Q is the diagonally embedded x, so we are done. The proof of the Main Theorem of Complex Multiplication will rest on the fact that for any CM type (L, Φ), the image of the reflex norm NΦ lies in a certain natural subtorus of ResL/Q (Gm ) that we now define. Let NmL/L0 : ResL/Q (Gm ) → ResL0 /Q (Gm ) denote the usual norm; its kernel is connected and hence is a Q-torus (as we verify via a calculation with character groups over Q). Hence, the preimage T = Nm−1 L/L0 (Gm ) is a Q-subtorus of ResL/Q (Gm ). This torus has the following important property. 2.1.4.5. Proposition. The image of NΦ is contained in T . Proof. Since the locus of Q-points is Zariski-dense in any Q-torus (due to the unirationality of tori, valid over any field), it suffices to prove that for any a ∈ ResE/Q (Gm )(Q) = E × , the element NΦ (a) ∈ L× lies in T (Q). In other words, we claim that NmL/L0 (NΦ (a)) ∈ Q× . In fact, we will prove that this norm coincides with NmE/Q (a). By definition, NΦ (a) = detL (a : tΦ → tΦ ). But NmL/L0 (c) = cc∗ where c 7→ c∗ is the complex conjugation involution of L over L0 , and this involution carries the CM type

2.1. INTRODUCTION

49

Φ on L to its complement Φ∗ in the set of embeddings of L into Q. Hence, inside of L we have NmL/L0 (NΦ (a)) = NΦ (a)NΦ∗ (a). This product of L-linear determinants is the L-linear determinant of multiplication by a on the L ⊗Q E-module direct sum tΦ ⊕ tΦ∗ . But this direct sum is a free L⊗Q E-module of rank 1, as we may check via scalar extension along the inclusion E ,→ Q, so we arrive at the L-linear determinant of the natural a-multiplication on L ⊗Q E. This is NmE/Q (a) ∈ Q× . 2.1.5. CM lifting obstruction. Returning to the topic of CM lifting problems, observe that by the normality of R, the inclusion of fields E ⊂ M implies the inclusion of rings OE ⊂ R. But R is local with residue characteristic p, so its maximal ideal contracts back to a prime ideal p of OE over p. Note that this p-adic place on E is also induced by the inclusion of E into Qp arising from the definition of E in terms of (L, Φ). By the construction of p in terms of R, we see that OE /p occurs as a subfield of the residue field Fq of R. Thus, if (IN) in 1.7.5 holds for B over Fq using some CM field L ⊆ End0 (B) then there is a p-adic CM type Φ on L whose associated reflex field E ⊂ Qp has residue field at the induced place over p that is not “too large” in the sense that it can be embedded into Fq . This is an arithmetic restriction on the p-adic CM type Φ. There is a further restriction on the p-adic CM type Φ ⊂ Homring (L, Qp ) = Homring (L, M ) of an L-linear CM abelian scheme over R that lifts the L-linear CM abelian variety B over Fq . This restriction will not require normality of the complete local noetherian domain R, and so it is a necessary condition on CM types arising in (I) in 1.7.5 when R there is noetherian and complete. To explain this restriction, first note that since L is its own centralizer in End0 (B), and End(B) ∩ L is an order in OL , there is a unique element FrB,q ∈ OL acting on B by the q-Frobenius endomorphism. Assume that there is an L-linear isogeny between B and A0 := AFq for some A over R as in (I) with R noetherian and complete. This L-linear isogeny is compatible with q-Frobenius endomorphisms, so the element FrA0 ,q ∈ OL is equal to FrB,q . The Shimura–Taniyama formula for A relates the factorization of FrA0 ,q OL = FrB,q OL to the p-adic CM type Φ ⊂ Homring (L, Qp ) = Homring (L, M ) arising from the L-action on Lie(AM ): for each p-adic place v on L, (2.1.5.1)

ordv (FrB,q ) #{φ ∈ Φ : φ induces v on L} = ; ordv (q) [Lv : Qp ]

the choice of normalization of ordv cancels out on the left side, and the right side is independent of the choice of Qp . The ratios on the left side (as v varies) are called the slopes of B (over Fq ). More generally, if C is an abelian variety over Fq with q = pf and if FC ∈ Z[T ] is the associated characteristic polynomial of FrC,q then as λi varies through the zeros of FC in an algebraic closure Qp of Qp , the numbers ordp (λi )/f (usually counted with multiplicity) are called the slopes of C over Fq ; here, ordp is normalized by the condition ordp (p) = 1. Obviously the formation of the slopes is invariant under isogeny and finite extension of the base field. In the case that R is a p-adic integer ring, the formula (2.1.5.1) was first proved by Shimura and Taniyama using global arguments, and a local proof was given by Tate using p-divisible groups [77, §5]. The role of a Dedekind base in these proofs is to make the CM order become maximal upon passing to an isogenous abelian scheme. That is, for Dedekind R, an abelian Rscheme A with an action by L in the isogeny category over R admits an isogeny to an abelian R-scheme A0 on which the induced L-action in the isogeny category arises from an action of OL on the abelian scheme A0 . This is proved by using Proposition 1.6.4.3 on the generic fiber, together with R-flatness of some scheme-theoretic closures from the generic fiber. Difficulties with R-flatness of scheme-theoretic closure for non-Dedekind R make it unclear if such an A0 can be found for general R. Nonetheless, we have:

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2.1.5.1. Proposition. The formula (2.1.5.1) holds for any complete local noetherian domain R. Proof. First we treat the case dim(R) = 1. Let κ be the residue field of the R-finite e of R and let e = [κ : Fq ] > 1. Clearly FrB ,#κ = Fre in OL since κ has size normalization R κ B,q e (equivalently, for Bκ ) q e . Thus, the Shimura–Taniyama formula for ARe over the Dedekind R implies the same for A (equivalently, for B). The general case can be reduced to the case of a 1-dimensional base by using an argument with universal deformation rings similar to what was done in 2.1.1, but here is a more concrete argument via direct limit and specialization arguments. Let M denote the fraction field of R, O ⊆ OL an order that acts on A over R, and O0 ⊆ Qp a p-adic integer ring that is large enough to contain φ(OL ) for all φ ∈ Φ. By applying standard direct limit arguments to the eigenline decomposition of Lie(A) ⊗R M over M via Φ, there is a nonzero r ∈ R and an O0 -subalgebra R0 ⊆ M that is finite ´etale over R[1/r] and contains O0 such that Lie(A) ⊗R R0 decomposes into a direct sum of free R0 -modules of rank 1 on which O acts by the maps φ : O → O0 ⊆ R0 . We can assume that p|r in R, so Spec(R[1/r]) is a dense open subscheme of Spec(R[1/p]). Since Spec(R[1/p]) is Jacobson (e.g., due to [45, Lemma 1], or [35, 7.1.9] and the Jacobson property of affinoid algebras [7, 5.2.6/3]), we can therefore choose a maximal ideal n ∈ MaxSpec(R[1/p]) that does not contain r. Let p be the corresponding prime ideal of R. The local noetherian quotient domain R0 = R/p is clearly contained in the valuation ring of the finite extension R0 [1/p] = R[1/r]/n of Frac(W (Fq )), so it is an order in a p-adic integer ring. The quotient R0 /p is finite ´etale over R0 , so upon choosing a local factor ring of R0 /p we see that A mod p over R0 equipped with its action by O has a CM type that is naturally identified with Φ and has special fiber A0 over Fq (compatibly with the O-actions). Hence, the Shimura–Taniyama formula for A over R follows from the settled case of the formula for A mod p over the 1-dimensional R0 .

2.1.6. Residual reflex condition. The considerations in 2.1.5 show that if (IN) in 1.7.5 holds for an isotypic abelian variety B over Fq with dimension g > 0, then there is a CM subfield L ⊆ End0 (B) with [L : Q] = 2g and a p-adic CM type Φ ⊂ Homring (L, Qp ) such that (L, Φ) satisfies the following residual reflex condition: (i) The slopes of B are given in terms of (L, Φ) by the Shimura–Taniyama formula (2.1.5.1). (ii) Let E ⊂ Qp be the reflex field attached to (L, Φ), and let v be the induced p-adic place of E. The residue field κv of OE,v can be realized as a subfield of Fq . For example, if g = 1 then p splits in L if and only if the elliptic curve B is ordinary, so it is straightforward to check that the residual reflex condition is satisfied for one choice (resp. both choices) of Φ when B is ordinary (resp. supersingular). That is, for elliptic curves over finite fields the residual reflex condition can be satisfied for any imaginary quadratic subfield of the endomorphism algebra. We do not consider (i) in the residual reflex condition to be a serious constraint because Remark 1.5.5.2 shows that any CM subfield L ⊆ End0 (B) with [L : Q] = 2g (which exists, by Corollary 1.5.2.5) admits a p-adic CM type Φ for which (i) holds. Condition (ii) is more interesting because in 2.3.5 we will use Honda–Tate theory to give absolutely simple examples such that L = End0 (B) is a CM field of degree 2g and every p-adic CM type Φ on L satisfying the Shimura–Taniyama formula violates (ii). Moreover, we will show by elementary methods (i.e., not requiring the general case in Chapter 4) that these examples satisfy (I) in 1.7.5. Thus, the (second part of the) residual reflex condition is a nontrivial obstruction to the validity of (IN) in 1.7.5 for a given triple (B, L, Φ) over Fq , and this obstruction is eliminated by passing

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to a finite extension Fq0 ⊇ Fq if and only if Fq0 is large enough to contain the residue field κv as in (ii). In these absolutely simple counterexamples to (IN) that satisfy (I), B is an `-dimensional abelian variety over Fp with two distinct slopes over Fp , where ` > 5 is an arbitrary prime and p lies in a certain non-empty set of congruence classes depending on `. It is natural to ask whether there are obstructions to (IN) other than the residual reflex condition. We show in 2.5 that the combinatorial residual reflex condition is the only obstruction for (IN). This amounts to a refinement of the general validity of (RIN) from 1.7.5 (see Remark 1.5.5.2), and it is the main result of this chapter: 2.1.7. Theorem. Let B be an isotypic abelian variety of dimension g > 0 over Fq and let L ⊆ End0 (B) be a CM field with [L : Q] = 2g. Let Φ ⊂ Homring (L, Qp ) be a p-adic CM type, and let E ⊂ Qp be the associated reflex field. Assume that (L, Φ) satisfies the residual reflex condition in 2.1.6. There exists a finite extension E 0 /E inside of Qp , a g-dimensional abelian variety A over E 0 with good reduction at the p-adic place v 0 on E 0 induced by Qp , and an inclusion L ,→ End0 (A) with associated p-adic CM type Φ such that the reduction of A at v 0 is L-linearly isogenous to B over an isomorphism of finite fields κv0 ' Fq . In particular, B satisfies (IN) from 1.7.5 using a lifting of the L-action over a p-adic integer ring with residue field Fq . Note that the choice of isomorphism κv0 ' Fq in Theorem 2.1.7 does not matter, since B is L-linearly isogenous to B (p) via the relative Frobenius isogeny (and the natural isomorphism B (q) ' B is L-linear). The main ingredient in the proof of Theorem 2.1.7 is a purely arithmetic result in Theorem 2.4.11 that gives a procedure to modify algebraic Hecke characters. (A review of algebraic Hecke characters is given in 2.4.) The arithmetic theory of complex multiplication will tell us which algebraic Hecke characters comes from abelian varieties with complex multiplication (see Theorem 2.5.2), so the algebraic Hecke character produced by Theorem 2.4.11 will give us a CM abelian variety over a number field E 0 /E. This will be a CM lifting of B (up to isogeny) as required in Theorem 2.1.7. After assembling the necessary background in 2.2–2.4, the statement of of Theorem 2.1.7 will be recalled again in Proposition 2.5.3, after which we give the proof. In 2.6 we give an alternative proof of Theorem 2.1.7 that uses a local analogue of Theorem 2.4.11 (see Lemma 2.6.1) but replaces the global arithmetic theory of complex multiplication with local arguments resting on p-divisible groups and p-adic Hodge theory. The preceding considerations, including the formulation of the residual reflex condition, can be generalized to the case of abelian varieties with sufficiently many complex multiplications by a CM algebra (i.e., a product of finitely many CM fields). We leave it to the reader to give a precise statement of this generalization because the proofs are immediately reduced to the case when the CM algebra is a CM field. 2.1.8. Example. To illustrate the necessity and sufficiency of the residual reflex condition, let B = E × E over Fp2 where E is an elliptic curve with πE = −p. By Corollary 1.5.2.2(1), such an E exists and its geometric endomorphisms are defined over Fp2 , with D := End0 (E) the unique quaternion division algebra over Q ramified at exactly p and ∞. In particular, B has endomorphism algebra Mat2 (D). Let L1 = Q(α) with α4 = p; this is a quartic CM field √ (via the real embedding of its unique quadratic subfield Q(α2 ) 1 defined via α2 7→ − p) and p is totally ramified in L1 . Let L2 be a Z/(4)-extension of Q that is CM and in which p is totally inert (i.e., Frobp generates the Galois group). Such fields L2 are easy to construct. In both cases there is only one p-adic place on Li , and DLi splits away from that place (the archimedean places being clear since Li is totally complex), so Mat2 (D) is split by each Li . Hence, by Proposition 1.2.3.1 each Li embeds Mat2 (D) as a maximal commutative subfield. 1Error: this is not a CM field, since it has a real place. Need to find a correct example.

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The only CM subfield of L2 is itself, at which the p-adic place has residue field Fp4 , so part (ii) of the residual reflex condition is √ violated and (IN) fails for (B, L2 ). On the other hand, the Galois closure of L1 over Q is L1 ( −1), and this Galois closure contains the reflex field E of any CM type on L1 , so the p-adic places of any such E embed into Fp2 . Thus, by Theorem 2.1.7, (B, L1 ) satisfies (IN).

2.2. Existence of CM lifting up to isogeny Let B be an isotypic abelian variety of dimension g > 0 over a finite field Fq with size q that is a power of a prime p. In this section we give a sufficient criterion for the condition (I) in 1.7.5 to be satisfied by B. (The general case is treated in Chapter 4.) By Corollary 1.5.2.5, there exists a CM subfield L ⊆ End0 (B) with [L : Q] = 2g. Such an L is its own centralizer in End0 (B), so there is an element π ∈ OL whose action on B is the q-Frobenius endomorphism FrB,q . As we noted in 2.1.6, there exists a p-adic CM type Φ on L that satisfies the Shimura–Taniyama formula: for each p-adic place v of L, ordv (π) #{φ ∈ Φ | φ induces v on L} = . ordv (q) [Lv : Qp ] Fix such a Φ. We want to prove that (I) holds for B using the CM type (L, Φ), subject to a restriction on the behavior of p in L (given in Proposition 2.2.1 below). Let O = L ∩ End(B) be the associated CM order in OL . Since OL is a finite O-algebra with the same fraction field as O, by Proposition 1.6.4.3 there is an L-linear isogeny B → B 0 where B 0 has CM order equal to OL . Thus, upon replacing B with B 0 it suffices to consider the case when O = OL . 2.2.1. Proposition. With notation and hypotheses as above, assume in addition to OL being the CM order that OL /p = Fp for every prime p of L over p. There exists an order R in a p-adic integer ring and an abelian scheme A over R equipped with an action by OL having CM type (L, Φ) as above such that R has residue field Fq and AFq is OL -linearly isomorphic to B. In particular, B satisfies (CML) and hence satisfies (I). Some hypothesis on L at p is required in this proposition, since otherwise Remark 2.3.4 gives counterexamples via a tangential obstruction (the Kottwitz invariant). Beware that this does not mean that such examples cannot satisfy (I), since we may just need to pass to another member of the L-linear Fq -isogeny class (perhaps with non-maximal CM order) to get the required lift over a possibly non-normal base; this is what will be done in Chapter 4. Proof. By Remark 1.5.5.2, we may find a number field F ⊂ Qp and a CM abelian variety A1 over F with good reduction at the induced p-adic place w on F such that: the CM type is (L, Φ), the CM order is OL , and there exists an OL -linear isogeny φ1 : A1 → Bκw . (In particular, κw is endowed with a structure of extension of Fq .) The kernel ker φ1 is a finite OL -submodule scheme of the g-dimensional A1 . We claim that there is a (unique) nonzero ideal I ⊆ OL such that ker φ1 = A1 [I]. This is obvious on `-primaryQparts for ` 6= p since the `-adic Tate module T` (A1 ) is an invertible module over OL,` = q|` OL,q . The case of p-primary parts requires more care because the (contravariant) Dieudonn´e module D(A1 [p∞ ]) is a module over W (κw ) ⊗Zp OL,p rather than over OL,p . Since the natural map Lp := Qp ⊗Q L ,→ Qp ⊗Q End(A1 ) → End(A1 [p∞ ] is injective (Proposition 1.2.5.1), by Proposition 1.4.5.5 (with F = Lp , κ1 = κ = κw ) we see that D(A1 [p∞ ]) is invertible over W (κ) ⊗Zp OL,p . Since D(ker φ1 ) is a quotient of D(A1 [p∞ ]) as modules over the Dieudonn´e ring Dκw over W (κw ), by W (κw ) ⊗Zp OL,p -module freeness of D(A1 [p∞ ]) the kernel of the projection to this quotient is J ·D(A1 [p∞ ]) for a uniquely determined ideal J ⊆ W (κw )⊗Zp OL,p that is invertible (since D(ker φ1 ) has finite length over W (κw )). Each local factor OL,p of OL,p has residue field

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Fp , so each the factor W (κw ) ⊗Zp OL,p of W (κw ) ⊗Zp OL,p is a local unramified extension of OL,p . Hence, J uniquely descends to an invertible ideal Ip ⊆ OL,p . 2 This provides the p-part of the desired ideal I ⊆ OL , so we have constructed I. Let A1 denote the N´eron model of A1 over the complete local ring OF,w of OF at w, so A1 is an abelian scheme over OF,w and the OL -action on A1 uniquely extends to an action on A1 . Since the algebraic localization OL,(`) is a principal ideal domain for every rational prime `, functorial consideration of primary parts yields that the torsion subscheme A1 [I] is a finite flat OF,w -subgroup scheme of A1 (even if I is not globally principal). Thus, A2 = A1 /(A1 [I]) is an abelian scheme over OF,w endowed with a natural action by OL . Moreover, the special fiber A2 of A2 is OL -linearly isomorphic to A1 /(A1 [I]) = A1 /(ker φ1 ) ' Bκw . Hence, by replacing A1 with A1 /(A1 [I]) we can arrange that φ1 is even an isomorphism. That is, A1 is an OL -linear deformation of Bκw . A polarization λ1 of A1 over OF,w induces a polarization on Bκw , but to get such a λ1 whose reduction over κw descends to B over Fq it is convenient (and also useful for later purposes in Theorem 2.2.3) to construct polarizations that interact well with the OL -action in rather more general situations: 2.2.2. Lemma. Let C be an abelian variety of dimension g > 0 over an arbitrary field k, and let L be a CM field of degree 2g over Q equipped with an action on C in the isogeny category over k. Let L0 ⊂ L be the maximal totally real subfield, and let L act on the dual abelian variety C t (in the isogeny category) via composition of the dual action with complex conjugation on L. (1) There exists an L-linear polarization λ : C → C t . t (2) Assume OL ⊂ End(C). The OL0 -module Homsym OL (C, C ) of symmetric OL -linear homomorphisms over k is invertible, and likewise the OL -module HomOL (C, C t ) of OL -linear homomorphisms over k is invertible over OL . By Remark 1.4.3.1, when char(k) = 0 the definition of the CM structure on the dual abelian variety preserves the CM type. Proof. By Proposition 1.6.4.3 and the functorial properties of polarizations, for the proof of (1) we may assume OL ⊂ End(C). Hence, we may and do make this assumption for the remainder of the argument. Consider the OL -module HomOL (C, C t ) of OL -linear maps over k. The additive endomorphism h 7→ h0 := ht ◦ ιC of HomOL (C, C t ) (where ιC : C ' C tt is the canonical isomorphism) is semi-linear over complex conjugation on OL due to how the OL -action on C t is defined, t 00 and its fixed set is the OL0 -module Homsym OL (C, C ). Moreover, h = h for all h because ιC t is inverse to ιtC , so to prove the two invertibility claims it suffices to prove that HomOL (C, C t ) is an invertible OL -module. It is clear that HomOL (C, C t ) is a finite flat OL -module, and by elementary rank considerations with `-adic Tate modules for ` 6= p we see that its rank is at most 1. Thus, invertibility is reduced to being nonzero. Step 1. Consider k with characteristic 0. For an algebraic closure k of k, we can descend the CM abelian variety Ck (with its CM structure) to a number field (Theorem 1.6.2.1), so there is a finite Galois extension k 0 /k such that Ck0 with its CM structure descends to a CM abelian variety (A, α) over a number field F ⊆ k 0 . The complex-analytic theory (Example 1.4.3) provides an L-linear polarization on a complex fiber of A (due to how we define the L-action on At ). By Lemma 1.2.1.2 this descends to the corresponding Q-fiber of A, and so also to a finite Galois extension F 0 /F . Thus, by increasing k 0 to split F 0 /F we can find an L-linear polarization λ0 : Ck0 → Ckt 0 of Ck0 . The sum of the Gal(k 0 /k)-conjugates of the L-linear map λ0 descends to the desired polarization over k, and in particular this is a nonzero symmetric map. Step 2. For k with characteristic p > 0 we will proceed in reverse, first proving that HomOL (C, C t ) is nonzero and then afterwards using this to infer the existence of an L-linear 2unramified descent

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polarization. Since C must be isotypic (Theorem 1.3.1.1), End0 (C) is a simple Q-algebra. Its center is identified with a subfield Z ⊆ L, and the image Z ∗ ⊆ L of Z under complex conjugation is identified with the center of End0 (C t ) when using the action of L on C t through duality and complex conjugation. Let h : C → C t be a polarization. The associated Rosati involution on End0 (C) restricts to an involution τ (perhaps trivial) on the center Z. By (the proof of) Lemma 1.3.6, either Z is totally real and τ is the identity or Z is a CM field and τ is its complex conjugation. Either way, Z ∗ = Z inside of L and the Rosati involution is complex conjugation on Z. This says that h is Z-linear via the definition of the L-action on C t . It also follows that the center of End0 (C t ) is the subfield Z of L ⊆ End0 (C t ). The finite-dimensional Q-algebras End0 (C) and End0 (C t ) are simple Q-algebras (by kisotypicity of C) in which the center is Z ⊆ L, so the Z-linear h induces a Z-algebra isomorphism ψ : End0 (C) ' End0 (C t ). This may not be L-linear, but we can use the Skolem–Noether theorem to find a unit u ∈ End0 (C)× such that composing ψ with conjugation by u is an Lalgebra isomorphism. Thus, after replacing u with a nonzero integral multiple that lies in End(C), h ◦ u is an OL -linear k-isogeny C → C t (perhaps not a polarization). So far, char(k) could have been arbitrary. Step 3. Now we construct L-linear polarizations (over k) in positive characteristic. We can assume that k is finitely generated over Fp , so k is the function field of an integral variety U over the finite field κ that is the algebraic closure of Fp in k. By shrinking U we can arrange that the CM abelian variety C over k spreads out to a CM abelian scheme C over U . Since Hom(C, C t ) is finitely generated, by further shrinking we may assume that every homomorphism C → C t over k = κ(U ) extends (uniquely) to a U -homomorphism C → C t , and so for a choice of closed point u ∈ U we get a specialization map Hom(C, C t ) → Hom(Cu , Cut ) that is injective. This induces an injection between the invertible OL -modules of OL -linear homomorphisms, so this latter injection is a finite-index inclusion. Hence, if the case of finite fields is settled then for a choice of L-linear polarization λu of Cu over the finite field κ(u) we can replace λu with a positive integral multiple so that it arises from an L-linear map λ : C → C t over k = κ(U ). By Lemma 2.1.2 this is a polarization because its extension over U specializes to a polarization on some fiber (namely, over u). It remains to construct an L-linear polarization of C when k is a finite field κ, and it suffices to do this after replacing κ with a finite extension κ0 (either due to the same finiteindex inclusion trick as above, or by using the more elementary Galois descent trick used in characteristic 0). By Remark 1.5.5.2, after making a finite extension on κ we can therefore assume that there is an L-linear κ-isogeny f : C → D onto the reduction D of an abelian scheme D over the valuation ring of a p-adic field F with residue field κ such that DF has CM by L lifting that on the reduction (but its CM order may not be OL ). By Proposition 1.6.4.3 we can choose an L-linear isogeny DF → DF0 onto an abelian variety 0 DF over F having CM order OL , and since OF is Dedekind this extends to an L-linear isogeny D → D 0 onto an abelian scheme D 0 over OF with CM order OL . We may compose f with the reduction of this latter isogeny to get to the case when D has CM order equal to OL . By functoriality properties of polarizations we can rename D as C, which is to say that we can assume that C equipped with its OL -action is identified with the reduction of an abelian variety CF equipped with an OL -action over a p-adic field F with residue field κ. Any L-linear polarization of CF has reduction on C that is the desired L-linear polarization over κ (using Lemma 2.1.2). To apply Lemma 2.2.2 in the proof of Proposition 2.2.1, let λ1 : A1 → At1 be an OL -linear polarization as in the setup preceding the statement of Lemma 2.2.2. Its Fw -fiber extends to an OL -linear homomorphism λ1,w : A1 → At1 over OF,w that is a polarization (Lemma 2.1.2).

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This reduces to an OL -linear polarization λ1 on Bκw . But the inclusion HomOL (B, B t ) → HomOL (Bκw , Bκt w ) between invertible OL -modules must be a finite-index inclusion, so by replacing λ1 with a positive integral multiple we can assume that the polarization λ1 over κw descends to a polarization λB of B. (This λB is also OL -linear, but that will not be needed.) Consider the deformation functor Def(B, iB , λB ) of the polarized abelian scheme (B, λB ) equipped with OL -action via the canonical inclusion iB : OL → End(B). This functor is the subfunctor of Def(B, λB ) defined by requiring that the OL -action on B lifts (necessarily uniquely) to the deformation of B. The deformation functor Def(B, iB , λB ) considered on the category of all complete local noetherian W (Fq )-algebras (with residue field possibly larger than Fq , such as κw ) is represented by a complete local noetherian W (Fq )-algebra R with residue field Fq . The polarized abelian scheme over OF,w provided by A1 equipped with its OL -action and polarization λ1,w (and residual isomorphism φ1 giving the deformation structure with respect to (B, iB , λB ) over Fq ) is classified by a local W (Fq )-algebra homomorphism f : R → OF,w . Let R = R/Ker(f ), so R is an order with residue field Fq in a p-integer ring contained in OF,w . The pullback to SpecR of the universal deformation, equipped with its pullback OL -action and residual isomorphism to B over Fq , is the desired CM lifting of (B, iB ) to characteristic 0 without increasing the residue field (and it has the same CM type Φ as does A1 ). This completes the proof of Proposition 2.2.1. In Corollary 4.4.14(ii) a mild generalization of Proposition 2.2.1 will be given. We conclude this section by using Lemma 2.2.2 to prove a tangential necessary and sufficient criterion for the algebraicity of a formal abelian scheme with complex multiplication. This criterion is only used in our alternative local proof of Theorem 2.1.7 in 2.6. 2.2.3. Theorem. Let R be a Zp -flat 1-dimensional complete local noetherian ring with residue characteristic p, and let A be a formal abelian scheme over R of relative dimension g > 0. Assume that the finite-dimensional Q-algebra Q ⊗Z EndR (A) contains a commutative semisimple CM subalgebra L with [L : Q] = 2g. Then A is algebraizable if and only if the R[1/p]-linear Laction on Lie(A)[1/p] is given by a CM type over each geometric point of Spec(R[1/p]). In such cases, the algebraization admits an ample line bundle giving rise to an L-linear polarization. Before we prove this theorem, we note that there are examples of p-adic integer rings R and such A over R for which the tangential hypothesis is violated and so A is not algebraizable. One way to make such A is by using the deformation theory of formal modules, applied to the examples discussed below in Remark 2.3.4. Also, in 4.1.2 we will build a nonalgebraizable Z[ζ5 ]-linear formal abelian surface A over W (Fp4 ) for any p ≡ 2, 3 mod 5. Any non-algebraizable example A is interesting because the rigid-analytic generic fiber G = Aan is a smooth, proper, connected rigid-analytic group over a p-adic field such that it admits complex multiplication in the evident sense but is not algebraic (and does not become so after any finite extension on the base field). This is a striking contrast with the classical fact that every complex-analytic torus of dimension g equipped with an action by an order in a CM field of degree-2g is an abelian variety (i.e., algebraic). Proof. The necessity of the tangential condition is a consequence of the complex-analytic theory of abelian varieties. Thus, we focus on sufficiency. Step 1. Let us first reduce to the case when R is a domain. Observe that Rred is Zp -flat. If b R Rred admits a formally ample line bundle L then by positivity of the residue characteristic A⊗ m there is an m > 1 (depending on the order of nilpotence of the nilradical of R) such that L⊗p lifts to a line bundle N on A. Such a lift is necessarily formally ample, and the associated formal polarization φN is L-linear if φL is. Hence, we can assume R is reduced. Assuming the domain case is settled, we shall now induct on the number of irreducible components via a gluing argument along closed subschemes.

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Let {p1 , . . . , pn } be the minimal primes of the reduced R, so ∩pi = (0). We may assume n > 1, as otherwise we are in the domain case that we are assuming is settled. For J = p1 and J 0 = ∩j>1 pj , the natural map R → (R/J) ×R/(J+J 0 ) (R/J 0 ) is an isomorphism. (Note that R/J and R/J 0 are Zp -flat since R is Zp -flat.) We now check rather generally that if R is an adic noetherian ring and V is any proper flat formal R-scheme with algebraizable pullbacks to Spf(R/J) and Spf(R/J 0 ) for ideals J, J 0 ⊆ R such that R ' (R/J) ×R/(J+J 0 ) (R/J 0 ) then V is algebraizable. Let V and V 0 be the respective proper flat algebraizations of V mod J and V mod J 0 over R/J and R/J 0 , so by formal GAGA there is a unique isomorphism f between their pullbacks over R/(J + J 0 ) respecting the identification of the formal completions of such pullbacks with V mod (J + J 0 ). The R-flatness allows us to “glue” along f to get a proper flat Rc over Spf(R) is the analogous “gluing” scheme W such that the associated formal scheme W c0 = V mod J 0 . Since V is R-flat, this gluing of formal schemes is of Vb = V mod J and V identified with V. Hence, W is an algebraization of V. It is clear that this gluing argument behaves well with respect to polarizations in the case that V is a formal abelian scheme. Step 2. We next reduce to the case when the 1-dimensional Zp -flat complete local noetherian domain R is a discrete valuation ring with algebraically closed residue field. By Lemma 2.2.2, applied to the isotypic factors of the special fiber A of A over the residue field k (all of which are CM abelian varieties over k with CM algebras given by the CM factor fields of L), t we can choose an L-linear polarization φL : A → A arising from an ample line bundle L on A. Let I be an mR -primary ideal admitting topologically nilpotent divided powers, such as pR if p > 2 or 4R if p = 2. Since R/I is 0-dimensional with residue characteristic p, if we replace L with a suitable p-power then we can assume that φL lifts to a homomorphism φ0 over R/I. Let Rn = R/I n+1 and An = A ⊗R Rn for n > 0. Granting the case when R is a discrete valuation ring, we claim that φ0 over R0 lifts to an R-homomorphism φ∞ : A → At , thereby providing a formally ample line bundle on A (the (1, φ∞ )-pullback of the formal Poincar´e bundle), so Grothendieck’s algebraization theorem [22, III1 , 5.1.4] applies to A. (See Example 1.4.5.8.) Let Γ = A[p∞ ] and Γ0 = At [p∞ ], and for n > 0 let Γn and Γ0n denote the reductions modulo I n+1 . Note that Rn  R0 has kernel I/I n+1 with canonical nilpotent divided powers. By the Serre–Tate deformation theorem (Theorem 1.4.5.7), to make the desired R-lift of φ0 it is equivalent to construct an R-lifting of the L-linear k-isogeny Γ0 → Γ00 induced by φ0 on p-divisible groups. By Grothendieck–Messing theory [47] and the divided power hypothesis on I, the data of the deformation Γ of Γ0 is covariantly functorially encoded as a pair (D0 , D) where D is a finite projective R-module of rank 2g and D0 is a finite projective R-submodule of rank g that is a direct summand. Explicitly, if we let Dn be the Lie algebra of the universal vector extension E(Γn ) of Γn by a vector group over Rn then naturally Dn+1 ⊗Rn+1 Rn ' Dn for all n > 0 and D is the inverse limit of the Dn ’s. The functoriality of D in Γ0 is determined by that of each Dn via the above specification of divided-powers on I. Also, D0 ⊆ D is the inverse limit of the subbundles Dn0 = ωΓ∗n ⊆ Dn given by the cotangent spaces along the identity to the dual p-divisible groups Γ∗n . There is a similar such pair (D0 0 , D0 ) for Γ0 as a deformation of Γ00 , and the k-map φ0 induces an R-linear map [φ0 ] : D → D0 . The existence of φ∞ is equivalent to the condition that [φ0 ] carries D0 into D0 0 . Since the R-modules D0 ⊆ D and D0 0 ⊆ D0 are subbundles over Spec(R), and R is a Zp -flat domain, it is equivalent to check the subbundle compatibility of [φ0 ] after extending scalars to an algebraic closure Q of the field Q = Frac(R) of characteristic 0. In particular, it is harmless to replace φ0 with pr φ0 for any r > 1. By functoriality, the Q-linear map 0 is also L-linear. We claim that D and D 0 are invertible Q ⊗ L-modules [φ0 ]Q : DQ → DQ Q Q Q 0 0 0 and that the submodules DQ and D Q are spanned over Q by the 1-dimensional eigenspaces 0 into D 0 0 , as for a common set of g embeddings L → Q. This will force [φ0 ]Q to carry DQ Q required.

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The definitions of (D0 , D) and (D0 0 , D0 ) as well as the action on these pairs by CM orders for Γ and Γ0 = Γ∗ in L are intrinsic to the system of universal vector extensions of Γn and Γ0n over Spec(Rn ) for all n > 0. We have thereby eliminated the need to pay attention to functoriality of these vector extensions with respect to the k-morphism φ0 that we do not yet know to have a lifting, since we have reduced ourselves to some module-theoretic properties of (D0 , D) and (D0 0 , D0 ) that have nothing to do with φ0 . By considering Q-ranks, the invertibility of DQ over Q⊗Q L is equivalent to its faithfulness as a Q⊗Q L-module. Since R is a noetherian local domain of dimension 1 we can choose a local injection R → O into a complete discrete valuation ring O, so Q injects into Frac(O) = O[1/p]. The formation of the universal vector extension of a p-divisible group (over a base on which p is locally nilpotent) commutes with any base change, so it suffices to consider our linear algebra problems after the base change Spf(O) → Spf(R). That is, we may assume that R is a discrete valuation ring, and we can also assume that the residue field k of R is algebraically closed. (Observe that this reduction step preserves the geometric tangential hypothesis whose sufficiency for algebraization we are trying to prove.) Step 3. We now focus on proving that DQ is an invertible Q ⊗Q L-module (so we ignore D0 ). This will rest on comparison isomorphisms between Dieudonn´e theory and crystalline cohomology after inverting p (to avoid restrictions on the absolute ramification degree). By [47, Ch. V, 2.1], there is a canonical R/(pn+1 )-linear isomorphism between the Lie algebras of the universal vector extensions of Γ mod pn+1 and A mod pn+1 for all n > 0, and this is compatible with change in n. But [46, Ch. I, 2.6.7, 3.2.3, 4.1.7, 4.2.1] gives a canonical R/(pn+1 )-linear isomorphism Lie(E(A mod pn+1 )) ' H1dR ((At mod pn+1 )/(R/(pn+1 ))) compatibly with change in n, so passage to the inverse limit gives D ' H1dR (At /R) due to the theorem on formal functions for hypercohomology of bounded C-linear complexes of coherent sheaves on a proper formal schemes over a complete local noetherian ring C. (This version of the theorem on formal functions is easily deduced from the special case of cohomology of coherent sheaves on proper formal schemes given in [22, III1 , 3.4.4].) Hence, we have a canonical Q-linear isomorphism (2.2.3.1)

t

DQ ' H1dR (At /R)[1/p] ' Q ⊗W H1cris (A /W ),

where W = W (k) and the final isomorphism comes from [6, (2.4.2)]. By canonicity with respect the action on A by an order in L, the composite isomorphism t (2.2.3.1) is Q ⊗Q L-linear when using the L-action on H1cris (A /W )[1/p] defined via pullback of the dual of the L-action on A in the isogeny category over k. Thus, the invertibility of DQ t over Q ⊗Q L is equivalent to that of H1cris (A /W )[1/p] over K0 ⊗Q L, where K0 = W [1/p]. The comparison of classical contravariant Dieudonn´e theory and crystalline cohomology for abelian varieties [5, 2.5.5–2.5.7, 3.3.7, 4.2.14] naturally identifies the W -modules H1cris (C/W ) and D(C[p∞ ])(p) for any abelian variety C over k, so our problem reduces to proving that D(C[p∞ ])[1/p] is an invertible module over K0 ⊗Q L = K0 ⊗Qp Lp for any such C of dimension g endowed with a CM structure by L. This follows from Proposition 1.4.5.5(1), and so completes 0 over Q ⊗ L. the proof of invertibility of DQ and DQ Q 0 and D 0 0 . More Step 4. It remains to determine the Q ⊗Q L-module structures of DQ Q 0 are viewed over Q, each Q-algebra precisely, when the invertible Q ⊗Q L-modules DQ and DQ 0 and D 0 0 are each spanned map L → Q has a 1-dimensional eigenspace and the submodules DQ Q over Q by the half of these eigenlines. We have to show that the eigencharacters arising in these submodules coincide. By its construction, D0 is the formal cotangent space of Γ∗ along the identity, or equivalently of At along the identity (see Example 1.4.5.3). This latter cotangent space is naturally identified with the R-linear dual of H1 (A, OA ) compatibly with the L-actions after inverting p (where L acts on H1 (A, OA )[1/p] via pullback of its given action on A in the

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isogeny category over R). Likewise, via the double duality isomorphism (At )t ' A of formal abelian schemes over R, D0 0Q is identified with H0 (A, Ω1A/R )[1/p] as a Q ⊗Q L-module where L acts through the complex conjugate of its given action on A in the isogeny category over R (due to how L acts on At ). Thus, it suffices to show that when we view H1dR (A/R)[1/p] as an invertible Q ⊗Q L-module using the given L-action on A then complex conjugation on L swaps the sets of eigencharacters arising on the two graded pieces H0 (A, Ω1A/R )[1/p] and H1 (A, OA )[1/p] of the Hodge filtration. By invertibility over Q⊗Q L and Q-rank considerations it is equivalent to check that there are no conjugate pairs arising in one of the graded parts, such as in H0 (A, Ω1A/R )[1/p]. Passing to the Q-linear dual, we are reduced to checking that the eigencharacters arising for the L-action on Lie(A)[1/p]Q constitute a (Q-valued) CM type of L. But this is exactly the initial tangential hypothesis.

2.3. CM lifting to a normal domain up to isogeny: counterexamples We now give two classes of counterexamples to the condition (IN) in 1.7.5, the second of which is more satisfying but the first of which is easier to explain (and will arise as the “toy model” in 4.1.2). 2.3.1. Supersingular counterexamples. Choose a prime number p such that p ≡ 2, 3 (mod 5), so p remains prime in Q(µ5 ). Consider an algebraic integer in Q(µ5 ) of the form π = pζ5 , where ζ5 is a primitive fifth root of unity in Q(µ5 ). This is a p2 -Weil number. By Honda–Tate theory in 1.5.2, there is a simple abelian variety B over Fp2 such that the center of End0 (B) may be identified with F := Q[π] = Q(µ5 ) via π 7→ FrB,p2 and the central division algebra D = End0 (B) over F is split away from the unique p-adic place of F . Thus, D is globally split over F , so p 2 dim(B) = [F : Q] [D : F ] = 4 and hence dim(B) = 2. It follows that the only CM structure on B (up to choosing an isomorphism Q(µ5 ) ' End0 (B)) is the canonical one provided by the action of Q(µ5 ). 2.3.2. Lemma. The reflex field of every CM type on Q(µ5 ) is non-canonically isomorphic to Q(µ5 ). Proof. If L is a CM field that is Galois over Q then any CM type Φ on L has reflex field E ⊂ Q that is contained in the canonical image of L in Q. But the reflex field of a CM type is a CM field as well (Remark 2.1.4.2), so if in addition L has no proper CM subfield then E ' L. An example of such an L is Q(µ` ) for a Fermat prime `, such as ` = 5. 2.3.3. Corollary. For p ≡ 2, 3 mod 5 and B over Fp2 as above, no member of the isogeny class of B can be lifted to an abelian scheme B over a normal local domain R with characteristic 0 and residue field Fp2 such that the generic fiber of B has sufficiently many complex multiplications. In particular, B does not satisfy the condition (IN) in 1.7.5. Proof. Assume that such a B exists, and let F be the fraction field of R. By faithfulness of reduction to the special fiber, we get an injection End0 (BF ) = End0 (B) ,→ End0 (B) = Q(µ5 ) into a quartic number field. (The initial equality holds by Lemma 1.7.4, since R is normal.) Hence, since dim(BF ) = dim(B) = 2, the only way that BF can have sufficiently many complex multiplications is if it has a CM structure given by a lifting of the Q(µ5 )-action on its special fiber (upon fixing an isogeny between B and the special fiber of B, the choice of which does not matter since End0 (B) = Q(µ5 ) is commutative). We therefore assume that there exists such a lifted action of Q(µ5 ) on BF in the isogeny category over F (or equivalently on B in the isogeny category over R).

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Choose an algebraic closure F of F and let Φ be the resulting p-adic CM type on Q(µ5 ). The congruence condition on p implies that there is a unique p-adic place on Q(µ5 ) and it is unramified over p. It follows that Φ satisfies the Shimura–Taniyama formula (2.1.5.1) for B over Fp2 . By Lemma 2.3.2, the reflex field of (Q(µ5 ), Φ) is isomorphic to Q(µ5 ). Hence, this reflex field likewise admits a unique (and unramified) p-adic place, and so this place has residue field Fp4 of degree 4 over Fp . By the necessity of the residual obstruction in 2.1.6 (since R is normal), the residue field of R contains Fp4 . But the residue field of R is Fp2 , so we have a contradiction and therefore no such B exists.

2.3.4. Remark. If we replace Q(µ5 ) with Q(µ` ) for any Fermat prime ` > 5 (so there is no proper CM subfield of Q(µ` ) and [Q(µ` ) : Q] > 2), then the preceding construction works over Fp2 for any prime p 6= ` with p 6≡ ±1 mod ` (so p mod ` in (Z/`Z)× has order exceeding 2). In such examples, B is simple with dimension (` − 1)/2 and has endomorphism algebra Q(µ` ) in which FrB,p2 = π := pζ` for a primitive `th root of unity ζ` in Q(µ` ). Thus, π ` = p` , so there is an isogeny between BFp2` and a power of a supersingular elliptic curve over Fp2` . Hence, these B are supersingular abelian varieties. Consider the special case when p is a generator of (Z/`Z)× and the CM order is Z[µ` ]. In this case the tangent space Lie(B) of dimension (` − 1)/2 over Fp2 is 1-dimensional over Z[µ` ]/(p) ' Fp`−1 . Since complex conjugation on Z[µ` ] induces an Fp2 -automorphism of Fp`−1 , B over Fp2 cannot satisfy the condition (CML) in 1.7.5 (necessarily using L = Q(µ` )) since the µ` -action on the tangent space in characteristic p forces the geometric tangential action in characteristic 0 to be given by a collection of eigencharacters that is stable under complex conjugation, contradicting the requirement to be a CM type. 2.3.5. Counterexamples with two slopes. The preceding supersingular counterexamples to the condition (IN) in 1.7.5 are not absolutely simple. We now give absolutely simple counterexamples. To make it easier to violate the necessary residual reflex condition from 2.1.6 we will work over Fp . Fix a prime ` > 5 and an integer r such that 1 < r < `/2. We shall construct counterexamples B to (IN) over Fp that satisfy (CML) via the elementary criterion in Proposition 2.2.1 (and hence satisfy (I)) and are absolutely simple abelian varieties of dimension `, with p satisfying some congruence conditions to be determined shortly. In these examples B will have slopes r/` and 1 − r/`, each with multiplicity `. Choose an imaginary quadratic field F whose discriminant is prime to ` (so in particular, F is linearly disjoint from Q(µ` ) over Q). Choose a prime p - ` · disc(F ) such that p mod ` is a generator of (Z/`Z)× , which is to say that p is inert in Q(µ` ). Assume that p splits in F and that the two prime ideals ℘ and ℘ of OF √ over p are principal ideals. For example, if we consider F with class number 1 (i.e., F = Q( −D) with D = 1, 2, 3, 7, 11, 19, 43, 67, 163, so at most one such F is ruled out by the condition ` - disc(F )), then the conditions on p say that it lies in some√non-empty set of congruence classes modulo ` · disc(F ); in the special case ` = 5 and F = Q( −1), the condition is p ≡ 13, 17 mod 20. Write ℘ = αOF . Let L = F (u) be an extension field of F such that u` = αr α ¯ `−r = pr α`−2r . The extension L/F is unambiguous because the polynomial X ` − αr α`−r ∈ F [X] is irreducible (as it is even ℘-adically irreducible). Likewise, F = Q(u` ), so L = Q(u). Observe that the algebraic integer u is a p-Weil number, and L = Q(u) is a CM field containing the imaginary quadratic field F in which p is split. Since (p/u)` = pr α`−2r , we see that L/F is totally ramified at both primes ℘ and ℘ of OF over p (so OL has residue field Fp at all primes over p). By Honda–Tate theory in 1.5.2, there is a simple `-dimensional abelian variety B over Fp endowed with an action by L in the isogeny category over Fp such that FrB,p = u. This abelian variety has two distinct slopes, namely r/` and (` − r)/` (each with multiplicity `), and its endomorphism algebra is Q(u) = L. We now check that B is absolutely simple. Consider BFpe with e > 1. This is isotypic since B is simple (Proposition 1.2.6.1), so the center of its endomorphism algebra is Q(ue ) with

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ue = FrBFpe ,pe . To prove absolute simplicity we may replace e with a multiple so that e = `e0 0

0

0

0

for some e0 ∈ Z. Since the element ue = αre α(`−r)e in OF generates the ideal ℘re ℘(`−r)e that does not come from Z (due to the unequal exponents), the inclusion Q(ue ) ⊆ Q(u` ) = F is an equality for degree reasons. Hence, if X is the unique (up to isogeny) simple factor of BFpe then the endomorphism algebra of X is a central division algebra D over F . The local invariants of D at the p-adic places v of F are the slopes of BFpe , which are r/` and (` − r)/`. These p have denominator `, and D is split at all places p of F away from p, so [D : F ] = `. Thus, Honda–Tate theory gives 2 dim(X) = [F : Q] [D : F ] = 2`, so dim(X) = ` = dim(BFpe ). This establishes the desired absolute simplicity. (The reader can check that if ` - e then Q(ue ) = Q(u) = L and that this is the endomorphism algebra of BFpe , as for e = 1.) Let M be a Galois closure of L over F . This is generated over Q by L = Q(u) and Q(µ` ), and it is Galois over Q because u and p/u are respective `th roots of the Q-conjugate elements pr α`−2r and pr α`−2r in F that generate F over Q. Since [L : Q] = 2`, so F is the unique quadratic subfield of L, it follows from the linear disjointness of F and Q(µ` ) over Q that L and Q(µ` ) are linearly disjoint over Q. Hence, [M : L] = ` − 1 and [M : Q] = 2`(` − 1). Since p splits in F and is inert in Q(µ` ), F (µ` )/F is inert at both primes over p. But L/F is totally ramified at both places, so by degree-counting we see that the composite field M of degree `(` − 1) over F has exactly two p-adic places, each with residual degree ` − 1 over Fp . In particular, complex conjugation on the CM field M switches the two p-adic places of M and both such places are inert over L. In addition to the preceding arithmetic properties of F and L at p, we need to record some group-theoretic properties before we can show that B does not satisfy (IN). Let Γ := Gal(M/Q), G := Gal(M/F ), H := Gal(M/L) = Gal(L(µ` )/L) ' (Z/`Z)× , and N := Gal(M/F (µ` )) ' Z/`Z, so #N = `, #H = ` − 1, and G = N o H ' (Z/`Z) o (Z/`Z)× with (Z/`Z)× acting on the additive group Z/`Z via its canonical multiplicative scaling action. The group G has the following properties. (i) Let `0 be a prime divisor of ` − 1 and let P be the `0 -Sylow subgroup of H. The normalizer subgroup NG (P ) of P in G is equal to H. (In particular, gHg −1 = H for g ∈ G if and only if g ∈ H.) (ii) The only proper subgroups H 0 of G such that N ·H 0 = G are the conjugates of H. (iii) Any two distinct conjugates of H intersect trivially. (iv) For any conjugate H 0 of H, there are exactly two orbits for the left H 0 -action on G/H. One orbit consists of a single coset and H 0 operates freely on the other H 0 -orbit. (v) Let Σ be a subset of G/H with 1 < #Σ < ` − 1 (i.e., Σ and its complement Σc are subsets of G/H with more than one element). The G-stabilizer StabG (Σ) ⊆ G of Σ has image in G/N = Gal(F (µ` )/F ) = (Z/`Z)× that is a proper subgroup of G/N . The statement (i) follows from an easy calculation, and statements (ii) and (iii) follow from (i). (Statements (i)–(iii) are also easily seen in field-theoretic terms via the Galois correspondence.) The statement (iv) follows from (iii) (or is a simple calculation for H 0 = H, and for a general H 0 = gHg −1 follows from the fact that left multiplication by g −1 on G/H carries H 0 -orbits to H-orbits). Finally, (v) follows from (ii) and (iv) because if StabG (Σ) maps onto G/N then by (ii) either StabG (Σ) = G or StabG (Σ) is a conjugate of H; the first option contradicts that Σ is a proper subset of G/H and (by (iv)) the second option contradicts the assumptions on #Σ. Let Φ be a CM type of L that is a collection of embeddings of L into its Galois closure M over Q. Such Φ are in bijective correspondence with H-invariant M -valued CM types on M , which is to say subsets ΦM ⊆ Γ = Gal(M/Q) satisfying ΦM · H = ΦM and ΦM = Ψ t ι ◦ Ψc , where ι denotes the complex conjugation on the CM field M , Ψ is a subset of G, and Ψc is the complement of Ψ in G. (Here we have used that Γ = G × C, where C is the subgroup generated by the central involution given by complex conjugation on the CM field M .) The H-stability condition on ΦM says precisely that Ψ · H = Ψ. If the above B over Fp is to

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satisfy (IN) using a particular CM lift (over a normal local domain R with characteristic 0 and residue field Fp , with R taken to be noetherian and complete without loss of generality) then the Shimura–Taniyama formula in the residual reflex condition from 2.1.6 is exactly the requirement that the generic fiber of this lift has p-adic CM type Φ on L corresponding to such a ΦM with #(Ψ/H) = r or #(Ψ/H) = ` − r (depending on which of the two p-adic places of M is induced by the choice of embedding of M into Qp so as to view M -valued CM types on L or M as p-adic CM types). Hence, to prove that B does not satisfy (IN), it suffices to consider those M -valued Φ such that 1 < #(Ψ/H) < ` − 1 (such Φ do exist) and to show that the second part of the residual reflex condition is violated at all p-adic places of the reflex field E of (L, Φ). Since Ψ and Ψc have distinct cardinalities (as r 6= `/2) and complex conjugation is central in Γ, an element of Γ stabilizes ΦM ⊆ Γ = Gal(M/Q) under the left action if and only if it stabilizes the non-empty subset Ψ ⊆ G = Gal(M/F ). In particular, F ⊆ E due to the definition of E. Hence, in terms of Galois theory, the subfield E ⊆ M corresponds to the subgroup Gal(M/E) = StabG (Ψ/H) using the left G-action on G/H = Homring (L, M ). Since the reflex field E of (L, Φ) is a CM field (as is the reflex field of any CM type, by Remark 2.1.4.2), complex conjugation acts nontrivially on this field. But complex conjugation on M switches the two p-adic places of M , and restriction to the imaginary quadratic field F in which p is split sets up a bijection between the p-adic places of M and of F . Thus, since F ⊆ E, we see that E admits exactly two p-adic places, these places are switched by complex conjugation on E, and each has decomposition group in Gal(M/E) equal to the full Galois group. Since M/F (µ` ) is totally ramified at both p-adic places whereas F (µ` )/F is unramified with full residual degree ` − 1 at both of its p-adic places (since p is inert in Q(µ` ) and split in F ), the natural map from Gal(M/E) to G/N = Gal(F (µ` )/F ) has image equal to the Galois group of the residue field extension for M/E at both p-adic places. Hence, the two p-adic places of E have residue field strictly larger than Fp if and only if Gal(M/E) has proper image in G/N . But Gal(M/E) = StabG (Ψ/H), so property (v) above (applied to Σ = Ψ/H ⊆ G/H) implies that the p-adic places of E do indeed have nontrivial residual degree over Fp . Thus, we have contradicted the second part of the residual reflex condition for the abelian variety B over Fp , so it does not satisfy the property (IN). To summarize, in these absolutely simple counterexamples to (IN) the endomorphism algebra over the finite field Fp is a CM field L, but its CM types that satisfy the requirement from the Shimura–Taniyama formula for a possible CM lifting (over a complete local noetherian domain as in the condition (I) from 1.7.5) do not satisfy the second part of the residual reflex condition. Moreover, since OL has residue field Fp at all p-adic places, by Proposition 2.2.1 these examples all satisfy (CML) and hence (I) over Fp . 2.3.6. Denote by N P NI (resp. N P p,NI ) the set of all symmetric Newton polygons N such that every isotypic abelian variety B over a finite field Fq (resp. over any finite field Fq of characteristic p) with Newton polygon N satisfies (IN). If dim(B) = g > 0 then this concave-up polygon is a concatenation of 2g segments whose slopes (by definition) are the slopes of B in the sense defined in 2.1.5 using suitably normalized p-adic ordinals of the roots of the characteristic polynomial FB of the q-Frobenius of B; these slopes lie in Q∩[0, 1]. The “symmetry” condition means that for every slope λ that occurs in N, the slopes λ and 1 − λ occur in N with the same multiplicity; the Newton polygons arising from abelian varieties are necessarily symmetric since π 7→ q/π is a permutation of the zeros of FB , For any symmetric Newton polygons N P1 , . . . , Nr with no slopes in common and any non-negative integers m1 , . . . , mr , we write mj Nj to denote the symmetric Newton polygon obtained by concatenation of the segments in mj copies of each Nj (with all segments arranged in order of increasing slope). For each n > 2, let ( n1 , n−1 n ) denote the symmetric Newton polygon of length 2n with 1 n−1 slopes n and n each occurring with multiplicity n. Analogously, we let ( 12 , 12 ) denote the symmetric Newton polygon with slope 21 occurring with multiplicity 2, and we let (0, 1) denote the symmetric Newton polygon with slopes 0 and 1 each occurring with multiplicity 1. In

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2.3.1 and 2.3.5 we have seen examples of symmetric Newton polygons that are not in N P NI , namely 2( 12 , 21 ) and ( r` , `−r ` ) for any odd prime ` > 5 and 1 < r < `/2. (In each case, the counterexamples were found in any characteristic p > 0 such that p satisfies a suitable nonvacuous congruence condition.) In the affirmative direction, the following are examples of symmetric Newton polygons N in N P NI . (i) An ordinary symmetric Newton polygon N = g(0, 1), which is to say one whose only slopes (each with multiplicity g > 0) are 0 and 1, is in N P NI . This is exactly the case when the abelian variety B over a finite field k has p-divisible group with ´etale part having the maximal possible height, namely g = dim(B), or in other words B is an ordinary abelian variety. To establish (IN) for any such B, choose a CM subfield L ⊆ End0 (B) with [L : Q] = 2g (Corollary 1.5.2.5) and pass to an isogenous abelian variety if necessary so that OL ⊆ End(B) (Proposition 1.6.4.3). By Theorem 1.4.5.7, to infinitesimally deform B with its OL -action is equivalent to doing the same for B[p∞ ] with its OL -action. Since k is perfect, B[p∞ ] uniquely splits as a direct product of its connected and ´etale parts (in the sense of Example 1.4.5.3). Thus, upon choosing an OL -linear polarization λ of B over k (as we may do, by Lemma 2.2.2), the functoriality and uniqueness of deformations of ´etale and multiplicative p-divisible groups provide an OL -linear deformation Γ of B[p∞ ] over W (k) and an OL -linear W (k)-isogeny Γ → Γt lifting the one induced by λ over k (where OL acts on B[p∞ ]t through the composition of Cartier duality and complex conjugation on OL ). This corresponds to a formal abelian scheme B over W (k) equipped with an OL -action and isogeny B → Bt that respectively lift the OL -action and polarization λ on B. Thus, by Grothendieck’s algebraization theorem [22, III1 , 5.1.4], B satisfies (IN). (ii) The Newton polygon ( 21 , 12 ) corresponds to a supersingular elliptic curve E over a finite field k, and these satisfy (IN). (The counterexample to (IN) in 2.3.1 has Newton polygon 2( 12 , 21 ).) To prove this, choose an imaginary quadratic field L ⊆ End0 (E) and pass to a k-isogenous elliptic curve if necessary so that OL ⊆ End(E). Since Zp ⊗Z OL acts faithfully on the 1-dimensional E[p∞ ] (Proposition 1.2.5.1) and this p-divisible group is local-local by supersingularity, there is a unique prime p in OL over p. Also, the action by OL on the tangent space to E defines an embedding of κ = OL /p into k. Using this embedding to make k into an OL,p -algebra, E[p∞ ] corresponds to a 1-dimensional formal OL,p -module G over k, and it has OL,p -height 1 (since the Zp -height is 2 and OL,p is finite free of rank 2 over Zp ). Clearly R := OL,p ⊗W (κ) W (k) is a p-adic integer ring unramified over OL,p with residue field k. Since elliptic curves are canonically projective, to lift E with its OL -action to R (having tangential action via OL,p ,→ R) it is equivalent to lift G to a formal OL,p -module over R. By Lubin–Tate theory (or more generally the deformation theory of formal modules with finite height over algebras over p-adic integer rings [32, 22.4.4]), for any p-adic integer ring C and complete local noetherian C-algebra A, a height-1 formal C-module with dimension 1 over the residue field over A admits (up to unique isomorphism) a unique deformation to a formal C-module over A. This provides the required lift of G over R as a formal OL,p -module. (iii) Every Newton polygon of the form N = ( n1 , n−1 n ) , n > 2, is in N P NI . Indeed, let B be an abelian variety over a finite field k (with size q) such that it has this Newton polygon; in particular, dim(B) = n. Let L ⊆ End0 (B) be a CM subfield with [L : Q] = 2n, and use a k-isogeny if necessary so that OL ⊆ End(B) (Proposition 1.6.4.3). The Frobenius element FrB,q ∈ OL has p-adic slopes ordw (FrB,q )/ordw (q) given by only 1/n and (n − 1)/n = 1 − 1/n. These are positive, so G = B[p∞ ] is a connected p-divisible group with connected dual. If w is a p-adic place of L with associated slope λw = ordQ w (FrB,q )/ordw (q) then λw = 1 − λw 6= λw , so w 6= w. The faithful action of OL,p ' w|p OL,v on G (Proposition 1.2.5.1) decomposes G into a

2.3. CM LIFTING TO A NORMAL DOMAIN UP TO ISOGENY: COUNTEREXAMPLES

63

Q product Gw where Gw is a nonzero connected p-divisible group equipped with an action by OL,w . Since X X [Lw : Qp ], height(Gw ) = height(G) = 2n = [L : Q] = w|p

w|p

it follows from Proposition 1.4.5.5(1) that Gw has height [Lw : Qp ] for each w. Tate’s method of proof of the Shimura–Taniyama formula in [77, §5] via pdivisible groups carries over to each Gw over the finite field k to yield the formula ordw (FrB,q ) dim(Gw ) dim(Gw ) = = . ordw (q) height(Gw ) [Lw : Qp ] (This equality also follows from general facts concerning slopes of p-divisible groups over algebraically closed fields, via the isoclinicity in Proposition 1.4.5.5(2) and the Dieudonn´e–Manin classification in Theorem 3.1.2.1(2).) But the left side is either 1/n or 1 − 1/n, and hence has denominator n. This forces n|[Lw : Qp ] for all w, yet [L : Q] = 2n. Hence, there are exactly two p-adic places on L, necessarily of the form v and v, say with respective slopes 1/n and 1 − 1/n > 1/n. We also get dim(Gv ) = 1 and dim(Gv ) = n − 1. The tangential action by OL,v on the 1dimensional Gv thereby defines an embedding κv ,→ k, and so Gv may be identified with a 1-dimensional formal OL,v -module with OL,v -height 1 over the OL,v -algebra k. Applying the same considerations to the dual abelian variety B t over k (with the usual CM structure as in Remark 1.4.3.1), we see that via the canonical isomorphism OL,v ' OL,v induced by complex conjugation, Gv is identified with the Cartier dual of a 1-dimensional formal OL,v -module G0v with OL,v -height 1 over k. Let R = OL,v ⊗W (κv ) W (k), a p-adic integer ring unramified over OL,v with residue field k. Using Lemma 2.2.2, choose an OL -linear polarization λ : B → B t . The associated OL,p -linear map on p-divisible groups Gv × (G0v )t → G0v × (Gv )t decomposes as the OL,p -linear product λv ×µtv of an OL,v -linear isogeny λv : Gv → G0v and the dual of another such isogeny µv : Gv → G0v . To prove (IN) for B, it suffices to deform λv and µv over R; we handle λv , and then µv will go in the same way. Arguing via the deformation theory of 1-dimensional formal C-modules of C-height 1 over complete local noetherian algebras over p-adic integer rings C, exactly as in (ii) we may uniquely deform Gv and G0v over R. There is at most one lift of λv , and so if k/k is an algebraic closure then by 3 it suffices to construct such a lift when k is replaced with k. But Gv and G0v become OL,v -linearly isomorphic over k, so λv becomes multiplication by some nonzero element of OL,v . Uniqueness of deformations in the height-1 case thereby settles the lifting problem for λv . It is easy to adapt the arguments underlying the preceding examples (i)–(iii) to show that any symmetric Newton polygon that can be written as a finite sum in the form   X 1 n−1 N = m1 (0, 1) + mn , , m1 > 0, mn ∈ {0, 1} for all n ≥ 2, n n n≥2

is in N P NI . We do not know whether there is an element of N P NI that is not of the above form. 2.3.7. Remark. (i) The method in 2.3.5 using Galois extensions whose Galois group is the standard semi-direct product G ' (Z/nZ) o (Z/nZ)× for prime n fails dramatically when instead n = 15. In this case we have N ' Z/15Z and H = (Z/15Z)× ' (Z/3Z)× × (Z/5Z)× . The left action of H on G/H is identified with the standard action of (Z/3Z)× × (Z/5Z)× on (Z/3Z) × (Z/5Z), and this has 4 orbits (namely, {0}, (Z/3Z)× , (Z/5Z)× , and the rest) with respective cardinalities 1, 2, 4, and 8. Therefore, for any integer r with 1 6 r 6 14 there exists 3completed unramified descent: give reference

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a subset of G/H with r elements that is stable under the left action by H, and so likewise for the left action on G/H by any conjugate of H. (ii) It would be interesting to determine which symmetric Newton polygons belong to N P NI , and which ones belong to N P p,NI as the prime number p varies. For a given symmetric Newton polygon N, the question as to whether N belongs to N P p,NI is easier when p does not divide the denominator of any slope of N. For instance it is not difficult to r 15−r check that ( 15 , 15 ) is in N P p,NI for r = 2, 4, 6, 7 if p 6= 3, 5. We do not know any plausible statistics about N P NI or N P p,NI among all symmetric Newton polygons.

2.4. Algebraic Hecke characters In this section we will first review the definition and basic properties of algebraic Hecke characters before establishing Theorem 2.4.11, which gives a procedure to modify an algebraic Hecke character by specifying its value on a uniformizer at a desired unramified place. The reader is referred to [81], [68, §7], [65, Chap. II], [19, §5], [50], and [49] for more information about algebraic Hecke characters. We sometimes write L× as shorthand for the Q-torus ResL/Q (Gm ) over Q, where ResL/Q denotes Weil restriction with respect to a finite extension L/Q. 2.4.1. Definition. Let k be an infinite field. For a finite separable extension F/k and an extension K/k, a homomorphism χ : F × → K × is algebraic (with respect to k) if one of the following equivalent conditions holds: (i) The map χ is induced by a homomorphism of algebraic groups φ

ResF/k (Gm )K → Gm over K. The relation between χ and φ is that the map (F ⊗k K)× → K × induced by φ on K-points restricts to χ on the subgroup F × ⊆ (F ⊗k K)× (inclusion defined by t 7→ t ⊗ 1). (ii) (K/k finite separable) The map χ is induced by a k-homomorphism ResF/k (Gm ) → ResK/k (Gm ). (iii) Let e1 , . . . , en be a k-basis of F . Then there exists a rational function f (X1 , . . . , Xn ) ∈ K(X1 , . . . , Xn ) such that every non-zero element of k n is in the domain of definition of f (X1 , . . . , Xn ) and ! n X χ xi ei = f (x1 , . . . , xn ) i=1

for every non-zero element (x1 , . . . , xn ) ∈ k n . (iv) Let I = Homk (F, Ks ) be the finite set of k-embeddings of F into a (fixed) separable closure Ks of K. Then there exists a Gal(Ks /K)-invariant function m : I → Z, necessarily unique, such that Y χ(x) = τ (x)m(τ ) τ ∈I

for all x ∈

F ×.

The equivalence of (i) and (ii) is due to functorial adjointness, and the equivalence of (i) and (iv) is clear, as is the implication (iv) ⇒ (iii). Finally, to deduce (i) from (iii) one uses that the set of k-points is Zariski-dense in any torus over k (such as ResF/k (Gm )), due to the unirationality of tori over fields, and that a rational map between algebraic groups over a field is a morphism if it is generically a homomorphism (in an evident sense). 2.4.2. Remark. In practice the ground field k is understood from context (such as Q or Qp ). The Galois group Gal(Ks /K) operates naturally on I in condition (iv), and the displayed

2.4. ALGEBRAIC HECKE CHARACTERS

65

equation in (iv) implies that m is constant on each Gal(Ks /K)-orbit of I. Conversely, for any function m : I → Z that is constant on every Gal(Ks /K)-orbit of I there is a unique algebraic homomorphism χ from F × to K × giving rise to m as in (iv). If K ⊆ F (over k) then the special case with m identically equal to 1 corresponds to χ = NmF/K . If K/k splits F/k then Gal(Ks /K) acts trivially on I, so in such cases the maps F × → K × induced by field embeddings over k are a Z-basis for the group of algebraic homomorphisms (with respect to k). 2.4.3. Definition. Let F be a number field, m a nonzero ideal of the ring of integers OF of F , Im the group of fractional ideals of OF that are relatively prime to m, and A× F the group of F -ideles. Let K be a field of characteristic 0. (a) A homomorphism χ : Im → K × is an algebraic Hecke character with conductor m if there exists a homomorphism χalg : F × → K × that is algebraic (with respect to Q) and satisfies χ((x)) = χalg (x) × lying in the for every x ∈ F × such that: x ≡ 1 (mod m) and x has image in F∞ × 0 identity component (F∞ ) (i.e., x is positive at all real places). Here, (x) := xOF is the principal fractional ideal of OF generated by x. × is an algebraic Hecke character if the (b) A continuous homomorphism  : A× F → K restriction |F × of  to the diagonally embedded subgroup F × ⊂ A× F is equal to an algebraic homomorphism alg : F × → K × (with respect to Q). Here, the target K × of  is given the discrete topology, so the kernel of  is an open subgroup of A× F.

An equivalence of these two definitions can be formulated and proved as follows. Suppose × that  : A× F → K is an algebraic Hecke character as in 2.4.3(b). Choose a nonzero ideal m of OF such that ker() contains the open subgroup Y Y × × 0 Um := (F∞ ) × (1 + mOF,v ) × OF,v v|m

v ∈S /

of A× F , where S = S(m) is the union of the set of archimedean places of F and the set of places of F dividing m. Denote by ASF the factor ring of AF consisting of factors away from S (i.e., Q Q S,× the restricted product 0v∈S = 0v6∈S Fv× . / Fv ), so its unit group is the restricted product AF The restriction |AS,× of  to AS,× factors through the canonical surjection πS : AS,×  Im F F F

and gives a homomorphism χ : Im → K × satisfying the conditions in 2.4.3(a). Both χ and  are said to have conductor ≤ m. Conversely, given a homomorphism χ : Im → K × satisfying the conditions in 2.4.3(a), one × × defines a continuous homomorphism  : A× F → K as follows. Every element x ∈ AF can be S,× written as a product x = u · y · z with u ∈ Um , y ∈ F × , and z ∈ AF , where S is as in the × previous paragraph. Define  : A× F → K by (x) = χalg (y)χ(πS (z)) for x = uyz as above. It is easy to see that  is well-defined and satisfies the conditions in 2.4.3(b). If χ and  correspond to each other as above, then the algebraic homomorphisms χalg and alg are equal; this algebraic homomorphism is called the algebraic part of χ or . As an example, if F 0 /F is a finite extension and m0 is a sufficiently divisible modulus on F 0 lying over m such that m|NmF 0 /F (m0 ), then  ◦ NmF 0 /F is an algebraic Hecke character with algebraic part alg ◦ NmF 0 /F and associated homomorphism Im0 → K × given by χ ◦ NmF 0 /F . Note that the concepts of algebraic character and algebraic Hecke character as discussed Q above make sense when K = Ki is a finite product of fields Ki of characteristic 0, an interesting example of which is L` := Q` ⊗Q L for a number field L. In such cases, the preceding assertions (and their proofs) concerning algebraic characters and algebraic Hecke characters carry over essentially verbatim, and these “algebraicity” properties hold if and only if they do after composing with projection to each factor field Ki .

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2.4.4. The concept of algebraic Hecke character can be expressed in terms of (possibly disconnected) linear algebraic Q-groups Sm defined in Chapter II of [65]. This will be very useful for our purposes, so we now recall the definition of Sm and, for any number field K, use it to formulate a bijective correspondence between K × -valued algebraic Hecke characters of a number field F and certain continuous representations Gal(F ab /F ) → K`× (with a fixed rational prime `). × The group Sm is defined as an extension of the finite group Cm := A× F /(F · Um ) by a certain Q-torus Tm . We now recall the constructions. By definition, Tm is the quotient of F × := ResF/Q (Gm ) by the Zariski closure of the image of the finitely generated Z-module F × ∩ Um in F × . The Q-torus Tm stabilizes as a quotient of F × when m is sufficiently divisible (since the closure of F × ∩ Um in F × has identity component that is independent of m); we denote it by SF , following the notation in [49]. (The notation for the Q-torus SF is S F in [50] and is FS in [20].) By definition, Sm is the middle term in the extension 1 → Tm → Sm → Cm → 1 that is the push-out of the short exact sequence 1 → F × /(F × ∩ Um ) → A× F /Um → Cm → 1 by the natural homomorphism F × /(F × ∩ Um ) → Tm (Q). In other words, we have a commutative diagram of locally finite type Q-groups 1

/ F × /F × ∩ Um

/ A× /Um F





iF,m

/1

=



/ Sm

/ Tm

1

/ Cm

/ Cm

/ 1,

with exact rows (and the top row consisting of constant groups). In particular, iF,m can equivalently be viewed as a map of abstract groups A× F /Um → Sm (Q), and its image is Zariskidense in Sm . These commutative diagrams form a projective system as m runs through all nonzero ideals of OF . By [80, 3.3], every Hopf algebra over a field is a direct limit of Hopf subalgebras finitely generated over the ground field, with faithfully flat transition maps. This property does not hold over more general rings R, so for general R we call an affine R-group scheme G = Spec(A) pro-algebraic if A is such a direct limit. This property implies flatness and is preserved by base change; we will be interested in pro-algebraic groups arises by base change from affine group schemes over a field. Taking the inverse limit of the short exact sequences 1 → Tm → Sm → Cm → 1 over nonzero ideals m of OF , we get an exact sequence 1 → SF → SF → Gal(F ab /F ) → 1

(2.4.4.1)

of affine Q-groups; the right term is a constant (profinite) Q-group and the exact sequence is the pullback of the exact sequence 1 → Tm → Sm → Cm → 1 by the canonical surjection Gal(F ab /F ) → Cm for m sufficiently divisible so that Tm does not change (as a quotient of F × ) when m is replaced with any nonzero multiple. Here we have used class field theory to identify lim Cm with Gal(F ab /F ) as profinite groups. We have a natural commutative diagram ←− 1

1

/ F×

/ A× F





/ SF (Q)

iF

/ SF (Q)

/ A× /F × F 

/1

recF

/ Gal(F ab /F )

/1

× ab where recF : A× F /F → Gal(F /F ) is the surjective Artin map with the classical (arithmetic) normalization.

2.4. ALGEBRAIC HECKE CHARACTERS

67

By definition of SF as a limit, any Q-rational homomorphism from SF to an affine finite type group over Q factors through the projection to some Sm . The following lemma gives another description of algebraic Hecke characters of F . The proof is omitted. (See §2.6 in Chapter II of [65] for details.) 2.4.5. Lemma. Let K be a number field. (i) Let ρ : SF → K × (resp. ρm : Sm → K × ) be a Q-rational homomorphism. The composition A× F

iF

/ SF (Q)

ρ

/ K × (Q) = K ×

is an algebraic Hecke character (resp. the composition S(m),×

Im = AF

/

Q

× OF,v v ∈S(m) /

/ A× /Um F

iF,m

/ Sm (Q) ρm / K × (Q) = K ×

is an algebraic Hecke character with conductor ≤ m). × (resp. algebraic Hecke (ii) Conversely, every algebraic Hecke character  : A× F → K × character χ : Im → K with conductor ≤ m) comes from a unique Q-rational homomorphism ρ : SF → K × (resp. ρm : Sm → K × ). 2.4.6. Remark. (i) In the situation of Lemma 2.4.5, the composition of the natural (quotient) map F × → SF with the restriction ρ|SF of ρ to SF ⊆ S F is the algebraic part alg of the algebraic Hecke character . (ii) For later reference, we record the following fact: any two algebraic Hecke characters × 1 , 2 : A× F ⇒ K with the same algebraic part coincide on an open subgroup of finite index × containing F . This is easily seen from either of the equivalent definitions of the concept of algebraic Hecke character in Definition 2.4.3. 2.4.7. Consider the exact sequence of A× f -groups obtained from (2.4.4.1) by base change along Spec Af → Spec Q. There exists a natural continuous section φ : Gal(F ab /F ) → SF (Af ) = lim Sm (Af ) over the subgroup of “constant points” in the “profinite” constant group scheme ←− over Spec Af associated to Gal(F ab /F ). We review its construction below; in Chapter II, §2.3 in [65] the associated continuous `-adic splittings φ` : Gal(F ab /F ) → SF (Q` ) = lim Sm (Q` ) ←− over Q` are constructed (or rather, the composite of this with projection to each Sm (Q` ) is constructed and is denoted ε` with m fixed). The splitting φ does not extend to a splitting as group schemes over Spec(Af ), since a map from an affine (flat) proalgebraic group scheme to a finitely presented affine group scheme factors through a finite type flat quotient of the source whereas the splitting maps Gal(F ab /F ) → Sm (Q` ) that we shall construct will not factor through any finite quotient Cm0 of the source. To construct φ it suffices to construct a compatible family of continuous homomorphisms φm : Gal(F ab /F ) → Sm (Af ), where φm lifts the canonical quotient map Gal(F ab /F )  Cm (whose target is viewed as Q-points within the Af -points of the finite constant Q-group associated to Cm ). Let πm : F × → Tm be the canonical quotient map over Q. Define φem : A× F → Sm (Af ) by φem (x) = iF,m (x mod Um ) · πm (xf )−1 , × and x ∈ A× where x = (x∞ , xf ) with x∞ ∈ F∞ f F,f denoting the archimedean and finite components of x respectively. The point iF,m (x) ∈ Sm (Q) is viewed as an element of Sm (Af ) in the natural manner, and πm (xf ) ∈ Tm (Af ) ⊆ Sm (Af ), so iF,m (x mod Um ) · πm (xf )−1 makes sense as an element of Sm (Af ). By construction, the map φem is continuous and its restriction × × to F × is trivial. Thus, φem factors continuously through the projection A× F  AF /F . Since the target group of φem is totally disconnected, φem factors through the topological quotient × × ab ab A× F  π0 (AF /F ) ' Gal(F /F ) to define a continuous homomorphism φm : Gal(F /F ) →

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2. CM LIFTING OVER A DISCRETE VALUATION RING

Sm (Af ). The compatibility with respect to change in m is easy to check, so we have constructed φ. For every prime number `, define the continuous map φ` := π` ◦ φ : Gal(F ab /F ) → SF (Q` ), where π` : SF (Af ) → SF (Q` ) is the homomorphism induced by the projection Af → Q` . The associated `-adic splitting Gal(F ab /F ) → Sm (Q` ) is the `-adic component of φm . × be an algebraic Hecke character corresponding to a Q2.4.8. Lemma. Let  : A× F → K rational homomorphism ρ : SF → K × , with K a number field. Let ` be a prime number. Then ψ` := ρ ◦ φ` : Gal(F ab /F ) → K × (Q` ) = K`×

is a continuous homomorphism with the following properties. (i) There exists a finite set Σ of places of F , including all archimedian places and all places above `, such that ψ` is unramified outside Σ and such that ψ` (Wv ) ⊂ K × for each place v ∈ / Σ of F , where Wv is the Weil subgroup of the decomposition group of ab v in Gal(F /F ). In fact, ψ` (Frobv ) = (πv ), where πv is any local uniformizer at v 6∈ Σ and Frobv ∈ Gal(F ab /F ) is an arithmetic Frobenius element at v. (ii) The composition Y Fλ× → K`× ψ` ◦ rF,` : F`× := λ|`

of ψ` with the finite product of local Artin maps Y Y rF,` : F`× := Fλ× → Gal(Fλab /Fλ ) → Gal(F ab /F ) λ|`

λ|`

coincides with the algebraic homomorphism × × × × −1 alg : F` = F (Q` ) → K (Q` ) = K`

on an open subgroup of F`× . In particular, ψ` is locally algebraic at ` in the sense that ψ` ◦ rF,` coincides near 1 with a homomorphism that is algebraic with respect to Q` . Moreover, if ρ factors canonically through a homomorphism ρm : Sm → K × then ψ` = ρm ◦ (φm )` , where (φm )` is the Q` -part of φm . Proof. The assertion in (i) is immediate from the definitions. Note that if  has conductor ≤ m, then the set Σ in (i) can be taken to be the union of the set of all archimedian places of F and the set of all places of F dividing m. (See Chapter II, §2.5 in [65] for a related discussion.) Also, the description of ψ` in terms of ρm if ρ factors through ρm is clear from the definitions. As for (ii), first observe that for x ∈ F × we have ψ` (rF,` (x)) = ψ` (rF ((x)(`) ))−1 where × (`) (x) ∈ A× F is the idele that is trivial at all v|` and is given by x at all v - ` and rF : AF → × × × with the Artin map rec × → Gal(F ab /F ) be the composition of A× F : AF /F F → AF /F ab ab Gal(F /F ). But for any finite place v - ` on F , the image of inertia at v in Gal(F /F ) has finite pro-` part. Thus, since ψ` is mapping continuously into K`× , for x sufficiently near 1 at all v ∈ Σ with v - ` we have ψ` (rF ((x)(`) )) = ψ` (rF ((x)(Σ) )) where (x)(Σ) ∈ AΣ,× is the idele that F (Σ) is trivial at Σ and given by x at all places away from Σ. By (i), ψ` (rF ((x) )) = ((x)(Σ) ). Taking x to also satisfy x ≡ 1 mod m where m is the modulus of , it follows from the algebraicity of  that ((x)(Σ) ) = alg (x) for such x. Hence, ψ` (rF,` (x)) = alg (x)−1 for x ∈ F × that is sufficiently close to 1 at all v ∈ Σ. Using these nearness conditions at v|` gives an open neighborhood V` of the identity in F`× on which the desired identity holds because it does so on the subset F × ∩ V` that is dense in V` .

2.4. ALGEBRAIC HECKE CHARACTERS

69

The following converse lemma includes a uniqueness result for ψ` as in Lemma 2.4.8, and it will be useful later when we need to check an equality of algebraic Hecke characters. 2.4.9. Lemma. Let F and K be number fields. Let ` be a prime number, and let Σ be a finite set of places of F including all archimedian places and all places above `. Let ψ` : Gal(F ab /F ) → K`× be a continuous homomorphism that is unramified outside Σ and for which ψ` (Wv ) ⊆ K × for the Weil subgroup Wv in the decomposition group Dv ⊂ Gal(F ab /F ) at each v 6∈ Σ. Assume moreover that there exists a Q-homomorphism χalg : F × → K × such that ψ` ◦ rF,` coincides with χalg on an open subgroup U` of F`× = F × (Q` ). × with algebraic part χ−1 There exists a unique algebraic Hecke character  : A× F → K alg such that  induces ψ` as in Lemma 2.4.8. In particular, if F 0 /F is a finite extension then  ◦ NmF 0 /F corresponds to ψ` |Gal(F 0 ab /F 0 ) . × Proof. Define  : A× F → K` by

(2.4.9.1)

(x) = ψ` (rF (x)) · χ−1 alg (x` ),

where x` ∈ F`× denotes the `-adic component of x ∈ A× F . Choose an open subgroup W of Q × such that (ψ ◦ r )(W ) = {1}; such an open subgroup W exists because ψ is F F ` ` p∈Σ, p6=`,∞ p continuous and takes values in a finite product of multiplicative groups of `-adic fields. It is clear that  is trivial on the open subgroup Y × × 0 U := (F∞ ) × U` × W × OF,v v ∈Σ / × −1 × × of A× F . It is also clear that  coincides with χalg on F . Since AF is generated by U , F −1 × × × and (AΣ F ) , we have (AF ) ⊆ K . Thus,  is an algebraic Hecke character with χalg as its algebraic part. Moreover,  is unramified away from Σ and ψ` (Frobv ) = (πv ) for all v 6∈ Σ, so by continuity of ψ` we see that this ψ` is the same as the one arising from  in Lemma 2.4.8. As for the uniqueness of , it has to be proved that if  is a K × -valued finite-order Hecke character of F with ρ ◦ φ` = 1 (where ρ : SF → K × corresponds to ) then  = 1. But (πv ) = 1 for all but finitely many v, so the result follows. The final part concerning ◦NmF 0 /F follows from the uniqueness and the relationship in class field theory between Galois restriction and norm maps.

The next lemma will play a technical role in the proof of the main arithmetic result in this section, Theorem 2.4.11. 2.4.10. Lemma. Let Γ0 be a normal subgroup of finite index in a group Γ. Let γ be an element of Γ of infinite order, and let Γ1 := Γ0 · γ Z be the subgroup of Γ generated by Γ0 and γ. Let n be the unique positive integer such that hγ n i = Γ0 ∩ γ Z . Let ρ0 : Γ0 → H be a homomorphism of groups. Let h ∈ H be an element such that hn = ρ0 (γ n ). Assume that ρ0 (γ x γ −1 ) = h ρ0 (x) h−1 for all x ∈ Γ0 . Then there exists a unique extension of ρ0 to a homomorphism ρ1 : Γ1 → H such that ρ1 (γ) = h. Proof. Let Γ0 be the semidirect product Γ0 o Z defined by the group law (x, a) · (y, b) = (x γ a y γ −a , a + b) for x, y ∈ Γ0 and a, b ∈ Z (i.e., 1 ∈ Z acts on Γ0 by y 7→ γyγ −1 ). Define a map ρ0 : Γ0 → H by ρ0 ((x, a)) = ρ0 (x)ha for all (x, a) ∈ Γ0 . The assumption on ρ0 guarantees that ρ0 is a homomorphism of groups. The kernel of the natural surjection π : Γ0 → Γ1 = Γ0 · γ Z is the cyclic subgroup generated by the element (γ n , −n) since on the normal subgroup Γ0 ⊆ Γ0 this surjection is the identity and modulo Γ0 it is the natural projection Z → Z/nZ (so its kernel

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cannot contain (γ0 , i) for γ0 ∈ Γ and 0 < i < n, and for i = 0 necessarily γ0 = 1). Since the restriction of ρ0 to ker(π) is trivial, ρ0 = ρ1 ◦ π for a unique homomorphism ρ1 : Γ1 → H. × be an algebraic Hecke character, and Let F and K be number fields. Let  : A× F → K × × let alg : F → K be the algebraic part of . Let E be a subfield of F such that alg factors through the norm map NmF/E : F × → E × ; i.e., there exists a Q-rational homomorphism χalg : E × → K × such that alg = χalg ◦ NmF/E . (Beware that this is weaker than saying × with algebraic part χ ,  = χ ◦ NmF/E for an algebraic Hecke character χ : A× alg E → K since an algebraic Hecke character is only determined by its algebraic part up to a finite order character. Moreover, χalg may not even be the algebraic part of an algebraic Hecke character.) We now construct a special kind of “local root extraction” of  at the expense of replacing F/E with another extension:

2.4.11. Theorem. Using notation and hypotheses as above, let v be a finite place of F such that  is unramified at v and let w be the place of E induced by v. Suppose that an element β ∈ K × satisfies β [κv :κw ] = (πv ), where πv is a local uniformizer of OF,v . There exists a finite e of E, a place w e above w, and an algebraic Hecke character δ : A× → K × extension E e of E e E with the following properties. (i) [κwe : κw ] = 1; i.e., the degree of the residue field extension for w e over w is 1. (ii) The algebraic part of δ is equal to χalg ◦ NmE/E . e (iii) The algebraic Hecke character δ is unramified at w, e and δ(πwe ) = β, where πwe is a uniformizer of OE, . ew e Proof. Let F 0 /F be a finite extension that is Galois over E, and choose a place v 0 on F 0 over v. The composition 0 of  with NmF 0 /F on ideles is an algebraic Hecke character that is unramified at v 0 and has algebraic part alg ◦ NmF 0 /F = χalg ◦ NmF 0 /E . If πv0 ∈ OF 0 ,v0 is a [κ 0 :κ ]

× uniformizer then NmF 0 0 /Fv (πv0 ) is an OF,v -multiple of πv v v , so 0 (πv0 ) = β [κv0 :κw ] . Thus, by v replacing F , , and v with F 0 , 0 , and v 0 we may and do assume that F is Galois over E. Choose an arithmetic Frobenius element σ in a decomposition Dve|w ⊂ Gal(F ab /E); i.e., σ induces x 7→ xqw on the residue field of an extension ve of v to F ab , with qw = #κw . Let ` be a prime number which is prime to v, and let ψ` : Gal(F ab /F ) → K`× be the `-adic character attached to the algebraic Hecke character  as in Lemma 2.4.8. As we saw in Lemma 2.4.8, ψ` is locally algebraic above ` with −1 alg as its algebraic part above `. Choose M > 1 such that f (v|w)M σ lies in the decomposition group Dve|v ⊂ Gal(F ab /F ) at ve, with f (v|w) := [κv : κw ]. This agrees modulo the inertia group Ive|v with the M th power of an arithmetic Frobenius element in Dve|v , so by unramifiedness of ψ` at v we compute ψ` (σ f (v|w)M ) = ψ` (FrobM v ) = M f (v|w)M (πv ) = β . Since F/E is Galois, the element σ ∈ Gal(F ab /E) acts on Gal(F ab /F ) by conjugation. Consider the conjugate of ψ` by σ; i.e., the continuous homomorphism σψ` : Gal(F ab /F ) → K`× defined by τ 7→ ψ` (σ −1 τ σ) . Clearly σψ` is unramified at all but finitely many places and is locally algebraic above `, with algebraic part above ` given by composing the algebraic part of ψ` above ` with the action by σ −1 |F on F`× . Because alg factors through NmF/E , it is invariant under the action of σ|F ∈ Gal(F/E). Hence, the algebraic part of σψ` above ` is also equal to −1 alg . Therefore, by Remark 2.4.6(ii), there exists a finite abelian extension field F1 of F which is Galois over E such that σ ψ` coincides with ψ` on Gal(F ab /F1 ). Replacing F1 by a finite extension that is abelian over F and Galois over E with sufficiently divisible degree over F1 (e.g., a sufficiently large ray class field of F ), we may assume (by the relationship between b that there exists a positive integer N1 finite-index subgroups of Z and open subgroups of Z) Z ab f (v|w)M N Z 1 . such that σ ∩ Gal(F /F1 ) = σ e Let E ⊆ F1 be the subfield over E that is the fixed field of the cyclic group hσ|F1 i ⊆ Gal(F1 /E) with order f (v|w)M N1 . Since ψ` (σ f (v|w)M N1 ) = β f (v|w)M N1 , by applying Lemma

2.4. ALGEBRAIC HECKE CHARACTERS

71

2.4.10 with Γ0 = Gal(F ab /F1 ), Γ = Gal(F ab /E), γ = σ, and n = f (v|w)M N1 , we see that e → K × that coincides with ψ` there is a unique continuous homomorphism ψe` : Gal(F ab /E) ` on Gal(F ab /F1 ) and such that ψe` (σ) = β. In particular, ψe` is unramified at all but finitely e since ψ` is unramified at all but finitely many places of F and the finite many places of E e is unramified at all but finitely many places. extension F1 /E e of the place ve on F ab , so σ ∈ Dve|we . Since the subgroup Let w e be the restriction to E e ⊆ Gal(F1 /E) is the subgroup generated by σ|F , we have that κw ⊆ κwe . But κwe Gal(F1 /E) 1 has trivial action by the automorphism x 7→ xqw induced by σ|F1 . Hence, κw = κwe , so (i) e is an arithmetic Frobenius holds. Thus, qwe = qw , so the element σ ∈ Dve|we ⊂ Gal(F ab /E) e we element in this decomposition group. Beware that even though σ|F1 generates Gal(F1 /E), e is ramified at the restriction v1 of ve to F1 do not see how to avoid the possibility that F1 /E (especially if F/E is ramified at v). Thus, we will have to do some work to check that the algebraic Hecke character δ that we shall construct is unramified at w. e e ab /E) e induced by the By construction, the continuous K`× -valued character ψe`0 of Gal(E × ab e e continuous homomorphism ψ` : Gal(F /E) → K` is locally algebraic above `, with algebraic part χ−1 because we can compute the algebraic part by working near the identity (such e alg ◦NmE/E as on the open image of Gal(F ab /F1 )) and the relative norm map goes over to “restriction” on Galois groups via class field theory. Since the continuous ψe`0 is unramified at all but finitely many places, the proof of Lemma 2.4.9 (especially the analysis of open kernels) therefore gives that the continuous homomorphism δ : A×e → K`× defined by the formula E

δ(x) = ψe`0 (rEe (x)) · χalg (NmE/E e (x` )) e × is an algebraic Hecke character of E e with values for x ∈ A×e with `-adic component x` ∈ E ` E in the finite product of fields K` , and that δ has algebraic part χalg ◦ NmE/E e . In particular, δ

has open kernel and hence is continuous with respect to the discrete topology on K`× . e × · Nm e (A× ) of finite index in A× , and let πwe be a Let U be the open subgroup E F1 e F1 /E E e ab /E) e uniformizer of O e chosen such that the arithmetic Frobenius element r e (πwe ) ∈ Gal(E E,w e

E

at w e is equal to the arithmetic Frobenius element σ|Eeab . Such a πwe can be found because the × ewe ). The idele group e ab /E profinite unit group Ow e e surjects onto the inertia subgroup of Gal(Ew × e is generated by σ|F . Thus, to show that δ A e is generated by U and πwe because Gal(F1 /E) 1 E × × takes its values in K ⊂ K` it suffices to prove that δ(πwe ) ∈ K × and δ|U is valued in K × . Since ` - w e (as ` - v), by the definitions of δ, ψe` , and πwe we have δ(πwe ) = ψe`0 (rEe (πwe )) = ψe`0 (σ|Eeab ) = ψe` (σ) = β ∈ K × . e× For any element x ∈ U , written in the form x = NmF1 /Ee (y) · z with y ∈ A× F1 and z ∈ E , e × ) = {1} we have since r e (E E

δ(x) = ψe`0 (rEe (x)) · χalg (NmE/E e (x` )) = ψ` (rF (NmF1 /F (y))) · χalg (NmF/E (NmF1 /F (y` ))) · χalg (NmE/E e (z` )) × = (NmF1 /F (y)) · χalg (NmE/E e (z)) ∈ K ,

where the final equality uses the construction of  from ψ` as in the proof of Lemma 2.4.9 (since alg = χalg ◦ NmF/E ). Thus, indeed δ(A×e ) ⊆ K × . E We have shown that δ is an algebraic Hecke character satisfying the required properties (i)–(iii), except for verifying that δ is unramified at w. e The key to settling this remaining e is ramified at w e and more specifically issue even if F1 /E e is to exploit the cyclicity of F1 /E, the Hasse Norm Theorem for cyclic extensions of number fields. For a local unit u ∈ O×e , E,w e

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we have to prove δ(u) = 1 when u is viewed in A×e in the natural manner. We can write E e × , and y ∈ A× . In particular, since the E-idele e u = z · Nm e (y) · π i for some i ∈ Z, z ∈ E F1 /E

−i uπw e

F1

w e

e × is a has trivial local component at every place away from w, e it follows that z ∈ E × e , z is a local local norm from F1 at every place away from w. e By the triviality of rEe on E norm at w. e Hence, by the Hasse Norm Theorem for cyclic extensions, z is a global norm: z = NmF1 /E (z1 ) for some z1 ∈ F1× . Thus, by replacing y with z1 y we can assume z = 1. e and Gal(F1 /E) e is There is a unique place v1 of F1 over w e because σ ∈ Dve|we ⊂ Gal(F ab /E) i generated by σ|F1 . We therefore have u = NmF1,v /Ee (y1 )πwe , where y1 is the v1 -component w e 1 of the F1 -idele y. Let πv1 ∈ OF×1 ,v1 be a uniformizer, so there are unique j ∈ Z and u1 ∈ OF×1 ,v1 such that y1 = πvj1 u1 . Since NmF1,v

e 1 /Ew

e

f (v1 |w) e

(πv1 ) is an O×e -multiple of πwe E,w e

, the condition ordwe (u) = 0

forces i = −f (v1 |w)j e = −f (v1 |w)j = −f (v1 |v)f (v|w)j. Hence, δ(u) = δ(NmF1 /Ee (y1 ))δ(πwe )i = (NmF1 /F (y1 ))β −f (v1 |v)f (v|w)j = (NmF1 /F (y1 ))(πv )−f (v1 |v)j where y1 is viewed as an F1 -idele supported at v1 (so NmF1 /F (y1 ) = NmF1,v1 /Fv (y1 ) as an F -idele supported at v). Since  is unramified at v, (NmF1,v1 /Fv (y1 )) = (NmF1,v1 /Fv (πv1 ))j . Thus, we just have to show (πv )f (v1 |v) = (NmF1,v1 /Fv (πv1 )), and this equality follows from the unramifiedness of  at v and the fact that the (normalized) v-adic ordinal of NmF1,v1 /Fv (πv1 ) is f (v1 |v). 2.4.12. Remark. In the final part of the proof of Theorem 2.4.11, to prove the unramifiedness of δ at w, e we had to do some extra work with the Hasse Norm Theorem to overcome e could be ramified at its unique place above w. the possibility that F1 /E e Let us indicate an alternative procedure to bypass this difficulty by using the Grunwald–Wang theorem [4, Ch. X, Thm. 6]. Let v1 be the unique place of F1 above w, e and let e(v1 |w) e be the ramification index. By the Grunwald–Wang theorem, there exists an cyclic extension F2 /F1 which is unramified at v1 such that the residual degree f (v1 , F2 /F1 ) is a multiple of e(v1 |w). e Let F3 /F2 be a Galois e closure of F2 over E, so F3 /F1 is abelian and unramified over v1 . Let v2 be a place of F2 above v1 , and let v3 be a place of F3 above v2 . Consider the short exact sequence 1 → Iv1 |we → Dv1 |we → Gal(κv1 /κwe ) → 1 for the decomposition group Dv1 |we . The short exact sequence 1 → Iv3 |we → Dv3 |we → Gal(κv3 /κwe ) → 1 for the decomposition group Dv3 |we is the pullback of the previous short exact sequence by the natural surjection Gal(κv3 /κwe )  Gal(κv1 /κwe ) because e(v3 |w) e = e(v1 |w). e This pullback sequence splits (as a semi-direct product) because Gal(κv3 /κwe ) is a cyclic group whose cardinality is a multiple of the cardinality of Dv1 |we ; here we use that F2 /F1 was chosen so that e(v1 |w)|[κ e v2 : κv1 ]. Thus, there exists σ3 ∈ Dv3 |w e that maps to the arithmetic Frobenius element in Gal(κv3 /κwe ) and for which the subgroup generated by σ3 is a lifting of Gal(κv3 /κwe ) in Dv3 |we . Let E3 be the subfield of F3 fixed by σ3 , so v3 is the only place on F3 over its restriction w3 in E3 . The extension F3 /E3 is unramified over w3 , by consideration of field degrees (residually e w) and generically). The statement of Theorem 2.4.11 holds with (E, e replaced by (E3 , w3 ); that δ is unramified at w3 is easy to see because F3 /E3 is unramified.

2.5. Theory of complex multiplication We now review part of the theory of complex multiplication due to Shimura and Taniyama, and then use it to translate Theorem 2.4.11 into a statement concerning CM liftings of abelian varieties up to isogeny. We will make extensive use of the reflex field E ⊂ Q associated to a

2.5. THEORY OF COMPLEX MULTIPLICATION

73

Q-valued CM type (K, Φ), as well as the reflex norm NΦ : ResE/Q (Gm ) → ResK/Q (Gm ) (see Definitions 2.1.4 and 2.1.4.3). For our present purposes, the following version of the Main Theorem of Complex Multiplication will suffice. A more refined version is stated and proved in Appendix A.2. 2.5.1. Theorem (Main Theorem of Complex Multiplication). Let (A, α) be a CM abelian variety over a number field F ⊂ Q, and let (K, Φ) be its CM type. (i) The reflex field E ⊂ Q of (K, Φ) is contained in F . × such that for every (ii) There exists a unique algebraic Hecke character  : A× F → K ab prime number `, the continuous homomorphism ψ` : Gal(F /F ) → K`× attached to  in Lemma 2.4.8 is equal to the `-adic representation of Gal(F ab /F ) attached to the `-power torsion points of A. (iii) The algebraic part of the algebraic Hecke character  is the map NΦ ◦ NmF/E , where NΦ : ResE/Q (Gm ) → ResK/Q (Gm ) is the reflex norm. (iv) Let v be a finite place of F . The abelian variety A has good reduction at v if and only if the algebraic Hecke character  is unramified at v. (v) If v is a finite place of F where A has good reduction, say with reduction A over κv , then (πv ) = FrA,qv , where πv is a uniformizer of OFv and qv = #κv . Part (i) is immediate from the definition of the reflex field (see Example 2.1.4.1), and in Appendix A.2 we use the torus from Proposition 2.1.4.5 to state and prove an alternative version of the Main Theorem (see Theorem A.2.4.4), from which we deduce Theorem 2.5.1. More precisely, parts (ii) and (iii) above correspond to Theorem A.2.5.3 and Remark A.2.5.4 (the uniqueness in (ii) being obvious), and parts (iv) and (v) correspond to the two respective parts of Theorem A.2.5.8. 2.5.1.1. Remark. Since Theorem 2.5.1 is insensitive to K-linear isogenies in A over F , by Proposition 1.6.4.3 there is no loss of generality in assuming that the CM order is OK , as is sometimes done in the literature on the Main Theorem. Parts (ii) and (iii) of Theorem 2.5.1 amount to a reformulation of the Main Theorem as stated in [70]. Part (iv) is proved in §7 of [68] as a consequence of this Main Theorem (and (v) follows from (ii) and (iv)). See [65, (II-25)–(II-28)] and [41, §1, Ch. 4] for further discussion of the Main Theorem and related constructions. The following converse to Theorem 2.5.1 is a consequence of the motivic theory of complex multiplication (see [20, Prop. E.1, pp. 273–275]) and is also proved in [41, §5, Ch. 5] using an older style of algebraic geometry and the traditional analytic formulation of the Main Theorem of Complex Multiplication (as stated in [41, Thm. 6.1, Ch. 3], and developed in a coordinate-free form in A.2.8). 2.5.2. Theorem. Let K be a CM field with [K : Q] = 2g. Let Φ be a CM type for K valued in a fixed algebraic closure Q and let E ⊂ Q be the reflex field of (K, Φ). Let F/E be a × finite extension. Let  : A× F → K be an algebraic Hecke character whose algebraic part is the composite map NΦ ◦ NmF/E , where NΦ is the reflex norm associated to (K, Φ). There exists a g-dimensional abelian variety A over F and a ring homomorphism α : K → End0 (A) such that Φ is the CM type of (A, α) and  is the algebraic Hecke character attached to (A, α) as in Theorem 2.5.1(ii). Moreover (A, α) is unique up to K-linear F -isogeny. In Appendix A.3 we provide a proof of Theorem 2.5.2 using the constructions in the proof of the Main Theorem in Appendix A.2. We are now in position to prove Theorem 2.1.7. Let us first recall the statement.

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2.5.3. Proposition. Let K be a CM field with degree 2g over Q. Let q = pr , where p is a prime number. Let B be an isotypic abelian variety of dimension g over a finite field Fq with size q, and let αB : K → End0 (B) be a ring homomorphism. Suppose that there exists a p-adic CM type Φ ⊂ Homring (K, Qp ) of K such that the residual reflex condition in 2.1.6 is satisfied for (K, Φ). Let E ⊂ Qp be the reflex field of (K, Φ), and let w be the induced p-adic place on E. e e and a ring There exists a finite extension E/E inside of Qp , an abelian variety A over E, homomorphism α : K → End0 (A) such that the following statements hold. e at the place w (i) The residue field κwe of E e induced from Qp is isomorphic to Fq . (ii) The p-adic CM type of (A, α) is Φ. e has good reduction at w. (iii) The abelian variety A over E e (iv) Via a suitable isomorphism Fq ' κwe , B is K-linearly isogenous to the reduction of A at w. e Note that in (iv) the choice of isomorphism Fq ' κwe does not matter, since B and B (p) are K-linearly isogenous over Fq via the relative Frobenius map for B. Proof. This proposition is a consequence of Theorems 2.4.11, 2.5.1, and 2.5.2, as we explain in the argument below. By Proposition 1.4.4.1 (and Proposition 1.6.4.3), there exists an abelian variety X1 of dimension g over a finite extension L1 of Qp within Qp and a homomorphism ξ1 : OK → End(X1 ) such that (X1 , ξ1 ) has CM type Φ. In particular, the reflex field E ⊂ Qp is a subfield of L1 such that the canonical p-adic absolute value on L1 induces the p-adic place w on E. By replacing L1 with a finite extension if necessary, we can assume that X1 has good reduction over OL1 and that the size q1 of the residue field κ1 of L1 is a power of q. Let π1 ∈ OK be the element whose action on the reduction X 1 over κ1 is the q1 -Frobenius endomorphism (so it is a q1 -Weil number). The Shimura–Taniyama formula (2.1.5.1) gives ordv (π1 ) #{φ ∈ Φ | φ induces v on K} = ordv (q1 ) [Kv : Qp ] for all p-adic places v of K. The first part of the residual reflex condition in 2.1.6 is the hypothesis on the p-adic CM type Φ that the right side of this identity is equal to ordv (FrB,q )/ordv (q) for each v (where FrB,q is identified with a nonzero element of OK via the CM structure on B over Fq ). Since q1 is a power of q, we can choose an embedding Fq ,→ κ to make sense of FrBκ ,q1 , [Fq :Fq ]

and as an element of K this q1 -Frobenius endomorphism is FrB,q1

. Hence,

ordv (FrB,q ) ordv (FrBκ ,q1 ) ordv (π1 ) = = , ordv (q1 ) ordv (q) ordv (q1 ) so the q1 -Weil numbers π1 , FrBκ ,q1 ∈ OK have the same order at each p-adic place of OK . Their ratio is therefore a root of unity, so by replacing L1 with a suitable unramified extension (so as to replace q1 with a suitable power) we can arrange that π1 = FrBκ ,q1 inside of OK . By Theorem 1.5.2.1, the g-dimensional isotypic abelian varieties X 1 and Bκ over κ are isogenous. Although a choice of such an isogeny X 1 → Bκ may not be K-linear, it at least carries π1 = FrX 1 ,q1 to FrBκ ,q1 . In other words, the induced isomorphism θ : End0 (X 1 ) ' End0 (Bκ ) of simple Q-algebras may not be K-linear but it is linear over the subfield Z ⊆ K generated by the q1 -Frobenius endomorphisms in each algebra. But Z is the center of these simple Q-algebras (Corollary 1.5.2.2) in which K is a maximal commutative subfield, so the Zlinearity allows us to use the Skolem–Noether theorem to find a unit u ∈ End0 (Bκ )× such that composing θ with conjugation by u is a K-algebra map. In other words, if we compose the initial choice of isogeny with a suitable isogenous endomorphism of Bκ (corresponding to a

2.6. LOCAL METHODS

75

nonzero Z-multiple of u) then we can arrange that the isogeny X 1 → Bκ is K-linear. Finally, by replacing L1 with a further finite extension if necessary we can ensure (Theorem 1.6.2.1) that the pair (X1 , ξ1 ) descends to a number field F ⊂ L1 , and by increasing F we can ensure that L1 is identified with the completion Fv at the induced p-adic place v on F . In particular, E ⊆ F and v on F restricts to w on E. To summarize, by using the first part of the residual reflex condition we have constructed a finite extension F/E inside of Qp and a CM abelian variety (A1 , α1 ) over F with good reduction at the induced p-adic place v of F such that the associated p-adic CM type is (K, Φ) and the reduction A1 at v is K-linearly isogenous to Bκv via a choice of embedding Fq ,→ κv . Such an isogeny carries FrA1 ,qv to FrBκv ,qv as endomorphisms, and so the corresponding elements of OK (via the CM structures) are equal, due to K-linearity of the isogeny. Pick a finite extension E1 of E that is linearly disjoint from F over E and which has a place w1 above w such that κw1 is isomorphic to Fq . Choose an E-embedding E1 → Qp inducing w1 . By replacing F with the composite field E1 ⊗E F ⊂ Qp , we may assume that F contains a subfield E 0 /E on which the induced place w0 from v satisfies κw0 ' Fq . By Theorem 2.5.1, × we get an algebraic Hecke character  : A× F → K attached to (A1 , α1 ) which has NΦ ◦ NmF/E as its algebraic part and is unramified at v with (πv ) = Fr(B,αB )κv . Apply Theorem 2.4.11 to (, E 0 , β) with χalg := NΦ ◦NmE 0 /E and β equal to the element of OK corresponding to FrB,q via αB (so β [κv :κw0 ] = β [κv :Fq ] = Fr(B,αB )κv = (πv )). This provides e 0 , a place w a finite extension E/E e above w0 , and an algebraic Hecke character δ : A×e → K × E with the properties stated in Theorem 2.4.11. Hence, δalg = χalg ◦ NmE/E e 1 = NΦ ◦ NmE/E e , δ e is unramified at w, e δ(πwe ) = β = FrB,q , and (E, w) e has property (i). 0 e By Theorem 2.5.2 (and Theorem e Fix an E -embedding of E into Qp inducing w e on E. e with the required 2.5.1(iv),(v)), the algebraic Hecke character δ comes from a pair (A, α) over E properties (ii)–(iv). Let us briefly explain how to get (iv). Let A denote the reduction of A at w e and let π ∈ OK denote δ(πwe ), so the respective CM structures α and αB on A and B satisfy α(π) = FrA,q and αB (π) = FrB,q . Since A and B are isotypic (due to the existence of CM structures; see Theorem 1.3.1.1), it follows from Honda–Tate theory (Theorem 1.5.2.1) that there is an isogeny between A and B over any isomorphism κwe ' Fq . We want to find such an isogeny that is K-linear. At least any isogeny respects the q-Frobenius endomorphisms, and so (via the CM structures α and αB ) is linear over the subfield Q(π) ⊆ K. But via α and αB the subfield Q(π) is the center of each of the endomorphism algebras of A and B (Corollary 1.5.2.2), so we can use the Skolem–Noether theorem as we did earlier in this proof to modify the isogeny by a suitable isogenous endomorphism of B to get a K-linear isogeny.

2.6. Local methods It is possible to prove Proposition 2.5.3 by replacing the global Theorem 2.5.2 with purely local methods (but still using the classical theory of complex multiplication, to construct a suitable abelian variety with good reduction over a local field). This rests on some basic concepts in p-adic Hodge theory, such as Tate’s work on p-divisible groups over discrete valuation rings (Theorem 1.4.5.6), crystalline representations, and the relationship between these two topics. (We will provide references for what we need.) After using local methods to prove a local version of Proposition 2.5.3 (in which the e is replaced with a p-adic field) we will formally deduce the global version of number field E the proposition (over number fields) from the local version. First we establish a local analogue of Theorem 2.4.11, using the argument in Remark 2.4.12. 2.6.1. Lemma. Let K be a number field, and φ an element of K × . Let Ew be a finite extension field of Qp , Fv a finite extension field of Ew , and v : Fv× → K × an unramified

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character such that v (πvn ) = φn[κv :κw ] for some fixed n > 1, with πv a chosen uniformizer of Fv . ewe ⊆ Feve over Ew such that the There exists a finite extension Feve/Fv and a subfield E following conditions hold. ewe is unramified. (i) The extension Feve/E ewe /Ew is totally ramified. (ii) The residue field extension κwe /κw is trivial; i.e., E [κ :κ ] w v e (iii) The value v (NmFe /Fv (πve)) is equal to φ . v e e e × → K × such that (iv) For any uniformizer πwe of Ewe , there is a unique character we : E w e v ◦ NmFe /Fv = we ◦ NmFe /Ee and we (πwe ) = φ. Moreover, we is unramified. v e

v e

w e

Proof. Let F1 be the unramified extension field of Fv of degree n, and let F2 be the Galois closure of F1 over Ew . Consider the short exact sequence of finite groups 1 → IF2 /Ew → Gal(F2 /Ew ) → Gal(κF2 /κw ) → 1 where IF2 /Ew is the inertia subgroup in Gal(F2 /Ew ). This is not necessarily a split extension, but it would be split if we replace F2 by the unramified extension field F3 of F2 of degree e(F2 /Ew ), or more generally by any finite unramified extension field F3 such that [F3 : F2 ] is a multiple of the ramification index e(F2 /Ew ), because the short exact sequence 1 → IF3 /Ew → Gal(F3 /Ew ) → Gal(κF3 /κw ) → 1 is the pullback of the exact sequence 1 → IF2 /Ew → Gal(F2 /Ew ) → Gal(κF2 /κw ) → 1 by the natural surjection Gal(κF3 /κw )  Gal(κF2 /κw ). Take Feve = F3 . Choose a splitting j : Gal(κve/κw ) → Gal(Feve/Ew ) of the short exact sequence 1 → IFe /Ew → Gal(Feve/Ew ) → Gal(κve/κw ) → 1 , v e

ewe be the subfield of Feve fixed by j(Gal(κve/κw )). The statements (i)–(iv) are all easy and let E to check (for (iv) we use that the local norm map is surjective on local unit groups in the unramified case, and for (iii) we use that n|[κve : κv ] by construction of F1 ). Our aim is to remove the global Theorem 2.5.2 from the proof of Proposition 2.5.3. The construction of a CM abelian variety (A1 , α1 ) with p-adic CM type (K, Φ) over a finite extension F/E inside of Qp goes as in the first two paragraphs of the proof of Proposition 2.5.3. In particular, A has good reduction at the place v on F over w induced by the inclusion F ⊂ Qp and moreover the residual extension κv /κw has Fq as an intermediate extension such that the reduction A1 over κv is K-linearly isogenous to Bκv . Let Ew0 0 /Ew be the intermediate unramified extension of Ew corresponding to Fq /κw , and let v be the restriction to Fv× of the algebraic Hecke character  attached to the CM abelian variety (A1 , α1 ) over F with CM order OK . The character × ψp,v : Gal(Fvab /Fv ) → OK,p attached to the p-divisible group (A1 )Fv [p∞ ] equipped with its Kp -action via α1 is related to the character v via the formula v (x) = ψp,v (rF (x)) · NΦ (NmFv /Ew (x)) Fv×

for all x ∈ by Lemma 2.4.8, Lemma 2.4.9 (see (2.4.9.1)), and Theorem 2.5.1(iii). Apply Theorem 2.5.1(iv) and Lemma 2.6.1 to (Ew0 0 , Fv , v , φ, n), where φ = αB (FrB,q ) ∈ K × and ewe in Feve n = 1. We obtain a finite extension Feve/Fv inside of Qp , an intermediate extension E × 0 × e satisfying the properties (i)–(iv) in over Ew0 , and a K -valued unramified character we of E w e e e Lemma 2.6.1. In particular, Fve/Ewe is unramified, κwe = κw0 = Fq , and we (πwe ) = αB (FrB,q ) ∈ K ×. Denote by A1 the abelian scheme over OFe extending the base change (A1 )Fe , and endow it v e v e with the unique K-action (in the isogeny category) extending that defined on its generic fiber

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via α1 . The p-divisible group G1 = A1 [p∞ ] over OFe is thereby endowed with a natural action v e by Kp in the isogeny category, and its generic fiber is given by the restriction to Gal(Feveab /Feve) of the Galois character ψp,v of Fv as above. e ab /E ewe ) → O× such that There exists a unique continuous character ψp,we : Gal(E K,p w e (2.6.1.1)

we (x) = ψp,we (rEe (x)) · NΦ (NmEe

w e /Ew

w e

(x))

e × . Indeed, uniqueness is clear, and for existence we have to check that the for all x ∈ E w e continuous homomorphism e × → Kp× we · (NΦ ◦ Nm e )−1 : E Ewe /Ew

w e

× has image contained in OK,p (and so it is of “Galois type”: it factors through the profinite completion of its source). This is a problem of analyzing the image of a single uniformizer × πwe up to OK,p -multiple. Thus, it suffices to check that the product FrB,q · NΦ (πw )−[κwe :κw ] in × Kp× lies in OK,p for some uniformizer πw in Ew . It suffices to check this after raising to the [κv : κwe ]th-power. Since κwe = κw0 = Fq , the desired result follows from the fact that v (πv ) [κv :κw0 ] [κ :κ ] × and is an OK,p -multiple of NΦ (πw v w ) (due to the Galois character ψp,v is equal to FrB,q × ). taking values in OK,p By using Lemma 2.6.1(iv), we get the identity

(ψp,v ◦ rFv ) ◦ NmFe

v e /Fv

= (ψp,we ◦ rEe ) ◦ NmFe w e

v e /Ew e

e

.

ewe of the generic fiber G1 of G1 over Thus, the character ψp,we defines a descent G0 over E ewe acquires “good Feve equipped with its natural OK,p -action. The p-divisible group G0 over E 0 e reduction” over the finite unramified extension Fve in the sense that G e ' G1 over Feve and Fve this extends to the p-divisible group G1 over OFe,ev . The resulting Galois descent data relative ewe on the generic fiber of G1 over Feve can be viewed as a collection of isomorphisms to Feve/E ewe )-twists (satisfying a cocycle condition). These isomorphisms between G1 and its Gal(Feve/E to the Galois twists uniquely extend to isomorphisms from G1 to its Galois twists, by Tate’s full ewe , we can use the resulting faithfulness result in Theorem 1.4.5.6. Since Feve is unramified over E equivalence of Galois descent and finite ´etale descent [8, 6.2B] at each torsion level over OEe w e 4 to construct a unique p-divisible group G 0 over OEe having generic fiber G0 compatibly with w e the descent data. Again using Tate’s full faithfulness theorem, the OK,p -action on G0 extends uniquely to one on G 0 . 2.6.2. Lemma. The q-Frobenius endomorphism of the reduction G00 := Gκ0 we is given by the action of we (πwe ) ∈ OK . This lemma (along with some of the ingredients in its proof) is a local version of Theorem 2.5.1(v) for p-divisible groups with complex multiplication. Proof. By decomposing G00 up to isogeny according to the factor fields of Kp , our task is a special case of the following problem. Let Γ be a p-divisible group of height d over a padic integer ring OF with finite residue field k of size q and fraction field F . Let L/Qp be an extension of degree d and assume that Γ is equipped with an action of L in the isogeny category over OF . By Proposition 1.4.5.5, L is its own centralizer in the endomorphism algebra of the reduction Γ0 over k. Hence, some nonzero element λ ∈ L induces the qFrobenius endomorphism of Γ0 . We wish to compute λ in terms of the continuous Galois × character ψ : Gal(F ab /F ) → OL associated to the generic fiber ΓF . It suffices to prove that there is a unique homomorphism χ : F × → L× algebraic with respect to Qp such that χ, ψ ◦ rF : F × ⇒ L× coincide on OF× and the character (ψ ◦ rF ) · χ−1 : 4unramified descent: give general reference for other use elsewhere

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F × → L× that is trivial on OF× carries uniformizers to λ. Indeed, this would give the desired result concerning G00 due to (2.6.1.1), since we is unramified. Our assertion refines the local part of Serre’s theory of locally algebraic abelian representations in [65, III.1] since we are asserting the algebraicity of ψ ◦ rF on the entire local unit group OF× of F × . The existence and uniqueness of χ is a special case of a general property of abelian semisimple crystalline p-adic representations of Galois groups of p-adic fields, and it is proved in [18, App. A]. Fix an isomorphism Fq ' κwe so as to view B as an abelian variety over κwe . The p-divisible groups B[p∞ ] and G00 over κwe are endowed with actions by Kp in the isogeny category. Since αB (π) = FrB,q and π = we (πwe ), by Lemma 2.6.2 the element π ∈ OK acts as the q-Frobenius endomorphism on each of these p-divisible groups. This is crucial in the proof of the following lemma. 2.6.3. Lemma. There is an Kp -linear isogeny B[p∞ ] → G00 over κwe . Proof. Let k = κwe . We will use contravariant Dieudonn´e theory, so let Dk = W (k)[F, V] denote the usual Dieudonn´e ring over the finite field k (non-commutative if k 6= Fp ). The category of left Dk [1/p]-modules that have finite dimension over W (k)[1/p] and admit a Dk stable W (k)-lattice is anti-equivalent to the isogeny category of p-divisible groups over k. Note that the center of Dk [1/p] = W (k)[1/p][F] is Qp [Fq ] where Fq = F [k:Fp ] . The Dieudonn´e modules D(B[p∞ ]) and D(G00 ) are left Dk -modules that are finite free over W (k) with rank 2g that is equal to the Zp -rank of OK,p , and each is equipped with a Dk -linear action by OK,p . The actions by OK,p on both p-divisible groups are faithful. Indeed, for B[p∞ ] the faithfulness is due to Proposition 1.2.5.1 with ` = p, and for G00 the faithfulness can be checked after the ground field extension κve/k where it becomes the p-divisible group of an abelian variety (the reduction of the CM abelian variety (A1 , α1 ) over Feve). In view of this faithfulness, we can argue exactly as in the beginning of the proof of Proposition 2.2.1 to see that both Dieudonn´e modules are invertible as W (k) ⊗Zp OK,p -modules. Let Q Z = Q(π), so if f ∈ Q[T ] is the minimal polynomial of π then Q[T ]/(f ) ' Z via T 7→ π. Let v0 |p fv0 be its monic irreducible factorization in Qp [T ], corresponding to the decomposiQ tion Zp ' v0 |p Zv0 . Here, Zv0 ' Qp [T ]/(fv0 ) in which π ∈ Z ⊂ Zv0 is the image of T . Since Fq is central in Dk [1/p], it makes sense to form the Zv0 -algebra Cv0 = Dk [1/p]/Dk [1/p]fv0 (Fq ) in which the element π ∈ Zv0 acts as Fq . By Theorem A.1.2.1, Cv0 is a central simple Zv0 algebra. Thus, the isomorphism class of a finitely generated left Cv0 -module is determined by its Zv0 -dimension. Both D(B[p∞ ])[1/p] and D(G00 )[1/p] are invertible modules over W (k) ⊗Zp Kp , and their Zv0 -factors are left Cv0 -modules because the q-Frobenius endomorphisms of B[p∞ ] and G00 are induced by the element π ∈ OZ ⊆ OK whose image in Zv0 is a root of fv0 ∈ Qp [T ]. When viewing these Dieudonn´e modules as left Cv0 -modules, they have the same Zv0 -dimension, namely that of the Zv0 -part of the Zp -algebra W (k) ⊗Zp Kp . Hence, the Zv0 -factors are isomorphic as Cv0 -modules, and forming the direct product of these over all v 0 gives a Zp -linear isomorphism between the underlying Dk [1/p]-modules. This provides a Zp -linear isogeny B[p∞ ] → G00 over k, but it may not be Kp -linear. Hence, the resulting isomorphism of endomorphism algebras End0 (B[p∞ ]) ' End0 (G00 ) is Zp -linear but perhaps not Kp -linear. These Zp -isomorphic endomorphism algebras are central simple Zp -algebras with Kp a maximal commutative subalgebra in each because Theorem 1.5.1.1 gives Qp ⊗Q End0 (B) ' End0 (B[p∞ ]) with End0 (B) a central simple Z-algebra having K as a maximal commutative Z-subalgebra. Hence, the Skolem–Noether theorem ensures that if we compose a choice of Zp -linear isogeny B[p∞ ] → G00 with a suitable Zp -linear isogenous endomorphism of G00 then we get a Kp -linear isogeny.

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Using Lemma 2.6.3, we may choose a Kp -linear κwe -isogeny B[p∞ ] → G00 . The kernel is identified with a finite subgroup scheme of B over κwe = Fq , so if we replace B with its quotient modulo this kernel then we gain the property that there is a Kp -linear isomorphism B[p∞ ] ' G00 of p-divisible groups (not just an isogeny). To summarize, the p-divisible group G 0 over OEe equipped with its K-action (in the isogeny category) is identified with a deformation w e of the p-divisible group of the abelian variety B over κwe equipped with its K-action (in the isogeny category). Hence, by working with CM orders we deduce via Theorem 1.4.5.7 that there is a unique formal abelian scheme A0 over Spf(OEe ) equipped with an action of K (in w e the isogeny category) that compatibly deforms B and has p-divisible group G 0 (respecting the K-actions). But the tangent space of A0 coincides with that of G 0 over OEe (Example w e 1.4.5.3), which in turn descends the tangent space Lie(G1 ) ' Lie(A1 ) over OFe , all respecting v e the actions by an order in OK . After inverting p this recovers Lie(A1 )Fe , on which the action v e of K is given by the p-adic CM type Φ of (A1 , α1 )Fv . Hence, by Theorem 2.2.3, A0 algebraizes to an abelian scheme A0 over OEe endowed with an action by K (in the isogeny category) w e giving rise to the CM type Φ. This algebraization has reduction B and its p-divisible group is the descent G 0 of G1 = A1 [p∞ ] compatibly with K-actions and the residual isomorphism B[p∞ ] ' G00 . ewe with its K-action, so (A0 , α0 ) has Let (A0 , α0 ) be the CM generic fiber of A0 over E good reduction that is K-linearly isomorphic to (B, αB ) over the initial choice of isomorphism κwe ' Fq and the p-adic CM type of (A0 , α0 ) is (K, Φ). We have just shown that (B, αB ) satisfies (IN) in 1.7.5, with a lift of p-adic CM type (K, Φ) over the integer ring of the subfield ewe rather than a number field. ewe ⊂ Qp . This is a local version of Proposition 2.5.3, using E E To complete the local proof of Proposition 2.5.3, it remains to carry out one global step: ewe (necessarily containing the reflex field E ⊂ Qp descend (A0 , α0 ) to a number field within E ewe so that its of the CM type (K, Φ)); we can then increase such a number field within E ewe ; in particular, this place has residue field κwe and completion at the place induced by w e is E 0 the descent of A has good reduction at this place. The required descent to a number field is given by the next lemma. 2.6.4. Lemma. Any CM abelian variety X with good reduction over a p-adic field L descends together with its CM structure to a CM abelian variety over a number field within L. Proof. Fix a positive integer n > 3 not divisible by p, and let X over R = OL be the N´eron model of X. The finite ´etale group scheme X [n] over R is uniquely determined by its special fiber. Let F ⊂ L be a dense subfield that is a number field and let v be the induced p-adic place of F , so Fv = L. The special fiber of X [n] can be uniquely lifted to a finite ´etale h group scheme G over the henselization OF,(v) of the algebraic local ring of OF at v, and by uniqueness of finite ´etale liftings over henselian local rings this descends X [n] via the canonical h local inclusion OF,(v) → R. The generic fiber of G lives over an algebraic extension of Q, so it descends to a finite ´etale group G over a number field F 0 ⊂ L. Choose a polarization of X, say with degree d2 . Let g = dim(X), and let O denote the CM order on X arising from the CM structure. Consider the moduli scheme M of finite type over F 0 classifying isomorphism classes of polarized abelian schemes of relative dimension g (over F 0 -schemes) endowed with the following extra structure: a degree-d2 polarization, an isomorphism of the n-torsion with the pullback of the F 0 -group G, and an action by O. Every geometric point of this moduli scheme M descends to Q (Theorem 1.6.2.1), so M is 0-dimensional. Hence, the F 0 -map Spec(L) → M corresponding to X with its additional structure factors through some point of M that is necessarily defined over a finite extension of F 0 . This is a number field within L to which X with its CM structure descends.

3

Although p-divisible groups are interesting enough in their own right, our main motivation for studying them has been their applications to abelian varieties. John Tate

CM lifting of p-divisible groups 3.1. Motivation and background We now study the questions (CML) and (R) in 1.7.5, and want to decide whether there exist abelian varieties over finite fields that do not admit a CM lift, even after an extension of the base field (but not applying any nontrivial isogeny). This problem was mainly settled in [59], where it was shown that there are “many” abelian varieties over an algebraic closure F of Fp that do not admit a CM lifting. However, that analysis was restricted to the case of abelian varieties over F whose endomorphism algebra is a commutative field. We seek to make non-liftable examples without this restriction on the endomorphism algebra, and to also get effective control on the finite fields over which such examples can be constructed. 3.1.1. Why p-divisible groups? In 1.4.5–1.4.5.3 we reviewed some basic notions related to p-divisible groups and Dieudonn´e modules. To improve upon CM lifting results in [59] (which we review below), we will use the deformation theory of p-divisible groups, as in Theorem 1.4.5.7 and Example 1.4.5.8. This encodes deformations of homomorphisms as well, so pdivisible groups are especially well-suited to the study of CM lifting problems for abelian varieties even though most p-divisible groups do not arise from abelian varieties. Among the advantages of working with p-divisible groups (apart from their “computaibility” via semi-linear objects such as Dieudonn´e modules) is that the isogeny classes are easily understood over an algebraically closed field (Theorem 3.1.2.1) and the constraints on the endomorphism algebras under consideration are much less severe than for abelian varieties. Since a simple abelian variety in characteristic p > 0 can have a p-divisible group that is isogenous to a nontrivial product, leaving the framework of abelian varieties opens the door to more operations. In this way, we can remove the restrictive hypothesis in [59] that the endomorphism algebra of the residual abelian variety in characteristic p is a commutative field, leading to many more non-liftable examples and the solution of some CM lifting problems for supersingular abelian surfaces (thereby answering [59, Question C]). Moreover, we provide an effective procedure to construct isogenies to non-liftable abelian varieties over specific finite fields (see 3.5.3). In 3.5.4 and 3.5.5 we illustrate how the method constructs explicit examples for which a CM lifting does not exist. To prove that certain abelian varieties do not admit a CM lifting to characteristic 0, it will suffice to show this for the associated p-divisible group (in a sense defined in 3.4.1). It will not be necessary to keep track of polarizations when proving such negative results. In contrast, when we allow isogenies and prove affirmative liftability results via the Serre–Tate deformation theorem (Theorem 1.4.5.7) in Chapter 4 then polarizations must be taken into account to ensure the algebraizability of the formal abelian schemes that we construct. 3.1.2. Slopes. In 2.1.5 we defined the concept of slope for abelian varieties over finite fields. For a p-divisible group G over a finite κ of size q = pf , the numbers ordp (λ)/f arising from the eigenvalues λ (counted with multiplicity) of the W (κ)-linear q-Frobenius on D(G) provide a notion of slope that is consistent with the case of abelian varieties over κ in the evident manner and is invariant under finite extension of the ground field. However, for infinite κ 81

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3. CM LIFTING OF p-DIVISIBLE GROUPS

this does not make sense. We need a notion of slope for p-divisible groups over any field κ, especially algebraically closed κ. To define the slopes of any p-divisible group G over any κ, we shall use the classification of isogeny classes of p-divisible groups over κ. This is really a problem in semilinear algebra. More precisely, there is a general result of Dieudonn´e and Manin that classifies all left Dκ [1/p]modules of finite dimension over W (κ)[1/p] when κ = κ, with Dκ = W (κ)[F, V] denoting the Dieudonn´e ring for κ. Not every such Dκ [1/p]-module arises from a Dκ -module that is finitely generated over W (κ) (i.e., arises from the isogeny class of a p-divisible group over κ). The existence of a W (κ)-lattice stable under F and V imposes non-trivial constraints on the F and V-actions. Upon working out the constraints, one arrives at: 3.1.2.1. Theorem (Dieudonn´e–Manin). Let k be an algebraically closed field of characteristic p > 0. (1) The isogeny category of p-divisible groups over k is semisimple. (2) For any λ = r/s ∈ Q ∩ [0, 1] in reduced form (with s > 1), the left Dκ -module Dκ /Dκ (F s−r −V r ) is finite free over W (κ) of rank s and the corresponding p-divisible group Gλ of height s has dimension r and is simple in the isogeny category. (3) The Gλ are pairwise non-isogenous, and every simple object in the isogeny category of p-divisible groups over k is isogenous to some Gλ . (4) The Cartier dual of Gλ is G1−λ . A proof Q of this result is given in 1. For general perfect κ and nonzero G over κ, we uniquely have Gκ ∼ Gλeii with ei > 1 and pairwise distinct λi . The slopes of G are defined to be the numbers λi , with respective multiplicities mi := ei · ht(Gλi ) = ei si , where ri /si is the reduced form of λi (so mi λi ∈ Z for all i). In particular, a nonzero G has only one slope (isoclinic) precisely when Gκ is isotypic (i.e., isogenous to a power of a simple object). We leave it to the reader to check that when κ is finite of size q, this definition of slopes agrees with the definition using eigenvalues of the W (κ)-linear q-Frobenius endomorphism of the Dieudonn´e module. 3.1.2.2. Example. The p-divisible group G is ´etale if and only if it is isoclinic of slope 0, and it is of multiplicative type (i.e., has ´etale Cartier dual) if and only if it is isoclinic of slope 1. Also, G is local-local (i.e., G and Gt are both connected) precisely when the slopes of G lie strictly between 0 and 1. All of these assertions are invariant under isogeny and can be checked over an algebraically closed extension of the ground field, so they are immediate from Theorem 3.1.2.1. 3.1.3. a-numbers. Let k be a field of characteristic p > 0. When k is perfect, a useful invariant of a p-divisible group over k is the maximal number of “independent” copies of αp that it contains. Before we make a precise definition along these lines, we prove a useful basic fact concerning powers of αp . 3.1.3.1. Proposition. Let H be a finite commutative k-group scheme. If k 0 /k is an extension field such that Hk0 ' αpr as k 0 -groups then H ' αpr as k-groups. Such an isomorphism exists if and only if the Frobenius and Verschiebung morphisms FH/k : H → H (p) and VH/k : H (p) → H vanish. To make the isomorphism criterion more concrete, we note that VH/k is the morphism dual to FH D /k , where H D denotes the Cartier dual of H.

1Need to give precise reference(s) for statements and proofs.

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83

Proof. Suppose such a k 0 /k exists. By standard direct limit and specialization arguments, we may arrange that k 0 is a finite extension of k. A direct calculation proves that the automorphism functor of αpr on the category of k-schemes is represented by GLr , so the descent datum describing H as a twisted form of αpr (relative to k 0 /k) defines a class in the cohomology set H1 (k 0 /k, GLr ). But this cohomology set vanishes (as it also classifies descent through Spec k 0 → Spec k for rank-r vector bundles, and all k-vector spaces have bases), so the descent datum can be trivialized. That is, H ' αpr . To check the equivalence with the vanishing of both FH/k and VH/k , we may now increase the ground field. Hence, we can assume k is perfect. Then a calculation with Dieudonn´e modules does the job. 3.1.3.2. Definition. A finite commutative k-group H is of α-type if it satisfies the equivalent conditions in Proposition 3.1.3.1. The characterization in terms of the vanishing of FH/k and VH/k implies that the property of being of α-type is inherited by subgroups and quotients. In particular, every finite commutative k-group H contains a unique maximal α-type subgroup, denoted H[F, V ]. By Galois descent, if k is perfect then the formation of H[F, V ] commutes with any extension of the base field. Likewise, for a p-divisible group G over a perfect field k of characteristic p > 0, we define G[F, V ] to be the maximal α-type subgroup of G[p], so G[F, V ] ' αpr where r = dimk Hom(αp , G). This leads us to: 3.1.3.3. Definition. The a-number of a p-divisible G over a perfect field k of characteristic p is dimk Hom(αp , G) = dimk D(G)/(F, V)D(G). It is denoted a(G). The formation of a(G) clearly commutes with any extension of the perfect ground field. It is not generally invariant under isogenies (a fact that we will exploit later on), though if dim G = 1 and G0 is local-local (i.e., its dual is not ´etale) then clearly a(G) = 1, so in such cases all members of the isogeny class of G have a-number equal to 1. The a-number behaves well with respect to duality: 3.1.3.4. Proposition. The a-numbers a(G) and a(Gt ) coincide. Proof. Let G0 denote the scalar extension G(1/p) by the geometric Frobenius automorphism of k, so G0 (p) = G. Consider the complex of p-divisible groups (3.1.3.1)

G

δ:=(FG/k ,VG0 /k )

/ G(p) × G0

VG/k −FG0 /k

/G

The kernel on the left is the maximal α-type subgroup G[F, V ], whose order is pa(G) by definition. We claim that the homology in the middle has the same order. Using that the terms in (3.1.3.1) are p-divisible groups, an elementary diagram chase identifies the homology in the middle with the cokernel of the inclusion (3.1.3.2)

δ(G[p]) ,→ ker VG/k × ker FG0 /k .

But ker FG0 /k is a scalar extension of ker FG/k = ker FG[p]/k = im(VG[p]/k ) (the final equality since G[p] is the p-torsion of a p-divisible group), so its order coincides with #G[p]/# ker VG/k . In other words, the right side of (3.1.3.2) has the same order as G[p], so the cokernel of (3.1.3.2) has the same order as ker δ = G[F, V ]. Note that the same conclusions hold if we replace the relative Frobenius morphisms with their negatives at the outset. In terms of (contravariant) Dieudonn´e modules, (3.1.3.1) yields a W (k)-linear complex of finite free W (k)-modules L0 → L → L00 that is exact on generic fibers and has homologies at the middle and right terms with respective lengths ` = a(G) and `00 = a(G). The dual to (3.1.3.1) is the analogue for Gt up to replacing

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the relative Frobenius morphisms with their negatives, so in terms of Dieudonn´e modules we conclude that the linear dual complex ∨

L00 → L∨ → L0

∨

that is exact on generic fibers has both homologies at the middle and right terms with length a(Gt ). But elementary calculations with duality for W (k)-modules show that the respective lengths are `00 and `. Hence, a(Gt ) = a(G).

3.2. Earlier non-liftability results and a new proof We recall the statement of [59, Thm. B] (whose proof uses CM theory over C): 3.2.1. Theorem. Let B be a non-ordinary abelian variety of dimension > 2 over an algebraic closure F of Fp . Assume that End0 (B) is a commutative field and that B is of p-rank at most dim(B)−2. Then there exists an abelian variety B 0 over F isogenous to B such that B 0 cannot be CM lifted to characteristic 0. For the convenience of the reader, we now sketch the idea behind the original proof of Theorem 3.2.1. (Below we will give a new proof via p-divisible groups which eliminates the hypothesis on End0 (B).) One first chooses an abelian variety A over a finite field κ = Fq such that there exists an isogeny AF → B with properties analogous to ones described in Proposition 3.3.2 (cf. (3.2) in [59]). These A naturally form a parametric family {Bb } with b varying through P1 (F) − P1 (κ0 ) for a certain finite extension κ0 /κ (as we will review in Proposition 3.3.2(iii)). The image of any such b in P1κ is a closed point, and we let κ(b) denote the residue field at that point. By construction, Bb descends to an isogenous quotient of Aκ(b) , and one proves that this Bb cannot be defined over a proper subfield of κ(b) over κ as an abstract abelian variety. One chooses an integer Q by a procedure similar to what we will do with p-divisible groups in 3.3.4, and proves that any Bb that admits a CM lifting can be defined over the maximal pro-p-extension of FqQ (see [59, §4]). Hence, by avoiding that pro-p extension there exist b ∈ F for which the associated isogenous quotient of Aκ(b) does not admit a CM lift even after scalar extension to F. 3.2.2. Instead of avoiding the maximal pro-p extension of some finite field FqQ as above, we can prove a much finer result: it suffices to avoid a specific finite extension of κ, the degree of which can be effectively computed. More specifically, in terms of a notion of CM lifting to characteristic 0 for p-divisible groups that is defined in 3.4.1, we will prove: 3.2.2.1. Theorem. Suppose that G is a non-ordinary p-divisible group over F such that G and its dual Gt each have dimension > 2. Then there exists a p-divisible group G0 over F isogenous to G that cannot be CM lifted to characteristic 0. A more precise version of this result is given in Theorem 3.5.3 (e.g., we can arrange that is defined over a finite field, though possibly of large degree), and there we also provide a proof. The basic idea is to prove that for any CM p-divisible group over a mixed characteristic local domain with residue field F, the special fiber can always be defined over a small finite field. Hence, if a p-divisible group over F cannot be defined over a small finite field then it does not admit a CM lift to characteristic 0. For an illustration, see 3.5.4 and 3.5.5. A consequence of Theorem 3.2.2.1 is that the restrictions on End0 (B) can be completely eliminated from Theorem 3.2.1, thereby answering [59, Question C]: G0

3.2.2.2. Corollary. Let B be an abelian variety over F of dimension > 2 such that the p-rank of B is at most dim B − 2. There exists an isogeny B → B 0 over F such that B 0 does not admit a CM lifting to characteristic 0.

3.3. FINER PROPERTIES OF THE a-NUMBER

85

This corollary is an optimal result in the following sense. Suppose instead that the prank of B (i.e., the height of the ´etale part of B[p∞ ]) is at least dim B − 1. Equivalently, B is ordinary or the local-local part of B[p∞ ] is isoclinic of height 2 and slope 1/2 (i.e., the pdivisible group of a supersingular elliptic curve). In all such cases, by [59, §2] any B 0 isogenous to B can be CM lifted to characteristic 0.

3.3. Finer properties of the a-number To prove Theorem 3.2.2.1 (after defining a suitable notion of CM lifting for p-divisible groups), we require deeper results concerning a-numbers. This section addresses those facts. For a p-divisible group G over a perfect field κ, there is a canonical isomorphism G ' G´et × Gmult × G(0,1) where G´et is etale, Gmult is of multiplicative type, and G(0,1) is local-local; cf. Example 3.1.2.2. The height of G´et is called the p-rank of G, out of analogy with the case of abelian varieties over κ. We say that G is ordinary if G(0,1) = 0. For example, if B is an abelian variety over κ then B is ordinary if and only if B[p∞ ] is ordinary. Observe also that passage to the dual p-divisible group swaps the ´etale and multiplicative parts and carries the local-local part back to itself (i.e., (Gt )(0,1) is the dual of G(0,1) ). The next lemma is an easy consequence of Theorem 3.1.2.1 and Proposition 3.1.3.4, so we leave the details to the reader. 3.3.1. Lemma. Let G be a p-divisible group over a perfect field κ of characteristic p. The following conditions are equivalent. (i) Every p-divisible group G0 over κ isogenous to Gκ satisfies a(G0 ) 6 1. (ii) Either G is ordinary or dim G(0,1) = 1 or dim Gt(0,1) = 1. We say that G is of extended Lubin–Tate type if G satisfies the equivalent conditions of the preceding lemma. For G over F not of extended Lubin–Tate type, a suitable isogeny always brings us to the case of a-number 2. This is part of the following result. 3.3.2. Proposition. Let G be a p-divisible group over F that is not of extended Lubin–Tate type. Then there exists a finite field κ, a p-divisible group G0 over κ, and a finite extension κ0 of κ with the following properties. (i) The base change G0F is isogenous to G, and the natural map End(G0 ) → End(G0F ) is an isomorphism. (ii) The a-number a(G0 ) is equal to 2. (iii) Fix an embedding j : αp × αp ' G0 [F, V ] ⊂ G0 over κ. For any extension k of κ, use Lie(jk ) to identify P1 (k) with the set of k-subgroups of G0k isomorphic to αp by assigning to any such subgroup its tangent line in k 2 = Lie(αp2 ) ⊂ Lie(G0k ). For every b ∈ P1 (F), let κ(b) denote the finite residue field at the closed image of b in P1κ , and let ab denote the copy of αp in G0κ(b) classified by b. For all b ∈ P1 (F) − P1 (κ0 ), the quotient Gb := G0κ(b) /j(ab ) satisfies a(Gb ) = 1. See [59, Prop. 3.2] for a proof of Proposition 3.3.2. Although [59, Prop. 3.2] is formulated for abelian varieties, its proof works verbatim for p-divisible groups. For any given G as in Proposition 3.3.2, it is not hard to find the desired κ, G0 , and κ0 .

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3.3.3. Lemma. Consider G, κ, G0 , κ0 , and Gb over κ(b) as in Proposition 3.3.2. For any b1 , b2 ∈ P1 (F) − P1 (κ0 ) such that there exists an isomorphism ψ : (Gb1 )F ' (Gb2 )F as abstract p-divisible groups over F then necessarily κ(b1 ) = κ(b2 ). Proof. Fix a choice of ψ. We claim that ψ uniquely lifts to an automorphism ψe of G0F . To see this, observe that dualizing the exact sequence 0 → ab1 → G0κ(b1 ) → Gb1 → 0 over κ(b1 ) yields G0 tκ(b1 ) ' Gtb1 /(ab1 )D (where H D denote the Cartier dual of a finite commutative group scheme H over a field). By Proposition 3.1.3.4 we have a(Gtb1 ) = a(Gb1 ) = 1, so t D (ab1 )D F is the unique copy of αp as an F-subgroup of (Gb1 )F . The same goes for (ab2 )F inside t t of (Gb2 )F , so ψ must respect these subgroups. Passing to the quotient and dualizing back e yields ψ. Clearly ψe induces an automorphism of G0F [F, V ] ' αp2 , and by construction it carries (ab1 )F to (ab2 )F . In particular, the linear automorphism of Lie(G0F [F, V ]) induced by ψe induces an F-automorphism of P1F sending b1 to b2 . It therefore suffices to prove that this automorphism of P1F is defined over κ. By the construction of G0 , every F-endomorphism of G0F is defined over κ. Applying this to ψe does the job (cf. proof of [59, Lemma 3.3]). 3.3.4. A construction. Let G0 be a p-divisible group over a finite field κ of characteristic p > 0 such that End(G0 ) = End(G0k ) for an algebraic closure k of κ. We define a non-negative integer Q = Q(G0 ) as follows. The functoriality of G0 [F, V ] in G0 yields a natural group homomorphism Aut(G0 ) → Aut(G0 [F, V ]) ' GLa(G0 ) (κ). The image of this map is a finite group ΓG0 , and we define Q(G0 ) to be the least common multiple of the orders of the elements in ΓG0 . 3.3.4.1. Remark. If we instead use the definition Q(G0 ) := #ΓG0 then what follows will work equally well. However, for constructing examples it is better to use an optimized definition such as we have given above. Using notation as in Proposition 3.3.2, the following result gives a lower bound uniform in b for the κ-degree of any field of definition κ0 ⊃ κ for Gb := G0κ(b) /j(ab ) as an abstract p-divisible group (i.e., this field of definition is not assumed to have any good relation with the quotient map G0κ(b)  Gb ). 3.3.4.2. Proposition. With notation as above, for any b ∈ P1 (F) − P1 (κ0 ) if (Gb )F descends to a p-divisible group H over a finite extension κ0 ⊃ κ inside of F then [κ0 (b) : κ0 ]|Q(G0 ). In particular, [κ(b) : κ] [κ0 (b) : κ] > , [κ0 : κ] = 0 0 [κ (b) : κ ] Q(G0 ) The main point of this result is that by choosing b with sufficiently large κ-degree we can ensure that such a finite field of definition over κ (if it exists!) cannot have small κ-degree. Proof. Since a(Gb ) = 1 and the formation of the a-number is invariant under extension of a perfect ground field, necessarily a(H) = 1. Thus, a(H t ) = 1 by Proposition 3.1.3.4. The map ξ 0 : H t → H t /(H t [F, V ]) must be a κ0 -descent of the quotient of HFt = (Gb )tF by its unique F-subgroup isomorphic to αp . But as we saw in the proof of Lemma 3.3.3, this latter quotient map over F is necessarily the dual of the natural quotient map G0F  (Gb )F . Hence, the dual of ξ 0 is an isogeny of p-divisible groups ξ : X → H over κ0 that is a κ0 -descent of the quotient map fb : (G0κ(b) )F → (Gb )F .

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The Gal(F/κ0 )-twisting that constructs X from G0 is through automorphisms of G0F , and in fact only those automorphisms that preserve ker fb . By construction of G0 we have Aut(G0F ) = Aut(G0 ). Thus, X[F, V ] is a Galois-twist of G0 [F, V ]κ0 through the image ΓG0 of Aut(G0 ) in Aut(G0 [F, V ]) (and really only the image of the subgroup of Aut(G0 ) that preserves ker fb over F). Thus, this latter Galois twist is classified by an element of H1 (Gal(F/κ0 ), ΓG0 ) = Homcont (Gal(F/κ0 ), ΓG0 ). Since Gal(F/κ0 ) is pro-cyclic, this cohomology class is split by a finite extension κ00 of κ0 of degree dividing Q(G0 ) (due to how Q(G0 ) was defined). In particular, the κ00 -structure imposed on G0 [F, V ]F from the κ0 -structure X[F, V ] coincides with the initial one arising from the κ-structure G0 [F, V ]. But as we have already noted, the twisting is even through Fautomorphisms that preserve the F-subgroup ker fb of G0 [F, V ]F whose field of definition over κ relative to the initial κ-structure G0 [F, V ] is κ(b). Hence, κ00 must contain κ0 (b) over κ0 inside of F, so [κ0 (b) : κ0 ] divides [κ00 : κ0 ], which in turn divides Q(G0 ). 3.3.4.3. Remark. The constant Q, which depends only on G0 , is effectively computable. Although the argument in [59, Prop. 3.2] for the existence of the finite field κ0 in Proposition 3.3.2(iii) does not provide an algorithm to compute κ0 , in any given situation it is usually not difficult to find a κ0 . We now work out some simple examples. Fix h > 2, and for κ = Fph consider G := H × H t over Fph where H is a 1-dimensional p-divisible group of height h (so its dual H t has dimension h − 1). Since H is local-local of dimension 1 we must have a(H) = 1, so likewise a(H t ) = 1 (Proposition 3.1.3.4). All endomorphisms of HF are defined over κ because 2, so the same holds for endomorphisms of GF since there are no nonzero homomorphisms between HF and its dual (due to simplicity and slope considerations, as h > 2). The finite group im (Aut(G) → Aut(G[F, V ])) , so Q(G) = ph − 1. × F× is isomorphic to F× ph ph The product structure of G induces an isomorphism j : αp × αp ' G[F, V ] with j(αp × 0) = H[F, V ],

j(0 × αp ) = H t [F, V ].

In this case the only parameters b for which GF /j(ab ) has a-number 2 are the two coordinate points [1, 0] and [0, 1] (as we may check over F via a computation with the Dieudonn´e modules as given in Theorem 3.1.2.1(2)), so we can take G0 = G and κ0 = Fph = κ in Proposition 3.3.2(iii), Lemma 3.3.3, and Proposition 3.3.4.2.

3.4. Complex multiplication for p-divisible groups 3.4.1. Let R be a complete local noetherian domain with fraction field M of characteristic 0 and residue field k of characteristic p > 0. For any nonzero p-divisible Z over R, the endomorphism ring is a Zp -subalgebra of End(Tp (ZM )). Thus, the Zp -algebra End(Z) is finite free as a Zp -module and End0 (Z) := Qp ⊗Zp End(Z) is a finite-dimensional Qp -algebra. We are interested in situations when this contains large commutative semisimple Qp -subalgebras. The following lemma is straightforward to prove. 3.4.1.1. Lemma. Let h be the height of Z. For any commutative semisimple Qp -subalgebra F ⊆ End0 (Z), [F : Qp ] 6 h with equality if and only if Vp (ZM ) is free of rank 1 as an F -module. 2we need to justify or give a reference

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3.4.1.2. Definition. We say that Z admits sufficiently many complex multiplications, or Z is a CM p-divisible group, if there exists a commutative semi-simple algebra F over Qp and an injective homomorphism i : F → End0 (Z) such that [F : Qp ] = height(Z). In such situations, we also say that Z has CM by F . A natural source of examples is abelian schemes: if R is normal and A is an abelian scheme of relative dimension g over R such that End0 (AM ) contains a commutative semisimple Q-algebra L of rank 2g then the equality End(AM ) = End(A) (Lemma 1.7.4) identifies the commutative semisimple Qp -algebra Lp := Qp ⊗Q L with a Qp -subalgebra of End0 (A[p∞ ]). Hence, the p-divisible group A[p∞ ] over R has CM by Lp . In general, since F is a Q-algebra and F ∩ End(Z) is an order in F , there is a natural F ⊗Qp M -module structure on the generic fiber Lie(Z)M of the Lie algebra of Z. Much like in the case of abelian varieties, we have: 3.4.1.3. Lemma. Let d = dim Z. The M -linear action of F on Lie(Z)M is given by a collection Φ = Φ(Z, F ) of d distinct Qp -embeddings of F into M (or equivalently, into Qp ). Proof. Since Lie(Z)M is a module over the semisimple M -algebra F ⊗Q M , the problem is to prove that the Qp -embeddings F → M with a nonzero eigenspace have a 1-dimensional eigenspace. That is, we have to prove that there are no multiplicities for the eigencharacters of the F -action. For the analogous fact concerning CM types arising from abelian varieties over fields of characteristic 0, the only known proofs use Hodge theory (or at least the Hodge filtration of deRham cohomology). Likewise, for p-divisible groups we will use the Hodge–Tate decomposition. First, we pass to the case of a simpler base ring, as follows. By 3, there is a local injective map of R into a complete discrete valuation ring. The formation of the identity component of a p-divisible group over a complete local noetherian ring (of residue characteristic p) commutes with local extension of scalars, so we may apply such a scalar extension to reduce to the case when R is a discrete valuation ring. We may likewise arrange that its residue field is algebraically closed. Let CM denote the completion of M . The Hodge–Tate decomposition (1.4.5.1) provides a functorial CM -linear isomorphism CM ⊗Qp Vp (ZM ) ' (CM ⊗M Lie(Z)M ) ⊕ (CM (−1) ⊗M Lie(Z t )M ). Now consider its functoriality with respect to the F -action (in the isogeny category over R). Since Vp (ZM ) is free of rank 1 over F (Lemma 3.4.1.1), the left side is free of rank 1 over CM ⊗Qp F . Hence, there is a 1-dimensional eigenspace over CM for each Qp -embedding of F into CM (or equivalently, into Qp ). Passing to the right side, both Lie algebras over M have no repeated eigencharacters for the F -actions on their geometric fibers. The (p-adic) CM type of (Z, F ) is the collection of embeddings in Lemma 3.4.1.3; note that these embeddings are valued in the algebraic closure Qp of Qp in M . It is equivalent to work with the Qp -linear trace F → M that classifies the M -linear F -action on Lie(Z)M up to isomorphism. More generally, we have: 3.4.1.4. Definition. Fix an algebraic closure Qp of Qp . For a finite-dimensional commutative semisimple Qp -algebra F , a p-adic CM type on F valued in Qp is a subset Φ ⊆ HomQp (F, Qp ). There is no risk of confusion with the notion of p-adic CM type on a CM field L as at the end of 2.1.3. The two notions are linked: since Hom(L, Qp ) = HomQp (Lp , Qp ) for Lp := Qp ⊗Q L, a p-adic CM type on L gives rise to one on Lp . 3should give EGA or other reference

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Now we consider the case of p-divisible groups in characteristic p. By using Dieudonn´e modules, we will obtain a suitable analogue of Lemma 3.4.1.1. First, we require a finiteness result for endomorphism algebras: 3.4.1.5. Lemma. Let Z be a nonzero p-divisible group over a field κ of characteristic p. The Zp -algebra End(Z) is finite free as a Zp -module. As usual, we will write End0 (Z) to denote Qp ⊗Zp End(Z). Beware that we do not claim that this is a semisimple Qp -algebra; such semisimplicity holds for algebraically closed κ but in general the isogeny category of p-divisible groups over κ is not semisimple. Proof. We may increase the ground field to be algebraically closed, and it is harmless to pass to an isogenous p-divisible group. Thus, by the Dieudonn´e–Manin classification (Theorem 3.1.2.1) it suffices to treat the simple objects Z = Gλ with slope λ ∈ Q ∩ [0, 1]. Equivalently, it suffices to prove Zp -finiteness for the Dκ -linear endomorphism algebra of each Dieudonn´e module Dκ /Dκ (F s−r − V r ) with relatively prime integers 0 6 r 6 s. Since W (κ) ∩ Qp = Zp inside of W (κ)[1/p] and this Dieudonn´e module is finite free as a W (κ)-module, it is harmless to invert p and reduce to proving Qp -finiteness of End0 (Z), or equivalently of the Dκ [1/p]linear endomorphism algebra of Dκ [1/p]/Dκ [1/p](F s − pr ). This is a classical fact. 4 (In A.1.2 we prove an analogous fact for finite κ.) 3.4.1.6. Proposition. Let Z be a p-divisible group of height h > 0 over a field κ of characteristic p. If F ⊆ End0 (Z) is a commutative field containing Qp with [F : Qp ] = h then Z is isoclinic. Proof. We may assume κ is algebraically closed, and then by the Dieudonn´e–Manin classifiQ cation (Theorem 3.1.2.1) we have a functorial decomposition in the isogeny category Z ∼ Zj for nonzero isoclinic Zj with pairwise distinct slopes. By functoriality, F preserves each Zj . By Proposition 1.4.5.5(1) each Zj has height at least [F : Qp ] = h, so there is only one such factor. That is, Z is isoclinic. Over fields κ of characteristic p it is now natural to adapt the definition of a CM abelian variety to the case of nonzero p-divisible groups Z: we say that Z admits sufficiently many complex multiplications if End0 (Z) contains a commutative semisimple Qp -subalgebra F with [F : Qp ] = height(Z). We then say that Z is a CM p-divisible group over κ, and that it has CM by F . 3.4.1.7. Example. A CM p-divisible group Z over a complete noetherian local domain (R, m) with generic characteristic 0 and residue characteristic p has special fiber Z that is a CM p-divisible group over the residue field. Also, if B is a CM abelian variety over a field of characteristic p > 0 then so is its p-divisible group, since Zp ⊗Q End(B) → End(B[p∞ ]) is injective (Proposition 1.2.5.1 with ` = p). 3.4.1.8. Definition. Let Z be a p-divisible group over a field κ of characteristic p. A CM lifting of Z over a complete local noetherian domain R with generic characteristic 0 is a triple (Z, φ, f ) where Z is a CM p-divisible group over R, φ : R → κ is a local map, and f : Zκ ' Z is an isomorphism. By the same deformation ring arguments as in 2.1.1 (but using deformation theory for p-divisible groups 5 rather than for polarized abelian schemes), if Z admits a CM lifting to such an R then there exists a finite extension κ0 of κ such that Zκ0 admits a CM lifting to a complete discrete valuation ring of characteristic 0 with residue field κ0 , and we can require κ0 = κ if we allow R to be an order in finite totally ramified extension of the Cohen ring of 4we should give a precise reference which also identifies its center and local invariant. 5we should give a precise reference for and statement of the pro-representability of the deformation functor

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κ rather than to be the entire valuation ring in such an extension. In particular, if κ is finite then the problem of CM lifting to orders in p-adic integer rings is essentially the general case. 3.4.2. Slopes, CM types, and reflex fields. Let (F, Φ) be the p-adic CM type of a CM pdivisible Q group Z over R as in 3.4.1. Let Z denote the special fiber. The product decomposition Q F = Q Fj with fields Fj induces product decompositions of p-divisible groups Z ∼ Zj and Z ∼ Zj in the respective isogeny categories. Each Zj has height [Fj : Qp ] and CM by Fj , so the same holds for each Zj . In particular, the Zj are isoclinic (by Proposition 3.4.1.6). Renaming Zj as Z, let us focus on the case when F is a field, so Z is isoclinic. Is there any relationship between the unique slope and the CM type? This question can be posed for a CM p-divisible group over κ in the absence of a CM lifting to R, and in that generality there is no simple answer. However, in the above situation that is a reduction from characteristic 0, there is a very simple relationship: 3.4.2.1. Proposition. When the F -action on Z is obtained by reduction from characteristic 0 then the slope is #Φ/[F : Qp ]. Proof. Since [F : Qp ] is equal to the height and #Φ = dimM Lie(Z)M = dim Z = dim Z, the assertion reduces to the claim that the slope of an isoclinic p-divisible group over a field of characteristic p is its dimension divided by its height. Passing to higher powers and extending the ground field are harmless, so Theorem 3.1.2.1(2) gives the result. This motivates: 3.4.2.2. Definition. Let Z be a p-divisible group of height h over a field κ of characteristic p > 0 and suppose Q Z is equipped with an action by a Zp -order in a commutative semisimple ` Qp -algebra F = Fj of rank h, with each Fj a field. A p-adic CM type (F, Φ) = (Fj , Φj ) is compatible with the F -action on Z if #Φj = dim(Zj ) for each j, or equivalently if the slope of the isoclinic Zj is #Φj /[Fj : Qp ] for each j. Our interest in the preceding definition is due to the fact (proved above) that if there exists a CM lift Z of Z over a mixed characteristic local noetherian domain (R, m) and it has CM type (F, Φ), then (F, Φ) is compatible with the F -action on Z. We need one more definition. As motivation, fix an algebraic closure M of M = Frac(R), and view the CM types as being M -valued. The CM type Φ of F decomposes into the disjoint union of the CM types Φj for the Zj , with Φj ⊆ HomQp (Fj , M ) for each j. Thus, the following definition is applicable. 3.4.2.3. Definition. Let F be a finite-dimensional commutative semisimple Qp -algebra, and ` {Fj } its factor fields. Fix Q an algebraic closure Qp of Qp , and let Φ = Φj be a Qp -valued p-adic CM type on F = Fj . The reflex field Ej = E(Fj , Φj ) ⊂ Qp over Qp is the subfield corresponding to the open subgroup of elements σ ∈ Gal(Qp /Qp ) such that σ(Φj ) = Φj inside of HomQp (Fj , Qp ). The reflex field E = E(F, Φ) ⊂ Qp is the compositum over Qp of the reflex fields Ej = E(Fj , Φj ) inside of Qp . In the special case that (F, Φ) arises from a CM p-divisible group over R as above and we take Qp to be the algebraic closure of Qp inside of a choice of algebraic closure M of M , it is clear that E(F, Φ) ⊆ M . The following proposition records some facts for CM p-divisible groups over discrete valuation rings, analogous to results discussed earlier for CM abelian varieties (as we review below). All but the last part are well-known.

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3.4.3. Proposition. Let F be a commutative semisimple Qp -algebra with [F : Qp ] = h > 0. Choose an algebraic closure K 0 of K0 = W (F)[1/p], and let Qp denote the algebraic closure of Qp in K 0 . (1) Let R be a mixed characteristic complete discrete valuation ring with residue field F, and let Y1 and Y2 be p-divisible groups over R with height h, each equipped with an F -action defining the same CM type (F, Φ). Then Y1 and Y2 are F -linearly isogenous over R. (2) Let K/K0 be a finite extension inside of K 0 , and let E be a subfield of K 0 of finite degree over Qp . Let Y1 and Y2 be p-divisible groups over OE and OK respectively, each equipped with a CM structure by F have the same K-valued CM type Φ. There exists a finite extension E 0 /E inside of K 0 , a finite flat OE 0 -subgroup C in (Y1 )OE0 , and an F -linear isomorphism ((Y1 )OE0 /C)OK 0 ' (Y2 )OK 0 where K 0 := E 0 K is the compositum of E 0 and K over Qp inside of K 0 . (3) Let Φ be a Qp -valued CM type on F , E its reflex field, and K := EK0 the compositum over Qp . There exists a p-divisible group Y over OK with CM type (F, Φ). (4) Suppose that F is identified with a maximal commutative semisimple Qp -subalgebra of the endomorphism algebra End0 (Z) of a p-divisible group Z of height h over a finite field κ. Choose a CM type Φ of F that is compatible with the F -action on Z, and let E 0 a finite extension field of Qp containing the reflex field E = E(F, Φ) such that the residue field κ0 of E 0 contains κ. There exists a p-divisible group Z over OE 0 of CM type (F, Φ) such that its special fiber is F -linearly isogenous to Zκ0 . Before we address the proof of this proposition, we record the analogies with CM theory for abelian varieties. In view of Tate’s full faithfulness theorem for p-divisible groups over mixedcharacteristic (0, p) discrete valuation rings, first note that in (1) it is equivalent to make the isogeny between the generic fibers and in (2) it is equivalent (using schematic closure from the generic fiber) to say that there is an F -linear isogeny (Y1 )K 0 → (Y2 )K 0 whose kernel is defined over E 0 . Hence, (1) is analogous to the isogeny aspect of Proposition 1.4.4.1 (2) is analogous to a hybrid of Theorem 1.6.2.1 and Theorem 1.6.2.5, (3) is analogous to Theorem 2.5.2, and (4) is analogous to Theorem 2.1.7 (and is ultimately a consequence of the arguments in 2.6). Proof. A proof is given in 3.6 (an appendix to this chapter) via the methods of p-adic Hodge theory. Briefly, (1) follows from Proposition 3.6.3.1(ii), (2) follows from (1), (3) follows from Proposition 3.6.3.1(i), and (4) follows from Proposition 3.6.4.2. As a convenience to the reader, we now sketch an alternative proof using CM theory for abelian varieties. To prove (1), by Tate’s theorem on extending homomorphisms between p-divisible groups, it suffices to construct an F -linear isomorphism between the corresponding Qp -linear Galois representations. These Galois representations have the form Gal(M /M ) ⇒ F × (with M = Frac(R)) and are algebraic relative to Qp 6 by [65, Chap. 3]. 7 Due to this algebraicity on the entire inertia group, the F -linear Galois representations arising from the Yi ’s in (1) are determined by their CM types [67, 2.3]. Hence, the assumed equality of the CM types settles (1). To prove the existence result in (3), we may and do assume that F is a field. By the Hilbert irreducibility theorem, 8 we can construct a CM field L and a CM type ΦL such that 6Since M has residue field F that is not finite, we cannot use local class field theory, so in what sense is this “algebraic”? Are we invoking Serre’s geometric local CFT? 7 Hmm, I am not sure this is right: the methods in [65] are usually too weak to nail down algebraicity on the entire inertia group. For finite residue field I address this refinement using some p-adic Hodge theory in [18, App. A]. In general? 8please say some more here

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(F, Φ) is a component of (L, ΦL ) ⊗ Qp . Fix an embedding i : L ,→ F such that the resulting p-adic place on L yields the completion L℘ ' F . Let E alg ⊂ F be the reflex field of (L, ΦL ), and v be the place of E alg induced by the embedding E alg ,→ F , so Evalg = E inside of F . By Theorem 2.5.2 there exists an OL -linear abelian variety A over E alg with CM type (L, ΦL ) having good reduction at v. 9 Let A denote the N´eron model of AE alg over OE alg = OE (so it v v is an abelian scheme). The ℘-divisible group AOK [℘∞ ] of AOK is a p-divisible group over OK with CM type (F, Φ), as required in (3). The deduction of (2) from (1) given in 3.6 is formal (not relying on arguments specific to p-adic Hodge theory), 10 so it remains to prove (4), which we will deduce from the proof of (3). As before, we may and do assume that F is a field. Repeating the argument for (3), Z := A[℘∞ ] is a p-divisible group over E of CM type (F, Φ). The composite field K 0 = E 0 K0 is a completed maximal unramified extension of E 0 , and a suitable OK 0 /OE 0 -twist Z 0 of ZOE0 has Frobenius endomorphism on its special fiber that equals the Frobenius endomorphism of Zk0 (in the sense of equality as elements of F in the endomorphism algebra). 11 Such a Z 0 has the required properties. 12 3.4.4. Proposition. Let E = E(F, Φ) be the reflex field of a p-adic CM type (F, Φ). Let E 0 be a finite extension of E. There exists a CM p-divisible group Y over OE 0 with action by OF 0 defining the CM type (F, Φ) such that Galois representation ρY : Gal(E /E 0 ) → OF× carries the inertia subgroup IE 0 onto the image of the entire Galois group. 13

Proof. This is proved later as Corollary 3.6.3.3. The idea is that p-divisible groups over OE 0 of CM type (F, Φ) correspond to continuous homomorphisms Gal(E 0 ab /E 0 ) → OF× such that the inertial restriction is induced by the algebraic (relative to Qp ) homomorphism E 0 × −→ F × attached to (F, Φ). Since Gal(E 0 ab /E 0 ) is the direct product of its inertial part and its unramified quotient Gal(F/κE 0 ), we can adjust a p-divisible group Y 0 over OE 0 of CM type (F, Φ) to make the image of the entire Galois group of E 0 coincide with the image of its inertia subgroup. 14 (Here we exploit the fact that p-divisible groups admit unramified twisting by characters valued in p-adic unit groups, in contrast with abelian varieties.) The corresponding p-divisible group Y has the required property

3.5. An upper bound for a field of definition 3.5.1. Let G be a p-divisible group over over finite field κ ⊂ F with h := height(G) > 0. It is a classical fact that there are only finitely many extensions of Qp of a given degree up to isomorphism, due to finiteness of the residue field, Krasner’s Lemma, and the compactness of the space of Eisenstein polynomials over a p-adic field (all of which are separable). Thus, by Lemma 3.4.1.5, up to isomorphism there are only a finite number of commutative semi-simple Qp -algebras that can be embedded in End0 (G). It follows from Proposition 1.4.5.5(2) and the Skolem–Noether theorem that there are only a finite number of End0 (G)× -conjugacy classes 9Is this right? We first have to produce the candidate algebraic Hecke character . Even for elliptic curves, cannot usually realize the CM type over the reflex field. Probably need to replace E alg with a finite extension without introducing more ramification. 10Please say exactly where in 3.6 the deduction is given so that a reader can find it. 11Where is this “twist” coming from? Where did we use the residue field hypothesis in (3)? Since K 0 /E 0 is not quite algebraic, we should be more careful about what kind of “twisting” we mean; e.g., completed unramified descent? 12Are we using Honda–Tate here? 13Why is this proposition being stated here? It looks unmotivated to the reader. 14reference for completed unramified descent?

3.5. AN UPPER BOUND FOR A FIELD OF DEFINITION

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of pairs (F ⊂ End0 (G), Φ) where F is a (necessarily maximal) commutative semisimple Qp -subalgebra of End0 (G) of Qp -degree h and Φ is a CM type of F compatible with G (in the sense of Definition 3.4.2.2) and valued in a fixed algebraic closure Qp (whose residue field we identify with F). Let (Fj ,→ End0 (G)× , Φj ), j = 1, . . . , N be a complete set of representatives of these conjugacy classes. Let Ej = E(Fj , Φj ) ⊂ Qp be the reflex field of (Fj , Φj ). For each j = 1, . . . , N , let Zj be an OFj -linear p-divisible group over the ring of integers OEj of Ej , such that (i) the CM type of Zj is (Fj , Φj ), (ii) The Galois representation ρZj : Gal(Qp /Ej ) → OF×j attached to Zj carries the inertia subgroup IEj onto its image. The existence of Zj is guaranteed by Theorem 3.4.3(3) (or by Corollary 3.6.3.3). Denote by κj the residue field of Ej , and let Zj be the closed fiber of Zj . We know by the construction of Zj that its CM type is compatible with Zj . Hence (Zj )F is isogenous to GF (using scalar extension of Zj and G along the respective given inclusions of κj and κ into F). Suppose that Z is a p-divisible group F-isogenous to GF such that Z admits a CM lift Z over the ring of integers of a finite extension field K of K0 = W (F)[1/p]. There exists an integer j with 1 6 j 6 N such that the CM type of Z is (Fj , Φj ). By Theorem 3.4.3(1),(2), there exists a finite extension field Ej0 of Ej , a finite flat subgroup scheme C ⊂ (Zj )OE0 over j

OEj0 , and an F -linear isomorphism ξ : ((Zj )OE0 /C)OL ' ZOL j

where L := Ej0 K is the compositum of Ej0 and K over K0 . Denote by M = Mj the minimal subfield of Ej0 over Qp over which the generic fiber Cη of C is defined; i.e., M is the subfield of Ej0 generated by the residue fields at the generic points of C. Thus, via schematic closure considerations over a Dedekind base we see that C descends to a finite flat subgroup scheme C 0 ⊂ (Zj )OM over OM . Let Z 0 := (Zj )OM /C 0 ; this is a p-divisible group over OM of CM type (Fj , Φj ). By construction there is an OL isomorphism Z 0 ×Spec(OM ) Spec(OL ) ' Z ×Spec(OK ) Spec(OL ). In particular the closed fiber Z 0 of Z 0 is a p-divisible group over the residue field κM of M and its geometric fiber ZF0 is isomorphic to Z. As Z varies over all p-divisible groups over F which admit CM lifts corresponding to a fixed j as above, the subfields M ⊂ Ej0 vary. However, their residue fields κM remain constant. In fact, the assumption that ρZj carries IEj onto the image of the entire Galois group of Ej implies that M is totally ramified over Ej , so κM is equal to the residue field κj of Ej . We summarize our conclusions in the following proposition. 3.5.2. Proposition. Using notation as above, let κ be the compositum over Fp of the residue fields κj of the reflex fields Ej = E(Fj , Φj ), j = 1, . . . , N . Let Z be a p-divisible group over F isogenous to GF . If Z admits a CM lifting to characteristic 0 of CM type (Fj , Φj ), then there exists a totally ramified finite extension M = Mj of Ej and a CM lifting of Z over OM of CM type (Fj , Φj ). In particular there exists a p-divisible group Z 0 over κ such that ZF0 is isomorphic to GF . By Proposition 3.3.4.2 and Proposition 3.5.2, we immediately obtain the following counterexamples to the conditions (CML) and (R) from 1.7.5.

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3.5.3. Theorem. Let G0 be a p-divisible group over a finite field κ. We suppose that a(G0 ) = 2 and that there exists an F-subgroup a ⊂ G0F isomorphic to αp such that a(G0F /a) = 1. Assume that κ contains the residue fields κj as in Proposition 3.5.2. Let Q be the integer as defined in 3.3.4. Assume that the minimal field of definition κ0 ⊃ κ for a ⊂ G0F has κ-degree not dividing Q. The resulting quotient G of G0κ0 modulo the κ0 -descent of a has no CM lifting to characteristic 0. In particular if G0 = A[p∞ ] is the p-divisible group attached to an abelian variety A over κ then the corresponding quotient of Aκ0 by an αp -subgroup does not admit a CM lift to characteristic 0 even after extension of scalars to a field containing κ0 . 3.5.4. To finish 3.5, we illustrate Theorem 3.5.3 √ with two examples. For our first example, consider the imaginary quadratic field L := Q( −7) that has class number one. Let p be a prime number which is split in L, or equivalently p ≡ 1, 2, 4 mod 7. Let ℘1 and ℘2 be the two prime ideals in OL above p. Because L has class number one, there exists an element π ∈ L× such that πOL = ℘1 · ℘22 . By the product formula, π is a Weil p3 -number. Thus, by Honda–Tate theory in 1.5.2 there exists a simple abelian variety B over Fp3 with π as its Weil number, and it has dimension 3. By Proposition 1.6.4.3, we may and do assume that OL ⊆ End(B), so the p-divisible group ∞ of B decomposes as the product of B[℘∞ 1 ] and B[℘2 ] with respective slopes 1/3 and 2/3. In particular, B[F, V ] ' αp × αp and we can take κ = Fp3 in the statement of Theorem 3.5.3. (We note that B is a hypersymmetric abelian variety in the sense of [13].)√The endomorphism algebra End0 (B) is a 9-dimensional central division algebra over L = Q( −7). For any CM structure F on B[p∞ `] and compatible CM type Φ in the sense of Definition 3.4.2.2, (F, Φ) has the form (F1 , Φ1 ) (F2 , Φ2 ), where F1 and F2 are cubic extension fields of Qp , #Φ1 = 1, and #Φ2 = 2. Thus, E(Fj , Φj ) = Fj for j = 1, 2, and we can take κ = Fp3 in the statement of Theorem 3.5.3. By Remark 3.3.4.3, we can take Q = p3 − 1 and k = Fp3 as in the statement of Theorem 3.5.3. Thus, for any extension κ0 of Fp3 not contained in Fp3(p3 −1) , Theorem 3.5.3 provides an αp -quotient of Bκ0 that does not admit a CM lifting to characteristic 0 (even after further increase of the residue field).

3.5.5. Here is a second √ example that moreover provides an effective answer to [59, Question C]. The Weil p-number −p corresponds to an elliptic curve E 0 over Fp , and E 00 := EF0 2 has all endomorphisms defined over Fp2 . Let E = EF0

p8

p

and define A = E × E and κ = κ0 = Fp8 .

Define G0 = A[p∞ ], so for any b 6∈ Fp2 the corresponding αp -quotient Gb of G0κ(b) satisfies a(Gb ) = 1. A CM algebra of rank 4 over Qp inside End0 (G0 ) either is a product of two quadratic extensions or a single quartic extension that contains a quadratic subextension. In the first case, the reflex field has residue degree at most 2 over Fp . In the second case, the Galois closure over Qp for the reflex field has degree at most 8, so the residual degree of the reflex field divides 8. We conclude that in either case, the hypotheses in Proposition 3.3.2 are satisfied. The group ΓG0 := im(Aut(G0 ) → Aut(G0 [F, V ]) is isomorphic to GL2 (Fp2 ), so its order is Q0 := 180. It follows that any b 6∈ Fp8·180 provides a p-divisible group isogenous to G0F 8·180 that does not admit a CM lifting to characteristic 0 p

(even after increasing the residue field). Thus, for any finite extension κ0 of Fp8 not contained in Fp8·180 , we have constructed an αp -quotient of A that does not admit a CM lifting to characteristic 0.

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Remark. The parts of 3.6 used to derive the results in this section, especially 3.6.3.1–3.6.4.2, can be proved based on the pre-1980 literature 15 without the notion of a crystalline representation. However, we do not know whether one can obtain these results using only classical CM theory for abelian varieties, without passing through CM p-divisible groups via a localization procedure.

3.6. Appendix: Crystalline representations of CM type In this appendix we summarize the local theory of complex multiplication from the perspective of p-adic Hodge theory, and supply proofs of Theorem 3.4.3 and Proposition 3.4.4. Some standard references for p-adic Hodge theory are [26], [27], [28], and [14]. 3.6.1. Some categories of p-adic representations. Fix an algebraic closure F of Fp , and let K0 := W (F)[1/p]. Also fix an algebraic closure K 0 of K0 , and let Qp denote the algebraic closure of Qp in K 0 . (It is common in p-adic Hodge theory to consider general perfect residue fields of characteristic p, but for our arithmetic purposes only subfields of F will be necessary and some of what we do is not true otherwise.) Let L ⊂ K 0 be either a finite extension field of K0 or a finite extension field of Qp , so its algebraic closure L ⊆ K 0 is either K 0 or Qp . Observe that the completion of the maximal unramified extension Lun of L is a finite extension of K0 , and so as such its algebraic closure is naturally identified with K 0 . Denote by ΓL the Galois group Gal(L/L) of L, and let IL denote the inertial subgroup (so IL is naturally identified with the Galois group of the completion of Lun ). Denote by Rep∗ (ΓL ) the Qp -linear tensor category of all continuous linear representations of ΓL on finite-dimensional Qp -vector spaces, where ∗ ∈ {crys, pcrys, st, pst, dR, HT} denotes the following respective concepts: crystalline, potentially crystalline, semistable, potentially semistable, de Rham, Hodge–Tate. The definitions of the notions of Hodge Tate, de Rham, semistable, and crystalline are all given in [28], and we will review a bit of this formalism below in the Hodge–Tate and crystalline cases. For now, we simply record the basic fact that these notions are insensitive to inertial restrction, and are inherited under passage to subrepresentations, quotients, tensor products, and duals (but not extensions). Also, we say that a property of a ΓL -representation holds potentially when it holds upon restriction to an open subgroup (i.e., it is satisfied after replacing L with a finite extension inside of L). Denote by Repab ∗ (ΓL ) the full tensor subcategory of semisimple objects in Rep∗ (ΓL ) for which the ΓL -action factors through the topological abelianization Γab L (equivalently, the splitting field of the representation is an abelian extension of L). In other words, the objects consist of finite direct sums of irreducible (over Qp ) representations that satisfy property ∗ and have abelian splitting field. The “abelian semisimple” condition is equivalent to the requirement that after extending scalars to Qp on the representation space, the representation decomposes into a direct sum of 1-dimensional characters. (Indeed, for an abelian irreducible object V , the Qp -subalgebra of EndQp (V ) generated by ΓL is a commutative division algebra, hence a finite extension of Qp , so it splits into a product of copies of Qp after scalar extension to Qp .) The plainer notations Rep(ΓL ) and Repab (ΓL ) have the evident analogous meaning: we impose no “extra property” as above. Denote by MFfL (φ) (resp. MFfL (φ, N )) the Qp -linear tensor category of weakly admissible filtered φ-modules (resp. weakly admissible filtered (φ, N )-modules) relative to L. One can regard MFfL (φ) as the full tensor subcategory consisting of objects (M, Fil, φ, N ) in MFfL (φ, N ) with N = 0. 15Are we sure of this? Some earlier footnotes give me pause here.

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Rather generally, we have the implications crystalline ⇒ semistable ⇒ de Rham ⇒ Hodge Tate. The properties of being de Rham or Hodge–Tate are coarse insofar as they are insensitive to restriction to an open subgroup. Hence, potentially semistable representations are always de Rham. In contrast, the crystalline and semistability properties are sensitive to ramification (e.g., both properties are destroyed by twisting by a finite-order ramified character), and it is a deep theorem (which we will not require) that the implication “potentially semistable ⇒ de Rham” is actually an equivalence. When restricting to abelian semisimple representations, many of these implications collapse to equivalences. Before stating such a result, we need to recall a notion of algebraicity for abelian semisimple representations of ΓL . 3.6.1.1. Definition. Let ρ : ΓL → GL(V ) be an object in Repab (ΓL ), and let F ⊂ EndQp (V ) denote the finite-dimensional commutative semisimple Qp -algebra generated by ρ(ΓL ). When L has finite residue field, we say that ρ is algebraic if the composite map r

L × L× → Γab L →F

is algebraic with respect to Qp in the sense of Definition 2.4.1. (Here, rL is the local Artin map with arithmetic normalization: uniformizers are carried to arithmetic Frobenius elements.) We likewise say that ρ is locally algebraic if the induced map L× → F × agrees with an algebraic homomorphism on a neighborhood of 1 in L× , and that ρ is algebraic on inertia if × → F × coincides with the restriction of an algebraic homomorphism from the restriction OL L× to F × . If L has residue field F, then we say that ρ is algebraic if 16 3.6.1.2. Proposition. Let L be either a finite extension field of Qp or a finite extension field of K0 . Let ρ : ΓL → GL(V ) be an continuous linear abelian semisimple representation of ΓL on a finite-dimensional Qp -vector space V . (i) The representation ρ is potentially crystalline if and only if it is Hodge–Tate, so these conditions are equivalent to ρ being potentially semistable or de Rham. Moreover, these properties are equivalent to the requirement that ρ be locally algebraic. (ii) The representation ρ is crystalline if and only if it is semistable. Moreover, these properties are equivalent to the requirement that ρ be algebraic on inertia in case L has finite residue field, and that ρ be algebraic when L has residue field F. Proof. First assume that L has finite residue field. Clearly (i) is a consequence of (ii), and a proof of (ii) (refining the arguments in [65, Ch. 3, App.]) is given in [18, App. A] in the case of finite residue fields. It should be emphasized, however, that much of (i) was known before the introduction of the notion of crystalline representations. Indeed, the essential points (which do not involve p-adic Hodge theory) are to prove that that Hodge–Tate property implies local algebraicity, and that locally algebraic representations are subquotients of tensorial constructions in Tate modules of Lubin–Tate formal groups (as then we can apply the fact, due to Fontaine, that the Tate module of the generic fiber of a p-divisible group over OL is always crystalline). These matters are explained in [65, Chap. 3, Appendix]; note that the applicability of those arguments requires making a preliminary finite extension of certain p-adic fields, which is harmless for the proof of (i). Now suppose that L has residue field F. 17 16Is asking that it extend to an abelian representation of the Galois group of a finite extension of Q really p

correct? For example, do all crystalline abelian semisimple representations on inertia really extend to abelian semisimple representations on such a field? 17So what is the argument here? First we need to make sure that our definition of “algebraic” for such cases is appropriate.

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For our purposes below (with p-divisible groups) it is the crystalline representations that are most relevant. Thus, we now state one of the fundamental results in p-adic Hodge theory only in the crystalline case. 3.6.1.3. Theorem. Let L be either a finite extension field of either Qp or K0 . Let L0 ⊆ K0 be the maximal absolutely unramified subfield in L (over which L is a finite totally ramified extension). Let Bcrys and BdR denote Fontaine’s period rings for crystalline and de Rham representations respectively (so BdR canonically contains K 0 and Bcrys is a subring canonically containing K0 ) in a ΓL -equivariant manner. (i) There is an equivalence of tensor categories Dcrys : Repcrys (ΓL ) ' MFfL (φ) where Dcrys (V ) := (Bcrys ⊗Qp V )ΓL is an L0 -vector space of dimension dimQp (V ) whose Frobenius-semilinear action φ is induced by the Frobenius on Bcrys , and for which the decreasing filtration Fil on Dcrys (V )L is defined by
Γ Fili (Dcrys (V )L ) = (Dcrys (V )L ) ∩ Fili BdR ⊗Qp V L . (ii) A functor inverse to Dcrys is given by Vcrys (D, φ, Fil) = Fil0 (Bcrys ⊗K0 D)φ=1 . (iii) A crystalline representation (V, ρ) of ΓL comes from a p-divisible group X over OL if and only if the Hodge-Tate weights of (V, ρ) are contained in {0, 1}. In such cases, dimQp V {1} = dim(X ) and dimQp V {0} = dimX t = ht(X ) − dim(X ), where V {r} := (CL ⊗Qp V (−r))ΓL using the completion CL of L. The most difficult parts of the proof are the essential surjectivity of the functor Dcrys in part (i) and the “if” implication in part (iii). The proof of the first is given in [14], and the second is proved in [10] when p > 2, and in [37] for any p. The dimension assertions in (iii) are due to Tate, in his original work on the Hodge–Tate decomposition (1.4.5.1)) for p-divisible groups. 3.6.2. Cocharacters and reflex tori. Let K be a finite extension of K0 inside of K 0 . For any (V, ρ) ∈ Repab crys (ΓK ), denote by H(V ) the Zariski closure of ρ(ΓK ) in the Qp -group ab to be the projective limit of all GL(V ); this has a torus as its identity component. Define HK HV as V runs through all abelian semisimple crystalline representations. This is an extension of a profinite group by a pro-algebraic torus. (We will see below that it is actually connected, ab is the Galois group of the neutral which is to say a pro-algebraic torus.) In fancier terms, HK Tannakian category Repab crys (ΓK ) when the latter is equipped with the forgetful functor to finite-dimensional Qp -vector spaces as a fiber functor over Qp , and H(V ) is the Galois group ab generated by V ; see [67, §1]. of the tensor subcategory of HK Since V is Hodge-Tate, consider the Hodge-Tate decomposition M (CK ⊗K V {i}) ' CK ⊗Qp V, i∈Z

where CK denotes the completion of K 0 and V {i} denotes the K-linear subspace of x ∈ CK ⊗Qp V such that ρ(g)(x) = χ(g)i x for all g ∈ ΓK , with χ : ΓK → Z× p the p-adic cyclotomic Γ K character. (In other words, V {i} is a copy of (CK ⊗Qp V (−i)) .) This CK -linear grading on CK ⊗Qp V defines a CK -cocharacter hV : Gm → GL(V )CK that factors through the inclusion HV ,→ GL(V ) viewed over CK ; see [67, 1.4]. Since this grading ab as a projective is functorial in V , hV is also functorial in V . Thus, using the definition of HK limit over V as above, the limit of the hV define a CK -cocharacter h of the pro-algebraic torus ab . identity component of HK

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3.6.2.1. Example. Now we make some examples using Lubin–Tate groups. For any pair (E, π) consisting of a finite extension E of Qp (inside of K 0 ) and a uniformizer π of E, let fE,π (x) := πx + xq , where q is the cardinality of the residue field of E, and let FE,π (x, y) ∈ OE [[x, y]] be the Lubin–Tate formal group law such that fE,π (x) is the endomorphism of FE,π attached to π. Let (VE,π , ρE,π ) be the 1-dimensional E-linear abelian representation of ΓE obtained from the p-adic Tate module of FE,π by inverting p, so it defines a continuous homomorphism × ρE,π : Γab E → OE . It is a classical fact that when the local reciprocity map rE : E × → Γab E is defined with the arithmetic normalization (carrying uniformizers to arithmetic Frobenius elements) then ρE,π ◦ rE (π) = 1,

ρE,π ◦ rE (u) = u−1

× for all u ∈ OE .

In particular the restriction of ρE,π to the inertia subgroup IE ⊂ ΓE is independent of the × choice of the uniformizer πE of E, since rE (OE ) is the image of IE in Γab E . We denote this restriction as ρE , and let VE denote its underlying Qp -vector space (which is a 1-dimensional E-vector space). Clearly the Zariski closure HVE of ρE (ΓK ) is TE := ResE/Qp Gm (since ΓK has finite index in IE ). The E-module structure of VE gives a CK -linear ΓK -equivariant decomposition M VE,τ CK ⊗Qp VE = τ :E→Qp

where the CK -vector space VE,τ has E-structure via τ . In terms of Hodge-Tate structure of the Qp -linear representation space VE for ΓK , we have Cp ⊗Qp VE {0} = VE,ι where ι : E → Qp is the canonical inclusion (using that E was chosen as a subfield of K 0 ), and Cp ⊗Qp VE {1} = ⊕τ 6=ι VE,τ . Let hE : Gm → TE /CK be the CK -cocharacter of TE corresponding to ι. The pairs (TE , hE ) form a projective system as E varies through subfields of K of finite degree over Qp , with transition maps (TE 0 , hE 0 ) → (TE , hE ) induced by the norm NmE 0 /E : TE 0 → TE for E ⊂ E 0 ⊂ K. 3.6.2.2. Theorem. Let K be a finite extension of K0 , and use notation as defined above. ab is connected; i.e., it is a pro-torus over Q . More (i) The pro-algebraic Qp -group HK p specifically, the natural homomorphism ab (HK , h) −→ lim (TE , hE ) ←− E⊂K

is an isomorphism. ab (ii) Consider (V, ρV ) in Repab crys (ΓK ) such that the corresponding homomorphism HK → GL(V ) of Qp -groups factors through a Qp -homomorphism δ : TE → GL(V ) for some finite extension E of Qp contained in K. Viewing ΓK as an open subgroup of IE , the Galois representation ρV : ΓK → GL(V ) is given by −1 ρV (γ) = δ(rE ([γ]))−1

for all γ ∈ ΓK , where [γ] denotes the image of γ in the inertia subgroup of Γab E (which × is isomorphic to OE via rE ). (iii) An object in Repab crys (ΓK ) comes from a p-divisible group over OK if and only if its Hodge–Tate weights lie in the set {0, 1}. Proof. See [67, §2] for the proof of (i). (Implicit in (i) is the fact 18 that Repab crys (ΓK ) is generated as a Qp -linear tensor category by the abelian semisimple representations VE,π of ΓK for varying E ⊂ K as above.) The assertion in (ii) follows from the corresponding statement 18do we have a reference for this fact? It seems to address my main concern about the definition of algebraicity in the case of infinite residue field.

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99

for the Tate modules of Lubin–Tate formal group laws. Finally, (iii) follows from Theorem 3.6.1.3(iii). 3.6.3. CM types and CM p-divisible groups. We now give two essentially inverse constructions for passing between p-adic CM types and CM p-divisible groups over OK (a finite extension of W (F)). First consider a CM p-divisible group X over OK , with CM structure given by a finite extension F of Qp inside of End0 (X ). Let V (X ) denote the Qp -Tate module of the generic fiber, a 1-dimensional F -vector space equipped with a continuous linear action by ΓK . The K-linear F -action on the generic fiber Lie(X )K of the Lie algebra of X defines the associated CM type Φ ⊆ HomQp (F, K 0 ). By the definition of the associated reflex field E (as in Definition 3.4.2.3), the K ⊗Qp F -module Lie(X )K canonically descends to a K ⊗Qp E-module tΦ . Since the CM type Φ contains no multiplicities (Lemma 3.4.1.3), it defines a canonical CK -linear grading of the CK -fiber of the Lie algebra, which is to say a CK -cochcaracter hΦ : Gm → TF/CK of the Qp -torus TF = ResF/Qp (Gm ). It is straightforward to check that E is the (unique) minimal field of definition over Qp for this cocharacter. ab −→ T ⊂ GL(V (X )) corresponding to the F -linear The homomorphism of Qp -groups HK F ab  T representation space V (X ) factors as the compsition of the canonical surjection HK E and a Qp -homomorphism δF,Φ : TE −→ TF . The homomorphism δF,Φ can be explicitly described in two ways. (i) It is the unique Qp -homomorphism that on Qp -points is the map E × → F × given by the F -linear determinant of the E × -action: x 7→ det(x : tΦ → tΦ ) ∈ F × . (ii) It is the composite map TE = ResE/Qp Gm

ResE/Qp (hΦ )

/ ResE/Q ((TF )E ) p

NE/Qp

/ TF ,

where NE/Qp is defined for general commutative affine Qp -groups in place of TF , as in Proposition 2.1.4.4. See [67, 2.3] for a proof of the the first description of δF,Φ . The implication “(i) ⇒ (ii)” is analogous to Proposition 2.1.4.4 and is proved by similar arguments. The viewpoint of (i) is more classical, and it is the viewpoint of (ii) (emphasizing the role of algebraic groups in the study of the reflex field, via the cocharacter hΦ with minimal field of definition E) that paves the way towards broad generalizations within the framework of Shimura varieties. Let’s now turn things around and begin with the p-adic CM type rather than with a CM p-divisible group. So let F be a finite extension field of Qp , and Φ ⊆ HomQp (F, Qp ) any subset. Let E = E(F, Φ) ⊂ Qp be the reflex field of the p-adic CM type (F, Φ), as in Definition P 3.4.2.3. Explicitly, it is the subfield of Qp generated over Qp by the “traces” ι∈Φ ι(x) ∈ Qp . Equivalently, it is the minimal field of definition over Qp of the CK -cochcaracter hΦ : Gm → TF/CK of TF attached to Φ. This essentially reverses the preceding construction of a p-adic CM type from a CM p-divisible group, in the following sense: 3.6.3.1. Proposition. Let (F, Φ) be a Qp -valued CM type as above. Let E 0 be a finite extension of E = E(F, Φ) inside of Qp . Let K denote the compositum K0 E 0 over Qp inside of K 0 ; i.e., the completion of the maximal unramified extension of E 0 in K 0 . (i) There exists a p-divisible group X of height [F : Qp ] over OK along with CM structure by F such that OF is the CM order and Φ is the associated p-adic CM type. (ii) The construction in (i) is unique in the sense that any two p-divisible groups over OK with CM type (F, Φ) are OF -linearly isomorphic.

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(iii) Given any p-divisible group X over OK satisfying the properties in (i), any u ∈ OF× , and any uniformizer π of E 0 , there exists a p-divisible group Y over OE 0 equipped with a CM structure by F such that the following properties hold: OF is the CM order on Y, YOK is OF -linearly isomorphic to X , and the abelian Galois representation × ρY : Γab E 0 → OF associated to Y satisfies ρY (rE 0 (π)) = u. Proof. Part (iii) is a consequence of 19. Since OF is a discrete valuation ring, it suffices to prove (i) and (ii) up to OF -linear isogeny. These respectively follow from Proposition 3.6.1.2(ii) and Theorem 3.6.1.3(iii). For example in (i) we take X to have as its Qp -Tate module the object in Repab crys (ΓK ) corresponding to the Qp -homomorphism ab HK

can

/ TE

δF,Φ

/ TF

where δF,Φ is the homomorphism between Qp -tori attached to the CM type (F, Φ) as in ab  T Example 3.6.3 and can : HK E is the canonical surjection defined by the Lubin–Tate formal group laws attached to uniformizers of E. 3.6.3.2. Remark. In terms of Galois representations, the isogeny statement corresponding to Proposition 3.6.3.1(iii) is the assertion that every algebraic Galois representation IE 0 → OF× can be extended to a continuous homomorphism ΓE 0 → OF× . This assertion is clear because 0 there is a (non-canonical) topological splitting ΓE 0 ' IE 0 × Gal(k /k 0 ) where k 0 is the residue 0 field of E . An immediate consequence of Proposition 3.6.3.1(iii) is: 3.6.3.3. Corollary. In the setup of Proposition 3.6.3.1, there exists a CM p-divisible group Y over OE 0 with CM order OF such that the associated Galois representation ρY : ΓE 0 → OF× carries IE 0 onto the entire image of ΓE 0 . 3.6.4. A lifting result for CM p-divisible groups over finite fields. Let E be a finite extension of Qp , and κ its residue field. Consider an object (V, ρ) in Repab crys (ΓE ). The representation ρ is valued in the Qp -points of a Qp -torus T = ResF/Qp (Gm ) ⊂ GL(V ) corresponding to the multiplicative group of the commutative semisimple Qp -subalgebra F ⊂ EndQp (V ) × yields a composite generated by ρ(ΓE ). The resulting homomorphism Γab E → T (Qp ) = F homomorphism rE ab E× → ΓE → T (Qp ) = F × that is algebraic (relative to Qp ) on inertia, due to the crystalline hypothesis (see Proposition 3.6.1.2(ii)). Hence, there is a unique Qp -homomorphism δ : ResE/Qp (Gm ) → T such that for all γ ∈ IE we have −1 δ(rE ([γ])) = ρ(γ)−1 , where [γ] denotes the image of γ in Γab E (and we recall that the local reciprocity map rE : × × ab E → ΓE carries OE isomorphically onto the inertia subgroup of Γab E ). Let D = Dcrys (V ). This is a finite-dimensional vector space over E0 := W (κ)[1/p] equipped with a bijective additive endomorphism φ that is semilinear over the absolute Frobenius on E0 . Thus, if r = [κ : Fp ] then φr := φr is an E0 -linear endomorphism of D. 3.6.4.1. Proposition. Let π ∈ E be a uniformizer. (i) With notation as above, the E0 -linear endomorphism φ−1 of D arises via Dcrys r functoriality applied to the Qp [ΓE ]-linear endomorphism ρ(rE (π)) · δ(π)−1 of V . 19give reference for completed unramified descent

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101

(ii) Assume that (V, ρ) is the Qp -Tate module arising from a p-divisible group X over OE . Then the q-Frobenius endomorphism of the special fiber X = Xκ is induced by the endomorphism of X associated to ρ(rE (π)) · δ(π)−1 ∈ GL(V ). We note that (ii) is an analogue of a standard fact in the theory of CM abelian varieties; see [68, §7]. Proof. We give a sketch of a proof below; see Lemma 2.6.2 for details. 20 First we note that the expression ρ(π) · δ(π)−1 is independent of the choice of the uniformizer π, due to the formula on units given in Theorem 3.6.2.2(ii). Also, the assertion in (i) is compatible with unramified twists (valued in the centralizer of the ΓE -action in GL(V )). Moreover, (i) is compatible with tensor products. Hence, it suffices to check (i) for Lubin–Tate formal group laws, which is a well-known fact recalled in the proof of Theorem 3.6.2.2(ii). 21 The statement (ii) follows immediately from (i). 22 As an application, we get the following lifting result (which is a reformulation of the main work in 2.6). 3.6.4.2. Proposition. Let X be a CM p-divisible group over a finite field Fq , with CM structure defined by a finite extension F of Qp (so X is isoclinic). Let Φ be a Qp -valued CM type on F such that Φ is compatible with the CM structure; i.e., the unique slope of X is #Φ/[F : Qp ]. Finally, let E = E(F, Φ) be the associated reflex field, and E 0 a finite extension of E inside of Qp , such that its residue field κ0 has size q r ; fix an embedding of Fq into κ0 . There exists a CM p-divisible group Y over OE 0 with CM type (F, Φ) and CM order OF such that its special fiber Yκ0 is F -linearly isogenous to Xκ0 . Proof. By Proposition 3.6.3.1 and Proposition 3.6.4.1, there exists an OF -linear p-divisible group Y over OE 0 with height [F : Qp ] such that the q r -Frobenius endomorphism of its special fiber is induced by π r . Hence this special fiber is F -linearly isogenous to Xκ0 due to Tate’s conjecture for p-divisible groups over finite fields. 23

20

We should be clearer about this reference to Lemma 2.6.2, since the surrounding setup there is not quite in suitable generality. Should that stuff be rewritten? 21This fact was not stated there; all that was discussed there was the situation on inertia. 22No, this requires a precise link between D e module of the special fiber. It should crys and the Dieudonn´ not be called “immediate”. 23I think we only discussed the case of abelian varieties over finite fields. Do we have a reference or discussion somewhere for this “Tate conjecture”?

I also see along which lines to look for a generalization of Tate’s theorem to crystals in the unequal characteristic case . . . the description of this functor remains however the most mysterious point . . . Maybe these statements will even turn out to be provable! Alexandre Grothendieck

4

CM lifting of abelian varieties up to isogeny 4.1. Introduction

4.1.1. Statement of the main result. In this chapter we prove the following strong form of Problem (I) in 1.7.5, CM lifting up to isogeny without increasing the base finite field. This problem has an affirmative answer, stated below. The proof is given in 4.4.7; the statement of 4.4.7 is essentially the same as Theorem (I). Theorem (I) (CM lifting up to isogeny). Let p be a prime number. Given an abelian variety B1 over a finite field κ ⊃ Fp , a CM field L with [L : Q] = 2dim(B1 ) and an embedding β1 : L → End0 (B1 ), there exist • an abelian variety B0 over κ isogenous to B1 over κ such that OL operates on B0 , i.e. β1 induces an embedding β0 : OL → End(B0 ), • a complete noetherian local integral domain R with generic characteristic 0 and residue field κ, • an abelian scheme B over R, an embedding α : Z + pOL −→ End(B), and an isomorphism ζ0 from the closed fiber of B to B0 such that ζ0 ◦ α0 (x) = β0 (x) ◦ ζ0 for all x ∈ Z + pOL , where α0 is the ring homomorphism from Z + pOL to End(B0 ) induced by α. Remark. (1) It is explained in [59] and in Chapter 3 that there is an obstruction of geometric nature, from the size of fields of definition of the p-divisible group B0 [p∞ ]×Spec(κ)Spec(F), to the CM lifting problem for B0 . Therefore it will be necessary to modify the given abelian variety B0 by a suitable isogeny in the statement of Theorem (I) to make sure that the resulting abelian variety B1 can be CM lifted. (2) Although the abelian variety B0 admits a CM lifting to characteristic 0, it may not be possible to lift the action of the whole ring of integers OL to characteristic 0. See the toy model below for a 2-dimensional example with L = Q(5 ), where the CM lifting B is constructed “by hand” using a trick. This toy model example turns out to be the key. Through a series of intermediate steps involving localization to p-divisible groups, classification of (OL ⊗Z Zp )linear p-divisible groups and Galois descent, the theorem is reduced to the case of the toy model! 4.1.2. A toy model. Let p be a prime number with p ≡ 2, 3 (mod 5), and let C0 be an abelian variety over Fp2 such that FrC0 = p·ζ5 and Z[ζ5 ] operates on C0 , where ζ5 is a nontrivial fifth root of unity. This is (one of) the supersingular counterexamle in 2.3.1. We first explain why it is not possible to lift C0 together with the Z[ζ5 ]-action to a local integral domain (R, m) with residue field Fp2 and generic characteristic 0. Suppose there were such a Z[ζ5 ]-linear abelian scheme C over R. The action of Z[ζ5 ] on Lie(C)⊗Z Q corresponds to a subset Φ of Homring (Q(ζ5 ), frac(R)alg ). Since p is inert in Q(ζ5 ), the subset Φ is determined by the action of Z[ζ5 ] on Lie(C/R ) ⊗R κ = Lie(C0 ). Because C0 is defined over Fp2 , we see that Φ is stable under the action of the element σ in Gal(Q(ζ5 )/Q) which induces the element 103

104

4. CM LIFTING OF ABELIAN VARIETIES UP TO ISOGENY 2

x 7→ xp on the residue field Z[ζ5 ]⊗Fp . But then σ is the complex conjugation on Q(ζ5 ) and Φ is stable under σ, contradicting the familiar self-duality condition for the CM type Φ of the generic fiber of C that Φ is disjoint with its complex conjugate. The above argument actually shows that for every algebraic extension field κ ⊃ Fp2 , the Z[ζ5 ]-linear CM abelian variety (C0 , Z[ζ5 ] ,→ End(C0 )) ×Spec(Fp2 ) Spec(κ) over κ cannot be lifted to a CM abelian scheme (C, Z[ζ5 ] ,→ End C)) over an integral local domain of characteristic 0 with residue field κ. However this pair (C0 , Z[ζ5 ] → End(C0 )) ×Spec(Fp2 ) Spec(Fp4 ) admits a lifting to a formal CM abelian scheme (C, Z[ζ5 ] → End(C)) over the ring W (Fp4 ) of Witt vectors with entries in Fp4 , because the deformation functor Def(C0 , Z[ζ5 ] → End(C0 )) is smooth over W (Fp4 ) of relative dimension 0, as can be seen by an easy calculation. This formal abelian scheme C over W (Fp4 ) is, of course, not algebraic as we just saw, and provides an illustration of the algebraicity criterion in 2.2.3. A CM lifting of C0 can be constructed as follows. An easy calculation shows that the a-number of the supersingular abelian variety C0 is 2. Moreover there is a Z[ζ5 ]-linear abelian variety A0 over Fp4 and a Z[ζ5 ]-linear isogeny δ : A0 → C0 ×Spec(Fp2 ) Spec(Fp4 ) with the following properties. (i) The subset Φ0 of Homring (Q(ζ5 ), frac(R)alg ) corresponding to the character of the action of Z[ζ5 ] on Lie(A0 ) is a CM type, i.e. Φ0 is disjoint from its complex conjugate. (ii) The kernel of δ is isomorphic to αp . (iii) The a-number of the supersingular abelian variety A0 is equal to 1, so Ker(δ) is the unique subgroup scheme of A0 [p] of order p; moreover this property persists after arbitrary extension of the (perfect) base field. The property (i) implies that there is a Z[ζ5 ]-linear abelian scheme A over W (Fp4 ) lifting the Z[ζ5 ]-linear abelian variety A0 . Let K be a finite extension of the fraction field Kp4 of W (Fp4 ) such that there exists a finite flat subgroup scheme G of A[p]×Spec(W (Fp4 )) Spec(OK ) of order p over OK . By (iii) the closed fiber of G coincides with the base change of the kernel of δ, so C := A/G is a lifting of C0×Spec(F 2 )Spec(Fp4 ) such that the action of the order O0 = Z + pZ[ζ5 ] p

on C0 extends to C. From the deformation theory for the pair (C0 , O0 → End(B0 )), we conclude that there is complete noetherian local subring R of OK with residue field Fp2 such that the pair (C, O0 → End(C)) descends to R. We have constructed a CM lifting of C0 to a ring R of characteristics (0, p). The following definition summarizes the main feature in the above example. We call a CM abelian surface (C, O ,→ End(C)) over a finite field κ a toy model if the following holds. - O is a subring of finite index in a quartic CM field L such that O ⊗Z Zp ∼ = W (Fp4 ). - The subset of Homring (O/pO, κalg ) corresponding to the action of O/pO on Lie(C) is stable under the action of the p2 -Frobinius for the finite field O/pO ∼ = Fp 4 . For any toy model (C, O ,→ End(C)) over a finite field κ, the same argument shows that the pair (C, Z+pO ,→ End(C)) admits a CM lift to characteristic 0, even though (C, O ,→ End(C)) does not. Below is a sketch of the main ideas of the proof of Theorem (I) and the tools involved. 4.1.3. Preliminaries. As already indicated in the previous paragraph, deformation theory tells us that the existence of CM lifting is a geometric property: a pair (A0 , β0 : L → End0 (A0 )) of an abelian variety over a finite field κ with complex multiplication by a CM field L can be lifted to characteristic 0 if and only if its base change to F can be CM lifted. In other words, in 1.7.5 the statement (CML) is equivalent to the apparently weaker statement (R); see step (5) of the proof of Thm. 4.4.7 for this equivalence. So to show Theorem (I), it suffices to find a suitable κ-isogeny B1 → B0 , where B0 is an abelian variety over κ with the CM structure β0 : L ,→ End0 (B0 )) induced from (B1 , β1 ), such that the base change of (B0 , β0 ) to F is the

4.1. INTRODUCTION

105

specialization of an abelian variety with CM by L over a finite extension field K of the fraction field K0 of W (F). Changing B1 by an isogeny over κ, we may and do assume that OL operates on B1 . Consider all OL -linear abelian varieties B0 over κ which are OL -linearly isogenous to (B1 , β1 ). We have to show that at least one among them admits an L-linear lift to characteristic 0. By the Serre–Tate deformation theorem (Theorem 1.4.5.7) the problem of constructing a formal abelian scheme B over OK which lifts B0 ×Spec(κ) Spec(F) such that the action of an order O1 lifts to B is equivalent to the problem of constructing a p-divisible group Y over OK which lifts p-divisible group B0 [p∞ ]×Spec(κ) Spec(F) such that the action of O1 lifts to Y. For the resulting formal abelian scheme to be algebraizable, it is sufficient (and also necessary) that the K-linear action of L on Lie(Y)⊗Z Q is given by a CM type of L; see 4.4.3. This is a self-duality condition on the (L⊗Q K)-module Lie(Y)⊗Z Q with respect to the involution for the quadratic extension L/L0 , where L0 is the maximal totally real subfield of L. This notion of self-duality can be localized to places v of L0 above p. For any p-adic place v of L0 , write Lv := L⊗L0 L0,v and let OLv := OL ⊗OL0 OL0 ,v be the maximal order in Lv ; the complex conjugation for the CM field L induces an involution on the ring Lv . For every Lv -linear CM p-divisible group Xv , a p-adic CM type (or a local CM type) of Xv is the subset Φv of HomQp -alg (Lv , K) which corresponds to the linear Lv -action on the Lie algebra of Xv ; Φv is said to be self-dual if Φv is disjoint from its complex conjugate. 4.1.4. The Lie type of OL,v -linear CM p-divisible groups in characteristic p. Let v be a place of L0 above p. An OL,v -linear p-divisible group is said to be a CM p-divisible group if its height is equal to [Lv : Qp ]. The Lie type of an OL,v -linear p-divisible group Zv over a field k ⊃ Fp is the class [Lie(Zv )] of the (OL,v⊗Zpk)-module Lie(Zv ) in a suitable Grothendieck group Rk (OL,v ). The slopes of Zv can be read off from its Lie type. The Lie type is a useful tool for classifying OL,v -linear CM p-divisible groups over fields of characteristic p and for constructing isogenies and OL,v -linear lifting to characteristic 0; see (L1)–(L3) below. (L1) Classification of CM p-divisible groups in characteristic p by their Lie types. The p-divisible group B1 [p∞ ] decomposes into the product of OL,v -linear CM p-divisible groups B1 [v ∞ ], where v runs through all places of L0 above p, and we will localize our problem accordingly. The classification result in [86] says that an Ov -linear CM p-divisible group X0 over F, or more generally an algebraically closed field k ⊃ Fp , is determined up to isomorphism by its Lie type; see Prop. 4.2.6. (L2) Galois descent—constructing isogenies from Lie types. There is a natural action of the Galois group Gal(F/Fp ) on the Grothendieck group RF (OL,v ). The Lie type of any OL,v -linear CM p-divisible group Zv over a subfield κ ⊂ F is a κ-rational element of RF (OL,v ), in the sense that it is fixed by Gal(F/κ). Moreover the following weak Galois descent statement holds; see Prop. 4.2.8. If Zv is a OL,v -linear CM p-divisible group over a subfield κ ⊂ F and δv is a κ-rational element of RF (OL,v ) with the same slopes (counted with multiplicities) as the Lie type of Zv , then there exists an OL,v -linear pdivisible group Yv over κ with Lie type δv which is OL,v -linearly isogenous to Zv . So the Lie type tells us exactly which OL,v -linear p-divisible groups over κ are OL,v -linearly isogenous to Zv over κ. (L3) Lifting by the OL,v -action and self-duality. For every place v of L0 above p, OL,v -linear CM p-divisible groups Xv over unspecified finite extension fields of K0 in K0 /K0 are determined up to isomorphism by its p-adic CM type Φv ,

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a subset of Hom(Lv , K0 ) which corresponds to the class in the Grothendieck group RK0 (Lv ) of the (Lv ⊗Qp K0 )-module Lie(Xv )[1/p]. With this in mind, the classification result (L1) becomes a method to construct Ov -linear CM liftings: To lift an OL,v -linear p-divisible group Zv over F to an OL,v -linear p-divisible group Zv over the ring of integers of some unspecified finite extension field with a self-dual p-adic CM type, it suffices to find a self-dual p-adic CM type Φv for Lv which maps to the Lie type of Zv under the natural reduction map RK0 (OL,v ) → RF (OL,v ) between Grothendieck groups. See Prop. 4.3.9 for the precise statement. An element of RF (O) which is obtained by reduction modulo p from a self-dual p-adic CM type of Lv is said to be a self-dual Lie type. There is a simple combinatorial description for self-dual Lie types; see 4.3.4, 4.3.7 (a) and 4.3.8. 4.1.5. Given a CM abelian variety (B1 , OL ,→ End(B1 )) over a finite field κ, the task of constructing a CM lifting after performing a κ-isogeny is easier for some p-adic places and not as easy for others. The following definition spells out the exact conditions. Definition. Let L be a CM field, let L0 be the maximal totally real subfield of L, and let κ be a finite field. Let ∆L0 = ∆(L/L0 , κ) be the set of all p-adic places v of L0 such that the following holds. (i) L/L0 is unramified and inert above v; denote by w the p-adic place of L above v. (ii) e(L/Q, w) = e(L0 /Q, v) is odd, where e(L/Q, w) and L0 /Q, v) are the ramification indices for w and v respectively. (iii) fw ≡ 0 (mod 4), where fw = [κw : Fp ] is the degree of the residue field of w. (iv) [κw : (κw ∩ κ)] is even. A p-adic place v of L0 in ∆L0 is said to be bad ; p-adic places of L0 not in ∆L0 are said to be good.1 Denote by ∆L the set of all p-adic places of L above ∆L0 . A p-adic place w of L is bad if w ∈ ∆L ; otherwise w is good. A related definition of badness in the local situation is given in 4.3.17.

4.1.6. Sketch of the proof of theorem (I) (a) Construction of a suitable OL -linear abelian variety B0 over κ isogenous B1 . Recall that we have assumed that the ring OL of integers of the CM field L operates on the given abelian variety B0 . We use (L1)–(L3) to find an abelian variety B0 which is OL -linearly isogenous to the given OL -linear abelian variety B1 over κ so that B0 [p∞ ] is “as close to being self-dual as possible”. More precisely, consider all places v of L0 lying above p. Whenever it is possible to change the OL,v -linear p-divisible group B0 [v ∞ ] by an OL,v -linear isogeny over κ to make it self-dual, we do so; this can be done precisely when v is good, i.e. v ∈ / ∆L0 . For every bad place w ∈ ∆L lying above a bad place v ∈ ∆L0 , we choose an Ow -linear CM p-divisible group B0 [w∞ ] = B0 [v ∞ ] such that its Lie type [Lie(B0 [w∞ ])] is maximally symmetric for the action of Gal(κw /Fp ). The Lie type of such an Ow -linear p-divisible groups B0 [w∞ ] is said to be striped in 4.3.13 (b). Combing these two cases, we obtain a κ-isogeny (B1 , OL ,→ End(B1 ) → (B0 , OL ,→ End(B0 ) such that B0 [v ∞ ] is self-dual for every good p-adic place v of L0 , and B0 [v ∞ ] is striped for every bad p-adic place v ∈ ∆L0 . 1The problem of constructing a CM lifting with self-dual p-adic CM type up to κ-isogeny for the p-divisible group B1 [v ∞ ] is easier if v ∈ / ∆L0 and more difficult if v ∈ ∆L0 .

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(b) Reduction of the CM lifting problem to the toy model. Let v be a bad p-adic place of the totally real field L0 and let w be the place of L above v. Then the Ow -linear p-divisible group B0 [w∞ ]×Spec(κ) Spec(F) has the form Ow ⊗W (κw ) Ytoy , a Serre tensor construction, where Ytoy is a 2-dimensional W (Fp4 )-linear CM p-divisible group over F isomorphic to the W (Fp4 )-linear p-divisible group C[p∞ ] attached to a toy model C in 4.1.2, while the tensor product is taken in the category of sheaves for the flat topology. See 4.3.23 and 4.3.26 (ii); the tensor product construction is explained in 4.2.10–4.2.11. Putting together the p-divisible groups B0 [v ∞ ] we picked for each place v of L0 above p, we get an OL -linear abelian variety B0 over κ whose components at every place v above p coincide with the OL,v -linear p-divisible groups B0 [v ∞ ] that have been chosen. We know by (L3) how to lift the OL,v -linear p-divisible group B0 [v ∞ ] to an OL,v -linear CM p-divisible group over a mixed characteristic complete discrete valuation ring with self-dual p-adic CM type after extending the base finite field to F, for each good p-adic place v of L0 . To prove that B0 admits a CM lifting to characteristic 0, it suffices to produce, for each bad place w ∈ ∆L , a lifting Zw of B0 [w∞ ]×Spec(κ) Spec(F) to characteristic 0 such that the action of an order of Ow on B0 [w∞ ]×Spec(κ) Spec(F) extends to Zw and the p-adic CM type of Zw is self-dual. We have seen in 4.1.2 how to construct a CM lifting for the toy model. Because for any bad place w of L the Ow -linear p-divisible group B0 [w∞ ] is isomorphic to the tensor product of the W (Fp4 )-linear p-divisible group of a toy model with Ow over W (Fp4 ), we get from the CM lifting of the toy model a CM lifting of B0 [w∞ ] with a self-dual p-adic CM type; see 4.3.28. Summary: How to construct a CM lifting of B0 ×Spec(κ) Spec(F). It suffices to construct, for each place v of L0 above p, a p-divisible group Zv over the ring of integers OK of some finite extension field K of K0 whose closed fiber is B0 [v ∞ ]×Spec(κ)Spec(F) such that the action of an order O1,v of OL ⊗OL0Ov on B0 [v ∞ ] lifts to Zv and the p-adic CM type of Zv is self-dual. There is no problem if v ∈ Σ0L0 ,p is a good place of L0 above p: we simply take Zv to be an (OL ⊗OL0 Ov )-linear lifting of B0 [v ∞ ] with self-dual p-adic CM type Φv whose existence is guaranteed by the self-duality of B0 [v ∞ ] at v. For a bad place v of L0 and the unique place w of L above v, B0 [w∞ ] is isomorphic to Ow ⊗Owur X0,w , we construct Zv in a way similar to the 2-dimensional example. Start with Ow ⊗Owur Xw , which is isomorphic to the product of several copies of Xw . The kernel of Ow ⊗Owur X0,w −→ Ow ⊗Owur Y0,w is also the product of the same number of copies of the kernel of X0,w → Y0,w , so it can be lifted to a finite flat subgroup Gw scheme of Ow ⊗Owur Xw . Let Zv := Ow ⊗Owur Xw /Gw . Clearly Zv is a lifting of B0 [v ∞ ] to characteristic 0 such that OK ⊗OL0 Ov operates on Zv via isogenies, and its p-adic CM type is self-dual because the CM type of Xw is. This finishes our sketch of the construction of CM lifting of B0 . Details can be found in the proof of Thm. 4.4.7. 4.1.7. Our original proof of theorem (I) is more complicated than the proof sketched above, in that the construction of a CM lifting of B0 [w∞ ] with self-dual p-adic CM type for a bad prime is first reduced, using a Serre tensor product construction, to the case when Ew is unramified over Qp . Under the assumption that Ew is unramified over Qp , the required CM lifting of B0 [w∞ ] is constructed directly using p-adic Hodge theory; later we replaced p-adic Hodge theory with Raynaud’s [62]. These two versions of the direct construction of CM lifting can be found in appendices B.2 and B.1 respectively; cf. 4.3.29. 4.1.8. In the proof of B.1.16 we need a formula of the closed fiber of the p-divisible groups or a finite flat group scheme over a complete discrete valuation ring finite flat over W (F), associated to some input linear algebra data in p-adic Hodge theory. Such a formula is provided in B.4.15 and B.4.17 in appendix B.4. In appendix B.3 we summarize the relation between different flavors of Dieudonn´e theory over a perfect field of characteristic p. We thank Bill Messing for useful discussion on the comparison between the crystalline and the classical theory.

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4.1.9. This chapter is notation-heavy, partly because of the need to decode the CM data explicitly. At the end of this chapter is a three-page summary of them; see page 133. 4.1.10. There remains a significant open question on CM lifting, whether one can kill all obstruction to the existence of a CM lifting by requiring that the whole ring of integers of the CM field operates on the abelian variety. This question, stated in 1.7.5 and recalled below, is essentially geometric, in the sense that it is equivalent to the apparently weaker version when κ is replaced by F; c.f. 4.4.11. We also formulate a closely related question. Question (sCML). Let L be a CM field. Let (B0 , β0 : OL → End(B0 )) be an abelian variety over a finite field κ of dimension [L : Q]/2 with endomorphisms by OL . Does there exist an abelian scheme B over a complete noetherian local ring R of generic characteristic 0 with residue field κ, and an embedding β : L → End0 (B), such that the closed fiber of (B, β) is equal to (B0 , β0 ⊗Q : End0 (B))? Question (LTI) Lie type of closed fiber of isogenous CM abelian schemes. Let L be a CM field, and let (B, β : L → End0 (B)) be a CM abelian scheme over a mixed characteristic complete discrete valuation ring R whose residue field is algebraically closed of characteristic p > 0. Consider the family I (B) of all abelian schemes B1 over R isogenous to B, such that the whole ring OL of integers of L operates on the closed fiber B1 of B1 via the CM structure β1 : L → End0 (B1 ) on the closed fiber B1 of B1 induced by β. Let LTI(B) be the set of all Lie types of the closed fibers of CM abelian schemes in I (B). What can one say about the subset LTI(B) of all Lie types LT(L, p) for the CM field L? We expect (sCML) does not hold in general. There are examples where LTI(B) 6= LT(L, p), but we do not know any structure or statistics for LTI(B).

4.2. Classification and Galois descent by Lie types We review in this section the classification of O-linear CM p-divisible groups over an algebraically closed field k of characteristic p, where O is the ring of integers of a finite extension field of Qp .2 As mentioned in §4.1, the statements 4.2.6, 4.2.7 are special cases of results in [86]. 4.2.1. Notation in the local case. • E is a finite extension field of Qp , OE is the ring of integers in E, and mE is the maximal ideal of OE . • e = e(E/Qp ) is the absolute ramification index of E. • κE = the residue field of OE . • f = fE = [κE : Fp ], q = pf . • k ⊃ Fp denotes an algebraically closed field. • E ur is the maximal subfield of E which is unramified over Qp ; it is isomorphic to the fraction field of W (κE ) and its ring of integers is denoted by OE ur . ' ' • I := HomZp -alg (OE ur , W (k)) −→ HomZp -alg (OE ur , k) −→ Homring (κE , k). The set I has a natural structure as a torsor for the finite cyclic group Gal(κE /Fp ) such that each element τ ∈ Gal(κE /Fp ) sends every i ∈ I to the embedding x 7→ i(τ (x)) ' of κE into k. We have a natural isomorphism Z/f Z −→ Gal(κE /Fp ), where the canonical generator 1 + f Z of Z/f Z is sent to the arithmetic Frobenius element x 7→ xp in Gal(κE /Fp ). • Rk (OE ) is the Grothendieck group for the abelian category of all finite (OE ⊗Zp k)modules. Similarly for any finite field κ ⊇ Fp , Rκ (OE ) is the Grothendieck group of the abelian category of all finite (O⊗Zp κ)-modules. 2Only the case k = F is needed for the main theorem 4.4.7 of this chapter.

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109

+

• Rk (OE ) (resp. Rκ (OE )) is the subset of effective elements in Rk (OE ) (resp. Rκ (OE ) ) consisting of classes of (OE ⊗Zp k)-modules (resp. (O⊗Zp κ)-modules) of finite length. • Let κ ⊂ k be a finite subfield of k. Tensoring with k over κ induces a natural homomorphism jκ : Rκ (OE ) −→ Rk (OE ). • We have a canonical decomposition
Y (OE ⊗(OEur ,i) W (k)) , OE ⊗Zp W (k) ∼ = = OE ⊗OEur OE ur ⊗Zp W (k) ∼ i∈I

where each factor OE ⊗(OEur ,i) W (k) is a complete discrete valuation ring of ramification index e over W (k). • The above decomposition induces a decomposition
Y OE ⊗(OEur ,i) k , OE ⊗Zp k ∼ = = OE ⊗⊗OEur OE ur ⊗Zp k ∼ i∈I

where each factor OE ⊗(OEur ,i) k is (non-canonically) isomorphic to k[t]/(te ). • Let   Rk (OE , i) := R OE ⊗(OEur ,i) k be the Grothendieck group for the category of finite (OE ⊗(OEur ,i) k)-modules. Denote by R+ k (OE , i) the subset of Rk (OE , i) consisting of classes of finite (OE ⊗(OE ur ,i) k)modules. • We have canonical isomorphisms ' i : Rk (OE , i) −→ R(k) ∼ = Z,

where R(k) is the Grothendieck group for the category of finite dimensional vec+ tor spaces over k, and i (Rk (OE , i)) = N. For a module M of finite length over OE ⊗(OEur ,i) k, the integer i ([M ]) is the length of M over the discrete valuation ring OE ⊗(OEur ,i) k. • The decomposition of the k-algebra OE ⊗OEur k induces a decomposition Rk (OE )

'

/

L

i∈I

R(OE , i)

+

⊕i i

/

L

i∈I

Z

under which the subset Rk (OE ) of effective classes corresponds to • Denote by  : Rk (OE ) → Z the homomorphism such that X i (δ) (δ) =

L

i∈I

N.

i∈I

for all δ ∈ Rk (OE ); (δ) is called the dimension of δ. We abuse notation and write  : Rκ (OE ) → Z for the composition of  with the natural map Rκ (OE ) → Rk (OE ) for any finite subfield κ ⊂ k. 4.2.2. Definition. (a) Let R be a commutative noetherian ring such that either p is nilpotent, or R is complete with respect to an ideal I containing p. An OE -linear (resp. E-linear) p-divisible group X over Spec(R) is a pair (X, α : OE ,→ End(X)) (resp. (X, α0 : E ,→ End0 (X) := End(X)⊗Zp Qp )), where X is a p-divisible group over Spec(R) and α (resp. α0 ) is a ring homomorphism which sends 1 to 1. (b) Let (X, α) be an OE -linear p-divisible group over Spec(R) as in (a). We say that (X, α) is an OE -linear CM p-divisible group if [E : Qp ] = height(X).

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(c) Suppose that (X0 , α : OE → End(X0 )) is an OE -linear p-divisible group over k. Define the Lie type of (X0 , α) to be the class [Lie(X0 )] := [Lie(X0 , α)] in Rk (OE ) of the (OE /pOE )-module Lie(X0 ), the Lie algebra of X0 . We have ([Lie(X0 )]) = dim(X0 ) , which justifies calling (x) the dimension of x for elements x ∈ Rk (OE ). (d) Denote by K0 the fraction field of W (k) and let K0 be an algebraic closure of K0 . Note that in §B.1 K0 denotes the fraction field of W (F) because k = F there. Let K be a finite extension field of K0 contained in K0 and let (Y, α : E → End0 (Y)) be an E-linear p-divisible group over OK such that height(Y) = [E : Qp ]. The character of the K0 -linear action of E on Lie(Y) ⊗OK K0 corresponds to a subset Ψ ⊂ HomQp -alg (E, K0 ) such that card(Ψ) = dim(Y). We call Ψ the p-adic CM type of (Y, α), or the CM type of (Y, α) for short. (e) Denote by 2Homring (E,K0 ) the set of all subsets of Homring (E, K0 ). Define a reduction map ξ : 2Homring (E,K0 ) −→ Rk (OE ) such that for any subset Φ ⊆ HomQp -alg (E, K0 ), ξ(Φ) is the element of Rk (OE ) characterized by the condition that i (ξ(Φ)) = # {σ ∈ Φ | σ ¯ = i}

∀i ∈ I ,

where σ ¯ denotes the ring homomorphism from κE to k induced by σ : E → K0 . Clearly (Φ) = #Φ for all Φ ⊆ HomQp -alg (E, K0 ). (f) An element δ ∈ Rk (OE ) is said to be a Lie type if 0 ≤ i (δ) ≤ e for all i ∈ I. It is easy to see that the image of the reduction map ξ is the subset of all Lie types in Rk (OE ). The proofs of Lemmas 4.2.3 and 4.2.4 are straight-forward and omitted. 4.2.3. Lemma. Let (X , α : OE → End(X )) be an OE -linear CM p-divisible group over OK , where K is a finite extension field of the fraction field K0 of W (k). Let Φ ⊂ HomQp -alg (E, K0 ) be the p-adic CM type of (X , α). Let (X0 , α0 ) be the closed fiber of X , an OE -linear p-divisible group over k. The element ξ(Φ) ∈ Rk (OE ) is equal to the Lie type [Lie(X0 , α0 )] of (X0 , α0 ). 4.2.4. Lemma. Let κ ⊂ k be a finite field. (i) The canonical homomorphism jκ : Rκ (OE ) −→ Rk (OE ) is injective. + + (ii) The inverse image of Rk (OE ) under jκ is equal to Rκ (OE ). (iii) Let σ be an automorphism of the field k, and let σ ¯ be its restriction to κE . Let V be a OE ⊗Zp k-module, and let σ V := V ⊗k,σ k with OE operation via the first factor in the tensor product. Then i ([σ V ]) = σ¯ ·i ([V ]) for all i ∈ I. (iv) The canonical morphism jκ identifies Rκ (OE ) as the set consisting of all elements in Rk (OE ) fixed by the action of the subgroup Gal((κκE )/κ) = Gal(κE /(κE ∩ κ)) ⊂ Gal(κE /Fp ) of Gal(κE /Fp ), where κκE is the compositum of the finite fields κ and κE , and the natural Galois action of Gal(κE /Fp ) on Rk (OE ) is described in (ii) above. Remark. Suppose that k = F and we identify RF (OE ) with ⊕i∈I Z. Lemma 4.2.4 (iii) says the natural action of σ on RF (OE ) is induced by the translation action on I by σ.

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111

4.2.5. Definition. (i) The slope of a Lie type δ ∈ Rk (OE ) is defined to be the rational number (δ)/(ef ) = (δ)/[E : Qp ]. Two Lie types in Rk (OE ) are said to be isogenous if they have the same slope. (ii) An element δ ∈ Rk (OE ) is said to be rational over a subfield κ ⊂ k if δ belongs to the image of the natural map jκ : Rκ (OE ) → Rk (OE ). By Lemma 4.2.4 (ii), this condition on δ coincides with the apparently weaker condition that δ is fixed under the natural action of Gal(F/κ) on RF (OE ). More explicitly, suppose that κ ∼ = Fpr , s = gcd(r, fE ). We identify the Galois group Gal(κE /F) with the cyclic group Z/fE Z and regard I as a (Z/fE Z)-torsor. The latter rationality condition on δ is i (δ) = s+i (δ)

∀i ∈ I .

Propositions 4.2.6 and 4.2.7 below are proved in [83]. Prop. 4.2.6 says that OE -linear CM p-divisible groups over an algebraically closed field k ⊃ Fp are determined up to isomorphism by their Lie type. 4.2.6. Proposition. Suppose that (X0 , α0 ) is an OE -linear CM p-divisible group over an algebraically closed field k ⊃ Fp . (i) For each i ∈ I, the torsion module Lie(X0 ) ⊗(OEur ,i) k over the discrete valuation ring OE ⊗(OEur ,i) W (k) is cyclic. (ii) We have 0 ≤ i ([(Lie(X0 ), α0 )]) ≤ e for all i ∈ I, and ([(Lie(X0 ), α0 )]) = dim(X0 ) . (iii) The p-divisible group X0 is isoclinic; its slope is equal to the slope of the Lie type [(Lie(X0 ), α0 )] of X0 . (iv) If Y0 is an OE -linear p-divisible group over k of height [E : Qp ] such that [Lie(Y0 )] = [Lie(X0 )] in Rk (OE ), then Y0 is OE -linearly isomorphic to X0 . 4.2.7. Proposition. Recall that k ⊃ Fp is an algebraically closed field. For any Lie type δ0 ∈ Rk (OE ), there exists an OE -linear p-divisible group X0 over k such that [Lie(X0 )] = δ0 . Proofs of 4.2.6 and 4.2.7. We use covariant Dieudonn´e modules. In the context of 4.2.6, let M be the Dieudonn´e module for X0 . The ring OE operates W (k)-linearly on M , commuting with the actions of F and V . The decomposition Y OE ⊗Zp W (k) = OE ⊗(OEur ,i) W (k) i∈I

of OE ⊗Zp W (k) induces a decomposition M = ⊕i∈I Mi , where each factor Mi is a free module of rank one over OE ⊗(OEur ,i) W (k). Moreover we have V ·Mi ⊂ Mi−1 ,

F ·Mi ⊂ Mi+1 ,

∀i ∈ I

where we have used the natural action of Gal(κE /Fp ) on I, and identified Gal(κE /Fp ) with Z/f Z using the arithmetic Frobenius x 7→ xp as the canonical generator of Gal(κE /Fp ). For each i ∈ I, V · Mi+1 is a non-zero (OE ⊗(OEur ,i)W (k))-submodule the Mi , hence is equal to meEi Mi for some natural number ei . We have 0 ≤ ei ≤ e for all i because F V = V F = p IdM . Because Lie(Y0 ) is naturally isomorphic to M/V M , ei is equal to i ([Lie(Y0 )]) for each i ∈ I. Choose a generator vi for each Mi . Then there exist units ai ∈ that V · vi+1 = ai π ei vi ∀i ∈ I .



× OE ⊗(OEur ,i) W (k) such

The action of F is uniquely determined by the above structural equations and the condition that F V = V F = p IdM . It is easy to see, using the fact that k is algebraically closed, that

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we can adjust the generators vi so that all the structural constants ai ’s are equal to 1. We have proved 4.2.6. To prove 4.2.7, let ei := i (δ0 ) and consider a OE -linear Dieudonn´e module M with generators vi for Mi satisfying the above equations with ai = 1 for all i ∈ I. The corresponding OE -linear p-divisible group X0 satisfies the required properties. The following proposition can be thought of as a weak form of Galois descent for OE -linear CM p-divisible groups over k in terms of their Lie type. In the proof of (I) in 4.4.7, CM lifting up to isogeny, we will use 4.2.8 to construct an abelian variety B0 over κ isogenous to the given E-linear abelian variety B1 over κ such that (B0 , E ,→ End0 (B0 )) admits a CM-lifting to characteristic 0. 4.2.8. Proposition. Let κ be a finite subfield of F. Let X0 be a OE -linear p-divisible group over κ of height [E : Qp ]. Let δ0 be a κ-rational Lie type in RF (OE ) with the same slope as X0 . Then there exists a p-divisible group Y0 over κ with [Lie(Y0 )] = δ0 and an OE -linear isogeny over κ from Y0 to X0 . Proof. We use the notation as in the proofs of 4.2.6 and 4.2.7. Let M be the OE -linear Dieudonn´e module of X0 ×Spec(κ) Spec(F), and let M = ⊕i∈I Mi be the decomposition of M . The fact that X0 is defined over κ means that M has a natural descent data for F/κ, which translates into a natural semi-linear action of Gal(F/κ). Moreover the submodule V M ⊂ M is stable under the action by Gal(F/κ). Choose and fix an element i0 ∈ I and an (OE ⊗OEur ,i0 W (F))-module Ni0 ⊂ Mi0 . The κ-rational element δ0 ∈ Rκ (OE ), together with Ni0 , determines, for every element i ∈ I, a free (OE ⊗(OEur ,i0 ) W (F))-submodule Ni ⊂ Mi ⊗Zp Qp , such that  (δ0 )

V Ni+1 = mEi

Ni .

To see this, notice that the above condition determines Ni0 +1 , Ni0 +2 , . . . , Ni0 +f −1 , Ni0 +f = Ni0 , Ni0 +f +1 = Ni0 + 1, . . . P successively, starting with the chosen Ni0 . The condition that dim(X0 ) = i∈I i (δ0 ) ensures the compatibility with the original Ni0 when we go from Ni0 +f −1 to Ni0 +f . Changing Ni0 to a smaller free submodule of rank one if necessary, we may assume that Ni ⊂ Mi for all i ∈ I. The fact that δ0 is stable under Gal(κE /(κE ∩ κ)) implies that the submodule N := ⊕i∈I Ni of M is stable under the natural action of Gal(F/κ). This means that the OE -linear isogeny γ : Y0,F −→ X0 ×Spec(κ) Spec(F) between OE -linear p-divisible groups over F corresponding to the inclusion map N ,→ M is compatible with the action of Gal(F/κ). So γ descends to an OE -linear isogeny Y0 → X0 by Galois descent. Remark. In the statement of 4.2.8, we can replace F by an algebraically closed field k and κ by a perfect subfield of k; the same proof works. The argument for 4.2.6, 4.2.7 and 4.2.8 also proves another property of Lie types. 4.2.9. Proposition. Let κ be a subfield of k. Let ξ : (Y0 , β0 ) → (X0 , γ0 ) be an OE -linear isogeny between OE -linear CM p-divisible groups. Suppose that the Lie types of Y0 and X0 are equal. Then there exists a non-zero element u ∈ OE such that Ker(ξ) = Ker(β0 (u)). Consequently Y0 and X0 are OE -linearly isomorphic over κ. Proof. It suffices to prove 4.2.9 when κ = k, so we assume that κ = k. Let N ,→ M be the homomorphism between the covariant Dieudonn´e modules associated to Y0 and X0 induced by ξ. Let N = ⊕i∈I Ni and M = ⊕i∈I Mi be the decomposition of N and M respectively, and

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we have Ni ⊂ Mi for each i ∈ I. Define natural numbers ci , ni for i ∈ I by ci := i ([Lie(M )]) = i (N )] ,

Ni = mnEi Mi .

Then we have V Ni+1 = mcEi Ni

and V Mi+1 = mcEi Mi

∀i ∈ I .

n )) for any generator So all the integers ni are equal to a fixed integer n, and Ker(ξ) = Ker(β0 (πE πE of mE .

We explain Serre’s tensor product construction for abelian varieties and p-divisible groups, and discuss the corresponding operation for Lie types 4.2.10–4.2.15. 4.2.10. Lemma. (i) Let κ be a field, not necessarily finite or of characteristic p. Let A be an abelian variety over κ. Let R be a subring of End(A) containing 1. Let S be a finitely generated right R-module. (a) The fppf sheaf S⊗R A over Spec(κ), whose value over any κ-scheme U is S⊗R A(U ) , is representable by an R-linear abelian variety A1 over κ. (b) Suppose in addition that S has a ring structure and the R-module structure on S is induced by a ring homomorphism R → S which sends 1 to 1. Then the abelian variety A1 over κ representing S ⊗R A has a natural left S-linear action, and the natural map γ : A → A1 induced by the ring homomorphism R → S is R-linear. Moreover the homomorphism γ : A → A1 has the following universal property: For every Slinear abelian variety (B, S −→ End(B)) over κ and every R-linear homomorphism γ 0 : A → B, there exists a unique S-linear homomorphism η : A1 → B over κ such that γ 0 = η ◦ γ. (c) Suppose that we have a ring homomorphism R → S as in (b) which is injective and the quotient S/R is a finite abelian group. Then γ : A → A1 is an isogeny. (ii) Similarly, let κ ⊃ Fp be a field, not necessarily finite. Let X be a p-divisible group over κ, let R be a Zp -subalgebra of End(X) and let S be a finitely generated right R-module. (d) The fppf sheaf S⊗R X over Spec(κ) is representable by an R-linear p-divisible group X1 over κ (e) Suppose in addition that S has a ring structure and the R-module structure on S is induced by a ring homomorphism R → S which sends 1 to 1. Then the p-divisible group X1 over k has a natural left S-action, and the natural map γ : X → X1 induced by the ring homomorphism R → S is R-linear. Moreover for every S-linear p-divisible group Y over κ and every R-linear homomorphism γ 0 : X → Y , there exists a unique S-linear homomorphism η : X1 → Y over κ such that γ 0 = η ◦ γ. (f) Assume that we have a ring homomorphism R → S as in (e) which is injective and the quotient S/R is Zp -module of finite length, then γ : X → X1 is an isogeny. Proof. We give only the proofs of (d), (e) and (f). The statement (a) for abelian varieties is proved in 1.6.4. The statement (b) is immediate from (a), and the proof of (c) is similar to that of (f). The statement (e) follows immediately from (d). Suppose that S/R is an Zp -module of finite length. Then there exists a positive integer N which is a power of p such that the map “multiplication by N ” on S factors as the composition of an R-linear map r : S → R and the inclusion of (the image of) R in S. The map r induces a homomorphism δ : A1 = S⊗RA −→ A such that γ ◦ δ = N · IdA1 and δ ◦ γ = N · IdA . So the statement (f) also follows from (d). It remains to prove the existence statement (d). h

Pick an exact sequence Rn −→ Rm −→ S of R-modules, so we have an exact sequence h⊗1

X n = Rn ⊗R X −→ X m = Rm ⊗R X −→ S ⊗R X := S −→ 0

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of fppf sheaves on Spec(κ). Clearly the fppf sheaf S is of p-power torsion and p-divisible. According to the definition of p-divisible groups it suffices to show that the kernel S [p] of [p] : S → S is representable by a finite group scheme over κ. Let K and F be the kernel and the image of h ⊗ 1 : X n → X m respectively. We have short exact sequences 0 −→ K −→ X n −→ F −→ 0

and

0 −→ F −→ X m −→ S −→ 0 .

The sheaf F is p-divisible because it is a quotient of the p-divisible sheaf X m . So we have a short exact sequence 0 −→ F [p] −→ X m [p] −→ S [p] −→ 0 . To show that S [p] is representable by a finite group scheme over κ, it suffices to show that F [p] is representable by a finite group scheme over κ. From the short exact sequence 0 −→ K −→ X n −→ F −→ 0 we get a 4-term exact sequence 0 −→ K [p] −→ X n [p] −→ F [p] −→ coker([p] : K → K ) −→ 0 . Clearly K [pj ] is representable by a finite group scheme over κ for each j, because it is the kernel of the homomorphism X n [pj ] → X m [pj ] induced by h ⊗ 1 : X n → X m . So it suffices to show that the fppf sheaf coker([p] : K → K ) on Spec(κ) is representable by a finite group scheme over κ. Because the fppf sheaf coker([p] : K → K ) is the direct limit of the cokernels Cj := coker(uj : K [pj+1 ]−→K [pj ]) induced by “multiplication by p”, where the transition maps ιj : Cj → Cj+1 are induced by the inclusions K [pj ] ,→ K [pj+1 ], it suffices so show that the direct system (C)j≥1 of commutative finite group schemes over κ is essentially constant. Consider the commutative diagram 0

/ K [pj+1 ] uj

0



/ K [pj ]

/ K [pj+2 ] uj+1



/ K [pj+1 ]

/ K [pj+2 ]/K [pj+1 ]   

/ X[pj+2 ]/X[pj+1 ]

vj

/ K [pj+1 ]/K [pj ]  

' wj

 / X[pj+1 ]/X[pj ]

where the horizontal arrows are induced by inclusions and the vertical arrows are induced by “multiplication by p”. Clearly Ker(vj ) = (0) for all j ≥ 1 because wj is an isomorphism, so Ker (ιj : Cj −→ Cj+1 ) = (0) for all j ≥ 1. The trivial bound rk(Cj ) ≤ rk(K [pj ]) ≤ rk(X n [p]) = pn·height(X) of rank of the finite group schemes Cj over κ implies that there exists an j0 ∈ N such that ∼ Cj coker([p] : K → K ) = for all j ≥ j0 . Remark. Suppose that κ is a perfect field. The existence of the cokernel of h ⊗ 1 : X n −→ X m in (d) is an easy consequence of classical Dieudonn´e theory. Moreover under the assumptions in (f), the statement (f) can also be proved by the familiar “saturation” procedure. Similarly the statement (c), using in addition the `-adic Tate modules for primes ` dividing card(S/R) and invertible in κ. 4.2.11. Remark. (i) In 4.2.10 (a), if we assume that A is an abelian scheme over a base scheme T and S is finitely generated projective right R-module, then the fppf sheaf S ⊗R A over T is representable by an S-linear abelian scheme over T . Similarly in 4.2.10 (ii), if we assume that X is a p-divisible group over a scheme T and S is a finitely generated projective

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R-module, then the fppf sheaf S⊗R X over T is representable by an S-linear p-divisible group over T . The proof is obvious and omitted. (ii) When the base scheme is not a field, we will use Serre’s tensor product construction only in the above situation where S is a finitely generated projective R-module. 4.2.12. Let E 0 /Qp be a subextension of E/Qp . In 4.2.14 we give a necessary and sufficient condition for an OE -linear CM p-divisible group over k to come from an OE 0 -linear CM pdivisible group over k via the Serre tensor product construction. We extend the general notation scheme to the local field E 0 . For instance • Rk (OE 0 ) is the Grothendieck group attached to the abelian category of all (OE 0⊗Zpk)modules of finite type, • κE 0 is the residue field of OE 0 , • f 0 = fE 0 = [κE 0 : Fp ], e0 = e(E 0 /Qp ), • f = f 0 · fE/E 0 , e = e0 · e(E/E 0 ), where e(E/E 0 ) is the ramification index for E/E 0 and fE/E 0 = [κE : κE 0 ] is the degree of the residue fields extension for E/E 0 , ' ' • I 0 = HomZp -alg (OE 0 ur , W (k)) −→ HomZp -alg (OE 0 ur , k) −→ Homring (κE 0 , k), We have a natural surjection ρE/E 0 : I  I 0 which is equivariant with respect to the natural surjective homomorphism Gal(κE /Fp )  Gal(κE 0 /Fp ) , the action of Gal(κE /Fp ) on I and the action of Gal(κE 0 /Fp ) on I 0 . The tensor product functor M 7→ OE ⊗OE0 M from the abelian category Mod(OE 0 ⊗Zp k) of all finite (OE 0 ⊗Zp k)-modules to the abelian category Mod(OE ⊗Zp k) of all finite (OE ⊗Zp k)modules induces a homomorphism τE/E 0 : Rk (OE 0 ) −→ Rk (OE ) between the Grothendieck groups. 4.2.13. Lemma. Notation as above. (i) For every element δ 0 ∈ Rk (OE 0 ), we have i (τE/E 0 (δ)) = eE/E 0 · ρ +

E/E 0

(i) (δ)

∀i ∈ I . +

An element δ 0 ∈ Rk (OE 0 ) is in Rk (OE 0 ) if and only if τE/E 0 (δ) ∈ Rk (OE ); moreover δ 0 is a Lie type in Rk (OE 0 ) if and only if τE/E 0 (δ) is a Lie type in Rk (OE ). (ii) The homomorphism τE/E 0 : Rk (OE 0 ) −→ Rk (OE ) between Grothendieck groups induced by tensoring with OE over OE 0 is injective. (iii) The image τE/E 0 (Rk (OE 0 )) of τE/E 0 consists of all elements δ ∈ Rk (OE ) such that – i (δ) ≡ 0 (mod eE/E 0 ) for all i ∈ I, and – i1 (δ) = i2 (δ) for all i1 = i2 ∈ I such that ρE/E 0 (i1 ) = ρE/E 0 (i2 ). Proof. A proof of the displayed formula in statement (i) is elementary and left as an exercise; the rest of statement (i) follows from the formula. The statements (ii) and (iii) are consequences of (i). 4.2.14. Proposition. An OE -linear CM p-divisible group Y over k is OE -linearly isomorphic to OE ⊗OE0 X for an OE 0 -linear CM p-divisible group X over k if and only if its Lie type [Lie(Y )] ∈ Rk (OE ) belongs to τE/E 0 (Rk (OE )). If so, then the OE 0 -linear p-divisible group X is uniquely determined by Y up to an OE 0 -linear isomorphism. Proof. It is clear that the Lie algebra of OE⊗OE0X is naturally isomorphic to OE⊗OE0 Lie(X), and the “only if” part of 4.2.14 follows. Conversely, suppose that [Lie(Y )] ∈ Rk (OE ) belongs

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to τE/E 0 (Rk (OE )). Then [Lie(Y )] = τE/E 0 (δ 0 ) for a unique Lie type δ 0 ∈ Rk (OE 0 ) by 4.2.13. By 4.2.7 there exists an OE 0 -linear CM p-divisible group X over k with Lie type δ 0 . Then Y is OE -linearly isomorphic to OE ⊗OE0 X by 4.2.13 and 4.2.6. The last sentence in the Proposition is a consequence of 4.2.13 and 4.2.6. 4.2.15. Remark. The statement of 4.2.14 will not be true if the algebraically closed field k is replaced by a finite field κ ⊃ Fp . Start with an OE 0 -linear CM p-divisible group Z over a × finite field κ. Then every homomorphism ρ : Gal(F/κ) −→ OE gives a κ-form of OE ⊗OE0 Z, × but only those attached to a homomorphism ρ with image in OE 0 are of the form OE ⊗OE0 X for some OE 0 -linear p-divisible group X over κ.

4.3. Self-duality and CM lifting The set-up of this section is semi-local, intermediate between the local situation in §4.2 and the global situation in §4.4. As before k ⊃ Fp is an algebraically closed field, and κ denotes a finite subfield of k unless otherwise specified. 4.3.1. Notation in the semi-local case. • E0 is a finite extension field of Qp , E is a commutative semisimple E0 -algebra with [E : E0 ] = 2. So either E is a quadratic extension field of E0 or is isomorphic to E0 × E0 . We refer to the two possibilities as the non-split and split case respectively. (In applications in §4.4, E0 will be the v-adic completion of a totally real field at a place v above p, and E will be the v-adic completion of a quadratic totally imaginary extension of that totally real field.) • Let OE = OE0 × OE0 in the split case. In the non-split case OE denotes the ring of elements in E integral over Zp as usual. • ι is a non-trivial automorphism of the E0 -algebra E of order 2 which fixes all elements of E0 . In the non-split case ι is nothing but the natural E0 -isomorphism between the two factors of E. • κE0 is the residue field of E0 , f0 = fE0 = [κE0 : Fp ]. Define κE := κ0 × κ0 in the split case. Let f = fE := [κE : Fp ]. • e0 := e(E0 /Qp ) is the absolute ramification index of E0 over Qp . Define e = e0 in the split case, and e = e(E/Qp ) in the non-split case. The usual identity [E : Qp ] = ef holds. • In the split case, let E ur := E0ur × E0ur and OE ur := OE ur × OE ur . Note that E ur and 0 0 OE ur have been defined in the non-split case. 0 • Let '

'

I = IE = HomZp -alg (OE ur , W (k)) −→ HomZp -alg (OE ur , k) −→ Homring (κE , k) '

as before. In the non-split case, we have a natural isomorphism IE −→ IE0 × IE0 , equivariant with respect to the natural action of Gal(κE0 /Fp ) × Gal(κE0 /Fp ), and IE is a torsor for Gal(κE0 /Fp ) × Gal(κE0 /Fp ). The involution ι on E induces a natural involution on IE , interchanging the two factors. • Let Rk (OE ) (resp. Rκ (OE )) be the Grothendieck group attached to the abelian category Mod(OE ⊗Zp k) of all finite (OE ⊗Zp k)-modules (resp. the abelian category Mod(OE ⊗Zp κ) of finite (OE ⊗Zp κ)-modules). The subset of effective classes (i.e. classes of objects of Mod(OE ⊗Zp k) or Mod(OE ⊗Zp κ)) in them will be denoted + R+ k (OE ) and Rκ (OE ) respectively. In the non-split case, with E = E0 × E0 , we have Rk (OE ) := Rk (OE0 ) × Rk (OE0 ) ,

Rκ (OE ) := Rκ (OE0 ) × Rκ (OE0 ) .

The involution ι on E induces an involution on the Grothendieck groups Rk (OE ) and Rκ (OE ).

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117

• Similar to the situation in 4.2.1, we have a product decomposition  Y OE ⊗Zp W (k) ∼ OE ⊗O(Eur ,i) W (k) . = i∈I




Let R(OE , i) := R OE ⊗OEur ,i k for each i ∈ I, and let '

i : R(OE , i) −→ Z be the natural isomorphism, under which the subset of effective classes maps to N. •  = E : Rk (OE ) → Z is the dimension function defined by X (δ) := i (x) ∀δ ∈ Rk (OE ). i∈I

The integer (δ) is called the dimension of δ. In the split case, we have E = E0 × E0 and Rk (OE ) = Rk (OE0 ) × Rk (OE0 ), and we have a dimension function E0 : Rk (OE0 ) → Z such that E (δ1 , δ2 ) = E0 (δ1 ) + E0 (δ2 ) for all (δ1 , δ2 ) ∈ Rk (OE0 )×Rk (OE0 ) = Rk (OE ). • The above product decomposition induces an isomorphism Rk (OE )

∼ =

/

L

i∈I

R(OE , i)

⊕i i

/

L

i∈I

Z

and the subset of effective classes maps to ⊕i∈I N. • As in 4.2.4, the natural map jκ : Rκ (OE ) −→ Rk (OE ) is an injection and identifies Rκ (OE ) with the subset of all elements of Rk (OE ) fixed by the natural action of Gal(F/κ). These elements are said to be κ-rational. When E = E0 ××E0 , an element (δ1 , δ2 ) ∈ Rk (OE0 ) × Rk (OE0 ) = Rk (OE ) is κ-rational if and only if the δ1 and δ2 are κ-rational in Rk (OE0 ). • An element δ of Rk (OE ) is said to be a Lie type if 0 ≤ i (δ) ≤ e for all i ∈ I. Clearly every Lie type is effective. 4.3.2. Definition. (a) Let R be either a noetherian ring such that p is nilpotent in R, or a complete noetherian ring such that p is topologically nilpotent in R. An OE linear (res. E-linear) p-divisible group over a ring R is a pair (X, α : OE → EndS (X)) (resp. (X, α : E → EndR (X)⊗Zp Qp ) where X is a p-divisible group over R and α is an injective Zp -linear ring homomorphism. Such a p-divisible group is said to be an E-linear CM p-divisible group if the height of X is equal to [E : Qp ]. (b) The Lie type of an OE -linear CM p-divisible group (X, α) over k (resp. κ is the class in Rk (OE ) (resp. Rκ (OE )) of the k-linear action (resp. the κ-linear action) of OE on Lie(X). 4.3.3. Definition. Let δ be a Lie type of Rk (OE ). Define the slopes of δ and their multiplicities as follows. (a) In the non-split case, (δ)/[E : Qp ] is the only slope of δ, with multiplicity [E : Qp ]. (b) In the split case, we have E = E0 × E0 and Rk (OE ) = Rk (OE0 )×Rk (OE0 ). Write δ = (δ1 , δ2 ) with (δ1 , δ2 ) ∈ Rk (OE0 ) × Rk (OE0 ) = Rk (OE ). Then (δ1 )/[E0 : Qp ] and (δ2 )/[E0 : Qp ] are the slopes of δ. If E0 (δ1 )/[E0 : Qp ] 6= (δ2 )/[E0 : Qp ], the multiplicity of each slope is [E0 : Qp ]. If the two slopes are equal, then the multiplicity of the only slope is [E : Qp ] = 2[E0 : Qp ].

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(c) An element δ ∈ Rk (OE ) is said to be supersingular if it is a Lie type with 1/2 as its only slope. (d) Two Lie types in Rk (OE ) are said to be isogenous if they have the same slopes and the same multiplicities. Notice that the requirement on having the same multiplicities is redundant by (b). 4.3.4. Definition.

(a) A Lie type δ ∈ Rk (OE ) is self-dual if δ + ι(δ) = [OE ⊗Zp k]

in Rk (OE ). In the non-split case this conditions means that i (δ) + i◦¯ι (δ) = e

∀i ∈ I ,

where ¯ι is the involution on κE induced by ι, while in the split case it means that δ = (δ1 , δ2 ) where δ1 , δ2 ∈ Rk (OE0 ) satisfy i (δ1 ) + i (δ2 ) = e0 = e(E0 /Qp )

∀i ∈ IE0 .

Note that if E is a field ramified over E0 , then e = e(E/Qp ) is even, and the selfduality condition for v means that w,i (δw ) = e/2 for all i ∈ Iw . (b) A subset Φ ⊂ HomQp -alg (E, K0 ) such that Hom(E, K0 ) is the disjoint union of Φ and ι(Φ) will be said to be self-dual. (c) The reduction map ξ : 2HomQp -alg (E,K0 ) −→ Rk (OE ) from the set of all subsets of HomQp -alg (E, K0 ) to the set of all Lie types is characterized by the requirement that i (ξ(Φ)) = #{σ ∈ Φ | σ = i}

∀i ∈ I ,

exactly as in 4.2.2 (e). (d) Let Φ be a subset of HomQp -alg (E, K0 ). In the split case, we have E = E0 ×E0 and a HomQp -alg (E0 , K0 ) . HomQp -alg (E, K0 ) = HomQp -alg (E0 , K0 ) So Φ is the disjoint union of its components Φ1 , Φ2 ⊂ HomQp -alg (E0 , K0 ). We extend the notion of reflex field in the obvious way, define the reflex field of Φ in K0 to be the subfield of K0 generated by the reflex fields of Φ1 and Φ2 . 4.3.5. Remark. (i) If E is a field, and δ is a self-dual Lie type, then δ is supersingular. (ii) If E is a ramified quadratic extension field of E0 and δ is a self-dual Lie type, then e(E/Qp ) 2 such a Lie type is called uniform in 4.3.13. i (δ) = e(E0 /Qp ) =

∀i ∈ I ;

4.3.6. Definition. Let K be a finite extension field of Qp or a finite extension field of K0 . Denote by K an algebraic closure of K0 containing K0 . (a) An E-linear (resp. OE -linear) CM p-divisible group over OK is a pair (Y, α : E → End0 (Y))

( resp. (Y, α : E → End(Y)) )

where Y is a p-divisible group over OK of height [E : Qp ] and αQ is a Zp -linear injective homomorphism of rings. (b) Let (Y, α) be an E-linear or OE -linear CM p-divisible group over OK . As in 4.2.2 (d), the p-adic CM type of (Y, α) is the subset Φ = Φ(Y, α) of HomQp -alg (E, K) corresponding to the K-linear action of E on Lie(Y)⊗OKK; we have Card(Φ(Y, α)) = dimOK (Y). If the p-adic CM type Φ(Y, α) is self-dual in the sense of 4.3.4, then we say that (Y, α) is self-dual up to isogeny.

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4.3.7. Lemma. Let K be a finite extension field of K0 . (i) Let (Y, α : OE → End(Y)) be an OE -linear p-divisible group over OK whose height is equal to [E : Qp ]. Let Φ be the p-adic CM type of (Y, α). Then the Lie type of the closed fiber (Y0 , α0 ) of (Y, α) is equal to the reduction ξ(Φ) of Φ. Moreover K contains the reflex field of (E, Φ), and (Y, OE → End(Y)) is determined by Φ up to isomorphism. (ii) Let Φ be a subset of HomQp -alg (E, K). If K contains the reflex field of Φ, then there exists an OE -linear p-divisible group Y over OK of height [E : Qp ] whose p-adic CM type is equal to Φ. Proof. The first part of (i) is a reformulation of 4.2.3 in the present set-up. The last sentence of (i) and the statement (ii) are well-known; see e.g. 3.2.4. 4.3.8. Lemma. Let K be a finite extension field of K0 . If Ψ ⊂ HomQp -alg (E, K) is a selfdual subset, then its reduction ξ(Ψ) is a self-dual element in Rk (OE ). Conversely, for every self-dual element δ ∈ Rk (OE ), there exists a self-dual subset Ψ ⊂ HomQp -alg (E, K) such that ξ(Ψ) = δ. Proof. The first statement in 4.3.8 is clear. Let δ ∈ Rk (OE ) be a self-dual element. We will prove 4.3.8 in the case when E is a field; the proof for the split case is similar but easier. For each i ∈ IE = Homring (κE , k), let  Si := τ ∈ HomZp -alg (OE , K) | τ induces j on κE , a subset of HomZp -alg (OE , K). If κE = κE0 , then i (δ) = i (ι · δ) = 1/2 for each i ∈ I. Pick a subset Ψi of representatives of Si /{1, ι} in Si for each i ∈ IE0 . Then the union Ψ of all the Ψj ’s has the required properties. Suppose now that E/E0 is unramified. Pick a subset J of representatives for I/{1, ι} in I. For each j ∈ J, pick a subset Ψj ⊂ Sj with #Ψj = j (δ). Then the union Ψ of all the Ψj ’s has the required properties. 4.3.9. Proposition. Let (Y0 , α0 ) be an OE -linear CM p-divisible group over k whose Lie type is self-dual. (i) There exists a self-dual subset Φ ⊂ HomQp -alg (E, K0 ) whose reduction ξ(Φ) is a equal to the Lie type of (Y0 , α0 ). (ii) For any self-dual subset Φ ⊂ HomQp -alg (E, K0 ) such that ξ(Φ) = [Lie(Y0 , α0 )] and any finite subextension field K/K0 in K0 /K0 which contains the reflex field of (E, Φ) in K0 , there exists an OE -linear p-divisible group (Y, α) over OK with Φ as its p-adic CM type such that the closed fiber of (Y, α) is isomorphic to (Y0 , α0 ). In particular there exists a finite extension field K of K0 and an OE -linear p-divisible group (Y, α) over OK whose closed fiber is isomorphic to (Y0 , α0 ). Proof. The statement (i) follows from 4.3.8. Let K be a finite extension field of K0 containing the reflex field of (E, Φ) as in (ii). By 4.3.7 (ii), there exists an OE -linear p-divisible group (Y, α) over OK of height [E : Qp ] whose p-adic CM type is equal to Φ. By Prop. 4.2.6 (iv), the closed fiber of (Y, α) is OE -linearly isomorphic to (Y0 , α0 ). 4.3.10. Remark. (i) Lemmas 4.3.7, 4.3.8 and Prop. 4.3.9 are special cases of results in [83]. (ii) In the case when E = E0 × E0 , a subset Φ ⊂ HomQp -alg (E, K0 ) = HomQp -alg (E0 , K0 )

a

HomQp -alg (E0 , K0 )

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is a disjoint union of a two subsets Φ1 , Φ2 in HomQp -alg (E0 , K0 ) so that (E, Φ) = (E0 , Φ1 ) × (E0 , Φ2 ). Then the reflex field of (E, Φ) is by definition the smallest subfield in K0 containing the two reflex fields in K0 attached to (E0 , Φ1 ) and (E0 , Φ2 ). 4.3.11. Lemma. Suppose that E = E0 × E0 . Then for any finite field κ ⊂ k and any κrational Lie type δ1 ∈ Rk (OE0 ), there exists a unique κ-rational Lie type δ2 ∈ Rk (OE0 ) such that the element δ = (δ1 , δ2 ) ∈ Rk (OE0 ) × Rk (OE0 ) = Rk (OE ) is a self-dual κ-rational Lie type in Rk (OE ). The proof of 4.3.11 is easy, therefore omitted. Lemma 4.3.11 will be used when we are given an OE -linear CM p-divisible group Z0 over a finite field κ. 4.3.12. Corollary. Suppose that E = E0 × E0 and (Z0 , γ0 ) is an OE -linear CM p-divisible group over κ. Then there exists a CM p-divisible group OE -linearly which is isogenous to (Z0 , γ0 ) over κ and admits a OE -linear CM lift to characteristic 0 with a self-dual p-adic CM type. Proof. By 4.3.11 and Prop. 4.2.8, Z0 is OE -linearly isogenous to an OE -linear p-divisible group Y0 over κ whose Lie type is self-dual. Apply 4.3.8 and Prop. 4.3.9, we see that Y0 can be OE -linearly lifted to a p-divisible group Z in characteristic 0 with self-dual p-adic CM type. In short, self-dual CM lifting up to isogeny for a CM p-divisible group over κ is easy to construct in the split case, and can be done OE -linearly. 4.3.13. Definition. Suppose that E is a field, i.e. we are in the non-split case. (a) A Lie type in x ∈ Rk (OE ) is said to be uniform if and only if all the integers i (x) for i ∈ I are equal. Supersingular uniform Lie types exist if only if e = e(E/Qp ) is even. If so then i (x) = e/2 for all i ∈ I and the unique supersingular uniform Lie type δunif is self-dual and rational over Fp . (b) Assume that f is even. A Lie type δ ∈ Rk (OE ) is striped if δ is Fp2 -rational and i (δ) ∈ {0, e} for all i ∈ I. 4.3.14. Remark. (i) Every striped Lie type is supersingular. There are exactly two striped Lie types in Rk (OE ); each is of the form δstp,J for a unique Gal(κE /Fp2 )-orbit J in I, characterized by the following property:  e if i ∈ J, i (δstp,J ) = 0 if i ∈ / J. (ii) If f ≡ 2 (mod 4), then both striped Lie types in Rk (OE ) are self-dual. If f ≡ 0 (mod 4), then no striped Lie type in Rk (OE ) is self-dual. (iii) The Lie type [Lie(Y0 )] of the p-divisible group Y0 in B.1.4, represented by the class of the OE -linear k-module N/V N , is striped. We will need the notion of striped Lie types only when e is odd and f ≡ 0 (mod 4). (iv) We will only use supersingular uniform Lie types; uniform Lie types with other slopes will not be used. 4.3.15. Lemma. Suppose that E is a field such that fE is even. Let E2 be the subfield of E 0 such that E 0 /Qp is unramified and of degree two. A Lie type δ ∈ Rk (OE ) is striped if and only if there exist a Lie type δ 0 ∈ Rk (OE2 ) such that Qp , δ 0 is striped and δ = τE/E 0 (δ 0 ) as in 4.2.12.

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Proof. The “if” part is easy to see from 4.3.13 and 4.2.13. Conversely, suppose that δ ∈ Rk (OE ) is a striped Lie type. By 4.3.14 (i), there exists a Gal(κ/Fp2 )-orbit J ⊂ I such that δ is equal to the Lie type δJ defined in 4.3.14 (i). Then there is an embedding i2 of OE2 to W (k) such that the restriction to OE 0 of every element of I is equal to i2 . Let δ{i2 } be the striped Lie type in Rk (OE2 ) attached to {i2 }. We see from 4.2.13 (i) that τE/E2 (δ{i2 } ) = δ. 4.3.16. Corollary. For every striped Lie type δ ∈ Rk (OE ) and every subfield E 0 of E containing the quadratic unramified subextension E2 /Qp of E/Qp , there exists a unique striped Lie type δ 0 ∈ Rk (OE 0 ) such that δ = τE/E 0 (δ 0 ). 4.3.17. Given a finite field κ ⊃ Fp and a κ-rational supersingular Lie type x ∈ R+ k (OE ), we want to know whether there exist a self-dual κ-rational Lie type δ which is isogenous to x. In lemmas 4.3.18, 4.3.19 and 4.3.21 below we will see that such a κ-rational self-dual supersingular Lie type δ can be found unless (1) (2) (3) (4)

E is a field, e is odd, f is divisible by 4, and [κE : κE ∩ κ] is even.

Following the definition in 4.1.5, we say that the pair (E/E0 , κ) is bad if the conditions (1)–(4) above hold; otherwise we say that the pair is good. In applications, x is the Lie type of a supersingular OE -linear CM p-divisible group Z0 over κ. If (E/E0 , κ) is good, then by 4.3.9 Z0 can be OE -linearly lifted to a p-divisible group to characteristic 0 with self-dual p-adic CM type and we are done. On the other hand, suppose that (E/E0 , κ) is bad, then κ-rational supersingular self-dual Lie types do not exist. In such situations we modify Z0 by an isogeny: Lemma 4.3.20 shows that κ ⊃ Fp2 , so by 4.2.8 Z0 is OE -linearly isogenous to an OE -linear p-divisible group Y0 whose Lie type is striped. 4.3.18. Lemma. Suppose that E is a field and the ramification index e of E is even. Let δunif be the supersingular uniform element in R+ k (OE ) such that i (δunif ) = e/2 for all i ∈ I. Then there exists a self-dual subset Φ0 ⊂ Hom(E, K0 ) such that ξ(Φ0 ) = δunif . Proof. As already remarked in 4.3.14 (iv), δunif is self-dual, so 4.3.18 follows from 4.3.8. Below we give an alternative detailed proof. Suppose first that E is ramified over E0 . For each embedding σ : E0 ,→ K0 , choose an extension σ ˜ : E ,→ K0 of σ. Let Φ0 be the set of all σ ˜ ’s. Then ξ(Φ0 ) = δunif . Assume now that E is inert over E0 . Then E = E0 ⊗E0ur E ur . For each σ : E0ur ,→ K0 , let τ1 , . . . , τe be the embeddings of E0 into K0 extending σ, and let σ ˜, σ ˜ 0 be the embeddings of ur E into K0 extending σ. For j = 1, . . . , e/2, let τ (σ)j : E ,→ K0 be the embedding given by ˜ 0 and σ ˜ and τ σj . For j = (e/2) + 1, , . . . , e, let τ (σ)j : E ,→ K0 be the embedding given by σ τ σj . Let Φ0 be the subset of Φ0 ⊂ Hom(E, K0 ) consisting of all τ (σ)j ’s, where σ runs through Hom(E0ur , K0 ) and j = 1, . . . , e. Then Φ0 is self-dual and ξ(Φ0 ) = δunif . 4.3.19. Lemma. Assume that E is a field, e is odd and f = 2f0 with f0 odd, so that E is unramified over E0 and ι induces a non-trivial involution on the residue field κE . Let κ be a finite field. Suppose that there exists a κ-rational supersingular Lie type in Rk (OE ). Then the following statements hold. (i) # Gal(κE /(κE ∩ κ)) is odd. (ii) There exists a subset Ψ1 ⊂ Hom(E ur , K0 ) such that – Hom(E ur , K0 ) is the disjoint union of Ψ1 and ι(Ψ1 ), and

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– Ψ1 is stable under the action of the subgroup of Gal(E ur /Qp ) which corresponds to Gal(κE /(κE ∩κ)) under the natural isomorphism Gal(E ur /Qp ) ∼ = Gal(κE /Fp ). (iii) The subset Φ1 of Hom(E, K0 ) induced by Ψ1 is self-dual and ξ(Φ1 ) is a κ-rational self-dual supersingular element in Rk (OE ). Proof. The statements (ii) and (iii) are of combinatorial nature and follow easily from (i) because (i) implies that # (Gal(κE /(κE ∩ κ))\I) is even. Suppose that # Gal(κE /(κE ∩ κ)) were even. Then each orbit of Gal(κE /(κ PE ∩ κ)) on I has even cardinality. Let x3 be a supersingular element of Rκ (OE ) . In the sum i∈I P i (x3 ) , those terms with i inPthe same orbit are equal, so the sum over each orbit is even and i∈I i (x3 ) is even. But  (x ) = ef because x is supersingular and ef is odd by assumption. 0 3 0 i∈I i 3 This is a contradiction. We have proved (i). 4.3.20. Lemma. Assume that E is a field, e is odd and f ≡ 0 (mod 4). Let J be an orbit for the action Gal(κE /Fp2 ) of on I, and let δstp,J be the element of Rk (OE ) such that  e if i ∈ J, i (δstp,J ) = 0 if i ∈ / J. Suppose that κ is a finite field and there exist a supersingular Lie type in Rκ (OE ). Then κ ⊃ Fp2 and δstp is a κ-rational supersingular element in Rk (OE ). Proof. Let x3 be a κ-rational element of Rk (OE ). Write f = 2a b where b is an odd integer, a ≥ 2. Fix an isomorphism Gal(κE /Fp ) ∼ = Z/2a bZ. The desired condition means that the Galois group Gal(κE /(κE ∩ κ)) is contained in the subgroup 2Z/2a bZ of Z/2a bZ. If not, then Gal(κE /(κE ∩ κ)) contains the 2-Sylow subgroup bZ/2a bZ of Z/2a bZ, The fact that x3 is fixed by the action of Gal(κE /(κE ∩ κ)) implies that P i∈I i (x3 ) ≡ 0 (mod # Gal(κE /(κE ∩ κ))), ≡ 0 (mod 2a ) , P which is a contradiction because i∈I i (x3 ) = f /2 = 2a−1 b is not divisible by 2a . 4.3.21. Lemma. Assume that E is a field, e is odd and f ≡ 0 (mod 4). Let κ be a finite field such that # Gal(κE /(κE ∩ κ)) is odd. Then there exists a self-dual κ-rational Lie type δ in Rk (OE ). Consequently there exists a self-dual subset Φ ⊂ Hom(E, K0 ) which is induced from a self-dual subset Ψ ⊂ Hom(E ur , K0 ) such that ξ(Φ) = δ Proof. Exercise. 4.3.22. Lemma. Suppose that E is a field. Let κ be a finite field and δ be a κ-rational Lie type in Rk (OE ). Then there exists a self-dual κ-rational Lie type in Rk (OE ) isogenous to δ if and only if at least one of the following two conditions is violated. • δ is supersingular, • (E/E0 , κ) is bad. Proof. This is a consequence of 4.3.11, 4.3.18, 4.3.19. 4.3.23. Proposition. Suppose that E is a field and f (E/Qp) ≡ 0 (mod 4). Let E 0 /Qp be the unramified subextension of E/Qp with [E 0 : Qp ] = 4. Let (Y0 , β0 ) be an OE -linear CM p-divisible group over an algebraically closed field k ⊃ Fp . Assume that the Lie type of Y0 is striped. (i) There exists an OE -linear isomorphism Y0 ∼ = OE ⊗O Y 0 E ur

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between OE -linear p-divisible groups of CM type, where (Y00 , β00 ) is an OE ur -linear CM p-divisible group over k whose Lie type is striped. (ii) There exists an OE 0 -linear CM p-divisible group (Ytoy , βtoy ) over F and an OE -linear isomorphism ' (Y0 , β0 ) −→ OE ⊗OE0 (Ytoy , βtoy ). Proof. Prop. 4.3.23 is a consequence of 4.2.14 and 4.3.16. In the next paragraph we give a direct proof. The assumption implies that [κE : Fp ] = [κE ur : Fp ] is even. Let J be the orbit of Gal(κE /Fp2 ) in I = IE such that [Lie(Y0 )] is equal to the striped Lie type δstp,J attached to J. Notice that IE ur is canonically identified with IE , so J can be also regarded as a Gal(κE ur /Fp2 )-orbit in IE ur . Let Y00 be an OE ur -linear CM p-divisible group over k whose Lie type is equal to the striped Lie type in Rk (OE ur ) attached to the subset J ⊂ IE ur ; such an OE ur -linear p-divisible group Y00 exists by Prop. 4.2.7. Clearly the Lie type of OE ⊗OEur Y00 is the class of the OE -module OE ⊗OEur Lie(Y00 ) in Rk (OE ), which is easily seen to be equal to δstp,J . By Prop. 4.2.6 OE ⊗OEur Y00 is OE -linearly isomorphic to Y0 . We have proved (i). The proof of (ii) is similar: compare the Lie types of (Y0 , β0 ) and OE ⊗OE0 (Ytoy , βtoy ). 4.3.24. Remark. Suppose that (Z00 , γ00 ) is a an OE ur -linear p-divisible group over k, and (Z0 , γ0 ) = OE ⊗OEur (Z00 , γ 0 ). Then (Z00 , γ00 ) is CM if and only if (Z0 , γ0 ) is, and the Lie type of Z00 , γ00 ) is striped if and only if the Lie type of (Z0 , γ0 ) is. In the proof of 4.3.23 above, the OE ur -linear p-divisible group (Y00 , β00 ) is known to CM with a striped Lie type by construction, before the relation between (Y00 , β00 ) and (Y0 , β0 ) is proved. However these properties of (Y00 , β00 ) are also consequence of the relation (Y0 , β0 ) = OE ⊗OEur (Y00 , β00 ). 4.3.25. Remark. A slightly stronger conclusion holds under the assumption in 4.3.23, by the same proof. For every subfield E 0 of E containing the quadratic unramified subextension E2 /Qp inside E/Qp , there exists an OE 0 -linear CM p-divisible group Z0 over k with striped Lie type and an OE -linear isomorphism Y0 ∼ = OE ⊗O Z0 . E0

The following proposition summarizes many results in §4.2 and §4.3; it reduces the proof of CM lifting up to isogeny for abelian varieties over finite fields to the question of finding a CM lifting with self-dual p-adic CM type for an OE -linear p-divisible group of the form OE ⊗OEur Y0 , where Y0 is an OE ur -linear p-divisible group over F whose Lie type is striped. 4.3.26. Proposition. Notation as in 4.3.1. Let (Y1 , β1 ) be an OE -linear CM p-divisible group over a finite field κ ⊂ F. We assume that either Y1 has two distinct slopes µ1 , µ2 with µ1 + µ2 = 1, which happens only when E = E0 × E0 and both slopes have multiplicity [E0 : Qp ], or that 1/2 is the only slope of Y1 . (i) Suppose either that E = E0 × E0 , or that E is a field (which implies that Y1 is supersingular in the sense that 1/2 is its only slope) and (E/E0 , κ) is good. Then there exists – an OE -linear p-divisible group (Y0 , β0 : OE → End(Y1 )) over κ whose Lie type is self-dual, – an OE -linear isogeny (Y0 , β0 ) −→ (Y1 , β1 ) over κ, – a finite extension K of the fraction field K0 of W (F), and – an OE -linear p-divisible group (Y, β : OE → EndOK (Y)) over OK such that the p-adic CM type of (Y, β) is self-dual and the closed fiber of (Y, β) is OE -linearly isomorphic to (Y0 , β0 )×Spec(OK ) Spec(F).

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(ii) Suppose that (E/E0 , κ) is bad. Then there exists – an OE -linear CM p-divisible group (Y0 , β0 : OE → End(Y0 )) over κ whose Lie type is striped, – an OE -linear isogeny (Y0 , β0 ) −→ (Y1 , β1 ) over κ, – an OE ur -linear CM p-divisible group (Y00 , β00 ) over F whose Lie type is striped, and – an OE -linear isomorphism '

(Y0 , β0 )×Spec(κ) Spec(F) −→ OE ⊗OEur (Y00 , β00 ) over F. (iii) Assumption as in (ii). Let E 0 /Qp be the unramified subextension of E/Qp such that [E 0 : Qp ] = 4. Then there exists – an OE -linear CM p-divisible group (Y0 , β0 : OE → End(Y0 )) over κ whose Lie type is striped, – an OE -linear isogeny (Y0 , β0 ) −→ (Y1 , β1 ) over κ, – an OE 0 -linear CM p-divisible group (Ytoy , βtoy ) of height 4 over F whose Lie type is striped, and – an OE -linear isomorphism '

(Y0 , β0 )×Spec(κ) Spec(F) −→ OE ⊗OE0 (Ytoy , βtoy ) over F. Proof. Under the assumption of (i), by 4.3.22 there exists a κ-rational self-dual Lie type δ0 ∈ RF (OE ). By 4.2.8 there exists an OE -linear p-divisible group (Y0 , β0 ) over κ whose Lie type is δ1 and an OE -linear isogeny (Y0 , β0 ) −→ (Y1 , β1 ) over κ. By 4.3.8, there exists a self-dual subset Ψ ⊂ HomQp -alg (E, K) whose reduction ξ(Ψ1 ) is equal to δ0 . By 4.3.9, there exists a finite extension field K over K0 and an OE -linear p-divisible group (Y, β) over OK whose closed fiber is isomorphic to (Y0 , β0 )×Spec(OK ) Spec(F). We have proved (i). Assume now that E is a field, e(E/Qp ) is odd, f (E/Qp ) ≡ 0 (mod 4), and [κE : (κE ∩ k)] is even. We know that Y1 is supersingular because E is a field. By 4.3.20, the finite field κ contains Fp2 and both striped Lie types in Rk (OE ) are κ-rational. Let δ0 be one of the striped Lie types. By 4.2.8 (i), there exists an OE -linear CM p-divisible group (Y0 , β0 ) over κ with δ0 as its Lie type and an OE -linear isogeny (Y0 , β0 ) −→ (Y1 , β1 ) over κ. We have proved (ii). The proof of (iii) is similar, using 4.2.8 (ii) instead of 4.2.8 (i). Remark. As noted in 4.3.24, that (Y00 , β00 ) in (ii) is a CM p-divisible group with a striped Lie type follows from the rest of 4.3.26 (ii). Similarly for (iii). 4.3.27. Lemma. Let (Ytoy , βtoy ) be a two-dimensional W (Fp4 )-linear CM p-divisible group over F whose Lie type is striped. Denote by E 0 the fraction field of W (Fp4 ). (1) There exists – a W (Fp4 )-linear two dimensional W (Fp4 )-linear CM p-divisible group (Xtoy , γtoy ) over F, – a W (Fp4 )-linear isogeny ρtoy : Xtoy → Ytoy such that Ker(ρtoy ) is subgroup scheme of Xtoy [p] isomorphic to αp stable under the action of W (Fp4 ), and – a W (Fp4 )-linear two dimensional W (Fp4 )-linear CM p-divisible group (Xtoy , γ˜toy ) over W (F) whose closed fiber is (Xtoy , γtoy ), such that (i) Ker(ρtoy ) is the unique subgroup scheme of Xtoy of order p. (ii) The Lie type of (Xtoy , γtoy ) is self-dual, and the p-adic CM type of (Xtoy , γ˜toy ) is also self-dual.

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(2) Let Gtoy be a finite flat subgroup scheme of Xtoy [p] ×Spec(W (F)) Spec(OK ) of order p over OK for a finite extension
field K of fraction field K0 of W (F). Then the quotient Xtoy [p]×Spec(W (F)) Spec(OK ) /Gtoy with the induced action of E 0 up to isogeny is an E 0 -linear CM lifting of (Ytoy , βtoy ) with a self-dual p-adic CM type. Proof. The proof of 4.3.27 (1) is a straight forward translation of the proof of for the toy model case in 4.1.2 to the situation of CM p-divisible groups. The statement (2) follows from (1). Remark. Up to isomorphism there are exactly two W (Fp4 )-linear isogenies ρtoy satisfying the requirements in (1). 4.3.28. Proposition. Suppose that E is a field and f ≡ 0 (mod 4). Let (Y0 , β0 ) be an OE linear p-divisible group of CM type over F whose Lie type is striped. Then there exists a finite tamely ramified extension field K of the fraction field of W (F) and a self-dual E-linear CM p-divisible group (Y, β : E → End0 (Y)) whose closed fiber is isomorphic to the E-linear p-divisible group (Y0 , β0⊗Zp Qp ) attached to the OE -linear p-divisible group (Y0 , β0 ). Moreover (Y, β) can be chosen so that β(Zp + pOE ) ⊂ End(Y). Proof. By 4.3.26(ii), the given OE -linear p-divisible group (Y0 , β0 ) over F with a striped Lie type can be written as a Serre tensor product: (Y0 , β0 ) ∼ = OE ⊗W (F ) (Ytoy , βtoy ) p4

where (Ytoy , βtoy ) is is a two-dimensional W (Fp4 )-linear CM p-divisible group over F whose Lie type is striped. Apply 4.3.27, we obtain • a W (Fp4 )-linear two dimensional W (Fp4 )-linear CM p-divisible group (Xtoy , γ˜toy ) over W (F) whose p-adic CM type is self-dual, and • a W (Fp4 )-linear isogeny ρtoy : (Xtoy , γtoy ) → (Ytoy , βtoy ), where (Xtoy , γtoy ) is the closed fiber of (Xtoy , γ˜toy ) such that Ker(ρtoy ) is subgroup scheme of Xtoy [p] isomorphic to αp stable under the action of W (Fp4 ) and is the unique subgroup scheme of Xtoy [p] of order p. Let Gtoy be a finite flat subgroup scheme of X [p] ×Spec(W (F) Spec(OK ) of order p for some finite extension K of the fraction field K0 of W (F) as in 4.3.27. Clearly the closed fiber of Gtoy is equal to Ker(ρ0 ). Let (X0 , γ0 ) := OE ⊗W (F) (Xtoy , γtoy ) and let (X , γ) : OE ⊗W (F) (Xtoy , γ˜toy ), twoOE -linear CM p-divisible group over F and W (F) respectively. Then ρ0 := OE ⊗W (Fp4 ) ρtoy is an OE -linear isogeny from (X0 , γ0 ) to (Y0 , β0 ). Notice that the Lie type of (X0 , γ0 ) is self-dual and the p-adic CM type of (X , γ) is self-dual, because the respective statements hold for Xtoy and Xtoy . Choose and fix an isomorphism OE ∼ = W (Fp4 )⊕r with r = [E : Qp ]/4, we get compatible isomorphisms X ∼ = (Xtoy )m , X0 ∼ = (Xtoy )r and Ker(ρ0 ) ∼ = Ker(ρtoy )r . Consider the lifting Gtoy of Ker(ρtoy ) over OK . Then G := (Gtoy )r is a finite flat subgroup scheme of X ×Spec(W (F)) Spec(OK ) which lifts Ker(ρ0 ). So the quotient Y := X ×Spec(W (F)) Spec(OK )/G is a p-divisible group over OK with an induced E-linear action β : E → End0 (X ) up to isogeny, whose closed fiber is isomorphic to (Y0 , β0 ), and the p-adic CM type of (Y, β) is selfdual because the p-adic CM type of (X , γ) is. The last assertion that End(Y) ⊇ β(Zp + pOE ) is clear because G is killed by p. 4.3.29. Remark. In appendices B.1 and B.2, we give two versions of the same proof of the special case of 4.3.28 when E is unramified over Qp . The version in appendix B.1 uses

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Raynaud’s theory [62], see B.1.16 and B.1.17, while the version in appendix B.2 uses p-adic Hodge theory, see B.2.16. In an earlier draft of this chapter 4.3.28 was proved by reducing to B.1.17 or B.2.16 using 4.3.26 (i). Our original proof of the existence of CM lifting up to isogeny followed this route, using p-adic Hodge theory to produce self-dual CM liftings in the case of bad primes.

4.4. Complex conjugation and CM lifting 4.4.1. Notation in the global case. • L is a CM field, a totally imaginary quadratic extension over a totally real number field L0 , OL is the ring of integers in L. Denote by ι the complex conjugation on L. • k ⊃ Fp is an algebraically closed field. • ΣL,p (resp. ΣL0 ,p ) is the set of all places of L (resp. L0 ) above p. • For any element v ∈ ΣL0 ,p , denote by ΣL,v the set of places in ΣL,p above v. The set ΣL,v has a single element if v is ramified or inert in L; it has two elements if v is split in L. ur be the ring of • For every w ∈ ΣL,p , let Ow be the w-adic completion of OL , let Ow ur integers of the maximal unramified subfield Lw in Lw . Let κw be the residue field of OL at w, and let fw = [κw : Fp ]. Denote by ew ramification index of Lw /Qp . • Similarly, for every v ∈ ΣQ L0 ,p , let ev be the ramification index of L0,v /Qp and let ur := ur fv = [κv : Fp ]. Let OL,v w∈ΣL,v Ow . • For every v ∈ ΣL0 ,p , let Ov be the ring of integers in the v-adic completion Q L0v of L0 , and let Lv = L⊗L0 L0v be the v-adic completion of L; it is isomorphic to w∈ΣL,v Lw . Let Y Y ∼ ∼ OL,v = OL := OL ⊗Z Ov = Ow and OL,p := OL ⊗Z Zp = Ow v

w∈ΣL,v

w∈ΣL0 ,p

be the v-adic and p-adic completion of OL respectively. Let Y κL,v = κLv := κw . w∈ΣL0 ,v

Note that the use of OLv and κLv is compatible with the notation scheme in 4.3.1. • Let Rk (OL,p ) be the Grothendieck group for the category of all (OL,p⊗Zp k)-modules of finite length. Similarly, let Rk (OL,v ) be the Grothendieck group attached to the category of (OL,v ⊗Zp k)-modules of finite length for any v ∈ ΣL0 ,p . • For any finite field κ, let Rκ (OL,p ) be the Grothendieck group for the category of all (OL,p⊗Zpκ)-modules of finite length, and let let Rκ (OL,v ) be the Grothendieck group attached to the category of (OL,v ⊗Zp κ)-modules of finite length for any v ∈ ΣL0 ,p . • For every w ∈ ΣL,p , let '

'

ur ur Iw := HomZp -alg (Ow , W (k)) −→ HomZp -alg (Ow , k) −→ HomFp -alg (κw , k) .

As before Iw has a natural structure as a torsor for the Galois group Gal(κw /Fp ) ∼ = Z/fw Z. Similarly for every v ∈ ΣL0 ,p , let ur ur Iv := HomZp -alg (OL,v , W (k)) := HomZp -alg (OL,v , k) .

• We have canonical decompositions Y O ⊗Z W (k) ∼ Ow ⊗Z W (k) ∼ = = L

p

Y

p

w∈ΣL,p

and

Y

Rk (OL,p ) ∼ =

M v∈ΣL0 ,p

Ow ⊗(Owur ,i) W (k)

w∈ΣL,p i∈Iw

Rk (OL,v ) ∼ =

M

M

v∈ΣL0 ,p w∈ΣL,v

Rk (Ow ) .

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127

• For each w ∈ ΣL,p and for each i ∈ Iw , let w,i : Rk (OL,p ) −→ Z prw,i

be the composition of the projection Rk (OL,p ) −→ Rk (Ow ) with the isomorphism
' ' ' Rk (Ow ) −→ Rk Ow ⊗(Owur ,i) W (k) −→ R(k) −→ Z defined in 4.2.1. Let X

w :=

w,i : Rk (OL,p ) −→ Z .

i∈Iw

• An OL,p -linear p-divisible group Z over a base scheme S decomposes as follows: Y Y Y Z= Z[v ∞ ] = Z[w∞ ] , v∈ΣL0 ,p

Z[v ∞ ]

v∈ΣL0 ,p w∈ΣL,v

Z[w∞ ])

where each factor (resp. is an OL,v -linear (resp. Ow -linear) p-divisible group over S. We say that Z is a OL,p -linear CM p-divisible group if the height of Z[w∞ ] is equal to [Lw : Qp ] for each w ∈ ΣL,p . For every place v of L0 above p, the ring OL,v is of the form considered in §4.3. So the definitions in §4.3 apply to the v-component δv of an element δ ∈ Rk (OL ). We repeat some of them in 4.4.2 for the convenience of the reader. 4.4.2. Definition. Let δ be an element of Rκ (OL ), and write X X X δ= δv = δw , δw ∈ Rk (Ow ) ∀w ∈ ΣL,p . v∈ΣL0 ,p

v∈ΣL0 ,p w∈ΣE,v

(1) An element δ ∈ Rk (OL ) is a Lie type if every component δw of δ is a Lie type, i.e. 0 ≤ w,i (δw ) ≤ ew for all w ∈ ΣL,p ; it is κ-rational for a finite field κ if δw ∈ Rk (Ow ) is κ-rational for every w ∈ ΣL,p . (2) A Lie type δ is self-dual if the component δv of δ is self-dual for all v ∈ ΣL0 ,p ; see 4.3.4. This means w1 ,i (δw1 ) + w2 ,i (δw2 ) = ev for all i ∈ Iv = Hom(κv , k) = Hom(κw1 , k) = Hom(κw2 , k) if v splits into two places w1 , w2 in ΣL,v , and w,i (δw ) + w,i·¯ι (δw ) = ew for all i ∈ Iw = Hom(κw , k), in the case when w is the unique place of L above v. Here ¯ι denotes the automorphism of κw induced by the complex conjugation ι on L. (3) Suppose that v ∈ ΣL0 ,p stays prime in L. Let w be the unique place of E above v. Assume that ew is even. Note that ew is even if L/L0 is ramified above v. As in 4.3.13 (a) we say that the v-component δv = δw ∈ Rk (OL,v of a Lie type δ ∈ Rk (OL ) is uniform if w,i (δw ) = e/2 ∀i ∈ Iw . (4) Suppose that v ∈ ΣL0 ,p stays prime in L. Let w be the unique place of L above v. Assume that fw is even. As in 4.3.13 (b) we say that the v-component δv = δw of a Lie type δ is striped if there exists a subset Jw ⊂ Iw which is an orbit for the action of the subgroup Gal(κw /Fp2 ) ⊂ Gal(κw /Fp ) on Iw , such that  ew,i if i ∈ Jw w,i (δw ) = 0 if i ∈ Iw , i ∈ / Jw If we choose a trivialization of the Gal(κw /Fp )-torsor Iw and identify Iw with Z/fw Z, then Jw is either the subset of all even elements in Iw or the subset of all odd elements in Iw .

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(5) Let v be a place of L0 above p. We say that a subset Φv ⊂ Homring Lv , K0
self-dual if Homring Lv , K0 is the disjoint union of Φv and ι · Φv .




is

Remark. (i) Suppose that v stays prime in L and w is the unique place P in L above v. If δv is self-dual, uniform or striped, then δw is supersingular, i.e. w (δw ) := i∈Iw w,i (δw ) = [Lw : Qp ]/2 and 0 ≤ w,i (δw ) ≤ ew ∀i ∈ Iw . (ii) If δv is uniform, then it is self-dual and rational over Fp . (iii) If δv is striped, then it is Fp2 -rational, but it is self-dual if and only if fw ≡ 2 (mod 4). It is convenient to recall in 4.4.3 a criterion for a formal abelian scheme over a complete one-dimensional noetherian local ring to be algebraic. 4.4.3. Proposition. Let (R, m) be a one-dimensional complete noetherian local ring flat over Zp whose residue field is of characteristic p. Let A0 be an abelian variety over the residue field R/m of R. Let (X , β : L → End0 (X )) be an Lp -linear CM p-divisible group; i.e. the height of X [v ∞ ] is equal to [Lv : Qp ] for every place v of L0 above p. Let (A, α : L → End0 (A)) be the L-linear formal abelian scheme over R attached to (X , β) according to Theorem 1.4.5.7. Then A is algebraizable if and only if the v-adic component (X [v ∞ ], β[v ∞ ]) of X is a self-dual Lv -linear p-divisible group over R for every place v ∈ ΣL0 ,p in the sense of 4.3.6 (b). Proof. This is a reformulation of 2.2.3. Proposition 4.4.4 below is proved in [86]; it is a consequence of Lemma 4.3.7, Lemma 4.3.8 and Proposition 4.4.3. See also 4.4.14 for a variant of 4.4.4. 4.4.4. Proposition. Let (A0 , α0 ) be an OL -linear abelian variety over k with 2dim(A0 ) = [L : Q]. Then there exists a finite extension field K of K0 and an OL -linear abelian scheme (A, α) over OK with (the base extension to the residue field of OK of ) the OL -linear abelian variety (A0 , α0 ) as its closed fiber if and only if the Lie type [Lie(A0 )] is self-dual in the sense that [Lie(A0 [p∞ ])] is self-dual at every place v ∈ ΣL0 ,p . Proof. By Prop. 4.3.8, for every place v of L0 above p, there exists a self-dual subset Φv ⊂ HomQp -alg (OL,v , K0 ), whose reduction ξv (Φv ) is equal to the v-component of the Lie type of A0 . Choose a finite extension K containing the reflex field of (Lv , Φv ) for all v ∈ ΣL0 ,v . By 4.3.7 (ii), for every v ∈ ΣL0 ,p there exists an OL,v -linear lift Yv of A0 [v ∞ ] over OK with p-adic CM type Φv . We conclude by Theorem 1.4.5.7 and Proposition 4.4.3. α

4.4.5. Definition. Let (Z0 , OL,p −→ End(Z0 )) be an OL,p -linear CM p-divisible group over a field κ ⊃ Fp . Define a homomorphism α∗ : OL,p −→ End(Z0t ) by α∗ (a) := α(ι(a)t )

∀ a ∈ OL,p .

α

We say that (Z0 , OL,p −→ End(Z0 )) is self-dual up to isogeny if there exists an OL,p -linear isogeny from (Z0 , α) to (Z0t , α∗ ); in other words there exists an isogeny λ : Z0 −→ Z0t such that λ ◦ α(a) = α(ι(a)t ) ◦ λ ∀ a ∈ OL,p . Remark. The self-duality condition for (Z0 , α) above implies the following. (i) Suppose that v ∈ ΣL0 ,p splits in L and w1 , w2 are the two places of L above v. Let (Z0 [wi∞ ], α[wi∞ ]), i = 1, 2 be the two factors of Z0 above v. Then there exists an isogeny λ : Z0 [w1∞ ] → Z0 [w2∞ ]t such that λ ◦ α[w1∞ ](ι(a)) = (α[w2∞ ](a))t ◦ λ

∀a ∈ Ow1 .

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129

In particular the sum of the slopes of the two isoclinic factors Z0 [w1∞ ] and Z0 [w2∞ ] is equal to 1. (ii) Suppose that w ∈ ΣL,p is either inert or ramified for L/L0 . Then the factor Z0 [w∞ ] of Z0 is supersingular. 4.4.6. Lemma. (i) Let A0 be an OL -linear abelian variety over a field κ ⊃ Fp such that 2dim(A0 ) = [L : Q]. Then the (OL ⊗Z Zp )-linear p-divisible group A0 [p∞ ] is self-dual up to isogeny. (ii) Let Z0 be an OL,p -linear p-divisible group over a field κ ⊃ Fp which is self-dual up to isogeny. Let v is be place of L0 above p. Assume either that v is split in L and κ ∩ κL0v = Fp , or that v stays prime in L and κ ∩ κw = Fp for the unique place w of L above p. Then the v-component [Lie(Z0 )]v of the Lie type of Z0 is self-dual. Proof. The statement (i) is obvious. Because the Lie type [Lie(Z0 )] is rational over κ, the assumption implies [Lie(Z0 )] is fixed under the action of Gal(κw /Fp ) for every place w of L above p. So for every w ∈ ΣL,v there exists a natural number cw such that w,i (δw ) = cw for all i ∈ Iw . Since the OL,v -linear p-divisible group B0 [v ∞ ] is self-dual up to isogeny, we see that • if v splits into two places w1 , w2 in L, then cw1 + cw2 = ev = e(L0v /Qp ), while • if w is the unique place of L above p, then ew = e(Lw /Qp ) is even, cw = ew /2, and δw is uniform. In both cases δv is self-dual. We have proved (ii). 4.4.7. Theorem (CM lifting up to isogeny). Let κ ⊃ Fp be a finite field. Let L be a CM field as before. Let B1 be an abelian variety over κ such that 2dim(B1 ) = [L : Q] and there exists an embedding β1 : L ,→ End0 (B1 ). There exists • an OL -linear abelian variety (B0 , β0 : OL → End(B0 )) which is L-linearly isogenous to B1 over κ, • a complete noetherian local integral domain (R, m) of generic characteristic 0 ' • an isomorphism R/m −→ κ, • an abelian scheme B over R, • a subring O1 ⊂ OL which contains Z + pOL , • a ring homomorphism β : O1 → End(B) such that (B, β)×Spec(R) Spec(κ) ∼ = (B0 , β0 |O1 ). Preliminary reduction step for 4.4.7. There exists by 4.2.10 (c) an isogeny γ : B1 → B2 over κ such that End(B2 ) ⊃ γ ◦ β1 (OL ) ◦ γ −1 . For instance one can take γ to be the isogeny γ : B1 −→ OL ⊗β −1 (End(B1 )) B1 =: B2 1

described in 4.2.10 (i) (c) using Serre’s tensor construction. Replacing the given abelian variety B1 by such a B2 if necessary, we may and do assume that OL operates on B1 . Outline of strategy for the proof of 4.4.7. (1) Change B1 by an OL -linear isogeny over κ to an OL -linear abelian variety (B0 , β0 ) over κ such that the Lie type of B0 is “as close to being self-dual as possible”. To be precise, we demand that the v-component δv of [Lie(B0 )] is self-dual for every good p-adic place v ∈ ΣL0 ,p − ∆L0 of L0 . For a bad place v ∈ ∆L0 , it is not possible for the κ-rational Lie type δv to be self-dual, so we ask instead that the δv is striped. (2) Use Cor. 4.4.3, Lemma 4.3.8, Prop. 4.3.9 and Cor. B.1.17 to construct an L-linear abelian scheme (B 0 , β 0 ) over the ring of integers OK for some finite extension field K of the fraction field of W (F) such that the closed fiber of (B 0 , β 0 ) is isomorphic to (B0 , β0 ⊗Z Q)×Spec(κ) Spec(F).

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(3) A deformation argument concludes the proof. The above steps are elaborated below. 4.4.8. Step 1. Let (B1 , β1 : OL ,→ End(B1 )) be an OL -linear abelian variety over a finite field κ ⊃ Fp . Then there exists an OL -linear abelian variety (B0 , β0 : OL ,→ End(B0 )) over κ which is OL -linearly isogenous to (B1 , β1 ), such that • There is an OL,p -linear isomorphism '

ψ1 : B0 [p∞ ] −→ Y0 such that ψ[p∞ ] = ψ0 ◦ ψ1 . • The Lie type δv of B0 [v ∞ ] is self-dual for all v ∈ ΣL0 ,p − ∆L0 . • For all v ∈ ∆L0 , the Lie type of B0 [v ∞ ] = B0 [w∞ ] is striped, where w is the place in ∆L above v. ur -linear CM p-divisible group Y 0 • There is an Ow 0,w over F whose Lie type is striped and an Ow -linear isomorphism 0 B0 [w∞ ] ×Spec(κw ) Spec(F) ∼ = Ow ⊗Owur Y0,w

between Ow -linear p-divisible groups of CM type over F. Proof of 4.4.8. Apply 4.3.26 to the OL,v -linear p-divisible group B1 [v ∞ ] over κ to each place v of L0 above p. Transporting the OL,v -linear isogenies Y0,v −→ B1 [v ∞ ] over κ in 4.3.26 to the present situation and put them together, we obtain an OL,p -linear isogeny Y Y ψ0 : Y0 := Y0,v −→ B1 [v ∞ ] = B1 [p∞ ] v∈ΣL0 ,p

v∈ΣL0 ,p

of OL,p -linear p-divisible groups over κ. This gives us an OL -linear abelian variety B0 over κand an OL -linear isogeny ψ : B0 → B1 over κ with the required properties. 4.4.9. Step 2. Let (B0 , β0 ) be the OL -linear abelian scheme over κ as in 4.4.8. There exists a finite extension field K over the fraction field of W (F) and an L-linear abelian scheme (B 0 , β 0 ) over OK such that the closed fiber (B 0 , β 0 ) is isomorphic to the L-linear abelian variety (B0 , β00 )×Spec(κ) Spec(F), the base change to F of the L-linear abelian variety attached to the OL -linear abelian variety (B0 , β0 ) over κ. Proof of 4.4.9. According to Cor. 4.4.3, it suffices to construct, for each place v of L0 above p, a finite extension field Kv of the fraction field of W (F) and a self-dual Lv -linear p-divisible group (Y[v ∞ ], βv ) over OKv whose close fiber is isomorphic to the Lv -linear p-divisible group B0 [v ∞ ]×Spec(W (F)) Spec(OKv ): Suppose we have constructed (Y[v ∞ ], βv ) over OKv . Let K be a finite extension field of the fraction field of W (F) which contains all Kv ’s. Then Y (Y, β) := (Yv , βv ) v∈ΣL0 ,p

is a self-dual Lp -linear CM p-divisible group over OK . Case when v ∈ ΣL0 ,p − ∆L0 , i.e. v is a good place of L0 above p. Since the Lie type δv of each component B0 [v ∞ ] is self-dual, by Lemma 4.3.8 and Prop. 4.3.9 there exists for each v ∈ ΣL0 ,p − ∆L0 • a self-dual p-adic CM type Φv ⊂ Hom(Lv , K0 ) such that ξv (Φv ) = δv , • a finite extension Kv of K0 , equal to the reflex field of (Lv , Φv ), • an OL,v -linear p-divisible group Yv over OKv with p-adic CM type Φv . We have finished the “easy case” when v is a good p-adic place of L0 .

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131

Case when v ∈ ∆L0 , i.e. v is a bad place of L0 above p. By 4.3.28, there exists a self-dual Lv -linear CM p-divisible group (Y [v ∞ ], βv ) over OKv , where Kv is a tamely ramified extension of the fraction field of W (F), such that the closed fiber of (Y [v ∞ ], βv ) coincides with the Lv -linear CM p-divisible group attached to the OL,v -linear p-divisible group (B0 [v ∞ ], β0 [v ∞ ])×Spec(κ) Spec(F) over F. Step 2 is finished. 4.4.10. Step 3. There exists a complete noetherian local integral domain R, m and an Llinear CM lifting of (B0 , β0 ⊗Z Q) over R. Proof of 4.4.10 and Theorem 4.4.7. Let (B 0 , β 0 ) be the L-linear CM lifting of (B0 , β0 ⊗Z Q)×Spec(κ) Spec(F) in 4.4.10. Let O1 be a subring in OL containing Z + pOL such that O1 operates on B 0 . Let Cκ be the category whose objects consists of all triples (S, mS , ε : κ → S/mS ), where (S, mS ) is an artinian local ring and κ is a ring homomorphism; morphisms in Cκ are local morphisms between artinian local rings which are compatible with the ε’s. Consider the deformation functor Def(B0 , β0 |O1 ) on the category Cκ . This deformation functor is represented by the formal spectrum of a complete discrete valuation ring R1 with residue field κ. We have just seen that this deformation space has a point h : R1 −→OK which corresponds to the O1 -linear abelian scheme B 0 over OK obtained in (4). Then the quotient ring R1 /Ker(h) is a complete noetherian local integral domain of generic characteristic 0. The tautological deformation of (B0 , β0 |O1 ) over R := R1 /Ker(h) gives us the a formal abelian scheme B over R which lifts B0 . Moreover we know that for a (or all) geometric point(s) R1 [1/p] ,→ K the action of O1 on Lie(B) of Spec(R1 [1/p]) is given by a CM type of L, because this is so for the abelian scheme B 0 over OK . By 4.4.3 the formal abelian scheme B over R is the formal completion of an abelian scheme B over R with action by O1 which lifts (B0 , β0 |O1 ). We record some corollaries of the proof of 4.4.7. 4.4.11. Corollary. Let F be a finite extension field of Qp . Let Y0 be an OF -linear p-divisible over a finite field κ with height(Y0 ) = [F : Qp ]. Let Φ be a subset of Hom(F, K0 ) such that the Lie type of [Lie(Y0 )] in Rκ (OF ) ⊂ RF (OF ) is equal to ξ(Φ). Then there exists an OF -linear p-divisible group Y over complete noetherian local integral domain (R, m) contained in the algebraic closure of frac(W (F)) with residue field κ such that Φ is the CM type of Y and the closed fiber of Y is OF -linearly isomorphic to Y0 . Proof. Let E be the reflex field of the p-adic CM type (F, Φ), and let K be the compositum of E and the fraction field K0 of W (F). There exists an OF -linear p-divisible group X over OK with CM type (F, Φ); see 3.4.3 (3). Then the closed fiber of X is an OF -linear p-divisible group over F with Lie type ξ(Φ), therefore it is OF -linearly isomorphic to Y0 by 4.2.6 (iii). We conclude by the deformation theory argument in 4.4.10. 4.4.12. Corollary. Let L be a CM field. Let B1 be an abelian variety over a finite field κ ⊃ Fp with 2 dim(B1 ) = [L : Q] such that there exists an embedding β : OL ,→ End(B1 ). Then there exists a complete noetherian local integral domain (R, m) with residue field κ and OL -linear m-adic formal abelian scheme B over Spf(R) whose closed fiber B×Spf(R) Spec(κ) is OL -linearly isogenous to the given OL -linear abelian variety B1 over κ. Proof. Construct a lifting of the OL,p -linear p-divisible group B1 [p∞ ] by 4.4.11, then apply Theorem 1.4.5.7. 4.4.13. Corollary. Notation and assumption as in 4.4.7. Assume that every p-adic place of L0 is good in the sense of 4.1.5. In other words, for every place w of L above p which is inert over L0 , we assume that at least one of the following conditions are satisfied: • the absolute ramification index ew = e(Lw /Qp ) is even, or

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• the residue field degree fw is not divisible by 4, or • # Gal(κw /(κ ∩ κw )) is odd. Then the statement in 4.4.7 holds with O1 = OL . In other words there exists an OL -linear abelian scheme B over a complete noetherian local integral domain (R, m) with residue field κ whose closed fiber is L-linearly isogenous to the given abelian variety B1 over κ. Proof. Examine the proof of 4.4.7 and we can skip the last part in the proof of 4.4.9 as all p-adic places of L0 are good. Remark. The condition that ∆L = ∅ is automatically satisfied if 1/2 is not a slope of B1 , or more generally if κw ∩ κ = Fp for every place w of L above p which is inert for L/L0 . 4.4.14. Corollary. Let L be a CM field. Let B0 be an OL -linear abelian variety over a finite field κ with 2 dim(B0 ) = [L : Q]. (i) If the Lie type [Lie(B0 )] ∈ Rκ (OL ) is self-dual, then there exists an OL -linear abelian scheme B over a complete noetherian local integral domain (R, m) with residue field κ whose closed fiber is OL -linearly isomorphic to B0 over κ. (ii) The self-duality assumption on [Lie(B0 )] in (i) is satisfied if κ ∩ κw = Fp for every place w of L above p. The above condition clearly holds if the residue field κw = Fp for every place w of L above p. Proof. Suppose first that the Lie type of B0 is self-dual. We know from 4.4.4 that there exists an OL -linear abelian scheme B 0 over a complete noetherian local integral domain (R0 , m0 ) with residue field F whose closed fiber is OL -linearly isomorphic to B0 ×Spec(κ) Spec(F). We conclude the proof of the main statement by the deformation argument in 4.4.10. We have proved (i). The statement (ii) is a restatement of 4.4.6 (ii). Remark. Corollary 4.4.14 is a strengthening of 4.4.4. The special case when fw = 1 for every place w of L above p in the last sentence of 4.4.14 (ii) has been proved in Proposition 2.2.1.

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133

Notation scheme for chapter 4 1. CM fields and their localizations. • In §4.4, L denotes a CM field, L0 is the maximal totally real subfield in L, g = [L0 : Q] = [L : Q]/2, and ι is the complex conjugation for L. • In the “local situation” of §4.2 and appendix B.1, E denotes a finite extension field of Qp , and we consider OE -linear p-divisible groups of height [E : Qp ]; E is unramified over Qp in §B.1. They arise from the global situation of §4.4 by localizing at places of a CM field L above p as follows. Suppose that A → S is an OL -linear abelian scheme with 2dim(A/S) = [L : Q] and w is a place of L above p, then A[w∞ ] → S is an Ow -linear p-divisible group of height [Lw : Qp ]. Here Lw is the completion of L at w and Ow is the ring of integers of Lw . • In the “semi-local” situation of §4.3, E denotes a commutative algebra of degree 2 over a finite extension field E0 of Qp . So E is either an quadratic extension field of E0 or E is isomorphic to E0 × E0 ; the latter is referred to as the “split case”. Denote by OE the integral closure of OE0 in E; in the split case OE = OE0 × OE0 . In either case the group of E0 -automorphisms of the ring E is cyclic of order 2, generated by a non-trivial element ι; in the split case ι interchanges the two copies of E0 . Such semi-local situations arise from localization at a place of L0 above p: Let L be a CM field, let L0 be the maximally totally real subfield of L and let v be a place of L0 above p. Then E = L⊗L0 L0,v is of rank two over E0 = L0,v , where L0,v is the completion of L0 at v. In §4.3 we consider p-divisible groups of height [E : Qp ] with faithful linear action by OE ; such OE -linear p-divisible groups arise from global situation as follows. If A → S is an OL -linear abelian scheme with dim(A/S) = [L : Q]/2, where L is a CM field, then A[v ∞ ] → S is a p-divisible group of height [OL⊗OL0 : Qp ] with a faithful action by OL ⊗OL0 OL0 ,v .

2. Base schemes. • In §4.2–§4.4 k ⊃ Fp is an algebraically closed field. For the main theorem (I) of this chapter we can take k = F. In §B.1 the base field of characteristic p is F and k does not appear. In other words k = F but we suppressed “k” from the notation in §B.1. • In §4.2 and §4.3 κ denotes a subfield of the algebraically closed base field k ⊇ F; often κ is a finite field. • K0 is for the fraction field of an absolutely unramified complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field; in §4.2 and §4.3 K0 is the fraction field of W (k). We use the symbol “K” for the fraction field of a complete discrete valuation ring OK with a perfect residue field; usually the residue field is either k or a finite subfield of k.

3. Places, residue fields and ramification. • Given a CM field L and its maximal totally real subfield L0 , we will use the symbol “w” (resp. “v”) to denote a place of L (resp. L0 ) above the prime number p. • The set of all places w (resp. v) of L (resp. L0 ) above p will be denoted by ΣL,p (resp. ΣL,p ); similarly the set of all places of L above a place v ∈ ΣL0 ,p is denoted by ΣL0 ,v . • Ow = OLw is the ring of integers of the completion of L at w; similarly Ov = OL0 ,v is the ring of integers of the completion of L0 at v. • OL,v := OL ⊗OL0 Ov ; it is a typical example of rings E in the semi-local situation in §4.3. Similarly, OL,p := OL ⊗Z Zp , OL0 ,p := OL0 ⊗Z Zp . • For every w ∈ ΣL,p (resp. every v ∈ ΣL0 ,p ), denote by κw (resp. κw ) the residue field of w (resp. v), fw := [κw : Fp ], fv := [κv : Fp ]; the ramification indices are ew := e(Lw /Qp ), ev := e(L0,v : Qp ), and ew/v = e(Lw /L0,v for w ∈ ΣL,v .

134

4. CM LIFTING OF ABELIAN VARIETIES UP TO ISOGENY

• Let κ be a finite field. Denote by ∆L,κ the set of all places w ∈ ΣL,p such that the following properties hold. – w is inert for the quadratic extension L/L0 , – ew is odd, – fw ∼ = 0 (mod 4), and – Card (Gal(κw /(κw ∩ κ))) is even. Here the intersection of κw ∩ κ takes place in a finite field (or an algebraic closure of Fp ) which contains both κw and κ. • In the local situation of §4.2, κE is the residue field of the complete local ring OE , f = fE = [κE : Fp ], e = e(E/Qp ). Denote by E ur /Qp the maximal (absolutely) unramified subextension field of E/Qp . • In the semi-local situation of §4.3, κ0 = κE0 is the residue field of the complete local ring OE0 , f0 = fE0 = [κE0 : Fp ]. If E is a field, then κE is the residue field of OE and E ur /Qp is the absolutely unramified subextension field of E/Qp . If E = E0 × E0 then κE = κ0 × κ0 and E ur = E0ur × E0ur .

4. Some Grothendieck groups.

'

• In the local situation of §4.2, I = IE := HomZp -alg (OE ur , W (k)) −→ Homring (κE , k). The natural action of Autfield (k) on I factors through its quotient Gal(κE /Fp ) and gives I a natural structure as a torsor over the cyclic group Gal(κE /Fp ). • In the semi-local situation of §4.3, I = IE := HomZp -alg (OE ur , W (k)) is defined the same way as in the local situation; the involution ι induces an involution on IE . In the ` split case we have E = E0 × E0 , IE = IE0 IE0 , a disjoint union, and the involution interchanges the two copies of IE0 . • In both the local situation of §4.2 and the semi-local situation of §4.3, we denote by Rk (OE ) the Grothendieck group attached to the category Mod(OE⊗Zpk) of all (OE⊗Zp k)-modules of finite length. Similarly for each i ∈ I Rk (OE , i) = Rk (OE ⊗(OEur ,i) k) is the Grothendieck group attached to the category Modk (OE ⊗(OEur ,i) k) of all
OE ⊗(OEur ,i) k -modules of finite length. In the semi-local situation the involution ι of LE induces an involution on Rk (OE ), compatible with the decomposition Rk (OE ) ∼ = i∈I Rk (OE , i) and the involution on I induced by ι. In the split case of §4.3 we have E = E0 × E0 and Rk (OE ) ∼ = Rk (OE0 ) ⊕ Rk (OE0 ). • The length function on Modk (OE ⊗(OEur ,i) k) induced a natural isomorphism '

Rk (OE , i) −→ Z for each i ∈ I, and we have a natural isomorphism M Rk (OE ) ∼ Rk (OE , i) ∼ = ZI . = i∈I

The natural action of Aut(k) on Rk (OE ) is compatible with the above isomorphism Rk (OE ) ∼ = ZI and the natural action of Aut(k) on I. This action factors through the canonical surjection Aut(k)  Gal(κE /Fp ) (resp. Aut(k)  Gal(κ0 /Fp ) in §4.2 and also in the case E is a field in §4.3 (resp. in the split case of §4.3). • For each i ∈ I, denote by i the composition of the i-th projection Rk (OE ) to P ' Rk (OE , i) with the natural isomorphism Rk (OE , i) −→ Z. The sum  = i∈I i is the surjection  : Rk (OE ) −→ Z . Rk (OE ) → Z induced by the length function on the category Mod(OE ⊗Zp k). • In both the local situation of §4.2 and the semi-local situation of §4.3, for every subfield κ ⊂ k, we have a Grothendieck group Rκ (OE ), defined by “replacing k by κ”. The natural map jκ : Rκ (OE ) → Rk (OE ) is an injection. The image of jκ is equal to the subset of all κ-rational elements in Rk (OE ); i.e. elements of Rk (OE ) fixed by all elements of Aut(k/κ).

4.4. COMPLEX CONJUGATION AND CM LIFTING

135

5. Lie types. • In §4.2 and §4.3, an element δ ∈ Rk (OE ) is said to be a Lie type if δ is the class of a subquotient (or equivalently a sub-module, or equivalently a quotient module) of the free rank one module (OE ⊗Zp k). • If E is a field, the slope of a Lie type δ ∈ Rk (OE ) is by definition the rational number (δ)/[E : Qp ] ∈ [0, 1]. If E = E0 × E0 and δ = (δ1 , δ2 ) ∈ Rk (OE0 ) × Rk (OE0 ) = Rk (OE ) is a Lie type for E, then the slopes of δ are by definition the rational numbers (δ1 )/[E0 : Qp ] and (δ2 )/[E0 : Qp ]. • In the local situation of §4.2 and the semi-local situation of §4.3, the Lie type of a p-divisible group X of height [E : Qp ] with faithful action by OE over a field κ ⊂ is the class [Lie(X)] ∈ Rκ (OE ) of the tangent space Lie(X) of X at the point 0. • In the semi-local situation, a Lie type δ ∈ Rk (OE ) said to be self-dual if δ + ι(δ) = [OE ⊗Zp k]. • If L is a CM field and A is an OL -linear abelian variety over a field κ ⊃ Fp , the Lie type of A is the class [Lie(A)] ∈ Rκ (OL,p ) of the tangent space Lie(A) of A at the point 0.

6. p-adic CM types and their reduction • In the set-up of §4.2 and §4.3, a p-adic CM type (or a local CM type) for E is a subset Φ of HomQp −alg (E, K0 ), such that where K0 is an algebraic closure of K0 . • If X is a p-divisible group of height [E : Qp ] with a faithful action by an order O0 of OE over OK for a finite extension field K/K0 ⊂ K0 /K0 , then the (E⊗Qp K0 )-module
structure Lie(X ) ⊗Zp Qp ⊗K K0 defines a p-adic CM type for E. • In the semi-local situation of §4.3, a p-adic CM type Φ for E is said to be self-dual if HomQp −alg (E, K0 ) is the disjoint union of Φ and ι(Φ). • We have a natural reduction map ξ : 2HomQp −alg (E,K0 ) −→ Rk (OE ) with the following properties. – ξ is additive, i.e. ξ(Φ ∪ Φ0 ) = ξ(Φ) + ξ(Φ0 ) if Φ ∩ Φ0 = ∅. – ξ is compatible with the natural action of ι on the source and target of ξ. – If E is a field, then the slope of ξ(Φ) is equal to card(Φ)/[E : Qp ]. – If E = E0 × E0 and Φ = (Φ1 , Φ2 ) ∈ HomQp −alg (E0 , K0 ) × HomQp −alg (E0 , K0 ) = HomQp −alg (E, K0 ) , then the slopes of ξ(Φ) are card(Φ1 )/[E : Qp ] and card(Φ2 )/[E : Qp ]. – The image ξ(Φ) of a self-dual p-adic CM type Φ for E is a self-dual element of Rk (OE ). – Suppose that X is a p-divisible group of height [E : Qp ] a faithful action by OE , and X is a lifting of X to OK for a finite extension field K/K0 such that the action of OE extends to X . The reduction ξ(Φ) of the p-adic CM type Φ of the OE -linear p-divisible group X is equal to the Lie type [Lie(X)] of the OE -linear p-divisible group X. • Suppose that L is a CM field. Fix an algebraic closure Q of L, and choose an embedding of Q in C and an embedding of Q in Q p . Then every CM type Φ of L determines a family {Φv | v ∈ ΣL0 ,v } of self-dual p-adic CM types Φv for L⊗L0 L0,v indexed by places v of L0 above p. Conversely every such family of self-dual p-adic CM types indexed by ΣL0 determines a CM type of L.

Although our discussion . . . is restricted to elliptic curves, the theory cannot be fully understood unless one considers them as special cases of abelian varieties. Goro Shimura

A

Some arithmetic results for abelian varieties A.1. The p-part of Tate’s work The proof of Tate’s isogeny theorem over finite fields of characteristic p > 0 (Theorem 1.5.1.1) consists of two separate parts: the case ` 6= p and the case ` = p. The first of these is easier insofar as the `-divisible groups are ´etale and hence can be treated within the more concrete framework of Galois modules. The case ` = p requires additional techniques from noncommutative algebra. An important application of this case is the complete characterization of endomorphism algebras of simple abelian varieties over finite fields as in Corollary 1.5.2.2 (more specifically, the determination of the local invariants of such division algebras at the p-adic places of the center). Tate’s proofs of these results were published by Milne and Waterhouse in [51]. The arguments use a lot of non-commutative algebra beyond what is generally familiar to algebraic number theorists. The exposition of the proof of Tate’s isogeny theorem for ` = p that is given below considerably reduces the input from non-commutative algebra to just basic properties of central simple algebras, and the application to computing local invariants of endomorphism algebras is also provided. This section is based on a set of notes by K. Eisentr¨ager, and we are grateful to her for permission to present part of her notes in a more permanent form. A.1.1. The isogeny theorem for ` = p: first steps. Let k be a finite field of characteristic p > 0, W = W (k), and L = W [1/p]. We refer the reader to 1.4.5 for a review of notation and basic facts in the theory of p-divisible groups and their Dieudonn´e modules over a perfect field of characteristic p. In analogy with the `-adic case we will now write Tp G to denote the (contravariant) Dieudonn´e module D(G) for a p-divisible group G over k, and Vp G := Qp ⊗Zp Tp G. We also write Tp A and Vp A when G = A[p∞ ] for an abelian variety A over k. Let Dk = W [F, V] denote the Dieudonn´e ring of k (non-commutative if k 6= Fp ). The left Dk -module Tp G is the replacement for the `-adic Tate module in the “classical” case, even though it is contravariant in A; its Dk -action is the analogue of the Galois action on `-adic Tate modules. In general Vp G is an L-module of rank equal to the height of G, and it also has a left module structure over the “Laurent polynomial ring” Dk [1/p] = L[F, 1/F] that is non-commutative if k 6= Fp . Our aim is to prove the following result, which is Theorem 1.5.1.1 when ` = p: A.1.1.1. Theorem. For any abelian varieties A and B over k, the Zp -linear map Zp ⊗Z Hom(A, B) → Hom(A[p∞ ], B[p∞ ]) is an isomorphism. The same methods as in the case ` 6= p show injectivity (Proposition 1.2.5.1) and reduce the proof of surjectivity to the case B = A after inverting p on both sides. That is, we aim to prove that the injective map (A.1.1.1)

Qp ⊗Q End0 (A) → End0 (A[p∞ ]) 137

138

A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

is an isomorphism for any abelian variety A over k. This will be proved by recasting the problem in terms of Dieudonn´e modules and using some module theory over non-commutative algebras. Let q = #k = pa . The q-Frobenius endomorphism π of A acts W (k)-linearly on D(A[p∞ ]) with the same characteristic polynomial (in Z[T ]) as it does as a Z` -linear endomorphism of the `-adic Tate module of A for any ` 6= p; this is true more generally for any element of End(A), by using Dieudonn´e modules in the role of Tate modules in the proof of the `-adic cases (` 6= p) in [53, Thm. 4, p. 180]. Hence, we may use Tate’s proof of surjectivity essentially verbatim (say, as in Appendix 1 of [53]) by replacing `-adic Tate modules with Dieudonn´e modules throughout to deduce (via [53, Lemma 6, App. 1]) that the Qp -dimension of the left side of (A.1.1.1) is equal to the L-dimension of the centralizer of the semisimple L-linear endomorphism F a = Vp (π) acting on the L-vector space Vp (A) (i.e., dimL EndL[F a ] (Vp (A))). Thus, it is necessary and sufficient to prove that this L-dimension is equal to the Qp -dimension of End0 (A[p∞ ]) = EndDk [1/p] (Vp (A))opp . The central subalgebra F = Q[π] in the finite-dimensional semisimple Q-algebra End0 (A) must be semisimple and hence a finite product of fields; we do not assume A to be k-simple, so in particular F may not be a field. Consider the decomposition of A and A[p∞ ] (in the isogeny sense) according to the idempotents of F . The action by F on Vp (A) (through its action on A in the isogeny category of abelian varieties over k) commutes with the action by Dk [1/p] on Vp (A), and the element F a ∈ Dk [1/p] acts as Vp (π) on Vp (A). Since there are no nonzero maps between pairs of factor abelian varieties or p-divisible groups over k on which the central subalgebra F acts through distinct quotient fields, we may reduce the proof to the case when F is a field. Let m ∈ Z[T ] be the common characteristic polynomial for V` (π) on the Q` -vector space V` (A) for all ` 6= p as well as for Vp (π) on the L-vector space Vp (A). By the faithfulness of Vp (A) as a Qp ⊗Q F -module, Q we see that rad(m) is the minimal polynomial of π ∈ F over Q. Hence, Qp ⊗Q F ' v|p Fv with Fv ' Qp [T ]/(mv ) for the monic irreducible factors mv ∈ Zp [T ] of m over Qp . Each mv (0) divides m(0) = q dimA , so all mv have nonzero constant term. Since Qp ⊗QQF acts Dk [1/p]-linearly on Vp (A), we get a decomposition of Dk [1/p]Q modules Vp (A) ' v|p Vp (Gv ) where v|p Gv is the isogeny decomposition of A[p∞ ] with respect to the idempotents of Qp ⊗Q F . Since the central element mv (F a ) ∈ Dk acts on Gv through the element mv (π) = 0 in Fv , Vp (Gv ) is a left module over the quotient algebra Cv = Dk [1/p]/Dk [1/p]mv (F a ). Using the compatible decompositions (as L-algebras and Qp algebras respectively) Y EndL[F a ] (Vp (A)) ' EndL[F a ] (Vp (Gv )) v|p

and EndDk [1/p] (Vp (A)) '

Y

EndCv (Vp (Gv )),

v|p

we are reduced to proving ?

dimL EndL[F a ] (Mv ) = dimQp EndCv (Mv ) for any v|p and any left Cv -module Mv with finite Qp -dimension (e.g., Mv = Vp (Gv )). This general equality is valid when Cv is replaced with Dk [1/p]/Dk [1/p]h(F a ) for any h ∈ Qp [T ] that is monic irreducible with nonzero constant term (i.e., h 6= T ); a proof will be given in Corollary A.1.2.6 as a consequence of some general calculations in non-commutative algebra (having no logical dependence on abelian varieties or p-divisible groups). A.1.2. Cyclic algebras. We now analyze the structure of the Qp -algebras Cv that arose above. The first key fact is that the Cv are always simple, with center Fv . More generally:

A.1. THE p-PART OF TATE’S WORK

139

A.1.2.1. Theorem. Let h ∈ Qp [T ] be a monic irreducible polynomial distinct from T , and let K = Qp [T ]/(h). The central K-algebra C = Dk [1/p]/Dk [1/p]h(F a ) (with T acting as F a ) is central simple. In order to prove Theorem A.1.2.1, we require some preparations. Recall that L denotes the fraction field of W (k). We have dimL C = deg(h(T a )) = a[K : Qp ] due to: A.1.2.2. Lemma. Let λ ∈ L[T ] be monic with positive degree d and non-vanishing constant term. Every g ∈ Dk [1/p] can be uniquely written g = h · λ(F) + (c0 + c1 F + · · · + cd−1 F d−1 ) with c0 , . . . , cd−1 ∈ L and h ∈ Dk [1/p]. Thus, the left Dk [1/p]-module Dk [1/p]/Dk [1/p] · λ(F) has dimension d as a left L-vector space. Proof. The uniqueness is proved by chasing “least-degree” coefficients (when nonzero). For existence it suffices to treat the cases g = F r with r ≥ d and r < 0. These go as in the commutative case (for r < 0 we use the non-vanishing of the constant term of λ). This completes the proof of the lemma. Since C is an L ⊗Qp K-module with finite Qp -dimension, Lemma A.1.2.2 implies dimK C = [K : Qp ]−1 dimQp C =

[L : Qp ] [L : Qp ] dimL C = · a[K : Qp ] = a[L : Qp ]. [K : Qp ] [K : Qp ]

Choose any π ∈ K × . Let (L ⊗Qp K)[F] denote the K-algebra defined by the relations F a = 1 ⊗ π and F · x = (σ ⊗ 1)(x) · F for x ∈ L ⊗Qp K, with σ ∈ Gal(L/Qp ) the absolute Frobenius. This is easily checked to be a central K-algebra with dimension a[L : Qp ] = a2 . There is a unique map of central K-algebras (L ⊗Qp K)[F] → C sending F to F and L to L by the identity map, and this is clearly surjective. By consideration of K-dimensions, it is an isomorphism. Theorem A.1.2.1 therefore will follow from: A.1.2.3. Theorem. The central K-algebra (L⊗Qp K)[F] is K-isomorphic to a matrix algebra over a cyclic K-algebra ∆. A.1.2.4. Remark. Cyclic algebras are a special class of central simple algebras, studied by Dickson before the advent of the general theory. Proof. We first construct the cyclic K-algebra ∆ over which the K-algebra (L ⊗Qp K)[F] will be proved to be a matrix algebra. Let f be the absolute residue degree of K, and let g := gcd(f, a). Let L ∩ K ⊆ K denote the maximal unramified subextension of K that embeds into L (over Qp ), so g = [L ∩ K : Qp ]. We arbitrarily choose one of the g embeddings L ∩ K → L as cyclic extensions of Qp , and let LK denote the resulting linearly disjoint compositum L ⊗L∩K K. Clearly LK/K is unramified with degree a/g. Let θ : L ⊗Qp K → LK be the resulting projection map. Let ∆ be the central K-algebra (LK)[F 0 ] defined with relations (F 0 )a/g = π ∈ K × and F 0 · x0 = σ 0 (x0 ) · F 0 where σ 0 : LK ' LK is the L ∩ Kautomorphism of order a/g induced by σ g on L and the identity on K. Thus, σ 0 (θ(x)) = θ((σ g ⊗ 1)(x)) for x ∈ L ⊗Qp K, and we have a natural isomorphism of K-algebras (A.1.2.1)

L ⊗Qp K '

g Y j=1

LK

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

defined by x 7→ (θ(x), θ((σ ⊗ 1)(x)), . . . , θ((σ g−1 ⊗ 1)(x))). The central K-algebra ∆ is a cyclic K-algebra; in classical notation, ∆ = (LK/K, σ 0 , π), where LK is a cyclic (unramified) extension of K and σ 0 is a chosen generator of its Galois group. Now consider the natural map of central K-algebras ξ : (L ⊗Qp K)[F] → Matg (∆) defined by    x 7→  

θ(x) 0 ... 0 θ((σ ⊗ 1)(x)) 0 .. .. .. . . . 0 0 ...

0 0 .. . θ((σ g−1 ⊗ 1)(x))

    

for x ∈ L ⊗Qp K and  0 1 0 ... 0  0 0 1 ... 0   . .  ..  . ..  F 7→  .. 0 . . .   . .   .. .. 0 . . . 1  F0 0 0 . . . 0 The map ξ is obviously a well-defined K-algebra homomorphism, and it is easy to check that both sides have the same Qp -dimension. To prove that ξ is an isomorphism, it suffices to check surjectivity. The decomposition (A.1.2.1) of L⊗Qp K as a product of copies of LK shows that the image of ξ contains the diagonal matrices whose entries are elements of LK ⊆ ∆. Moreover, ξ(F g ) is the diagonal matrix whose diagonal entries are all equal to F 0 ∈ ∆. Thus, by the definition of ∆ we see that the image of ξ contains all diagonal matrices in Matg (∆). Left-multiplication by ξ(F) carries the set of diagonal matrices onto the set of matrices of the form   0 δ2 0 . . . 0 0 δ3 . . . 0   0  . .  ..  . . ..  0 ...  .   .  ..  .. . 0 . . . δg  F 0 δ1 0 0 ... 0 

with arbitrary δ1 , . . . , δg ∈ ∆, and so since F 0 ∈ ∆ has a multiplicative inverse (as (F 0 )a/g = π ∈ K × ) it follows that the image of ξ contains the matrix   0 1 0 ... 0  0 0 1 ... 0   . .  ..  . . ..   . 0 ... .  . .   .. .. 0 . . . 1  1 0 0 ... 0 Powers of this matrix, together with arbitrary diagonal matrices in Matg (∆), generate Matg (∆) under the operations of multiplication and addition. This establishes surjectivity. A.1.2.5. Remark. For later purposes, it is convenient to describe the cyclic algebra ∆ in Theorem A.1.2.3 in terms of an arithmetic Frobenius generator φ of the Galois group of the unramified extension LK/K. With notation as used above, we claim that ∆ := (LK/K, σ 0 , π) is K-isomorphic to the cyclic K-algebra (LK/K, φ, π f /g ). This amounts to checking that σ 0 f /g = φ in Gal(LK/K), and so we just have to compare these on the residue field of L g (viewed inside that of LK). Since σ 0 reduces to x 7→ xp on the residue field of L, σ 0 f /g

A.1. THE p-PART OF TATE’S WORK

141

f

reduces to x 7→ xp on this residue field. But f is the residual degree for K over Qp , so we are done. It is an elementary result in the theory of simple rings that (up to isomorphism) over a simple ring there is a unique simple left module and all finitely generated left modules are isomorphic to a direct sum of copies of the simple module. Thus, by Theorem A.1.2.1 there is (up to isomorphism) a unique simple left Dk [1/p]-module V killed by the central element h(F a ), and for any nonzero left C-module M with finite Qp -dimension we have M ' V ⊕r for some r ≥ 1 (with C denoting the quotient of Dk [1/p] modulo the central element h(F a )). In particular, EndDk [1/p] (M ) is a matrix algebra over the division ring D := EndC (V ). Since there is an isomorphism of left C-modules C ' V ⊕r0 for some r0 > 1, so Matr0 (D) = EndC (C) = C opp , D has center K and in the Brauer group of K the classes of D and C opp coincide. Hence, the class of EndDk [1/p] (M )opp in Br(K) is the same as that of C for any nonzero finitely generated left C-module M . For any such M we have the following result that completes the final step in the proof of Theorem A.1.1.1: A.1.2.6. Corollary. Let h ∈ Qp [T ] be monic irreducible with h 6= T , and define C := Dk [1/p]/Dk [1/p]h(F a ). For any finite left C-module M , (A.1.2.2)

dimL EndL[F a ] (M ) = dimQp EndC (M ).

Since L[F, 1/F] = Dk [1/p] and EndC (M ) = EndL[F ] (M ) (F acts invertibly on M , as h(0) ∈ Q× p ), the equality [L : Qp ] = a suggests the possibility that one may be able to deduce (A.1.2.2) by some general nonsense with Galois descent. However, it seems that the proof requires fine structural information concerning C (such as Theorem A.1.2.1). Proof. Let V be a simple left C-module and let K = Qp [T ]/(h). We have M ' V ⊕r for some r > 0. The two sides of (A.1.2.2) for M are each r2 times the two sides of (A.1.2.2) with V in the role of M . Hence, the truth of (A.1.2.2) is independent of the choice of r, so it suffices to verify this identity for one M 6= 0. We choose M = C. Since EndC (C) = C opp as Qp -algebras, dimQp EndC (C) = [L : Qp ]dimL (C opp ) = [L : Qp ] deg h(T a ) by Lemma A.1.2.2 (with λ := h(T a ) satisfying λ(0) 6= 0 since the irreducible monic h is not T ). Because deg h = [K : Qp ] and [L : Qp ] = a we may rewrite this as dimQp EndC (C) =

[L ⊗Qp K : Qp ]a2 , [L : Qp ]

and so by expressing an L-dimension as [L : Qp ]−1 times a Qp -dimension it is equivalent to prove dimQp EndL[F a ] (C) = [L ⊗Qp K : Qp ]a2 . It therefore suffices to show that EndL[F a ] (C) is isomorphic as a Qp -algebra to an a × a matrix algebra over L ⊗Qp K. Let t ∈ K = Qp [T ]/(h) denote the nonzero residue class of T , so K = Qp [t]. We saw above Theorem A.1.2.3 that as L[F a ]-algebras C ' (L ⊗Qp K)[F] where the right side has relations F a = 1 ⊗ t and F · x = (σ ⊗ 1)(x) for all x ∈ L ⊗Qp K. Hence, the L[F a ]-endomorphism ring of C is the same as the L ⊗Qp K-endomorphism ring of C, and this in turn is indeed an a × a matrix algebra over L ⊗Qp K. This completes the proof of Theorem A.1.1.1.

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A.1.3. Local invariants. As an application of Theorem A.1.1.1 we wish to now prove the formula for local invariants in Corollary 1.5.2.2(3). Let A be a simple abelian variety over a finite field k of size q = pa , and let E denote its endomorphism algebra, so E is a central simple algebra over the subfield F := Q[π]Q(with π denoting the q-Frobenius endomorphism of A). Under the decomposition Qp ⊗Q E ' v|p Ev with Ev := Fv ⊗F E, we get a corresponding decomposition of A[p∞ ] in the isogeny category of p-divisible groups over k, as a product of nonzero p-divisible groups Gv on which Qp ⊗Q F acts through its quotients Fv (here we use the elementary injectivity aspect of Theorem A.1.1.1 to know that each Gv is nonzero). The Dieudonn´e functor is fully faithful and contravariant, and (A.1.1.1) is an isomorphism for finite k, so we may identify the central F ⊗Q Qp -algebra E ⊗Q Qp with the opposite F ⊗Q Qp -algebra to EndDk [1/p] (Vp A), where F = Q[π] acts through functoriality via its action on A in the isogeny category over k. In particular, Vp (π) is the action of the central element F a ∈ Dk [1/p], and so we get Ev ' End0 (Gv ) ' EndDk [1/p] (Vp Gv )opp .

(A.1.3.1)

as finite-dimensional Fv -algebras. We conclude that the right side is a central simple Fv algebra. Our problem is to compute its invariant. Since π in the ring of integers of F acts on A as the q-Frobenius over k, on the finitedimensional Qp -vector space Vp (Gv ) the Qp -linear operator F a = Vp (π) acts with a characteristic polynomial that divides the common characteristic polynomial m ∈ Q[T ] ⊂ Qp [T ] for the q-Frobenius on each V` (A) over Q` (for ` 6= p) and Vp (A) over L. (Explicitly, for all n ∈ Z the integer m(n) is the degree of n−π acting on A.) As we noted already, m isQ a power of the minimal polynomial of π ∈ F over Q, and so the decomposition Qp ⊗Q F ' v|p Fv corresponds to the pairwise distinct monic irreducible factors mv of m over Qp . That is, Fv = Qp (π) with π ∈ Fv having minimal polynomial mv over Qp . In particular, mv (0) divides m(0) = q dimA , so each mv has a nonzero constant term. For each v|p, the Fv -action on the nonzero Vp (Gv ) commutes with the Dk [1/p]-action (as it arises from an action of Fv on the p-divisible group Gv in the isogeny category over k), and the central element mv (F a ) ∈ Dk [1/p] acts as multiplication by the element mv (π) ∈ Fv that is zero. In other words, Vp (Gv ) is a nonzero module over the ring Cv := Dk [1/p]/Dk [1/p]mv (F a ) that was described as a matrix algebra over a specific cyclic algebra in Theorem A.1.2.3 and Remark A.1.2.5. Now it remains to compute the local invariants for certain cyclic algebras over nonarchimedean local fields. We begin by recording a general lemma. A.1.3.1. Lemma. Let K be a field and let K 0 /K be a cyclic extension. Fix a generator s0 of Γ = Gal(K 0 /K). Let χs0 : Γ → Q/Z be the unique homomorphism sending s0 to 1/[K 0 : K], and let θs0 = δ(χs0 ) ∈ H2 (Γ, Z). (1) For c ∈ K × , the class of the cyclic algebra (K 0 /K, s0 , c) in Br(K) is the image of c mod NmK 0 /K (K 0 × ) under the Tate periodicity isomorphism ×

× ∪θs

×

b 0 (Γ, K 0 ) ' H2 (Γ, K 0 ) ⊆ Br(K). K × /NmK 0 /K (K 0 ) = H (2) If K00 /K is a subextension of K 0 /K and s00 = s0 |K00 then 0

0

[(K00 /K, s00 , c)] = [(K 0 /K, s0 , c[K :K0 ] )] in Br(K). Proof. The first part is an explicit cocycle calculation via unwinding definitions and usb • (Γ, ·) (using Tate ing the cup-product compatibility of the δ-functorial map H• (Γ, ·) → H

A.1. THE p-PART OF TATE’S WORK

143

cohomology only in non-negative degrees). The second part follows from the first via the commutativity of the diagram K × /NmK00 /K (K00 × )

'



K × /NmK 0 /K (K 0 × )

/ H2 (K 0 /K, K 0 × ) 0 0 

'

inf

/ H2 (K 0 /K, K 0 × )

(which in turn follows from the definition of the horizontal isomorphisms and both the δfunctoriality and cup product compatibilities of inflation maps). Take K to be a non-archimedean local field, and define invK : Br(K) ' Q/Z according to the conventions in 1.2.2. By Lemma A.1.3.1 and the existence of unramified splitting fields for all Brauer classes of K, it follows that every element of Br(K) is represented by a cyclic algebra of the form (K 0 /K, φ, c) with K 0 /K a finite unramified extension, φ ∈ Gal(K 0 /K) the arithmetic Frobenius generator, and c ∈ K × an element. (We could also work with geometric Frobenius generators, but for the applications to abelian varieties it is the cyclic algebras resting on arithmetic Frobenius that naturally arise in the analysis of the Dieudonn´e modules; see Remark A.1.2.5.) The main result is: A.1.3.2. Theorem. Let K be a non-archimedean local field and K 0 /K an unramified finite extension. Let φ ∈ Gal(K 0 /K) be the arithmetic Frobenius element. For any c ∈ K × , the cyclic K-algebra (K 0 /K, φ, c) has local invariant in Q/Z represented by ordK (c)/[K 0 : K]. Proof. The normalized valuation for K identifies K × /NmK 0 /K (K 0 × ) with Z/[K 0 : K]Z, so [(K 0 /K, φ, c)] ∈ Br(K) only depends on the ratio ordK (c)/[K 0 : K]. In particular, we may change c by a unit multiple so as to reduce to the case when c is a power of a local uniformizer of K, and by passing to a suitable subextension over K we may reduce to the case when ordK (c) is relatively prime to [K 0 : K]. In this case the class of c in K × /NmK 0 /K (K 0 × ) has order [K 0 : K] and so the cyclic K-algebra (K 0 /K, φ, c) of degree [K 0 : K]2 is necessarily a division algebra (due to period dividing the index). A central division algebra D over K admits a unique valuation extending the normalized valuation on K [66, Ch. XII, §2], and there is a classical procedure that uses this valuation to compute invK ([D]) as follows. Let ordD = ordK ◦ NrdD/K : D× → Z (with NrdD/K denoting the reduced norm), so if [D : K] = n2 then (1/n)ordD restricts to the normalized valuation on K × . By the Skolem–Noether theorem, there exists γ ∈ D× such that γxγ −1 = φ−1 (x) for all x in a copy of the degree-n unramified extension Kn /K within D (so γ n ∈ Kn× ). Pick any such γ; in the case of cyclic algebras such a γ is provided as part of the construction. Since we use the method of non-abelian cohomology (as in [66, Ch. X, §5]) to identify H2 (L/K, L× ) with the subgroup of classes in Br(K) split by a Galois extension L/K and we take arithmetic Frobenius as the preferred topological generator for the Galois group of a finite field, after some explicit unwinding of definitions and calculation with 2-cocycles one finds ordD (γ n ) ordD (γ) =− . invK ([D]) = − n n2 Now take D to be the cyclic algebra (K 0 /K, φ, c) equipped with its canonical choice of γ (so γ n = 1/c). We thereby get invK ([D]) = −

ordD (1/c) ordK (c) mod Z = mod Z. n2 n

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

A.1.4. Applications to abelian varieties. Let A be a simple abelian variety over a finite field k of size q, and E := End0 (A), so E is a central division algebra over the number field F = Q(π) with π ∈ E the q-Frobenius endomorphism. We seek to compute the local invariant invv (Ev ) for every p-adic place v of F . By (A.1.3.1) and the discussing preceding Corollary A.1.2.6, the class of Ev in Br(Fv ) is the same as that of the cyclic Fv -algebra ∆v = (LFv /Fv , φ, π fv /gv ) that rests on the arithmetic Frobenius generator φ of Gal(LFv /Fv ) and the element π fv /gv in Fv× (where fv = f (v|p) and gv = gcd(fv , a)). Since [LFv : Fv ] = a/gv , by the formula for local invariants in Theorem A.1.3.2, we get 1 fv invv (∆v ) = · v(π fv /gv ) = · v(π). a/gv a Let ev be the ramification index of Fv over Qp , so ev fv = [Fv : Qp ]. We have v(q) = a · v(p) = a · ev , so a = v(q)/ev . This implies that invv (∆v ) =

ev fv v(π) fv · v(π) = · v(π) = · [Fv : Qp ]. a v(q) v(q)

This establishes the formula for local invariants in Corollary 1.5.2.2(3).

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

145

A.2. The Main Theorem of Complex Multiplication In [72, Ch. IV, §18] the Main Theorem of Complex Multiplication is proved in terms of a style of algebraic geometry that has been largely abandoned. To help a wider audience appreciate many of the beautiful ideas in that proof, in this section we give a scheme-theoretic account of the argument. Our exposition centers on an algebraic statement of the Main Theorem over Q (Theorem A.2.4.4), without analytic uniformizations or the automorphism group of C. In A.2.8 we use this version of the Main Theorem to recover a coordinate-free version of the traditional formulation of the Main Theorem in terms of analytic uniformizations and Riemann forms of certain polarizations. Complex-analytic methods certainly have their place in the theory, but we prefer to minimize their appearance and keep proofs as algebraic as possible. The reason that we can achieve this is because of our systematic use of Serre’s tensor construction (see 1.6.4), especially in its relative incarnation over base schemes with possibly mixed characteristic. A.2.1. Notation. Here are some notational conventions in A.2. We let Q denote the algebraic closure of Q in C, and the kernel of exp : C → C× is denoted Z(1). We also write M (1) to denote Z(1) ⊗Z M for any Z-module M . For a scheme X over a field K and an automorphism σ : K → K, X σ denotes the K-scheme Spec(K) ×Spec(σ),Spec(K) X obtained by base change. The same notation is used for K-morphisms. We indulge in one serious abuse of notation: if σ is an automorphism of K that restricts to an automorphism σ0 of a subfield K0 (the case of most interest being K = Q and K0 a number field) and if Z is a K-scheme for which a K0 -descent Z0 is specified, we write Z0σ to denote the K0 -scheme Z0σ0 descending the K-scheme Z σ . Hopefully this will not cause confusion. For a CM field K, a CM abelian variety (A, i : K → End0 (A)) over a field k is principal if the CM order i−1 (End(A)) is equal to OK . If A is an abelian variety over a field k of characteristic 0 and k/k is an algebraic closure then Tf (A) denotes the “total Tate module” Q lim A[n](k) ' ` T` (A) and Vf (A) denotes the finite free Af -module Q ⊗Z Tf (A). The total ←− Weil pairing for A is denoted h·, ·iA : Vf (A) × Vf (At ) → Af (1). A.2.2. Overview. Let (A, i) be an abelian variety over Q with CM type (K, Φ). Let φ : A → At be a K-linear polarization, where At has the K-action it (c) = i(c∗ )t , with c∗ denoting complex conjugation on the CM field K (so (At , it ) also has CM type (K, Φ); see Remark 1.4.3.1). Such a φ exists, by the complex-analytic theory (see Lemma 2.2.2). Let E ⊂ Q be the reflex field (Definition 2.1.4), and pick σ ∈ Gal(Q/E) and a finite idele s ∈ A× E,f that maps to σ|E ab under the global Artin map. The Main Theorem describes the AK,f linear isomorphism σ : Vf (A) ' Vf (Aσ ) and the Weil pairing eφσ on Vf (Aσ ) in terms of adelic operations on Vf (A) and eφ using s, and constructs (Aσ , iσ ) from (A, i) by an adelic procedure that does not use Galois automorphisms. Specializing the arguments below to the 1-dimensional case gives a proof for elliptic curves that exhibits a different flavor from the traditional one (as in [40]). Some technical issues simplify tremendously in the 1-dimensional case because elliptic curves have unique polarizations of each positive square degree. In A.2.3 we develop some concepts that are required in the proof of the Main Theorem, and in A.2.4 we give the setup for and statement of the Main Theorem. Before delving into the proof, in A.2.5 we work out a few consequences of the Main Theorem. The proof of the Main Theorem occupies A.2.6–A.2.7, Finally, in A.2.8 we use the “algebraic” version of the Main Theorem over Q to deduce the traditional version in terms of analytic uniformizations and we also translate analytic refinements back into algebraic language (by using a variant on the Serre tensor construction in the absence of CM by the maximal order) and give algebraic proofs of the latter.

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

A.2.3. Q-polarizations. Let K be a CM field, with maximal totally real subfield K0 of degree g over Q. We write c 7→ c∗ to denote complex conjugation on K. Let Φ ⊂ Hom(K, Q) be a CM type; i.e., a set of representatives for the quotient of Hom(K, Q) by the free action of the complex conjugation of K. Clearly Φ has size g and [K : Q] = 2g. Finally, let E ⊂ Q be the reflex field of (K, Φ) (Definition 2.1.4); this is the fixed field of the open subgroup of Gal(Q/Q) consisting of those elements σ such that σΦ = Φ. (That is, composition with σ permutes the set of embeddings Φ.) We shall be interested in studying pairs (A, i) where A is an abelian variety over Q and i : K → End0 (A) is a ring homomorphism making A an abelian variety of CM type (K, Φ). That is, the eigenspace decomposition for the Q-linear action of K on the g-dimensional tangent space of A consists of eigenlines on which K acts through the mappings ϕ : K → Q given by the g elements ϕ ∈ Φ. The dual At is endowed with the K-action it : K → End0 (At ) defined by it (c) = i(c∗ )t , so the CM type for (At , it ) is Φ (Remark 1.4.3.1). Since Q is algebraically closed, so Hom(X, Y ) = Hom(XC , YC ) for any abelian varieties X and Y over Q (Lemma 1.2.1.2), a polarization of A over Q is the “same” as a polarization of AC over C. Thus, in terms of the analytic theory, such a polarization is encoded as a skew-symmetric bilinear pairing ψZ : H1 (A(C), Z) × H1 (A(C), Z) → Z(1) such that with respect to the complex structure induced by the R-linear isomorphism H1 (A(C), R) ' Lie(A(C)) (defined by σ 7→

R σ

∈

Ω1 (A(C))∨

= Cot0 (A(C))∨ ) the R-linear extension ψR satisfies ψR (cx, y) = ψR (x, cy)

for all c ∈ C and the resulting R-valued symmetric bilinear form √ √ (A.2.3.1) (2π −1)−1 ψR ( −1 · x, y) is positive-definite. In terms of the algebraic theory, ψZ may be encoded as a symmetric isogeny φ : A → At such that (1, φ)∗ (P) is ample on A, where P is the Poincar´e bundle over A × At . For our purposes it will inconvenient to impose “integrality” conditions, so the following notion is useful. A.2.3.1. Definition. A Q-polarization of an abelian variety over a field is a positive rational multiple of a polarization. For an abelian variety A over Q, the positive rational multiple in this definition may be viewed within the Q-vector space Hom0 (A, At ) or as a Q(1)-valued skew-symmetric form on H1 (A(C), Q) via the analytic theory. That is, a Q-polarization of A is a positive rational multiple of a pairing ψZ as above or is a mapping φ : A → At in the isogeny category of abelian varieties over Q such that φ admits a positive integral multiple that is a polarization. By Lemma 2.2.2(1), there exists a polarization φ : A → At that is K-linear, or equivalently the associated Rosati involution on End0 (A) restricts to complex conjugation on K. The concept of K-linear Q-polarization is defined in the evident manner. A useful formulation of the K-linearity condition on a Q-polarization φ : A → At is this: under the associated skew-symmetric Weil pairing (A.2.3.2)

h·, ·iφ,Q : H1 (A(C), Q) × H1 (A(C), Q) → Q(1),

for any c ∈ K the adjoint to H1 (i(c)) is H1 (i(c∗ )). In a special case, K-linearity is automatic: A.2.3.2. Example. Suppose A is simple. In this case i : K → End0 (A) is an isomorphism, by Proposition 1.3.7.3(1). Moreover, by Lemma 1.3.6, the Rosati involution for any polarization of A induces complex conjugation on K.

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

147

In the general (possibly non-simple) case, there is a useful refinement: A.2.3.3. Lemma. The set of elements in Hom0 (A, At ) associated to K-linear Q-polarizations is a principal homogeneous space for the action on A (in the isogeny category) by the subgroup of totally positive elements α ∈ K0× . Proof. Since K is its own centralizer in End0 (A), any two K-linear isogenies from A to At (in the isogeny category) are related through the K-action on A. Hence, given a single K-linear Q-polarization φ : A → At , we just have to work out the condition on c ∈ K × that φ ◦ i(c) is a Q-polarization. Since φ is a symmetric K-linear isogeny, the symmetry condition on φ ◦ i(c) says exactly that i(c) = i(c∗ ) in End0 (A); i.e., c ∈ K0 . In [53, §22] there are given Hermitian forms that describe explicit K-linear polarizations on analytic models in Example 1.4.3 for a set of representatives of the K-linear isogeny classes of abelian varieties with each CM type (K, Φ) over C. These explicit Hermitian forms show that the positivity condition on φ ◦ i(c) says precisely that H1 (i(c))R preserves the positive-definiteness property of (A.2.3.1). This translates into the condition that for each ϕ ∈ Φ the totally real algebraic number ϕ(c) ∈ Q ⊂ C is positive. This is equivalent to the condition that c ∈ K0 is totally positive, since Φ|K0 is the set of all embeddings of K0 into R (as Φ is a CM type on K). A.2.3.4. Lemma. Let (A0 , i0 ) and (A00 , i00 ) be abelian varieties with CM type (K, Φ) over a number field L. Assume L is so large that Hom0 ((A0 , i0 ), (A00 , i00 )) is nonzero (see Proposition 0 1.4.4.1). If P is a prime of good reduction for A0 and A00 , and A0 and A0 denote the associated reductions over κ(P), then the injective reduction mapping 0

0

Hom0 ((A0 , i0 ), (A00 , i00 )) → Hom0 ((A0 , i0 ), (A0 , i0 )) is bijective. Proof. Since the abelian varieties all have dimension g with [K : Q] = 2g, the source and target Hom0 ’s are each at most 1-dimensional over K due to the injectivity in Proposition 1.2.5.1. The reduction mapping is K-linear, so the assumption of non-vanishing in characteristic 0 gives the result. Consider triples (A, i, φ) where (A, i) is a CM abelian variety of type (K, Φ) over Q and φ : A → At is a Q-polarization of A that is K-linear. If σ ∈ Gal(Q/E) (where E is the reflex field for (K, Φ)) then scalar extension along the isomorphism σ : Q ' Q gives another triple (Aσ , iσ , φσ ) with iσ (c) := i(c)σ ∈ End0 (Aσ ). The Lie algebras of A and Aσ satisfy Lie(Aσ ) ' σ ∗ (Lie(A)) as K ⊗Q Q-modules, so (Aσ , iσ ) has CM type σΦ = Φ. Hence, by Proposition 1.4.4.1, there exists a K-linear isogeny A → Aσ . A fundamental fact is that such isogenies can be found that respect K-linear Q-polarizations up to rational homothety: A.2.3.5. Theorem. The triples (A, i, φ) and (Aσ , iσ , φσ ) are isogenous up to rational homethety. More precisely, if φ : A → At is a K-linear Q-polarization then there is a K-linear isogeny ξ : A → Aσ in the isogeny category over Q such that ξ t ◦ φσ ◦ ξ = qφ,ξ,σ · φ with 1/g . qφ,ξ,σ ∈ Q× >0 , and necessarily qφ,ξ,σ = deg(ξ) The notion of degree for maps in the isogeny category is defined by Z-homogeneity of degree for morphisms of abelian varieties. The intervention of the Q× >0 -multiplier in this theorem is the primary reason that we need to work with Q-polarizations and not just polarizations (also see Remark A.2.3.7). Proof. Fix a choice of φ. Once we find the desired ξ then computing the degree of both 2g sides will then give qφ,ξ,σ = deg(ξ) deg(ξ t ) = deg(ξ)2 , so qφ,ξ,σ = deg(ξ)1/g .

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Let L/E be a finite Galois subextension of Q that is a field of definition for (A, i, φ) in the sense that: (i) A descends to an abelian variety A0 over L, (ii) the K-action on A (in the isogeny category over Q) descends to a ring homomorphism i0 : K → End0 (A0 ), and (iii) there is a K-linear Q-polarization φ0 : A0 → At0 whose Q-scalar extension is φ. We also take L so big that (A.2.3.3)

Hom((A0 , i0 ), (Aσ0 , iσ0 )) = Hom((A, i), (Aσ , iσ )).

Choose a prime P of L such that P is unramified over its contraction p in E, P is a prime of good reduction for A0 , and σ|L = ( L/E P ). In particular, since σ acting on OL fixes P, it follows that P is also a prime of good reduction for Aσ0 . Let A0 denote the N´eron model for A0 over OL,P , so this is an abelian scheme. Viewing σ as an automorphism of OL,P , the abelian scheme A0σ is the N´eron model of its generic fiber Aσ0 , and likewise the dual abelian scheme A0t is the N´eron model of its generic fiber At0 . By the N´eron mapping property for abelian schemes (i.e., the valuative criterion for properness and Weil’s extension lemma), φ0 extends uniquely to a mapping of abelian schemes A0 → A0t . Lemma 2.1.2 ensures that this OL,P -map induces a polarization on the closed fiber. (Since we will be free to choose among infinitely many P, we can instead arrange for the reduction of φ0 modulo P to be a polarization by “denominator-chasing”, thereby avoiding some of the preceding technical issues with N´eron models and abelian schemes.) Let q = qp be the size of the residue field at p, so σ induces the qth-power map on the residue field κ(P) at P. Because we chose L so large that (A.2.3.3) holds, it follows from Lemma A.2.3.4 that there exists a (unique) K-linear isogeny ξ : A0 → Aσ0 in the isogeny (q)

category over L such that its reduction ξ : A0 → A0 over κ(P) is the relative q-Frobenius map FrA0 /κ(P),q , where A0 denotes the closed fiber of A0 . Of course, ξ depends on the choice of P. We claim that ξQ works in the theorem, with the rational multiplier q. Equivalently, we claim that ξ t ◦ φσ0 ◦ ξ = qφ0 . To check this equality as maps in the isogeny category of abelian varieties over L with good reduction at P, it is equivalent to check the corresponding equality for reductions over κ(P). The reduction of φ0 is a Q-polarization on A0 , so it suffices to prove more generally that if X is an abelian variety over a field k of characteristic p > 0 and if q is a power of p then for any Q-polarization φ : X → X t there is an equality FrtX/k,q ◦ φ(q) ◦ FrX/k,q = qφ. We may extend scalars so that k is algebraically closed, and we may multiply φ by a sufficiently divisible nonzero integer so that it is a polarization. Hence, φ = φL for an ample line bundle L on A, where φL : A → At is x 7→ t∗x (L) ⊗ L−1 . Since the formation of φL is compatible with base change, we have φ(q) = φL(q) with L(q) denoting the pullback of L along the projection A(q) → A (or equivalently, along the base change Spec(k) → Spec(k) given by the qth-power map on k). By the functorial properties of the “Mumford construction” L φL , FrtX/k,q ◦ φ(q) ◦ FrX/k,q = φFr∗

(q) ) X/k,q (L

.

Since qφL = φL⊗q , it now suffices to prove Fr∗X/k,q (L(q) ) ' L⊗q for any invertible sheaf L on X, where X is any Fp -scheme. The composite of the relative q-Frobenius FrX/k,q : X → X (q) and the projection X (q) → X is the absolute q-Frobenius map FrX,q : X → X that is the identity on topological spaces and the qth-power map on structure sheaves. Hence, Fr∗X/k,q (L(q) ) ' Fr∗X,q (L). Our problem is therefore to prove Fr∗X,q (L) ' L⊗q for any invertible sheaf L on X. Since FrX,q is the identity on the topological space X and the qth-power × ˇ map on OX , we conclude the proof by a calculation with OX -valued Cech 1-cocycles (for a trivialization of L over X). In the language of the bilinear pairings (A.2.3.2), Theorem A.2.3.5 says hH1 (ξ)(·), H1 (ξ)(·)iφσ ,Q = deg(ξ)1/g · h·, ·iφ,Q

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

149

and in particular we may choose a K-linear isomorphism ϕσ : Aσ → A in the isogeny category over Q such that ϕσ carries φ back to φσ up to Q× >0 -multiple. This isogeny is unique up to the action on A by elements c ∈ K × such that simultaneous multiplication by c on both factors of H1 (A(C), Q) preserves h·, ·iφ,Q up to Q× >0 -multiple. That is, for a K-linear Q-polarization t t ∗ φ : A → At the condition on c is that i(c)t ◦ φ ◦ i(c) is a Q× >0 -multiple of φ. But i(c) = i (c ) and φ is K-linear, so the condition on c is that ?

φ ◦ i(cc∗ ) = qφ = φ ◦ i(q) × for some q ∈ Q× >0 . Equivalently, this says NmK/K0 (c) = q ∈ Q (such a norm is necessarily positive). We conclude:

A.2.3.6. Corollary. The K-linear isogeny ϕσ is unique up to the action of T (Q) ⊆ K × on A, where T is the Q-group T := ker(ResK/Q (Gm )

NmK/K0



ResK0 /Q (Gm )/Gm ).

This T is the torus in Proposition 2.1.4.5. A.2.3.7. Remark. Since the choice of φ is unique up to the action on A by totally positive elements of K0 (by Lemma A.2.3.3), and the action by these elements commutes with the action by T (Q) ⊆ K × , we conclude that the defining condition on ϕσ in Corollary A.2.3.6 is independent of the choice of φ. That is, such a ϕσ carries φ0 to a positive rational multiple of φ0 σ for every K-linear Q-polarization φ0 of A. A.2.3.8. Remark. The preceding argument applies over `-adic fields and adelically, so the elements of AutAK,f (Vf (A)) = A× K,f preserving the A× f -homothety class of the Af -bilinear skew-symmetric Weil pairing h·, ·iφ : Vf (A) × Vf (A) → Af (1) are precisely the elements of T (Af ). A.2.4. Algebraic formulation of the Main Theorem. We retain the same setup as above: we fix the triple (A, i, φ) over Q with (A, i) of type (K, Φ) and φ a K-linear Qpolarization of A. We also choose σ ∈ Gal(Q/E) and we pick a K-linear isogeny ϕσ : Aσ → A carrying φ back to φσ up to Q× -multiple (determined by the degree of ϕσ ); this condition on ϕσ is independent of the choice of φ (by Remark A.2.3.7). Consider the AK,f -linear isomorphism (A.2.4.1)

[σ]

Vf (A) ' Vf (Aσ )

Vf (ϕσ )

'

Vf (A).

The map [σ] in (A.2.4.1) is the standard Galois-action on Q-points, and so in terms of skewsymmetric Af (1)-valued Weil pairings it carries the bilinear form associated to φσ to the bilinear form associated to φ up to multiplication by the total cyclotomic character value χ(σ) ∈ A× f giving the action of σ on Af (1). The second step in (A.2.4.1) is equivariant for the bilinear forms defined by φσ and φ up to positive rational multiple, due to the defining condition on ϕσ . Hence, (A.2.4.1) is an AK,f -linear automorphism of the free rank-1 AK,f -module 1/g . Vf (A) preserving the bilinear form associated to φ up to the A× f -multiplier χ(σ)(deg ϕσ ) × The composite (A.2.4.1) is multiplication by an element µσ,ϕσ ∈ AK,f = ResK/Q (Gm )(Af ) that therefore preserves φ up to an A× f -multiple and so lies in T (Af ) (by Remark A.2.3.8). The T (Q)-ambiguity in the choice of ϕσ implies that the T (Q)-congruence class of the multiplier µσ,ϕσ ∈ T (Af ) does not depend on the choice of ϕσ , and by Lemma A.2.3.3 it does not depend on the choice of φ. Since the pair (A, i) of type (K, Φ) is unique up to K-linear

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isogeny (Proposition 1.4.4.1), it follows that the choice of (A, i) does not affect the element we have just built in T (Q)\T (Af ). Hence, we get a map of sets Gal(Q/E) → T (Q)\T (Af )

(A.2.4.2)

by sending σ ∈ Gal(Q/E) to the common T (Q)-congruence class of elements µσ,ϕσ ∈ T (Af ) for which the composite map in (A.2.4.1) is multiplication by µσ,ϕσ with ϕσ as above. The map (A.2.4.2) only depends on (K, Φ) rather than on (A, i). A.2.4.1. Lemma. The map (A.2.4.2) is a continuous group homomorphism. Proof. We first check that it is a group homomorphism. Choose σ, σ 0 ∈ Gal(Q/E), and pick ϕσ and ϕσ0 . We need to express the isomorphism [σ]

Vf (A) ' Vf (Aσ )

[σ 0 ]

Vf (ϕσ )

0

Vf (A) ' Vf (Aσ )

'

Vf (ϕσ0 )

'

0

Vf (A) 0

(ϕσ )σ

0 0

ϕ

0

σ as ϕσ0 σ ◦ [σ 0 σ] for a suitable choice of isogeny ϕσ0 σ . The isogeny Aσ σ ' (Aσ )σ → Aσ → A 0 carries φ back to φσ σ up to a positive rational multiple and is K-linear, so we may take this composite isogeny as our choice of ϕσ0 σ . Thus, it suffices to prove the commutativity of the outside edge of the diagram

Vf (A)

[σ]

/ Vf (Aσ )

JJ JJ JJ [σ0 ] J [σ 0 σ] JJ%  '

0

Vf (Aσ σ )

ϕσ

(ϕσ

0 )σ

/ Vf (A) 

[σ 0 ]

/ V (Aσ0 ) f

KKK KKK K 0 KKK ϕσ0 ◦(ϕσ )σ %



ϕσ0

Vf (A) Since the left and bottom triangles clearly commute, it is therefore enough to prove that the inner square commutes. This amounts to the obvious identity hτ (τ (x0 )) = τ (h(x0 )) for any Q-map h : X 0 → X between Q-schemes and any x0 ∈ X 0 (Q). We have verified the group homomorphism condition, so to check continuity it is enough to do so near the identity. Thus, we can restrict attention to σ ∈ Gal(Q/L) for a number field L ⊂ Q over which (A, i, φ) is “defined” (any such L must contain E). Let (A0 , i0 , φ0 ) be a descent of (A, i, φ) to such an L. This choice of descent naturally identifies Aσ with A carrying iσ to i and φσ to φ. Taking ϕσ to be the resulting canonical identification Aσ ' A associated to σ ∈ Gal(Q/L), the element µσ,ϕσ ∈ T (Af ) ⊆ A× K,f is the multiplier for the AK,f -linear action by σ ∈ Gal(Q/L) on Vf (A0 ). Thus, the continuity of (A.2.4.1) on the open subgroup Gal(Q/L) ⊆ Gal(Q/E) follows from the continuity of the Galois representation on the total Tate module of any abelian variety over a field of characteristic 0. A.2.4.2. Lemma. The quotient space T (Q)\T (Af ) is Hausdorff. This technical lemma is the key reason for working with the torus T rather than with × the ambient torus ResK/Q (Gm ) (for which the analogue of the lemma is false whenever OK is infinite; i.e., whenever the CM field K is not imaginary quadratic). Proof. We shall prove that T (Q) is a discrete subgroup of T (Af ). Since T (Af ) is a second countable Hausdorff space, we can test discreteness by studying sequences that converge to the identity. The canonical Q-subgroup Gm ⊂ ResK/Q (Gm ) clearly is contained in T . Consider the resulting short exact sequence of Q-groups 1 → Gm → T → T → 1.

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

151

Granting for a moment that T (Q) is discrete in T (Af ), it follows that a sequence in T (Q) tending to 1 in T (Af ) has image in T (Q) that stabilizes at 1. Such a sequence is eventually a sequence in Gm (Q) = Q× tending to 1 in T (Af ) and hence tending to 1 in Gm (Af ) = A× f. × × × b But Q is discrete in A (as it meets the compact open subgroup Z in {±1}), so we would f

be done. It remains to prove that T (Q) is discrete in T (Af ). Since T (Q) is discrete in T (A) = T (R) × T (Af ) (due to the discreteness of Q in A), it is enough to prove that T (R) is compact. (Note that T (R) is not compact.) By Hilbert’s Theorem 90 we have H1 (R, Gm ) = 1, so the natural open map T (R)/R× → T (R) is a topological group isomorphism. It is therefore equivalent to prove that T (R)/R× is compact. By definition of T , the group T (R) is the closed subgroup of elements Y (z1 , . . . , zg ) ∈ (K ⊗K0 R)× K0 ,→R

whose image in K0 ,→R R× under NmK/K0 lies in the diagonal R× . This is identified with the subgroup of points (z1 , . . . , zg ) ∈ (C× )g such that |zj | is independent of j, and by working modulo the diagonally embedded subgroup R× we conclude that T (R)/R× is a quotient of (S 1 )g and so is compact. Q

By Lemma A.2.4.2, the continuous group homomorphism Gal(Q/E) → T (Q)\T (Af ) to an abelian target must factor through the topological abelianization of Gal(Q/E). The continuous ab Artin map E × \A× E → Gal(E /E) is a surjection and it identifies the abelianized Galois group × × with the topological group quotient of E × \A× E by the closure in E \AE of the image of the × identity component of the archimedean part (R ⊗Q E) [4, Ch. the reflex field E Q IX]. Since × × is a CM field and hence has no real places, (R ⊗Q E) ' w|∞ (Ew ) is connected. The infinite divisibility of (Ew )× for each w|∞ on E therefore implies that the restricted Artin ab map E × \A× E,f → Gal(E /E) is surjective and identifies the abelianized Galois group with the maximal Hausdorff group quotient of E × \A× E,f (i.e., the quotient by the closure of the identity point). Hence, we have built a natural continuous composite mapping of topological groups (A.2.4.3)

ab E × \A× E,f  Gal(E /E) → T (Q)\T (Af )

that only depends on the CM type (K, Φ) and not on the paritcular triple (A, i, φ) used to build it in the first place. It is therefore natural to demand a direct description of (A.2.4.3) in terms of (K, Φ). A.2.4.3. Remark. Note that any continuous group homomorphism h : E × \A× E,f → T (Q)\T (Af ) ab must factor continuously through the surjective Artin quotient map E × \A× E,f  Gal(E /E). Indeed, h is a mapping to a Hausdorff target group T (Q)\T (Af ), and so it factors continuously through the maximal Hausdorff group quotient of the source.

We may rewrite the composite map (A.2.4.3) in the form ResE/Q (Gm )(Q)\ResE/Q (Gm )(Af ) → T (Q)\T (Af ). To the CM type (K, Φ) we associate the reflex norm NΦ : ResE/Q (Gm ) → T (Definition 2.1.4.3 and Proposition 2.1.4.5). Hence, it is reasonable to ask if the maps induced by the reflex norm on finite-adelic points and on rational points have any relation to the composite mapping (A.2.4.3). This is the content of the following formulation of the Main Theorem of Complex Multiplication:

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

A.2.4.4. Theorem (Main Theorem of Complex Multiplication; algebraic form). The map× ping (A.2.4.3) is induced by s 7→ NΦ (s)−1 ∈ T (Af ) ⊆ A× K,f for s ∈ AE,f . There is a further important aspect to the Main Theorem: giving an adelic description of (Aσ , iσ , φσ ) in terms of (A, i, φ). We shall address this aspect in Theorem A.2.7.2 in the case of CM order OK and in Theorem A.2.8.8 in the general case. A.2.4.5. Remark. The reason NΦ (s)−1 rather than NΦ (s) appears in the statement of the Main Theorem is due to our convention for defining the Artin map: it carries local uniformizers to arithmetic Frobenius elements. (See above Remark A.2.6.3.) If we were to use the alternative convention (common in algebraic geometry) that the Artin map carries local uniformizers to geometric Frobenius elements then the identification of Gal(E ab /E) with the maximal Hausdorff group quotient of E × \A× E,f would be modified by inversion and so the formula in the Main Theorem would lose the inversion. A.2.5. Some applications. Before we take up the proof of Theorem A.2.4.4, we deduce several interesting arithmetic consequences. Pick σ ∈ Gal(Q/E) and choose s ∈ A× E,f such ab that its image rE (s) ∈ Gal(E /E) under the Artin map is σ|E ab . By Remark A.2.4.3, NΦ (s)−1 ∈ T (Af ) has class modulo T (Q) that only depends on rE (s) = σ|E ab ∈ Gal(E ab /E) and not on the choice its lifting s through the Artin map. By the Main Theorem, the element NΦ (s)−1 ∈ A× K,f = AutAK,f (Vf (A)) agrees modulo T (Q) with the composite in (A.2.4.1) for any choice of K-linear isogeny ϕσ : Aσ → A carrying φ back to a (necessarily positive) rational multiple of φσ . Since this ϕσ is unique up to precisely the T (Q)-action on A, we may uniquely choose ϕσ to get the exact agreement: NΦ (s)−1 = Vf (ϕσ ) ◦ [σ]. To summarize: A.2.5.1. Proposition. For all σ ∈ Gal(Q/E) and s ∈ A× E,f such that rE (s) = σ|E ab , there is a unique K-linear isogeny λσ,s : (A, i) → (Aσ , iσ ) such that NΦ (s−1 ) · Vf (λσ,s ) = [σ], where [σ] : Vf (A) ' Vf (Aσ ) is the natural map defined by σ-action on torsion points. By the construction, λσ,s satisfies deg(λσ,s )1/g ∈ Q× and carries φσ back to to deg(λσ,s )1/g φ for all K-linear polarizations φ of (A, i). Since [σ] induces an isomorphism on underlying total Tate modules, the rational multiplier deg(λσ,s )1/g is easy to compute by formation of adelic degree on the identity NΦ (s−1 )Vf (λσ,s ) = [σ]: it is the gth root of the generalized adelic lattice index [Tf (A) : NΦ (s)Tf (A)] = NmK/Q (NΦ (s)) = NmK0 /Q (NΦ (s)NΦ∗ (s)) = NmK0 /Q (NE/Q (s)) = NmE/Q (s)g (the third equality due to NΦ factoring through the Q-subtorus T ⊂ ResK/Q (Gm )). In other words, if we let qs ∈ Q× >0 be the unique positive generator of the fractional Q-ideal associated to the finite Q-idele NE/Q (s) then λσ,s carries φσ back to qs φ. We can describe qs b× as a finite idele rather directly: qs = χ(σ)NE/Q (s) inside A× f , where χ : Gal(Q/Q) → Z is the total cyclotomic character. Indeed, passing from Q-ideles to fractional Q-ideals gives the × result, provided that χ(σ)NE/Q (s) ∈ Q× >0 inside of Af . By functoriality of the Artin map with respect to the extension Q → E, the image of NE/Q (s) in Gal(Qab /Q) is σ|Qab . Since our local Artin map convention implies that the global Artin map for Q identifies Gal(Qab /Q) with b × = Q× \A× /R× = Q× \A× via the reciprocal of the cyclotomic character, we conclude Z >0 >0 Q Q,f × that NmE/Q (s) and χ(σ)−1 represent the same coset in Q× >0 \AQ,f , as desired. A.2.5.2. Remark. If we replace s with cs for c ∈ E × , then since NΦ (cs) = NΦ (c)NΦ (s) with NΦ (c) ∈ T (Q) ⊆ K × , we conclude by uniqueness that λσ,cs = NΦ (c)λσ,s .

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

153

Let L ⊂ Q be a number field that is a field of definition for (A, i), so E ⊆ L. We let (A0 , i0 ) be a descent of (A, i) to L. The `-adic representations attached to A0 (for all primes `) define a continuous group homomorphism ρ : Gal(Q/L) → AutAK,f (Vf (A0 )) ' A× K,f , so this uniquely factors through a continuous homomorphism ρab : Gal(Lab /L) → A× K,f . Pick × 0 0 ab ab 0 s ∈ AL,f , so rL (s ) ∈ Gal(L /L) acts on Vf (A0 ) via ρ (rL (s )). We want to describe this action: × such that ρab (r (s0 )) = A.2.5.3. Theorem. For s0 ∈ A× L L,f there is a unique λs0 ∈ K × 0 −1 0 × NΦ (NmL/E (s )) · λs0 in AK,f . Moreover, s 7→ λs0 ∈ K is continuous for the discrete topology on K × .

Proof. By functoriality of the Artin symbol, the restriction of rL (s0 ) to E ab ⊆ Lab is rE (NmL/E (s0 )). By working over Q we may find some φ as above, and upon choosing σ 0 ∈ Gal(Q/L) ⊆ Gal(Q/E) lifting rL (s0 ) we get that σ 0 |E ab = rE (s) for s = NmL/E (s0 ). Using the L-structure on (A, i) provided by the identification A ' (A0 )Q , we get a canonical K0 0 linear isomorphism Aσ ' A and hence a K-linear isogeny λσ0 ,s : A → Aσ ' A such that NΦ (s)−1 ·Vf (λσ0 ,s0 ) = ρ(σ 0 ) as endomorphisms of Vf (A). Since ρ(σ 0 ) = ρab (rL (s0 )), we conclude that λσ0 ,s0 only depends on s0 and not on σ 0 . But K is its own centralizer in End0 (A), so the K-linear λσ0 ,s0 is multiplication by an element λs0 ∈ K × . This completes the construction of λs0 ∈ K × such that ρab (rL (s0 )) = NΦ (NmL/E (s))−1 · λs0 in A× K,f . 0 It remains to prove continuity of s 7→ λs0 with respect to the discrete topology on K × . That is, we want λs0 = 1 for s0 ∈ A× L,f sufficiently near 1. Since λs0 = ρab (rL (s0 ))NΦ (NL/E (s0 )) ∈ A× K,f , the mapping s0 7→ λs0 ∈ K × is continuous for the topology on K × induced by its inclusion into × 0 the group A× K,f of finite K-ideles. In particular, for s sufficiently near 1 in AL,f and a fixed × choice of positive integer M > 3 we have several properties: λs0 ∈ OK , λs0 ≡ 1 mod M , and λs0 lies in the CM order OK ∩ End(A). Hence, for such s0 the element λs0 is an automorphism of the abelian variety A acting trivially on the M -torsion. If we can prove that λs0 has finite order then since M > 3 such an automorphism must be the identity and so we will be done. Pick a K-linear Q-polarization φ for (A, i). Taking 0 ab s0 ∈ A× L,f sufficiently near 1, we can arrange that the element rL (s ) ∈ Gal(L /L) has a lift to Gal(Q/L) that acts trivially on a number field of definition for φ over L (after applying a base change to (A0 , i0 )). It therefore follows from the construction of λs0 that for such s0 the element λs0 ∈ K × acts as an automorphism of A (not merely in the isogeny category) preserving φ up to a positive rational multiple. Degree considerations force this positive rational multiplier to be 1, and the automorphism group of a Q-polarized abelian variety is finite. Hence, λs0 has finite order as desired. ab × A.2.5.4. Remark. Since the global Artin map A× L → Gal(L /L) kills L , by Theorem × A.2.5.3 we get an algebraic Hecke character A× L → K in the sense of Definition 2.4.3(b) via × s 7→ λsf (trivial on L∞ ). Its algebraic part is NΦ ◦ NmL/E .

We next use the Main Theorem (or rather, its consequence in Theorem A.2.5.3) to deduce some results concerning L-functions. First, we eliminate the intervention of Q by working with an “abstract” number field as the base field, as follows. Let A0 be an abelian variety of dimension g over a number field L and let i0 : K → End0L (A0 ) be a CM structure on A over L. The extension L((A0 )tor )/L is abelian, so it uniquely embeds into Lab over L. The field L has no real places because it contains a CM field: upon embedding L into Q to define a

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

Q-valued CM type Φ for (A0 , i0 ), we know that L contains the reflex field E for (K, Φ) and E × is a CM field. Hence, L× ∞ = (R ⊗Q L) is connected and infinitely divisible. The associated CM character is the map × ab α(A0 ,i0 )/L : A× L  Gal(L /L) → AK,f ab defined by the Galois action on torsion of A. The image of L× ∞ in Gal(L /L) is trivial, so × α(A0 ,i0 )/L (s) only depends on the finite component sf ∈ AL,f for any s ∈ A× L. The reflex norm NΦ : ResE/Q (Gm ) → T ⊆ ResK/Q (Gm ) was defined in Definition 2.1.4.3 (and lands in T by Proposition 2.1.4.5). The specification of the CM type Φ and the reflex subfield E ⊆ L depend on a choice of embedding of L into Q, but we have:

A.2.5.5. Lemma. The composite Q-homomorphism NΦL : ResL/Q (Gm )

NmL/E

→

N

ResE/Q (Gm ) →Φ ResK/Q (Gm )

is independent of the embedding of L into Q. This lemma will be especially useful when applied to points valued in the Q-algebra Af . Proof. By the Zariski-density of the Q-points in any Q-torus (due to the unirationality of tori over any field), it suffices to check that the composite map on Q-points is independent of the Q-embedding of L. Thus, it suffices to prove that Nτ Φ (τ x) = NΦ (x) in K × for any x ∈ E × and τ ∈ Gal(Q/E). By definition of the reflex norm, it is equivalent to prove ?

detK (τ (x) : tτ Φ ' tτ Φ ) = detK (x : tΦ ' tΦ ) in K for all x ∈ E. Q Q The natural map hτ i : ϕ∈Φ Q ' ψ∈τ Φ Q defined by (xϕ ) 7→ (τ (xτ −1 ψ )) carries the “descent datum” action (xϕ ) 7→ (g(xg−1 ϕ )) by g ∈ Gal(Q/E) over to the action by τ gτ −1 ∈ Gal(Q/τ (E)) and respects the K-actions (but carries multiplication by y ∈ E over to multiplication by τ (y) ∈ τ (E)). Thus, the isomorphism hτ i descends to an isomorphism tΦ ' tτ Φ linear over the ring isomorphism 1 ⊗ τ : K ⊗Q E ' K ⊗Q τ (E) (viewing E and τ (E) as subfields of Q). The equality of K-determinants for the x-action on tΦ and the τ (x)-action on tτ Φ therefore drops out (for any x ∈ E). × A.2.5.6. Remark. By Theorem A.2.5.3, for each s ∈ A× L there is a unique λs ∈ K depending only on sf such that α(A0 ,i0 )/L (s) = λs NΦL (sf )−1 ,

and we have λs = 1 if sf is sufficiently close to 1. Since α(A0 ,i0 )/L kills L× , it also follows that for c ∈ L× , λc = NΦL (c) in K × . Consideration of quadratic twists shows that α(A0 ,i0 )/L generally depends on (A0 , i0 ) over L and not just on its Q-fiber (i.e., not just on (K, Φ)). A.2.5.7. Remark. Two elementary properties of the construction s 7→ λs are (i) for s ∈ A× L, × the fractional K-ideal [NΦL (sf )]K associated to the idele NΦL (sf ) ∈ AK,f is λs OK , and (ii) ∗ λs λ∗s = NL/Q (sf )α(A,i)/L (s)α(A,i)/L (s) in A× f . Identity (ii) follows from picking an embedding of L into Q (so as to get a CM type Φ) and using the identity NΦ · NΦ∗ = NmE/Q (see the proof of Proposition 2.1.4.5) and the transitivity of norms. To prove (i), first note that the automorphism Vf (λs · NΦL (s−1 f )) = [σ] of Vf (A0 ) is an automorphism of the total Tate module T (A ) and hence is an automorphism of the free rank-1 module it generates over the ring 0 Qf O of integral K-adeles. Hence, the fractional K-ideals λs OK and [NΦL (sf )]K have v-∞ Kv the same order at all finite places of K, so these ideals coincide.

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

155

A.2.5.8. Theorem. Let (A0 , i0 ) be a CM abelian variety over a number field L ⊂ Q with × be the character s 7→ λ i0 : K → End0 (A0 ) for a CM field K. Let λ : A× sf that is L → K × × trivial on L∞ and continuous for the discrete topology on K . Pick a prime P of L. (1) The abelian variety A0 has good reduction at P if and only if λP := λ|L× is trivial P

× on OL . P (2) For a prime P of good reduction and a uniformizer πP at P, the element λP (πP ) ∈ K × lies in OK and in the isogeny category over κ(P) it acts on the reduction of A0 over κ(P) as the qP -Frobenius endomorphism.

Proof. To check good reduction at P, we choose a rational prime ` distinct from the residue characteristic of P and we need to determine if the action of an inertia group IP at P is trivial × on V` (A0 ). The image of IP in Gal(Lab /L) is the image of OL under the Artin map, so it P × comes from ideles s ∈ AL with trivial `-part. Hence, the formula (A.2.5.1)

α(A0 ,i0 )/L (s) = λs NΦL (sf )−1

× implies that α(A0 ,i0 )/L (s) has trivial `-part for all s ∈ OL if and only if the value λs ∈ K × P × viewed in K`× is trivial for all s ∈ OL . That is, A0 has good reduction at P if and only if λP P × is trivial on OLP . Now choose P at which there is good reduction, and pick ` as above. The preceding calculation shows that for a local uniformizer πP , the arithmetic Frobenius element at P acts on V` (A0 ) via the action of λP (πP ) ∈ K × . Passing to the reduction A0 at P, it follows that the action by λP (πP ) on V` (A0 ) agrees with the action by the qP -Frobenius endomorphism, and in End0 (A0 ) the element λP (πP ) ∈ K × coincides with this Frobenius endomorphism. In particular, λP (πP ) acts as a genuine endomorphism of A0 (not just in the isogeny category) and so is integral over Z. Thus, as an element of K it lies in OK .

For any embedding τ : K ,→ C× , let λτ = τ ◦ λ with λ as in Theorem A.2.5.8. Define × × × × α∞ = N−1 ΦL ,∞ · λ : AL → K∞ , where NΦL ,∞ is the composite of NΦL : AL → AK and the × × projection A× K  K∞ . In particular, NΦL ,∞ kills AL,f . Clearly α∞ is continuous, and it kills L× due to Remark A.2.5.6. Hence, for each τ : K ,→ C and the corresponding archimedean place v of K, the composite map τ

α

∞ × × × ατ : A× L → K∞  Kv ' C

τ and λτ of ατ and λτ to L× ⊂ A× coincide for all is a Hecke character and the restrictions αP P L P primes P of L.

A.2.5.9. Remark. The restriction of the Hecke character ατ to L× ∞ is the continuous extension of the map NΦ

τ

L× →L K × → C× that is algebraic with respect to Q in the sense of Definition 2.4.1. More specifically, the C× -valued ατ (or, better, the K × -valued λ) is an algebraic Hecke character, as we noted in Remark A.2.5.4. Let us make this explicit by unraveling the proof of the equivalence of Definitions 2.4.3(a) and 2.4.3(b) in this case. × is trivial on the open Let f be an integral nonzero ideal of OL such that λ : A× L → K × subgroup Uf ⊂ AL,f of integral finite ideles congruent to 1 modulo f, so the group I(f) of τ fractional ideals of L relatively prime to f is naturally a quotient of A× L,f /Uf . Hence, α induces a well-defined homomorphism [ατ ] : I(f) → τ (K)× ⊂ C× that sends a prime P to τ (π ) = τ (λ (π )) for any local uniformizer π αP P P P P at P. On the subgroup P (f) ⊆ I(f) of

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principal fractional ideals of the form xOL with x ∈ L× satisfying x ≡ 1 mod f, we have [ατ ](xOL ) = ατ (xf ) = ατ (x−1 ∞ ) = τ (NΦL ,∞ (x)). Let E ⊂ L be the reflex field of (K, Φ). For each embedding σ : E ,→ Q, pick σ e ∈ Gal(Q/Q) lifting σ via the canonical inclusion of E into Q given in the definition of the reflex field E. e ◦ ϕ for some (necessarily unique) Let Σ(Φ, τ ) be the set of σ ∈ Hom(E, Q) such that τ = σ ϕ ∈ Φ. For xOL ∈ P (f) with x ≡ 1 mod f we have Y Y σ(NL/E (x)) = [ατ ](xOL ) = τ (NΦL ,∞ (x)) = ψ(x) σ∈Σ(Φ,τ )

ψ:L→Q,ψ|E ∈Σ(Φ,τ )

and (as we noted in Remark A.2.5.7) the set Ψτ of ψ’s in this final product satisfies Y ψ(x)ψ(x) = NL/Q (x) ψ∈Ψτ

for all x ∈ L× . It follows (e.g., by Artin’s lemma on linear independence of characters, or arguments with algebraic tori) that Ψτ is a set of representatives for the archimedean places of L. For each τ , Theorem A.2.5.8(1) says that A0 over L has good reduction at P if and only if is unramified at P. By Theorem A.2.5.8(2) and the Riemann Hypothesis for abelian −1/2 varieties over finite fields, the Hecke character || · ||L ατ takes values in the unit circle. This unitary character is non-trivial, for otherwise by working with degree-1 primes of L it would follow from the Riemann Hypothesis for abelian varieties that K contains square roots of infinitely many odd rational primes, an absurdity. (It can happen for CM elliptic curves that this non-trivial unitary character has finite order, and even order 2.) Hence, the Euler products that define each L(s, ατ ) are absolutely and uniformly convergent in half-planes Re(s) > 3/2+ε for all ε > 0, and extend to holomorphic functions on C. These Hecke L-functions compute the L-function of A0 : ατ

A.2.5.10. Theorem. For s ∈ C with Re(s) > 3/2, Y L(s, A0 ) = L(s, ατ ). τ :K→C

In particular, the L-function for A0 has an analytic continuation to C. Note that in the product in the theorem, we do repeat conjugate pairs of embeddings. Proof. We compare Euler factors at good and bad primes separately. Let P be a prime of good reduction for A0 , and pick ` distinct from the residue characteristic of P. Let λ(P) ∈ K × be the common value of λ on local ideles coming from uniformizers at P, and likewise for λτ . By Theorem A.2.5.8, we get detQ` (1 − λ(P)X) = NK` /Q` (1 − λ(P)X) = NK/Q (1 − λ(P)X) = NC⊗Q K/K (1 − (1 ⊗ λ(P))X), Q Q and under the natural decomposition C⊗Q K ' τ C we get that this norm is τ (1−λτ (P)X). −s For X = qP this yields the product of the Euler factors at P for the right side of the proposed identity because ατ and λτ have the same value on any idele with trivial archimedean component. Now we turn to the bad primes for A0 . At such primes we know by Theorem A.2.5.8(1) × that λ is non-trivial on OL and hence all Hecke characters ατ are ramified at P. Thus, the P Euler factor at P for the L-function of each ατ is trivial. We therefore have to prove that for ` as above, the abelian `-adic representation V` (A0 ) for Gal(Q/L) has vanishing subspace × of inertial invariants at P. Since λ(OL ) ⊂ K × is nontrivial, it contains some element P × × x ∈ K × − {1}. But λ(OL ) viewed in K`× is the image of the `-part of α(A0 ,i0 )/L on OL P P

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(due to (A.2.5.1)), so the subspace of inertial invariants at P is contained in the subspace of V` (A0 ) killed by x − 1 ∈ K × . Hence, this subspace is 0.

A.2.5.11. Remark. Theorem A.2.5.10 can be generalized to the case of L-functions of abelian varieties that are merely potentially CM (i.e., acquire a CM structure over a finite extension of the base field). Note that if there is a CM structure over the base field then a CM reflex field lies in the base field, so a potentially CM abelian variety over a real number field (such as Q) cannot have a CM structure over the base field. Next, we record how λ = λ(A0 ,i0 )/L behaves with respect to the action by Aut(K/Q). A.2.5.12. Theorem. For γ ∈ Aut(K/Q), λ(A0 ,i0 ◦γ −1 )/L = γ ◦ λ(A0 ,i0 )/L . Proof. Pick an embedding of L into Q, so we get a Q-valued CM type Φ for (A0 , i0 ) and a reflex field E ⊆ L. In this way, we canonically identify Lab with a subfield of Q over L. Pick s ∈ A× L and σ ∈ Gal(Q/L) with σ|Lab = rL (s). We need to prove ?

Vf (λ(A0 ,i0 ◦γ −1 )/L (s)) = Vf (γ(λ(A0 ,i0 )/L (s))) = Vf (γ)(Vf (λ(A0 ,i0 )/L (s))) in A× K,f when the left and right sides are respectively viewed as endomorphisms of Vf (A0 ) with AK,f -module structures defined by i0 ◦ γ −1 and i respectively. In other words, for NΦL := NΦ ◦ NmL/E and N(Φ◦γ)L := NΦ◦γ ◦ NmL/E we need to prove ?

N(Φ◦γ)L (s) · (Vf (i0 ) ◦ γ −1 )−1 ([σ]) = Vf (γ)(NΦL (s) · Vf (i0 )−1 ([σ])) = Vf (γ)(NΦL (s)) · Vf (γ)(Vf (i0 )−1 ([σ])) (where Vf (i0 ) : AK,f ' EndAK,f (Vf (A0 )) is the natural isomorphism). The second factors on the left and right sides clearly agree, so we just have to show γ −1 (N(Φ◦γ)L (s)) = NΦ (s) × × × in A× K,f for all s ∈ AL . Thus, we just have to prove that the isomorphism γ : AK ' AK carries NΦ◦γ to NΦ as maps from A× E. In view of the algebraic definition of the reflex norm as a K-determinant, the problem reduces to constructing a K ⊗Q E-module isomorphism K ⊗γ −1 ,K tΦ◦γ −1 ' tΦ . We apply the extension of scalars E → Q and just have to compare dimensions of isotypic factors for the K-action. By definition via descent, the K ⊗Q Q-module Y K ⊗γ −1 ,K (tΦ ⊗E Q) ' K ⊗γ −1 ,K Q ψ∈Φ◦γ −1

is Q-linearly isomorphic to a product of copies of Q indexed by ψ ∈ Φ ◦ γ −1 such that K acts on the ψ-th copy via ψ ◦ γ. Hence, the isotypic pieces are 1-dimensional over Q and the eigencharacters are the elements of the set (Φ ◦ γ −1 ) ◦ γ = Φ. In some references one also sees a discussion of the behavior of λ(A0 ,i0 )/L behavior with respect to the action by Gal(Q/Q), namely that λ(Aτ0 ,iτ0 )/τ (L) (sτ ) = λ(A0 ,i0 )/L (s). This says × exactly that the character λ(A0 ,i0 )/L : A× L → K is independent of the initial embedding of L into Q used to define Φ, which in turn is a consequence of Lemma A.2.5.5 (due to Theorem A.2.5.3).

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A.2.6. Hom-modules and fractional ideals. We now begin the proof of the Theorem A.2.4.4. By Remark A.2.3.7 the choice of K-linear Q-polarization φ for (A, i) does not matter, and any two pairs (A, i) and (A0 , i0 ) of type (K, Φ) over Q are K-linearly isogenous, so in fact the choice of triple (A, i, φ) of type (K, Φ) does not matter. Thus, by the final part of Proposition 1.6.4.3, we may and do take (A, i) to be principal (i.e., i−1 (End(A)) = OK ); cf. Remark 2.5.1.1. This opens the door to applying Serre’s tensor construction from 1.6.4. Unlike adelic multiplication operations in the isogeny category that we shall develop in A.2.8 when the CM order is not necessarily maximal, the Serre tensor construction is applicable at torsion levels in the principal case. This is why we have passed to the principal case. Pick σ ∈ Gal(Q/E). Choose a finite Galois extension L/E inside of Q such that (A, i) descends to a pair (A0 , i0 ) over L and such that the K-linear Q-polarization φ of A descends to a K-linear Q-polarization φ0 of A0 . (We will have no need for φ0 until after (A.2.7.3), so the reader may safely forget about it until then.) Make L big enough so that it splits K over Q (i.e., it contains the Galois closure of K in Q) and so that (A.2.6.1)

Hom((A0 , i0 ), (Aσ0 , iσ0 )) = Hom((A, i), (Aσ , iσ )).

In particular, this module of K-linear mappings over L is an invertible OK -module. We write aσ to denote this abstract invertible OK -module. It will soon be naturally identified with a fractional K-ideal (once we choose a suitable auxiliary prime of L). Let N be the product of all primes ` of Q that arise as the residue characteristic of a prime factor of disc(L/Q) or a prime of bad reduction for A0 . (Note that if ` is ramified in K then ` is ramified in L and hence divides N , as L is assumed to split K over Q.) We will be interested in primes P of OL [1/N ] such that σ|L = ( L/E P ). Before we pick such a P, we discuss an important feature of working over Z[1/N ]: the K ⊗Q E-module tΦ that is implicit in the definition of the reflex norm (Definition 2.1.4.3) has a natural integral structure over Z[1/N ]. To be precise, first note that L contains all embeddings of K into Q Q (due to the “largeness” of the choice of L), so tΦ ⊗E Q descends the K ⊗Q L-module ϕ∈Φ L. Since the generically Galois extension OE [1/N ] → OL [1/N ] is finite ´etale, we may use finite Q ´etale descent in place of Galois descent to descend the OK [1/N ] ⊗Z[1/N ] OL [1/N ]-module ϕ∈Φ OL [1/N ] to an OK [1/N ] ⊗Z[1/N ] OE [1/N ]-module tΦ,Z[1/N ] such that Q ⊗Z[1/N ] tΦ,Z[1/N ] ' tΦ as K ⊗Q Emodules. In particular, tΦ,Z[1/N ] is a finite locally free module over each of OK [1/N ] and OE [1/N ]. Thus, the reflex norm NΦ between Q-tori extends to a homomorphism between Z[1/N ]-tori (A.2.6.2)

NΦ,Z[1/N ] : ResOE [1/N ]/Z[1/N ] (Gm ) → ResOK [1/N ]/Z[1/N ] (Gm )

defined on R-points (for a Z[1/N ]-algebra R) by x 7→ detOK [1/N ]⊗Z[1/N ] R (x : tΦ,Z[1/N ] ⊗Z[1/N ] R ' tΦ,Z[1/N ] ⊗Z[1/N ] R) ∈ (OK [1/N ] ⊗Z[1/N ] R)× for x ∈ (OE [1/N ] ⊗Z[1/N ] R)× . (The map NΦ,Z[1/N ] factors through the closed subtorus TZ[1/N ] = ker(ResOK [1/N ]/Z[1/N ] (Gm )

NOK [1/N ]/OK



0

[1/N ]

ResOK0 [1/N ]/Z[1/N ] (Gm )/Gm )

because it does so on generic fibers over Q, but we do not need to use TZ[1/N ] and so we will not mention it any further.) Q Q × × × It follows that NΦ : A× w-N OEw into v-N OKv and also respects E,f → AK,f carries integrality at the collection of places over any rational prime ` - N . Thus, for any place w of E away from N and any local uniformizer πw of Ew , if we let ιw : Ew× → A× E,f be the natural × inclusion map then the element NΦ (ιw (πw )) ∈ AK,f has local component 1 at all places v of K not over p and is integral at all places v of K over p. Moreover, the idele NΦ (ιw (πw )) changes by an integral unit multiple in all components at places of K over p when πw is replaced with × an OE -multiple. Hence, the reflex norm induces a well-defined homomorphism on fractional w

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

159

ideal groups away from N : if p is a prime ideal of E away from N then we may define the fractional K-ideal Y (A.2.6.3) NΦ (p) := qordq (NΦ (ιp (πp ))) q|pOK

for any uniformizer πp of OEp . Now pick a prime P of L over a rational prime p - N such that σ|L = ( L/E P ). Let p be the prime of E below P. By Lemma A.2.3.4 and (A.2.6.1), there is a unique K-linear morphism ξσ,P : A0 → Aσ0 over L whose reduction is the relative q-Frobenius morphism (q)

FrA0 /κ(P),q : A0 → A0 , where q = #κ(p). The nonzero element ξσ,P ∈ aσ := Hom((A0 , i0 ), (Aσ0 , iσ0 )) endows the 1-dimensional K-vector space K ⊗OK aσ with a distinguished basis and so identifies aσ with a fractional K-ideal aσ,P ⊂ K that contains OK and depends on P. Equivalently, composition with ξσ,P defines an OK -linear embedding of invertible OK -modules OK = Hom((A0 , i0 ), (A0 , i0 ))

ξσ,P ◦(·)

→

Hom((A0 , i0 ), (Aσ0 , iσ0 )) = aσ .

Of course, if we change the choice of the prime P of OL [1/N ] such that ( L/E P ) = σ|L then this inverse-integral fractional K-ideal aσ,P is replaced with another such ideal. Note that the fractional ideal aσ,P has nothing to do with φ. Rather important for us is the fact that it is given by a simple formula that only depends on (K, Φ) and P: A.2.6.1. Theorem. With notation as above, aσ,P = NΦ (p)−1 with p = P ∩ E. Proof. Since ξσ,P has p-power degree, OK has p-power index in aσ,P . Hence, the fractional ideal aσ,P is a unit away from p. By construction, the same holds for NΦ (p). It is therefore enough to fix a place v of K over p and to compare ordv ’s of aσ,P and NΦ (p)−1 . Rather than make such a comparison directly, it will be convenient to compare after raising both ideals to the f (P|p)th-power. To make effective use of this, we need to describe anσ,P for positive integers n. n First of all, note that for any positive integer n, all K-linear maps A → Aσ descend n n to K-linear maps A0 → Aσ0 . This says that the Gal(Q/L)-action on Hom(A, Aσ ) (via the n L-structures A0 and Aσ0 ) is trivial on the subgroup of K-linear maps, and it suffices to check this in the isogeny category. The subspace of K-linear maps in the isogeny category is 1dimensional as a K-vector space, and the K-action on A over Q descends to A0 over L, so it n suffices to exhibit a single nonzero K-linear mapping A0 → Aσ0 . There is a K-linear isogeny n−1 h0 : A0 → Aσ0 (e.g., ξσ,P ), so the composite map hσ0 ◦ · · · ◦ hσ0 ◦ h0 does the job. We conclude that for all n > 1 n n aσn := Hom((A0 , i0 ), (Aσ0 , iσ0 )) is an invertible OK -module equal to the module of such mappings over Q. Hence, by Example 1.6.4.1, the natural evaluation mapping aσn ⊗OK A0 → Aσ0

n

over L is an isomorphism (as it becomes an isomorphism over Q). Next, observe that the Hom-module aσn contains a unique element ξσ,n,P that lifts the (q n )

K-linear relative q n -Frobenius morphism A0 → A0 over κ(P). This identifies aσn with a fractional K-ideal aσ,n,P . For example, aσ,1,P = aσ,P . What is the ideal aσ,n,P in general? Since FrA0 /κ(P),qn is roughly an n-fold composite of FrA0 /κ(P),q , the following answer is not a surprise:

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A.2.6.2. Lemma. For all n > 1, aσ,n,P = anσ,P . Proof. Since all K-linear maps between A and Aσ are defined over L relative to the respective descents A0 and Aσ0 , we have a canonical evaluation morphism aσ ⊗OK A0 → Aσ0 over L and it is an isomorphism. Hence, for any τ ∈ Gal(Q/E), applying scalar extension along the isomorphism τ : L ' L and using the base-change compatibility of Serre’s construction yields a natural K-linear isomorphism aσ ⊗OK Aτ0 ' (aσ ⊗OK A0 )τ ' (Aσ0 )τ ' Aτ0 σ

(A.2.6.4)

over L, carrying 1 ⊗ iτ0 to iτ0 σ . By the definition of higher relative Frobenius maps, for any τ = σ n with n ∈ Z+ the composite OK -linear module isomorphism aτ σ := Hom((A0 , i0 ), (Aτ0 σ , iτ0 σ ))

(A.2.6.4)

' ' ' '

Hom((A0 , i0 ), (aσ ⊗OK Aτ0 , 1 ⊗ iτ0 )) Hom((A0 , i0 ), (aσ ⊗OK aτ ⊗OK A0 , 1 ⊗ 1 ⊗ i0 )) aσ ⊗OK aτ ⊗OK Hom((A0 , i0 ), (A0 , i0 )) aσ ⊗OK aτ

carries ξσ,n+1,P to ξσ,P ⊗ξσ,n,P . (The reader should check this assertion!) In terms of fractional ideals resting on the distinguished elements ξσ,m,P for m ∈ Z+ , this says aσ,n+1,P = aσ,P aσ,n,P . Hence, by induction on n > 1 we conclude that aσ,n,P = anσ,P for all n > 1. f (P|p)

As a special case of Lemma A.2.6.2, since σ|L has order f (P|p), the fractional ideal aσ,P is associated to the invertible OK -module EndL (A0 , i0 ) = OK endowed with the distinguished element π0 = ξσ,f (B|p),P that lifts the Frobenius qP -endomorphism of the abelian variety A0 f (P|p)

over the finite field κ(P) with size qP = qp so for any place v of K over p we have

f (P|p)

ordv (aσ,P

f (P|p)

. Hence, we conclude that aσ,P

= π0−1 OK ,

) = −ordv (π0 ).

Now recall the Shimura-Taniyama formula from (2.1.5.1): ordv (π0 ) = ordv (qP ) ·

#Φv [Kv : Qp ]

with Φv ⊆ Φ ⊂ HomQ (K, L) equal to the subset of elements ϕ ∈ Φ such that ϕ : K → L carries the P-adic place back to v. Since ordv (qP ) f (P|p)ordv (p) f (P|p) = = , [Kv : Qp ] [Kv : Qp ] f (v|p) ?

the problem ordv (aσ,P ) = ordv (NΦ (p)−1 ) may be rephrased as −

f (P|p) ? · #Φv = ordv (NΦ (p)−f (P|p) ), f (v|p)

or equivalently as (A.2.6.5)

?

f (p|p)#Φv = f (v|p)ordv (NΦ (p)).

We will analyze the right side and eventually transform it into the left side. Choose πp ∈ OE that has order 1 at p and order 0 at all other places of E over p, so by (A.2.6.3) the fractional K-ideal NΦ (p) is the “p-part” of the principal fractional ideal generated by detK (πp : tΦ ' tΦ ) ∈ K × . Hence, passing to the v-part of this K-determinant gives f (v|p)ordv (NΦ (p)) = f (v|p)ordv (detKv (πp : Kv ⊗K tΦ ' Kv ⊗K tΦ )).

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161

For any x ∈ Kv× , f (v|p)ordv (x) = ordp (NKv /Qp (x)) since p is unramified in K. Also, for any Kv -linear endomorphism µ : V → V of a finite-dimensional Kv -vector space V we have NmKv /Qp (detKv (µ)) = detQp (µQp ) with µQp denoting the underlying Qp -linear endomorphism. Taking µ to be multiplication by πp on Kv ⊗K tΦ , f (v|p)ordv (NΦ (p)) = ordp (detQp (πp : Kv ⊗K tΦ ' Kv ⊗K tΦ )). Recall that tΦ as a K ⊗Q E-module is the “generic fiber” of an OK [1/N ] ⊗Z[1/N ] OE [1/N ]module tΦ,Z[1/N ] . Extending scalars by Z[1/N ] → Zp and using the decomposition Kv ⊗K (K ⊗Q E) '

Y

Kv ⊗Qp Ew

w|pOE

(and the analogous one with the Dedekind rings OKv ⊗Zp OEw ), we get a decomposition of Kv ⊗K tΦ into a product of w-parts over all w|pOE . The πp -action respects the integral structure on this module decomposition. Thus, this action has integral unit Zp -determinant × on w-factors for all w 6= p since πp ∈ OE for all w|pOE with w 6= p, so f (v|p)ordv (NΦ (p)) is w equal to ordp of the Qp -determinant of the πp -action on the p-part of Kv ⊗K tΦ . This latter action is Ep -linear, and for any Ep -linear endomorphism µ of a finite-dimensional Ep -vector space V we have ordp (detQp (µQp )) = f (p|p)ordp (detEp (µ)), where µQp is the underlying Qp -linear endomorphism. We conclude that f (v|p)ordv (NΦ (p)) is equal to f (p|p)ordp (detEp (µ)) with µ equal to scalar multiplication by πp ∈ Ep× on the p-part of Kv ⊗K tΦ . The Ep -determinant of such a scalar d action is just πp p , where dp is the Ep -dimension of the p-part of Kv ⊗K tΦ . Hence d

f (v|p)ordv (NΦ (p)) = f (p|p)ordp (πp p ) = f (p|p)dp . It follows that (A.2.6.5) is equivalent to the assertion #Φv = dp . Under the extension of scalars E → L applied to tΦ , the p-part of the Kv ⊗Q E-module Kv ⊗K tΦ is carried by the base change Ep → LP over to the P-part of the Kv ⊗Q L-module Kv ⊗K (tΦ ⊗E L). Thus, this P-part has LP -dimension dp . But by construction of the K ⊗Q E-module tΦ via Galois Q descent, tΦ ⊗E L ' ϕ∈Φ L as K ⊗Q L-modules. Hence, our problem is to prove that the Q Q P-part of the module Pv = Kv ⊗K ( ϕ∈Φ L) over the ring Kv ⊗Q L ' w0 |p Kv ⊗Qp Lw0 has LP -dimension #Φv . Q The module Pv decomposes as ϕ∈Φ (Kv ⊗K,ϕ L), so it suffices to show that Pv,ϕ := Kv ⊗K,ϕ L has vanishing P-part if ϕ 6∈ Φv and that it has P-part with LP -dimension 1 for ϕ ∈ Φv . By Q standard facts concerning completions of global fields, Kv ⊗K,ϕ L ' w0 ∈Σ Lw0 where Σ is the set of places on L lifting v via ϕ. Thus, the P-part of Pv,ϕ is at most 1-dimensional over LP , and it is 1-dimensional precisely when ϕ pulls the P-adic place back to v. That is, the set of ϕ for which there is a nonzero P-part in Pv,ϕ is precisely Φv . This completes the proof of Theorem A.2.6.1. By applying functoriality of the Serre tensor construction with respect to the natural OK -linear inclusion OK → NΦ (p)−1 , we may restate Theorem A.2.6.1 in the following more convenient manner: for any prime P of L over p - N such that σ|L = ( L/E P ), there is a unique OK -linear isomorphism θσ,P : NΦ (p)−1 ⊗OK A ' Aσ0

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

of abelian varieties over L such that the diagram

(A.2.6.6)

ξσ,P

/ Aσ A0 NN O0 NNN NNN NNN ' θσ,P N& NΦ (p)−1 ⊗OK A0

commutes, where the top map is the unique lifting of the relative qp -Frobenius for A0 over κ(P) and the diagonal map is induced by the inclusion of OK into NΦ (p)−1 . The isomorphism θσ,P depends on the choice of P. The isomorphism θσ,P uses a non-canonical L-descent (A0 , i0 ) of (A, i) as well as a noncanonical choice of P. Our task in A.2.7 will be to improve this situation by constructing a canonical K-linear isomorphism

(A.2.6.7)

[NΦ (s)−1 ]K ⊗OK A ' Aσ

of abelian varieties over Q for any σ ∈ Gal(Q/E) and s ∈ A× E,f such that rE (s) = σ|E ab (with (A, i) having CM order OK ), where [·]K denotes the fractional K-ideal associated to a finite K-idele. We consider θσ,P to be an approximation to (A.2.6.7) “at level L” since the Artin map × AE,f  Gal(L/E) carries the idele sp = ιp (πp ) to σ|L (due to our convention on local uniformizers and arithmetic Frobenius elements under the Artin map!) and [NΦ (sp )]K = NΦ (p). The map θσ,P generally does not descend the canonical isomorphism in (A.2.6.7) that will be constructed in A.2.7. Rather, θσ,P over Q will be related to (A.2.6.7) via the action by an element of NΦ (E × ) ⊆ T (Q) ⊆ K × . A.2.6.3. Remark. The most important applications of the Main Theorem are to abelian varieties over number fields, as in A.2.5. However, the situation is very much simplified by first proving the Main Theorem in the form stated over Q (or more traditionally, over C) and only later deducing arithmetic consequences over number fields. The main way in which this simplification manifests itself it is in the task of improving the construction of θσ,P over L to become canonical by passing up to Q. More specifically, to eliminate the intervention of P we will have to apply certain procedures involving passage to a field that splits A0 [M ] for several relatively prime M > 3. By working over Q from the outset we thereby avoid the unpleasant task of having to increase the base field at several places in the middle of an argument. A.2.7. Polarizations and torsion. In A.2.6 the number field L ⊂ Q was a rather general “sufficiently big” finite Galois extension of E, and we now need a few more “largeness” conditions on L. Choose a fixed auxiliary integer M > 1 (later we will take M > 3) and impose the additional “largeness” condition on L that the finite ´etale L-group A0 [M ] is constant. Thus, Aσ0 [M ] is constant, as is (a ⊗OK A0 )[M ] ' a ⊗OK (A0 [M ]) for any fractional ideal a of K. Recall that the choice of the prime P of L was controlled by σ|L (via the condition ( L/E P ) = σ|L ) and by the geometry of A0 and the arithmetic of L and K (since we required the residue characteristic p of P to not divide the product N of the residue characteristics of the primes of bad reduction for A0 and the ramified primes for L and K over Q). We impose the further condition on the residue characteristic p of P that p - M (in addition to the condition p - N ).

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163

The natural map A0 → NΦ (p)−1 ⊗OK A0 has p-power degree and so induces an isomorphism on M -torsion. Hence, we get a diagram on L-points (A.2.7.1)

A0 [M ](L)

[σ]

/ Aσ [M ](L) 0 O SSSS SSSS SS ' θσ,P ' SSSSS )

(NΦ (p)−1 ⊗OK A0 )[M ](L)

that is the same as the associated diagram on Q-points (due to the constancy property of these M -torsion groups over Spec L). The top map in this diagram is Galois-theoretic, and the other two sides arise from K-linear morphisms of abelian varieties over L. We claim that the diagram (A.2.7.1) commutes. Since p - M and the three abelian varieties involved all have good reduction at P, we may identify these L-point groups with the corresponding κ(P)-point groups in the reductions of the abelian-scheme N´eron models. The diagram (A.2.7.1) viewed on κ(P)-points is the same as the diagram induced on M -torsion by the reduction of (A.2.6.6) because the morphism ξσ,P reduces to the morphism FrA0 /κ(P),q whose effect on κ(P)-points is exactly the action by the arithmetic Frobenius element in Gal(κ(P)/κ(p)) on such points. (Here, we use that the Galois-theoretic arithmetic Frobenius automorphism is given by the algebraic formula t 7→ tqp .) This proves the commutativity of (A.2.7.1) as a consequence of the commutativity of (A.2.6.6). To go further, we need to impose one final “largeness” condition on L/E before choosing P: the extension L/E must contain a certain class field for E to now be described. Consider × the reflex norm mapping NΦ : A× E,f → AK,f on Af -points. We have seen via (A.2.6.2) that for rational primes ` - N , the induced mapping on Q` -points Y Y NΦ,` : Ew× = (E ⊗Q Q` )× → (K ⊗Q Q` )× = Kv× w|`

v|`

× × carries OE into OK (where E` := Q` ⊗Q E, OE` := Z` ⊗Z OE ' ` ` Q × K). Define U ⊂ AE,f to be the open subgroup ` U` where

Q

× w|` OEw ,

and similarly for

× × U` = OE ∩ N−1 Φ,` ({u ∈ OK` | u ≡ 1 mod M }) ` × for all rational primes ` (so U` = OE for ` - N M ). We require L to contain the class field ` × × ab has kernel for the open subgroup E E∞ U , so the restricted Artin map A× E,f  Gal(L/E) contained in E × U . −1 to be in the The conditions σ|L = ( L/E P ) and rE (s) = σ|E ab force the idele s · ιp (πp ) × ab × open kernel of the Artin map E × \A× E,f  Gal(L/E) , so s = ιp (πp )uc in AE,f where c ∈ E and u ∈ U , so the element NΦ (u) ∈ A× K,f is everywhere a local unit and satisfies NΦ (u) ≡ 1 mod M as an integral adele for K. (For later purposes, it is convenient to consider u as being determined by s, πp , and c.) In particular, since p - M the finite Q K-idele NΦ (s−1 c) = −1 −1 NΦ (ιp (πp ) u ) that is possibly non-integral at p has component in `|M K`× that is an integral unit congruent to 1 modulo M . Consider the commutative diagram of abelian varieties over L:

(A.2.7.2)

NΦ (p)−1 ⊗OK A0 NΦ (c)−1

θσ,P

/ Aσ 0

' pp8

p ppp p ' p p '  ppp

NΦ (cp)−1 ⊗OK A0

where the diagonal mapping is defined to force commutativity; this diagonal mapping is a K-linear isomorphism. Letting [x]K denote the fraction K-ideal associated to a finite idele

164

A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

x ∈ A× K,f , we have NΦ (cp) = NΦ (c) · NΦ (p) = NΦ (c) · [NΦ (ιp (πp ))]K = [NΦ (c · ιp (πp ))]K = [NΦ (su−1 )]K = [NΦ (s)]K Q × since NΦ (u) ∈ w OE ⊂ A× E,f . Thus, the K-linear isomorphism of abelian varieties over L w given by the diagonal map in (A.2.7.2) may be expressed as an abstract isomorphism (A.2.7.3)

θσ,P,πp ,c,s,M : [NΦ (s−1 )]K ⊗OK A0 ' Aσ0 .

We shall prove that if M > 3 then the K-linear isomorphism [NΦ (s−1 )]K ⊗OK A ' Aσ obtained by applying the base change L → Q to (A.2.7.3) only depends on σ ∈ Gal(Q/E) and s ∈ A× E,f (linked by the condition rE (s) = σ|E ab ), and not on the choice of L/E, the choice of Ldescent (A0 , i0 , φ0 ) of (A, i, φ), or the choices of P, πp , c, or M . We will also show that these canonical isomorphisms over Q are well-behaved with respect to composition in Gal(Q/E) and multiplication in A× E,f . These matters will be settled in Theorem A.2.7.2 and Remark A.2.7.3. Up to now, in A.2.6 and A.2.7 the K-linear Q-polarization φ and its L-descent φ0 on A0 have played no role. We have arrived at the point where φ0 will be used. We must endow [NΦ (s−1 )]K ⊗OK A0 with a natural K-linear Q-polarization φ0,s over L such that θσ,P,πp ,c,s,M carries φ0,s to a positive rational multiple of φσ0 (and the rational multiplier is made explicit in terms of s). In view of how θσ,P,πp ,c,s,M is defined, and the fact that the mapping ξσ,P in (A.2.6.6) that lifts FrA0 /κ(P),qp respects the Q-polarizations φ0 and φσ0 up to a positive rational multiple (as we showed in the proof of Theorem A.2.3.5), our task comes down to proving: A.2.7.1. Lemma. Let (A, i) be a CM abelian variety of type (K, Φ) over Q with CM order OK , and let φ be a K-linear Q-polarization on A. There is a unique way to assign to each α ∈ A× E,f a K-linear Q-polarization φα to [NΦ (α)]K ⊗OK A over Q such that: (1) φ1 = φ via the canonical identification OK ⊗OK A = A, (2) φα only depends on [NΦ (α)]K , (3) if h : [NΦ (α)]K → [NΦ (β)]K is an OK -linear map induced by multiplication by c ∈ NΦ (E × ) ⊆ K × then the K-linear isogeny h⊗1A : [NΦ (α)]K ⊗OK A → [NΦ (β)]K ⊗OK A carries φα to qc,α,β φβ with qc,α,β ∈ Q× >0 the unique positive generator of the fractional Q-ideal [NmE/Q (β/cα)]Q . Moreover, deg φα = deg φ and if L ⊆ Q is a subfield such that (A, i, φ) descends to a triple (A0 , i0 , φ0 ) over L then each φα uniquely descends to a K-linear Q-polarization φ0,α of A0 . Proof. Galois descent works for Q-polarizations because descent theory is effective for polarizations. Hence, the compatibility of the Serre construction with respect to change of the base field (such as scalar extension along automorphisms of Q) and the uniqueness aspect in the lemma imply the final descent claim in the lemma via Galois descent. Thus, we may and do focus on the existence, uniqueness, and degree problems over Q. The uniqueness aspect is obvious, via parts (1) and (3). For the existence aspect, let qs ∈ Q× >0 be the unique positive generator of [NmE/Q (s)]Q for × each s ∈ AE,f . Working in the isogeny category and using the evident AK,f -linear isomorphism Vf ([NΦ (α)]K ⊗OK A) = Vf (A) carrying Tf ([NΦ (α)]K ⊗OK A) to NΦ (α) · Tf (A), there is a unique Q-polarization φα on [NΦ (α)]K ⊗OK A satisfying the identity of adelic Weil pairings eφα = qα−1 eφ . By definition of the target torus T for the reflex norm NΦ : ResE/Q (Gm ) → T ⊆ ResK/Q (Gm ),

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

165

the product mapping NΦ N∗Φ = NΦ NΦ∗ factors through the subtorus Gm and as such is equal to NmE/Q (see Proposition 2.1.4.5 and its proof). Thus, eφα is uniquely determined by Iα = [NΦ (α)]K because the product of Iα and its complex conjugate is the fractional K-ideal qα OK with qα as its unique positive rational generator. The degree of a Q-polarization can be detected by studying the failure of perfectness and integrality on the total Tate module, so this φα clearly satisfies all of the requirements. Lemma A.2.7.1 and the preceding discussion provide K-linear isomorphisms σ θσ,P,πp ,c,s,M : [NΦ (s)]−1 K ⊗OK A0 ' A0

of abelian varieties over L carrying φ0,1/s to a positive rational multiple of φσ0 ; the rational multiplier must be 1 because deg(φσ ) = deg φ = deg φ1/s . We shall now prove that the isomorphism obtained from θσ,P,πp ,c,s,M after extension of scalars to Q only depends on σ and s, so it can be called θσ,s , and we will prove that θσ,s is well-behaved with respect to composition in Gal(Q/E) and multiplication in A× E,f . Since NΦ (c) · NΦ (p) = [NΦ (s)]K , we can define the K-linear isogeny θp,c : A0 → [NΦ (s)]−1 K ⊗OK A0 over L to be the composite mapping A0

/ NΦ (p)−1 ⊗O A0 K

NΦ (c)−1 ⊗1 '

/ [NΦ (s)]−1 ⊗O A0 . K K

This isogeny has p-power degree, so it induces an isomorphism on M -torsion. Since NΦ (sc−1 ) = NΦ (ιp (πp )u) with p - M and NΦ (u) ≡ 1 mod M , it follows that on the level of geometric points the composite isomorphism θp,c

−1 A0 [M ] ' ([NΦ (s)]−1 K ⊗OK A0 )[M ] ' [NΦ (s)]K ⊗OK A0 [M ]

' ([NΦ (s)−1 ]K /M [NΦ (s)−1 ]K ) ⊗OK /M OK A0 [M ] of constant finite ´etale OK /M OK -modules over Spec L is induced by multiplication by the idele NΦ (s)−1 that is naturally a representative of a basis for the invertible OK /M OK -module [NΦ (s)−1 ]K /M [NΦ (s)−1 ]K . Consider induced maps on Q-points of M -torsion subgroups: A0 [M ](Q) θp,c '



[σ]

/ Aσ [M ](Q) 0 ' k k 5 ' kkkkk

kkkk kkkk θσ,P,πp ,c,s,M

([NΦ (s)]−1 K ⊗OK A0 )[M ](Q)

Since these Q-point groups coincide with L-point groups, the commutativity of this diagram follows from the commutativity of (A.2.7.1) and (A.2.7.2). Via the isomorphism (A0 )Q ' A (carrying (i0 )Q and (φ0 )Q to i and φ respectively, and likewise after base change by σ), the Klinear isomorphism θσ,P,πp ,c,s,M of abelian varieties over L must therefore induce the following composite of canonical maps on the Q-points of M -torsion subgroups: (A.2.7.4) NΦ (s)

[σ]

−1 σ ([NΦ (s)]−1 K ⊗OK A)[M ](Q) ' [NΦ (s )]K ⊗OK (A[M ](Q)) ' A[M ](Q) ' A [M ](Q).

The steps in this composite are independent of all choices (including L/E and the descent (A0 , i0 , φ0 ) over L) aside from the choices of σ, s, M , and for any M 0 |M it is clear that on M 0 torsion subgroups this intrinsic map on M -torsion restricts to the analogously described map arising from θσ,P,πp ,c,s,M 0 . In other words, θσ,P,πp ,c,s,M and θσ,P,πp ,c,s,M 0 agree on M 0 -torsion subgroups for M 0 |M .

166

A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

−1 The automorphism θσ,P0 ,π0 0 ,c0 ,s,M ◦ θσ,P,π lies in Aut(Aσ0 , iσ0 , φσ0 ) = Aut(Aσ , iσ , φσ ), p ,c,s,M p where the preservation of the polarization is due to the fact that both θ’s carry φ to φσ . We allow for the possibility P0 = P and c0 = c but πp0 6= πp . The preceding argument via (A.2.7.4) shows that this automorphism of Aσ0 induces the identity on M -torsion. Since the automorphism group of a Q-polarized abelian variety is finite, this automorphism of Aσ0 has finite order. However, it acts as the identity on M -torsion, so as long as we restrict attention to M > 3 this finite-order automorphism is trivial. Thus, if M > 3 then the K-linear isomorphism σ (θσ,P,πp ,c,s,M )Q : [NΦ (s)]−1 K ⊗OK A ' A

of abelian varieties over Q only depends on σ, s, M ; we therefore denote this isomorphism θσ,s,M . For M 0 |M with M 0 > 3, the same argument now shows that the K-linear automorphism −1 σ over Q is trivial on M 0 -torsion and hence is trivial. θσ,s,M ◦ θσ,s,M 0 of the abelian variety A Thus, for any M1 , M2 > 3 we have θσ,s,M1 = θσ,s,M1 M2 = θσ,s,M2 , so θσ,s,M is independent of M > 3. (See Remark A.2.6.3.) We summarize the conclusion of our efforts as follows: A.2.7.2. Theorem. Let (A, i) be a CM abelian variety of type (K, Φ) over Q, and assume (A, i) has CM order OK . For any σ ∈ Gal(Q/E) and s ∈ A× E,f such that rE (s) = σ|E ab there is a unique K-linear isomorphism σ θσ,s = θσ,s,(A,i) : [NΦ (s)]−1 K ⊗OK A ' A

such that for all M > 1 the isomorphism [σ] : A[M ](Q) ' Aσ [M ](Q) is the composite (A.2.7.5)

NΦ (s−1 )

A[M ](Q)

'

/ ([NΦ (s−1 )]K /M [NΦ (s−1 )]K ) ⊗O /M O A[M ](Q) K K '



([NΦ (s)]−1 K ⊗OK A)[M ](Q) θσ,s '



Aσ [M ](Q). For a K-linear Q-polarization φ on A, the Q-polarization φ1/s of [NΦ (s)]−1 K ⊗OK A and σ σ the Q-polarization φ of A are intertwined by the isomorphism θσ,s . Also, for any c ∈ E × we have θσ,cs = NΦ (c)θσ,s , and the formation of θσ,s,(A,i) is natural in the principal CM abelian variety (A, i). Proof. The uniqueness follows from the specification of the induced map on every torsion level (for all M ). The existence is the content of the entire preceding analysis, and the proof of existence also gives the asserted behavior with respect to K-linear Q-polarizations. The behavior with respect to relacing s with cs for c ∈ E × comes from the uniqueness (or, less elegantly, from the construction). The naturality in (A, i) follows from the explicit canonical description (A.2.7.5) on every torsion level.

A.2.7.3. Remark. We briefly address the behavior of θσ,s with respect to composition in σ and multiplication in s. Pick σ, σ 0 ∈ Gal(Q/E) and s, s0 ∈ A× E,f satisfying rE (s) = σ|E ab and rE (s0 ) = σ 0 |E ab , so rE (s0 s) = (σ 0 σ)|E ab . We claim that θσ0 σ,s0 s,(A,i) is the composite

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

167

isomorphism [NΦ (s0 s)]−1 K ⊗OK A

' 1⊗θσ,s,(A,i)

'

θσ0 ,s0 ,(Aσ ,iσ )

−1 [NΦ (s0 )]−1 K ⊗OK ([NΦ (s)]K ⊗OK A) σ [NΦ (s0 )]−1 K ⊗OK A

'

(Aσ )σ

'

Aσ σ .

0

0

This equality may be checked by comparing what happens on M -torsion for arbitrary M > 3. This is a straightforward calculation because [σ] : A(Q) ' Aσ (Q) is K-linear with respect to i and iσ . The descent down to models over number fields has served its purpose, and for the rest of the proof of Theorem A.2.4.4 we will work exclusively over Q. Passing to the inverse limit on (A.2.7.5) and tensoring with Q gives a commutative diagram (A.2.7.6)

' / ' / Vf ([NΦ (s)]−1 [NΦ (s)]−1 K ⊗OK A) K ⊗OK Vf (A) QQQ 4 i i i QQQ i ' iii QQQ mult. ' ' Vf (θσ,s ) iiii Q i i i NΦ (s)−1 QQQ( i   i i i / Vf (Aσ ) Vf (A)

Vf (A) Q

'

in which the composite across the top and right sides is [σ] and the right diagonal and lower horizontal maps are defined by commutativity. Since NΦ (s) ∈ T (Af ) ⊆ A× K,f , the multiplication map by NΦ (s)−1 respects the A× -homothety class of the skew-symmetric bilinear form f induced by φ. Suppose for a moment that the diagonal mapping Vf (A) → Vf ([NΦ (s)]−1 K ⊗OK A) in −1 0 (A.2.7.6) is induced by a K-linear isogeny ψs ∈ Hom (A, [NΦ (s)]K ⊗OK A) that respects the K-linear Q-polarizations φ and φ1/s up to a (necessarily positive) rational multiple. The K-linear isogeny λσ,s = θσ,s ◦ ψs ∈ Hom0 (A, Aσ ) therefore carries φ to φσ up to a positive rational multiple and satisfies Vf (λσ,s ) · NΦ (s)−1 = [σ] in HomAK,f (Vf (A), Vf (Aσ )), completing the proof of the Main Theorem! (Note that this map λσ,s coincides with the one introduced with the same notation in the initial part of A.2.5, where we saw that λσ,s carries φσ back to qs φ with qs the unique positive generator of [NE/Q (s)]Q .) To construct ψs , first recall that for any c ∈ E × the multiplier NΦ (c) ∈ T (Q) ⊆ K × acting on A (in the isogeny category) preserves φ up to a rational multiple. Since θσ,cs = NΦ (c)θσ,s , it follows that we only need to construct ψcs for some c ∈ E × (and then NΦ (c) · ψcs serves as ψs ). Choosing c ∈ OE − {0} that is sufficiently divisible (depending on s), we can replace s with cs to reduce to the case when NΦ (cs) = NΦ (c)NΦ (s) ∈ A× K,f is everywhere integral as an −1 adele. Hence, OK ⊆ [NΦ (s)]K . In this case the canonical mapping Vf (A) → [NΦ (s)]−1 K ⊗OK Vf (A) is inverse to the vertical multiplication mapping in (A.2.7.6), so the map A → [NΦ (s)]−1 K ⊗OK A induced by the canonical inclusion OK → [NΦ (s)]−1 may be taken to be ψ . (The compatibility s K of this map with K-linear Q-polarizations follows from property (3) in Lemma A.2.7.1, taking α = 1 and β = 1/s.) The Main Theorem of Complex Multiplication is now proved. A.2.7.4. Remark. Without recourse to replacing s with cs for a sufficiently divisible nonzero c ∈ OK as in the preceding proof, the map ψs can always be described by the same recipe as at the end of the proof: it is the element in −1 0 Hom0K (A, [NΦ (s)]−1 K ⊗OK A) ' [NΦ (s)]K ⊗OK HomK (A, A) ' K

168

A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

corresponding to 1 (where the second isomorphism is induced by multiplication, due to Example 1.6.4.1). A.2.8. Analytic version of Main Theorem and generalized Serre construction. To complete A.2, we explain how to express the Main Theorem in terms of adelic operations and analytic uniformizations. This recovers the traditional statement of the Main Theorem, except that we formulate things in more intrinsic terms. For example, our formulation of the analytic version of the Main Theorem omits all mention of bases of tangent spaces, in contrast with [72, 18.6]. We also explain how to translate such adelic operations into algebraic language by generalizing the Serre tensor construction from 1.6.4 within the framework of certain abelian varieties of CM type over a field, and this provides an algebraic generalization of Theorem A.2.7.2 to the case of any CM order. For CM abelian varieties over C with a fixed CM type (K, Φ), where Φ ⊂ Hom(K, C) = Hom(K, Q), morphisms among such objects uniquely and functorially descends to the subfield Q (Lemma 1.2.1.2, Theorem 1.6.2.1). Thus, to avoid far-out things such as Aut(C) we shall consider our CM abelian varieties (A, i) with CM type (K, Φ) to be given over Q. Consider the canonical analytic exponential uniformization V /Λ ' A(C). Clearly Λ is a module for the CM order i−1 (End(A)) ⊆ OK , and A(C)tor = ΛQ /Λ. Hence, ΛQ is a 1dimensional K-vector space and Q the R-vector space R ⊗Q ΛQ ' V with its complex structure is isomorphic to (R ⊗Q K)Φ ' ϕ∈Φ Cϕ = Cg as a C ⊗Q K-module (where K acts on Cϕ = C through ϕ). Traditionally, an identification of V with (R ⊗Q K)Φ as C ⊗Q K-modules is chosen in the analytic formulation of the Main Theorem because this identifies the “abstract” Λ with something more concrete, namely a Z-lattice in (R ⊗Q K)Φ = Cg that is stable under some order of OK . However, there is no essential benefit obtained in this way, so we avoid such a choice. Given σ ∈ Aut(Q/E), our goal is to describe an analytic uniformization of Aσ (C) in terms of the canonical one for A such that the composite map σ

A(C)tor = A(Q)tor ' Aσ (Q)tor = Aσ (C)tor is easily described via an adelic operation on lattices arising from these analytic uniformizations. We need to first discuss adelic operations on ΛQ /Λ. Somewhat more abstractly, let W be a finite-dimensional nonzero K-vector space and let Λ ⊂ W be a Z-lattice that is stable under some order O0 of OK . We call such a lattice Λ an order lattice in W . There are many orders O0 that preserve Λ, but there is a unique largest one, namely EndZ (Λ) ∩ K ⊆ OK inside of EndQ (W ), and it is called the endomorphism order for Λ in K. The quotient W/Λ is a torsion Z-module, so it is a torsion O0 -module. Under contraction each maximal ideal of OK gives rise to one of O0 (though several in OK may give rise to the same one in O0 ), and for each maximal ideal v 0 of O0 we write Ov0 0 to denote the Q corresponding completion of O0 , so this completion is a local order in the semi-local product v|v0 Ov of the local rings for K at places Q over v 0 . In particular, Ov0 0 has total ring of fractions Kv0 := v|v0 Kv . For all but finitely many v on K we have that v is the only prime of OK over its contraction v 0 on O0 , and then Ov0 0 = Ov inside of Kv . There is a canonical isomorphism M (Wv0 /Λv0 ) W/Λ ' v0

Q Q with Wv0 := v|v0 (Kv ⊗K W ) a free module of rank 1 over Kv0 = v|v0 Kv and Λv0 := Ov0 0 ⊗O0 Λ. Concretely, Wv0 /Λv0 is identified with the m0v0 -power torsion submodule of W/Λ. It is easy the to check that the subset Λv0 ⊂ Wv0 is the closure of Λ in Wv0 , and that if we shrink Q order O0 to some O00 ⊆ O0 then for each maximal ideal v 00 of O00 we have Wv00 = v0 |v00 Wv0 Q and Λv00 = v0 |v00 Λv0 . In this sense, the description of the primary decomposition of W/Λ is

A.2. THE MAIN THEOREM OF COMPLEX MULTIPLICATION

169

well-behaved with respect to change in the choice of order O0 in K preserving Λ (as there is a unique maximal such O0 containing all others). A.2.8.1. Lemma. Let O0 be the endomorphism order for Λ in K. (1) For any s ∈ A× order K,f , there is a unique order lattice sΛ in W with endomorphism Q 0 0 0 0 0 0 0 0 O such that (sΛ)v = sv Λv inside of Wv for all v on O , where sv ∈ v|v0 Ov denotes the v 0 -part of the idele s. (2) There is a unique O0 -linear isomorphism W/Λ ' W/(sΛ) such that on v 0 -factors it is the mapping Wv0 /Λv0 ' Wv0 /(sv0 Λv0 ) induced by multiplication by sv0 on Wv0 . 0 0 (3) If s, s0 ∈ A× K,f then s (sΛ) = (s s)Λ inside of W . b A quick of Lemma A.2.8.1 can be given by working with Af -modules and Z-algebras Q proof 0 0 (such as v0 Ov0 ), but we prefer to give a proof that is “intrinsic” to the order O and avoids the crutch of the subring Z over which O0 is finite and flat. Proof. For all but finitely many v 0 , the endomorphism order O0 for Λ is maximal at v 0 and × sv0 ∈ OK . Hence, by multiplying Λ by a sufficiently divisible nonzero element of OK we get v0 Q an OK -lattice Λ0 ⊆ Λ such that v|v0 Λ0v ⊆ sv0 Λv0 for all v 0 . Moreover, for all but finitely many Q v 0 we have v|v0 Λ0v = sv0 Λv0 = Λv0 inside of Wv0 . Our construction problem therefore takes place inside of the OK -module W/Λ0 , and we need to look inside of the I-torsion submodule for a sufficiently divisible nonzero ideal I of OK . Such a torsion submodule is of finite length over OK , and so decomposing along the finitely many primes in its support gives a solution to our O0 -module existence problem via finitely many local constructions. The uniqueness is seen in the same way. The uniqueness implies s0 (sΛ) = (s0 s)Λ. By working locally, we see that the endomorphism orders for Λ and sΛ coincide. The existence and uniqueness of the desired isomorphism W/Λ ' W/(sΛ) is proved by using the decomposition into v 0 -components for each v 0 . A.2.8.2. Example. If Λ ⊂ W is a finitely generated OK -submodule then sΛ = [s]K Λ, where [s]K is the fractional K-ideal associated to s ∈ A× K,f . We can now prove the Main Theorem in its analytic guise as originally stated by Shimura and Taniyama in a coordinatized manner (but we avoid their “coordinates” on tangent spaces). There are two parts to this theorem: the first gives a description of the analytic uniformization of a Galois twist preserving the reflex field and the second describes the Riemann form of a Galois twist of a K-linear polarization (providing an analytic version of the observation in A.2.5 that λσ,s carries φσ back to qs φ with qs ∈ Q× >0 the unique positive generator of [NmE/Q (s)]Q ). A.2.8.3. Theorem (Main Theorem of Complex Multiplication; analytic form). Pick σ ∈ ab Gal(Q/E) and s ∈ A× E,f such that rE (s) = σ|E ab in Gal(E /E). Let φ be a K-linear Qpolarization of A. There is a unique C ⊗Q K-linear identification of V = Lie(A(C)) with Lie(Aσ (C)) under which the canonical analytic uniformization of Aσ (C) is identified with V /NΦ (1/s)Λ and the diagram ΛQ /Λ

'

/ A(C)tor o

'

NΦ (1/s) '



ΛQ /NΦ (1/s)Λ

A(Q)tor ' [σ]

/ Aσ (C)tor o ' '



Aσ (Q)tor

commutes. Under the identification of H1 (Aσ (C), Q) with ΛQ , the Q(1)-valued Riemann form Ψφσ is identified with qs Ψφ , where qs is the unique positive generator of the fractional Q-ideal [NmE/Q (s)]Q .

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See Remark A.2.4.5 for the reason 1/s rather than s intervenes in this diagram. Proof. Any two such isomorphisms V ' Lie(Aσ (C)) are related through the action of an automorphism of V that preserves Λ and induces on V /Λ ' A(C) an analytic endomorphism that is the identity on all torsion points and so is the identity. This establishes uniqueness. For existence, we use the algebraic form of the Main Theorem over Q in Theorem A.2.4.4. As we saw at the start of A.2.5, this provides a (necessarily unique) K-linear isomorphism in the isogeny category λσ,s ∈ Hom0 (A, Aσ ) (possibly not a genuine morphism of abelian varieties) such that the diagram (A.2.8.1)

[σ]

/ Vf (Aσ ) u: uu u u NΦ (1/s) ' uu  uu Vf (λσ,s )

Vf (A)

'

Vf (A) commutes and λσ,s carries φσ back to qs φ. Letting Vσ /Λσ be the canonical analytic uniformization of Aσ (C), there is a well-defined K-linear map H1 (λσ,s ) : ΛQ ' (Λσ )Q on rational homologies. The R-scalar extension of this Q-vector space isomorphism is the C-linear isomorphism V ' Vσ induced by λσ,s on tangent spaces at the identity. I claim that H1 (λσ,s ) carries NΦ (1/s)Λ isomorphically onto Λσ . It suffices to check this inside of the Af -modules obtained through extension of scalars, but the commutative diagram (A.2.8.1) can be rewritten as a commutative diagram (A.2.8.2)

NΦ (1/s)



[σ]Af

/ Af ⊗Z Λσ 8 qqq q q qq qqq H1 (λσ,s )

Af ⊗Z Λ

'

Af ⊗Z Λ

b → Af to the isomorphism where [σ]Af is obtained by applying the scalar extension Z b ⊗Z Λ = Tf (A) [σ] b ⊗Z Λσ Z ' Tf (Aσ ) = Z b b ⊗Z (NΦ (1/s)Λ) of Z b ⊗Z Λ along the left side of of finite free Z-modules. Hence, the image Z (A.2.8.2) (check this really is the image!) is carried by the Af -module map Af ⊗Q H1 (λσ,s ) b ⊗Z Λσ . This confirms that H1 (λσ,s ) carries NΦ (1/s)Λ onto Λσ , since isomorphically onto Z b Z → Z is faithfully flat. In this way, we get an analytic isomorphism V /NΦ (1/s)Λ ' Vσ /Λσ = Aσ (C). This identifies Aσ (C)tor with ΛQ /NΦ (1/s)Λ. Due to how this identification has been constructed, when it is used in conjunction with the identification ΛQ /Λ ' A(C)tor we see that the map [σ]

A(C)tor = A(Q)tor ' Aσ (Q)tor = Aσ (C)tor is identified with the isomorphism ΛQ /Λ ' ΛQ /NΦ (1/s)Λ induced by NΦ (1/s)-multiplication on primary components (as in Lemma A.2.8.1). This completes the proof of Theorem A.2.8.3.

A.2.8.4. Remark. For a principal CM abelian variety (A, i) of type (K, Φ) over Q, Theorem A.2.8.3 and Example A.2.8.2 provide a K-linear analytic isomorphism Aσ (C) ' V /NΦ (1/s)Λ ' [NΦ (1/s)]K ⊗OK (V /Λ) ' ([NΦ (1/s)]K ⊗OK A)(C), and by GAGA and Lemma 1.2.1.2 this is induced by a K-linear isomorphism Aσ ' [NΦ (1/s)]K ⊗OK A

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over Q. This isomorphism is precisely the isomorphism θσ,s that is uniquely characterized in Theorem A.2.7.2. In this sense, Theorem A.2.8.3 may be considered to be a generalization of Theorem A.2.7.2 to the non-principal case. The interested reader may check as an exercise that the identification of Ψφσ with qs Ψφ in Theorem A.2.8.3 is equivalent to the description of Ψφσ in terms of Ψφ given in [72, 18.6] using the classical description of Riemann forms of K-linear Q-polarizations via 4-tuples (K, Φ, a, ζ) (resting on choosing bases for tangent spaces as rank-1 free modules over (R ⊗Q K)Φ ). We now finish the discussion of adelic operations by using a generalization the Serre tensor construction to give an algebraic formulation and proof of the adelic multiplication formalism in Theorem A.2.8.3. Let (A, i) and (A0 , i0 ) be CM abelian varieties over an algebraically closed field k of characteristic 0, with CM field K. Thus, Hom((A0 , i0 ), (A, i)) is a lattice in the K-vector space Hom0 ((A0 , i0 ), (A, i)) that has dimension 6 1. Hence, for any finite idele s ∈ A× K,f another such lattice is s Hom((A0 , i0 ), (A, i)). For a fixed (A, i), consider the following functor F = F(A,i) from the category of such pairs (A0 , i0 ) to the category of order lattices in finite-dimensional K-vector spaces: F (A0 , i0 ) = s Hom((A0 , i0 ), (A, i)). In the case k = C, if A(C) = V /Λ and A0 (C) = V 0 /Λ0 then working locally over Z shows that F (A0 , i0 ) = Hom((V 0 /Λ0 , i0 ), (V /sΛ, is )) inside of HomK (Λ0Q , ΛQ ), with is denoting the action map on sΛ by an order in K. Thus, an algebraic version of the adelic operation Λ sΛ on lattices is provided by: A.2.8.5. Lemma. Let (A, i) over k be as above, and work in the category of K-linear maps over k in what follows. The functor F(A,i) is represented by a pair (sA, is ) and there is a canonical K-linear equality Hom(A0 , sA) = Hom0 (A0 , A) ∩ HomAf [Gal(k/k)] (Tf (A0 ), sTf (A)) inside of HomAf [Gal(k/k)] (Vf (A0 ), Vf (A)). In particular, there is a canonical isomorphism (sA)t ' s−1 At and the formation of the representing object sA commutes with extension of the base field k 0 /k in the sense that the canonical map (sA)k0 → s · Ak0 is an isomorphism. Proof. It is harmless to multiply s by a sufficiently divisible nonzero integer so that it lies in the profinite completion O∧ of a common order O ⊆ OK acting on A and A0 . Hence, suppressing explicit mention of the K-action maps, the subset s Hom(A0 , A) ⊆ Hom(A0 , A) is given by local conditions: its `-adic completion is s` (Z` ⊗Z Hom(A0 , A)) for all primes ` (with s` ∈ (Q` ⊗Q K)× ). Under the injection of Hom(A0 , A) into the O` -module Hom(A0 [`∞ ], A[`∞ ]) such elements are precisely those that kill the kernel of the isogeny s` on A0 [`∞ ]. By O` -linearity of the maps under consideration, it is equivalent to require that the dual map of `-divisible groups kills the kernel of s` acting on At [`∞ ] through duality (without the intervention of complex conjugation on K). Hence, if we let G` ⊂ At [`∞ ] be the k-finite kernel of the isogeny given by the dual action of s` then G` = 0 for all but finitely many ` and so there is a unique k-finite subgroup G in At whose `-component is G` for all `. Dualizing again, F(A,i) (A0 , i0 ) consists of those O-linear maps A0 → A over k that lift (necessarily uniquely) through the isogeny (At /G)t → A. That is, (At /G)t represents the functor F(A,i) . It is clear from this construction that the other properties hold. The proof of Lemma A.2.8.5 also provides a canonical AK,f -linear and Gal(k/k)-equivariant isomorphism Vf (sA) = Vf (A) within which Tf (sA) goes over to sTf (A), and this respects duality and extension of the base field. There is likewise a unique K-linear isomorphism

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s0 (sA) ' (s0 s)A over k compatible with the total Tate module description (so these isomorphisms are associative with respect to any three ideles). By the construction, in the case b k = C these Tate module descriptions are the Z-scalar extensions arising from unique analytic isomorphisms V /(sΛ) ' s · (V /Λ). In this sense, the formation of sA is an algebraic substitute for adelic operations on uniformization lattices and so (via the Lefschetz principle) it follows that the CM type of sA coincides with that of A. A.2.8.6. Example. In the case of CM order OK , there are canonical K-linear isomorphisms sA ' [s]K ⊗OK A over k that respect iteration of the idelic multiplication and Serre tensor operations. In the case s = NΦ (1/s0 ) for s0 ∈ A× E,f , the resulting identification Vf (A) = Vf (sA) ' Vf ([s]K ⊗OK A) is Vf (ψs0 ) for ψs0 as in A.2.7, due to Remark A.2.7.4. The following lemma is obvious, and by the proof of Lemma A.2.7.1 it recovers the construction in Theorem A.2.7.2 in the principal case: A.2.8.7. Lemma. Let φ be a K-linear polarization on a pair (A, i) over Q with CM type (K, Φ). For all s ∈ A× E,f there is a unique K-linear polarization φs on NΦ (s)A such that the AK,f -linear identification Vf (A) = Vf (NΦ (s)A) carries eφs to qs−1 eφ as total Weil pairings, where qs ∈ Q× >0 is the unique positive generator of [NmE/Q (s)]Q . The algebraic version of Theorem A.2.8.3 is: A.2.8.8. Theorem. Let (A, i) be a CM abelian variety over Q with CM type (K, Φ). Choose σ ∈ Gal(Q/E) and s ∈ A× E,f mapping to σ|E ab under the Artin map. There is a unique K-linear isomorphism θσ,s : NΦ (1/s) · A ' Aσ with respect to which the composite map σ

−1 θσ,s

Tf (A) ' Tf (Aσ ) ' Tf (NΦ (1/s) · A) = NΦ (1/s)Tf (A) is multiplication by NΦ (1/s) ∈ A× K,f . Moreover, for any K-linear polarization φ the isomorphism θσ,s carries φσ to φ1/s . The analogue of Remark A.2.7.3 carries over by essentially the same argument (chasing actions on Vf ’s rather than on torsion subgroups). Proof. This can be deduced from the analytic version in Theorem A.2.8.3, but let us instead give an algebraic proof. In the special case that A has CM order OK , the theorem is exactly Theorem A.2.7.2. In the general case, uniqueness for θσ,s is clear. For existence, first observe that if we construct θσ,s merely in the isogeny category subject to the condition that −1 ) and [σ] is multiplication by N (1/s) on V (A) then consideration with composing Vf (θσ,s Φ f total Tate modules forces θσ,s to be a genuine isomorphism of abelian varieties. Hence, it is enough to carry out the existence proof in the isogeny category. Moreover, if A0 → A is a Klinear isomorphism in the isogeny category then it is equivalent to solve the existence problem for A or for A0 (by functoriality of the idelic multiplication operation on abelian varieties with a fixed CM type). But every A of type (K, Φ) over Q is K-linearly isogenous to one with CM order OK (Proposition 1.6.4.3). Hence, we are done.

Theorem A.2.8.8 accomplishes something remarkable: without Galois theory or complex analysis, it gives an adelic construction of (Aσ , iσ , φσ ) from (A, i, φ): this σ-twisted triple is (NΦ (1/s) · A, iNΦ (1/s) , φ1/s ) with the canonical AK,f -linear identification Vf (NΦ (1/s)A) = Vf (A) carrying eφ1/s to qs eφ , and the K-linear isomorphism θσ,s : NΦ (1/s)A ' Aσ is uniquely

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173

characterized by the fact that it fits into the commutative diagram [σ]

Vf (A)

'

NΦ (1/s) '

' Vf (θσ,s )



Vf (A)

/ Vf (Aσ ) O

'

/ Vf (NΦ (1/s)A)

with the bottom side given by the canonical identification (that in turn arises from a unique K-linear isogeny ψs : A → NΦ (1/s)A).

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A. SOME ARITHMETIC RESULTS FOR ABELIAN VARIETIES

A.3. A converse to the Main Theorem of Complex Multiplication In this section we prove Theorem 2.5.2, which is a converse to the Main Theorem. The argument will use constructions from the proof of the Main Theorem in A.2. The notation and hypotheses from the statement of Theorem 2.5.2 will be freely used below without comment. A.3.1. First we check the uniqueness of (A, α) up to K-linear isogenies. If (A, α) and (A0 , α0 ) are two such pairs, then since they have the same CM type (K, Φ) there is (by Proposition 0 . The invertible K -modules V (A) and V (A0 ) have 1.4.4.1) a K-linear isogeny h : AQ → AQ ` ` ` Gal(Q/F ) acting through the same K`× -valued character (namely the continuous ψ` that is uniquely attached to ), so all K` -linear maps V` (A) → V` (A0 ) are automatically Gal(Q/F )equivariant. Thus, for all σ ∈ Gal(Q/F ) we have V` (σ ∗ (h)) = ρA0 ,` (σ) ◦ V` (h) ◦ ρA,` (σ)−1 = V` (h), so σ ∗ (h) = h. That is, h is defined over F (and its F -descent is a K-linear isogeny). To prove the existence of (A, α), which is the hard part, we will use Galois descent.

A.3.2. By Proposition 1.4.4.1, there is a finite extension F 0 /F inside of Q and a CM abelian variety (A0 , α0 ) over F 0 with CM type (K, Φ). Its associated algebraic Hecke character 0 : × has algebraic part N ◦ Nm 0 A× Φ F /E = alg ◦ NmF 0 /F . Thus, the algebraic Hecke F0 → K 0 0 characters  and  ◦ NmF /F have the same algebraic part. It follows from Remark 2.4.6(ii) that they are related through multiplication by a finite-order Hecke character of F 0 , so upon replacing F 0 with a finite extension we can arrange that 0 =  ◦ NmF 0 /F . We may and do also assume that F 0 /F is Galois. By means of a suitable isogeny over F 0 , we can arrange that OK is the CM order (i.e., OK acts on A0 as an abelian variety, not only in the isogeny category); see Proposition 1.6.4.3. We shall construct an OK -linear descent datum on (A0 , α0 )Q with respect to Q/F , and the resulting F -descent (A, α) of (A0 , α0 )Q will then be checked to have associated algebraic Hecke character ; it necessarily has CM type (K, Φ). For any finite idele ξ ∈ A× K,f , let [ξ]K denote the associated fractional ideal of OK . In × particular, if s ∈ AE,f then the reflex norm NΦ : ResE/Q (Gm ) → ResK/Q (Gm ) yields a fractional ideal [NΦ (s)]K . Recall from 1.6.4 that if X is any OK -module scheme over a scheme S and if M is any finite flat OK -module then there is an associated OK -module scheme M ⊗OK X(S 0 ) on the category of SM ⊗OK X over S that represents the functor S 0 schemes. In the special case that M is invertible and X is an abelian scheme with constant relative dimension g then M ⊗OK X is another such abelian scheme; if X with its OK -action is a CM abelian variety over a field of characteristic 0 then M ⊗OK X is another such (for invertible M ) and it has the same CM type. Choose σ ∈ Gal(Q/E) and s ∈ A× E,f satisfying rE (s) = σ|E ab , and let Is denote the fractional ideal [NΦ (1/s)]K , so for all n > 1 the finite idele NΦ (1/s) ∈ A× K,f defines a canonical generator of the OK /(n)-module Is /nIs . By the Main Theorem of Complex Multiplication in its adelic form for CM abelian varieties with maximal CM order (Theorem A.2.7.2), there is a unique K-linear isomorphism of abelian varieties σ

0 θσ,s : [NΦ (1/s)]K ⊗OK A0Q = Is ⊗OK AQ ' (A0Q )

0 [n](Q) ' (A0 )σ [n](Q) is given over Q such that for all n > 1 the natural isomorphism [σ] : AQ Q by the composite map NΦ (1/s)

0 [n](Q) AQ

'

/ (Is /nIs ) ⊗OK A0 [n](Q) Q

'

/ (Is ⊗OK A0 )[n](Q) Q

θσ,s '

/ (A0 )σ [n](Q) Q

By uniqueness, for any c ∈ E × we have that θσ,cs is the composition of θσ,s and the natural multiplication map by NΦ (c) ∈ K × carrying Ics = [NΦ (1/(cs))]K isomorphically to Is = [NΦ (1/s)]K as fractional ideals of OK . The intervention of NΦ (1/s) rather than NΦ (s) in the definition of θσ,s is due to the use of the arithmetic normalization of the Artin map (implicit

A.3. A CONVERSE TO THE MAIN THEOREM OF COMPLEX MULTIPLICATION

175

in the requirement rE (s) = σ|E ab ). We are interested in applying this with σ ∈ Gal(Q/F ) ⊆ Gal(Q/E) and s = NmF/E (ξ) for any ξ ∈ A× F,f such that rF (ξ) = σ|F ab (so rE (s) = σ|E ab ). A.3.3. For ξ ∈ A× F,f we claim that [NΦ (NmF/E (ξ))]K is the principal fractional ideal generated by (ξ) ∈ K × . To prove this, first recall that NΦ ◦ NmF/E = alg . Thus, it suffices to prove 0 × is an algebraic Hecke character more generally that if L is a number field and χ : A× L → L valued in the multiplicative group of some number field L0 then for any s ∈ A× L , χ(s) is a × × generator of the fractional ideal [(χalg )A (s)]L0 , where (χalg )A : AL → AL0 is the map induced by χalg : ResL/Q (Gm ) → ResL0 /Q (Gm ) on A-points. Note that χ annihilates the infinitely divisible identity component of the archimedean −1 may be viewed as a continuous homomorphism A× /L× → A× . factor L× ∞ , and (χalg )A · χ L L0 In particular, it carries the compact norm-1 idele class group of L into the maximal com× pact subgroup of A× L0 . But composing this map with projection to AL0 ,f kills the image of × the identity component of the archimedean factor L× ∞ and so carries AL,f into the maximal Q × compact subgroup v-∞ OL 0 ,v (since any finite idele of L can be realized as the finite part of a norm-1 idele of L whose archimedean factor is in the identity component). This says that [(χalg )A (s)]L0 = χ(s)OL0 as fractional ideals of OL0 for any s ∈ A× L , as desired. ab such that σ| For σ ∈ Gal(Q/F ) ⊆ Gal(Q/E), choose ξσ ∈ A× F ab = rF (ξσ ) in Gal(F /F ) F,f and let sσ = NmF/E (ξσ ) ∈ A× E,f . Thus, [NΦ (sσ )]K = [(alg )A (ξσ )]K = (ξσ )OK . A.3.3.1. Lemma. The composite K-linear isomorphism 0 AQ

(ξσ )−1

'

θσ,sσ

0 ' (A0Q )σ [NΦ (1/sσ )]K ⊗OK AQ

is independent of the choice of ξσ . In A.3.4 and A.3.5 we will prove that these isomorphisms for varying σ constitute a Galois descent datum on (A0 , α0 )Q with respect to Gal(Q/F ) such that the resulting descent (A, α) over F is the desired CM abelian variety. × × 0 −1 ∈ ker r Proof. Let ξσ0 ∈ A× F = F∞ F . Thus, the F,f be another such choice, so ξσ ξσ × × × in A× , so for any open subset element ξσ0 ξσ−1 ∈ A× F,f F,f ' AF /F∞ is in the closure of F × such that ξ 0 ∈ λξ U . Since ker  meets U ⊆ A× around the identity there exists λ ∈ F σ σ F,f × AF,f in an open subgroup and [NΦ (sσ )]K only depends on sσ modulo the maximal compact × 0 open subgroup in A× E,f , it suffices to check two things: (a) if ξσ = λξσ for some λ ∈ F then the isomorphism in the statement of the Lemma and the analogous isomorphism for ξσ0 0 ' (A0 )σ , and (b) for any σ ∈ Gal(Q/E) and define the same composite isomorphism AQ Q

s ∈ A× E,f such that rE (s) = σ|E ab , the isomorphism θσ,s is unaffected by replacing s with us for u ∈ A× E,f ∩ ker rE sufficiently near the identity (perhaps depending on s). For (a), s0σ := NmF/E (ξσ0 ) is equal to NmF/E (λ)·sσ , so NΦ (s0σ ) = (λ)NΦ (sσ ). The required 0 ' (A0 )σ therefore reduces to a general identity already noted: equality of Q-isomorphisms AQ Q for any c ∈ E × (such as c = NmF/E (λ)), θσ,cs is the composition of θσ,s and the action of NΦ (c) ∈ K × . For (b), to study the general behavior of θσ,s under multiplication of s against any u ∈ (ker rE ) ∩ A× near to the identity, we can restrict our attention to E,f that is sufficiently Q × multiplication by u ∈ (ker rE ) ∩ v-∞ OE,v . If s0 = us for such a u then [NΦ (s0 )]K = [NΦ (s)]K , −1 ◦ θ 0 is an automorphism of [N (1/s)] ⊗ 0 so θσ,s Φ K OK AQ . Moreover, by taking u sufficiently σ,s near to 1 we can ensure that θσ,s and θσ,s0 induce the same map on m-torsion for a fixed −1 ◦ θ 0 is trivial on the m-torsion. m > 3, so θσ,s σ,s 0 . It follows from Lemma By Lemma 2.2.2(1) there exists a K-linear polarization φ of AQ −1 ◦ θ 0 commutes with a certain A.2.7.1 and the subsequent construction of θσ,s that θσ,s σ,s

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polarization φs = φs0 canonically associated to φ and the fractional ideal [NΦ (s)]K = [NΦ (s0 )]K . But an automorphism of a polarized abelian variety has finite order and hence is trivial when it is trivial on the m-torsion for some m > 3. Lemma A.3.3.1 provides a K-linear Q-isomorphism 0 c(σ) : AQ ' (A0Q )σ

depending only on σ ∈ Gal(Q/F ). A.3.4. Since the isomorphisms θσ,s are K-linear and we can use ξσ ξτ as ξστ , it is easy to check ?

that the cocycle relation c(στ ) = σ ∗ (c(σ)) ◦ c(τ ) for σ, τ ∈ Gal(Q/F ) reduces to the general ?

identity θστ,st = σ ∗ (θτ,t ) ◦ (1 ⊗ θσ,s ) as isomorphisms 0 0 τ σ [NΦ (1/st)]K ⊗OK A0Q ' [NΦ (1/t)]K ⊗OK ([NΦ (1/s)]K ⊗OK AQ ) ⇒ ((AQ ) ) ' (A0Q )στ

for σ, τ ∈ Gal(Q/E) and s, t ∈ A× E,f such that rE (s) = σ|E ab and rE (t) = τ |E ab . This identity is immediately deduced from the unique characterization of θστ,st . The Gal(Q/F )-cocycle σ 7→ c(σ) valued in Q-isomorphisms is a descent datum provided that it satisfies a continuity condition, as follows. By the construction of 0 0 σ θσ,s : [NΦ (1/s)]K ⊗OK AQ ' (AQ )

in A.2.6–A.2.7, there is a finite extension L/F 0 inside of Q that is Galois over F such that the K-linear θσ,s descends to an isomorphism [NΦ (1/s)]K ⊗OK A0L ' (A0L )σ|L for all σ ∈ Gal(Q/E) 0 and s ∈ A× E,f such that rE (s) = σ|E ab . (We do not expect that we can take L = F .) Hence, for every σ ∈ Gal(Q/F ) the isomorphism c(σ) over Q descends to a K-linear isomorphism cL (σ) : A0L ' (A0L )σ|L . Increase L to split A0 [m] for a fixed m > 3, so cL (σ) is the identity on m-torsion when σ|L is trivial. Choose a K-linear polarization φ of A0 ; these exist by Lemma 2.2.2(1). By Lemma A.2.7.1 there is an associated K-linear polarization φα of [NΦ (α)]K ⊗OK A0 for all α ∈ A× E,f . Using the properties of the φα as in Lemma A.2.7.1, the definition of c(σ) via Lemma A.3.3.1 implies that cL (σ) carries φL to (φL )σ|L for all σ ∈ Gal(Q/F ). Hence, if σ|L is the identity then cL (σ) is an automorphism of a polarized abelian variety that is trivial on the m-torsion with some m > 3, so cL (σ) is the identity for such σ. It follows (using the cocycle relation) that cL (σ) only depends on σ|L , so we can view cL as a function on Gal(L/F ) that defines a K-linear Galois descent datum on A0L with respect to L/F . In other words, if we rename L as F 0 (as we may), then we get to the case where σ

σ 7→ cF 0 (σ) ∈ Isom(A0 , A0 ) is a Galois descent datum on (A0 , α0 ) with respect to F 0 /F . By descent theory, this uniquely determines an abelian variety A over F equipped with a ring homomorphism α : OK → End(A) and a K-linear isomorphism AF 0 ' A0 respecting the Galois descent data on both sides. × be the algebraic Hecke character associated to (A, α). To prove A.3.5. Let A : A× F → K that A = , it suffices (by Lemma 2.4.9) to show that for a fixed choice of rational prime ` the continuous `-adic characters ψ` , ψ`0 : Gal(F ab /F ) ⇒ K`× that are respectively uniquely associated to  and A coincide. By Theorem 2.5.1(ii), ψ`0 computes the action of Gal(F ab /F ) on V` (A). Also, by construction, ψ` (rF (ξ)) = (ξ)NΦ (NmF/E (1/ξ))` for all ξ ∈ A× F,f . Thus, we want each σ ∈ Gal(Q/F ) to act on V` (A) as multiplication by (ξσ )NΦ (NmF/E (1/ξσ ))` for any ξσ ∈ A× F,f such that rF (ξσ ) = σ|F ab . Fix such a σ and ξσ , and let sσ = NmF/E (ξσ ). It is evident that σ acting on V` (A) is the composition of two maps: the canonical σ-pullback map [σ] : V` (AQ ) ' V` ((AQ )σ ) and the isomorphism on `-adic Tate spaces induced by the descent-data isomorphism (AQ )σ ' AQ . But 0 with respect to the 1-cocycle σ 7→ c(σ) provides the definition of A by Gal(Q/F )-descent of AQ

A.3. A CONVERSE TO THE MAIN THEOREM OF COMPLEX MULTIPLICATION

177

a canonical isomorphism AQ ' A0Q with respect to which the first step, [σ], is identified with the composition of V` (θσ,sσ ) and the isomorphism V` (AQ ) ' [NΦ (1/sσ )]K ⊗OK V` (AQ ) ' V` ([NΦ (1/sσ )]K ⊗OK AQ ) induced by multiplication by the `-part NΦ (1/sσ )` of the finite idele NΦ (1/sσ ) ∈ A× K,f . The σ 0 descent data isomorphism (AQ ) ' AQ is (by definition in terms of AQ ) inverse to the composition of θσ,sσ and the multiplication map induced by the principal generator (1/ξσ ) ∈ K × of [NΦ (1/sσ )]K . By definition, the induced map V` ((AQ )σ ) ' V` (AQ ) is the second step in the description of the σ-action on V` (A). Composing both steps, the two contributions from the mysterious θσ,sσ cancel out and what remains is the product of multiplication by NΦ (1/sσ )` = NΦ (NmF/E (1/ξσ ))` ∈ K`× and multiplication by (ξσ ) ∈ K × on V` (A), exactly as desired.

. . . the theory of moduli spaces has the wonderful feature of having many doors, many techniques by which this theory can be developed. . . . This is another part of its charm. . . . one fact stands out: the story is not finished. David Mumford

B

CM lifting via p-adic Hodge theory B.1. A generalization of the toy model

We consider a class of higher dimensional generalization of the p-divisible group over F attached the toy model in 4.1.2. These are CM p-divisible groups over F of height 4m with action by W (Fp4m ) whose Lie type is striped in the sense of 4.3.13 (b). We will show that such a CM p-divisible group admits a CM lift to characteristic 0, using Raynaud’s method in [62]. Note that CM lifting constructed this way is different from the one constructed in 4.3.28 using the toy model. B.1.1. In this section m is a fixed positive integer, O = W (Fp4m ), and (Y0 , β0 ) is an O-linear CM p-divisible group over F whose Lie type [Lie(Y0 , β0 )] is striped in the sense of 4.3.13 (b). We recast the structure of the proof of 4.3.28 for (Y0 , β0 ) under the above conditions in the following framework. (a) Find a suitable O-linear p-divisible group (X0 , γ0 ) and an O-linear isogeny ρ : (X0 , γ0 ) → (Y0 , β0 ) . (b) Denote by (X , γ) a O-linear p-divisible group over W (F) whose closed fiber is (X0 , γ). An easy application of deformation theory of abelian varieties shows that such an Olinear p-divisible group (X , γ) exists and is unique up to unique isomorphism. (c) Find a suitable finite extension field K of the fraction field K0 of W (F) and a suitable finite flat subgroup scheme G ⊂ X ×Spec(O) Spec(OK ) such that the closed fiber of G is equal to Ker(ρ). The proof of 4.3.28 fits the above template, in which the O-linear p-divisible groups (X0 , γ0 ) and (X , γ), over F and W (F) respectively, are both Serre tensor constructions, from W (Fp4 )linear p-divisible groups Xtoy and Xtoy over F and W (F) respectively. The finite flat group scheme G over OK is isomorphic to the direct sum of m copies of a finite flat subgroup scheme of Xtoy of order p. We will construct a CM lifting of the O-linear p-divisible group (Y0 , β0 ) with striped Lie type, with the same logical structure as outlined in B.1.1 but with a different O-linear O-linear isogeny ρ. In particular the O-linear p-divisible groups (X0 , γ0 ) and (X, γ) considered in this section will not be Serre tensor constructions from a smaller discrete valuation ring R contained in O and an R-linear CM p-divisible group. The finite flat group schemes X [p] over W (F), one for each m ≥ 1, are “irreducible generalizations” of the finite flat group scheme Xtoy [p]; in particular X [p] is not a product of two non-trivial finite flat group schemes. B.1.2. We will follow the notation scheme in §4.2 and §4.3. Recall that we have fixed a positive integer m which serves as mostly invisible parameter. • E is an unramified extension field of Qp of degree 4m. Let O = OE be the ring of integers in E, and let κE be the residue field of O. • Gal(κE /Fp ) is identified with Z/4mZ such that the generator 1 (mod 4m) of the cyclic group Z/4mZ corresponds to the arithmetic Frobenius x 7→ xp in Gal(κE /Fp ). 179

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B. CM LIFTING VIA P -ADIC HODGE THEORY

• σ is the Frobenius automorphism on W (F), which sends every element (x0 , x1 , x2 , . . .) in W (F) to (xp0 , xp1 , xp2 , . . .). ' • I = HomZp ,ring (O, W (F)) −→ Homring (κE , F); we choose and fix an identification of I with Z/4mZ, compatible with the natural Gal(κE /Fp )-module structure for I and ' the isomorphism Gal(κE /Fp ) ←− Z/4mZ. B.1.3. Construction of the O-linear isogeny ρ0 : X0 → Y0 . • Let (Y0 , β0 ) be an O-linear p-divisible group over F whose Lie type is striped in the sense of 4.3.13 (b). In particular Y0 is isoclinic of slope 1/2. The Lie type Lie(Y0 , β0 ) of (Y0 , β0 ) corresponds to half of I with a fixed parity. After changing the trivialization of the (Z/4mZ)-torsor I, we may and do assume that the Lie type of (Y0 , β0 ) is the subset J := {2, 4, 6, . . . , 4m} ⊂ I of all even elements of I. • ρ0 : X0 → Y0 is an O-linear isogeny between O-linear p-divisible groups over F such that – the Lie type of X0 is the subset J 0 := {1, 3, 5, . . . , 2m−1, 2m+2, 2m+4, . . . , 4m}, – G0 := Ker(ρ0 ) is a subgroup scheme of X0 [p] of rank pm stable under the action of O, and – the character of the O-action on Lie(G0 ) corresponds to the subset {1, 3, . . . , 2m − 1} of all odd elements in J 0 . – The O-linear isogeny ρ0 will be made more explicit in B.1.4 and B.1.5 below. The notation in B.1.4 and B.1.5 is consistent with the notation in the proof of lemmas 4.2.6 and 4.2.7. B.1.4. Explicit description of (Y0 , β0 ). To describe the O-linear p-divisible group Y0 explicitly, we will write down its covariant Dieudonn´e module N = D∗ (Y0 ), naturally isomorphic to the Cartier module of the smooth formal group attached to Y0 .1 It is a free W (F)-module of rank 4m, with a σ-linear endomorphism F and a σ −1 -linear endomorphism V , such that F V = p IdM = V F . In addition the ring O = OE operates on M . We have a decomposition Y W (F)⊗Zp O ∼ W (F)i = i∈I

where each W (F)i is a copy of W (F), such that the composition of obvious embedding Y O ,→ W (F)⊗Zp O ∼ W (F)i = i∈I

Q

with the i-th projection i∈I W (F)i  W (F)i is equal to the embedding i : O ,→ W (F) for each i ∈ I. This decomposition of W (F)⊗Zp O induces a decomposition N∼ = ⊕i∈I Ni , where each Ni is a free W (F)-module of rank one stable under the action of O, and ! X X [η] ni = i(η) ni ∀η ∈ O i∈I

i∈I

P for all elements n ∈ N written in the form n = i∈I ni with ni ∈ Ni for all i ∈ I. Here we have written [η] : N → N for the W (F)-linear action on N of an element η ∈ O. We have p Ni+1 ⊆ F · Ni ⊆ Ni+1

and

p Ni−1 ⊆ V · Ni ⊆ Ni−1

1Here we used the notation in appendix B.3, which contains a summary of relations between different flavors of Dieudonn´e theories over a perfect field of characteristic p.

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181

for all i ∈ I, and 

p Ni if i ∈ J , Ni if i ∈ /J where J = {2, 4, . . . , 4m} is the set of all even elements in I. Consequently we have  Ni+1 if i ∈ J F · Ni = . p Ni+1 if i ∈ /J V · Ni+1 =

To summarize: (a) The Cartier module D∗ (Y0 ) of Y0 , isomorphic to the covariant crystalline Diedonn´e module of Y0 , is ⊕i∈I Ni . i

(b) The ring O operates on the factor Ni via the embedding O ,→ W (F) for each i ∈ I, and (c) F and V operate according to the above displayed formulas. B.1.5. Explicit description of (X0 , γ) and ρ0 : (X0 , γ0 ) → (Y0 , β0 ). Let M be the submodule of N defined by X X X M= Nb ⊂ N . N2a−1 + p N2a + 1≤a≤m

1≤a≤m

2m+1≤b≤4m

It is easy to see that M is stable under F , V and the action of O, and the Lie type J 0 of M is J 0 := {1, 3, 5, . . . , 2m − 1, 2m + 2, 2m + 4, . . . , 4m} , consisting of all odd integers between 1 and 2m and all even integers between 2m + 1 and 4m. Clearly the Lie type J 0 is self-dual, in the sense that I is the disjoint union of J 0 and the J 0 + 2m, the translation of J 0 by the unique element of order 2 in Z/4mZ. See 4.3.4 for the general definition of self-dual elements of the Grothendieck group RF (O). Let X0 be the O-linear p-divisible group over F with M as its Cartier-Dieudonn´e module, and let ρ0 : X0 → Y0 be the O-linear isogeny corresponding to the inclusion map M ,→ N . Then G0 := Ker(ρ0 ) is a finite group scheme over F killed by p, with an action by O. The covariant Dieudonn´e module of G0 is the submodule D∗ (G0 ) = (p−1 M2 + p−1 M4 + · · · + p−1 M2m + M )/M ⊂ p−1 M/M of the covariant Dieudonn´e module M/pM of X0 [p]. Remark. The class of the (O ⊗Zp F)-module Lie(G0 ) in RF (O) corresponds to the subset {1, 3, . . . , 2m − 1} of I, while the class of the (O ⊗Zp F)-module M (G0 ) corresponds to the subset {2, 4, . . . , 2m}. This reflects the general fact that for every finite group scheme H0 over a perfect field k ⊃ Fp we have a natural isomorphism   V Lie(H0 ) = Ker D∗ (H0 ) −→ D∗ (H0 )(p) ⊗(k,σ−1 ) k , where σ −1 is the automorphism x 7→ x1/p of k and D∗ (H0 ) is the crystalline covariant Dieudonn´e module of H0 ; see B.3.10.2. B.1.6. Denote by Def(X0 , γ0 ) the deformation functor which to every augmented artinian ' local ring (R, m, R/m −→ F) over W (F) associates the set of all isomorphism classes of triples   ' X, O −→ EndR (X), δ : X ×Spec(R) Spec(F) −→ X0 where (X, O −→ EndR (X) is an O-linear p-divisible group over R, and δ is an O-linear isomorphism. By the crystalline deformation theory of Grothendieck-Messing ([31], [47]), the deformation functor Def(X0 , γ0 ) is ´etale over W (F). Let X be the universal O-linear p-divisible group over W (F) extending the O-linear p-divisible group X0 .

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B. CM LIFTING VIA P -ADIC HODGE THEORY

Let K0 be the field of fractions of W (F). Let K be the tame totally ramified extension field of K0 of degree p4m − 1. We will construct a finite flat subgroup scheme G ⊂ X [p] ×Spec(W (F)) Spec(OK ) over OK whose closed fiber is the subgroup scheme G0 = Ker(ρ0 ) in X0 [p]. Then
Y := X ×Spec(OK ) Spec(W (F)) / G is a lifting of Y0 such that the action of the subring Zp + pO of O on Y0 extends to Y. B.1.7. In the rest of this section we will use the method of [62] to construct the desired subgroup G of X [p] ×Spec(W (F)) Spec(OK ). According to [62, Thm. 1.4.1 and Cor. 1.5.1],
there exist “structural constants” γi , δi i∈I for the κE -linear finite locally free group scheme X [p] over W (F), with γi , δi ∈ W (F) and γi · δi = w ∈ p · W (F)×

∀i ∈ I

such that the bi-algebra structure of the coordinate ring R of X [p] can be explicitly described in term of these constants. Here w ∈ p · W (F)× is the universal constant defined in [62, p. 249 (11 bis)]. It can be expressed in terms of either Jacobi sums or Gauss sums, depends only on κE , and satisfies w ≡ p! (mod p2 ); see [62, Prop. 1.3.1]. The only other condition for the δi ’s is that  δi ∈ p W (F)× if i ∈ J 0 − 1 = {4m, 2, 4, . . . , 2m − 2, 2m + 1, 2m + 3, . . . , 4m − 1} δi ∈ W (F)× if i ∈ {1, 3, 5, . . . , 2m − 3, 2m − 1, 2m, 2m + 2, . . . , 4m − 2} This condition is a consequence of [62, 1.5.4] and the fact that the J 0 is the Lie type of X0 ; see B.1.8. It is clear that for any other family of constants (γi0 , δi0 ) satisfying the same conditions, one can find a family of units ( ui ∈ W (F)× )i∈I such that δi0 = upi δi u−1 i+1

and γi = upi γi0 u−1 i+1

∀i ∈ I .

So by [62, 1.5.1] (γi0 , δi0 ) can also serve as a family of structural constants for X [p]. The Hopf algebra structure of the coordinate ring R of X [p] and the action of κE on R are described in B.1.8 below. B.1.8. Description of the κE -linear bi-algebra R. B.1.8.1. The algebra structure for the coordinate ring R of the κE -linear p-divisible group X [p] over W (F) is R = W (F)[X1 , . . . , X4m ]/(Xip − δi Xi+1 )i∈I . We write xi for the image of Xi in R . B.1.8.2. The co-algebra structure of R is determined by the formula

c(xi ) = xi ⊗ 1 + 1 ⊗ xi +

X χ0 χ00 =χi

Y
γi−h · · · γi−1 Y a0j
a00 j x ⊗ j j∈I j∈I xj wχ0 wχ00

for each i ∈ I, to be read as follows. (1) The summations runs through all pairs (χ0 , χ00 ) of µp4m −1 (W (F))-valued non-trivial 0 00 characters of κ× E such that χ χ = χi . × (2) χi : κE −→ µp4m −1 (W (F)) is the fundamental character of κE indexed by i, namely × with the restriction the composition of the Teichm¨ uller lifting κ× E → W (κE ) i|W (κE )× : W (κE )× → W (F)× of i to W (κE )× ⊂ W (κE ). We have χpi = χi+1 for all i ∈ I. (3) a0 = (a0j )j∈I and a00 = (a00j )j∈I are elements of {0, 1, . . . , p − 1}I corresponding to χ0 and χ00 respectively: Y a00j Y a0j χ0 = χj , χ00 = χj , a0 , a00 6= (0, . . . , 0) or (p − 1, . . . , p − 1) . j∈I

j∈I

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183

(4) h = h(χ0 , χ00 ) is the unique integer such that 0 < h ≤ 4m and  0  ai−h + a00i−h = p a0 + a00 = p − 1 for 0 < k < h  0i−k 00 i−k aj = aj = 0 otherwise (5) wχ0 and wχ00 are units of W (F) defined in [62, p. 248 (11)]. They can be written in terms of Jacobian or Gauss sums and depends only on κE , χ0 and χ00 . Moreover we have Y Y

wχ0 ≡ a0j ! (mod p) , wχ00 ≡ a00j ! (mod p) . j∈I

j∈I

See [62, Prop. 1.3.1]. The co-unity  : R −→ W (F) is given by (xi ) = 0 for all i ∈ I. B.1.8.3. The κE -action on R is the homomorphism κE −→ Endbi-alg (R) such that for every

element η ∈ κE , the corresponding endomorphism [η]∗ of the bi-algebra R is  [η]∗ (xi ) = χi (η) xi ∀ i ∈ I if η ∈ κ× E ∗ [0] (xi ) = 0 ∀i ∈ I B.1.9. The Cartier dual of X [p].

B.1.9.1. Let R0 := HomW (F) (R, W (F)) be the W (F)-linear dual of R, endowed with the

natural bi-algebra structure coming from the bi-algebra structure of R. Then Spec(R0 ) is the Cartier dual of X [p] over W (F). According to [62, 1.5.3], the structural constants for X [p]t are (δi , γi )i∈I . Below is a quick review to establish the notation for later use. B.1.9.2. Let ya a ∈ {0, 1, . . . , p − 1}I be the W (F)-basis of R0 dual to the W (F)-basis ) ( Y a xi i a = (ai )i∈I , 0 ≤ ai ≤ p − 1 ∀ i ∈ I xa := i∈I

of R, indexed by finite sequences a ∈ {0, 1, . . . , p − 1}I . Here xi is the image of Xi in R as in B.1.8.1 for each i ∈ I. For each i ∈ I, let ei be the element of {0, 1, . . . , p − 1}I whose i-th coordinate is 1 and all other coordinates are 0, and let yi := yei . The action of κE on R dualizes to an action of κE on R0 such that [η]∗ (yi ) = χi (η) yi and [0]∗ (yi ) = 0

∀ i ∈ I , ∀ η ∈ κ× E.

B.1.9.3. It is easy to see that

y b :=

Y

yiai = wχa · yb

i∈I

for each b ∈ {0, 1, . . . , p − 1}, where χb :=

Q

i∈I

yip = γi · yi+1

χbi i . Moreover we have ∀i ∈ I .

The co-algebra structure of R0 is given by c0 (yi ) = yi ⊗ 1 + 1 ⊗ yi +

X χ0 χ00 =χi

read in the same way as B.1.8.2.

Y
δi−h · · · δi−1 Y a0j
a00 j ⊗ , j∈I yj j∈I yj wχ0 wχ00

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B. CM LIFTING VIA P -ADIC HODGE THEORY

B.1.10. We will need to examine the closed fiber X0 [p] = X [p]×Spec(W (F)) Spec(F) of X [p] and its subgroup scheme G0 . By definition we have
X [p] ×Spec(W (F)) Spec(F) = Spec R ⊗W (F) F , and
¯1, . . . , X ¯ 4m ]/ X ¯ p − δ¯i X ¯ i+1 R ⊗W (F) F = F[X i i∈I ¯ ¯ where δi is the image of δi in F and Xi is the image in W (F)[X1 , . . . , X4m ]⊗W (F) F of the element Xi ∈ W (F)[X1 , . . . , X4m ]. Recall that δ¯i = 0 if i ∈ {4m, 2, 4, . . . , 2m − 2, 2m + 1, 2m + 3, . . . , 4m − 1}, δ¯i ∈ F× if i ∈ {1, 3, . . . , 4m − 1, 2m, 2m + 2, . . . , 4m − 4, 4m − 2}. B.1.10.1. The kernel of the Frobenius homomorphism

FrX0 [p] : X0 [p] −→ X0 [p](p) is the closed subscheme   p p  p ¯ ,X ¯ ,...,X ¯p ¯p ¯p ¯p X 1 3 2m−1 , X2m+2 , X2m+4 , . . . , X4m−2 , X4m ¯1, . . . , X ¯ 4m ] Spec F[X =: Λ ¯2, X ¯4, . . . , X ¯ 2m−2 , X ¯ 2m , X ¯ 2m+1 , X ¯ 2m+3 . . . , X ¯ 4m−1 X ¯1, . . . , X ¯ 4m ]/(X ¯ p − δ¯i X ¯ i+1 )i∈I . of F[X i
B.1.10.2. Denote by HJ 0 −{4m} the subscheme of Ker FrX0 [p] defined by principal ideal (x4m ). It is easy to see that HJ 0 −{4m} is a finite subgroup scheme of X0 [p] of order p2m−1 , isomorphic to the product of 2m − 1 copies of αp . More explicitly, the coordinate ring of HJ 0 −{4m} is   ¯p ¯p ¯p ¯p ¯p ¯ p, . . . , X ¯ p, X X 2m−3 , X2m−1 , X2m+2 , X2m+4 , . . . , X4m−2 3 1 ¯1, . . . , X ¯ 4m ] F[X ¯2, X ¯4, . . . , X ¯ 2m−2 , X ¯ 2m , X ¯ 2m+1 , X ¯ 2m+3 . . . , X ¯ 4m−1 , X ¯ 4m X The co-multiplication on R induces a co-multiplication x ¯i 7→ x ¯i ⊗ 1 + 1 ⊗ x ¯i

for i = 1, 3, . . . , 2m − 3, 2m − 1, 2m + 2, 2m + 4, . . . , 4m − 2  on the coordinate ring Γ HJ 0 −{4m} , OH 0 of HJ 0 −{4m} , where x ¯i denotes the image J −{4m} ¯ of Xi in the above quotient ring. The character of the action of κE on the Lie algebra of HJ 0 −{4m} corresponds to the subset J 0 − {4m}. As before we say that J 0 − {4m} is the Lie type of HJ 0 −{4m} . B.1.10.3. For each subset S ⊂ J 0 − {4m} = {1, 3, . . . , 2m − 1, 2m + 2, 2m + 4, . . . , 4m − 2}, there exists a unique subgroup scheme HS of HJ 0 −{4m} over F stable under the action of κE whose Lie type is equal to S. Namely HS is the subscheme of HJ 0 −{4m} defined by the ideal generated by { x ¯i | i ∈ J 0 − S }. In particular H{1,3,...,2m−1} = G0 . 

B.1.11. Lemma. Let K0 be an algebraic closure of K0 . Suppose that z is a non-zero point in X [p](K0 ). Let zi = xi (z) be the i-th coordinate of z , i ∈ I. Let ordp be the p-adic valuation on K0 normalized by ordp (p) = 1. Then ordp (zi ) =

4m−1 4m−1 X X 1 1 4m−1−a ord (δ ) p = ordp (δi−1−b ) pb , p i+a 4m m p −1 p −1 a=0

b=0

where each integer δj above is either 0 or 1 as indicated in B.1.7. Moreover we have ordp (z2m−1 ) < ordp (z2m−3 ) < · · · < ordp (z3 ) < ordp (z1 ) < ordp (z2m ) < ordp (z2m+2 ) < · · · < ordp (z4m−4 ) < ordp (z4m−2 ) < ordp (z2m−2 ) < ordp (z2m−4 ) < · · · < ordp (z2 ) < ordp (z4m ) < ordp (z2m+1 ) < ordp (z2m+3 ) < · · · < ordp (z4m−3 ) < ordp (z4m−1 ).

B.1. A GENERALIZATION OF THE TOY MODEL

185

Proof. The first statement on the valuation of the coordinates zi of z is [62, p. 269 (6)]. The inequality for these valuations is immediate from the fact that  1 if i ∈ {4m, 2, 4, . . . , 2m − 2, 2m + 1, 2m + 3, . . . , 4m − 1} = J 0 − 1, ordp (δi ) = 0 if i ∈ {1, 3, . . . , 2m − 3, 2m − 1, 2m, 2m + 2, . . . , 4m − 4, 4m − 2}. B.1.12. Recall that K is the tame totally ramified extension of the fraction field K0 of W (F) of degree p4m − 1. Lemma B.1.11 tells us that X [p]×Spec(W (F)) Spec(K) is isomorphic to the constant group scheme over K attached to (κE , +). Choose and fix a non-trivial point z(0) of X [p](K). Let G(0) be the schematic closure in X [p]×Spec(W (F)) Spec(K) of the subgroup generated by z(0) . Then it is easy to see from B.1.11 that the closed fiber of G(0) is the subgroup scheme H{2m−1} of G0 which is stable under the action of κE with Lie type {2m − 1} ⊂ I. We want to choose another m − 1 non-trivial points z(1) , . . . , z(m−1) of X [p](K) such that the closed fiber of the schematic closure G(m−1) in X [p] ×Spec(W (F)) Spec(K) of the subgroup generated by z(0) , . . . , z(m−1) is equal to the kernel G0 of the isogeny ρ0 : X0 → Y0 . Choose and fix a generator ζ of the finite cyclic group κ× E . For a = 1, . . . , m − 1, let (a) a a z = [ζ ](z(0)), a non-trivial point of X [p](K). Here [ζ ] denotes the action of the element (a) ζ a on X [p]. Write zi = xi (z(a) ) for the i-th coordinate of z(a) , i ∈ I. Then we have (a) zi = χi (ζ)a for all i ∈ I and all a = 0, 1, 2, . . . , m − 1. B.1.13. Lemma. The elements z(0) , . . . , z(m−1) in the Fp -vector space X [p](K) are linearly independent over Fp . Proof. For suppose that f0 , . . . , fm−1 are elements of Fp , not all equal to 0, such that [f0 ](z(0) ) + . . . + [fm−1 ](z(m−1) ) = 0. Then we have "m−1 # m−1 X X (a) a 0= [fa ](z ) = fa ζ (z(0) ) , a=0

Pm−1

a=0

ζa

so a=0 fa = 0. Here [fa ] denotes the action of the element fa ∈ Fp ⊂ κE on the κE -linear Pm−1 a group scheme X [p]. The equality a=0 fa ζ = 0 implies that κE = Fp (ζ) has degree at most m over Fp , a contradiction. B.1.14. Definition. Notation as above. For each a = 0, 1, . . . , m − 1, denote by G(a) the schematic closure in X [p] ×Spec(W (F)) Spec(OK ) of the subgroup of X [p](K) generated by z(0) , . . . , z(a) . By Lemma B.1.13, G(a) is a finite flat subgroup scheme of X [p]×Spec(W (F)) Spec(OK ) of rank pa+1 over OK . Moreover the generic fiber G(a) ×Spec(OK ) Spec(K) of G(a) is isomorphic to the product of a + 1 copies of the constant group scheme Z/pZ over K. Note that G(a) depends on the choice of the generator ζ of κE . B.1.15. Remark. It is easy to see that G(a) descend uniquely to a closed subgroup scheme 0 G(a) of X [p]×Spec(W (F)) Spec(OK1 ), where K1 is the tamely ramified subextension of K/K0 such that [K1 : K0 ] = (pm − 1)/(p − 1). The statements of B.1.16 and B.1.17 below holds 0 . with K replaced by K1 and G(a) replaced by G(a) B.1.16. Theorem. For a = 0, 1, . . . , m − 1, the closed fiber of G(a) is equal to the subgroup scheme H{2m−2a−1,2m−2a+1,...,2m−1} of G0 defined in B.1.10, the subgroup scheme of G0 stable under the action of κE with Lie type {2m − 2a − 1, 2m − 2a + 1, . . . , 2m − 3, 2m − 1}. In particular the closed fiber of G(m−1) is equal to the kernel G0 of the O-linear isogeny ρ0 : X0 → Y0 .

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B.1.17. Corollary. Let Y be the quotient p-divisible group
Y := X ×Spec(W (F)) Spec(OK ) /G(m−1) over OK . (i) Y is a p-divisible group over OK whose closed fiber is Y0 . (ii) The action of the subring Zp + pO ⊂ O on Y0 extends to the p-divisible group Y over OK . (iii) The p-adic CM type of Y is the self-dual subset J 0 ⊂ I. We recall that I is identified with {1, 2, . . . , 4m}, and J 0 = {1, 3, . . . , 2m − 1, 2m + 2, 2m + 4, . . . , 4m − 2, 4m}. Proof of Theorem B.1.16.  D
D Denote by X [p]/G(a) (resp. X0 [p]/H{2m−2a−1,2m−2a+1,...,2m−1} ) the Cartier dual of the quotient X [p]/G(a) (resp. X0 [p]/H{2m−2a−1,2m−2a+1,...,2m−1} ), naturally embedded in the Cartier dual X [p]D (resp. X0 [p]D ) of X [p] (resp. X0 [p]) as a closed subgroup scheme. We will prove the following statement, equivalent to B.1.16, on the closed subgroup scheme (X [p]/G(a) )D : The closed fiber of (X [p]/G(a) )D , as a subscheme of X0 [p]D , is equal to  D X0 [p]/H{2m−2a−1,2m−2a+1,...,2m−1} . Note that (X [p]/G(a) )D ⊂ X [p]D is the schematic closure, computed inside X [p]D , of its own generic fiber, because X [p]/G(a) is finite flat over OK . In the rest of the proof we will compute the schematic closure of (X [p]/G(a) )D (K) in X [p]D . Step 1. Compute



X0 [p]/H{2m−2a−1,2m−2a+1,...,2m−1}

D

.

It is easy to see from B.1.10.1–B.1.10.3 that the inclusion H{2m−2a−1,2m−2a+1,...,2m−1} ,→ X0 [p] corresponds to the surjection

h R⊗W (F) F  R⊗W (F) F x ¯2 , x ¯4 , . . . , x ¯4m , x ¯1 , x ¯3 , . . . , x ¯2m−2a−3
p p ∼ = F[Z¯2m−2a−1 , Z¯2m−2a+1 , . . . , Z¯2m−1 ]/ Z¯2m−2a−1 , Z¯2m−2a+1 , . . . , Z¯2m−1 ¯ Z¯ p ) =: F[Z]/( of bi-algebras, where the co-multiplication for the last bi-algebra is determined by Z¯i 7→ Z¯i ⊗ 1 + 1 ⊗ Z¯i for i = 2m − 2a − 1, 2m − 2a + 1, . . . , 2m − 3, 2m − 1 . ¯ Z¯ p ) is self-dual, the transpose of the above surjection is an inclusion Since the bi-algebra F[Z]/( ¯ Z¯ p ) −→ R0 ⊗W (F) F ht : F[Z]/( such that ht (Z¯i ) = y¯i for i = 2m − 2a − 1, 2m − 2a + 1, . . . , 2m − 1. Here y¯i denotes the image in R0 ⊗W (F) F of the element yi ∈ R0 defined in B.1.9.2. Notice that for each i = 2m − 2a − 1, 2m − 12a + 1, . . . , 2m − 1 we have y¯ip = 0

in R0 ⊗W (F) F

because γi ≡ 0 (mod p), and c0 (¯ yi ) = y¯i ⊗ 1 + 1 ⊗ y¯i

in R0 ⊗W (F) R0 ⊗W (F) F

because δi−1 ≡ 0 (mod p), as expected for the co-algebra homomorphism ht . The fact that  D X0 [p]/H{2m−2a−1,2m−2a+1,...,2m−1} is the kernel of the surjective homomorphism  D X0 [p]D −→ H{2m−2a−1,2m−2a+1,...,2m−1} means:

B.1. A GENERALIZATION OF THE TOY MODEL

The subgroup scheme



X0 [p]/H{2m−2a−1,2m−2a+1,...,2m−1}

D

187

,→ X0 [p]D is

the closed scheme of X0 [p]D defined by the ideal generated by the a + 1 elements y¯2m−2a−1 , y¯2m−2a+1 , . . . , y¯2m−1 in R0 ⊗W (F) F .

D X [p]×Spec(W (F)) Spec(OK ) /G(a) .

Step 2. Equations for the generic fiber of

There are three equivalent ways to think about the (a + 1) points z(0) , . . . , z(a) : • as a + 1 homomorphisms ξ(0) , . . . , ξ(a) from the constant group scheme Z/pZ to X [p] ×Spec(W (F)) Spec(OK ) , t , . . . , ξt D • as a+1 homomorphisms ξ(0) (a) from X [p] ×Spec(W (F)) Spec(OK ) to the group scheme

µp × Spec(OK ) = Spec(OK [T ]/(T p − 1)) , • as a + 1 elements ψ(0) , . . . , ψ(a) in R0 ⊗W (F) OK such that p ψ(b) =1

and

c0 (ψ(b) ) = ψ(b) ⊗ ψ(b) ,

t , for all where ψ(b) is the function on X [p]D ×Spec(W (F)) Spec(OK ) induced by ξ(b) b = 0, 1, . . . , a.

D We know that X [p]×Spec(W (F)) Spec(OK ) /G(a) is the kernel of the natural homomorphism
D D h(a) : X [p]×Spec(W (F)) Spec(OK ) −→ G(a) .

Restricting h(a) to the generic fiber over Spec(K) we see that the generic fiber of

D X [p]×Spec(W (F)) Spec(OK ) /G(a) (a)

is the closed subscheme of X [p]D ×Spec(W (F)) Spec(K) defined by the ideal IQ ⊂ R0 ⊗W (F) K generated by the a + 1 elements ψ(c) − 1 , c = 0, 1, . . . , a . (a)

Explicitly, the generators ψ(c) − 1 of IQ

are


P Q cbi z bi · y ψ(c) − 1 = b6=0 i∈I χi (ζ) i b
Q P Q bi −1 · cbi z bi · = w χ (ζ) χb b6=0 i∈I i i∈I yi i where the indexing element b runs through all non-zero elements in {0, 1, . . . , p − 1}I , and bj = the i-th component of b. Here zi ∈ OK is the xi -coordinate of the non-zero point z as in B.1.11, and the elements yi ∈ R0 are defined in B.1.9.2. As already noted, (X [p]/G(a) )D ⊂ X [p]D is the schematic closure of its own generic fiber. In other words (X [p]/G(a) )D ⊂ X [p]D is the subscheme of X [p]D defined by the ideal (a)

I (a) := IQ ∩ R0 ⊗W (F) OK in R0 ⊗W (F) OK . We need to determine the ideal I (a) in R0 ⊗W (F) OK .

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B. CM LIFTING VIA P -ADIC HODGE THEORY

Step 3. For each c = 0, 1, . . . , a , let fc (T ) = fc (T1 , . . . , T4m ) ∈ OK [T1 , . . . , T4m ] be the polynomial in 4m variables defined by ! X Y Y b cbi · χ (ζ) · Ti i . wχ−1 fc (T ) := i b b6=0

i∈I

i∈I

We have ψ(c) − 1 = fc (z1 y1 , . . . , zm ym ) ∈ IQ for c = 0, 1, . . . , a. Consider the (a + 1) × (a + 1) minor   ∂fc Jac(a) (f , T ) := ∂Ti c=0,1,...,a; i=2m−2a−1,2m−2a+1,...,2m−1 of the Jacobian matrix for the polynomials f0 (T ), . . . , fa (T ). Since wχi = 1 for each i ∈ I, the value Jac(a) (f , T ) T =0 of the Jacobian matrix Jac(a) (f , T ) evaluated at Ti = 0 for all i ∈ I is the (a + 1) × (a + 1) van der Monde matrix attached to the a + 1 (p4m − 1)st roots of unity χ2m−2a−1 (ζ), χ2m−2a+1 (ζ), . . . , χ2m−1 (ζ) . × OK

because the χi (ζ)’s have distinct images in F× . By the implicit function So Jac(a) (f , T ) ∈ theorem, there exist a + 1 elements  gj (T ) : j ∈ J(a) , J(a) := {2m − 2a − 1, 2m − 2a + 1, . . . , 2m − 1} with constant terms equal to 0 in the power series ring OK [[T1 , T2 , . . . , T2m−2a−2 , T2m−2a , T2m−2a+2 , . . . , T2m−2 , T2m , T2m+1 , . . . , T4m−1 , T4m ]] such that zj yj − gj (zy) ∈ I (a) Here gj (zy) denotes the “value” gj (T )

∀ j ∈ J(a) .

Ti =zi yi ∀ i∈I−J(a)

of the power series gj (T ) when the

4m − a − 1 variables Ti with i ∈ I − J(a) are substituted by zi ·yi ; the resulting power series converges in R0 ⊗W (F) OK because zi is an element of the maximal ideal mK of OK for every i ∈ I. We recall that zi ∈ mK is the xi -coordinate of the non-zero point z as in B.1.11 for each i, while the elements yi ∈ R0 are defined in B.1.9.2. Step 4. Because the gj (T )’s have no constant term, we see from B.1.11 that
zj−1 · gj (zy) ∈ mK · R0 ⊗W (F) OK ∀ j ∈ J(a) . For each j ∈ J(a) we have (a)

yj − zj−1 gj (zy) ∈ IQ ∩ R0 ⊗W (F) OK = I (a) ,
an element of I (a) which is congruent to yj modulo m · R0 ⊗W (F) OK . Consider the natural surjection  
π : R0 ⊗W (F) F /(¯ y2m−2a−1 , y¯2m−2a+1 , . . . , y¯2m−1 )  R0 ⊗W (F) OK )/I (a) ⊗OK F .  D We saw in Step 1 that the left hand side is the coordinate ring of X0 [p]/HJ(a) , so it has dimension p4m−a−1 over F. The right hand side is the coordinate ring of the closed fiber of the Cartier dual of X /G(a) , so it also has dimension p4m−a−1 over F. Therefore the surjection π is an isomorphism. Consequently the elements yj − zj−1 gj (zy) with j ∈ J(a) generate the ideal I (a) by Nakayama’s lemma. We conclude by Step 1 that the closed fiber of the Cartier dual of X /G(a) is equal to the Cartier dual of X0 [p]/HJ(a) .

B.2. CONSTRUCT CM LIFTING BY P -ADIC HODGE THEORY

189

B.2. Construct CM lifting by p-adic Hodge theory In this section we give another proof of the main result in § B.1 using p-adic Hodge theory. The strategy remains the same; we replace [62] with tools from recent progress in p-adic Hodge theory. Because we want to include the case p = 2, we will use Kisin’s version in [38]. See also B.4.2 and B.4.5 where the definitions of some category of p-linear algebra data are recalled, and B.4.15 and B.4.17 for the formulas for the closed fibers constructed from such p-linear algebra data. B.2.1. We will follow the notation scheme in § B.1, • m ≥ 1 is a positive integer. • E is an unramified extension field of Qp of degree 4m, O = OE is the ring of integers in E, κE is the residue field of O. • Gal(κE /Fp ) is identified with Z/4mZ such that the generator 1 (mod 4m) of the cyclic group Z/4mZ corresponds to the arithmetic Frobenius x 7→ xp in Gal(κE /Fp ). • σ is the Frobenius automorphism on W (F), which sends every element (x0 , x1 , x2 , . . .) in W (F) to (xp0 , xp1 , xp2 , . . .). ' • I = HomZp ,ring (O, W (F)) −→ Homring (κE , F); we choose and fix an identification of I with Z/4mZ, compatible with the natural Gal(κE /Fp )-module structure for I and ' the isomorphism Gal(κE /Fp ) ←− Z/4mZ. • J is the subset of I ∼ = Z/4mZ with {2,4,. . . , 4m} as a set of representatives; i.e. J is the subset of all even elements in I. • J 0 ⊂ I is the subset of I with {1, 3, . . . , 2m − 1, 2m + 2, 2m + 4, . . . , 4m} as a set of representatives, i.e. J 0 is the subset of I ∼ = Z/4mZ represented by the set consisting of all odd integers between 1 and 2m − 1 and all even integers between 2m + 2 and 4m. • X0 is an O-linear p-divisible group of height 4m over F whose Lie type [Lie(X0 )] ∈ RF (O) corresponds to the subset J 0 ⊂ I. • Y0 is an O-linear p-divisible group of height 4m over F whose Lie type [Lie(X0 )] ∈ RF (O) corresponds to the subset J ⊂ I. • ρ0 : X0 → Y0 is an O-linear isogeny over F whose kernel G0 = Ker(ρ0 ) is the finite subgroup scheme over F of the OE -linear commutative group scheme X0 [p] over F characterized by the following properties. – G0 has order pm and dimF (Lie(G0 )) = m. In particular the relative Frobenius (p) (p) homomorphism FG0 : G0 → G0 and the Verschiebung map VG0 : G0 → G0 are both equal to 0. – G0 is stable under the action of O such that Lie(G0 ). – The Lie type [Lie(G0 )] ∈ RF (O) of G0 corresponds to the subset {1, 3, . . . , 2m−1} of all odd elements in J 0 . • X is the OE -linear p-divisible group over W (F) whose closed fiber is the OE -linear p-divisible group X0 over F. Equivalently, X is the OE -linear p-divisible group over W (F) whose p-adic CM type is the subset J 0 ⊂ I. • K0 is the fraction field of W (F), and K is the tame totally ramified extension of K0 of degree p4m − 1. • e = e(K/K0 ) = p4m − 1. B.2.2. In B.1.7–B.1.16 we used the method in [62] to construct a finite flat subgroup scheme G ⊂ X [p]×SpecW (F) Spec(OK ) over OK , denoted by G(m−1) in B.1.16, such that the closed fiber of G is equal to the kernel G0 of the isogeny ρ0 : X0 → Y0 . Then
X×SpecW (F) Spec(OK ) /G

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B. CM LIFTING VIA P -ADIC HODGE THEORY

is a p-divisible group over OK with action by the order Zp + pOK ⊂ OK which is a lifting of (Y0 , Zp + pOK ,→ End(Y0 )), whose p-adic CM type is self-dual. In the rest of this section we will construct finite flat subgroup schemes G(0) , G(1) , . . . , G(m−1) ⊂ X [p]×SpecW (F) Spec(OK ) similar to those B.1.16, but with Raynaud’s theory in [62] substituted by the p-adic Hodge theory in [38]. See B.2.16 for the statement and B.2.14 for the definition of the finite flat group schemes G(b) . For the question in hand there is no obvious advantage of using p-adic Hodge theory, except that the computation of the closure in X [p]×SpecW (F) Spec(OK ) of a subgroup scheme of the generic fiber X×SpecW (F)Spec(K) is a little easier. However it is likely that p-adic Hodge theory will be useful for the question (sCML) in 4.1.10. At a technical level, an important step is to identify the Kisin module (M, φM , ψM ) corresponding to the (O/pO)-linear finite flat group scheme X [p]×Spec(OK0 ) Spec(OK ). The rest of the computation is similar to but easier than those in B.1.16. The steps for finding the desired Kisin module (M, φM , ψM ) are indicated in B.2.3–B.2.4. B.2.3. We will need two p-adic base fields, K0 and K, when applying p-adic Hodge theory. • Choose p as the generator of the maximal ideal of OK0 = W (F), whose Eisenstein polynomial in the variable t is E0 (t) = t − p. • Let S0 = W (F)[[t]], and let 0 : S0  OK0 = W (F) be the W (F)-linear ring homomorphism such that 0 (t) = p. • Let φS0 : S0 → S0 be the continuous ring homomorphism such that φS0 (t) = tp and φS0 |W (F) = σ, where σ is the canonical lifting of Frobenius on W (F). • Choose a generator π of the maximal ideal of OK such π e − p = 0; recall that e = p4m − 1. Let E(u) = ue − p, the minimal polynomial of π with respect to K0 in the variable u. • Let S = W (F)[[u]], and let  : S  OK be the W (F)-linear ring homomorphism such that (u) = π. Let φS the continuous ring endomorphisms of S such that φS (u) = up and φS |W (F) = σ. • Let h : S0 → S be the injective continuous W (F)-linear ring homomorphism such that h(t) = ue . It is clear that  ◦ h = 0 and φS ◦ h = h ◦ φS . We will abbreviate both φS0 and φS to φ unless this abuse leads to a problem. B.2.4. How to find the Kisin module for X [p] 1. We will first find an O-linear object (N, φN , ψN ) in BTφ,f /S0 which corresponds to the Olinear p-divisible group X over OK0 . 2. Let M0 = N/pN, let φM0 : M0 → M0 be the φ-linear map induced by φN , and let   ψM0 : M0 → φ∗ M0 be the S-linear map induced by ψN . Then M0 , φM0 , ψM0 is an (O/pO)linear object in the category(Mod/S0 )c ofconnected Kisin modules, and the (O/pO)-linear finite flat group scheme GKis M0 , φM0 , ψM0 over OK0 is (O/pO)-linearly isomorphic to X [p] over OK0 .   3. Let (M, φM , ψM ) be the object of (Mod/S)c obtained from the object M0 , φM0 , ψM0 of   (Mod/S0 ) via the arrow S0 , φS0 −→ (S, φS ). Then GKis (M, φM , ψM ) is an (O/pO)-linear finite flat group scheme over OK which is (O/pO)-linearly isomorphic to X [p]×Spec(OK0 ) Spec(OK ) . B.2.5. Definition. For a given a subset Λ $ I of I, let (NΛ , φΛ , ψΛ ) be an O-linear object in the category BTφ,f S0 with the following properties.

B.2. CONSTRUCT CM LIFTING BY P -ADIC HODGE THEORY

191

(1) The S0 -module NΛ is a direct sum NΛ = ⊕i∈I Ni of free rank-one S0 -modules Ni indexed by the set I ∼ = Z/4mZ. (2) The ring O operates on each of the S0 -modules Ni via the ring homomorphism i : O → W (F) ⊂ S0 . In other words the action [η] : NΛ → NΛ of any element η ∈ O is given by ! X X [η] i(η) · xi ∀ xi ∈ Ni , ∀ i ∈ I . xi = i∈I

i∈I

 (3) S0 · φΛ (Ni ) =

E0 (t) · Ni+1 Ni+1

if i ∈ Λ if i ∈ /Λ

Remark. (a) It is clear that for any subset Λ ⊂ I there exists an O-linear S0 -module NΛ and a φ-linear map φΛ : NΛ → NΛ satisfying the conditions (1)–(3) above. (b) The condition that Λ 6= I guarantees that the 4mth iterate ψΛ4m := φ4m−1 ∗ ψΛ ◦ φ4m−2 ∗ ψΛ ◦ · · · φ∗ ψΛ ◦ ψ : NΛ −→ φ4m ∗ NΛ of the map ψΛ : NΛ → S0 ⊗φ,S0 NΛ =: φ∗ NΛ defined by (NΛ , φΛ ) satisfies ψΛ4m (NΛ ) ⊂ E0 (t) · φ4m ∗ NΛ , hence the triple (NΛ , φΛ , ψΛ ) is an object in BTφ,f /S0 which has the properties in (a). Such an object in BTφ,f /S0 is determined by the subset Λ ⊂ I up to non-unique isomorphisms. (c) Any two objects in BTφ,f S0 satisfying the properties (1)–(3) are O-linearly isomorphic. The proof is easy and is left as an exercise. B.2.6. Lemma. Notation as in B.2.5. Let XΛ = p-DivKis (NΛ , φΛ , ψΛ ) be the O-linear formal p-divisible group over OK0 = W (F) attached to (NΛ , φΛ , ψΛ ). Then the p-adic CM type of XΛ is the subset c Λ + 1 = c (Λ + 1) of I, translation by 1 of the complement c Λ in I of the subset Λ ⊂ I, equal to the complement of the translation by 1 of Λ ⊂ I. Equivalently the Lie type of the closed fiber X0,Λ of XΛ corresponds to the subset c Λ + 1 ⊂ I. Proof. According to B.4.15, the Lie algebra Lie(X0,Λ ) is Lie(X0,Λ ) ∼ = Coker

NΛ /tNΛ

ψΛ

(mod t)

/ φ∗ N /tφ∗ N Λ Λ

! .

It follows that for any i ∈ I, the (i + 1)st component Lie(X0,Λ )i+1 of Lie(X0,Λ ) is non-zero if and only if i ∈ / Λ. So the p-adic CM type of XΛ is c Λ + 1 = c (Λ + 1). Remark. One can compute the p-adic CM type of XΛ directly using the formula in B.4.16, to give an alternative proof of B.2.6. B.2.7. Definition. has

(1) Let Λ0 := c J 0 − 1 = c (J 0 − 1), the subset of I ∼ = Z/4mZ which

{1, 3, 5, . . . , 2m − 3, 2m − 1, 2m, 2m + 2, 2m + 4, . . . , 4m − 2} as a set of representatives. (2) Let X := XΛ0 , and let X0 be the closed fiber of X , an O-linear p-divisible group whose Lie type [Lie(X0 )] corresponds to the subset J 0 ⊂ I according to B.2.6.

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B. CM LIFTING VIA P -ADIC HODGE THEORY


(3) Define an (O/pO)-linear object M0 , φM,0 , φM,0 in the category (Mod/S0 )c as follows. – M0 := NΛ0 as an S0 -module, – φM,0 : M0 → M0 is the φ-linear map induced by φΛ0 , – ψM,0 : M0 → φ∗ M0 is the S0 -linear map induced by ψΛ0 . – The action of κE = O/pO on NΛ0 induces an action of κ on M0 , compatible with φM0 and ψM0 . (4) Define an object (M, φM , φM ) in (Mod/S)c as follows.2 – M := S ⊗h,S0 M0 as an S-module, – φM : M → M is the φ-linear map induced by φM0 – ψM : M → φ∗ M is the S-linear map induced by ψM0 . – The action of κE on (M0 , φM0 , ψM0 ) induces an action of κE on (M, φM , ψM ).
In other words (M, φM ,φM ) is obtained from M0 , φM0 , φM0 by base change with  respect to the map h :

S0 , φS0

−→ (S, φS )

Remark. The finite flat group scheme over OK associated to the Kisin module (M, φM , φM ) is isomorphic to X [p]×SpecW (F) SpecOK , by functoriality of the functor GKis with respect to the base complete discrete valuation ring of characteristics (0, p). B.2.8. Lemma.

(1) The Kisin module M has a direct sum decomposition M = ⊕i∈I Mi ,

where each component Mi is a free F[[u]]-module of rank one, and  E(u) · Mi+1 = ue · Mi+1 if i ∈ Λ0 F[[u]] · φM (Mi ) = Mi+1 if i ∈ / Λ0 where Λ0 ⊂ I is defined in B.2.7 (1). (2) There exists a family of F[[u]]-bases vi of the free rank-one F[[u]]-modules Mi , indexed by i ∈ I, such that  e u · vi+1 if i ∈ Λ0 φMi (vi ) = vi+1 if i ∈ / Λ0 B.2.9. We would like to find all subgroup schemes of X [p]×Spec(W (F)) Spec(OK ) of order p. Equivalently, we want to find all F[[u]]-submodules of M of the form F[[u]]·y with the following properties. (i) φM (y) ∈ F[[u]]·y (ii) F[[u]]·φM (y) ⊇ ue F[[u]]·y (iii) F[[u]]·y is an F[[u]]-direct summand of M. P Write y = i∈I fi (u) · vi and φM (y) = g(u) y, the above conditions is translated to the following system of equations with 4m + 1 unknown elements g(u), f1 (u), f2 (u), . . . , f4m (u) in F[[u]] subject to certain constraints on their orders: There exist elements fi (u) ∈ F[[u]] indexed by i ∈ I and an element g(u) ∈ F[[u]] which satisfy the following properties.  e u · φ(fi (u)) = ue · fi (u)p if i ∈ Λ0 (a) g(u) · fi+1 (u) = φ(fi (u)) = fi (u)p if i ∈ / Λ0 (b) ordu (g(u)) ≤ e. In particular g(u) 6= 0 and fi (u) 6= 0 for all i ∈ I. (c) There exists an element i1 ∈ I such that fi1 (u) ∈ F[[u]]× . 2The S -module M here should not be confused with the 0th component of the κ -linear S-module M. 0 0 E

B.2. CONSTRUCT CM LIFTING BY P -ADIC HODGE THEORY

193

Write µ = ordu (g(u)) and di = ordu (fi (u)) for the “unknown” elements g(u) and fi (u), i ∈ I, we arrive at the following system of linear equations for 4m + 1 unknown integers µ and d1 , d2 , . . . , d4m , subject to certain linear inequalities.  p di + e if i ∈ Λ0 (1) di+1 + µ = p di if i ∈ / Λ0 (2) 0 ≤ µ ≤ e, and di ≥ 0 for all i ∈ I. (3) There exists an element i1 ∈ I such that di = 0. The system of equations (1)–(3) give the order of the components of any solution for the system of equations of equations (a)–(c), equivalent to the system of equations (i)–(iii). We will see that the system of equations (1)–(3) has a unique solution. B.2.10. • • • •

We proceed to solve the system of equations B.2.9 (1)–(3). Let U be the Q-vector space with basis {ui | iP ∈ I}. Consider the di ’s as “variables”, and let d = i∈I di ui . Let S be P the Q-linear endomorphism P of V such that S(ui ) = ui−1 for all i ∈ I. Let 1 := i∈I ui , and let uΛ0 := j∈Λ0 uj .

The equations B.2.9 (1)–(3) can be written in the following vector form (p · IdU − S) · d = µ · 1 − e · uΛ0 , where µ appears as an integer-valued parameter between 0 and µ, all components di of the solution d are natural numbers, and at least one of the components di0 is equal to 0. Because S 4m = IdU , we have (p · IdU − S)−1 = (p4m − 1)−1 · (S 4m−1 + p S 4m−2 + · · · + p4m−2 S + p4m−1 ) . Using the fact that S · 1 = 1 and recall that e = p4m − 1, we get d = (p4m − 1)−1 · (p · IdU − S)−1 · (µ · 1 − e · uΛ0 ) " ! ! # 4m−1 4m−1 X X = (p4m − 1)−1 · µ · pa · 1 − e · pa · S 4m−1−a · uΛ0 a=0

=

=

X e µ · 1 − 4m · p4m−1−a uj−a p−1 p −1 i∈I a=0 j∈Λ0   X X µ  ·1− p4m−1−a  · ui  p−1 0≤a≤4m−1 i∈I

=

a=0

X 4m−1 X

a∈Λ0 −i

X µ ci · ui ·1− p−1 i∈I

where ci :=

X

p4m−1−a

∀i ∈ I .

0≤a≤4m−1 a∈Λ0 −i

In other words we have

µ − ci ∀i ∈ I . p−1 The p-adic expansions of the numbers ci , consisting of 0’s and 1’s, are cyclic permutations of each other; c2m−1 is the largest and c4m−1 is the smallest among these 4m numbers. The p-adic expansions of c2m−1 and c4m−1 are depicted below. di =

c2m−1 : 11 0101 . . . 01} 00 |1010{z . . . 10} ; | {z 2m-2 digits

2m-2digits

c4m−1 : 00 1010 . . . 10} 11 |0101{z . . . 01} | {z 2m-2 digits

2m-2 digits

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B. CM LIFTING VIA P -ADIC HODGE THEORY

B.2.11. Lemma.    µ

(i) The equations B.2.9 (1)–(3) have a unique solution given by X = (p − 1) · c2m−1 = (p − 1) · p4m−1−a µ p−1

  d = i

0≤a≤4m−1 a∈Λ0 −2m+1

− ci

= c2m−1 − ci

∀i ∈ I

where ci :=

X

p4m−1−a

∀i ∈ I .

0≤a≤4m−1 a∈Λ0 −i

(ii) The integers di satisfy the following inequalities 0 = d2m−1 < d2m−3 < · · · < d3 < d1 < d2m < d2m+2 < · · · < d4m−4 < d4m−2 < d2m−2 < d2m−4 < · · · < d2 < d4m < d2m+1 < d2m+3 < · · · < d4m−3 < d4m−1 . In particular d1 < d2m = e − µ . Proof. We have seen the formulas for the ci ’s and the di ’s in B.2.10. The value of the parameter µ is determined by the condition that Min ( di | i ∈ I ) = 0. The ordering of the di ’s is easily determined from the p-adic expansion of the ci ’s. The equality d2m = e − µ is the special case i = 2m − 1 of equation B.2.9 (1), included here for future reference. One can also show by direct computation that e − µ − d1 = p2m − 1. B.2.12. With the solution of equations B.2.9 (1)–(3) available, we can solve the equations B.2.9 (a)–(c). Their solutions are far from unique, because one can scale the fi (u)’s simultaneous by a unit c(u) ∈ F[[u]]× and multiply g(u) by φ(c(u)); that scaling corresponds to multiplying the corresponding solutions of equations B.2.9 (i)–(iii) by the unit c(u). So we normalize and look for solutions of equations B.2.9 (1)–(3) with f2m−1 (u) = 1. We solve the equations B.2.9 (i)–(iii) recursively, starting with f2m−1 (u) = 1, and get fj (u) =

usj g(u)rj

j = 2m, 2m + 1, . . . , 4m, 1, 2, . . . , 2m − 2

where  X  e· pj−i    2m≤i≤j   i−1∈Λ0     sj =     X j−i   e· p +     1≤i≤j i−1∈Λ0

for 2m ≤ j ≤ 4m,

 X

 pj+4m−i 

for 1 ≤ j ≤ 2m − 2,

2m≤i≤4m i−1∈Λ0

and

rj =

          

X

pj−2m−a

for 2m ≤ j ≤ 4m,

pj+2m−a

for 1 ≤ j ≤ 2m − 2.

0≤a≤j−2m

X 0≤a≤j+2m

For later reference, we set s2m−1 = 0 = s2m−1 .

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195

The above formulas depend on the yet-unknown element g(u) ∈ F[[u]] with ordu (g(u)) = µ, where µ is given in B.2.11. The system of equations B.2.9 (i)–(iii) is now reduced to a single equation in the unknown element g(u), obtained from B.2.9 (i) by setting i = 2m − 2, namely g(u) = f2m−2 (u)p because 2m − 2 ∈ / Λ0 , where f2m−2 (u) is given in the preceding paragraph. One can check that the above equation for g(u) is compatible with the requirement that X ordu (g(u)) = µ = (p − 1) c2m−1 = (p − 1) · p4m−1−a . 0≤a≤4m−1 a∈Λ0 −2m+1

Write the unknown element g(u) ∈ F[[u]] as 

 g(u) = uµ · h(u) ,

h(u) = ξ · 1 +

X

cj uj  ,

j∈N>0

the above equation for g(u) is transformed to the following equation P

h(u)

0≤a≤4m−1

pa

=1

in the unknown element h(u) ∈ F[[u]]× . It follows that h(u) = ξ ∈ µe/(p−1) (F), i.e. h(u) is a constant in F[[u]], equal to a (p4m − 1)/(p − 1)-th root of 1 in F. We have found all solutions of the system of equations B.2.9 (i)–(iii), summarized in B.2.13 below. B.2.13. Lemma. Let N = e/(p − 1) = (p4m − 1)/(p − 1). Choose and fix a generator ξ of the group µN (F) of all N -th roots of unity in F. There are exactly N admissible rankone sub-objects of the object in the exact category (M, φM , ψM ) in (Mod/S)c , corresponding to the N finite flat subgroups of X [p]×Spec(W (F)) Spec(OK ) of order p over OK . Each of these N admissible sub-objects is given by a free rank-one F[[u]]-direct summand of the free F[[u]]-module M of the form F[[u]] · ya a ∈ Z/N Z , where ya :=

X

ξ −a rj · usj −µ rj · vj =

X

ξ −a rj · udj · vj

j∈I

j∈I

and the natural numbers µ, sj and rj are defined in B.2.11 and the second paragraph of B.2.12. Note that the action of O/pO ∼ /Fp× on the = κE on M induces a simply transitive action of κ× E collection { F[[u]]·ya | a ∈ Z/N Z } of F[[u]]-submodules of M. B.2.14. Definition. Fix a generator ζ of the cyclic group κ× . For any element η ∈ κE , write E [η] : M → M for the action of η on the κE -linear F[[u]]-module M. Recall that [η](y) = i(η) · y

∀ η ∈ κE , ∀ i ∈ I, ∀ y ∈ Mi ,

and (i + 1)(η) = i(η)p

∀ i ∈ I, ∀ η ∈ κE .

(1) Denote by M(0) the free F[[u]]-submodule generated by the element ! X y0 = udi · vi i∈I

in B.2.13. (2) For b = 1, 2, . . . , m − 1, let M[b] be the F[[u]]-submodule of M generated by the b + 1 elements y0 , [ζ](y0 ), . . . , [ζ b ](y0 ) in M.

196

B. CM LIFTING VIA P -ADIC HODGE THEORY

(3) For b = 1, 2, . . . , m − 1, let M(b) := {x ∈ M | um x ∈ M[b] for some m ∈ N} = (F((u)) · M[b] ) ∩ M . be the saturation in M of the F[[u]]-submodule M[b] ⊂ M. B.2.15. Lemma. Let b be an integer with 0 ≤ b ≤ m − 1. (i) The F[[u]]-submodule M[b] of M = ⊕i∈I F[[u]]·vi is a free F[[u]]-module of rank b + 1 with a basis of the form X λ0,i · vi z0 = ud2m−1 · v2m−1 + i∈{2m−2b−1,2m−2b+3,...,2m−1} /

.. .

.. .

.. .

za

= ud2m−2a−1 · v2m−2a−1 +

X

λa,i · vi

i∈{2m−2b−1,2m−2b+3,...,2m−1} /

.. . zb

.. .

.. .

= ud2m−2b−1 · v2m−2b−1

X

+

λb,i · xi

i∈{2m−2b−1,2m−2b+3,...,2m−1} /

where each λa,i is an element of F[[u]] with ordu (λa,i ) ≥ di

∀0 ≤ a ≤ b, ∀i ∈ / {2m−2b−1,2m−2b+1,...,2m−1}

and the di ’s are the natural numbers defined in B.2.11. (ii) The F[[u]]-submodule M(b) of M = ⊕i∈I F[[u]]·vi is a free F[[u]]-module of rank b + 1 with basis   −d 2m−2a−1 u · za a=0,1,...,b.

−d2m−2b−1

(iii) M(b) ⊂ u

· M[b] .

Proof. The statements (ii) and (iii) are immediate consequences of (i) and the inequalities in B.2.11 (ii): d2m−a−1 < ordu (λa,i )

∀0 ≤ a ≤ b, ∀i ∈ / {2m−2b−1,2m−2b+1,...,2m−1} .

It remains to prove (i). We have [ζ a ](y0 ) =

X

i(ζ a ) · udi · vi

i∈I

for a = 0, 1, . . . b, a system of b + 1 linear equations with coefficients Ca,i := i(ζ)a

0 ≤ a ≤ b, i ∈ I

indexed by {0, 1, . . . , b} × I. Look at the maximal minor C(b) of the (b + 1) × (4m)-matrix C where the second index i belongs to the subset {2m−2b−1, 2m−2b−3, . . . , 2m−3, 2m−1 } ⊂ I. This square matrix is a van der Monde matrix given by the b + 1 distinct roots of unity in F, hence C(b) is an invertible matrix with entries in F. Multiplying the system of linear equations by the inverse of the matrix C(b) , we get a system of generators of the F[[u]]-module M[b] as described in (i). B.2.16. Proposition. Let b be an integer with 0 ≤ b ≤ m − 1. (1) φM (M(b) ) ⊂ M(b) .

B.2. CONSTRUCT CM LIFTING BY P -ADIC HODGE THEORY

197


(2) ψM (M(b) ) ⊂ φ∗ (M(b) ) ⊆ φ∗ M ∩ F((u)) · φ∗ (M(b) ) , where the “φ∗ (M(b) )” in the last term is identified with its image in φ∗ M via the natural injection φ∗ (M(b) ) −→ φ∗ M. (3) Let φM(b) : M(b) → M(b) be the φ-linear map induced by φM , and let ψM(b) : M(b) → φ∗ (M(b) ) be the F[[u]]-linear map induced by ψM . The triple   M(b) , φM(b) : M(b) → M(b) , ψ (b) : M(b) → φ∗ (M(b) ) M

defines an admissible subobject of the exact category (M, φM , ψM ), corresponding to a finite flat subgroup scheme G(b) ⊂ X×Spec(W (F)) Spec(OK ) over OK of order pb+1 . (4) The classical covariant Diedonn´e module M∗ (G(b) ) of the closed fiber G(b) of G(b) , identified as a Dieudonn´e submodule of the Dieudonn´e module M∗ (X0 [p]) = ⊕i∈I M∗ (X0 [p])i ∼ = ⊕i∈I (Mi /uMi ) of the p-torsion subgroup scheme X0 [p] of the O-linear p-divisible group X0 over F κE = O/pO, and we have M∗ (G(b) ) = M∗ (X0 [p])2m−2b−1 ⊕ M∗ (X0 [p])2m−2b−3 ⊕ · · · ⊕ M∗ (X0 [p])2m−1 . In particular the subgroup G(b) of X0 is stable under the action of O. (5) The closed fiber G(m−1) of the finite flat subgroup scheme G(m−1) ⊂ X [p]×Spec(W (F)) Spec(OK ) is equal to the kernel of the O-linear isogeny ρ0 : X0 → Y0 . Proof. The statement (1) is clear. To see (2), recall first from B.2.9 and B.2.11 that ψM (F[[u]]· y0 ) = ue−µ · φ∗ (F[[u]]· y0 ) , hence ψM (M[b] ) = ue−µ · φ∗ (M[b] ) . We know from B.2.15 (iii) that M(b) ⊂ u−d2m−2b−1 · M[b] and we have d2m−2b−1 < e − µ by B.2.11 (ii). So we get ψM (M(b) ) ⊆ ψM (u−d2m−2b−1 · M[b] ) ⊆ ue−µ−d2m−2b−1 · φ∗ (M[b] ) ⊂ φ∗ (M[b] ) ⊆ φ∗ (M(b) ) . We have proved (2). The statement (3) is immediate from (1) and (2). The statement (4) is a consequence of B.2.15 (ii) and the inequalities in B.2.11 (ii). We have O-equivariant isomorphisms M∗ (X0 [p]) ∼ = M/uM and Lie(Ker(ρ0 )) ∼ = Ker

 M/uM

(1/c)ψM

/ φ∗ M/uφ∗ M



from B.4.17. Recall that Ker(ρ0 ) is the unique subgroup of X0 [p] of order pm whose Lie type corresponds to the subset {1, 3, 5, . . . , 2m − 1} of I. So M∗ (Ker(ρ0 )) is the sum of the eigensubspaces M∗ (X0 [p])i of M∗ (X0 [p]) for i = 1, 3, . . . , 2m − 1. The statement (5) now follows from the special case b = m − 1 of the statement (4). 4m

−1 B.2.17. Remark. Let K1 /K0 ⊂ K/K0 be the subextension of K/K0 with [K1 : K0 ] = p p−1 and [K : K1 ] = p − 1. It is clear from the formulas in B.2.11 that the integers di ’s and µ in are all divisible by p − 1. From this congruence it is not difficult to deduce that the admissible subobject (M(b) , φM(b) , ψM(b) ) of (M, φM , ψM ) in the category (Mod/OK )c is the base

change, from OK1 to OK , of an admissible subobject of (M1 , φM1 , ψM1 ) in (Mod/OK1 )c , where (M1 , φM1 , ψM1 ) is the base extension from OK0 to OK1 of (M0 , φM0 , ψM0 ). Correspondingly, the finite flat group scheme G(b) in B.2.16 is the base change of a finite flat subgroup scheme

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B. CM LIFTING VIA P -ADIC HODGE THEORY

over OK1 of X [p]×Spec(W (F)) Spec(OK1 ). Alternatively, one can replace K by K1 and redo the constructions and proofs in this section. See B.1.15 for a similar situation with Raynaud’s method.

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199

B.3. Dieudonn´ e theories over a perfect field of characteristic p In this section we compare various Dieudonn´e theories over a perfect base field k ⊃ Fp . B.3.1. Notation. Let W = W (k) be the ring of p-adic Witt vectors with entries in the base field k ⊃ Fp , and let K0 = frac(W (k)) be its fraction field. Let σ be the Frobenius automorphism on W (k) such that σ(x0 , x1 , x2 , . . .) = (xp0 , xp1 , xp2 , . . .) for every Witt vector (x0 , x1 , x2 , . . .) ∈ W (k). For any of the Dieudonn’e theories reviewed below we will get, for any finite group scheme (or a p-divisible group or a finite dimensional commutative smooth formal group) over k, a W (k)-module M of finite type, together with two semi-linear operators F and V , F : M (1) := W (k)⊗σ,W (k) M −→ M,

V : M −→ M (1) .

For any n ∈ Z, define M (n) := W (k)⊗σn ,W (k) M . Note that the map y 7→ 1 ⊗ y from M to M (n) is a bijection. We can use this map to identify M (n) with M as an abelian group, but M (n) has a new W (k)-module structure such that ξ ·new y = σ −n (ξ) ·old y

∀ξ ∈ W (k), ∀y ∈ M .

We also have Frobenius twist of semi-linear maps, for instance    (−1) Ker FM (−1) : M → M (−1)) ∼ . = Ker(FM : M (1) → M ) B.3.2. Witt covectors. We recall the definition and some basic properties of Witt covectors; more information can be found in [25, Ch. II]. B.3.2.1. For every commutative ring R with 1, the set CW (R) of all p-adic Witt covectors

with entries in R is the set consisting of all sequences a = (. . . , a−n , . . . , a−1 , a0 ) such that (i) a−n ∈ R for all n ∈ N (ii) there exists an r ∈ N (which depends on a) such that the ideal generated by all a−n ’s with n ≥ r is nilpotent. More generally, we can pass to the limit and define, for any complete separated topological commutative ring R with 1 such that there is a fundamental system of open neighborhoods of 0 consisting of open ideals, the set CW (R), by replacing condition (ii) by (ii)0 For every open ideal I in R, there exist positive integers r and N such that the ideal Jr = (a−n )n≥r generated by all a−n ’s with n ≥ r satisfies JrN ⊆ I. B.3.2.2. The group law on CW (R) is defined as follows: For any two elements

a = (. . . , a−n , . . . , a−1 , a0 )

and b = (. . . , b−n , . . . , b−1 , b0 )

in CW (R), their sum c = (. . . , c−n , . . . , c−1 , c0 ) is defined by cn = lim Sm (a−n−m , . . . , a−n ; b−n−m , . . . , b−n ) , m→∞

where Sm (a−n−m , . . . , a−n ; b−n−m , . . . , b−n ) on the right hand side is constant for m sufficiently large if R has discrete topology, and the polynomials Sm (x0 , x1 , . . . , xm ; y0 , y1 , . . . , ym ) ∈ Z[x0 , x1 , . . . , xm ; y0 , y1 , . . . , ym ] for m ∈ N are the universal polynomials which define the group laws on the p-adic Witt vectors; i.e. (x0 , x1 , . . . , xn , . . .) +Witt (y0 , y1 , . . . , yn , . . .) = (Sm (x0 , x1 , . . . , xm ; y0 , y1 , . . . , ym ))m∈N

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B. CM LIFTING VIA P -ADIC HODGE THEORY

characterized by m X i=0

a

p

m−a xpa

+

m X

p

a

m−a yap

=

i=0

m X

pa · (Sa (x0 , x1 , . . . , xm ; y0 , y1 , . . . , ym ))p

m−a

∀m ∈ N.

i=0

B.3.2.3. The group law for the functor CW is encoded in one single formal power series

S(x; y) = S(. . . , x−2 , x−1 , x0 ; . . . , y−2 , y−1 , y0 ) ∈ Z[[x, y]] in infinitely many variables {x−n , y−n | n ∈ N}, where n runs through all natural numbers, characterized by the equation X X X n pn p−n xp−n + p−n y−n = p−n pn S−n (x; y) , n∈N

n∈N

n∈N

where for each n ∈ N the power series S−n (x; y) , is defined in terms of S by shifting the variables: S−n (. . . , x−2 , x−1 , x0 ; . . . , y−2 , y−1 , y0 ) := S(. . . , x−n−2 , x−n−1 , x−n ; . . . , y−n−2 , y−n−1 , y−n ) . The group law for the Witt covectors is a +CW b = (. . . , S−m (a; b), . . . , S−1 (a; b), S0 (a; b)) = (. . . , S(. . . , a−2 , a−1 ; . . . , b−2 , b1 ), S(. . . , a−1 , a0 ; . . . , b−1 , b0 )) Alternatively S(x; y) is the limit S(x; y) = lim Sm (x−m , x−m+1 , . . . , x−1 , x0 ; y−m , y−m+1 , . . . , y−1 , y0 ) ; m→∞

this limit makes sense because Sm+1 (0, x−m , x−m+1 , . . . , x−1 , x0 ) = Sm (x−m , x−m+1 , . . . , x−1 , x0 )

∀m ∈ N .

Note that each term of the power series S(x; y) are homogeneous of weight one if x−n and y−n are given the weight p−n for every n ∈ N. B.3.2.4. Restricting the group-valued functor CW to the category of commutative k-algebras

(with 1), we get an infinite dimensional smooth group CWk over k. For every k-algebra R, the group CWk (R) has a natural W (k)-module structure: For any u = (u0 , u1 , u2 , . . . , un , . . .) ∈ W (k) and any a = (. . . , a−n , . . . , a−1 , a0 ) ∈ W (R), the product b = (. . . , b−n , . . . , b−1 , b0 ) = u · a of u and a is given by b−n = Pm (σ −n−m (u0 ), σ −n−m (u1 ), . . . , σ −n−m (un ); a−n−m , a−n−m+1 , . . . , a−1 , a0 ) for m sufficiently large, where the polynomials Pm (x0 , x1 , . . . , xm ; y0 , y1 , . . . , ym ) ∈ Z[x0 , x1 , . . . , xm ; y0 , y1 , . . . , ym ] indexed by m ∈ Z are the universal polynomials which define the product law on the p-adic Witt vectors. In particular (ξ, 0, 0, . . .) · (. . . , b−n , . . . , b−1 , b0 ) = (. . . , σ −n (ξ)b−n , . . . , σ −1 (ξ)b−1 , ξb0 ) for all ξ ∈ k and all . . . , b−n , . . . , b−1 , b0 ) ∈ CW (R).

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Define endomorphisms F and V on CWk by F (. . . , a−m , . . . , a−2 , a−1 , a0 ) = (. . . , ap−m , . . . , ap−2 , ap−1 , ap0 ) and V (. . . , a−m , . . . , a−2 , a−1 , a0 ) = (. . . , a−m , . . . , a−2 , a−1 ) for all k-algebra R and all (. . . , a−m , . . . , a−2 , a−1 , a0 ) ∈ CW (R). We have F ◦ V = p · IdCWk = V ◦ F,

F ◦ [a]CWk = [σ(a)]CWk ◦ F,

[a]CWk ◦ V ] = V ◦ [σ(a)]CWk

for all a ∈ W (k), where [a] denotes the endomorphism “multiplication by a” of the formal group CWk . B.3.3. Let CW u be the subfunctor of CW such that for any commutative ring R with 1, CW u (R) is the subset of CW (R) consisting of all elements (. . . , a−n , . . . , a−1 , a0 ) in CW (R) such that a−n = 0 for all but finitely many n’s. In other words   V V V CW u = lim W1 −→ W2 −→ · · · Wn −→ Wn+1 · · · , n→∞

the direct limit of truncated Witt groups Wn with respect to transition maps V : Wn → Wn+1 , which sends (x0 , x1 , . . . , xn−1 ) to (0, x0 , x1 , . . . , xn−1 ) as usual. It is easy to see that CW u is a subgroup of CW , and CWku is stable under multiplication by elements of W (k), B.3.4. Classical Dieudonn´ e theory. B.3.4.1. For any finite group scheme G over k of p-power order, the classical contravariant Dieudonn´e module M∗ (G) is by definition the W (k)-module M∗ (G) = Homk (G, CWk ) together with W (k)-linear maps FM∗ (G) : M∗ (G)(1) → M∗ (G) ,

VM∗ (G) : M∗ (G) → M∗ (G)(1)

induced by the relative Frobenius homomorphism FrG : G → G(p) and the Verschiebung homomorphism VerG : G(p) → G. The W (k)-module structure on M∗ (G) comes from the W (k)-module structure on the functor CWk . We have VM∗ (G) ◦ FM∗ (G) = p · IdM∗ (G)(1)

and FM∗ (G) ◦ VM∗ (G) = p · IdM∗ (G) .

B.3.4.2. If we identify M∗ (G)(1) with M∗ (G) by the bijection 1 ⊗ x 7→ x for all x ∈ M∗ (G),

then M∗ (G) becomes a left module over the ring Dk := W (k)[F, V ], where W (k)[F, V ] is the non-commutative ring generated by F and V over W (k), subject to the relations F V = p = V F,

F a = σ(a) F ∀a ∈ W (k),

V σ(a) = a V ∀a ∈ W (k).

∗

Equivalently, the action of F and V on M (G) are induces by the endomorphism F and V on CWk . B.3.4.3. The following facts are well-known; see [25, Ch. III] and also [21] and [44].

(i) M∗ (G) is a W (k)-module of finite length for every commutative finite group scheme G over k of p-power order. (ii) The functor M∗ induces an anti -equivalence between the abelian category of commutative finite groups schemes of p-power order over k and the abelian category of Dk -modules of finite length over W (k). The inverse functor is 

M

/ HomD (M, CW ) , k

which sends a Dk -module M of finite length over W (k) to the group-valued functor R



/ HomD (M, CW (R)) k

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on the category of commutative k-algebras with 1, or the category of complete separated commutative topological rings with a fundamental system of neighborhoods of 0 consisting of open ideals. (iii) The cotangent space ωG of G at the zero-section is naturally isomorphic to   Coker FM∗ (G) : M∗ (G)(1) → M∗ (G) . When G is a BT1 group over k, i.e. G ∼ = X[p] for some p-divisible group X over k, we have an isomorphism   ' ωG −→ Coimage VM∗ (G) : M∗ (G) → M∗ (G)(1) , because the sequence M∗ (G)(1)

FM∗ (G)

/ M∗ (G)

VM∗ (G)

/ M∗ (G)(1)

is exact at the middle term. B.3.4.4. The example of the case G = µpn illustrates the difference between CW and CW u .

We have Homk (µpm , CW u ) = (0) because Homk (µpm , Wn ) = (0) for all n ≥ 1.3 However '

M∗ (µpn ) = Homk (µpm , CWk ) is isomorphic to W/pm W , and V : M∗ (µpn ) −→ M∗ (µpn )(1) . We will write down an injection ξ : µpm → CWk corresponding to a generator of the W -module M∗ (µpn ) with VCW ◦ ξ = ξ. Let ! X Y pm Y µ(n) n n = exp − ∈ Z(p) [[Y ]] E(Y ) = (1 − Y ) pm n∈N m∈N

≥1 gcd(n,p)=1

be the classical Artin-Hasse exponential in the variable Y , and let X ψ(T ) = T + bi T i ∈ Z(p) [[T ]] i≥2

be the inverse of E(Y ), in the sense that 1 − E(ψ(T )) = T ,

and ψ(1 − E(Y )) = Y .

The map ξ : µpn −→ CW over k is given by µpn (R) 3 1 − t  n

/ (. . . , ψ(t), ψ(t), ψ(t)) ∈ CW (R) n

for all t ∈ R with tp = 0, i.e. (1 − t)p = 1. To see that ξ respects the group law, it suffices to check that the inverse map CW (R) 3 (. . . , y, y, y, y) 

/ E(y) ∈ G∧ (R) m

which is defined for every topologically nilpotent elements y in a complete separated topological ring R over Z(p) with a fundamental system of neighborhoods of 0 consisting of open ideals of R, respects the group law for CW and the multiplicative formal group G∧ m . The last assertion follows immediately from the description of the group law on CW in terms of the power series S(x; y).

3This fact caused problems in treatments of covariant Dieudonn´ e theory using lim Wn ∼ = CW u before

−→ [25], where the case of finite group schemes of multiplicative type over k are treated separately.

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B.3.4.5. We have a functorial isomorphism M∗ (GD ) ∼ = HomW (M ∗ (G), K0 /W ) , where GD is

the Cartier dual of G. Under this isomorphism, the map FM∗ (GD ) : M∗ (GD )(1) → M∗ (GD ) corresponds to the transpose of VM∗ (G) : M∗ (G) → M∗ (G)(1) . Alternatively, there is a functorial pairing h , iM ∗ (G) : M∗ (G) × M∗ (GD ) −→ K0 /W such that ∀ x ∈ M∗ (G)(1) , ∀ y ∈ M∗ (G) ,

hF x, iM∗ (G) = hx, V yi

M∗ (G)(1)

where h , i

M∗ (G)(1)

is the composition

(W ⊗σ,W M∗ (G)) × (W ⊗σ,W M∗ (GD ))

W⊗σ,W h , iM∗

/ W⊗ σ,W K0 /W

w⊗a 7→ w σ(a)

/ K0 /W .

'

Let jG : G −→ (GD )D be the canonical isomorphism from G to its double Cartier dual. The diagram M∗ (G) × M∗ (GD ) h , iM ∗ (G)

u⊗v 7→ v⊗jG (u)

/ M∗ (GD ) × M∗ ((GD )D ) h , i





K0 /W

M ∗ (GD )

/ K0 /W

=

anti -commutes. B.3.4.6. Define the classical covariant Dieudonn´ e module M∗ (G) of a commutative finite group

scheme G over k of p-power order by M∗ (G) := M∗ (GD ) with FM∗ (G) = FM∗ (GD ) and VM∗ (G) = VM∗ (GD ) . Alternatively, one can define M∗ (G) := HomW (M∗ (G), K0 /W ) , define FM∗ (G) to be the dual of VM∗ (G) , and define VM∗ (G) to be the dual of FM∗ (G) . B.3.4.7. The functor M∗ gives an equivalence from the abelian category commutative finite group scheme G over k to the category of W (k)-modules of finite length over W (k). We have a natural isomorphism   Lie(G) ∼ = Ker VM∗ (G) : M∗ (G) → M∗ (G)(1) .

When G is a BT1 -group over k, we have an isomorphism   ' Lie(G) ←− Image FM∗ (G) : M∗ (G)(1) → M∗ (G) because the sequence M∗ (G)(1) is exact at the middle term.

FM∗ (G)

/ M∗ (G)

VM∗ (G)

/ M (G)(1) ∗

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B. CM LIFTING VIA P -ADIC HODGE THEORY

B.3.4.8. We can extend the classical contravariant Dieudonn´ e theory M∗ to p-divisible groups

over k, as follows. For any p-divisible group X over k, define M∗ (X) := lim M∗ (X[pn ]) , ←− n→∞

∗

where the transition maps M −→ M∗ (X[pn ]) are induced by the inclusions X[pn ] ,→ n+1 X[p ]. The F and V operators for M∗ (X[pn ]) pass to the limit and defines (X[pn+1 ])

FM∗ (X) : M∗ (X)(1) → M∗ (X)

andVM∗ (X) : M∗ (X) → M∗ (X)(1)

It is known that M∗ (X) is a free W (k)-module of finite rank, equal to the height of X. Moreover the functor M∗ induces an anti-equivalence of categories, from the category of pdivisible groups over k to the category of left W (k)[F, V ]-modules free of finite rank over W (k). B.3.4.9. Similarly, define

M∗ (X) := lim M∗ (X[pn ]) , ←− n→∞

(X[pn+1 ])

where the transition maps M∗ −→ M∗ (X[pn ]) are induced from the faithfully flat n+1 n surjections X[p ]  X[p ]. The functor M∗ induces an equivalence of categories, from the category of p-divisible groups over k to the category of left W (k)[F, V ]-modules free of finite rank over W (k). We have a functorial isomorphism ∼ HomW (k) (M∗ (X), W (k)) ; M∗ (X) = the operators FM∗ (X) and VM∗ (X) corresponds to the transpose of VM∗ (X) and FM∗ (X) respectively. B.3.4.10. One can recover the Dieudonn´ e theory for finite flat group schemes over k from the Dieudonn´e theory for p-divisible groups (or abelian varieties), as follows. A theorem of Raynaud asserts that every finite locally free group scheme G over a local ring R can be embedded into the Jacobian of an abelian scheme A over R; see [5, 3.1.1]. ρ Starting with a short exact sequence 0 → G → X −→ Y → 0, where G is a given finite group scheme over k of p-power order, X and Y are p-divisible groups over k, and ρ is an isogeny over k. Then we have a natural isomorphism     M∗ (ρ) 1⊗M∗ (ρ) ' / M∗ (X) / (K/W )⊗W M∗ (X) −→ Coker M∗ (Y ) Ker (K/W )⊗W M∗ (Y ) ∗

and we take M (G) to be Coker



∗

M (Y )

M∗ (ρ)

/ M∗ (X)

 .

B.3.5. Cartier theory. Cartier’s theory [11] for commutative formal groups was systematically documented in [42]; we recommend [88] and refer to it for more information. Although Cartier’s theory is valid for commutative smooth formal groups, we will only discuss the special case when the base ring R is an algebra over Z(p) . ˆ plays a key role, a sort of “free B.3.5.1. The following commutative smooth formal group W R generator” for the category of commutative smooth formal groups over k, For any commutative complete separated topological ring S over R (with 1), with a fundamental system of open ˆ (S) is the subgroup of W (S), consisting of neighborhoods of 0 consisting of open ideals, W R all p-adic Witt vectors a = (a0 , a1 , a2 , . . .) with all ai ∈ S with the following property: For every open ideal I ⊂ S, there exists an integer m ∈ N which depends on a and I, such that the ideal of S generated by all ai ’s with i ≥ c(m) is contained in I.

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ˆ ) of B.3.5.2. Define the Cartier ring Cartp (R) to be the opposite ring of the ring EndR (W ˆ R. endomorphisms of W (1) The ring Cartp (R) contains two elements V and F and is complete with respect to the filtration by right ideals V m Cartp (R). Moreover Cartp (R) contains a family of elements { [a] | a ∈ R} indexed by R. (2) The following identities hold. – [0] = 0Cartp (R) , [1] = 1Cartp (R) , – [a] · [b] = [ab] for all a, b ∈ R, – F [a] = [ap ]F for all a ∈ R, – [a]V = V [ap ] for all a ∈ R, – F V = p. (2) There is an injective ring homomorphism j : W (R) → Cartp (R) such that X V n [an ]F n for all a = (a0 , a1 , a2 , . . .) ∈ W (R) . j(a0 , a1 , a2 , . . .) = n≥0

(3) Every element of Cartp (R) has a unique expression as a convergent sum X V m [am,n ]F n , am,n ∈ R ∀ m, ∀ n , m,n≥0

with the property that for every m ∈ N, there exists an integer c(m) ∈ N such that am,n = 0 for all n ≥ c(m). (4) A left Cartp (R)-module M is V -reduced if V : M → V is injective and the natural map M → limm→∞ M/V m · M is an isomorphism. A V -reduced left Cartp (R)-module M ←− is V -flat if M/V M is a flat R-module. B.3.5.3. The Cartier module of a commutative smooth formal group X over R is defined to

be ˆ , X) C∗ (X) := HomR (W R endowed with the left Cartp (R)-module structure from the right Cartp (R)-module structure ˆ . We have a natural isomorphism on W R '

C∗ (X)/V · C∗ (X) −→ Lie(X) . B.3.5.4. The main theorem of Cartier theory asserts that the functor X 7→ C∗ (X) is an

equivalence, from the category of commutative smooth formal groups over R, to the category of V -reduced V -flat left Cartp (R)-modules. The inverse functor is   ˆ (S)⊗ ˆ Cartp (R) M , M 7→ S 7→ W ˆ denotes where S denotes an arbitrary complete commutative topological R-algebra and ⊗ a suitably defined completed tensor product; see [88, 4.23, 3.28] for details. The completed ˆ can be replaced by the usual tensor product ⊗ if the left Cartp (R)-module tensor product ⊗ M is finitely generated. B.3.5.5. In the case R = k is a perfect field of characteristic p, Cartp (k) is the completion of

the Dieudonn´e ring W (k)[F, V ] with respect to the filtration by the right ideals V n ·W (k)[F, V ]. The V -reduced V -flat left Cartp (k)-modules corresponding to some p-divisible smooth formal group over k are exactly those left W (k)[F, V ]-modules M free of finite rank over W (k) such that there exits an integer m ∈ N with V m · M ⊆ p · M .

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B. CM LIFTING VIA P -ADIC HODGE THEORY

B.3.5.6. We can use the procedure in B.3.4.10 to define a covariant Dieudonn´ e theory for

finite group schemes over k, as follows. For a given finite connected group scheme G over k of p-power order, embed G in a p-divisible ρ formal group X over k to get a short exact sequence 0 → G → X −→ Y → 0. Then we have a natural isomorphism     1⊗C∗ (ρ) C∗ (ρ) ' / (K0 /W )⊗W C∗ (Y ) −→ / C∗ (Y ) Ker (K0 /W )⊗W C∗ (X) Coker C∗ (X)  and we define C∗ (G) to be Ker

(K0 /W )⊗W C∗ (X)

1⊗C∗ (ρ)

/ (K0 /W )⊗W C∗ (Y )

 .

For a general finite group scheme G over k, one decomposes G into a product G = Gconn × Get , and define
C∗ (G) := C∗ (Gconn ) ⊕ HomW (C∗ (Get )D ), K0 /W . B.3.5.7. The functor X 7→ C∗ (X) gives an equivalence, from the category p-divisible groups

over k, to the category of left W (k)[F, V ]-modules which are free of finite rank over W (k). The functor G 7→ C∗ (G) gives an equivalence, from the category of finite group schemes over k of p-power order, to the category of left W (k)[F, V ]-modules of finite length over W (k). Similarly the contravariant functors X 7→ C∗ (X) and G 7→ C∗ (G) give anti -equivalences of categories. B.3.5.8. Similar to the case of classical Dieudonn´ e theory, we have functorial isomorphisms

C∗ (GD ) ∼ = HomW (C∗ (G), K0 /W ) respecting the F and V operators. We define a contravariant theory over k by C∗ (G) := HomW (C∗ (G), K0 /W ) for any finite group scheme G over k, and define C∗ (X) := lim C∗ (X[pn ]) ←− n→∞

for any p-divisible group X over k, where the transition maps in the projective systems are induced by the inclusions X[pn ] ,→ X[pn+1 ]. One can verify the compatibilities, for instance the C∗ (X) defined above for a formal p-divisible group X over k is naturally isomorphic to the W (k)-linear dual of the Cartier module C∗ (X ∧ ) of the smooth formal group X ∧ attached to X. B.3.6. Zink’s theory of displays. We refer to [90] for the theory of displays; see also B.4.7 for the notation for displays. Let R be an excellent base commutative ring, and we assume either that p is nilpotent in R or that R is complete with respect to an ideal and p is topologically nilpotent. What we need is the relation between • a display   P = P, Q, FP : P → P, F1,Q : Q → P, V ] : P → W (R) ⊗F,W (R) P over R, and • the Cartier module C∗ (BT(P)) of the formal completion of the formal p-divisible group BT(P) over R attached to P.

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The answer is provided in [90, p.131] as follows.
C∗ (BT(P)) ∼ = Cartp (R) ⊗W (R) P/ F ⊗ x − 1 ⊗ FP (x), V ⊗ F1,Q (y) − 1 ⊗ y x∈P, y∈Q
where F ⊗ x − 1 ⊗ FP (x), V ⊗ F1,Q (y) − 1 ⊗ y x∈P, y∈Q is the submodule over the Cartier ring Cartp (R), of the projective Cartp (R)-module Cartp (R) ⊗W (R) P , generated by elements of the form F ⊗ x − 1 ⊗ FP (x) for some x ∈ P , or of the form V ⊗ F1,Q (y) − 1 ⊗ y for some y ∈ Q. In the case when R = k ⊃ Fp is a perfect field, write X = BT(P), the above formula simplifies to • C∗ (X) ∼ =P • The W (k)-linear operator FC∗ (X) : W (k) ⊗σ,W (k) C∗ (X) −→ C∗ (X) is FC∗ (X) (w ⊗ x) = w · FP (x)

∀ w ∈ W (k), ∀ x ∈ P

after identifying C∗ (X) with P . • The W (k)-linear operator VC∗ (X) : C∗ (X) −→ W (k) ⊗σ,W (k) C∗ (X) is identified with V ] . B.3.7. Crystalline theory. We will restrict ourselves to the case of finite commutative group schemes G of p-power order over a perfect field k ⊃ Fp , or p-divisible groups X over k. Moreover we will take an overly simplistic view about crystalline theory and look only at the “values” of the Dieudonn´e crystals when evaluated at surjections W (k)  k with the standard divided power structure on the kernel pW (k). Subtle points such as whether the divided power structure is nilpotent, which appears when p = 2, will be ignored. B.3.7.1. For a p-divisible group X over k, a crystal D∗ (X) attached to X is constructed in

[47] from crystalline nature of the universal vector extension of (deformations of) X. One then get a W (k)-module4 D∗ (X) := D∗ (X)Spec(k),→Spec(W (k)) and W (k)-linear map FD∗ (X) : D∗ (X)(1) → D∗ (X)

and VD∗ (X) : D∗ (X) → D∗ (X)(1)

by functoriality, from the relative Frobenius morphism FrX/k : X → X (p) for X and the Verschiebung homomorphism VerX/k : X (p) → X. B.3.7.2. For a commutative finite group scheme G over k of p-power order, one has a crystal

D∗ (G) := E xt1k/W (k) (G, Ok/W (k) ) and functorial morphisms F : σ ∗ D∗ (G) → D∗ (G)

and

V : D∗ (G) → σ ∗ D∗ (G)

defined in [5, 3.1.5]. Let D∗ (G) := D∗ (G)Spec(k),→Spec(W (k)) and let FD∗ (G) : D∗ (G)(1) → D∗ (G)

and

VD∗ (G) : D∗ (G) → D∗ (G)(1)

be the W (k)-linear maps induced by the respective morphism F and V for the crystal D∗ (G). 4Strictly speaking, we need to assume p > 2 so that the divided power structure on the ideal pW (k) ⊂ W (k) is nilpotent.

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B.3.7.3. For a p-divisible group X over k, define

D∗ (X) = lim D∗ (X[pn ]) ←− n→∞

as before. It is shown in [5, §3.2 ,§3.3] that the above definition coincides with the W (k)linear dual of D∗ (X), defined using the universal vector extension. So we will stick to the construction in [5]. As before, define a covariant theory for finite commutative group schemes of p-power order by D∗ (G) := HomW (k) (D∗ (G), K0 /W ) , which is naturally isomorphic to D∗ (GD ) and compatible with the induced operators F and V . Similarly define D∗ (X) := lim D∗ (X[pn ]) , ←− n→∞

for any p-divisible group X over k, with the induced operators F and V . We get both a covariant and a contravariant theory, for finite commutative group schemes over k of p-prime power and also for p-divisible groups over k. B.3.8. Comparison of crystalline and Cartier theory. The left W (k)[F, V ]-module C∗ (X) of a formal p-divisible group X over k obtained from Cartier’s theory, naturally isomorphic to the W (k)[F, V ]-module underlying a display P over k which “produces” X as we saw in B.3.6, also coincides with the crystalline theory when both are applicable: We have a functorial isomorphism C∗ (X) ∼ = D∗ (X)Spec(k),→Spec(W (k)) for any formal p-divisible group X over k; see (5.8) of [9] on p. 56 and also lemma 93 on p. 214–215 of [90]. Therefor the covariant Dieudonn´e theory C∗ (·) obtained from the Cartier theory coincides with the theory D∗ (·) obtained from the crystalline theory. Consequently the contravariant theories C∗ (·) and D∗ (·) also coincide. From now on we will suppress the notations C∗ (·) and C∗ (·) replacing them with D∗ (·) and D∗ (·). Note that we have deliberately confused formal p-divisible groups and their associated smooth formal groups in the notation; we will continue this practice as long as such abuse does not lead to problems. B.3.9. Comparison of the classical and crystalline theory. The basic comparison result between the classical and crystalline theory is found in [5, 4.2.14]: There exist functorial isomorphisms D∗ (G) ∼ = M∗ (G)(1) , D∗ (G) ∼ = M∗ (G)(1) for every commutative finite group scheme G over k. Taking the limit, we have functorial isomorphisms D∗ (X) ∼ = M∗ (X)(1) , D∗ (X) ∼ = M∗ (X)(1) for every p-divisible group X over k. '

Remark. The isomorphism M∗ (X) −→ D∗ (X) for any p-divisible group X over k given in [46, (15.3), p. 123] is σ-linear, for the W (k)-module structures explained previously. (The W (k)-module structure for either side of the isomorphism was not specified in [46, (15.3), p. 123].) B.3.10. Basic invariants.

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B.3.10.1. Let G be a commutative finite group scheme over the base perfect field k. G

• Write `G for the co-Lie complex of G over k, defined as the pull-back ∗ L •/k of the G cotangent complex L •/k of G/k along the zero section  of G. ∨ • The Lie complex `G of G/k is defined to be the k-linear dual of `G . Their cohomologies are ωG := H 0 (`G ), nG := H −1 (`G );

∨

∨

Lie(G) := H 0 (`G ), νG := H 1 (`G ) .

Clearly we have natural isomorphisms νG ∼ = n∨ G.

∨ Lie(G) ∼ , = ωG

B.3.10.2. Formulas for the basic invariants. Let G be a finite group scheme over the perfect

base field k ⊃ Fp . The invariants for G and its Cartier dual GD are given in terms of D∗ (G) and D∗ (G) as follows. !!(−1) FD∗ (G) / D∗ (G) (1) ω = Lie(G)∨ ∼ = Coker D∗ (G)(1) G

(2) νGD ∼ =

(3) Lie(G) = (4) nGD ∼ =

(5) Lie(G ) ∼ = (6) nG ∼ =

∼ =

Ker

Ker

(7) ωGD ∼ = (8) νG ∼ =

Ker

D∗ (G)

D∗ (G)(1)

Ker

D

/ D∗ (G)(1)

D∗ (G)

Coker ∨ ωG

VD∗ (G)

D (G)

D∗ (G)(1)

Coker

D∗ (G)

/ D (G)(1) ∗

/ D∗ (G)

/ D∗ (G)(1)

FD∗ (G)

/ D∗ (G)

!!(−1)

!!(−1)

VD∗ (G)

D∗ (G)(1)

Coker

VD∗ (G)

FD∗ (G)

∗

!!(−1)

!!(−1)

!!(−1)

FD∗ (G)

/ D∗ (G)

VD∗ (G)

/ D (G)(1) ∗

!!(−1)

!!(−1)

Note that (5)–(8) are dual to (1)–(4) and can be obtained from (1)–(4) with G replaced by its Cartier dual. B.3.10.3. We have a 6-term exact sequence

0

/ nG

α

/ D∗ (G)

FD∗ (G) (−1)

/ Lie(GD )

δ

/ ωG

VD∗ (G) (−1)

/ D∗ (G)

β

/ νGD

/0

where α is the natural injection, β is the natural surjection and δ is induced by IdD∗ (G)(−1) . Dually we have a similar 6-term exact sequence 0

/ nGD

/ D∗ (G)

FD∗ (G) (−1)

/ Lie(G)

/ ωGD

VD∗ (G) (−1)

/ D∗ (G)

/ νG

/ 0.

Remark. The two exact sequences in B.3.10.3 are special cases of the 6-term exact sequences for finite commutative group schemes given in 3.2.10 on p. 132 and Remarque (i) on p. 133 of [5],

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B. CM LIFTING VIA P -ADIC HODGE THEORY

B.3.11. Remark. When G is a BT1 group, or equivalently G ∼ = X[p] for some p-divisible group over k, the first exact sequence in B.3.10.3 splits into two short exact sequences / ωG

0

VD∗ (G) (−1)

β

/ D∗ (G)

/ νGD

/0

and 0

/ nG

/ D∗ (G)

α

FD∗ (G) (−1)

/ Lie(GD )

/ 0,

/ νGD

/0

which are isomorphic to each other: 0

/ ωG

VD∗ (G) (−1)

0

VD∗ (G) (−1)

=



/ nG

α

β

/ D∗ (G) 

/ D∗ (G)

FD∗ (G) (−1)



FD∗ (G) (−1)

/ Lie(GD )

/ 0.

Similarly the second 6-term exact sequence in B.3.10.3 splits into two isomorphic short exact sequences 0

/ ωGD

VD∗ (G) (−1)

VD∗ (G) (−1)

/ D∗ (G) =



/ nGD 0 when G is a BT1 -group over k.



/ D∗ (G)

/ νG FD∗ (G)

(−1)



/0

FD∗ (G) (−1)

/ Lie(G)

/0

B.3.12. An example. Suppose that G is a commutative finite group scheme over k of αtype, in the sense that both the relative Frobenius homomorphism FrG : G → G(p) and the Verschiebung homomorphism VerG : G(p) → G are zero. The 6-term exact sequence for D∗ (G) gives isomorphisms ' ' nG −→ D∗ and D∗ (G) −→ νGD . On the other hand we have a natural isomorphism D∗ (G) ∼ = k ⊗σ,k ω = ω (1) G

G

because ωG ∼ = M∗ (G). We can see the above natural isomorphisms “in action” in the context of one-dimensional abelian varieties. Let E be a supersingular elliptic curve over k. Let G = Ker(FrE : E → E (p) ), the unique subgroup scheme of order p in E. Because the relative Frobenius FrE induces the zero map on the Lie algebras, the co-Lie complex `G• of G is represented by the 2-term complex ωE (1) = ω

0

E (p)

−→ ωE

with ω(p) at degrees −1 and ωE at degree 0. In particular, the actions of the endomorphism E

ring End(E) of E on ωG = H0 (`G• ) and nG = H−1 (`G• ) differ by a Frobenius twist.

B.4. P-ADIC HODGE THEORY AND A FORMULA FOR THE CLOSED FIBER

211

B.4. p-adic Hodge theory and a formula for the closed fiber The goal of this section is an explicit formula for the functor MCF in B.4.3, which sends an input linear algebra data (a Kisin module) (M, φM , ψM ) in p-adic Hodge theory to the classical covariant Dieudonn´e module M∗ (Gk ) of the closed fiber Gk of the finite flat group scheme G attached to (M, φM , ψM ). B.4.1. Notation. Throughout this section k ⊃ Fp is a perfect field. • K is a finite totally ramified extension field of K0 = W (k)[1/p], the fraction field of W (k). • e = e(K/K0 ) is the ramification index of K/K0 ; π ∈ OK is a chosen/fixed generator of the maximal ideal of OK . • E(u) = ue + ae−1 ue−1 + . . . + a1 u + a0 is the minimal polynomial of π over K0 . In particular E(u) is an Eisenstein polynomial, ai ∈ p W (k) for all i = 0, 1, . . . , e − 1, and c := a0 /p ∈ W (k)× . • S = W (k)[[u]], and φ : S → S is the ring endomorphism of S such that φ(u) = up and φ|W (k) = σ, the Frobenius automorphism of W (k). • S is the p-adic completion of W [u, E(u)n /n!]n∈N , the subring of K0 [u] generated by W (k)[u] and all elements of the form E(u)n /n! for some n ∈ N. It is easy to see that the natural inclusion W (k)[u] ,→ S extends to a natural inclusion S ,→ S. • The ring endomorphism φ of S extends uniquely to a ring endomorphism of S, again denoted by φ. • Fil1 S is the kernel of the surjective W (k)-linear ring homomorphism  : S  OK such that (u) = π. It is clear that Fil1 S is the p-adic completion of the ideal of S generated by all elements of the form E(u)n /n! for some n ∈ N. Notice that φ(E(u))/p ∈ S × because (φ(E(u))/p) − σ(c) ∈ Fil1 S + uW (k)[u] + pW (k)[u]. B.4.2. Recall from [38] that (Mod/S)c is the additive category whose objects are triples (M, φM : M → M, ψM : M → φ∗ M) satisfying the following properties. (1) M is a finitely generated S-module killed by some power of p and proj. dimS (M) = 1. (2) φM : M → M is a φ-linear endomorphism of M, in the sense that φM (am) = φ(a)φM (m) for every a ∈ S and every m ∈ M.5 (3) Coker (1 ⊗ φM : φ∗ M → M) is killed by E(u). (4) The following diagram ψM

M

E(u)·

* / φ∗ M t tt tt t t  zttt 1⊗φM

/ E(u) M _

γ

M

commutes. In other words γ : E(u)M → φ∗ M is the restriction to E(u)M of the inverse of the linearization 1 ⊗ φ of φ, and ψ is the composition of γ with “multiplication with E(u)”. In particular the S-linear map ψ : M → φ∗ M is uniquely defined by the pair (M, φ) for any pair satisfying the conditions (1)–(3) above. (4) There exists a positive integer n such that the cokernel of the composition n ψM := φn−1 ∗ ψ M ◦ φn−2 ∗ ψ M ◦ · · · ψ M : M −→ φn ∗ M

is killed by the maximal ideal (p, u) of S. A morphism in (Mod/S)c from an object (M1 , φ1 , ψ1 ) to an object (M1 , φ2 , ψ2 ) is an Smodule homomorphism h : M1 → M2 such that φM2 ◦ h = h ◦ φM1 ; the last condition implies that ψM2 ◦ h = φ∗ h ◦ ψM1 . The category (M/S)c is an exact category. 5We often shorten φ to φ and ψ to ψ if there is no danger of confusion. M M

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B. CM LIFTING VIA P -ADIC HODGE THEORY

It is proved in [38, 1.3.9] that there is an equivalence '

GKis : (Mod/S)c −→ (p-Gr/OK )c of exact categories, where (p-Gr/OK )c is the exact category of connected commutative finite locally free group schemes over OK . B.4.3. Let (p-Gr/k)c be the abelian category of connected commutative finite group schemes of p-power order over k. Let (Mod/Dk )c be the abelian category of left Dk -modules of finite length on which V acts as a unipotent σ −1 -linear operator. Consider the composition of functors (Mod/S)c GKis

MCF

/ (Mod/Dk )c O M∗



(p-Gr/OK )c

CF

/ (p-Gr/k)c

where CF is the functor which sends every connected commutative finite locally free group scheme G of p-power order over OK to the closed fiber of G, a connected commutative finite group scheme over k. We need an explicit “formula” for the functor MCF : (M, φM , ψM ) 7→ M∗ (closed fiber of G) , where G is the finite flat group scheme attached to a connected Kisin module (M, φM , ψM ). B.4.4. The functor GKis is constructed from a similar equivalence '

f p -DivKis : BTφ,f /S −→ (p-div/OK ) ,

where (p-div/OK )f is the category of formal p-divisible groups over OK and BTφ,f /S is a category of linear algebra data designed for describing p-divisible groups whose definition is recalled in B.4.5 below. The functor p-DivKis in [38, 1.2.8] is a composition of two functors p -DivKis

BTφ,f /S

∼ = II II ∼ I= II S-Win II$

/ (p-div/O )f . K pp8 ∼ = ppp ppp ppp BT ◦ Dsp

(Win/S) We proceed to recall the definition of the categories BTφ,f /S , (Win/S), the functor S-Win, and the two functors BT and Dsp. c B.4.5. The definition of the exact category BTφ,f /S is similar to that of (Mod/OK ) . Its objects are quadruples (M, φM : M → M, ψM : M → φ∗ M) satisfying the following properties. (1) M is a free S-module of finite rank. (2) φM : M → M is a φ-linear endomorphism of M, in the sense that φM (am) = φ(a)φM (m) for every a ∈ S and every m ∈ M. (3) Coker (1 ⊗ φM : φ∗ M → M) is killed by E(u). (4) The diagram ψM

M

E(u)·

* / φ∗ M tt tt tt t  zttt 1⊗φM

/ E(u) M _

M commutes.

γ

B.4. P-ADIC HODGE THEORY AND A FORMULA FOR THE CLOSED FIBER

213

(4) There exists a positive integer n such that the cokernel of the composition n ψM := φn−1 ∗ ψ M ◦ φn−2 ∗ ψ M ◦ · · · ψ M : M −→ φn ∗ M

is killed by the maximal ideal (p, u) of S. A morphism in (Mod/S)c from an object (M1 , φ1 , ψ1 ) to an object (M1 , φ2 , ψ2 ) is an Smodule homomorphism h : M1 → M2 such that φM2 ◦ h = h ◦ φM1 ; the last condition implies that ψM2 ◦ h = φ∗ h ◦ ψM1 . B.4.6. We recall the definition of the category (Win/S) of S-windows in [89]. An S-window is a quintuple (M, M1 , Φ, Φ1 , Ψ) with the following properties. (1) M is a finitely generated projective S-module. (2) M1 is an S-submodule of M which contains Fil1 S·M such that M/M1 is a projective OK -module. (3) Φ : M → M and Φ1 : M1 → M are φ-linear maps of S-modules such that Φ(m1 ) = p Φ1 (m1 )

and

Φ1 (am) = (φ(a)/p) · Φ(m)

for all m1 ∈ M, all a ∈ Fil1 S and all m ∈ M. (Recall that φ(Fil1 S) ⊂ p · S.) (4) M is generated by Φ1 (M1 ) and Φ(M) as an S-module. (5) Ψ : M → S ⊗φ,S M is an S-linear map such that Ψ(Φ1 (m1 )) = 1 ⊗ m1

and

Ψ(Φ(m)) = p ⊗ m

6

for all m1 ∈ M1 and all m ∈ M. (6) There is a natural number n ∈ N such that Ψn (M) ⊂ Fil1 S · (S ⊗φn ,S M), where Ψn := φn−1 ∗ Ψ ◦ φn−2 ∗ Ψ ◦ · · · ◦ Ψ : M −→ φn ∗ M = S ⊗φn ,S M . B.4.7. We recall the definition of the category (Dsp/R) of displays over a commutative ring R (with 1) in [90, 1.1]. We will only be interested in the case where R is excellent, and either p is nilpotent, or R is complete with respect to an ideal which contains p. Only the cases R = OK and R = k will be used later. We fix some notation for p-adic Witt vectors. • Write F : W (R) → W (R) for the “Frobenius” ring endomorphism for W (R), characterized by wn (F (x)) = wn+1 (x) where

∀ x = (x0 , x1 , x2 , . . .) ,

n

+ · · · + pn−1 xn−1 + pn xn ) wn (x) = xp0 + pxn−1 1 is the n-th Witt polynomial. • Denote by IR the kernel of the surjective ring homomorphism w0 : W (R) → R, i.e. IR consists of all Witt vectors of the form (0, x1 , x2 , . . .) in W (R). • Let V : W (R) → W (R) be the additive endomorphism which sends a typical element x = (x0 , x1 , x2 , . . .) of W (R) to (0, x0 , x1 , x2 , . . .). A display over a commutative ring R with 1 is a quintuple P = (P, Q, FP , F1,Q , V ] ) with the following properties. (1) P is a finitely generated projective W (R)-module. (2) Q is a submodule of P which contains IR ·P . (3) The quotient P/Q, with its natural R-module structure, is a projective R-module. (Consequently there exists a decomposition P = L ⊕ T of P as a direct sum of two projective W (R)-submodules L and P such that Q = L ⊕ IR · T .) 6It is shown in [89, p. 497] that for any quadruple (M, M , Φ, Φ ) with properties (1)–(4), there exists a 1 1 unique map Ψ which satisfies (5).

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B. CM LIFTING VIA P -ADIC HODGE THEORY

(4) Both FP and F1,Q are F -linear homomorphisms of W (R)-modules in the sense that FP (w · x) = F (w) · FP (x)

and F1,Q (w · y) = F (w) · F1,Q (y)

for all w ∈ W (R), all x ∈ P and all y ∈ Q. (5) F1,Q (V w · x) = w · FP (x) for every w ∈ W (R) and every x ∈ P , where V w is the image of w under V : W (R) → W (R). (6) P is generated by the subset F1,Q (Q) as a W (R)-module. (7) V ] : P → W (R) ⊗F,W (R) P is the W (R)-linear homomorphism such that V ] (w · FP (x)) = pw ⊗ x

and V ] (w · F1,Q (y)) = w ⊗ y

for all w ∈ W (R), all x ∈ P and all y ∈ Q. It is shown in [90, Lemma 10, p.139] that for every quadruple (P, Q, FP , F1,Q ) satisfying (1)–(6) above, there exists a unique W (R)-linear map V ] with the above properties. (7) There exists a natural number n ∈ N such that the W (R)-module homomorphism V ] n := F n−1 ∗ V ] ◦ F n−2 ∗ V ] ◦ · · · ◦ V ] : P −→ W (R) ⊗F n ,W (R) P is congruent to 0 modulo the ideal IR + p W (R). Note that the Lie algebra Lie(BT(P)) of the p-divisible group BT(P) attached to a display P as above is naturally isomorphic to P/Q; see [90] (157) on p. 207 in the proof of theorem (81). B.4.8. We have a commutative diagram of functors p-DivKis

Dsp

/ (Dsp/OK ) / BTφ,f /S UUUU (Win/S) UUUU UUUU UUUU BCDsp UUUU CFdsp UU*  (Dsp/k) S-Win

BT

& / (p-div/O )f K 

BTk

BCp-div

/ (p-div/k)f 6

CFp-pdvi

where • BT and BTk are the functors which associate to a display over a ring R = OK or k a formal p-divisible group over R; see in [90, Thm. 81, p. 204]. • BCDsp is the base change functor, from displays over OK to displays over k, defined in [90, Def. 20, p. 1444] and recalled in B.4.11 below. • Dsp is the functor which to every S-window over OK associates a display over OK , defined in [89, pp. 497–498] and recalled in B.4.10 below. • S-Win is the functor which to every object in BTφ,f /S an S-window, defined in [38, 1.2.5] and recalled in B.4.9 below. • BCp-div is the base change functor from OK to k, which associates to every formal p-divisible group X over OK its closed fiber X0 over k. Our intermediate goal, before scooping up a formula for the functor MCF, is to find an explicit formula for the functor CFdsp = BCDsp ◦ Dsp ◦ S-Win .

B.4. P-ADIC HODGE THEORY AND A FORMULA FOR THE CLOSED FIBER

215

B.4.9. The functor S-Win : BTφ,f /S −→ (Win/S) is defined in [38, 1.2.5] as follows. Given an object (M, φM , ψM ) in BTφ,f /S , the associated window S-Win(M, φM , ψM ) for the frame S/OK is the quintuple (M, M1 , Φ, Φ1 , Ψ) defined as follows. • M := S ⊗φ,M M • M1 is defined by the Cartesian diagram7  M1 

/ M = S ⊗φ,M M 2



Fil1 S · (S ⊗S M)





1⊗φM

/ S ⊗S M E(u) ·

/ OK ⊗S M Notice that Fil1 S ·(S ⊗S M) ←− Fil1 S ⊗S M because OK ⊗S M is injective. • Φ : M → M is the φ-linear map which sends s ⊗ m to φ(s) ⊗ φM (m) for any s ∈ S and any m ∈ M. • Φ1 = p1 · (Φ|M1 ), well-defined because φ(Fil1 S) ⊂ p S. Note that Φ1 (M1 ) generates M over S because p1 φ(E(u)) ∈ S × . • Ψ : M → S ⊗φ,S M is the S-linear map such that Ψ(Φ1 (x1 )) = 1 ⊗ x1 for all x1 ∈ M1 and Ψ(Φ(x)) = p ⊗ x for all x ∈ M. '

'

Note that M/M1 −→ φ∗ M/Fil1 φ∗ M, where φ∗ M = S ⊗φ,S M and Fil1 φ∗ M is defined by the Cartesian diagram8  / φ∗ M = S ⊗φ,S M Fil1 φ∗ M 2





 E(u) · M 

1⊗φM

/ M

B.4.10. The functor Dsp : (Win/S) −→ (pDsp/OK ) is defined in [89, pp. 497–498]. It sends a quintuple (M, M1 , Φ, Φ1 , Ψ) to the quintuple P = (P, Q, FP , F1,Q , V ] ), as follows. • P := W (OK ) ⊗κ,S M, where – κ : S → W (OK ) is the composition of two ring homomorphisms β

δ

κ : S −→ W (S) −→ W (OK ) , – β is induced by the surjective continuous W (k)-linear ring homomorphism  : S → OK such that α(u) = π, – δ : S → W (S) is the Cartier homomorphism, characterized by wn (δ(a)) = φn (a) ∀a ∈ S, ∀n ∈ N . • Q := Ker (W (OK ) ⊗κ,S M −→ M/M1 ), where the map is (a0 , a1 , a2 , . . .) ⊗ x 7→ a0 · x ¯

∀a = (a0 , a1 , a2 , . . .) ∈ W (OK ), ∀x ∈ M

and x ¯ is the image of x in M/M1 . • FP : P −→ P is the map FP : ξ ⊗ x 7→ F (ξ) ⊗ Φ(x)

∀ ξ ∈ W (OK ), ∀ x ∈ M .

• F1,Q : Q −→ P is the unique map such that F1,Q (ξ ⊗ x1 ) = F (ξ) ⊗ Φ1 (x1 )

and F1,Q (V (ξ) ⊗ x) = ξ ⊗ Φ(x)

for all ξ ∈ W (OK ), all x1 ∈ M1 and all x ∈ M. 7The S-module M is denoted by Fil1 M in [38, 1.2.5] 1 ' 8The formula M/Fil1 M −→ φ∗ M/(E(u)·M) in the third paragraph of the proof of [38, 1.2.5] should be

interpreted as indicated in this Cartesian diagram; i.e. the term “E(u) · M” in “φ∗ M/(E(u) · M)” should be understood to be the submodule Fil1 φ∗ M of φ∗ M in our notation.

216

B. CM LIFTING VIA P -ADIC HODGE THEORY

• V ] : W (OK )⊗κ,S M −→ W (OK )⊗F,W (OK ) W (OK )⊗κ,S M is the unique W (OK -linear map such that V ] (w ⊗ Φ(x)) = p w ⊗ 1 ⊗ x

and V ] (w ⊗ Φ1 (x1 )) = w ⊗ 1 ⊗ x1

for all x ∈ M and all x1 ∈ M1 . B.4.11. The base change functor BCDsp from OK to k is given in [90, Def. 20, p. 144]. It ˜ F˜ , F˜1 , V˜ ] ) over k, sends a display P = (P, Q, F, F1 , V ] ) over OK to a display P˜ = (P˜ , Q, defined as follows. • P˜ = W (k) ⊗W (OK ) P ,   ˜ = Ker W (k) ⊗ • Q P −→ k ⊗ (P/Q) , R W (OK ) • F˜ (a ⊗ x) = F (a) ⊗ FP (x) for all a ∈ W (k) and all x ∈ P , ˜ → P˜ is the unique F -linear homomorphism such that • F˜1 : Q F˜1 (V (w) ⊗ x) = w ⊗ F (x)

and F˜1 (w ⊗ y) = w ⊗ F1 (y)

for all w ∈ W (k), all x ∈ P and all y ∈ Q , • V˜ ] : W (k) ⊗W (OK ) P = P˜ −→ W (k) ⊗W (OK )W (OK ) ⊗F,W (OK ) P is defined by V˜ ] (w ⊗ x) = w ⊗ V ] (x)

∀ w ∈ W (k), ∀ x ∈ P .

B.4.12. We have the following ring homomorphisms. • Recall that δ : S → W (S) is the Cartier homomorphism with respect to the Frobenius lifting φ : S → S. W ()

/ W (S) / W (OK ) is the composition of δ with the surjec• κ = W () ◦ δ : S tion W () : W (S)  W (OK ) induced by the W (k)-linear surjection  : S  OK with (u) = π. • Let W (can) be the map from W (OK ) to W (k) induced by the natural can surjection OK → k which identifies k with the OK /πOK . • Let jS,→S : S → S be the natural inclusion map from S to S. • Let ρ := W (can) ◦ κ ◦ φ ◦ jS,→S : S −→ W (k) be the composition of the ring homomorphisms φ ◦ jS,→S , κ and W (can). δ

B.4.13. Lemma. Notation as above. (1) (2) (3) (4)

δ(u) = (u, 0, 0, 0 . . .), the Teichm¨ uller representative of u in W (S). ρ(u) = 0. W (can) ◦ κ|W (k) = IdW (k) . ρ|W (k) = σ, the canonical lifting of Frobenius on W (k). n

Proof. The element δ(u) ∈ W (S) is characterized by the property that wn (δ(u)) = up for n all n ∈ N, and the Teichm¨ uller element [u] := (u, 0, 0, . . .) ∈ W (S) satisfies wn ([u]) = up . This proves (1). It follows that ρ(u) = W (can)(W ()(up )) = W (can)(π p )) = 0 . We have proved (2). The restriction to W (k) ⊂ S of the Cartier homomorphism δ is the Cartier homomorphism δW (k) : W (k) −→ W (W (k)), characterized by the condition that wn (δ(a)) = F n (a) for all a ∈ W (k). By the same argument for (1), for any Teichm¨ uller element [a] = (a, 0, 0, . . .) ∈ W (k), we have δ([a]) = ([a], 0, 0, . . .), the Teichm¨ uller representative of [a] in W (W (k)). So W (can)(κ([a])) = W (can)([a], 0, 0, 0, . . .) = (a, 0, 0, 0, . . .)

∀a ∈ k.

B.4. P-ADIC HODGE THEORY AND A FORMULA FOR THE CLOSED FIBER

217

Because the set of all Teichm¨ller elements of the form [a] generate a p-adically dense subring of W (k), we conclude that the restriction to W (k) of the composition W (can) ◦ κ is equal to IdW (k) . We have proved (3). The statement (4) follows immediately. B.4.14. We are ready to compute the functor CFdsp BTφ,f /S

SWin /

(Win/S)

Dsp

/ (Dsp/OK )

BCDsp

/ (Dsp/k) . 5

CFdsp

˜ ˜ ˜ ˜ ˜ ˜] Let (M, φM , ψM ) be a given object in BTφ,f /S . Let P = (P , Q, F , F1 , V ) be the display CFdsp (M, φM , ψM ) over k of the p-divisible group X := CFp-div (M, φM , ψM ) over k. From the recipes in B.4.9–B.4.11, we see that P˜ = W (k) ⊗W (can),W (OK )W (OK )⊗κ,S S ⊗φ,S M ∼ = W (k) ⊗ρ,S M ∼ = W (k) ⊗S φ∗ M . The last isomorphism in the above displayed formula follows from B.4.13; the implied ring homomorphism S → W (k) in the tensor product W (k) ⊗S φ∗ M is the W (k)-linear quotient map which sends u to 0. Tracing through the recipe in B.4.9–B.4.11, we see that F˜ : P˜ → P˜ is induced by the φ-linear map φM : M → M. The rest in the data of the display P˜ are determined by P˜ and F˜ . We summarize the statement in B.4.15 below. B.4.15. Proposition. For any object (M, φM , ψM ) in BTφ,f /S , the display ˜ F˜ , F˜1 , V ] ) CFdisp (M, φM , ψM ) = (P˜ , Q, over k of the closed fiber of the p-divisible group p-divKis (M, φM , ψM ) attached to (M, φM , ψM ) is explicitly given as follows. ∼ W (k) ⊗ρ,S M = ∼ W (k) ⊗S φ∗ M = ∼ φ∗ M/(u · φ∗ M), (1) P˜ = (2) F˜ : P˜ → P˜ is the map induced by 1 ⊗ φ. In other words F˜ sends the image in φ∗ M of the element 1 ⊗ x ∈ φ∗ = S⊗φ,S M to the image in φ∗ M/(u · φ∗ M) of the element 1 ⊗ φ(x) ∈ S⊗φ,S M.   F˜ / ˜ / / ˜ ˜ ˜ ˜ (3) Q = Ker P P /(p · P ) , where the second arrow is the quotient map P from P˜ = φ∗ M/(u· φ∗ M) to P˜ /(p · P˜ ) = φ∗ M/((p, u) · φ∗ M). ˜ −→ P˜ . (4) F˜1 = (1/p) · F˜ |Q˜ : Q (5) V˜ ] : P˜ → W (k) ⊗σ,W (k) P˜ is the unique W (k)-linear map such that   ˜. V˜ ] w · F˜1 (y) = w ⊗ y ∈ W (k) ⊗σ,W (k) P˜ ∀y ∈ Q Moreover we have   V˜ ] w · F˜ (x) = pw ⊗ x ∈ W (k) ⊗σ,W (k) P˜

∀ x ∈ P˜ .

(6) Alternatively, using the isomorphism P˜ ∼ = W (k) ⊗ρ,S M, the map V˜ ] is induced by 1 c ψM : V˜ ] = 1 ⊗ ((1/c)ψM ) : P˜ = W (k) ⊗ρ,S M

1⊗( 1c · ψM )

/ W (k) ⊗ρ,S φ∗ M .

(Recall that pc is the constant term of the Eisenstein polynomial E(u) = ue + ae−1 ue−1 + . . . + a1 u + p c for the generator π of the maximal ideal of OK .)

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B. CM LIFTING VIA P -ADIC HODGE THEORY

In particular, the Lie algebra of the p-divisible group X over k is !  ψM ∗ ∼ ∼ / Lie(X) = W (k) ⊗S Coker M φ M = Coker M/uM

ψ (mod u)

/ φ∗ M/uφ∗ M

 .

Proof. The statements (1) and (2) have been proved in B.4.14. Statements (3)–(5) follow from (1)–(2); for instance the map V˜ ] : P˜ → W (k) ⊗σ,W (k) P˜ is uniquely determined by the linearization 1 ⊗ F˜ : W (k) ⊗σ,W (k) F˜ → F˜ of F˜ because (1 ⊗ F˜ )(V˜ ] (x)) = p x and V˜ ] (F˜ (x)) = p ⊗ x

∀ x ∈ P˜ .

B.4.16. Remark. For any object (M, φM , ψM ) in BTφ,f /S , we have a natural isomorphism Lie (p-divKis (M, φM , ψM )) ∼ = φ∗ M/Fil1 φ∗ M , where Lie (p-divKis (M, φM , ψM )) is the Lie algebra  of the p-divisible group p-divKis (M, φM , ψM ) 1⊗φ

attached to (M, φM , ψM ), and Fil1 φ∗ M = Ker φ∗ M −→ M  M/(E(u) · M) as in B.4.9. To see this, we proceed as in B.4.15, with the same notation: • (M, M1 , Φ, Φ1 , Ψ) = S-Win(M, φM , ψM ), • (P, Q, F, F1 , V ] ) = Dsp(M, M1 , Φ, Φ1 , Ψ), • X = BT(P, Q, F, F1 , V ] ) = p-divKis (M, φM , ψM ). It is proved in pp. 207–208 of [90] that Lie(X) ∼ = P/Q. We have seen that P/Q ∼ = M/M1 in 1 ∗ ∗ ∼ B.4.10 and M/M1 = φ M/Fil φ M in B.4.9. The assertion follows. B.4.17. Proposition. For any object  (M, φM , ψM ) in (Mod/S)c , the classical covariant Dieudonn´e module MCF (M, φM , ψM ) = M∗ (Gk ), FM∗ (G ) , VM∗ (G ) of the closed fiber Gk of k k the connected finite flat group scheme G = GKis (M, φM , ψM ) attached to (M, φM , ψM ) can be explicitly described as follows. (1) We have functorial W (k)-module isomorphisms M∗ (Gk ) ∼ = W (k) ⊗S M ∼ = M/u·M . (2) FM∗ (G ) : W (k) ⊗σ,W (k) (M/uM) = W (k) ⊗σ,W (k) M∗ (Gk ) −→ M∗ (Gk ) = M/uM k is the map FM∗ (G ) (w ⊗ [x]) = [w· φM (x)] k

∀ w ∈ W (k), ∀ x ∈ M ,

where [x] and [φM (x)] are the images in M/uM of the elements x and φM (x) respectively. (3) VM∗ (G ) : M/uM = M∗ (Gk ) −→ W (k) ⊗σ,W (k) M∗ (Gk ) = φ∗ M/(u · φ∗ M) k is the map VM∗ (G ) ([x]) = [(1/c) · ψM (x)] ∀x ∈ M, k

where [x] is the image of x in M/uM and [(1/c) · ψM (x)] is the image of (1/c) · ψM (x) in φ∗ M/(u · φ∗ M). In particular the Lie algebra of the finite group scheme Gk over k is ! (1/c)ψM ∗ ∗ / φ M/uφ M . Lie(Gk ) ∼ = Ker M/uM Proof. The equivalence GKis between (Mod/S)c and (p-Gr/OK )c in [38, 1.3.9] is deduced f from the equivalence p-DivKis between BTφ,f /S and (p-div/OK ) as follows: Suppose that a connected finite flat group scheme G over OK of p-power order is the kernel of an isogeny γ : X → X 0 between formal p-divisible groups over OK , and this isogeny γ corresponds to a 0 morphism f in the category BTφ,f /S , from an object (N, φN , ψN ) to an object (N , φN0 , ψN0 ).

B.4. P-ADIC HODGE THEORY AND A FORMULA FOR THE CLOSED FIBER

219

Then G corresponds to the object (M, φM , ψM ), where M = Coker(f ) is the cokernel of the S-module homomorphism f : M → N, φM is induced by φN0 , and ψM is induced by ψN0 . ˜ F˜ , F˜1 , V˜ ] ) over k, the Recall that for a p-divisible group X over k attached to a display (P˜ , Q, crystalline Dieudonn´e module   D∗ (X), FD∗ (X) : W (k) ⊗σ,W (k) D∗ (X) → D∗ (X), VD∗ (X) : D∗ (X) → W (k) ⊗σ,W (k) D∗ (X) is canonically isomorphic to   P˜ , 1 ⊗ F˜ : W (k) ⊗σ,W (k) P˜ , V˜ ] : P˜ → W (k) ⊗σ,W (k) P˜ . Using this fact and trace through the construction recalled in the previous paragraph, we see from B.4.15 the following: • The W (k)-module underlying the crystalline Dieudonn´e module D∗ (Gk ) of Gk is D∗ (Gk ) ∼ = W (k) ⊗ρ,S M ∼ = φ∗ M/uM . • The map FD∗ (G ) : W (k) ⊗σ,W (k) W (k) ⊗ρ,S M ∼ = W (k) ⊗σ,W (k) D∗ (Gk ) −→ D∗ (Gk ) ∼ = W (k) ⊗ρ,S M k

is given by FD∗ (G ) (w1 ⊗ w2 ⊗ x) = (w1 · σ(w2 )) ⊗ φM (x) k

∀w1 , w2 ∈ W (k), ∀x ∈ M .

• The map DD∗ (G ) : W (k) ⊗ρ,S M ∼ = D∗ (Gk ) −→ W (k) ⊗σ,W (k) D∗ (Gk ) ∼ = W (k) ⊗ρ,S S ⊗φ,S M k

is given by DD∗ (G ) (w ⊗ x) = w ⊗ ((1/c)ψM (x)) k

∀w ∈ W (k), ∀x ∈ M .

The statements B.4.17 (1)–(3) follow from the above recipe and the fact that     D∗ (Gk ), FD∗ (G ) , VD∗ (G ) ∼ = W (k) ⊗σ,W (k) M∗ (Gk ), FM∗ (G ) , VM∗ (G ) . k

k

k

k

The last statement about Lie(Gk ) follows (3) and the general formula for the Lie algebra  of a finite group scheme over k in terms of its classical covariant Dieudonn´e module D(Gk ), FD(G ) , VD(G ) . k

k

Bibliography [1] A. A. Albert, On the construction of Riemann matrices, I, II. Ann. Math. 35 (1934), 1–28; 36 (1935), 376–394. [2] A. A. Albert, A solution of the principal problem in the theory of Riemann matrices. Ann. Math. 35 (1934), 500–515. [3] A. A. Albert, Involutorial simple algebras and real Riemann matrices. Ann. Math. 36 (1935), 886–964. [4] E. Artin & J. Tate, Class field theory. AMS Chelsea Publishing, Providence, 2009. [5] P. Berthelot, L. Breen & W. Messing, Th´eorie de Dieudonn´e cristalline II. Lect. Notes Math. 930, SpringerVerlag 1982. [6] P. Berthelot & A. Ogus, F -isocrystals and de Rham cohomology I. Inv. Math. 72 (1983), 159–199. [7] S. Bosch, U. G¨ untzer & R. Remmert, Non-archimedean analysis. Grundl. 261, Springer-Verlag, New York, 1983. [8] S. Bosch & W. L¨ utkebohmert & M. Raynaud, N´eron models. Ergebnisse der Mathematik 21, SpringerVerlag, New York, 1990. [9] L. Breen, Rapport sur la th´eorie de Dieudonn´e. Ast´erisque 63 (1979), 39–66. [10] C. Breuil, Groupes p-divisibles, groupes finis et modules filtr´es. Ann. of Math. 152 (2000), 489–549. [11] P. Cartier, Modules associ´es a ` un groupe formel commutatif. Courbes typiques. C. R. Acad. Sci. 265 (1967), 129–132. [12] J. W. S. Cassels, Global fields. In Algebraic Number Theory (Cassels-Fr¨ ohlich, ed.), Academic Press, New York, 1967. [13] C.-L. Chai, & F. Oort, Hypersymmetric abelian varieties. Quaterly Journ. Pure Applied Math. 2 (Coates Special Issue) (2006), 1–27. [14] P. Colmez & J-M. Fontaine, Construction des repr´esentations p-adiques semi-stables. Invent. Math. 140 (2000), 1–43. [15] B. Conrad, Chow’s K/k-image and K/k-trace, and the Lang–N´eron theorem. Enseign. Math (2) 52, no. 1–2 (2006), 37–108. [16] B. Conrad, Gross-Zagier rivisited. In Heegner points and Rankin L-series, MSRI Publ. 49, Cambridge Univ. Press, Cambridge, 2004. [17] B. Conrad & O. Gabber & G. Prasad, Pseudo-reductive groups. New Mathematical Monographs 17, Cambridge Univ. Press, 2010. [18] B. Conrad, Grunwald–Wang for global character groups, 2010 preprint. [19] P. Deligne, Application de la formule des traces aux sommes trigonom´etriques. In Cohomologie Etales, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie SGA4 21 , Lect. Notes Math. 569, Springer-Verlag 1977, 168–232. [20] P. Deligne, Motifs et groupes de Taniyama. In Hodge Cycles, Motives, and Shimura Varieties, Lect. Notes Math. 900, Springer-Verlag, 1982, 261–279. [21] M. Demazure, Lectures on p-divisible groups. Lect. Notes Math. 302, Springer-Verlag, 1972. ´ ements de g´eom´etrie alg´ebrique. Publ. Math. IHES 11, 1961. [22] J. Dieudonn´e & A. Grothendieck, El´ [23] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenk¨ orper. Abh. Math. Sem. Hamburg 14 (1941), 197 – 272. [24] G. Faltings, Finiteness theorems for abelian varieties over number fields. In Arithmetic geometry (Ed. G. Cornell & J. Silverman), Springer-Verlag, New York, 1986, pp. 9–27. [25] J-M. Fontaine, Groupes p-divisible sur les corps locaux. Ast´erisque 47–48 (1977). [26] J-M. Fontaine, Module galoisiens, modules filtr´es et anneaux de Barsotti-Tate. Ast´erisque 65 (1979), 3–80. [27] J-M. Fontaine, Sur certains types de repr´esentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti–Tate. Annals of Math. 115, 1982, 529–577. [28] J-M. Fontaine, Repr´esentations p-adiques semi-stables. Ast´erisque 223 (1994), 113–184. [29] G. van der Geer & B. Moonen, Abelian varieties. In preparation. [30] L. Gerritzen, On multiplication algebras of Riemann matrices. Math. Ann., 194 (1971), 109–122. [31] A. Grothendieck, Groupes de Barsotti–Tate et cristaux de Dieudonn´e. S´em. Math. Sup. 45, Presses de l’Univ. de Montreal, 1970. [32] M. Hazewinkel, Formal groups and applications. Academic Press, New York, 1978. 221

222

BIBLIOGRAPHY

[33] T. Honda, Isogeny classes of abelian varieties over finite fields. Journ. Math. Soc. Japan 20 (1968), 83 – 95. [34] N. Jacobson, Basic algebra II. W.H. Freeman & Co., 1989. [35] A. J. de Jong, Crystalline Dieudonn´e module theory via formal and rigid geometry. Publ. Math. IHES 82, 1995, 5–96. [36] N. M. Katz, Serre–Tate local moduli. Lect. Notes Math. 868, Springer-Verlag 1981, pp. 138–202. [37] M. Kisin, Crystalline representations and F -crystals. In Algebraic Geometry and Number Theory, Progr. Math., 253, Birkhser Boston, Boston, MA, 2006, pp. 459–496. [38] M. Kisin, Modularity of 2-adic Barsotti–Tate representations. Inv. Math. 178 (2009), 587–634. [39] A. Kresch, Hodge-theoretic obstruction to the existence of quaternion algebras. Bull. London Math. Soc. 35 (2003), 109–116. [40] S. Lang, Elliptic functions (2nd ed.). Graduate Texts in Math. 112, Springer-Verlag, 1987. [41] S. Lang, Complex multiplication. Grundlehren mathematischen Wissenschaften 255, Springer-Verlag, 1983. [42] M. Lazard, Commutative Formal Groups. Lect. Notes Math. 443, Springer-Verlag, 1975. [43] H. W. Lenstra jr & F. Oort, Simple abelian varieties having a prescribed formal isogeny type. Journ. Pure Appl. Algebra 4 (1974), 47 - 53. [44] Y. Manin, The theory of commutative formal groups over fields of finite characteristic. Usp. Math. 18 (1963), 3–90; Russ. Math. Surveys 18 (1963), 1–80. [45] H. Matsumura, Commutative ring theory. Cambridge studies in advanced mathematics Vol. 8, Cambridge Univ. Press, 1989. [46] B. Mazur & W. Messing, Universal extensions and one-dimensional crystalline cohomology. Spring Lect. Notes Math. 370, Springer-Verlag, 1974. [47] W. Messing, The crystals associated to Barsotti–Tate groups: with applications to abelian schemes. Lect. Notes Math. 264, Springer-Verlag, 1972. [48] J. Milne, Abelian varieties. In Arithmetic geometry ( Ed. G. Cornell & J. Silverman), Springer-Verlag, New York, 1986. [49] J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In Automorphic Forms, Shimura Varieties, and L-functions, Vol. I, Perspectives in Math. Vol. 10, Academic Press, 1990, 283–414. [50] J. Milne & K.-Y. Shih, Langlands’ construction of the Taniyama group. In Hodge Cycles, Motives, and Shimura Varieties. Lect. Notes Math. 900, Springer-Verlag, 1982, 229–260. [51] J. Milne & W. Waterhouse, Abelian varieties over finite fields. In 1969 Number Theory Institute (Sympos. Pure Math., Vol. XX, SUNY Stony Brook, NY, 1969 ), AMS, Providence, 1971, 53–64. [52] L. Moret-Bailly, Pinceaux de vari´et´es ab´eliennes. Ast´erisque 129, 1985. [53] D. Mumford, Abelian varieties. Tata Inst. Fund research studies in mathematics Vol. 5, First printing 1970; second edition TIFR and Oxford Univ. Press 1970; latest printing 2008. [54] D. Mumford, Geometric invariant theory. First printing 1965, Ergebn. Math. und ihre Grensgebiete, Springer-Verlag; 3rd enlarged edition by D. Mumford, J. Fogarthy & F. Kirwan, Springer-Verlag, 1994. [55] P. Norman & F. Oort, Moduli of abelian varieties. Annals of Math., 112 (1980), pp. 413–439. [56] F. Oort, The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field. Journ. Pure Appl. Algebra 3 (1973), 399–408. [57] F. Oort, Isogenies of formal groups. Indag. Math., 37 (1975) 391–400. [58] F. Oort, Endomorphism algebras of abelian varieties. In Algebraic Geometry and Commut. Algebra in honor of M. Nagata (Ed. H. Hijikata et al), Kinokuniya Cy Tokyo, Japan, 1988, Vol. II; pp. 469–502. [59] F. Oort, CM-liftings of abelian varieties. Journ. Alg. Geom. 1 (1992), 131–146. [60] F. Oort, Abelian varieties over finite fields. Summer School on ”Varieties over finite fields”, G¨ ottingen, 25-VI — 6-VII-2007. Higher-dimensional geometry over finite fields, Proceedings of the NATO Advanced Study Institute 2007 (Editors: Dmitry Kaledin Yuri Tschinkel). IOS Press, 2008, pp. 123–188. [61] R. Pink, The Mumford–Tate conjecture for Drinfeld modules. Publ. RIMS, Kyoto Univ. 33 (1997), 393–425. [62] M. Raynaud, Sch´emas en groupes de type (p, . . . , p). Bull. Soc. math. France 102 (1974), 241–280. [63] J-P. Serre, Local class field theory. In Algebraic Number Theory (Cassels-Fr¨ ohlich, ed.), Academic Press, New York, 1967, pp. 129–161. [64] J-P. Serre, Complex multiplication. In Algebraic Number Theory (Cassels-Fr¨ ohlich, ed.), Academic Press, New York, 1967, pp. 292–296. [65] J-P. Serre, Abelian `-adic representations and elliptic curves. W. A. Benjamin, 1968. [66] J-P. Serre, Local fields. Grad. Texts Math. 67. Springer-Verlag, New York, 1979. [67] J-P. Serre, Groupes alg´ebriques associ´es aux modules de Hodge–Tate. Ast´erisque 65 (1979) 155–188. [68] J-P. Serre & J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1965), 492–517. [69] S. Shatz, Group schemes, formal groups, and p-divisible groups. In Arithmetic geometry (Cornell/Silverman, ed.), Springer-Verlag, New York, 1986. [70] G. Shimura & Y. Taniyama, Complex multiplication of abelian varieties and its applications to number theory. Publ. Math. Soc. Japan, 6, 1961.

BIBLIOGRAPHY

223

[71] G. Shimura, On analytic families of polarized abelian varieties and automorphic functions. Ann. Math. 78 (1963), 149–193. [72] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton Univ. Press, 1994. [73] J. Tate, Algebraic cycles and poles of zeta functions. Arithmetical algebraic geometry (Proc. Conf. Purdue Univ., 1963). Harper & Row, New York, 1965, pp. 93–110. [74] J. Tate, Endomorphisms of abelian varieties over finite fields. Inv. Math. 2 (1966), 134–144. [75] J. Tate, p-divisible groups. In Proc. Conf. on Local Fields (Ed. T. A. Springer), Springer-Verlag, 1967, 158–183. [76] J. Tate, Global class field theory. In Algebraic Number Theory (Cassels-Fr¨ ohlich, ed.), Academic Press, New York, 1967, pp. 163–203. [77] J. Tate, Class d’isogenie des vari´et´es ab´eliennes sur un corps fini (d’apr`es T. Honda). S´eminaire Bourbaki, 1968/69, no. 352. Lect. Notes Math. 179, Springer-Verlag, 1971, pp. 95–110. [78] J. Tate & F. Oort, Group schemes of prime order. Ann. Sci. ENS 3 (1970), pp. 1–21. [79] A. Vasiu & T. Zink, Breuil’s classification of p-divisible groups over regular local rings of arbitrary dimension. Preprint July 15, 2008. [80] W. Waterhouse, Affine group schemes. Grad. Texts Math. 66 Springer-Verlag, New York, 1979. [81] A. Weil, On a certain type of character of the id`ele-class group of an algebraic number field. In Proc. Intern. Symp. on Algebraic Number Theory, Tokyo-Nikko, 1955, pp. 1–7. [82] A. Weil, Basic number theory. Springer-Verlag, New York, 1974. [83] C.-F. Yu, Lifting abelian varieties with additional structure. Math. Z. 242 (2002), 427–441. [84] C.-F. Yu, On the supersingular locus in Hilbert-Blumenthal 4-folds. Journ. Alg. Geom. 12 (2003), 653–698. [85] C.-F. Yu, On reduction of Hilbert-Blumenthal varieties. Annales de l’Institut Fourier (Grenoble) 53, (2003) 2105–2154. [86] C.-F. Yu, The isomorphism classes of abelian varieties of CM-type. Journ. of Pure and Applied Algebra 187 (2004), 305–319. [87] Y. Zarhin, Abelian varieties in characteristic p. Mat. Zametki 19 (1976), no. 3, 393–400. [88] T. Zink, Cartiertheorie kommutative formaler Gruppen. Teubner, 1984. [89] T. Zink, Windows for displays of p-divisible groups. In Moduli of Abelian Varieties, Progress in Math. 195, Birkh¨ auser, 2001, pp. 491–518. [90] T. Zink, The display of a formal p-divisible group. Ast´erisque 278 (2002), 127–278.

Notes Sources of quotes at the beginning of each chapter. (0) The Zariski quote on page 1 is from the Preface of Oscar Zariski: Collected Papers, page xii of volumes I, II and page xiv of volumes III, IV. (1) The Hilbert quote on page 7 is from page 182 of Hilbert’s obituary by Olga Taussky in Nature, vol. 152, pp. 182–183. Taussky, who heard Hilbert’s comment, recorded it as follows. It is interesting to recall that, in connexion with a lecture by Prof. R. Fueter at the 1932 Zurich Congress, Hilbert asserted that the theory of complex multiplication (of elliptic modular functions) which forms a powerful link between number theory and analysis, is not only the most beautiful part of mathematics but also of all science. (2) The Igusa quote on page 45 is from page 614 of Igusa’s article Arithmetic variety of moduli for genus two, Annals of Math. vol. 72, 1960, pp. 612–649. (3) The Tate quote on page 81 is from page 158 of Tate’s original article [75] on p-divisible groups. (4) The Grothendieck quote on page 103 is in a letter from Grothendieck to Mumford dated September 4, 1968. The letter is included in David Mumford Selected Papers vol. II, Springer, 2010, pp. 735–737. The passage quoted is on page 736. (5) The Mumford quote on page 179 is from the Preface of David Mumford Selected Papers, On the Classification of Varieties and Moduli Spaces, Springer, 2004. (6) The Shimura quote on 137 is from a footnote on the first page of Chapter 4 of Shimura’s book Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1994.

225

Glossary of Notations

F ab Yb G1,h−1 Gh−1,1 (F, Φ) E(Fi , Φi ) E(F, Φ) ΓL L0 IL REP? (ΓL ) Dcrys Vcrys FE,π (x, y) VE,π ρE,π recE TE δF,Φ L L0 v OL,v [Lie(Zv )] ∆L0 , ∆(L/L0 , κ)) ∆L E0 E κE f , fE e, e(E/Qp ) e, e(K/K0 ) ι E ur k I κ Rk (OE ) Rκ (OE ) + R+ k (OE ), Rκ (OE ) Rk (OE , i)  i [Lie(X0 , α0 )]

an algebraic closure of Fp , 81–198 the subgroup of X[F, V ] ∼ = αp × αp isomorphic to αp attached to b, 85–85, 94 (X ⊗κ κ(b))/β(ab ) the αp -quotient of X parametrized by b, 85–85, 94 a one-dimensional p-divisible formal group of height h in characteristic p, 87 the Serre dual of a G1,h−1 , 87 p-adic CM type, decomposed as (F1 , Φ1 ) × · · · × (Fc , Φc ), 90, 92 reflex field of the p-adic CM type Φi , 90, 92 the compositum of the reflex fields E(Fi , Φi ), 90, 92 the local Galois group Gal(L/L) of a p-adic local field L, 95 the maximal absolutely unramified subfield in L, 97 the inertia subgroup of ΓL , 95 the tensor category of all a? -representations of ΓL , 95 Fontaine’s functor, from REPcrys (ΓL ) to MFfL (φ), 97 the inverse of Dcrys , 97 a Lubin-Tate formal group law for a local field E with local parameter π, 98 the p-adic Tate module attached to FE,π (x, y), 98 × the OE -valued character of ΓE attached to FE,π (x, y), 98 Artin map for the local field E, 98 the induced torus for an extension field E/Qp of finite degree, 98 local Shimura reciprocity law homomorphism, 99 a CM field, 103, 126–132 the maximal totally real subfield of L, 105, 126–132 a p-adic place of L0 above p, 105, 126–132 OL ⊗OL0 OL0,v , 105, 126–132 the Lie type of an OL,v -linear CM p-divisible group Zv , 105 the set of all bad p-adic places of L0 with respect to κ, 106, 130 the set of all good p-adic places of L, 106 a finite extension field of Qp , 116–126 a finite extension field of Qp , 108–116 a commutative semisimple quadratic algebra over E0 , 116–126 residue field of E, 108–116 either the quadratic extension field of κE0 or κE0 × κE0 , 116–126 dimFp (κE ), 108–116, 116–126 the ramification index of E, equals to e(E0 /Qp ) if E ∼ = E0 × E0 , 108–116, 116–126 the ramification degree of K over K0 , 189–198, 211–198, the involution for E/E0 , 116–126, 126–132 the maximal absolutely unramified Qp -subalgebra of E, 108–116, 116–126 an algebraically closed field of characteristic p > 0, 108–132 a perfect field of characteristic p > 0, 199–219 HomZp -alg (OE ur , W (k)), 108–116, 116–126 a finite field contained in k, 108–132 the Grothendieck group of (OE ⊗Zp k)-modules of finite length, 108–116, 116–126 κ-rational elements in Rk (OE ), 108–116, 116–126 effective elements in Rk (OE ) and Rκ (OE ) respectively, 108–116, 116–126 the i-th component of Rk (OE ) for any i ∈ I, 109–116, 117–126 the degree map for Rk (OE ), 109–116, 117–126, the i-th component of the degree map , 109–116, 117–126, the Lie type of an OE -linear CM p-divisible group (X0 , α0 ), 110, 117 227

228

ξ δunif δstr,J (Ytoy , βtoy ) ΣL,p ΣL0 ,p ΣL,v w Ow ur Ow κw Iw Iv Rk (OLv ) Rκ (OL,v ) Rk (OL,p ) w w,i I J J0 O (Y0 , β0 ) ρ0 (X0 , γ0 ) N , D∗ (Y0 ) M , D∗ (X0 ) G0 HJ 0 −{4m} HS K0 K K1 /K0 X , XΛ0
M0 , φM,0 , φM,0 (M, φM , φM ) D∗ ( · ), D∗ ( · ) G X C∗ (X) D∗ (G), D∗ (X) D∗ (G), D∗ (X) `G• nG νG E(u) S S0 S c (Mod/S) c (Mod/S0 ) c (p-Gr/k) c (Mod/Dk ) f (p-div/OK )

GLOSSARY OF NOTATIONS

reduction map, from p-adic CM types to Lie types, 110, 118 the supersingular uniform Lie type, 120 the striped Lie type attached to a Gal(κE /Fp2 )-orbit J, 120 a CM p-divisible group with a striped Lie type, 123–125 the set of all p-adic places of L, 126–132 the set of all p-adic places of L0 , 126–132 the set of all places of L above a p-adic place v of L0 , 126–132 a p-adic place of L above a place v of L0 , 126–132 the ring of integers in the w-adic completion Lw of w, 126–132 the maximal absolutely unramified subfield of Lw , 126–132 the residue field of w, 126–132 ur HomZp -alg (Ow , W (k)), 126–132 ur HomZp -alg (OL,v , W (k)), 126–132 the Grothendieck group of (OLv ⊗Zp k)-modules of finite length, 126–132 κ-rational elements of Rk (OLv ), 126–132 the Grothendieck group of (OL,p ⊗Zp k)-modules of finite length, 126–132 the degree map for Rk (Ow ), 127–132 the i-th component of w , 127–132 identified with {1, 2, 3, . . . , 4m}, 180–188, 189–198 the subset {2, 4, 6, . . . , 4m} of I, 180–198 the subset {1, 3, 5, . . . , 2m − 1, 2m + 2, 2m + 4, . . . , 4m} of I, 180–198 W (Fp4m ), 179–198 an O-linear CM p-divisible group over F with Lie type J, 180–198 an O-linear isogeny from X0 to Y0 of degree pm , 180–198 an O-linear CM p-divisible group over F with Lie type J 0 , 180–198 the O-liner Cartier module of (Y0 , β0 ), 180–198 the O-liner Cartier module of (X0 , γ0 ), 181–198 Ker(ρ0 ), a finite group scheme over F of order pm 181–198 a subgroup scheme of X0 isomorphic to αp4m−1 with Lie type J 0 − {4m}, 184–198 the subgroup scheme of HJ 0 −{4m} with Lie type S ⊂ J 0 − {4m}, 184–198 the fraction field of W (F), 182–198 the fraction field W (k)[1/p] of W (k), 211–219 the tame totally ramified extension field of K0 of degree p4m − 1, 182–198 a totally ramified finite extension field of K0 = W (k)[1/p], 211–219 the subextension of K/K0 such that [K : K1 ] = p − 1 the O-linear lifting of X0 over W (F), 181–198, 191–198 the Kisin module for X [p] over W (F), 192–198 the Kisin module for the base change of X [p] to OK , 192–198 the covariant and contravariant crystal in crystalline Dieudonne theory, 207–210 a finite commutative group scheme over the perfect base field k, 199–198 a smooth formal group or a p-divisible group over the perfect base field k, 199–198 the Cartier module of a smooth formal group X over k, 205–210 covariant Dieudonne module for G and X, 207–210 contravariant Dieudonne module for G and X, 207–210 the co-Lie complex of a finite group scheme G (over a perfect field k), 209–210 H −1 (`G ), 209–210 ∨ H 1 (`G ), 209–210 the Eisenstein polynomial over K0 for an local parameter π of K, 211–219 W (k)[[u]], 190–198, 211–219 W (F)[[t]], 190–198, the p-adic completion of W [u, E(u)n /n!]n∈N , 211 the category of connected Kisin modules for finite flat group schemes over OK , 190, 211 the category of connected Kisin modules for finite flat group schemes over W (F), 190 the category of connected commutative finite group schemes of p-power order over k, 212 Dieudonne modules for commutative finite groups schemes over k of local type, 212 the category of formal p-divisible groups over OK , 212

GLOSSARY OF NOTATIONS

BTφ,f /S0 BTφ,f /S (Win/S) (Dsp/R)

the the the the

category category category category

of of of of

Kisin modules for formal p-divisible groups over W (F), 190 Kisin modules for formal p-divisible groups over OK , 212–219 S-windows, 213–219 displays over a p-adic ring R, 213–219

229