The Bloch-Kato Conjecture for the Riemann Zeta Function

London Mathematical Society Lecture Note Series 418

The Bloch–Kato Conjecture for the Riemann Zeta Function Edited by John Coates, A. Raghuram, Anupam Saikia, and R. Sujatha

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London Mathematical Society Lecture Note Series: 418

The Bloch–Kato Conjecture for the Riemann Zeta Function Edited by

JOHN COATES University of Cambridge A. RAGHURAM Indian Institute of Science Education and Research, Pune, India ANUPAM SAIKIA Indian Institute of Technology Guwahati, India R. SUJATHA University of British Columbia, Vancouver, Canada

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107492967 c Cambridge University Press 2015  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United Kingdom by Clays, St lves plc A catalogue record for this publication is available from the British Library ISBN 978-1-107-49296-7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

List of contributors Preface 1

2 3

4

5

6 7 8 9

Special Values of the Riemann Zeta Function: Some Results and Conjectures A. Raghuram

page vii viii

1

K-theoretic Background R. Sujatha

22

Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory John Coates

45

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps Anupam Saikia

65

The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture Manfred Kolster

97

Regulators and Zeta-functions Stephen Lichtenbaum

121

Soul´ e’s Theorem Stephen Lichtenbaum

130

Soul´ e’s Regulator Map Ralph Greenberg

140

On the Determinantal Approach to the Tamagawa Number Conjecture T. Nguyen Quang Do

154

vi

Contents

10 Motivic Polylogarithm and Related Classes Don Blasius

193

11 The Comparison Theorem for the Soul´ e–Deligne Classes Annette Huber

210

12 Eisenstein Classes, Elliptic Soul´ e Elements and the ` -Adic Elliptic Polylogarithm Guido Kings

239

13 Postscript R. Sujatha

297

Contributors

Don Blasius University of California at Los Angeles, USA. John Coates University of Cambridge, UK. Ralph Greenberg University of Washington, Seattle, USA. Annette Huber Albert-Ludwigs-Universit¨ at Freiburg, Germany. Guido Kings University of Regensburg, Germany. Manfred Kolster McMaster University, Hamilton, Canada. Stephen Lichtenbaum Brown University, Providence, USA. T. Nguyen Quang Do University of Franche-Comt´e, France. A. Raghuram Indian Institute of Science Education and Research, Pune, India. Anupam Saikia Indian Institute of Technology Guwahati, India. R. Sujatha University of British Columbia, Vancouver, Canada.

Preface

These are the proceedings of a week-long workshop entitled ‘The Bloch– Kato conjectures for the Riemann zeta function at the odd positive integers’, which was held at the Indian Institute of Science Education and Research (IISER), Pune, India, in July 2012. This workshop was immediately followed by the 2012 Pan Asian Number Theory (PANT) conference, and there was a considerable overlap between the participants of the two events. The workshop was organized by J. Coates, S. Maity, A. Raghuram, A. Saikia and R. Sujatha, and had a total of 18 lectures given by D. Blasius, J. Coates, R. Greenberg, G. Kings, S. Lichtenbaum, A. Raghuram, A. Saikia and R. Sujatha. These proceedings contain not only expanded versions of all the lectures given at the workshop, but also several invited articles on related material. The Riemann zeta function ζ(s) is the oldest L-function, and it remains one of the most important functions in modern number theory. The values of ζ(s) at even positive integers and odd negative integers have been known since the time of Euler. However, its values at odd positive integers remain mysterious to this day. The Bloch–Kato conjectures, also known as the Tamagawa number conjectures, provide a conjectural framework to understand the special values of all motivic L-functions. The principal aim of the workshop was to prove the Bloch– Kato conjecture in the simplest non-critical example of the values of ζ(s) at odd positive integers s > 1. On behalf of all the organizers, it gives me great pleasure to thank Dr Soumen Maity and Ms Suvarna Gharat, who went far beyond the call

Preface

ix

of duty in handling all the local logistics for both the workshop and the conference. As a co-editor of this volume, I am grateful to Ms Ayesha Fatima and Mr Jeeten Patel, both of whom are IISER alumni, for providing critical help with LaTeX, and for putting together different tex files into one coherent volume. I also thank Professor K.N. Ganesh, Director of IISER Pune, and Dr V.S. Rao, the then Registrar of IISER Pune, for their generous and untiring support. Indeed, the Bloch–Kato workshop and the PANT conference went a long way towards establishing IISER Pune as an important centre for mathematics in India. Finally, on behalf of all the lecturers and participants, I express my gratitude to the International Center for Theoretical Sciences for its financial support for the workshop and the conference. A. Raghuram Coordinator for Mathematics Indian Institute of Science Education and Research Pune, February 6th, 2014.

1 Special Values of the Riemann Zeta Function: Some Results and Conjectures A. Raghuram

Abstract These notes are based on two lectures given at the instructional workshop on the Bloch–Kato conjecture for the values of the Riemann ζfunction at odd positive integers. The workshop was held at IISER, Pune, in July 2012. The aim of these notes is to give a brief introduction to (i) Borel’s results and Lichtenbaum’s conjectures on the special values of the Riemann ζ-function, and (ii) Deligne’s conjecture and the Tamagawa number conjecture of Bloch and Kato on the special values of motivic L-functions as applied to Tate motives.

1.1 Values of the Riemann ζ-function and K-groups of Z 1.1.1 Definition and basic analytic properties of ζ(s) Definition of ζ(s) The Riemann zeta function is defined by the series ζ(s) =

∞ X 1 , ns n=1

σ > 1,

where s = σ + it is a complex variable with σ = <(s) and t = =(s). The series converges absolutely for σ > 1, Rwhich may be seen using the ∞ integral test by comparing the series with 1 1/xσ dx. Indian Institute of Science Education and Research (IISER), Pune, Maharashtra 411021, India. e-mail : [email protected]

2

A. Raghuram

Euler product, analytic continuation and functional equation Theorem 1.1.1 (Basic Analytic Properties) The Riemann zeta function has the following properties: 1. (Euler Product) For σ > 1 we have −1 Y 1 ζ(s) = 1− s , p p where the product runs over all primes p. 2. (Analytic continuation) The function ζ(s), which is defined for σ > 1, extends to a meromorphic function to all of C with only one pole which is located at s = 1 and is a simple pole with residue 1. Around s = 1, we have ∞ X 1 ζ(s) = +γ + γk (s − 1)k s−1 k=1

where γ is Euler’s constant. 3. (Functional equation)   s 1−s π −s/2 Γ ζ(s) = π (1−s)/2 Γ ζ(1 − s), 2 2 where Γ(s) is the usual Γ-function. See, for example, Ivic [Iv85, Chapter 1].
s −s/2 Let ζ∞ (s) = π Γ 2 , and define the completed zeta function by s Λ(s) := ζ∞ (s)ζ(s) = π −s/2 Γ ζ(s). (1.1) 2 Then Λ(s) has a meromorphic continuation to all of C with simple poles at s = 0, 1 and is holomorphic elsewhere. The functional equation looks like Λ(s) = Λ(1 − s). In analytic number theory, one also completes ζ(s) as Λ∗ (s) = s(1 − s)Λ(s). We still have the same functional equation Λ∗ (s) = Λ∗ (1 − s), but Λ∗ (s) has the virtue of being an entire function. However, from the motivic or automorphic perspective, the completed zeta function is always taken to be Λ(s) and not Λ∗ (s). An easy consequence of the functional equation is: 1 ζ(0) = − . 2

(1.2)

(For the interested reader, here is a quote from Ramanujan’s Notebooks: The constant of a series has some mysterious connection with the given infinite series and it is like the centre of gravity of a body. Mysterious

ζ-values

3

because we can substitute it for the divergent infinite series. Now the constant of the series 1 + 1 + 1 + &c is − 12 . See p.79 of ‘Notebooks of Srinivasa Ramanujan’, Volume 1, Published by TIFR, Mumbai 2012.)

1.1.2 Euler’s Theorem Critical points Definition 1.1.2 An integer n is said to be critical for ζ(s) if both ζ∞ (s) and ζ∞ (1 − s) are regular (i.e., no poles) at s = n. The set of all critical integers is called the critical set. Observe that, by definition, the critical set is symmetric, i.e., invariant under s 7→ 1 − s. Proposition 1.1.3 The critical set for ζ(s) consists of all even positive integers and all odd negative integers, i.e., critical set for ζ(s) = {. . . , 1 − 2m, . . . , −5, −3, −1} ∪ {2, 4, 6, . . . , 2m, . . . } . Proof

Let n be critical for ζ(s). This means two conditions on n:

1. ζ∞ (s) = π −s/2 Γ(s/2) does not have a pole at s = n; exponentials are entire and non-vanishing and so this means Γ(s/2) has no pole at n, i.e., n/2 ∈ / {. . . , −3, −2, −1, 0}, which means that n is not a non-positive even integer; and 2. ζ∞ (1 − s) = π −(1−s)/2 Γ((1 − s)/2) does not have a pole at s = n; this translates to Γ((1 − s)/2) having no pole at n, i.e., (1 − n)/2 ∈ / {. . . , −3, −2, −1, 0}, or n ∈ / {1, 3, 5, . . . }, which means that n is not an odd positive integer.

The critical values of ζ(s) The Bernoulli numbers are defined by the formal power series expansion of z/(ez − 1): ∞ X z zk = B . (1.3) k ez − 1 k! k=0

Some easy values are: B0 = 1, B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, . . . . Indeed, we have B2k+1 = 0 for k ≥ 1.

4

A. Raghuram

Theorem 1.1.4

The critical values for ζ(s) are given by:

1. The critical values to the right of the centre of symmetry: ζ(2m) =

(−1)m+1 (2π)2m B2m . 2(2m)!

2. The critical values to the left of the centre of symmetry: B2m . 2m See Neukirch [Ne99, Chapter VII, Section 1] for a detailed proof. ζ(1 − 2m) = −

Remark 1.1.4.1 Let us note the special case ζ(−1) = −1/12 was ‘proved’ by Euler (and later rediscovered by Ramanujan) via the following intriguing calculation: S = 1 + 2 + 3 + 4 + 5 + 6 + ··· 4S = 4 + 8 + 12 + · · · −3S = 1 − 2 + 3 − 4 + 5 − 6 · · · =

1 = 1/4 (1 + 1)2

=⇒ S = −1/12.

The non-critical values of ζ(s) The non-critical values of ζ(s) are its values at non-critical integers, i.e., the values {ζ(2m + 1) : m ≥ 1} and {ζ(−2m) : m ≥ 1}. The values at the odd positive integers are mysterious and the purpose of this workshop is to understand these values via the Bloch–Kato conjectures. However, the values at the negative even integers are trivial: Lemma 1.1.5 (Trivial zeros) zero at s = −2m. Proof

For any integer m ≥ 1, ζ(s) has a simple

Put s = −2m into the functional equation to get:   1 + 2m π m Γ(−m) ζ(−2m) = π (1+2m)/2 Γ ζ(1 + 2m). 2

The right hand side is finite and non-zero, therefore so is the left hand side; but Γ(−m) is a simple pole, hence ζ(−2m) is a simple zero. What is mysterious about ζ(s) at s = −2m is not so much the value, but rather the leading term: ζ ∗ (−2m) :=

lim ζ(s)(s + 2m).

s→−2m

The mystery about ζ(2m+1) is equivalent, via the functional equation, to the mystery about ζ ∗ (−2m). We mention the following transcendental statements only for the sake of completeness:

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5

1. ζ(3) is irrational. (See Ap´ery [Ap79].) 2. The Q-span of {ζ(2m+1) : m ≥ 1} is an infinite-dimensional subspace of R, i.e., infinitely many of the zeta values at odd positive integers are Q-linearly independent. (See Ball and Rivoal [BR01].)

1.1.3 Borel’s Theorem Tamagawa number of SLn /Q and ζ-values Let n ≥ 2. Fix an isomorphism Top exterior : (sln )/Z → Z. This induces the Tamagawa measure on SLn (A) as follows: The meaQ sure dg on SLn (A) is the product of local measures dg = v dgv and locally everywhere dgv is the Haar measure determined by the above isomorphism. By definition, the Tamagawa number of SLn /Q is: τ (SLn /Q) := vol (SLn (Q)\SLn (A)) . Theorem 1.1.6 The Tamagawa number of SLn /Q is 1, i.e., τ (SLn /Q) = 1. See, for example, Weil [We58]. Corollary 1.1.7 n Y

ζ(m) = vol (SLn (Z)\SLn (R)) .

m=2

Proof

The strong approximation theorem gives:

vol (SLn (Q)\SLn (A)) = vol (SLn (Z)\SLn (R))

Y

vol(SLn (Zp ));

p

the left hand side is 1 by the above theorem, and for the right hand side we have n Y vol(SLn (Zp )) = (1 − p−m ), m=2

where all the volumes are with respect to the Tamagawa measures. It is rather piquant to note that ζ(3)π 2 /6 = vol (SL3 (Z)\SL3 (R)) .

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A. Raghuram

Statement of Borel’s results In this section we give a very brief sketch of some results of Borel [Bo77]. The serious reader should consult Borel for all details. Consider the following diagram of cohomology groups: µ•

H • (SU(n); C) c

α•

/ H • (SLn (R)/Γn ; C) :

β•

H • (sln (C); C) where Γn is an arithmetic torsion-free subgroup of SLn (Z). The morphisms α• and β • are defined in terms of invariant forms and α• is an isomorphism; now define µ• := β • ◦ α• −1 . All the cohomology groups in sight are exterior algebras and we can talk of indecomposable elements. (Let A be an algebra over a field which is graded by subspaces {Ap }p∈N . The space of indecomposable elements of A of degree p, denoted I p (A), is defined to be the quotient of Ap by the subspace of decomposable elements, i.e., subspace generated by products of elements of degree less than p.) We can also work with rational cohomology, and let us record the following results of Borel: For an even positive integer m, the spaces I 2m+1 (sln (Q); Q) and I 2m+1 (SU(n); Q) are 1-dimensional Q-subspaces of the ambient complex spaces. (For brevity, we have let I 2m+1 (sln (Q); Q) to stand for I 2m+1 (H • (sln (Q); Q)), etc.) Borel studied the effect of the maps α• and β • on these one-dimensional lines, and proved α• (I 2m+1 (sln (Q); Q)) = (πi)m+1 I 2m+1 (SU(n); Q).

(1.4)

See [Bo77, Proposition 5.4]. Similarly, one has β • (I 2m+1 (sln (Q); Q)) = ζ(m+1) I 2m+1 (SLn (R)/Γn ; Q), (n > 8m+5). (1.5) This involves a calculation of integrating a top-degree rational form on a ‘modular symbol’ H/(Γ ∩ H) ,→ G/Γ where H is a suitable SL1 (D) inside G = SLn ; this is a Tamagawa number calculation, the simplest case of which is briefly described in 1.1.3. For more details, see Borel’s [Bo77, Th´eor`eme 5.5] and its proof. The heart of Borel’s paper is to construct and analyze a certain canonical morphism in a relative context (i.e., mod-maximal-compact) for real

ζ-values

7

cohomology: jΓ• : H • (SU(n)/SO(n); R) −→ H • (SLn (Z)\SLn (R)/SO(n); R). (1.6) Observe that the right hand side is also group cohomology: H • (SLn (Z)\SLn (R)/SO(n); R) ' H • (SLn (Z); R). The main result of Borel, stemming from (1.4), (1.5) and (1.6), is the following: Theorem 1.1.8 8m + 5. Then

Let m = 2r be an even positive integer and let n >

jΓ• (I 4r+1 (SU(n)/SO(n); Q)) =

ζ(2r + 1) 4r+1 I (SLn (Z); Q). π 2r+1

See [Bo77, Th´eor`eme 6.2]. This result has an interpretation in terms of K-groups, which we discuss in the next subsection after introducing K-groups.

1.1.4 K-groups of Z For us, Km is a functor from the category of commutative rings to the category of abelian groups; Sujatha’s lectures in this workshop go more deeply into algebraic K-theory that is necessary to study the Riemann ζ-function. One calls K0 (R) the projective module group and it is defined to be the quotient of the free abelian group on [P ] where P runs over isomorphism classes of a finitely generated projective module by the normal subgroup generated by the relations [P ⊕Q]−[P ]−[Q]. Since Z is a PID, and every finitely generated projective module is free, we get K0 (Z) = Z. The group K1 (R) is called the Whitehead group and it is defined to be the quotient GL(R)/E(R) where GL(R) := limn GLn (R), the limit taken −→ over the maps GLn (R) → GLn+1 (R) given by g 7→ diag(g, 1); and E(R) is the subgroup generated by all elementary matrices. Since we have taken R to be commutative, the determinant homomorphism is defined and one has K1 (R) ∼ = R× ⊕ (SL(R)/E(R)). The group SL(R)/E(R) is often denoted SK1 (R) and is called the reduced Whitehead group. If R is a Euclidean domain, one knows that SK1 (R) = {1}. Hence K1 (Z) = Z× ∼ = Z/2. (See, for example, Milnor [Mi71].) For a ring R, one defines Km (R) := πm (BGL(R)+ ),

(1.7)

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A. Raghuram

i.e., the m-th homotopy group of Quillen’s plus construction applied to the classifying space of the limit GL(R) of general linear groups. Computing K-groups is a highly non-trivial problem, and even for K-groups of Z not everything is known. In the next subsection we give a summary of some more precise results on Km (Z). What is known about Km (Z)? The following brief summary on Km (Z) is taken from Weibel [We05]. We begin with two general finiteness results: 1. Km (Z) is a finitely generated abelian group. (Quillen [Qu73].) 2. Rank of Km (Z) is 1 if m ≥ 5 is 1 mod 4; in all other cases Km (Z) is a finite group, i.e., has rank 0. (Borel [Bo74].)

K0 (Z) K1 (Z) K2 (Z) K3 (Z) K4 (Z) K5 (Z) K6 (Z) K7 (Z)

= = = = = = = =

Z Z/2 Z/2 Z/48 0 Z 0 Z/240

K8a (Z) = 0? K8a+1 (Z) = Z ⊕ Z/2 K8a+2 (Z) = Z/2c2a+1 K8a+3 (Z) = Z/2w4a+2 K8a+4 (Z) = 0? K8a+5 (Z) = Z K8a+6 (Z) = Z/c2a+2 K8a+7 (Z) = Z/w4a+4

(1.8)

The question marks mean that it is expected K4a (Z) = 0. This is proven for a = 1 and is open as yet for a ≥ 2. The numbers cm and wm are defined as follows: cm = numerator of (−1)m+1 B2m /4m.

(1.9)

(It is understood that if we talk of the numerator a of a rational number a/b then one has taken the rational to be in its lowest form, i.e., a and b are relatively prime.) Let W be the group of all roots of unity in Q. Then W is naturally a GQ = Gal(Q/Q)-module, since if w ∈ W and g ∈ GQ then g(w) ∈ W. For any integer m ≥ 1 we let W(m) stand for the GQ module where g ∈ GQ acts on w ∈ W(m) = W via g ·m w := g m (w). One says W(m) is the Galois module W with a Tate twist by m. Now define wm := |{w ∈ W : g m (w) = w, ∀g ∈ GQ }|,

(1.10)

i.e., it is the cardinality of the set of those roots of unity which are fixed by GQ under the m-twisted action.

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9

Borel regulators and non-critical values Let us go back to Theorem 1.1.8: recall that m = 2r is an even positive integer and n  m, then jΓ• (I 2m+1 (SU(n)/SO(n); Q)) =

ζ(m + 1) 2m+1 I (SLn (Z); Q). π m+1

Now pass to the limit over n. Define Xu := limn SU(n)/SO(n) and −→ SL(Z) := limn SLn (Z). Then it is known that we have a duality: −→ I 2m+1 (Xu ; Q) × (π2m+1 (Xu ) ⊗Z Q) −→ Q, and similarly, I 2m+1 (SL(Z); Q) × (K2m+1 (Z) ⊗Z Q) −→ Q. (Recall that m = 2r is even, and so K2m+1 (Z) ⊗Z Q is a one-dimensional ∗ Q-vector space.) Fix a basis x∗m for π2m+1 (Xu ), and ym for K2m+1 (Z). Let xm and ym be the dual basis. Definition 1.1.9 (Borel Regulators) jΓ• (xm ) = Rm (Q) ym . From Theorem 1.1.8 and Definition 1.1.9 we get the following beautiful result of Borel on the non-critical values of Riemann zeta function: Theorem 1.1.10 (Borel)

Let r ≥ 1. Then

ζ(2r + 1) ∼ R2r (Q), π 2r+1 where ∼ means up to a non-zero rational number.

1.1.5 Lichtenbaum’s conjecture Critical values and K-groups Reference: Lichtenbaum [Li73, Conjecture 2.4]. Theorem 1.1.11 (Critical values on the left) Up to 2-torsion for any odd integer m ≥ 1 |K2m (Z)| |ζ(−m)| = . |K2m+1 (Z)| Proof

Follows from Theorem 1.1.4 and (1.8).

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The caveat ‘up to 2-torsion’ is necessary as can already be seen from the case m = 1: B2 1 ζ(−1) = − = − , 2 12 whereas |K2 (Z)| = 2 and |K3 (Z)| = 48 giving |K2 (Z)| 1 = . |K3 (Z)| 24 K-theoretic Lichtenbaum’s conjecture In [Li73, Question 4.2] Lichtenbaum formulated a conjecture for ζ-values at any negative integer −m in terms of certain higher regulators R0m (Q) which are essentially the same as Borel’s regulators Rm (Q). For Licht0 enbaum’s definition of Rm (Q) see [Li73, p.498–499]. Conjecture 1.1.12 (Any special value on the left) m ≥ 1, possibly up to 2-torsion, we have ζ ∗ (−m) = ±

For any integer

|K2m (Z)| 0 · Rm (Q). |K2m+1 (Z)tors |

A slightly modified version of Conjecture 1.1.12 was proved for any abelian number field by Kolster, Nguyen Quang Do and Fleckinger [KNF96, Theorem 6.4]; some errors on Euler factors in [KNF96] have been corrected in [BN02]. For a result character by character, see [HK03]. The reader is also referred to the survey article by Flach [Fl04] for a historic introduction. Cohomological Lichtenbaum’s conjecture The K-theoretic Lichtenbaum’s conjecture can be restated in the language of Galois cohomology. The connection is provided by the Quillen– Lichtenbaum conjecture which relates K-groups to Galois cohomology. (The reader should look at Lichtenbaum [Li73, Conjecture 2.5] and Huber–Kings [HK03, p.410].) Indeed, the formulas in (1.8) cited from [We05] use this connection between K-groups and Galois cohomology. Give Spec(Q) the ´etale topology. Then the category of discrete GQ modules is equivalent to the category of sheaves of abelian groups over Spec(Q), and Galois cohomology is the same as sheaf cohomology which in this case is called ´etale cohomology. For any m ≥ 1, the GQ -module W(m) gives a sheaf on Spec(Q). Denote this sheaf also by W(m). Fix a prime `. Let X` := Spec(Z[1/`]), and let j : Spec(Q) ,→ X`

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11

be the canonical inclusion. Now consider the cohomology of X` with coefficients in the direct image j∗ W(m) sheaf on X` . Conjecture 1.1.13 (Critical values on the left) m ≥ 1, we have `-part of |ζ(−m)| =

For any odd integer

|H 1 (X` , j∗ W(m + 1)| . |H 0 (X` , j∗ W(m + 1)|

For m = 1 this was conjectured by Birch and Tate. See [Li73, Conjecture 1.5]. More generally, for any special value ζ ∗ (−m) with m ≥ 1, the cohomological Lichtenbaum’s ‘conjecture’ (now a theorem; see below) states: Conjecture 1.1.14 (Any special value on the left) Let m ≥ 1. Let 00 Rm (Q) be the m-th Beilinson regulator of Q. Then, up to powers of 2, and up to sign, we have 00 ζ(−m)∗ = Rm (Q)

Y |H 2 (X` , j∗ Z` (m + 1)| . |H 1 (X` , j∗ Z` (m + 1)tors | `

The reader is referred to the excellent monograph by Burgos Gil [Bu02] 00 for a precise relation between the Beilinson regulator Rm (Q) and the Borel regulator Rm (Q); the main theorem therein says that a renormal00 ized version of Rm (Q) is twice Rm (Q). For m > 1, Conjecture 1.1.14 above is equivalent to the Bloch–Kato conjecture for the Tate motive Z(1 − m) (see next section), which is in turn equivalent, via the calculation of the Tamagawa numbers performed in the proof of [BK90, Theorem 6.1], to the Bloch–Kato conjecture for Z(m) which is the focus of the workshop.

1.2 The Bloch–Kato conjecture for the Riemann ζ-function A motive is a piece in the cohomology of a smooth projective variety, although we will deal exclusively at the level of realizations (Betti, de Rham and `-adic) of the motive. A motivic L-function is the Artin Lfunction of the Galois representation on the `-adic realization of the motive. An integer is critical for a motivic L-function if the Γ-factors at infinity on both sides of the functional equation are regular at that integer. Deligne’s conjecture says that a motivic critical L-value is rational

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A. Raghuram

up to a period/regulator. Beilinson’s conjecture looks roughly the same as Deligne except for any L-value, critical or not. Bloch and Kato give a recipe for that rational number in terms of Tamagawa numbers.

1.2.1 The Riemann zeta function as a motivic L-function Axiomatic definition of a motive and Deligne’s conjecture References: Deligne [De79] and Blasius [Bl87]. (Under the assumption that the motive has coefficients in Q which will be the case for the motives for the Riemann zeta function.) A motive over Q with coefficients also in Q is a five tuple of data M = (MB , MdR , Mf , I∞ , If ) where MB and MdR are finite-dimensional Q-vector spaces both of dimension d, and Mf is an Af -module of rank d. The maps I∞ and If are comparison isomorphisms: I∞ : MB ⊗Q C −→ MdR ⊗Q C, If : MB ⊗Q Af −→ Mf . Hodge Decomposition and Complex Conjugation: Under the assumption that the motive M is pure and has a purity weight w, one has a Hodge decomposition M MB ⊗ C = M p,q , p+q=w

where the action of complex conjugation ρ on the second factor in MB ⊗C interchanges M p,q with M q,p . Recall that a motive is supposed to be a piece in the cohomology H • (X) of an algebraic variety X, and so MB is • a piece in HBetti (X(C), Z). This gives an action of complex conjugation on MB , denoted F∞ , arising via functoriality from the action of complex conjugation on the ‘real’ variety X(C). Let MB± := Ker(F∞ ∓ 1MB ) be the ±1-eigenspace for the action of F∞ . Let d± := dimQ (MB± ). Assume that F∞ acts on M w/2,w/2 via the scalar  (= ±1) whenever M w/2,w/2 6= 0. (When the middle Hodge piece M w/2,w/2 is non-trivial, this assumption is necessary for the existence of critical points for the L-function.)

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13

Hodge filtration: For all p, there exists a Q-subspace F p MdR ⊂ MdR such that M p0 ,q −1 I∞ (F p MdR ⊗ C) = MB . p0 ≥p

Using the Hodge filtration, we identify the analogues of MB± for MdR as follows: 1. If w ∈ 2Z,  = 1 then let F + MdR := F w/2 MdR and F − MdR := F 1+w/2 MdR . 2. If w ∈ 2Z,  = −1 then let F + MdR := F 1+w/2 MdR and F − MdR := F w/2 MdR . 3. If w is odd then let F + MdR = F − MdR := F (w+1)/2,(w+1)/2 MdR . Now define ± MdR := MdR /F ∓ MdR .

The comparison isomorphism I∞ induces isomorphisms I ± as: MB O ⊗ C ? MB± ⊗ C

I∞

I±

/ MdR ⊗ C O ? / M± ⊗ C dR

Deligne’s periods attached to the motive M are defined as: c± (M ) := det(I ± ),

δ(M ) := det(I)

(1.11)

where the determinants are computed relative to rational bases for MB , ± MdR , MB± and MdR . The finite part of the motivic L-function is an Euler product: Lf (s, M ) Q := p Lp (s, M ), where we fix a prime `, and the Euler factor at p 6= ` is: Lp (s, M ) := det(I − ϕp |M Ip p−s )−1 ; here M` = M ⊗ Q` which con` jecturally carries a strictly compatible system of `-adic representations, Ip is the inertia subgroup of GQ at p, ϕp is the (arithmetic) Frobenius at p which is a coset of Ip that has a well-defined action on the inertia I invariants M` p , and det(I − ϕp |M Ip p−s ) is, a priori, a polynomial with ` Q` -coefficients but, conjecturally, it is `-independent. For more details the reader is referred to Deligne [De79, §1.1, 1.2]. The product of Γ-factors at infinity, denoted L∞ (s, M ), is defined by a recipe of Serre [Se69, 3.2], which in our simplified situation goes as follows: define the Hodge numbers hp,q := dim(M p,q ). And when M w/2,w/2 6= 0, recall our assumption that the complex conjugation acts

14

A. Raghuram w/2,w/2

on M w/2,w/2 as . If  = ±(−1)w/2 then put M± = M w/2,w/2 and w/2,w/2 w/2,w/2 ± M∓ = 0. Then we put hw/2 := dim(M± ). Now define: Y h+ h− L∞ (s, M ) := ΓR (s−w/2) w/2 ΓR (s−w/2+1) w/2 ΓC (s−p)hp,w−p , p<w/2

(1.12) −s/2

−s

where ΓR (s) = π Γ(s/2) and ΓC (s) = 2(2π) Γ(s). The completed motivic L-function is defined as L(s, M ) := L∞ (s, M ) Lf (s, M ). Even though it is defined a priori only in a half-plane, one expects that it has a meromorphic continuation to all of C, and that it satisfies a functional equation of the form L(s, M ) = ε(s, M )L(1 − s, M v ), 1

where ε(s, M ) is the epsilon-factor (usually written as W (M )N (M ) 2 −s ; here W (M ), called the root number of M , is a complex number of absolute value 1; and N (M ) may be called the conductor of M ), and M v is the dual motive. Conjecture 1.2.1 (Deligne) Suppose s = 0 is critical for L(s, M ) then L(0, M ) ∼ c+ (M ), where ∼ means up to an element of Q. More generally, if m ∈ Z is a critical integer for L(s, M ) then ±

Lf (m, M ) ∼ (2πi)md c± (M ),

± = (−1)m .

Motives for ζ(s) The following description is taken from certain informal lecture notes of G¨ unter Harder. For n ≥ 1 define the motive Q(−n) := H 2n (Pn ).

(1.13)

We will work through the previous subsection for this motive: . Betti realization: Q(−n)B = HB2n (Pn (C), Q) = Q · 1B , (−1)n 1B .

F∞ (1B ) =

. Purity: Q(−n) is pure of weight w = 2n. . Hodge numbers: hn,n = 1 and all other hp,q = 0. − n . Middle Hodge numbers: h+ n = 1 and hn = 0, since  = (−1) . 2n . de Rham realization: Q(−n)dR = HdR (Pn /Q) = Q · 1dR .

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. Comparison: I∞ (1B ) = (2πi)−n 1dR . . Hodge filtration: F n Q(−n)dR = Q · 1dR , F n+1 Q(−n)dR = 0. . The ±-spaces: . When n is even: − . Q(−n)+ B = Q · 1B and Q(−n)B = 0. + . Q(−n)dR = Q · 1dR and Q(−n)− dR = 0.

. When n is odd: − . Q(−n)+ B = 0 and Q(−n)B = Q · 1B . − . Q(−n)+ dR = 0 and Q(−n)dR = Q · 1dR .

. The periods: . When n is even: c+ (Q(−n)) = δ(Q(−n)) = (2πi)−n , c− (Q(−n)) = 1. . When n is odd: c− (Q(−n)) = δ(Q(−n)) = (2πi)−n , c+ (Q(−n)) = 1. . `-adic realization: H 2n (Pn (Q), Q` ) = Q` (−n). . Galois action: the arithmetic Frobenius ϕp at p acts on Q` (−n) as: ϕp · x = pn x. Q . Finite part of the L-function: Lf (s, Q(−n)) = p (1 − pn p−s )−1 = ζ(s − n).
. Gamma factors at infinity: L∞ (s, Q(−n)) = π −(s−n)/2 Γ s−n . 2 . Q(−n) is critical: if and only if n is odd. (Recall that n ≥ 1.) . Deligne: If n is odd then L(0, Q(−n)) ∼ c+ (Q(−n)) translates to ζ(−n) ∼ 1, i.e., ζ(odd negative integer) ∈ Q. . Euler: Use the functional equation to get ζ(even positive integer 2m) ∈ π 2m Q. The Tate motive Q(1) Reference: Deligne [De79, Section 3.1]. One may say that Q(1) is the motive that is dual (in the category of Grothendieck motives) to Q(−1) = H 2 (P1 ). Or, the simplest way is to define it directly as Q(1) := H1 (Gm ). Here are some details about this motive:

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. Betti realization: Q(1)B = H1 (C∗ , Q) = Q·1B , F∞ (1B ) = −1B , d+ = 0, d− = 1. . Purity: Q(1) is pure of weight w = −2. . Hodge numbers: h−1,−1 = 1 and all other hp,q = 0. − . Middle Hodge numbers: h+ −1 = 1 and h−1 = 0, since  = −1. 1 . de Rham realization: Q(1)dR = Q · 1dR which is dual to HdR (Gm ) ' ∗ ∗ ∗ Q · 1dR where 1dR is the differential form dz/z on C .

. Comparison: I∞ (1B ) = (2πi)1dR . − . The ±-spaces: Q(1)+ B = 0 and Q(1)B = Q · 1B .

. The periods: c− (Q(1)) = δ(Q(1)) = (2πi), c+ (Q(1)) = 1. . `-adic realization: Q(1)` = Q` (1). . Galois action: the arithmetic Frobenius ϕp at p acts on Q` (1) as: ϕp · x = p−1 x. Q . Finite part of the L-function: Lf (s, Q(1)) = p (1 − p−1 p−s )−1 = ζ(s + 1). . Deligne: For example, s = 1 is critical for Lf (s, Q(1)). Conjecture 1.2.1 would give: −

ζ(2) = Lf (1, Q(1)) ∼ (2πi)d c− (Q(1)) = (2πi)2 ∼ π 2 , which is Euler’s classic ζ(2) = π 2 /6 up to rational multiplies.

The Tate motives Q(n) For n = 0, define Q(0) := H • (Point). For n ≥ 1, the motive Q(n) may be defined in one of these two ways: 1. Q(n) is the motive that is dual to Q(−n). 2. Q(n) = ⊗n Q(1), i.e., it is the n-fold tensor product of the Tate motive Q(1).

1.2.2 Tamagawa number conjecture of Bloch and Kato In this section, I only hope to convey a very general feeling for the Bloch–Kato conjecture. This is partly because defining all notation, and writing everything down, would be tantamount to repeating everything in their paper. I have followed the notation of Bloch and Kato [BK90] as much as possible.

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Given a motive, for example Q(n), a crucial ingredient in the Bloch– Kato conjecture is a Galois stable lattice M. (It is notationally unfortunate that Bloch and Kato used the letter M for a lattice, unlike in Deligne [De79] where M is always a motive; Bloch–Kato do not really need to talk about the motive per se, but they only need their data to come from a motive.) This lattice M will be used to define certain groups A(Qp ) and A(Q) in terms of Galois cohomology; these groups are to behave like the Qp -points and the Q-points of the intermediate Jacobian of a cohomology group. The A in the A(Q) or A(Qp ) is merely a symbolic notation. For a suitable Tamagawa measure, the volume of A(Qp ) is the local L-value The Tamagawa num Q at 0 almost everywhere.  ber Tam(M) := Vol ( p A(Qp ))/A(Q) is conjectured to be a rational number which is interpreted in terms of a Tate–Shaferevic group for M . This is the essence of the Bloch–Kato conjecture. Motivic pairs A motivic pair (see [BK90, Definition 5.5]) is a pair (V, D) of finitedimensional Q-vector spaces with extra structure: 1. (Galois representation) V ⊗Af has an Af -linear continuous GQ -action such that V ⊂ V ⊗ Af is Galois stable. 2. (Hodge filtration) D has a decreasing filtration (Di )i∈Z of Q-subspaces such that D i = D for i  0 and Di = 0 for i  0. 3. (Comparison isomorphisms) For finite p, there is an isomorphism of Qp -vector spaces ∼

θp : Dp −→ DR(Vp ) := H 0 (Qp , BdR ⊗ Vp ), where BdR is the field constructed by Fontaine; see [BK90, Section 1]. Similarly, one has a comparison map ∼

θ∞ : D∞ −→ (V∞ ⊗R C)+ . For the right hand side, σ ∈ Aut(C/R) acts as σ ⊗ σ on V∞ ⊗R C and the + means we have taken invariants under Aut(C/R). A motivic pair is assumed to satisfy the following axioms: 1. For almost all p, V` is unramified at p if ` 6= p and Vp is crystalline. 2. There exist Z-lattices M ,→ V and L ,→ D with several properties; an important one being that M ⊗ Zp is Galois stable in Vp and corresponds to L ⊗ Zp across the isomorphism θp .

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A. Raghuram

3. The local Artin factors are `-independent. 4. There is a Galois stable lattice T in V ⊗ Af such that the invariants under inertia subgroup at p in T ⊗ Q` /Z` are divisible for all `. One should mention that the isomorphism θ∞ , which is essentially the same as I∞ , comes from classical de Rham theorem that de Rham cohomology is isomorphic to singular cohomology. Its p-adic counterpart, namely the existence of θp for a motive of the form H • (X) was conjectured by Fontaine and proved by Faltings. L-functions and weights Given a motivic pair (V, D) as above one defines the partial motivic L-function as an Euler product Y LS (s, V ) := Lp (s, V ) p∈S /

where Lp (s, V ) := det(I − ϕp |V Ip p−s )−1 as in 1.2.1, while keeping the `

axiom 3 above in mind. Suppose that the polynomial det(I − ϕp |V Ip X) ` Qdim(V ) looks like i=1 (1 − αi X). We say that (V, D) has weights ≤ w if i + |αi | ≤ pw/2 for all p (including p ∈ S), and D∞ ∩ V∞ = 0. The Euler product converges for <(s) ≥ w/2 + 1. Hence, to talk about LS (0, V ), we need to assume w < −2. The Tamagawa number conjecture Define certain abelian groups: A(Qp ) :=

ˆ Hf1 (Qp , M ⊗ Z),

A(R) := (D∞ ⊗R C)/(D∞ ⊗R C + M )+ .
Here Hf1 (Qp , N ) := Ker H 1 (Qp , N ) → H 1 (Qp , Bcrys ⊗ N ) is called the finite part of the Galois cohomology group for the Galois module N = ˆ and Bcrys is a ring defined and studied by Fontaine. M ⊗Z ˆ as a finitely Now define the ‘global’ group A(Q) ⊂ Hf1 (Q, M ⊗ Z) ˆ = H 1 (Q, M ⊗ Z) ˆ and A(Q)⊗ generated abelian group such that A(Q)⊗ Z f 0 1 Q is a suitable Q-structure Φ on H (Q, V ⊗ Af ). Definition 1.2.2 (Tamagawa number of M ) Q  p A(Qp ) Tam(M) := Vol , A(Q)

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19

where the volume is with respect to the Tamagawa measure as in [BK90, Definition 5.9].

Definition 1.2.3 (Tate–Shafarevich group of M )   1 M H 1 (Qp , M ⊗ Q/Z) H (Q, M ⊗ Q/Z) . X(M ) := Ker  −→ A(Q) ⊗ Q/Z A(Qp ) ⊗ Q/Z p≤∞

We can now state the Tamagawa number conjecture of Bloch–Kato (see [BK90, Conjecture 5.15]). Conjecture 1.2.4 Assume that the motivic pair (V, D) indeed comes from a motive, and that we are given a suitable Q-structure as in the ˆ ⊂ V ⊗ Af definition of A(Q). Let M be a Z-lattice in V such that M ⊗ Z is Galois stable. Then X(M ) is finite and the Tamagawa number is given by |H 0 (Q, M ∗ ⊗ Q/Z(1))| , |X(M )|

Tam(M ) =

where M ∗ = Hom(M, Z) is the dual lattice. The local Tamagawa number Vol(A(Qp )) is L(0, Vp )−1 for almost all primes p; hence we may also write LS (0, V ) =

|X(M )| · Vol(A(R)/A(Q)) ⊗ Q/Z(1))|

|H 0 (Q, M ∗ Y

Vol(A(Qp )).

p∈S−{∞}

1.2.3 Results on the Tamagawa number conjecture for the motive Q(n) The purpose of these proceedings is to give a detailed proof of the Tamagawa number conjecture for the motive Q(n).

If n ≥ 2 is an even integer then the Tamagawa number conjecture is true up to powers of 2 for Q(n). This was proved by Bloch and Kato; see [BK90, Theorem 6.1, (i)].

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A. Raghuram

If n ≥ 2 is an odd integer then the Tamagawa number conjecture up to powers of 2 is true for Q(n) provided another conjecture holds. See Bloch–Kato [BK90, Theorem 6.1, (ii)]. This other conjecture (see [BK90, Conjecture 6.2]) concerns the equality of certain Deligne–Soul´e cyclotomic elements in a Galois cohomology group. The lectures by Blasius and Kings in this workshop will be devoted to proving [BK90, Conjecture 6.2].

References [Ap79] Ap´ery, R. 1979. Irrationalit´e de ζ(2) et ζ(3). Ast´erisque, 61, 11–13. [BR01] Ball, K., and Rivoal, T. 2001. Irrationalit´e d’une infinit´e de valeurs de la fonction zeta aux entiers impairs. (French) [Irrationality of infinitely many values of the zeta function at odd integers]. Invent. Math., 146, no. 1, 193–207. [BN02] Benois, D., and Nguyen Quang Do, T. 2002. Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs Q(m) sur un corps ab´elien. (French) [Local Tamagawa numbers and the Bloch– Kato conjecture for the motives Q(m) over an abelian field]. Ann. Sci. ´ Ecole Norm. Sup., 35, no. 5, 641–672. [Bl87] Blasius, D. 1987. Appendix to Orloff Critical values of certain tensor product L-functions. Invent. Math., (4) 90, no. 1, 181–188. [Bo74] Borel, A. 1974. Stable real cohomology of arithmetic groups. Ann. Sci. ´ Ecole Norm. Sup., (4) 7, 235–272. [Bo77] Borel, A. 1977. Cohomologie de SLn et valeurs de fonctions zeta aux points entiers (French). Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 4, no. 4, 613–636. (Errata: “Cohomologie de SLn et valeurs de fonctions zeta aux points entiers”. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 7 (1980), no. 2, 373.) [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I. Progr. Math., 86, 333–400. Birkh¨ auser Boston, Boston, MA. [Bu02] Burgos Gil, Jos´e I. 2002. The regulators of Beilinson and Borel. CRM Monograph Series, 15. American Mathematical Society, Providence, RI. [De79] Deligne, P. 1979. Valeurs de fonctions L et p´eriodes d’int´egrales (French). With an appendix by N. Koblitz and A. Ogus. Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, 313–346, Amer. Math. Soc., Providence, R. I. [Fl04] Flach, M. 2004. The equivariant Tamagawa number conjecture: a survey. With an appendix by C. Greither. Contemp. Math., 358, 79–125. Stark’s conjectures: recent work and new directions, Amer. Math. Soc., Providence, RI.

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[Fo82] Fontaine, J.-M. 1982. Sur Certains Types de Representations p-Adiques du Groupe de Galois d’un Corps Local; Construction d’un Anneau de Barsotti-Tate. Ann. of Maths., 2nd Series, 115, no. 3, 529–577. [Fo92] Fontaine, J.-M. 1992. Valeurs sp´eciales des fonctions L des motifs, Ast´erisque, 206, no. 4, 205–249, S´eminaire Bourbaki, Vol. 1991/92, exp. 751. [HK03] Huber, A., and Kings, G. 2003. Bloch–Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters. Duke Math. J., 119, no. 3, 393–464. [Iv85] Ivi´c, A. 1985. The Riemann zeta-function. The theory of the Riemann zeta-function with applications. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York. [KNF96] Kolster, M., Nguyen Quang Do, T., and Fleckinger, V. 1996. Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J., 84, no. 3, 679–717. [Li73] Lichtenbaum, S. 1973. Values of zeta-functions, ´etale cohomology, and algebraic K-theory. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lect. Notes in Math., 342, 489–501. Springer, Berlin. [Mi71] Milnor, J. 1971. Introduction to algebraic K-theory. Annals of Mathematics Studies, no. 72. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo. [Ne99] Neukirch, J. 1999. Algebraic number theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322, xviii-571. Springer-Verlag, Berlin. [Qu73] Quillen, D. 1973. Finite generation of the groups Ki of rings of algebraic integers. Lecture Notes in Math., 341, Springer-Verlag, 179–198. [Se69] Serre, J.-P. 1969–1970. Facteurs locaux des fonctions zeta des vari´et´es alg´ebriques (d´efinitions et conjectures). S´eminaire Delange-Pisot-Poitou. Th´eorie des nombres, 11, no. 2, exp. no. 19, 1–15. [We05] Weibel, C. 2005. Algebraic K-theory of rings of integers in local and global fields. Handbook of K-theory. 1-2, 139–190, Springer, Berlin. [We58] Weil, A. 1958–1960. Ad`eles et groupes alg´ebriques. (French) [Ad`eles and algebraic groups] S´eminaire Bourbaki. 5, exp. no. 186, 249–257.

2 K-theoretic Background R. Sujatha

Abstract The aim of these lectures is to cover the requisite background from Algebraic K-theory needed to prove the main result of Soul´e [So79, Th´eor`eme 5], which is dealt with in detail in [Li15]. This result of Soul´e which, along with an observation of Schneider, proves the vanishing of the ´etale cohomology group He´2t (Spec Z[1/l], Ql /Zl (m)), m ≥ 2, is equivalent to the finiteness of He´2t (Spec Z[1/l], Zl (m)) and plays a crucial role in the proof of the Tamagawa number conjecture due to [BK90]. In this article, l will denote an odd prime, all the rings considered will be commutative, associative, unital rings, unless otherwise mentioned, and all categories are small. We shall mainly consider Quillen’s K-theory referring the reader to [Q73a] for details. We would like to thank the referee for the comments which helped improve the exposition.

2.1 Quillen’s K-theory Let G be a topological group. Recall that the classifying space BG of G is the quotient of a weakly contractible space (i.e., all of whose homotopy groups are trivial) EG by a free action of G. This gives a principal Gbundle p : EG −→ BG which is universal in the sense that if X is of the homotopy type of a CW-complex, and P → X is a principal G-bundle, then there is a University of British Columbia, Vancouver, Canada. e-mail: [email protected]

K-theoretic Background

23

continuous map f : X → BG (defined up to homotopy) such that P occurs as the pullback f ∗ EG. The spaces BG and EG are defined up to homotopy equivalence, and are connected. The topology of G is crucial when considering the classifying space. If G is discrete, then BG is a K(π, 1)-space, so that π1 (BG) = G, πn (BG) = 0 for n ≥ 2, where πn (X) denotes the n-th homotopy group of a topological space X. Similarly, Hn (X, ∗) denotes the n-th homology group of the space X with coefficients ∗. Quillen defines the higher K-groups of a ring R using the ‘plus construction’ [Q71], whose main properties are encapsulated in the following theorem. Theorem 2.1.1 (Quillen) Let (X, x) be a path connected space, N a perfect normal subgroup of π1 (X, x). Then there exists a continuous map of pairs φ : (X, x) → (X + , x+ ) such that the following hold: a) There exists an exact sequence φ∗

0 → N → π1 (X, x) −→ π1 (X + , x+ ) → 0. b) For any local coefficient system L on X + , the induced homomorphism φ∗ : Hn (X, φ∗ L) → Hn (X + , L) is an isomorphism for any n ≥ 0. c) If g : (X, x) → (Y, y) is a continuous map such that N ⊂ Ker (g∗ : π1 (X, x) → π1 (Y, y)), then there exists a continuous map h : (X + , x+ ) → (Y, y), unique up to homotopy, such that the diagram (X, x) g

φ

/ (X + , x+ )

h

 y (Y, y) commutes. Let R be a ring and let GLn (R) denote the group of (n × n) matrices over R that are invertible. Each such matrix g is naturally viewed as an

24

R. Sujatha   g 0 (n + 1) × (n + 1) matrix given by and thus gives an embedding 0 1 of GLn (R) into GLn+1 (R). The group GL(R) is defined as the union [ GLn (R), n

and is called the infinite linear group over R. Consider it as a topological group with the discrete topology. Take X = BGL(R), the classifying space of GL(R). Then BGL(R) is a connected space with π1 (BGL(R)) ' GL(R), πi (BGL(R)) = 0 for i ≥ 2. These properties characterize BGL(R) up to homotopy equivalence. Let En (R) be the subgroup of (n × n) elementary matrices, and put E(R) = lim En (R). Then E(R) is a perfect −→ n

normal subgroup of GL(R) and the space X + as in Theorem 2.1.1 is denoted by BGL(R)+ . Definition 2.1.2 For i ≥ 1, the i-th Quillen K-group of R, denoted Ki (R), is defined as Ki (R) = πi (BGL(R)+ ). For i ≥ 1, these groups are clearly abelian, since higher homotopy groups are. That they are abelian for i = 0, 1 follows from the discussion below (see Defn. 2.2.5). The groups K0 (R) and K1 (R) are dealt with below in greater detail. We now discuss some of the features of Quillen K-theory. For i ≥ 0, it comes equipped with a product structure Ki (R) ⊗ Kj (R) → Ki+j (R), so that we in fact have a ring K∗ (R). Further, we have Ki (R) ⊗Z Q ⊆ Hi (BGL(R), Q).

2.2 Ki (R) for small i Let C be an abelian category. Recall that a non-empty full subcategory D of C is a Serre subcategory if given an exact sequence f

g

A0 → A → A00 in C, we have that A0 and A00 are objects in D if and only if A is an object in D. The group K0 (C) is the abelian group generated by isomorphism classes [A], where A runs over the objects of C, and subject to the relation [A] = [A0 ] + [A00 ],

K-theoretic Background

25

for each short exact sequence 0 → A0 → A → A00 → 0 in C. The group K0 (C) is universal with respect to this property in the following sense. Suppose we are given an abelian group G and a map f : Ob(C) → G, such that for each short exact sequence 0 → A0 → A → A00 → 0 in C, we have f (A) = f (A0 ) + f (A00 ) in G. Then there exists a unique homomorphism φ : K0 (C) → G making the diagram Ob(C)

φ

/ K0 (C)

φ

f

 z G

commutative. More generally, the group K0 can be defined analogously for any exact category, which we will not define rigorously (see [Q73a, §2]); intuitively, it is an additive category in which exact sequences are well defined. Then the map C → K0 (C) is a covariant functor from the category of exact categories and exact functors to the category of abelian groups. If, instead of exact sequences, we declare the relation [A] = [A0 ] ⊕ [A00 ] for any split, short exact sequence 0 → A0 → A → A00 → 0, then the corresponding group is denoted by K0 (C, +). The following proposition is well known [Sw68, II, 1.10], see also [We13, IV(1.1)]. Proposition 2.2.1 Let A, B be objects of C. If [A] = [B] in K0 (C, +), then there is an object C ∈ C such that A ⊕ C ' B ⊕ C.

Let R be a ring and CR be the abelian category of finitely generated (left) R-modules and let PR be the full subcategory of CR consisting of finitely generated projective left R-modules. The category PR is an exact

26

R. Sujatha

category, and the natural inclusion PR ⊂ CR induces a homomorphism K0 (PR ) → K0 (CR ). Definition 2.2.2 Let R be a ring. The group K0 (R) is defined to be the group K0 (PR ). Proposition 2.2.3 (see [Sw68, II.1]) Let R be a ring. Then the following assertions hold: i) The group K0 (PR ) is generated by [P ], where P is a finitely generated projective R-module with the relations [P ] = [Q] if P ' Q and [P ⊕ Q] = [P ] + [Q]. ii) Any element of K0 (R) can be written as [P ] − [Q] for finitely generated projective R-modules P and Q. Further, [P ] − [Q] = [P 0 ] − [Q0 ] in K0 (R) if and only if there is a finitely generated projective R-module M such that P ⊕ Q0 ⊕ M = P 0 ⊕ Q ⊕ M. A ring homomorphism η : R → R0 , along with the base change map P 7→ R0 ⊗R P, induces a group homomorphism K0 (R) → K0 (R0 ) as η induces an exact functor from the category PR to PR0 . Definition 2.2.4 Let f : R → R0 be a ring homomorphism. The category Cf consists of triplets (P, a, Q) as objects, where P and Q are finitely generated projective R-modules, and a is an isomorphism a

a : R0 ⊗R P −→ R0 ⊗R Q. Morphisms between two objects (P, a, Q) and (P 0 , a0 , Q0 ) consist of a pair of R-morphisms g : P → P 0 and h : Q → Q0 such that a0 ◦ (1R0 ⊗ g) = (1R0 ⊗ h) ◦ a. It is an isomorphism if both g and h are isomorphisms. A sequence of morphisms in Cf (see [Sw68, II.13], [We13, II(2.10)]) 0 → (P 0 , a0 , Q0 ) → (P, a, Q) → (P 00 , a00 , Q00 ) → 0

(2.1)

is a short exact sequence if the underlying sequences 0 → P 0 → P → P 00 → 0,

0 → Q0 → Q → Q00 → 0

are short exact. The group K0 (f ) (also called the relative K0 ) is an abelian group defined by generators [(P, a, Q)] with (P, a, Q) an object in the category Cf subject to the following relations:

K-theoretic Background

27

• [(P, a, Q)] = [(P 0 , a0 , Q0 )] + [(P 00 , a00 , Q00 )] for every short exact sequence as in (2.1). • [(P1 , b ◦ a, P3 )] = [(P1 , a, P2 )] + [(P2 , b, P3 )]. We now discuss the group K1 (R) for a ring R. There are various (equivalent) definitions though checking the equivalence involves several technicalities and is hence beyond the scope of these lectures. Recall the commutator subgroup [G, G] of a group G, which is always a normal subgroup of G and is universal with respect to the quotient group being abelian. The first definition of K1 (R) (also called the Whitehead group of R) is as the quotient of GL(R) by its commutator subgroup. Definition 2.2.5

K1 (R) is the abelian group GL(R)/[GL(R), GL(R)].

The abelian group K1 (R) thus has the universal property that every homomorphism from GL(R) to an abelian group must factor through K1 (R). A ring homomorphism R → R0 induces a homomorphism from K1 (R) to K1 (R0 ) and K1 is thus a covariant functor from the category of rings to Ab. Example: Since R is commutative, the determinant homomorphism is a group homomorphism from GL(R) onto the (abelian) group R× of units in R, and we denote the induced surjective map from K1 (R) by det, det : K1 (R) → R× .

The classical Whitehead’s lemma asserts that the group generated by elementary matrices E(R) is the commutator subgroup of GL(R) and there is an isomorphism K1 (R) ' GL(R)/E(R). The group K1 (R) measures the obstruction to taking an arbitrary invertible matrix over R and reducing it to the identity via a series of elementary operations. Let us now turn to an alternative definition of K1 (R) using projective modules. Let P be a finitely generated projective R-module. Choosing an isomorphism P ⊕ Q ' Rn gives a group homomorphism from Aut(P ) to GLn (R), by sending α to α ⊕ 1Q . This homomorphism is well

28

R. Sujatha

defined when viewed as a homomorphism into GL(R), up to inner automorphisms of GL(R), and thus there is a well-defined homomorphism Aut(P ) → K1 (R). Definition 2.2.6 For a ring R, the group K1 (R) is an abelian group which consists of generators [P, a] where P is a finitely generated projective R-module and a is in Aut(P ), subject to the relations • [P, a] = [Q, b] for f : P ' Q such that bf = f a. • [P, a ◦ b] = [P, a][P, b]. • [P ⊕ Q, a ⊕ b] = [P, a][Q, b]. The equivalence of this definition with the earlier one is seen using the well-defined homomorphism from Aut(P ) to K1 (R) as discussed above. There is a homological interpretation of K1 (R) given by K1 (R) = H1 (GL(R), Z),

(2.2)

the first group homology of GL(R) with Z-coefficients. A further topological interpretation elucidates K1 (R) as π1 (K(R)) where K(R) is a certain topological space built using the category PR .

2.3 K-theory of exact categories More generally, Quillen defines the K-groups for a (small) exact category C. We give a brief outline of this definition and refer the reader to [Q73a] for more details. One first defines a new (small) category QC, having the same objects as C but with morphisms between objects defined using the notion of admissibility (see [Q73a, §2]). There is also the important notion of the classifying space BC of C, which is a pointed topological space. These together enable one to define the classifying space BQC of the category QC. We also need the double Q construction Q2 C := QQC of Waldhausen, and the pairing BQC ∧BQC → BQ2 C, which is technical, and is given in [Wa78, IV(6.6)]. Definition 2.3.1 The K-groups Ki (C), for i ≥ 0, are defined as Ki (C) = πi+1 (BQC). We are being sloppy here by neglecting the base point, see [Sr93, Chap. 4] for details. When R is a ring, then, working with the category PR , it can be proved that the earlier definition of Ki (R) coincides with the definition above, viz., πi+1 (BQPR ) (see [Sr93, Chap. 7]. We shall simply

K-theoretic Background

29

write BQR for BQPR . A deep theorem of Quillen proves that there are natural isomorphisms for i ≥ 1, πi+1 (BQR) ' πi (BGL(R)+ ),

(2.3)

and hence we have isomorphisms for i ≥ 0 Ki (R) ' πi+1 (BQR) ' πi (BGL(R)+ ). We shall also need the important localization sequence in K-theory (see [Q73a, Theorem 5], [We13, Chap. V], [Gr68, §7]). Let B be a Serre subcategory of an abelian category A and consider the canonical functors i : B → A and η : A → A/B, where A/B is the quotient category. Then there is a long exact sequence η∗

δ

i

∗ · · · → Ki (A/B) → Ki−1 (B) → Ki−1 (A) → Ki−1 (A/B) → · · · .

(2.4)

As a special case of this, consider a commutative ring R and let S be a multiplicatively closed set whose elements are non-zero divisors of R. S Consider the abelian subcategory CR of CR , which consists of finitely generated R-modules that are S-torsion. This is a Serre subcategory and the quotient category is equivalent to CS −1 R , the category of finitely generated modules over the ring S −1 R. The category PR acts on the above categories compatibly and this makes the corresponding long exact sequence in K-theory an exact sequence of K∗ (R)-modules (see [We13, V (2.6.3)]); the full details of such an action require products in Quillen’s K-theory, (see [Gr68, §7]). A special example is that of a discrete valuation ring R with quotient field F and finite residue field k. In this case, the long exact localization sequence actually breaks up into short exact S sequences by ‘devissage’, we have isomorphisms Kn (k) ' Kn (CR ) (see [Ge73], [We13, V (6.9.2)]) 0 → Kn (R) → Kn (F ) → Kn−1 (k) → 0.

(2.5)

Another interesting example is when R = k[t], the polynomial ring over a field k. In this case we get short exact sequences that are split [We13, V (6.7.1)] 0 → Kn (k) → Kn (k(t)) → ⊕Kn−1 (kv ) → 0,

(2.6)

v

where the last direct sum is taken over the residue fields corresponding to the discrete valuations given by the closed points of the affine line Spec(k[t]). For a general Dedekind domain with quotient field F a global

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R. Sujatha

field, we get short exact sequences (due to Soul´e)[We13, V(6.8)] 0 → Kn (R) → Kn (F ) → ⊕Kn−1 (kv ) → 0

(2.7)

v

where the direct sum is taken over height one prime ideals Pv of R and kv is the residue field R/Pv . The conjectural existence of such exact sequences for functors goes under the rubric of Gersten’s conjecture. We shall also need the norm or transfer maps defined in K-theory, which we recall below. Definition 2.3.2 Let R be a ring and R0 a ring containing R such that R0 is finitely generated and projective as an R-module. Let PR0 and PR denote the categories of finitely generated projective modules over R0 and R respectively. Then the natural forgetful functor PR0 → PR induces a homomorphism Kn (R0 ) → Kn (R), which is called the norm or transfer homomorphism.

2.4 K-theory with finite coefficients Let q be a positive integer. In this section, for a ring R (not necessarily commutative), we define the K-groups Ki (R; Z/q) with finite coefficients and relate them to the K-groups K∗ (R). These K-groups were first introduced by Browder (see [Br78]) and Karoubi. We first recall some notions from homotopy theory. A good reference for this is [Sel97]. Let π : E → B be a continuous map between topological spaces, and I be the unit interval [0, 1]. The map π is said to have the homotopy lifting property with respect to a space Y if given a homotopy H : Y × I → B and a lift f 0 : Y → E of H0 = H|Y ×0 , there 0 exists a lift H 0 : Y × I → E of H such that H00 = f 0 , where H00 = H|Y ×0 . A surjective map π : E → B is called a (Hurewicz) fibration if it has the homotopy lifting property with respect to Y for all spaces Y . The space B is called the base space of the fibration, E the total space and the inverse image π −1 (b) of a point b ∈ B is called the fibre of π over b. When we are dealing with pointed spaces and maps, the fibre over the base point of B is called the fibre of π. Given a fibration π : E → B π of pointed spaces with fibre F , we shall use the phrase ‘F → E → B is a fibration sequence’. The usual projection map F × B → B is called the ‘trivial fibration’. A sequence W → X → Y will be called a homotopy fibration if it is equivalent to a fibration sequence up to homotopy.

K-theoretic Background

31

Explicitly, this means that there is a commutative diagram W −−−−→   y

X −−−−→   y

Y   y

F −−−−→ E −−−−→ B in which the vertical maps are homotopy equivalences with the bottom row being a fibration sequence. Given a fibration sequence as above, there is a long exact sequence · · · → πk+1 (B, b) → πk (F, x) → πk (E, x) → πk (B, b) → · · · .

(2.8)

The dual notion is that of a cofibration. An inclusion i : A ,→ X is said to have the homotopy extension property with respect to a space Y if given a homotopy H : A × I → Y and an extension f 0 : X → Y of H0 , there exists an extension H 0 : X × I → Y of H such that H00 = f 0 . An inclusion i : A ,→ X is called a cofibration if it has the homotopy extension property with respect to Y for all Y . If A is a subspace of X, we write X/A for the quotient space X/ ∼, where a ∼ a0 for all a, a0 ∈ A. If i : A ,→ X is a cofibration, then X/A is called the cofibre i of i and we write ‘A → X → C is a cofibration’, to indicate that C is the cofibre of i. Similar to (2.8), a cofibration sequence A → X → C of pointed spaces gives rise to a long exact sequence · · · → [S n C, V ] → [S n X, V ] → [S n A, V ] → [S n−1 C, V ] → · · · ,

(2.9)

where for a space X, S n X denotes the n-fold suspension of X and [X, Y ] denotes the set of based homotopy classes of based maps from the pointed space X to the pointed space Y . Note that [X, Y ] has no canonical group structure but the set [S n X, Y ] does have a group structure for n > 0. For any integer n ≥ 2, define Yqn by Yqn = S n−1

[

en ,

(2.10)

×q

where S n−1 is the unit (n−1)-sphere, en is the n-unit disc and the glueing is done by the map of degree q on the sphere. We shall abbreviate Yqn to Y n in what follows. There is a homotopic cofibration (see [Br78]) ×q

S n−1 −→ S n−1 → Y n .

(2.11)

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Definition 2.4.1 The homotopy groups of a pointed space X with coefficients in Z/q are defined by πn (X; Z/q) = [Y n , X], n ≥ 2 where we remark that the set [Y n , X] has a canonical group structure. Definition 2.4.2 The K-groups of a ring R with coefficients in Z/q, denoted Ki (R; Z/q), for i ≥ 2 are defined as Ki (R; Z/q) = πi (BGL(R)+ ; Z/q) = [Y i , BGL(R)+ ]. For i = 0, define K0 (R, Z/q) = K0 (R)/qK0 (R), and for i = 1, K1 (R; Z/q) = π2 (BQR; Z/q). Note that, in analogy with (2.3), we have πi+1 (BQR; Z/q) = πi (BGL(R)+ ; Z/q) = Ki (R; Z/q) if i ≥ 2. Further, the cofibration sequence (2.11) clearly gives a long exact sequence ×q

×q

· · · → Ki (R) −→ Ki (R) → Ki (R; Z/q) → Ki−1 (R) −→ · · · K0 (R; Z/q) → 0. In particular, if K0 (R) is without torsion, then we recover K1 (R; Z/q) = K1 (R)/qK1 (R). Recall that if (X, x) and (Y, y) are pointed topological spaces, then the wedge product and smash product denoted respectively by X ∨ Y and X ∧ Y are defined by X ∨Y X ∧Y

= X ×y∪x×Y ⊆X ×Y = X × Y /X ∨ Y.

There is a natural pairing BQR∧BQR → BQ2 PR , see [We13, IV (6.6.2)], that induces a Z/q-algebra structure on K∗ (R; Z/q) which is a graded anti-commutative structure for odd integers q. The product structure can also be deduced via a pairing BGL(R)+ ∧ BGL(R)+ → BGL(R)+ which is defined up to weak homotopy equivalence. We refer the reader to [Br78, §1], [Lo76] for more details. Let B be an abelian category, A be a Serre subcategory and A/B be the corresponding quotient category. There is also a long exact sequence

K-theoretic Background

33

for the K-theory groups with coefficients (see [We13, V (5.2)]) analogous to the localization sequence in K-theory (2.4), → Kn (A; Z/q) → Kn (B; Z/q) → Kn (A/B; Z/q) → Kn−1 (A; Z/q) → · · · . If R is a regular Noetherian ring and H(R) is the (abelian, since R is regular) category of finitely generated R-modules with finite projective dimension, and S is a central, multiplicatively closed set of non-zero divisors in R, we may take A = H(R) and B to be the Serre subcategory HS (R) of S-torsion modules in H(R). The quotient category A/B is identified with finitely generated S −1 R-modules with finite projective dimension. The natural action of the category PR then gives a long exact sequence of K∗ R-modules (cf. [We13, Chap. V, §6]) → Kn (HS (R); Z/q) → Kn (H(R); Z/q) → Kn (H(S −1 R); Z/q) → Kn−1 (HS (R); Z/q) · · · . There are also analogues of (2.5) and (2.6) for the K-theory groups with Z/q-coefficients. For a field F with a discrete valuation v, with completion Fv and residue field kv , there are naturally defined residue maps δ¯ : Kn (F ; Z/q) → Kn−1 (kv ; Z/q), ∂v : H n (F, Z/q) → H n−1 (kv , Z/q).

2.5 Hurewicz homomorphisms Let (X, x) be a pointed topological space. The exact sequence q

0 → Z → Z → Z/q → 0 gives a long exact (Bockstein) sequence ×q

×q

· · · → πn (X) −→ πn (X) → πn (X; Z/q) → πn−1 (X) −→ of homotopy groups and ×q

×q

· · · → Hn (X) −→ Hn (X) → Hn (X, Z/q) → Hn−1 (X) −→ of homology groups, where Hi (X) denotes the i-th homology group of X with Z-coefficients. For every element α in πn (X; Z/q) = [Y n , X] (see (2.4.1)) let α∗ denote the induced maps on the homology groups Hi (Y n ) → Hi (X). Recall the classical Hurewicz homomorphisms hn :

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R. Sujatha

πn (X; Z/q) → Hn (X, Z/q) (n ≥ 2, and q odd) are defined by hn (f ) = f∗ (n ) where n is a chosen generator of the homology group Hn (S n ). There is a natural commutative diagram (for n ≥ 2), ×q

−−−−→ πn (X) −−−−→ πn (X) −−−−→ πn (X; Z/q) −−−−→ πn−1 (X)        q  yh yh yh yh ×q

−−−−→ Hn (X) −−−−→ Hn (X) −−−−→ Hn (X, Z/q) −−−−→ Hn−1 (X). It is easily seen that for Y n , the cohomology groups Hi (Y n , Z/q) equal Z/q if i = 0, n − 1, n and is zero otherwise. Let nq be the generator of Hn (Y n , Z/q) whose image in Hn−1 (Y n , Z) under the Bockstein homomorphism is the class of Sn−1 ⊂ Y n . Definition 2.5.1

The Hurewicz map with Z/q-coefficients, hq : πn (X; Z/q) → Hn (X, Z/q),

is defined by the formula hq (α) = α∗ (nq ). It is a homomorphism except when n = q = 2. If q is odd, then (see [So79, Lemme 2]) any pairing X ∧Y → Z induces a commutative diagram πm (X; Z/q) ⊗ πn (Y ; Z/q) −−−−→ πm+n (Z; Z/q)     hq ×hq y hq y Hm (X, Z/q) ⊗ Hn (Y, Z/q) −−−−→ Hm+n (Z, Z/q). If ∆ : X → X ∧ X is the diagonal morphism, with ∆∗ : Hn (X, Z/q) → Hn (X ∧ X, Z/q) the induced morphism on homology, and if n ≥ 2, then ∆∗ ◦ hq = 0 (see [So79, Lemme 3]).

´ 2.6 Etale cohomology and equivariant cohomology For details on ´etale sheaves and ´etale cohomology groups on a scheme X, the reader is referred to [Mi80]. We shall briefly summarize some of the results that will be needed. Let A be a commutative domain in which the prime number l is invertible. We shall consider two important ´etale sheaves, viz. Gm , the ´etale multiplicative group sheaf and the sheaf µl of l-th roots of unity. Given an ´etale sheaf F on Spec A, the ´etale

K-theoretic Background

35

cohomology groups H n (Spec A, F) (n ≥ 0) will be denoted simply by H n (A, F ). If A is a field F , an ´etale sheaf is a continuous GF -module for the Galois group GF = Gal(Fs /F ), where Fs denotes a separable closure of F and the ´etale cohomology groups are simply the Galois cohomology groups. For the sheaf Gm on Spec F , the corresponding Galois module is Gm = Fs× and we have H 0 (F, Gm ) ' F × , H 1 (F, Gm ) = 0 and H 2 (F, Gm ) ' Br(F ),

(2.12)

where for a commutative ring A, Br(A) is the Brauer group of A consisting of equivalence classes of central simple algebras. Let q = ln be invertible on A. The ´etale sheaf µ⊗i q denotes the i-th Tate twist of the ´etale sheaf µq ; the underlying object is the same as µq but the Galois group acts via the i-th cyclotomic character. The long exact sequence in ´etale cohomology associated to the Kummer exact sequence of ´etale sheaves ×q

1 → µq → Gm −→ Gm → 1,

(2.13)

along with (2.12), gives isomorphisms H 0 (F, µq ) ' µq (F ), q H 1 (F, µq ) ' F × /F ×

(the q th roots of unity in F ) and H 2 (F, µq ) ' q Br(F )

(2.14)

where q Br(F ) denotes the q-torsion subgroup of the Brauer group of F . If the group of units A× in A contains a non-zero element of order q, then the sheaf µ⊗i q is isomorphic to the constant sheaf Z/q. There is a cup-product j ⊗n i+j H i (A, µ⊗m (A, µ⊗m+n ) q ) ⊗ H (A, µq ) → H q

which is an isomorphism for i = 0 if µq ⊂ A× . If A is a Dedekind domain with quotient field F and j : A ,→ F is the canonical inclusion, then the natural morphism of ´etale sheaves ⊗i (µ⊗i q )A → j∗ ((µq )F )

is an isomorphism. For a field F with a discrete valuation v, with completion Fv and residue field kv , there are naturally defined residue maps (see [So79, III.3]), k−1 ∂v : H k (F, µ⊗i (kv , µ⊗(i−1) ). q )→ H q

If A is a Dedekind domain with quotient field F , then we have a long

36

R. Sujatha

exact localization sequence in ´etale cohomology [So79, Proposition 1] δ

⊗(i−1)

1 ⊗i 0 0 → H 1 (A, µ⊗i q ) → H (F, µq ) → ⊕H (kv , µq v

) → H 2 (A, µ⊗i q )···

⊗(i−1)

k ⊗i k−1 → H k (A, µ⊗i (kv , µq q ) → H (F, µq ) → ⊕H v

) → H k+1 (A, µ⊗i q )

··· , where the direct sum is taken over the discrete valuations v of F associated to height one prime ideals of A, kv denotes the residue field at v, and the connecting maps are obtained from the residue maps. It is well known that the residue map lands in the direct sum; this follows from an analysis of the definition of the residue map and the fact that given a non-zero element x in F , there are only finitely many valuations v such that v(x) is non-zero. Let B be a finite ´etale extension of a ring A [Mi80] and f be the injection of A into B. There is an induced morphism k ⊗i f ∗ : H k (Spec A, µ⊗i q ) → H (Spec B, µq )

and also an associated transfer morphism (see [So79, III.1.5]) k ⊗i f∗ : H k (Spec B, µ⊗i q ) → H (Spec A, µq )

which satisfies the projection formula f∗ (x ∪ f ∗ (y)) = f∗ (x) ∪ y. A − →   y

B − → A/B     y y

A0 − → B0 − → A0 /B0 induces the following commutative diagrams of the corresponding localization sequences in K-theory and ´etale cohomology:

δ

−−−→ Kn (B; Z/q) −−−→ Kn (L; Z/q) −−−→ ⊕Kn−1 (lw ; Z/q) −−−→ w       y y y δ

−−−→ Kn (A; Z/q) −−−→ Kn (F ; Z/q) −−−→ ⊕Kn−1 (kv ; Z/q) −−−→ , v

and

K-theoretic Background

37

∂

⊗(i−1)

k ⊗i k−1 −−−−→ H k (B, µ⊗i (lw , µq q ) −−−−→ H (L, µq ) −−−−→ ⊕H w       y y y ∂

⊗(i−1)

k ⊗i k−1 −−−−→ H k (A, µ⊗i (kv , µq q ) −−−−→ H (F, µq ) −−−−→ ⊕H

)

)

w

Here w varies over the discrete valuations of L (as before, associated to height one prime ideals of the corresponding Dedekind domains) and the vertical arrows are given by the transfer maps. In particular, for a number field F with ring of integers A, and for a field k with characteristic different from l, we obtain exact sequences (see [So79, III.2]) ∂

k ⊗i k−1 → H k (A, µ⊗i (kv , µ⊗i−1 ) q ) → H (F, µq ) → ⊕ H q w

∂

n ⊗i n−1 0 → H n (k, µ⊗i (kw , µ⊗i−1 ) → 0, q ) → H (k(T ), µq ) → ⊕ H q w

where the maps ∂ are given by the sum of the residue maps defined earlier. The surjectivity of the residue map in the first sequence is known to be true in certain cases (see [So79, III.2, III.3]). Note that if k > 2, the groups H k−1 (kv , µ⊗i−1 ) are zero for finite fields. Finally we mention q that the morphisms of sheaves µ⊗j → µ⊗j ln , n ≥ 1, ln+1 allow us to define H k (A, Zl (j)) := lim H k (A, µ⊗j ln ). n ←−

We now briefly recall the equivariant cohomology groups. Let X be a topological space and G a sheaf of groups on X. Denote by BG the classifying topos of G, which is identified with the category of sheaves of sets on X with an action of G (see [Gr68, Defn. 1.9]). The category of abelian group objects in BG is equivalent to the category of Z[G]-modules where Z[G] is the sheaf of integral group rings associated to Z[G]. If F is an abelian object of BG , then the G-equivariant cohomology groups of F are defined as H i (X, G, F ) = Ri (Γ(X, F G )) ' Ext iZ[G] (Z, F ), where Z is the constant sheaf with trivial G-action associated to Z. These cohomology groups can be computed using Z[G]-injective resolutions of F. Equivariant cohomology groups will play a key role in the theory of Chern classes to be discussed below.

38

R. Sujatha

2.7 K-groups of rings of integers in number fields The K-theory of the rings of integers of number fields is an important topic which has deep connections with arithmetic. We refer the reader to [Ko04] for a broad survey of this topic. Let F be a number field and denote the ring of integers in F by OF . We have K0 (OF ) ' Z ⊕ Cl(F ), K1 (OF ) ' OF× , where Cl(F ) is the Class group of F . Quillen proved that the higher K-groups are finitely generated [Q73b] and Borel [Bo77] showed that for n ≥ 2, the K-groups K2n−1 (OF ) are of the form K2n−1 (OF ) ' Zkn ⊕ H where H is a finite group and    r1 + r2 − 1 if n = 1 k n = r1 + r2 if n ≥ 3 is odd   r if n ≥ 2 is even. 2 We remark that kn is precisely the order of vanishing of the Riemann zeta function ζF (s) at s = 1 − n, n ≥ 2. Borel defined higher regulator maps kn ρB n (F ) : K2n−1 (OF ) → R

and showed that the kernel is finite and the image is a lattice of rank kn . The covolume of this lattice is the Borel regulator which is related to special zeta values. Soul´e showed that the inclusion map OF ,→ F in fact induces isomorphisms K2n−1 (OF ) ' K2n−1 (F ) for all integers n ≥ 2. This follows from (2.7) and the vanishing of K2n (kv ). In the simplest case when F = Q, it is also known that K4k+1 (Z) is Z for odd k, and is isomorphic to Z ⊕ Z/2 for positive even k. Let He´∗t (Z[1/l], Zl (n)) denote the ´etale cohomology groups of the ring Z[1/l]. The exact localization sequence gives a long exact sequence ⊗n ⊗n−1 1 0 0 → He´1t (Z[1/l], µ⊗n ) lm ) → He´t (Q, µlm ) → ⊕ He´t (Fp , µlm p6=l

→

He´2t (Z[1/l], µ⊗n lm )

→ ··· .

K-theoretic Background

39

Passing to the projective limit and using the fact that He´0t (Fp , Zl (n − 1)) = 0 for n 6= 1, we obtain isomorphisms He´1t (Z[1/l], Zl (n)) ' He´1t (Q, Zl (n)), n 6= 1.

(2.15)

2.8 Chern classes ´ a la Grothendieck In this section, we shall briefly discuss the theory of equivariant l-adic Chern classes as developed by Grothendieck in [Gr58], [Gr68]. We shall largely follow Soul´e’s exposition in [So79], see also [Gr68]. The basic property that we shall need of the Chern class maps are that they connect K-theory to ´etale cohomology and further commute with the relevant boundary maps as well as the residue maps of the localization sequences in K-theory and ´etale cohomology. Let l be an odd prime number and P be a projective module of finite type over A of bounded rank r over every residue field of A. Suppose that we are given a representation ρ : G → Aut(P ). This defines a locally free sheaf on Spec A with a G-action. Grothendieck associates Chern classes to such a fibre bundle, which are denoted ci (ρ) ∈ H 2i (Spec A, G; µ⊗i lv ), 0 ≤ i, v ≥ 1. They are zero for i ≥ r, and the groups are the ´etale equivariant cohomology groups where G acts trivially on Spec A. We have c0 (ρ) = 1 and, under the canonical projections µ⊗i → lv+1 ⊗i µlv , the Chern classes form a projective system. The total Chern class, denoted c(ρ), is defined by c(ρ) = 1 + c1 (ρ) + c2 (ρ) + · · · + cr (ρ). It has the following properties: 1. (Functoriality): If f : A → A0 , G → G0 and P → P 0 are compatible morphisms, then ci (f ∗ (ρ)) = f ∗ (ci (ρ)). 2. (Normalisation): Let det(ρ) be the determinant bundle of rank one viewed as an element in H 1 (Spec A, G; Gm ). Then c1 (ρ) = βv (det(ρ)), where βv is the connecting Bockstein map H 1 (Spec A, G; Gm ) → H 2 (Spec A, G; µlv )) associated to the Kummer sequence (2.13). 3. (Additivity): If 0 → P 0 → P → P 00 → 0 is an exact sequence of G-modules which are projective as A-modules and ρ0 , ρ , ρ00 are the corresponding representations, then c(ρ) = c(ρ0 ) ∪ c(ρ00 )

40

R. Sujatha where ∪ denotes the cup-product in equivariant cohomology.

4. (Multiplicativity): Let Qi be the universal polynomial with integral coefficients associated to the Chern classes. If ρ and ρ0 are two representations as above, then we have ci (ρ ⊗ ρ0 ) = Qi (c1 (ρ), c2 (ρ), · · · ; c1 (ρ0 ), c2 (ρ0 ), · · · ). 5. Identity: If idn denotes the natural representation of GLn (A) on An , m we get classes ci (idn ) ∈ H 2i (Spec A, GLn (A); µ⊗i q ), with q = l , l a prime. These classes stabilize in the sense that under the natural inclusion in of GLn (A) (resp. An ) into GLn+1 (A) (resp. An+1 ), we have i∗n (c(idn+1 )) = c(i∗n (idn+1 )) = c(idn ⊕ 1) = c(idn ) ∪ c(1) = c(idn ), (2.16) where 1 denotes the trivial representation of GLn (A) on A. These Chern classes can be used to obtain maps on the homology groups of G. More precisely, if G is a group that operates trivially on Spec A, then there exists a natural morphism [So79, Lemma 1] Φ

2i

k ⊗i H 2i (SpecA, G; µ⊗i ln ) −→ ⊕ Hom (H2i−k (G; Z/q), H (SpecA; µq )). k=0

Thus to the Chern classes ci (ρ), one may associate homomorphisms ci,k (ρ) : H2i−k (G; Z/q) → H k (SpecA; µ⊗i q ) defined by t 7→ Φ(ci (ρ)(t)). In particular, for the natural representation of GLn (A) above, taking limits over n, which is possible by the stability property (2.16). One obtains the morphism ci,k (id) : H2i−k (GL(A); Z/q) −→ H k (Spec A, µ⊗i q ), 0 ≤ k ≤ 2i. (2.17) These maps are used to construct Chern class maps on the K-theory groups which take values in the ´etale cohomology groups. Fix the ring A and integers i , k ≥ 0, q = lv . We define morphisms c¯i,k : K2i−k (A; Z/q) → H k (SpecA; µ⊗i q ),

(2.18)

as follows: c¯i,k

• ci,k is the composite K2i−k (A) → K2i−k (A; Z/q) −→ H k (SpecA, µ⊗i q ).

K-theoretic Background

41

• If 2i − k ≥ 2, then c¯i,k = ci,k (id) ◦ hq , where ci,k (id) is as in (2.17) and hq is the Hurewicz homomorphism. hq

K2i−k (A; Z/q) = π2i−k (BGL(A)+ ; Z/q) −→ H2i−k (GL(A); Z/q) ci,k (id)

−→ H k (Spec A, µ⊗i q ).

• If 2i = k, then ci,2i is given by the Chern classes of the trivial group, and it factors through K0 (A; Z/q) = K0 (A)/qK0 (A). • If 2i − k = 1, then the definition is rather ad hoc and we only consider the special case of c¯1,1 when A is a Dedekind domain. In this case, if F is the quotient field, we then have a commutative diagram where the exact rows are consequences of the localization sequences, K1 (A; Z/q)  c¯ y 1,1

−−−−→ K1 (F ; Z/q) −−−−→ ⊕Z/q v   '  y y

0 −−−−→ H 1 (SpecA, µq ) −−−−→ H 1 (F, µq ) −−−−→ ⊕Z/q. v

The centre vertical map c¯1,1 is an isomorphism on Spec F , both groups q being isomorphic to F × /F × . The first vertical arrow is then defined to be the obvious map arising from this commutative diagram. We mention that in all cases, ci,k is always defined on K1 (A) using ci,k (id) and the morphisms (2.2) K1 (A) = H1 (GL(A); Z) → H1 (GL(A); Z/q). When K0 (A) has no q-torsion, this permits us to define c¯i,k by reduction modulo q. • The Chern class maps commute with transfers and the residue maps in K-theory and Galois cohomology. The following theorem [So79, Th´eor`eme 1] tells us how we can bootstrap the Chern classes. Theorem 2.8.1 i) Let a ∈ Km (A; Z/q) and b ∈ Kn (A; Z/q), with m ≥ 2 (resp. n ≥ 2) where a ∈ K1 (A)/qK1 (A) when m = 1 (resp. b ∈ K1 (A)/qK1 (A)), when n = 1). Let a.b ∈ Km+n (A; Z/q) be the product of a and b. Then we have c¯i,k (a.b) = Σ

−(i − 1)! c¯i0 ,k0 (a) ∪ c¯i00 ,k00 (b). (i0 − 1)!(i00 − 1)!

In this formula, the sum is taken over all integers i0 , i00 , k 0 , k 00 verifying

42

R. Sujatha

i = i0 + i00 , k = k 0 + k 00 , 2i0 − k 0 = m, 2i00 − k 00 = n. The element ci0 ,k0 (a) ∪ ci00 ,k00 (b) is obtained by the cup-product 0

0

00

00

k H k (Spec A, µ⊗i (Spec A, µ⊗i ) → H k (SpecA, µ⊗i q )×H q q ).

ii) If a ∈ K0 (A; Z/q) and b ∈ Km (A; Z/q), with m ≥ 2, or b ∈ K1 (A)/qK1 (A), under the condition that ci,2i (a) = 0 for i ≥ 2, X 00 −(i − 1)! c¯i,k (a.b) = (ci1,2 )(a) ∪ c¯i0 ,k0 (b), 0 − 1)!(i00 − 1)! (i 0 00 0 i ,i ,k

00

00

00

with i0 + i00 = i, and k 0 + 2k 00 = k, and ci1,2 (a) ∈ H 2i (Spec A, µ⊗i ) q being the i00 -th power of c1,2 (a). Soul´e proved [So79, Th´eor`eme 4] that the Chern class map c¯i,k : K2i−k (A; Z/q) → H k (Spec A[1/l], µq ) is surjective for i < l and k ≤ 2, and under certain (mild) additional hypotheses, which certainly hold if A is the ring of integers of a number field, except possibly for k = 1. By [DF85], it is also an isomorphism for k = 1. In particular, it can be deduced that the map c¯1,0 : K2 (A; Z/q) → H 0 (Spec A, µq ) is surjective for Spec A, when A is the ring of integers of a number field.

References [Ba68] Bass, H. 1968. Algebraic K-theory, W.A. Benjamin Inc., New YorkAmsterdam. [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, vol. 1. Progress in Math., 86, 333-400. Birkh¨ auser, Boston, MA. [Bo77] Borel, A. 1977. Stable real cohomology of arithmetic groups. Ann. Sci. ´ Ecole Norm. Sup., 7, 613–636. [Br78] Browder, W. 1978. Algebraic K-theory with coefficients Z/p, in “Geometric applications of Homotopy theory I”. Lecture Notes in Math., 657, 40–84. Berlin-Heidelberg-New York. [Co15] Coates, J. 2015. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 45–64. [DF85] Dwyer, W. G., and Friedlander, E. M. 1985. Algebraic and ´etale Ktheory. Trans. AMS., 292, 247–280.

K-theoretic Background

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[Ge73] Gersten, S. M. 1973. Some exact sequences in the higher K-theory of rings, in Higher K-theories. Lecture Notes in Math., 341, 211–243. [Gi81] Gillet, H. 1981. Riemann-Roch Theorems for Higher Algebraic Ktheory. Advances in Math. 40, 203–281. [Gr76] Grayson, D. 1976. Higher Algebraic K-theory II (After Daniel Quillen), in Algebraic K-theory (Proc. Conf. Northwestern Univ., Evanston III). Lecture Notes in Math., 551, 217–240. [Gr58] Grothendieck, A. 1958. La th´eorie des classes de Chern. Bull. Soc. Math. France, 86, 137–154. [Gr68] Grothendieck, A. 1968. Classes de Chern et r´epresentations lineaires des groupes discrets, (French), in Dix Expos´es sur la Cohomologies des Sch´emas, North-Holland, Amsterdam; Masson, Paris, 215–305. [Ka93] Kahn, B. 1993. On the Lichtenbaum-Quillen conjecture, Algebraic Ktheory and algebraic topology (Lake Louise, AB, 1991). NATO Adv. Sci. Inst. Ser. C Math.Phys.Sci., 407, 147–166. Kluwer Acad Pub., Dordrecht. [Ko04] Kolster, M. 2004. K-theory and arithmetic. Contemporary developments in algebraic K-theory, ICTP Lecture Notes XV, Trieste, 191–258. [Ko15] Kolster, M. 2015. The Norm residue homomorphism and the QuillenLichtenbaum conjecture, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 97–120. [Li15] Lichtenbaum, S. 2015. Soul´e’s theorem, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 130–139. [Lo76] Loday, J.-L. 1976. K-th´eorie et repr´esentations de groupes. Ann. Sci´ Norm. Sup. , 4 i´eme s´erie, 9, 309–377. ent. Ec. ´ [Mi80] Milne, J. S. 1980. Etale cohomology. Princeton Math. Series, 33, Princeton University Press, Princeton, New Jersey. [Ng15] Nguyen Quang Do, T. 2015. On the determinantal approach to the Tamagawa Number Conjecture, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 154–193. [Q71] Quillen, D. 1971. Cohomology of Groups. ICM Proceedings, Nice 1970, Gauthier Villars, Vol. II, 47-51. [Q73a] Quillen, D. 1973. Higher Algebraic K-theory I, in Higher K-theories, Proc. Conf. Battelle Memorial Inst., Seattle, Was., (1972), Lecture Notes in Math., 341, 85–147. Springer-Verlag, Berlin-New York. [Q73b] Quillen, D. 1973. Finite generation of the groups Ki of rings of algebraic integers, in Higher K-theories. Proc. Conf. Battelle Memorial Inst., Seattle, Was. (1972), Lecture Notes in Math., 341, 179–198, SpringerVerlag, Berlin-New York. [Ra15] Raghuram, A. 2015. Special values of the Riemann zeta function: some results and conjectures, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 1–21. [Sel97] Selick, P. 1997. Introduction to Homotopy theory. Field Institute Monographs, 9, American Mathematical Society, Providence, RI.

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[So79] Soul´e, C. 1979. K-th´eorie des anneaux d’entiers de corps de nombres et cohomogie ´etale. Invent. Math., 55, 251–295. [Sr93] Srinivas, V. 1993 Algberaic K-theory, 2nd. Ed. Progress in Math. vol. 90, Birkh¨ auser. [Sw68] Swan, R. G. 1968. Algebraic K-theory. Lecture Notes in Math., 76, Springer-Verlag, Berlin-New York. [Wa78] Waldhausen, F. 1978. Algebraic K-theory of generalized free products I, II. Ann. of Math., (2), 108, no.2, 135–204, 205–256. [We13] Weibel, C. 2013. The K-book: An introduction to Algebraic K-theory. Graduate Studies in Math., 145, AMS.

3 Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory John Coates

3.1 Introduction The modest aim of the present paper is to give a detailed account of the intriguing p-adic argument, which, when combined with the motivic argument presented in [Hu15] and [Ki15], enables one to complete the proof of the original Bloch–Kato conjecture for the values of the Riemann zeta function at the odd integers > 1, at least up to an unknown power of 2. This p-adic argument is based on deep results from the Iwasawa theory of cyclotomic fields (we use both the main conjecture and Iwasawa’s theorem, in the simplest case when the Galois extension of Q given by the maximal real subfield of the field obtained by all p-power roots of unity to Q; see, for example, [CS06] for an account of this classical material), as well as important theorems of Borel and Soul´e on the K-theory of Z, and its relation to Galois cohomology. We stress that it is unknown at present how to carry out this p-adic argument using Iwasawa theory alone. We follow the original arguments of [BK90], section 6, but in addition make use of the deep motivic results of [Hu15] and [Ki15] included in this volume, which establish, in a form suitable for our needs, Conjecture 6.2 of [BK90], namely Theorem 3.6.2 of this paper. This Conjecture 6.2 was unproven at the time when Bloch and Kato wrote their original paper [BK90]. For an account of much further related work on all of these questions, see the article [Ng15] in the present volume.

University of Cambridge, UK. e-mail : [email protected]

46

John Coates

3.2 Notation The following notation will be used throughout this paper. Always, p will denote an odd prime number. We write Zp and Qp for the ring of p-adic integers, and the field of p-adic numbers, respectively, and define Dp = Qp /Zp , S = {p, ∞}. Write QS for the maximal extension of Q which is unramified outside S. If K is any intermediate field between Q and QS , GS (K) will denote the Galois group of QS over K. Let µp∞ denote the group of all p-power roots of unity, and write χ : GS (Q) → Z× p for the character giving the action of GS (Q) on µp∞ . For each integer m, let Zp (m) be the free Zp module of rank 1 on which GS (Q) acts via χm . More generally, if M is any Zp -module on which GS (Q) acts, M (m) will, as usual, denote the Zp -module M ⊗Zp Zp (m), endowed with the diagonal action of GS (Q). If A is a discrete or compact Zp -module, we write A∗ = Homcont (A, Dp ) for its Pontrjagin dual, which again we endow with the natural action of GS (Q) when A is itself a GS (Q)-module. As usual, we endow GS (Q) with the profinite topology. If M is a topological GS (Q)-module, H i (GS (Q), M )(i ≥ 0) will denote the cohomology groups of M formed with continuous cochains. Similarly, if M is a topological module for the absolute Galois group G(Q¯p /Qp ) of the field Qp of p-adic numbers, we write H i (Qp , M )(i ≥ 0) for the cohomology groups of M formed with continuous cochains. If M is a finitely generated Zp -module, rZp (M ) will denote the Zp -rank of M modulo its torsion submodule.

3.3 The Poitou–Tate sequence We begin by using the Poitou–Tate sequence to establish the following result. For brevity, we put GS = GS (Q). Theorem 3.3.1

For each integer m 6= 0, we have the exact sequence

0 → H 2 (GS , Zp (m))∗ → H 1 (GS , Dp (1 − m)) → H 1 (Qp , Dp (1 − m)) → → H 1 (GS , Zp (m))∗ → H 2 (GS , Dp (1 − m) → 0. Proof Let M be any finite GS -module, and define M D = Hom(M, µp∞ ). Then, since p is odd, by the theorem of Poitou and Tate, we have the exact sequence

Zeta Values at the Odd Positive Integers and Iwasawa theory

47

0 → H 0 (GS , M ) → H 0 (Qp , M ) → H 2 (GS , M D )∗ → H 1 (GS , M ) → H 1 (Qp , M ) → H 1 (GS , M D )∗ → H 2 (GS , M ) → H 2 (Qp , M ) → H 0 (GS , M D )∗ → 0. We apply this sequence with M = Z/pr Z(1 − m), and r any integer ≥ 1 (so that M D = Z/pr Z(m)). Note that H 0 (GS , M ) = H 0 (Qp , M ) because p is totally ramified in the field obtained by adjoining all ppower roots of unity to Q. Also, since the H j (GS , M D ) are finite for all j ≥ 0, it is well known (see [Ta76]) that, for each j ≥ 0, H j (GS , Zp (m)) = lim H j (GS , Z/pr Z(m)), ←

where the projective limit is taken over all r ≥ 0. It is clear that H 0 (GS , Zp (m)) = 0 because m 6= 0. Thus, taking the inductive limit of the above sequences over all r ≥ 1, we obtain the exact sequence 0 → H 2 (GS , Zp (m))∗ → H 1 (GS , Dp (1 − m)) → H 1 (Qp , Dp (1 − m)) → → H 1 (GS , Zp (m))∗ → H 2 (GS , Dp (1 − m) → H 2 (Qp , Dp (1 − m)) → 0. It follows from Tate local duality that H 2 (Qp , Z/pr Z(1 − m)) is dual to H 0 (Qp , Z/pr Z(m)). Hence, passing to the inductive limit, it follows that H 2 (Qp , Dp (1 − m)) is dual to H 0 (Qp , Zp (m)) = lim H 0 (GS , Z/pr Z(m)), ←

and this latter group is equal to 0 because m 6= 0. This completes the proof. We next relate the H i (GS , Dp (1−m)) to the relevant Iwasawa module, and now it is vital that we assume that m is an odd integer. Definition 3.3.2 F∞ is the maximal real subfield of the field Q(µp∞ ), and G = Gal(F∞ /Q). We further define K∞ to be the maximal abelian p-extension of F∞ , which is unramified outside p (note that K∞ /F∞ is automatically unramified outside the infinite places because it is pro-p and we have assumed that p is odd), and put X∞ = Gal(K∞ /F∞ ).

(3.1)

Since K∞ is clearly Galois over Q, the group G acts continuously on

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X∞ via inner automorphisms in the usual fashion. As X∞ is also a Zp module, this action extends to an action of the whole Iwasawa algebra of G, which we will always denote by Λ(G) in what follows. Theorem 3.3.3 morphisms

For each odd integer m 6= 1, we have canonical iso-

H 1 (GS , Dp (1−m))∗ = X∞ (m−1)G , H 2 (GS , Dp (1−m))∗ = X∞ (m−1)G , where, as above, GS = GS (Q). Proof Recall that GS (F∞ ) denotes the subgroup of GS fixing F∞ . Now, since m is odd, both G and GS (F∞ ) act on Dp (1 − m). Also G has pcohomological dimension equal to 1. Hence the Hochschild–Serre spectral sequence E2i,j = H i (G, H j (GS (F∞ ), Dp (1 − m))) =⇒ H i+j (GS , Dp (1 − m)) degenerates into the two short exact sequences 0 → H 1 (G, H i−1 (GS (F∞ ), Dp (1 − m))) → H i (GS , Dp (1 − m)) → H i (GS (F∞ ), Dp (1 − m))G → 0 with i = 1, 2. As m is odd, GS (F∞ ) acts trivially on Dp (1 − m), and so we have H 1 (GS (F∞ ), Dp (1 − m)) = Hom(X∞ , Dp (1 − m)) = (X∞ (m − 1))∗ . Moreover, since m 6= 1, we have H 1 (G, Dp (1 − m)) = 0 (indeed, G is topologically generated by one element, say g, and multiplication by g−1 is an automorphism of Qp (1 − m) since m 6= 1, whence it follows that multiplication by g − 1 on Dp (1 − m) must be surjective). Finally, the weak Leopoldt conjecture, proven by Iwasawa, is the assertion that H 2 (GS (F∞ ), Dp (1 − m)) = H 2 (GS (F∞ ), Dp )(1 − m) = 0. The assertions of the theorem now follow from these remarks and the above two exact sequences. This completes the proof. We next give the entirely analogous local argument at the prime p. Let [ Φr = Qp (µpr+1 )+ , Φ∞ = Φr . r≥0

Of course, the Galois group of Φ∞ over Qp can be identified with the

Zeta Values at the Odd Positive Integers and Iwasawa theory

49

global Galois group G. Define Ω∞ to be the Galois group over Φ∞ of the maximal abelian p-extension of Φ∞ . Again, there is a natural action of G on Ω∞ by inner automorphisms. Local class field theory then yields a G-isomorphism ¯ × , with Φ ¯ × = lim Φ× /(Φ× )pt , Ω∞ ' lim Φ r r ←− r ←− r where the first projective limit is taken with respect to the norm maps as r varies, and the second with respect to the natural maps as t varies. Proposition 3.3.4

H 1 (Qp , Dp (1 − m))∗ ' (Ω∞ (m − 1))G = Zp .

Proof For the first isomorphism, the proof is entirely analogous to the global case given above, but using the Hochschild–Serre spectral sequence for the extension Φ∞ /Qp . For the second equality, one uses the theorem of Iwasawa (see [Iw73], Theorem 25) that Ω∞ ' Λ(G). This completes the proof. Combining Theorems 10.1.1, 10.1.2, and Proposition 3.3.4, we immediately obtain, for each odd integer m 6= 1, the first fundamental exact sequence 0 → X∞ (m − 1)G → H 1 (GS , Zp (m)) → Ω∞ (m − 1)G → → X∞ (m − 1)G → H 2 (GS , Zp (m)) → 0.

(3.2)

By a fundamental theorem of Iwasawa [Iw73], X∞ is a finitely generated torsion Λ(G)-module (which means that it has a non-zero annihilator which is not a divisor of zero in the Iwasawa algebra Λ(G)). Hence, by a well-known algebraic argument, for all odd integers m, we have rZp (X∞ (m − 1)G ) = rZp (X∞ (m − 1)G ).

(3.3)

We remark that the H i (GS , Zp (m)) are all easily seen to be finitely generated Zp -modules, and they vanish for i > 2 because GS has pcohomological dimension equal to 2 since p is odd. Combining (3.3) with the exact sequence (3.2), and Proposition 3.3.4, we immediately obtain: Corollary 3.3.5

For all odd integers m 6= 1, we have

rZp (H 1 (GS , Zp (m))) − rZp (H 2 (GS , Zp (m))) = 1.

(3.4)

We also note the following lemma. Lemma 3.3.6 For all odd integers m 6= 1, the following assertions are equivalent: (i) H 2 (GS , Zp (m)) is finite, (ii) rZp (H 1 (GS , Zp (m))) = 1 and (iii) H 2 (GS , Dp (m)) = 0.

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Proof The equivalence of (i) and (ii) is immediate from (3.4). Taking GS -cohomology of the exact sequence 0 → Zp (m) → Qp (m) → Dp (m) → 0,

(3.5)

we obtain the exact sequence H 1 (GS , Dp (m)) → H 2 (GS , Zp (m)) → H 2 (GS , Qp (m)) → H 2 (GS , Dp (m)) → 0. Since Qp (m) is a vector space over Qp , so is H 2 (GS , Qp (m)), and hence it contains no non-zero Zp -torsion; in particular, the image of the torsion subgroup of H 2 (GS , Zp (m)) in it is zero. On the other hand, we know that H i (GS , Dp (m)) is a torsion abelian group for i = 1, 2. Thus, if we assume that, H 2 (GS , Zp (m)) is finite, it would follow from the above exact sequence that H 2 (GS , Qp (m)) must be torsion, and hence 0, whence also H 2 (GS , Dp (m)) = 0. Conversely, if H 2 (GS , Dp (m)) vanishes then H 2 (GS , Zp (m)) would have to map to onto H 2 (GS , Qp (m)), hence this latter group would have to be zero because no non-zero divisible group can be a finitely generated Zp -module. Hence the torsion group H 1 (GS , Dp (M )) would map onto H 2 (GS , Zp (m)), and so this latter group would have to be finite because it is a finitely generated Zp module. This completes the proof. Before proceeding further, we make a brief digression to discuss the relationship of the exact sequence (3.2) to the values of the p-adic analogue of the Riemann zeta function, using the so-called main conjecture of Iwasawa theory (which is not a conjecture at all, but a proven theorem in this case – for a full discussion of this theorem, and a more detailed explanation of the notions in what follows, see, for example, [CS06]). Let us denote the common value of the two Zp -ranks appearing in (3.3) by α(m − 1). When m = 1 it is easy to see that α(0) = 0. In fact, it is one of the standard conjectures on cyclotomic fields that α(m − 1) = 0 for all odd integers m, and, as we shall now explain, this is proven for odd m < 0, but unknown for odd m > 1. Indeed, since X∞ is a finitely generated torsion Λ(G)-module, the structure theorem for such modules enables us to define its characteristic ideal C(X∞ ) in Λ(G), which is a principal ideal. Let us write η for any generator of the ideal C(X∞ ). Recall that χ denotes the character giving the action of the absolute Galois group of Q on µp∞ , so that even powers of χ can be viewed as characters of G, and therefore define Zp -algebra homomorphism from

Zeta Values at the Odd Positive Integers and Iwasawa theory

51

Λ(G) to Zp . It is then not difficult to see that, for all odd integers m, α(m − 1) = 0 if and only if χ1−m (η) 6= 0.

(3.6)

On the other hand, the main conjecture gives an analytic version of this last statement. Let us write ρ for the p-adic analogue of the Riemann zeta function. Then ρ belongs to the fraction field of Λ(G), and has the property that (σ − 1)ρ belongs to Λ(G) for all σ in G. It is uniquely determined by the property that, for all odd integers m < 0, we have χ1−m (ρ) = (1 − p−m )ζ(m),

(3.7)

where ζ(s) denotes the classical complex Riemann zeta function. Let I(G) denote the kernel of the augmentation map from Λ(G) to Zp . Then the classical main conjecture is the assertion that C(X∞ ) = ρI(G).

(3.8)

Now ζ(m) 6= 0 for all odd integers m < 0. Hence, combining (3.6), (3.7) and (3.8), we have established: Proposition 3.3.7 Let m be any odd integer. Then α(m − 1) = 0 when m ≤ 1, and, when m > 1, we have α(m − 1) = 0 if and only if χ1−m (ρ) 6= 0. It is a folklore conjecture that χ1−m (ρ) 6= 0 for all odd integers m > 1, but no proof is known at present. Note also that, for any fixed odd integer m < 0, (3.3.7) shows that χ1−m (ρ) is a p-adic unit provided p is sufficiently large compared to |m|. It seems very reasonable to similarly conjecture that, for any fixed odd integer m > 1, χ1−m (ρ) is a p-adic unit provided p is sufficiently large compared to m. Again, nothing has been proven in this direction at present. However, the question of the possible truth of such a conjecture was first posed to me by Frank Adams many years ago, and more recently by Lars Hesselholt.

3.4 The comparison diagram We now compare (3.2) with a second fundamental exact sequence, arising from the Λ(G)-module which originally led Iwasawa to the main conjecp ture. Let ζr be a primitive pr+1 -th root of unity, such that ζr+1 = ζr −1 for all r ≥ 0. Then the elements cr = (1 − ζr )(1 − ζr ) in Φr are norm compatible, and so give rise to an element c∞ = (cr ) ∈ Ω∞ .

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John Coates

We then define C∞ to be the ideal C∞ = Λ(G)c∞ in Ω∞ ' Λ(G), and introduce the Λ(G)-module Y∞ = Ω∞ /C∞ . In particular, for each odd integer m, we have the exact sequence of Λ(G)-modules 0 → C∞ (m − 1) → Ω∞ (m − 1) → Y∞ (m − 1) → 0. Since Ω∞ ' Λ(G), we have Ω∞ (m − 1)G = 0. Thus, as G has phomological dimension equal to 1, on taking G-homology we obtain the exact sequence 0 → Y∞ (m − 1)G → C∞ (m − 1)G → Ω∞ (m − 1)G → Y∞ (m − 1)G → 0. (3.9) By global class field theory, there is a Λ(G)-homomorphism δ ∞ : Y∞ → X ∞ ,

(3.10)

whose cokernel is Gal(L∞ /F∞ ), where L∞ denotes the maximal abelian p-extension of F∞ , which is everywhere unramified. Twisting m−1 times, where m is any odd integer, we obtain a homomorphism φ∞ : Y∞ (m − 1) → X∞ (m − 1),

(3.11)

which, in turn, gives rise to two Zp -homomorphisms G G φG ∞ : Y∞ (m − 1) → X∞ (m − 1) , φ∞,G : Y∞ (m − 1)G → X∞ (m − 1)G . (3.12) We can then compare the exact sequences (3.2) and (3.9), thanks to the following proposition.

Proposition 3.4.1 homomorphism

For each odd integer m 6= 1, there exists a Zp -

ψm : C∞ (m − 1)G → H 1 (GS , Zp (m))

(3.13)

such that we have the commutative diagram with exact rows 0 − → Y∞ (m − 1)G − →  G φ∞ y

C∞ (m − 1)G   ψm y

− → Ω∞ (m − 1)G − →   =y

Y∞ (m − 1)G   φ∞,G y

− → 0

λ

0 − → X∞ (m − 1)G − → H 1 (GS , Zp (m)) − → Ω∞ (m − 1)G − → X∞ (m − 1)G ,

where λ is the map appearing in the exact sequence (3.2). Proof For each integer r ≥ 0, let Fr = Q(µpr+1 )+ , and write Jr for the group of S-units of Fr . Put t J¯r = lim Jr /Jrp , J¯∞ = lim J¯r ,

←

←

Zeta Values at the Odd Positive Integers and Iwasawa theory

53

where the latter projective limit is taken with respect to the norm maps. Now Kummer theory gives a canonical isomorphism jr : J¯r → H 1 (GS , Zp (1)).

(3.14)

Define Z∞ = lim H 1 (GS (Fr ), Zp (1)), ←

where the projective limit is taken with respect to the co-restriction maps. Passing to the projective limit over r for the maps (3.14), we then obtain a canonical Λ(G)-isomorphism j∞ : J¯∞ → Z∞ .

(3.15)

Now, as the Galois group GS (F∞ ) acts trivially on Zp (m − 1), it is well known that there is a canonical Λ(G)-isomorphism Z∞ (m − 1) ' R∞ = lim H 1 (GS (Fr ), Zp (m)). ←

Thus, twisting the map (3.15) by the Tate module Zp (m − 1), we obtain a Λ(G)-isomorphism l∞ : J¯∞ (m − 1) → R∞ . The map ψ is derived from l∞ as follows. The natural injective map from C∞ (m − 1) to J¯∞ (m − 1) gives rise, on taking G-coinvariants, to a Zp -homomorphism (C∞ (m − 1))G → (J¯∞ (m − 1))G .

(3.16)

There is also the natural map (J¯∞ (m − 1))G → (R∞ )G → H 1 (GS , Zp (m)),

(3.17)

which is given by the composition of the map on G-coinvariants derived from l∞ with the map induced from the projection map from R∞ onto H 1 (GS , Zp (m)). Then ψm is given by composing the two maps (3.16) and (3.17). It can then be verified that this definition leads to the commutative diagram given in the proposition. This completes the proof. Corollary 3.4.2 H 2 (GS , Zp (m)).

For all odd integers m 6= 1, we have Coker(φ∞,G ) =

Proof This is immediate from the above diagram, and the fact that Coker(λ) = H 2 (GS , Zp (m)) by the exact sequence (3.2).

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We now invoke the main conjecture for the field F∞ to prove the following proposition. As above, let L∞ be the maximal unramified abelian p-extension of F∞ , and define W∞ = Gal(L∞ /F∞ ).

(3.18)

It is a finitely generated torsion Λ(G)-module, and so by the structure theory of such modules, it has a characteristic ideal, which is principal. We define θ to be any generator of the characteristic ideal of W∞ . Proposition 3.4.3 For all odd integers m 6= 1, the following assertions are equivalent: (i) H 2 (GS , Dp (m)) = 0, (ii) H 2 (GS , Zp (m)) is finite, (iii) 1−m Coker(φ∞,G ) is finite, (iv) Ker(φG (θ) 6= 0. ∞ ) = 0 and (v) χ Proof The equivalence of (i) and (ii) and (iii) is given by Lemma 3.3.6 and Corollary 3.4.2. Note also that G Ker(φG ∞ ) = (Ker(φ∞ )) , Coker(φ∞,G )

= (Coker(φ∞ ))G = W∞ (m − 1)G .

(3.19) (3.20)

Also, by Iwasawa’s explicit determination of the Λ(G)-module Y∞ (see [CS06]), it is clear that Y∞ has no non-zero finite Λ(G)-submodule, whence the same is also true for Y∞ (m−1) and its submodule Ker(φ∞ ). Hence we conclude that Ker(φG ∞ ) is zero if and only if it is finite. Now, by the main conjecture on cyclotomic fields, Y∞ and X∞ have the same characteristic ideal, which is given by (3.8), whence it follows by the multiplicativity of characteristic ideals in exact sequences that Ker(φ∞ ) and Coker(φ∞ ) = W∞ (m − 1) also have the same characteristic ideals. As both Ker(φ∞ ) and Coker(φ∞ ) are finitely generated torsion Λ(G)modules, their G-invariants and G-coinvariants have the same Zp -rank, and so it now follows easily that assertions (iii), (iv) and (v) are equivalent. Corollary 3.4.4 For all odd integers m < 0, the equivalent assertions of Proposition 3.4.3 are valid. This is immediate from the equation (3.7), and the fact that the characteristic element θ of the Λ(G)-module W∞ must divide the characteristic ρ of X∞ , again by the multiplicativity of characteristic ideals in exact sequences.

Zeta Values at the Odd Positive Integers and Iwasawa theory

55

3.5 Theorem of Soul´ e As has been explained in the article [Li15], Soul´e [So79] has deduced the following result from Borel’s theorem on the ranks of the higher Kgroups of Z, by proving that, for all odd integers m > 1, there is an isomorphism K2m−2 (Z) ⊗ Qp ' H 2 (GS , Zp (m)) ⊗ Qp . Theorem 3.5.1 is finite.

(Soul´ e) For all odd integers m > 1, H 2 (GS , Zp (m))

This result, together with Corollary 3.4.4, shows that the equivalent assertions of Proposition 3.4.3 hold for all odd integers m 6= 1. However, we stress that, in our present state of knowledge, no proof of Soul´e’s theorem has ever been found which uses only the Iwasawa theory of cyclotomic fields. At the same time, it should be pointed out that, in all known numerical examples to date, it has always been found that L∞ = F∞ , so that we can take θ = 1, and thus the equivalent assertions of Proposition 3.4.3 are certainly true. We assume for the remainder of this section that m 6= 1 is an odd integer, and we now establish the following basic result. Theorem 3.5.2

For all odd integers m 6= 1, the map ψm : C∞ (m − 1)G → H 1 (GS , Zp (m))

is injective, and has finite cokernel of the same order as H 2 (GS , Zp (m)). Proof Thanks to Soul´e’s theorem (when m > 1), and the main conjecture (when m < 0), we know that the equivalent assertions of Proposition 3.4.3 are valid for m. The injectivity of ψm then follows immediately from the comparison diagram in Proposition 3.4.1, on noting that φG ∞ is injective. Since ψm is injective, and H 1 (GS , Zp (m)) has Zp -rank 1, it is easy to see that the cokernel of ψm must be finite. However, as we shall now explain, it is more delicate to compute the exact order of this cokernel. We can break up the commutative diagram appearing in Proposition 3.4.1 into two separate commutative diagrams with exact rows 0 −−−−→ Y∞ (m − 1)G −−−−→   φG ∞y

C∞ (m − 1)G   ψm y

−−−−→ A −−−−→ 0   βy

0 −−−−→ X∞ (m − 1)G −−−−→ H 1 (GS , Zp (m)) −−−−→ B −−−−→ 0 ,

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John Coates

and 0 −−−−→ A −−−−→ Ω∞ (m − 1)G −−−−→ Y∞ (m − 1)G −−−−→ 0       =y τy βy 0 −−−−→ B −−−−→ Ω∞ (m − 1)G −−−−→

D

−−−−→ 0 .

Moreover, from the diagram in Proposition 3.4.1, it is clear that Ker(τ ) = Ker(φ∞,G ). Note that (Coker(φ∞ ))G is finite, since (Coker(φ∞ ))G is finite, whence we deduce easily that Coker(φG ∞ ) is also finite. Similarly, the finiteness of both (Ker(φ∞ ))G and (Coker(φ∞ ))G imply the finiteness of Ker(φ∞,G ). Hence we conclude from the above diagrams that #(Coker(ψm )) = #(Coker(φG ∞ ))#(Ker(φ∞,G )).

(3.21)

Recall that G = ∆ × Γ, where ∆ is a cyclic group of order (p − 1)/2, and Γ is isomorphic to the additive group of Zp . Moreover, the action of ∆ on a Zp -module is semisimple because it has order prime to p. Thus, as the main conjecture and Iwasawa’s theorem together show that Y∞ and X∞ have the same characteristic ideals as Λ(G)-modules, we can apply the purely algebraic Lemma 3.5.5 proven below to the Γ-homomorphism ∆ ∆ φ∆ ∞ : Y∞ (m − 1) → X∞ (m − 1) .

It follows that G #(Ker(φG ∞ ))/#(Coker(φ∞ )) = #(Ker(φ∞,G ))/#(Coker(φ∞,G )).

Hence the formula (3.21) can be rewritten as #(Coker(ψm )) = #(Coker(φ∞,G ))#(Ker(φG ∞ )).

(3.22)

But we have shown earlier (see Corollary 3.4.2 and Proposition 3.4.3) that 2 Ker(φG ∞ ) = 0, Coker(φ∞,G ) = H (GS , Zp (m)),

and so the proof of the theorem is complete. We now deduce an interesting corollary of this theorem. Let ω denote the Teichmuller character modulo p. For an even integer i, and a Zp [∆] i module U , we write U (ω ) for the eigenspace of U on which ∆ acts via ω i , where, as earlier, ∆ denotes the Galois group of the maximal real subfield of Q(µp ) over Q.

Zeta Values at the Odd Positive Integers and Iwasawa theory

57

Corollary 3.5.3 For all odd integers m 6= 1, the cokernel of the map ψm has the same order as the G-coinvariants of Gal(L∞ /F∞ )(m − 1), where L∞ denotes the maximal unramified abelian p-extension of F∞ . In particular, if p is an odd prime number such that H 2 (GS , Zp (m)) = 0 for some odd integer m 6= 1, then the ω1−m -component of the p-primary subgroup of the ideal class group of the maximal real subfield of Q(µp ) is zero. Proof We have shown above that H 2 (GS , Zp (m)) and Coker(φ∞,G ) have the same order. But Coker(φ∞,G ) = (W∞ (m − 1))G = (W∞ (ω

1−m

)

(m − 1))Γ ,

where, as before, W∞ = Gal(L∞ /F∞ ). Now a compact Λ(Γ) module is zero if and only if its Γ-coinvariants vanish. Hence the remaining assertion of the corollary follows from the well-known fact that (W∞ (ω

1−m

)

)Γ = A(ω

1−m

)

,

where A denotes the p-primary subgroup of the ideal class group of the maximal real subfield of Q(µp ). This completes the proof. For all odd integers m > 1, it is shown in [DF85] that there is a surjective homomorphism from K2m−2 (Z) ⊗ Zp onto H 2 (GS , Zp (m)). But by Borel’s generalization of Garland’s theorem, K2m−2 (Z) is a finite abelian group. Hence the above corollary implies the following result. Corollary 3.5.4 Let m be a fixed odd integer > 1. Then, for all sufficiently large prime numbers p, the ω 1−m -component of the p-primary subgroup of the ideal class group of the maximal real subfield of Q(µp ) is zero. Results of this kind are well known (see, for example [So99], where a very large lower bound in terms of m for such primes p is established). We remark that, in all known numerical examples, the class number of the maximal real subfield Q(µp )+ of Q(µp ) is prime to p, and this in turn implies that Gal(L∞ /F∞ ) = 0 for such primes p. Thus, at present, we do not know a single numerical example of an odd prime number p, and an odd integer m 6= 1, for which H 2 (GS , Zp (m)) 6= 0. We also remark that, if it is indeed true (see the question raised by Adams and Hesselholt discussed at the end of Section 3) that, for fixed

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John Coates

odd m > 1, χ1−m (ρ) is a p-adic unit for all sufficiently large primes p (here, as earlier, ρ denotes the p-adic analogue of the Riemann-zeta function), then it would follow that the ω 1−m -component of the Galois group of the maximal abelian p-extension of Q(µp )+ unramified outside p is zero for all sufficiently large primes p. Finally, we establish the following purely algebraic lemma (see also [KNF96]), which was used in the proof of the last theorem. Let Γ = Zp , and, as usual, we identify the Iwasawa algebra of Γ with the ring of formal power series R = Zp [[T ]] by mapping some fixed topological generator of Γ to 1 +T . Let π : A → B be any R-homomorphism of finitely generated torsion R-modules. Write π Γ : AΓ → B Γ and πΓ : AΓ → BΓ for the maps induced by π. Note that, if A and B have the same characteristic ideals as R-modules, so also do Ker(π) and Coker(π), by the multiplicativity of characteristic ideals in exact sequences. Lemma 3.5.5 Assume that A and B are finitely generated torsion Rmodules, which have the same characteristic ideals. Suppose that we are given a homomorphism π : A → B such that a characteristic element of Ker(π), or equivalently a characteristic element of Coker(π), does not vanish at T = 0. Then both of the maps π Γ and πΓ have finite kernels and cokernels, and we have #(Ker(π Γ ))/#(Coker(π Γ )) = #(Ker(πΓ ))/#(Coker(πΓ )).

(3.23)

Proof Put D = Ker(π), and E = Coker(π). Since, by our hypotheses, D and E both have the same characteristic element which does not vanish at T = 0, a very well-known lemma asserts that #(DΓ )/#(DΓ ) = #(E Γ )/#(EΓ ),

(3.24)

where all the groups appearing in this formula are finite. Moreover, it is clear that Ker(π Γ ) = DΓ , Coker(πΓ ) = EΓ . Hence, in view of (3.24), we must show that Coker(π Γ ) and Ker(πΓ ) are both finite, and that #(DΓ )#(E Γ ) = #(Coker(π Γ ))#(Ker(πΓ )).

(3.25)

To establish this, we recall that, quite generally, if δ = β ◦ α is the composition of any two homomorphisms α and β of Zp -modules, then

Zeta Values at the Odd Positive Integers and Iwasawa theory

59

we always have the exact sequence 0 → Ker(α) → Ker(δ) → Ker(β) → Coker(α) → Coker(δ) → Coker(β) → 0.

(3.26)

Put I = Im(π), so that we clearly have π Γ = β1 ◦ α1 , πΓ = β2 ◦ α2 ,

(3.27)

where α1 : AΓ → I Γ , β1 : I Γ → B Γ , α2 : AΓ → IΓ , β2 : IΓ → BΓ are the natural maps. These fit into the exact sequences of Γ-homology given by α1 α2 AΓ −−−− → I Γ −−−−→ DΓ −−−−→ AΓ −−−− → IΓ −−−−→ 0, (3.28)

and β1 β2 0 −−−−→ I Γ −−−−→ B Γ −−−−→ EΓ −−−−→ IΓ −−−−→ BΓ . (3.29)

Since β1 is injective, we obtain from (3.26) the exact sequence 0 → Coker(α1 ) → Coker(π Γ ) → Coker(β1 ) → 0,

(3.30)

and since α2 is surjective, (3.26) gives the exact sequence 0 → Ker(α2 ) → Ker(πΓ ) → Ker(β2 ) → 0.

(3.31)

The desired finiteness of Coker(π Γ ) and Ker(πΓ ) follows immediately from (3.28), (3.29), (3.30) and (3.31), as also does (3.25) on computing orders using these exact sequences. This completes the proof.

3.6 The Bloch–Kato conjecture We assume from now on that m is always an odd integer such that m > 1. We begin by giving a slightly more explicit description of the cokernel of the map ψm defined by (3.13). We know that H 1 (GS , Zp (m)) has rank 1 as a Zp -module, but in fact it is a free Zp -module of rank 1. Indeed, taking GS -cohomology of the exact sequence (3.5), we see immediately that the torsion subgroup of H 1 (GS , Zp (m)) is equal to H 0 (GS , Dp (m)), and this latter group is zero because m is odd and p is odd. Recall that C∞ is the free Λ(G)-module of rank 1, which is generated

60

John Coates

by the element c∞ = (cr ), where cr = (1 − ζr )(1 − ζr−1 ); here ζr denotes p a primitive pr+1 -th root of unity satisfying ζr+1 = ζr for all r ≥ 0. For each r ≥ 0, let Cr be the group of cyclotomic units in Fr generated by ⊗(m−1) cr and its conjugates over Q. We write µpr+1 for the (m − 1)-th fold tensor product of µpr+1 with itself, endowed with the diagonal action of G (of course, G acts on it because m − 1 is even). We can then define the element c∞ (m − 1) = (cr ⊗ ζr ⊗ . . . ⊗ ζr )r , which belongs to ⊗(m−1)

C∞ (m − 1) = lim (Cr ⊗ µpr+1 ←

),

with the projective limit being taken with respect to the norm maps. It is easily seen that, in fact, we have C∞ (m − 1) = Λ(G)c∞ (m − 1). Let c∞^ (m − 1) denote the image of c∞ (m − 1) in (C∞ (m − 1))G , and define the Soul´e element sm,p by sm,p = ψm (c∞^ (m − 1)).

(3.32)

Since (C∞ (m − 1))G is a free Zp -module of rank 1, we see that Theorem 3.5.2 can be rewritten in the following equivalent form: Theorem 3.6.1 For each odd integer m > 1, let sm,p be the Soul´e element defined by (3.32). Then sm,p 6= 0, and the index of Zp sm,p in the free Zp -module of rank 1 given by H 1 (GS , Zp (m)) is equal to the order of H 2 (GS , Zp (m)). To proceed further, we need Conjecture 6.2 of [BK90], which is proven in the articles [Hu15], [Ki15] in this volume. By Borel’s theorem, K2m−1 (Z) is a finitely generated abelian group of rank 1. For each odd integer m > 1, and for each odd prime number p, let λm,p : K2m−1 (Z) ⊗ Qp → H 1 (GS , Qp (m))

(3.33)

be Soul´e’s Chern class isomorphism. Note that the natural map from H 1 (GS , Zp (m)) to H 1 (GS , Qp (m)) is injective because m is odd and p is odd, and so we can view the Soul´e elements sm,p as lying in H 1 (GS , Qp (m)). Let r∞ : K2m−1 (Z) ⊗ Q → R

(3.34)

be Beilinson’s regulator map defined in [Hu15]. The following result,

Zeta Values at the Odd Positive Integers and Iwasawa theory

61

which is absolutely crucial for the rest of our argument, is then proven in [Hu15] (see Corollary 11.7.2 of [Hu15]). Theorem 3.6.2 For each odd integer m > 1, there exists bm ∈ K2m−1 (Z) ⊗ Q, such that λm,p (bm ⊗ 1) = −(1 − pm−1 )−1 sm,p , and r∞ (bm ) = (m − 1)!ζ(m), (3.35) where ζ(s) denotes the Riemann zeta function. Now we know that K2m−1 (Z) is a finitely generated abelian group, which has rank 1 by Borel’s theorem because m > 1 is odd. Write K2m−1 (Z)tor for its torsion subgroup. Proposition 3.6.3

Let m > 1 be an odd integer. Then we have

r∞ (K2m−1 (Z)/K2m−1 (Z)tor ) = ((m − 1)!ζ(m)/θ(m))Z,

(3.36)

where θ(m) = 2a

Y

#(H 2 (GS , Zp (m))),

(3.37)

p>2

for some (unknown) integer a. Proof Let zm denote any generator of K2m−1 (Z)/K2m−1 (Z)tor . Then we know that there exist relatively prime integers u, v such that ubm = vzm . It follows immediately from the second equality in (3.35) that r∞ (K2m−1 (Z)/K2m−1 (Z)tor ) = ((m − 1)!ζ(m)u/v)Z. On the other hand, it is shown in [Li15] that the Chern class map from K2m−1 (Z) ⊗ Zp to H 1 (GS , Zp (m)) is surjective, and thus λm,p (zm ⊗ 1) must be a Zp -generator of H 1 (GS , Zp (m)), which is free of rank 1 over Zp . But λm,p (zm ⊗ 1) = λm,p (bm ⊗ 1)uv −1 = −(1 − pm−1 )−1 uv−1 sm,p . Note that (1−pm−1 )−1 is a p-adic unit since m > 1. We conclude immediately from Theorem 3.6.1 that the exact power of p occurring in the nonzero rational number v/u must be precisely the order of H 2 (GS , Zp (m)). In particular, it follows firstly that H 2 (GS , Zp (m)) is zero for all but a finite number of p, and secondly that v/u must be equal to θ(m) up to sign. This completes the proof. Q ˆ For an odd integer m > 1, put Z(m) = p≥2 Zp (m), and, following

62

John Coates

Bloch–Kato [BK90], define the Tamagawa measure of the motive Z(m) by the convergent infinite product T am(Z(m)) = µ∞ (R/r∞ (K2m−1 (Z)/K2m−1 (Z)tor )) Y ˆ µp (H 1 (Qp , Z(m)), p≥2

where the measures µp are defined in [BK90] section 4, and µ∞ arises from the usual measure on the real line. It is shown in ([BK90], Theorem 4.2), using the explicit reciprocity law of Bloch–Kato (see [Sa15] in this volume), that ˆ µp (H 1 (Qp , Z(m)) = (1 − p−m )|(m − 1)!|p #(H 0 (Qp , Qp /Zp (1 − m)). Note that, since the prime p is totally ramified in the fields obtained by adjoining p-power roots of unity to Q, we have Y #H 0 (Q, Q/Z(1 − m)) = #(H 0 (Qp , Qp /Zp (1 − m)). p≥2

Combining these results with Theorem (3.6.2), we immediately deduce the following result: Proposition 3.6.4

For every odd integer m > 1, we have Y T am(Z(m)) = #H 0 (Q, Q/Z(1 − m))/(2a #(H 2 (GS , Zp (m))), p>2

for some integer a. To complete the proof of the Bloch–Kato conjecture for Z(m), we only have to relate the group H 2 (GS , Zp (m)) to the analogue of the Tate– Shafarevich group for Z(m), as defined in section 5 of [BK90]. We write X(m) for this Tate-Shafarevich group of Z(m). Lemma 3.6.5 For every odd integer m > 1, and every odd prime p, the p-primary subgroup of the Tate–Shafarevich group X(m) of Z(m) is finite, and its order is equal to the order of H 2 (GS , Zp (m)). Proof We only sketch the argument. The essential point is that Theorem 10.1.1 shows that the p-primary part of the Tate–Shafarevich group of the motive Z(1 − m) is equal to the dual of the finite group H 2 (GS , Zp (m)). We then invoke a Theorem of Flach [Fo82], which proves that there is a dual pairing between the p-primary subgroups of the Tate–Shafarevich group of the motives Z(m) and Z(1 − m).

Zeta Values at the Odd Positive Integers and Iwasawa theory

63

We also remark that, while for simplicity we have always assumed that p is an odd prime, a similar argument to the above, but involving only Galois cohomology and Iwasawa theory, also enables one to show that, for all odd integers m > 1, the 2-primary subgroup of the Tate– Shafarevich group of X(m) is finite. Hence, combining the last lemma and proposition, we finally obtain the following result, which is the original Bloch–Kato conjecture for the values of the Riemann zeta function at the odd positive integers > 1. Theorem 3.6.6 For every odd integer m > 1, the Tate–Shafarevich group of the motive Z(m) is finite, and T am(Z(m)) = #(H 0 (Q, Q/Z(1 − m))2b /#(X(m)), for some (unknown) integer b.

References [BK90] Bloch, S., and Kato, K. 1990. Tamagawa numbers of motives. The Grothendieck Festschrift Volume 1. Progress in Mathematics, 86, 333– 400. [CS06] Coates, J., and Sujatha, R. 2006. Cyclotomic fields and zeta values. Monographs in Mathematics, Springer. [DF85] Dwyer, W., and Friedlander, E. 1982. Etale K-theory and arithmetic. Bulletin Amer. Math. Soc., 6 (1982), 453–455. [Fo82] Flach, M. 1990. A generalisation of the Cassels-Tate pairing. J. Reine Angew. Math., 412, 113–127. [Hu15] Huber, A. 2015. The comparison theorem for the Soule-Deligne classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 210–238. [Iw73] Iwasawa, K. 1973. On Zl -extensions of algebraic number fields. Annals Math., 98, 246–326. [Ki15] Kings, G. 2015. The l-adic realisation of the elliptic polylogarithm and the evaluation of Eisenstein classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 239–296. [KNF96] Kolster, M., Nguyen Quang Do, T., and Fleckinger, V. 1996. Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J., 84, 679–717. [Li15] Lichtenbaum, S. 2015. Soule’s theorem, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 130–139.

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[Ng15] Nguyen Quang Do, T. 2015. On the determinental approach to the Tamagawa number conjecture, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 154–192. [Sa15] Saikia, A. 2015. Explicit reciprocity law of Bloch-Kato and exponential maps, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 65–96. [So79] Soul´ e, C. 1979. K-theorie des anneaux d’entiers de corps de nombres et cohomologie etale. Inventiones Math., 55, 251–295. [So99] Soul´ e, C. 1999. Perfect forms and the Vandiver conjecture. J. Reine Angew. Math., 517, 209–221. [Ta76] Tate, J. 1976. Relations between K2 and Galois cohomology. Inventiones Math., 36, 257–274.

4 Explicit Reciprocity Law of Bloch–Kato and Exponential Maps Anupam Saikia

Abstract The first goal of this expository article is to give a detailed proof of Bloch–Kato’s reciprocity law in (theorem 2.1 in [BK90]) in the simplest case, where it follows from the classical reciprocity law. The second goal is to explain Bloch–Kato’s exponential and dual exponential maps, providing the background material of Fontaine’s theory of p-adic representations. Thirdly, we show how we can compute the image of the Soul´e– Deligne cyclotomic elements under the dual exponential map (theorem 3.2.6 in [HK03]). The purpose of this article is to provide a self-contained and transparent account of some of the key ingredients required in the proof of the Bloch–Kato conjecture for the Riemann zeta function.

4.1 Introduction Bloch–Kato’s explicit reciprocity law is a crucial ingredient in their work on Tamagawa number of motives ([BK90]). Their reciprocity law can be viewed as an extension of the classical explicit reciprocity law. In the second section of this article we are going to review the classical explicit reciprocity law due to Iwasawa. In the third section we introduce Fontaine’s theory of p-adic representations, which is essential for understanding Bloch–Kato’s explicit reciprocity law as well as the exponential maps of Kato. In the fourth section, we explain how Bloch–Kato’s reciIndian Institute of Technology Guwahati, India. e-mail : [email protected]

66

Anupam Saikia

procity law in the simplest case can be proved from the classical explicit reciprocity law. In the final section, we discuss Bloch–Kato’s exponential and dual exponential maps. In particular we look into the image of Soul´e–Deligne cyclotomic elements under the dual exponential map (cf. theorem 3.2.6 of [HK03]). In order to make the material accessible to beginners, we provide the explicit details throughout the article.

4.2 The classical explicit reciprocity law Let p be an odd prime and µpn be the group of all pn -th roots of unity. Let L be a finite extension of Qp containing µpn and L denote an algebraic closure of L and GL denote the Galois group Gal(L/L). The surjective n × × map x 7→ xp from L to L results in the Kummer sequence × pn

×

0 −→ µpn −→ L −→ L −→ 0. From the Galois cohomology sequence of the above, we obtain n

κ

×

0 −→ L× /(L× )p −→ H 1 (GL , µpn ) −→ H 1 (GL , L ) = 0, where κ is the Kummer map sending given by 1
σ−1 a 7−→ κ(a) : σ 7→ a pn . Let Lab denote the maximal abelian extension of L in its algebraic closure L. The Kummer map gives rise to a pairing known as the Kummer pairing given by
pn 1
σ−1 < , >n : L× / L× × Gal(Lab /L) −→ µpn , < α, σ >n = α pn . On the other hand, we have the reciprocity map of local class field theory (see chapter VI, [CF67]) given by b7→σb =(b,Lab /L)

L× −−−−−−−−−−−−−−−→ Gal(Lab /L). It is well known that Lab is the composite of the maximal unramfied extension Lur of L and the totally ramified Lubin Tate division tower S L∞ = L(Wπn ), where Wπn denotes all the [π]n -torsion points of the n

Lubin–Tate formal group associated with the local parameter π of L. For any b = uπ n ∈ L× , σb acts as the n-th power of the Frobenius on Lur and as the automorphism [u]−1 π associated with the Lubin–Tate group

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 67 on Wπn (loc. cit.). We refer to σb as the Artin symbol. The Kummer pairing can be interpreted as a pairing h , in :

L× / L×


pn

× L× −→ µpn ,

hα, βin =

1

α pn


σβ −1

.

The Kummer pairing behaves well in the cyclotomic tower of Qp , and the classical explicit reciprocity law describes it analytically. Before stating this law, let us fix certain notation. For each n ≥ 1, let ε(n) = ζpn denote a primitive pn -th root of unity such that ζppn+1 = ζpn . Let Kn = Qp (ζpn ),

B∞ := lim Kn× , ←−

K∞ =

[

n

Kn ,

U∞ := lim U , ←− n n

n

(4.1) where Un denotes the principal units in Kn . Let α ∈ Un ,

β = (βn ) ∈ B∞


NKn+1 /Kn βn+1 = βn ∀n ≥ 1 .

Let us define [α, β]n ∈ Zp /pn Zp by [α,β]n

ζpn

= hα, βin =



1

α pn

σβ −1

.

(4.2)

The classical explicit reciprocity law due to Iwasawa (see [Iw86] or chapter I of [Sh87]) can be stated as  
[α, β]n = p−n T rKn /Qp log(α) gβ0 /gβ (ζpn − 1) mod pn .

(4.3)

Here, gβ denotes the Coleman power series in T v(β) Zp [[T ]]× associated with β such that gβ (ζpn − 1) = βn ∀n ≥ 1. (v(β) denotes the normalized p-adic valuation of β.) We discuss Bloch and Kato’s generalization to crystalline and De Rham representations. Starting with the ‘fundamental exact sequence’ in Fontaine’s ring of + periods Bcris , one can construct a canonical class in H 1 (Qp , Qp (r)), and Bloch–Kato’s reciprocity law identifies this class essentially as the r-th Coates–Wiles homomorphism. Their work has been vastly generalized by Kato, Tsuji and others.

68

Anupam Saikia

4.3 Fontaine’s rings: Bcris and BdR In this section, we will cover the basic tools that we need from the theory of p-adic representations. For an elegant and comprehensive account, see [FO] or [BC09]. Let K be a finite extension of Qp , GK = Gal(K/K) and V be a p-adic representation, i.e., a finite dimensional Qp vector space with a continuous linear action . of GK . Let χ denote the cyclotomic character given by χ : GK −→ Z× p,

χ(σ)

σ(ζpn ) = ζpn .

For a p-adic representation and r ∈ Z, V (r) will denote the p-adic representation with the same underlying space V with the Galois action ? given by g ? v = χr (g)g.v

∀v ∈ V.

We will be mostly interested in V = Qp (r) in this article. Fontaine’s idea of studying p-adic representations was to construct rings of periods, which are topological Qp -algebras B, with a continuous linear action of GK such that B GK is a field, with some additional structures (e.g., a Frobenius φ, a filtration Fil etc) such that the B GK -module DB (V ) = B ⊗Qp V


GK

,

which inherits the additional structures, is a useful invariant of V . One can show that (theorem 5.2.1, [BC09]) dimBK G DB (V ) ≤ d = dimQ (V ). p

(4.4)

Definition 4.3.1 The p-adic representation V is called B-admissible if equality holds in (4.4). Equivalently, V is B-admissible if we have an isomorphism of GK -modules B ⊗Qp V ∼ = Bd. The canonical B-linear GK -equivariant map αV : B ⊗B GK DB (V ) −→ B ⊗BGK (B ⊗BGK V ) = (B ⊗B GK B) ⊗Qp V −→ B ⊗Qp V can be shown to be an isomorphism if and only if V is B-admissible (loc. cit.).

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 69 The coefficients of a matrix of the above isomorphism αV in two bases of DB (V ) and V are called the periods of V . Two natural choices for B c in p-adic are the algebraic closure K = Q and its completion C = Q p

p

p

topology. But V is K admissible if and only if the action of GK on V factors through a finite quotient, and V is Cp -admissible if and only if the action of inertia subgroup IK on V factors through a finite quotient (§I.2.3 in [Be04]). For example, Qp (r) is not Cp -admissible: DCp (Qp (r)) = Cp ⊗Qp Qp (r)


GK

= H 0 (GK , Cp (r)) = 0

∀r 6= 0,

by a result of Sen–Tate (theorem 4.5.1). Two of the most useful rings of periods are Bcris and BdR . For construction of these rings, we need the theory of Witt vectors.

4.3.1 Witt vectors For this subsection, an ideal reference is chapter II of [Se79]. Let R be a perfect ring of characteristic p, i.e., the Frobenius ring-endomorphism φ : x 7→ xp is bijective. Then there exists a unique (up to isomorphism) ring W (R) in which p is not nilpotent, which is separated and complete for the topology defined by pn W (R) with R as residue ring. W (R) is called the ring of Witt vectors over R. There is a unique system of representatives [ ] : R → W (R),

−n

n

[y] = lim yˆnp where yˆn ≡ y p n→∞

modulo p

satisfying [yz] = [y][z]. The representative [y] of y ∈ R is called the Teichmuller representative. Any element x of W (R) can be uniquely written as ∞ X x= pn [xn ], xn ∈ R. n=0

There exist polynomials Sn and Zn in variables x0 , y0 , . . . , xn , yn with integral coefficients such that ∞ X

pn [xn ]+

n=0

∞ X

pn [yn ] =

n=0

∞ X

pn [sn ],

n=0

∞ X n=0

pn [xn ].

∞ X

pn [yn ] =

n=0

∞ X

pn [zn ],

n=0

with 1 n

1 n

sn = Sn (x0p , y0p , . . . , xn , yn ),

1 n

1 n

zn = Zn (x0p , y0p , . . . , xn , yn ),

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Anupam Saikia

and Sn , Zn are given by wpn (S0 , . . . , Sn ) = wpn (X0 , . . . , Xn ) + wpn (Y0 , . . . , Yn ), wpn (Z0 , . . . , Zn ) = wpn (X0 , . . . , Xn ) . wpn (Y0 , . . . , Yn ), where n

wpn ((X0 , . . . , Xn ) = X0p + . . . + pn Xn .

For example, if R = Fp , then W (R) = Zp . If R = k = Fq is the finite field of q = pf elements, then W (R) is the ring of integers of the unique unramified extension of degree f over Qp . If R = Fp , then W (R) = OQd nr . p

The construction of W (R) is functorial, thus any homomorphism f : R −→ R0 of perfect rings of characteristic p can be lifted to a homomorphism W (f ) : W (R) −→ W (R0 ) so that W (f )

∞ X

pn [xn ]



n=0

=

∞ X

pn [f (xn )].

n=0

In particular, one can lift the Frobenius automorphism φ : R → R, x 7→ xp to an automorphism (also denoted by) φ : W (R) −→ W (R). Proposition 4.3.2 The elements of W (R) satisfying φ(α) = αp are precisely the Teichmuller representatives [a], a ∈ R. P∞ Proof: Let us consider any element α = n=0 pn [an ] in W (R). We will show that a1 = a2 = · · · = 0: φ(α) = αp =⇒

∞ X

pn [apn ] = [a0 ]p + p2 [a0 ][a1 ] + · · ·

n=0

=⇒ [ap0 ] + p[ap1 ] ≡ [a0 ]p modulo p2 , =⇒ ap1 = 0 =⇒ a1 = 0. Similarly, we can show that a2 = a3 = · · · are all 0.

4.3.2 Witt vectors over perfection of OQp /pOQp The ring OQp /pOQp is not perfect as the Frobenius endomorphism is not an automorphism. But we can construct a perfect ring out of it as

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 71 follows. Consider xpi = xi−1 .

 R = lim OQp /pOQp = (x0 , x1 , · · · ) | xi ∈ OQp /pOQp , ←− x7→xp

The Frobenius endomorphism R −→ R, x 7→ xp is clearly bijective: x = (x0 , x1 , · · · , xn , xn+1 , · · · ) = (x1 , · · · , xn , xn+1 , · · · )p , xp = 0 =⇒ (xp0 , x0 , x1 , · · · ) = (0, 0, · · · ) =⇒ x = 0. Hence R is a perfect ring of characteristic p. We have another useful description of R. Let x ˆi be any representative of xi in OQp , and define n

x(i) = lim x ˆp ∈ OCp . −→ i+n n

n

The above limit exists in OCp as it is easily seen that the sequence (ˆ xpi+n ) is Cauchy: k

k

n+k

n

x ˆpi+n+k ≡ xpn+i+k ≡ xi+n ≡ x ˆi+n mod p =⇒ x ˆpi+n+k ≡ x ˆpi+n mod pn+1 The association x = (x0 , x1 , . . .) 7→ (x(0) , x(1) , . . .)
identifies R as a subp set of all infinite sequences in OCp such that x(i) = x(i−1) . Thus, we can also write R = lim OCp . ←− x7→xp

Observe that k

k

x ˆpi+k ≡ xpi+k ≡ xi ≡ x ˆi mod p =⇒ x(i) ≡ x ˆi ≡ xi mod p. We can define addition and multiplication in R in the new representation as follows. For x = (x(i) ) and y = (y (i) ), we must have pn

n

n

= lim x ˆp . lim yˆp = x(i) .y (i) , n→∞ i+n n→∞ i+n
pn \ = lim xi+n + yi+n . n→∞
pn = lim x ˆi+n + yˆi+n .

(x.y)(i) = lim xi+n \ .yi+n n→∞

(x + y)(i)

n→∞

Note that x ˆi+n + yˆi+n ≡ x(i+n) + y (i+n) mod p
pn
pn =⇒ x ˆi+n + yˆi+n ≡ x(i+n) + y (i+n) mod pn+1 n
p
pn =⇒ lim x ˆi+n + yˆi+n = lim x(i+n) + y (i+n) n→∞ n→∞
pn (i) =⇒ (x + y) = lim x(i+n) + y (i+n) . n→∞

72

Anupam Saikia

We can define a valuation on R by setting vR (x) = vp (x(0) ), where vp is the p-adic valuation normalized by vp (p) = 1. Then R is a complete valuation ring, whose residue field is Fp . There is a canonical element ε = (ε(0) = 1, ε(1) = ζp , ε(2) = ζp2 , · · · ) ∈ R. The canonical surjective map
x(i) → 7 x(0) mod p

θ : R → OCp /pOCp ,

is a homomorphism. By functoriality of Witt construction, we have a (homomorphism) lift θ : W (R)  OCp ,

∞ X

pn [xn ] 7→

n=0

∞ X

pn x(0) n .

n=0

Let us denote the kernel of θ by J. Then it can be shown that J is P∞ principal ideal of W (R), generated by any α = n=0 pn [an ] for which (0) θ(α) = 0 and vp (a0 ) = 1 (proposition 2.4 in [Fo82]). In particular, we can take ω = [˜ p] − p

where

1

1

p˜ = (p, p p , p p2 , · · · ) ∈ R

(4.5)

as a generator of J. Observe that [ε] − 1 ∈ J as θ([ε] − 1) = ε(0) − 1 = 0. The Frobenius of W (R) preserves J + pW (R) but not J: φ(β) ≡ β p mod pW (R),

θ(β p ) = θ(β)p = 0 ∀β ∈ J, 1

φ(ω) = φ([˜ p] − p) = [˜ pp ] − p = [(pp , p, p p , · · · )] − p =⇒ θ(φ(ω)) = pp − p 6= 0. The Galois group acts by functoriality on R and W (R) and commutes with φ and θ. Proposition 4.3.3 θφ−n ([ε]) = ζpn .

For a ∈ R, θφ−n ([a]) = a(n) . In particular,

Proof: Observe that for x ∈ R, φ−n (x)(m) = x(n+m) . + 4.3.3 The rings BdR and BdR

Consider W (R)

h1i p

=

n X k>>−∞

o pk [xk ] | xk ∈ R .



Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 73 The homomorphism θ : W (R)  OCp can be extended to a homomorphism h1i X X (0) θQ : W (R) −→ Cp , pk [xk ] 7−→ pk xk . p k>>−∞

k>>−∞

+ The ring BdR is defined as + BdR = lim W (R) ←− n

h1i /(ker(θQ ))n . p

It is a complete discrete valuation ring with residue field Cp , and any gen+ erator of ker(θQ ) is a uniformizer in BdR (proposition h4.4.6 i in [BC09]). T + One can show that (ker(θQ ))j = {0}, hence W (R) 1p ⊂ BdR (loc. cit.). As ker(θQ ) = (ω), we can write + BdR

=

∞ nX

h 1 io yn ωn | yn ∈ W (R) . p n=0

Period t for the cyclotomic character: We now construct a + very useful element in the ring BdR which not only acts as a period for the cyclotomic character but also gives a filtration. Note that as θQ ([ε] − 1) = 0, the series ∞ X (−1)n−1 ([ε] − 1)n n n=1 + converges in BdR . We denote the sum of this convergent series by t. We can think of t as log([ε]). If σ ∈ GK , we now have

σ(t) = σ(log([ε])) = log([σ(1, ε(1) , · · · )]) = log([ε]χ(σ) ) = χ(σ) log([ε]) = χ(σ)t so t is a period for the cyclotomic character (see [BC09] p.61). It is the + p-adic analogue of 2πi ∈ C. Though exp(t) = [ε] 6= 1 ∈ BdR , θ([ε]) = 1 ∈ Cp . One can show that t is in fact a generator of the maximal ideal + in BdR . All we need is to check that [ε] − 1 ∈ W (R)ω − W (R)ω2 . But vp ([ε] − 1) = vp (ε − 1) = vp lim (ε(j) − 1)


pj

j→∞


pj = vp lim (ζpj − 1) = j→∞

p < 2. p−1

74

Anupam Saikia

Definition 4.3.4

We define the ring BdR as h i + 1 BdR = BdR . t

BdR is clearly a field endowed with a decreasing filtration defined by + i Fili BdR = BdR = ti BdR

∀i ∈ Z.

+ 0 In this notation, Fil0 BdR = BdR = BdR .

We also have i+1 i BdR /BdR = Cp (i). GK + GK It can be deduced from theorem 4.5.2 that BdR = (BdR ) = K, so that

DdR (V ) = (BdR ⊗Qp V )GK is a natural filtered K-vector space, with filtration given by i i DdR (V ) = Fili (DdR V ) = H 0 (GK , BdR ⊗Qp V )

∀i ∈ Z.

Definition 4.3.5 We say V is de Rham if V is BdR -admissible, i.e., the dimension of V as a Qp -vector space equals the dimension of DdR (V ) as a K-vector space. In other words, V is called a de Rham representation if ∼

BdR ⊗Qp V → BdR ⊗K DdR (V ). For example, Qp (r) is a de Rham representation for any integer r. Sub-quotients, duals and tensor products of de Rham representations are also de Rham (see §II.6 in [BC09]).

+ 4.3.4 The rings Acris and Bcris + One shortcoming of BdR is that the Frobenius automorphism of W (R)[ 1p ] does not preserve ker(θQ ), so there is no natural extension of φ on BdR : 1

1

1

1

+ ∈ BdR [˜ p ]−p   1 1 + =⇒ φ = ∈ BdR , 1 [˜ p ] − p p [˜ p ]−p

θQ ([˜ p p ] − p) = p p − p 6= 0 =⇒ [˜ pp ] − p ∈ / JQ =⇒

But θQ ([˜ p] − p) = 0 =⇒

1 + ∈ / BdR . [˜ p] − p

1 p

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 75 + Therefore, we introduce an auxiliary subring A0cris ⊂ BdR that is φstable and gives rise to a large subring Bcris of BdR on which there is a natural φ-action. Define N nX

o ωn | N < ∞, an ∈ W (R) n! n=0 h ωm i h1i = W (R) ⊂ W (R) . m! m≥1 p

A0cris =

an

Observe that A0cris is closed under multiplication: ωm ωm (m + n)! ωm+n = ∈ A0cris . m! m! m!n! (m + n)! The subring A0cris is stable under the action of the Frobenius of W (R)

h i 1 p :

 ωp  φ(ω) = ω p + pη = p η + (p − 1)! p!  p m ω ⇒ φ(ω m ) = pm η + (p − 1)! p!  ω m  pm  h ωp i ω p m pm ⇒φ = η + (p − 1)! ∈ W (R) ⊂ A0cris as ∈ Z. m! m! p! p! m! Definition 4.3.6 i.e.,

We define Acris to be the p-adic completion of A0cris , Acris = lim A0 /pn A0cris . ←− cris n

Acris is p-adically separated and complete, and A0cris embeds in Acris . The Frobenius φ on A0cris extends by continuity to Acris . One can show + that Acris is a subring of BdR given by ∞ n X ωn + Acris = x ∈ BdR |x= xn , n! n=0

o xn → 0 in the p-adic topology of W (R) . We define + Bcris := Acris

∞ n X ωn + = x ∈ BdR |x= xn , p n! n=0

h1i

xn → 0 in the p-adic topology of W (R)

h 1 io p

.

76

Anupam Saikia

Observe that t = log([ε]) =

∞ ∞ X X (−1)n−1 ([ε] − 1)n ωn = (−1)n−1 (n − 1)!ξ n , n n! n=1 n=1

for some ξ ∈ W (R) with [ε] − 1 = ωξ ∈ ker(θ). In other words, the + cyclotomic period t ∈ BdR belongs to Acris . Definition 4.3.7 The crystalline period ring Bcris for K is defined as + 1 + 1 the GK -stable W (R)[ p1 ]-subalgebra Bcris [ t ] = Acris [ 1t ] inside BdR [t] = BdR . One can show (proposition 9.1.3 in [BC09]) that tp−1 ∈ pAcris , so inverting t makes p a unit, and hence the equality. The filtration on Bcris is i Fili Bcris = Bcris ∩ BdR = Bcris ∩ ti BdR ,

i ∈ Z.

+ The Frobenius φ extends to Bcris as well as Bcris , and

φ(t) = p.t.

(4.6)

The Galois action also carries over to Acris , and θ also extends to Acris . If K0 is the maximal totally unramified subfield of K, then  h 1 iGK GK + GK + GK W (R) = K0 ⊂ (Bcris ) = K0 = Bcris ⊂ (BdR ) p GK = BdR = K. GK GK In particular, if K/Qp is unramified, then Bcris = BdR = K.

4.3.5 Ideals of Acris We noted that θ : W (R) → OCp can be extended to A0cris , and by continuity to Acris . Let Jcris denote the kernel of θ in Acris . The ideal Jcris is not principal, it is not even finitely generated. Its divided powers are the ideals  ωr  ωr+1 [r] Jcris = Acris , , ··· r ≥ 1. r! (r + 1)! We also define [r]

Jcris = {α ∈ Jcris | φ(α) ∈ pr Acris }.

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 77 For any β ∈ 1 + J, we can use the power series expansion of log(1 + x) to obtain log(β) =

∞ X

(−1)n−1 (n − 1)!

n=1

(β − 1)n [1] ∈ Jcris . n!

<1> Observe that log(β) is an element of Jcris as

φ(log(β)) = log(φ(β)) ≡ log(β p ) ≡ p log(β) mod pAcris =⇒ φ(log(β)) ∈ pAcris . Therefore we have the logarithm map log

<1> <1> log : 1 + J → Jcris , 1 + J ⊃ [ε]Zp −−−−−→ Zp .t = Zp (1) ⊂ Jcris . (4.7)

4.3.6 The fundamental commutative diagram The following commutative diagram will be crucial in our proof of the explicit reciprocity law:

p−φ

0 −−−−→ [ε]Zp −−−−→ 1 + J −−−−→ 1 + pW (R) −−−−→ 0       −1 logy logy yp log 1−p−1 φ

<1> 0 −−−−→ Zp (1) −−−−→ Jcris −−−−−→

Acris

−−−−→ 0

We will prove the exactness of the top row only; for the exactness of the bottom row, see theorem 6.24 of [FO]. Also note that the third vertical arrow maps 1 + pW (R) onto W (R). Let us first show the surjectivity of p − φ. If β ∈ W (R)× , then clearly β ∈ 1 + pW (R). Consider any element α = 1 + p[a1 ] + p2 [a2 ] + · · · in 1 + pW (R). We show that by successive approximation using the Teichmuller representation, we can find b1 , b2 , · · · ∈ R such that β = 1 + p[b1 ] + p2 [b2 ] + · · · ∈ 1 + pW (R) satisfies β p−φ = α. For the first p−φ

78

Anupam Saikia

approximation, (1 + p[b1 ] + p2 [b2 ] + · · · )p = (1 + p[b1 ] + p2 [b2 ] + · · · )φ (1 + p[a1 ] + p2 [a2 ] + · · · ) =⇒ 1 ≡ 1 + p[bp1 ] + p[a1 ] mod p2 W (R) =⇒ [bp1 ] = −[a1 ] mod pW (R) 1

=⇒ b1 = (−a1 ) p in R

(R is perfect).

Having chosen b1 , we can proceed to b2 , b3 etc. and obtain a preimage β of α in 1 + pW (R). To have a pre-image of α inside 1 + J, we pick a ∈ R such that θ(β) = a(0) , and then replace the preimage β ∈ 1 + pW (R) by β/[a]. Then θ(β/[a]) = 1 and β/[a] ∈ 1 + J. But (β/[a])p−φ = β p−φ = α. Let us now show the exactness at the middle term. If β ∈ 1 + J such that β p−φ = 1, then β = [b] for some b ∈ R by proposition 4.3.2. Now, θ(β) = 1 = b(0) , hence we must have b = [ε]y for some y ∈ Zp .  There is a generalization of the bottom row for r ≥ 1 [(2.5.1) in [BK90])]. There exists an integer c(r) such that the following sequence is exact: 1−p−r φ

0 −→ p−c(r) Zp (r) −→ Jcris −−−−−→ Acris −→ 0.

By tensoring the above sequence with Qp , we obtain an exact sequence [(1.13) of [BK90]] 1−p−r φ

+ + 0 −→ Qp (r) −→ Jcris ⊗ Q = Filr Bcris −−−−−→ Bcris −→ 0. (4.8)

4.4 Explicit reciprocity law of Bloch and Kato Let K be a finite unramified extension of Qp . We now consider the connecting homomorphism ∂ r of GK -comohomology of the exact sequence (4.8) + ∂ r : K = H 0 (K, Bcris ) −→ H 1 (K, Qp (r)).

Bloch–Kato’s explicit reciprocity law gives a concrete description of the cohomology class ∂ r (a) in H 1 (K, Qp (r)) for a ∈ K. We first identify the cohomology group H 1 (K, Qp (r)) with a readily recognizable homomorphism group.

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 79

4.4.1 Description of H 1 (K, Qp (r)) Rather than just working with Qp , let us consider a finite unramified extension K of Qp and denote [ Kn = K(ζpn ), K∞ = Kn , Γ = Gal(K∞ /K), ∼

ab B∞ := lim Kn× −→ Gal(K∞ /K∞ ) = P∞ , U∞ = lim U , ←− ←− n n

n

where Un denotes the principal units of Kn . The notation in (4.1) is the special case of above with K = Qp . By local class field theory we can identify U∞ with the inertia subgroup of P∞ , and observe that ˆ with the trivial action of Γ. Then, P∞ /U∞ ∼ =Z H 1 (K∞ , Qp (r)) = Hom(P∞ , Qp (r)), ∼ Hom(Z, ˆ Qp (r)) = ∼ Qp (r). Hom(P∞ /U∞ , Qp (r)) = Moreover, 0 −→ U∞ −→ P∞ −→ P∞ /U∞ −→ 0 =⇒ 0 −→ Hom(P∞ /U∞ , Qp (r)) ∼ = Qp (r) −→ Hom(P∞ , Qp (r)) −→ Hom(U∞ , Qp (r)) → 0 res

=⇒ 0 −→ H 0 (Γ, Qp (r)) −→ HomΓ (P∞ , Qp (r)) −→ HomΓ (U∞ , Qp (r)) −→ H 1 (Γ, Qp (r)). We also have the inflation-restriction sequence of cohomology res

0 −→ H 1 (Γ, Qp (r)) −→ H 1 (K, Qp (r)) −→ H 1 (K∞ , Qp (r))Γ −→ H 2 (Γ, Qp (r)) = 0. Combining all these, we obtain res

H 1 (K, Qp (r)) −→ H 1 (K∞ , Qp (r))Γ ∼ = HomΓ (P∞ , Qp (r)) res

−→ HomΓ (U∞ , Qp (r)).

(4.9)

We further have H q (Γ, Qp (r)) = 0 for q ≥ 1 and r 6= 0, and both the restriction maps in (4.9) become isomorphisms. Therefore, ∼

H 1 (K, Qp (r)) −→ HomΓ (U∞ , Qp (r))

r 6= 0.

Next, we will show that the latter module has a canonical element coming from the Coates–Wiles homomorphism.

80

Anupam Saikia

4.4.2 The Coates–Wiles homomorphism φrCW for a positive integer r We have mentioned that for any β ∈ B∞ , there is a uniquely associated Coleman power series gβ (T ) ∈ T −v(β) OK [[T ]]× such that (see theorem 2.2 in [Sh87]) (φ−n gβ )(ζpn − 1) = βn ∀n ≥ 1. For u ∈ U∞ and a positive integer r, the r-th Coates–Wiles homomorphism is defined as (see §2 of [BK90]) φrCW : U∞ −→ K(r) = K ⊗ Q(r),  d r u 7−→ (1 + T ) log gu (T ) |T =0 ⊗tr ∈ K ⊗ Q(r). dT It is often convenient to use the variable d d z = log(1 + T ) so that (1 + T ) = . (4.10) dT dz Then we can express  d r  g0  φrCW (u) = log gu (ez −1) |z=0 ⊗tr ∈ K⊗Q(r) = Res z −r u ⊗tr . dz gu Observe that guσ (T ) = gu ((1 + T )χ(σ) − 1) =⇒ φrCW (uσ ) = χr (σ)φrCW (u), and hence φrCW is a Γ-homomorphism from U∞ to K(r). By composing with the trace map T rK/Qp , we obtain a canonical element in HomΓ (U∞ , Qp (r)) = H 1 (K, Qp (r)).

4.4.3 The explicit reciprocity law The explicit reciprocity law of Bloch and Kato can be expressed as follows (cf. theorem 2.1 in [BK90] and §6 in [Sh95]): Theorem 4.4.1 Let K be a finite unramified extension of Qp for an odd prime p. Then the connecting homomorphism of the exact sequence (4.8) + ∂ r : K = H 0 (K, Bcris ) → H 1 (K, Qp (r)) = HomΓ (U∞ , Qp (r))

is given by ∂ r (a) = −

1 T rK/Qp (a.φrCW ) (r − 1))!

r ≥ 1.

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 81 In particular, if K = Qp , we have ∂ r (1) = −

1 φr . (r − 1))! CW

In the rest of this section, we will show that the above reciprocity law for r = 1 and K = Qp follows from the classical explicit reciprocity law. The proof for r ≥ 2 can be reduced to the case r = 1, but the reduction steps are quite technical, and will not be discussed in this article (see §8, 9 in [Sh95]). In the following, we will present a detailed proof of the above theorem (cf. [Sh95]). We will first show that it is enough to work with cohomology over K∞ , and then descend to Qp by taking the Γ-invariants in the following commutative diagram:

p−φ

0 −−−−→ [ε]Zp −−−−→ 1 + J −−−−→ 1 + pW (R) −−−−→ 0       −1 logy logy yp log 1−p−1 φ

<1> 0 −−−−→ Zp (1) −−−−→ Jcris −−−−−→

Acris

−−−−→ 0

Commutative Diagram: 1 Let T = [ε] − 1, and observe that Zp [[T]] ⊂ H 0 (K∞ , Acris ). As explained at the beginning of this section, we have H 1 (K∞ , Zp (1))  Hom(U∞ , Zp (1)). Restricting the connecting homomorphism of GK∞ cohomology of the bottom row to Zp [[T]], we obtain a continuous pairing of Γ-modules ∂ 1 : Zp [[T]] × U∞ → Zp (1),

∂ 1 (f (T)), u) = ∂ 1 (f (T))(u).

Proposition 4.4.2 ∂ 1 (f (T), u) = −Res



1 g 0 (T )  f (T ) u ⊗ t. log(1 + T ) gu (T )

The above proposition can be regarded as key to the reciprocity law of Bloch and Kato. We have ∂1

+ (H 0 (K∞ , Bcris ))Γ −−−−→ H 1 (K∞ , Qp (1))Γ



+ H 0 (Qp , Bcris )

∂1

−−−−→ H 1 (Qp , Qp (1)),

82

Anupam Saikia

where the second equality comes from the inflation-restriction sequence 0 −→ H 1 (K∞ /Qp , Qp (1)) = 0 −→ H 1 (Qp , Qp (1)) −→ H 1 (K∞ , Qp (1))Γ −→ H 2 (K∞ /Qp , Qp (1)) = 0. By taking α(T) = 1, we can deduce from the proposition that   1 d ∂ 1 (1)(u) = ∂ 1 (1, u) = −Res log gu (T ) ⊗ t = −φ1CW (u). log(1 + T ) dT

4.4.4 Proof of the key proposition 4.4.2 For u ∈ U∞ , consider the local Artin symbol σu ∈ GK∞ ab /K . We want ∞ to determine how the cocycle corresponding to the image of an element α ∈ H 0 (K∞ , 1 + pW (R)) under the connecting homomorphism of the top row of the commutative diagram 1 acts on σu . Pick β ∈ 1 + J such that β p−φ = α in the top row. As (β σu −1 )p−φ = ασu −1 = 1, we must have β σu −1 = [a] for some a ∈ R by proposition (4.3.2). But θ(β) = 1 implies that a(0) = 1, and hence β σu −1 = [ε](α,u)

for some (α, u) ∈ Zp .

(4.11)

Then, the connecting homomorphism of the GK∞ -cohomology of the top row sends α to the cocycle σu 7→ β σu −1 = [ε](α,u) . Thus, we have α7→(σu 7→β σu −1 =[ε](α,u) )

H 0 (K∞ , 1 + pW (R)) −−−−−−−−−−−−−−−−→ H 1 (K∞ , [ε]Zp )     p−1 logy [ε](α,u) 7→log([ε](α,u) )=(α,u)⊗tylog H 0 (K∞ , Acris )

∂1

−−−−→

H 1 (K∞ , Zp (1)),

and ∂ 1 (p−1 log(α), u)) = ∂ 1 (p−1 log(α))(u) = log([ε](α,u) ) = (α, u) ⊗ t.

(4.12)

We will now relate (α, u) to the logarithmic derivative of the Coleman power series of u using the classical explicit reciprocity law. Lemma 4.4.3   (α, u) = p−n T rKn /Qp log(α(n) )(gu0 /gu ) (ζpn − 1) mod pn . Proof: Applying θφ−n in the definition (4.11) of (α, u), we obtain (α,u)

θφ−n (β σu −1 ) = θφ−n [ε](α,u) = ζpn

[see lemma 4.3.3].

(4.13)

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 83 Next, we compute θφ−n (β σu −1 ) using Kummer theory. As β p−φ = α we have βp

n

−φn

n−1

= αp

+pn−2 φ+···+φn−1

. n

Now apply θφ−n on both sides and observe that θφ−n (β φ ) = θ(β) = 1 as β ∈ 1 + J. Therefore, n

n−1

θφ−n (β)p = θφ−n (α)p

n−2

. · · · . θφ−1 (α).

. θφ1−n (α)p

Now take α = α(T) ∈ 1 + pZp [[T]]. As θφ−n ([ε] − 1) = ζpn − 1, we have θφ−n α(T) = α(ζpn − 1). Therefore, n

θφ−n (β)p

n−1

n−2

= α(ζpn − 1)p

. α(ζpn −1 − 1)p

. · · · . α(ζp − 1) n−1

= α(ζpn − 1)α((1 + ζpn − 1)p − 1) . · · · α((1 + ζpn − 1)p

− 1).

Let us denote α(n) (T) = α(T)p

n−1

n−2

. α((1 + T)p − 1)p

n−1

. · · · . α((1 + T)p

− 1). (4.14)

Then,
1n θφ−n (β) = α(n) (ζpn − 1) p .

(4.15)

Therefore, θφ−n (β σu −1 ) = θφ−n (β)σu −1 
1n σu −1 = α(n) (ζpn − 1) p [α(n) (ζpn −1),u]n

= ζpn (α,u)

=⇒ ζpn

−n

p

= ζpn



T rKn /Qp log(α

(n)

[by (4.15)]

[by (4.2)] 

0 )(gu /gu ) (ζpn −1)

[by (4.13) and (4.3)].

Hence the lemma follows. We now simplify the right hand side of the above equality. Lemma 4.4.4 

For n ≥ 1, we have 

0 T rKn /Qp (log α(n) )(gu /gu ) (ζpn −1)=pn−1

P

0




0 (log α)(gu /gu ) (ζpi n −1)

Proof: For the Coleman power series gu (T ) of a norm compatible unit u, we have (§2.1 in [Sh87]) Y gu ((1 + T )p − 1) = gu ((1 + T )η − 1). η∈µp

84

Anupam Saikia

d We now take logarithm on both sides and apply the operator (1+T ) dT = d [see (4.10)] to obtain dz

p(gu0 /gu )(ζpm − 1) =

X

(gu0 /gu )(ζpm η − 1)

η∈µp

=

X

(gu0 /gu )(ξ − 1).

(4.16)

ξ p =ζpm

Now, computing the trace down one step at a time, we find that T rKn /Kn−1 (log(α(n) (ζpn −1 )(gu0 /gu )(ζpn − 1) X h = (pn−1 log α(ξ − 1) + (pn−2 log α(ξ p − 1) + · · · ξ p =ζpn−1

i − 1) (gu0 /gu )(ξ − 1) [by (4.14)]    pn−1 (log α)(gu0 /gu ) (ξ − 1) + (pn−2 log α(ζpn−1 − 1)

n−1

=

+ log(ξ p X ξ p =ζpn−1

+ · · · + log(ζp − 1)



X

(gu0 /gu )(ξ − 1)

ξ p =ζpn−1

X

=

   pn−1 (log α)(gu0 /gu ) (ξ − 1) + (pn−1 log α(ζpn−1 − 1)

ξ p =ζpn−1

 + · · · + p log α(ζp − 1) (gu0 /gu )(ζpn−1 − 1) [by (4.16)]. Next, we take trace down to Kn−2 and obtain 0 T rKn /Kn−2 log(α(n) (ζpn −1)(gu /gu )(ζpn −1))

h = T rKn−1 /Kn−2

P ξp =ζ n−1 p








0 pn−1 (log α)(gu /gu ) (ξ−1)





0 + (pn−1 log α(ζpn−1 −1)+···+p log α(ζp −1) (gu /gu )(ζpn−1 −1)

P

=

2 ξp =ζ n−2 p





0 pn−1 (log α)(gu /gu ) (ξ−1)+

η p =ζ n−2 p

+ pn−2 log α)(ζn−2 −1)+···+p log α(ζp −1) P

=

2 ξp =ζ n−2 p





P







0 (pn−1 (log α)(gu /gu ) (ξ−1)+



i

P η p =ζ n−2 p

P

ξp =ζ n−2 p



0 pn−1 (log α)(gu /gu ) (η−1)

0 (gu /gu )(η−1)





0 pn−1 (log α)(gu /gu ) (ξ−1)+



0 pn−1 log α(ζpn−2 −1)+···+p2 log(ζp −1) (gu /gu )(ζpn−2 −1).

Continuing in this fashion, we obtain the lemma.

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 85 In view of lemmas 4.4.3 and 4.4.4, we have p−n T rKn /Qp log(α(n) )(gu0 /gu )(ζpn − 1) mod pn X = (p−1 log(α)(gu0 /gu )(ζpi n − 1) mod pn . (4.17)

(α, u) =

0
For any h(T ) ∈ Zp [[T ]] and n ≥ 1,   X 1 h(ζpi n − 1) = −Res h(T ) mod pn . log(1 + T ) 0
Lemma 4.4.5

Proof: It is enough to check the above for h(T ) = (1 + T )m , m ≥ 0, as it will then hold for Zp [T ] and, by continuity, for Zp [[T ]] as well. For h(T ) = (1 + T )m , the left hand side is clearly pn − 1 if pn | m. If pn - m, then the left hand side is −1. The right hand side can be computed as −Res(T −1 (1 + T /2 − · · · )(1 + T )m ) = −1.



By combining the previous lemma with (4.17), we find that X   (α, u) = (p−1 log(α)(gu0 /gu ) (ζpi n − 1) mod pn ∀n 0
=

 Res

 1 p−1 log(α)(gu0 /gu ) mod pn log(1 + T )

(4.18) ∀n

By (4.12) and (4.18), we have  ∂ 1 (p−1 log(α))(u) = (α, u) ⊗ t = Res

 1 p−1 log(α)(gu0 /gu ) ⊗ t. log(1 + T )

As α = α(T) ∈ 1 + Zp [[T]] was arbitrary, so is p−1 log(α) ∈ Zp [[T]], the proof of proposition 4.4.2 is complete. The key proposition 4.4.2 generalizes for any β ∈ B∞ , and not just for u ∈ U∞ . For β ∈ B∞ with vp (β) = d, we have gβ0 /gβ ∈ T −1 Zp [[T ]], and the proof still works with minor modifications.

4.5 The exponential and the dual exponential maps For this section, we refer the readers to chapter II of [Kt93a]. Let V be a de Rham representation. We are interested in the cohomology groups H q (GK , BdR ⊗Qp V ). We have a BdR -linear GK -equivariant isomorphism

86

Anupam Saikia

between BdR ⊗Qp V and BdR ⊗K DdR (V ). It follows that H 1 (GK , BdR ⊗Qp V ) = H 1 (GK , BdR ⊗K DdR (V )) ∼ = H 1 (GK , BdR ) ⊗K DdR (V ). Therefore, it is enough to look into the cohomology of BdR which we do next.

4.5.1 Galois cohomology of BdR We have the following well-known theorem (theorem 5.2.4 of [FO]). Theorem 4.5.1

Let K be a finite extension of Qp . Then

1. H q (GK , Cp (r)) = 0 unless q = 0, 1 and r = 0. 2. H 0 (GK , Cp ) = K. 3. H 1 (GK , Cp ) = K. log χ where χ is the cyclotomic character, and the ∼ cup-product x 7→ x ∪ log χ gives an isomorphism H 0 (GK , Cp ) −→ H 1 (GK , Cp ). Theorem 4.5.2

Suppose i < j ∈ Z ∪ {±∞}.

j i 1. If i ≥ 1 we have H q (GK , BdR /BdR )=0 ∀j ∀q ≥ 0. j i 2. If j ≤ 0 we have H q (GK , BdR /BdR )=0 ∀i ∀q ≥ 0. 3. If i ≤ 0 and j > 0, then x 7−→ x ∪ log χ gives an isomorphism ∼ j j i i H 0 (GK , BdR /BdR )(∼ /BdR ). = K) −→ H 1 (GK , BdR i+1 i Proof: 1. When i ≥ 1, H q (GK , BdR /BdR ) = H q (GK , Cp (i)) = 0 by part 1 of the theorem above. If the result holds for j, then it also holds for j + 1 by the long exact sequence of GK -cohomology j j+1 j+1 j i i 0 −→ BdR /BdR −→ BdR /BdR −→ BdR /BdR −→ 0,

noting that j j+1 ∼ H q (GK , BdR /BdR ) = H q (GK , Cp (j)) = 0 for j > 0.

2. When 0 ≥ j(> i), then the result holds for i = j − 1, as j−1 j H q (GK , BdR /BdR ) = H q (GK , Cp (j − 1)) = 0 for j − 1 < 0.

If the result holds for i, then it also holds for i − 1 by the long exact sequence of GK -cohomology j j i−1 i−1 i i 0 −→ BdR /BdR −→ BdR /BdR −→ BdR /BdR −→ 0,

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 87 i−1 i noting that H q (GK , BdR /BdR ) = H q (GK , Cp (i − 1)) = 0 for i < 0.

3. When i ≤ 0 < j, we combine the long exact sequences of GK cohomology from j j 0 i i 0 0 −→ BdR /BdR −→ BdR /BdR −→ BdR /BdR −→ 0, j j 1 0 0 1 0 −→ BdR /BdR −→ BdR /BdR −→ BdR /BdR −→ 0

and use the previous two parts to deduce that j j i 0 0 1 H q (GK , BdR /BdR ) = H q (GK , BdR /BdR ) = H q (GK , BdR /BdR )

= H q (GK , Cp ). The rest follows from the previous theorem. When i or j is infinite, we can prove it by taking limits.  In particular, it follows that + H q (GK , BdR /BdR ) = 0,

+ H q (GK , BdR ) = H q (GK , BdR )

∀q. (4.19)

4.5.2 The dual exponential By virtue of theorem 4.5.2 we obtain an isomorphism induced by cupproduct ∼

K = H 0 (GK , BdR ) −→ H 1 (GK , BdR ),

x 7−→ x ∪ log χ.

(4.20)

By (4.19) and (4.20), we obtain an isomorphism ∼ DdR (V ) −→ H 1 (GK , BdR ) ⊗K DdR (V ) ∼ = H 1 (GK , BdR ⊗Qp V ),

x 7−→ x ∪ log χ. The dual exponential map exp∗ is defined as the composite ∼

H 1 (GK , V ) −→ H 1 (GK , BdR ⊗Qp V ) −→ DdR (V ), where the second map is the inverse of the isomorphism given by ∪ log χ. Proposition 4.5.3

0 The image of exp∗ lies in DdR (V ).

Proof: It is clear from the previous section that the image of any element + c of H 1 (GK , V ) in H 1 (GK , BdR /BdR ⊗Qp V ) is 0. But we have injective maps x7→x∪log χ

+ 0 DdR (V )/DdR (V ),−→H 0 (GK ,BdR /BdR ⊗Qp V )

,−→

+ H 1 (GK ,BdR /BdR ⊗Qp V )

88

Anupam Saikia

where the first injection comes from the cohomology of the exact sequence + + 0 −→ BdR ⊗Qp V −→ BdR ⊗Qp V −→ BdR /BdR ⊗Qp V −→ 0. 0 Hence, exp∗ (c) ∈ DdR (V ).

The dual exponential indeed turns out to be the dual of a natural map (to be called the exponential map) associated with the cohomology of the representation V .

4.5.3 The exponential map The fundamental exact sequence of Bloch and Kato [(1.17.1) of [BK90]] is given by α

β

φ=1 + 0 −→ Qp −→ Bcris ⊕ BdR −→ BdR −→ 0

α(x) = (x, x),

(4.21)

β(x, y) = x − y.

By tensoring with a p-adic representation V of GK , and taking the long exact sequence of cohomology, we obtain δ

0 0 −→ H 0 (GK , V ) −→ Dcris (V )φ=1 ⊕DdR (V ) −→ DdR (V ) −→ H 1 (K, V ).

The connecting homomorphism above induces the exponential map to be denoted by exp. From the exactness of the above sequence, it is clear 0 that δ factors through DdR (V ) and the resulting map is known as the exponential map. In other words, 0 exp : DdR (V )/DdR (V ) −→ H 1 (K, V ),

0 exp(x mod DdR (V )) = δ(x).

Our immediate goal is to show that the maps exp and exp∗ are related via Tate duality.

4.5.4 Tate duality Let V ∗ = HomQp (V, Qp ) be the dual representation of V , with GK action given by h 7→ h.σ −1 . The cup-product gives a perfect pairing (corollary 2.3 of [Mi86]) H q (GK , V ) × H 2−q (GK , V ∗ (1)) −→ H 2 (GK , Qp (1)) ∼ = Qp ,

q = 0, 1, 2.

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 89 We also have a canonical perfect K-bilinear pairing given by cup-product (theorem 5.2.1 of [BC09]) ∪

h , i : DdR (V ) × DdR (V ∗ (1)) −→ H 0+0 (GK , BdR ⊗Qp V ⊗Qp V ∗ (1)) ∼ DdR (Qp (1)) ∼ (4.22) = = K. Proposition 4.5.4 The above paring h , i followed by the trace map T rK/Qp : K −→ Qp induces an isomorphism between DdR (V ) and HomQp (DdR (V ∗ (1)), Qp ). Proof: Consider the Qp -linear map
f : DdR (V ) −→ HomQp DdR (V ∗ (1)), Qp , x 7−→ fx : y 7→ T rK/Qp (hx, yi)

∀y ∈ DdR (V ∗ (1)).

The map f must be injective as follows. If x belongs to the kernel of f , then T rK/Qp (hx, kyi) = T rK/Qp (khx, yi) = 0 ∀ k ∈ K. If x 6= 0 then there exists y such that hx, yi = β 6= 0 as h, i is nondegenerate. Then, T rK/Qp (kβ) must be 0 for all k. Taking k = β −1 , we would obtain T rK/Qp (β −1 β) = [K : Qp ] = 0, which is absurd. Hence x = 0 and f must be
injective. But the vector spaces DdR (V ) and HomQp DdR (V ∗ (1)), Qp have the same dimension over Qp as DdR (V ) and DdR (V ∗ (1)) have the same dimension over K. So f gives an
isomorphism which identifies DdR (V ) with HomQp DdR (V ∗ (1)), Qp . 

4.5.5 The relation between exp and exp∗ The following theorem establishes a relation between the exponential map and its dual. Theorem 4.5.5 tial map

(cf. theorem 1.4.1 (4) in [Kt93a]) The dual exponen-

exp∗V ∗ (1) : H 1 (GK , V ∗ (1)) −→ DdR (V ∗ (1)) coincides with the composite map ∼


∼

H 1 (GK ,V ∗ (1))→HomQp (H 1 (GK ,V ),Qp )→HomQp DdR (V ),Qp →DdR (V ∗ (1)).

where the first map is by Tate duality, and the second one by the Qp -dual of the exponential map of V , and the third is induced by the canonical pairing mentioned above. In other words, for a ∈ DdR (V ) and

90

Anupam Saikia

b ∈ H 1 (GK , V ∗ (1)) we have expV (a) ∪ b = −T rK/Qp (ha, exp∗V ∗ (1) (b)i). In order to prove the theorem, we need the lemma below. + 1 Lemma 4.5.6 (Lemma 1.4.3 in [Kt93a]) Let ψ : H 1 (GK , BdR /BdR )→ 2 H (GK , Qp (1)) be the connecting homomorphism of the exact sequence φ=p + + 1 0 −→ Qp (1) −→ Bcris ∩ BdR −→ BdR /BdR −→ 0.

Then for any de Rham representation V of GK , a ∈ DdR (V ) and b ∈ H 1 (GK , V ∗ (1)), we have expV (a) ∪ b = ψ(ha, exp∗V ∗ (1) (b)i log χ) ∈ H 2 (GK , Qp (1)), where h , i : DdR (V ) × DdR (V ∗ (1)) → K, is the pairing considered in (4.22). Proof: It is well known that the connecting homomorphism of cohomology is compatible with cup-product (proposition 1.4.3 in [NSW08]). Hence we have the following commutative diagram H 1 (GK ,V )

×

x a7→exp (a)  V H 0 (GK ,BdR ⊗Q V ) p

×

H 1 (GK ,V ∗ (1))

∪

− −−−− →

 b7→exp∗ ∗ (b)∪log χ y V (1) H 1 (GK ,BdR ⊗Q V ∗ (1)) p

− −−−− →

H 2 (GK ,Qp (1))∼ =Qp

x ψ  + 1 1 H 1 (GK ,BdR )∼ =H (GK ,BdR /BdR ),

where the middle arrow comes from the very definition of exp∗ . By associativity and compatibility of cup-product with connecting homomorphism of cohomology, we have expV (a) ∪ b = ψ(a ∪ (exp∗V ∗ (1) b ∪ log χ)) = ψ(ha, exp∗V ∗ (1) (b)i log χ).  Proof of theorem 4.5.5: In view of the previous lemma, it is enough to show that the map + 1 ψ : H 1 (GK , BdR /BdR ) = H 1 (GK , Cp ) = K log χ → H 2 (GK , Qp (1)) = Qp

sends c log χ to −T rK/Qp (c). First assume that K = Qp and take V = Qp (1) in the commutative diagram above. Then by local class field theory of Qp , the pairing H 1 (GQp , Qp (1)) × H 1 (GQp , Qp ) → H 2 (GQp , Qp (1)) ∼ = Qp

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 91 sends (exp(c), log χ) to −c (lemma I.4.5 in [Kt93a]). Now for any finite extension K of Qp we use compatibility of the corestriction map with the connecting homomorphism ψ (proposition 1.5.2 in [NSW08]) as well as cup-product (proposition 1.5.3 in [NSW08]) in the following commutative diagram. ψ

∪

H 0 (GK , BdR ) × H 1 (GK , BdR ) − → H 1 (GK , BdR ) − → H 2 (GK , Qp (1)) (∼ (∼ = K) = K log χ)  x  res cory=T rK/Qp 

(∼ = K log χ)  cor y

(. ∼ = Qp )

ψ

∪

H 0 (GQp , BdR )×H 1 (GQp , BdR ) − → H 1 (GQp , BdR ) − → H 2 (GQp , Qp (1)) (∼ = Qp )

(∼ = Qp log χ)

(∼ = Qp log χ)

(∼ = Qp )



4.5.6 Relation between exp and Coates–Wiles homomorphism Let K be a finite unramified extension of Qp . By tensoring the fundamental sequence (4.21) with Qp (r) = Qp .tr , we obtain α

r

βr

r φ=p r 0 −→ Qp (r) −→ Bcris ⊕ BdR −→ BdR −→ 0

αr (x) = (x, x),

(4.23)

βr (x, y) = x − y.

From the Galois cohomology sequence with respect to GK , we obtain a connecting homomorphism expr : K = DdR (Qp (r)) −→ H 1 (GK , Qp (r)) = HomΓ (U∞ , Qp (r)). On the other hand, the exact sequence 1−p−r φ

+ + 0 −→ Qp (r) −→ Jcris ⊗ Q = Filr Bcris −−−−−→ Bcris −→ 0 (4.24)

gives another connecting homomorphism + ∂ r : K = H 0 (GK , Bcris ) −→ H 1 (GK , Qp (r)) = HomΓ (U∞ , Qp (r)).

In this subsection we establish a relation between these two connecting homomorphisms. As a consequence, we will be able to express the exponential map in terms of the Coates–Wiles homomorphism in view of theorem 4.4.1. We have the following result. Theorem 4.5.7 expr (a)(g) = −∂ r (1 − p−r ap )(g),

a ∈ K, g ∈ GK .

92

Anupam Saikia r

φ=p r Proof: Let (x, y) ∈ Bcris ⊕ BdR be a pre-image of an element a ∈ K = 0 H (GK , BdR ) under βr in (4.23), i.e., x − y = a. By definition,

expr (a)(g) = c, where (c, c) = (x, y)g − (x, y) = (xg − x, y g − y). Note that βr ((xg − x, y g − y)) = (x − y)g − (x − y) = ag − a = 0. Thus, we can take expr (a)(g) = y g − y for g ∈ GK . + It is now enough to show that y ∈ Filr (Bcris ), as the connecting homor morphism ∂ of the exact sequence (4.24) will then send (1 − p−r φ)(y) to g 7→ y g − g, and

expr (a)(g) = y g − g = ∂ r ((1 − p−r φ)(y))(g)   = ∂ r (x − a) − p−r (φ(x) − φ(a))(g) = −∂ r (1 − p−r φ)(a)(g)

r

φ=p as x ∈ Bcris .

r Now, y = x − a ∈ BdR ∩ Bcris = Filr (Bcris ). It is enough to show that S + + + y ∈ Bcris . Observe that Bcris = s≥0 t−s Bcris , hence ts y ∈ Bcris for some s ≥ 0. If s = 0, we are done. Suppose s ≥ 1. Then, m

m

m

y φ = (x − a)φ = pmr x − aφ = pmr (y + a) − aφ

m

∈ Filr Bcris + Fil0 (Bcris ) ⊂ Fil0 Bcris . As φ(t) = pt (see (4.6)), we have m

m

(ts y)φ = psm ts y φ ∈ Fils Bcris ⊂ Fil1 Bcris . By a result of Fontaine (Prop 4.14 in [Fo82]), it now follows that + + ts y ∈ tBcris =⇒ ts−1 y ∈ Bcris . + By repeating the process, we arrive at y ∈ Bcris .

4.5.7 The image of Soul´ e–Deligne elements under the dual exponential The reference for this section is §3 of [HK03] (cf. §6 in [BK90]). The Soul´e–Deligne cyclotomic elements form an Euler system and hence are related to p-adic L-functions. Let r be a positive integer as in sections

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 93 3.2 and 3.3. Let N be an integer coprime to p, and K = Qp (ζN ). Let cp ≡ 1 modulo N , so that n

c φ−n (ζN ) = ζN .

Consider the tower of fields Kn = Qp (ζN , ζpn ) = K(ζpn ), where ζpn is a primitive pn -th power of unity such that ζppn = ζpn−1 for all n ∈ N. Let B∞ = lim Kn× . ←− n

The Soul´e–Deligne cyclotomic element c1−r (ζN ) ∈ H 1 (GK , Zp (1 − r)) is obtained from the norm-compatible element u in B∞ given by 1 − ζ  N c c2 cn n, · · · 2, · · · , 1 − ζ u= , 1 − ζ ζ , 1 − ζ ζ ζ N 6= 1, p p p N N N c 1 − ζN   u = p, 1 − ζp , 1 − ζp2 , · · · , 1 − ζpn , · · · N = 1, K = Qp ,

as follows (cf §2.1.5 in [Kt93a]): ⊗(−r)

⊗ζpn

Kummer

n

c u = (1 − ζN ζpn ) ∈ lim←− Kn× −−→ lim H 1 (GKn , Zp (1)) −−→ ← n



lim←− H 1 GKn , n

Zp (1 pn Zp

n



cores

  Zp 1 − r) −−→ lim H G , (1 − r) K n ←− p Zp n ∼

→ H 1 (GK , Zp (1 − r)). We have the following commutative diagram:

DdR (Qp (r))  exp y r

T rK/Qp

×DdR (Qp (1 − r)) → K −−−−−→ x exp∗  1−r

H 1 (GK , Qp (r)) ×   y Hom(U∞ , Qp (r))×

Qp



∪ H 1 (GK , Qp (1 − r)) − → H 2 (GK , Qp (1)) ∼ = Qp x x   ⊗(−r) n u=(1−ζNc ζpn )7→c1−r (ζN ) ⊗t

U∞

→ −

Qp (r).

The commutativity of the upper-half follows from theorem 4.5.5. In order to examine the commutativity of the lower-half, let us view it at the finite level n. It is well known that restriction, co-restriction and connecting homomorphisms of cohomology behave well with respect to cup-product

94

Anupam Saikia

(propositions 1.4.3 and 1.5.3 in [NSW08]). Hence we have the following commutative diagram:

H1

GK ,(Zp /pn Zp )(r)




×

H1

GK ,(Zp /pn Zp )(1−r)




∪

− →

H2

GK ,(Zp /pn Zp )(1)




n (∼ =Zp /p Zp )

  resy H1

GK

n

x cores 

,(Zp /pn Zp )(r)




×

H1

GK

n

x cores 

,(Zp /pn Zp )(1−r)




∪

− →

H2

GK ,(Zp /pn Zp )(1) n




n (∼ =Zp /p Zp )



Hom

GK

n



,(Zp /pn Zp )(r)




× H1




⊗(−r) GK ,(Zp /pn Zp )(1) ⊗ζ n n p



Hom

n Gab Kn ,(Zp /p Zp )(r)




×

H1

GK

n

,(Zp /pn Zp )(1)

× ,(Z /pn Z )(r) Kn p p




×

H0

GK

n

,Kn ×




× =Kn

∪

− →




×

Un

⊗(r) Zp /pn Zp ⊗ζ n p

x   − →

x  

Un ,(Zp /pn Zp )(r)

Zp /pn Zp

x ⊗ζ⊗(−r)  pn

x Kummer 


  resy Hom

∪

− →

x ⊗ζ⊗(−r)  pn



Hom



(Zp /pn Zp )(r)

x   − →

(Zp /pn Zp )(r).

For a ∈ K = DdR (Qp (r)), we can compute exp∗1−r (c1−r (ζN )) as −T rK/Qp (a exp∗1−r (c1−r (ζN ))) = expr (a) ∪ (c1−r (ζN )) r

= −∂ (1 − p

−r φ

a )(u) ⊗ t

(by theorem 4.5.5) ⊗(−r)

(by theorem 4.5.7 and the previous commutative diagram)

i 1 h = T rK/Qp aφrCW (u) − p−r T rK/Qp aφ φrCW (u) ⊗ t⊗(−r) (r − 1)! (by theorem 3.1) i
−1
1 h = T rK/Qp aφrCW (u) − p−r T rK/Qp aφrCW (u)φ ⊗ t⊗(−r) (r − 1)!  i 1 h d
r = T rK/Qp a(1 − p−r φ−1 ) (1 + T ) log gu (T ) |T =0 , (r − 1)! dT

Explicit Reciprocity Law of Bloch–Kato and Exponential Maps 95 where gu (T ) is the Coleman power series of u given by gu (T ) = 1 − ζN (1 + T ). Using the modified Soul´e–Deligne cyclotomic element c˜1−r (ζN ) = −

∞ X

i

p ) (pr )i c1−r (ζN

i=1

=−

∞ X

(pr φ)i c1−r (ζN ),

i=1

we can removes the Euler factor (1 − p−r φ−1 ) and obtain T rK/Qp (a exp∗1−r (˜ c1−r (ζN ))) h i  −1 d
r = log gu (T ) |T =0 .  T rK/Qp a (1 + T ) (r − 1)! dT The above result allows us to relate the cyclotomic elements c˜1−r (ζN ) to zeta values. It further shows that these cyclotomic elements are nontorsion as cohomology classes (§3.2 in [HK03]). As a consequence, the Bloch–Kato conjecture for the Riemann zeta function up to powers of 2 can be reduced to a statement about the Soul´e–Deligne cyclotomic elements (loc. cit.).

References [Be04] Berger, L. 2004. An introduction to the theory of p-adic representations. Geometric aspects of Dwork theory, Vol. I, II, 255292, Walter de Gruyter GmbH Co. KG, Berlin. [BO78] Berthelot, P., and Ogus, A. 1978. Notes on Crystalline Cohomology. Princeton University Press. [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I. Progr. Math., 86, 333400. Birkh¨ auser Boston, Boston, MA. [BC09] Brinon, O., and Conrad, B. 2009. Notes on p-adic Hodge Theory. CMI Summer School, available online at www.math.stanford.edu/conrad/papers/notes.pdf [CF67] Cassels, J. W. S., and Frohlich, A. 1967. Algebraic Number Fields. Academic Press. [Co] Colmez, P. Fontaine’s rings and p-adic L-functions. Course given at Tsinghua University (www.math.jussieu.fr/∼colmez/tsinghua.pdf). [Fo82] Fontaine, J. M. 1982. Sur certains types de repr´esentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. of Math. (2) 115, no. 3, 529–577.

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[FO] Fontaine, J. M., and Ouyang, Y. Theory of p-adic Galois Representations. Book in preparation (www.math.upsud.fr/∼fontaine/galoisrep.pdf). [Gu99] Guo, L. 1999. Hecke characters and formal group characters. Topics in number theory (University Park, PA, 1997). Math. Appl., 467, 181192. Kluwer Acad. Publ., Dordrecht. [HK03] Huber, A., and Kings, G. 2003. Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters. Duke Math. J., 119, no. 3, 393464. [Iw86] Iwasawa, K. 1986. Local Class Field Theory. Oxford Science Publications. [Kt93a] Kato, K. 1993. Lectures on the approach to Iwasawa theory for HasseWeil L-functions via BdR. I. Arithmetic algebraic geometry (Trento, 1991). Lecture Notes in Math., 1553, 50163, Springer, Berlin. [Kt93b] Kato, K. 1993. Iwasawa theory and p-adic Hodge theory. Kodai Math. J., 16, no. 1, 131. [Mi86] Milne, J. S. 1986. Arithmetic Duality Theorems. Academic Press. [NSW08] Neukirch, J., Schmidt, A., and Wingberg, K. 2008. Cohomology of Number Fields. Springer. [Se79] Serre, J. P. 1979. Local fields. Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin. [Sh87] Shalit, E. 1987. Iwasawa theory of elliptic curves with complex multiplication. Perspectives in Mathematics, 3. Academic Press, Inc., Boston, MA. [Sh95] Shalit, E. 1995. The explicit reciprocity law of Bloch-Kato. Columbia University Number Theory Seminar (New York, 1992). Astrisque no. 228, 4, 197221.

5 The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture Manfred Kolster

5.1 Introduction The ‘other’ Bloch–Kato Conjecture, also referred to as the ‘Norm Residue Theorem’ (Theorem 5.5.6), relates Milnor K-theory to Galois cohomology, and has been proven now by Voevodsky [Vo11] with major contributions from Rost, Suslin and Weibel. This article attempts to describe the historic development of the conjecture and some of the major achievements that led to the final proof. We try to describe some of the key ideas behind the proof and some consequences of the result, in particular on the relationship between algebraic K-theory, motivic cohomology and ´etale cohomology for rings of integers in number fields in the framework of the Quillen–Lichtenbaum Conjecture.

The final part of the proof of the theorem has appeared in [Vo11], but the method had already been outlined in [Vo03], where Voevodsky proved the Milnor Conjecture (the special case that l = 2). A good introduction to the sophisticated and deep methods developed by Voevodsky for motivic cohomology, which are used in the proof and which are beyond the scope of this article, can be found in the books [VSF00] and [MVW06], the last one being based on lectures given by Voevodsky in 1999/2000. The book [HW12], which is supposed to appear soon, provides a comprehensive treatment of the whole material surrounding the proof.

McMaster University, Hamilton, Canada. e-mail : [email protected]

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5.2 Background L∞ We recall that for a field F the Milnor ring K∗M (F ) = n=0 KnM (F ) of F was defined by Milnor in [Mi70] as follows: K0M (F ) = K0 (F ) = Z and K1M (F ) = K1 (F ) = F ∗ , the elements in K1M (F ) being written additively as symbols l(a), a ∈ F ∗ , satisfying l(ab) = l(a) + l(b). Let T (F ∗ ) denote the tensor algebra over Z generated by K1M (F ), and let I denote the two-sided ideal generated by tensors of the form l(a)⊗l(1−a), a 6= 0, 1. Then the Milnor ring is defined as K∗M (F ) = T (F ∗ )/I. The graded piece KnM (F ) is then the quotient of F ∗ ⊗· · ·⊗F ∗ (n factors) by the subgroup generated by those symbols l(a1 )⊗l(a2 )⊗· · ·⊗l(an ) satisfying ai + ai+1 = 0 for some i, 1 ≤ i ≤ n − 1. The generators of KnM (F ) are called symbols of length n and denoted by a = {a1 , · · ·, an }. In the special case n = 2 the group K2M (F ) is isomorphic to the Kgroup K2 (F ), which had been previously defined by Milnor for arbitrary rings, as a consequence of Matsumoto’s presentation of K2 (F ) for a field F in terms of generators and relations [Mi71]. The relation between the Milnor ring and the graded Galois cohomoL∞ logy ring n=0 H n (F, µ⊗n l ), l a prime number 6= char F , is based on the following Kummer isomorphism: We apply Galois cohomology to the Kummer sequence ·l

∗ ∗ 0 → µl → Fsep → Fsep → 0,

where Fsep denotes a separable closure of F and µl denotes the group of ∗ l-th roots of unity, and use the fact that H 1 (F, Fsep ) = 0, which is the cohomological version of Hilbert’s Theorem 90, to obtain the Kummer isomorphism ∼ H 1 (F, µl ). F ∗ /F ∗l = This isomorphism extends now to a homomorphism from the Milnor ring to the graded Galois cohomology ring, by sending a symbol {a1 , · · ·, an } to the cup-product a1 ∪ a2 · · · ∪ an ∈ H n (F, µ⊗n l ), where we identify the elements ai ∈ F ∗ with their images in H 1 (F, µl ). This homomorphism is well defined, since Tate observed that for n = 2 the cup-product a ∪1 −a vanishes in H 2 (F, µ⊗2 l ). The resulting homomorphism KnM (F )/l → H n (F, µ⊗n l )

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is called Norm Residue Homomorphism or Galois symbol. This homomorphism is clearly an isomorphism for n = 0, since both groups are isomorphic to Z/l, and is equal to the Kummer isomorphism for n = 1. For n = 2 and µl ⊂ F the cohomology group H 2 (F, µ⊗2 l ) is isomorphic to µl ⊗ l Br(F ), where for any abelian group A we denote by l A the subgroup of elements of exponent l in A. The map K2 (F )/l → µl ⊗ l Br(F ) sends {a, b} to ζ ⊗ [Aζ (a, b)], where ζ is a primitive l-th root of unity and [Aζ (a, b)] is the Brauer class of the central simple algebra Aζ (a, b), which is generated by two elements x and y, subject to the relations xl = a, y l = b and yx = ζxy. The class [Aζ (a, b)] is called norm residue symbol, because its value is 1 if and only if b is a norm from the field √ F ( n a) (for details see [Mi71, section 15]). For local or global fields the norm residue homomorphisms are isomorphisms for all integers n ≥ 0 and all primes l. For n = 2 and all primes l this was shown by Tate in [Ta76, Theorem 5.1] and for n ≥ 3 this follows – as Milnor observed – from the computations of the groups KnM (F ) for n ≥ 3 by Bass and Tate [BT73], which showed that for n ≥ 3 and a global field F one has KnM (F ) ∼ = (Z/2Z)r1 (F ) , where r1 (F ) is the number of real embeddings of F . In [Mi70] Milnor focussed on the Milnor ring k∗ (F ) := K∗M (F )/2 modulo 2 and its relationship both to the graded Galois cohomology L∞ ring n=0 H n (F, Z/2) and to the graded Witt ring gr W (F ) of a field F of odd characteristic. The latter is obtained as follows: Let W (F ) denote the Witt ring of F and let I(F ) denote the fundamental ideal in W (F ) consisting of all classes of bilinear forms of even rank. Then the graded Witt ring gr W (F ) is defined as gr W (F )) = ⊕n≥0 I n (F )/I n+1 (F ), where I 0 (F ) = W (F ). Given a1 , · · ·, an ∈ F ∗ /F ∗2 we denote by < a1 , · · ·, an > the quadratic Pn form i=1 ai x2i , and define the so-called Pfister form << a1 , · · ·, an >> via

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<< a1 , · · ·, an >>:=< 1, −a1 > ⊗ · · · ⊗ < 1, −an > . Milnor showed [Mi70, Theorem 4.1] that there is a unique homomorphism sn (F ) : kn (F ) → I n (F )/I n+1 (F ) for each n ≥ 0, which sends a symbol {a1 , a2 , · · ·, an } to the class of the Pfister form << a1 , · · ·, an >> mod I n+1 (F ), and that these homomorphisms are isomorphisms for n = 0, 1, 2 and surjective for all n. In the introduction to his paper [Mi70] Milnor conjectured not only that the sn are isomorphisms for all n, but also that the norm residue homomorphisms for l = 2 are isomorphisms: ‘Section 4 studies the conjecture that the associated graded ring (W/I, I/I 2 , I 2 /I 3 , · · · ) is canonically isomorphic to K∗ F/2K∗ F ’, and a few lines later he writes: ‘Section 6 describes the conjecture that K∗ F/2K∗ F is canonically isomorphic to the cohomology ring H ∗ (GF ; Z/2Z) · · · ’ . The second conjecture is now known as the Milnor Conjecture. It was proven by Voevodsky in [Vo03], and with the use of methods from [Vo03] it was then shown by Orlov, Vishik and Voevodsky [OVV07] that the Milnor Conjecture implies the first conjecture, namely that all homomorphisms sn are isomorphisms. These results answered some fundamental open questions in the algebraic theory of quadratic forms. In particular one obtains isomorphisms I n (F )/I n+1 (F ) ∼ = H n (F, Z/2Z), which define new cohomological invariants for bilinear forms. We refer to [OVV07] for more applications of the Milnor Conjecture to quadratic forms.

It seems that the general conjecture on the norm residue homomorphisms for arbitrary primes l was first explicitly formulated by Kato in the 1980 paper [Kt80, p.608]: ‘Concerning this homomorphism [the norm residue homomorphism] the experts perhaps have the following Conjecture [the Norm Residue Isomorphism Conjecture] in mind: · · · ’

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Bloch [Bl10, p.xvii] reflects about his contribution to the conjecture in the preface to the second edition (‘30 Years later’) of the reprint of his book ‘Lectures on Algebraic Cycles’ [Bl80] as follows: ‘My own contribution to this, which is explained in Lecture 5, is a proof that KnM (F )/l → H n (F, µ⊗n l ) is surjective when F has cohomological dimension n.’ In 1986 the conjecture was mentioned again in their joint paper [BK90, p.118], and since then has been referred to as the Bloch–Kato Conjecture. To avoid confusing this conjecture with the other Bloch–Kato Conjecture (the Tamagawa Number Conjecture) it is also referred to as The Norm Residue Isomorphism Conjecture.

5.3 Merkurjev–Suslin The first general result (for arbitrary fields) was obtained by Merkurjev [Me82] in 1981. He proved the Milnor conjecture for n = 2: Theorem 5.3.1 (Merkurjev) norm residue homomorphism

For a field F of odd characteristic the

K2 (F )/2 → H 2 (F, Z/2) is an isomorphism. This result gives a presentation of the subgroup 2 Br(F ) of Br(F ) of elements of exponent 2 in terms of generators and relations, and shows – in particular – that every element of exponent two in the Brauer group is similar to a tensor product of quaternion algebras. This answered an open question in the theory of algebras. Shortly afterwards the result was generalized to all primes l ([MS83]): Theorem 5.3.2 (Merkurjev–Suslin) For a field F and a prime l 6= char F , the norm residue homomorphism K2 (F )/l → H 2 (F, µ⊗2 l ) is an isomorphism. ∼ If F contains µl we have a (non-canonical) isomorphism H 2 (F, µ⊗2 l )= l Br(F ) by choosing a primitive l-th root of unity. Theorem 5.3.2 then

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implies again that elements of order l in the Brauer group are similar to tensor products of cyclic algebras of order l. The novelty of the ideas of Merkurjev and Suslin behind the proofs of Theorem 5.3.1 and Theorem 5.3.2 – in particular the use of Severi– Brauer varieties associated to central simple algebras, and Hilbert’s Theorem 90 for K2 for cyclic extensions – had a major influence on all the subsequent work on the Bloch–Kato Conjecture, and we will explain some of the ideas in more detail, once we have discussed some preliminary issues. We first note that both the Galois cohomology groups and the Milnor K-theory groups have natural restriction and corestriction maps for finite extensions E/F of fields, which we denote in both cases by iE/F resp. NE/F . For Milnor K-theory the corestriction is usually called transfer and has been – under certain assumptions – constructed in [BT73, section 5]. The full construction was done by Kato [Kt80]. Both homomorphisms commute with the norm residue homomorphisms, and the compositum NE/F ◦ iE/F is multiplication by the degree [E : F ]. This immediately implies: Lemma 5.3.3 If the norm residue homomorphism is an isomorphism for a prime l and a field F , then this is true also for any finite extension E of F of degree prime to l. We see that in order to prove the Bloch–Kato Conjecture we can assume without loss of generality that F contains the l-th roots of unity, and – if necessary – we can even assume that F does not have finite extensions of degree prime to l, in particular we can assume that F is perfect. Fields that do not have finite extensions of degree prime to l are called l-special [SJ06]. We note that for an l-special field all extensions of degree l are cyclic. The following result is due to Voevodsky ([Vo96, Lemma 5.2], [HW12, Lemma 1.1]) and shows that we can even restrict attention to fields of characteristic 0, if necessary, to prove the Bloch–Kato Conjecture. Lemma 5.3.4 If the norm residue homomorphism is an isomorphism for all fields of characteristic 0, then this is true for all fields in positive characteristic as well. Proof We sketch a proof, since the source [Vo96] will probably not be published and the other source [HW12] is still only available as a preprint.

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Let k be a field of positive characteristic 6= l, which we may assume to be perfect. Then the ring R of Witt vectors over k is a complete discrete valuation ring, and the natural homomorphism KnM (R)/l → KnM (k)/l is an isomorphism, since R is henselian. Let K denote the field of fractions of R. Then KnM (K) is generated by symbols of the form {π m1 a1 , π m2 a2 , · · ·, πmn an }, where π is a uniformizer in K and the ai ’s are units in R. The specialization map sπ : KnM (K) → KnM (k), which depends on the choice of π, maps the symbol {π m1 a1 , π m2 a2 , · · ·, π mn an } to {¯ a1 , a ¯2 , · · ·, a ¯n )}. This map induces a corresponding map sπ : KnM (K)/l → KnM (k)/l, ∼ K M (k)/l. The same works which is split surjective, since KnM (R)/l = n for the Galois cohomology groups and these split surjections commute with the norm residue homomorphisms. Since K has characteristic 0, we can assume the conjecture for K, and the result follows. After these preliminaries let us turn now to some of the main ideas behind the proof of Theorem 5.3.2: For a given central simple F -algebra A of degree m the associated Severi–Brauer variety X is a projective F -variety of dimension m − 1, which has a rational point in F if and only if A splits. Moreover, if F (X) denotes the function field of X, then the kernel of the map Br(F ) → Br(F (X)) is cyclic and generated by the class of A (for details see [GS06, Chapter 5], [Se79]). In dimension 1 the Severi–Brauer varieties are conics corresponding to quaternion algebras. The symbol a = {a, b} ∈ K2 (F )/2 maps to the class of the quaternion algebra (a, b), which in turn corresponds to the conic with equation ax2 + by 2 = z 2 . In general, if a = {a, b} ∈ K2 (F )/l is a symbol, then we denote by Xa the Severi–Brauer variety associated to a and by Fa the function field F (Xa ) of Xa . One of the fundamental ideas of Merkurjev and Suslin was to study the norm residue homomorphism on the level of function fields of Severi– Brauer varieties, which allowed a geometric approach, for which several K-theoretic tools were available: Quillen’s computation of the K-theory

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of Severi–Brauer varieties [Q73] and the Brown–Gersten–Quillen spectral sequence (also referred to as co-niveau spectral sequence). For details we refer to the survey article by Suslin [Su85, Sections 7–10] and the book of Srinivas [Sr91, section 8]. Using these techniques Merkurjev and Suslin were able to prove the following result [MS83, Prop. 13.5], which – as we will see – turned out to be fundamental for the general case of the Bloch–Kato Conjecture as well: Theorem 5.3.5 (Hilbert 90 for K2 ) Let E/F be a cyclic extension of degree l and let Gal(E/F ) = < σ > . Then the sequence σ−1

NE/F

K2 (E) → K2 (E) → K2 (F ) is exact. ‘Proof ’ of Theorem 5.3.2: To simplify the notation we denote again the quotient K2M (F )/l by k2 (F ). We also assume without loss of generality that F contains the ∼ l-th roots of unity, so that H 2 (F, µ⊗2 l ) = l Br(F ), and the norm residue homomorphism can be viewed to take values in l Br(F ). Any element x in the kernel of the norm residue homomorphism is Qm of the form x = i=1 {ai , bi }. If m = 1, say x = {a, b}, then it is well known that x must be trivial: The algebra Aζ (a, b) splits if and only if b √ is a norm from E = F ( l a). We therefore have in our case b = NE/F (c) for some c ∈ F ∗ . Hence √ x = {a, b} = {a, NE/F (c)} = NE/F {a, c} = (NE/F { l a, c})l = 1. We proceed now by induction on the number m of factors for all fields of characteristic 6= l. We can assume that the algebra Aζ (am , bm ) is not split, and we denote by X the corresponding Severi–Brauer variety. Then {am , bm } = 0 in k2 (F (X)), hence by induction x = 0 in k2 (F (X)). One of the non-trivial results [MS83, Corollary (9.3)] of Merkurjev– Suslin is that the kernel of the map k2 (F ) → k2 (F (X)) is cyclic and generated by {am , bm }. Under the norm residue homomorphism this kernel, which contains x, is mapped isomorphically to the cyclic group generated by [Aζ (am , bm )]. Hence x = 0.

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This proves the injectivity of the norm residue homomorphism. We now want to show that the cokernel V (F ) of the norm residue homomorphism k2 (F ) → l Br(F ) is trivial. The idea is to construct a large field extension F˜ of F , for which l Br(F˜ ) = 0 and V (F ) injects into V (F˜ ). The first property implies that V (F˜ ) vanishes, and the second property shows that the same holds for V (F ). This will finish the proof of Theorem 5.3.2. Here is the construction of F˜ : For each symbol a = {a, b} the map V (F ) → V (Fa ) is injective ([MS83, Corollary 11.3]) – this is a key result and again highly non-trivial. Let F 0 denote the compositum of all fields Fa , where a runs through the generators of k2 (F ). Let F1 be the maximal algebraic prime-to-l-extension of F 0 . Repeating the procedure with F1 instead of F etc., we obtain an increasing tower of fields Fi . Let F˜ = ∪Fi . Then by construction we get an injective map V (F ) ,→ V (F˜ ) and clearly k2 (F˜ ) = 0. This implies that l Br(F˜ ) has no non-trivial elements represented by cyclic algebras. We have to show that l Br(F˜ ) is trivial. We first show that in general the vanishing of k2 (K) implies the vanishing of k2 (L) for a cyclic extension of degree l: Hilbert’s Theorem 90 σ−1 implies that in this situation the map k2 (L) → k2 (L) is surjective, since k2 (K) = 0. However the l-th power (σ − 1)l of σ − 1 is trivial, hence k2 (L) is trivial as well. Since F˜ is l-special, this shows that every finite extension E of F˜ has trivial k2 (E). Assume now that there is a non-trivial central simple algebra A over F˜ , which is split by an extension L/F˜ of minimal degree. Since F˜ is l-special, the degree [L : F˜ ] is a power of l. Therefore, there exists a subfield E of L with [L : E] = l, so that A does split over L, but not over E. Let AE denote the extension of A to E. The algebra AE is cyclic, because L/E is a cyclic extension, and non-trivial, but this is impossible, because k2 (E) is trivial. Therefore l Br(F˜ ) is trivial.

Merkurjev and Suslin succeeded to use the same set of ideas to prove

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that the norm residue homomorphism K3M (F )/2 → H 3 (F, Z/2) is an isomorphism ([MS91]) by proving Hilbert’s Theorem 90 for K3M (F ) and using a result of Swan’s, who computed the K-groups of quadric hypersurfaces [Sw85]. The difficult part of the approach was to describe the deviation between Milnor’s K3M and Quillen’s K3 [MS91]. It became clear that for higher K-groups the methods based on Kcohomology groups could not succeed, since the deviation between Quillen K-theory groups and Milnor K-theory groups becomes extremely complicated for n ≥ 4. What was needed was a new ‘cohomology theory’, which was more closely related to Milnor’s K-theory groups for fields, and allowed to calculate these groups for function fields of certain algebraic varieties. This turned out to be ‘motivic cohomology’.

5.4 Motivic cohomology Motivic cohomology was envisioned by Beilinson and Lichtenbaum in the early 1980s ([Li83, section 3], [Be87, 5.10.D]): They conjectured the existence of a complex of sheaves Z(n) – called motivic complexes and n is called the weight – in the ´etale cohomology (Lichtenbaum) or the Zariski topology (Beilinson), so that the hypercohomology groups of a scheme X over a field F with coefficients in Z(n) – called motivic cohomology groups – satisfy a list of desired properties. We will denote i these motivic cohomology groups by HM (X, Z(n)) and HLi (X, Z(n)), respectively, depending on whether we consider the Zariski site or the ´etale site. In the case X = spec (F ) we replace spec F simply by F in the cohomological notation. Here is a selection of the properties conjectured by Beilinson and Lichtenbaum (F is a field and l a prime 6= charF ): (0) Z(0) = Z;

Z(1) = Gm [−1]

n (1) HM (F, Z/l(n)) ∼ = KnM (F )/l

(2) Beilinson–Lichtenbaum Conjecture:

The natural map

i HM (F, Z/l(n)) → HLi (F, Z/l(n))

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is an isomorphism for i ≤ n. (3) HLi (F, Z/l(n)) ∼ = H i (F, µ⊗n l ). (4) Hilbert 90:

HLn+1 (F, Z(n)) = 0

for all n ≥ 0.

Remark: Conjecture (3) has actually been formulated more generally for smooth varieties X over F . Conjecture (4) had been christened by Lichtenbaum as ‘Hilbert’s Theorem 90’, since for n = 1 one has HL2 (F, Gm [−1]) = HL1 (F, Gm ), and the vanishing of HL1 (F, Gm ) is exactly the cohomological version of Hilbert’s Theorem 90. We will use two definitions of motivic cohomology: The first one is due to Bloch [Bl86] (see also the survey article [Ge05]): Bloch introduced cycle complexes for quite general schemes X and defined higher Chow groups CH q (X, p) as the cohomology groups of the associated complex. The motivic cohomology groups are then obtained by re-indexing the Chow groups: i HM (X, Z(n)) := CH n (X, 2n − i).

These groups behave like a cohomology theory only for smooth schemes X over a field F . If X is smooth (and equi-dimensional), then we have the i important fact that HM (X, Z(n)) vanishes for i > min(2n, n + dim X) ([Ge05, section 1.2.1]). In particular Lemma 5.4.1

i HM (F, Z(n)) = 0,

if i > n.

We will use Bloch’s definition in the last section, where the base ring is a Dedekind ring. Another definition of motivic cohomology with good properties even for non-smooth schemes over a field was – according to Voevodsky ([MVW06, page viii]) – first envisioned by Suslin in 1985, and then has been formalized by Suslin and Voevodsky ([SV00]): They define for any scheme of finite type X over F a motivic complex of sheaves in the Zariski or Nisnevic or ´etale topology. The motivic cohomology groups are then defined to be the Nisnevic hypercohomology groups of this complex of sheaves. They show ([SV00, Corollary 1.1.1])

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that in the case of smooth schemes these groups are isomorphic to the ones obtained using Zariski hypercohomology instead (see also [Vo00a, Theorem 5.7], [Vo00b] and [Ge05, section 1.2.4]). For more details we refer to the survey article by Kahn [Ka05]. Both definitions – using Bloch’s higher Chow groups or the complexes of Suslin–Voevodsky – of motivic cohomology coincide [Vo02] for smooth varieties over a field. To establish some functorial properties the Suslin– Voevodsky approach has to assume resolution of singularities, and hence these properties are only conjecturally true in characteristic p. In these cases the Chow group approach is often used, e.g., in the work of Levine [Le01] and Geisser–Levine ([GL01], [GL00]). The relation between motivic cohomology and Milnor K-theory for a field F is given by the following fundamental result, which is due independently to Totaro [To92] and Nesterenko–Suslin [NS90] (see also [MVW06, Theorem 5.1]): Theorem 5.4.2 isomorphisms

For any field F and any integer n ≥ 0 there are n KnM (F ) ∼ (F, Z(n). = HM

Using Lemma 5.4.1 we immediately obtain the conjectural property (1) in the list of Beilinson and Lichtenbaum: Corollary 5.4.3 an isomorphism

For all fields F and all primes l 6= char F there is n HM (F, Z/l(n)) ∼ = KnM (F )/l.

The conjectural property (3) in the list of Beilinson and Lichtenbaum was proved by Voevodsky in [Vo03, Theorem 6.1] – for a more ‘elementary’ proof see [MVW06, 10.2]: Theorem 5.4.4 then

If m is relatively prime to the characteristic of F , HLi (F, Z/m(n)) ∼ = H i (F, µ⊗n m ).

Combining this result with Theorem 5.4.2 shows that the Bloch–Kato Conjecture is a consequence of the Beilinson–Lichtenbaum Conjecture. One of the main results of Suslin and Voevodsky [SV00, Theorem 7.4] implies that the two conjectures are in fact equivalent:

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Theorem 5.4.5 If the Bloch–Kato Conjecture holds for the prime l, then the same is true for the Beilinson–Lichtenbaum Conjecture. The proof used resolution of singularities, but this assumption was later removed by Geisser and Levine in [GL01]. Another proof is given in [HW12, section 2.5]. Remark: It is worth noting that Suslin and Voevodsky only assumed the surjectivity of the norm residue homomorphisms.

5.5 Voevodsky We now try to present some of the structural ingredients in Voevodsky’s proof of the Norm Residue Theorem. We first look more closely at the homomorphism i HM (F, Z(n)) → HLi (F, Z(n))

induced from the change of sites. The kernel and cokernel of this map are torsion, since i HM (F, Q(n)) ∼ = HLi (F, Q(n)) n+1 [MVW06, Proposition 14.23]. Lemma 5.4.1 shows that HM (F, Z(n)) n+1 vanishes, and therefore HL (F, Z(n)) is a torsion group, and the ltorsion is then given by HLn+1 (F, Z(l) (n)).

We see that the 4th property in our list of conjectures of Beilinson and Lichtenbaum, Hilbert 90, is true for a field F if and only if HLn+1 (F, Z(l) (n)) vanishes for all primes l and all weights n. The only additional result needed here is due to Geisser and Levine [GL00, Theorem 8.6], who showed that for a field of characteristic l > 0, the group HLn+1 (F, Z(l) (n)) vanishes. Voevodsky’s idea was to show that the Bloch–Kato (and the Beilinson– Lichtenbaum) Conjecture is in fact equivalent to Hilbert 90, and therefore can be formulated in terms of the vanishing of a cohomology group. To make the relationship more precise for a fixed prime l and a fixed weight n, we introduce the following abbreviations (due to Voevodsky): We write H90(l, n) for the statement that for all fields F of characteristic other than l and for all weights k ≤ n the cohomology groups

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HLk+1 (F, Z(l) (k)) vanish. Similarly, we write BK(l, n) for the statement that for all fields F of characteristic 6= l and all k ≤ n the norm residue homomorphisms KkM (F )/l → H k (F, µ⊗k l ) are isomorphisms. We want to compare H90(l, n) and BK(l, n). The easy implication is: Proposition 5.5.1

BK(l, n) =⇒ H90(l, n).

Proof Multiplication by l and BK(l, n) produce the following commutative diagram for all k ≤ n: ·l

→ KkM (F ) → KkM (F ) → KkM (F )/l → 0      ∼ y y y= ·l

→ HLk (F, Z(k))→HLk (F, Z(k))→ HLk (F, Z/l(k)) → l HLk+1 (F, Z(k)) → 0 This implies that HLk+1 (F, Z(k)) has no l-torsion. The result follows, since HLk+1 (F, Z(l) (k)) is the l-torsion subgroup of HLk+1 (F, Z(k)). The final implication, which yielded that all three conjectures are equivalent, was proven by Voevodsky in [Vo11, section 6]: Theorem 5.5.2

H90(l, n) =⇒ BL(l, n).

The following result, which again is due to Voevodsky [Vo03, Lemma 6.11], explains why the central role played by the cohomological version of Hilbert 90 in proving the Bloch–Kato Conjecture is similar to the role played by the K-theoretic version of Hilbert 90 in the work of Merkurjev– Suslin: Proposition 5.5.3 If H90(l, n) holds, then Hilbert 90 holds for the Milnor K-groups KkM for all cyclic extensions E/F of degree l and all k ≤ n.

Voevodsky’s approach to prove H90(l, n) was to use induction on the weight n. A crucial step is the following result ([Vo03, Theorem 5.9]): Theorem 5.5.4 Assume that F is l-special and that KnM (F )/l = 0. If H90(l, n − 1) holds, then HLn+1 (F, Z(l) (n)) = 0.

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It is obvious that we can imitate now the method of Merkurjev–Suslin, which we sketched above to prove Theorem 5.3.2, to finish the proof of Hilbert 90, provided the following key problem can be solved: Given a symbol a ∈ KnM (F ) there exists a smooth projective variety Xa , so that the function field Fa := F (Xa ) has the following properties: a) a vanishes in KnM (Fa )/l. b) The map HLn+1 (F, Z(l) (n)) → HLn+1 (Fa , Z(l) (n)) is injective. For l = 2 good candidates for these varieties are the so-called Pfister quadrics, which generalize the Severi–Brauer varieties (the case n = 2): Given a1 , · · ·, an ∈ F ∗ /F ∗2 the associated Pfister quadric is given by the equation << a1 , · · ·, an−1 >>= an · t2 . It has dimension 2n−1 − 1. The main result in [Vo03] showed that the Pfister quadrics satisfy the two properties above. Voevodsky combined some unpublished work of Rost on properties of Pfister quadrics with motivic techniques he had developed earlier. This finished the proof of the Milnor Conjecture ([Vo03]). Theorem 5.5.5 (The Milnor Conjecture) For any field F of odd characteristic and any n ≥ 0 the norm residue homomorphism KnM (F )/2 → H 2 (F, Z/2) is an isomorphism. The construction of analogous varieties for odd primes l satisfying properties a) and b) above was again based on work of Rost ([Ro98], [HW09]) and extended by Suslin and Joukhovitski [SJ06] and Weibel [We09]. These varieties are now called Rost varieties. The last key part of Voevodsky’s proof [Vo11]) of the ‘other’ Bloch-Kato Conjecture was to verify that the Rost varieties indeed have the right properties: Theorem 5.5.6 (The Norm Residue Theorem) For any field F , any prime l 6= char F and any n ≥ 0 the norm residue homomorphism KnM (F )/l → H n (F, µ⊗n l ) is an isomorphism.

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5.6 Quillen–Lichtenbaum Let X be a smooth scheme over a field F . Another conjectural property in the list of Beilinson and Lichtenbaum was the existence of a natural spectral sequence of the form p−q E2p,q = HM (X, Z(−q)) =⇒ K−p−q (X),

which relates motivic cohomology to Quillen K-theory. This was proven by Bloch and Lichtenbaum [BL95] for fields and generalized to smooth schemes of finite type over a field by Friedlander and Suslin [FS02]. It seems that there was a gap in the construction of the spectral sequence in [BL95], however in the meantime other constructions of this motivic spectral sequence have been done by Grayson [Gr95] and Levine [Le01] and shown to be equivalent to those of Bloch and Lichtenbaum. Details about the various constructions of the motivic spectral sequence can be found in Grayson’s article [Gr05]. It was well known to the experts that the Bloch–Kato Conjecture in combination with the motivic spectral sequence implies the QuillenLichtenbaum Conjecture (formulated in Theorem 5.6.8 below). By the end of the 1990s Kahn [Ka97] and independently Rognes–Weibel [RW00] studied the much harder case of the prime 2, where they used the Milnor Conjecture, which had just been proven by Voevodsky, and the motivic spectral sequence to obtain the relation between the K-theory of the ring of integers oF in a number field F (tensored by Z2 ) and the ´etale cohomology groups of spec oF [1/2]. Their techniques combined with the Norm Residue Isomorphism can also be used to give a proof of the Quillen–Lichtenbaum Conjecture. In [Le01], [Le99] Levine generalized the motivic spectral sequence to smooth schemes of finite type over a commutative 1-dimensional base ring. This made it possible to compare K-theory, motivic cohomology and ´etale cohomology directly on the level of rings of integers in a number field without analyzing the motivic spectral sequence for number fields and then descend to rings of integers, and simplified the proof of the Quillen–Lichtenbaum Conjecture. Levine followed this approach in his (unpublished) paper [Le99, section 14]. We combine ideas of Levine and of Kahn from his (unpublished) paper [Ka97] to sketch a proof of the Quillen–Lichtenbaum Conjecture. Let F be a number field, and let oF denote the ring of integers of F . We fix an arbitrary prime l and want to show first that an analogue of the

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∗ Beilinson–Lichtenbaum Conjecture holds for oF . We write HM (oF , •) ∗ instead of HM (spec oF , •) and similarly for ´etale cohomology. We also write o0F for oF [1/l].

Lemma 5.6.1

There are isomorphisms i HM (oF , Z/l(n)) → H´eit (o0F , µ⊗n l )

for all i ≤ n, and there is an isomorphism n+1 ⊗n n+1 ⊗n 0 HM (oF , Z/l(n)) ∼ = ker (H´en+1 t (oF , µl ) → H´ et (F, µl )).

Proof This follows from the commutative diagram of localization sequences for motivic cohomology – due to Geisser [Ge04] – and for ´etale cohomology – due to Soul´e [So79] – using the Beilinson–Lichtenbaum n+1 Conjecture for fields for n ≤ i and the fact that HM (F, Z(n)) = 0: i−2 i i · · · ⊕v H M (kv , Z/l(n − 1))→HM (oF , Z/l(n))→HM (F, Z/l(n)) → · · ·       ∼ ∼ =y =y y i i · · · ⊕v H´ei−2 et (oF , Z/l(n)) → H´ et (F, Z/l(n)) → · · · t (kv , Z/l(n − 1))→ H´

We now look at Levine’s spectral sequence for spec oF : p−q E2p,q = HM (oF , Z(−q)) =⇒ K−p−q (oF ).

In [Le99, Theorem 9.7] Levine shows that this integral motivic spectral sequence has properties similar to the field case with respect to the action of the Adams operations ψ k , k > 0, which are due to Soul´e ([So95], [GS99, Theorem 7]). The action of ψ k on the spectral sequence converges to the action of ψ on K−p−q (oF ), and ψ k acts on Erp,q , r ≥ 2, by multiplication by k −q . k

As usual this implies that the differentials of the spectral sequence are torsion, hence n−2i Kn (oF ) ⊗ Q ∼ (oF , Q(n − i)). = ⊕i∈Z HM

However – as Kahn observed ([Ka97, Corollary 2.2], [Ka05, section 4.3.2]) – there is a stronger implication:

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Proposition 5.6.2 Up to groups of finite exponent the K-group Kn (oF ) is a direct sum of motivic cohomology groups M n−2i Kn (oF ) = HM (oF , Z(n − i)). i∈Z

A fundamental result of Borel [Bo77] states that the K-groups Kn (oF ) are finite for even n ≥ 2 and finitely generated for odd n ≥ 1. Combining this with Proposition 5.6.2 we obtain in particular: n−2i Corollary 5.6.3 The motivic cohomology groups HM (oF , Z(n − i)) have no non-trivial divisible subgroups.

We combine this now with Lemma 5.6.1 to obtain the first main result comparing motivic cohomology groups of rings of integers to ´etale cohomology groups: Theorem 5.6.4 Let oF denote the ring of integers in a number field F , and let n ≥ 2. For each prime l and for all k ≤ n there are isomorphisms k HM (oF , Z(n)) ⊗ Zl ∼ = H´ekt (o0F , Zl (n)).

Proof Given a prime l we compare the long exact sequences in motivic cohomology (tensored with Zl ) and ´etale cohomology, associated to the short exact sequence of complexes ·l

0 → Z(n) → Z(n)→Z/l(n) → 0. We obtain the following commutative diagram, where the sequences end at k = n ≥ 2: ·l

·l

k+1 k k · · · →HM (oF , Z(n)) ⊗ Zl →HM (oF , Z/l(n))→HM (oF , Z(n)) ⊗ Zl → · · ·       ∼ fk+1 y fk y =y ·l

··· →

H´ekt (o0F , Zl (n))

·l

0 → H´ekt (o0F , Z/l(n)) → H´ek+1 t (oF , Zl (n)) → · · ·

The following Lemma, which is easily proved, provides information about the kernels and cokernels of the maps fk : Lemma 5.6.5

Let ·r

C −−−−→ A −−−−→ A −−−−→    o   fy fy y ·r

B  o y

C 0 −−−−→ A0 −−−−→ A0 −−−−→ B 0

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be a commutative diagram of abelian groups, such that the first and the last vertical arrows are isomorphisms and the map f is given by multiplication by a positive integer r. Then multiplication by r is surjective on ker f and injective on coker f . Returning now to the maps fk we immediately see that all maps fk are injective, since the motivic cohomology groups do not contain any nontrivial divisible subgroups, and the kernel of fk is l-divisible. Moreover, there is an isomorphism ∼

coker fk−1 /l → l ker fk , so that the vanishing of ker fk implies that coker fk−1 is l-divisible, hence trivial. This finishes the proof. As a consequence we obtain the following results about the structure k of the motivic cohomology groups HM (oF , Z(n)) for n ≥ 2: Theorem 5.6.6 Let F be a number field with ring of integers oF . Let r1 (F ) (resp. r2 (F )) denote the number of real (resp. pairs of conjugate complex) embeddings of F . For any n ≥ 2 we have: k 1. HM (oF , Z(n)) = 0 if k ≤ 0 or k > n.

2. For 3 ≤ k ≤ n:

k HM (oF , Z(n))

∼ =

( (Z/2Z)r1 0

if k + n is even if k + n is odd.

2 3. HM (oF , Z(n)) is torsion. 1 4. The rank of HM (oF , Z(n)) is equal to

1 rkZ HM (oF , Z(n))

=

( r1 (F ) + r2 (F ) r2 (F )

if n is odd if n is even

and the torsion subgroup is isomorphic to H 0 (F, Q/Z(n)). Proof The results of Theorem 5.6.4 allow us to use the corresponding results for ´etale cohomology groups (see e.g., [Ko04, section 2] for details about the structure of the ´etale cohomology groups of rings of integers).

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This proves items 1., 2. and 3. except if k = n + 1. But the localization sequence in ´etale cohomology shows that H´eit (o0F , Zl (n)) ∼ = H´eit (F, Zl (n)) n+1 for i ≥ 3, so that HM (oF , Z(n)) = 0 as claimed. k We see in particular that all motivic cohomology groups HM (oF , Z(n)) are torsion for k 6= 1, and therefore the motivic spectral sequence im1 plies that the rank of HM (oF , Z(n)) is equal to the rank of the K-group K2n−1 (oF ), which has been determined by Borel [Bo77] for n ≥ 2 to be equal to r1 (F ) + r2 (F ), if n is odd, and to r2 (F ), if n is even. 1 Again we get from Theorem 5.6.4 that the l-torsion in HM (oF , Z(n)) 1 0 is equal to the torsion subgroup of H´et (oF , Zl (n)), which in turn is isomorphic to H 0 (F, Ql /Zl (n)), hence we obtain the statement about the 1 torsion in HM (oF , Z(n)) as well.

Let us now consider an odd prime number l. We tensor Levine’s integral motivic spectral sequence by Zl . From Theorem 5.6.6 we obtain that the E2 -terms E2p,q ⊗ Zl vanish except possibly for k := p − q = 1, 2. Therefore after tensoring by Zl the motivic cohomology sequence degenerates and we obtain: Theorem 5.6.7

For odd primes l we have

k K2n−k (oF ) ⊗ Zl ∼ (oF , Z(n)) ⊗ Zl = HM

for k = 1, 2.

Remark: An important consequence is the fact that the motivic coho2 mology group HM (oF , Z(n)) is finitely generated as an abelian group, a result that improves Theorem 5.6.6. Combining the results of Theorem 5.6.7 and Theorem 5.6.4 we finally obtain: Theorem 5.6.8 (Quillen–Lichtenbaum Conjecture) natural isomorphisms for any odd prime l and n ≥ 2: K2n−k (oF ) ⊗ Zl ∼ = H´ekt (o0F , Zl (n))

There are

for k = 1, 2.

We note that the surjectivity of the maps in Theorem 5.6.8 was proven by Soul´e for n even [So79, Theorem 6 (iii)] and – using ´etale K-theory – by Dwyer–Friedlander for all n [DF85]. The Quillen–Lichtenbaum Conjecture was explicitly formulated by

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Quillen in 1974 [Q74, section 9, p.175], although he must have talked about it at least 2 years earlier, because Lichtenbaum [Li73, Conjecture 2.5] stated the conjecture in 1972 and attributed it to Quillen.

References [Be87] Beilinson, A. A. 1987. Height pairing between algebraic cycles. Ktheory, arithmetic and geometry (Moscow, 1984–1986). Lecture Notes in Math., 1289, 1–25. Berlin: Springer. [Bl80] Bloch, S. 1980. Lectures on Algebraic Cycles. Duke University Mathematics Series IV. Duke University. [Bl86] Bloch, S. 1986. Algebraic cycles and higher K-theory. Adv. Math., 61, 267–304. [Bl10] Bloch, S. 2010. Lectures on Algebraic Cycles, Second Edition. New Mathematical Monographs, 16. Cambridge, UK: Cambridge University Press. [BL95] Bloch, S., and Lichtenbaum, S. 1995. A spectral sequence for motivic cohomology. Unpublished. [BK90] Bloch, S., and Kato, K. 1986. p-adic cohomology. Publ.Math.de ´ l’I.H.E.S., 63, 107–152. [Bo77] Borel, A. 1977. Stable real cohomology of arithmetic groups. Ann. ´ Sci. Ecole Norm. Sup., 7, 613–636. [BT73] Bass, H., and Tate, J. 1973. The Milnor ring of a global field (with an appendix on euclidean quadratic imaginary fields, by J. Tate). Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., 342, 349–446. Berlin: Springer. [DF85] Dwyer, W, and Friedlander, E. 1985. Algebraic and ´etale K-theory. Trans. Amer. Math. Soc., 292 (1), 247–280. [FS02] Friedlander, E., and Suslin, A. 2002. The spectral sequence relating ´ algebraic K-theory to motivic cohomology. Ann. Sci. Ecole Norm. Sup. (4), 35 (6), 773–875. [Ge04] Geisser, Th. 2004. Motivic cohomology over Dedekind rings. Math. Z., 248 (4), 773–794. [Ge05] Geisser, Th. 2005. Motivic Cohomology, K-Theory and Topological Cyclic Homology. Handbook of K-Theory, 193–234. Berlin, New York: Springer. [GL00] Geisser, Th., and Levine, M. 2000. The p-part of K-theory of fields in characteristic p. Invent. Math., 139, 459–494. [GL01] Geisser, Th., and Levine, M. 2001. The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math., 530, 55–103. [GS06] Gille, Ph., and Szamuely, T. 2006. Central Simple Algebras and Galois Cohomolog. Cambridge Studies in Advanced Mathematics, 101. Cambridge UK: Cambridge University Press.

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[GS99] Gillet, H., and Soul´e, C. 1999. Filtrations on higher algebraic Ktheory. Algebraic K-theory (Seattle, WA, 1997). Proc. Sympos. Pure Math., 67, 89–148. Providence, RI: Amer. Math. Soc. [Gr95] Grayson, D. 1995. Weight filtrations via commuting automorphisms. K-Theory, 9 (2), 139–172. [Gr05] Grayson, D. 2005. The Motivic Spectral Sequence. Handbook of KTheory, 39–70. Berlin, New York: Springer. [HW12] Haesemeyer, C., and Weibel, Ch. The Norm Residue Theorem in Motivic Cohomology. Preprint 2012. [HW09] Haesemeyer, C., and Weibel, Ch. 2009. Norm varieties and the chain lemma (after Markus Rost). Abel Symposium, 95–130. Berlin: Springer. [Ka97] Kahn, B. 1997. The Quillen-Lichtenbaum Conjecture at the prime 2. Unpublished. [Ka05] Kahn, B. 2005. Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry. Handbook of K-Theory, 351–428. Berlin, New York: Springer. [Kt80] Kato, K. 1980. A generalization of local class field theory by using K-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (3), 603–683. [Ko04] Kolster, M. 2004. K-theory and arithmetic. Pages 191–258 (electronic) of: Contemporary developments in algebraic K-theory. ICTP Lect. Notes, XV. Trieste. [Le99] Levine, M. 1999. K-theory and motivic cohomology of schemes. Unpublished. [Le01] Levine, M. 2001. Techniques of localization in the theory of algebraic cycles. J. Alg. Geom, 10, 299–363. [Li73] Lichtenbaum, S. 1973. Values of zeta-functions, ´etale cohomology, and algebraic K-theory. Algebraic K-theory, II: “Classical” algebraic Ktheory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., 342, 489–501. Berlin: Springer. [Li83] Lichtenbaum, S. 1983. Values of zeta-functions at negative integers. Number Theory. Lecture Notes in Math., 1068, 127–138. Berlin: Springer. [MVW06] Mazza, C., Voevodsky, V., and Weibel, Ch. 2006. Lecture notes on motivic cohomology. Clay Mathematics Monographs, 2. Providence, RI: American Mathematical Society. [Me82] Merkurjev, A. S. 1982. On the norm residue symbol of degree 2. Soviet Math. Dokl., 1546–1551. [MS83] Merkurjev, A. S., and Suslin, A. A. 1983. K-cohomology of SeveriBrauer varieties and the norm residue homomorphism (in Russian). English trans.: Math. USSR Izv., 21, 307–340. [MS91] Merkurjev, A. S., and Suslin, A. A. 1991. The norm residue homomorphism of degree 3 (in Russian). English trans.: Math. USSR Izv., 36, 349–367. [Mi70] Milnor, J. 1970. Algebraic K-theory and quadratic forms. Invent. math., 9, 318–344. [Mi71] Milnor, J. 1971. Introduction to algebraic K-theory. Princeton, NJ: Princeton University Press. Annals of Mathematics Studies, no. 72.

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[NS90] Nesterenko, Yu., and Suslin, A. A. 1990. Homology of the general linear group over a local ring, and Milnor’s K-theory (in Russian). English trans.: Math. USSR Izv., 34, 121–146. [OVV07] Orlov, D., Vishik, A., and Voevodsky, V. 2007. An exact sequence for K∗M /2 with applications to quadratic forms. Ann. of Math., 165 (1), 1–13. [Q73] Quillen, D. 1973. Higher algebraic K-theory I. Algebraic K-theory, I: Higher K-Theories, (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., 341, 81–147. Berlin: Springer. [Q74] Quillen, D. 1974. Higher Algebraic K-theory. Proc. of the International Congress of Mathematicians, 1, 171–176. Vancouver, BC. [RW00] Rognes, J., and Weibel, C. 2000. Two-primary algebraic K-theory of rings of integers in number fields. J. Amer. Math. Soc., 13 (1), 1–54. Appendix A by M. Kolster. [Ro98] Rost, M. 1998. Chain lemma for splitting fields of symbols. Unpublished. [Se79] Serre, J.-P. 1979. Local fields. Graduate Texts in Mathematics, 67. New York: Springer. [So79] Soul´e, C. 1979. K-th´eorie des anneaux d’entiers de corps de nombres et cohomologie ´etale. Invent. Math., 55 (3), 251–295. [So95] Soul´e, C. Feb. 8, 1995. Letter to Bloch and Lichtenbaum. [Sr91] Srinivas, V. 1991. Algebraic K-Theory. Progress in Mathematics, 90. Boston: Birkh¨ auser. [SV00] Suslin, A., and Voevodsky, V. 2000. Bloch-Kato conjecture and motivic cohomology with finite coefficients. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998). NATO Sci. Ser. C Math. Phys. Sci., 548, 117–189. Dordrecht: Kluwer Acad. Publ. [Su85] Suslin, A. A. 1985. Algebraic K-theory and the norm residue homomorphism. J. of Math. Sciences, 30 (6), 2556–2611. [SJ06] Suslin, A. A., and Joukhovitski, S. 2006. Norm varieties. J. Pure Appl. Algebra, 206 (1–2), 245–276. [Sw85] Swan, R. 1985. K-Theory of Quadric Hypersurfaces. Ann. of Math., 122 (1), 113–153. [Ta76] Tate, John. 1976. Relations between K2 and Galois cohomology. Invent. Math., 36, 257–274. [To92] Totaro, B. 1992. Milnor’s K-theory is the simplest part of algebraic K-theory. K-theory, 6, 177–189. [Vo96] Voevodsky, V. 1996. The Milnor Conjecture. Unpublished. [Vo00a] Voevodsky, V. 2000a. Cohomological theory of presheaves with transfers. Cycles, transfers and motivic homology theories. Annals of Mathematics Studies, 143, 87–137. Princeton: Princeton University Press. [Vo00b] Voevodsky, V. 2000b. Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories. Annals of Mathematical Studies, 143, 87–137. Princeton: Princeton University Press. [Vo02] Voevodsky, V. 2002. Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Mat. Res. Notices, 7, 331–355.

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[Vo03] Voevodsky, V. 2003. Motivic cohomology with Z/2-coefficients. Publ. ´ Math. de l’I.H.E.S., 98, 59–104. [Vo11] Voevodsky, V. 2011. On motivic cohomology with Z/l-coefficients. Ann. of Math., 174, 401–438. [VSF00] Voevodsky, V., Suslin, A., and Friedlander, E. M. 2000. Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies, 143. Princeton: Princeton University Press. [We09] Weibel, Ch. 2009. The norm residue isomorphism theorem. J. Topology, 2, 346–372.

6 Regulators and Zeta-functions Stephen Lichtenbaum

6.1 Introduction This is just a short note recalling the different definitions of higher Ktheory regulators, what the relations are between them, and various formulas in the literature relating them to special values of Dedekind zeta-functions. We will explain the relationships among these formulas, and correct several inaccurate statements in the literature. Let F be a number field, OF the ring of integers in F , ζ(F, s) the Dedekind zeta function of F , and dF the discriminant of F . Let ζ ∗ (F, u) be the limit as s goes to u of ζ(F, u)(s−u)−cu , where cu is the order of the zero of ζ(F, s) at s = u. Let n = [F : Q] = r1 + 2r2 .. Let m be an integer ≥ 2. Let am = r2 if m is even and r1 +r2 if m is odd, and let bm = n−am . Note that Borel [Bo77] proved that the rank of K2m−1 (OF ) is equal to am . Recall that the natural map from K2m−1 (OF ) to K2m−1 (F ) is an isomorphism. Let x and y be in C∗ . Define x ∼ y if and only if x/y is a non-zero rational number. If V is a finite-dimensional vector space, let Λ(V ) denote the highest exterior power of V . We recall the functional equation of the zeta-function from [La70], p.254: Let φ(F, s) = Γ(s/2)r1 Γ(s)r2 (2−r2 |dF |1/2 π −n/2 )s ζ(F, s). Then we have the functional equation φ(F, s) = φ(F, 1 − s). It is an immediate consequence of the functional equation that ζ(F, m) ∼ π nm−am Brown University, Providence, USA. e-mail : [email protected]

p

|dF |

−1 ∗

ζ (F, 1 − m).

(6.1)

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Stephen Lichtenbaum

6.2 Regulators We start by discussing the general regulator framework. On the one hand we have groups K2m−1 (OF ), which have rank am by Borel’s theorem. Let x1 , . . . xam be a basis of K2m−1 (OF ) modulo torsion. Let σi run through the embeddings of F into C, but choose only one from each pair of conjugate embeddings, as usual. The σ’s induce natural maps from K2m−1 (OF ) to K2m−1 (C). We should have ‘higher logarithm’ maps Tm (sometimes called ‘regulator maps’, but we would like to reserve the word ‘regulator’ for the determinant, as in the classical case) from K2m−1 (C) to C. (These maps should be in some sense polylogarithms, but since the actual relation is unknown we will call them ‘higher logarithms’ to avoid confusion). If m is even, combining the map Tm with the r2 non-real embeddings σj induces a map Tm,F from K2m−1 (OF ) ⊗ C to Cr2 = Cam . If m is odd, we take all the am = r1 + r2 embeddings to again get a map into Cam . We will consider three different ways of defining such higher logarithm maps. The first one is Beilinson’s Chern class map. We will define it in general, and then specialize to the number field setting. To define this map we need to consider Deligne cohomology. Let Y be a smooth and projective scheme of finite type over R, and let YC = Y ×R C. There exists a natural period isomorphism β from the Betti cohomology HB (YC , C) to the de Rham cohomology HDR (YC ). The involution F∞ of YC induced by complex conjugation on C induces involutions F∞,B on HB (YC ) and F∞,DR on HDR (YC ). We may identify HDR (Y ) with HDR (YC )F∞,DR , and this is identified by β with the subspace of elements c of HB (YC , C) such that c = F∞,B (¯ c) ([De79], Proposition 1.4). Let M be the motive H i (YC , (Z(m)), and assume that the weight i − 2m ≤ −3. In this case we can define the motivic cohomology group i+1 H(M ) to be Hmot (YC , Q(m)), which we may take as grm (K2m−i−1 (YC )⊗ Q). (Here grm refers to Adams weight.) We next have the Betti cohomoli ogy HB (M ) = HB (YC , Z), and the tangent space tM , which is defined to be the hypercohomology H i (YC , OYC → ΩYC → Λ2 ΩYC . . . → Λm−1 ΩYC → 0) (ΩY = Ω1 (Y /Q)). We map HB (M ) to tM by composing the natural map from HB (YC , Z) to tM with multiplication by (2πi)m , and define the integral (resp. real) Deligne cohomology HD (M, Z) (resp. HD (M, R)) to be the cokernel tM /HB (M ) (resp. tM /HB (M ) ⊗ R)). As described in [RSS88] Chapter 1, Beilinson defines Chern class maps

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from H(M ) to HD (M, Z), and hence to HD (M, R). Beilinson shows that + the image of the Chern class maps lies in HD (M, R), which is defined to + F∞,DR F∞,B be (tM ) /(HB (Y ), R) . Let αM be the map from HB (M, C) to tM . We may identify HD (M, R) with a subspace of Coker αM such that (HD (M, R)) ⊗R C naturally identifies with Coker αM . Now let X be a projective and smooth scheme over Spec F , and ˜ for X which is regular and projective assume that there exists a model X over Spec OF . Let σ run through the embeddings of F into C. Let Xσ be X ×F C, where σ : F → C and Y = X ×Q R, so X ×Q C = YC . Let MX be the motive over Q given by H i (X, Z(m)). We define the motivic ˜ ⊗ Q in cohomology group H(MX ) to be the image of grm K2m−i−1 (X) ˜ grm K2m−i−1 (X) ⊗ Q. This is independent of the choice of X. Then Beilinson conjectures that the Chern class maps on H(Xσ ) induce an isomorphism from the motivic cohomology group H(MX ) ⊗ R to HD (Y, R), and hence from H(MX ) ⊗ C to Coker αM,X , where αM,X L + maps HB (MX ) to σ tMσ , where Mσ is the motive H i (Xσ , Z(m)). Now, let i = 0, and let X be a point, where we have the integral Deligne complex Z(m) : (2πi)m Z → C, and the real Deligne complex R(m) : (2πi)m R → C. Taking the first cohomology groups of these 1 complexes we obtain the integral Deligne cohomology HD (C, Z(m)) = m 1 C/(2πi) Z and the real Deligne cohomology HD (C, R(m)) = C/(2πi)m R. Then Beilinson Chern class maps go from K2m−1 (C) to C/(2πi)m Z, and hence to C/(2πi)m R, which we identify with im+1 R ⊆ C. We denote this Be map from K2m−1 (C) to C by Tm . Bo Borel also defines a higher logarithm map Tm , which ([Bu02], p.95) Bo Be satisfies Tm = 2Tm . We next look at the higher logarithm maps considered by Zagier [Za91] (which in this case are actually polylogarithm maps). We first consider the case m = 2, and let F be any field. In this case, by a theorem of Suslin [Su87], the indecomposable part of K3 (F ) maps with finite kernel onto a group B(F ) known as the Bloch group. The Bloch group is obtained by dividing the free abelian group on F ∗ by an infinite collection of relations coming from the functional equation for the dilogarithm, and then taking a certain subgroup of this. Consequently the single-valued version P2 of the dilogarithm (which has values in iR) induces a map from B(C) to iR and hence also from K3 (C) to iR, which as above we view as contained in C. We would like to be able to play this game in general, but we don’t know enough. We do, however, have a conjecture of Zagier ([Za91]). He first considers a single-valued version Pm of the m-th polylogarithm,

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which maps C to R. It is more natural to modify the definition by setting Qm equal to Pm if m is odd and iPm if m is even. In both cases we will view Qm as having values in C. Zagier defines analogous groups Bm (F ), (B2 (F ) = B(F )), which are subquotients of the free abelian group Z[F ∗ ] and such that Pm is well defined on Bm (C). Zagier then conjectures that we can identify Bm (F ) ⊗ Q with K2m−1 (F ) ⊗ Q, so that Qm induces a Z map Tm from K2m−1 (C) to C. Zagier also conjectures (Conjecture Z1) that his polylogarithm map is, up to a rational number, the one defined by Borel and Beilinson. In order to define a regulator, we need, in addition to the polylogarithm map T , a rational basis for Cam or, more precisely, a rational basis b for the exterior power Λ(Cam ). The map Tm,F induces a map ΛTm,F from Λ(K2m−1 (OF ) ⊗ C) to Λ(Cam ). If we let a be the basis of Λ(K2m−1 (OF ) ⊗ C) coming from K2m−1 (OF ), then the corresponding regulator is T (a)/b, well defined up to a non-zero rational number. Z The Zagier regulator Rm (F ) is obtained by choosing the standard basis for Cam . Based on considerable numerical evidence ([Gr81] and [Za91]), Zagier conjectures (Conjecture Z2) that we have the formula Z ζ(F, m) ∼ (2πi)mbm Rm

p

dF

−1

.

(6.2)

(Actually Zagier’s conjecture involves Pm rather than Qm , but it is easily seen to be equivalent to our statement above.) By the functional equation, Zagier’s conjecture is equivalent to: ζ ∗ (F, 1 − m) ∼ (2πi)am (1−m) RZ m.

(6.3)

Bo As renormalized by Burgos-Gil ([Bu02]), Borel’s regulator Rm is m−1 obtained by multiplying the standard basis by (2πi) , which implies, assuming Zagier’s conjecture on the equality of polylogarithms, am (1−m) Z RBo Rm , so Borel’s theorem: m = (2πi) Bo ζ ∗ (F, 1 − m) ∼ Rm

(6.4)

together with Conjecture Z1, implies Conjecture Z2. Borel’s theorem implies, using the functional equation, that Bo ζ(F, m) ∼ (2πi)mn−am Rm .

(6.5)

Remark: In [Za91], p.393, Zagier misapplies the functional equation,

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stating that 6.2 is equivalent to: p ζ(F, m) ∼ π mbm ( |dF |)−1 RBo m ,

(6.6)

which is not correct. Zagier thinks this should be true because he thinks his regulator is the same as Borel’s, but in fact a renormalization as indicated above is necessary. Note: In Borel’s original paper, the basis in effect is chosen to be (2πi)m (rather than (2πi)m−1 ) times the standard basis. We leave it to the reader to compute Borel’s original regulator formulas. We now go back to Beilinson’s regulator and Beilinson’s conjectures on special values of L-functions. We will give our version of Fontaine’s presentation ([Fo92]) of Beilinson’s conjecture and generalizations thereof. Let X be a regular scheme, projective over Spec F , and as above we ˜ projective over Spec OF . Let assume that X has a regular model X XC = X ×Q C. We will examine the conjectures for the motive M = H i (X, Z(m)). We will assume that the weight i − 2m is either ≤ −3 or ≥ 1, and we will give conjectures up to a non-zero rational number. We first consider the case when the weight is ≤ −3. In this case recall that i+1 we defined the motivic cohomology group H(M ) to be Hmot (X, Q(m)), ˜ which is the image of grm (K2m−i−1 (X) ⊗ Q) in grm (K2m−i−1 (X) ⊗ i Q). We next have the Betti cohomology HB (M ) = HB (XC , Q), and the tangent space tM.X , which is defined to be the hypercohomology H i (X, OX → ΩX → Λ2 ΩX . . . → Λm−1 ΩX → 0) (ΩX = Ω1 (X/Q)). We next recall the period map from HB (M ) ⊗ C to HDR (M ) ⊗ C, which induces a map which we call αM,X from (HB (M ) ⊗ C)+ to tM ⊗ C, where (HB (M ) ⊗ C)+ is the subgroup of HB (M ) left fixed by the automorphism induced by complex conjugation on the scheme XC and complex conjugation acting on the coefficients. As we have seen, Beilinson defines a Chern class map from H(M ) to + the integral Deligne cohomology H(D,Z) (M ), and hence also to the real + Deligne cohomology H(D,R) (M ). This is naturally a real subspace of + Coker αM , such that HD (M ) ⊗ C is naturally isomorphic to Coker αM . Beilinson’s conjecture implies that the induced map θM from H(M )⊗R C to Coker αM is an isomorphism of complex vector spaces. We would like to go one step further. We have the cohomology sequence arising from integral Deligne cohomology, as described on page 8 of [RSS88]. 0 → HB (M ) → tM ⊗ C → H(D,Z) (M ) → 0.

(6.7)

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Recall that the map from HB (M ) to tM ⊗ C is given by the usual period map multiplied by (2πi)m . The involution given by complex conjugation on the scheme on Betti cohomology and complex conjugation on the coefficients is compatible with the involution on the coefficients of de Rham cohomology. Applying this we have the exact sequence: + + 0 → HB (M ) → tM ⊗ R → H(D,Z) (M ) → 0.

(6.8)

Taking the pull-back of this exact sequence by means of the Beilinson + Chern class map from H(M ) to H(D,Z) (M ), we obtain + 0 → HB (M ) → Hζ1 (M ) → H(M ) → 0,

(6.9)

where this sequence serves to define Hζ1 (M ). This immediately defines a map from Hζ1 (M )⊗R to tM ⊗R, and hence a map θM from Hζ1 (M ) ⊗ C to tM ⊗ C. The above Beilinson conjecture + implies that this map is also an isomorphism. Since HB (M ) and H(M ) are finite dimensional Q-spaces, so is Hζ1 (M ), so both of the C-spaces Hζ1 (M ) ⊗ C and tM ⊗ C have natural Q-structures. Conjecture 6.2.1 The determinant of θM taken with respect to the bases coming from the natural Q-structures is, up to a non-zero rational number, L∗ (M ). In the number field case, let M = H 0 (X, Z(m)) with m ≥ 2 and o we will call RF m (F ) the Fontaine regulator, i.e., the determinant of the map induced by the Chern class map from Λ(H(M ) ⊗ C) to Λ(HD (M )), where we use the rational basis of H(M ) ⊗ Q, and we identify HD (M ) with Cam and choose the standard basis of Cam . (The rational basis of (HB (M ) ⊗ C)+ is given by multiplying a rational basis of HB (X)+ by (2πi)m .) It is an easy computation from Conjecture 6.2.1 that p (−1) Fo ζ(F, m) ∼ (2πi)bm Rm (F ) dF . (6.10) L Note that tM ⊗C is isomorphic to σ tMσ , but the natural Q-structures L are different. In the number field case, tM is OF ⊗ Q, and σ tMσ is L L Q-structure σ Q. The difference between σ C, which has the natural √ these two Q-structures is dF . Note also that for these weights Beilinson does not give a conjecture about the special values of L-functions. Bloch and Kato do in [BK90], and Fontaine does also in [Fo92]; this conjecture should be compatible with both of these. Both of these conjectures are meant to be valid up

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to sign, not just up to rational numbers, but we will discuss the more precise compatibility in a later paper. Finally, we consider the special value conjectures if the weight is ≥ 1. Here there is no Bloch–Kato conjecture, so we will make a conjecture and discuss its relation to those of Fontaine and Beilinson. In the previous weight range, we had Ker αM = 0 and in this dual weight range we have Coker αM = 0. Let N = M ∗ (1) be the dual motive to M , so that as in [Fo92] Ker αM is the dual complex vector space to Coker αN . We define the motivic cohomology Q-space Hc (M ) to be Hom (H(N ), Q). The dual of Beilinson’s higher logarithm map for N is now a map γM from Ker αM to Hc (M ) ⊗ C. In order to extend our conjecture, we need something like a Q-structure on Ker αM . This time we start with the exact sequence + 0 → Ker αM → HB (M ) → tM ⊗ C → 0,

(6.11)

and we take the push-out coming from the map γM . This produces an exact sequence 0 → Hc (M ) ⊗ C → Hζ2 (M ) → tM ⊗ C → 0,

(6.12)

and this sequence serves to define Hζ2 (M ). We have natural Q-structures on Hc (M ) ⊗ C and tM ⊗ C, but these do not combine to produce a Q-structure on Hζ2 (M ). However, they do give rise to Q-structures on the determinants Λ(Hc (M ) ⊗ C) and 2 Λ(tM ⊗ C), which gives rise to a Q-structure on Λ(Hζ(M ) ). Coming from our push-out diagram, we get a natural map, which we again call θM + 2 from HB (M ) ⊗ C to Hζ(M ) . Using this Q-structure, we again have: Conjecture 6.2.2 The determinant of Λ(θM ) taken with respect to 2 the bases coming from the natural Q-structures on Λ(Hζ(M ) ) and on + ∗ Λ(HB (M )) is, up to a non-zero rational number, L (M ). This conjecture in principle agrees with that of Fontaine, although Fontaine does not give enough details in this case for us to be absolutely sure that this is what he is saying. Fontaine also claims that his conjecture here is consistent with that of Bloch–Kato, so ours should be also. Fontaine says that his conjecture agrees with Beilinson’s, but misquotes Beilinson’s to have it say that L∗ (N) is equal to the determinant of Λ(θN ).

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Fontaine, in a footnote, restates Beilinson’s description of how to obtain a rational basis of Deligne cohomology, but this is not compatible with his own formalism (which we have explained above). As for Beilinson, his conjecture ([RSS88], p.31, conjecture II), would at first glance seem to say that L∗ (M ) is equal to the determinant of Λ(θN ), which is obviously false. However, Beilinson puts a different Qstructure on the Deligne cohomology, coming from the relation + i Λ(F m (HDR (XR ) ⊗R ΛHD (X, R(n) ' Λ(H i (X(C), R(n − 1)−1)

n−1

. (6.13) Jannsen shows in [Ja88] 4.9 that this conjecture is consistent with Conjecture 6.2.1, but the more natural way to view the picture is via Fontaine’s analysis as reflected in our conjecture 6.2.2. Even though Beilinson’s conjecture is in fact correct (or at least equivalent to that of Fontaine, which should be correct), it involves making, as Jannsen says, an ‘ad hoc’ choice of basis for Deligne cohomology. In the number field case, with M = H 0 (X, Z(1 − m)), m ≥ 2. we have + tM = 0, so we may identify Hζ2 (M ) with HB (M ) ⊗ C. This may in turn am be identified with C but with the basis equal to the standard basis for Cam multiplied by (2πi)1−m . Conjecture 6.2.2 then says that for m ≥ 2 we have Fo ζ(F, 1 − m) ∼ (2πi)am (1−m) Rm ,

(6.14)

which is compatible with the hypothesis that the Fontaine regulator is the Zagier regulator, and of course with Conjecture 6.2.1 in the number field case and the functional equation.

References [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives in The Grothendieck Festschrift. vol. 1. Prog. in Math., 86. 333–400. Birkh¨ auser, Boston. [Bo77] Borel, A. 1977. Cohomologie de SLn et valeurs de fonctions zeta aux points entiers. Ann. Scuola Normale Superiore (4), 613–636. [Bu02] Burgos Gil, J. I. 2002. The Regulators of Beilinson and Borel. American Math. Society, Providence, RI. [De79] Deligne, P. 1979. Valeurs de fonctions L, in Automorphic Forms, Representations, and L-functions, II (Oregon State Univ. Corvallis, OR,

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1977). Proc. Sympos. Pure Math. 33, 313–346, Amer. Math. Soc., Providence, RI. [Fo92] Fontaine, J.-M. 1992. Valeurs speciales des fonctions L des motifs. Seminaire Bourbaki, 1991/92, Asterisque no. 206, Exp. no. 751, 4, 205–249. [Gr81] Grayson, D. 1981. Dilogarithm computations for K3, Algebraic Ktheory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980). Lecture Notes in Math., 854, 168–178, Springer, Berlin. [Ja88] Jannsen, U. 1988. Deligne Homology, Hodge D-conjecture, and Motives in Beilinson’s Conjectures on Special Values of L-functions (edited by M. Rapoport, N. Schappacher, and P. Schneider), Academic Press. [La70] Lang, S. 1970. Algebraic Number Theory. Addison-Wesley, Reading, Mass. [Li75] Lichtenbaum, S. 1975. Values of zeta-functions, `etale cohomology, and algebraic K-theory. Algebraic K-theory II Springer Lecturer Notes in Mathematics, 342, 489-501. [RSS88] Rapoport, M., Schappacher, N., and Schneider, P. 1988. Beilinson’s Conjectures on Special Values of L-functions (edited by M. Rapoport, N. Schappacher, and P. Schneider), Academic Press. [Su87] Suslin, A.-A. 1987. Algebraic K-theory of fields. Proceedings of the International Congress of Mathematicians, 1–2, 222–244 (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI. [Za91] Zagier, D. 1991. Polylogarithms, Dedekind zeta-functions, and algebraic K-theory of fields, in Arithmetic Algebraic Geometry, Prog. in Math. 89. Birkh¨ auser, Boston, p.3

7 Soul´e’s Theorem Stephen Lichtenbaum

7.1 Introduction Notation: Let p be an odd prime number, F be a number field and OF be the ring of integers in F . Let S be the set of primes of OF lying over p and A = (OF )S = OF [1/p]. Let j be the natural inclusion of Spec F in X = Spec A. Let kv be the residue field of OF at the discrete valuation v, let FS be the maximal extension of F unramified outside of S, and let GS be the Galois group of FS over F . Let Wp (n) denote the GF -module consisting of all p-power roots of unity, with GF acting by σ ∗ (ζ) = σ n (ζ), this latter action being the usual one. Note that the p-power roots of unity Wp in this notation are Wp (1). In this chapter, we are going to sketch the proofs of two important theorems which will be needed later in this volume. Theorem A: H 2 (GS , Wp (n)) = 0 for all n ≥ 2. Theorem B: The natural map from K-theory to ´etale K-theory induces et a surjective map from K2n−1 (A)⊗Zp to K2n−1 (A, Zp ) ∼ = H 1 (Xet , Zp (n)). The proof of Theorem A is taken from [So79], with an emendation from [So84], while the proof of Theorem B follows [DF85], with the deep topological parts omitted. We refer to [Su15] for the definitions of the various Chern class maps used in Lemma 7.2.1 and for her Theorem 2.8.1. Note that k below only has the values 1 and 2. Let OF be a discrete valuation ring with fraction field F and residue field kv . Brown University, Providence, USA. e-mail: [email protected]

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7.2 Proof of Theorem A Lemma 7.2.1

The following diagram commutes: ∂¯

K2n−1 (OF , Z/pm ) −−−−→ K2n−1 (F, Z/pm ) −−−v−→ K2n−2 (kv , Z/pm )    c¯ c¯  (1−n)¯ cn−1,k−1 y y n,k y n.k H ! (OF , Wp (n))

−−−−→

H 1 (F, Wp (n))

∂

−−−v−→ H 0 (kv , Wp (n − 1)).

Proof: It is clear that the left hand square commutes, and consequently that the right hand square commutes for any element in the kernel of ∂¯v . We may assume that F is complete for v. Let q = pm . Let π be a uniformizing parameter for F . Let π˙ be the image of π in K1 (F, Z/q). Let rv denote reduction modulo v. Let s denote a section of rv in either K-theory or ´etale cohomology. (The existence of such a section follows from [HS75] p.27.) Let sπ (x) = π.s(x), ˙ for x in K2n−2 (kv , Z/q), so sπ (x) is in K2n−1 (F, Z/q). We first observe that ∂(sπ (x)) = ∂(π.r ˙ v (s(x))) = ∂(π.x) ˙ = x. Since we know the commutativity of the right hand square for elements in the kernel of ∂v , it suffices to prove it for elements in the image of sπ . a) c¯n,k (sπ (x)) = c¯n.k (π.s(x)) ˙ = (n − 1)π˙ ∪ c¯n−1,k−1 s(x)), by b) c¯n−1.k−1 (s(x) = rv−1 (rv c¯n−1.k−1 (s(x) = rv−1 (¯ cn−1,k−1 x)). (Note that rv is an isomorphism on ´etale cohomology, so rv−1 is well defined, and also that the Chern classes commute with rv .) This implies c) c¯n,k (sπ (x) = (n − 1)sπ (¯ cn−1,k−1 (x)), which immediately implies the commutativity of the right hand square for elements in the image of sπ . Theorem 7.2.2 The long exact localization sequence for the algebraic K-theory of a Dedekind domain A breaks up, if A is the ring of integers in a number field, into short exact sequences 0 → K2n (A) → K2n (F ) → ` v K2n−1 (kv ) → 0 and isomorphisms K2n−1 (A) ' K2n−1 (F ). Proof: This is now well known. However, the original proof given in [So79] is incomplete, as was pointed out by Soul´e himself, who filled the gap in [So84]. Lemma 7.2.3 Let E be a finite field, ` an odd prime number different from the characteristic of E, q = `m a power of ` and k = 0 or 1. Then

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the Chern class maps c¯i,k : K2i−k (E, Z/q) → H k (E, µ⊗i q ) have kernel and cokernel killed by i!. Proof: By using the surjectivity of the norm map on the multiplicative groups of finite fields, we may assume that E contains a q-th root of unity. In this case the cup-product map on ´etale cohomology is easily seen to be surjective, so Soul´e’s Chern class product formula ([Su15], Theorem 2.8.1) and induction give the required result.  Let P∞ be the unique Zp -extension of Q, and let F∞ = F P∞ . Let G∞ = GF∞ . Let Γ = Gal(F∞ /F ). Lemma 7.2.4 and n 6= 0.

(Tate’s Lemma) H 1 (Γ, Wp (n) ⊗ N ) = 0 for N discrete

Proof: Let γ be a topological generator of Γ. We can write N as the direct limit of Nα where each Nα is finitely generated, and so fixed by m m γ p for some m. So we may assume N is fixed by γ p . Let T be the Tate module of W = Wp (n), and let V = T ⊗Zp Qp . Then W ⊗ N is the quotient of the finite-dimensional Qp -vector space V ⊗ N . The eigenvalues of γ on Qp ⊗ N are pm -th roots of unity, so if n 6= 0 the eigenvalues of γ on V ⊗ N are not roots of unity, so the image of Γ in the automorphism group Aut(V ) is not finite, which implies that γ − 1 acts bijectively on V ⊗ N , hence surjectively on W ⊗ N . If A is a torsion Γ-module, H 1 (Γ, A) is the cokernel of γ − 1, so we are done. Proposition 7.2.5

H 2 (GF , Wp (n)) = 0 if n ≥ 2.

Proof: Let Mp = Q(µp ). Since [F Mp : F ] is prime to p, we have; H 2 (GF , Wp (n)) = H 2 (GF Mp , Wp (n))G(F Mp /F ) so we may assume that Mp ⊆ F . The cohomological dimension of Γ is equal to 1, so the spectral sequence H i (Γ, H j (G∞ , Wp (n))) ⇒ H i+j (GF , Wp (n)) yields the short exact sequence 0→H 1 (Γ,H 1 (G∞ ,Wp (n)))→H 2 (GF ,Wp (n))→H 2 (G∞ ,Wp (n))Γ →0. ∗ ∗ Now H 1 (G∞ , µpm ) = F∞ ⊗ Z/pm , so H 1 (G∞ , Wp (1)) = F∞ ⊗ Qp /Zp , 1 ∗ which in turn implies H (G∞ , Wp (n)) = F∞ ⊗ Wp (n − 1). Lemma 7.2.4

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∗ implies that H 1 (Γ, F∞ ⊗ Wp (n − 1)) = 0 for n ≥ 2. It remains only to 2 show that H (G∞ , Wp (n)) = 0. Since Wp ⊆ F∞ , we may assume n = 1. But H 2 (G∞ , Wp ) = Br(F∞ )(p) = 0. (See, for example, [Se74] Prop. II.9.)

Theorem 7.2.6 The group H 2 (X, j∗ Wp (n)) is annihilated by (n−1)n!. Proof: Starting with the exact sequence of relative K-theory for pm and taking the direct limit over m, we obtain the exact sequence: 0 → Kr (A) ⊗ Qp /Zp → Kr (A, Qp /Zp ) → Kr−1 (A)(p) → 0, where Kr (A, Qp ./Zp ) is the direct limit of the relative K-groups Kr (A, Z/pm ). Lemma 7.2.1 implies that the following diagram commutes: ¯ ∂

K2n−1 (F, Qp /Zp ) −→  cn,1 y H 1 (F, Wp (n))

β

−→

`

K2n−2 (kv , Qp /Zp )  (n−1)c y n−1,0

`

H 0 (kv , Wp (n − 1)) −→ H 2 (X, Wp (n)) −→ H 2 (F, Wp (n))

v

v

Of course, H 2 (F, Wp (n)) = 0 if n ≥ 2, by Proposition 7.2.5. We have the following diagram which commutes up to sign, and where the vertical maps are given by the Bockstein maps for K-theory with coefficients: ∂¯

K2n−1 (F, Qp /Zp ) −−−−→  γ y K2n−2 (F )(p)

∂

−−−−→

`

K2n−2 (kv , Qp /Zp )  α y ` v K2n−3 (kv )(p)

v

We know that K2n−2 (F ) is torsion by Borel ([Bo74]) and the localization sequence for K-theory. Since ∂ would be surjective by Theorem 7.2.2 if we replaced K2n−2 (F )(p) by K2n−2 (F ), it follows that ∂ is surjective. Since α is an isomorphism by Quillen’s computation of the K-theory of a finite field [Q72], and γ is surjective, it follows that ∂¯ is also surjective. We know by Lemma 7.2.3 that the image of cn,0 contains the image of multiplication by n!. Hence the first commutative diagram shows that the image of β contains the image of multiplication by (n − 1)n! and hence that (n − 1)n! kills H 2 (X, j∗ Wp (n)), which completes the proof of Theorem 7.2.6. Corollary 7.2.7

H 2 (X, j∗ Wp (n)) = 0 if n ≥ 2.

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Proof: Since H 3 (X, µ⊗n pm ) = 0 by Artin–Verdier duality ([M04] p.177), it follows that H 2 (X, Wp (n)) is divisible, hence zero by Theorem 7.2.6. H 2 (GS , Wp (n)) = 0 for n ≥ 2.

Theorem 7.2.8

Proof : This follows immediately from the fact ([M04] p.170) that if E is a locally constant sheaf on X corresponding to the finite GS -module M , and the order of M is invertible on X, then H i (X, E) ' H i (GS , M ).

7.3 Proof of a lemma of Soul´ e Let p be an odd prime, and let q = pm . Let A be a Dedekind ring whose field of fractions is a number field F . Assume that p is invertible in A. Let X = Spec A. Let K(n, m, A) be the statement that the Chern class map from Kn (A)/m to H n (X, µm ⊗n ) is an isomorphism. (Here n will be 1 or 2.) Lemma 7.3.1

K(1.p, A) is true if P ic(A)(p) = 0.

Proof: Standard. Lemma 7.3.2 Let B be a Galois extension of A of degree r. If m is prime to the characteristic of A and to r, then K(2, m, B) implies K(2, m, A). Proof: This is immediate from the existence of the norm map. Lemma 7.3.3

K(2, p, A) is true.

Proof: Lemma 7.3.2 shows that we may reduce to the case where A contains a primitive p-th root of unity. Let α be an element of K2 (A, Z/p) whose image in K1 (A) = A∗ is of exact order p. If a runs through the entire class group Pic(A) ⊆ K0 (A) the products α.a run through a subgroup of K2 (A, Z/p) which we call Ω. Let i be the injection of A in F , and i∗ the morphism induced by i. We have i∗ (α.a) = i∗ (α) ∪ i∗ (a) = 0 because Pic(F ) = 0. Hence i∗ (βp (α.a)) = βp (i∗ (α.a)) = 0 (where βp is the Bockstein morphism, and as i∗ : K1 (A) → K1 (F ) is injective, the group Ω is in the kernel K2 (A)/pK2 (A) of βp : K2 (A, Z/p) → K1 (A). It is therefore a subgroup of the kernel of i∗ : K2 (A)/pK2 (A) → K2 (F )/pK2 (F ) ⊆ K1 (F, Z/p). In addition, Soul´e’s multiplication formula ([Su15], Theorem 2.8.1), shows that c¯2,2 (α.a) = c¯1,0 (α) ∪ c1,2 (a). We have seen in [Su15] that c¯1,0 (α) is a generator of H 0 (X, µ⊗2 p ), and that c1,2 (a) runs through the

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subgroup Pic(A)/pPic(A) of H 2 (SpecA, µp )). Hence c¯2,2 (Ω) is the subgroup βp (H 1 (X, Gm )) ∪ H 0 (X, µp ) ' P ic(A) ⊗ µp of H 2 (X, µ⊗ p 2). Tate has shown in [T76] that the kernel of the map from K2 (A)/pK2 (A) to K2 (F )/pK2 (F ) is isomorphic to Pic(A) ⊗ µp . As the class group is finite, if follows that this kernel is equal to Ω. We finish by observing that the exact sequence a 0 → Ω → K2 (A)/p → K2 (F )/p → kv∗ /(kv∗ )p , v

where the last three terms come from the localization sequence in Ktheory, maps to the sequence a 2 ⊗2 0 → P ic(A) ⊗ µp → H 2 (X, µ⊗2 H 1 (kv , µp ), p ) → H (F, µp ) → v

where the last three terms come from the localization sequence in ´etale cohomology. Since four of the five maps (counting the zero map!) between the sequences are isomorphisms, the five lemma completes the proof.  Lemma 7.3.4

K(1, p, A) + K(2, p, A) implies K(2, pm , A).

Proof: We have the following commutative diagram: K1 (A)/p − → K2 (A)/pm − → K2 (A)/pm+1 − →    γ1  γ2 γ3 y y y

K2 (A)/p   γ4 y

− → 0

β

H 1 (A, µp ) − → H 2 (A, µ⊗2 → H 2 (A, µ⊗2 ) − → H 2 (A, µ⊗2 pm ) − p ) pm+1 The lower sequence is exact and the upper sequence is exact except at K2 (A)/pm . Diagram-chasing shows that γ1 surjective, γ2 and γ4 injective implies γ3 is injective. So by induction the map γ on K2 (A)/pm is injective for all m. Another induction now shows that γ is also surjective. So we have proved K(2, pm , A) for all m if Pic(A)(p) = 0. Lemma 7.3.5 Let OF be the ring of integers in a number field F , and let A = OF [1/p]. Let B be a ring obtained from A by inverting a finite number of primes. Then K(2, pm , B) for all m implies K(2, pm , A) for all m. Proof: Let X = Spec A and Y = Spec B. We remark that we have shown K(2, p, A) for all such A, and consequently that the map from

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K2 (A)/pm to H 2 (X, µ⊗2 pm ) is surjective for all A and an isomorphism for those A containing a primitive p-th root of unity. ⊗2 m 2 By taking inverse limits, we have lim lim ←−K2 (B)/p ' ←−H (Y, µpm ) m and, since K2 (A) is finite, K2 (A)(p) ' limK2 (A)/p , which surjects ←− onto limH 2 (X, µ⊗2 pm ). ←− Now the localization sequence and the vanishing of even K-groups of finite fields implies that K2 (A)(p) injects into K2 (B)(p), and hence limK2 (A)/pm maps isomorphically to limH 2 (X, µ⊗2 pm ). ←− ←− Finally we have the two exact sequences: K2 (A)/ps → K2 (A)/pm+s → K2 (A)/pm → 0 ⊗2 2 2 ⊗2 H 2 (X, µ⊗2 ps ) → H (X, µpm+s ) → H (X, µpm ) → 0.

Take inverse limits of these with respect to s and map the first to the second to finish the proof of the lemma.  Since we can find a localization B of A such that P ic(B)(p) = 0, the preceding lemmas provide a proof of Theorem 7.3.6

K(2, pm , A) is true.

(This is Lemma 10 in [So79], but his proof there is incomplete. I thank R. Sujatha and B. Kahn for helping to fill the gap.)

7.4 The theorem of Dwyer and Friedlander We give a sketch of the proof in [DF85] that the Dwyer–Friedlander map from K2n−1 (A) to H 1 (Xet , Zp (n)) is surjective, leaving out the deep topological arguments and confining ourselves to the algebraic and arithmetic side. Let R be Z[1/p]. We begin with the secondary transfer: Let g : Y → X be a map of noetherian schemes with Y a finite Galois extension of X with Galois group Γ. (Note that g is ´etale.) We recall that there exists usual transfer, or norm maps, which Dwyer and Friedlander call g!et (resp. g! ) mapping Knet (Y ) → Knet (X) (resp. Kn (Y ) → Kn (X), and also the analogous maps with coefficients. They construct a ‘secondary transfer’ mapping et g!! : (Ki−1 (Y, Z/pm ))Γ → Kiet (X, Z/pm )/Imageg!et ,

and prove

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Lemma 7.4.1 Let X be a scheme over R of finite mod p ´etale cohomological dimension. Suppose that X ← X1 ← X2 ← · · · ← Xn . . . is an infinite sequence of maps such that each Xn is Galois over X, each group Γn = Gal(Xn /X) is cyclic and the p-adic valuation of the order of Γn goes to ∞. Let Y = X1 and g be the map from Y to X. Then the secondary transfer maps defined above are surjective for all i ≥ 1. (In the application we choose Xn to be the Spec of the ring of integers in the pn -th cyclotomic extension of X, where X = Spec OF .) We next consider the following spectral sequence. Theorem 7.4.2 Let X be a connected scheme over R of finite Z/p cohomological dimension. Then there exists a strongly convergent fourthquadrant spectral sequence et E2i,−j = H i (Xet , Z/pm (j/2)) ⇒ Kj−i (X, Z/pm ).

(Here the convention is that H i (X, Z/n(j/2)) = 0 if j is odd.) Corollary 7.4.3 If H i (Xet , Z/pm (r)) = 0 for i > 2, then et K2r+1 (X, Z/pm ) is naturally isomorphic to H 1 (Xet , Z/pm (r + 1)). From now on, as in the introduction, let F be a number field, OF the ring of integers in F . Let p be a rational odd prime, A = OF [1/p], and X = Spec A. Lemma 7.4.4 The natural map from Kn (A, Z/pm ) to Knet (A, Z/pm ) is an isomorphism for n = 1, 2. The proof is a not very difficult consequence of Theorem 7.3.6 and Theorem 13.3.2. Lemma 7.4.5 If A contains a primitive pm -th root of unity ζ then taking cup-product with the class corresponding to ζ in K2et (A, Z/pm ) et induces an isomorphism from Knet (A, Z/pm ) to Kn+2 (A, Z/pm ). Proof: This is Theorem 5.6 of [DF85]. Lemma 7.4.6 If A contains a primitive pm -th root of unity, the map γn from Kn (A, Z/pm ) to Knet (A, Z/pm ) is naturally split surjective and remains so after taking invariants by a group of automorphisms of A.

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Proof: This follows easily from Lemmas 7.4.4 and 7.4.5 and the multiplicative properties of γn . Theorem 7.4.7 The natural maps from Kn (X, Z/pm ) to Knet (X, Z/pn ) are surjective for all n > 0, even if A does not contain a primitive p-th root of unity. Theorem 7.4.8 The natural map from K-theory to ´etale K-theory et induces a surjective map from K2n−1 (A) ⊗ Zp to K2n−1 (A, Zp ) 1 ∼ = H (Xet , Zp (n)). Proof This follows easily from Lemmas 7.4.1 and 7.4.6, by using the cyclotomic Zp -extension of A. 1. The natural map from Kn (A) ⊗ Zp to limKn (A, Z/pm ) is an isomor←− phism. This follows from the long exact sequence for relative K-theory and Quillen’s theorem that Kn (A) is finitely generated. 2. The natural map from limKn (A, Z/pm ) to limKnet (A, Z/pm ) is sur←− ←− jective. This follows immediately from Theorem 7.4.7. 3. If n = 2r + 1 is odd then we have limKnet (A, Z/pm ) ' limH 1 (Xet , Z/pm (r + 1)) = H 1 (Xet , Zp (r + 1)). ←− ←− This is an immediate consequence of Corollary 7.4.3, and the three statements above together yield the result we wanted to prove.

References [Bo74] Borel, A. 1974. Stable real cohomology of arithmetic groups. Ann. Scient. Ec. Norm. Sup. 4e s´erie, 7, 235–272. [DF85] Dwyer, W. and Friedlander E. 1985. Algebraic and Etale K-Theory. TAMS, 292, 247–280. [HS75] Harris, B. and Segal, G. 1975. Ki of rings of algebraic integers. Ann. of Math., 101, 20–33. [Ka93] Kahn, B. 1993. Descente galoisienne et K2 des corps de nombres. KTheory, 7, 55–100. [L72] Lichtenbaum, S. 1972. On the values of zeta and L-functions. Ann. of Math., 96, 338–360. [M04] Milne, J. 2004. Arithmetic Duality Theorems (2nd Edition). Booksurge, LLC. [Q72] Quillen, D. 1972. On the cohomology and K-theory of the general linear group over a finite field. Ann. of Math., 96, 552–586.

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[Se74] Serre, J.-P. 1974. Cohomologie Galoisienne SLN, 5, Springer. [So79] Soul´e, C. 1979. K-th´eorie des anneaux d’entiers de corps de nombres et cohomologie ´etale. Inventiones Math., 55, 251–295. [So84] Soul´e, C. 1984. Groupes de Chow et K-thorie de varits sur un corps fini. Math. Annal., 268, 317–345. [Su15] Sujatha, R. 2015. K-theoretic Background, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 22–44. [T76] Tate, J. 1976. Relations between K2 and Galois cohomology. Inventiones Math., 36, 257–274.

8 Soul´e’s Regulator Map Ralph Greenberg

8.1 Introduction This article is an exposition of part of Soul´e’s paper On higher p-adic regulators. We will describe the proof of one of his main results, but only over Q. This assumption makes the argument simpler, but the idea for the general proof is quite similar. The result concerns the Chern class maps ci,k : K2i−k (A, Zp ) −→ Hk (Spec(A), Zp (i)) for an odd prime p, where A = Z[ p1 ], k ∈ {1, 2} and 2i − k > 1. A conjecture of Quillen asserts that the map ci,k is an isomorphism. The surjectivity of these maps has been proven by Dwyer and Friedlander in [DF85]. Soul´e’s result is that the map is an isomorphism after tensoring with Qp . First of all, we recall the definitions of the above objects. For any n ≥ 1, let µpn denote the group of pn -th roots of unity and let µp∞ denote ∪n≥1 µpn . The action of GQ on µp∞ is given by a continuous homomorphism χ : GQ → Z× p . For any i ≥ 1, we can define the GQ module Zp (i) by Zp (i) = lim µ⊗i . ←− pn n≥1

It is a free Zp -module of rank 1 and GQ acts by χi . Let S = {p, ∞} and let QS be the maximal extension of Q which is unramified at all primes q 6∈ S. The action of GQ on Zp (i) factors through Gal(QS /Q). The ´etale cohomology group Hk (Spec(A), Zp (i)) is actually the Galois University of Washington, Seattle, USA. e-mail: [email protected]

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cohomology group Hk (QS /Q, Zp (i)), which can be defined by Hk (QS /Q, Zp (i)) = lim H k (QS /Q, µ⊗i pn ) . ←− n≥1

In [So79], Soul´e defines maps ci,k,n : K2i−k (A, Z/pn Z) −→ Hk (Spec(A), µ⊗i pn ) for all n ≥ 1. Taking inverse limits, one obtains the Chern class map ci,k , where one defines K2i−k (A, Zp ) = lim K2i−k (A, Z/pn Z) . ←− n≥1

Now a well-known theorem of Borel gives the dimension of the Q-vector space Km (F ) ⊗ Q for any number field F and any m ≥ 2. (Of course, K1 (F ) ∼ = F × and tensoring with Q gives an infinite dimensional vector space.) For F = Q, this dimension is 1 if m ≡ 1 (mod 4) and m ≥ 5, and is otherwise 0 for m ≥ 2. The rank of Km (A) is the same. For i > 1, it follows from this that the Zp -rank of K2i−k (A, Zp ) is 1 precisely when k = 1 and i is odd. Apart from that case, K2i−k (A, Zp ) is finite. The article of Lichtenbaum [Li15] proves the same statement about the Zp rank of Hk (QS /Q, Zp (i)), as we point out in section 2. Soul´ e’s Theorem. The map ci,k has finite kernel and cokernel. We need only consider the case where the Zp -rank of K2i−k (A, Zp ) is 1. The proof, which we describe in section 4, is based on Iwasawa theory.

8.2 The Zp -ranks of H 1 (QS /Q, Zp (i)) and H 2 (QS /Q, Zp (i)) Soul´e proof that H2 (QS /Q, Zp (i)) is finite for all i ≥ 2 is given in [Li15]. Here we explain how this gives the results about the Zp -rank of H1 (QS /Q, Zp (i)) mentioned in the introduction. This is based on the formula for the Euler–Poincar´e characteristic for the Gal(QS /Q)-module Zp (i)-module, which
is defined to be the alternating sum of the Zp -ranks of Ht QS /Q, Zp (i) for 0 ≤ t ≤ 2. It depends only on the parity of i,
and is equal to −1 if i
is odd, 0 if i is even. Since both H0 QS /Q, Zp (i)
and H2 QS /Q, Zp (i) have Zp -rank 0, it follows that H1 QS /Q, Zp (i) has Zp -rank 1 if i is odd and Zp -rank 0 if i is even.

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We also consider the discrete Galois module Qp /Zp (i). This is defined to be Qp /Zp as a group on which Gal(QS /Q) acts by χi . Thus, Qp /Zp (i) is the direct limit of the finite Galois modules µ⊗i pn . The corresponding cohomology groups are also discrete Zp -modules. One has an exact sequence 0 −→ Zp (i) −→ Qp (i) −→ Qp /Zp (i) −→ 0 of Gal(QS /Q)-modules, where Qp (i) = Zp (i) ⊗Zp Qp , a 1-dimensional Qp -vector space on which the Galois action is given by χi .
One deduces easily that, for any t, the Qp -dimension of Ht QS /Q, Qp (i) is the same
t as the Zp -rank of H QS /Q,
Zp (i) . If r is that rank, then it also follows that Ht QS /Q, Qp /Zp (i) contains a subgroup of finite index isomorphic to (Qp /Zp )r , namely its maximal divisible
subgroup. Thus, we refer to r as the ‘Zp -corank’ of Ht QS /Q, Qp /Zp (i) .
Since H2 QS /Q, Z
p (i) is finite for any i ≥ 2, it follows easily that H2 QS /Q, Qp /Zp (i) is also finite. However, if we assume that p is odd,
then it is known that H3 QS /Q, Zp (i) vanishes for any i. This is so because the p-cohomological dimension of Gal(QS /Q) is 2. The long exact cohomology sequence derived
from the above exact sequence then 2 shows that H QS /Q, Qp /Zp (i) is divisible and therefore vanishes for all i ≥ 2. Although it won’t have any bearing on the topic of this
article, we want to point out that the finite group H2 QS /Q, Zp (i) can be nontrivial when i is odd. To explain this, we use the local and global duality theorems of Tate and Poitou. First of all, the local Galois cohomology group H 2 (Qp , Zp (i)) is dual to H 0 (Qp , Qp /Zp (1 − i)) and that group vanishes unless i ≡ 1 (mod p − 1). Thus, for any i 6≡ 1 (mod p − 1), we have
H 2 (QS /Q, Zp (i)) = ker H 2 (QS /Q, Zp (i)) −→ H2 (Qp , Zp (i)) . That kernel is often denoted by

X2 (QS /Q, Zp (i)) and is dual to

X1 (QS /Q, Qp /Zp (1 − i)), which is the subgroup of H 1 (QS /Q, Qp /Zp (1 − i)) defined in a similar way. Furthermore, still assuming that i 6≡ 1 (mod p − 1), the maps H 1 (QS /Q, µ⊗(1−i) ) −→ H 1 (QS /Q, Qp /Zp (1 − i))[p] , p H 1 (Qp , µ⊗(1−i) ) −→ H 1 (Qp , Qp /Zp (1 − i))[p] p

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are both isomorphisms. As a consequence, it follows that 1 X1 (QS /Q, µ⊗(1−i) )∼ = X (QS /Q, Qp /Zp (1 − i))[p] , p

the subgroup of elements of order dividing p. Therefore, for i 6≡ 1 (mod p − 1), the vanishing of H 2 (QS /Q, Zp (i)) is equivalent to the van1 ⊗(1−i) ishing of X (QS /Q, µp ). Let L be the p-Hilbert class field of F = Q(µp ). The restriction map

X1 (QS /Q, µ⊗(1−i) ) −→ Hom∆ (Gal(L/F ), µ⊗(1−i) ) p p is also easily verified to be an isomorphism. It follows that the vanish1 ⊗(1−i) ing of X (QS /Q, µp ) is equivalent to the vanishing of the ω(1−i) component of the Zp [∆]-module Gal(L/F ), or equivalently, the vanishing (1−i) ) of Cl(F )[p](ω , where Cl(F ) denotes the ideal class group of F . Note 1−i p−i that ω =ω . 1−i

If i is even, then Cl(F )[p](ω ) can be nontrivial. Assuming that i is positive, this happens precisely when p divides the numerator of the rational number ζ(1 − i). If p is an irregular prime, this will indeed occur when i is in certain congruence classes modulo p − 1. Thus, for such i, H 2 (QS /Q, Zp (i)) will be nontrivial, and hence so will K2i−2 (Z). Here is one specific example. Suppose p = 37. In that case, p divides the numerator of the rational number ζ(1 − i) whenever i ≡ 32 (mod 36). 1−i Correspondingly, it is known that Cl(F )[p](ω ) is nontrivial, and hence so is H 2 (QS /Q, Zp (i)). In contrast, when i is odd, then a conjecture of Vandiver asserts that 1−i Cl(F )[p](ω ) is trivial, and hence that H 2 (QS /Q, Zp (i)) is also trivial. It is interesting to note that if we assume that Quillen’s conjecture is true, then, for any odd prime p and odd integer i ≥ 3, H 2 (QS /Q, Zp (i)) is trivial if and only if K2i−2 (A, Zp ) is trivial. A theorem of Lee and Szczarba [LS78] states that the order of the finite group K4 (Z) is a power of 2. Hence K4 (A, Zp ) is trivial for all odd primes p. As a consequence of the surjectivity of ci,2 , which is proved in p−3 [DF85], it follows that Cl(F )[p](ω ) is trivial, a result which is pointed out in [Ku92]. One interesting application concerns the Jacobian variety Ja,p of the curve y p = xa (1− x) , where a is an integer not divisible by p. This curve has genus (p −1)/2. It is proved in [Gr81] that |J(K)[p]| ≥ p3 p−3 and that the triviality of Cl(F )[p](ω ) implies that |J(K)[p]| = p3 . As this illustrates, the behavior of the K-groups has a close connection with various arithmetic questions.

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8.3 Some results from Iwasawa theory Let Fn = Q(µpn ) and Gn = Gal(Fn /Q) for n ≥ 1. Let [ F∞ = Fn = Q(µp∞ ), n≥1

a subfield of QS . Let G∞ = Gal(F∞ /Q). For any i ∈ Z, let Qp /Zp (i) be the corresponding Tate twist, as in section 2. The inflation-restriction sequence defines an isomorphism

G∞ H1 QS /Q, Qp /Zp (i) −→ H 1 QS /F∞ , Qp /Zp (i) . Furthermore, let X∞ be the maximal, abelian, pro-p quotient of the Galois group Gal(QS /F∞ ). In fact, X∞ = Gal(M∞ /F∞ ), where M∞ is the maximal, abelian, pro-p extension of F∞ unramified outside S. We then have

H 1 QS /F∞ , Qp /Zp (i) = Hom X∞ , Qp /Zp (i) . Thus, the restriction map defines a canonical isomorphism

H1 QS /Q, Qp /Zp (i) −→ HomG∞ X∞ , Qp /Zp (i) .

(8.1)

+ + Let F∞ denote the maximal real subfield of F∞ and let E∞ be the p ∞ p + + group of p-units in F∞ . Let N∞ = F∞ ( E∞ ), the field generated by + all p-power roots of the elements in E∞ . It is clear that N∞ ⊆ M∞ . Let Y∞ = Gal(N∞ /F∞ ). Note that G∞ acts on X∞ and Y∞ by conjugation.

If we let Q∞ be the cyclotomic Zp -extension of Q, then Q∞ ⊂ F∞ and ∆ = Gal(F∞ /Q∞ ) is cyclic of order p − 1. Let ω = χ|∆ . One can identify ∆ with Gal(F1 /Q) and ω gives the action of ∆ on µp . Furthermore, one can identify G∞ with ∆ × Γ, where Γ = Gal(F∞ /F1 ). One has a decomposition M (ω i ) Y∞ = Y∞ (8.2) 0≤i≤p−2 i

(ω )

where Y∞ is the subgroup of Y∞ on which ∆ acts by ω i . We call (8.2) the ∆-decomposition of Y∞ . However, the definition of N∞ and (ωi )

Y∞ shows that Y∞ = 0 when i is even. This follows from the following isomorphism which is defined by Kummer theory: + Y∞ ∼ ⊗ (Qp /Zp ), µp∞ ) = Hom(E∞

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which is equivariant for the action of G∞ , and hence of ∆. The action + + of ∆ on E∞ ⊗ (Qp /Zp ) factors through the quotient Gal(F∞ /Q∞ ). Let Λ = Zp [[Γ]], the completed group ring for Γ over Zp . The action of Γ on Y∞ commutes with the action of ∆ and so Γ acts on each (ω i ) component in the ∆-decomposition of Y∞ . Then, for each i, Y∞ is a Λ-module. It is finitely generated. It vanishes for even i, but when i is odd, the following theorem of Iwasawa [Iw73] tells us something more precise about its structure. (ω i )

Iwasawa’s Theorem. For odd i, Y∞ of Λ of finite index.

is isomorphic to a submodule

If p does not divide the class number of the maximal real subfield of (ω i ) Q(µp ), then Y∞ is itself a free Λ-module of rank 1 for odd values of (ω i )

(ω i )

(ω i )

i. Actually, in that case, we also have X∞ = Y∞ and so X∞ is also free of rank 1. We still have no examples where that class number is divisible by p, and it is possible that no such examples exist. This is the conjecture of Vandiver referred to in section 2. If we don’t assume that conjecture, then we can still say that Gal(M∞ /N∞ ) is a torsion (ω i ) Λ-module and hence Y∞ is the maximal torsion-free quotient of the (ω i ) Λ-module X∞ . By Kummer theory, as we stated above, there is a perfect pairing
+ Y∞ × E ∞ ⊗ Qp /Zp −→ µp∞ which is equivariant for the action of G∞ . For y ∈ Y∞ and α = η ⊗ ( p1t + + Zp ) in E∞ ⊗ Qp /Zp , this pairing is defined by √
 √ (y, α) 7→ y pt η pt η . Thus, Iwasawa’s theorem tells us something useful about the structure
(ωj ) + of E∞ ⊗Qp /Zp as a discrete Λ-module when j is even, namely that its Pontryagin dual is isomorphic to a submodule of Λ of finite index.
Furthermore, it follows that HomG∞ Y∞ , Qp /Zp (i) has Zp -corank 1 for any odd integer i. The maximal divisible subgroup of this group is isomorphic to Qp /Zp and hence must coincide with
the image of the maximal divisible subgroup of H1 QS /Q, Qp /Zp (i) under the restriction map (8.1). This is because the latter group also has Zp -corank 1. Thus, the restriction map identifies the maximal divisible subgroup of
H1 (QS /Q, Qp /Zp (i)) with a certain subgroup of HomG∞ Y∞ , Qp /Zp (i) .

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That subgroup can be described as Hom Gal(D∞ /F∞ ), Qp /Zp (i) , where D∞ is a certain extension of F∞ contained in N∞ . It has the following properties: Gal(D∞ /F∞ ) ∼ = Zp , D∞ is Galois over Q, and G∞ acts on Gal(D∞ /F∞ ) by χi . (ω i )

Iwasawa’s theorem is equivalent to the assertion that Y∞ is a torsionfree Λ-module of rank 1 for any odd value of i. For that assertion implies (ω i ) that the reflexive hull of Y∞ is a free Λ-module of rank 1. The quotient is a pseudo-null Λ-module and hence finite. Iwasawa gives an interpretation of that quotient. For any torsion-free Λ-module Y , we let Ye denote its reflexive hull. Let H = Ye∞ Y∞ . For any m ≥ n ≥ 0, one has a natural map Cl(Fn ) → Cl(Fm ) defined by mapping the class of an ideal In in Fn to the class of the ideal generated by In in the ring of integers of Fm . One defines Cl(F∞ ) to be the direct limit under these maps. For any n ≥ 0, let
Jn = ker Cl(Fn ) → Cl(F∞ ) . It is clear that Jn is a finite p-group. Iwasawa proves that the Jn ’s have bounded order as n varies. More precisely, let L∞ be the maximal, abelian, pro-p extension of F∞ which is everywhere unramified. We can regard Gal(L∞ /F∞ ) as a Λ-module. It is a torsion Λ-module according to one of Iwasawa’s basic theorems. Let J denote the maximal finite Λsubmodule of Gal(L∞ /F∞ ). Then one shows that Jn ∼ = J for sufficiently large n. The isomorphism is G∞ -equivariant. In addition, Iwasawa shows i that J (ω ) = 0 for odd i. Thus, Jn can be identified with the kernel of + the natural map Cl(Fn+ ) → Cl(F∞ ). Furthermore, Iwasawa shows that H ∼ = Hom(J, µp∞ ). In particular, for any odd i, we have i 1−i ] (ω i )  (ωi ) ∼ Y∞ Y∞ = H (ω ) ∼ = J (ω )

where the last isomorphism is just a group isomorphism. This makes it (ω i ) clear that if p doesn’t divide the class number of F1+ , then Y∞ is free for all odd i. For that assumption implies that, for all n ≥ 0, the class number of Fn+ is not divisible by p, and hence certainly Jn = 0 too. Thus (ω i )

Vandiver’s conjecture implies that the Λ-modules Y∞ odd values of i, as we mentioned earlier.

are free for all

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8.4 The proof of Soul´ e’s theorem (ωi )

To simplify the rest of the discussion, we will assume that Y∞ is actually a free Λ-module for all odd i. Later on, we will point out how to modify the argument if this is not the case. For any n ≥ 1, let Γn = Gal(F∞ /Fn ). Of course, G∞ /Γn ∼ = Gn . Consider the maps En+ ⊗ Qp /Zp −→ H 1 (FS /Fn , µp∞ ) −→ H 1 (FS /F∞ , µp∞ )Γn = HomΓn (X∞ , µp∞ ) . (8.3) Here we are denoting E(Fn+ ) by En+ . The first map is the Kummer homomorphism and is injective. The second map is the restriction map and is an isomorphism. The last group in (8.3) contains HomΓn (Y∞ , µp∞ ).

(8.4)

The freeness assumption about the odd ∆-components of Y∞ has the following consequences. Note that Gn acts on (8.4) and that action fac+ tors through the quotient G+ n = Gal(Fn /Q). The freeness assumption implies that the Pontryagin dual of (8.4) must be a free Zp [G+ n ]-module of rank 1. Furthermore, the image of the composite map (8.3) is contained in HomΓn (Y∞ , µp∞ ). But since En+ ⊗ Qp /Zp has Zp -corank equal to [Fn+ : Q], which is the same as the Zp -corank of (8.4), it follows that (8.3) induces an isomorphism En+ ⊗ Qp /Zp −→ HomΓn (Y∞ , µp∞ ) and hence that the Pontryagin dual of En+ ⊗ Qp /Zp is a free Zp [G+ n ]+ + pt t module of rank 1. It also follows that En /(En ) is a free (Z/p Z)[G+ n ]module of rank 1 for all n, t ≥ 1. t

+ + p Consequently, one sees that the norm map Nm/n from Em /(Em ) to + + pt En /(En ) is surjective. Furthermore, the above remarks imply that one can choose a sequence {ηn }, where ηn ∈ En+ , such that the image of ηn t in En+ /(En+ )p is a generator of that (Z/pt Z)[G+ n ]-module and such that Nm/n (ηm ) = ηn for m ≥ n ≥ 1. To explain the second property, first note that if the image of ηn in En+ /(En+ )p generates that (Z/pZ)[G+ n ]+ + pt module, then the image of ηn in ηn in En /(En ) is a generator of that + + (Z/pt Z)[G+ n ]-module for any t ≥ 1. In fact, the Z[Gn ]-submodule of En generated by ηn will have index not divisible by p.

Now take t = 2. To argue inductively, suppose that ηn has been chosen 2 2 + + as above. The map Nn+1,n : En+1 /(En+1 )p → En+ /(En+ )p is surjective. 2 + 0 0 Therefore, for some ηn+1 ∈ En+1 , we have Nn=1,n (ηn+1 ) = ηn δ p , where

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0 δ ∈ En+ . Let ηn+1 = ηn+1 δ −p . Then Nn+1,n (ηn+1 ) = ηn and the image + + p of ηn+1 in En+1 /(En+1 ) is a generator. This shows the existence of a norm-compatible sequence {ηn } of p-units with the properties stated above.

We remark in passing that, if we assume that p does not divide the class number of F1+ , then p also does not divide the class number of Fn+ for all n ≥ 1. Thus, for each n, the analytic class number formula implies that the index of the subgroup of cyclotomic p-units in the group En+ has index prime to p. That subgroup is a free Z[G+ n ]-module and one could choose ηn to be a generator. In contrast, the approach described above is purely algebraic. One can use the ηn ’s to construct an element in H 1 (QS /Q, Zp (i)) which generates a subgroup of finite index. First of all, one has the Kummer map n

En+ /(En+ )p

−→ H 1 (QS /Q, µpn ),

which is injective and Gn -equivariant. Let ζn be a generator of µpn , p chosen so that ζn+1 = ζn . We write ζn⊗i for the corresponding element ⊗i ⊗i of µpn . Then the ζn ’s define an element of Zp (1) which generates that Zp -module. Tensoring with ζn⊗i−1 , we obtain an injective Gn -equivariant map n

En+ /(En+ )p ⊗µ⊗i−1 −→ H 1 (QS /Qn , µpn )⊗µ⊗i−1 = H 1 (QS /Qn , µ⊗i pn pn pn ), where the last identification arises because Gal(QS /Qn ) acts trivially on µpn . We write ηn ∪ ζn⊗i−1 for the image of ηn ⊗ ζ ⊗i−1 under the above map, thinking of it as the cup-product map H 1 (QS /Qn , µpn ) ⊗ H 0 (QS /Qn , µpn )⊗i−1 −→ H 1 (QS /Qn , µ⊗i pn ) . n

It is important to note that since En+ /(En+ )p is a free Z/pn Z[Gn ]n module, then the same is true for En+ /(En+ )p ⊗ µ⊗i−1 . It is generated pn by ηn ⊗ ζ ⊗i−1 . In particular, the image of ηn ⊗ ζ ⊗i−1 under the norm map Nn has order pn . Here we write Nn for the norm map for any Zp [Gn ]-module. Gn Now Nn (ηn ∪ζn⊗i−1 ) is an element of H 1 (QS /Qn , µ⊗i . The restricpn ) tion maps 1 ⊗i Gn H 1 (QS /Q, µ⊗i pn ) −→ H (QS /Qn , µpn )

are injective. (One uses the fact that i is odd, and so not divisible by

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⊗i−1 p−1, in order to verify that H 1 (Gn , µ⊗i ) pn ) vanishes.) Hence Nn (ηn ∪ζn ⊗i 1 n defines an element in H (QS /Q, µpn ) of order p .

The element Nn (ηn ∪ ζn⊗i−1 ) is the image of Nn (ηn ⊗ ζ ⊗i−1 ), which is
Gn n contained in En+ /(En+ )p ⊗ µ⊗i−1 . It has the form εn ⊗ ζ ⊗i−1 , where pn + + + pn ε ∈ En . The image of εn in En /(En ) is a generator of the subgroup √ on which Gn acts by χ1−i . By Kummer theory, Fn ( pn εn ) is Galois over √ Q and Gn acts on Gal(Fn ( pn εn )/Fn ) by χi . Furthermore, the field D∞ √ n defined earlier is F∞ ({ p εn | n ≥ 1 }). Following Soul´e, we will define an element in K2i−1 (A, Z/pn Z) which is almost mapped to the above element by ci,1,n . This will suffice for proving Soul´e’s theorem. One first defines an element αn in K2 (A, Z/pn Z) as follows. Let An be the integral closure of A in Fn . One has an isomorphism

π2 BGL1 (An ); Z/pn Z −→ π1 BGL1 (An ) pn = GL1 (An )pn = µpn , where BGL1 (An ) is the classifying space for GL1 (An ) = A× n . (For any abelian group X, the notation Xpn denotes the subgroup of elements of order dividing pn .) The inclusion map GL1 (An ) → GL(An ) induces a map

π2 BGL1 (An ); Z/pn Z −→ π2 BGL(An ); Z/pn Z = K2 (An ; Z/pn Z) and αn is defined as the element corresponding to ζn ∈ µpn . Note that K1 (An ) = A× n and so K1 (An )pn = µpn . The Bockstein map K2 (An , Z/pn Z) −→ K1 (An )pn sends αn to ζn . Now K1 (An ) = A× n contains ηn . Under the product operation for K-groups, one can consider ηn · αi−1 n , which is an element in K2i−1 (An ; Z/pn Z). The norm map for Fn /Q then n defines an element Nn (ηn · αi−1 n ) in K2i−1 (A; Z/p Z), which we denote by βn . In fact, these elements form a projective system with respect to the maps K2i−1 (A; Z/pn+1 Z) −→ K2i−1 (A; Z/pn Z) defining K2i−1 (A; Zp ). Soul´e verifies this as follows. Let rn generically denote various maps induced by the reduction map

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Z/pn+1 Z → Z/pn Z. We want to show that rn (βn+1 ) = βn . Let jn generically denote maps induced by the inclusion An → An+1 . Then one has jn (αn ) = rn (αn+1 ). This can be seen from the commutative diagram:

/ A× π2 BGL1 (An+1 ); Z/pn+1 Z n+1 pn+1 z→z p

rn


×



/ A n+1 O


π2 BGL1 (An+1 ); Z/pn Z O jn

pn

id

/ A× n


π2 BGL1 (An ); Z/pn Z


pn

Furthermore, letting Nn+1/n denote generically the maps induced from the norm map from Fn+1 to Fn , we have

i−1 rn Nn+1/n (ηn+1 · αi−1 n+1 ) = Nn+1/n ηn+1 · rn (αn+1 )
= Nn+1/n ηn+1 · jn (αni−1 ) = Nn+1/n (ηn+1 ) · αi−1 = ηn · αni−1 . n Applying the map Nn , one then obtains the claimed equality rn (βn+1 ) = βn . One can define the above elements starting from any norm compatible sequence ηn of p-units. Thus, we have defined maps ϕn
and ψn from En ⊗ µ⊗i−1 to K2i−1 (A; Z/pn Z) and to H 1 QS /Q, µ⊗i pn pn , respectively. Theorems 1 and 2 in [So79], which describe general properties of the Chern class maps, show that the following diagram commutes for all n. En ⊗ µ⊗i−1 pn

ϕn

/ K2i−1 (A; Z/pn Z)

i!ψn

(

ici,1,n


H 1 QS /Q, µ⊗i pn Since the images of the ψn ’s contain elements of arbitrarily high order, and i! is fixed, it follows that the image of ici,1
has infinite order. The 1 image is a Zp -submodule of H QS /Q, Zp (i) which must then have finite index, proving Soul´e’s theorem under our simplifying assumption (ωi ) that the Λ-modules Y∞ are free for all odd i. (ω i )

We now discuss the argument without the assumption that the Y∞ ’s

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are free. We will show that one can always find a norm compatible se+ quence {ηn } such that ηn ∈ En+ , the Z[G+ n ]-submodule of En generated by ηn is free, and the power of p dividing the corresponding index is bounded as n varies. That suffices for carrying out the above argument. Iwasawa’s theorem implies that Y∞ is isomorphic to a Λ-submodule p−1 of Λ 2 with finite index. More precisely, considering it as a Zp [[G∞ ]]module, Y∞ is isomorphic to a submodule of A∞ = Zp [[G∞ ]]− with finite index. The Kummer duality pairing then implies a similar statement + about E∞ ⊗ Qp /Zp . Let Φ∞ denote the Pontryagin dual of Zp [[G+ ∞ ]]. Thus, Φ∞ is a Λmodule with Λ-corank (p − 1)/2 on which ∆+ acts. Furthermore, for any (ω j ) ˆ as a Λ-module, where Λ ˆ is the Pontryagin dual of Λ. even j, Φ∞ ∼ =Λ It follows that ΦΓ∞n is isomorphic to the Pontryagin dual of Zp [G+ n ] as a module over that group ring. In particular, ΦΓ∞n is a divisible Zp -module whose Zp -corank is pn (p − 1)/2 and, for every t ≥ 1, ΦΓ∞n [pt ] is a free module over (Z/pt Z)[G+ n ] of rank 1. A consequence of Iwasawa’s theorem is that there is a surjective homomorphism + Φ∞ −→ E∞ ⊗ Qp /Zp k of discrete Zp [[G+ ∞ ]]-modules whose kernel is finite. Let p be the order of the kernel. It follows that, for every n ≥ 0, there is a Zp [G+ n ]-module homomorphism
Γn + ΦΓ∞n −→ E∞ ⊗ Qp /Zp (8.5)

with finite kernel of order bounded by pk . p−1

Since Y∞ is a submodule of Λ 2 with finite index, it follows that (Y∞ )Γn is a finitely generated Zp -module of rank (p − 1)pn /2. Con
Γn + sequently, the maximal divisible subgroup of E∞ ⊗ Qp /Zp is ison morphic to (Qp /Zp )(p−1)p /2 . Since ΦΓ∞n is divisible and has the same Zp -corank, it is clear that the image of the map (8.5) is precisely the
Γn + maximal divisible subgroup of E∞ ⊗ Qp /Zp . Now consider the map En+ ⊗ Qp /Zp −→

+ E∞ ⊗ Qp /Zp


Γn

.

+ Since the restriction map H 1 (Fn+ , µp∞ ) → H 1 (F∞ , µp∞ ) is injective, so is the above map. Both groups have the same Zp -corank and the first group is divisible. Hence the image of the map is again the
Γabove n + maximal divisible subgroup of E∞ ⊗ Qp /Zp . Thus, the map (8.5)

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can be identified with a G+ n -equivariant map ΦΓ∞n −→ En+ ⊗ (Qp /Zp )

(8.6)

with finite kernel whose order is bounded by pk . For a fixed t ≥ 1, the subgroups Φn,t = ΦΓ∞n [pt ] form an increasing sequence of subgroups of Φ∞ [pt ]. Suppose that m ≥ n. Regarding Φm,t + as a Gal(Fm /Fn+ )-module, the norm map Nm,n maps Φm,t onto Φn,t . That is, the groups {Φn,t } form a norm-compatible sequence. For each n ≥ 1, let Ψn,t denote the image of Φn,t under the map (8.6), which is a subgroup of ∼ E + /(E + )pt . (E + ⊗ Qp /Zp )[pt ] = n

n

n

The sequence {Ψn,t } also forms a norm-compatible sequence. Since the t groups Φn,t and En+ /(En+ )p have the same order, the index of Ψn,t in t En+ /(En+ )p has maximal order pk as n varies. t

Suppose that ηn ∈ En+ is chosen so that its image in En+ /(En+ )p is in Ψn,t and generates that (Z/pt Z)[G+ n ]-module for t = k + 1. Then it follows easily that the same statement is true for all t ≥ k + 1. Further+ more, the Z[G+ n ]-submodule of En generated by ηn has the same Z-rank + as En itself and therefore has finite index. Furthermore, the power of p dividing that index will be at most pk . To show that we can find a norm-compatible sequence of such ηn ’s, we argue inductively as before. Take t = k + 2. If ηn has been chosen, k+2 + 0 then pick ηn+1 ∈ En+1 so that its image in En+ /(En+ )p generates k+2 0 Ψn,k+2 and such that Nn+1,n (ηn+1 ) = ηn δ p , where δ ∈ En+ . Then k+1 0 ηn+1 = ηn+1 δ −p has norm ηn . Also, one sees easily that the image + + pk+1 of ηn+1 in En /(En ) is a generator of Ψn,k+1 as a (Z/pk+1 Z)[G+ n ]module.

References [DF85] Dwyer, W. G., and Friedlander, E. M. 1985. Algebraic and etale Ktheory. Trans. Amer. Math. Soc., 292, 247–280. [Gr81] Greenberg, R. 1981. On the Jacobian variety of some algebraic curves. Comp. Math., 42, 345–359. [Iw73] Iwasawa, K. 1973. On Z` -extensions of algebraic number fields. Ann. of Math., 98, 246–326. [Ku92] Kurihara, M. 1992. Some remarks on conjectures about cyclotomic fields and K-groups over Z. Comp. Math., 81, 223–236.

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[Li15] Lichtenbaum, S. 2015. Soul´e’s theorem. in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 130–139. [LS78] Lee, R., and Szczarba, R. H. 1978. On the torsion in K4 (Z) and K5 (Z), with addendum by C. Soul´e. Duke Math. Jour., 45, 101–132. [So79] Soul´ e, C. 1979. K-th´eorie des anneaux d’entiers de corps de nombres et cohomologie ´etale. Inv. Math., 55, 251–295. [So81] Soul´ e, C. 1981. On higher p-adic regulators. Lect. Notes in Math., 854, 372–401. Springer-Verlag.

9 On the Determinantal Approach to the Tamagawa Number Conjecture T. Nguyen Quang Do

Abstract We give a survey of Fontaine and Perrin-Riou’s formulation of the Tamagawa number conjecture on special values of the L-functions of motives in terms of determinants and Galois cohomology. Following Fontaine’s Bourbaki talk, we show its equivalence with the original formulation of Bloch–Kato. As an illustration, we sketch a proof for the Dedekind zeta function of an abelian number field. The conjecture of Bloch and Kato [BK90] on the special values of the L-functions of motives was originally expressed – in analogy with the theory of semi-simple algebraic groups – in terms of Haar measures and Tamagawa numbers. Hence its usual other name, the Tamagawa number conjecture (TNC for short), to which we shall stick in these notes, in order to avoid confusion with another Bloch–Kato conjecture (on K-theory and Galois cohomology; see [Ko15] in this volume). Later on, Fontaine and Perrin-Riou [FPR94] proposed another formulation in terms of determinants of perfect complexes and Galois cohomology. Although the arithmetic becomes less apparent in the new formalism, it allows more flexibility and generality, as illustrated for instance by the subsequent development of the equivariant version of the conjecture (ETNC for short), which ‘provides a coherent overview and refinement of many existing “equivariant” conjectures, including for example the refined Birch–Swinnerton-Dyer conjecture for CM elliptic curves formulated by Gross, the conjectural congruences of Dirichlet L-functions formulated by Gross and Tate, the conjectures formulated by Chinburg et University of Franche-Comt´ e, France. e-mail: [email protected]

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al. in the area of Galois module theory’ (see [BG03, Introduction, p.303]). As for the TNC proper, Fontaine and Perrin-Riou note that ‘the complicated formulas bringing in Tamagawa numbers, orders of Shafarevich groups, are only the consequence of the explicit calculation of an “intrinsic” formula making use of certain Euler–Poincar´e characteristics’ [FPR94, p. 600]. Being among the arithmeticians who regret the occultation of these ‘complicated formulas’ in the style of the analytic class number formula, we were too happy to accept the proposal of the organizers of the Pune workshop to write these notes on the comparison and (at least when the field of coefficients is Q) the equivalence between the two formulations, that of Bloch–Kato and that of Fontaine–PerrinRiou. There is of course nothing new under the sky, this task having been already performed by Fontaine in his Bourbaki talk [Fo92], which we shall follow closely, resorting occasionally to [BK90] and [FPR94]. Our only contribution will be, with the zeta function in view, to develop systematically the examples of the Tate motives as we go along. These notes will culminate in a sketched proof (following [BN02] and parallel to [BK90]) of the conjecture on special values of the Dedekind zeta function ζF (s) of an abelian number field F . See also a refined version for Dirichlet’s motives in [HK03]. Some explanations are in order: the scope of the workshop was on purpose limited to the Riemann-zeta function (i.e., F = Q); but we think that within these limits, the Fontaine-PerrinRiou formulation could not actually show its whole versatility. Hence the compromise on ζF (s) : when F 6= Q, real problems occur, which were masked when staying in Q (the K-groups and the X-groups no longer coincide, Euler factors and regulators become troublesome...); but, paradoxically, the ‘motivic process’ at work can be made clearer we hope.

9.1 Motives and avatars In this section, we set up the minimal motivic formalism necessary to formulate the Bloch–Kato conjecture in terms of complexes and determinants. Let us fix two a priori independent number fields E and F . To make a clearer distinction between the primes (finite or infinite) of these two fields, we shall systematically use the notation l for E and p for F .

9.1.1 Let M be a mixed motif defined over F , with coefficients in E. Since the category MMF (E) of such motives is not yet fully known, in the sequel,

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we shall only deal with a system of ‘realizations’ of M (the Sanskrit word ‘avatars’ is certainly more suggestive) together with comparison isomorphisms. • The ´etale realizations Ml for all finite primes l | l of E, are l-adic representations of the absolute Galois group GF (or more generally of a specific profinite group), i.e., finite dimensional vector spaces over the completed field El with a continuous action of GF . • The Betti realizations MB,p , for all infinite primes p | ∞ of F, are finite dimensional E-vector spaces with an action of complex conjugation if p + is real. We denote by MB, p the subspace fixed by complex conjugation if p is real, and MB,p itself if not. • The de Rham realization MdR is a finitely generated module over E⊗Q F equipped with a decreasing filtration (Filj MdR )j∈Z of sub- E ⊗ F -modules, the Hodge filtration. The tangent space tM of M is by definition the quotient MdR /Fil0 MdR .

For more detailed definitions, in particular concerning the comparison isomorphisms, we refer to [Fo92, §6], and also to [Ra15] (for E = F = Q) in this volume. The fact is that we do not actually need these details in the general case, and we need them only in the particular case of the Tate motives that we are working with. Example 9.1.1 The Tate motives Q(m), m ∈ Z. Here we take E = Q. The Tate motif Q(m) over F (usually also denoted by h0 (F )(m)) is a pure motif of weight −2m whose avatars are: - for any prime number l, the ´etale realization Ql (m) = Zl (m) ⊗ Ql , with the natural action of GF . Here ( . )(m)denotes the m-th Tate twist, i.e., the Galois action induced by σ 7→ κcyc (σ)m (σ), where κcyc is the l-cyclotomic character. - for any infinite prime p of F , the Betti realization Q(m)B,p which Q is defined as being (2πi)m Q, and Q(m)B := Q(m)B,p , with GF p|∞ acting on the set of infinite primes and on the local factors. - the de Rham realization is Q(m)dR = F, equipped with the filtration ( F if j ≤ −m j FildR Q(m) = 0 otherwise. We shall write Z(m)dR = OF , the ring of integers of F . As for the comparison isomorphisms, they are defined as follows:

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- between Betti and de Rham, the isomorphism Q(m)B,p ⊗Q C ' Q(m)dR ⊗F C is induced by the natural inclusion (2πi)m Q ⊂ C. - between Betti and l-adic, the isomorphism Q(1)B,p ⊗Q Ql ' Ql (1) is induced by the family of maps x ∈ 2πiZ 7→ exp(x/n) ∈ µpn . The shift from Q(1) to Q(m) is obvious. Remark: Taking Q for the field of coefficients E is no longer sufficient to tackle extended problems such as the TNC for Dirichlet motives [HK03] or the ETNC for Tate motives [BG03], for which it becomes necessary to enlarge E. For instance, if F/Q is a Galois extension with group G, then the realizations of M = Q(0)E = h0 (F )E are 0 MB := HB (Spec(F ⊗ C), E) '

⊕

E ' E[G]

τ :F →C

0 MdR := HdR (F/Q) ⊗Q E ' F ⊗Q E and

¯ l) ' Ml := He´0t (Spec F ⊗Q Q.E

⊕

El ' El [G]

¯ τ :F →Q

where l is a prime number, El ' E ⊗Q Ql . The shift from 0 to m is ‘obvious’ as far as the ´etale realization is concerned: if Mm = Q(m)E , then (Mm )l ' El [G](m) (the group algebra with twisted coefficients). For the other realizations, the precise definitions are more involved.

9.2 l-adic representations and the fundamental line Let us leave for a moment the abstract realm of motives and venture into the concrete land of l-adic representations, armed with [Fo92] as a guidebook. For a finite place l (resp. p) of E (resp. F ), let us write El,p = El if l 6= p and El,p = (Fp )0 ⊗Q Ep if l = p, where (Fp )0 is the p maximal unramified extension of Qp contained in Fp .

9.2.1 A hierarchy of l-adic representations We shall make free use of the Fontaine rings Bcrys,p ⊂ BdR,p (see [Fo82]); the field BdR,p is an F¯p -algebra equipped with a decreasing filtration (Filj BdR,p )j∈Z and a natural action of Gp = Gal(F¯p /Fp ); the ring Bcrys,p is Gp -stable, and equipped with an endomorphism φ commuting with the action of Gp and semi-linear with respect to the automorphism

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σp of F¯p which lifts the p-th power map of the residual field of (F¯p )0 . So σp is just a suitable p-primary power of the Frobenius automorphism G G Frobp at the prime p. Note that (BdR,p ) p ' Fp and Bcrys,p p = (Fp )0 . Let V be an l-adic representation of Gp of dimension b(V ). Define I Dp (V ) = V p if l 6= p (where Ip is the inertia subgroup), Dp (V ) = G Dcrys,p (V ) = (Bcrys,p ⊗ V ) p if l = p. These are free El,p -modules of rank bp (V ) ≤ b(V ), with a linear action by Frob−1 p defined as follows: if −1 l 6= p, Frobp lives in Gp /Ip , hence acts on Dp (V ); if l = p, φ acts on Bcrys,p ⊗ V via b ⊗ v 7→ φb ⊗ v, the module Dp (V ) is φ-stable and the rp action of Frob−1 p is defined as the action of φ , with rp = [Fp : Qp ]. We say that V has good reduction if bp (V ) = b(V ); if l 6= p, this simply means that V is unramified; if l = p, V is then called crystalline. Finally, let us recall the definition of a de Rham representation. If l = p, put G DdR,p (V ) = (BdR,p ⊗Q V ) p , which is an (Fp ⊗Q El )-module equipped p

p

G with a decreasing filtration Filj DdR,p (V ) := (Filj BdR,p ⊗Q V ) p . The p

dimension of the Fp -vector space DdR,p (V ) is ≤ [El : Qp ].b(V ), and V is called de Rham if there is equality, in which case DdR,p (V ) is free of rank b(V ) over Fp ⊗ El and BdR,p ⊗Fp DdR,p (V ) ' BdR,p ⊗Q V. p If l = p, then the tangent space of V is by definition the Fp ⊗ El module  Gp tV,p := (BdR,p / Fil0 BdR,p ) ⊗Q V . p

When V is de Rham, tV,p can be identified with DdR,p (V )/ Fil0 DdR,p (V ). If l 6= p, tV,p = 0 by convention.

9.2.2 The f -cohomology Let V be an l-adic representation of Gp . Denote by Hf1 (Fp , V ) the subEl -vector space of H 1 (Fp , V ) consisting of all extension classes W of El by V such that the natural map Dp (W ) → Dp (El ) = El is surjective (this amounts to ask that if V has good reduction, then W has too). It can be checked that ( Hf1 (Fp , V

)=

ker(H 1 (Fp , V ) → H 1 (Ip , V )) 1

1

ker(H (Fp , V ) → H (Fp , Bcrys,p ⊗Q V )) p

if l 6= p if l = p.

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I

Hence, if l 6= p, Hf1 (Fp , V ) ' H 1 (kp , V p ) and we have an exact sequence (recall that kp is the residual field and Dp = Dcrys,p ): φ−1

0 → H 0 (Fp , V ) → Dp (V ) −→ Dp (V ) → Hf1 (F, V ) → 0.

(9.1)

Things get more complicated when l = p. The rings BdR,p and Bcrys,p are related by the so-called fundamental exact sequence (see [BK90, Prop. 1.7]):
0 → Qp → Bcrys,p → Bcrys,p ⊕ BdR,p / Fil0 BdR,p → 0, (9.2) where the leftmost arrow is the inclusion and the middle arrow is given by (φ−1)⊕π, π being the projection of Bcrys,p ⊂ BdR,p to BdR,p / Fil0 BdR,p . Upon tensoring with V and taking Gp -invariants, one gets another exact sequence (recall that Dp (V ) = Dcrys,p (V ))): δ

V 0 → H 0 (Fp , V ) → Dp (V ) → Dp (V ) ⊕ tV,p −→ Hf1 (Fp , V ) → 0. (9.3)

The map δV induces a homomorphism expV,p : tV,p −→ Hf1 (Fp , V ) called the Bloch–Kato exponential map. Note that if H 0 (Fp , V ) = 0, then expV,p is an isomorphism. Example 9.2.1

(continuation of example 9.1.1)

Take E = Q, l = p and V = Qp (m). The module Dcrys (Qp (m)) is a one-dimensional vector space over (Fp )0 . Let ε = (ζpn )n≥0 be a coherent system of primitive pn -th roots of unity. Let [ε] ∈ BdR,p be its Teichm¨ uller representative and t = log[ε](see [Fo82]). Then the element em = t−m ⊗ ε⊗m ∈ DdR,p (Qp (m)) does not depend on the choice of ε and yields a canonical basis of Dp (Qp (m)). The map φ acts on em by φ(em ) = p−m em . The module DdR,p (Qp (m)) is an Fp -vector space of dimension one generated by em and equipped with the filtration ( DdR,p (Qp (m)) ' Fp if j ≤ −m j FildR (Qp (m)) = 0 otherwise. From the exact sequence (9.3) for V = Qp (m), we deduce immediately that for m 6= 0, the map φ − 1 : Dp (Qp (m)) → Dp (Qp (m))

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is bijective and the Bloch–Kato exponential expp,m : tQ (m),p → Hf1 (Fp , Qp (m)) p is an isomorphism. This can be made more explicit: the exponential seP∞ ries E(x) = n=0 xn /n! yields an isomorphism Fp ' Op× ⊗Qp ; composing it with the Kummer homomorphism Fp× → H 1 (Fp , Zp (1)), we get a map Fp → H 1 (Fp , Zp (1)) which coincides with expp,1 (see [BK90], section 3.10). In particular the image of the induced map Op × ⊗ Qp → H 1 (Fp , Qp (1)) coincides with Hf1 (Fp , Qp (1)). The following result is well known (see e.g., [BK90, §3]): Lemma 9.2.2 For E = Q, we have (i) For m 6= 1, we have  1   H (Fp , Qp (m)) 1 Hf (Fp , Qp (m)) = H 1 (kp , Qp )   0

if m ≥ 2 if m = 0 if m ≤ −1.

(ii) For l 6= p, ( Hf1 (Fp , Ql (m))

=

H 1 (kp , Ql )

if m = 0

0

otherwise.

9.2.3 Pseudo-geometric representations and the fundamental line We now replace Gp by GF and consider an l-adic representation. It is called pseudo-geometric if it is unramified outside a finite set of places of F , and is de Rham at all places dividing l. The property of being pseudogeometric is compatible with taking sub-objects, quotient objects, direct sums, tensor products, duals and Tate twists. 9.2.3.1 Define Hf1 (F, V ) as the inverse image of map H 1 (F, V ) →

Y p

Q p

Hf1 (Fp , V ) by the localization

H 1 (Fp , V ).

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This is a finite dimensional El -vector space. Let us introduce some more notation. For a finite dimensional vector space W over a field K, detK W is the maximal exterior power of W over K; if W ∗ is the dual space, detK W ∗ will be denoted by det∗K W or det−1 K W (this ‘inverse’ notation will soon be explained below); if K = El , let us abbreviate detE as detl . l For an l-adic pseudo-geometric representation V of GF , put V + = ⊕ H 0 (Gp , V ), tV = ⊕ tV,p p|∞ p|l where, as before, the ps are places of F , and 1 Lf (V ) := detl H 0 (F, V ) ⊗ det−1 l Hf (F, V ).

The fundamental line of V is the one-dimensional El -vector space + ∆f (V ) := Lf (V ) ⊗ Lf (V ∗ (1)) ⊗ det−1 l V ⊗ detl tV .

Example 9.2.3

(continued from example 9.2.1)

We go back to the example E = Q, l = p, V = Qp (m) (obviously a pseudo-geometric representation of GF ). The computation of Lf (Qp (m)) is non-trivial, in that it uses p-adic regulators: Let p 6= 2, and let chi,m : K2m−i (OF ) ⊗ Zp → He´it (OF [1/p], Zp (m)) m ≥ 1, i = 1, 2, be the p-adic Chern class map constructed by Soul´e [So79] and Dwyer– Friedlander [DF85]. The Quillen–Lichtenbaum conjecture (now a theorem, see [Ko15] in this volume) states that chi,m is an isomorphism, but we shall only need the following properties: • the map ch1,m is surjective, with finite kernel [DF85] • the map ch2,m is surjective [So79], [DF85] and split [Ka93] • then the finiteness of K2m−2 (OF ) implies the vanishing of the cohomology group He´2t (OF [1/p], Qp /Zp (m)). For all m ∈ Z, m 6= 1, the ‘m-twisted Leopoldt Conjecture’, (Leopm ) for short, predicts the vanishing of He´2t (OF [1/p], Qp /Zp (m)) (see [Sc75, p.192]). The classical Leopoldt conjecture is (Leop0 ). Now the localization exact sequence in ´etale cohomology [So79, p.268] 0 → He´1t (OF [1/p], Qp (m)) → H 1 (F, Qp (m)) → ⊕ H 0 (Fp , Qp (m − 1)) p-p

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shows that He´1t (OF [1/p], Qp (m)) ' H 1 (F, Qp (m)) for m 6= 1. Besides, the Poitou–Tate exact sequence (for all m) ⊕ H 2 (Fp , Qp /Zp (1 − m))∗ (= 0) → He´2t (OF [1/p], Qp /Zp (1 − m))∗ → p|p He´1t (OF [1/p], Zp (m)) → ⊕ H 1 (Fp , Zp (m)) p|p shows that, for m ≤ −1, the map H 1 (F, Zp (m)) → ⊕ H 1 (Fv , Zp (m)), v|p

hence also the map H 1 (F, Qp (m)) → ⊕ H 1 (Fv , Qp (m)), is injective bev|p

cause (Leop1−m ) holds. As Hf1 (F, Qp (m)) = 0 for m ≤ −1 (see Lemma 9.2.2), we have Hf1 (F, Qp (m)) = 0, as well. If m = 0, we have Hf1 (F, Qp ) = 0 because there is no everywhere unramified Zp -extension by finiteness of the class number. Thus Hf1 (F, Qp ) is isomorphic to ⊕ H 1 (kp , Qp ), according to Lemma 9.2.2. Let now p|p m ≥ 2; by Lemma 9.2.2, Hf1 (Fp , Qp (m)) = H 1 (Fp , Qp (m)) and, besides, ch1,m gives an isomorphism K2m−1 (F ) ⊗ Qp ' H 1 (F, Qp (m)). Finally, if m = 1, the composite map ch1,m

K1 (OF ) ⊗ Qp −→ He´1t (OF [1/p], Zp (1)) ⊗ Qp → H 1 (F, Qp (1)) is easily seen to coincide with the Kummer map OF× ⊗Qp → H 1 (F, Qp (1)), the image of which is Hf1 (F, Qp (1)) by Lemma 9.2.2. Summarizing, we have

Lemma 9.2.4

We have  × 1   Image (OF ⊗ Qp → H (F, Qp (1))

Hf1 (F, Qp (m)) =

K2m−1 (F ) ⊗ Qp

  0

if m = 1 if m > 1 if m ≤ 0.

Since H 0 (F, Qp (m)) = 0 if m 6= 0 and Qp if m = 0, we immediately get Lf (Qp (m)) and also Lf (Qp (1 − m)), which is equal to Lf (Qp (m)∗ (1)). The passage from m to 1 − m is significant, since dimQp K2m−1 (F ) ⊗ Qp , for m 6= 1, is equal to r1 (F ) + r2 (F ) + δm (resp. r2 (F ) + δm ) if m is odd (resp. even), whereas Hf1 (F, Qp (1 − m)) = 0. Here δm denotes the defect 2 of the conjecture (Leopm ) i.e., the Zp corank of Het (OF [1/p], Qp /Zp (m)).

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9.2.4 Local Tamagawa numbers These numbers appear with the introduction of integral structures, for E = Q. 9.2.4.1 Let T be a GF -equivariant Zl -sublattice of the l-adic representation V, and define Hf1 (F, T ) as the inverse image of Hf1 (F, V ) in H 1 (F, T ). Consider the integral analogue of Lf (V ), namely 1 Lf (T ) := detl H 0 (F, T ) ⊗ det−1 l Hf (F, T ).

Coming back to §9.2.2, we can see that: - if l 6= p, the exact sequence (9.1) induces an isomorphism ιv : Lf (V ) ' Ql . Define then Tam0p (T ) ∈ lZ by the formula ιv (Lf (T )) = Zl Tam0p (T ). - if l = p, the exact sequence (9.3) induces an isomorphism ιV : Lf (V ) ' −1 det−1 p tV,p . Let wp be a basis of detp tV,p , wp its dual basis and define Tam0p,w (T ) ∈ pZ by the formula p ιv (Lf (T )) = Zp .Tam0p,w (T )wp−1 . p

Example 9.2.5 [continued] Take E = Q, V = Ql (m), T = Zl (m). For the basis wp of detp tV,p choose an element of detp DdR,p (Ql (m)) which generates the lattice detp (Op em ) (see Example 9.2.1); write Tam0p for Tam0p,w and define p Y Tam0p (Z(m)) = Tam0p (Zl (m)). l prime

Lemma 9.2.6 (continued) Let q = #kp and write expp,m for the Bloch–Kato exponential of the representation Qp (m). Then  m i   q [H (Fp , Zp (m)) : expp,m (OFp )] 0 Tamp (Z(m)) = 1   w(p) := #H 0 (F , Q /Z (m)) m p p p

if m ≥ 2 if m = 0, 1 if m ≤ −1.

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Proof. If l 6= p, the representation Ql (m)) is unramified and Tam0p (Zl (m)) = 1 (see [FPR94, I, Prop. 4.2.2]; one can also prove this easily by computing the order of H 1 (Fp , Zl (m)). If m ≤ −1, then Hf1 (Fp , Qp (m)) = 0 and Hf1 (Fp , Zp (m)) = H (Fp , Zp (m))tor . The exact sequence 1

0 → Zp (m) → Qp (m) → Qp /Zp (m) → 0 gives an isomorphism H 1 (Fp , Zp (m))tor ' H 0 (Fp , Qp /Zp (m)), hence (p)

Tam0p (Z(m)) = wm (Fp ). If m = 0, we have an exact sequence 1−φ

0 → H 0 (Fp , Zp ) → O(Fp )0 −→ O(Fp )0 → Hf1 (Fp , Zp ) → 0, hence Tam0p (Zp (0)) = 1. If m ≥ 1, H 0 (Fp , Zp (m)) = 0 and 1 − φ maps em to (1 − p−m )em , hence Tam0p (Zp (m)) = q m · [Hf1 (Fp , Zp (m)) : expp,m (OFp )]. If m = 1, in particular, the isomorphism Fp ' H 1 (Fp , Qp (1)) (see Example 9.2.1.) gives q · [Hf1 (Fp , Zp (1)) : expp,1 (OFp )] = q 1−e(p|p) · [UF1 : E(pOFp )] = 1, p where we recall that E(∗) here is the usual exponential map as mentioned in Example 9.2.1. In section 9.5.2, we shall push these calculations further in the abelian case.

9.2.5 The fundamental line and Galois cohomology The fundamental line ∆f (V ) is a one-dimensional vector space over El , hence it can be equipped with a norm induced by the l-adic valuation, once a basis is chosen. However, what we need is a canonical norm, which will be provided by ´etale (or Galois) cohomology.

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9.2.5.1 Determinants of perfect complexes We shall freely use the vocabulary of complexes and derived categories (see for example [GM]). For the TNC proper, we shall only deal with determinants over a ring R which is a field or a principal ideal domain. For such a ring R, let us denote by D perf (R) the derived category of perfect complexes of R-modules. Recall that such a complex is by definition quasi-isomorphic to a bounded complex P • of finitely generated, projective (hence free) R-modules · · · → P i−1 → P i → P i+1 → · · · . For each P i , the usual determinant module detR (P i ) is defined as the top exterior power ΛrRi (P i ) where ri is the rank of P i , and is an invertible (i.e., projective rank one) module over R, and one can define (−1)i

detR (P • ) = ⊗ detR i∈Z

(P i ),

−1 where det−1 . Note that the parity convention R (.) stands for detR (.) i (−1) is arbitrary, but it bears no consequence on our subsequent calculations. Any quasi-isomorphism induces an isomorphism of determinants, which allows one to define the determinant of an object of D perf (R). We shall constantly use the following properties:

1. if the perfect complex C • is acyclic, then there is a canonical isomorphism detR (C • ) ' R. 2. if C1• → C2• → C3• is a distinguished triangle in D perf (R), then there is a canonical isomorphism detR (C2• ) ' detR (C1• ) ⊗ detR (C3• ). 3. any finitely generated R-module N has finite projective dimension (because R is principal), hence N can be considered as a perfect complex concentrated in degree 0 and detR (N ) is well defined. Then, for any perfect complex C • , there is a canonical isomorphism detR (C • ) ' (−1)i

⊗ detR i∈Z

H i (C • ).

N.B. The more general notion of the Knudsen–Mumford determinant of perfect complexes over an arbitrary noetherian ring is needed in more general conjectures such as the ETNC (see [BG03, §2]). We shall evoke it punctually in the proof of section 5.2.4 below.

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9.2.5.2 Various complexes of ´ etale cohomology In the sequel, let us fix a finite set S of primes of F which contains the set S∞ (F ) of infinite primes and the set Sl (F ) of primes above l and let Sf = S \S∞ (F ). Write OFS for the ring of S-integers of F and GSF for the Galois group of the maximal algebraic extension of F which is unramified outside S. For any Ol (resp. El )-sheaf T which is constructible for the ´etale topology, over the complement of Sf in Spec(OF ), the generic fibre of T is an Ol (resp. El )-representation of GSF , and conversely ([Fo92, footnote, p.213]). Thus the complex RΓe´t (Spec OFS , T ), or RΓe´t (OFS , T ) for short, can be identified with the complex of continuous cochains ∗ which computes the continuous Galois cohomology Hcont (GSF , T ). This perf perf is an object of D (Ol ) (resp. D (El )), to which are canonically attached two distinguished triangles: • The cohomology with compact support of T is that of the complex RΓc (OFS , T ) which is defined as C • [−1], where C • is the cone of RΓe´t (OFS , T ) → ⊕ RΓe´t (Fp , T ), p∈S so that we have a distinguished triangle RΓc (OFS , T ) → RΓe´t (OFS , T ) → ⊕ RΓe´t (Fp , T ). p∈S

(9.4)

The complex RΓc (OFS , T ) is an object of D perf (Ol ) (resp. D perf (El )), and is acyclic outside degrees 1, 2, 3. Define the Euler–Poincar´e line of T to be ∆l,EP (T ) = detl RΓc (OFS , T ). This determinant depends only on the generic fibre of T . Moreover, Q i if T is torsion, one can check that i #Hci (OFS , T )(−1) = 1, hence ∆l,EP (T ) = Ol (see property (c)). This shows that if V is an El sheaf and T an Ol -sheaf (both constructible for the ´etale topology) such that V = El ⊗ T, then ∆l,EP (T ) is a lattice of ∆l,EP (V ) which is independent of the choice of T . One denotes by | |l,EP the unique norm on ∆l,EP (V ) such that, if δ is a basis of ∆l,EP (T ), then |δ|l,EP = 1. • Let V be an El -sheaf as above. The f -cohomology of V is that of the complex RΓf (OFS , V ) which is defined as C • [−1], where C • is the cone of the composite morphism RΓe´t (OFS , V ) → ⊕ RΓe´t (Fp , V ) → ⊕ RΓ/f (Fp , V ), p∈Sf p∈Sf

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where the complexes RΓ/f are defined as follows. Recall the groups Hfi (Fp , V ) (i = 0, 1) that we defined in §2.2 and showed to be the cohomology of the two-term complex Dp (V ) → Dp (V )⊕ tV,p (see the exact sequences (1) and (3), with the convention that tV,p = 0 if l 6= p). Denoting that complex by RΓf (Fp , V ), the complex RΓ/f (Fp , V ) is defined tautologically by the distinguished triangle RΓf (Fp , V ) → RΓe´t (Fp , V ) → RΓ/f (Fp , V ) and we get another distinguished triangle RΓf (OFS , V ) → RΓe´t (OFS , V ) → RΓ/f (OFS , V ).

(9.5)

It can be shown directly ([Fo92, p.215]) that the complex RΓf (OFS , V ) is acyclic outside degrees 0, 1, 2, 3 and its cohomology groups are: Hf0 (OFS , V ) = H 0 (F, V ),

Hf1 (OFS , V ) = Hf1 (F, V )

Hf2 (OFS , V ) ' H 1 (F, V ∗ (1)))∗ , Hf3 (OFS , V ) ' H 0 (F, V ∗ (1))∗ . It can also be shown by using Poitou–Tate duality ([Fo92, pp.215–216]) that detl Hf2 (OFS , V ) = detl Hf1 (OFS , V ∗ (1)), hence that detl RΓf (OFS , V ) = Lf (V ) ⊕ Lf (V ∗ (1)). But the combination of (4) and (5) gives a distinguished triangle: RΓc (OFS , V ) → RΓf (OFS , V ) → ⊕ RΓf (Fp , V ) ⊕ ( ⊕ RΓ(Fp , V )). p∈Sf p∈S∞ (F ) Hence, by property (b), we have + detl RΓf (OFS , V ) = ∆l,EP (V ) ⊗ det−1 l tV ⊗ detl V .

Finally, we have: Proposition 9.2.7 If V is a pseudo-geometric El -representation, the lines ∆f (V ) and ∆l,EP (V ) are canonically isomorphic. This isomorphism allows to transport the canonical norm | . |l,EP from ∆l,EP (V ) to ∆f (V ). It also shows that ∆l,EP (V ) does not depend on the finite set S ⊃ Sp ∪ S∞ .

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9.3 Motivic cohomology and conjectures on special values The rule of motives is that all avatars, archimedean or not, should be treated alike, so we must introduce infinite primes before going any step further. For l | ∞, the mixed Hodge structures play the role of the previous l-adic representations (or the other way around).

9.3.1 Mixed Hodge structures ([Fo92, §5]; see also [Ra15], this volume). According to Deligne, an Rmixed Hodge structure over C is a finite dimensional R-vector space V , together with an increasing weight filtration (Wn V )n∈Z by sub-R-vector spaces such that Wn V = V for n >> 0, Wn V = 0 for n << 0, and a decreasing Hodge filtration (Film VC )m∈Z by sub-C-vector spaces of VC = C ⊗R V, with properties that need not be detailed here ([Fo92, §5.1]). The integers m for which grW m VC 6= 0 are the weights of V , and V is called pure if it has a single weight. An R-mixed Hodge structure over R is an R-mixed Hodge structure over C with an action of complex conjugation. This is equivalent to giving a direct sum decomposition V = V + ⊕ V − which is compatible with the weight filtration and such that the Hodge filtration is obtained by extending scalars from a filtration defined on Gal(C/R) VdR = V = V + ⊕ iV − . C For the rest of this section, let us fix some convenient notation: K, L ∈ ¯ = C, GK = Gal(C/K), V = an L-mixed Hodge structure {R, C}, K over K. Concretely, K will be Fp and L will be El , for p ∈ S∞ (F ) and l ∈ S∞ (E). If K = C, let V + = V and VdR = VC , so that V + = V GK in all cases. It can be verified that the L-mixed Hodge structures over K form an abelian category HSK (L) which admits a unit object 1K,L (= the unique pure object of weight 0 such that 1K,L = (1K,L )+ = L) and a Tate object 1K,L (1) (= the unique pure object of weight–2 such that 1K,L (1) = L.(2πi) and 1K,L (1)− = 1K,L (1) if K = R). For i ∈ N, define i HH (K, V ) = ExtiHSK (L) (1K,L , V ).

Example 9.3.1 [continued] ([Fo92, §6], [FPR94, III.1]) We aim to compute the Hodge cohomology group 1 HH (F, Q(m)B ⊗ R) :=

1 ⊕ HH (Fp , Q(m)B,p ⊗ R), p∈S∞ (F )

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⊕ Q(m)B,p = ⊕ (2πi)m Q. Then p∈S∞ (F ) p∈S∞ (F ) 1 HH (F, Q(m)B ⊗ R) = 0 if m ≤ 0 and, if m ≥ 1, this group sits in an exact sequence with Q(m)B :=

1 0 → Q(m)+ B ⊗Q R → Q(m)dR ⊗Q R → HH (F, Q(m)B ⊗ R) → 0.

But for m ≥ 1, the twisted duality between Q(m) and Q(1 − m) gives an exact sequence + 0 → Q(m)+ B ⊗Q R → Q(m)dR ⊗Q R → Q(m − 1)B ⊗Q R) → 0. (9.6)

Noting that C ' R(m − 1) ⊕ R(m), the comparison between these two exact sequences gives that ( R(m − 1)r1 if m ≥ 1 is even + 1 HH (F, Q(m)B ⊗R) ' Q(m−1)B ⊗R ' R(m − 1)r1 +r2 if m ≥ 1 is odd.

9.3.2 Motivic structures and L-functions All the cohomologies which appeared previously should be realizations i of the motivic cohomology HMM ( · , · ) attached to the conjectural category MMF (E) (MM for short) of mixed motives defined over F with coefficients in E. If 1F,E is the ‘unit object’ of MMF (E), then, for all i i ∈ N, one should define HMM (F, M ) := ExtiMM (1F,E , M ). To give a suggestive image of motivic cohomology, let us pass from Sanskrit to Greek, and recall Plato’s myth of the cave: the multiple cohomologies which we have encountered are but shadows cast on the walls of the cave by the light of the sun in our back; the relations between the shadows cannot be understood unless we return to the object which projects them, which Plato calls the ‘archetype’ (here, motivic cohomology). Section 6 of [Fo92] is devoted to the ‘hand made’ construction of an ad hoc category MSF (E) of ‘motivic structures for the working mathematician. Without giving details (for which we refer to [Fo92, pp.218–220]), let us say that an object M of this category is defined by a de Rham realization MdR ; a Betti realization MB,v for any v ∈ S∞ (F ); an f realization which is an Af (E)-free module of finite type Mf on which GF acts linearly and continuously (here Af ( · ) denotes the ring of finite ad`eles). This category MSF (E) is abelian, E-linear, is equipped with a ⊗ structure and Hom(·, ·), contains a unit object 1F,E as well as a Tate object 1F,E (1); it is Tannakian, i.e., stable by taking subobjects, quotients, direct sums, tensor products and duals (recall that

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M ∗ = Hom(M, 1F,E ).) It is expected to be equivalent, via a ‘realization functor, to the essential image of the (still hypothetical) category MMF (E). Add two more properties: 9.3.2.1 f -admissibility 0 For any object M of MS F (E), let HMS (F, M ) = Hom(1F,E , M ) and 1 HMS (F, M ) be the E-vector space of isomorphism classes of extensions of 1F,E by M in MSF (E). For any finite place l of E, for any x ∈ 1 HMS (F, M ), write xl for the image of x in Hf1 (F, Ml ), where Ml is the El -component of Mf , which is a ‘pseudo-geometric’ l-adic representation 0 0 of GF . Put HMS,f (F, M ) = HMS (F, M ) and 1 1 HMS,f (F, M ) = {x ∈ HMS (F, M ) | xl ∈ Hf1 (F, Ml ) for all finite l}.

The category MSF (E) is called f -admissible if for any object M, for any place l of E (finite or infinite), for i ∈ {0, 1}, the following conditions Cli (M ) are satisfied: i 1. If l is finite, then the natural map El ⊗ HMS,f (F, M ) → Hfi (F, Ml ) is an isomorphism. 0 1 2. If l is infinite and if HMS,f (F, M ∗ (1)) = HMS,f (F, M ∗ (1)) = 0, then the natural map i El ⊗E HMS,f (F, M ) →

⊕ H i (Fp , Ml ) p∈S∞ (F )

is an isomorphism. These conditions imply that MSF (E) is stable by M 7→ M ∗ (1) = M ⊗ 1F,E (1) 1 and by taking extensions of 1F,E by M classified by HMS,f (F, M ). The isomorphisms in 1. and 2. we might call ‘regulators’ (e.g., §4.1 below). They are the rays of light in Plato’s apologue of the cave.

The reader is warned and should be aware that the existence of the category MSF (E) is still partly hypothetical. Actually, what is well defined by the above considerations is the category denoted by CF (E) in [Fo92, 6.3], and MSF (E) should be the smallest Tannakian subcategory of CF (E) which contains any object which is isomorphic to the realization of a motif [Fo92, p.221].

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9.3.2.2 Existence of L-functions Only now do we come to the main objects of our study. Let V be an l-adic representation of GF . For any finite place p of F , define Pp (V, T ) = detEl,p (1 − Frob−1 p T | Dp (V )) ∈ El [T ]. Note that the degree of this polynomial is ≤ b(V ) := dimEl (V ), with equality if and only if V has good reduction at p. For any finite set S of places of F containing S∞ (F ), we say that V admits a function LS over E if and only if, for any p 6∈ S, the polynomial Pp (V, T ) ∈ E[T ] Q and the infinite product LS (V, s) = p6∈S Pp (V, N (p)−s )−1 converges in C ⊗Q E for Re(s) >> 0. If S = S∞ (F ), we say simply that V admits an L-function on E, and write L(V, s) = LS∞ (F ) (V, s). Back now to the category MSF (E). For an object M of MSF (E), for a finite set S of places of F containing S∞ (F ), we say that M admits an LS -function if for any finite place l of E, Ml admits an LS -function which is independent of l; if S = S∞ , we say that M admits an Lfunction. These functions will be denoted resp. LS (M, s) and L(M, s), taking values in C ⊗Q E for general s and in R ⊗Q E for s real. Example 9.3.2 (continued) One concrete example is the category MMF,1 (E) of Deligne’s 1-motives ([Fo92, §8-1]). Recall the category CF (E) that was mentioned at the end of 9.3.2.1. It is known that there exists a functor Real : MMF,1 (E) → CF (E) which is fully faithful, hence induces an equivalence between MMF,1 (E) and its essential image, which will be denoted by MSF,1 (F ) and is conjectured to be f -admissible. In the particular case of E = Q, the unit object M = 1F,Q is Q(0) and the Tate object 1F,Q (1) = M ∗ (1) is Q(1). For m ∈ N, the Tate motif Q(m) is Q(1)⊗m and Q(−m) is its dual (so for m ∈ Z, Q(m) is no longer a mere notation!). It is easily checked that the function L(Q(m), s) is the shifted Dedekind zeta function ζF (s + m).

9.3.3 Formulation of the conjecture We are now ready to formulate the TNC on special values of the motivic L-function defined in 9.3.2. From now on, let M denote the category of mixed motives MMF (E) if we are ready to accept ‘conjectural conjectures’; if not, take M to be MSF (E) (admitting f -admissibility and the

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existence of L-functions). Replacing, if necessary, the motif M by a Tate twist of M , we can restrict to the special value 0. Let rM be the (unique) integer such that s−rM L(M, s) has a finite non-null limit L∗ (M, 0)when s → 0. i 1 Recall that HM (F, M ) = ExtiM (1F,E , M ) and HM,f (F, M ) is the sub1 E-vector space of H (F, M ) which classifies the extensions of 1F,E by M which have good reduction at all finite places of F . In the sequel, for simplification, we drop the index M if there is no possible confusion. The TNC actually consists of a corpus of three conjectures. The first is the ‘rank conjecture’:

Conjecture Cr (M ): rM = dimE (Hf1 (F, M ∗ (1)) − dimE (H 0 (F, M (1)). Remark rM is a priori a Z-valued function on Spec(E ⊗ R) since the vanishing orders of the L-functions need not be the same. The conjecture implies that rM actually comes from a Z-valued function on Spec(E), i.e., is a single integer. Once the rank rM is determined, the two other conjectures are concerned with the special value L∗ (M, 0). Analogously to what was done for local representations, let us define: 1 Lf (M ) = detE H 0 (F, M ) ⊗ det−1 E Hf (F, M ) 1 ∗ Lf (M ∗ (1)) = detE H 0 (F, M ∗ (1)) ⊗ det−1 E Hf (F, M (1))

and the motivic fundamental line: ∗

∆f (M ) = Lf (M ) ⊗ Lf (M (1)) ⊗

det−1 E



⊕ M+ p∈S∞ (F ) B,p

 ⊗ detE tM

(recall that the tangent space tM is MdR / Filo MdR ). The following archimedean ‘trivialization’ of the fundamental line is a crucial preliminary result. 9.3.3.1 Theorem There is a canonical isomorphism ιM : R ⊗Q ∆f (M ) ' R ⊗Q E. Proof. See [Fo92, §6.10]; we outline the proof below. Let M be a motivic structure and l ∈ S∞ (E). If p ∈ S∞ (F ), Mp,l is

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an El -mixed Hodge structure on Fp . The inclusion of Mp,l in (Mp,l )C induces an El -linear map + αp,l : (MB, p ) l → tM l ,

hence also an El -linear map  α Ml =

⊕

αp,l :

p∈S∞ (F )

⊕

+ MB, p

p∈S∞ (F )

 → tM l . l

Using the f -admissibility, one can show the existence of an exact sequence 0 → H 0 (F, M )l → ker αl → Hf1 (F, M ∗ (1))∗l → H 1 (F, Ml ) → Coker αl → H 0 (F, M ∗ (1))∗l → 0

(9.7)

which yields a canonical isomorphism (Lf (M ) ⊗E Lf (M ∗ (1)))l ' detEl (Ker αMl ) ⊗El det−1 El (Coker αMl ) (this is because Ker αMl and Coker αM ∗ (1)l are in duality, and similarly for Coker αMl and Ker αM ∗ (1)l ). Besides, the tautological exact sequence   + 0 → Ker αMl → ⊕ M → tMl → Coker αMl → 0 p∈S∞ (F ) B,p yields an isomorphism det−1 E (

⊕ M + )⊗E detE tMl ' det−1 El (Ker αMl )⊗E detE (Coker αMl ). p∈S∞ (F ) B,p

The tensor product of the two isomorphisms gives a third isomorphism ιM,l : ∆f (M )l → El . Since R ⊗Q ∆f (M ) = map ιM =

⊕ l∈S∞ (E)

⊕ l∈S∞ (E)

∆f (M )l and R ⊗Q E =

⊕

El , the

l∈S∞ (E)

ιM,l is an isomorphism R ⊗Q ∆f (M ) ' R ⊗Q E.

Using the isomorphism ιM , we can enunciate the Deligne–Beilinson conjecture. Conjecture CDB (M ) : There exists a basis (necessarily unique) b of ∆f (M ) such that ιM (1 ⊗ b) = L∗ (M, 0)−1 ∈ R ⊗Q E. This determines the special value L∗ (M, 0) up to multiplication by an

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element of E ∗ . The more precise Bloch–Kato conjecture (or TNC) will × give L∗ (M, 0) up to multiplication by an element of OE . Conjecture CBK (M ) : The conjecture CDB (M ) is true and for any finite place l of E, and one has |1 ⊗ b|l,EP = 1, or, equivalently, ∗ −1 |ι−1 )|l,Ep = 1. M (L (M, 0)

Here we use the l-adic realization Ml of M to equip El ⊗E ∆f (M ) with a canonical norm | · |l,EP . This formulation clearly shows that the special value L∗ (M, 0) is the meeting point of the archimedean and the non-archimedean, two apparently antagonistic worlds (one is connected, the other totally disconnected).

9.4 Bloch and Kato versus Fontaine and Perrin–Riou In this whole section, take E = Q. As was noticed in the introduction, the original Bloch–Kato conjecture in [BK90] was expressed analogously to the theorems on the Tamagawa numbers of semi-simple algebraic groups. These numbers were calculated by introducing Shafarevich–Tate groups and regulator maps on Hf1 (Q, M ). In this section, we shall follow [Fo92, §11] to show the equivalence of the two formulations of the TNC when E = Q. While we are at it, let us make a quick digression on regulators ‘ `a la’ Beilinson; this can be considered as a complement to Theorem 9.3.3.1.

9.4.1 Regulators of Tate motives and archimedean Tamagawa numbers In Example 9.3.2, we saw that the Tate motives Q(m) are objects of the category M = MMF (Q) of mixed motives or, more concretely, of the category M = MSF (Q) of motivic structures. But, after all, the ambient category does not matter, we only need ‘global’ cohomoli ogy groups HM (F, Q(m)), which satisfy the f -admissibility conditions i Cl (Q(m)) of §9.3.2.1. Many definitions of motivic cohomology groups i HM (F, Z(m)), i = 0, 1 which all coincide modulo 2-torsion (see [Ko15]), have been given by various authors (Bloch, Levine, Voevodsky...), see i [Ko15]. Let us pick up Beilinson’s definition and put HM (F, Q(m)) = i HM (F, Z(m) ⊗ Q). Then:

Determinantal Approach to the TNC

0 HM (F, Q(m)) =

1 HM (F, Q(m))

=

175

( Q if m = 0. 0 otherwise.

( K2m−1 (F ) ⊗ Q if m ≥ 1 0 otherwise,

(the nullity for m < 1 is due to considerations of weight.) It is known that K1 (OF ) = OF× and K2m−1 (OF ) = K2m−1 (F ) for 1 m ≥ 2. The image of K2m−1 (OF ) in HM (F, Q(m)) is a Z-lattice, denoted 1 1 1 by HM,f (F, Q(m))Z . Define HM,f (F, Q(m)) = HM,f (F, Q(m))Z ⊗ Q. It 1 1 is clear that HM,f (F, Q(m)) = HM (F, Q(m)) for all m 6= 1. It does 1 not matter whether the HM,f here coincide with the ones defined in §9.3.2.1 because, in order to compute determinants, all we need is to construct suitable ‘realization’ isomorphisms. These will be given by regulator maps: 1. The ´etale regulator maps 1 ρp,m : HM (F, Q(m)) → K2m−1 (F ) ⊗ Qp → He´1t (OF [1/p], Qp (m))

have already been introduced in Example 9.2.3.1. They induce the f admissibility condition (a) of §9.3.2.1 for m ≥ 1 according to Lemma 9.2.4. 2. As for the archimedean regulator maps, we have the choice between Borel’s and Beilinson’s (they are known to differ only by a factor 2), but for motivic reasons, we must decide for Beilinson’s which goes 1 into Deligne’s cohomology, identified here with HH (F, Q(m)B ⊗ R) ' ( ⊕ R(m − 1)) as vector spaces (see Example 9.3.1): Hom(F,C)+ 1 ρ∞,m : HM (F, Q(m)) → K2m−1 (F ) ⊗ Q → (

⊕

R(m − 1)).

Hom(F,C)+

We must distinguish two cases: 1 • If m > 1, ρ∞,m induces an isomorphism HM,f (F, Q(m)) ⊗ R= 1 HM (F, Q(m))⊗R ' ( ⊕ R(m−1)). The vector space R(m−1) Hom(F,C)+ has a canonical basis (2πi)m−1 , hence a canonical basis for the Hodge 1 HH -group. The Beilinson regulator number is defined as Rm (F ) = 1 det(ρ∞,m (HM,f (F, Q(m))Z ) with respect to this basis.

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• if m = 1, ρ∞,1 is but the Dirichlet regulator map x ∈ OF× ⊗ R 7→ (log kxkv )v , which gives rise to an exact sequence ρ∞,1

sum

0 → OF× ⊗Q R −→ Rr1 +r2 −→ R → 0. If (uj ) is a Z-basis of OF× /tors, the Dirichlet regulator number R1 (F ) is the usual determinant constructed from the numbers log kuj kv . A unified formulation is that, for all m ≥ 1, one has an isomorphism 1 0 detHM,f (F, Q(m)) ⊗ det−1 HM (F, Q(1 − m)) ⊗ R 1 ' detHH (F, Q(m))B ⊗ R)

(9.8)

and the regulator number Rm (F ) is the determinant of this isomorphism calculated with respect to the canonical basis. Consider now the fundamental line + ∆f (Q(m)) = Lf (Q(m)) ⊗ Lf (Q(1 − m)) ⊗ det−1 Q Q(m)B ⊗ detQ tQ(m)

and let us reprove quickly the isomorphism ιm : R ⊗ ∆f (Q(m)) ' R 0 1 of Theorem 9.3.3.1. For m ≥ 1, HM (F, Q(m)) = HM (F, Q(1 − m)) = 0, hence 1 0 ∆f (Q(m)) = det−1 HM (F, Q(m)) ⊗ detHM (F, Q(1 − m))

⊗det−1 Q(m)+ B ⊗ det Q(m)dR and we get the desired isomorphism ιm by combining the isomorphism (9.8) with the exact sequence (9.6) of Example 9.3.1. Still for m ≥ 1, we have 1 0 ∆f (Q(1 − m)) = det−1 HM (F, Q(m)) ⊗ detHM (F, Q(1 − m))

⊗det−1 Q(m − 1)+ B, hence the isomorphism ι1−m : R ⊗ ∆f (Q(1 − m)) ' R by combining the isomorphism (9.8) and Example 9.3.1. 1 The lattices Z(m)B , Z(m)dR and HM (F, Z(m))Z determine a canonical Z-lattice ∆f (Z(m)) of ∆f (Q(m)) and we can define the archimedean Tamagawa number Tam0∞ (Z(m)) := |ιm (∆f (Z(m))|−1 .

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Lemma 9.4.1 Let n be the degree of F and dF its discriminant. For any m ≥ 1, we have: (i) Tam0∞ (Z(1 − m)) = Rm (F ) (the Beilinson–Dirichlet regulator number) (ii) Tam0∞ (Z(m))

=

( (2π)mn−r2 |dF |−1/2 Rm (F ) (2π)

mn−r2

π

−r1

|dF |

−1/2

Rm (F )

if m is even if m is odd.

Proof. Formula (i) is obvious. To show (ii), just note that the map 1 Q(m)dR ⊗Q R → HH (F, Q(m)B ⊗ R) (see example 9.3.1) sends a Z-basis P (w1 , . . . , wi ) of OF on the parallelotope ( w1τ , . . . , wiτ ), τ ∈ Hom(F, C), τ

the volume of which is ±det(wiτ ) = |dF |1/2 .

9.4.2 Global Tamagawa numbers Let us recall the definition of the p-adic Tamagawa numbers in §9.2.4. Let Vl be a pseudo-geometric representation of GF , Tl be a GF -equivariant Zl -sublattice. • if p | p and l 6= p, one has an isomorphism ιv : Lf (Vl ) ' Ql , and Tam0p (T ) ∈ lZ is defined by ιV (Lf (Tl )) = Zl .Tam0p (Tl ). −1 • If l = p, one has an isomorphism ιV : Lf (Vl ) ' det−1 p tVl ,p . Let wp be a basis of detp tVl ,p and define Tam0p,w ∈ pZ by ιV (Lf (Tl )) = p Zp · Tamp,wp (Tl )wp−1 . Modulo some Euler factors, Bloch and Kato ([BK90, §4]) take exactly the same definitions, but expressed in terms of Haar measures. They relate the p-adic Tamagawa numbers to values of local Lp -functions in the following way. Suppose that Pp (Vl , 1) 6= 0. Then H 0 (Fp , Vl ) = 0 and • if l 6= p, Hf1 (Fp , Vl ) = 0, hence Hf1 (Fp , Tl ) = H 1 (Fp , Tl )tor • if l = p, the Bloch–Kato exponential tVl ,p → Hf1 (Fp , Vl ) is an isomorphism. In particular, the choice of a basis wp of detp tVl ,p allows, by transport of structure, to define a Haar measure µ0p,w on Hf1 (Fp , Vl ). One can p check that • if l 6= p, Tam0p (Tl ) = |Lp (Vl , 0)|l · #H 1 (Fp , Tl )tor

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• if l = p, Tam0p,w (Tl ) = |Lp (Vl , 0)|p · µ0p,w (Hf1 (Fp , Tl )). p p One key technical property is that all these local Tamagawa numbers are often trivial outside a finite set of places (this finite set need not be specified). Let us call this property (P1). • for l 6= p, and Tam0p (Tl ), the property (P1) follows from elementary (non-)ramification properties [BK90, Theorem 4.1(i)] • for l = p and Tam0p,w (Tl ), one needs to impose some conditions on p the comparison isomorphisms, but these are always met in practice, in particular for Zl (m) [BK90, Theorem 4.1(iii)]. Consider now a motivic structure, i.e., an object M of the category MSF (Q) defined in §13.3.2. Recall that it comes equipped in particular with an f -realization Mf which is an Af (Q)-free module of finite type with continuous linear action of GF . In order to define global Tamagawa numbers, it is necessary that the ´etale realizations Ml (= the l-components of Mf ) which are l-adic pseudo-geometric representations of GF , also contain GF -equivariant Zl -sublattices which are ‘coherent’ in some sense. This is the meaning of the introduction by Bloch and Kato [BK90, Def. 5.5] of a motivic pair (M, Θ), where M is an object of MSF (Q) and Θ is a free Z-module of finite type with continuous ˆ ⊗ Θ, together with an isomorphism ˆ := Z linear action of GF on Θ Q ⊗ Θ ' MB,∞ and the compatibility conditions inherited from the axioms defining the category MSF (Q) in §13.3.2 (see [BK90, §5], [Fo92, §11.6]; see also [Ra15] in this volume). Note that here we do not suppose that F = Q, because we need a TNC for the Dedekind zeta function ζF (s). As for the definitions pertaining to the motivic pair (M, Θ), Bloch and Kato note that ‘as usual one reduces to the case F = Q by Weil restriction of scalars, anyway’ [BK90, p.371]. Recall that the tangent space tM is here the F -vector space MdR / Fil0 . If p is a finite place of F above p and l = p, then tMl ,p can be identified with Fp ⊗F tM ; if p is infinite, then the C-vector space C ⊗R Mp,∞ can be identified with C ⊗F MdR (these follow from the axioms defining MSF (Q); see [Fo92, p.213]). Using the compatibility between de Rham and Betti ([Fo92, footnote 16, p.219]), one gets a natural composite map g : Θ → C ⊗ MdR → C ⊗ tM . The motivic pairs (M, Θ) form an abelian category and we can define as usual Hf1 (F, Θ) inside H 1 (F, Θ) := H 1 (F, (M, Θ)) [Fo92, p.238], the

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latter group being the Ext1 group in this category. Using the notation AΘ ( . ) as in [BK90] (this is just a notation), let us define Y ˆ = AΘ (F ) = Hf1 (F, Θ), AΘ (Fp ) = Hf1 (Fp , Θ) Hf1 (Fp , Θl ) l

Q ˆ = Θl ), AΘ (R) = AΘ (C)G(C/R) , AΘ (C) = C ⊗ tM /g(Θ). (where Θ l

Assume one additional property for the motivic structure M : (P2) For any finite prime p of F , Pp (M, 1)(= Pp (Ml , 1) for any l) is not zero. Recall that the non-vanishing of Pp (Ml ) implies that H 0 (Fp , Ml ) = 0 and that • if p - l, H 1 (Fp , Ml ) = 0, hence Hf1 (Fp , Ml ) = Hf1 (Fp , Ml )tor • if p | l, then the Bloch–Kato exponential map tMl ,p → Hf1 (Fp , Ml ) is an isomorphism. Property (P2) also implies that, if l 6= p, then Hf1 (Fp , Θl ) is finite, trivial for almost all l (see (P1) at the beginning of this section). Then AΘ (Fp ) is compact if p - ∞, locally compact if p | ∞, Q and (AΘ (Fp) )/AΘ (F ) is compact [Fo92, p.239]. all p Choose now a basis w over F of detF tM . For any place p of F , finite for infinite, w can be viewed as a basis of detFp tMp . If p is finite, we already introduced at the beginning of this section the measure µ0p,w on Hf1 (Fp , Qp ) = Q ⊗Z AΘ (Fp ); if p is infinite, then the basis w defines a Haar measure on AΘ (R), denoted also by µ0p,w . The property (P1) of Bloch–Kato cited at the beginning of this section asserts that (modulo the conditions of [BK90, Theorem 4.1(iii)]), for almost all p, µ0p,w (AΘ (Fp )) = Pp (M, 1). Let us assume further a weight condition (which holds for example if M is pure of weight ≤ −3) (P3) W−3 M = M ([Fo92, p.239]; [BK90]), to ensure that the infinite Q 0 product µp,w (AΘ (Fp )) is convergent, thus defining a product meaall p

sure which happens to be independent of w because of the product formula. This is the Tamagawa global measure, denoted by µ0BK . The global Tamagawa number is then Y Tam0F (M ) := µ0BK (( AΘ (Fp ))/AΘ (F )) all p

=

Y p|∞

µp,w (AΘ (Fp )/AΘ (F )) ·

Y p-∞

µp,w (AΘ (Fp )).

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For the motif Q(m), we prefer the notation YY Tam0F (Z(m)) = Tam0p,w (Zl (m)) · Tam0∞ (Z(m) (see §5 below). p l p

9.4.3 Shafarevich–Tate groups. Let V be an l-adic representation of GF , T a GF -equivariant Zl -sublattice. The f -group Hf1 (F, V ) has been defined in §9.2.3. Let us introduce the Selmer group of T as Hf1 (F,V /T ):={x∈H 1 (F,V /T )| xp ∈Hf1 (Fp ,V /T ) for any place

p

of F },

and the Bloch–Kato Shafarevich–Tate group of T as 1 XF (T ) : = Hf1 (F,  V /T )/Image of Hf (F, V )

= ker

H 1 (F,V /T ) H 1 (F,V /T )⊗Ql /Zl

H 1 (Fp ,V /T )

−→ ⊕ H 1 (F ,V /T )⊗Q /Z l l all p p

 .

9.4.3.1 Proposition (i) The group XF (T ) is finite. (ii) Let T ∗ = Hom(T, Zl ) be the dual representation. Then there is a perfect duality (called Flach’s duality) XF (T ) × XF (T ∗ (1)) → Ql /Zl . In particular, XF (T ) and XF (T ∗ (1)) have the same order. Proof. See [FPR94, Prop. 5.4.2] for (i) and [Fl90, Prop. 5.3.5] for (ii). We are particularly interested in the case V = Ql (m), T = Zl (m), m ∈ Z. 9.4.3.2 Proposition Let l be an odd prime. Then: (i) For m ≥ 2, XF (Zl (m)) is canonically isomorphic to the kernel X2F (Zl (m)) of the localization map He´2t (OF [1/l], Zl (m)) → ⊕ H 2 (Fp , Zl (m)). p|l (ii) For m ≥ 2, (l)

w (F ) #XF (Zl (m)) = Q l−m · #He´2t (OF [1/l], Zl (m)), (l) w1−m (Fp) p|l

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(l)

where w1−m (K) = #H 0 (K, Ql /Zl (1 − m)) for a field K. (iii) For m = 1, XF (Zl (1)) is canonically isomorphic to cl(F ) ⊗ Zl , the l-part of the class group of F . Proof. Let m ≥ 2. By definition, it is easily seen that XF (Zl (m)) is equal to the kernel of the localization map Y H 1 (F, Ql /Zl (m))/Div → H 1 (Fp , Ql /Zl (m))/Div, all p

where Div denotes the maximal divisible subgroup. A calculation of P. Schneider [Sc75, §4] shows that this kernel coincides with the kernel of the localization map He´1t (OF [1/l], Ql /Zl (m))/Div → ⊕ H 1 (Fp , Ql /Zl (m))/Div. p|l

But the surjectivity of the l-adic Chern class map and the finiteness of K2m−2 (OF ) imply the finiteness of He´2t (OF [1/l], Zl (m)), which is then isomorphic to He´1t (OF [1/l], Ql /Zl (m))/Div by a well-known property of l-adic ´etale cohomology. An analogous conclusion holds for the local cohomology groups, by local duality. Thus XF (Zl (m)) ' X2F (Zl (m)) for m ≥ 2. To compute the order of X2F (Zl (m)), we use the Poitou–Tate exact sequence 0 → X2F (Zl (m)) → He´2t (OF [1/l], Zl (m)) → ⊕ H 2 (Fp , Zl (m)) p|l

→

He´0t (OF [1/l], Ql /Zl (1

− m)) → 0

(9.9)

and local duality. Finally, for the zeroth twist, Hf1 (Fp , Ql ) coincides with Hom(Gal(Fpnr /Fp ), Ql ), where Fpnr denotes the maximal unramified extension of Fp . It follows that XF (Zl (0)) = Hom(Gal(F nr /F ), Ql /Zl ) = Hom(cl(F ), Ql /Zl ), and we get XF (Zl (1)) by using Flach’s duality. Remarks: i) Proposition 9.4.3.2 determines the groups XF (Zl (m)) for all m ∈ Z, thanks to Flach’s duality. But beware that the isomorphisms in (i) do not extend to m ≤ 0, in other words, that Flach’s duality and Poitou–Tate’s duality are not compatible for all m; the main reason being that Poitou–Tate’s duality behaves symmetrically with respect to m (when passing from m to 1 − m), whereas Flach’s duality does not (because of the asymmetrical expression of Hf1 (F, Qp (m)). Note also the singular role of the twists m = 0, 1, which correspond

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to special values of the function ζF (s) at s = 0, 1 (on the frontier of the critical strip), see §9.5.1 below. ii) The following remark is the common fruit of numerous discussions with R. Sujatha on the question whether it is necessary or not to appeal to the surjectivity of the Chern class map, in order to show the finiteness of He´2t (OF , Zl (m)) (or equivalently of X2F (Zl (m)), for all m ≥ 2. This finiteness was used in the proof of Prop. 9.4.3.2 given above, when asserting the isomorphism between He´1t (OF , Ql /Zl (m))/Div and He´2t (OF [1/l], Zl (m)). It is also used in the proof of the isomorphism between X2F (Zl (m)) and XF (Zl (m)) [BN02, Lemma 4.3.1 and Proposition 4.4]. Note that the finiteness of XF (Zl (m)) is a general property. iii) A related question is the vanishing of XF (Zl (m)) for almost all l. Assuming the surjectivity of the Chern class map, this happens for all m ≥ 2, hence for all m by Flach’s duality. Without assuming the surjectivity, one could appeal to a convergence argument as in [Su15]. 9.4.3.3 Let us now define the global Shafarevich–Tate groups. • In the context of motivic pairs (M, Θ), Bloch and Kato conjecture that the finite groups XF (Θl ) are trivial for almost all l, and define naturally Y XF (Θ) := XF (Θl ). l prime

• In the particular case of the Tate motives, Proposition 9.4.3.2 ensures that the above conjecture holds (the case l = 2 requires a special study, not difficult, but not clean). Better, we can give a formula for the order of Y XF (Z(m)) := XF (Zl (m)). l prime

Let us first introduce some notation: for m ≥ 2, l 6= 2, the l-adic Chern class maps are split surjective (and are even isomorphisms if one admits the Quillen–Lichtenbaum conjecture), so that we can identify He´2t (OF [1/l], Zl (m)) with a subgroup of K2m−2 (OF ) ⊗ Zl , and define Y coh K2m−2 (OF ) := He´2t (OF [1/l], Zl (m)). l6=2

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It is then straightforward to derive from Proposition 9.4.3.2 the following formula (for m ≥ 2): w1−m (F ) 2 coh #XF (Z(1 − m)) = #XF (Z(m)) = QQ (l) · #K2m−2 (OF ). w1−m (Fp ) l p|l

(9.10) 2

Here the notation = means equality up to powers of 2. 9.4.3.4 Equivalence of the two formulations The original Bloch–Kato conjecture for motivic pairs (M, Θ) relates the Tamagawa number Tam0F (M ) and the Shafarevich–Tate groups XF (Θ) in the style of theorems on algebraic groups. It reads Conjecture CTam (M ) : L∗ (M, 0) =

Tam0F (Θ) · #XF (Θ) . #H 0 (F, M ∗ (1)/Θ∗ (1))

To compare CTam (M ) and CBK (M ), let us go back to the setting of an l-adic pseudo-geometric representation V equipped with a GF equivariant Zl -sublattice T . Recall the definition of the fundamental line + ∆f (V ) = Lf (V ) ⊗ Lf (V ∗ (1)) ⊗ det−1 ⊗ detl tV l V

and consider similarly ˜ f (T ) = Lf (T ) ⊗ Lf (T ∗ (1)) ⊗ det−1 T + . ∆ l To compute Tamagawa numbers, we must choose a basis of ∆f (V ) : let ˜ f (T ) and w = ⊗wp a basis of detl tV . Then w w ˜f be a basis of ∆ ˜f ⊗w is a p|l basis of ∆f (V ) and the problem in CBK (M ) is to compute |w˜f ⊗w|l,EP ∈ lZ . Since Tam0p (T ) = 1 for almost all finite p - l, the product Y Y Tam0w (T ) := Tam0p (T ) × Tamp,wp p-l p|l is finite and depends only on w. Lemma 9.4.2

(|w˜f ⊗ w|l,EP )−1 = Tam0w (T ) · #X(T ∗ (1)).

Proof. See [FPR94, Theorem 5.3.6]. Using this lemma, it is just a question of bookkeeping to show the

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equivalence between the conjectures CBK (M ) and CTam (M ) for motivic pairs.

9.5 The TNC for the Tate motives over abelian number fields We want to sketch in this section a proof of the Bloch–Kato conjecture for the special values ζF∗ (m), m ∈ Z, F being an abelian number field. We shall follow the approach of [BN02], which is actually parallel to Bloch and Kato’s original one (see also [Co15]; that of [HK03] for Dirichlet motives is quite different). Of course, serious technical difficulties will appear if F 6= Q, into which we do not intend to go in detail, our goal being rather to delineate the main lines and the articulation between the arguments. This will not be a gratuitous exercise provided we succeed in showing the pertinence of the Platonician apologue. The reader will also see as (s)he goes along where the hypothesis that F is abelian is needed. In order not to let the technical difficulties hide the articulation of the arguments, let us give a brief preliminary summary: (i) First, compute the local Tamagawa numbers and use them to show the compatibility of the Bloch–Kato conjecture with the functional equation of ζF (s), more precisely show the equivalence of the conjectures CTam (Z(m)) and CTam (Z(1 − m)) (see definitions below). It then remains to prove, for m ≥ 2, the conjecture CTam (Z(1 − m)), which is none other than the (cohomological version of) the Lichtenbaum conjecture (see [BN02]). (ii) A first analytical step consists in expressing the special value ζF∗ (1 − m) as a linear combination of special values of polylog functions. For the remaining part of the proof, the functorial properties of the Tamagawa number conjecture allow us to replace the abelian field F by a cyclotomic field Q(ζN ). (iii) Apply then the ‘motivic philosophy’. The Platonician archetype will be a K-theoretic special element, the so-called Beilinson element bm (ζN ). The rays of light will be the Beilinson archimedean regulator ρ∞,m and the p-adic ´etale regulators ρp,m . The archimedean ρ∞,m sends bm (ζN ) to a linear combination of polylog values, which is an archimedean shadow (in Plato’s sense) of bm (ζN ). The ´etale regulator ρp,m sends bm (ζN ) to the so-called Deligne–Soul´e cyclotomic element, hence by the Iwasawa main conjecture, ζ(1 − m)∗ is related to a p-adic shadow of bm (ζN ). In Plato’s words, the relations between the archimedean and

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non-archimedean shadows are explained by the archetype bm (ζN ), and they give the proof of the Tamagawa number conjecture.

9.5.1 Reformulation of the conjecture Let F be a number field (not necessarily abelian), E = Q, Tam0F (Z(m)) =

YY p

l

Tam0p.w (Zl (m)) · Tam0∞ (Z(m) (see §9.4.2), p

XF (Z(m)) =

Y XF (Zl (m)) (§9.4.3.3). l

The Bloch–Kato conjecture then reads: CTam (Z(m)) : For all m ∈ Z, m 6= 0, 1, we have 2

ζF∗ (m) = ±

#XF (Z(m)) · Tam0F (Z(m)). w1−m (F )wm (F )

As we already pointed out, the ‘singular’ values m = 0, 1 require a separate formulation, which will be: 2

ζF∗ (m) = ±

XF (Z(m)) · Tam0F (Z(m)) for m = 0, 1. w1 (F )

2

ζF∗ (0) = ±h(F )R1 (F )/w1 (F ), 2

ζF∗ (1) = ±(2π)r2 · | dF |−1/2 ·h(F )R1 (F )/w1 (F ). These are weakened versions of the two equivalent forms (via the functional equation) of Dedekind’s theorem. One can modify the general formulation (for m 6= 0, 1) of CTam (Z(m)) to come closer to Dedekind Q 2 coh by introducing the group K2m−2 (OF ) := He´t (OF [1/l], Zl (m)) (see l6=2 coh §9.4.3.3), writing hm (F ) := #K2m−2 (OF ) and using formula (9.10) at the end of §9.4.3. We now sketch the proof for m 6= 0, 1.

9.5.2 First step Compatibility with the functional equation: Let m ≥ 2. The formulations of CTam (Z(m)) and CTam (Z(1 − m)) are

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not the same, because of the non-symmetry with respect to m which we pointed out in the remark following Proposition 9.4.3.2. Recall that coh hm (F ) := #K2m−2 (OF ). 2

CTam (Z(1−m)) : ζF∗ (1−m) = ± 2

hm (F ) ·Rm (F )(see Lem. 9.2.6 & 9.4.1) wm (F )

CTam (Z(m)) : ζF∗ (m) = ζF (m) = ±

hm (F ) ·Tam0F (Z(m)). QQ (l) wm (F ) w1−m (Fp ) l p|l

As a first reduction step, let us show that the two conjectures imply each other, i.e., that the Tamagawa numbers are compatible with the functional equation. For this we need to make more precise the general formulas of Lemma 9.2.6 for m ≥ 2 : Theorem 9.5.1 For an abelian number field F , for all m ≥ 2, for any finite prime p of F dividing p, np (p) Tam0F (Z(m)) = w1−m (Fp )·|(m−1)!|p |dFp |m−1 , where np = [Fp : Qp ]. p p For the (very technical) proof, see [BN02, §2], and the comments therein. This formula generalizes the original Bloch–Kato formula for F = Q [BK90, Theorem 4.2]. We naturally conjecture its validity for non-abelian F . Q Assume now the validity of CTam (Z(1 − m)). As |dFp |p = |dF |p , p|p Theorem 9.5.1 gives that Y Y −n Tam0p (Z(m)) = w1−m (Fp )|dF |1−m Γ(m) p . p-∞ p-∞ Combining with the formula giving Tam0∞ (Z(m)) in Lemma 9.4.1(ii), we get that CTam (Z(m)) is equivalent to 2

ζF (m) = ±

hm (F )|dF |1/2−m sm π , wm (F )Γ(m)n

where n = [F : Q] and sm = mn − r2 (resp. mn − r1 − r2 ) if m is even (resp. odd). The veracity of CTam (Z(m)) is then an immediate consequence of CTam (Z(1 − m)) and of the functional equation ζF (m) = 1/2−m dF Γ(m)−n π sm ζF∗ (1 − m). The compatibility above is a particular case of the so-called CEP,Fp (V ) conjecture of Fontaine and Perrin-Riou [FPR94, III.4.5].

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9.5.3 Second step: The Lichtenbaum conjecture Thus we are reduced to prove conjecture CTam (Z(1 − m)). Under this form, it is the cohomological version of the Lichtenbaum conjecture for F (and is equivalent to it if we admit the Quillen–Lichtenbaum conjecture). It is worth mentioning that this is exactly how Bloch and Kato proceed in [BK90, §6], for F = Q, although their Theorem 6.1 is stated only for m ≥ 2 (which means that the motif Q(m) has weight ≤ −4). The abelian cohomological Lichtenbaum conjecture was proved in [KNF96], but with some errors which were subsequently corrected in the appendix of [BN02]. We sketch its proof here in a way which underlines the main features of the motivic approach. See also a refinement character by character in [HK03]. 9.5.3.1 Analytic part Let G = Gal(F/Q), supposed to be abelian. Classically Y ζF (s) = L(s, ψ), ˆ ψ∈G

where ψ runs through the characters of G, or equivalently, through the primitive Dirichlet characters belonging to F . Also, Y Y ζF∗ (1 −m) = L0 (1 −m, ψ) · L(1− m, ψ). (9.11) ψ(−1)=(−1)m

ψ(−1)=(−1)m−1

It follows that, denoting by F + the maximal totally real subfield of F , we have ( Y ζF + (1 − m) if m is even L(1 − m, ψ) = ζF∗ (1 − m)/ζF + (1 − m) if m is odd. ψ(−1)=(−1)m To go on further, we need an analytic expression of L0 (1 − m, ψ) in P∞ terms of polylogarithms. For m ≥ 1, define Lim (z) = k=1 z k /k m . This series is convergent for |z| < 1 and admits an analytic prolongation to C \ [1, ∞); note that for m ≥ 2, Lim (z) converges for |z| = 1. For any Dirichlet character ψ : Gal(Q(ζN )/Q) ' (Z/N Z)× → C× , define lm (ψ, ζN ) =

Σ g∈(Z/N Z)×

g ψ(g)Lim (ζN ).

Then we have Proposition 9.5.2 (see [Ne98])

(i) Let ψ be a Dirichlet character mod

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N , with conductor fψ , and let ψ˜ : (Z/fψ Z)× → C× be the corresponding primitive character. Then Y ˜ m−1 )f m−1 lm (ψ, ˜ ζN ) = N m−1 lm (ψ, ζN ). (1 − ψ(l)l ψ l|N

(ii) If ψ is a primitive character mod N such that ψ(−1) = (−1)m−1 , 1 (m−1)! m−1 then L0 (1 − m, ψ) = (2πi) lm (ψ, ζN ). m−1 · N · We now adopt the motivic philosophy in order to relate this analytic information to algebraic objects. 9.5.3.2 Special motivic elements 1 Recall that HM (Q(ζN ), Q(m)) = K2m−1 (Q(ζN ))⊗Q. There exists a spe1 cial element bm (ζN ) ∈ HM (Q(ζN ), Q(m)), called the Beilinson element, which verifies the following remarkable properties.

Theorem 9.5.3 (see [HW98], [Ne98], theorem 11.1.1 in [Hu15]) (i) The Beilinson regulator 1 ρ∞,m : HM (Q(ζN ), Q(m)) → R(m − 1)+ B

sends bm (ζN ) to −N m−1 (m − 1)!

Σ g∈(Z/N Z)×

g Lim (ζN )g.

(ii) The p-adic regulator 1 ρp,m : HM (OF [1/p], Q(m)) → He´1t (Q(ζN ), Qp (m)), where F = Q(ζN ), sends bm (ζN ) to the ‘cyclotomic element’ c˜m (ζN ) defined as follows: Fix k a system (ζn )n∈N of roots of unity such that ζnk = ζn and define ⊗(m−1)

cm (ζN )n = coresQ(ζN pn )/Q(ζN ) (δN pn (1 − ζN pn ) ⊗ ζpn 1

)

n

∈ H (Q(ζN ), Z/p Z(m)), where δN pn is the Kummer map Q(ζN pn )× → H 1 (Q(ζN ), Z/pn Z(1)). Let cm (ζN ) = proj lim cm (ζN )n ∈ H 1 (Q(ζN ), Zp (m)) (this is the socalled Deligne–Soul´e element). Then c˜m (ζN ) = cm (ζN ) if p | N, and c˜m (ζN ) = (1 − pm−1 Frob−1 p )cm (ζN ) if p - N. Remark: The Deligne–Soul´e elements are norm compatible elements which define Euler systems, hence are related to p-adic L-functions, but they miss the Euler factor at p. Property (ii) in the theorem is Conjecture

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6.2 of [BK90], proved by Huber and Wildeshaus [HW98] (see the articles by Blasius, Huber and Kings, [Bl15], [Hu15], [Ki15]) in this volume. In the sequel, take N to be the conductor of the abelian field F , which is considered as a subfield of Q(ζN ). We shall denote by Bm (Q(ζN )) the 1 Galois submodule of HM (Q(ζN ), Q(m)) generated by the Beilinson ele1 ment bm (ζN ) and by Bm (F ) ∈ HM (F, Q(m)) the image of Bm (Q(ζN )) under corestriction (recall that N is the conductor of F ). 9.5.3.3 Going to the archimedean world via the Beilinson regulator Property (i) of Theorem 9.5.3 immediately gives the covolume of ρ∞,m (Bm (F )) in R(m − 1)+ , Y

covol(ρ∞,m (Bm (F )) = Em (F )

ψ(−1)=(−1)

where the Euler factor Em (F ) =

QQ

(m − 1)! m−1 f lm (ψ, ζfψ ), m−1 ψ (2πi) m−1

(1 − ψ −1 (l)lm−1 ) accounts for the

ψ l|N

deviation between primitive and imprimitive characters. Then Proposition 9.5.2 yields Y Em (F ) · L0 (1 − m, ψ) = covol(ρ∞,m (Bm (F )). (9.12) ψ(−1)=(−1)m−1

But covol(ρ∞,m (Bm (F ))) i. Rm (F ) = h K2m−1 (F ) : Bm (F ) tors Indeed, in §9.4.1, we defined the regulator number Rm (F ) as the determinant of the regulator map. This is also classically equal to the covolume of ρ∞,m (K2m−1 (F )/tors). Since Bm (F ) is a sublattice of K2m−1 (F )/tors and regulators are ‘linear’, we obviously have here the above formula for Rm (F ). Our next task will be to compute the index in the denominator. We do this by 9.5.3.4 Going to the p-adic world via the ´ etale regulator For the totally real field F + , one has the p-adic equivalence (i.e., equality up to p-adic units, and denoted by ∼ ) p

ζF + (1 − m) ∼ ζF + ,p (1 − m) for even m, p

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and the classical Main Conjecture (here, the theorem of Mazur–Wiles) immediately gives the formulas: ζF + (1 − m) ∼ # H 2 (OF + [1/p], Zp (m))/#H 1 (OF + [1/p], Zp (m)) p

for even m, ∗

ζF (1 − m) ∼ # H 2 (OF [1/p], Zp (m))− /#H 1 (OF [1/p], Zp (m))− ζF + (1 − m)∗ p for odd m. Applying formula (9.11), one obtains easily (for details, see [KNF96, p.710]) Y 2 coh L(1 − m, ψ) = #(K2m−2 (OF )± )/wm (F ), (9.13) ψ(−1)=(−1)m 2

where one takes the ‘plus’ part if and only if m is even, and = denotes equality up to powers of 2. The formulas (9.11), (9.12), (9.13) put together give: 2

ζF∗ (1 − m) =

coh #(K2m−2 (OF )± ) ([K2m−1 (F )/tors : Bm (F )] · Rm (F ) · wm (F ) Em (F )

and it remains only to compute the last index. Here serious technical difficulties appear in the non-semisimple case (i.e., when p | |G|). The usual calculations character by character can be salvaged only after showing the nullity of the µ-invariants of certain Iwasawa modules, especially of the units modulo circular units (see [BN02, p.667]). Another way – particularly well adapted to Dirichlet motives – consists in twisting the cyclotomic elements of Theorem 9.5.3 by the characters of G ([HK03, 3.1.3]). Also, to circumvent the problem of characters, we could squarely appeal to an (abelian) Equivariant Main Conjecture which is the ‘limit theorem of [BG03, Theorem 7.1]; see also [Wi06, Theorem 7.4] (but there is no miracle, the hard work on characters being hidden in the proof of the EMC). In any case, denote by Dp,m (Q(ζN )) the Galois submodule of He´1t (OF [1/p], Zp (m)) generated by cm (ζN ) and Dp,m (F ) = TrQ(ζN )/F Dp,m (Q(ζN )). By Theorem 9.5.3(ii) [K2m−1 (F )/tors : Bm (F )] ∼ [(He´1t (OF [1/p], Zp (m))/tors : Dp,m (F )], p

and any one of the approaches mentioned above allows one to show that the latter index is p-adically equivalent to Em ·#(He´2t (OF [1/p], Zp (m))± .

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Finally, coh [K2m−1 (F )/tors : Bm (F )] = Em (F ) · #(K2m−2 (OF )∓ ),

and the desired formula for ζF∗ (1 − m) follows. This ends the proof of the TNC for Tate motives over abelian fields. Remark: The 2-part of the TNC (and even of the ETNC) for abelian fields F has been settled recently by M. Flach [Fl11]. It appears that the difficulties do not come only from the complications of the 2-adic Chern class map, but also from deep arithmetic properties of F at the prime 2.

References [Bl15] Blasius, D. 2015. Motivic polylogarithm and related classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 193–209. [BG03] Burns, D., and Greither, C. 2003. On the equivariant Tamagawa number conjecture for Tate motives. Invent. Math., 153, 303–359. [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift. Progr. Math., 86, 33–400. Vol. I, Birkh¨ auser Boston. [BN02] Benois, D., and Nguyen Quang Do, T. 2002. Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs Q(m) sur ´ un corps ab´elien. Ann. Scient. Ecole Norm. Sup., 35, 4, 641–672. [Co15] Coates, J. 2015. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 45–64. [DF85] Dwyer, W. G., and Friedlander, E. 1985. Algebraic and ´etale K-theory. Trans. AMS, 292, 247–250. [Fl90] Flach, M. 1990. A generalization of the Cassels-Tate pairing. Jour. reine angew. Math., 412, 113–127. [Fl11] Flach, M. 2011 On the cyclotomic main conjecture for the prime 2. Jour. reine angew. Math., 661, 1–36. [Fo82] Fontaine, J.-M. 1982. Sur certains types de repr´esentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. of Math., 115, 529–577. [Fo92] Fontaine, J.-M. 1992. Valeurs sp´eciales de fonctions L de motifs. S´em. Bourbaki exp. 751, Ast´erisque, 206, 205–249. [FPR94] Fontaine, J.-M., and Perrin-Riou, B. 1994. Autour des conjectures de Bloch et Kato, Cohomologie Galoisienne et valeurs de fonctions L, in Motives. Proc. Symp. in Pure Math., 55, 599–706. [GM03] Gelfand, S., and Manin, Y. 2003. Methods of homological Algebra. Springer Monographs in Math. 2nd ed.

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[Hu15] Huber, A. 2015. A motivic construction of the Soul´e-Deligne classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 210–238. [HK03] Huber, A., and Kings, G. 2003. Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters. Duke Math. J., 119, 3, 353–464. [HW98] Huber, A., and Wildeshaus, J. 1998. Classical motivic polylogarithm according to Beilinson and Deligne. Doc. Math., 3, 27–133. [Ka93] Kahn, B. 1993. On the Lichtenbaum - Quillen conjecture, in Algebraic K-theory and Algebraic Toplogy. NATO Proc. Lake Louise, 407, 147– 166. [Ki15] Kings, G. 2015. Sheaves of Iwasawa modules, Moment maps and the l-adic elliptic polylogarith, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 239–296. [Ko15] Kolster, M. 2015. The Norm residue homomorphism and the QuillenLichtenbaum conjecture, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 97–120. [KNF96] Kolster, M., Nguyen Quang Do, T., and Fleckinger, V. 1996. Twisted S-units, p-adic class number formulas and the Lichtenbaum conjectures. Duke Math. J., 84, 679–717. [Ne98] Neukirch, J. 1998. The Beilinson conjecture for algebraic number fields, in Beilinson’s conjectures on special values of L-functions. Perspectives in Math., 4, 153–247. [Ra15] Raghuram, A. 2013. Special values of the Riemann zeta function: Some results and conjectures, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 1–21. ¨ [Sc75] Schneider, P. 1975. Uber gewisse Galoiscohomologiegruppen. Math. Zeit., 168, 181–205. [So79] Soul´e, C. 1979. K-th´eorie des anneaux d’entiers de corps de nombres et cohomologie ´etale. Invent. Math., 55, 251–295. [Su15] Sujatha, R. 2015. Postscipt, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 297–305. [Wi06] Witte, M. 2006. On the equivariant main conjecture of Iwasawa theory. Acta Arith., 122, 275–296.

10 Motivic Polylogarithm and Related Classes Don Blasius

Abstract This chapter is an expanded reprise of two lectures given by the author at the Workshop on Bloch–Kato Conjectures held at IISER, Pune, July 17–21, 2012. The author thanks the organizers of the workshop for the opportunity to participate in this worthwhile program. This research was also partially supported by NSF Grant DMS-0854949. The author thanks the NSF for its generous support.

10.1 Introduction This chapter provides background for the contribution [Hu15] of Huber to the volume. Our first main task is to review the construction of the motivic (i.e., class in K-theory) version PolkM , for k ≥ 0, of the (absolute) cohomological elliptic polylogarithms associated to universal elliptic curves. This is done in Section 10.2 below. The discussion is barebones: just the main steps and facts, which require mastery of many of the properties of motivic cohomology, are given. Main references for the paper are the foundational paper [BL94], which introduced the elliptic polylogarithm, as well as [Wi02], and [Ki99]. We mostly follow [Ki99], which actually generalizes the construction of the elliptic polylogarithm to families of abelian schemes. On the other hand, [Wi02] gives a detailed discussion which focuses on just the case needed here, also using the same auxiliary non-proper schemes in order to construct the sought classes. i

University of California at Los Angeles, USA. e-mail : [email protected]

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The motivic elliptic polylogarithm PolkM is introduced in Section 10.2. Its function is to give, via a linear combination of specializations, followed by degeneration to the cusp at infinity, some single class in Ktheory of a cyclotomic field whose image under the archimedean regulator can be identified with an L-value times a canonical generator of the Deligne cohomology, and whose image under the l-adic regulator is a Soul´e class. In its precise formulation, for Dirichlet L-series, this is Conjecture 6.2 of the foundational paper of Bloch and Kato. It is this conjecture that Huber and Kings prove in their contributions here. The motivic Eisenstein classes EiskM (ψ) are introduced in Section 10.3. They depend on the choice of a divisor ψ supported on torsion and are the result of specialization, starting from the polylogarithm. These classes should not be confused with those of [Be86], which were constructed without use of the motivic polylogarithm. Nevertheless, Beilinson’s construction (‘Eisenstein symbol’) retains a role in the discussion. For this aspect, see Section 10.4.6 below and the sketch of the proof of Theorem 11.6.5 of [Hu15], and Section 1 of [HK99b]. Perhaps ideally one ought to review constructions of the polylogarithm, and associated constructions of Eisenstein classes, in the three settings of motivic, l-adic and Deligne (also known as absolute Hodge) cohomology, checking that the latter two constructions give rise to classes which are images under regulators of the Eisenstein class in motivic cohomology. However, we have given here only two-thirds of the discussion, restricting to the motivic and l-adic case. The point is that, for our purposes, the archimedean theory, although very differently defined at the outset and extremely rich for study in detail, has formal properties so parallel to those of the l-adic one that a separate detailed discussion would seem pedantic. We refer the reader to Section 5 of Huber’s contribution, and to the original sources cited there, for some archimedean review. In Section 10.4 we introduce l-adic versions Polkl and Eiskl (ψ) of the motivic classes and sketch their comparison using the l-adic regulator. The key facts of the paper, to be quoted in [Hu15], are the identities Theorem 10.1.1 rl (EiskM (ψ)) = Eiskl (ψ)

(10.1)

rH (EiskM (ψ)) = EiskH (ψ).

(10.2)

and

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The first equation takes place in ´etale cohomology. The second equation takes place in Deligne cohomology and is stated, as per the above caveat, without discussion. The first equation is, in fact, ambiguous since we give two constructions in 10.4.4 and 10.4.5 of classes Eiskl . However, Prop. 10.4.2 asserts that these classes are the same, so the ambiguity disappears. The polylogarithmic identities which underlie Theorem 10.1.1 are Theorem 10.1.2 rl (PolkM ) = Polkl

(10.3)

rH (PolkM ) = PolkH .

(10.4)

and

The above compatibilities for the Eisenstein classes in Theorem 10.1.1 follow from Theorem 10.1.2 and the functoriality of regulators relative to the parallel processes of construction of the Eisenstein classes.

10.2 Motivic polylogarithm classes This part follows closely Section 2 of [Ki99]. For more details see that paper, [BL94] and [Wi02].

10.2.1 Motivic cohomology Let X be a smooth variety over Q. The motivic cohomology of X is the bigraded family of rational vector spaces i HM (X, j) = (K2j−i (X) ⊗ Q)(j) .

Here the superscript (j) denotes, as usual, the generalized simultaneous eigenspace of all Adams operators ψ k , k ≥ 1 belonging to the eigenvalues k j . The groups K∗ (X) are the usual Quillen K-groups associated to the exact category of locally free coherent sheaves of finite rank on X.

10.2.2 Basic schemes This section follows closely Section 2 of [Ki99]. For further background, see [BL94] and [Wi02]. Let E → Y be an abelian scheme over the base Y , of relative dimension 1, i.e., a family of elliptic curves; we may also

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denote it E/Y . Normally, Y = Y (N ), the modular scheme over Q with full level N structure, for some N we will not need to specify. (Thus, over C, the points of Y (N ) are parametrized by (Z/N Z)∗ × Γ(N )\H.) Let e : Y → E be the identity section; an alternative notation for e(Y ) is 0. Regard E ×E as an E-scheme via the second projection, and let (E ×Y E)n be n-fold fiber product over E. Let U denote the scheme U = (E r e(Y )) ×S (E r e(Y )). Then U is an E r e(Y )-scheme via projection on the second factor. Let V be the complement of the diagonal ∆(E r e(Y )) in U , i.e., V = U r (∆(E r e(Y )). For W = U or W = V , let, for n ≥ 1, W n = W n−1 ×Ere(Y ) W, where we understand W 0 = E r e(Y ). Evidently, V n ⊂ U n ⊂ (E ×Y E)n . Let pri : W n → E r e(Y ) be the projection on the i-th factor. For a subset I of {1, ..., n} define V I ⊂ U n to be the subscheme satisfying pri (u1 , ..., un ) ∈ V if i ∈ I, and pri (u1 , ..., un ) ∈ ∆(E r e(Y )) for i ∈ / I.

10.2.3 Sign character Let Σn (resp. ΣI ) be the permutation group on [1, ..., n] (resp. I) with sign character sgnn (resp. sgnI ). For any Q-vector space V on which Σn (resp. ΣI ) acts, let Vsgnn (resp. VsgnI ) be the largest quotient on which Σn (resp. ΣI ) acts by sgnn (resp. sgnI ).

10.2.4 The operator tr[a] For an integer a and any abelian scheme, let [a] denote the usual morphism of multiplication by a. Then [a]−1 U ⊂ U (resp. [a]−1 V ⊂ V ) is defined and is an open subscheme of U (resp. V ). Let j denote the canonical inclusion morphism in these cases.

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Definition a. For W = U or W = V , as above, define · tr[a] ∈ End(HM (W, ∗))

(10.5)

as the composition tr[a] = [a]∗ ◦ j ∗ . b. Letting [a]n be the product morphism, i.e., multiplication by a on the product abelian scheme, define n · tr[a] = [a]n∗ ◦ j ∗ ∈ End(HM (W n , ∗)).

(10.6)

· c. Let r ∈ Z and denote by HM (W n , ∗)(r) the largest subspace on which n all operators tr[a] − ar Id are nilpotent.

Note that [a]n commutes with the action of Σn on W n . Hence Σn acts on (r) · · HM (W n , ∗)(r) , so HM (W n , ∗)sgnn is defined. It is key to what follows.

10.2.5 Residue sequence Let k be a field. The localization long exact sequence in motivic cohomology is attached to a pair (X, Z) consisting of a scheme of finite type X and a closed subscheme Z of pure codimension d, both defined over k. Putting Y = X r Z, the sequence is: res

i+1−2d i i · · · → HM (X, ∗) → HM (Y, ∗) −→ HM (Z, ∗ − d) → · · ·

(10.7)

The labelled arrow is called the residue map. Taking X = U n , Z = V n , then d = 1 and Y has a stratification by the V I for I a proper subset of {1, . . . , n}. Taking sgnn eigenspaces, the sequence becomes: res

·−1 · · → HM (U n , ∗)sgnn → HM (V n , ∗)sgnn −→ HM (Y, ∗ − 1)sgnn → . (10.8) Let J ⊂ [1, ..., n] with |J| = n − 1. In [Ki99], Section 2.1, it is shown how to simplify the last term of (10.7). Indeed, the proof of Prop. 2.1.2 of [Ki99] provides a natural isomorphism ·−1 ·−1 HM (Y, ∗ − 1)sgnn ∼ (V J , ∗ − 1)sgnJ . = HM

(10.9)

Therefore we have the long exact residue sequence ([Ki99], Cor. 2.1.4):

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Proposition 10.2.1 res

·−1 · · → HM (U n , ∗)sgnn → HM (V n , ∗)sgnn −→ HM (V J , ∗ − 1)sgnJ → . (10.10)

The proof of (10.9) has two steps. First, the sgnn condition forces, via Frobenius reciprocity, the vanishing of the contribution to cohomology ·−1 of HM (Y, ∗ − 1)sgnn made by all the V I for |I| ≤ n − 2. Thus, by the localization sequence attached to the pair (Y, ∪|I|=n−1 V I ), we have ·−1 ·−1 HM (Y, ∗ − 1)sgnn ∼ (V I , ∗))sgnn . = (⊕|I|=n−1 HM ·−1 Finally, the Σn module ⊕|I|=n−1 HM (V I , ∗) is the one induced from ·−1 J the ΣJ module HM (V , ∗), and so formula (10.9) follows by Frobenius reciprocity.

10.2.6 Residue isomorphism n Now it is time to bring in the operators tr[a] . Using them, one finds the basic vanishing result:

Proposition 10.2.2

For 0 ≤ r ≤ n and n ≥ 1, · HM (U n , ∗)(r) = 0.

This is ([Ki99], Thm. 2.2.3). As a consequence, we get the following key. Theorem 10.2.3

Residue Isomorphism Theorem

· ∼ ·−1 n−1 , ∗ − 1)(1) ∼ HM (V n , ∗)(1) sgnn = HM (V sgnn−1 = · · · ∼ = H ·−n (E r e(Y ), ∗ − n)(1) .

(10.11)

M

This result is ([BL94], 6.3.3; [Ki99], Cor. 2.2.5; [Wi02], Cor. 1.6.4). It is an immediate consequence of Prop. 10.2.1, combined with Prop. 10.2.2 and the easily proved (See Lemma 2.2.2 of [Ki99]) equivariance of n (10.9) with respect to the tr[a] actions.

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10.2.7 Base changed version If we base change each V j to E × V j , we obtain another chain of isomorphisms from which we conclude what we actually need: Proposition 10.2.4 · (1) ∼ ·−n HM (E × V n , ∗)(1) . sgnn = HM (E × (E r e(Y )), ∗ − n)

(10.12)

10.2.8 Definition of motivic polylogarithm This is the class (taking · = k + 3, ∗ = k + 2, so n = k + 1) k+1 k+3 k+1 Polk+1 , k + 2)(1) sgnk+1 M = PolM,E ∈ HM (E ×Y V

(10.13)

corresponding under the isomorphism (10.11) to the class of the diagonal ∆(E r e(Y )) ⊂ E ×Y (E r e(Y ) in 2 HM (E ×Y (E r e(Y )), 1)(1) .

This is Definition 2.2.6 of [Ki99]. Notation is adjusted to application to Eisenstein classes in [Hu15].

10.3 Motivic Eisenstein classes For each positive integer k > 0 and each torsion divisor ψ ∈ Q[E[N ]r0], N > 1, there are defined motivic, l-adic and Hodge Eisenstein classes. The l-adic and the Hodge Eisenstein classes are studied in [Hu15] and our first task here is to recall the construction of the motivic class. In each case, the class is defined by pulling back the corresponding l-adic or Hodge polylog class along ψ, then implementing successively projection and contraction operations. Thus, in the l-adic case we have ([Hu15], Definition 11.4.2; [Ki99] Definition 8.4.1; see also 10.4.2 below): 1 Eiskl (ψ) ∈ Ext1Y (N) (Ql , Symk H(1)) = Het (Y (N ), Symk H(1))

defined by the formula Eiskl (ψ) = −N k−1

X

ψ(x)contr(%x x∗ Pol)k+1

(10.14)

x∈E[N ]r0

whose motivic analogue we now describe. To start, note that since k+1 1 Het (Y (N ), Symk H(1)) ⊆ Het (E k , Ql )(k + 1),

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we should seek a motivic Eisenstein class k+1 EiskM (ψ) ∈ HM (E k , k + 1)

(10.15)

constructed by a similar formula. In fact, it will be an element of the (k) k+1 k Σk × tr[a] -subspace HM (E k , k + 1)sgnk of this group. The construction of EiskM (ψ) proceeds as follows; we give the motivic version first and sketch the parallel l-adic version in Section 10.4.4 below.

10.3.1 Definition of x∗ PolM Fix N > 1 and let E(Y )[N ] denote the set of sections x ∈ Hom(Y, E) of π E → Y of order N . Then subschemes Ux ⊂ U and Vx ⊂ V are defined with Ux ∼ = E r e(Y ) and Vx ∼ = E r {e(Y ), x(Y )}; these are the evident cartesian products over E r e(Y ) of pr2 and x. Likewise, Vxk+1 is defined pr2 as the cartesian product of V k+1 → E r e(Y ) and x. Thus, we have a natural restriction map: x∗

k+3 k+3 HM (E ×Y V k+1 , k + 2) −→ HM (E ×Y Vxk+1 , k + 2)

(10.16)

k+1 and the restriction x∗ Polk+1 M of PolM is defined.

10.3.2 A key eigenspace n Note that since x has order N , [N + 1]x = x and the operators tr[N +1] · n · n are well defined on HM (Ux , ∗), and HM (Vx , ∗). Also, we have the comk · k muting actions of tr[a] × Id and of Id × tr[N +1] on HM (E ×Y Ux , ∗)sgnk . Each commutes with the evident action on cohomology induced by that of Σk on the second factor of E ×Y Uxk . Thus the eigenspace · HM (E ×Y Uxk , ∗)(1,k) sgnk

(10.17)

is defined. Here the pair (1, k) means the generalized joint eigenspace of k k tr[a] × Id (for eigenvalue a) and of Id × tr[N +1] (for eigenvalue (N + 1) ).

10.3.3 Splitting lemma and consequence Lemma 10.3.1 1. The restriction map · · HM (Uxk , ∗)sgnk → HM (Vxk , ∗)sgnk

is canonically split.

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2. There exists a canonical projection operator · · k (1,k) · k Π : HM (E ×Y Vxk , ∗)(1,k) sgnk → HM (E ×Y Ux , ∗)sgnk ⊂ HM (E ×Y E , ∗). (10.18)

The first part of the lemma is proven in [Ki99] using the tr[N +1] - equivariant localization long exact sequence for the pair (Ux , Vx ) and an analysis of the weights by which tr[N +1] acts. The result of the first part holds after base change of both Uxk and Vxk by E, so the second part of the lemma follows at once.

10.3.4 Contraction We now consider the motivic analogue of the contraction operation contr in Formula (10.14).The diagonal translation map σ : E ×Y E k+1 → E k+1 defined by (e, e1 , . . . , ek+1 ) → (e1 + e, . . . , ek+1 + e) for e, e1 , · · · , en ∈ E is Σk+1 -invariant. The associated Gysin map is k+3 k+1 σ∗ : HM (E ×Y E k+1 , k + 2)sgnk+1 → HM (E k+1 , k + 1)(k) sgnk .

On the other hand, letting I = [1, ..., k] ⊂ [1, ..., k + 1], there is a restriction morphism k+1 k+1 k+1 ρI : HM (E k+1 , k + 1) → HM (E I , k + 1) = HM (E k , k + 1).

The composition pr = ρI ◦ σ∗

(10.19)

is the sought motivic contraction. It is easy to see that k+3 k+1 k+1 pr(HM (E×Y E k+1 ,k+2)(1,k+1) )⊆HM (E I ,k+1)=HM (E k ,k+1)(k) sgn sgn . k+1

k

10.3.5 Motivic Eisenstein class For x as above, define EiskM (x) = −N k−1 pr ◦ Π ◦ x∗ (Polk+1 M ),

(10.20)

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so that k+1 EiskM (x) ∈ HM (E k , k + 1)(k) sgnk .

For each torsion divisor ψ ∈ Q[E[N ] r 0], of degree 0, i.e., X ψ= nx · x, x∈Ere(Y )

define EiskM (ψ) =

X

nx · EiskM (x).

(10.21)

x∈E[N]re(Y )

By analogy with Huber ([Hu15], Definition 11.4.2), we call this class EiskM (ψ) the motivic Eisenstein element for ψ. We note again that this construction via polylogarithms is very different from the original one of Beilinson in [Be86]. Nevertheless the value of this class under the regulators coincides with those of Beilinson’s class.

10.4 Comparison of motivic, l-adic and archimedean classes 10.4.1 l-adic polylog This polylog Poll is usually defined directly as an infinite rank sheaf on E r e(Y ) with a canonical (weight) filtration W · by subsheaves of finite codimension. The quotients are the P ollk and these are the sheaves whose classes correspond to the PolkM under the regulator. These sheaves are unipotent, which means that the associated graded sheaf of successive quotients has graded terms each of which is a pullback from the base Y . In fact the graded sheaf Gr. (Poll ) of Poll is   0, if k ≥ 0    π ∗ H, if k = 1 Gr−k (Poll ) = Ql (1), if k = 2     ∗ k−2 π Sym (H)(1), if k ≥ 3 Here H is the usual local system of Tate modules R1 π∗ F (1) = (R1 π∗ F )∨ on Y . The sheaf W −2 Poll is Logl (1), the Tate twist of the l-adic logarithm sheaf Logl . Poll is reviewed in 11.3 of [Hu15]. We have  Logl W k+1 Log = Symk (Log (1) ), l l

Motivic Polylogarithm and Related Classes (1)

and Logl

203

is the Kummer sheaf 0 → π ∗ H → Log → Ql → 0

on E whose stalk at a point P of E is the extension of π ∗ H by Ql defined by the H torsor of all projective systems of l-power roots of P . Thus, Pol is defined up to isomorphism by an Ext1 class, also denoted Poll , in Ext1E\0 (π ∗ H, LogEr0 (1)), and Logl is defined up to isomorphism by an Ext1 class, also denoted Logl , in Ext1E (Ql , π ∗ H). In fact, each of the groups has a simple structure with an obvious canonical element. In the case of Logl , we have a split exact sequence: π∗

0 → Ext1Y (Ql , HE ) → Ext1E (Ql , π∗ H) → HomY (Ql , H∨ ⊗ H) → 0, where the splitting e∗ of the left arrow is induced by the 0-section denoted by e (or sometimes 0). The canonical class of the rightmost group is the map sending 1 ∈ Ql to IdH ∈ End(H) = H∨ ⊗ H. The class of Logl is the unique preimage in Ext1E (Ql , π ∗ H) whose restriction e∗ via the 0-section is trivial in Ext1Y (Ql , HE ). In the case of Poll , we have a canonical isomorphism: Ext1E\0 (π ∗ H, LogEr0 (1)) = HomY (H,

Y

Symk (H)).

(10.22)

k≥1

In terms of the group on the right, the class of Poll is the element sending h ∈ H to the tuple (h, 0, . . .). The proof of these isomorphisms follows from an analysis of the Leray spectral sequence in both cases. See [BL94], Section 1.2, [Ki99], Section 1, or [HK99b], Appendix A. (k)

10.4.2 Alternative construction of Poll as parallel to (k) the construction of PolM The key point in the construction of the motivic polylogarithms is the ladder of isomorphisms gained from the localization sequence (10.10)

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plus the vanishing result Prop. 10.2.2. These facts have analogues in absolute ´etale (= Jannsen’s continuous ´etale cohomology [Ja88]) and Deligne cohomologies which enable fully parallel constructions. Thus, the family of (residue) isomorphisms ∼ n+1 (E ×Y V n−1 , Ql (n))(1) Hln+2 (E ×Y V n , Ql (n + 1))(1) sgnn = Hl sgnn−1 follows as before from the localization sequence in cohomology plus the vanishing of the terms Hln (U n , Ql (N + 1))(1) , proven as for motivic cohomology. It is also essential to identify the l-adic classes constructed by the residue ladder with those defined above using the logarithm sheaves and some structural results for associated Ext groups not available in the motivic setting. Assume we have defined (k+1)

Poll

(k+1)

= Poll,E

∈ Hlk+3 (E ×Y V k+1 , k + 2)(1) sgnk+1

(10.23)

via the ladder. The connection to the construction of Section 10.4.1 is established by the following result: Proposition 10.4.1

There is an isomorphism

(n) 2 ∗ (1) ∼ Hln+2 (E ×Y V n , Ql (n + 1))(1) sgnn = ExtE×Y Ere(Y ) (Ql , pr2 LogE (1))

and the residue map gives, for all n ≥ 1, isomorphisms (n)

Ext2E×Y Ere(Y ) (Ql , pr2∗ LogE (1))(1) (n−1) ∼ (1))(1) = Ext2E×Y Ere(Y ) (Ql , pr2∗ LogE (n)

(n−1)

induced by the natural map LogE (1) → LogE

(1).

The proof of this result (which is Proposition 2.3.1 of [Ki99]) has several aspects. Working without imposing the sgnn condition, the Leray spectral sequence for the morphism p : E ×Y V n → E ×Y E r e(Y ) is shown to degenerate via the vanishing of all terms except Ext2E×Y Ere(Y ) (Ql , pn∗ Ql,V n ), thus giving an isomorphism of the type of the first line. Second the sheaf pn∗ Ql,V n on E ×Y E r e(Y ) has to be shown to be equal to (Log (1) )⊗n ; from this point, one sees easily, in view of the anticommutative nature of cup-product, that the projection of this sheaf to its sgnn eigenspace (Σn acting on the V n factor of E ×Y V n ) is Symn (Log (1) ) = Log (n) . The vanishing follows from an analysis of the 1 n weights by which tr[a] × tr[a] acts.

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Considering the first term of this Ext2 ladder, one notes that Ext1Er0 (π ∗ H, LogEr0 (1)) is a summand of Ext2E×Y Ere(Y ) (Ql , Ql (1)), and the class defined geometrically by the diagonal is identified with the extension class previously defined. Using the ladder, this class defines for each n ≥ 0, a class (n)

Poll

(n)

∈ Ext1Er0 (π ∗ H, LogEr0 (1)),

(10.24)

which necessarily coincides with that defined in 10.4.1.

10.4.3 l-adic regulator rl The term regulator is an alternative term for the Chern class morphisms between the motivic and the l-adic cohomologies. For some introduction to the theory, see [Sc88] or [So84]. They exist very generally and are compatible with the exact sequences and the residue morphisms. Sometimes denoted cn,k when the source is a bi-indexed motivic cohomology group n HM (X, k), we follow the custom of denoting them all as rl . Evidently (1)

(1)

the identity rl (PolM ) = Poll holds because the classes are given by the class of the diagonal in the two theories and rl is just the first Chern class of E ×E composed with the map induced by restriction to E ×Y E r0. For higher terms, (k)

(k)

rl (PolM ) = Poll

holds because the residue maps are isomorphisms which commute with the rl ’s. This proves Theorem 10.1.2 of the Introduction.

10.4.4 l-adic Eisenstein class In parallel with Section 10.3, a class Eiskl (ψ) can also be defined in (k) Hlk+1 (E k , k + 1)sgnk . To make this explicit, we start with the l-adic (k+1) polylogarithm Poll of (10.23). Any x ∈ E(Y )[N ], defines a morphism x∗

l Hlk+3 (E ×Y V k+1 , k + 2) −→ Hlk+3 (E ×Y Vxk+1 , k + 2).

(k+1)

Hence as before the restriction x∗l Poll as in (10.15) the eigenspace

(k+1)

of Poll

· HM (E ×Y Uxk , ∗)(1,k) sgnk

(10.25)

is defined. Next,

(10.26)

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is defined. Then, as in (10.16), we have a projection operator · k (1,k) · k Πl : Hl· (E ×Y Vxk , ∗)(1,k) sgnk → Hl (E ×Y Ux , ∗)sgnk ⊂ Hl (E ×Y Ex , ∗). (10.27) Finally, as in (10.17), the action σ induces

σ∗,l : Hlk+3 (E ×Y E k+1 , k + 2)sgnk+1 → Hlk+1 (E k+1 , k + 1)(k) sgnk , and, for I = [1, ..., k], we have a restriction morphism ρI,l : Hlk+1 (E k+1 , k + 1) → Hlk+1 (E I , k + 1) = Hlk+1 (E k , k + 1). Put prl = ρI,l ◦ σ∗,l ,

(10.28)

Eiskl (x) = −N k−1 prl ◦ Πl ◦ x∗l (Polk+1 M ),

(10.29)

and define

and for ψ=

X

nx · x ∈ Q[E[N ] r 0],

x∈Ere(Y )

of degree 0, Eiskl (ψ) =

X

nx · Eiskl (x).

(10.30)

x∈E[N]re(Y )

Since the regulators rl commute with the operations defining the Eisenstein classes, we get the analogue of Theorem 1 of the Introduction: rl (EiskM ) = Eiskl for this construction of Eiskl .

10.4.5 Another construction of Eiskl and proof of Theorem 1 On the other hand, a variant construction which commences from the sheaf-theoretic construction of Polk+1 in 10.4.1 is given in Section 11.4 l of [Hu15] and also Section 12.4.2 of [Ki15]. This variant constructs a class

Motivic Polylogarithm and Related Classes

Eiskl ∈ Ext1Y (Ql , Symk H) = H 1 (Y, Symk H).

207

(10.31)

Also, the extension of the well-known ‘Lieberman’s trick’ to the l-adic cohomology of fiber powers of E → Y shows that the Leray spectral sequence for this fibration degenerates, and in fact we have a canonical identification H 1 (Y, Symk H) = Hlk+1 (E k , k + 1)(k) sgnk .

(10.32)

The following Construction-Proposition concludes the proof of Theorem 1: Proposition 10.4.2 Via the identification (10.32), the constructions of classes both denoted Eiskl , in Section 10.4.4 of this paper and Section 11.4.2 of [Hu15], agree. Sketch of Proof. We review the steps of [Hu15], 11.4.2, in view of (10.32). Since (k+1)

Poll

(k+1)

x∗l Poll

(k+1)

∈ Ext1Er0 (π ∗ H, LogEr0 (1)), j ∈ Ext1Y (H, ⊕k+1 j=0 Sym H(1)),

(10.33)

(10.34)

j using Corollary 11.3.6 of [Hu15] which asserts x∗l Log (k+1) ∼ = ⊕k+1 j=0 Sym H, ∗ ∗ canonically; here our xl is the same operator as ρt t of [Hu15] 11.4.2. Now j=k+1 Ext1Y (H, ⊕0≤j≤k+1 Symj H(1)) ∼ = ⊕j=0 Ext1Y (H, Symj H(1)), (10.35) and let (x∗ Poll )k+1 be the class so defined in the term on the right of (10.35) for j = k + 1. Noting that

Ext1Y (H, Symk+1 H(1)) ∼ = Ext1Y (Ql , H∨ ⊗ Symk+1 H(1)) and that we have a unique decomposition: H∨ ⊗ Symk+1 H(1) ∼ = Symk+2 H ⊕ Symk H(1),

(10.36)

define contrl : Ext1Y (Ql , H∨ ⊗ Symk+1 H(1)) → Ext1 (Ql , Symk H(1)) (10.37)

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Don Blasius

via projection onto the second factor of (10.36). Since H∨ = H(−1), in view of (10.32), (10.36) and (10.37) we can regard contrl as a map k+1 contrl : Hlk+3 (E ×Y E k+1 , k + 2)(1,k+1) (E k , k + 1)(k) sgnk+1 → Hl sgnk , (10.38)

and it is not hard to see that (10.38) coincides with the map prl (10.28). Similarly, the classes denoted (x∗ Poll )k+1 ∈ Hlk+3 (E ×Y E k+1 , k + 2)(1,k+1) sgnk+1 , here and in 10.4.4, coincide. The claim follows.

10.4.6 Eisenstein symbol As we have already noted, the original construction of Eisenstein classes (k) k+1 in HM (E k , k +1)sgnk was given by Beilinson in [Be86] (see also [SS91]). The classes so obtained are called the Eisenstein symbols, and we can denote them EiskM,B (x) for x as in 10.3.1. Their relation to the classes defined in 10.3.5 is not known to the author. However, it is known that the images of these classes under the regulators coincide with those defined starting from polylogarithms in Section 10.3, i.e., we have ([HK99b], Appendix C, Theorem C.2.2) rl (EiskM,B (x)) = rl (EiskM (x)) = Eiskl (x)

(10.39)

and ([Hu15], Lemma 11.6.2) r∞ (EiskM,B (x)) = r∞ (EiskM (x)) = Eisk∞ (x).

(10.40)

These results follow from the rigidity principle ([Hu15], 11.4.7), according to which such a class is determined by its image under the residue homomorphism ([Hu15], 11.4.4) associated to the cusp at infinity, and the explicit computations of the images under the residue homomorphism of the classes. Part 1 (archimedean case), of the Main Theorem of [Hu15] relies on known results relating the r∞ (EiskM,B (x)) to Eisenstein series. It might be interesting to provide a unified treatment, proving the needed results directly from the r∞ (EiskM (x)).

References [Be86] Beilinson, A. 1986. Higher Regulators of Modular Curves. Contemporary Math., 55.

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[BL94] Beilinson, A., and Levin, A. 1994. The elliptic polylogarithm. Motives (Seattle, WA, 1991). Proc. Sympos. Pure Math., Part 2, 55, 123–190. Amer. Math. Soc., Providence, RI. [Hu15] Huber, A. 2015. The Comparison Theorem for the Soul´e-Deligne Classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 210–238. [HK99b] Huber, A., and Kings, G. 1999. Degeneration of l-adic Eisenstein classes and of the elliptic polylog. Invent. Math., 135, no. 3, 545–594. [Ja88] Jannsen, U. 1988. Continuous ´etale cohomology. Math. Annalen., 280, 207–245. [Ki99] Kings, G. 1999. K-theory elements for the polylogarithm of abelian schemes. J. Reine Angew. Math., 517, 103–116. [Ki15] Kings, G. 2015. Eisenstein classes, elliptic Soul´e elements and the `adic elliptic polylogarithm, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 239–296. [SS91] Schappacher, N., and Scholl, A. J. 1991. The boundary of the Eisenstein symbol. Math. Ann., 290, 303–320. [Sc88] Schneider, P. 1988. Introduction to the Beilinson Conjectures. Beilinsons’s Conjectures on Special Values of L-Functions. Perspectives in Mathematics, 4, 1–35, Academic Press, San Diego. [So84] Soul´e, C. 1984–1985. R´egulateurs, Sem. N. Bourbaki, exp. no , 644, 237–253. [Su15] Sujatha, R. 2015. K-theoretic Background, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 22–44. [Wi02] Wildeshaus, J. 2002. On the Eisenstein symbol. Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998). Int. Press Lect. Ser., 3, 291–414, I, Int. Press, Somerville, MA.

11 The Comparison Theorem for the Soul´e–Deligne Classes Annette Huber

Abstract There is an element in motivic cohomology of cyclotomic fields whose image both in Galois cohomology and Deligne cohomology can be described explicitly. We explain a proof of this comparison theorem using the elliptic polylogarithm and the degeneration from the modular curve to the cusps. The result is needed in the proofs of the Tamagawa number conjecture of Bloch and Kato for Dirichlet characters, in particular for the Riemann ζ-function. Corollary 11.7.2 is the input for the article [Co15] by Coates in the present volume.

11.1 Introduction 11.1.1 Statement of the Main Theorem In this expository note we are going to explain a proof of a comparison theorem crucial to the proof of the Tamagawa number conjecture of Bloch and Kato for Dirichlet characters (and hence for cyclotomic fields). Theorem 11.1.1 Let N ≥ 1 and k ≥ 1 be integers, µN the group of N -th roots of unity in C. For every α ∈ µN there is an element (see Definition 11.5.12) 1 bk+1 (α) ∈ HM (Q(µN ), k + 1)

such that: University of Freiburg, Germany. e-mail: [email protected]

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1. The image of bk+1 (α) under the Beilinson regulator (see Section 11.2) M 1 r∞ : H M (Q(µN ), k + 1) → (2πi)k R σ:Q(µN )→C

is given by Dir∞ (α) = (−Lik+1 (σ(α))σ , where Lik+1 is the polylogarithm function (see Definition 11.6.4). 2. For any prime l, the image of bk+1 (α) under the Soul´e regulator (see Section 11.2) 1 rl : H M (Q(µN ), k + 1) → H 1 (Q(µN ), Ql (k + 1))

is given by the modified Soul´e–Deligne element (see Definition 11.5.8) 1 c˜k+1 (α) N k k!   X 1 = k lim  (1 − β) ∪ (β N )∪k  . N k! ←− r

Dirl (bk+1 (α)) =

r

β l =α

r

Note that in this range of indices r∞ and rl induce isomorphisms after tensoring the domain with R or Ql , respectively. The simpler case k = 0 is also true mutatis mutandis. Main Theorem 11.1.1 was formulated as Conjecture 6.2 by Bloch and Kato in [BK90]. The case N = 1 and α = 1 (see Corollary 11.7.2) implies r? (bk+1 (1)) =

( −ζ(k + 1) 1 ˜k+1 (1) k! c

? = ∞, ? = l.

This special case is needed in the proof by Bloch and Kato for their Tamagawa number conjecture for the Riemann ζ-function. It also enters the exposition of their argument in the article [Co15] by Coates in the present volume. The general case is used by Fleckinger, Kolster and Nguyen Quang Do in their work on the Lichtenbaum conjecture for cyclotomic fields [KNF96], by Kings and the author in the proof of the conjecture for Dirichlet characters [HK03] and by Burns and Greither for the proof of the equivariant conjecture for cyclotomic fields [BG03]. In the present volume it enters as Theorem 9.5.3 the proof in the abelian case given by Nguyen Quang Do [Ng15].

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11.1.2 Proofs of the Main Theorem There are a couple of different proofs of the Main Theorem in the literature. We now want to explain how they are related and how the one we are giving here fits in. All are based on the polylogarithmic extension in different ways. The classical polylog is an object living on Gm r {1} = P1 r {0, 1, ∞}. It can be specialized at torsion points, i.e., at roots of unity, defining the so-called cyclotomic elements. These objects exist in a motivic, an l-adic and a Hodge theoretic incarnation. (The motivic and Hodge-theoretic incarnation of the cyclotomic element is also called Beilinson element. Its l-adic incarnation is the Deligne–Soul´e element.) Main Theorem 11.1.1 asserts that they are compatible under the regulator maps. There is an analogue, the elliptic polylog, on E r {0} where E/S is a relative elliptic curve. In the universal case, where S = Y (N ) is the modular curve, we can consider the degeneration of the elliptic polylog at the cusps and retrieve the classical polylog. Specializing the elliptic polylog at torsion sections we obtain the so-called Eisenstein elements. All objects are summed up in the commutative diagram: degenerates

elliptic polylog on E r {0} −−−−−−−→ classical polylog on Gm     roots of unityy6= 1 torsion sectionsy6= 0 Eisenstein classes on Y (N ) −−−−−−−→ degenerate

cyclotomic elements

Main Theorem 11.1.1 is about the lower right hand corner. We do not know how to prove the result directly. Rather motivic elements in the other corners are constructed and their images under regulators computed. The diagram then yields the Main Theorem. In all cases, the bulk of the computations is carried out on the sheaf theoretic level. This needs to be carried out both in the Hodge-theoretic and in the l-adic setting. Parts of the arguments are completely in parallel, but at some point explicit knowledge on the realization has to enter in order to get the explicit formulas. In detail: • Beilinson [Be84] (with additional work by Esnault [Es89] and Neukirch [Ne88]) proved assertion 1. (comparison of motivic and Hodge theoretic classes) directly. • The first full proof also including the l-adic part appeared in [HW98] by Wildeshaus and the author, based on an unpublished preprint by

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Beilinson and Deligne. The motivic classical polylog is constructed. Using the rich structure theory of polylog, its image under regulators is identified with an a explicit candidate in [Wi97] or [De89].(Indeed,this paper of Deligne started the whole subject.) The argument is explained (mostly in Hodge-theoretic language) in the series of talks [HW97]. This proof is conceptually very elegant, however, technically it is rather demanding. Relative cohomology, cohomology and algebraic K-theory of simplicial schemes, perverse sheaves and Hodge modules are used. One reason is that Gm is not proper. • Harder [Ha93] and Anderson were the first to realize that it should be possible to generate mixed Artin motives from the geometry of modular curves even though the splitting principle seems to say that the cohomology of the cusps is only a direct summand in the cohomology of the modular curve. In [HK99a] Kings and the author study this construction and give a motivic version (the ’cup-product construction’). It uses Beilinson’s motivic Eisenstein classes in [Be86] and gives an alternative proof of assertion 1. This is completed to a full proof of the Main Theorem in the sequel [HK99b]. The cup-product construction is given an interpretation as degeneration of sheaves. The study of the degeneration of the Eisenstein classes is reduced to the study of the degeneration of the elliptic polylog. (This is a bit more subtle than this sketch sounds.) The explicit formula comes from explicit knowledge of the classical l-adic polylog as in the first approach. The technical advantage is that the definition of the motivic element is simpler because elliptic curves are proper. On the other hand, we need to understand the geometry of modular curves. • The present approach as explained in Pune is another variant. It relies on the construction of the motivic elliptic polylog by Kings [Ki99]. The motivic Eisenstein elements are then defined by specializing the elliptic polylog at torsion sections. On the other hand, Kings in [Ki15] computes the degeneration of the l-adic Eisenstein elements explicitly. This computation relies on an explicit description of the elliptic polylog instead of the classical polylog. Again together with [HK99a] this gives a full proof of the Main Theorem. We expect that the Hodgetheoretic part could also be done directly with arguments similar to [Ki15]. This maze of arguments makes it a matter of taste which argument is the simplest and best.

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11.1.3 Structure of the paper Our exposition concentrates on the l-adic part, i.e., assertion 2. of the Main Theorem. We start by defining the elliptic polylogarithm in Section 11.3. In Section 11.4 it is specialized at torsion sections in order to define Eisenstein elements. These objects have a motivic counterpart, see [Bl15] in the present book. In Section 11.5 we degenerate the Eisenstein elements at the cusp ∞. In the motivic setting this is the cup-product construction. Its l-adic analogue has a direct sheaf theoretic description which can be computed explicitly, see [Ki15] in the present book. We then proceed to prove assertion 2. of the Main Theorem. The last but one section sketches the Hodge-theoretic counterpart and the proof of assertion 1. At the very end we specialize to the case of the Riemann ζ-function.

Acknowledgment Guido Kings prepared and delivered the actual talks at the Pune meeting when I was not able to attend myself. I am very thankful for his standing in. The present write-up owes a lot to his notes.

11.2 Notation Let X be a smooth variety over Q. Motivic cohomology of X is defined as (j)

i HM (X, j) = K2j−i (X)Q

the j-th eigenspace of the Adams operators on higher algebraic K-theory of X tensored by Q. Alternatively, we can define it as i HM (X, j) = HomDMgm (M (X), Q(j)[i])

where DMgm is Voevodsky’s triangulated category of geometric motives over Q and with rational coefficients (see [Vo00] Section 2). The two definitions are known to agree (e.g., [MVW06] Theorem 19.1 together with [Le94]). Let l be a prime. A smooth Ql -sheaf F on X is represented by a projective system (Fr )r≥0 of finite locally constant ´etale Z/lr -sheaves on X satisfying additional conditions. Equivalently, it can be described

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as a continuous representation of the ´etale fundamental group π1 (X, x ¯) on a finite dimensional Ql -vector space. We denote H i (X, F) its continuous l-adic cohomology in the sense of Jannsen [Ja88]. In the case X = SpecK where K is a number field, it agrees by [Ja88] Theorem (2.2) with continuous Galois cohomology ¯ H i (SpecK, F ) = H i (K, F) = H i (Gal(K/K), F) . Let Ql (j) be a smooth l-adic sheaf corresponding to the representation ¯ of Gal(Q/Q) on Ql given by the j-th power of the cyclotomic character. For j > 1 we have ! i

H (K, Ql (j)) = Ql ⊗Zl

lim H ←−

i

(K, µ⊗j lr )

.

r

The Soul´e regulator i rl : HM (X, j) → H i (X, Ql (j))

is defined as the Chern class (see [So81], also [Gi81] and [HW98] Proposition B.4.6). We are going to need pro-objects (F n )n≥0 with all F n smooth Ql sheaves. In this case we use the ad hoc definition H i (X, F) = lim H i (X, F n ) . ← − n We also need a general notion of residues. Let X be a smooth variety, j : U → X an open subvariety such that the closed complement i : Y → X is also smooth and of codimension 1. In this case j∗ has cohomological dimension 1 and we have a distinguished triangle of functors i∗ j∗ → i∗ Rj∗ → i∗ R1 j∗ [−1] . Definition 11.2.1 Let F and G be smooth Ql -sheaves on U and f ∈ ∗ ∗ 1 ExtqU (F , G). We define the residue of f in Extq−1 Y (i j∗ F , i R j∗ G) as res(f ) : i∗ j∗ F → i∗ Rj∗ F → i∗ Rj∗ G[q] → i∗ R1 j∗ G[q − 1] . In the case F = Ql , G = Ql (k+1) this turns by purity into the familiar res : H q (U, Ql (k + 1)) → H q−1 (Y, Ql (k)) . We denote by X(C) the complex valued points viewed as a complex manifold. Complex conjugation on C induces a continuous map F∞ :

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X(C) → X(C). Let j ≥ 0. Real Deligne cohomology of X is defined (see [Be84]) as hypercohomology of the complex of sheaves on X(C) i + HD (X/R, R(j)) = H i (X, [(2πi)j R → OX(C) → Ω1X(C) → . . . → Ωj−1 X(C) ])

where ΩpX(C) is the sheaf of holomorphic p-forms and + refers to invariants under the operation of complex conjugation both on X(C) and on the coefficients. If X = SpecK with K a number field and j > 0, we ` have X(C) = ? and hence σ:K→C

!+ 1 HD (SpecK/R, R(j))

=

M

j

C/(2πi) R

σ:K→C

!+ ∼ =

M

(2πi)

j−1

R

σ:K→C

with complex conjugation operating on the σ’s and on coefficients. (Note that C ∼ = (2πi)j R ⊕ (2πi)j−1 R.) The Beilinson regulator i i r∞ : HM (X, j) → HD (X/R, R(j))

is defined as the Chern class (see [Be84], also [Gi81] and [HW98] Theorem B.5.8).

11.3 The elliptic polylogarithm We introduce the polylogarithm on a relative elliptic curve in its l-adic version. Its construction is due to Beilinson and Levin, [BL94]. Our exposition follows largely [HK99b], Appendix A. Using [Bl15] we establish that it is motivic.

11.3.1 Logarithm Let S be a base scheme, say of finite type over Q. Let π : E → S be an elliptic curve and e : S → E the unit section. We work in the categories of smooth Ql -sheaves on E and S where l is a fixed prime. Put H = (R1 π∗ Ql )∨ .

(11.1)

Its stalk in s ∈ S is nothing but the Tate-module Vl Es of the fibre. The Leray spectral sequence for π∗ induces a short exact sequence 0 → Ext1S (Ql , H) → Ext1E (Ql , π∗ H) → HomS (Ql , H∨ ⊗ H) → 0 (11.2) which is split by e∗ .

The Comparison Theorem for the Soul´e–Deligne Classes Definition 11.3.1

217

Let Log(1) be the unique extension class 0 → π ∗ H → Log(1) → Ql → 0

such that its image in HomS (Ql , H ⊗ H∨ ) is the standard morphism Ql → H∨ ⊗ H and e∗ Log(1) is split. Let Log(k) := Symk Log(1) . Remark 11.3.1.1 If S is of finite type over a finite field or over Q, we can use weight arguments to deduce that Log(1) is even uniquely determined as a sheaf. The Log(k) organize into a projective system with transition maps Symk+1 Log(1) → Symk+1 (Log(1) ⊕ Ql ) → Symk Log(1) ⊗ Ql where the first map is induced by the identity and Log(1) → Ql and the second map is the canonical projection in the symmetric algebra of a direct sum. Hence, there are exact sequences 0 → π ∗ Symk+1 H → Log(k+1) → Log(k) → 0. Definition 11.3.2

(11.3)

The pro-sheaf Log := lim Log(k) ←−

is called logarithm sheaf on E/S. The splitting e∗ of Log(1) induces a splitting Y e∗ Log = Symk H . k≥0

Remark 11.3.2.1 Recall that in the case S = Speck with k a field, the sheaf H has the explicit description as Tate module ! H = Vl (E) = Ql ⊗

lim E[lr ] ←−

.

r

In a similar way, the Zl -sheaf underlying Log can be described explicitly, see [Ki15] Definition 12.4.14 and Theorem 12.4.15. The above definition is quick but looks arbitrary. However, Log is in fact characterized by a universal property. We are not going to need this universal property later on but explain it as background.

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Definition 11.3.3 A smooth Ql -sheaf F on E/S is called unipotent of length n if there exists a filtration F = A0 F ⊃ A1 F ⊃ · · · ⊃ An+1 F = 0 such that for all i = 0, . . . , n there is a smooth Ql -sheaf Gi on S such that Ai F/Ai+1 F ∼ = π ∗ Gi . Recall that a smooth Ql -sheaf is equivalent to a continuous representation of the fundamental group. The functor maps a sheaf to the stalk in a geometric point with the operation of monodromy. There is a version of this equivalence for unipotent sheaves and moreover relative to a base. This is what we want to formulate. Recall that the relative Tate module H is fibrewise the pro-l-part of the fundamental group of E tensored by Ql . ˆ H be the completion of the universal envelopDefinition 11.3.4 Let U ing algebra of the abelian Lie algebra H with respect to the augmentation ideal ˆ H → Ql ) , I = ker(U ˆ H = lim U (H)/I n . U ←− n

As pro-Ql -sheaves on S, we have Y ˆH ∼ U Symk H = e∗ Log . = k≥0

Theorem 11.3.5 (Beilinson–Levin) The functor F 7→ e∗ F induces an equivalence of the category of unipotent sheaves on E/S and the category ˆ H -modules on S. of U Proof. [HK99b] Theorem A.2.5, compare [BL94] 1.2.10 v). Log is the universal object for this equivalence of categories. It correˆ H as a U ˆ H -module. sponds to U Remark 11.3.5.1 There is a Hodge-theoretic counterpart of this statement for E/SpecC. The category of unipotent variations of Hodge strucˆ π of ture is equivalent to modules under the pro-unipotent completion U the group ring of the fundamental group of E (see [PS08] Chapter 8). The main step in the proof is the construction of a pro-Hodge structure ˆ π via Chen’s theory of iterated integrals. on U

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Corollary 11.3.6 Let f : E → E 0 be an isogeny of elliptic curves over S. Then H → H0 is an isomorphism and LogE ∼ = f ∗ LogE 0 . In particular, Log is invariant under translation by torsion sections t : S → E and there is a canonical isomorphism Y ρt : t∗ Log ∼ Symk H . = e∗ Log = k≥0

ˆ H on e∗ Log . As e∗ f ∗ = Proof. LogE is characterized by the action of U E 0∗ e the claim follows. If f is an isogeny which maps t to e, then ρt is given by the composition t∗ LogE ∼ = e∗ LogE 0 ∼ = e∗ LogE . The isomorphism can also be deduced from the explicit description of Log, see [Ki15] Proposition 12.4.3.

11.3.2 Polylogarithm As before let π : E → S be an elliptic curve with relative Tate module H. We will need the following computation: Lemma 11.3.7

We have i

R π∗ Log =

( 0

i 6= 2, 2

R π∗ Ql = Ql (−1)

i = 2.

Proof. This follows from the computation of Ri π∗ Log(k) and their transition maps in [HK99b] Lemma A.1.4. By abuse of notation we also write π for the restriction of π : E → S to E r 0. Lemma 11.3.8

We have

Ext1Er0 (π ∗ H, LogU (1)) = HomS (H, R1 π∗ LogU (1)) Y = HomS (H, Symk H) . k≥1

Proof. By the Leray spectral sequence 0 → Ext1S (H, π∗ (LogU (1))) → Ext1Er0 (π ∗ H, LogU (1)) → HomS (H, R1 π∗ (LogU (1))) → 0 . The first term vanishes because π∗ LogU (1) = π∗ Log(1) = 0

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by Lemma 11.3.7. This implies the first equality. By purity, there is an exact sequence 0 → R1 π∗ Log(1) → R1 π∗ (LogU (1)) → e∗ Log → R2 π∗ Log(1). By Lemma 11.3.7, R1 π∗ Log(1) = 0 and R2 π∗ Log(1) = Ql . Moreover, the morphism e∗ Log → Ql is the projection to the zeroth component. Definition 11.3.9

Let

Pol ∈ Ext1Er0 ((π ∗ H)Er0 , LogU (1)) Q be the preimage of the natural inclusion H → k≥1 Symk H under the identification in Lemma 11.3.8. This class is called the (small) elliptic polylogarithmic extension. By abuse of notation we also denote Pol the sheaf representing the extension class. It is unique up to unique isomorphism by weight arguments (which are available because we work over a base of finite type over Q). By definition, we have an exact sequence on E r 0 0 → LogU (1) → Pol → π ∗ H → 0 . It is compatible with base extensions. Remark 11.3.9.1 As mentioned in Remark 11.3.2.1 the pro-Ql -sheaf Log has an integral model LogZl . The sheaf Pol does not allow such an explicit description. However, it is true for a modified polylogarithm on E r E[c] where c is an appropriate positive integer, see [Ki15] Definition 12.4.8 and loc. cit. Theorem 12.4.21. The effect is well known in Iwasawa theory: certain classes only become integral after removing a suitable Euler factor. Remark 11.3.9.2 As explained by Blasius in [Bl15] Section 10.2.8 there is a system of classes k+3 Polk+1 (E ×S V k+1 , k + 2)(1) sgnk+1 M ∈H

compatible under residue maps. (For the definition of V and the character sgn see loc. cit. sections 10.2.2 and 10.2.3.) By loc. cit Theorem 10.1.2, the motivic polylog is mapped to the l-adic polylog under the Soul´e regulator. We also refer to loc. cit. for the details of the identification of the cohomology groups.

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11.4 Eisenstein classes We specialize the geometric setting to the case where the base is the universal one, i.e., the modular curve of elliptic curves. Pull-back of the elliptic polylogarithm at torsion sections defines the Eisenstein classes. Our main aim is to characterize theses classes by their residues at the cusps of the modular curve. The notation and normalizations are the same as in [Kt04] and [Ki15]. Note that they differ from [HK99b]. All sheaves are Ql -sheaves as before.

11.4.1 Geometric setup We fix an integer N ≥ 3. Let ζN = exp(2πi/N ) ∈ C, B = SpecQ(ζN ). Let l be a prime number. Let Y (N ) be the modular curve parametrizing elliptic curves with full level-N -structure viewed as a variety over Q. Remark 11.4.0.3

We have Y (N)(C) = (Z/N Z)∗ × Γ(N )\H

where H = {τ ∈ C|Imτ > 0} is the upper half plane and Γ(N ) ⊂ SL2 (Z) the congruence subgroup of matrices congruent to the identity mod N . The point (ν, τ ) corresponds to the elliptic curve C/Zτ + Z with levelN-structure ντ 1 (Z/N Z)2 → E[N ] ; (a, b) 7→ a +b . N N Moreover, let j : Y (N ) → X(N ) be the compactification and Cusp = X(N ) r Y (N ) the subscheme of cusps. The standard-N -gon over B with level-N -structure Z/N Z × Z/N Z → Z/N × Gm via (a, b) 7→ (a, ζ b ) defines a section ∞ : B → Cusp. We have a unique GL2 (Z/N )-equivariant isomorphism a Cusp ∼ SpecQ(µN ) (11.4) = ∗ GL2 (Z/N )/±( ∗ 0 1) mapping ∞ to the component of the identity matrix. Let E/Y (N ) be the universal elliptic curve and e : Y (N ) → E be the unit section. Recall from Section 11.3.1, Equation (11.1), the smooth Ql -sheaf H on Y (N ), the relative Tate module of E/Y (N ).

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Remark 11.4.0.4 Over the complex numbers, H can be described as a representation of the fundamental group Γ(N ) of Y (N)(C). It operates by the standard operation of Γ(N ) ⊂ SL2 (Z) on Q2l . Similarly, the symmetric powers Symk H are the standard representation spaces of dimension (k + 1), i.e., as on homogeneous polynomials in two variables of degree k. These local systems are well known from the theory of modular forms of weight k + 2.

11.4.2 The l-adic Eisenstein class Recall from Definition 11.3.9 the elliptic polylogarithm Pol ∈ Ext1Er0 (π ∗ H, LogEr0 (1)) , where we specialize to S = Y (N ) the modular curve. Let t : Y (N ) → E[N ] be a non-zero torsion-section. Recall also from Corollary 11.3.6 that there is a natural isomorphism Y %t : t∗ Log → e∗ Log = Symk H . k≥0

Hence we have %t t∗ Pol ∈ Ext1Y (N ) (H,

Y

Symk H) ∼ = Ext1Y (N) (Ql , H∨ ⊗

k≥0

Y

Symk H) .

k≥0

We write (ρt t∗ Pol)k ∈ Ext1Y (N) (H, Symk H) = Ext1Y (N ) (Ql , H∨ ⊗ Symk H) for the component in degree k. For k ≥ 1 let contr : H∨ ⊗ Symk H → Symk−1 H   k X 1 ˆ j . . . ⊗ hk  (h∨ ⊗ h1 ⊗ . . . ⊗ hk ) 7→  h∨ (hj )h1 ⊗ . . . h k + 1 j=1 be the contraction. Lemma 11.4.1

contr induces an isomorphism on Ext1Y (N ) (Ql , ·).

Proof. See [Wi97] 3.19 b). Use the Leray spectral sequence and weight ¯ arguments for the cohomology of Symk H over Q. This yields an element 1 contr(%t t∗ Pol)k+1 ) ∈ Ext1Y (N ) (Ql , Symk H(1)) = Het (Y (N ), Symk H(1)).

More generally:

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Let k ≥ 0, ψ ∈ Q[E[N ] r 0]. Then we put X Eiskl (ψ) = −N k−1 ψ(t)contr(%t (t∗ Pol)k+1 )

Definition 11.4.2

t∈E[N ]r0

as element of 1 Ext1Y (N ) (Ql , Symk H(1)) = Het (Y (N ), Symk H(1)) .

It is called the l-adic Eisenstein element for ψ. Remark 11.4.2.1 These Eisenstein classes were first constructed in motivic cohomology by Beilinson [Be86] by different arguments. Their realization in Deligne cohomology is described by Eisenstein series. Remark 11.4.2.2 Our definition is identical to the one in [Ki15] Definition 12.4.6. An explicit description in terms of elliptic units is then given in loc. cit. Theorem 12.4.22. Recall from [Bl15] Section 10.3.5, the motivic Eisenstein elements k+1 Eisk (ψ) ∈ HM (E k , k + 1)(k) sgnk

(see loc. cit. for the definition of the character sgn). The l-adic regulator map (i.e., the Chern class) i i rl : H M (X, n) → Het (X, Ql (n))

X a scheme , i, n ∈ Z

allows to compare it with the l-adic Eisenstein elements. Theorem 11.4.3

There is a natural isomorphism

k+1 HM (E k , k

k 1 + 1)(k) sgnk → Het (Y (N ), Sym H(1)) .

For all ψ ∈ Q[E[N ]r 0], the l-adic regulator maps the motivic Eisenstein element Eisk (ψ) to Eiskl (ψ). Proof. This is [Bl15] Section 10.4.4 and Theorem 10.1.1. The motivic polylog is mapped to the l-adic polylog. By functoriality, the construction of the motivic Eisenstein classes is mapped to the same construction in terms of l-adic cohomology. In sheaf theoretic terms it translates into Definition 11.4.2.

11.4.3 The residue of the Eisenstein class Recall the compactification j : Y (N ) → X(N ) with complement i : Cusp → X(N ) and the distinguished cusp ∞ : B → Cusp with B = SpecQ(µN ).

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Definition 11.4.4 sition

The residue homomorphism is given as the compo-

1 res : Het (Y (N ), Symk H(1)) → H 0 (Cusp, R1 j∗ Symk H(1))

= H 0 (Cusp, Ql ). We write res∞ for ∞∗ ◦ res. Recall the Bernoulli polynomial Bk defined by the expansion ∞

X tetx tk = Bk (x) . t e −1 x! k=0

Let k ≥ 0, N ≥ 3 and ψ ∈ Q[E[N ] r 0], then     k X N t 1  res∞ (Eiskl (ψ)) = Bk+2  ψ(t) (k + 2)k! N

Proposition 11.4.5

t∈E[N]r0

where t1 is the first component of t under the fixed identification E[N ] ∼ = (Z/N Z)2 and where we identify x ∈ R/Z with its representative in [0, 1). Proof. There are a number of different proofs for this fact. It is originally due to Beilinson and Levin [BL94] prop. 2.2.3. A different proof is given by Wildeshaus in [Wi97] cor. III 3.26. A third version can be found in [HK99b] Appendix C. (It is also due to Beilinson and Levin from an earlier preprint version of [BL94].) Finally, there is a new l-adic argument by Kings in [Ki15] Theorem 12.1.1. Remark 11.4.5.1 Using the equivariance of the situation under the standard operation of GL2 (Z/N ) this easily allows to write down the residue not only in the distinguished cusp ∞ but in all cusps, see [HK99b] Theorem C.1.1. We omit the statement because it would mean introducing more notation.

11.4.4 Rigidity 1 We are going to show that elements of Het (Y (N ), Symk H(1)) are uniquely determined by their residues at all cusps. We do not need this fact in the sequel but it allows to compare our definition of the Eisenstein classes to others in the literature.

Lemma 11.4.6

For k ≥ 0 we have

j∗ Symk H|Cusp = Ql (k),

R1 j∗ Symk H = Ql (−1)Cusp .

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Proof. This is well known from the theory of modular forms. For a detailed argument see [HK99b] Appendix B or [Ki15] Corollary 12.5.4. Proposition 11.4.7 (Rigidity) morphism for k > 0.

The residue homomorphism is an iso-

Proof. We use the Leray spectral sequence for a : Y (N ) → B. As Y (N ) is an affine curve, it consists of short exact sequences 1 1 0 → Het (B, a∗ Symk H(1)) → Het (Y (N ), Symk H(1)) 0 → Het (B, R1 a∗ Symk H(1)) → 0 .

As the sheaf Symk H(1) has no global sections for k > 0 (e.g., check on the local system over C), the term on the left vanishes. The vector space underlying R1 a∗ Symk H(1) is l-adic cohomology over the algebraic 1 ¯ induces a short closure Het (Y (N )Q¯ , Symk H(1)). The residue map over Q exact sequence 1 1 0 → Het (X(N )Q¯ , j∗ Symk H(1)) → Het (Y (N )Q¯ , Symk H(1)) 0 → Het (CuspQ¯ , R1 j∗ Symk H(1)) → 0

This sequence is well known from the theory of modular forms. The term on the left hand side is the cuspidal cohomology corresponding to cusp forms. As a Galois module it is known to be pure of weight −k − 1. More abstractly, j∗ Symk H(1) is a perverse sheaf on X(N ). It is pure of weight −k −2, hence its first cohomology is pure of weight −k −1. Either way, it does not have any global sections over B. Together with Lemma 11.4.6 we have established 1 0 0 Het (Y (N ), Symk H(1)) ∼ (B, Het (CuspQ¯ , Ql )) ∼ = Het = H 0 (Cusp, Ql ) .

11.5 The cup-product construction Our aim is to use Eisenstein elements in order to construct elements in 1 the space HM (Q(µN ), Q(k +1)) and in parallel in H 1 (Q(µN ), Ql (k +1)). We continue in the geometric setting of the last section. We fix N ≥ 3 and k > 0. Let B = SpecQ(µN ).

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11.5.1 The motivic cup-product construction Definition 11.5.1

Let φ∞ ∈ Q[E[N ] r 0] such that res∞ Eisk (φ∞ ) = 1

and that the residue vanishes at all other cusps. Remark 11.5.1.1 By the explicit formula for the residues of Eisenstein classes, it is possible to write down φ∞ explicitly in terms of the horosperical map, see [SS91] equation (7.3.1) or [HK99b] Definition 1.2.3 and [HK99a] Lemma 7.6. (Because we use a different normalization, a sign change is necessary in this formula.) In particular such an element exists. We are not going to need this explicit shape. Let E k = E ×Y (N) · · · ×Y (N ) E

k factors).

By abuse of notation we also write π : E k → Y (N ). Recall from [Bl15] Section 10.3.5 the motivic Eisenstein element k+1 Eisk (ψ) ∈ HM (E k , k + 1)

for ψ ∈ Q[E[N ] r {0}]. We consider the composition ∪Eisk (φ∞ )

k+1 HM (E k , k + 1) −−−−−−−→ π

res

∗ 2k+2 2 1 HM (E k , 2k + 2) −→ HM (Y (N), k + 2) −−−∞ → HM (B, k + 1) .

Definition 11.5.2 We put

Let ψ ∈ E[N ] r 0 be such that res∞ Eisk (ψ) = 0. 1 Dir(ψ) ∈ HM (B, k + 1))

the image of Eisk (ψ) under the above composition. Recall that B = SpecQ(µN ). Hence this yields elements in motivic cohomology of cyclotomic fields. We call them Dirichlet motives because their Hodge regulator is related to Dirichlet series. Remark 11.5.2.1 Harder and Anderson (see [Ha93]) were the first to point out that the splitting principle notwithstanding, the geometry of modular curves can be used to construct mixed Dirichlet motives. In [HK99a] their construction was translated into the above cup-product construction. The special shape of φ∞ and the assumption on the residue for ψ is not needed in the cup-product construction. It only becomes necessary in

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its translation to a degeneration in Theorem 11.5.5. However, it follows from the explicit computation in Theorem 11.5.13 that all elements of 1 HM (B, k + 1) can already be obtained by these special choices. We now turn to the l-adic version of the construction in terms of cohomology on Y (N ). We consider the composition ∪Eisk (φ∞ )

H 1 (Y (N ), Symk H(1)) −−−−l−−−→ H 2 (Y (N ), Symk H(1) ⊗ Symk H(1)) r

∞ → H 2 (Y (N ), Ql (k + 2)) −− → H 1 (B, Ql (k + 1))

induced by the Weil pairing H ⊗ H → Ql (1). Definition 11.5.3 We put

Let ψ ∈ E[N ] r 0 be such that res∞ Eisk (ψ) = 0. Dirl (ψ) ∈ H 1 (B, Ql (k + 1))

the image of Eisl (ψ) under the above composition. Lemma 11.5.4

We have rl (Dir(ψ)) = Dirl (ψ) .

Proof. This follows from the compatibility of motivic and l-adic Eisenstein classes (see Theorem 11.4.3) and the fact that rl is compatible with cup-products. Finally, cup-product on E k is compatible with cup-product for cohomology on Y (N ) with coefficients, see [HK99b] Lemma 2.1.5.

11.5.2 Sheaf theoretic interpretation We are going to give an alternative sheaf theoretic description of Dirl (ψ). Recall that Eisl (ψ) ∈ H 1 (Y (N ), Symk H(1)) = Ext1Y (N ) (Ql , Symk H(1)) , Dirl (ψ) ∈ H 1 (B, Ql (k + 1)) = Ext1B (Ql , Ql (k + 1)) . Theorem 11.5.5 Let ψ ∈ Q[E[N ] r 0] be such that res∞ Eisk (ψ) = 0. Let Eisk (ψ) be the sheaf representing the extension class Eiskl (ψ) ∈ Ext1Y (N) (Qp , Symk H(1)). Then the sequence 0 → ∞∗ j∗ Symk H(1) → ∞∗ j∗ Eisk (ψ) → Ql → 0 is exact and it represents Dirl (ψ) via the isomorphism ∞∗ j∗ Symk H(1) = Ql (k + 1) of Lemma 11.4.6.

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This is [HK99b] Theorem 2.1.4. The proof of this theorem will take the rest of this section. Note that the short exact sequence 0 → Symk H(1) → Eisk (ψ) → Ql → 0 induces a long exact sequence 0 → ∞∗ j∗ Symk H(1) → ∞∗ j∗ Eisk (ψ) → Ql → ∞∗ R1 jSymk H(1) → and the boundary map is nothing but the residue map res∞ . So the first statement is obvious. The point of the theorem is the identification with Dirl (ψ). The situation is described in abstract terms in the following lemma of homological algebra. Lemma 11.5.6 Let X be a smooth variety, j : U → X an open subvariety with closed complement j : Y → X smooth and of pure codimension 1. Let f ∈ Ext1U (Ql , F ) and g ∈ Ext1U (F, G) where F and G are Ql sheaves on U . Let Ef be a sheaf representing f . We assume that res(f ) ∈ HomY (i∗ j∗ Ql , i∗ R1 j∗ F) (see Definition 11.2.1) vanishes. Then the residue of g ∪ f ∈ Ext2U (Ql , G) is given by the push-out of the short exact sequence 0 → i∗ j∗ F → i∗ j∗ Ef → Ql → 0 induced by f via the map i∗ j∗ F → i∗ R1 j∗ G induced by g. Proof. See [HK99b] Lemma 2.1.6 where the simple argument is given in terms of morphisms in the derived category. Proof of Theorem 11.5.5.. We specialize the lemma to the situation of the theorem and put U = Y (N ) , F = Symk H(1) , G = Ql (k + 2) , f = Eiskl (ψ) , g = Eiskl (φ∞ ) , or more precisely, using the Weil pairing Eisk (φ∞ )⊗id

g : Ql ⊗ Symk H(1) −−−l−−−−−→ Symk H(1) ⊗ Symk H(1) → Ql (k + 1) .

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Hence res∞ (Eiskl (ψ) ∪ Eiskl (φ∞ )) is given by the push-out of the short exact sequence 0 → ∞∗ j∗ Symk H(1) → ∞∗ j∗ Eisk (ψ) → Ql → 0 via the map Ql (k + 1) = ∞∗ j∗ Symk H(1) → ∞∗ R1 j∗ Ql (k + 2) = Ql (k + 1) induced by Eiskl (φ∞ ). The latter map is the identity because it is equal to res∞ Eiskl (φ∞ ) = 1.

11.5.3 The Main Theorem, l-adic part In order to state the l-adic part of the Main Theorem precisely, we need to define the Soul´e–Deligne elements, also known as cyclotomic elements. As before, for any positive integer M let µM be the group of M -th roots of unity. We fix integers N ≥ 1, k > 0 and a prime l. r Fix r ≥ 1 for the moment and let β ∈ C such that β l = α. Recall that the Kummer map ∂ : Q(µlr N )× → H 1 (Q(µlr N ), µlr ) is the connecting morphism of the short exact sequence [lr ]

1 → µlr → Gm −−→ Gm → 1 in Galois cohomology of Q(µlr N ). In particular, the cyclotomic unit 1−β gives rise to an element in Galois cohomology. Note that the field is chosen such that the coefficient module is trivial. The number β N can be viewed as an element of H 0 (Q(µlr N ), µlr ). Taking cup-products, we get the class ∂(1 − β) ∪ (β N )∪k ∈ H 1 (Q(µlr N ), µ⊗k+1 ). lr Let N ≥ 1 be fixed and α ∈ µN . We put
ck+1,r (α) = coresQ(µlr N )/Q(µN ) ∂(1 − β) ∪ (β N )∪k

Definition 11.5.7

r

in H 1 (Q(ζN ), µ⊗k+1 ) where β is such that β l = α. lr As Soul´e realized, the ck+1,r (α) are compatible for varying r. Definition 11.5.8 as

Let α ∈ µN . The Soul´e–Deligne element is defined

ck+1 (α) = lim ck+1,r (α) ∈ H 1 (Q(µN ), Zl (k + 1)) . ←r−

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The modified Soul´e–Deligne element is defined as follows: if l|N , then c˜k+1 (α) = ck+1 (α) ; if l - N , then c˜k+1 (α) = (1 − lk Frl )−1 ck+1 (α) ∞ X −i = lki ck+1 (αl ) . i=0

Note that the series converges. The modified Soul´e–Deligne element also has a unified description. Lemma 11.5.9 ([HK03] Lemma 3.1.7) The image of the element c˜k+1 (α) in H 1 (Q(µlr N ), µ⊗k+1 )Gal(Q(µlr N )/Q(µN )) is given by lr X
c˜k+1,r (α) = ∂(1 − β) ∪ (β N )∪k . r

β l =α

In particular, our elements c˜k+1 (α) are the same as the ones defined by Kings in [Ki15] in the present volume. Remark 11.5.9.1 It is not apparent from the notation, but the elements ck+1 (α) and c˜k+1 (α) depend (mildly) on the choice of level N . They are not compatible under restriction of fields. The Soul´e–Deligne elements are the ones used in Iwasawa theory. They were introduced by Soul´e in [So81]. As we will see, the modified ones are the natural classes in motivic cohomology. They were introduced by Deligne in [De89] in the presentation of Lemma 11.5.9. They occur in the same shape in the comparison conjecture 6.2 in [BK90]. The elegant presentation of our definition is due to Kato in [Kt93]. We now turn to the construction of the Beilinson elements in motivic cohomology. Definition 11.5.10 ψuk :=

For u ∈ Z/N r 0 let

X −1 N2 (0, u) + (v, u) ∈ Q[E[N ] r 0] . k−1 k+1 N 1−N v6=0

Lemma 11.5.11 res∞ Eisk (ψuk ) = 0 . Proof. This follows from Proposition 11.4.5 and the well-known distribution properties of the Bernoulli polynomials.

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This allows the definition of our main object. u Definition 11.5.12 Let N ≥ 3, k ≥ 1 and α = ζN 6= 1. The Beilinson element is defined as 1 bk+1 (α) = Dir(ψuk ) ∈ HM (Q(µN ), k + 1) .

For N = 2, α = −1 we put 1 coresQ(i)/Q bk+1 (−1) . 2 For α = 1 and N ≥ 1 we put bk+1 (−1) =

bk+1 (1) =

2k bk+1 (−1) . 1 − 2k

Remark 11.5.12.1 In contrast to the modified Soul´e–Deligne elements, the Beilinson elements are compatible under restriction to bigger fields, i.e., the element bk+1 (α) ∈ Q(ζN ) is mapped to bk+1 (α) ∈ Q(ζN M ). This is easily seen from the description of its image under one of the regulators. Theorem 11.5.13 Let N ≥ 3, k ≥ 1 and l a prime. For α ∈ µN r{1}, the image of the Beilinson element under the Soul´e regulator is given by 1 c˜k+1 (α) ∈ H 1 (Q(µN ), Ql (k + 1)) . N k k! Assertion 2. of Main Theorem 11.1.1 holds. rl (bk+1 (α)) =

u Proof. We have α = ζN with u ∈ Z/N r 0. By Lemma 11.5.4 the image of Dir(ψuk ) is

rl (Dir(ψuk )) = Dirl (ψuk ) . By Theorem 11.5.5 it is represented by ∞∗ j∗ Eisk (ψuk ) . By [Ki15] Theorem 12.1.3 it is equal to X −1 b ψuk (0, b)˜ ck+1 (ζN ). N k! b∈Z/N r0

In the definition of ψuk only the first summand (with b = u) contributes and hence −1 −1 u Dirl (ψuk ) = c˜k+1 (ζN ). N k! N k−1 This is assertion 2. of the Main Theorem for N ≥ 3 and α 6= 1. The

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case N = 2, α = −1 follows from the case N = 4 because c˜k+1 (−1) ∈ H 1 (Q, Ql (k+1)) restricts by definition to 2k c˜k+1 (−1) ∈ H 1 (Q(i), Ql (k + 1)). For N = 1 and α = 1 it is elementary to check from the description in Lemma 11.5.9 that c˜k+1 (−1) + 2k c˜k+1 (1) = c˜k+1 (1) . (Note that the element for −1 has level 2 whereas the element for 1 has level 1.) The case α = 1 and N > 1 follows from this by restriction to the bigger field.

11.6 The Hodge theoretic story In order to prove the first assertion of our Main Theorem, we need to work in the Hodge theoretic setting. We only sketch the arguments. We replace the setting of smooth Ql -sheaves by admissible variations of mixed Q-Hodge structure on the complex manifold X(C). See [PS08] for this notion and its properties.

11.6.1 The construction We start again with the universal elliptic curve E/Y (N ) with N ≥ 3 as in Section 11.4. Definition 11.6.1

Put H = (R1 π∗ Q)∨

as variation of mixed Hodge structure on Y (N )(C). Remark 11.6.1.1 This variation is arguably the most basic and best studied example of a variation of Hodge structures. In the fibre at τ ∈ Y (N )(C), we have H1 (Eτ , Q) with its natural pure Hodge structure of weight −1. The underlying local system is the standard representation of Γ(N ) ⊂ SL2 (Z) on a vector space of dimension 2. The constructions of Log (see Definition 11.3.2) and Pol (see Definition 11.3.9) from H and the assertions about its monodromy work in the same way as in the l-adic situation, see Section 11.3.1. By pull-back

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of Pol along a torsion section we again obtain the Eisenstein class for ψ ∈ Q[E[N ] r 0] X

Eisk∞ (ψ) = −N k−1

ψ(t)contr(ρt (t∗ Pol)k+1 )

t∈E[N ]r0

as an element of 1 HD (Y (N)/R, Symk H(1)) .

As in the l-adic case, we have r∞ (Eisk (ψ)) = Eisk∞ (ψ) . Lemma 11.6.2 For k > 0, the element Eisk∞ (ψ) agrees with the element defined by Beilinson in [Be86]. Proof. By rigidity (see Lemma 11.4.7; also true in the Hodge theoretic setting by the same proof), the Eisenstein class is uniquely determined by its residue in the cusps. As the class is motivic, the residue can be computed motivically. By compatibility under rl it suffices to compute the residues l-adically for some prime l. The was achieved in Proposition 11.4.5. It agrees with the residues of the original Eisenstein symbol of Beilinson’s. Remark 11.6.2.1 There is an explicit description of Eisk∞ (ψ) in terms of real analytic Eisenstein classes, see [Be86] 3.1.7 and 2.2. Recall the element φ∞ ∈ Q[E[N ] r 0] from Definition 11.5.1 . Definition 11.6.3 We put

Let ψ ∈ Q[E[N ]r0 be such that res∞ Eisk∞ (ψ) = 0.

Dir∞ (ψ) = res∞ (Eis∞ (ψ) ∪ Eis∞ (φ∞ )) . By construction r∞ (Dir(ψ)) = Dir∞ (ψ)) . It is an element of 1 HD (SpecQ(µN )/R, R(k + 1)) =

M

(2πi)k R .

σ:Q(µN )→C

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11.6.2 The Main Theorem, Hodge theoretic part Definition 11.6.4 Let k ≥ 1. The k-th polylogarithm function is defined as the analytic continuation of Lik (z) =

∞ X zn nk n=1

to P1 (C) r {0, 1, ∞} = C∗ r {1}. Note that Li1 = − log(1 − z). Remark 11.6.4.1 The starting point of the subject was Deligne’s insight in [De89] that the monodromy properties of these polylogarithm functions can be read as saying that they define a variation of mixed Q-Hodge structure on C∗ r {1}. Theorem 11.6.5 Let N ≥ 3, k ≥ 1. For α ∈ µN r {1}, the image of the Beilinson element (see Definition 11.5.12) under the Beilinson regulator is given by M u r∞ (bk+1 (α)) = (−Lik+1 (σ(ζN ))σ ) ∈ (2πi)k R . σ:Q(µN )→C

Assertion 1. of Main Theorem 11.1.1 holds. u Proof. (Sketch) Recall that r∞ (bk+1 (α)) = Dir∞ (ψuk ) for α = ζN and k a certain ψu . Using explicit formulas for the cup-product of classes in Deligne cohomology, the element can be computed directly from the description of Eis∞ (ψ) by Eisenstein series. This is carried out in [HK99a] Section 7. Alternatively, as in Section 11.5.2, the cup-product construction can be reinterpreted as degeneration of an extension class with residue 0. This class is described by an Eisenstein series, see [Be86] or [Den97]. To compute the degeneration means to evaluate the Eisenstein series at the cusp. This is assertion 2. of the Main Theorem for N ≥ 3 and α 6= 1. The case N = 2 and α = −1 follows from any even N because the Beilinson elements are compatible under restriction. For α = 1 note that there is an elementary identity of Dirichlet series

Lik+1 (−1) + Lik+1 (1) =

1 Lik+1 (1) 2k

which allows to reduce to the case α = −1.

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11.7 The case of the Riemann ζ-function We want to specialize our Main Theorem to the case of the field Q. Let p be a prime. We start by a lemma on cyclotomic elements. Lemma 11.7.1

If m is odd, then cm (1) = sm,p

with cm (1) as in Definition 11.5.8 and sm,p as defined by Coates in [Co15] Section 3.6. Proof. We choose a compatible system of primitive pr -th roots of unity ζpr . By definition, cm (1) = limr cm,r (1) with ←− cm,r (1) = coresQ(ζpr )/Q (1 − ζpr ) ∪ ζp⊗m−1 ∈ H 1 (Q, µ⊗m r pr ) . It m is odd, then ζp⊗m−1 is an element of H 0 (Q(ζpr )+ , µ⊗m−1 ). By conr pr struction, sm,p = limr sm,p,r with ←− ⊗m−1 sm,p,r = coresQ(ζpr )+ /Q) (1 − ζpr )(1 − ζp−1 . r ) ∪ ζpr

By the projection formula ⊗m−1 coresQ(ζpr )/Q(ζpr )+ (1 − ζpr ) ∪ ζp⊗m−1 = (1 − ζpr )(1 − ζp−1 . r r ) ∪ ζpr

Hence cm,r (1) = sm,p,r for all r. Corollary 11.7.2 For each odd integer m > 1, there exists bm ∈ K2m−1 (Q) ⊗ Q such that r∞ (bm ) = (m − 1)!ζ(m) and rp (bm ) = −˜ cm (1) = −(1 − pm−1 )−1 sm,p with sm,p as defined by Coates in [Co15] Section 3.6. Remark 11.7.2.1

This is precisely Theorem 3.6.2 in [Co15].

Proof. We have to specialize our Main Theorem 11.1.1 to the case N = 1, α = 1 and m = k + 1 odd. Recall that for m > 1 we have (m)

1 HM (SpecQ, m) = K2m−1 (Q)Q

= K2m−1 (Q)Q .

Moreover, as Q has only one embedding into Q, the Beilinson regulator r∞ takes values in (2πi)m−1 R = R. We put (using the Beilinson element from Definition 11.5.12) bm := −(m − 1)!bm (1) ∈ K2m−1 (Q)Q .

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By Main Theorem 11.1.1, we have r∞ (bm ) = (m − 1)!Lim (−1) = (m − 1)!

∞ X 1 = (m − 1)!ζ(m) . m n n=1

Again by the Main Theorem 11.1.1 and using Definition 11.5.8 and Lemma 11.7.1, we have rp (bm ) = −˜ cm (1) = (1 − pm−1 )−1 cm (1) = (1 − pm−1 )−1 sm,p .

References [BG03] Burns, D., and Greither, C. 2003. On the equivariant Tamagawa number conjecture for Tate motives. Invent. Math., 153, no. 2, 303–359. [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I. Progr. Math., 86, 333–400. Birkh¨ auser Boston, Boston, MA. [BL94] Beilinson, A., and Levin, A. 1994. The elliptic polylogarithm. Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., Part 2, 55, 123–190. Amer. Math. Soc., Providence, RI. [Be84] Beilinson, A. A. 1984. Higher regulators and values of L-functions. Current Problems in Mathematics, 24, 181–238, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. [Be86] Beilinson, A. A. 1986. Higher regulators of modular curves. Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983). Contemp. Math., 55, 1–34. Amer. Math. Soc., Providence, RI. [Bl15] Blasius, D. 2015. The motivic Eisenstein classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 193–209. [Co15] Coates, J. 2015. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 45–64 [De89] Deligne, P. 1989. Le groupe fondamental de la droite projective moins trois points. Galois groups over Q (Berkeley, CA, 1987). Math. Sci. Res. Inst. Publ., 16, 79–297. Springer, New York. [Den97] Deninger, C. 1997. Extensions of motives associated to symmetric powers of elliptic curves and to Hecke characters of imaginary quadratic fields. Arithmetic geometry (Cortona, 1994), 99–137. Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge. [Es89] Esnault, H. 1989. On the Loday symbol in the Deligne-Beilinson cohomology. K-Theory, 3, no. 1, 1–28. [Gi81] Gillet, H. 1981. Riemann-Roch theorems for higher algebraic K-theory. Adv. in Math., 40, no. 3, 203–289.

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[HK99a] Huber, A., and Kings, G. 1999. Dirichlet motives via modular curves. Ann. Sci. Ecole Norm. Sup., 32, no. 3, 313–345. [HK99b] Huber, A., and Kings, G. 1999. Degeneration of l-adic Eisenstein classes and of the elliptic polylog. Invent. Math., 135, no. 3, 545–594. [HK03] Huber, A., and Kings, G. 2003. Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet Characters. Duke Math. J., 119, no. 3, 393–464. [HW97] Huber, A., and Wildeshaus, J. May 1-4, 1997. The Classical Polylogarithm. Abstract of a series of lectures given at the workhsop on polylogs in Essen. arXiv:1210.2358 [math.NT] [HW98] Huber, A., and Wildeshaus, J. 1998. Classical motivic polylogarithm according to Beilinson and Deligne. Doc. Math. 3, 27–133; and Correction Doc. Math., 3, 297–299. [Ha93] Harder, G. 1993. Eisensteinkohomologie und die Konstruktion gemischter Motive. Lecture Notes in Mathematics, 1562, Springer-Verlag, Berlin. [Ja88] Jannsen, U. 1988. Continuous ´etale cohomology. Math. Ann., 280, no. 2, 207–245. [KNF96] Kolster, M., Nguyen Quang Do, T., and Fleckinger, V. (1996). Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J., 84, no. 3, 679–717. [Ki99] Kings, G. 1999. K-theory elements for the polylogarithm of abelian schemes. J. Reine Angew. Math., 517, 103–116. [Ki08] Kings, G. 2008. Degeneration of polylogarithms and special values of L-functions for totally real fields. Doc. Math., 13, 131–159. [Ki15] Kings, G. 2015. Eisenstein classes, elliptic Soul´e elements and the `adic elliptic polylogarithm, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 239–296. [Kt93] Kato, K. 1993. Iwasawa theory and p-adic Hodge theory. Kodai Math. J., 16, no. 1, 131. [Kt04] Kato, K. 2004. p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithm´etiques. III. Ast´erisque, 295. [Le94] Levine, M. 1994. Bloch’s higher Chow groups revisited. Proc. of the Congress of K-theory (Strasbourg, 1992), Ast´erisque No., 226, 10, 235– 320. [MVW06] Mazza, C., Voevodsky, V., and Weibel, C. 2006. Lecture notes on motivic cohomology. Clay Mathematics Monographs, 2. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA. [Ne88] Neukirch, J. 1988. The Beilinson conjecture for algebraic number fields. Beilinson’s conjectures on special values of L-functions. Perspect. Math., 4, 193247 Academic Press, Boston, MA. [Ng15] Nguyen Quang Do, T. 2015. On the determinantal approach to the Tamagawa Number Conjecture, in The Bloch-Kato Conjecture for the

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Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 154–192. [PS08] Peters, C., and Steenbrink, J. 2008. Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 52. Springer-Verlag, Berlin. [SS91] Schappacher, N., and Scholl, A. 1991. The boundary of the Eisenstein symbol. Math. Ann., 290, no. 2, 303–321, Erratum: Math. Ann. 290, no. 4, 815. [So81] Soul´e, C. 1981. On higher p-adic regulators. Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980). Lecture Notes in Math., 854, 372–401, Springer, Berlin-New York. [Vo00] Voevodsky, V. 2000. Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories. Ann. of Math. Stud., 143, 188–238, Princeton Univ. Press, Princeton, NJ. [Wi97] Wildeshaus, J. 1997. Realizations of polylogarithms. Lecture Notes in Mathematics, 1650. Springer-Verlag, Berlin.

12 Eisenstein Classes, Elliptic Soul´e Elements and the ` -Adic Elliptic Polylogarithm Guido Kings

Abstract In this paper we study systematically the `-adic realization of the elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads to a description of the elliptic polylogarithm in terms of elliptic units. As an application we prove a precise relation between `-adic Eisenstein classes and elliptic Soul´e elements. This allows to give a new proof of the formula for the residue of the `-adic Eisenstein classes at the cusps and the formula for the cup-product construction in [HK99], which relies only on the explicit description of elliptic units. This computation is the main input in the proof of Bloch–Kato’s compatibility conjecture 6.2 needed in the proof of Tamagawa number conjecture for the Riemann zeta function.

Introduction The purpose of this paper is twofold: on the one hand we prove a new and precise relation between `-adic Eisenstein classes and elliptic Soul´e elements using a description of the integral `-adic elliptic polylogarithm in terms of elliptic units. On the other hand this relation will be used to give a new proof for the cup-product construction formula, which is the main result of [HK99] and is the main input in [Hu15] to obtain a proof of Bloch–Kato’s compatibility conjecture 6.2. This new proof uses only elementary properties of elliptic units. The explicit description of the integral `-adic elliptic polylogarithm in University of Regensburg, Germany. e-mail : [email protected]

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terms of elliptic units was already one of the main results in the paper [Ki01]. There we used an approach via one-motives to treat the logarithm sheaf. But the main application of the `-adic elliptic polylogarithm is in the context of Iwasawa theory, which makes it desirable to approach the elliptic polylogarithm systematically in this context. That such an approach is possible is already suggested in the ground-breaking paper [BL94]. In Iwasawa theory Kato, Perrin-Riou and Colmez pointed out the usefulness to work with ‘Iwasawa cohomology’, which is continuous Galois cohomology with values in an Iwasawa algebra. We generalize this idea to treat families of Iwasawa modules under a family of Iwasawa algebras. The main example for this is the family of Iwasawa algebras on the moduli scheme of elliptic curves, where one has in each fibre the Iwasawa algebra of the Tate module of the corresponding elliptic curve. It is the fundamental idea of Soul´e [So81] that twisting of units can be used to produce interesting cohomology classes. Already in Kato’s paper [Kt93] it is implicit that this twisting is related to the Iwasawa cohomology. Later Colmez used this explicitly in [Co98], where he used moment maps of Q` -measure algebras. For our approach it is crucial to develop this further by constructing the moment map at finite level. We show that in the cyclotomic case one obtains the elements defined and studied by Soul´e and Deligne. Work by Soul´e in the CM elliptic case and Kato’s work in [Kt04] suggest that one should carry out Soul´e’s twisting construction also in the modular curve case to obtain elliptic Soul´e elements. One of the main results in this paper is that these elliptic Soul´e elements are essentially the `-adic Eisenstein classes in [HK99]. With the general theory of sheaves of Iwasawa modules, we obtain a concrete description of the elliptic polylogarithm in terms of the norm compatible elliptic units defined and studied by Kato [Kt04]. This gives strong ties of the elliptic polylogarithm to recent developments in Iwasawa theory and also allows many explicit computations with the `-adic elliptic polylogarithm. As an application of the concrete description of the elliptic polylogarithm, we give a new proof of the residue computation for `-adic Eisenstein classes on the moduli scheme for elliptic curves (Corollary 12.5.10). A second application is the evaluation of the cup-product construction used in [HK99] (and explained in this volume in [Hu15]) to obtain elements in the motivic cohomology of cyclotomic fields and to prove Conjecture 6.2 in [BK90]. The approach taken here does not need any

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computations of the cyclotomic polylogarithm as in [HK99]. It relies only on the concrete evaluation of the elliptic units at the cusps. An overview of the main results in this paper is given in Section 12.1. Acknowledgements: It is a pleasure to thank the organizers of the Pune workshop for invitation and the audience for their interest and their questions. Further I would like to thank several people for helpful comments on an earlier version of this paper: Annette Huber’s detailed reading led to many improvements and corrections. Ren´e Scheider pointed out a missing factor in a formula. The referee pointed out several improvements in the exposition and suggested to work with symmetric tensors in the construction of the moment map. This led to a complete rewriting of the earlier version.

Notations We fix an integer N ≥ 3 and let Y (N ) the moduli space of elliptic curves E with a full level N -structure α : (Z/N Z)2 ∼ = E[N ]. We denote by π : E → Y (N ) the universal elliptic curve. We let X(N ) be the smooth compactification of Y (N ) and denote by j : Y (N ) ,→ X(N ) the open immersion. If we fix an N -th root of unity ζN := e2πi/N ∈ C and consider the Tate curve Eq b with the level structure α : (Z/N Z)2 → Eq [N ] given by (a, b) 7→ q a ζN . This induces a map of schemes SpecQ(ζN )((q 1/N )) → Y (N ), which extends to SpecQ(ζN )[[q 1/N ]] → X(N ) and a hence a map ∞ : SpecQ(ζN ) → X(N ), whose image we call the cusp ∞. Define ´etale sheaves on Y (N ) by Hr := (R1 π∗ Z/`r Z)∨ ∼ = R1 π∗ Z/`r Z(1) H := (R1 π∗ Z` )∨ ∼ = R1 π∗ Z` (1) HQ := (R1 π∗ Q` )∨ ∼ = R1 π∗ Q` (1)

(12.1)

`

∨

where (.) denotes the Z/`r Z, Z` and Q` dual respectively. We denote by Symk Hr , Symk H and Symk HQ` the k-th symmetric power as Z/`r Z-, Z` - and Q` -modules respectively. In the same way we denote by TSymk Hr , TSymk H and TSymk HQ` the functor of symmetric ktensors as Z/`r Z-, Z` - and Q` -modules respectively. Note that there is

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a canonical map Symk Hr → TSymk Hr ,

(12.2)

which extends to a homomorphism of graded algebras Sym· Hr → TSym· Hr and similarly for H and HQ` . As we will not only deal with `-adic sheaves, we work in the bigger abelian category of inverse systems F = (Fr )r≥1 of ´etale sheaves modulo Mittag-Leffler-zero systems (which means to work in the procategory) and define the continuous ´etale cohomology in the sense of [Ja88]. This means that H i (S, F ) is the i-th derived functor of F 7→ lim H 0 (S, Fr ). ←− r

More generally, one defines R i π∗ F for a morphism π : S → T to be the i-th derived functor of F 7→ limr π∗ Fr . For `-adic sheaves, we also consider Ext-groups ←− ExtiS (F , G ), which are the right derived functors of HomS (F , −). Of crucial importance is the following lemma: Lemma 12.0.3 Let F = (Fr )r≥1 be a projective system with H 0 (S, Fr ) finite, then H 1 (S, F ) = lim H 1 (S, Fr ). ←− r

Proof. This follows from [Ja88, Lemma 1.15, Equation (3.1)] as the H 0 (S, Fr ) satisfy the Mittag-Leffler condition. The quotient category of the `-adic sheaves (or Z` -sheaves) by the torsion sheaves is the category of Q` -sheaves. In the case of projective systems of Q` -sheaves F = (Fr )r≥0 we use the ad hoc definitions H i (S, F ) := lim H i (S, Fr ) ←− r

ExtiS (G , F ) := lim ExtiS (G , Fr ), ←− r

where G is just a Q` -sheaf.

(12.3)

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12.1 Statement of the main results For better orientation of the reader we give an overview of the main results in this paper and the strategy and main ingredients of the proof.

12.1.1 The residue at ∞ of the Eisenstein class We identify sections t : Y (N ) → E with elements in (Z/N Z)2 via the universal level-N -structure on E. For any (0, 0) 6= t ∈ (Z/N Z)2 one can define a so-called Eisenstein class EiskQ` (t) ∈ H 1 (Y (N ), Symk HQ` (1)) (cf. Definition 12.4.6). It is convenient to introduce the following notation: For any map ψ : (Z/N Z)2 \ {(0, 0)} → Q we put X EiskQ` (ψ) := ψ(t)EiskQ` (t). t6=e

It is shown in [Bl15] that EiskQ` (ψ) is in fact the image of a class in motivic cohomology under the regulator map. We are interested in the image of EiskQ` (ψ) under the residue map res∞ : H 1 (Y (N ), Symk HQ` (1)) → H 0 (∞, Q` ) ∼ = Q` as defined in Definition 12.5.5. The following result was first proved in [BL94] by a completely different method: Theorem 12.1.1 (See Corollary 12.5.10) res∞ (EiskQ` (ψ)) =

−N k (k + 2)k!

One has

X

ψ(a, b)Bk+2 ({

(a,b)∈(Z/N Z)2 \{(0,0)}

a }), N

a where Bk+2 denotes the k + 2 Bernoulli polynomial and { N } is the repa resentative in [0, 1[ of N .

12.1.2 Evaluation of the cup-product construction For two maps φ, ψ : (Z/N Z)2 \ {(0, 0)} → Q we can consider the cup-product EiskQ` (φ) ∪ EiskQ` (ψ) ∈ H 2 (Y (N ), Symk HQ` ⊗ Symk HQ` (2)).

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The cup-product pairing HQ` ⊗ HQ` → Q` (1) induces a pairing Symk HQ` ⊗ Symk HQ` → Q` (k) and we can consider the image of the cup-product EiskQ` (ψ) ∪ EiskQ` (φ) in H 2 (Y (N ), Q` (k + 2)). EiskQ` (ψ) ∪ EiskQ` (φ) ∈ H 2 (Y (N ), Q` (k + 2)). Let res∞ : H 2 (Y (N), Q` (k + 2)) → H 1 (∞, Q` (k + 1)) be the edge morphism in the Leray spectral sequence for Rj∗ using the isomorphism ∞∗ R1 j∗ Q` (k + 2) ∼ = Q` (k + 1). Definition 12.1.2 Let φ∞ , ψ : (Z/N Z)2 \ {(0, 0)} → Q be two maps and suppose that res∞ (EiskQ` (φ∞ )) = 1 and res∞ (EiskQ` (ψ)) = 0. Then Dir` (ψ) := res∞ (EiskQ` (ψ) ∪ EiskQ` (φ∞ )) ∈ H 1 (∞, Q` (k + 1)) is called the cup-product construction (compare [Hu15, Definition 11.5.2.]). Note that Dir` (ψ) does not depend on the choice of φ∞ (as follows from the formula in Theorem 12.6.1). The main result of this paper is: Theorem 12.1.3 (see Corollary 12.6.10) Q be a map such that

Let ψ : (Z/N Z)2 \ {(0, 0)} →

res∞ (EiskQ` (ψ)) = 0. Then one has Dir` (ψ)) =

−1 N k!

X

b ψ(0, b)e ck+1 (ζN ) ∈ H 1 (∞, Q` (k + 1)),

06=b∈Z/N Z

b where e ck+1 (ζN ) is the modified cyclotomic Soul´e–Deligne element from Definition 12.3.4.

It is explained in [Hu15] how this theorem settles the compatibility conjecture 6.2 in [BK90]. The main idea in this paper (building upon our former work [Ki01]) is to describe a Z` -version of EiskQ` (t) as Soul´e’s twisting construction applied to elliptic units. Then all explicit computations with the Eisenstein classes are reduced to computations with the elliptic units.

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12.1.3 Eisenstein classes and elliptic units We explain how the Eisenstein classes are related to elliptic units and in particular how one can define these classes integrally. For this we need to introduce some notation. Recall that the Eisenstein class is associated to a non-zero N -torsion section t : Y (N ) → E[N ]. Let ` be a prime number. We define the E[`r ]torsor E[`r ]hti on the modular curve Y (N ) by the cartesian diagram E[`r ]hti −−−−→ E[`r N ]    [`r ] pr,t y y t

Y (N ) −−−−→ E[N ]. Definition 12.1.4

Define the ´etale sheaf Λr (Hr hti) on Y (N ) by Λr (Hr hti) := pr,t∗ Z/`r Z.

If t = e is the identity section we write Λr (Hr ). The sheaves Λr (Hr hti) form an inverse system with respect to the trace map Λr+1 (Hr+1 hti) → Λr (Hr hti) and we denote the resulting pro-system by Λ(H hti) := (Λr (Hr hti))r≥1 .

(12.4)

For t = e we write Λ(H ) := (Λr (Hr ))r≥1 . Remark. The sheaves Λ(H hti) form the main example of sheaves of Iwasawa modules mentioned in the title of this paper. The connection is explained in Lemma 12.2.10. Note that by Lemma 12.0.3 we have H 1 (Y (N ), Λ(H hti)(1)) ∼ H 1 (Y (N ), Λr (Hr hti)(1)) = lim ←− r

∼ H 1 (E[`r ]hti, Z/`r Z(1)). = lim ←−

(12.5)

r

Fix an auxiliary integer c > 1, which is prime to 6`N . Then Kato has defined a norm-compatible unit c ϑE on E \ E[c] (cf. Theorem 12.3.8). Note that for an N -torsion point t 6= e one has E[`r ]hti ⊂ E \ E[c] by our condition on c. Thus, we can restrict c ϑE to an invertible function on E[`r ]hti. The Kummer map (see 12.2.16) gives a class hti ESc,r := ∂r (c ϑE ) ∈ H 1 (E[`r ]hti, Z/`r Z(1))

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and by the norm-compatibility we can define: Definition 12.1.5

Let

ESchti := lim ∂r (c ϑE ) ∈ H 1 (S, Λ(H hti)(1)). ←− r

In 12.2.12 we define a moment map momkt : Λ(H ) → TSymk H which gives rise to a map momk

t H 1 (S, Λ(H hti)(1)) −−−−→ H 1 (S, TSymk H (1)).

(12.6)

We show in Proposition 12.2.21, inspired by a result of Colmez [Co98]: Proposition 12.1.6 (see 12.2.21) c ek (t)

The element

:= momkt (ESchti ) ∈ H 1 (S, TSymk H (1))

coincides with Soul´e’s twisting construction (see 12.2.18) applied to the norm compatible elliptic units c ϑE and is called the elliptic Soul´e element. hti

Consider the image of momkt (ESc ) in H 1 (S, TSymk HQ` (1)). The isomorphism Symk HQ` → TSymk HQ` induces H 1 (S, Symk HQ` (1)) ∼ = H 1 (S, TSymk HQ` (1)),

(12.7)

which allows us to consider k

mom ] t (ESchti ) :=

1 momkt (ESchti ) ∈ H 1 (S, Symk HQ` (1)). Nk

Theorem 12.1.7 (see Theorem 12.4.22) equality

With the above notation the

1 −1 k ] t (ESchti ) = k−1 (c2 EiskQ` (t) − c−k EiskQ` ([c]t)) c ek (t) = mom k N N holds in H 1 (S, Symk HQ` (1)). In particular, if c ≡ 1 mod N one has c ek (t)

= −N (c2 − c−k )EiskQ` (t).

This is the desired relation between EiskQ` (t) and the elliptic Soul´e element.

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12.2 Sheaves of Iwasawa modules and the moment map In this section we consider sheaves of Iwasawa modules and define the moment map. As a motivation we start by looking at the case of modules under the Iwasawa algebra.

12.2.1 Iwasawa algebras Fix a prime number `. Let X be a totally disconnected compact topological space of the form X = lim Xr ←− r

with Xr finite discrete. We denote by C(X, Z` ) := {f : X → Z` | f continuous} the continuous Z` -valued functions on X together with the sup-norm || − ||∞ . Definition 12.2.1

The space of Z` -valued measures on X is Λ(X) := HomZ` (C(X, Z` ), Z` ).

We also write Λr (X) := HomZ` (C(X, Z` ), Z/`r Z) for the Z/`r Z-valued measures on X. For each µ ∈ Λ(X) we write Z f µ := µ(f ) X

and for x ∈ X we let δx ∈ Λ(X) be the Dirac distribution characterized by δx (f ) = f (x). As every continuous function in C(X, Z` ) is the uniform limit of locally constant functions, we have Λ(X) = lim HomZ` (C(Xr , Z` ), Z` ) = lim Λ(Xr ) = lim Λr (Xr ). ←− ←− ←− r

r

r

For a continuous map φ : X → Y one has a homomorphism φ! : Λ(X) → Λ(Y )

(12.8)

defined by (φ! µ)(f ) := µ(f ◦ φ). If U ⊂ X is open compact one has Λ(X) ∼ = Λ(U ) ⊕ Λ(X \ U ).

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Define b Λ(X)⊗Λ(Y ) := lim(Λ(Xr ) ⊗Z` Λ(Yr )) ←− r

then one has a canonical isomorphism b Λ(X × Y ) ∼ ). = Λ(X)⊗Λ(Y Let now X = H = limr Hr be a profinite group. Then Λ(Hr ) is the ←− group algebra of Hr and Λ(H) inherits a Z` -algebra structure. This algebra structure can also be defined directly by using the convolution of measures µ ∗ ν := mult! (µ ⊗ ν), where mult : H × H → H is the group multiplication. As δg ∗ δh = δgh the map δ : H → Λ(H)× , h 7→ δh is a group homomorphism. Definition 12.2.2 Λ(H) with the above Z` -algebra structure is called the Iwasawa algebra of H. The following situation will frequently occur in the applications in this paper. Suppose that q

0→H →G− →T →0 is an exact sequence of profinite groups with T finite discrete. Define for t∈T Hhti := q −1 (t) S so that G = t∈T Hhti and each Hhti is an H-torsor, i.e., has a simply transitive H-action. Then M Λ(G) ∼ Λ(Hhti) = t∈T

is a Λ(H)-module and Λ(Hhti) is a free Λ(H)-module of rank one.

12.2.2 The moment map In this section we consider the profinite group H ∼ = Zd` and we write Hr :=H ⊗Z` Z` /`r Z` ∼ = (Z` /`r Z` )d ∼ Qd . HQ :=H ⊗Z Q` = `

`

`

The moment map will be a Z` -algebra homomorphism ·

\ H, Λ(H) → TSym

(12.9)

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249

·

\ H is the completion of the Z` -algebra of symmetric tensors where TSym with respect to the divided powers of the augmentation ideal. We start be recalling some facts about the algebra of symmetric tensors TSym· H. We remark right away that the right framework for the moment map is the divided power algebra Γ· H, which in our case is isomorphic to TSym· H. As we are interested in the relation with the symmetric algebra in the end, we found it more intuitive to work with TSym· H. The algebra TSym· H is graded M TSym· H = TSymk H k≥0

and for each h ∈ H one has the symmetric tensor h[k] := h⊗k ∈ TSymk H. This gives a divided power structure on TSym· H and one has the formulas X (g + h)[k] = g [m] h[n] m+n=k

h

[m] [n]

h

(12.10)

(m + n)! [m+n] = h . m!n!

The map H → TSym1 H, h 7→ h[1] is an isomorphism. By the universal property of the symmetric algebra this induces an algebra homomorphism Sym· H → TSym· H,

(12.11)

which is an isomorphism after tensoring with Q` . From the isomorphism Γ· H ∼ = TSym· H it follows directly that · TSym H is compatible with base change (TSym· H) ⊗Z` Z/`r Z ∼ = TSym· Hr and with direct sums TSym· (H ⊕ H) ∼ = TSym· H ⊗ TSym· H. If (e1 , . . . , ed ) is a basis of H, then [n1 ]

(e1

[n ]

· · · ed d | n1 + . . . + nd = k)

is a basis of TSymk H. Note that under the homomorphism Symk H → TSymk H one has [n1 ]

en1 1 · · · end d 7→ n1 ! · · · nd !e1

[nd ]

· · · ed

.

Let H ∨ := HomZ` (H, Z` ) be the dual Z` -module then one has a canonical isomorphism Symk H ∨ ∼ = (TSymk H)∨ .

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Let TSym+ H := note by

L

k>0

TSymk H be the augmentation ideal. We de-

·

\ H := lim TSym· H/(TSym+ H)[n] TSym ←− n

the completion of TSym· H with respect to the divided powers of the aug· · \ Hr and TSym \ HQ` mentation ideal. Similarly, we also denote by TSym the completions with respect to the divided powers of the augmentation ideal. Lemma 12.2.3

One has ·

·

\ H∼ \ Hr . TSym TSym = lim ←− r

Proof. As TSymk H is a free Z` -module, one has TSym· H/(TSym+ H)[n] ∼ TSym· Hr /(TSym+ Hr )[n] = lim ←r− for all n ≥ 1. Taking the inverse limit over n, the result follows. Proposition 12.2.4

There is a unique homomorphism of Z` -algebras ·

\ H, mom : Λ(H) → TSym P which maps δh 7→ k≥0 h[k] and is called the moment map. It is the limit mom = limr momr of moment maps at finite level ←− ·

\ Hr momr : Λr (Hr ) → TSym X X µr 7→ ( µr (h)h[k] ). k≥0 h∈Hr

Let (e1 , . . . , ed ) be a basis of H and (x1 , . . . , xd ) the dual basis considered as Z` -valued functions xi : H → Z` . In terms of measures the moment map is given by ! Z X X [n1 ] [nd ] n1 nd mom(µ) = ( x1 · · · xd µ)e1 · · · ed . k≥0

n1 +...+nd =k

H

The projection onto the k-th component is denoted by momk : Λ(H) → TSymk H and by momkr : Λr (Hr ) → TSymk Hr respectively.

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Remark. The formula for the moment map in terms of measures justifies the name. For the application to sheaves of Iwasawa algebras it is the formula on finite level which is important. ·

\ Hr in the proposition is Proof. The map momr : Λr (Hr ) → TSym the algebra homomorphism induced by the group homomorphism h 7→ P [k] and the universal property of the group algebra Λr (Hr ). Takk≥0 h · \ H. ing the inverse limit gives mom : Λ(H) → TSym Write µ ∈ Λ(H) as µ = limr µr with µr ∈ Λr (Hr ). The dual basis ←− x1 , . . . , xd considered as Z/`r Z-linear maps xi : Hr → Z/`r Z induce ni polynomial functions xi : Hr → Z/`r Z and by definition Z

n

Hr

n

n1 1 d d xn 1 · · · xd µr = µr (x1 · · · xd ) =

X

µr (h)x1 (h)n1 · · · xd (h)nd .

h∈Hr

If we observe that [n1 ]

P

n1 +...+nd =k

x1 (h)n1 ···xd (h)nd e1

[nd ]

···ed

=(x1 (h)e1 +...+xd (h)ed )[k] =h[k]

we get X

[n1 ]

µr (xn1 1 · · · xnd d )e1

[n ]

· · · ed d =

n1 +...+nd =k

X

µr (h)h[k]

h∈Hr

= momkr (µr ). This implies that for the measure µ = limr µr ∈ Λ(H) we get ←− momk (µ) = lim momkr (µr ) ←− r Z X [n ] [n ] = lim ( xn1 1 · · · xnd d µr )e1 1 · · · ed d , ←− H r n1 +...+nd =k

r

which implies the desired formula for mom(µ). Note that the moment map is functorial. If ϕ : H → G is a group homomorphism one has a commutative diagram ·

mom \ H Λ(H) −−−−→ TSym    \· φ! y yTSym (φ) ·

mom \ G Λ(G) −−−−→ TSym

It is a fact from classical Iwasawa theory that Λ(H) is isomorphic to a power series ring over Z` in d variables. In particular, it is a regular local ring. Let R

I(H) := ker(Λ(H) −−H → Z` )

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be the augmentation ideal. Then from the regularity of Λ(H) it follows that I(H)k /I(H)k+1 ∼ = Symk H and the k-th moment map factors momk : Λ(H) → Λ(H)/I(H)k+1 → TSymk H. Lemma 12.2.5

The k-th moment map induces k

mom Symk H ∼ = I(H)k /I(H)k+1 ,→ Λ(H)/I(H)k+1 −−−−→ TSymk H,

which is just the canonical map. Proof. The morphism momk maps an element (δh1 − 1) · · · (δhk − 1) ∈ I(H)k to the corresponding product taken in TSym· H. This implies the result. Consider again the exact sequence q

0→H →G− →T →0 of profinite groups with T a finite N -torsion group. Definition 12.2.6

For the H-torsors Hhti = q −1 (t) define momkt : Λ(Hhti) → TSymk H

to be the composition [N ]!

momk

momkt : Λ(Hhti) −−→ Λ(H) −−−−→ TSymk H, where [N ] : G → G is the N -multiplication, which factors through H. Remark. This moment map is not independent of the choice of N such that t is an N -torsion point. To remedy this defect consider the composition momk

t Λ(Hhti) −−−−→ TSymk H → TSymk HQ` .

Definition 12.2.7

(12.12)

The modified moment map k

mom ] t : Λ(Hhti) → Symk HQ` is the map (12.12) composed with the inverse of the isomorphism Symk HQ` ∼ = TSymk HQ` divided by N k , i.e., 1 momkt . Nk The following lemma is obvious from the definition. k

mom ] t :=

Lemma 12.2.8 and not on N .

k

The modified moment map mom ] t depends only on t

Eisenstein Classes and Elliptic Soul´e Elements

253

´ 12.2.3 Etale sheaves of Iwasawa modules Consider a projective system of finite ´etale schemes pr : Xr → S and let X := limr Xr . We denote by λr : Xr+1 → Xr the finite ´etale maps ←− in the projective system. We denote by Xr the ´etale sheaf associated to Xr and define an ´etale sheaf on S by Λr (Xr ) := pr∗ Z/`r Z.

(12.13)

The trace map with respect to λr induces a morphism of sheaves λr∗ Z/`r+1 Z → Z/`r+1 Z which gives rise to Λr+1 (Xr+1 ) = pr∗ λr∗ Z/`r+1 Z → pr∗ Z/`r+1 Z → pr∗ Z/`r Z = Λr (Xr ), where the last map is reduction modulo `r . Definition 12.2.9

Define an inverse system of ´etale sheaves on S by Λ(X ) := (Λr (Xr ))r≥1 ,

with the above transition maps. Remark. Note that Λ(X ) is not an `-adic sheaf in general. This construction is functorial in the sense that for a morphism of inverse systems (fr : Xr → Yr )r≥1 the trace map induces fr! : Λr (Xr ) → Λr (Yr )

(12.14)

and hence a map f! : Λ(X ) → Λ(Y ). We want to explain in which sense Λ(X ) is a sheafification of the space of measures Λ(X). Let us choose a geometric point s : SpecK → S and let Xr,s be the stalk of Xr at s. We consider Xr,s as a finite set with a continuous Galois action. Immediately from the definitions we have: Lemma 12.2.10

The stalk of Λr (Xr ) at s is Λr (Xr )s ∼ = Λr (Xr,s ).

In particular, if we define Λ(X )s := limr Λr (Xr )s and Xs := limr Xr,s ←− ←− we get Λ(X )s ∼ = Λ(Xs ), which is the space of measures on Xs with a Galois action.

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Guido Kings pr

In the case where each Xr = Hr −→ S is a finite ´etale group scheme over S, so that H := limr Hr is a pro-´etale group scheme, the sheaves ←− Λr (Hr ) become sheaves of Z/`r Z-algebras. In fact one has (pr × pr )∗ Z/`r Z ∼ = Λr (Hr ) ⊗ Λr (Hr ) and the group multiplication induces a ring structure on Λr (Hr ).

12.2.4 The case of torsors The following situation will occur very frequently in this paper. Suppose we have an inverse system of finite ´etale group schemes on S qr

0 → Hr → Gr −→ T → 0

(12.15)

where T = Tr for all r is an N -torsion group. For each section t : S → T we define an Hr -torsor Hr hti by the cartesian diagram Hr hti −−−−→   pr,t y

Gr   y

(12.16)

t

−−−−→ T.

S Denote by H := lim Hr , ←− r

G := lim Gr and Hhti := lim Hr hti ←− ←− r

r

the associate pro-´etale group schemes and by Hr , Gr , Hr hti and H , G , H hti the associated sheaves. In particular one has an exact sequence q

0→H →G− →T →0

(12.17)

and a cartesian diagram Hhti −−−−→   pt y S

G   y

(12.18)

t

−−−−→ T.

Each Λr (Hr hti) is a Λr (Hr )-module of rank one and consequently the same is true for the Λ(H )-module Λ(H hti). The sheaves Λ(H hti) are sheaves of Iwasawa modules under the sheaves of Iwasawa algebras Λ(H ).

Eisenstein Classes and Elliptic Soul´e Elements

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12.2.5 The sheafified moment map In this section we describe a sheaf version of the moment maps from Proposition 12.2.4. Let pr : Hr → S be a finite ´etale group scheme which is ´etale locally of the form Hr ∼ = (Z/`r Z)d for d ≥ 1. As in (12.16) we consider Hr -torsors pr,t : Hr hti → S associated to an exact sequence 0 → Hr → Gr → T → 0 and to an N -torsion section t of T . As T is an N -torsion group by assumption, the N -multiplication map [N ] : Gr → Gr factors through Hr and we get a map of schemes [N]

τr,t : Hr hti ,→ Gr −−→ Hr .

(12.19)

We interpret this as a section τr,t ∈ H 0 (Hr hti, p∗r,t Hr ). Definition 12.2.11

We let [k]

τr,t ∈ H 0 (Hr hti, p∗r,t TSymk Hr ) be the k-th tensor power τr,t . This will also be viewed as a map of sheaves [k]

τr,t : Z/`r Z → pr,t∗ p∗r,t TSymk Hr . Recall that for sheaves F , G on Hr hti one has the morphism (given by the projection formula and adjunction) pr,t,! F ⊗ pr,t∗ G ∼ = pr,t,! (F ⊗ p∗r,t pr,t∗ G ) → pr,t,! (F ⊗ G )

(12.20)

and that pr,t,! = pr,t∗ as pr,t is finite. Definition 12.2.12

The sheafified moment map

momkr,t : Λr (Hr hti) → TSymk Hr is the composition (p := pr,t ) [k]

id⊗τr,t

(12.20)

p∗ Z/`r Z −−−−→ p∗ Z/`r Z ⊗ p∗ p∗ TSymk Hr −−−−→ tr p∗ (Z/`r Z ⊗ p∗ TSymk Hr ) ∼ → TSymk Hr , = p∗ p∗ TSymk Hr −

where tr is the trace map with respect to p.

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From the definition it follows that the moment maps momkr,t are compatible with respect to the trace map for varying r. Definition 12.2.13

Let momkt : Λ(H hti) → TSymk H

be the inverse limit of momkr,t . We also let k

mom ] t :=

1 momkt : H 1 (S, Λ(H hti)(1)) → H 1 (S, Symk HQ` (1)) Nk

be the composition of the map induced by the inverse of the canonical isomorphism

1 momkt Nk

in cohomology with

H 1 (S, Symk HQ` (1)) ∼ = H 1 (S, TSymk HQ` (1)). On stalks the sheafified moment map coincides with the one defined in Definition 12.2.6: Lemma 12.2.14 moment map

Let s be a geometric point of S, then the stalk of the

(momkr,t )s : Λr (Hr hti)s → TSymk Hr,s coincides with the moment map momkr,t defined in Definition 12.2.6. Proof. We have Λr (Hr hti)s = Λr (Hr htis ) and we let X µr = mx δx x∈Hr htis

be an element in Λr (Hr htis ). We identify (p := pr,t ) p∗ p∗ TSymk Hr,s ∼ = Λr (Hr,s ) ⊗ TSymk Hr,s so that (p∗ Z/`r Z ⊗ p∗ p∗ TSymk Hr )s ∼ = Λr (Hr,s ) ⊗ Λr (Hr,s ) ⊗ TSymk Hr,s . [k]

With this identification the image of µr under id ⊗ τr,t is given by X X [k] ( mx δx ) ⊗ ( δy ⊗ τr,t (y)). (12.21) x∈Hr htis

y∈Hr htis

The homomorphism (12.20)

Λr (Hr,s ) ⊗ Λr (Hr,s ) ⊗ TSymk Hr,s −−−−→ Λr (Hr,s ) ⊗ TSymk Hr,s

Eisenstein Classes and Elliptic Soul´e Elements maps the element in (12.21) to X [k] mx δx ⊗ τr,t (x) =

X

x∈Hr htis

257

[k]

µr (x)δx ⊗ τr,t (x)

x∈Hr htis

and the trace of this is X [k] µr (x)τr,t (x) = x∈Hr htis

X

µr (x)([N ]x)[k]

x∈Hr htis

which is the desired formula.

12.2.6 Soul´ e’s twisting construction and the moment map Let us consider the situation in (12.17) q

0→H→G− →T →0 and recall the inverse system of Hr -torsors Hr hti. Denote by λr : Hr+1 hti → Hr hti the transition maps and by pr,t : Hr hti → S the structure map. Definition 12.2.15 A norm-compatible function θ = (θr )r≥1 on Hhti = (Hr hti)r≥1 is an inverse system of global invertible sections θr ∈ Gm (Hr hti) such that λr∗ (θr+1 ) = θr , where λr∗ is the norm map with respect to λr . Definition 12.2.16

The Kummer map

∂r : Gm (Hr hti) → H 1 (Hr hti, µ`r ) is the boundary map for the exact sequence [`r ]

0 → µ`r → Gm −−→ Gm → 0. [k]

Recall the section τr,t ∈ H 0 (Hr hti, p∗r,t TSymk Hr ) from Definition 12.2.11. Soul´e’s twisting construction is now as follows. Definition 12.2.17

Let [k]

s(r, k, t) := pr,t∗ (∂r (θr ) ∪ τr,t ) ∈ H 1 (S, TSymk Hr (1)), where we have written TSymk Hr (1) := TSymk Hr ⊗ µ`r as usual.

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Recall from Lemma 12.2.3 that limr TSymk Hr = TSymk H and de←− note by redr : TSymk Hr+1 → TSymk Hr the reduction modulo `r . Proposition 12.2.18 (Soul´e)

Under the transition maps

redr : H 1 (S, TSymk Hr+1 (1)) → H 1 (S, TSymk Hr (1)) one has redr (s(r + 1, k, t)) = s(r, k, t). In particular, one gets an element s(k, t) := lim s(r, k, t) ∈ H 1 (S, TSymk H (1)). ←− r

We refer to this construction as Soul´e’s twisting construction. Remark. In general Soul´e’s twisting construction allows also to construct elements in other Galois representations and it depends on the choice of elements in the Galois representation. Here we have fixed the tautological [k] sections τr,t of TSymk Hr to define this twist. In [HK06] one can find more general twisting constructions. Proof. By abuse of notation we also denote by redr any map on cohomology which reduces the coefficient module modulo `r . Then one has [k] [k] redr ◦ λr∗ = λr∗ ◦ redr . We have redr (τr+1,t ) = λ∗r (τr,t ) and by assumption redr ◦ λr∗ (θr+1 ) = λr∗ ◦ redr∗ (θr+1 ) = θr . Then [k]

redr (s(r + 1, k, t)) = redr ◦ pr+1,t∗ (∂r+1 (θr+1 ) ∪ τr+1,t ) [k]

= pr+1,t∗ (redr (∂r+1 (θr+1 )) ∪ redr (τr+1,t )) [k]

= pr,t∗ ◦ λr∗ (redr (∂r+1 (θr+1 )) ∪ λ∗r (τr,t )) [k]

= pr,t∗ (λr∗ ◦ redr (∂r+1 (θr+1 )) ∪ τr,t ) [k]

= pr,t∗ (∂r (θr ) ∪ τr,t ) = s(r, k, t). The following identification is fundamental for the whole paper. Lemma 12.2.19 Let pr,t : Hr hti → S be the Hr -torsor as above, then one has a canonical isomorphism H i (Hr hti, µ`r ) ∼ = H i (S, Λr (Hr hti)(1)). Proof. As pr,t is finite this follows from the Leray spectral sequence. With this identification we can rewrite the Kummer map and one gets

Eisenstein Classes and Elliptic Soul´e Elements

259

a commutative diagram ∂r+1

Gm (Hr+1 hti) −−−−→ H 1 (S, Λr+1 (Hr+1 hti)(1))    λ λr∗ y y r∗ Gm (Hr hti)

∂

−−−r−→

(12.22)

H 1 (S, Λr (Hr hti)(1)),

where the λr∗ on the right hand side is induced by the trace map λr! : Λr+1 (Hr+1 hti) → Λr (Hr hti). This diagram allows to consider the inverse limit of the ∂r (θr ): Definition 12.2.20 fine an element

The norm-compatible functions θ = (θr )r≥1 de-

S hti := lim ∂r (θr ) ∈ H 1 (S, Λ(H hti)(1)) = lim H 1 (S, Λr (Hr hti)(1)). ←− ←− r

r

hti

We also let Sr := ∂r (θr ). With these preliminaries we can finally explain the crucial relation between the moment map and Soul´e’s twisting construction. Proposition 12.2.21

The homomorphism

momkr,t : H 1 (S, Λr (Hr hti)(1)) → H 1 (S, TSymk Hr (1)) induced by the moment map momkr,t coincides with the composition ∪τ [k]

r H 1 (S, Λr (Hr hti)(1)) ∼ = H 1 (Hr hti, Z/`r Z(1)) −−−→

pr,t∗

H 1 (Hr hti, p∗r,t TSymk Hr (1)) −−−→ H 1 (S, TSymk Hr (1)). hti

In particular, one has momkr,t (Sr ) = s(r, k, t) and in the limit momkt (S hti ) = s(k, t). Proof. Let p := pr,t then the result follows from the commutative diagram H 1 (Hr hti, µ`r ) × H 0 (Hr hti, p∗ TSymk Hr )

∪

/

H 1 (Hr hti, p∗ TSymk Hr (1)) ∼ =



∼ =

H 1 (S, p∗ (Z/`r Z ⊗ p∗ TSymk Hr (1)))

O

(12.20)



H 1 (S, p∗ µ`r ) × H 0 (S, p∗ p∗ TSymk Hr )

∪

/ H 1 (S, p∗ Z/`r Z ⊗ p∗ p∗ TSymk Hr (1))

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12.3 Three examples 12.3.1 The Bernoulli measure and its moments Let N > 1 and t ∈ Z/N Z and consider for each r ≥ 0 the exact sequence qr

0 → Z/`r Z → Z/`r N Z −→ Z/N Z → 0, where qr is reduction modulo N . We let Zr := Z/`r Z and Z := Z` . In the notation of (12.17) we have Hr = Z/`r Z, Gr = Z/`r N Z and T = Z/N Z. We define Zr hti := qr−1 (t) = {x ∈ Z/`r N Z | x ≡ t mod N } so that Zr hti = Hr hti in the notation of (12.17). We denote by Zhti := lim Zr hti ←− r

the inverse limit. As before, each Zr hti is a Zr -torsor. Recall that Zh0i = Z = Z` and that Λ(Zhti) is a free rank one Λ(Z)-module. Let us define the Bernoulli measure in Λ(Zhti). We choose, as usual, an auxiliary c ∈ Z with (c, `N ) = 1 to make the Bernoulli distribution integral (for the properties of the Bernoulli numbers we refer to [La90, Ch. 2, §2]). Definition 12.3.1 The map

Denote by Bk (x) the k-th Bernoulli polynomial.

hti

B2,c,r : Zr hti → Z/`r Z x 7→

`r N 2 x cx (c B2 ({ r }) − B2 ({ r })), 2 ` N ` N

(12.23)

where for an element x ∈ R/Z we write {x} for its representative in [0, 1[, defines an element hti

B2,c,r ∈ Λr (Zr hti). By the distribution property of the Bernoulli polynomials the B2,c,r are compatible under the trace map Λr+1 (Zr+1 hti) → Λr (Zr hti) and give rise to a measure hti

hti

B2,c := lim B2,c,r ∈ Λ(Zhti). ←−

(12.24)

r

We want to compute the moments of the Bernoulli measure. Choose e = 1 ∈ Z` as a basis and let x = id : Z` → Z` be the dual basis. By standard

Eisenstein Classes and Elliptic Soul´e Elements

261

congruences for Bernoulli polynomials (see e.g., [La90, Theorem 2.1]) we have Z hti hti k momt (B2,c ) = xk dB2,c Zhti

=

N k+1 t ct (ck+2 Bk+2 ({ }) − Bk+2 ({ })). k c (k + 2) N N

(12.25)

Note that if c ≡ 1 mod N we get hti

momkt (B2,c ) =

N k+1 (ck+2 − 1) t Bk+2 ({ }). ck (k + 2) N

(12.26)

12.3.2 Modified cyclotomic Soul´ e–Deligne elements We review the cyclotomic elements defined by Soul´e [So81] and Deligne [De89] from our perspective. In the literature two kinds of Soul´e–Deligne elements are in use. There are the ones used in Iwasawa theory obtained by using the norm of the field extension Q(µ`r N )/Q(µN ) and the ones which come from motivic cohomology via the regulator map. These are obtained by the trace map from [`r ] : µ`r N → µN . The relation between these two elements is essentially an Euler factor (see the discussion in [Hu15]). We treat here only the later elements originating from motivic cohomology. Let N > 1 and consider the exact sequence of finite ´etale group schemes over a base S 0 → µ`r → µ`r N → µN → 0. With the notations in (12.15) we have Hr = µ`r , Gr = µ`r N and T = µN . For each 1 6= α ∈ µN (S) we define as in (12.16) the µ`r -torsor µ`r hαi by the cartesian diagram µ`r hαi −−−−→ µ`r N     pr,α y y S

α

−−−−→ µN .

The inverse limit of these µ`r -torsors is denoted by T hαi := lim µ`r hαi ←− r

and we use the same notation for the associated sheaves.

(12.27)

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On Gm \ {1} we have the invertible function Ξ : Gm \ {1} → Gm z 7→ 1 − z and it is well known that Ξ is norm-compatible: if [`r ] denotes the `r multiplication on Gm one has [`r ]∗ (Ξ) = Ξ.

(12.28)

Thus we can restrict Ξ to µ`r hαi to get norm-compatible functions θr in the notation of Definition 12.2.15. Definition 12.3.2

We let

CSrhαi := ∂r (Ξ) ∈ H 1 (S, Λr (µ`r hαi)(1)) and define CS hαi := lim CSrhαi ∈ H 1 (S, Λ(T hαi)(1)). ←− r

The section τr,α from Definition 12.2.11 is the map [N ]

τr,α : µ`r hαi ,→ µ`r N −−→ µ`r and its k-th tensor power gives [k] τr,α ∈ H 0 (µ`r hαi, Z/`r Z(k)).

Definition 12.3.3

(12.29)

Let 1 6= α ∈ µN (S). We denote by

[k] e ck+1,r (α) := pr,α∗ (∂r (Ξ) ∪ τr,α ) ∈ H 1 (S, Z/`r Z(k + 1))

the element s(r, k, α) obtained by Soul´e’s twisting construction. [k]

Note that for S := SpecQ(µ`r N ) the section τr,α is given by β 7→ (β N )⊗k for β ∈ µ`r hαi(S). Moreover, one has r

µ`r hαi(S) = {β ∈ µ`r N (S) | β ` = α}. It follows that over S := SpecQ(µ`r N ) the element e ck+1,r (α) is given explicitly by X ck+1,r (α) = e ∂r (1 − β) ∪ (β N )⊗k . (12.30) β∈µ`r hαi(S)

Eisenstein Classes and Elliptic Soul´e Elements

263

Definition 12.3.4 For 1 6= α ∈ µN (S) and k ≥ 1 the modified cyclotomic Soul´e–Deligne element is e ck+1 (α) := lim e c (α) ∈ H 1 (S, Z` (k + 1)). ←− k+1,r r

Moreover, for a function ψ : µN (S) → Z` we let X e ck+1 (ψ) := ψ(α)e ck+1 (α). α∈µN (S)

From the general result in Proposition 12.2.21 we get the following relation between CS hαi and e ck+1 (α) under the moment map momkα : H 1 (S, Λ(T hαi)(1)) → H 1 (S, Z` (k + 1)). In H 1 (S, Z/`r Z(k + 1)) one has

Proposition 12.3.5

momkr,α (CSrhαi ) = e ck+1,r (α) and in the limit momkα (CS hαi ) = e ck+1 (α). hαi

For later use we need a variant of CSr . Fix an integer c > 1 which is prime to `N . The function cΞ

: Gm \ µc → Gm

(12.31)

c2

2

c ∗ −1 defined by z 7→ (1−z) . 1−z c is norm-compatible and one has c Ξ = Ξ ([c] Ξ) Note that the [c]-multiplication maps

[c] : µ`r hαi ∼ = µ`r hαc i. Definition 12.3.6

Let

hαi CSc,r := ∂r (c Ξ) = c2 CSrhαi − [c]∗ CSrhα

c

i

in H 1 (S, Λr (µ`r hαi)(1)) and define hαi CSchαi := lim CSc,r ∈ H 1 (S, Λ(T hαi)(1)). ←r− hαi

We compute the moments of CSc . Proposition 12.3.7

Let k ≥ 1 then

momkα (CSchαi )

= c2 e ck+1 (α) − c−k e ck+1 (αc ).

In particular, for c ≡ 1 mod N one has momkα (CSchαi ) =

ck+2 − 1 e ck+1 (α). ck

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Proof. There is a commutative diagram Λr (µ`r hαi)

[c]!

momk r,α

 Z/`r Z(k)

/ Λr (µ`r hαc i) momk r,αc

[c]k

 / Z/`r Z(k)

and as [c] : µ`r hαi ∼ = µ`r hαc i is an isomorphism, one has [c]∗ [c]∗ = id and one computes c

c

c

ck momkr,α ([c]∗ CSrhα i ) = momkr,αc ([c]∗ [c]∗ CSrhα i ) = momkr,αc (CSrhα i ) =e ck+1,r (αc ) with Proposition 12.3.5 and the result follows.

12.3.3 Elliptic Soul´ e elements We use the theory of norm-compatible elliptic units as developed by Kato. First we fix an analytic uniformization of Y (N )(C) which is the same as in [Kt04]. Note that σ ∈ GL2 (Z/N Z) acts from the left on Y (N ) by σα(v) := α(vσ) for all v ∈ (Z/N Z)2 . Let H := {τ ∈ C | Imτ > 0} be the upper half plane, then one has an analytic uniformization ∼ =

ν : (Z/N Z)× × (Γ(N )\H) − → Y (N )(C) (a, τ ) 7→ (C/(Zτ + Z), α),

(12.32)

where Γ(N ) := ker(SL2 (Z) → SL2 (Z/N Z)) and α is the level structure τ +v2 given by (v1 , v2 ) 7→ av1N . Recall the Main Theorem from [Kt04]. Theorem 12.3.8 (Kato [Kt04] 1.10.) Let E be an elliptic curve over a scheme S and c be an integer prime to 6, then there exists a unit × c ϑE ∈ O(E \ E[c]) such that 1. divc ϑE = c2 (0) − E[c] 2. [d]∗c ϑE = c ϑE for all d prime to c 3. If ϕ : E → E 0 is an isogeny of elliptic curves over S with deg ϕ prime to c, then ϕ∗ (c ϑE ) = c ϑE 0

Eisenstein Classes and Elliptic Soul´e Elements

265

4. For τ ∈ H and z ∈ C \ c−1 (Zτ + Z) let c ϑ(τ, z) be the value at z of c ϑE for the elliptic curve E = C/(Zτ + Z) over C. Then c ϑ(τ, z)

=

2

2 − qz )c γ eqτ (qz )c γ eqτ (qzc )−1 c 1 − qz

2 2 (1 qτ(c −1)/12 (−qz )(c−c )/2

where qτ := e2πiτ , qz := e2πiz and Y Y γ eqτ (t) := (1 − qτn qz ) (1 − qτn qz−1 ). n≥1

n≥1

Note that γ e differs from Kato’s γ. b Corollary 12.3.9 Let t = aτ N + N ∈ C/(Zτ + Z) be an N -torsion 2πi point, a, b ∈ Z and let ζN := e N , then a

c ϑ(τ, t)

2 1 a 2 (c B2 ({ N

= qτ

})−B2 ({ ca N }))

2

b c−c (−ζN ) 2

a

2

2

b c b c (1 − qτN ζN ) γ eqτ (qτN ζN ) ca N

ca N

cb ) e cb ) (1 − qτ ζN γqτ (qτ ζN

.

a

b Proof. This follows from Theorem 12.3.8 by writing qz = qτN ζN and a 1 2 straightforward computation using B2 (x) = x − x + 6 , so that

c2 B2 ({

a ca (c − c2 )a c2 − 1 }) − B2 ({ }) = + . N N N 6

For the elliptic curve π : E → S and an integer N > 1 consider the exact sequence of finite ´etale group schemes 0 → E[`r ] → E[`r N ] → E[N ] → 0. In the notation of (12.15) we have Hr = E[`r ], Gr = E[`r N ] and T = E[N ]. For each section t ∈ E[N ](S) one has the E[`r ]-torsor Hr hti = E[`r ]hti defined by the cartesian diagram / E[`r N]

E[`r ]hti pr,t

 S

t

 / E[N ]

/E [`r ]

 / E.

We denote by Hr and Hr hti the sheaves associated to E[`r ] and E[`r ]hti respectively. We also define H := (Hr )r≥1 H hti := (Hr hti)r≥1 .

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Let c > 1 be an integer with (c, 6`N ) = 1 and consider the function c ϑE

: E \ E[c] → Gm .

By Theorem 12.3.8 this function is norm-compatible [`r ]∗ (c ϑE ) = c ϑE . Note that for t 6= e one has E[`r ]hti ⊂ E \ E[c], by our condition on c. Thus, we can restrict c ϑE to an invertible function, called θr in Definition 12.2.15, on E[`r ]hti. Definition 12.3.10

Let

hti ESc,r := ∂r (c ϑE ) ∈ H 1 (S, Λr (Hr hti)(1))

and in the limit ESchti := lim ESrhti ∈ H 1 (S, Λ(H hti)(1)). ←− r

The section τr,t from Definition 12.2.11 is given by [N]

τr,t : E[`r ]hti ,→ E[`r N ] −−→ E[`r ]. Its k-tensor power gives [k]

τr,t ∈ H 0 (E[`r ]hti, TSymk Hr ). Soul´e’s twisting construction allows now to define: Definition 12.3.11 c ek,r (t)

With the above notations let [k]

:= pr,t∗ (∂r (c ϑE ) ∪ τr,t ) ∈ H 1 (S, TSymk Hr (1))

and c ek (t)

:= lim c ek,r (t) ∈ H 1 (S, TSymk H (1)). ← − r

We call c ek (t) the elliptic Soul´e element. For a function ψ : (E[N ](S) \ {e}) → Z` we let X ψ(t)c ek (t). c ek (ψ) := t∈E[N ](S)\{e}

Suppose that S is a scheme such that the group scheme E[`r N ] is isomorphic to (Z/`r N Z)2 (for example S = Y (`r N )). Then one has Hr ∼ = (Z/`r Z)2 and the pull-back of c ek,r (t) to S is given explicitly by X ∂r (c ϑE (Q)) ∪ ([N ]Q)⊗k ∈ H 1 (S, TSymk (Z/`r Z)2 (1)). c ek,r (t) = [`r ]Q=t

Eisenstein Classes and Elliptic Soul´e Elements

267

The moment map is in our context momkr,t : H 1 (S, Λr (Hr hti)(1)) → H 1 (S, TSymk Hr (1)) or in the limit momkt : H 1 (S, Λ(H hti)(1)) → H 1 (S, TSymk H (1)). From the general result Proposition 12.2.21 we get: Proposition 12.3.12

One has hti momkr,t (ESc,r ) = c ek,r (t)

and momkt (ESchti ) = c ek (t). For later use we note: Lemma 12.3.13

With the above notations, one has the relation c ek (−t)

= (−1)k c ek (t).

Proof. The norm-compatibility of c ϑE implies [−1]∗c ϑE = c ϑE and hence h−ti hti [−1]∗ ESc,r = ESc,r .

The claim follows from the commutative diagram [−1]!

Λr (Hr h−ti) −−−−→ Λr (Hr hti)     momk ymomkt −t y (−1)k

TSymk Hr −−−−→ TSymk Hr .

12.4 Eisenstein classes, elliptic Soul´ e elements and the integral `-adic elliptic polylogarithm In this section we compare the elliptic Soul´e elements with the Eisenstein classes. The idea consists in writing the Eisenstein classes as specializations of the elliptic polylogarithm and then to define an integral version of the elliptic polylogarithm which is directly related to the elliptic Soul´e elements.

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12.4.1 A brief review of the elliptic logarithm sheaf We give a brief review of the elliptic polylogarithm and refer for more details to [Hu15], the original source [BL94] or to the appendix A in [HK99]. Let π : E → S be a family of elliptic curves with unit section e : S → E. We let Λ := Z/`r Z, Z` , Q`

(12.33)

and we consider lisse sheaves of Λ-modules. A Λ-sheaf G is unipotent of length n with respect to π, if it has a filtration G = A0 G ⊃ AG ⊃ . . . ⊃ An+1 G = 0 such that Ak G /Ak+1 G ∼ = ∗ k k π F for a lisse Λ-sheaf F on S. Beilinson and Levin show: Proposition 12.4.1 ([BL94] Proposition 1.2.6) There is an n-unipotent (n) (n) sheaf LogΛ together with a section 1(n) ∈ Γ(S, e∗ LogΛ ) such that for any n-unipotent Λ-sheaf G the homomorphism (n)

π∗ HomE (LogΛ , G ) → e∗ G φ 7→ φ ◦ 1(n) (n)

is an isomorphism. The pair (LogΛ , 1(n) ) is unique up to unique isomorphism. Obviously, as any n − 1-unipotent sheaf is also n-unipotent, one has (n) (n−1) transition maps LogΛ → LogΛ which map 1(n) 7→ 1(n−1) . (n)

Definition 12.4.2 The pro-sheaf (LogΛ , 1) := (LogΛ , 1(n) ) with the above transition maps is called the elliptic logarithm sheaf. We review some facts about LogΛ . Let HΛ := HomS (R1 π∗ Λ, Λ), then one has exact sequences (n)

(n−1)

0 → π ∗ Symn HΛ → LogΛ → LogΛ

→ 0,

(12.34)

which in the case that Λ = Q` induce an isomorphism (n) e∗ LogQ` ∼ =

n Y

Symk HQ` ,

(12.35)

k=0

which maps 1(n) to 1 ∈ Q` . Also in the case Λ = Q` the sheaf LogQ` admits an action of HQ` mult : π ∗ HQ` ⊗ LogQ` → LogQ` ,

(12.36)

Eisenstein Classes and Elliptic Soul´e Elements

269

which on the associated graded pieces π ∗ Symk HQ` is just the usual multiplication with HQ` π ∗ HQ` ⊗ π ∗ Symk HQ` → π ∗ Symk+1 HQ` . The most important fact about the logarithm sheaf is the vanishing of its higher direct images except the second one. Proposition 12.4.3 ([BL94], Lemma 1.2.7) One has ( 0 if i 6= 2 Ri π∗ LogΛ = 2 ∼ R π∗ Λ = Λ(−1) if i = 2. Another important fact about the logarithm sheaf is the splitting principle, which we formulate as follows: Proposition 12.4.4 ([BL94] 1.2.10 (vi), [HK99] Corollary A.2.6.) Let ϕ : E → E 0 be an isogeny and denote by Log0Q` the logarithm sheaf of E 0 . Then one has an isomorphism LogQ` ∼ = ϕ∗ Log0Q` . In particular, for each section t ∈ kerϕ(S) one has a canonical isomorphism Y ∗ t∗ LogQ` ∼ Symn HQ` . = e0 Log0Q` ∼ = e∗ LogQ` = n≥0

Note that in the case where ϕ = [N ] the isomorphism in the proposition induces the multiplication by [N ]k on the graded pieces π ∗ Symk HQ` of LogQ` .

12.4.2 The elliptic polylogarithm and Eisenstein classes The Leray spectral sequence together with Proposition 12.4.3 and the localization sequence give an isomorphism Y Ext1E\{e} (π ∗ HQ` , LogQ` (1)) ∼ Symn HQ` ) (12.37) = HomS (HQ` , n≥1

(see [HK99, A.3] or [Hu15]). Definition 12.4.5

The (small) elliptic polylogarithm is the class

pol ∈ Ext1E\{e} (π ∗ HQ` , LogQ` (1)), Q which maps to the canonical inclusion HQ` ,→ n≥1 Symn HQ` under the above isomorphism (12.37).

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Guido Kings

Consider a non-zero N -torsion section t ∈ E(S). If we use the isomorQ phism t∗ LogQ` ∼ = n≥0 Symn HQ` from Proposition 12.4.4 we get Y t∗ pol = (t∗ poln )n≥0 ∈ Ext1S (HQ` , Symn HQ` (1)). n≥0

To get classes in H (S, Sym HQ` (1)) we use the map n

1

contrHQ :Ext1S (HQ` , `

Q

n≥0

Symn HQ` (1))→Ext1S (Q` ,

Q

n≥1

Symn−1 HQ` (1))

(12.38)

defined by first tensoring an extension with HQ∨` , where HQ∨` is the dual of HQ` , and then compose with the contraction map HQ∨` ⊗ Symn HQ` → Symn−1 HQ` Pn 1 ∨ b mapping h∨ ⊗ h1 ⊗ · · · ⊗ hn to n+1 j=1 h (hj )h1 ⊗ · · · hj · · · ⊗ hn . Definition 12.4.6 Let N > 1 and t ∈ E[N ](S) be a non-zero N-torsion point, then EiskQ` (t) := −N k−1 contrHQ` (t∗ polk+1 ) ∈ H 1 (S, Symk HQ` (1)) is called the k-th Eisenstein class. If ψ : (E[N ](S) \ {e}) → Q is a map, we define X EiskQ` (ψ) := ψ(t)EiskQ` (t). t∈E[N ](S)\{e}

Remark. The factor −N k−1 is for historical reasons as the Eisenstein classes were originally defined in a different way by Beilinson (see [Be86, Theorem 7.3])

12.4.3 A variant of the elliptic polylogarithm For the comparison of the Eisenstein classes with the elliptic Soul´e elements a slight variant of the elliptic polylogarithm is useful. The localization sequence for LogQ` on E and the closed subscheme E[c] for c > 1 gives res

0 → H 1 (E \ E[c], LogQ` (1)) −−→ H 0 (E[c], LogQ` |E[c] ) → Q` → 0 (12.39) 2 ∼ because H 1 (E, LogQ` (1)) = 0 and H 2 (E, LogQ` (1)) ∼ H (E, Q = ` ) = Q` 0 by Proposition 12.4.3. In H (E[c], LogQ` |E[c] ) we have an element which maps to 0 in Q` as follows: We have H 0 (E[c], Q` ) ⊂ H 0 (E[c], LogQ` |E[c] )

Eisenstein Classes and Elliptic Soul´e Elements

271

and consider q : E[c] → S and the section e : S → E[c]. These morphisms induce e∗ : H 0 (S, Q` ) → H 0 (E[c], Q` ) and q ∗ : H 0 (S, Q` ) → H 0 (E[c], Q` ). Definition 12.4.7 Let 1 ∈ H 0 (S, Q` ) be the constant section which is identically 1 on S. Then we let c2 e∗ (1) − q ∗ (1) ∈ H 0 (E[c], Q` ) ⊂ H 0 (E[c], LogQ` |E[c] ). Note that c2 e∗ (1) − q ∗ (1) maps to 0 ∈ Q` under the map in (12.39). We can now define the variant of the elliptic polylogarithm. Definition 12.4.8 The elliptic polylogarithm polc associated with c2 e∗ (1) − q ∗ (1) is the cohomology class polc ∈ H 1 (E \ E[c], LogQ` (1)) with res(polc ) = c2 e∗ (1) − q ∗ (1) ∈ H 0 (E[c], LogQ` |E[c] ). This cohomology class is related to pol as follows. Write H 1 (E \ E[c], LogQ` (1)) ∼ = Ext1E\E[c] (Q` , LogQ` (1)) and define a map multHQ` : Ext1E\E[c] (Q` , LogQ` (1)) → Ext1E\E[c] (π ∗ HQ` , LogQ` (1)) (12.40) ∗ by first tensoring an extension with π HQ` and then push-out with mult : π ∗ HQ` ⊗ LogQ` → LogQ` from Equation (12.36). This gives multHQ` (polc ) ∈ Ext1E\E[c] (π ∗ HQ` , LogQ` (1)). On the other hand consider [c]∗ pol ∈ Ext1E\E[c] (π ∗ HQ` , [c]∗ LogQ` (1)) ∼ [c]∗ Log from Proposition 12.4.4 and use the isomorphism LogQ` = Q` 1 to obtain a class in ExtE\E[c] (π ∗ HQ` , LogQ` (1)). Restriction of pol to E \ E[c] gives another class in Ext1E\E[c] (π ∗ HQ` , LogQ` (1)). Proposition 12.4.9

There is an equality

multHQ` (polc ) = c2 pol |E\E[c] −c[c]∗ pol in Ext1E\E[c] (π ∗ HQ` , LogQ` (1)).

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Guido Kings

Proof. As in Equation (12.37) we have Ext1E\E[c] (π ∗ HQ` , LogQ` (1)) ⊂ HomE[c] (HQ` ,

Y

Symn HQ` )

n≥0

and we have to show that the images of the elements multHQ` (polc ) and c2 pol |E\E[c] −c[c]∗ pol in the right hand side are the same. One has two maps e∗ : HomS (HQ` , HQ` ) → HomE[c] (HQ` , HQ` ) and q ∗ : HomS (HQ` , HQ` ) → HomE[c] (HQ` , HQ` ). It follows from the definition of multHQ` (polc ) and c2 pol |E\E[c] −c[c]∗ pol that both elements map to c2 e∗ (id) − q ∗ (id) ∈ HomE[c] (HQ` , HQ` ) (note that the identification LogQ` ∼ = [c]∗ LogQ` is multiplication by c ∗ on HQ` so that the residue of [c] pol is 1c idE[c] ).

12.4.4 The variant of the elliptic polylogarithm and Eisenstein classes We are going to explain how specializations of polc are related to the Eisenstein classes. Let (c, N ) = 1 and recall from Definition 12.4.8 the class polc ∈ Ext1E\E[c] (Q` , LogQ` (1)). If we pull this back along a non-zero N -torsion section t ∈ E[N ](S), we Q get, using again t∗ LogQ` ∼ = n≥0 Symn HQ` , Y t∗ polc ∈ Ext1S (Q` , Symn HQ` (1)) n≥0

and the k-th component gives a class t∗ polkc ∈ H 1 (S, Symk HQ` (1)). Proposition 12.4.10

In H 1 (S, Symk HQ` (1)) we have the equality

−1 (c2 EiskQ` (t) − c−k EiskQ` ([c]t)). N k−1 In particular, for c ≡ 1 mod N one has t∗ polkc =

t∗ polkc =

−1 ck+2 − 1 k EisQ` (t). N k−1 ck

Eisenstein Classes and Elliptic Soul´e Elements

273

Proof. According to Proposition 12.4.9 we have multHQ` (polkc ) = c2 polk+1 |E\E[c] −c[c]∗ polk+1 . Taking the pull-back along t of the right hand side and applying the map contrHQ` gives −1 (c2 EiskQ` (t) − c−k EiskQ` ([c]t)) N k−1 ∼ [c]∗ Log is multiplication by ck+1 (note that the isomorphism Log = Q`

Q`

on Symk+1 HQ` by the remark after Proposition 12.4.4 so that we have to divide by ck+1 ). Thus it remains to show that contrHQ` (t∗ multHQ` (polkc )) = t∗ polkc . But obviously we have contrHQ` ◦ t∗ multHQ` = contrHQ` ◦ multHQ` t∗ , Q where the last multHQ` is now on n≥0 Symn HQ` , which gives contrHQ` (t∗ multHQ` (polkc )) = contrHQ` ◦ multHQ` (t∗ polkc ). A direct computation shows that contrHQ` ◦multHQ` is the identity map. This gives the desired result.

12.4.5 Sheaves of Iwasawa modules and the elliptic logarithm sheaf In this section we relate the elliptic logarithm sheaf to a certain sheaf of Iwasawa modules. Write Er := E with structure map πr∗ : Er → S and identity section er : S → Er . Let pr := [`r ] : Er → E be the `r -multiplication map. Definition 12.4.11 Let Λn = Z/`n Z, then the geometric elliptic logarithm sheaf with coefficients in R is the inverse system LΛn := (pr∗ Λn ) where the transition maps are the trace maps pr+1∗ Λn → pr∗ Λn . Define a ring sheaf by RΛn := e∗ LΛn and let 1n ∈ RΛn be the identity section.

274

Guido Kings

As Er is an E[`r ]-torsor over E, the sheaf LΛn is a π ∗ RΛn -module, which is locally free of rank one. Denote by IΛn ⊂ RΛn the augmentation ideal and by IΛkn its k-th power. We define (k)

LΛn := LΛn ⊗π∗ RΛn π ∗ (RΛn /IΛk+1 ). n

(12.41)

The first main result in this section is the following theorem: Theorem 12.4.12

There is a canonical isomorphism (k) ∼ =

(k)

LΛn − → LogΛn (k)

(k)

which maps 1n to 1Λn . Here LogΛn denotes the constant inverse system. For the proof we need a characterization of lisse Λn -sheaves on E. Proposition 12.4.13 an integer s such that

Let G be a lisse Λn -sheaf on E. Then there is π∗ HomE (ps∗ Λn , G ) ∼ = e∗ G .

In particular, the functor G 7→ e∗ G induces an equivalence between the category of lisse Λn -sheaves on E and lisse Λn -sheaves on S with a continuous action of RΛn . Proof. As pr is finite ´etale, one has π∗ HomE (pr∗ Λn , G ) = πr∗ HomEr (Λn , p∗r G ) = πr∗ p∗r G . As G is a lisse Λn -sheaf, there is an s such that p∗s G comes from S, which means p∗s G ∼ = πs∗ e∗s p∗s G . This implies that π∗ HomE (ps∗ Λn , G ) = πs∗ πs∗ e∗s p∗s G ∼ = e∗ G . This isomorphism allows to define a continuous action of RΛn on e∗ G (where continuous means that the action factors through some e∗ ps∗ Λn ). The inverse functor is given by F 7→ π ∗ F ⊗π∗ RΛn LΛn . Proof of Theorem 12.4.12. From Proposition 12.4.13 we get a morphism (k) (k) of pro-sheaves LΛn → LogΛn corresponding to 1Λn . It also follows (k)

(k)

that LΛn is k-unipotent because the RΛn -module structure on e∗ LΛn factors through RΛn /IΛk+1 . In particular, the above morphism factors n (k)

(k)

through LΛn . Moreover, by the definition of LogΛn we get also a mor(k)

(k)

phism LogΛn → LΛn corresponding to 1n . It is straightforward to check that these two morphisms are inverse to each other.

Eisenstein Classes and Elliptic Soul´e Elements

275

Define the pro-sheaf L by

Definition 12.4.14

L := (pr∗ Λr )r≥1 where the transition maps are induced by the trace maps and the reduction modulo `r . We also let R := e∗ L with unit section 1 and denote by I ⊂ R the augmentation ideal. Finally, let L (k) := L ⊗π∗ R π ∗ (R/I k+1 ). Theorem 12.4.15

There is a canonical isomorphism ∼ =

(k)

L (k) − → LogZ` (k)

(k)

which maps 1 to 1(k) , where LogZ` is (LogΛr )r≥1 . Proof. There is a surjective morphism L → LΛn and we get with Theorem 12.4.12 a map (k) (k) L → LΛn → LΛn ∼ = LogΛn , (k)

(k)

which induces a morphism L → LogZ` . As already the map L → LΛn (k)

factors through L (k) , we get the desired morphism L (k) → LogZ` , which is surjective by construction. From the isomorphism L (k) ⊗Z` Λn ∼ = (k) (k) (k) LΛn one deduces that L is k-unipotent. By the definition of LogZ` (k)

we get a morphism in the other direction LogZ` → L (k) and one checks (k)

directly that this is inverse to L (k) → LogZ` . We now discuss the relation between the sheaves L and the sheaves of Iwasawa modules. Proposition 12.4.16

Let t : S → E be an N -torsion section, then t∗ L ∼ = Λ(H hti).

Proof. From the commutative diagram E[`r ]hti −−−−→   pr,t y S

Er  [`r ]=p r y

t

−−−−→ E

we get t∗ pr∗ Λr ∼ = pr,t∗ Λr and the result follows from the definitions. Finally, we relate the moment map for Λ(H hti) to the splitting principle

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Guido Kings

for LogQ` . It follows from Proposition 12.4.16 and Theorem 12.4.15 that we have a morphism (k)

(k)

H 1 (S, Λ(H hti)(1)) → H 1 (S, t∗ LogZ` (1)) → H 1 (S, t∗ LogQ` (1)), (12.42) where the last morphism is the canonical map. Definition 12.4.17 The comparison map is the inverse limit over k of the maps in (12.42): comp : H 1 (S, Λ(H hti)(1)) → H 1 (S, t∗ LogQ` (1)). Recall from Proposition 12.4.4 the isomorphism Y t∗ LogQ` ∼ Symn HQ` . = n≥0

Proposition 12.4.18 Let t be an N -torsion section t : S → E. There is a commutative diagram comp

H 1 (S, Λ(H hti)(1))

/ H 1 (S, t∗ LogQ (1)) ` ∼ =

mom ]k t

 Q H 1 (S, n≥0 Symn HQ` (1)) prk

%  H 1 (S, Symk HQ` (1)), k

with mom ] t as in Definition 12.2.13. Proof. As the diagram (k)

Λ(H hti) −−−−→ t∗ LogZ`    [N ] [N]! y y ∗ Λ(H )

(k)

−−−−→ e∗ LogZ`

commutes, it suffices to treat the case t = 0. But recall that the identiQ fication t∗ LogQ` ∼ = n≥0 Symn HQ` is the composition [N ]∗

[N]∗

=

=

t∗ LogQ` −−− −→ e∗ LogQ` ←−∼ −−− ∼

Q

n≥0

Symn HQ` .

This introduces a factor N1k in front of Symk HQ` . Let I(H ) ⊂ Λ(H ) be the augmentation ideal. Then the isomorphism (k) Λ(H )/I(H )k+1 ∼ = e∗ LogZ`

Eisenstein Classes and Elliptic Soul´e Elements

277

induces isomorphisms of the associated graded pieces, which are the Symn H for n = 0, . . . , k. Therefore H 1 (S, Symk H (1)) → H 1 (S, Λ(H )/I(H )k+1 (1)) ∼ = H 1 (S, e∗ LogZ` ) (k)

(k)

pr

k → H 1 (S, e∗ LogQ` ) −−→ H 1 (S, Symk HQ` (1))

is just the comparison map for Symk H . It therefore follows from Lemma 12.2.5 that the diagram / H 1 (S, e∗ LogQ (1)) `

H 1 (S, Λ(H )(1)) mom ]k t

prk

)  H 1 (S, Symk HQ` (1))

commutes.

12.4.6 The elliptic polylogarithm and elliptic units In this section we describe the elliptic polylogarithm in terms of Kato’s norm compatible elliptic units. This will result in a comparison of the Eisenstein classes with the elliptic Soul´e elements. Let c be a positive integer with (c, 6`N ) = 1. We continue to write Λr := Z/`r Z and Er := E which we consider as an ´etale cover over E via pr := [`r ] : Er → E. This induces a morphism pr : Er \ E[`r c] → E \ E[c]. On E \ E[`r c] we have the elliptic unit c ϑE from Theorem 12.3.8. We denote by Θc,r := ∂r (c ϑE ) ∈ H 1 (E \ E[`r c], Λr (1)) ∼ = H 1 (E \ E[c], LΛr (1)) the image of c ϑE under the Kummer map ∂r . As the functions c ϑE are norm-compatible, we can pass to the inverse limit. Definition 12.4.19

We denote by

Θc := lim Θc,r ∈ lim H 1 (E \ E[c], LΛr (1)) ←− ←− r

r

= H 1 (E \ E[c], L (1)) the inverse limit of the classes Θc,r . Recall from Definition 12.3.10 the class ESchti ∈ H 1 (S, Λ(H hti)(1))

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Guido Kings

and from Proposition 12.4.16 the isomorphism t∗ L ∼ = Λ(H hti). Let t : S → E be an N -torsion section. Then

Lemma 12.4.20

t∗ Θc = ESchti ∈ H 1 (S, Λ(H hti)(1)). Proof. We have t∗ Θc = ∂r (c ϑE |E[`r ]hti ) so that the formula is clear from the definitions. As in Definition 12.4.17 one can define a comparison homomorphism comp : H 1 (E \ E[c], L (1)) → H 1 (E \ E[c], LogQ` (1)).

(12.43)

Recall from Definition 12.4.8 the class polc ∈ H 1 (E \ E[c], LogQ` (1)). Theorem 12.4.21

Let (c, 6`N ) = 1, then

comp(Θc ) = polc ∈ H 1 (E \ E[c], LogQ` (1)). Proof. Consider the commutative diagram H 1 (E \ E[c], L (1))

/ H 1 (E \ E[c], LogQ (1)) ` _

comp

res

 H 0 (E[c], L |E[c] )

res

comp

 / H 0 (E[c], LogQ |E[c] ). `

By definition of polc its image in H 0 (E[c], LogQ` |E[c] ) is the element c2 e∗ (1) − q ∗ (1) ∈ H 0 (E[c], Q` ) ⊂ H 0 (E[c], LogQ` |E[c] ). To conclude the proof of Theorem 12.4.21 it suffices to compute the image of comp(Θc ) in H 0 (E[c], LogQ` |E[c] ). For this we work at finite level and use the commutative diagram H 1 (E \ E[c`r ], Λr (1))

res

/ H 0 (E[c`r ], Λr )

res

 / H 0 (E[c], LΛ ). r

∼ =

 H 1 (E \ E[c], LΛr (1))

∼ =

The residue of c ϑE is c2 e∗ (1) − q ∗ (1) ∈ H 0 (E[c], Λr ) ⊂ H 0 (E[c], LΛr ) and taking the inverse limit over r shows that comp(Θc ) agrees with polc in H 0 (E[c], LogQ` |E[c] ).

Eisenstein Classes and Elliptic Soul´e Elements

279

12.4.7 Eisenstein classes and elliptic Soul´ e elements In this section we finally prove the comparison result between Eisenstein classes and elliptic Soul´e elements which is fundamental for the whole paper. It follows from Theorem 12.4.21 that for t ∈ E[N ](S) \ {e} one has Y t∗ comp(Θc ) = t∗ polc ∈ H 1 (S, Symn HQ` (1)). n≥0

Denote by prk :

Y

H 1 (S, Symn HQ` (1)) → H 1 (S, Symk HQ` (1))

n≥0

the projection onto the k-th component. Theorem 12.4.22 Let t ∈ E[N ](S) be a non-zero N -torsion section. Then one has 1 −1 k ] t (ESchti ) = k−1 (c2 EiskQ` (t) − c−k EiskQ` ([c]t)). c ek (t) = mom k N N Proof. As prk (t∗ comp(Θc )) = prk (t∗ polc ) by Theorem 12.4.21, this follows from Lemma 12.4.20 together with Proposition 12.4.10. Lemma 12.4.23 one has

The Eisenstein class EiskQ` (t) is of parity (−1)k , i.e., EiskQ` (−t) = (−1)k EiskQ` (t).

In particular, EiskQ` (ψ) = 0 if ψ is not of parity (−1)k , where we say that ψ has parity (−1)k , if ψ(−t) = (−1)k ψ(t). Proof. This follows from Lemma 12.3.13 and Theorem 12.4.22 for c ≡ 1 mod N .

12.5 The residue at ∞ of the elliptic Soul´ e elements In this section we will compute the residue at ∞ of the elliptic Soul´e elements and hence of the Eisenstein classes.

12.5.1 Definition of the residue at ∞ We are going to describe several variants of the map res∞ . Let ζN = e2πi/N ∈ C and consider over SpecQ(ζN )((q 1/N )) the Tate

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curve Eq with the level structure α : (Z/N Z)2 → Eq [N ] given by (a, b) 7→ b q a ζN . The corresponding map of schemes SpecQ(ζN )((q 1/N )) → Y (N ) induces SpecQ(ζN )[[q 1/N ]] → X(N) and a hence a map ∞ : SpecQ(ζN ) → X(N ), whose image we call the cusp ∞. b )∞ be the completion of X(N ) at ∞, which can be identified Let X(N via the above map with SpecQ(ζN )[[q 1/N ]]. We denote by Yb (N )∞ the b )∞ so that Yb (N )∞ ∼ generic fibre of X(N = SpecQ(ζN )((q 1/N )). One has a commutative diagram j b )∞ ←−∞ Yb (N )∞ −−−−→ X(N −−−     y y j

∞  = y

(12.44)

∞

Y (N ) −−−−→ X(N ) ←−−−− ∞. Note that by purity one has a canonical isomorphism ∞∗ R1 j∗ Z/`r Z(1) ∼ = r 2 r Z/` Z and that R j∗ Z/` Z(1) = 0. Definition 12.5.1

We define the residue map

res∞ : H i (Yb (N ), Z/`r Z(1)) → H i−1 (∞, Z/`r Z) to be the morphism induced by the edge morphism of the Leray spectral sequence for Rj∗ . Consider the Tate curve Eq over Yb (N )∞ . For each r ≥ 1 one has an exact sequence and a commutative diagram 0

/ µ`r N

/ E[`r N ]

pr

[`r ]

 E[N ]

/ Z/`r N Z

/0

(12.45)

[`r ]

p

 / Z/N Z

and pr induces a finite morphism pr : E[`r ]hti → Zr hp(t)i

(12.46)

and hence a morphism of sheaves pr! : Λr (Hr hti) → Λr (Zr hp(t)i). b )∞ so that we can Note that Λr (Zr hp(t)i) is a constant sheaf over X(N

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consider the composition pr∗

H 1 (Yb (N )∞ , Λr (Hr hti)(1)) −−→ H 1 (Yb (N )∞ , Λr (Zr hti)(1)) res −−−∞ → H 0 (∞, Λr (Zr hp(t)i)) ∼ = Λr (Zr hp(t)i). (12.47)

Definition 12.5.2

We define

b (N)∞ ,Λr (Hr hti)(1))→H 0 (∞,Λr (Zr hp(t)i))∼ res∞ :H 1 (Y =Λr (Zr hp(t)i)

to be the composition of the maps in (12.47). We also denote by res∞ : H 1 (Yb (N )∞ , Λ(H hti)(1)) → Λ(Zhp(t)i) the inverse limit. Over Yb (N )∞ one has also the exact sequence ι

p

0 → Z` (1) → − H − → Z` → 0.

(12.48)

Proposition 12.5.3 The subsheaf ι(Z` (1)) ⊂ H are the invariants of monodromy. In particular, ι : Z` (1) → H induces an isomorphism Z` (1) ∼ = ∞∗ j∗ H and p : H → Z` induces ∞∗ R1 j∗ H (1) ∼ = ∞∗ R1 j∗ Z` (1) ∼ = Z` . Proof. That ι(Z` (1)) ⊂ H are the invariants of monodromy is [GRR72, Expos´e IX, Proposition 2.2.5 and (2.2.5.1)]. From the long exact sequence for ∞∗ Rj∗ we get 0 → ∞∗ j∗ Z` → ∞∗ R1 j∗ Z` (1) → ∞∗ R1 j∗ H → ∞∗ R1 j∗ Z` → 0. As ∞∗ j∗ Z` ∼ = Z` and Z` ∼ = ∞∗ R1 j∗ Z` (1), the first map is an isomor∗ 1 phism and one gets ∞ R j∗ H ∼ = ∞∗ R1 j∗ Z` ∼ = Z` (−1). Corollary 12.5.4 Over Yb (N )∞ the maps Symk Z` (1) → Symk H induced by ι and Symk H → Symk Z` induced by p give rise to isomorphisms Z` (k) ∼ = ∞∗ j∗ Symk H

and

∞∗ R1 j∗ Symk H (1) ∼ = Z` .

Proof. This follows by induction on k from Proposition 12.5.3 and the exact sequence ι

0 → Z` (k) → − Symk H → Symk−1 H → 0. Definition 12.5.5

The residue at ∞ is the morphism

H i (Yb (N )∞ , Symk HQ` (1)) → H i−1 (∞, Q` )

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induced from the edge morphism of the Leray spectral sequence for Rj∗ and the isomorphism ∞∗ R1 j∗ Symk HQ` (1) ∼ = Q` . In the same way one defines a residue at ∞ res∞ : H i (Y (N), Symk HQ` (1)) → H i−1 (∞, Q` ),

(12.49)

which factors through the residue map on H i (Yb (N )∞ , Symk HQ` (1)). The residue maps defined in Definition 12.5.2 on finite level and in Definition 12.5.5 with Q` -coefficients are compatible in the following sense. Lemma 12.5.6

There is a commutative diagram res

∞ H 1 (Yb (N )∞ , Λ(H hti)(1)) −−−− → H 0 (∞, Λ(Zhp(t)i))    mom mom ]k y ] kp(t) ty

res

∞ H 1 (Yb (N )∞ , Symk HQ` (1))) −−−− → H 0 (∞, Q` ) ∼ = Q` .

Moreover, if one uses the isomorphism H 0 (∞, Λ(Zhp(t)i)) ∼ = Λ(Zhp(t)i) k the map mom ] p(t) is the composition momk p(t)

k

1 k

mom ] p(t) : Λ(Zhp(t)i) −−−−−→ Z` −N−−k!→ Q` . Proof. The functoriality of the moment map gives Λ(H hti)   momk ty

p!

−−−−→

Λ(Zhp(t)i)  momk y p(t)

k

TSym p TSymk H −−−−−→ TSymk Z` ∼ = Z`

and the lemma follows from the definitions if one observes that the canonical map Symk Z` → TSymk Z` maps the generator of Symk Z` to k! times the generator of TSymk Z` . Finally, we treat the compatibility of the Kummer map and the residue map. The scheme Zr hp(t)i over Yb (N )∞ is the disjoint union of copies of SpecQ(ζN )((q 1/N )). An invertible function on Zr hp(t)i is therefore just a collection of units in Q(ζN )((q 1/N )) and one can speak of the order of the unit in the uniformizing parameter q 1/N . If we denote by Z[Zr hp(t)i] the abelian group of maps ϕ : Zr hp(t)i → Z one gets a homomorphism ord∞ : Gm (Zr hp(t)i) → Z[Zr hp(t)i].

(12.50)

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283

The norm with respect to the finite morphism pr : E[`r ]hti → Zr hp(t)i induces a homomorphism pr∗ : Gm (E[`r ]hti) → Gm (Zr hti). With these notations we have: Lemma 12.5.7

The following diagram commutes: Gm (E[`r ]hti)

ord∞ ◦p∗

∂r

 H 1 (Yb (N )∞ , Λr (Hr hti)(1))

/ Z[Zr hp(t)i]  / Λr [Zr hp(t)i].

res∞

Here the right vertical arrow reduces the coefficients modulo `r . Proof. Compatibility of the Kummer map with traces and residues.

12.5.2 Computation of the residue at ∞ of the elliptic Soul´ e element Recall the residue map b (N)∞ ,Λr (Hr hti)(1))→H 0 (∞,Λr (Zr hp(t)i))∼ res∞ :H 1 (Y =Λr (Zr hp(t)i)

from Definition 12.5.2 and the elements hti ESc,r ∈ H 1 (Yb (N )∞ , Λr (Hr hti)(1))

defined in 12.3.10 and hp(t)i

B2,c,r ∈ Λr (Zr hp(t)i) defined in 12.3.1. The residue of the elliptic Soul´e elements will be deduced from the following fundamental result. Theorem 12.5.8

With the above notation one has hp(t)i

hti res∞ (ESc,r ) = B2,c,r .

In particular, taking the inverse limit one has hp(t)i

res∞ (ESchti ) = B2,c

.

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Guido Kings hti

Proof. Recall that ESc,r = ∂r (c ϑE ) so that Lemma 12.5.7 implies that we have to compute ord∞ ◦p∗ (c ϑE ). In order to do this we perform a base change from Yb (N)∞ to Yb (`r N )∞ . We introduce the shorter notation r Tr := Yb (`r N )∞ = SpecQ(ζ`r N )((q 1/` N ))

for r ≥ 0. Over Tr the scheme E[`r ]hti is isomorphic to the constant scheme Zr2 hti := {(x, y) ∈ (Z/`r N )2 | [`r ](x, y) = t} and the map p : Zr2 hti → Zr hp(t)i is simply given by the projection pr1 onto the first coordinate: (x, y) 7→ x. The base change from T0 to Tr induces a commutative diagram pr1∗

Gm (Zr2 htiTr ) O

/ Gm (Zr hpr1 (t)iT ) r O

ord∞

/ Z[Zr hpr1 (t)i] O

(12.51)

`r

Gm (E[`r ]htiT0 )

p∗

/ Gm (Zr hp(t)iT ) 0

ord∞

/ Z[Zr hp(t)i],

where the right vertical map is the multiplication of the coefficients with `r . The commutativity follows from the fact that the morphism Tr → T0 is ramified of degree `r in q 1/N . Moreover, as pr1 : Zr2 hti → Zr hpr1 (t)i is unramified one has a commutative diagram Gm (Zr2 htiTr ) pr1∗

 Gm (Zr hpr1 (t)iTr )

ord∞

/ Z[Zr2 hti] pr1!

 ord∞ / Z[Zr hpr (t)i]. 1

For each (x, y) ∈ Zr2 hti we now have to calculate the order of (x, y)∗ c ϑE at ∞. For this we can work on Y (`r N )(C). By Corollary 12.3.9 the function (x, y)∗ c ϑE is explicitly given by x

2 1 x 2 (c B2 ({ `r N

qτ

})−B2 ({ `cx r N }))

c−c (−ζ`yr N ) 2

2

x

2

(1 − qτ`r N ζ`yr N )c γ eqτ (qτ`r N ζ`yr N )c cx `r N

(1 − qτ

ζ`cy rN )

cx `r N

γ eqτ (qτ 1/`r N

As the uniformizing parameter for Y (`r N )(C) at ∞ is qτ ord∞ ((x, y)∗ c ϑE ) =

2

ζ`cy rN )

.

, we get

`r N 2 x cx hp(t)i (c B2 ({ r }) − B2 ({ r })) = B2,c,r (x). 2 ` N ` N

To compute pr1! ◦ord∞ ((x, y)∗ c ϑE ) observe that for a fixed x ∈ Zr hpr1 (t)i

Eisenstein Classes and Elliptic Soul´e Elements

285

there are `r elements y with (x, y) ∈ Zr2 hti. As ord∞ ((x, y)∗ c ϑE ) is independent of y this gives hp(t)i

pr1! ◦ ord∞ ((x, y)∗ c ϑE ) = `r B2,c,r (x). With diagram (12.51) we finally get hp(t)i

ord∞ ◦ p∗ (c ϑE ) = B2,c,r ∈ Z[Zr hp(t)i]. From Theorem 12.5.8 we will deduce a formula for the residue of the elliptic Soul´e elements. Recall the elliptic Soul´e element c ek (t)

= momkt (ESchti ) ∈ H 1 (Y (N ), TSymk H (1))

from Definition 12.3.11 and consider 1 k ] t (ESchti ) ∈ H 1 (Y (N ), Symk HQ` (1)). c ek (t) = mom k N In Definition 12.5.5 and (12.49) we have defined the residue map res∞ : H 1 (Y (N ), Symk HQ` (1)) → H 0 (∞, Q` ) ∼ = Q` . In the next theorem we identify E[N ] ∼ = (Z/N Z)2 . Theorem 12.5.9

Let t = (a, b) ∈ E[N ](Y (N )) \ {e}, then

res∞ (c ek (t)) =

N k+1 a ca (c2 Bk+2 ({ }) − c−k Bk+2 ({ })). k!(k + 2) N N

In particular, if c ≡ 1 mod N one gets res∞ (c ek (t)) =

N k+1 ck+2 − 1 a Bk+2 ({ }). k!(k + 2) ck N

Proof of Theorem 12.5.9. By Proposition 12.3.12, Lemma 12.5.6 and Theorem 12.5.8 one has k

res∞ (c ek (t)) = N k res∞ (mom ] t (ESchti )) 1 = momka (res∞ (ESchti )) k! 1 hai = momka (B2,c ) k! N k+1 a ca = (c2 Bk+2 ({ }) − c−k Bk+2 ({ })), k!(k + 2) N N where the last equality is formula (12.25).

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Corollary 12.5.10

Let S = Y (N ) and consider the residue map from

res∞ : H 1 (Y (N ), Symk HQ` (1)) → H 0 (∞, Q` ) ∼ = Q` , then if t = (a, b) ∈ (Z/N Z)2 \ {(0, 0)} one has res∞ (EiskQ` (t)) =

−N k a Bk+2 ({ }). k!(k + 2) N

Proof. This is Theorem 12.4.22 together with Corollary 12.5.9 in the case c ≡ 1 mod N . Remark. The formula differs by a minus sign from the one in [HK99] as we have a different uniformization of the elliptic curve.

12.6 The evaluation of the cup-product construction for elliptic Soul´ e elements 12.6.1 A different description of the cup-product construction Consider over Yb (N )∞ the sheaf Symk HQ` (1) and the diagram j ∞ b )∞ ← Yb (N )∞ − → X(N − ∞.

Recall from Corollary 12.5.4 the isomorphisms Q` (k + 1) ∼ = ∞∗ j∗ Symk HQ` (1)

and

∞∗ R1 j∗ Symk HQ` (1) ∼ = Q` .

The Leray spectral sequence for Rj∗ induces an exact sequence b )∞ , j∗ Symk HQ (1)) → H 1 (Yb (N )∞ , Symk HQ (1)) 0 → H 1 (X(N ` ` res

−−−∞ → H 0 (∞, Q` ) → 0

(12.52)

and we consider the Eisenstein class EiskQ` (ψ) ∈ H 1 (Yb (N )∞ , Symk HQ` (1)). The next result gives a different description of the cup-product construction. Theorem 12.6.1 ([HK99] Theorem 2.4.1, [Hu15] Theorem 11.5.5) sume that res∞ (EiskQ` (ψ)) = 0 so that one can consider b )∞ , j∗ Symk HQ (1)). EiskQ` (ψ) ∈ H 1 (X(N `

As-

Eisenstein Classes and Elliptic Soul´e Elements

287

Then Dir` (ψ) = ∞∗ EiskQ` (ψ) in H 1 (∞, ∞∗ j∗ Symk HQ` (1)) ∼ = H 1 (∞, Q` (k + 1)). Recall that EiskQ` (ψ) =

X

ψ(t)EiskQ` (t)

t∈E[N ]\{e}

and note that it is not possible to evaluate the individual classes EiskQ` (t) at ∞ as res∞ (EiskQ` (t)) 6= 0. The idea for the evaluation is as follows: Using Theorem 12.4.22 we have c ek (t)

k

= mom ] t (ESchti ) = −N (c2 EiskQ` (t) − c−k EiskQ` ([c]t)). hti

Although ESc still can not be evaluated at ∞, we will define an auxilhti hti iary class BSc in Definition 12.6.3 which has the same residue as ESc . The difference MESchti := ESchti − BSchti has then residue zero and can be evaluated at ∞. For this we use the hti description of MESc by an explicit function via the Kummer map. The evaluation at ∞ is then just the evaluation of the function at q = 0, where q is the local parameter at ∞. We conclude by comparing the resulting function with the one defining the Soul´e–Deligne classes.

hti

12.6.2 The auxiliary class BSc Recall from (12.46) the finite morphism pr : E[`r ]hti → Zr hp(t)i and recall that Yb (N )∞ = SpecQ(ζN )((q 1/N )). On E[`r ]hp(t)i consider the function hti

Bp2,c,r (x) :=

N 2 pr (x) cpr (x) 1 hp(t)i (c B2 ({ r }) − B2 ({ r })) = r B2,c,r (pr (x)), 2 ` N ` N `

which defines an element in Λr (Hr hti), hence a global section hti Bp2,c,r ∈ H 0 (Yb (N )∞ , Λr (Hr hti)).

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Lemma 12.6.2

The elements hti Bp2,c,r ∈ H 0 (Yb (N )∞ , Λr (Hr hti))

are norm-compatible, i.e., one can define hti

hti

Bp2,c := lim Bp2,c,r ∈ H 0 (Yb (N )∞ , Λ(H hti)). ← − r Proof. Consider the push-out by p1 0 −−−−→ E[`] −−−−→ E[`r+1 N ] −−−−→ E[`r N ] −−−−→ 0       p1 y %y =y 0 −−−−→ Z/`Z −−−−→   =y

Eer+1   y

σ

−−−−→ E[`r N ] −−−−→ 0   y

0 −−−−→ Z/`Z −−−−→ Z/`r+1 NZ −−−−→ Z/`r N Z −−−−→ 0. This induces on the fibres over t ∈ E[N ] %! σ! Λr+1 (Hr+1 hti) −→ Λr+1 (Hfr+1 hti) −→ Λr+1 (Hr+1 hti)

where Hfr+1 hti is the sheaf associated to the fibre of Eer+1 over t. As hti Bp2,c,r+1 is a pull-back from Z/`r+1 N Z the map %! multiplies the element with the cardinality of the fibres of p1 , which is `. Application hti hp(t)i of σ! to `Bp2,c,r+1 gives by the norm-compatibility of B2,c,r+1 exactly hti

Bp2,c,r , which is the desired result. Lemma 12.6.3

Let ηr be the invertible function on E[`r ]hti hti

hp(t)i

1

ηr (x) := (q 1/N )Bp2,c,r (x) = q `r N B2,c,r

(pr (x))

.

Then the class hti BSc,r := ∂r (ηr ) ∈ H 1 (Yb (N )∞ , Λr (Hr hti)(1))

is the image of hti ∂r (q 1/N ) ⊗ Bp2,c,r ∈ H 1 (Yb (N )∞ , Λr (1)) ⊗ H 0 (Yb (N )∞ , Λr (Hr hti))

under the cup-product. In particular, one can define hti BSchti := lim BSc,r ∈ H 1 (Yb (N )∞ , Λ(H hti)(1)). ←− r

Lemma 12.6.4

One has hp(t)i

res∞ (BSchti ) = B2,c

.

(12.53)

Eisenstein Classes and Elliptic Soul´e Elements

289

Proof. The residue res∞ factors through pr! : Λr (Hr hti) → Λr (Zr hp(t)i) and by definition hti

hti

hp(t)i

pr! (Bp2,c,r ) = `r Bp2,c,r = B2,c,r . hti

Using the cup-product representation of BSc,r gives the desired result. Consider the moment maps momkt : H 1 (Yb (N )∞ , Λ(H hti)(1)) → H 1 (Yb (N )∞ , TSymk H (1)). Definition 12.6.5 c bk (t)

We define

:= momkt (BSchti ) ∈ H 1 (Yb (N )∞ , TSymk H (1)).

For a function ψ : E[N ] \ {e} → Q we let X ψ(t)c bk (t). c bk (ψ) := t∈E[N ]\{e}

Proposition 12.6.6 then

Let ψ be a function such that res∞ (c bk (ψ)) = 0, c bk (ψ)

=0

in H 1 (Yb (N )∞ , TSymk H (1)). Proof. From Lemma 12.6.3 we see that c bk (t) is a cup-product c bk (t)

hti

= (lim ∂r (q 1/N )) ∪ momkt (Bp2,c ), ←− r

hti momkt (Bp2,c )

where ∈ H 0 (Yb (N )∞ , TSymk H ). The map p : H → Z` induces an isomorphism H 0 (Yb (N )∞ , TSymk H ) ∼ = H 0 (Yb (N )∞ , Z` ) ∼ = Z` P hti because of weight reasons. The image of momkt ( t∈E[N]\{e} ψ(t)Bp2,c ) under this isomorphism is just res∞ (c bk (ψ)), which is zero by assumption.

12.6.3 Evaluation at ∞ of the modified elliptic Soul´ e element We modify the elliptic Soul´e element c ek (t) by subtracting the element c bk (t). The resulting element has no residue at ∞ and hence can be evaluated.

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Definition 12.6.7

We let

hti hti hti MESc,r := ESc,r − BSc,r ∈ H 1 (Yb (N )∞ , Λr (Hr hti)(1)) hti

and MESc that

hti

:= limr MESc,r . Define εr := c ϑE ηr−1 ∈ Gm (E[`r ]hti) so ←− hti ∂r (εr ) = MESc,r .

Let f k (t) c me

k

:= mom ] t (MESchti ) 1 = k (c ek (t) − c bk (t)) ∈ H 1 (Yb (N )∞ , Symk HQ` (1)). N

By construction, the residue at ∞ of c me f k (t) is zero: Lemma 12.6.8

hti

One has res∞ (MESc ) = 0 hence res∞ (c me f k (t)) = 0,

so that one can consider c me f k (t) as a class in b )∞ , j∗ Symk HQ (1)). H 1 (X(N ` Proof. This follows from Lemma 12.5.6, Theorem 12.5.8 and Lemma 12.6.4. We now want to evaluate ∞∗ (c me f k (t)) ∈ H 1 (∞, Q` (k + 1)) in terms of Soul´e–Deligne elements. Recall that over Yb (N )∞ one has an exact sequence ι

p

0 → µN → − E[N ] − → Z/N Z → 0. Theorem 12.6.9 Let t be a non-zero N -torsion section of E. Let f k (t) be the element defined in 12.6.7. If p(t) 6= 0, then one has c me ∞∗ (c me f k (t)) = 0. If p(t) = 0, t is in the image of ι and will be considered as an N -th root of unity. Then the formula ∞∗ (c me f k (t)) 2 1 = 2k!N ck+1 (t)+(−1)k e ck+1 (t−1 ))+c−k (e ck+1 (ct)+(−1)k e ck+1 (ct−1 ))) k (c (e

holds in H 1 (∞, Q` (k + 1)). The proof of this theorem is given in the next section.

Eisenstein Classes and Elliptic Soul´e Elements

291

Remark. In fact one can show that in H 1 (∞, Q(k + 1)) the identity e ck+1 (t−1 ) = (−1)k e ck+1 (t) holds (see for example [De89] 3.14.) but we do not need this fact. The consequences for the evaluation of the cup-product construction are as follows. Identify E[N ] ∼ = (Z/N Z)2 and recall that X EiskQ` (ψ) = ψ(a, b)EiskQ` (a, b). (a,b)∈(Z/N Z)2 \{(0,0)}

Corollary 12.6.10 With the above notations suppose that ψ is a function with res∞ (EiskQ` (ψ)) = 0. Then Dir` (ψ) = ∞∗ (EiskQ` (ψ)) =

−1 k!N

X

b ψ(0, b)e ck+1 (ζN )

b∈Z/N Z\{0}

where ζN = e2πi/N . Proof. By assumption we have res∞ (EiskQ` (ψ)) = 0 which implies res∞ (c bk (ψ)) = 0. It follows from Proposition 12.6.6 that X 1 ∞∗ (c ek (ψ)) = ψ(a, b)∞∗ (c me f k (a, b)). k N 2 (a,b)∈(Z/N Z) \{(0,0)}

We now apply Theorem 12.6.9 and observe that ∞∗ (c me f k (a, b)) = 0, if a 6= 0. By Lemma 12.4.23 we can also assume right away that ψ(−t) = (−1)k ψ(t). Then one has X X −b b ψ(0, b)e ck+1 (ζN )= ψ(0, b)(−1)k e ck+1 (ζN ) b∈Z/N Z\{0}

b∈Z/N Z\{0}

and X b∈Z/N Z\{0}

X

cb ψ(0, b)e ck+1 (ζN )=

−cb ψ(0, b)(−1)k e ck+1 (ζN )

b∈Z/N Z\{0}

by substituting b 7→ −b. If we use this in the formula of Theorem 12.6.9 in the case of c ≡ 1 mod N we get 1 c2 − c−k ∞∗ (c ek (ψ)) = k N k!N k

X

b ψ(0, b)e ck+1 (ζN ).

b∈Z/N Z\{0}

On the other hand by Theorem 12.4.22 for c ≡ 1 mod N 1 −(c2 − c−k ) k ∗ ∞ ( e (ψ)) = EisQ` (ψ), c k Nk N k−1 which gives the desired result.

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12.6.4 Proof of Theorem 12.6.9 We need to introduce some more notation. Over Yb (N)∞ one has ι

pr

ι

p

µ`r N −−−r−→ E[`r N ] −−−−→ Z/`r N Z     [`r ] [`r ] [`r ]y y y µN −−−−→ E[N ] −−−−→ Z/N Z. ι

r Definition 12.6.11 We denote by µ`r hti −→ E[`r ]hti the fibre over the N -torsion section t. We let Tr hti be the sheaf associated to µ`r hti and define

T hti := lim Tr hti. ←− r

In the case t = 0 we write T := T h0i. Note that µ`r hti is empty if p(t) 6= 0. The maps ιr induce a map of sheaves ι! : Λ(T hti) → Λ(H hti).

(12.54)

On the other hand, pull-back by ιr gives a map of sheaves ι∗ : Λ(H hti) → Λ(T hti)

(12.55)

which is a splitting of ι! . On the other hand the maps pr,t : E[`r ]hti → Zr hpr (t)i give rise to p! : Λ(H hti) → Λ(Zhp(t)i). Proposition 12.6.12

(12.56)

The morphisms ι! and p! induce isomorphisms Λ(T ) ∼ = ∞∗ j∗ Λ(H )

and ∞∗ R1 j∗ Λ(H ) ∼ = Λ(Z). Proof. Let I(T ), I(H ) and I(Z) be the augmentation ideals of Λ(T ), Λ(H ) and Λ(Z) respectively and Λ(T )(k) etc. the quotient by the k +1power of the augmentation ideal. Then by induction on k and Corollary 12.5.4 one has a commutative diagram 0 −→ ∞∗ j∗ Symk H −→ ∞∗ j∗ Λ(H )(k) −→ ∞∗ j∗ Λ(H )(k−1) x x x ∼ ι ∼ = ! = 0 −→

Z` (k)

−→

Λ(T )(k)

−→

Λ(T )(k−1)

−→ 0

which implies that the injective morphism ι! is also surjective. In the same way one shows p! : ∞∗ R1 j∗ Λ(H ) ∼ = Λ(Z).

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293

With this result the Leray spectral sequence for Rj∗ gives H 1 (Yb (N )∞ , Λ(H hti)(1)) [N ]!

0

/ H 1 (X(N b )∞ , j∗ Λ(H )(1))

/ H 0 (∞, Λ(Zhti)) [N ]!



/ H 1 (Yb (N )∞ , Λ(H )(1))

res∞

 / H 0 (∞, Λ(Z)).

Recall that we want to compute k

k

∞∗ (c me f k (t)) = ∞∗ mom ] t (MESchti ) = ∞∗ mom ] ◦ [N ]! (MESchti ). From Lemma 12.6.8 we get that b )∞ , j∗ Λ(H )(1)). [N ]! (MESchti ) ∈ H 1 (X(N

(12.57)

Consider the commutative diagram ι! b )∞ , Λ(T )(1)) −−− H 1 (X(N −→   mom ] ky

b )∞ , j∗ Λ(H )(1)) H 1 (X(N   ymom ]k

b )∞ , Q` (k + 1)) −−−−→ H 1 (X(N b )∞ , j∗ Symk HQ (1)) H 1 (X(N `    ∗  ∗ y∞ y∞ H 1 (∞, Q` (k + 1)).

H 1 (∞, Q` (k + 1))

As ι! has the splitting ι∗ it follows that we have k

k

∞∗ (c me f k (t)) = ∞∗ mom ] ◦ ι∗ ◦ [N ]! (MESchti ) = mom ] t ◦ ∞∗ ι∗ (MESchti ). (12.58) At finite level we have Gm (E[`r ]hti)   ∂r y

ι∗

−−−−→

Gm (µ`r hti)  ∂ y r

∗

ι H 1 (Yb (N )∞ , Λr (Hr hti)(1)) −−−−→ H 1 (Yb (N )∞ , Λr (µ`r hti)(1))

which implies that we have with the notation in Definition 12.6.7 hti ι∗ (MESc,r ) = ∂r (ι∗ (εr )).

Lemma 12.6.13 The function ι∗ (εr ) ∈ Gm (µ`r hti) extends to a func∗ b tion on µ`r hti over all of X(N) ∞ (also denoted by ι (εr )). The special ∗ ∗ fibre ∞ ι (εr ) is the function ∞∗ ι∗ (εr ) : µ`r hti → Gm β 7→ (−β)

c−c2 2

2

(1 − β)c . (1 − β c )

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Guido Kings

Proof. Note first that there is nothing to show if p(t) 6= 0. So we assume that p(t) = 0. That ι∗ (εr ) extends can be checked after a base extension which adjoins all `r N -th roots of unity. Then by definition and Corollary 12.3.9 ι∗ (εr ) has the form ι∗ (εr )(β) = (−β)

c−c2 2

2

2

(1 − β)c γ eq (β)c , (1 − β c ) γ eq (β c )

where β ∈ µ`r hti and γq (β) = e

Y

(1 − q n β)(1 − q n β −1 ).

n>0

From this formula it is clear that ι∗ (εr ) makes sense for q = 0, i.e., b )∞ . Putting q = 0 gives the explicit extends to µ`r hti over all of X(N ∗ ∗ form of ∞ ι (εr ) as claimed. Lemma 12.6.14 Assume p(t) = 0 so that t is in the image of ι and will be considered as an N -th root of unity. In H 1 (∞, Λr (Tr hti)(1)) one has the identity −1

hti hti ht 2∞∗ ι∗ (MESc,r ) = CSc,r + [−1]∗ CSc,r

i

,

which gives in the limit −1

2∞∗ ι∗ (MESchti ) = CSchti + [−1]∗ CScht

i

in H 1 (∞, Λ(T hti)(1)). Proof. A direct computation gives !2 2 2 c2 c−c2 (1 − β) (1 − β)c (1 − β −1 )c (−β) 2 = , (1 − β c ) (1 − β c )(1 − β −c ) which implies ∞∗ ι∗ (ε2r ) = c Ξ · (c Ξ ◦ [−1]), where c Ξ is the function defined in (12.31). Thus, applying the Kummer map ∂r one has −1

hti hti ht 2∞∗ ι∗ (MESc,r ) = CSc,r + [−1]∗ CSc,r

i

because [−1] : µ`r hti ∼ = µ`r ht−1 i. To conclude the proof of Theorem 12.6.9, we have to compute k

−1

mom ] t (CSchti + [−1]∗ CScht In the definition of

k mom ]t

i

).

we use the inverse of the identification Q` (k) ∼ =

Eisenstein Classes and Elliptic Soul´e Elements

295

Symk Q` (1) ∼ = TSymk Q` (1) ∼ = Q(k) which maps 1 7→ k!. Note also that, as in the proof of Proposition 12.3.7, one has −1

momkt ([−1]∗ CScht

i

) = (−1)k e ck+1 (t−1 ).

With formula (12.58) and Proposition 12.3.7 one now gets: Corollary 12.6.15

One has in H 1 (∞, Q` (k + 1)) the identity

∞∗ (c me f k (t)) 2 1 = 2k!N ck+1 (t)+(−1)k e ck+1 (t−1 ))+c−k (e ck+1 (ct)+(−1)k e ck+1 (ct−1 ))). k (c (e

This proves Theorem 12.6.9.

References [BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I. Progr. Math., 86, 333–400. Birkh¨ auser Boston, Boston, MA. [BL94] Beilinson, A., and Levin, A. 1994. The elliptic polylogarithm. 55, 123– 190. Amer. Math. Soc. [Be86] Beilinson, A. A. 1986. Higher regulators of modular curves. 55, 1–34. Amer. Math. Soc. [Bl15] Blasius, D. 2015. Motivic Polylogarithm and related classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 193–209. [Co98] Colmez, P. 1998. Th´eorie d’Iwasawa des repr´esentations de de Rham d’un corps local. Ann. of Math., 148, no. 2, 485–571. [De89] Deligne, P. 1989. Le groupe fondamental de la droite projective moins trois points. In Galois groups over Q. Math. Sci. Res. Inst. Publ. 16 79– 297. Springer. [GRR72] Grothendieck, A., Raynaud, M., and Rim, D. S. 1972. Groupes de monodromie en g´eom´etrie alg´ebrique. I. Lect. Notes in Math., 288, Springer-Verlag. [HK99] Huber, A., and Kings, G. 1999. Degeneration of l-adic Eisenstein classes and of the elliptic polylog. Invent. Math., 135, 545–594. [HK06] Hornbostel, J., and Kings, G. 2006. On non-commutative twisting in ´etale and motivic cohomology. Ann. Inst. Fourier (Grenoble), 56, 1257– 1279. [Hu15] Huber, A. 2015. A Motivic Construction of the Soul´e Deligne Classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 210–238. [Ja88] Jannsen, U. 1988. Continuous ´etale cohomology. Math. Ann., 280, 207– 245. [Kt93] Kato, K. 1993. Iwasawa theory and p-adic Hodge theory. Kodai Math. J., 16, 1–31.

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[Kt04] Kato, K. 2004. p-adic Hodge theory and values of zeta functions of modular forms. Ast´erisque, ix, 117–290. [Ki01] Kings, G. 2001. The Tamagawa number conjecture for CM elliptic curves. Invent. Math., 143, 571–627. [La90] Lang, S. 1990. Cyclotomic fields I and II. Graduate Texts in Mathematics., 121, Springer-Verlag, New York, second edition, 1990. With an appendix by Karl Rbun. [So81] Soul´e, C. 1981. On higher p-adic regulators. Lect. Notes in Math., 854, 372–401. Springer-Verlag.

13 Postscript R. Sujatha

Abstract The aim of this postscript is to provide a Leitfaden between the articles in this volume and their interlinkages, thereby clearly delineating the results from Galois cohomology and K-theory that are used in proving the main results in [BK90]. One reason to do this is to explicitly spell out the K-theoretic results that are used in [BK90], especially those of Soul´e. We shall also indicate a proof of the finiteness of the global Tate– Shafarevich groups as considered in [BK90, 5.13] for M = Z(m), and note its relation to the Tate–Shafarevich groups considered by Fontaine and Perrin-Riou [FP94], as well as the Tate–Shafarevich groups that can be defined from the Poitou–Tate sequence. For simplicity, we only consider the base field Q, indicating briefly how the results generalize to an arbitrary totally real abelian number field. As in the previous articles, p will denote an odd prime. All other notation is as in [Su15]. Nguyen Quang Do’s contribution in putting this note together is gratefully acknowledged.

13.1 Leitfaden In the article [Li15], Lichtenbaum used K-theory and ´etale cohomology along with the Chern class maps to prove Soul´e’s result on the vanishing 2 of the cohomology groups Het (Z[1/p], Qp /Zp (m)) for all m ≥ 2. Soul´e [So79] proved that these groups are torsion and Schneider [Sc75] proved that they are divisible. The Quillen–Lichtenbaum conjecture asserts that University of British Columbia, Vancouver, Canada. e-mail : [email protected]

298

R. Sujatha

the Chern class maps k cpk,m : K2m−k (Z) ⊗ Zp → Het (Z[1/p], Zp (m))

(13.1)

are isomorphisms for p odd, k = 1, 2 and m ≥ 2. Soul´e in [So79, Th´eor´eme 6 (iii)] proved the surjectivity for even positive integers m. Dwyer and Friedlander [DF85] proved it for all integers m with k = 1 or 2 or 2m − k > 1, using ´etale K-theory. The reader should also see [Ka93]. 2 The vanishing of Het (Z[1/p], Qp /Zp (m)) for all m ≥ 2 is equivalent to 2 the finiteness of Het (Z[1/p], Zp (m)) for m ≥ 2 and this will play an important role in the proof of the Bloch–Kato conjecture, as explained in the article by Coates [Co15] in this volume. We remark that Iwasawa 2 theory gives an independent proof of the finiteness of Het (Z[1/p], Zp (m)) for all odd negative integers m (cf. [Co15]). It is still unknown whether 2 Het (Z[1/p], Zp (m)) vanishes for all even negative integers m. Also, for 2 all odd integers m 6= 1, the finiteness of the group Het (Z[1/p], Zp (m)) 1 is equivalent to Het (Z[1/p], Zp (m)) having rank one (see [Co15, Lemma 3.3.6]). The structure of the K-groups for Z, along with the surjectivity of the Chern class maps in (13.1) for k = 1, 2 would enable us to even conclude that the groups H´e2t (Z[1/p], Zp (m)) vanish for almost all p, a fact that is needed to prove the Tamagawa number conjecture. The group A(Q) that occurs in [BK90] in this context, and which is already mentioned in Raghuram’s article [Ra15] in this volume is an abelian group of rank one when m is odd and positive. The group Φ occurring in [BK90] is the group K2m−1 (Z)⊗Q and the ´etale regulator map which maps this group to H´e1t (Z[1/p], Qp (m)) is none other than the Chern class map cm,1 . The group H´e1t (Z[1/p], Qp (m)) is one dimensional for odd integers m > 2 and the proof of the Tamagawa number conjecture studies the image of the Chern class map, making use of Borel’s theorem as well as Iwasawa theory. The articles in this volume by A. Saikia ([Sa15]), A. Huber ([Hu15]), D. Blasius ([Bl15]) and G. Kings ([Ki15]) are all related to the study of the Chern class map and the Borel regulator. We remark that the surjectivity of the maps in (13.1) was proved by Dwyer–Friedlander [DF85] and Kahn [Ka93] proved that they are split surjective. The injectivity follows from the (other) Bloch–Kato conjecture that relates algebraic K-theory of fields and Galois cohomology groups. This is now known to be true, thanks to deep work of Voevodsky, Rost et al. and the article by Kolster [Ko15] in this volume exposes this relationship. Fontaine and Perrin-Riou gave another equivalent formulation of the Bloch–Kato

Postscript

299

conjecture using determinants, and this is explained in the article by Nguyen Quang Do [Ng15] in this volume. The first result we record is the following. Theorem 13.1.1 For any integer m ≥ 2, the following equivalent assertions hold: (i) rkZp H´e1t (Z[1/p], Zp (m)) = 1 (resp. rkZp H´e1t (Z[1/p], Zp (m)) = 0) if m is odd (resp. even). (ii) H´e2t (Z[1/p], Zp (m)) is finite. (iii) H´e2t (Z[1/p], Qp /Zp (m)) is finite. (iv) The group H´e2t (Z[1/p], Qp /Zp (m)) is zero. Proof For an odd integer m > 2, this is already proven in [Co15, Lemma 3.3.6]. The equivalence of assertions (ii) and (iii) are standard for any m. Soul´e proved the finiteness of H´e2t (Z[1/p], Qp /Zp (m)) for any m ≥ 2 in [So79] which, along with Schneider’s result that this group is divisible, gives (iv). The equivalence of (i) and (ii) for an even integer m ≥ 2 follows from the global Euler characteristic formula (see [Gr81, §8.2]). Assertion (iii) is designated Leopm in [Ng15]. Soul´e proved [So81, Theorem 1] that the Chern class map cm,k is rationally an isomorphism, i.e., cm,k : K2m−k (Z[1/p]) ⊗ Qp ' H´ekt (Z[1/p], Qp (m)) =

H´ekt (Z[1/p], Zp (m))

(13.2) ⊗ Qp (13.3)

for k = 1 or 2 and m ≥ 2. He also proved [So79, Th´eor`eme 6 (iii)] that cm,k : K2m−k (Z[1/p]) ⊗ Zp → H´ekt (Z[1/p], Zp (m))

(13.4)

is surjective for even m ≥ 2 and k = 1 or 2.

13.2 Tate–Shafarevich groups Recall the global Bloch–Kato Shafarevich–Tate group defined in [BK90, §5] (see also [Ng15, 9.4.3.3]) in the context of motivic pairs, which we denote by XBK (Z(m)) for our particular case. For the motive Z(m), we have [BK90, p.377], [Ng15, 9.4.3], XBK (Z(m)){p} = XBK (Zp (m))

(13.5)

300

R. Sujatha

where for an abelian group W , W {p} denotes its p-primary part, and XBK (Zp (m)) := ker



H 1 (Q, Qp /Zp (m))) p /Zp (m)) ⊗ Qp /Zp

H 1 (Q, Q

 H 1 (Ql , Qp /Zp (m)) . 1 all l H (Ql , Qp /Zp (m)) ⊗ Qp /Zp

−→ ⊕

(13.6)

This group coincides with the Tate–Shafarevich group considered by Fontaine and Perrin-Riou in [FP94, Chap. II, 5.3.4]. Theorem 13.2.1 [FP94, Prop. 5.3.5] For any integer m, the group XBK (Zp (m)) is finite. In particular, XBK (Z(m)){p} is finite for all primes p. Proof The proof follows from the finite generation of H´e2t (Z[1/p], Zp (m)) along with a diagram chase (see the diagram at the top of p.665, loc. cit.) Note that one does not need any K-theoretic result here. The second statement is a plain consequence of (13.5). We remark that the above theorem holds for any pair (V, T ) where V is a p-adic Galois representation of GQ as in [BK90], and T is a Galois equivariant lattice. Denote the Poitou–Tate kernel X2 (Zp (m)) by XP T (Zp (m)) so that XP T (Zp (m)) := ker(H´e2t (Z[1/p], Zp (m)) → H 2 (Qp , Zp (m))).

(13.7)

The following result is proved in [BK90, Lemma 6.7] but we give a detailed proof. We remark that the proof in [BK90, p.386] uses the finiteness result of Soul´e for H´e2t (Z[1/p], Zp (m)) without explicitly stating its usage. Our proof here follows [BN02, Rem 4.5], see also [BN02, Prop. 4.4]. Theorem 13.2.2 Assume that any of the equivalent statements of Theorem 13.1.1 hold. Then XBK (Zp (m)) is canonically isomorphic to XP T (Zp (m)). Proof

It follows from [Sc75, §4] that XBK (Zp (m)) ' ker →

H´e1t (Z[1/p], Qp /Zp (m)) 1 H´et (Z[1/p], Qp /Zp (m)) ⊗ Qp /Zp  H 1 (Qp , Qp /Zp (m)) . 1 H (Qp , Qp /Zp (m)) ⊗ Qp /Zp



From the long exact sequence in Galois cohomology associated to the

Postscript

301

short exact sequence 0 → Zp (m) → Qp (m) → Qp /Zp (m) → 0, and the well-known description of the connecting boundary homomorphisms, this latter kernel is isomorphic to
ker H´e2t (Z[1/p], Zp (m))tors → H 2 (Qp , Zp (m))tors , where the subscript tors denotes the corresponding torsion subgroup. Since our hypothesis (or Soul´e’s theorem) implies that H´e2t (Z[1/p], Zp (m)) is torsion, the result follows. We remark that the proof of the above theorem in [BN02, Prop. 4.4] uses the statement (iii) of Theorem 13.1.1. Our next goal is to prove that the finite groups H´e2t (Z[1/p], Zp (m)) are zero for almost all primes p, when Conjecture 6.2 of [BK90] holds. The proof needs Soul´e’s finiteness theorem. Theorem 13.2.3 For a fixed integer m ≥ 2, H´e2t (Z[1/p], Zp (m)) is trivial for almost all primes p. In particular, the global Tate–Shafarevich group XBK (Z(m)) is finite. Proof Let us first consider the case when m is odd. Recall the BeilinsonSoul´e element (bm ⊗ 1) in K2m−1 (Q). The validity of Conjecture 6.2 guarantees that the p-adic realization of this element under the p-adic regulator (which, as we have noted before, is none other than the first Chern class map cm,1 tensored with Qp , and is denoted by λm,p in [Co15, Theorem 3.6.1]) is a multiple of sm,p ⊗ 1 (cf. [Co15, Theorem 3.6.1]). Here sm,p is the Soul´e element that generates the rank one free abelian group H´e1t (Qp , Zp (m)) (see [Co15]). For each p, let Bm,p and Sm,p denote respectively the Zp -modules generated by bm ⊗ 1 and by sm,p . By Conjecture 6.2 of Bloch–Kato and the surjectivity of cm,1 (see (13.4), and also [Li15]), we get a surjective homomorphism from K2m−1 (Q) onto H´e1t (Z[1/p], Zp (m))/Sm,p . This implies that the target on the right, whose order is equal to # H´e2t (Z[1/p], Zp (m)), is trivial for all p not dividing the finite global index [K2m−1 (Q) : Zbm ], and the assertion follows. For m even, the main conjecture of Iwasawa theory (theorem of Mazur– Wiles) proves that the p-part ζ(1 − m) and that of #H´e1t (Z[1/p], Zp (m))/#H´e2t (Z[1/p], Zp (m)) are equal and hence the assertion.

302

R. Sujatha

Alternately, by Borel’s theorem, the group K2m−1 (Z[1/p]) is itself finite for m even, and hence by the surjectivity of the Chern class map cm,2 (see (13.4)), the assertion follows. We remark that the proof given above avoids the use of the surjectivity of cm,2 , which is not studied in this volume and uses technicalities from ´etale K-theory for odd m (see [DF85]), but was proved by Soul´e for even m. This is the algebraic (or motivic) version of the convergence argument used in [BK90], which is made possible thanks to the validity of Conjecture 6.2 of [BK90].

13.3 The case of a totally real base field In this section, we sketch a generalization of the results needed to prove the Tamagawa number conjecture along the lines of [BK90], as presented in [Co15] and the previous sections. We shall concentrate on the index formula proved in Theorem 13.2.2. When the base field F is no longer Q, some Iwasawa theoretic objects which intervene need to be changed. So let F be any number field, E = F (µp ), and let F∞ (resp. E∞ ) be the cyclotomic Zp -extension of F (resp. E) (note that this is different from the notation in [Co15]). Put U¯0 n = p − adic completion of Un0 = Un0 ⊗ Zp , where Un0 = group of p − units 0 ¯ ¯ 0 the projective limit being taken with respect to norms U ∞ = lim U n n ←−

0

X∞ = Gal(L0 ∞ /E∞ ), where L0 ∞ is the maximal abelian pro − p extension of E∞ in which all finite primes are totally split. General algebraic descent: The main algebraic results are Theorem 13.3.1 0 X∞ (m

[Sc75, §6, Lemma 1] For m 6= 1, we have

− 1)G∞ ' XP  T (F, Zp (m)) := ker

H´e2t (OF [1/p], Zp (m))

2



→ ⊕ H (Fv , Zp (m)) . v|p

The Poitou–Tate sequence along with local duality gives 0 Leopm for F ⇔ the finiteness of X∞ (m − 1)G∞ ⇔ the finiteness of XP T (F, Zp (m)) ⇔ the finiteness of H 2 (GS (F ), Zp (m)).

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303

This immediately gives [Co15, Prop. 3.4.3], without the intervention of cyclotomic units. We mention that conjecture Leopm for F is known to be true for all m > 1, by Soul´e [So79], and for all odd m < 0 from the main conjecture (cf. [Co15, Cor. 3.4.4]). The case of even m < 0 is still open and is related to the question of Adams–Hesselholt mentioned in [Co15]. Yet another formulation of Leopm uses the module X∞ , defined as the Galois group over E∞ of the maximal abelian pro-p-extension of E∞ unramified outside p. Conjecture Leopm is equivalent to the finiteness (hence the nullity, because X∞ has no non-trivial finite submodule) of X∞ (−m)G∞ for any m 6= 1. When F = Q and E = Q(µp ), we mention the equivalent notation in [Co15] in the right hand side of the list below. Let Ω∞ be the Galois group over the field Qp (µp∞ ) of the maximal abelian p-extension of this field. The + sign indicates the part fixed by complex conjugation. 0+ ¯ 0+ ¯∞ U /C∞

↔

ker δ∞

X+ ∞

↔

X∞

¯ 0+ Ω+ ∞ /C∞

↔

Y∞

↔

W∞

+

X 0∞

XP T (F, Zp (m)) ↔ H 2 (GS (F ), Zp (m)). Theorem 13.3.2 [KNF96, Theorem 3.2] For all m ∈ Z, m 6= 0, 1, we have a canonical exact sequence ¯ 0 (m − 1))G → H 1 (OF [1/p], Zp (m)) → X 0 ∞ (m − 1)G∞ → 0. 0 → (U ∞ ∞ ´ et Recall that Leopm for F is equivalent to the finiteness of X 0 ∞ (m − ¯ 0 (m−1) 1)G∞ . Theorem 13.3.2 shows that any ‘interesting’ element in U ∞ will give by the above exact sequence, a ‘special element’ in the group H´e1t (OF [1/p], Zp (m)). For m odd and F abelian over Q, the so-called Deligne–Soul´e elements (as defined for example, in [Ng15, 9.5.3.2]) arise in this way. Intervention of the circular units: We assume from now on that the base field F is abelian and totally real, and m ≥ 2. We consider separately the cases m even and m odd. For m even, the Iwasawa main conjecture (Wiles’ theorem) immediately implies that the p-part of the rational value ζF (1 − m) equals that of the quotient #H 1 (O[1/p], Zp (m)/#H 2 (O[1/p]/Zp (m)). For m odd, to get

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R. Sujatha

special elements in the manner described above, we introduce circular p-units in the sense of Sinnott (see [KNF96, §5]) and use the Iwasawa main conjecture. If N is the conductor of F (considered as a subfield of Q(ζN )) let cm (ζN ) ∈ H´e1t (Z[ζN ][1/p], Zp (m)) be the Deligne–Soul´e element, and Dp,m (Q(ζN )) the Galois submodule generated by cm (ζN ), Dp,m (F ) = TrQ(ζN )/F Dp,m (Q(ζN )) (see [Ng15, §9.5.3.4]). The problem mainly consists of computing the index [H´e1t (OF [1/p], Zp (m)) : Dp,m (F )] by using the main conjecture (= Wiles’ theorem). But Wiles’ theorem applies characterwise so that technical difficulties arise in the nonsemisimple case (i.e., when p | [F : Q])). The way out is described at the end of the proof of [Ng15, §9.5.3.4].

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