Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1471 3 Berlin Hei...

Author: Michel Courtieu | Alexei A. Panchishkin

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Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1471

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Michel Courtieu Alexei Panchishkin

Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms Second, Augmented Edition

13

Authors Michel Courtieu Lyc´ee Anna de Noailles 2, Avenue Anna de Noailles 74500 Evian les Bains, France e-mail: [email protected]

Alexei A. Panchishkin Institut Fourier Universit´e Grenoble I BP 74 38402 Saint-Martin d’H`eres, France e-mail: [email protected]

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): primary: 11F, 11R, 11S secondary: 19K, 46F, 46G ISSN 0075-8434 ISBN 3-540-40729-4 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-54137-3 1st edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 1991, 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10951675

41/3142/du - 543210 - Printed on acid-free paper

Preface

The present book is a updated version of the LNM 1471 ”Non-Archimedean L-Functions of Hilbert and Siegel Modular Forms” by Alexei Panchishkin, appeared in 1991. A part of this new book uses the results of the PhD Thesis of Michel Courtieu (Grenoble, Institut Fourier, 2000). The main subject of the book is the p-adic theory of L-functions of Siegel modular forms. In the case of the Riemann zeta functionand of the Dirichlet L-functions, this theory goes back to the classical Kummer congruences for Bernoulli numbers, and to their p-adic interpretation given by Kubota and Leopoldt, and by Mazur. Using the techniques of the p-adic integration, and of the Rankin-Selberg convolution method, we construct a p-adic analytic continuation of the standard L-functions of Siegel modular forms in a very general case, that is, for any non-zero Satake p-parameter. This second version has three basic new features, found recently by Alexei Panchishkin: 1. The use of arithmetical nearly holomorphic Siegel modular forms, viewed as certain formal expansions of many variables with algebraic coeﬃcients. 2. The use of arithmetical diﬀerential operators acting on nearly holomorphic Siegel modular forms. 3. The method of canonical projection allowing to deduce congruences between special L-values from congruences between arithmetical nearly holomorphic Siegel modular forms, via a systematic use of Atkin’s Up -operator (the Frobenius operator Π + (p) of A.N.Andrianov). This new method gives a conceptual explanation for the formulas of Manin type in the Siegel modular case, obtained in LNM 1471. In fact, the previously used distributions admit a canonical lift to distributions with values in an appropriate subspace of arithmetical Siegel modular forms. This lift depends on a choice of a non-zero Satake parameter. The book is intended for researches, to postgraduate students, and to professors, interested in representation theory, functional analysis and arithmetic algebraic geometry. It contains, together with new results, much background

VI

Preface

information about p-adic measures, their Mellin transforms, Siegel modular forms, Hecke operators acting on them, Euler products etc. It seems that the general methods developed in the book may have a number of other applications, in particular, to families of Siegel modular forms and L-functions. Saint-Hilaire du Touvet, August 2003

Contents

1

Non-Archimedean analytic functions . . . . . . . . . . . . . . . . . . . . . . . 1.1 p-adic numbers and the Tate ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Continuous and analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Distributions, measures, and the abstract Kummer congruences 1.4 Iwasawa algebra and the non-Archimedean Mellin transform . . 1.5 Admissible measures and their Mellin transform . . . . . . . . . . . . . 1.6 Complex valued distributions, associated with Euler products .

2

Siegel modular forms and the holomorphic projection operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Siegel modular forms and Hecke operators . . . . . . . . . . . . . . . . . . 2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series . 2.4 Holomorphic projection and Maass operator . . . . . . . . . . . . . . . . 2.5 Explicit description of diﬀerential operators . . . . . . . . . . . . . . . . .

13 13 17 21 28 35 38

45 46 62 70 83 91

3

Arithmetical diﬀerential operators . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1 Description of the Shimura diﬀerential operators . . . . . . . . . . . . 95 3.2 Nearly holomorphic Siegel modular forms . . . . . . . . . . . . . . . . . . . 99 3.3 Algebraic diﬀerential operators of Maass and Shimura . . . . . . . . 106 3.4 Arithmetical variables of nearly holomorphic forms . . . . . . . . . . 117

4

Admissible measures for standard L–functions . . . . . . . . . . . . . 127 4.1 Congruences between modular forms and p-adic integration . . . 127 4.2 Algebraic diﬀerential operators and Siegel-Eisenstein distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.3 A general result on admissible measures . . . . . . . . . . . . . . . . . . . . 144 4.4 The standard L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.5 Algebraic linear forms on modular forms . . . . . . . . . . . . . . . . . . . 165 4.6 Congruences and proof of the Main theorem . . . . . . . . . . . . . . . . 174

VIII

Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Introduction

Kummer Congruences and Bernoulli Numbers The starting point in the theory of zeta functions is the expansion of the Riemann zeta-function ζ(s) into the Euler product: ζ(s) =

∞ (1 − p−s )−1 = n−s p

(Re(s) > 1).

n=1

The set of arguments s for which ζ(s) is deﬁned can be extended to all s ∈ C, s = 1, and we may view C as the group of all continuous quasicharacters × C = Homcont (R× + , C ),

y −→ y s

of R× + . The special values ζ(1 − k) at negative integers are rational numbers: ζ(1 − k) = −

Bk , k

where Bk are Bernoulli numbers, which are deﬁned by the formal power series equality ∞ tet Bn tn Bt = t , e = n! e −1 n=0 and we know (by the Sylvester-Lipschitz theorem, see [Mi-St], [Kat3]) that c ∈ Z implies ck (ck − 1)

Bk ∈ Z. k

The theory of non-Archimedean zeta-functions originates in the work of Kubota and Leopoldt [Ku-Le] containing p-adic interpolation of these special values. Their construction turns out to be equivalent to classical Kummer congruences for the Bernoulli numbers, which we recall here in the following form (see [Kat3]):

M. Courtieu and A. Panchishkin: LNM 1471, pp. 1–12, 2004. c Springer-Verlag Berlin Heidelberg 2004

2

Introduction

Let p be a ﬁxed prime number, c > 1 an integer prime to p. Put (c)

ζ(p) (−k) = (1 − pk )(1 − ck+1 )ζ(−k) n and let h(x) = i=0 αi xi ∈ Zp [x] be a polynomial over the ring Zp of p-adic integers such that x ∈ Zp =⇒ h(x) ∈ pm Zp . Then we have that

n

(c)

αi ζ(p) (−i) ∈ pm Zp .

i=0 (c)

This property expresses the fact that the numbers ζ(p) (−k) depend continuously on k in the p-adic sense. This can be deduced from the known formula for the sum of k-th powers: Sk (N ) =

N −1 n=1

nk =

1 [Bk+1 (N ) − Bk+1 ] k+1

k in which Bk (x) = (x + B)k = i=0 ki Bi xk−i denotes the Bernoulli polynomial. Indeed, all summands in Sk (N ) depend p-adic analytically on k, if we restrict ourselves to numbers n, prime to p, so that the desired congruence (c) follows if we express the numbers ζ(p) (−k) in terms of Bernoulli numbers (see [Kat3]). p–adic L–Functions and Mellin Transforms The domain of deﬁnition of p-adic zeta functions is the p-adic analytic Lie group × Xp = Homcont (Z× p , Cp ) of all continuous p-adic characters of the proﬁnite group Z× p , where Cp = Qp denotes the Tate ﬁeld (completion of an algebraic closure of the p-adic ﬁeld Qp ), so that all integers k can be viewed as the characters xkp : y → y k . The construction of Kubota and Leopoldt is equivalent to the existence a p-adic analytic function ζp : Xp → Cp with a single pole at the point x = x−1 p , which becomes a bounded holomorphic function on Xp after multiplication by the elementary factor (xp x − 1), (x ∈ Xp ), and is uniquely determined by the condition ζp (xkp ) = (1 − pk )ζ(−k) (k ≥ 1). This result has a very natural interpretation in framework of the theory of non-Archimedean integration (due to B. Mazur): there exists a p-adic measure (c) xkp µ(c) = ζ(p) (−k). Indeed, if we µ(c) on Z× p with values in Zp such that Z× p integrate h(x) over Z× p we exactly get the above congruence. On the other

Introduction

3

hand, in order to deﬁne a measure µ(c) satisfying the above condition it suﬃces for any continuous function φ : Z× → Z to deﬁne its integral φ(x)µ(c) . p p Z× p For this purpose we approximate φ(x) by a polynomial (for which the integral is already deﬁned), and then pass to the limit. The important feature of the construction is that it equally works for primitive Dirichlet characters χ modulo a power of p. Throughout the book we ﬁx an embedding ip : Q → Cp

(0.1)

and we shall identify Q with a subﬁeld of C and of Cp . Then a Dirichlet × character χ : (Z/ZpN )× → Q becomes an element of the torsion subgroup × Xptors ⊂ Xp = Homcont (Z× p , Cp )

and the above equality also holds for the special values L(−k, χ) of the Dirichlet L-series ∞ L(s, χ) = χ(n)n−s = (1 − χ(p)p−s )−1 , n=1

p

so that we have ζp (χxkp ) = ip (1 − χ(p)pk )L(−k, χ) (k ≥ 1, k ∈ Z, χ ∈ Xptors ). (0.2) The original construction of T. Kubota and H.W. Leopoldt [Ku-Le] was successesfully used by K. Iwasawa [Iw] for the description of the class groups of cyclotomic ﬁelds. Since then the class of functions admitting p-adic analogues has gradually extended. Zeta-functions (of complex variable) can be attached as certain Euler products to various objects such as diophantine equations, representations of Galois groups, modular forms etc..., and they play a crucial role in modern number theory. Deep interrelations between these objects discovered in last decades are based on identities for the corresponding zeta functions which presumably all ﬁt into a general concept of Langlands of L-functions associated with automorphic representations of a reductive group G over a number ﬁeld K. From this point of view the study of arithmetic properties of these zeta functions is becoming especially important. The theory of modular symbols (due to B. Mazur and Y.I. Manin, see [Man1], [Man3], [Maz-SD]) provided a non-Archimedean construction of functions, which correspond to the case of the group G = GL2 over K = Q. Several authors (including P. Deligne, K.A. Ribet, N.M.Katz, P.F. Kurˇcanov and others, see [De-Ri], [Kat1], [Kat2], [Kurˇc1], [Kurˇc2], [Sho], [Vi1], [Vi2]) investigated this problem for the case G = GL1 and GL2 over totally real ﬁelds and ﬁelds of CM-type (i.e. totally imaginary quadratic extentions of totally real ﬁelds). But the case of more general reductive groups remained unclear until the mid-eighties although important complex analytic properties of the

4

Introduction

Langlands L-functions had been proved. In last decades a general approach to construction of non-Archimedean L-functions associated with various classes of automorphic forms was developed, in particular, for the case of symplectic groups of even degree over K = Q and the group G = GL2 × GL2 over a totally real ﬁeld K. The main tool of the appearing theory is the systematic use of the RankinSelberg method for obtaining both complex-valued and p-adic distributions as certain integrals involving cusp forms and Eisenstein series. By this method we constructed non-Archimedean analogues of the standard zeta functions attached to Siegel cusp forms of even degree and of suﬃciently large weight. p–adic L–Functions of Siegel Modular Forms. For a Siegel modular form f (z) of degree m and weight k, which is an eigenfunction of the Hecke algebra, and for each prime number p one can deﬁne the Satake p-parameters of f denoted by αi (p) with i = 0, 1, · · · , m. In this introduction we assume for simplicity that f is a modular form with respect to the whole Siegel modular group Γ m = Spm (Z), but in Chapter 3 we treat the greneral case of forms of level C with a Dirichlet character ψmodC. The standard zeta function of f is deﬁned by means of the Satake p-parameters as the following Euler product:

m −1 χ(p) χ(p)αi (p) χ(p)αi (p)−1 , 1− s D(s, f, χ) = 1− 1− p ps ps p i=1 where χ is an arbitrary Dirichlet character. According to A.N.Andrianov and V.L.Kalinin [An-Ka], this function can be represented in a form of an integral convolution of f and a theta series with a Siegel-Eisenstein series as a kernel. The construction of its p-adic analytic continuation is based on explicit formulas for the special values of the standard zeta function, and this result is equivalent to the existence of some generalized Kummer congruences for these values. Before giving the precise statements of our results we introduce the following normalized zeta functions   m Γ (s + k − j) D(s, f, χ) D (s, f, χ) = (2π)−m(s+k−(m+1)/2) Γ ((s + δ)/2)  j=1

D− (s, f, χ) = Γ ((s + δ)/2)−1 D (s, f, χ) iδ π 1/2−s D (s, f, χ) D− (s, f, χ) = Γ ((1 − s + δ)/2) where δ = 0 or 1 according as χ(−1) = 1 or χ(−1) = −1, and let k a(ξ)em (ξz) ∈ Sm f (z) = ξ>0

Introduction

5

be the Fourier expansion of the Siegel cusp form f (z) of weight k, the sum is extended over all positive deﬁnite half integral m × m-matrices, z ∈ Hm , Hm = {z ∈ GLm (C) | tz = z,

Im(z) is positive deﬁnite}

is the Siegel upper half plane of degree m and em (z) = exp(tr(2πiz)). Assume that k > 2m + 2 and that m is even. Theorem A (Algebraic properties of the special values of the standard zeta functions). (a) For all integers s with 1 ≤ s ≤ k − δ − m and Dirichlet characters χ such that χ2 is non-trivial for s = 1, we have that f, f −1 D+ (s, f, χ) ∈ K = Q(f, Λf , χ), where K = Q(f, Λf , χ) denotes the ﬁeld generated by Fourier coeﬃcients of f , by the eigenvalues Λf (X) of Hecke operators X on f , and by the values of the character χ. (b) For all integers s with 1 − k + δ + m ≤ s ≤ 0 we have that f, f −1 D− (s, f, χ) ∈ K. We may suppose without loss of generality that a(ξ) ∈ Q for all ξ > 0, and let us assume also that a(ξ0 ) = 1 for some ξ0 > 0 with det(2ξ0 ) = 1. Moreover, we make the essential assumption that a0 (ξ0 ) = 0 for the Fourier coeﬃcient a0 (ξ0 ) of the p-auxiliary form f0 (z) = a0 (ξ)em (ξz) ξ>0

deﬁned in the ﬁrst section of chapter 2 by the equality(2.57). On the contrary, it is possible that α0 (p) is not a p-adic unit (see Theorem B below). We call the case |α0 (p)|p < 1 supersingular (i.e. non-p-ordinary). Note that in the previous book ([Pa6], p 4), only the p-ordinary case was treated. According to its deﬁnition, this form satisﬁes the following multiplicativity property a0 (pN ξ) = α0 (p)N a0 (ξ). The importance of the above non-vanishing condition of a0 (ξ0 ) was pointed out to the second author by S. B¨ ocherer. Theorem B (non-Archimedean interpolation of the standard zeta functions). Under the assumptions as above for each integer c > 1 prime to p there exist two Cp -analytic functions Dc+ (x, f ), Dc− (x, f ) : Xp → Cp , with the following properties:

6

Introduction

(i) for all pairs (s, χ) such that χ ∈ Xptors is a non-trivial Dirichlet character, s ∈ Z with 1 ≤ s ≤ k − δ − m, s ≡ δ (mod 2), and for s = 1 the character χ2 is non-trivial, the following equality holds m(s+k−1−m) + Cχs Gm (χ)Cχ ¯ c+ s 2 −2s D (s, f, χ) (1 − χ ¯ (c)c ) D (χxp , f ) = ip α0 (Cχ )2 G(χ) ¯ f, f

(ii) for all pairs (s, χ) such that χ ∈ Xptors is a non-trivial Dirichlet character and s ∈ Z with 1 − k + δ + m ≤ s ≤ 0, s ≡ δ (mod 2), the following equality holds m(s+k−1−m) − ¯ Gm (χ)Cχ c− s 2 2s−2 D (s, f, χ) (1 − χ (c)c ) D (χxp , f ) = ip α0 (Cχ )2 f, f

where Gm (χ) =

χ(det(h))em (h/Cχ )

h∈Mm (Z)modCχ

denotes the Gauss sum of degree m of the primitive Dirichlet character χ mod Cχ , Cχ = pNχ , α0 (Cχ ) = α0 (p)Nχ , G(χ) = G1 (χ). (iii) if ordp α0 (p) = 0 then the holomorphic functions in (i), (ii) are bounded Cp -analytic functions; (iv) in the general case (but assuming that α0 (p) = 0) the holomorphic functions in (i), (ii) belong to the type o(log(xhp )) with h = [2ordp α0 (p)] + 1 and they can be represented as the Mellin transform of h-admissible measures; (v) if h ≤ k − m then the functions Dc± are uniquely determined by the above conditions (i) and (ii). (see Main Theorem in §4.1.5 and Theorem 4.23). Distributions with Values in Siegel Modular Forms. One of the purposes of this book is to give a new conceptual construction of admissible measures (in the sense of Amice-V´elu) attached to a standard L-function of a Siegel cusp eigenform. For this purpose we use the theory of p-adic integration with values in spaces of arithmetical nearly holomorphic Siegel modular forms (in the sense of Shimura, see [Sh9]) deﬁned over an Oalgebra A where O is the ring of integers in a ﬁnite extension K of Qp . Often we simply assume that A = Cp . Arithmetical nearly holomorphic Siegel modular forms admit two diﬀerent descriptions: a(ξ, Ri,j )q ξ ∈ Q[[q Bm ]][Ri,j ] such that • as formal power series g = ξ∈Bm

z − z))−1 and z ∈ Hm in for all R = (Ri,j )i,j=1,··· ,m = (4πIm(z))−1 = (2πi(¯ the Siegel upper half plane of degree m the series converges to a C∞ -Siegel modular form of a given weight k and character ψ;

Introduction

7

• as certain C∞ -Siegel modular forms g taking values in Q at all CM-points (up to a factor independnt of a concrete form). In fact, we need to ﬁx throughout the book two embeddings i∞ : Q → C,

i p : Q → Cp ,

and we shall often view the ﬁeld Q as a subﬁeld of both C and Cp (the complex and the p–adic numbers) via these embeddings, omitting the symbols i∞ and is the Tate ﬁeld (the completion of a ﬁxed algebraic closure i . Here C = Q p

p

p

Qp of Qp ), which is endowed with a unique norm | · |p such that | p |p = p−1 . We describe the action of certain arithmetical diﬀerential operators on algebraically deﬁned nearly holomorphic Siegel modular forms (in the sense of Shimura, see [Sh9] over an O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp . Often we simply assume that A = Cp . The important property of these arithmetical diﬀerential operators is their commutation with the Hecke operators (under an appropriate normalization, see Theorem 3.13). The reason of these nice arithmetical properties is that these (non-holomorphic) operators could be algebraically interpreted in terms of the Gauss-Manin connection acting on the de Rham cohomology sheaves of the arithmetical compactiﬁcation of Siegel modular varieties (see [Fa-Ch90], [Ha2], [Ga-Ha]). Denote by M = M(A) the A-module (or simply the Cp -vector space) of nearly holomorphic Siegel modular forms . We show in Chapter 4 that the action of these arithmetical diﬀerential operators produces natural families of distributions on a proﬁnite group Y = lim Yi with values in these A-modules ←− of nearly holomorphic Siegel modular forms inside a formal q-expansion ring like A[[q B ]][Ri,j ] where B = Bm = {ξ = t ξ ∈ Mm (Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms), and the nearly holomorphic parameters (Ri,j ) = R correspond to the matrix R = (4πIm(z))−1 in the Siegel modular case. We apply to these distributions the canonical projector πα : M → Mα onto the primary (characteristic) A-submodule associated to a non-zero eigenvalue α ∈ A× of the Frobenius operator U = Π + (p) = Up (the Atkin-Lehner operator in the Siegel modular case; Mα is the maximal A-submodule on which U − αI is nilpotent). This operator acts similar to the trace operator lowering the level of modular forms. On the other hand, U is invertible on Mα if α ∈ A× so that one can glue its action on forms of various levels. In this way one obtains the desired distributions with values in a ﬁnite dimensional vector space starting from naturally deﬁned distributions with values in spaces of modular forms (like Siegel-Eisenstein distributions, theta distributions etc.). In order to obtain from them numerically valued distributions interpolating critical values attached to standard L-functions of Siegel modular forms one applies a suitable linear form coming from the Petersson scalar product (using the Andrianov identity of Rankin type for the standard L-function).

8

Introduction

In previous works [Pa7], [Pa6] a non–Archimedean interpolation of these special values was constructed ﬁxing a non zero Satake p–parameter α0 (p) of the cusp eigenform f . It follows that the normalized critical values D(s, f, χ) of the standard L-function can be explicitely rewritten in terms of certain Cp –valued integrals of admissible p–adic measures (over a proﬁnite group of the type Z× p of p–adic units) provided the character χ is non–trivial. More explicit description of these special values was given by M.Courtieu in his PhD thesis ([Cour], Institut Fourier, 2000) using detailed study of the action of the Shimura diﬀerential operator on Siegel modular forms. In the present book we give a conceptual explanation of the p–adic properties satisfyied by the special values of the standard L-function D(s, f, χ), where f is a Siegel cusp form of an even degree m and of weight k > 2m + 2, χ is a varying Dirichlet character. We show that these admissible measures can be lifted to arithmetical nearly holomorphic Siegel modular forms studied by G.Shimura [Sh9]. This lifting is given by a universal sequence Φ± s (χ) of distributions with values in arithmetical nearly holomorphic Siegel modular forms (for critical pairs (s, χ), see Proposition 4.21). It would be interesting to extend these lifting results to Siegel cusp eigenforms of odd degree, using the method of B¨ ocherer-Schmidt [B¨ oSch]. Families of Siegel Modular Forms. Note that the problem of construction of families of modular forms is closely related to the context of Wiles’ proof [Wi] which is based on a Galois cohomological construction of p-adic families of classical elliptc modular forms. It seems that a natural thing would be to try to extend constructions of such families to other classes of modular forms; the present book gives an example: a canonical lift of previously known p–adic distributions to distributions with values in an appropriate subspace of arithmetical Siegel modular forms. This lift depends on a choice of a non-zero Satake parameter, and it produces families by integration of arithmetical characters. A construction of rigid– analytic families of modular distributions is given recently in [PaInv]. Applications of this theory to constructions of certain new p-adic families of modular forms (families of Siegel-Eisenstein series, families of theta-series with spherical polynomials. . .) is based on the following main sources: • • • •

Serre’s theory of p-adic modular forms as certain formal q-expansions, [Se2]. Shimura’s theory of arithmeticity for nearly holomorphic forms, [Sh9]. Hida’s theory of p-adic modular forms and p-adic Hecke algebras, [Hi5]. Constructions of p-adic Siegel-Eisenstein series, and of p-adic Klingen– Eisenstein series by the author, [Pa8].

Note that the eigenspaces M(α) of U are contained in the primary subspaces Mα , and they where used by D. Kazhdan, B. Mazur, C.-G. Schmidt,

Introduction

9

see [KMS2000], in the p-ordinary case via a p–adic limit procedure. Notice that we do not need a p–adic limit procedure, and we treat the general case of any positive slope. Remarks on modular forms of positive slope According to R.Coleman, F. Gouvˆea and B. Mazur, the structure of modular forms of a given positive slope is more complicated than in the ordinary case, even for elliptic modular forms (see the theory of ”ferns” in [Gou-Ma], and [Co-Ma]). Our results mean that the p-adic behaviour of the standard p-adic L-functions depends not only on the slope of a Siegel modular form but also the eigenvalue itself, see also [PaInv]. This would be important for constructions of rigid–analytic families of Coleman type [CoPB] in the Siegel modular case, and the corresponding families of p-adic L-functions. Notice that the structure of weights is more complicated in the Siegel modular case due to vector-valued modular forms. It seems that besides ﬁxing just a positive slope one needs some additional and more subtle parameters, given probably by an analogue of ”ferns”, which could provide a good understanding of overconvergency in the Siegel modular case. In the future, it would be interesting to combine our approach with geometric methods of Faltings–Chai [Fa-Ch90] and of Coleman–Mazur [Co-Ma]. For a ﬁxed positive integer N ∈ N consider the proﬁnite group Yv , where Yv = (Z/N pv Z)× . Y = lim ← v

There is a natural projection yp : Y → Z× p. Deﬁnition 0.1. a) For h ∈ N, h ≥ 1 let C h (Y, A) denote the A-module of × locally polynomial functions of degree < h of the variable yp : Y → Z× p → A ; in particular, C 1 (Y, A) ⊂ C loc−const (Y, A) (the A-submodule of locally constant functions). We adopt the notation Φ(U) := Φ(χU ) for the characteristic function χU of an open subset U ⊂ Y . Let also denote C loc−an (Y, A) the A-module of locally analytic functions and C(Y, A) the A-module of continuous functions so that C 1 (Y, A) ⊂ C h (Y, A) ⊂ C loc−an (Y, A) ⊂ C(Y, A). b) For a given positive integer h an h-admissible measure on Y with values in M is an A-module homomorphism Φ˜ : C h (Y, A) → M such that for all a ∈ Y and for v → ∞

10

Introduction

(yp − ap )j dΦ˜ a+(N pv )

= o(p−v(j−h) ) for all

j = 0, 1, · · · , h − 1,

p,M

where ap = yp (a). We adopt the notation (a)v = a + (N pv ) for both an element of Yv and the corresponding open compact subset of Y . U (p)–Operator and Method of Canonical projection. We explain in Section 4.3 of Chapter 4, how to construct an h-admissible α : C h (Y, A) → M(A) out of a sequence of distributions measure Φ Φj : C 1 (Y, A) → M with values in an A-module M = M(A) of nearly holomorphic modular forms over A (for all j ∈ N with j ≤ h − 1, where A is an O-algebra, and α ∈ A× is a ﬁxed non-zero eigenvalue of the Frobenius operator U = U (p) = Π + (p) acting on g= a(ξ, n)Rn q ξ ∈ A[[q Bm ]][Ri,j ], ξ∈Bm ,n≥0

by

g|U (p) =

a(pξ, n)(pR)n q ξ

ξ∈Bm ,n≥0

(over the complex numbers R = (4π)−1 Im(z)−1 and this notation corresponds to 1 u f |k m f |U (p)(z) = = 0m p1m t u=u∈Mm (Z) mod p

p−κm

tu=u∈M

m (Z)

f ((z + u)/p), mod p

Im ((z + u)/p) = Im(z/p) = Im(z)/p = (4π)−1 (pR)−1 (we use the Petersson-Andrianov notation for the action of matrices, Chapter 2). Then we consider (for an α ∈ A× ) the canonical projection operator πα : M(A) → M(A)α . We deﬁne an A-linear map α : C h (Y, A) → M Φ on local monomials ypj by

α = πα (Φj ((a)v )) ypj dΦ

(a)v

where Φj : C 1 (Y, A) → M(A) are certain M(A)-valued distributions on Y for all j = 0, 1, . . . , h − 1.

Introduction

11

Automorphic L–Functions The standard zeta function D(s, f, χ) provides an example for the general deﬁnition of the Langlands L-functions. For a reductive group G over a number ﬁeld K this deﬁnition is based on the notion of the Langlands L-group L G; this group is a complex analytic reductive group such that the lattice of characters of its maximal torus and the lattice of its one-parameter subgroups (cocharacters) are obtained from the analogous objects of the group G by inversion. The important fact in the representation theory of reductive groups over local ﬁelds is that semisimple conjugacy classes hv of L G for a place v of K correspond to certain inﬁnite dimensional representations πv of the group G(Kv ) over the local ﬁeld Kv (the completion of K at v). It is known that for groups of the type An and Dn this construction preserves their types, and interchanges the types Bn and Cn , so that if G = GLn then L G = GLn (C), and if G = GSpm then L G = Spin2m+1 (C), the universal covering of the orthogonal group SO2m+1 (C). For example, if G = GL 2 and v a non-Archimedean place α 0 1 then L G = GL2 (C) and for hv the corresponding representation 0 α2 πv is the representation IndG T (µ1 ⊗ µ2 ), which is induced from the maximal torus T = GL1 × GL1 , µ1 , µ2 : Kv× → C× being unramiﬁed quasicharacters of Kv× with µi (πv ) = αi , i = 1, 2. Let π be an automorphic representation of the group G (which is an irreducible subrepresentation of the smooth regular representation of the adelic group G(AK )). Then there is the decomposition of π into the inﬁnite tensor product: π = ⊗v πv where πv is a representation of G(Kv ) which correspond to certain classes hv from L G for almost all v (i.e for v ∈ S where S is a ﬁnite set of places of K). For a ﬁnite dimensional representation r : L G → GLt (C) of the L-group we deﬁne automorphic L-functions L(s, π, r) = LS (s, π, r) = det(1t − (N v)−s r(hv ))−1 v∈S

where N v is the number of elements of the residue class ﬁeld of v (which is a power of its characteristic), and the product is taken over all non-Archimedean places v, v ∈ S. In the Siegel modular case we consider, associated with f , the automorphic representation πf , which is generated by a function on GSpm (A) inﬂated from the cusp form f on Hm (as a subrepresentation of the regular representation of G(AQ ) = GSpm (A)). The irreducibility of πf is equivalent to the fact that f is an eigenfunction for the Hecke algebra Hm = ⊗p Hpm of the Siegel modular group Γm of degree m. In this case the corresponding character of Hp on f is completely determined by its Satake p-parameters, and for the universal covering r : Spin2m+1 (C) → SO2m+1 (C) with Spin2m+1 (C) ⊂ GL2m (C) we have that the classes hv and r(hv ) are represented by the matrices

12

Introduction

hv = Sp(hv ) diag{α0 (p)αi1 (p) · · · αir (p) | 0 ≤ r ≤ m, 1 ≤ i1 < · · · < ir ≤ m} r(hv ) = St(hv ) diag{1, α1 (p), · · · , αm (p), α1 (p)−1 , · · · , αm (p)−1 } where Sp and St are called, respectively, spinor and standard representations of the Langlands group L G. Therefore the standard zeta function D(s, f, χ) coincides with the L-function L(s, πf , St). The function   L(s, πf , Sp) =  p

−1  (1 − χ(p)α0 (p)αi1 (p) · · · αir (p)p−s )

0≤r≤m

1≤i1 <···
is the spinor zeta function of f . Its analytic properties were studied by A.N. Andrianov in the case m = 2 but still little is known about algebraic and arithmetic properties of the special values of this function; however, from the general Deligne conjecture on critical values of L-functions [De] it follows that the properties analogous to those given in Theorem A could exist only for s = k − 1. Acknowledgement Both authors are very grateful to the Institut Fourier (UJF, Grenoble-1) for the permanent excellent working conditions and atmosphere. A part of these results was exposed by Alexei Panchishkin in a lecture “A new way of constructing p-adic L functions” on July 4, 2001, in the Oberseminar of the Mathematical Institute in Heidelberg; it develops some techniques of [Pa9] found during his visit to the Institute for Advanced Study in Princeton in 1999-2000. The new proof of the main result was obtained during a visit of the second author at MPIM in 2002. Let these institutions be thanked for their kind hospitality. We combine the use of the Frobenius operator in order to prove congruences between arithmetical nearly holomorphic Siegel modular forms with results of the PhD Thesis of the ﬁrst author (Grenoble, 2000) [Cour], where the holomorphic forms where used. It is a great pleasure for us to thank S.B¨ ocherer, E.Freitag, and G.Robert, for valuable discussions and observations. The second author is especially grateful to Prof. Yu. I. Manin, for providing since 1973 with fruitful motivations and advice, and for inviting to Bonn in 2002, where the ﬁrst version of this renewed book was completed. Many thanks to Dr. Catriona M. Byrne, Mrs. Stefanie Zoeller, and Mr.Frank Holzwarth (Springer Verlag) for their help in editing and typesetting of the book, and to the referees for their careful reading and useful suggestions.

1 Non-Archimedean analytic functions, measures and distributions

In this chapter we give an exposition of some standard facts from the theory of continuous and analytic functions over a non-Archimedean local ﬁeld. We start by recalling the deﬁnitions and notations concerning p-adic and Sadic numbers. Then we discuss the theory of continuous p-adic functions and their p-adic interpolation, and also the basic properties of p-adic analytic functions. In the section 1.3 we introduce distributions and measures, and give a general criterion for the existence of a non-Archimedean measure with given values of integrals of functions belonging to certain dense family (“generalized Kummer congruences“). The next section is devoted to a description of the algebra of bounded measures in terms of their non-Archimedean Mellin transforms (Iwasawa isomorphism). The chapter is completed with an exposition of a general construction of measures, attached to rather arbitrary Euler products. This construction provides a generalization of measures ﬁrst introduced by Yu.I. Manin (see [Man2]), B. Mazur and H.P.F. Swinnerton-Dyer (see [Maz-SD]). Our construction (see [Pa2] and [Pa3]) was already successfully used in several problems concerning the p-adic analytic interpolation of special values of Dirichlet series (see [Ar], [Co-Schm], [Co-Schn] and [Sc]).

1.1 p-adic numbers and the Tate ﬁeld 1.1.1 p-adic numbers Let p be a prime number, Qp the ﬁeld of p-adic numbers, (i.e. the completion of the ﬁeld of rational numbers Q with respect to the p-adic metric), given by the p-adic valuation | · |p : Q −→ R+ a/b −→ |a/b|p = pordp (b)−ordp (a) 0 −→ |0|p = 0

M. Courtieu and A. Panchishkin: LNM 1471, pp. 13–44, 2004. c Springer-Verlag Berlin Heidelberg 2004

14

1 Non-Archimedean analytic functions

where ordp (a) is the highest power of p dividing the integer a. The function | · |p is multiplicative, since ordp (xy) = ordp (x) + ordp (y)

(1.1)

and satisﬁes the non-Archimedean property |x + y|p ≤ max (|x|p , |y|p ) .

(1.2)

If K is a ﬁnite algebraic extension of Qp , then K is generated over Qp by a primitive element α ∈ K, so that α is a root of an irreducible polynomial of degree d = [K : Qp ], f (x) = xd + ad−1 xd−1 + · · · + a0 ∈ Qp [x]. The valuation | · |p admits a unique extension to K deﬁned by 1/d , |β|p = |NK/Qp (β)|p

(1.3)

where NK/Qp (β) ∈ Qp is the algebraic norm of an element β ∈ K. The formula (1.3) deﬁnes a unique extension of |·|p to the algebraic closure Qp of Qp , which satisﬁes (1.2) (see [Kob2]). The function ordp is then also extended to Qp by ordp (α) = logp (|α|p ). The formula (1.3) implies that ordp (K × ) is an additive subgroup in 1d Z, hence ordp (K × ) = 1e Z for a certain positive integer e dividing d which is called the ramiﬁcation index of the extension K/Qp . Put OK = {x ∈ K

| |x|p ≤ 1} ,

MK = {x ∈ K

| |x|p < 1} .

(1.4)

Then MK is the maximal ideal in OK and the residue ﬁeld OK /MK is a ﬁnite extension of degree f of Fp and there is the relation d = ef , in which f is called the inertial degree of the extension. For each x ∈ OK , its Teichm¨ uller representative is deﬁned by fn

ω(x) = lim xp , n→∞

ω(x) ≡ x (mod MK ),

(1.5)

f

and satisﬁes the equation ω(x)p = ω(x). The map ω provides a homomorphism of the group of invertible elements × = {x ∈ K OK

| |x|p = 1}

of OK onto the group of roots of unity of degree pf − 1 in K, denoted by µpf −1 , and the isomorphism ∼

× (OK /MK )× −→ µpf −1 ⊂ OK .

(1.6)

For example, if e = 1 then the extension K is called unramiﬁed. In this case f = d and the Teichm¨ uller representatives generate K over Qp , therefore

1.1 p-adic numbers and the Tate ﬁeld

15

K = Qp (11/N ) for N = pd − 1. On the other hand, if e = d then the extension K is called totally ramiﬁed. For example, if ζ is a primitive root of unity of degree pn , then Qp (ζ) is totally ramiﬁed of degree d = pn − pn−1 , and we have that 1 . (1.7) ordp (ζ − 1) = n p − pn−1 1.1.2 Topology of p-adic ﬁelds For a ﬁeld K with the non-Archimedean valuation | · |p let us deﬁne Da (r) Da (r− )

= Da (r; K) = = Da (r− ; K) =

{x ∈ K {x ∈ K

| |x − a|p ≤ r}, | |x − a|p < r},

(1.8) (1.9)

(“closed” and “open” discs of radius r with center at the point a ∈ K, r ≥ 0). Then we have that Db (r) = Da (r) for any b ∈ Da (r) and Db (r− ) = Da (r− ) for any b ∈ Da (r− ). Note that in the topological sense both discs (1.8) and (1.9) are open and closed subsets of the topological ﬁeld K. The important property of the ﬁeld Qp and of its ﬁnite extensions is that they are locally compact, so that the discs (1.8) and (1.9) are compact. The disc Zp = D0 (1, Qp ) is a compact topological ring (the ring of p-adic integers), which is isomorphic to the projective limit of residue rings Z/pn Z with respect to the homomorphisms of reduction modulo pn : Zp = lim (Z/pn Z) . ←− n

Analogously, we have that n OK = lim (OK /MK ), ←− n

and

×

×

× n OK = lim (OK /MK ) . ←−

(Z/pn Z) , Z× p = lim ←− n

n

× 1.1.3 The structure of the multiplicative group Q× p and K

Put ν = 1 for p > 2 and ν = 2 for p = 2, and deﬁne U = Up = {x ∈ Zp

| x≡1

(mod pν )}.

∼

(1.10)

Then there is an isomorphism U → Zp of the multiplicative group Up and of the additive group Zp which is provided by combining the natural homomorphism ∼ pn U/U , U −→ lim ←− n

n

∼

and by special isomorphisms αpn : U/U p −→ Z/pn Z, given by

16

1 Non-Archimedean analytic functions

αpn ((1 + pν )a ) = a mod pn

(a ∈ Z).

(1.11)

One easily veriﬁes that (1.11) is well deﬁned and gives the desired isomorphism. Therefore, the group U is a topological cyclic group, and (1 + pν ) can be taken as its generator. There are the following decompositions ν × ∼ Z× p = (Z/p Z) × U.

Qp = pZ × Z× p,

(1.12)

Similarly, if [K : Qp ] = d, then × , K × = π Z × OK

× ∼ OK = (OK /MK )× × UK ,

where π is a generator of the principal ideal MK = πOK (i.e. any element π ∈ K × with ordp (π) = 1/e), × UK = {x ∈ OK

| |x − 1|p < 1} = D1 (1− ; K).

The structure of the group UK is then described as a direct product of d copies of the additive group Zp and a ﬁnite group consisting of all p-primary roots of unity contained in K (see [Ca-Fr], [Kob2]). 1.1.4 The S-adic numbers Let S be a ﬁnite set of prime numbers. For a positive integer M , denote by S(M ) its support (i.e. the set of all prime numbers dividing M ). Let us consider the projective limit ZS =

lim ←−

Z/M Z,

M|S(M)⊂S

which is taken over all positive integers M with the condition S(M ) ⊂ S with respect to the homomorphisms of reduction. From the Chinese Remainder Theorem (see [Se1], [Weil]) it follows then that Zq . ZS ∼ = q∈S

Put: QS =

q∈S

Qq ,

M0 =

q.

q∈S

1.1.5 The Tate ﬁeld For the purposes of analysis it is convenient to embed Qp into a bigger ﬁeld, which is already complete both in the topological and in the algebraic ˆ of an algebraic sence. This ﬁeld is constructed as the completion Cp = Q p closure Qp of Qp with respect to its single valuation satisfying the condition

1.2 Continuous and analytic functions

17

|p|p = 1/p. The proof of the fact that Cp is algebraically closed is not diﬃcult and is based on the Krasner’s lemma (see [Kob1],[Kob2], [Wa]). We shall use the notations Op = {x ∈ Cp | |x|p ≤ 1},

Mp = {x ∈ Cp | |x|p < 1}

(1.13)

Note that Op and Mp are no longer compact, and therefore the ﬁeld Cp is not locally compact. We also have that Op /Mp = Fp

(1.14)

is the algebraic closure of Fp .

1.2 Continuous and analytic functions over a non-Archimedean ﬁeld 1.2.1 Continuous functions Let K be a closed subﬁeld of the Tate ﬁeld Cp . For a subset W ⊂ K we consider continuous functions f : W → Cp . The standard examples of continuous functions are provided by polynomials, by rational functions (at points where they are ﬁnite), and also by locally constant functions. If W is compact then for any continuous function f : W → Cp and for any ε > 0 there exists a polynomial h(x) ∈ Cp [x] such that |f (x) − h(x)|p < ε for all x ∈ W . If f (W ) ⊂ L for a closed subﬁeld L of Cp then h(x) can be chosen so that h(x) ∈ L[x] (see [Kob1], [Wa]). Interesting examples of continuous p-adic functions are provided by interpolation of functions, deﬁned on certain subsets, such as W = Z or N with K = Qp . Let f be any function on non-negative integers with values in Qp or in some (complete) Qp -Banach space. In order to extend f (x) to all x ∈ Zp we can use the interpolation polynomials x x(x − 1) · · · (x − n + 1) . = n n! Then we have that nx is a polynomial of degree n of x, which for x ∈ Z, x ≥ 0 gives the binomial coeﬃcient. If x ∈ Zp then x is close (in the p-adic topology) x to a positive integer, hence the value of n is also close to an integer, therefore x . ∈ Z p n The classical Mahler’s interpolation theorem says that any continuous function f : Zp → Qp can be written in the form (see [La2], [Wa]): x f (x) = an , n n=0 ∞

(1.15)

18

1 Non-Archimedean analytic functions

with an → 0 (p-adically) for n → ∞. For a function f (x) deﬁned for x ∈ Z, x ≥ 0 one can write formally f (x) =

∞

an

n=0

x , n

where the coeﬃcients can be founded from the system of linear equations f (n) =

n

am

m=0

that is am =

m j=0

(−1)m−j

n , m

m f (j). j

The series for f (x) is always reduced to a ﬁnite sum for each x ∈ Z, x ≥ 0. If an → 0 then this series is convergent for all x ∈ Zp . As was noticed above, the inverse statement is also valid (“Mahler’s criterion”). If convergence of an to zero is so fast that the series deﬁning the coeﬃcients of the x-expansion of f (x) also converge, then f (x) can be extended to an analytic function, see 1.2.2 below. Unfortunately, for an arbitrary sequence an with an → 0 the attempt to use (1.15) for continuation of f (x) out of the subset Zp in Cp may fail. However, in the sequel we mostly consider anlytic functions, that are deﬁned as sums of power series. 1.2.2 Analytic functions and power series of convergence of a series ∞(see [Kob1], p 13). The well known criterion a is that the following partial sums n N ≤n≤M an are small for large n=0 N , M with M > N . In view of the non-Archimedean property (1.2) in Cp this occurs if and only if |an |p → 0 or ordp (an ) → ∞ for n → ∞. Therefore the convergence of the power series n≥0 an xn depends only on |x|p but not on the precise value of x, hencethere is no “conditional convergence” in this case. Thus, for any power series n≥0 an xn we can deﬁne its radius of convergence r such that only one of the following holds: ∞ n=0 ∞

an xn converges ⇐⇒

x ∈ D0 (r− ),

(1.16)

an xn converges ⇐⇒

x ∈ D0 (r).

(1.17)

n=0

An example of the ﬁrst alternative is n≥0 xn , where (1.16) is satisﬁed with n r = 1, and an example for the second is n≥0 pn xp −1 , where (1.17) is satisﬁed also with r = 1.

1.2 Continuous and analytic functions

19

The important examples of analytic functions are exp(x) and log(x), which are given as the power series exp(x) =

∞ xn , n! n=0

log(1 + x) =

∞ (−1)n+1 n x , n n=1

(1.18)

r = p−1/(p−1) ,

(1.19)

and we have that exp(x) converges on D0 (p−1/(p−1)− ), and

log(1 + x) converges on D0 (1− ),

r = 1,

(1.20)

ˇ (see [Kob1], [Bo-Sa]), so that exp(x) converges in a disc smaller than the unit disc, and log(1 + x) has better convergence that exp(x). Since the identity log(xy) = log(x) + log(y) (1.21) holds as a formal power series identity, it follows that (1.21) holds in Cp as long as |x − 1|p < 1 and |y − 1|p < 1. In particular, since |ζ − 1|p < 1 for ζ any pn -root of unity, we can obviously apply (1.21) to conclude that log(ζ) = 0. Also we have that for all x ∈ D0 (p−1/(p−1)− ) the following identities hold exp(log(1 + x)) = 1 + x, and log(exp(x)) = x, which are deduced from the corresponding properties of the formal series and can be used for establishing isomorphisms between certain additive and ν− multiplicative subgroups in Cp and C× ; Qp ) (with ν as in p ; for U = D1 (p 1.1.3) there are the isomorphisms ∼

n

exp : pν+n Zp −→ U p

(with n ≥ 0).

(1.22)

Theorem 1.1 (On analyticity of interpolation series). Let r < p−1/(p−1) < 1, and let ∞ x an f (x) = , n n=0 be a series with the condition |an |p ≤ M rn for some M > 0. Then f (x) is expresible as a certain power series whose radius of convergence is not less than R = (rp1/(p−1) )−1 > 1. Proof. (see [Wa], p. 53) As an example let us consider the function a x which is deﬁned for a ∈ Z× p by means of the decomposition a = ω(a) a where ω(a) is the Teichm¨ uller representative of a for p > 2 and ω(a) = ±1 for p = 2 with ω(a) ≡ a (mod 4), and the exponentiation is given by the binomial formula

20

1 Non-Archimedean analytic functions ∞ x a = (1 + a − 1) = ( a − 1)n . n n=0 x

x

(1.23)

Since | a − 1|p ≤ pν we may put in the above theorem r = p−ν and get that the function a x is a power series in x with the radius of convergence not less than pν−1/(p−1) > 1 and the following equality holds: ∞ x (1.24) exp(x log( a )) = ( a − 1)n . n n=0 in which both parts are analytic in x and coincide for x ∈ N. 1.2.3 Newton polygons (see [Kob1], p 21). The Newton polygon Mf for a power series f (x) =

∞

an xn ∈ Cp [[x]]

n=1

is deﬁned as the convex hull of the points (n, ordp (an )) (where we agree to take ordp (0) = ∞). It is not hard to prove the following. Proposition 1.2. If a segment of Mf has slope λ and horizontal length N (i.e. it extends from (n, ordp (an )) to (n + N, λN + ordp (an )) then f has precisely N roots rn with ordp (rn ) = −λ (counting multiplicity). The following theorem is the p-adic analog of the Weierstrass Preparation theorem (see [Kob1], p. 21). Theorem 1.3. Let f (x) = am xm + am+1 xm+1 + · · · ∈ Cp [[x]], am = 0 be a power series which converges on D0 (pλ ; Cp ). Let (N, ordp (aN )) be the right endpoint of the last segment of Mf with slope ≤ λ, if this N is ﬁnite. Otherwise, there will be a last inﬁnitely long segment of slope λ and only ﬁnitely many points (n, ordp (an )) on that segment. In that case let N be the last such n. Then there exists a unique polynomial h(x) of the form bm xm + bm+1 xm+1 + · · · + bN xN with bm = am and a unique power series g(x) which converges and does not vanish on D0 (pλ ; Cp ) such that f (x) =

h(x) on D0 (pλ ; Cp ). g(x)

In addition Mh coincide with Mf as far as the point (N, ordp (aN )).

1.3 Distributions, measures, and the abstract Kummer congruences

21

Corollary 1.4. A power series which converges everywhere and has no zeroes is a constant. A simple proof of the Weierstrass Preparation Theorem for power series of the type ∞ f (x) = an xn ∈ Op [[x]] n=1

is based on a generalization of the Euclid algorithm (see [Man1]). There exists another deﬁnition (dual) of the Newton polygon (see [Kob1], [Vi1]) of a series ∞ an xn ∈ Cp [[x]]. f (x) = n=1

Instead of the points (n, ordp (an )) let us look at the lines ln : y = nx+ordp (an ) f is deﬁned as the graph of the function minn ln (x). The with slope n. Then M f give x-coordinate of the points of intersection of the ln which appear in M ordp of the zeroes, and the diﬀerence between the slopes n of the successive ln f give the number of zeroes with given ordp . This deﬁnition which appear in M f coincides with the graph of the function is explained by the fact that M Mf (t) = log sup |f (x)|p p

|x|p
which is piecewise linear in the variable t = logp (r). This function is given on the intervals of linearity by the formula logp |ai |p + it where ai xi is the leading term (i.e. maximum in module) in the expansion of f . The ﬁrst obvious application of this other type of Newton polygon is that a function ∞ f (x) = an xn ∈ Cp [[x]] n=1

is bounded in the open disc D0 (1− ; Cp ) if and only if the coeﬃcents an are uniformly bounded.

1.3 Distributions, measures, and the abstract Kummer congruences 1.3.1 Distributions Let us consider a commutative associative ring R, an R-module A and a proﬁnite (i.e. compact and totally disconnected) topological space Y . Then Y is a projective limit of ﬁnite sets:

22

1 Non-Archimedean analytic functions

Y = lim Yi ←− I

where I is a (partially ordered) inductive set and for i ≥ j, i, j ∈ I there are surjective homomorphisms πi,j : Yi → Yj with the condition πi,j ◦ πj,k = πi,k for i ≥ j ≥ k. The inductivity of I means that for any i, j ∈ I there exists k ∈ I with the condition k ≥ i, k ≥ j. By the universal property we have that for each i ∈ I a unique map πi : Y → Yi is deﬁned, which satisﬁes the property πi,j ◦ πi = πj (for each i, j ∈ I). Let Step(Y, R) be the R-module consisting of all R-valued locally constant functions φ : Y → R. Deﬁnition 1.5. A distribution on Y with values in a R-module A is a R-linear homomorphism µ : Step(Y, R) −→ A. For ϕ ∈ Step(Y, R) we use the notations ϕdµ = ϕ(y)dµ(y). µ(ϕ) = Y

Y

Each distribution µ can be deﬁned by a system of functions µ(i) : Yi → A, satisfying the following ﬁnite-additivity condition µ(j) (y) = µ(i) (x) (y ∈ Yj , x ∈ Yi ). (1.25) −1 x∈πi,j (y)

In order to construct such a system it suﬃces to put µ(i) (x) = µ(δi,x ) ∈ A

(x ∈ Yi ),

where δi,x is the characteritic function of the inverse image πi−1 (x) ⊂ Y with respect to the natural projection Y → Yi . For an arbitrary function ϕj : Yj → R and i ≥ j we deﬁne the functions ϕi = ϕj ◦ πi,j ,

ϕ = ϕj ◦ πj ,

ϕ ∈ Step(Y, R),

πi,j

ϕi : Yi −→ Yj −→ R.

A convenient criterion of the fact that a system of functions µ(i) : Yi → A satisﬁes the ﬁnite additivity condition (1.25) (and hence is associated to some distribution) is given by the following condition (compatibility criterion): for all j ∈ I, and ϕj : Yj → R the value of the sums ϕi (y)µ(i) (y), (1.26) µ(ϕ) = µ(i) (ϕi ) = y∈Yi

is independent of i for all large enough i ≥ j. When using (1.26), it suﬃces to verify the condition (1.26) for some “basic” system of functions. For example, if

1.3 Distributions, measures, and the abstract Kummer congruences

23

Y = G = lim Gi ←− i

is a proﬁnite abelian group, and R is a domain containing all roots of unity of the order dividing the order of Y (which is a “supernatural number”) then it suﬃces to check the condition (1.26) for all characters of ﬁnite order χ : G → R, since their R ⊗ Q -linear span coincides with the whole ring Step(Y, R ⊗ Q) by the orthogonality properties for characters of a ﬁnite group (see [Kat3], [Maz-SD]). Example: Bernoulli distributions (see [La1]). Let M be a positive integer, f : Z → C is a periodic function with the period M (i.e. f (x + M ) = ˇ f (x), f : Z/M Z → C). The generalized Bernoulli number (see [Le1], [Saf]) k Bk,f is deﬁned as k! times the coeﬃcient by t in the expansion in t of the following rational quotient M−1 f (a)teat , eMt − 1 a=0 that is, ∞ Bk,f k=0

k!

tk =

M−1 a=0

f (a)teat . eMt − 1

(1.27)

Now let us consider the proﬁnite ring Y = ZS = lim (Z/M Z) ←− M

(S(M ) ⊂ S), the projective limit being taken over the set of all positive integers M with support S(M ) in a ﬁxed ﬁnite set S of prime numbers. Then the periodic function f : Z/M Z → C with S(M ) ⊂ S may be viewed as an element of Step(Y, C). We claim that there exists a distribution Ek : Step(Y, C) → C which is uniquely determined by the condition Ek (f ) = Bk,f for all f ∈ Step(Y, C).

(1.28)

In order to prove the existence of this distribution we use the above criterion (1.26) and check that for every f ∈ Step(Y, C) the right hand side in (1.28) (i.e. Bk,f ) does not depend on the choice of a period M of the function f . It follows directly from the deﬁnition (1.27); however we give here a diﬀerent proof which is based on an interpretation of the numbers Bk,f as certain special values of L-functions. For the function f : Z/M Z → C let L(s, f ) =

∞

f (n)n−s

n=1

be the corresponding L-series which is absolutely convergent for all s with Re(s) > 1 and admits an analytic continuation over all s ∈ C. For this series we have that

24

1 Non-Archimedean analytic functions

Bk,f . (1.29) k For example, if f ≡ 1 is the constant function with the period M = 1 then we have that ∞ t Bk k Bk t = t , ζ(1 − k) = − , k k! e −1 L(1 − k, f ) = −

k=0

Bk being the Bernoulli number. The formula (1.29) is established by means of the contour integral discovered by Riemann (see [La1]). This formula apparently implies the desired independence of Bk,f on the choice of M . We note also that if K ⊂ C is an arbitrary subﬁeld, and f (Y ) ⊂ K then we have from the formula (1.27) that Bk,f ∈ K hence the distribution Ek is a K-valued distribution on Y . 1.3.2 Measures Let R be a topological ring, and C(Y, R) be the topological module of all R-valued functions on a proﬁnite set Y . Deﬁnition 1.6. A measure on Y with values in the topological R-module A is a continuous homomorphism of R-modules µ : C(Y, R) −→ A. The restriction of µ to the R-submodule Step(Y, R) ⊂ C(Y, R) deﬁnes a distribution which we denote by the same letter µ, and the measure µ is uniquely determined by the corresponding distribution since the R-submodule Step(Y, R) is dense in C(Y, R). The last statement expresses the well known fact about the uniform continuity of a continuous function over a compact topological space. Now we consider any closed subring R of the Tate ﬁeld Cp , R ⊂ Cp , and let A be a complete R-module with topology given by a norm | · |A on A compatible with the norm | · |p on Cp so that the following conditions are satisﬁe: • for x ∈ A the equality |x|A = 0 is equivalent to x = 0, • for a ∈ R, x ∈ A: |ax|A = |a|p |x|A , • for all x, y ∈ A: |x + y|A < max(|x|A , |y|A ). Then the fact that a distribution (a system of functions µ(i) : Yi → A) gives rise to a A-valued measure on Y is equivalent to the condition that the system µ(i) is bounded, i.e. for some constant B > 0 and for all i ∈ I, x ∈ Yi the following uniform estimate holds: |µ(i) (x)|A < B.

(1.30)

This criterion is an easy consequence of the non-Archimedean property |x + y|A ≤ max(|x|A , |y|A )

1.3 Distributions, measures, and the abstract Kummer congruences

25

of the norm | · |A (see [Man2], [Vi1]). In particular if A = R = Op = {x ∈ Cp | |x|p ≤ 1} is the subring of integers in the Tate ﬁeld Cp then the set of Op -valued distributions on Y coincides with Op -valued measures (in fact, both sets are R-algebras with multiplication deﬁned by convolution, see section 1.4). Below we give some examples of measures based on the following important criterion of existence of a measure with given properties. Proposition 1.7 (The abstract Kummer congruences). (see [Kat3]). Let {fi } be a system of continuous functions fi ∈ C(Y, Op ) in the ring C(Y, Op ) of all continuous functions on the compact totally disconnected group Y with values in the ring of integers Op of Cp such that Cp -linear span of {fi } is dense in C(Y, Cp ). Let also {ai } be any system of elements ai ∈ Op . Then the existence of an Op -valued measure µ on Y with the property fi dµ = ai Y

is equivalent to the following congruences, for an arbitrary choice of elements bi ∈ Cp almost all of which vanish bi fi (y) ∈ pn Op for all y ∈ Y implies bi ai ∈ pn Op . (1.31) i

i

Remark 1.8. Since Cp -measures are characterised as bounded Cp -valued distributions, every Cp -measures on Y becomes a Op -valued measure after multiplication by some non-zero constant. Proof of proposition 1.7. The nessecity is obvious since b i ai = (pn Op − valued function)dµ = Y i = pn (Op − valued function)dµ ∈ pn Op . Y

In order to prove the suﬃciency we need to construct a measure µ from the numbers ai . For a function f ∈ C( Y, Op ) and a positive integer n there exist elements bi ∈ Cp such that only a ﬁnite number of bi does not vanish, and f− bi fi ∈ pn C(Y, Op ), i

according to the density of the Cp -span of {fi } in C(Y, Cp ). By the assumption (1.31) the value i ai bi belongs to Op and is well deﬁned modulo pn (i.e. does not depend on the choice of bi ). Following N.M. Katz ([Kat3]), we denote this value by “ Y f dµ mod pn ”. Then we have that the limit procedure f dµ = lim “ f dµ mod pn ” ∈ lim Op /pn Op = Op , ←− Y

gives the measure µ.

n→∞

Y

n

26

1 Non-Archimedean analytic functions

1.3.3 The S-adic Mazur measure Let c > 1 be a positive integer coprime to M0 = q q∈S

with S being a ﬁxed set of prime numbers. Using the criterion of the proposition 1.7 we show that the Q -valued distribution deﬁned by the formula Ekc (f ) = Ek (f ) − ck Ek (fc ),

fc (x) = f (cx),

(1.32)

turns out to be a measure where Ek (f ) are deﬁned in 1.3.1, f ∈ Step(Y, Qp ) and the ﬁeld Q is regarded as subﬁeld of Cp . Deﬁne the generelized Bernoulli (M) polynomials Bk,f (X) as ∞

(M)

Bk,f (X)

k=0

M−1 tk te(a+X)t = , f (a) Mt k! e −1 a=0

(1.33)

and the generalized sums of powers Sk,f (M ) =

M−1

f (a)ak .

a=0

Then the deﬁnition (1.33) formally implies that 1 (M) (M) [B (M ) − Bk,f (0)] = Sk−1,f (M ), k k,f

(1.34)

and also we see that k k (M) Bk,f (X) = Bi,f X k−i = Bk,f + kBk−1,f X + · · · + B0,f X k . (1.35) i i=0 The last identity can be rewritten symbolically as Bk,f (X) = (Bf + X)k . The equality (1.34) enables us to calculate the (generalized) sums of powers in terms of the (generalized) Bernoulli numbers. In particular this equality implies that the Bernoulli numbers Bk,f can be obtained by the following p-adic limit procedure (see [La1]): Bk,f = lim

n→∞

1 Sk,f (M pn ) M pn

(a p-adic limit),

(1.36)

where f is a Cp -valued function on Y = ZS . Indeed, if we replace M in (1.34) by M pn with growing n and let D be the common denominator of all (M) coeﬃcients of the polynomial Bk,f (X). Then we have from (1.35) that

1.3 Distributions, measures, and the abstract Kummer congruences

! 1 1 2 (Mpn ) (M) Bk,f (M ) − Bk,f (0) ≡ Bk−1,f (M pn ) (mod p n). k kD

27

(1.37)

The proof of (1.36) is accomplished by division of (1.37) by M pn and by application of the formula (1.34). Now we can directly show that the distribution Ekc deﬁned by (1.32) are in fact bounded measures. If we use (1.31) and take the functions {fi } to be all of the functions in Step(Y, Op ). Let {bi } be a system of elements bi ∈ Cp such that for all y ∈ Y the congruence bi fi (y) ≡ 0 (mod pn ) (1.38) i

holds. Set f = i bi fi and assume (without loss of generality) that the number n is large enough so that for all i with bi = 0 the congruence 1 Sk,fi (M pn ) (mod pn ) M pn

Bk,fi ≡

(1.39)

is valid in accordance with (1.36). Then we see that Mp −1 n

n −1

Bk,f ≡ (M p )

i

bi fi (a)ak

(mod pn ),

(1.40)

a=0

hence we get by deﬁnition (1.32): Ekc (f ) = Bk,f − ck Bk,fc ≡ (M pn )−1

(1.41)

n Mp −1

i

bi fi (a)ak − fi (ac)(ac)k

(mod pn ).

a=0

Let ac ∈ {0, 1, · · · , M pn − 1}, such that ac ≡ ac (mod M pn ), then the map a −→ ac is well deﬁned and acts as a permutation of the set {0, 1, · · · , M pn − 1}, hence (1.41) is equivalent to the congruence Ekc (f )

= Bk,f −c Bk,fc ≡ k

ak − (ac)k c

i

M pn

n Mp −1

bi fi (a)ak

(mod pn ). (1.42)

a=0

Now the assumption (1.37) formally inplies that Ekc (f ) ≡ 0 (mod pn ), completing the proof of the abstact congruences and the construction of measures Ekc . Remark 1.9. The formula (1.41) also implies that for all f ∈ C(Y, Cp ) the following holds f) (1.43) Ekc (f ) = kE1c (xk−1 p where xp : Y −→ Cp ∈ C(Y, Cp ) is the composition of the projection Y −→ Zp and the embedding Zp → Cp .

28

1 Non-Archimedean analytic functions

Indeed if we put ac = ac + M pn t for some t ∈ Z then we see that akc − (ac)k = (ac + M pn t)k − (ac)k ≡ kM pn t(ac)k−1

(mod (M pn )2 ),

and we get that in (1.42): akc − (ac)k ac − ac ≡ k(ac)k−1 n Mp M pn

(mod M pn ).

The last congruence is equivalent to saying that the abstract Kummer congruences (1.31) are satisﬁed by all functions of the type xk−1 fi for the measure p E1c with fi ∈ Step(Y, Cp ) establishing the identity (1.43).

1.4 Iwasawa algebra and the non-Archimedean Mellin transform 1.4.1 The domain of deﬁnition of the non-Archimedean zeta functions In the classical case the set on which zeta functions are deﬁned is the set of complex numbers C which may be viewed equally as the set of all continuous characters (more precisely, quasicharacters) via the following isomorphism: ∼

× C −→ Homcont (R× +, C ) s s −→ (y −→ y )

(1.44)

The construction which associates to a function h(y) on R× + (with certain growth conditions as y → ∞ and y → 0) the following integral dy Lh (s) = h(y)y s × y R+ (which converges probably not for all values of s) is called the Mellin transform. For example, if ζ(s) = n≥1 n−s is the Riemann zeta function, then the function ζ(s)Γ (s) is the Mellin transform of the function h(y) = 1/(1 − e−y ): ζ(s)Γ (s) =

∞ 0

1 dy ys , −y 1−e y

(1.45)

so that the integral and the series are absolutely convergent for Re(s) > 1. For an arbitrary function of type f (z) =

∞ n=1

a(n)e2iπnz

1.4 Iwasawa algebra and the non-Archimedean Mellin transform

29

with z = x + iy ∈ H in the upper half plane H and with the growth condition a(n) = O(nc ) (c > 0) on its Fourier coeﬃcients, we see that the zeta function L(s, f ) =

∞

a(n)n−s ,

n=1

essentially coincides with the Mellin transform of f (z), that is ∞ Γ (s) dy L(s, f ) = f (iy)y s . (2π)s y 0

(1.46)

Both sides of the equality (1.46) converge absolutely for Re(s) > 1 + c. The identities (1.45) and (1.46) are immediately deduced from the well known integral representation for the gamma-function ∞ dy Γ (s) = (1.47) e−y y s , y 0 × where dy y is a measure on the group R+ which is invariant under the group translations (Haar measure). The integral (1.47) is absolutely convergent for Re(s) > 0 and it can be interpreted as the integral of the product of an additive character y → e−y of the group R(+) restricted to R× + , and of the multiplicative character y → y s , integration is taken with respect to the Haar measure dy/y on the group R× +. In the theory of the non-Archimedean integration one considers the group Z× S (the group of units of the S-adic completion of the ring of integers Z) ˆ (the completion of instead of the group R× , and the Tate ﬁeld C = Q p

+

p

an algebraic closure of Qp ) instead of the complex ﬁeld C. The domain of deﬁnition of the p-adic zeta functions is the p-adic analytic group × × XS = Homcont (Z× S , Cp ) = X(ZS ),

where: and the symbol

(1.48)

× ∼ Z× S = ⊕q∈S Zq ,

X(G) = Homcont (G, C× p)

(1.49)

denotes the functor of all p-adic characters of a topological group G (see [Vi1]). 1.4.2 The analytic structure of XS Let us consider in detail the structure of the topological group XS . Deﬁne Up = {x ∈ Z× p

| x ≡ 1 (mod pν )},

where ν = 1 or ν = 2 according as p > 2 or p = 2. Then we have the natural decomposition

30

1 Non-Archimedean analytic functions

 XS = X (Z/pν Z)× ×

  × X(Up ). Z× q

(1.50)

q=p

The analytic dstructure on X(Up ) is deﬁned by the following isomorphism (which is equivalent to a special choice of a local parameter): ∼

ϕ : X(Up ) −→ T = {z ∈ C× | |z − 1|p < 1}, p where ϕ(x) = x(1 + pν ), 1 + pν being a topoplogical generator of the multiplicative group Up ∼ = Zp . An arbitrary character χ ∈ XS can be uniquely represented in the form χ = χ0 χ1 where χ0 is trivial on the component Up , and χ1 is trivial on the other component (Z/pν Z)× × Z× q . q=p

The character χ0 is called the tame component, and χ1 the wild component of the character χ. We denote by the symbol χ(t) the (wild) character which is uniquely determined by the condition χ(t) (1 + pν ) = t with t ∈ Cp , |t|p < 1. In some cases it is convenient to use another local coordinate s which is analogous to the classical argument s of the Dirichlet series: Op −→ XS s −→ χ(s) , where χ(s) is given by χ(s) ((1 + pν )α ) = (1 + pν )αs = exp(αs log(1 + pν )). The character χ(s) is deﬁned only for such s for which the series exp is padically convergent (i.e. for |s|p < pν−1/(p−1) ). In this domain of values of the argument we have that t = (1 + pν )s − 1. But, for example, for (1 + n t)p = 1 there is certainly no such value of s (because t = 1), so that the scoordonate parametrizes a smaller neighborhood of the trivial character than the t-coordinate (which parametrizes all wild characters) (see [Man2], [Man3]). | |z − 1|p < Recall that an analytic function F : T −→ Cp (T = {z ∈ C× p 1}), is deﬁned as the sum of a series of the type i≥0 ai (t − 1)i (ai ∈ Cp ), which is assumed to be absolutely convergent for all t ∈ T . The notion of an analytic function is then obviously extended to the whole group XS by shifts. The function ∞ F (t) = ai (t − 1)i i=0

is bounded on T iﬀ all its coeﬃcients ai are universally bounded. This last fact can be easily deduced for example from the basic properties of the Newton

1.4 Iwasawa algebra and the non-Archimedean Mellin transform

31

polygon of the series F (t) (see [Kob1], [Vi1], [Vi2]). If we apply to these series the Weierstrass preparation theorem (see [Kob2], [La1], [Man1] and theorem 1.3) we see that in this case the function F has only a ﬁnite number of zeroes on T (if it is not identically zero). In particular, consider the torsion subgroup XStors ⊂ XS . This subgroup is discrete in XS and its elements χ ∈ XStors can be obviously identiﬁed with primitive Dirichlet characters χmodM such that the support S(χ) = S(M ) of the conductor of χ is containded in S. This identiﬁcation is provided by a ﬁxed embedding denoted ×

ip : Q → C× p if we note that each character χ ∈ XStors can be factored through some ﬁnite factor group (Z/M Z)× : × ip

× × χ : Z× S → (Z/M Z) → Q → Cp ,

and the smallest number M with the above condition is the conductor of χ ∈ XStors . The symbol xp will denote the composition of the natural projection Z× S → × × Zp and of the natural embedding Z× p → Cp , so that xp ∈ XS and all integers k can be considered as the characters xkp : y −→ y k . Let us consider a bounded Cp -analytic function F on XS . The above statement about zeroes of bounded Cp -analytic functions implies now that the function F is uniquely determined by its values F (χ0 χ), where χ0 is a ﬁxed character and χ runs through all elements χ ∈ XStors with possible exclusion of a ﬁnite number of characters in each analyticity component of the decomposition (1.50). This condition is satisﬁed, for example, by the set of characters χ ∈ XStors with the S-complete conductor (i.e. such that S(χ) = S), and even for a smaller set of characters, for example for the set obtained by imposing the additional assumption that the character χ2 is not trivial (see [Man2], [Man3], [Vi1]). 1.4.3 The non-Archimedean Mellin transform Let µ be a (bounded) Cp -valued measure on Z× S . Then the non-Archimedean Mellin transform of the measure µ is deﬁned by Lµ (x) = µ(x) = xdµ, (x ∈ XS ), (1.51) Z× S

which represents a bounded Cp -analytic function Lµ : XS −→ Cp .

(1.52)

Indeed, the boundedness of the function Lµ is obvious since all characters x ∈ XS take values in Op and µ also is bounded. The analyticity of this function

32

1 Non-Archimedean analytic functions

expresses a general property of the integral (1.51), namely that it depends analytically on the parameter x ∈ XS . However, we give below a pure algebraic proof of this fact which is based on a description of the Iwasawa algebra. This description will also imply that every bounded Cp -analytic function on XS is the Mellin transform of a certain measure µ. 1.4.4 The Iwasawa algebra (see [La1]). Let O be a closed subring in Op = {z ∈ Cp | |z|p ≤ 1}, (i ∈ I),

G = lim Gi , ←− i

πij

a proﬁnite group. Then the canonical homomorphism Gi ←− Gj induces a homomorphism of the corresponding group rings O[Gi ] ←− O[Gj ]. Then the completed group ring O[[G]] is deﬁned as the projective limit O[[G]] = lim O[[Gi ]], ←−

(i ∈ I).

i

Let us consider also the set Dist(G, O) of all O-valued distributions on G which itself is an O-module and a ring with respect to multiplication given by the convolution of distributions, which is deﬁned in terms of families of functions (i) (i) µ1 , µ2 : Gi −→ O, (see the previous section) as follows: (i) (i) µ1 (y1 )µ2 (y2 ), (µ1 µ2 )(i) (y) =

(y1 , y2 ∈ Gi )

(1.53)

y=y1 y2

Recall also that the O-valued distributions are identiﬁed with O-valued measures. Now we describe an isomorphism of O-algebras O[[G]] and Dist(G, O). In this case when G = Zp the algebra O[[G]] is called the Iwasawa algebra. Theorem 1.10. (a) Under the same notation as above there is the canonical isomorphism of O-algebras ∼

Dist(G, O) −→ O[[G]].

(1.54)

(b) If G = Zp then there is an isomorphism ∼

O[[G]] −→ O[[X]],

(1.55)

where O[[X]] is the ring of formal power series in X over O. The isomorphism (1.55) depends on a choice of the topological generator of the group G = Zp .

1.4 Iwasawa algebra and the non-Archimedean Mellin transform

33

1.4.5 Formulas for coeﬃcients of power series We noticed above that the theorem 1.10 would imply a description of Cp -analytic bounded functions on XS in terms of measures. Indeed, these functions are deﬁned on analyticity components of the decomposition (1.50) as certain power series with p-adically bounded coeﬃcients, that is, power series, whose coeﬃcients belong to Op after multiplication by some constant from C× p . Formulas for coeﬃcients of these series can be also deduced from the proof of the theorem. However, we give a more direct computation of these coeﬃcients in terms of the corresponding measures. Let us consider the component aUp of the set Z× S where a ∈ (Z/pν Z)× × Z× q , q=

and let µa (x) = µ(ax) be the corresponding measure on Up deﬁned by restric∼ tion of µ to the subset aUp ⊂ Z× S . Consider the isomorphism Up = Zp given by: y = γ x (x ∈ Zp , y ∈ Up ), with some choice of the generator γ of Up (for example, we can take γ = 1+pν ). Let µ a be the corresponding measure on Zp . Then this measure is uniquely determined by values of the integrals x dµ a (x) = ai , (1.56) i Zp with the interpolation polynomials xi , since the Cp -span of the family " # x (i ∈ Z, i ≥ 0) i is dense in C(Zp , Op ) according to the Mahler’s interpolation theorem for continuous functions on Zp (see 1.2.1 and [Mah]). Indeed, from the basic properties of the interpolation polynomials it follows that x bi ≡ 0 (mod pn ) (for all x ∈ Zp ) =⇒ bi ≡ 0 (mod pn ). i i We can now apply the abstract Kummer congruences (see proposition 1.7), which imply that for arbitrary choice of numbers ai ∈ Op there exists a measure with the property (1.56). On the other hand we state that the Mellin transform Lµa of the measure µa is given by the power series Fa (t) with coeﬃcients as in (1.56), that is ∞ x χ(t) (y)dµ(ay) = (1.57) dµa (x) (t − 1)i i Up Z p i=0

34

1 Non-Archimedean analytic functions

for all wild characters of the form χ(t) , χ(t) (γ) = t, |t − 1|p < 1. It suﬃces to show that (1.57) is valid for all characters of the type y −→ y m , where m is a positive integer. In order to do this we use the binomial expansion x

γ mx = (1 + (γ m − 1)) =

∞ x (γ m − 1)i , i i=0

which implies that

m

y dµ(ay) =

γ Zp

up

mx

dµ a (x)

=

x dµa (x) (γ m − 1)i , i Zp

∞ i=0

establishing (1.57). 1.4.6 Example. The S-adic Mazur measure and the non-Archimedean Kubota-Leopoldt zeta function (see [La1], [Ku-Le], [Le2], [Wa]). Let us ﬁrst consider a positive integer c ∈ Z× S ∩ Z, c > 1 coprime to all primes in S. Then for each complex number s ∈ C there exists a complex distribution µcs on GS = Z× S which is uniquely determined by the following condition µcs (χ) = (1 − χ−1 (c)c−1−s )LM0 (−s, χ),

(1.58)

$ where M0 = q∈S q (see 1.3.1). Moreover, the right hand side of (1.58) is holomorphic for all s ∈ C including s = −1. If s is an integer and s ≥ 0 then according to criterion of proposition 1.7 the right hand side of (1.58) belongs to the ﬁeld Q(χ) ⊂ Qab ⊂ Q generated by values of the character χ, and we get a distribution with values in Qab . If we now apply to (1.58) the ﬁxed embedding ip : Q → Cp we get a Cp -valued distribution µ(c) = ip (µc0 ) which turns out to be an Op -measure in view of proposition 1.7, and the following equality holds µ(c) (χxrp ) = ip (µcr (χ)). This identity relates the special values of the Dirichlet L-functions at diﬀerent non-positive points. The function −1 L(x) = 1 − c−1 x(c)−1 Lµ(c) (x)

(x ∈ XS )

(1.59)

is well deﬁned and it is holomorphic on XS with the exception of a simple pole at the point x = xp ∈ XS . This function is called the non-Archimedean zetafunction of Kubota-Leopoldt. The corresponding measure µ(c) will be called the S-adic Mazur measure.

1.5 Admissible measures and their Mellin transform

35

1.4.7 Measures associated with Dirichlet characters Let ω mod$M be a ﬁxed primitive Dirichlet character such that (M, M0 ) = 1 with M0 = q∈S q. This section gives a construction of the direct image of × ¯ the Mazur measure under the natural map Z× ¯ → ZS where S = S ∪ S(M ), S $ ¯0 = M ¯ q. This construction is used in Chapter III. We show that for any q∈S ¯ 0 ) = 1, c > 1, there exist Cp -measures µ+ (c, ω), positive integer c with (c, M × − µ (c, ω) on ZS which are determined by the following conditions, for s ∈ Z, s > 0: Cωχ¯ −1 s + −s · (1.60) χxp dµ (c, ω) = (1 − χω(c)c ¯ ) ip × G(ω χ) ¯ ZS 1 − χ¯ ω (q)q s−1 · ¯ L+ M0 (s, χω), −s 1 − χω(q)q ¯ q∈S\S(χ)

and for s ∈ Z, s ≤ 0, i−1 p

Z× S

χxsp dµ− (c, ω)

= (1 − χ¯ ω (c)cs−1 )L− ¯ M0 (s, χω),

(1.61)

where δ ¯ = LM¯ (s, χω)2i ¯ L+ M0 (s, χω)

Γ (s) cos(π(s − δ)/2) (2π)s

¯ = LM¯ (s, χω) ¯ L− M0 (s, χω)

(1.62) (1.63)

are the normalized Dirichlet L-functions with δ ∈ {0, 1}, and χω(−1) ¯ = (−1)δ . The function G(ω χ) ¯ denote the Gauss sum of the Dirichlet character ω χ. ¯ The functions (1.60), (1.61) satisfy the functional equation 1 − χ¯ ω (q)q s−1 (1 − s, χω) ¯ = ¯ L− L+ M0 M0 (s, χω). −s 1 − χω(q)q ¯ q∈S\S(χ)

¯ Indeed, by the deﬁnition of the S-adic Mazur measure µc on Z× S , (1.60) and (1.61) are given by def −1 dµ+ (c, ω) = Z× xx−1 dµc , p ω Z× ¯ S S def xdµ− (c, ω) = Z× x−1 ωdµc , Z× S

¯ S

where x ∈ XS and XS¯ is viewed as a subgroup of XS .

1.5 Admissible measures and their Mellin transform 1.5.1 Non-Archimedean integration This construction was nicely explained by J. Coates and B. Perrin-Riou [Co-PeRi] using the Fourier transform of distributions. Let S be a ﬁnite set

36

1 Non-Archimedean analytic functions

of primes containing p. The set on which our non-Archimedean zeta functions are deﬁned is the Cp -adic analytic Lie group XS = Homcont (GalS , C× p ),

(1.64)

where GalS is the Galois group of the maximal abelian extension of Q unramiﬁed outside S and inﬁnity. Now we recall the notion of h-admissible measures on GalS and properties of their Mellin transform. These Mellin transforms are certain p-adic analytic functions on the Cp -analytic group XS . Recall that by class ﬁeld theory the group GalS is described as the projective limit (Z/M Z)× = Z× GalS ∼ = lim S, ←−

(1.65)

M

where M runs over integers with support in the set of primes S (i.e. S(M ) ⊂ S). The canonical Cp -analytic structure on XS is obtained by shifts from the × obvious Cp -analytic structure on the subgroup Homcont (Z× p , Cp ) ⊂ XS . We tors regard the elements of ﬁnite order χ ∈ XS as Dirichlet characters whose conductor c(χ) may contain only primes in S, by means of the decomposition × i

∞ × × × χ : A× Q /Q −→ ZS −→ Q −→ C ,

(1.66)

where i∞ is a ﬁxed embedding. The character χ ∈ XStors form a discrete subgroup XStors ⊂ XS . We shall need also the natural homomorphism × × xp : Z× S −→ Zp −→ Cp ,

xp ∈ XS ,

(1.67)

so that all integers k ∈ Z can be regarded as characters of the type xkp : y −→ yk . Recall that a p-adic measure on Z× S may be regarded as a bounded Cp linear form µ on the space C(Z× , C ) of all continuous Cp -valued functions p S C(Z× S , Cp ) −→ Cp

ϕ −→ µ(ϕ) =

Z× S

ϕdµ

which is uniquely determined by its restriction to the subspace Step(Z× S , Cp ) of locally constant functions. We denote by µ(a + (M )) the value of µ on the characteristic function of the set a + (M ) = {x ∈ Z× S

| x ≡ a (mod M )} ⊂ Z× S.

The Mellin transform Lµ of µ is a bounded analytic function Lµ : XS −→ Cp χ −→ Lµ (χ) =

Z× S

χdµ

on XS , which is uniquely determined by its values Lµ (χ) for the characters χ ∈ XStors .

1.5 Admissible measures and their Mellin transform

37

1.5.2 h-admissible measure A more delicate notion of an h-admissible measure was introduced by Y. Amice, J. V´elu and M.M. Viˇsik (see [Am-Ve], [Vi1]). Let C(Z× S , Cp ) denote the space of Cp -valued functions that can be locally represented by polynomials of degree less than a natural number h of the variable xp ∈ XS introduced above. Deﬁnition 1.11. A Cp -linear form µ : C h (Z× S , Cp ) −→ Cp is called an hadmissible measure if for all r = 0, 1, · · · , h − 1 the following growth condition is satisﬁed . (xp − ap )r dµ = o |M |r−h sup p a∈Z× a+(M) S

p

Note that the notion of a bounded measure is covered by the case h = 1, but the set of 1-admissible measures is bigger; it consists of the so called measures of bounded growth (see [Man2], [Vi1]), which are characterized by the property that they grow on open compact sets a + (M ) ⊂ Z× S slower than o(|M |−1 elu and M.M. Viˇsik) p ). We know (essentially due to Y. Amice, J. V´ that each h-admissible measure can be uniquely extended to a linear form on the Cp -space of all locally analytic functions so that one can associate to its Mellin transform Lµ : XS −→ Cp χ −→ Lµ (χ) = χdµ Z× S

which is a Cp -analytic function on XS of the type o(log(xhp )). Moreover, the measure µ is uniquely determined by the special values of the type Lµ (χxrp ) with χ ∈ XStors and r = 0, 1, · · · , h−1. First example of h-admissible measures and their L-functions concerned the L-function Lf (s, χ) =

∞

χ(n)an n−s

n=1

of an elliptic normalized Hecke cusp eigenform f (z) =

∞

an exp(2iπz)

(Im(z) > 0)

n=1

with ap divisible by p (the supersingular case). This notion is used in Chapter 4. We can consider another example, provided by the analytic function log(x). For anys ∈ N let us deﬁne a distribution µs by µs (χ) = µ(χxsp ) = s log(1 + pν ),

for all χ ∈ XStors

and s ∈ N.

38

1 Non-Archimedean analytic functions

This sequence of distributions turns out to be a 2-admissible measure. Indeed, for r = 0, 1 (xp − ap )r dµ = χmodM χ−1 (a) Z× χ(x)(xp − ap )r dµ S a+(M) " 0, if r = 0, = log(1 + pν ) χmodM χ−1 (a), if r = 1 But, in the case r = 1 the last sum χmodM χ−1 (a) is equal to 0 if a = 1, and if a = 1 then this sum is equal to ϕ(M ). So we see that the conditions of the deﬁnition 1.11 are satisﬁed, so that we obtain a 2-admissible measure. If we look now at its Mellin transform xdµ, Lµ (x) = Z× S

we see immediately that Lµ (χxp ) = log(xp ) which is a Cp -analytic function on XS× of the type o(log(xp )2 ).

1.6 Complex valued distributions, associated with Euler products In this section we give a general construction of distributions, attached to rather arbitrary Euler products. This construction provides a generalization of measures, which were ﬁrst introduced by Y.I. Manin, B. Mazur and H.P.F. Swinnerton-Dyer (see [Man2], [Maz-SD]). Our construction ([Pa2], [Pa3]) was already successfully used in several problems concerning the p-adic analytic interpolation of Dirichlet series (see [Ar], [Co-Schm], [Sc]). 1.6.1 Dirichlet series Let S be a ﬁxed ﬁnite set of primes and D(s) =

∞

an n−s

(s, an ∈ C)

(1.68)

n=1

be a Dirichlet series with the following multiplicativity property of its coeﬃcients an : Fq (q −s )−1 an n−s , (1.69) D(s) = q∈S

n=1

(n,S)=1

where the condition (n, S) = 1 means that n is not divisible by any prime in S, and Fq (X) are polynomials with the constant term equal to 1:

1.6 Complex valued distributions, associated with Euler products mq

Fq (X) = 1 +

Aq,i X i .

39

(1.70)

i=1

We assume also that the series (1.68) is absolutely convergent in some right half plane Re(s) > 1 + c (c ∈ R). This assumption is satisﬁed in most cases, for example, when the coeﬃcients an satisfy the estimate |an | = O(nc ). For a Dirichlet character χ : (Z/M Z)× → C× modulo M ≥ 1 the twisted Dirichlet series is deﬁned by ∞ D(s, χ) = χ(n)an n−s . (1.71) n=1

For all s ∈ C such that the series (1.68) is absolutely convergent, let us deﬁne the function Ps : Q → C by the equality ∞

Ps (x) =

e(nx)an n−s

(e(x) = exp(2iπx)).

(1.72)

n=1

Using the functions (1.72) we construct C-valued distributions µs on the compact group Z× S such that for every primitive Dirichlet character χ viewed × the value of the series D(s, χ) ¯ at s with as a homomorphism χ : Z× S → C Re(s) > 1 + c is canonically expressed in terms of the integral def χdµs = χ(a)µs (a + (M )). Z× S

amodM

(a,S)=1

Let for every q ∈ S, α(q) denote a ﬁxed root of the inverse polynomial X

mq

Fq (X

−1

)=X

mq

+

mq

Aq,i X mq −i

i=1

(that is, an inverse root of Fq (X)). We suppose that α(q) = 0 for every q ∈ S, and let us extend the deﬁnition of numbers α(n) to all positive integers whose support is contained in S (by multiplicativity): α(q)ordq (n) (S(q) ⊂ S). α(n) = q∈S

Let us deﬁne an auxiliary polynomial mq −1

Hq (X) = 1 +

Bq,i X i

(1.73)

i=1

by means of the relation Fq (X) = (1 − α(q)X)Hq (X),

(1.74)

40

1 Non-Archimedean analytic functions

which implies the identities Bq,i = −

mq

(i = 1, · · · , mq − 1)

Aq,j α(q)i−j

(1.75)

j=I+1

for the coeﬃcients of the polynomial (1.73). Let us also introduced the following ﬁnite Euler product B S (n)n−s = Hq (q −s ), (1.76) q∈S

S(n)⊂S

in which the coeﬃcients B S (n) are given by means of (1.75), namely, B S (n) = B S (q ordq (n) ) (S(n) ⊂ S), q∈S

"

with

Bq,i , for i < mq 0, otherwise. Now we state the main result of the section. S

i

B (q ) =

(1.77)

Theorem 1.12. (a) For any choice of the inverse roots α(q) = 0 (q ∈ S), and for any s from the convergency region of the series (1.68) there exists a distribution µs = µs,α on Z× S whose values on open compact subsets of the are given by the following type a + (M ) ⊂ Z× S a M s−1 n µs (a + (M )) = n−s , B S (n)Ps (1.78) α(M ) M S(n)⊂S

so that the sum in (1.78) is ﬁnite and the numbers B S (n) are deﬁned by (1.77). (b) For any primitive Dirichlet character χ viewed as a function χ : Z× S → × C the following equality holds Cχs−1 −s G(χ)D(s, χ), ¯ 1 − χ(q)α(q)−1 q −s Hq χ(q)q ¯ χdµs = α(Cχ ) Z× S q∈S\S(χ)

(1.79) with G(χ) =

χ(a)e

amodCχ

a Cχ

being the Gauss sum, Cχ the conductor, and S(χ) the support of the conductor of χ. Remark 1.13. The distribution µs can be obtained as the Fourier transform of the standard zeta-distribution attached to the Diriclet series D(s, χ) = Hq (χ(q)q −s ) = χ(n)a0 (n)n−s , q∈S

n≥1

where a0 (qn) = α(q)a0 (n) for each n ∈ N and q ∈ S (see [Co-PeRi]).

1.6 Complex valued distributions, associated with Euler products

41

Proof of theorem 1.12 The following proof of this theorem diﬀers from that given in [Pa2] and is based on the compatibility criterion (proposition 1.7). We check that the sum χ(a)µs (a + (M )), (1.80) amodM

(a,S)=1

does not depend on the choice of a positive integer M with the condition Cχ |M , S(M ) = S. This will be provided by a calculation which also implies that (1.80) coincides with the right hand side of (1.79) (and therefore is independent of M ). Lemma 1.14. For an arbitrary positive integer n and Cχ |M put χ(a)e(an/M ). Gn,M = amodM

(a,S)=1

Then the following holds

M Gn,M (χ) = G(χ) Cχ

−1

µ(d)d

χ(d)δ

d|(M/Cχ )

dn (M/Cχ )

χ ¯

dn (M/Cχ )

,

in which µ denotes the M¨ obius function, δ(x) = 1 or 0 according as x ∈ Z or not, and we assume that the character χ is primitive modulo Cχ . Proof. The proof of this lemma is deduced from the well known property of the M¨ obius function: " 1, if n = 1; µ(d) = 0, if n > 1. d|n

Consequently, Gn,M takes the following form: µ(d)χ(a)e(an/M ) = d|(a,M )

amodM

=

d|M

=

µ(d) µ(d)d

χ(d)

d|M

=

d|(M/Cχ )

χ(da1 )e(da1 n/M )

a1 modM/d −1

−1

µ(d)d

a1 modM

χ(d)δ

χ(a1 )e(da1 n/M ) dn (M/Cχ )

since χ(a1 ) depend only on a1 modCχ , and

a1 modM

χ(a1 )e(da1 n/M ),

42

1 Non-Archimedean analytic functions

e(a1 dn/M ) = e((a1 /Cχ )(dn/(M/Cχ ))). In the above equality we changed the order of summation, then we replaced the index of summation a by da1 and extended a system of residue classes a1 modM/d to a system a1 modM . Now we transform the summation into that one modulo Cχ . It remains to use the well known property of Gauss sums (see, for example, [Sh1], lemma 3.63): Gn,Cχ (χ) = χ(n)G(χ), ¯ establishing the lemma. In order to deduce the theorem, we now transform (1.80), taking into account the deﬁnition (1.78) and lemma 1.14: M s−1 α(M )

χ(a)

B S (n)n−s

n

amodM

an1 e

ann

n1

(a,M)=1

1

M

n−s 1 =

(1.81)

M s−1 S B (n)n−s an1 n−s 1 Gnn1 ,M (χ) α(M ) n n 1 Ms = G(χ) B S (n)n−s an1 n−s 1 · α(M )Cχ n n1 µ(d) dnn1 dnn1 χ(d)δ · χ ¯ . d (M/Cχ ) (M/Cχ ) =

d|(M/Cχ )

From the last formula we see that non-vanishing terms in the sum over n and n1 must satisfy the condition (M/Cχ d)|nn1 . Let us now split n1 into two factors n1 = n 1 · n 1 so that S(n 1 ) ⊂ S, and (n 1 , S) = 1. Then nn 1 dnn1 ¯ χ ¯ = χ(n ¯ 1 )χ (1.82) (M/Cχ ) (M/Cχ d)

and

dnn1 (M/Cχ )

(1.83)

since (n 1 , M ) = 1. According to (1.69) one has an1 n 1 −s = Fq (q −s )−1 .

(1.84)

S(n1 )⊂S

=δ

nn 1 (M/Cχ d)

,

δ

q∈S

Now we use the deﬁnition of the ﬁnite Euler product (1.76) and of the polynomials Hq (X) which we rewrite here in the form B S (n)n−s Fq (q −s )−1 = (1 − α(q)q −s )−1 . n

q∈S

q∈S

1.6 Complex valued distributions, associated with Euler products

43

Consequently,

B S (n)n−s

S(n1 )⊂S

n

an1 n 1 −s =

an2 n−s 2

S(n2 )⊂S

and for S(n2 ) ⊂ S we have that

α(n2 ) =

n2 =nn1

B S (n)an1 .

(1.85)

Keeping in mind (1.82) and (1.83) we transform (1.81) to the following µ(d) χ(d) χ(n ¯ 1 )an1 n 1 −s · (1.86) d d|(M/Cχ ) (n1 ,S)=1 nn1 nn 1 · B S (n)an1 (nn 1 )−s δ χ ¯ . (M/Cχ d) (M/Cχ d) Ms G(χ) α(M )Cχ n

n,n1

Now we transform (1.86) with the help of the relation (1.85), taking into account that non-zero summands can only occur for such n2 = nn 1 which are divisible by M/(Cχ d), (i.e. we put n2 = (M/Cχ d)n3 , S(n3 ) ⊂ S). We also note that by the deﬁnition of our Dirichlet series we have −s B S (n)an1 (nn 1 )−s = D(s, χ) ¯ Fq (χ(q)q ¯ ). n,n1

q∈S\S(χ)

Therefore (1.86) transforms to the following Ms G(χ) α(M )Cχ

−s Fq (χ(q)q ¯ )

q∈S\S(χ)

d|(M/C )

µ(d) χ(d)· d

χ −s n3 M n3 M · χ(n ¯ 3 )α = Cχ d cχ d S(n3 )⊂S Cχs−1 −s G(χ)D(s, χ) ¯ = Fq (χ(q)q ¯ ) µ(d)ds−1 χ(d)α(d)−1 · α(Cχ ) q∈S\(χ) d|(M/Cχ ) χ(n ¯ 3 )α(n3 )n−s . · 3

S(n3 )⊂S

The proof of the theorem is accomplished by noting that µ(d)ds−1 χ(d)α(d)−1 = (1 − χ(q)α(q)−1 q s−1 ),

d|(M/Cχ )

q∈S\S(χ)

χ(n ¯ 3 )α(n3 )n−s 3

=

S(n3 )⊂S

−s −1 (1 − χ(q)α(q)q ¯ )

q∈S\S(χ)

=

q∈S\S(χ)

−s −1 −s Fq (χ(q)q ¯ ) Hq (χ(q)q ¯ ).

44

1 Non-Archimedean analytic functions

1.6.2 Concluding remarks This construction admits a generalization [Pa4] to the case of rather general Euler products over prime ideals in algebraic number ﬁelds. These Euler products have the form an N (n)−s = Fp (N (p)−s )−1 , D(s) = n

p

where n runs over the set of integrals ideals, and p over the set of prime ideals of integers OK of a number ﬁeld K, with N (n) denoting the absolute norm of an ideal n, and Fp ∈ C[X] being polynomials with the condition Fp (0) = 1. In [Pa4] we constructed certain distributions, which provide integral representations for special values of Dirichlet series of the type χ(n)an N (n)−s , D(s, χ) = n

where χ denote a Hecke character of ﬁnite order, whose conductor consists only of prime ideals belonging to a ﬁxed ﬁnite set S of non-Archimedean places of K. The main result of [Pa4] provides a generalization of theorem 4.2 of the earlier work of Yu.I. Manin [Man3]. In the construction of non-Archimedean convolutions of Hilbert modular forms given in chapter 4, we give another approach to local distributions, which diﬀers from the given above and is applicable only to certain Dirichlet series (namely, to convolutions of Rankin type). However, there is an interesting link between these two approaches, which is based on a general construction of p-adic distributions attached to motives. It turns out that both types of distributions can be obtained using the Fourier transform in the distribution space (see [Co-PeRi], [Co-Schn]).

2 Siegel modular forms and the holomorphic projection operator

This chapter contains some preparatory facts which we shall use in the construction of non-Archimedean standard zeta-functions in the next chapter. We recall main properties of Siegel modular forms and of the action of the Hecke algebra on them, as well as the deﬁnitions of spinor zeta-functions and standard zeta-functions, see also [An1], [An3]. Then in the second section, we present some standard results on theta series with Dirichlet character (see [An-Ma1], [An-Ma2], [St2]), and recal the deﬁnitions of Siegel-Eisenstein series, and of Rankin type convolutions of Siegel modular forms. Also, we recall their relation with the standard zeta functions (Andrianov’s identity). In the third section, we give an exposition of some results of G. Shimura and P. Feit on real analytic Siegel-Eisenstein series and their analytic continuation in terms of conﬂuent hypergeometric functions (see [Fe], [Sh5], [Sh7]). These results extend previous results of V.L. Kalinin [Kal] and R. Langlands [Ll]. In the ﬁnal section, a detailed study of holomorphic projection operator and its basic properties is given. Note that the most complete and elegant exposition of properties of the holomorphic projection operator on symplectic and unitary groups is contained in [Sh9]. The formula of theorem 2.18 provides an explicit formula for computing the holomorphic projection onto the space of holomorphic (not necessary cusp) modular forms for functions belonging to a wide class of C ∞ -Siegel modular forms. Previously the holomorphic projection operator onto the space of cusp form was studied by J. Sturm [St1], [St2], B. Gross and D. Zagier [Gr-Za] under some assumptions on the growth of modular forms. Theorem 2.16 gives an explicit description of the action of this operator in terms of Fourier expansions. Here we also give a very explicit formula (2.121) for the special (critical) values of the conﬂuent hypergeometric function. Notations Let A be a commutative ring with identity, then Mr,s (A) denote the set of all (r × s)-matrices with coeﬃcients in A. For z ∈ Mr (C) put er (z) = e(tr(z)) with e(u) = exp(2iπu) for u ∈ C. We denote by tz ∈ Mr,s (A) the matrix, which

M. Courtieu and A. Panchishkin: LNM 1471, pp. 45–93, 2004. c Springer-Verlag Berlin Heidelberg 2004

46

2 Siegel modular forms and the holomorphic projection operator

is transpose to z ∈ Mr,s (A), and write ξ[η] for tηξη. For a non-degenerate square matrix ξ we put ξ = tξ −1 . If ξ is a hermitian matrix then we write ξ ≥ 0 or ξ > 0 according as ξ is non negative or positive deﬁnite. Let Hm denote the Siegel upper half plane on the degree m, Hm = {z ∈ Mm (C) |

t

z = z = x + iy, y > 0},

so that Hm is a complex analytic variety whose dimension is denoted by m = m(m + 1)/2. Let the symbol Am denote the lattice of all half integral symetric matrices in the vector space V = {y ∈ Mm (R) | ty = y}. This lattice is dual to the latice L = Mm (Z) ∩ V with respect to the pairing given by (u, v) −→ em (uv). For a function f : Hm −→ C of the form c(ξ)em (ξz) (z ∈ Hm ) f= ξ∈Am

and for a positive integer N we use the notations c(ξ)em (N ξz), f |V (N )(z) = f (N z) = f |U (N )(z) = fρ =

ξ∈A m

ξ∈Am

c(N ξ)em (ξz), c(ξ)em (ξz),

ξ∈Am

as well as the notation of A.N. Andrianov for the action of the Frobenius elements Π + (q), Π − (q) given in 2.1.8. Moreover, for N ≥ 1 and an integer k we put √ f |W (N )(z) = f |k W (N )(z) = det( Nz)k f (−(N z)−1 ), so that (f |W (N ))|W (N ) = (−1)mk f.

2.1 Siegel modular forms and Hecke operators 2.1.1 Symplectic group and Siegel upper half plane (see [An1], [An5], [Sh3], [Si2], [Fr], [Maa]). Let G = GSpm be the algebraic subgroup of GL2m deﬁned by GA = {γ ∈ GL2m (A) |

γJm γ = ν(γ)Jm , ν(γ) ∈ A× },

t

for any commutative ring A, where 0m −1m Jm = . 1m 0m

(2.1)

2.1 Siegel modular forms and Hecke operators

47

The elements of GA are characterized by the conditions bta − atb = dtc − ctd = 0m ,

and if γ=

ab cd

dta − ctb = 1m ,

∈ GA then γ −1 = ν(γ)−1

d −tb −tc ta t

(2.2) .

The multiplier ν deﬁnes a homomorphism ν : GA → A× so that ν(γ)2m = det(γ)2 and ker(ν) is denoted by Spm (A). We also put G∞ = GR ,

G+ ∞ = {γ ∈ G∞ | ν(γ) > 0},

+ G+ Q = G∞ ∩ GQ .

(2.3)

The group G+ ∞ acts transitively on the upper half plane Hm by the rule ab −1 + z −→ γ(z) = (az + b)(cz + d) γ= ∈ G∞ , z ∈ Hm cd so that the scalar matrices acts trivially, and Hm can be identiﬁed with a homogeneous space of the group Spm (R). Let Km denote the stabilizer of the point i1m ∈ Hm in the group Spm (R), Km = {γ ∈ Spm (R) | γ(i1m ) = i1m }, then there is a bijection Spm (R)/Km Hm and Km = Spm (R) ∩ SO2m . The group Km is a maximal compact subgroup of the Lie group Spm (R) and it can be identiﬁed with the group U(m) of all unitary m × m-matrices via the ab map γ = −→ a + ib. We adopt also the notations cd dxij , dy = , dz = dxdy, (2.4) dx = i≤j

d× y = det(y)−κ dy,

i≤j

d× z = det(y)−κ dz,

where z = x + iy, x = (xij ) = tx, y = (yij ) = ty > 0. Then d× z is a diﬀerential × on Hm invariant under the action of the group G+ ∞ , and the measure d y is invariant under the action of elements a ∈ GLm (R) on Y = {y ∈ Mm (R) |

t

y = y > 0}

deﬁned by the rule y −→ taya. 2.1.2 Siegel modular forms Let us consider the Siegel modular group denoted Γ m = Spm (Z) and let Γ ⊂ G+ Q be an arbitrary congruence subgroup. This means that Γ is commeasurable with Γ m in G+ Q modulo its center (i.e. as a group of transformations of Hm ), and Γ ⊃ Γ m (N ) for some N ∈ N, where

48

2 Siegel modular forms and the holomorphic projection operator

Γ m (N ) =

" ab γ= ∈ Γ m | γ ≡ 12m cd

# (mod N )

is the main congruence subgroup of level N in Γ m . In order to give the general deﬁnition of modular forms we consider ρ : GLm (C) → GLr (C) a rational ab representation which will also be denoted by ρ. For γ = ∈ G+ ∞ and cd for any complex valued function f : Hm → Cr we use the notation f |ρ γ(z) = ρ(cz + d)−1 f (γ(z)).

(2.5)

Deﬁnition 2.1. A function f : Hm → Cr is called a holomorphic modular form of weight ρ on Γ if the following conditions (2.6) - (2.8) are satisﬁed: f |ρ = f, f is holomorphic on Hm ,

(2.6) (2.7)

if m = 1 then f is holomorphic at cusps of Γ.

(2.8)

Let Mρ (Γ ) be the complex vector space of functions satisfying the above conditions. For each f ∈ Mρ (Γ ) there is the following Fourier expansion f (z) = c(ξ)em (ξz), ξ

where c(ξ) ∈ Cr , ξ run over all ξ = tξ ∈ Mm (Q), ξ > 0 (for m > 1 the last condition automatically follows by the Koecher principle). More precisely, let M be the smallest integer such that # " 1m M u t Γ ⊃ | u ∈ Mm (Z), u = u 0m 1m and we put A = Am = {ξ = (ξij ) ∈ Mm (R) | ξ = tξ, ξij , 2ξii ∈ Z} , B = Bm = {ξ ∈ A | ξ ≥ 0} , C = Cm = {ξ ∈ A | ξ > 0} . Then Am is a lattice of half-integral matrices in the R-vector space of | tx = x} dual to the lattice symmetric matrices V = {x ∈ Mm (R) L = Mm (Z) ∩ V with respect to the action (ξ, x) −→ em (ξx) and for each f ∈ Mρ (Γ ) there is the following Fourier expansion f (z) = c(ξ)em (ξz), (2.9) ξ∈M −1 B

Moreover, for each γ ∈ G+ Q we have that f |ρ γ ∈ Mρ (Γ (γ)), where Γ (γ) is a congruence subgroup,

2.1 Siegel modular forms and Hecke operators

(f |ρ γ)(z) =

cγ (ξ)em (ξz),

49

(2.10)

ξ∈Mγ−1 B

with cγ (ξ) ∈ Cr , Mγ ∈ N. A form f is called a cusp form if for all ξ with det(ξ) = 0 in expansion (2.10) one has cγ (ξ) = 0 for all γ ∈ G+ Q that is

(f |ρ γ)(z) =

cγ (ξ)em (ξz).

ξ∈Mγ−1 C

We denote by Sρ (Γ ) ⊂ Mρ (Γ ) the subspace of cusp forms. Deﬁnition of the vector spaces M(N, ψ). Let us consider congruence subgroups Γ1m (N ) ⊂ Γ0m (N ) ⊂ Γ m (N ) = Spm (Z), deﬁned by " # ab m Γ0 (N ) = γ = ∈ Spm (Z) | c ≡ 0m (mod N ) , " c d ab Γ1m (N ) = γ = ∈ Spm (Z) | c ≡ 0m (mod N ), cd # det(a) ≡ 1 (mod N ) , and let r = 1, ρ(x) = ρk (x) = det(x)k ( k ∈ N ). Then the vectore space M(Γ1m (N )) has already been deﬁned, and we set " Mkm (N, ψ) = f ∈ Mρ (Γ1m (N )) | f |ρ γ = ψ(det(a))f (2.11) # ab for all γ = ∈ Γ0m (N ) , cd where ψ is a Dirichlet character modulo N . Elements f ∈ Mkm (N, ψ) admits a Fourier expansion of the form f (z) = c(ξ)em (ξz), (2.12) ξ∈Bm

with z ∈ Hm , c(ξ) ∈ C, and the condition c(uξ tu) = ψ(det(u))det(u)k c(ξ)

(ξ ∈ Bm , u ∈ GLm (Z)).

(2.13)

Put k (N, ψ) = Mkm (N, ψ) ∩ Sρk (Γ1k (N )). Sm k (N, ψ) and h ∈ Mkm (N, ψ) The Petersson scalar product. For f ∈ Sm the Petersson scalar product is deﬁned by f, h N = f (z)h(z)det(y)k d× z, (2.14) Φ0 (N )

where Φ0 (N ) = Γ0m (N )\Hm is a fundamental domain for the group Γ0m (N ).

50

2 Siegel modular forms and the holomorphic projection operator

The Siegel operator connects the vector spaces Mkm (N, ψ) for diﬀerent values of m. If f ∈ Mkm (N, ψ), z ∈ Hm−1 and λ > 0 then we have that z 0 ∈ Hm , 0 iλ and it follows from (2.12) that there exists the limit z 0 ξ 0 (Φf )(z ) = lim f c = em−1 (ξ z ), 0 iλ 0 0 λ→∞

(2.15)

ξ∈Bm−1

where c(ξ) are the Fourier coeﬃcients of f . Then Φf ∈ Mkm−1 (N, ψ), (we put Mk0 (N, ψ) = 0. We then have that % & k Sm (N, ψ) ⊆ ker Φ = f ∈ Mkm (N, ψ) | Φf = 0 , (for N = 1 both sets coincide, see [Maa]). k (N, ψ) then there is the Estimates for Fourier coeﬃcients. If f ∈ Sm following upper estimate (z = x + iy ∈ Hm ), |f (z)| = O det(y)−k/2 (2.16) which provides us also with the estimate |c(ξ)| = O det(ξ)k/2 .

(2.17)

For modular (not necessary cusp) forms f (z) = c(ξ)em (ξz) ∈ Mkm (N, ψ) ξ∈Bm

there is the upper estimate of their growth: |c(ξ)| ≤ c1

m

(1 + λkj ),

(2.18)

j=1

with λ1 , · · · , λm being eigenvalues of the matrix y, z = x + iy (see [St2], p 335). In this situation one has also the following estimate |c(ξ)| ≤ c2 det(ξ )k

(2.19)

in which c2 is a positive constant depending only on f , ξ = tu

ξ 0 0 0

u, where

u ∈ SLm (Z), ξ ∈ Br , det(ξ ) > 0, r < m. We refer the reader to [Fo], [Ki], [Ra1], [Ra3] for a more detailed discussion of various estimates for Fourier coeﬃcients and for growth of Siegel modular forms, as well as for some intersting applications to quadratic forms.

2.1 Siegel modular forms and Hecke operators

51

2.1.3 The Hecke algebra (see [An5], [B¨o2], [Sat]). Let q be a prime, q N , ' ab + −1 ∆ = ∆m (N ) = γ = ∈ G ∩ GL (Z[q ]) 2m q Q cd ( ν(γ)± ∈ Z[q −1 ], c ≡ 0m (mod N ) m be a subgroup in G+ Q containing Γ = Γ0 (N ). The following Hecke algebra m over Q: L = Lq (N ) = DQ (Γ, ∆) is then deﬁned as a Q-linear space generated by the double cosets (g) = (Γ gΓ ), g ∈ ∆ of the group ∆, with respect to the subgroup Γ for which multiplication is deﬁned by the standard rule (see [An5], [Sh1] and in 2.1.7 below). We recall the description of the structure of L = Lm q (N ), (q N ); for each j, 1 ≤ j ≤ m let us denote by ωj an ±1 ±1 automorphism of the algebra Q[x±1 0 , x1 , · · · , xm ] deﬁned on its generators by the rule:

x0 −→ x0 xj , xj −→ x−1 j , xi −→ xi

(where 1 ≤ i ≤ m, i = j).

Then the automorphisms ωj and the permutation group Sm of the variables xi (1 ≤ i ≤ m) generate together the Weyl group W = Wm , and there is the Satake isomorphism : ∼

±1 ±1 Wm Sat : L −→ Q[x±1 0 , x1 , · · · , xm ]

(2.20)

where Wm indicates the subalgebra of elements ﬁxed by Wm . For any commutative Q-algebra A the group Wm acts on the set (A× )m+1 , therefore any homomorphism of Q-algebras λ : L → A can be identiﬁed with some element (α0 , α1 , · · · , αm ) ∈ [(A× )m+1 ]Wm .

(2.21)

An explicit description of the Satake isomorphism is given below in 2.1.7. 2.1.4 Hecke operators Any double coset (g) = (Γ gΓ ) (g ∈ ∆ = ∆m q (N )) can be represented as a disjoint union of left cosets: )

t(g)

(g) =

Γ gi ,

i=1

therefore any element X ∈ L of the Hecke algebra L takes the form of a ﬁnite linear combination t(X) X= µi (Γ gi ), i=1

52

2 Siegel modular forms and the holomorphic projection operator

withµi a g= c

∈ Q, gi ∈ ∆. In order to deﬁne the Hecke operators we put for any b ∈∆ d (f |k,ψ g)(z) = det(g)k−κ ψ(det(a))det(cz + d)−k f (g(z))

(2.22)

(this convenient notation, suggested by Petersson and Andrianov, is especially useful when dealing with Hecke operators in their normalized form, compare with (2.5)). With this notation the automorphy condition can be rewritten as follows (2.23) (f |k,ψ γ)(z) = f for all γ ∈ Γ = Γ0m (N ). In this case for any

t(X)

X=

µi (Γ gi ) ∈ L

i=1

we have that the expression

t(X)

f |X =

µi f |k,ψ gi ,

(2.24)

i=1

is well deﬁned and f |X ∈ Mkm (N, ψ) so that the formula (2.24) gives a representation of the Hecke algebra L = Lm q (N ) on the complex vector space k Mm (N, ψ) (q N ). 2.1.5 Hecke polynomials ˜ Following A.N.Andrianov (see [An5]), let us consider polynomials Q(z) ∈ ±1 ˜ ∈ Q[x±1 , · · · , x ][z] : Q[x0 , · · · , xm ][z] and R(z) m 0 ˜ ˜ 0 , x1 , · · · , xm ; z) Q(z) = Q(x m = (1 − x0 z)

(2.25) (1 − x0 xi1 · · · xir z),

r=1 1≤i1 <···
˜ R(z) =

m

(1 − x−1 i z)(1 − xi z).

(2.26)

i=1

From the deﬁnition it follows that the coeﬃcients of the powers of the vari±1 Wm able z all belong to the subring Q[x±1 . Therefore by the Satake 0 , · · · , xm ] isomorphism (2.20) we have that there exist polynomials m

Q(z) =

2 i=0

(−1)i Ti z i ,

R(z) =

2m

(−1)i Ri z i ∈ L[z],

i=0

over the associative commutative ring L = Lm q (N ) such that

(2.27)

2.1 Siegel modular forms and Hecke operators m

˜ Q(z) =

2

(−1)i T˜i z i ,

˜ R(z) =

i=0

2m

53

˜izi, (−1)i R

i=0

˜ = SatX, X ∈ L. The polynomials ∆ ±1 , Ri (1 ≤ i ≤ m − 1) and T1 with X M with ˜ ±1 = x20 x1 · · · xm , ∆ M −1 Ri = Si (x1 , · · · , xm ; x−1 1 , · · · , xm ), m m ˜ T1 = x0 Si (x1 , · · · , xm ) = x0 (1 + xi ), i=1

i=1

can be taken as generators of the Hecke algebra , where Si denotes the elementary symmetric polynomial of degree i of the corresponding set of variables. 2.1.6 The spinor zeta function and the standard zeta function Let f ∈ Mkm (N, ψ) be an eigenfunction of all Hecke operators f −→ f |X, X ∈ Lm q (N ) with q being prime numbers, q N , so that f |X = λf (X)f . Then the numbers λf (X) ∈ C deﬁne a homomorphism λf : L −→ C (see 2.1.3), which is uniquely determined by a (m + 1)-tuple of numbers Wm (α0 , α1 , · · · , αm ) = (α0,f (q), α1,f (q), · · · , αm,f (q)) ∈ (C× )m+1 (2.28) which are called the Satake q-parameters of the modular form f . Now let the variables x0 , x1 , · · · , xm in (2.25), (2.26) be equal to the corresponding Satake q-parameters α0,f (q), α1,f (q), · · · , αm,f (q), then Qf,q (z) = (1 − α0 z)

m

(1 − α0 αi1 · · · αir z),

(2.29)

r=1 1≤i1 <···
Rf,q (z) =

m

±1 ±1 (1 − α−1 i z)(1 − αi z) ∈ Q[α0 , · · · , αm ][z].

(2.30)

i=1

From the description of the Hecke polynomials in 2.1.5 it follows then that the coeﬃcients of (2.29) can be expressed in terms of the eigenvalues λf (X) of the Hecke operators X = Ti , Ri from (2.27). Next let us put Z (q) (s, f ) = Qf,q (q −s )−1 =  m (1 − α0 (q)q −s )

−1 (1 − α0 (q)αi1 (q) · · · αir (q)q −s  (2.31)

r=1 1≤i1 <···
and deﬁne the spinor zeta function Z(s, f ) of the modular form f ∈ Mkm (N, ψ) by the following product Z(s, f ) = Z (q) (s, f ). (2.32) qN

54

2 Siegel modular forms and the holomorphic projection operator

Complex analytic properties of the zeta functions Z(s, f ) were studied by A. N. Andrianov (see [An1] in the case m = 2. For m = 1 we have that α0 (q) + α0 (q)α1 (q) = a(q), α20 (q)α1 (q) = ψ(q)q k−1 , where f (z) = n≥0 a(n)e(nz) is the Fourier expansion of a normalized elliptic cusp Hecke eigenform, so that the zeta function Z(s, f ) =

−1 1 − a(q)q −s + ψ(q)q k−1−2s = qN

∞

a(n)n−s

n=1

(n,N )=1

coincides essentially with the Mellin transform of f L(s, f ) =

∞

a(n)n−s .

n=1

Notice that all of the inverse roots of the Hecke polynomial Qf,q (z) can be obtained by the action of the Weyl group Wm on α0 (q), and : (mk−m(m+1)/2) 2 z = Qf,q (z) = 1 − λf (q)q −s + · · · + q 2 m 1 − α0 (q)αi1 (q) · · · αir (q)z , (1 − α0 (q)z) m−1

m

(2.33)

r=1 1≤i1 >···>ir ≤m

λf (q) = λf T (q), T (q) = (Γ gΓ ) ν(g)=q

Γ0m (N ), tgJm g

= ν(g)Jm . are the Hecke operator for the group Γ = According to the generalized Ramanujan-Petersson conjecture for a cusp k eigen-form f ∈ Sm (N, ψ) the absolute values of all inverse roots of the polynomial (2.33) should be equal to each other (and therefore be equal to q −(2mk−m(m+1)/4) ). In the case m = 1 this conjecture is valid; it was deduced by Deligne [De1] from the Weil conjectures, also proven by him [De2]. However, for larger values of m the analogous conjecture is no longer true. Various counterexamples are known (see [Kuro], [Ho-PSh1], [Ho-PSh2]), which for the case m = 2 reﬂect the fact that some cusp forms of degree 2 (namely those belonging to the Maass subspace) are lifted from elliptic cusp forms (i.e. of degree 1) via the Saito-Kurokawa lifting (see [An5], [Koj]). According to the modiﬁed Ramanujan-Petersson conjecture these properties of the inverse roots and polynomials Qf,q should hold for those cusp forms, which can not be obtained from the forms of smaller degree by a lifting of the above type. The standard zeta function D(s, f, χ) of f ∈ Mkm (N, ψ) is the product D(q) (s, f, χ) D(s, f, χ) = d |N

2.1 Siegel modular forms and Hecke operators

with

55

D(q) (s, f, χ) = (1 − χ(q)ψ(q)q −s )−1 Rf,q (χ(q)ψ(q)q −s )−1 ,

that is : * m +−1 χψ(q) χψ(q)αl (q) χψ(q)αl (q)−1 D(s, f, χ) = , 1− 1− 1− qs qs qs q l=1

where χ is a Dirichlet character modulo M . Analytic properties of the standard zeta functions werestudied by A.N. Andrianov and V.L. Kalinin [An-Ka] in the case of even degree m, and more recently S. B¨ ocherer [B¨o2] extended these results to the case of arbitrary degree using a diﬀerent approach. For m = 1 ∞ k and a normalized cusp eigenform f (z) = n=0 a(n)e(nz) ∈ M1 (N, ψ) we have that D(s, f, χ) = L2,f (s + k − 1, χ), where L2,f (s, χ) = LN M (2s − 2k + 2, χ2 ψ 2 )

∞

χ(n)a(n2 )n−s

(2.34)

n=1

is the symmetric square of the modular form f (see [An2], [An-Ka], [Pa1], [Sc], [Sh2] and [Za]). 2.1.7 Non-commutative extension of the Hecke algebra and the Satake isomorphism Now we recall that the multiplication in the Hecke algebra denoted by L = DQ (Γ, ∆) is deﬁned by use of the larger vector space V = VQ (Γ, ∆) over Q consisting of all Q-linear combinations of left cosets of the form (Γ g) where g ∈ ∆ = ∆m q (N ). If γ ∈ Γ and

t(X)

X=

µj (Γ gj ) ∈ V

j=1

then the formula

t(X)

X ·γ =

µj (Γ gj γ) ∈ V

j=1

deﬁnes a right action of the group Γ on V so that the algebra L = DQ (Γ, ∆) coincides with the subspace of all elements in VQ (Γ, ∆) pointwise ﬁxed by Γ via the inclusion L → V which sends a double cosets t(g)

(g) = (Γ gΓ ) = ∪j=1 Γ gj (g ∈ ∆ = ∆m q (N )) to the formal sum

t(g) j=1

Γ gj . Let us consider two elements

56

2 Siegel modular forms and the holomorphic projection operator

t(X)

X=

t(Y )

aj (Γ gj ),

Y =

j=1

bl (Γ hl )

(2.35)

l=1

in DQ (Γ, ∆) ⊂ V then the element

t(X) t(Y )

X ·Y =

aj bl (Γ gj hl ) ∈ VQ (Γ, ∆)

j=1 l=1

is well deﬁned and also belongs to DQ (Γ, ∆) ⊂ V. This construction can be applied to a large variety of couples (Γ, ∆), for which Γ is a subgroup of a (semi)group ∆ such that any double coset (g) = (Γ gΓ ) (g ∈ ∆ = ∆m q (N )) can be represented as a disjoint union of left cosets )

t(g)

(g) =

Γ gj

j=1

(the theory of Hecke couples, see [An5], Chapter 1). In particular, for " # ab Γ0 = Γ0m = g = ∈Γ |c=0 , " c d # ab m ∆0 = ∆0,q = g = ∈ ∆m | c = 0 , q cd we thus obtain an associative (but no longer commutative) ring L0 = Lm 0,q (N ) = DQ (Γ0 , ∆0 ). From the theory of elementary divisors it follows that Γ ∆0 = ∆; therefore we get the homomorphism ε : L → L0 deﬁned as follows, for each

t(X)

X=

µj (Γ gj ) ∈ L ⊂ V

j=1

we may assume that

gj =

q νj dj bj 0m dj

,

where q νj = ν(gj ), and we put

t(X)

ε0 (X) =

j=1

ν q j dj bj µj Γ0 . 0m dj

(2.36)

Similar arguments show also that the matrix dj can be chosen in the form

2.1 Siegel modular forms and Hecke operators



q δj (1)  0 q δj (2) dj =   ··· ··· 0 0

 ··· ···   ··· ···  · · · q δj (m)

(δj (1) ≤ · · · ≤ δj (m))

57

(2.37)

with the exponents δj (l), which are uniquely determined by dj . Now for

t(X)

X=

µj (Γ gj ) ∈ L0

j=1

we put by deﬁnition

t(X)

Φ(X) =

ν

µj x0j

j=1

m (xl q −l )δj (l) .

(2.38)

l=1

The Satake isomorphism is then described as the composition of homomorphisms (2.36) and (2.38) : ∼

±1 ±1 Wm Sat = Φε0 : L −→ Q[x±1 0 , x1 , · · · , xm ]

(2.39)

As a consequence of the construction we get that for an arbitrary homomorphism of algebras λ : L → C (λ = λq , q N , L = Lm q (N )), attached to the (m + 1)-tuple of q-parameters (α0 , α1 , · · · , αm ) ∈ [(C× )m+1 ]Wm , and for any element

t(X)

X=

µj (Γ gj ) ∈ L with gj =

j=1

q νj dj bj 0m dj

and dj is chosen in the form (2.37), the following equality holds:

t(X)

λ(X) =

j=1

ν

µj α0j

m (αl q −l )δj (l) .

(2.40)

l=1

If λ = λf,q for a modular form f ∈ Mkm (N, ψ) then the numbers α0 , α1 , · · · , αm satisfy the following relation α20 α1 · · · αm = ψ(q)m q km−m(m+1)/2 ,

(2.41)

which follows directly from the formula 2.40 and the deﬁnition 2.22 applied to the operator X = (g) with g = q12m (see [An5]).

58

2 Siegel modular forms and the holomorphic projection operator

2.1.8 Action of the Hecke operators on Fourier expansions Let f (z) =

c(ξ)em (ξz) ∈ Mkm (N, ψ)

(ξ ∈ Bm )

ξ∈Bm

and for a ﬁxed prime q

t(X)

X=

µj (Γ0 gj ) ∈ L0

(Γ0 = Γ0m , L0 = Lm 0,q )

j=1

be an arbitrary element of the extended Hecke algebra L0 with ν q j di bj gj = ∈ ∆m 0,q . 0m dj Let coeﬃcients (c|X)(ξ) be deﬁned by the equality (c|X)(ξ)em (ξz). (f |X)(z) =

(2.42)

(2.43)

ξ∈Bm

Then we have that

t(X)

(c|X)(ξ) =

µj q νj (mk− m) det(dj )−k ψ(q mνj det(dj )−1 ) ·

(2.44)

j=1

·em (q −νj ξ tbj dj )c(q −νj dj ξ tdj ), (here we assume that c(ξ) = 0 for ξ ∈ Bm (see [An7], §3.2)). We remark that for X ∈ L0 not beloning to the subalgebra ε0 (L) ⊂ L0 we have, in general, ), where that f |X ∈ Mkm (N, ψ). However we claim that f |X ∈ Mkm (ΓX = ∩i gi−1 Γ1m (N )gi ΓX

(see (2.22)). As an example we consider the Frobenius elements (see [An7], §2.1): q1m 0m m Π − = Π− (q) = Γ0 Γ0 , (2.45) 0m 1m 1m 0m m (q) = Γ0 Γ0 . (2.46) Π+ = Π+ 0m q1m Then for f (z) = ξ∈Bm c(ξ)em (ξz) ∈ Mkm (N, ψ) we get from (2.44) (c|Π− )(ξ) = ψ(q)m q m(k−κ) c(q −1 ξ), (ξ ∈ Bm ⊂ Am ). (c|Π+ )(ξ) = c(qξ),

(2.47) (2.48)

Note that the operator Π+ = Π+ (q) is deﬁned by the formula (2.48) also for q|N , when Π+ (q) sends Mkm (N, ψ) to itself. However we assume in the next proposition that q N .

2.1 Siegel modular forms and Hecke operators

59

Proposition 2.2. If f ∈ Mkm (N, ψ) then for a ﬁxed prime q we have that f |Π− (q), f |Π+ (q) ∈ Mkm (qN, ψ).

(2.49)

Proof. Note that m Π− (q) = (Γ0

q1m 0m 0m 1m

thetrefore f |Π− (q) = f |k,ψ

Γ0 ),

q1m 0m 0m 1m

,

if we now take into account that −1 q1m 0m q1m 0m Γ0m (N ) Γ0m (qN ) ⊂ , 0m 1m 0m 1m then we get that for all γ ∈ Γ0m (qN ) the following equality holds f |Π− γ = (f |Π− )|k,ψ γ = f |Π− . In order to prove the analogous statement about Π+ we use the antiisomorphism (involution) X → X of the algebra L0 (see [An5], p. 75) which has the property Π− = Π+ . We recall that by deﬁnition if

t(X)

X=

µi (Γ0 gi ) ∈ L0

(µi ∈ Q, gi ∈ ∆0 ),

i=1

then

t(X)

X =

µi (Γ0 q νi gi−1 )

(q νi = ν(q)).

i=1

Next, let the equality XΓ = X be valid for a given X ∈ L0 where Γ0 ⊂ Γ Γ . Put ∆ = Γ ∆0 , ∆0 ⊂ ∆ ⊂ ∆. Consider as in the paragraph 2.1.3, the Hecke algebra L = DQ (Γ , ∆ ). Then by deﬁnition of the homomorphism ε : L → L0 we have again that its image ε 0 (L ) coincides with the subalgebra of all elements in L0 ﬁxed by Γ . Consequently, for some Y ∈ L we have that X = ε 0 (Y ) and we get X = ε 0 (Y ) by the deﬁnition of the antiautomorphism . In other words, the operator X ∈ L turns out to be Γ invariant. In particulary, we get Π+ Γ0m (qN ) = Π+ by the already proven equality Π− Γ0m (qN ) = Π− , and by the fact that f |Π+ ∈ Mkm (qN, ψ). Now let us consider again the Hecke polynomials, deﬁned by (2.25) - (2.27). A. N. Andrianov has discoverd factorization of the polynomials Q(z), R(z) ∈ L[z] over the non-commutative algebra L0 = Lm 0,q in terms of the Frobenius elements Π+ (q), Π− (q) (see 2.1.8). This factorization is essentially used in our non-Archimedean construction.

60

2 Siegel modular forms and the holomorphic projection operator

Proposition 2.3. There are the following factorizations of the Hecke polynomials introduced above: 2m −1 2m −1 + i − i Q(z) = Vi z (1 − Π+ z) = (1 − Π− z) Vi z , (2.50) i=1

i=1

where Vi+ =

i

i−j (−1)j Tj Π+ ∈ L0 ,

(2.51)

i−j (−1)j Tj Π− ∈ L0 .

(2.52)

j=0

Vi− =

i j=0

Proof. (see [An5], §2.2). Now let c(ξ)em (ξz) ∈ Mkm (N, ψ) f (z) = ξ∈Bm

be an eigenfunction of the algebra L = Lm p (N ) whose eigenvalue is a homomorphism λf : L → C, f |X = λf (X)f for all X ∈ L which is given by the (m + 1)-tuple α0 = α0 (p), α1 = α1 (p), · · · , αm = αm (p) of the Satake p-parameters. Put f0 =

m−1 ˜

α0 (p)−i f |Vi+ (p)

(m ˜ = 2m ).

(2.53)

i=1 ˜ Proposition 2.4. (a) The function f0 belongs to Mkm (N pm−1 , ψ) with m m ˜ =2 . (b) The following equality holds

f0 |Π+ (p) = α0 (p)f0 . Remark 2.5. Using the action of the Weyl group in the construction of ˜ , ψ) of the Proposition 2.4, we can obtain 2m eigenfunctions g ∈ Mkm (N pm−1 m local Hecke algebras L = Lq (N ) for q pN with the same eigenvalues as those of the original modular form f ∈ Mkm (N, ψ) such that for q = p we have that g|Π+ (p) = αg where α is any of the inverse roots α0 αi1 · · · αir

(1 ≤ i1 < · · · < ir ≤ m)

of the local factor Z (p) (s, f ) of the spinor zeta function. As was pointed out to the second author by S.B¨ ocherer, these functions may vanish if we take for f a certain Eisenstein seies (see the remark before the Main theorem in §1 of Chapter 3). He mentioned also that a stronger statement about the level of the above form f0 is valid. However we do not give here this interesting result because it is not directly needed for our proof.

2.1 Siegel modular forms and Hecke operators

61

Proof (of Proposition 2.4). The statement (a) follows directly from (2.51), (2.52) and from the deﬁnition (2.53). In order to prove (b) we note that there is the equality (with m ˜ = 2m ): m ˜

˜ 0 (p), α1 (p) · · · , αm (p); z)f ; f |Ti (p) · (−z)i = Q(α

i=0

in addition, by deﬁnition of the polynomials (2.25) we have the identity: m ˜

(−α0 (p))−i f |Ti (p) = 0.

(2.54)

i=0

We want to prove that f0 |(Π+ (p) − α0 (p)) = 0. But we have from (2.50), (2.53) that α−1 f0 |(Π+ (p) − α0 (p)) = −α0 (p)f |(1 0 (p)Π+ (p)) − m ˜ = −α0 (p)f | (−α0 (p))−i f |Vi+ (p) · i=0

·(1 − α−1 0 (p)Π+ (p)) m ˜ = f |Ti (p)(−α0 (p))−i

= 0,

i=0

and the statement (b) follows. An analogous statement is valid in the case of a ﬁnite set S of prime numbers q coprime to N . For this purpose it is convenient to consider the global Hecke algebras L(N ) = ⊗qN Lm q (N ),

L0 (N ) = ⊗qN Lm 0,q (N ).

Then the deﬁnition of the operators Π+ (M ), Π− (M ), Vi+ (q), Vi− (q) ∈ Lm 0,q (N ) and of the numbers α0 (q) (q N ) can be extended by multiplicativity to all positive integers M coprime with N ; more precisely, operators V + (M ), V − (M ) are deﬁned by the identities + *˜ m−1 −s + −s + M V (M ) = q Vi (q) , (2.55) ˜ M|M0m−1

qN

M

−s

−

V (M ) =

˜ M|M0m−1

i=1

*˜ m−1 qN

q

Vi− (q)

i=1

where the notation V ± (q i ) = Vi± (q),

+ −s

M0 =

q∈S

q.

,

(2.56)

62

2 Siegel modular forms and the holomorphic projection operator

is used. We then put

f0 = f0,S =

α0 (M )−1 f |V + (M ).

(2.57)

˜ M|M0m−1

Proposition 2.6. (a) Let f ∈ Mkm (N, ψ), then ˜ f0 = f0,S ∈ Mkm (N M m−1 , ψ).

(b) For all positive integers M with support S(M ) in the set S we have that (2.58) f0 |Π+ (M ) = α0 (M )f0 . (c) Let f0,S (z) =

˜ a0 (ξ)em (ξz) ∈ Mkm (N M m−1 , ψ)

(ξ ∈ Bm )

ξ∈Bm

be the Fourier expansion of the function f0,S (z) then there is the following multiplicativity property of its Fourier coeﬃcients: for all M ∈ N with S(M ) ⊂ S we have that a0 (M ξ) = α0 (M )a0 (ξ)

(ξ ∈ Am , ξ ≥ 0).

(2.59)

Proof. The proof of the proposition is carried out in a very similar way as that of the previous one if we take into account formulas (2.48) for the action of Π+ (M ) on Fourier expansions.

2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function 2.2.1 Theta series (see [An2], [An-Ma1], [An-Ma2], [St2]). Let F ∈ 2Cm be an even symmetric positive deﬁnite matrix, and q0 its level (i.e. the smallest positive integer such that q0 F −1 ∈ 2Cm , and χ a Dirichlet character modulo Q (not necesarily primitive). Put ν = 1 or 0 and deﬁne the theta function (ν) χ(det(ξ))det(ξ)ν em (zF [ξ]/2). (2.60) θ(χ) = θF (z, χ) = ξ∈Mm (Z)

Proposition 2.7. (a) If γ =

ab cd

∈ Γ0m (q0 Q2 ), then the following trans-

formation formula holds (ν)

(m)

(ν)

θF (γ(z), χ) = χ(det(d))χQ2 F (γ)det(cz + d)(m/2)+ν θF (z, χ),

(2.61)

2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function

63

(m)

where χF (γ) is a root of unity of the eighth degree, and if m is even, then (−1)m/2 det(F ) (m) χF (γ) = . (2.62) det(d) 0m −1m . If χ is primitive (b) Let J(M ) denote for M > 0 the matrix M 1m 0m 2 modulo Q then the action of the involution J(Q q0 ) on (2.60) is given by (ν) θF (J(Q2 q0 )z, χ) = χ(−1)m (iQ)mν GQ (1m , χ)det(Fˆ )m/2+ν ·

(2.63)

(ν) ¯ ·det(−iz)m/2+ν θFˆ (z, χ)

with Fˆ = q0 F −1 and

GQ (ξ, χ) =

χ(det(h))em (tξh/Q)

h∈Mm (Z)modQ

is the Gauss sum of degree m of the character χ. Proof. The proof of (a) is given in [An-Ma2] for primitive characters χ, and in [St2] in the general case by use of the generalized theta series (m) θF (z; X, Y ) = em (zF [ξ − Y ] + 2tξX − tXY )/2 , (2.64) ξ∈Mm (Z)

where z ∈ Hm , X,Y a erties: for all γ = c

∈ Mm (C). The series (2.64) satisfy the following propb ∈ Γ0m (q0 ) we have that d

θF (γ(z); X ta + F Y tb, F −1 X tc + Y td) = (m)

=

(m) χF (γ)det(cz

+

(m) d)m/2 θF (z; X, Y

(2.65) ),

θF (z; X, Y ) = det(F )−m/2 det(−iz)−m/2 θF −1 (−z −1 ; Y, −X), (m)

(m)

(2.66)

(see [An-Ma2], theorem 1 and lemma 2). For ν = 0, the statements (a) and (b) follow immediately from (2.65), (2.66). For example, in order to prove (b) it suﬃces to put in (2.66) F equal to Q2 F , x = 0, Y equal to −Q−1 Y then multiply the result by χ(det(Y )) and carry out summation over Y ∈ Mm (Z)modQ. From the left hand side of (2.66), we get (m) χ(det(Y ))θQ2 F (z; 0, −Q−1Y ) = (2.67) Y ∈Mm (Z)modQ

=

Y ∈Mm (Z)modQ

and from the right hand side

χ(det(Y ))

ξ∈Mm (Z)

em (zF [Qξ + Y ]/2),

64

2 Siegel modular forms and the holomorphic projection operator

χ(det(Y ))det(Q2 F )−m/2 det(−iz)−m/2 ·

(2.68)

Y ∈Mm (Z)modQ

·θQ−2 F −1 (−z −1 ; −Q−1 Y, 0) = (m)

= det(Q2 F )−m/2 det(−iz)−m/2 ·

·

Y ∈Mm (Z)modQ

em (−(zQ2 q0 )−1 (q0 F −1 )[ξ] − 2Q−1 ξY )/2 .

ξ∈Mm (Z)

Now we note that for any Dirichlet character χ the deﬁnition (2.60) can be rewritten as follows: (ν) χ(det(Y )) det(ξ)ν em (zF [ξ]/2). (2.69) θF (z, χ) = Y ∈Mm (Z)modQ

ξ≡Y modQ

If we take into account the primitivity of χ, then we get (see [An4], (5.12)) (ν) θF (z, χ) = GQ (1m , χ) ¯ −1 χ(det(Y ))· (2.70) ·

Y ∈Mm (Z)modQ

det(ξ)ν em (zF [ξ]/2).

ξ≡Y modQ

Let us use the standard relation ¯ = χ(−1)m Qm GQ (1m , χ)GQ (1m , χ)

2

(2.71)

(0)

for Gauss sums. We see that (2.67) becomes θF (z, χ), and using (2.68) and (2.70) we get θF (z, χ) = χ(−1)m GQ (1m , χ)det(Q2 F )−m/2 · (0) ¯ . ·det(−iz)−m/2 θFˆ J(q0 Q2 ), χ (0)

(2.72)

If we substitute z with J(q0 Q2 ) and simplify the power of Q, we get exactly (2.63) for ν = 0. In order to prove the statement (a) and (b) in the case ν = 1, we take a matrix F1 = tF1 ∈ Mm (R) such that F1 > 0, F12 = F , and let η = (ηij ) ∈ Mm (C) be a matrix variable. If we set in (2.65) X = 0, Y = F1−1 η and apply the diﬀerential operator Lη = det(∂/∂ηij ) to both parts of the resulting equality, then after some simpliﬁcations we get the equality (2.61) with ν = 1. The diﬀerential identity Lη em (P tηη + 2tT η + R)/2 = det 2iπ(ηP + T ) em (P tηη + 2tT η + R)/2 , (2.73) is particulary useful in this calculation, where P, T, R ∈ Mm (C), tP = P (see [An5], lemma 5.1).

2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function

65

We give a more detailed proof of (b). Let us set in (2.66) X = 0 and Y = F1−1 η and apply the operator Lη . Then the expression in the exponent of the summand corresponding to an integral matrix ξ in the left hand side of (2.66) becomes iπtr zF [ξ − F1−1 η] = iπtr z tηη − 2t(F1 ηz)η + zF [ξ] . This is easily deduced from the deﬁnition (2.64), if we notice that tr(AB) = tr(BA) = tr(tAtB). Let us apply to this term the operator Lη . According to (2.73), this term will then be multiplied by det 2iπ(ηz − F1 ηz) = (2iπ)m det(F )1/2 det (F1−1 η − ξ)z . Similarrly, after application of Lη to each summand corresponding to a matrix ξ in (m) det(F )−m/2 det(−iz)−m/2 θF −1 (−z −1 ; F1−1 , 0), this summand will get the factor det(F )−1/2 (2iπ)m det(ξ). If we remember that Y = F1−1 η, then we see that the result of the action of Lη on the expression (2.66) will be given by the following: (2.74) det(Y − ξ)em zF [ξ − Y ]/2 = ξ∈Mm (Z)

= im det(F )−m/2−1 det(−iz)−m/2−1

det(ξ)·

ξ∈Mm (Z)

·em (−z −1 F −1 [ξ] + 2tξY )/2 .

Now again we set in (2.74) F equal to Q2 F , Y equal to −Q−1 Y , ξ equal to −ξ, multiply by χ(det(Y )) and carry out summation over Y modQ, keeping in mind the relation (2.71); as a result we get the equality (2.63) with ν = 1. An (m) explicit calculation of the multipier χF (γ) for even m is given for example in [An-Ka]. 2.2.2 Siegel-Eisenstein series We start by recalling the deﬁnition of these series. We call matrices c, d ∈ M(Z) coprime if {G ∈ M(Q) | Gc, Gd ∈ M(Z)} = M(Z). A couple (c, d) is called a symmetric couple if ctd = dtc. Two symmetric couples of coprime matrices are called equivalent, iﬀ for some unimodular matrix U ∈ GLm (Z) we have (c1 , d1 ) = (U c2 , U d2 ).

66

2 Siegel modular forms and the holomorphic projection operator

Let ∆ = ∆m denote the set of equivalence classes of symmetric couples of coprime matrices. Then the set can be identiﬁed with the set of right coset classes Γ0m \Γ m of the group Γ m = Spm (Z) with respect to its parabolic subgroup " # ab Γ0m = γ = γ ∈ Γm , 0d via the map Γ0m \Γ m −→ ∆m ab m Γ0 −→ class of (c, d) cd

(2.75)

We see also that via this map the set {(c, d) ∈ ∆m | c ≡ 0

(mod N )}

is identiﬁed with the set of right coset classes Γ0m \Γ0m (N ). Now let k, N be positive integers, s a complex number and χ a Dirichlet character modulo N such that χ(−1) = (−1)k . For z ∈ Hm the SiegelEisenstein series is deﬁned by E(z, s; k, χ, N ) = E(z, s) = det(y)s χ(det(d))det(cz + d)−k−|2s| , (2.76) where the summation is taken over all (c, d) ∈ ∆ with the condition c ≡ 0 (mod N ) and we adopt the convenient notation by Deligne and Ribet [De-Ri]: z −k−|2s| = z −k |z|−2s def

for z ∈ C× .

The series (2.76) is absolutely convergent for k + 2Re(s) > m + 1 and it admits a meromorphic analytic continuation over the whole complex s-plane. ab Put j(α, z) = det(cz + d) for α = and z ∈ Hm , then it follows from cd the description of ∆m given above that E(z, s; k, χ, N ) = det(y)s χ(det(d)j(α, z)−k−|2s| , (2.77) α∈P ∩Γ \Γ

ab , and P denotes the subgroup of P ⊂ G∞+ , cd consisting of elements α with the condition c = 0.

where Γ = Γ0m (N ), α =

2.2.3 The Rankin zeta function Let f and g be two holomorphic modular forms of weights k and l on the congruence subgroup Γ0m (N ) ⊂ Γ m , and with Dirichlet characters ψ and ω. More precisely we suppose that

2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function

f (z) =

67

k a(ξ)em (ξz) ∈ Sm (N, ψ),

(2.78)

b(ξ)em (ξz) ∈ Mlm (N, ω),

(2.79)

ξ∈Cm

g(z) =

ξ∈Bm

with Bm being the set of half integral non negative matrices of size m × m, and Cm ⊂ Bm the subset of all positive deﬁnite matrices (see section 2.1). Deﬁne an equivalence relation on Bm by ξ1 ∼ ξ2 iﬀ ξ1 = tuξ2 u for some matrix u ∈ SLm (Z). Then the Rankin zeta function (convolution of two Siegel modular forms) is deﬁned as the series [St2]: ε(ξ)−1 a(ξ)b(ξ)det(ξ)−s (s ∈ C), (2.80) L(s, f, g) = ξ∈Cm /∼

where ε(ξ) = |Aut(ξ)| is the order of the unit group ( ' Aut(ξ) = η ∈ SLm (Z) | tηξη = ξ[η] = ξ . This series is well deﬁned due to basic properties of Fourier coeﬃcients of Siegel modular forms; Cm / ∼ denotes the orbit space with respect to ∼ and Re(s) is supposed to be large enough: Re(s) ≥ 0. To be more precise, we note that the series det(ξ)−s (2.81) ξ∈Cm /∼

is absolutely convergent for Re(s) > m. It is easily seen from the fact that the cardinality of the set ' ( SLm (Z)\ γ ∈ Mm (Z) | det(γ) = a > 0 is estimated by the ﬁnite sum (2.81) admits the upper bound

m−1

, hence the Dirichlet series in a2 a23 · · · am−1 m

ζ Re(s) − i ,

i=0

with ζ(s) being the Riemann zeta function (see [An5], p.133). Keeping in mind estimates (2.17) and (2.18) for Fourier coeﬃcients of Siegel modular forms we get b(ξ) = O(det(ξ)l ), a(ξ) = O(det(ξ)k/2 ), so that the Dirichlet series (2.80) is absolutely convergent for Re(s) > m + k/2 + l + ε, (ε > 0).

68

2 Siegel modular forms and the holomorphic projection operator

Proposition 2.8 (Integral representation for the Rankin zeta function). (see [St2], proposition 6). For s with Re(s) 0 there is the following integral representation. (4π)−ms Γm (s)L(s, f, g) / . , = f ρ , gE(z, s − k + (m + 1)/2; k − l, ψω, N )

(2.82)

N

where the inner product is deﬁned by (2.14), k ¯ f ρ (z) = a(ξ)em (ξz) ∈ Sm (N, ψ), ξ∈Bm

and Γm (s) denotes the Γ -function of degree m, i.e. Γm (s) = π m(m−1)/4

m−1

Γ (s − (j/2)),

j=0

(see also (2.94)). 2.2.4 The standard zeta function D(s, f, χ) as the Rankin convolution Now let f (z) =

k a(ξ)em (ξz) ∈ Sm (N, ψ),

ξ∈Cm

be a cusp form of even degree m which is an eigenfunction of the Hecke algebra Lm q (N ) for q not dividing N , and let (α0 , α1 , · · · , αm ) = (α0 (q), α1 (q), · · · , αm (q))) ∈ [(A× )m+1 ]Wm be the corresponding (m+1)-tuple of the Satake q-parameters of f (see (2.21)). Fix a matrix ξ0 ∈ Cm such that a(ξ0 ) = 0 and consider the primitive quadratic Dirichlet character χξ0 deﬁnied for positive integers d with (d, 2det(2ξ0 )) = 1 by the formula (−1)m/2 det(2ξ0 ) χξ0 (d) = , d and when det(2ξ0 ) is odd, then by deﬁnition χξ0 (2) = 1 or −1 according as to which of the two following quadratic forms x1 x2 + x3 x4 + . . . xm−1 xm x1 x2 + · · · + xm−3 xm−2 + x2m−1 + xm−1 xm + x2m our quadratic form ξ0 is equivalent over the ﬁeld F2 of two elements. Assume also that χ mod M is chosen so that (−1)ν = χ(−1). The main result of the

2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function

69

A.N.Andrianov’s work [An3] can be stated as a certain identity expressing the standard zeta function D(s, f, χ) as a Rankin zeta function, namely the convolution of the given form f and a theta function with the Dirichlet character χ mod M . The precise statement of the result is given in the following proposition. Proposition 2.9. Under the notation and assumptions as above for the suﬃciently large values of Re(s) the following identity holds a(ξ0 )R(s, f, χ) = 2−1 det(ξ0 )(s+k−1+ν)/2 L((s + k − 1 + ν)/2, f, θ2ξ0 (z; χ)), (2.83) where the function R(s, f, χ) is deﬁned by the following equality   m/2−1 L(2s + 2i, χ2 ψ 2 ) R(s, f, χ), D(s, f, χ) = L(s + (m/2), χψχξ0 )  (ν)

i=0

(2.84) and it is assumed that the modulus M of the character χ is divisible by all (ν) prime divisors of the number N det(2ξ0 ), where θ2ξ0 (z, χ) is the theta function as in 2.2.1. More explicitly, the right hand side of (2.83) can be represented as the series det(ξ0 )(s+k−1+ν)/2 χ(det(ξ))det(ξ)ν a(ξ0 [ξ])det(ξ0 [ξ])−(s+k−1+ν)/2 = ξ

=

χ(det(ξ))a(ξ0 [ξ])det(ξ)−(s+k−1+ν)/2 ,

(2.85)

ξ

where the summation is taken over the set of equivalence classes ξ ∈ SLm (Z)\M+ m (Z) of the form SLm (Z)ξ. Now we set q0 to be equal to the level of the quadratic form with the matrix 2ξ0 (see 2.2.1). In order to get the integral representation of the standard zeta function we apply to it the result of proposition 2.8. For this purpose we put in the notation of proposition 2.8 l = (m/2) + ν, ω = χχξ0 with χ being a Dirichlet character modulo M , and take the number N q0 M 2 as N . We note also that both parts of (2.83) and (2.84) converge absolutely for Re(s) > m (see also [An5], p. 133).

70

2 Siegel modular forms and the holomorphic projection operator

Proposition 2.10 (integral representation of the standard zeta function). For the notations and assumptions as above about Re(s), the following equality holds: 2a(ξ0 )det(ξ0 )−(s+k−1+ν)/2 (4π)−ms Γm ((s + k − 1 + ν)/2)R(s, f, χ) = . / (ν) = f ρ , θ2ξ0 (z, χ)E(z, (s − k + m + ν)/2) , (2.86) N M 2 q0

where the Siegel-Eisenstein series in the right hand side E(z, s) = E(z, s; k − ν − (m/2), χψχξ0 , N M 2 q0 ) is deﬁned by (2.76).

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series 2.3.1 Rationality properties of Fourier coeﬃcients For the full symplectic modular group Γ = Sp(Z) Siegel has deﬁned the series [Si1], [Si2] (see section 2.2, equality (2.76) ) (m) j(γ, z)−k , (2.87) E(z) = Ek (z) = γ∈P ∩Γ m \Γ m

where z ∈ Hm is the point ofthe Siegel upper half plane of degree m, ab j(γ, z) = det(cz + d) for γ = ∈ Γ m , P denotes the subgroup of cd ∗∗ m . In the original deﬁnition Γ consisting of block matrices of the form 0∗ by Siegel the number k is even and k > m + 1 so that the series (2.87) is absolutely convergent and is referred to as Siegel-Eisenstein series. The rationality property of its Fourier coeﬃcients were established by Siegel himself (although it was certainly known earlier in the case m = 1: (1)

∞ 2k σk−1 (n)e(nz) Bk n=1 ∞ 2σk−1 (n) = 1+ e(nz), ζ(1 − k) n=1

Ek (z) = 1 −

where σk−1 (n) =

(2.88)

dk−1 ,

d|n

and Bk are Bernoulli numbers, ζ(s) being the Riemann zeta function). After the original Siegel’s work his calculation was generalized in various directions:

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series

71

(m)

to the case of congruence subgroups of Γ0 (N ) ⊂ Γ m [St2] to non-convergent series deﬁned by analytic continuation over an additional parameter (Hecke’s method) [He], [Fe], to other classes of algebraic groups and symmetric domains [Ba], [Ha1], [Ha3], [Fe], [Sh5], [Sh8]. It was discovered that the rationality property remains valid even for more complicated series, which themselves are deﬁned by some inductive process of inducing from other cusp forms of lower degree (the Klingen-Eisenstein series). More precisely, let f ∈ Srk be a cusp form of degree r (with respect to the group Γ r ). If k > m + r + 1 and m ≥ r then the Klingen-Eisenstein series is deﬁned as the following absolutely convergent series Ekm,r (z, f ) = f ((γz)(r) )j(γ, z)−k , (2.89) γ∈∆m,r \Γ m

with z ∈ Hm , z (r) being the upper left corner of zof size r × r, and ∆m,r ∗∗ m denotes the set of elements in Γ having the form [Kl1]. This series 0∗ turns out to be a modular form of degree m on the group Γ m . M. Harris proved in [Ha1] - [Ha3] the validity of the Garrett’s conjecture: all the Fourier coeﬃcients of the modular form Ekm,r (z, f ) belong to the ﬁeld Q(f ) generated by the Fourier coeﬃcients of f . Explicit formulas for Fourier coeﬃcients of the series Ekm,r (z, f ) were given by B¨ocherer [B¨o3] in general for k > m + r + 1 and by Kurokawa and Mizumoto [Ku-Mi], [Miz1], [Miz2] who treated the case m = 2, r = 1. It turned out that the most signiﬁcant term in these formulas involves the special values of the standard zeta function of f twisted with a certain quadratic Dirichlet character attached to the matrix index ξ of a Fourier coeﬃcient; as it was noticed above, these functions reduce to the (twisted) symmetric squares of the form f if m = 2. The formulas of B¨ ocherer and Mizumota may be regarded as a vast generalization of the classical formulas (2.88), if we assume that cusp forms of degree 0 are constants and their zeta functions reduce to the Riemann zeta function and to Dirichlet L-series. However, in what follows we are interested only in Siegel-Eisenstein series which were deﬁned in §2.2.2 by: E(z, s; k, χ, N ) = E(z, s) χ(det(dα ))j(α, z)−k−|2s| , = det(y)s

(2.90)

α∈P ∩Γ \Γ

for k + 2Re(s) > m + 1 (we use again the notation z −k−|2s| = z −k |z|−2s for z, s ∈ C, k ∈ Z) and by analytic continuation over s for other values of s ∈ C (see [Kal], [Sh8] and §2.3.4 below). It is assumed in the identity (2.90) that N > 1, χ is a Dirichlet character mod N (not necessarily primitive, e.g. trivial modulo N > 1), and aα b α ∈ Γ = Γ0m (N ) ⊂ Γ m . α= cα dα

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2 Siegel modular forms and the holomorphic projection operator

In the following study of certain arithmetic properties of Fourier coeﬃcients we use an explicit computation of the Fourier expansion of the series E (z, s) = E(−z −1 , s)det(z)−k ,

(2.91)

obtained from (2.90) by applying the involution 0m −1m . Jm = 1m 0m However for k > m + 1 and N = 1 both series coincide and reduce to the series originally studied by Siegel: E(z) = Ekm (z) = E(z, 0) = E (z, 0). The detailed study of these series was made by G. Shimura [Sh8] and P. Feit ([Fe], §10) in a more general situation, in particular, for the case of Eisenstein series attached to the group Sp over a totally real ﬁeld. For convenience we reproduce only a specialization of these results to the case of F = Q. The remarkable summation method by means of the analytic continuation of the function E(z, f ) over s to the point s = 0 was ﬁrst discovered by E. Hecke [He]. Quite recently this method was essentially extended by R. Weissauer [Weis2] to the case of Klingen-Eisenstein series (2.89) and then it was used for a construction of Siegel modular forms of “small” weight (k = m, m + 1), which play a signiﬁcant role in studying geometric invariants of Siegel modular varieties [Weis1], [Weis3]. 2.3.2 Preparation: the conﬂuent hypergeometric function For a detailed description of the Fourier expansion of the series (2.91) we need some additional notation. Let V = Vm = {h ∈ Mm (R) |

t

h = h}

(2.92)

be the set of all real symmetric matrices of size m × m, and Y = V + = {h ∈ V

| h > 0}

(2.93)

the subset of its positive deﬁnite elements. For each matrix T ∈ Mm (R) let δ+ (T ) denote the product of all positive eigenvalues of T , δ− (T ) = δ+ (−T ) and δ+ (T ) = 1 if T does not have positive eigenvalues. For h ∈ V let p = p(h) denote the number of positive eigenvalues of h counted with their multiplicities, and q = q(h) the number of negative eigenvalues. Then r = r(h) = p + q is the rang of h. Let also m−1 Γ (s − (j/2)), (2.94) Γm (s) = π m(m−1)/4 j=0

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series

73

be the Γ -function of degree m. This function generalizes the usual Γ -function in view of the following integral representation Γm (s) = det(y)s e−tr(y) d× y, (2.95) Y

which is valid for s ∈ C with Re(s) > (m − 1)/2, and dy = dyij , d× y = det(y)−(m+1)/2 dy. i≤j

Recall that d× y is a measure on Y which is invariant with respect to the action of a ∈ GLm (R) given by d× (taya) = d× y. For complex numbers α and β we deﬁne the numbers κ = (m + 1)/2, (2.96) τ = τ (h, α, β) = (2p − m)α + (2q − m)β + m + (m − r)κ + pq/2,(2.97) σ = σ(h, α, β) = pα + qβ + m − r + (m − r)(m − r − 1)/2.

(2.98)

In [Sh6] G. Shimura constructed the function ω(y, h; α, β),

(2.99)

which is deﬁned for all (y, h; α, β) ∈ Y × V × C2 and is a holomorphic function of (α, β) ∈ C2 . It can be used for computing the Fourier expansion of the series det(z + a)−α det(¯ z + a)−β (z ∈ Hm ) (2.100) S(z, L; α, β) = a∈L

which is obtained by summation over a lattice L ⊂ V and is absolutely convergent for Re(α + β) > m. Let L = {h ∈ V

| tr(hL) ∈ Z}

be the lattice dual to L with respect to the pairing given by (u, v) → em (uv). In particular there is the equality µ(V /L)S(z; α, β) = ξ(y, h; α, β)em (hx) (2.101) h∈L

in which µ(V /L) =

dy V /L

denotes the volume of the fundamental domain V /L, ξ(y, h; α, β) = imβ−mα 2τ π σ Γm−r (α + β − κ)Γm−q (α)−1 Γm−p (β)−1 · ·det(y)κ−α−β δ+ (hy)α−κ+q/4 δ− (hy)β−κ+q/4 ω(2πy, h; α, β),

(2.102)

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2 Siegel modular forms and the holomorphic projection operator

and is additionaly assumed that Re(α) > m/2, Re(β) > m/2 (for the regularity of Γ -functions in (2.102) (see [Sh6], (4.34.K)) and we adopt the standard choice of branch for the exponentiation, namely, v α = eα log(v) ,

−π ≤ Im(log(v)) < π.

The function ξ(y, h; α, β) admits the following integral representation: for y ∈ Y , h ∈ V , (α, β) ∈ C2 ξ(y, h; α, β) = em (−hx)det(x + iy)−α det(x − iy)−β dx, (2.103) V

with the integral being absolutely convergent for Re(α + β) > 2κ + 1 (see [Sh6], (1.25)). Applying the equality (2.101) to the lattice L = S = V ∩ Mm (Z) when L = A = Am is the lattice of all symmetric half integral matrices and also to the lattice L = N S, L = N −1 Am with α = k, β = 0, k > m and Cm = Am ∩Y we get the classical equality: det(z + a)−k = (2πi)mk Γm (k)−1 det(h)k−κ em (hz), (2.104) a∈S

h∈Cm

(see, for example, the book of H. Maass [Maa]). Indeed, we notice that the only terms in the equality (2.104), correspond to p = m, q = 0 because of the poles of the Γ -functions in the denominator of (2.102). Also, the function ω(2πy, h; α, β) reduces to the exponent em (iyh) in view of the formulas: ξ(y, h; α, 0) = i−mα 2(1−κ)m (2π)mα Γm (α)−1 det(h)α−κ em (iyh), (2.105) (2.106) ξ(y, 0; α, β) = imβ−mα 2m(κ+1−α−β) π mκ Γm (α + β − κ) · ·Γm (α)−1 Γm (β)−1 , lim ξ(y, h; κ + s, s) = i−mκ 2θ π mκ Γm (κ)−1 em (iyh),

ξ→0

(2.107)

with q = 0 and θ = [(m + p)/2]. The formulas (2.105)-(2.107) are easily deduced from [Sh6], (1.31), (4.35.K); see also [Sh8], (7.11) - (7.14). The conﬂuent hypergeometric function ω(2πy, h; α, β) can be used for analytic continuation of the Siegel-Eisenstein series [Kal], [Fe], [Sh8] by means of the term by term analytic continuation of their Fourier coeﬃcients. This coeﬃcients can be expressed in terms of the functions (2.99) (see theorem 2.12 below). We list also some other properties of these functions, which are useful for the analytic continuation (see [Sh6], theorem 4.2): functional equation ω(2πy, h; α, β) = ω(2πy, h; κ + (t/2) − β, κ + (t/2) − α);

(2.108)

a uniform upper bound on compact subsets |ω(2πy, h; α, β)| ≤ C1 e−τ (hy)(1 + µ(hy)−C2 ),

(2.109)

with α, β varying in a ﬁxed compact subset T ⊂ C and the constants C1 , C2 depending only on T , τ (x) being the sum of eigenvalues of a matrix x, µ the minimum of their absolute values. 2

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series

75

2.3.3 Critical values of the conﬂuent hypergeometric function Now we give formulas, which express the function ω(2πy, h; α, β) in terms of certain polynomials of the entries of the matrix y = (yij ) provided h > 0 and either α − κ ∈ Z, α − κ ≥ 0 or β ∈ Z, β ≤ 0 (κ = (m + 1)/2). The pairs (α, β) satisfying these conditions will be called critical : as we shall see, the critical values of s for the standard zeta function correspond to certain critical pairs. The following calculation of the special values is based on properties of the function ζ(z; α, β) deﬁned for z ∈ H m = {z ∈ Mm (C) | iz ∈ Hm } by the integral ζ(z; α, β) = e−tr(zx) det(x + 1m )α−κ detxβ−κ dx, (2.110) Y

which is absolutely convergent for Re > κ − 1 and deﬁnes a holomorphic function of (z, α, β). Let ω(z; α, β) = Γm (β)−1 det(z)β ζ(z; α, β).

(2.111)

It was established by Shimura ([Sh6], theorem 3.1) that the function (2.111) can be analytically continued to a holomorphic function over H m × C2 satisfying the functional equation ω(z; κ − β, κ − α) = ω(z; α, β).

(2.112)

For an arbitrary compact subset T ⊂ C2 there exist positive constants A, B > 0 depending only on T such that |ω(z; α, β)| ≤ A(1 + µ(y)−B ) for

y ∈ Y ⊂ H m ,

(α, β) ∈ T.

(2.113)

It is known also (see [Sh6], (4.19)) that ω(y, 1m ; α, β) = 2mκ e−tr(y) ω(2y; α, β),

(2.114)

and that for all a ∈ GLm (R) one has ω(ta−1 ya−1 ; α, β) = ω(y; α, β), ω(y, −h; α, β) = ω(y, h; β, α), ω(y, h; α, β) = 1.

(2.115) (2.116) (2.117)

Comparison of (2.114) and (2.115) shows that for h > 0 there is the identity ω(y, h; α, β) = ω(a(hy)a−1 , 1m ; α, β) = 2mκ e−tr(y) ω(2a(hy)a−1 ; α, β), (2.118) This is done by taking a matrix a ∈ GLm (R) with the condition ahta = 1m . Now let us consider the diﬀerential operator ∆m (the Maass diﬀerential operator) acting on C ∞ -functions over V ⊗ C of degree m, which is deﬁned by the equality:

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2 Siegel modular forms and the holomorphic projection operator

∂ij = 2−1 − (1 + δij )∂/∂ij .

∆m = det(∂ij ),

(2.119)

For an integer n ≥ 0 and a complex number β consider the polynomial ! R(z; n, β) = (−1)mn etr(z) det(z)n+β ∆nm e−tr(z) det(z)−β , (2.120) with z ∈ V ⊗ C, where the exponentiation is well deﬁned by det(y)β = exp β log[det(y)] , for det(y) > 0, y ∈ Y ⊗ C. According to deﬁnition (2.120) the degree of the polynomial R(z; n, β) is equal to mn and the term of the highest degree coincides with det(z)n . We have also that for β ∈ Q the polynomial R(z; n, β) has rational coeﬃcients. Proposition 2.11. (See [Sh6] proposition 3.2). For any non negative integer n the functions det(z)n ω(z; n + κ, β) and det(z)n ω(z; α, −n) are polynomial functions of z. More precisely, we have that ω(z; n + κ, β) = det(z)−n R(z; n, β),

(2.121)

ω(z; α, −n) = ω(z; n + κ, κ − α) = det(z)−n R(z; n, κ − α). (2.122) Proof. The proof is carried out using the following diﬀerentiation rule: (−1)mn ∆nm e−tr(z) det(z)−β ω(z; α, β) = e−tr(z) det(z)−β ω(z; α + n, β), (2.123) which follows immediately from the deﬁnition (2.110), (2.111) and provides also an analytic continuation of the function ω(z; α, β). The formulas (2.111) and (2.122) follow then from the identities (2.123) and ω(z; κ, β) = ω(z; α, 0) = 1, Notice also that for m = 1 one has n n R1 (z, n, β) = β(β + 1) · . . . · (β + k − 1)z n−k . k k=0

In §2.5 we establish the following more explicit expression for the function Rm (z; r, β) of arbitrary m (see also [Cour], Th´eor`eme 3.1.6): r r R(z; r, β) = det(z)r−t t t=0

RL (β)λl1 (z) · . . . · λlt (z),

(2.124)

|L|≤mt−t

where L goes over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 +· · ·+lt ≤ mt−t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series

77

β of degree (mt−|L|) and with coeﬃcients in the ring Z[1/2]. The polynomials λi (z) are deﬁned by det(t1m + X) =

m

λi (z)tm−i ,

(2.125)

i=0

In other words λi (z) is the sum of all diagonal minors of size i×i of the matrix z. If we apply this to functions ω(2πy, h; α, β) from 2.121 then we get for h > 0 the following identity (we use the fact that the polynomials Rm are invariant by conjugation): ω(2πy, h; n + κ, β) = ω(2πy, h; κ − β, −n) −mκ

(2.126)

−1

=2 em (ihy)ω(4πa (hy)a; n + κ, β) −mκ em (ihy)det(4πhy)−n Rm (4πhy; n, β). =2 Proposition 2.12 (Fourier expansion of the Siegel-Eisenstein series E (z, s)). (see [Fe], §10). The series deﬁned by (2.91) have the following Fourier expansion E (z, s) = b(h, y, s)em (hx), (2.127) h∈N −1 Am

in which coeﬃcients have the form of the product b(h, y, s) = N −mκ W (y, h, s)Γ (h, s)RL (h, χ, k + 2s)M (h, χ, k + 2s), (2.128) with the factors described as follows ((a) - (d)): (a) The conﬂuent hypergeometric function W (y, h, s) = i−mk 2τ π σ ω(2πy, h; k + s, s)det(y)κ−k−s · ·δ+ (hy)k+s−κ+q/4 δ− (hy)s−κ+p/4 ,

(2.129)

with (compare with (2.102)) τ = (2p − m)(k + s) + (2q − m)s + m + (m − r) + pq/2 = 2(r − m)s + (2p − m)k + m + (m − r)(m + 1)/2 +!pq/2,

σ = p(k + s) + qs + m − r + (m − r)(m − r − 1) − pq /2 ! = rs + pk + (m − r)(m − r − 1) − pq /2;

(b) Gamma factor Γ (h, s). Let for the integer r the symbol ε(r) denote its parity: ε(r) = 0, 1 with r ≡ ε(r) (mod 2). Put δ = ε(k), µ = ε((r/2) + q + k) and then deﬁne: for ε(r) = 0 Γ (h, s) =

Γm−r (k + 2s − κ)Γ (s + (k + δ))/2 · Γm−q (k + s)Γm−p (s)Γ (s + (k − m + r/2 + µ)/2) $[m/2] Γ (k + 2s − i) · $[(m−r)/2] i=1 , Γ (k + 2s − m + i + (r − 1)/2) i=1

78

2 Siegel modular forms and the holomorphic projection operator

and for ε(r) = 1 $[m/2] Γm−r (k + 2s − κ)Γ (s + (k + δ)/2) i=1 Γ (k + 2s − i) Γ (h, s) = , $[(m−r−1)/2] Γm−q (k + s)Γm−p (s) i=1 Γ (k + 2s − m + i + r/2) (c) The ratio of Dirichlet L-functions RL . Let for a Dirichlet character χ modulo N of a parity δ = 0 or 1 (1 − χ(q)q −s )−1 (2.130) LN (s, χ) = Γ ((k + δ)/2)LN (s, χ) = Γ ((k + δ)/2) q |N

denote the normalized Dirichlet L-function, which is regular for all s ∈ C, s = 1, including s = 0 ( due to the condition N > 1). Next we deﬁne an additional quadratic Dirichlet character χh depending on h ∈ Am and deﬁned only for even r = 0. Namely, for h = 0 let χh = χ0 be trivial; for h = 0 we know that for some matrix u ∈ GLm (Q) h1 0 t uhu = (2.131) with det(h1 ) = 0, 0 0 then let 0 χh denote the quadratic character attached to the following quadratic ﬁeld Q( det(h1 ))/Q (this deﬁnition does not depend on choice of a matrix u). Under these notation we set: for an even r (i.e. for ε(r) = 0) RL (h, χ, k + 2s) =

LN (k + 2s − m + (r/2), χχh ) · LN (k + 2s, χ) $[(m−r)/2] LN (2k + 4s − 2m + r − 1 + 2i, χ2 ) , · i=1 $[m/2] LN (k + 2s, χ) i=1 LN (2k + 4s − 2i, χ2 )

and for an odd r (i.e. with ε(r) = 1) $[(m−r−1)/2]

RL (h, χ, k + 2s) =

i=1 LN (k

LN (2k + 4s − 2m + r − 1 + 2i, χ2 ) . $[m/2] + 2s, χ) i=1 LN (2k + 4s − 2i, χ2 )

(d) The integral factor M (h, χ, k + 2s) =

Mq (h, χ(q)q −k−2s ),

(2.132)

q∈P (h)

is a ﬁnite Euler product, extended over primes q in the set P (h) of prime divisors of the number N and of all elementary divisors of the matrix h. The important property of the product is that for each q we have that Mq (h, t) ∈ Z[t] is a polynomial with integral coeﬃcients. The explicit form of it is insigniﬁcant for our purposes; however, one can ﬁnd interesting explicit formulas for these coeﬃcients in [Ra2], [Ki].

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series

79

2.3.4 Normalized Siegel-Eisenstein series We introduce here three types of normalized Siegel-Eisenstein series in order to give a precise statement on their holomorphy properties with respect to the variable s, the properties of positivity of matrices ξ by which their Fourier coeﬃcients are indexed, and also algebraic properties of these Fourier coeﬃcients: G (z, s) = G∗ (z.s; k, χ, N ) = N m(k+2s)/2 imk 2−m(k+1) π −m(s+k) Γ (1m , s)−1 LN (k + 2s, χ) ·   [m/2] √ LN (2k + 4s − 2i, χ2 ) E(−(N z)−1 , s)det( N z)−k (2.133) · i=1





[m/2]

= N m(k+s) Γ t(k, s)LN (k + 2s, χ) 

LN (2k + 4s − 2i, χ2 ) ·

i=1 ∗

·E (N z, s), with E (N z, s) = E(−(N z)−1 , s)det(N z)−k = N −km/2 E|W (N ), and Γ t(k, s) = imk 2−m(k+1) π −m(s+k) Γ (1m , s)−1 Γ ((k + 2s + δ)/2) · (2.134)

[m/2]

·

Γ (k + 2s − j)

j=1

= imk 2−m(k+1) π −m(s+k) × " Γm (k + s)Γ (s + (k − m/2 + µ)/2), if m is even; × Γm (k + s), otherwise. If m is odd then we put G+ (z, s) = G− (z, s) = G (z, s). If m is even then we deﬁne (with µ = ε((m/2) + k)) −1 G (z, s), G− (z, s) = Γ (k + 2s − (m/2) + µ)/2 iµ π (1/2)−k−2s+(m/2) G (z, s) Γ (1 − k − 2s + (m/2) + µ)/2 2iµ Γ k + 2s − (m/2) · = (2π)k+2s−(m/2) · cos π(k + 2s − (m/2) − µ)/2 G− (z, s).

G+ (z, s) =

(2.135) (2.136)

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2 Siegel modular forms and the holomorphic projection operator

We shall see in §4.4 of chapter 4 that the normalizing factors in formulas (2.133), (2.135) and (2.136) are closely related to those of the Dirichlet Lseries and the standard zeta functions (for even m). Now we reformulate proposition 2.12 for the normalized series G (z, s) G (z, s) = b (h, y, s)em (hz), (2.137) h∈Am

where b (h, y, s) = W (y, h, s)Γ (h, s)LN (h, χ, k + 2s)M (h, χ, k + 2s), with Γ (h, s) = Γ (1m , s)−1 Γ (h, s), W (y, h.s) = N m(s+k−κ) imk 2−m(k+1) π −m(s+k) em (−ihy)W (N y, N −1 h, s) = imk 2−m(k+1) π −m(s+k) em (−ihy)W (y, h, s).

The factor M (h, χ, k + 2s) is given by (2.132), and for r even we have that LN (h, χ, k + 2s) = LN (k + 2s − m + (r/2), χχh )· [(m−r)/2] LN (2k + 4s − 2m + r − 1 + 2i, χ2 ), · i=1

and for r odd

[(m−r−1)/2]

LN (h, χ, k + 2s) =

LN (2k + 4s − 2m + r − 1 + 2i, χ2 ).

i=1

Theorem 2.13 (On holomorphy properties of Fourier coeﬃcients of the normalized Siegel-Eisenstein series). (see [Fe], theorem 9.1). Let 2|N , N > 1, then (a) If χ2 = 1 then the function G (z, s) is an entire function of the variable s; (b) Suppose that χ2 is trivial, then we have • (b1 ) if either 2k ≥ m and m odd or 2k ≥ m and m is even, but (m/2) + k is odd (i.e. µ = 1), then the function G (z, s) is an entire function of the variable s; • (b2 ) if 2k ≥ m and both numbers m and (m/2)+k are even (i.e. µ = ε(m) = 0) then the function G (z, s) is an entire function of the variable s with the exclusion of a possible simple pole at the point s = (m + 2 − 2k)/4; • (b3 ) if m > 2k ≥ 0 then the function G (z, s) is an entire function of the variable s with possible exclusion of simple poles at those points s for which 2s is an integer and [(m − 2k + 3)/2] ≤ 2s ≤ (m + 1 − 2k)/2; • (b4 ) if k = 0 then the function G (z, s) has a simple pole at the point s = (m + 1)/2 iﬀ χ is trivial, and in this case we have that the function Ress=(m+1)/2 G (z, s; 0, 1, N ) of the variable z is a non-zero constant.

2.3 Formulas for Fourier coeﬃcients of the Siegel-Eisenstein series

81

Theorem 2.14 (On positivity properties of Fourier expansions of the normalized Siegel-Eisenstein series). Assume that 2k > m and deﬁne the numbers A(χ), B(χ), C(χ, k) as follows: (a) If χ2 is not trivial, m is even and µ = ε(k + (m/2)) then put C(χ, k) = (m − 2k + 2 − 2µ)/4;

A(χ) = B(χ) = 1 + (m/2),

(b) If χ2 is trivial, m is even and µ = ε(k + (m/2)) then put A(χ) = B(χ) = (m/2),

C(χ, k) = (m − 2k + 2 − 2µ)/4;

(c) If χ2 is not trivial, m is odd,then put A(χ) = B(χ) = (m + 3)/2,

C(χ, k) = [(1 + m − 2k)/4];

(d) If χ2 is trivial, m is odd, then put A(χ) = (m + 5)/2,

B(χ) = (m + 1)/2,

C(χ, k) = [(3 + m − 2k)/4];

under these notation and assumptions there are the following positivity properties of matrices h ∈ Am , by which the Fourier coeﬃcients of the series G (z, s) are indexed: (1) if s ≤ 0, s ∈ Z and k + 2s ≥ A(χ) then b (h, y, s)em (hz), (2.138) G (z, s) = Am h>0

(2) if k + s − κ ∈ Z, k + s − κ ≥ 0, (κ = (m + 1)/2), s ≤ C(χ, k) then b (h, y, s)em (hz), (2.139) G (z, s) = Am h≥0

Proof. The proof of this theorem is contained in the book of P.Feit [Fe], theorems 14.1.A – 14.1.F; it is based on a detailed study of poles and residues of the Γ -factor Γ (h, s) and of the Dirichlet L-function L (h, χ, k + 2s), which was carried out in [Fe] in terms of the functions f (n, s) = Γ (n + s)/Γ (s); for positive integers n these functions turn out to be polynomials with zeros given by s ∈ Z, s ≤ 0, s + n > 0. It was shown in [Fe] that the factor Γ (g, s) = Γ (h, s)/Γ (1m , s) is equivalent (up to multiplication by an invertible entire function) to a certain explicitly given polynomial in C[s] (see [Fe], §11). It follows also from this calculation that (a) if χ2 is trivial, m is odd and s = s0 = (m + 2 − 2k)/4 then the function G (z, s) has a pole at the point s = s0 such that the residue Ress=s0 G (z, s) has a non negative Fourier expansion;

(2.140)

([Fe], theorem 14.1.C); (b) if χ2 is trivial and m is odd then the function G (z, s) is ﬁnite at the point s = s0 and has a non negative Fourier expansion.

82

2 Siegel modular forms and the holomorphic projection operator

It is essential for our purposes to reformulate the corresponding statements for the series G+ (z, s) and G− (z, s) (see (2.135), (2.136)), which are obtained from G (z, s) by an additional normalization. The following theorem is an immediate consequence of the theorem 2.13 on holomorphy and the properties (2.138)-(2.140). Theorem 2.15 (On Fourier coeﬃcients with positive matrix numbers). Let m be an even integer such that 2k > m, then: (a) For 2s an integer, s ≤ 0, k + 2s ≥ 1 + m/2 there is the following Fourier expansion b+ (h, y, s)em (hz), (2.141) G+ (z, s) = Am h>0

where for s > (m + 2 − 2k)/4 in (2.141) non-zero terms only occur for positive deﬁnite h > 0, and for all s from (a) with h > 0, h ∈ Am the following identity holds b+ (h, y, s) = W (y, h, s)L+ N (k + 2s − (m/2), χχh )M (h, χ, k + 2s), with L+ N (s, χ) =

2iδ Γ (s) cos(π(s − δ)/2) LN (s, χ) (2π)s

is the normalized Dirichlet L-function, δ = 0 or 1 according to χ(−1) = (−1)δ , the factor M (h, χ, k + 2s) is deﬁned by (2.132), W (y, h, s) = 2−mκ det(h)k+2s−κ det(4πy)s R(4πhy; −s; κ − k − s), provided s is an integer, where R(y; n, β) is deﬁned by (2.120), and b+ (h, y, s) = 0 otherwise (if s ∈ Z). (b)If 2s is an integer with k + 2s ≤ m/2, k + s ≥ κ then there is the following Fourier expansion b− (h, y, s)em (hz), (2.142) G− (z, s) = Am h≥0

and for all s from (b) with h > 0, h ∈ Am the following identity holds b− (h, y, s) = W (y, h, s)L− N (k + 2s − (m/2), χχh )M (h, χ, k + 2s), where L− N (s, χ) = LN (s, χ), W (y, h, s) = 2−mκ det(4πy)κ−k−s R(4πhy; k + s − κ, s), provided s + k − κ is an integer, and b− (h, y, s) = 0 otherwise.

2.4 Holomorphic projection and Maass operator

83

Proof. The proof is deduced from the expansions (2.137) if we remember the deﬁnition of the normalizing factors and the positivity property in the theorem 2.14. We also note that by (2.129) W (y, h, s) = em (−ihy)ω(2πy, h; k + s, s)det(y)κ−k−s · ·δ+ (hy)k+s−κ+q/4 δ− (hy)s−κ+p/4 , and then take into account the formula (2.126) for the critical values of the function ω. In case of the odd parity 2s ∈ Z we get vanishing of the Fourier coeﬃcients because of the Γ -factors in (2.135), (2.136).

2.4 Holomorphic projection operator and the Maass diﬀerential operator 2.4.1 Holomorphic projection operator We start with describing a vector space on which this operator acts. A function F : Hm → C, F ∈ C ∞ (Hm ) is called a C ∞ -modular form of weight k on the group Γ0m (N ) with a Dirichlet character ψmodN if the following automorphy condition is satisﬁed: F ((az + b)(cz + d)−1 ) = ψ(det(d))det(cz + d)k F (z)

for all γ=

ab cd

∈ Γ0m (N )

(compare with (2.11)). The space of functions F with the above condition is ˜ k (N, ψ) there is the following Fourier ˜ k (N, ψ). For all F ∈ M denoted by M m m expansion F (z) = A(y, h)em (hx), (2.143) h∈Am

where A(y, h) are some C ∞ -functions on Y . The Petersson inner prodk (N, ψ) and uct is deﬁned for an arbitrary holomorphic cusp form f ∈ Sm k ˜ F ∈ Mm (N, ψ) by f, F N = f (z)F (z)det(y)k−m−1 dxdy, Φ0 (N )

where Φ0 (N ) = Γ0m (N )\Hm is a fundamental domain for the group Γ0m (N ). ˜ k (N, ψ) a function of a bounded growth if for We call a function F ∈ M m each ε > 0 the following integral converges: |F (z)|det(y)k−1−m e−εtr(y) dydx < ∞ (2.144) X

Y

84

2 Siegel modular forms and the holomorphic projection operator

where

X = {x ∈ Mm (R) | Y = {y ∈ Mm (R) |

x = x, |xij | ≤ 1/2 for all i, j}, y = y > 0}.

t t

˜ k (N, ψ) a function of a moderate Respectively, we call a function F ∈ M m growth if for all z ∈ Hm and for all suﬃciently large values of Re(s) 0 the integral F (w)det(w ¯ − z)−k−|2s| det(Im(w))k+s d× w (2.145) Hm

is absolutely convergent and admits an analytic continuation over s to the point s = 0. The last deﬁnition may diﬀer from a traditional one; its meaning is clariﬁed by the following result (Theorem 2.16), which provides a reﬁnement of theorem 1 of the Sturm’s paper [St2]. It will follow from the proof that all functions of bownded growth automatically turn out to be of a moderate growth in the sense of deﬁnitions (2.144), (2.145) given above. ˜ k (N, ψ) and k > 2m. Put for h > 0, h ∈ Am Theorem 2.16. Let F ∈ M m a(h) = c(k, m)−1 det(4h)k−(m+1)/2 A(y, h)em (ihy)det(y)k−1−m dy, Y

(2.146) with

c(t, m) = Γm (t − (m + 1)/2)π −m(t−(m+1)/2) ,

and A(y, h) being coeﬃcients of the expansion (2.143) and suppose that the integral (2.146) is absolutely convergent. Deﬁne the function a(h)em (hz). (2.147) Hol (F )(z) = Am h>0

Then ˜ k (N, ψ) is of a bounded growth then Hol (F )(z) ∈ (a) if the function F ∈ M m k Sm (N, ψ). ˜ k (N, ψ) is of a moderate growth and the ex(b) If the function F ∈ M m pansion (2.143) contains only terms with positive deﬁnite matrices h ∈ Am , k (N, ψ) the following then Hol (F )(z) ∈ Mkm (N, ψ). In both cases for all g ∈ Sm equality holds: g, F N = g, Hol (F ) N . (2.148) Remark 2.17. The cusp form Hol (F ) is uniquely determined by (2.148) under the assumptions of (a), but in (b) this equality is not suﬃcient to identify the modular form Hol (F ). For example, (2.148) does not change if we replace this modular form by adding to it an Eisenstein series (of Siegel or of Klingen type). Part (a) of the theorem 2.16 was established by Sturm [St2].

2.4 Holomorphic projection and Maass operator

85

2.4.2 Poincar´ e series of two variables (of exponential type) of higher level In order to prove the theorem 2.16 we use this kind of Poincar´e serie introduced by H. Klingen [Kl1] and used by S. B¨ ocherer in [B¨ o2] instead of Poincar´e series of one variable from [St2], [Gr-Za]. We consider an element (Γ gΓ ) of the Hecke algebra Lm (N ) with Γ = Γ0m (N ), g ∈ ∆ = ∆m q with ' ab ∩ GL2m (Z[q −1 ]) ∆= γ= ∈ G+ Q cd ( ν(γ)± ∈ Z[q ± ], c ≡ 0m (mod N ) , where tgJm g = ν(g)Jm , ν(g) > 0. Put k Pm (z, w, g, s) = ψ(det(a))j(γ, z)−k−|2s| det(γ(z) + w)k−|2s| ,

(2.149)

γ∈Γ gΓ

ab def ∈ Γ gΓ and uk−|2s| = u−k |u|−2s for u ∈ C× , s ∈ C. The cd series in (2.149) converges absolutely and uniformly on products of the type Vm (d) × Vm (d) for k + Re(2s) > 2m + 1, d > 0, % & Vm (d) = z = x + iy ∈ Hm | y ≥ d1m , tr(txx) ≤ 1/d . γ =

We also put k k (z, w, g, s) = det(Im(z))s det(Im(w))s Pm (z, w, g, s). Pm

(2.150)

The following properties of these series were established by B¨ocherer in [B¨o2] (a) symmetry k k (z, w, g, s) = Pm (w, z, g, s); (2.151) Pm (b) automorphy with respect to both arguments: k Pm (γ(z), γ(w), g, s) k (z, w, g, s), = ψ(det(d1 ))ψ(det(d2 ))j(γ1 , z)k j(γ2 , w)k Pm ai b i ∈ Γ , i = 1, 2; where γi = ci di (c) action of Hecke operators:

(2.152)

k k Pm (z, w, 1m , s)|k,ψ (Γ gΓ )z = Pm (z, w, g, s) k = Pm (z, w, 1m , s)|k,ψ (Γ gΓ )w ,

(2.153)

where the subscript indicates to which of the variables the Hecke operator is being applied with the action deﬁned by (2.23).

86

2 Siegel modular forms and the holomorphic projection operator k (d) the integral representation: for all f ∈ Sm (N, ψ) we have that k (−¯ z , w, g, s), f (w) N,w = µ(m, k, s)f |k,ψ (Γ gΓ )(z), Pm

(2.154)

with µ(m, k, s) = 2m+m(m+1)/2−2ms+1 i−mk π m(m+1)/2

Γm (k + s − (m + 1)/2) . Γm (k + s)

The proof of the properties (2.151), (2.153) is easily deduced from the symmetry relation j(γ, z)det(γ(z) + w) = j(˜ γ , w)det(˜ γ (w) + z),

(2.155)

which is valid for all γ ∈ Spm (R) with 1 2 1m 0m γ˜ = γ 0m −1m and Γ gΓ = Γ g˜Γ for g ∈ ∆. Then (2.152) is immediately deduced from the deﬁnition (2.149). The proof of the integral formula (2.154) is carried out similarly to that in Klingen’s article [Kl2]; for this purpose we may assume that g = 1m . The integration in the left hand side of (2.154) can be reduced by the standard unfolding procedure to that over the whole Siegel upper half plane: notice that Hm = ∪γ γ(Φ0 (N )),

γ ∈ Γ = Γ0m (N ),

where Φ0 (N ) = Γ0m (N )\Hm is a fundamental domain for the group Γ0m (N ). The required property follows then from the integral representation f (w)det(w¯ − z)−k−|2s| det(Im(w))k+s d× w = (2.156) Hm

= imk 2m(m+1)−2ms−mk Im (k + s − m − 1)f (z)det(Im(z)), where

det(1m − ww) ¯ s dudv

Im (s) =

(2.157)

Em

=

π mκ Γ (s + 1 + (m − 1)/2) m 2 Γm (s + m + 1)

denotes the integral studied by Hua Lo-Ken [HuLK]. This integral is absok (N, ψ), the integration in (2.156) is lutely convergent for Re(s) > −1, f ∈ Sm extended over the generalized unit disc Em = {w = u + iv ∈ Mm (C) |

w = w, 1m − ww ¯ > 0}

t

of the degree m, the image of Hm via the Cayley transform

2.4 Holomorphic projection and Maass operator

w → (w − i1m )(w + i1m )−1

87

(w ∈ Hm ).

k (N, ψ) then there is the In order to prove (2.156) we note that if f ∈ Sm following upper estimate

f (z) ≤ cdet(Im(w))−k/2 , and the integral in (2.156) is majorated by |det(w ¯ − z)|−k−2Re(s) det(Im(w))(k/2)+Re(s) d× w, Hm

which provides the absolute convergence of (2.156) in this domain for (k/2) + Re(s) > m. Next we rewrite the integrand in (2.156) in the form g(w)det(w ¯ − z)−k−s det(Im(w))(k+s) with the holomorphic function g(w) = f (w)det(w − z¯)−s , which is integrated using the Cayley transform, so that (2.156) follows. 2.4.3 Reduction of theorem 2.16 to properties of Poincar´ e series We restrict ourselves to the case of functions F satisfying assumptions of (b). Set in formulas (2.152), (2.154) g = 1m and deﬁne a function of two k variables Km (z, w, s) by the equality k k (z, w, s) = µ(m, k, s)−1 Pm (−¯ z , w, 12m , s). Km

(2.158)

We show that the function k Hol (F )(z, s)|s=0 = Km (z, w, s), F (w) N,w |s=0

(2.159)

obtained by analytic continuation of the right hand side to the point s = 0 satisﬁes all conditions of the theorem, i.e. it coincides with the function deﬁned by (2.146), (2.147): Hol (F )(z, s)|s=0 ∈ Mkm (N, ψ) and the equality g(z), F (z) N = g(z), Hol (F )(z, s)|s=0 N holds for all g ∈ Skm (N, ψ). For this purpose we note that for suﬃciently large value of Re(s) the right hand side of (2.159) can be rewritten in the form of an integral over the whole Siegel upper half plane Hm of degree m: k Km (z, w, s), F (w) N,w = µ(m, k, s)−1 det(Im(z))s · F (w)det(w ¯ − z)−k−|2s| det(Im(w))k+s d× w, Hm

(2.160)

88

2 Siegel modular forms and the holomorphic projection operator

(due to the assumption on the growth of F the integral (2.160) is absolutely convergent for all Re(s) 0). Next let us consider the subgroup # " 1m b 0 Γ∞ = γ = ± | γ ∈ Γ ⊂ Γ = Γ0m (N ). 0m 1m Then the set {w = u + iv ∈ Hm | u ∈ X, v ∈ Y } 0 is a fundamental domain for the action of Γ∞ on Hm , and we see that for Re(s) 0 the right hand side of (2.160) takes the form −1 s 2µ(m, k, s) det(Im(z)) F (w) det(w ¯ − z + b)−k−|2s| · X

Y

b∈L

·det(Im(w))k+s d× w = (2.161) = 2µ(m, k, s)−1 det(Im(z))s F (w)S(w¯ − z, L; k + s, s)· X

Y

·det(Im(w))k+s d× w, where L = Mm (Z) ∩ V is a lattice in V = {x ∈ Mm (R) | function det(z + b)−k−|2s| S(z, L; k + s, s) =

t

x = x}, and the

b∈L

admits an analytic continuation to all s ∈ C by means of the Fourier expansion of (2.156). Moreover, for k > m we have that S(w ¯ − z, L; k + s, s)|s=0 = (−2πi)mk Γm (k)−1

(2.162) k−κ

det(h)

em (f (z − w)). ¯

Am h>0

Under the assumption on growth of F the integral admits an analytic continuation to the point s = 0. This analytic continuation can be explicitly given in the form of a Fourier expansion by means of (2.161), (2.162) using the positivity of h. As a result the function Hol (F )(z, s)|s=0 takes the form (4π)m(k−(m+1)/2) Γm (k − (m + 1)/2)−1 Hol (F )(z, s)|s=0 = k × det(h)k−κ · F (z)em (h(z − w))det(Im(w)) ¯ d w. Am h>0

X

Y

Then the formula (2.146) follows from the obvious equality: k × F (z)em (h(z − w))det(Im(w)) ¯ d w= X Y em (hz) Y A(v, h)em (ihv)det(v)k−1−m dv. In order to prove the remaining statements we note that the function

2.4 Holomorphic projection and Maass operator

89

Hol (F )(z, s)|s=0 deﬁned by (2.159) is holomorphic and it satisﬁes the automorphic properties with respect to Γ0m (N ) of weight k with the Dirichlet character ψ (mod N ). k (z, w, s) and conseIndeed, these properties are satisﬁed by the function Km quently by (2.160) for Re(s) 0. But the identity expressing the automorphy condition (2.11) does not change by analytic continuation, and we get Hol F (z, s)|s=0 ∈ Mm k (N, ψ) (note for m > 1 also the Koecher principle is applicable). The equality (2.148) is then deduced from deﬁnitions (2.158) and (2.159), and from the automorphy property (2.154) of the Poincar´e series. For Re(s) 0 we get k g, Km (z, w, s), F (w) N,w N,z = g, F N

=

(2.163)

k (z, w, s), g(z)

Km N,z , F (z) N,w .

These equalities remain valid by the analytic continuation and we get (2.148). In the equality (2.163) the property k (z, w, s) = K k (z, w, s Km ¯). m

was taken into account. 2.4.4 Fourier expansion of the holomorphic projection of special modular forms When applying the formula (2.146), it is convenient to use the integral representation (2.95) for the Γ -function of degree m, Γm (s) = π m(m−1)/4

m−1

Γ (s − (j/2)),

j=0

This integral representation can be rewritten in the equivalent form Γm (ν − κ)det(u)κ−ν = det(y)ν−m−1 e−tr(uy) dy (2.164) Y = det(y)ν−κ em (i(2π)−1 uy)d× y. Y

Moreover, if R(y) ∈ C[yij ] is a polynomial of y = (yij ), i ≤ j then for all ν ∈ Z, ν > m we have that ! R(y)det(y)ν−κ e−tr(uy) d× y = R(−∂/∂u)e−tr(uy) det(y)ν−κ d× y Y Y (2.165) = R(−∂/∂u) Γm (ν − κ)det(u)κ−ν , where ∂/∂u = ∂ij , ∂ij = 2−1 (1 + δij ∂/∂uij ). Indeed, it suﬃces to verify the statement (2.165) for monomials of the form

90

2 Siegel modular forms and the holomorphic projection operator

R(y) =

a(i,j)

yij

,

a(i, j) ∈ Z, a(i, j) ≥ 0.

i≤j

In this particular case this is done by application of the diﬀerential operator a(i,j) ∂/∂uij i≤j

to both sides of the equality (2.164). We shall need formulas (2.146) and (2.165) in a special situation, which is described in the next theorem below. m Theorem 2.18. Let F ∈ Mm k (N, ψ) be a C-modular form F ∈ Mk (N, ψ) which has the form of a product of the type F (z) = g(z)G(z), where B(h)em (hz), g(z) =

G(z) =

Am h>0

C(h)det(4πy)−n R(4πhy; n, β)em (hz),

Am h≥0

F (z) satisﬁes one of the two assumptions (a) or (b) of the theorem 2.16 and R(z; n, β) is the polynomial (2.120) deﬁned for any integer n ≥ 0, β ∈ C and z = tz ∈ Mn (C) by ! R(z; n, β) = (−1)mn etr(z) det(z)n+β ∆nm etr(z) det(z)−β , where ∆m = det(∂ij ),

(∂ij = 2−1 (1 + δij )∂/∂ij z, i ≤ j)

is the Maass diﬀerential operator. Then the following equality holds Hol F (z) = B(h1 )C(h2 )P (h2 , h; n, β)em (hz),

(2.166)

Am h=h1 +h2 >0

where P (v, u) = P (v, u; n, β) denotes a polynomial of u = tu = (uij ) and v = tv = (vij ) with the property P (v, u; n, β) ≡ (−1)mn det(v)n

(mod uij ),

(2.167)

and P (v, u; n, β) ∈ Q[u, v] for any β ∈ Q. Proof ( of 2.18). The proof of 2.18 is carried out by a straightforward application of the integral formula (2.146) for the action of Hol on each of the Fourier coeﬃcients of the function F (z): B(h1 )C(h2 )det(4πy)−n R(4πh2 y; n, β)em (ihz). A(y, h) = Am h=h1 +h2 >0

As a result we get

2.5 Explicit description of diﬀerential operators

A(h) =

91

B(h1 )C(h2 )P (h2 , h; n, β),

Am h=h1 +h2 >0

where P (v, u) (2.168) det(4πu)k−κ = R(4πvy; n, β)det(4πy)−n det(y)k−κ em (2iuy)d× y Γm (k − κ) Y det(4πu)k−κ = R(4πvy; n, β)det(4πy)−n+k−κ em (2iuy)d× y Γm (k − κ) Y det(4πu)k−κ R(vy; n, β)det(y)−n+k−κ e−tr(uy) d× y = Γm (k − κ) Y Γm (k − n − κ) det(u)k−κ R(v · ∂/∂u; n, β) det(u)κ−k+n , = Γm (k − κ) with n ∈ Z, n ≥ 0, β ∈ C. In order to accomplish the proof it suﬃces to show that the function P (v, u) = P (v, u; n, β) is a polynomial with the desired properties (2.167). This last fact is deduced from the last of the equalities (2.168) and some general properties of the diﬀerential operator ∂/∂u which are given below (see also [Kl2]).

2.5 Explicit description of diﬀerential operators 2.5.1 The polynomial R(z; r, β) Here we want to describe explicitely the polynomial R(z; r, β), introduced in the section 2.3 (see (2.120)). We show that this is a polynomial invariant for the action of conjugation on the space of symetric matrices y ∈ V = {h ∈ Mm (R) | th = h}. Let us consider the natural representation (0 ≤ r ≤ m) ρr : GLm (C) −→ GL(Λr Cm ) z −→ ρr (z) of the group GLm (C) on the vector space Λr Cm with respect to the basis {ei1 ∧ · · · ∧ eir : i1 < · · · < ir }. Thus ρr (z) is a matrix of size m r composed of the subdeterminants of z of degree r. Put ρr (z) = det(z)ρm−r (tz)−1

(r = 0, 1, · · · , m),

in other words, ρm−r (z) is the matrix representing the action of z on Λm−r Cm with respect to the basis dual to the above basis of Λr Cm . Then the representations ρr and ρr turn out to be polynomial representations so that for each z ∈ Mm (C) the linear operators ρr (z), ρr (z) are well deﬁned. We consider

92

2 Siegel modular forms and the holomorphic projection operator

the diﬀerential operators ρr (∂/∂z) and ρr (∂/∂z) which associate to each C valued function on Hm a certain Mt (C) valued function on Hm with t = m r . In particular, the Maass diﬀerential operator is done by ∆ = ρm (∂/∂z) = ρm (∂/∂z) = det(2−1 (1 + δij )∂/∂zij ). We deﬁne the invariant polynomials λr (z) = tr(ρr (z)). They are the coeﬃcients of the caracteristic polynomial of z det(t1m − z) =

m

(−1)j λj (z)tm−j .

j=0

The following diﬀerentiation rules are valid (see [Sh8], lemma 9.1): ∆(f g) =

m

tr tρr (∂/∂z)f · ρm−r (∂/∂z)g ,

(2.169)

r=0

ρr (∂/∂z)det(z)α = cr (α)det(z)α−1 ρm−r (z), ρr (∂/∂z)det(z)α = cr (α)det(z)α−1 ρm−r (z), ρr (∂/∂z)etr(uz) = ρr (u)etr(uz) , m ∆ etr(uz) det(z)α = etr(uz) det(z)α−1 cm−r (α)λr (uz),

(2.170) (2.171) (2.172) (2.173)

r=0

for α ∈ C with cr (α) =

r−1

(α + j/2) =

j=0

Γr (α + (r + 1)/2) Γr (α + (r − 1)/2)

(2.174)

Now we generalized formula (2.169) to the application of the Maass operator ∆ on n functions f1 , · · · , fn .

2.5 Explicit description of diﬀerential operators

93

Lemma 2.19. Let f1 , · · · , fn : Hm → C be n C ∞ -functions, then we get ∆(f1 · · · fn ) = (2.175) L L t (−1)ε(σK,L ) ρt1 (∂/∂z)Kt1 (f1 ) · · · ρtn (∂/∂z)Kttnn (fn ), 1

|T |=m K,L

where T goes through the multi-indices T = {0 ≤ t1 , · · · , tn ≤ m} which satisfy |T | = t1 + · · · + tn = m. The second sum is made on the multiindices K and L of multi-indices strictely ordonnered Ktj and Ltj such that card(Ktj ) = card(Ltj ) and ∪nj=1 Ktj = ∪nj=1 Ltj = m = {0, · · · , m} are two partitions of m. The permutation σK,L of m is determined by: for any h ∈ m, there is tj such that h = hi ∈ Ktj with i the rank of h in Ktj with respect to the natural order, then σK,L (hi ) = h i where h i is the element of rank i of Ltj . We note ε(σK,L ), the sign of the permutation σK,L . Proof. We prove this lemma by induction on the number n of functions. • If n = 2, this is the formula (2.169). • We assume that (2.175) is true for (n − 1). By hypothesis we have ∆(f1 , · · · , fn ) = (2.176) Ltn−1 Lt1 ε(σK,L ) (−1) ρt1 (∂z )Kt (f1 ) · · · ρtn−1 (∂z )Kt (fn−1 fn ). 1

|T |=m K,L

n−1

Lt

But the operator ρtn−1 (∂z )Ktn−1 coincides with the operator ∆tn−1 (z1 ) for the n−1 variable z1 = (zij )i∈Ktn−1 ,j∈Ltn−1 . Thus formula (2.176) give Lt

ρtn−1 (∂z )Ktn−1 (fn−1 fn ) = n−1

Lt

(−1)ε(σK,L ) ρtn (∂z )Ktn (fn−1 )·

tn−1 +tn =tn−1 Lt ·ρtn (∂z )Ktn (fn ),

n

n

(see also Section 3.3 of Chapter 3 where a more detailed proof of the Lemma 2.19 is given); compare with the proof of Lemma 3.6)

3 Arithmetical diﬀerential operators on nearly holomorphic Siegel modular forms

3.1 Description of the Shimura diﬀerential operators Let p be a prime number (we often assume p ≥ 5). The purpose of this chapter is to describe the action of certain arithmetical diﬀerential operators on algebraically deﬁned nearly holomorphic Siegel modular forms (in the sense of Shimura, see [Sh9] over an O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp . Often we simply assume that A = Cp . We view such modular forms as certain formal expansions. The important property of these arithmetical diﬀerential operators is their commutation with Hecke operators (under an appropriate normalization, see Theorem 3.13). Their action produces natural families of distributions on a proﬁnite group Y = lim Yi with values in these A-modules of nearly holomorphic Siegel mod←− ular forms inside a formal q-expansion ring like A[[q B ]][Ri,j ] where B is an additive semi-group, q B = {q ξ |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bm = {ξ = tξ ∈ Mm (Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms), and the nearly holomorphic parameters (Ri,j ) = R correspond to the matrix R = (4πIm(z))−1 in the Siegel modular case. There exist nice applications of this theory to construction of certain new p-adic families of modular forms (families of Siegel-Eisenstein series, families of theta-series wth spherical polynomials. . .). Main sources of this theory are: Serre’s theory of p-adic modular forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zˆeta p-adiques, LNM 350 (1973) 191268) [Se2]. Shimura’s theory of arithmeticity for nearly holomorphic forms (Shimura, G., Arithmeticity in the theory of automorphic forms. AMS, 2000, [Sh9]).

M. Courtieu and A. Panchishkin: LNM 1471, pp. 95–125, 2004. c Springer-Verlag Berlin Heidelberg 2004

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3 Arithmetical diﬀerential operators

Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi5]). Construction of p-adic Siegel-Eisenstein series by the author, appeared in Israel Journal of Mathematics in 2000, [Pa8]. Nerarly holomorphic Siegel modular forms provide additional arithmetical parameters. Moreover they admit an algebraic interpretation in terms of sections of certain de Rham cohomology sheaves on Siegel modular varieties (see [Ha3]), and a typical example of the arithmetical diﬀerential operators is given by the Gauss-Manin connection. On the other hand, diﬀerential operators preserving rational structures naturally come from the action of universal envelopping algebras of corresponding Lie algebras on automorphic forms. subsection Arithmetical nearly holomorphic Siegel modular forms They were studied by G.Shimura [Sh9] and descriptions: they admitξtwo diﬀerent a(ξ, Ri,j )q ∈ Q[[q Bm ]][Ri,j ] such that • as formal power series g = ξ∈Bm

for all R = (Ri,j )i,j=1,·...·,m = (4πIm(z))−1 = (2πi(¯ z − z))−1 and z ∈ Hm in the Siegel upper half plane of degree m the series converges to a C∞ -Siegel modular form of a given weight k and character ψ; • as certain C∞ -Siegel modular forms g taking values in Q at all CMpoints (up to a factor independnt of a concrete form) and satisfying certain reciprocity laws at these points. The ﬁrst description motivates our choice of arithmetical variables: q ξ , (r) R = (Ri,j ) = (4πIm(z))−1 = 2πi(¯ z − z)−1 . We describe the action of δk in terms of these arithmetical variables. Consider the diﬀerential operator ∆m (the Maass diﬀerential operator) of degree m, acting on complex C ∞ -functions on Hm , deﬁned by the equality: ∆m = det(∂˜ij ),

∂˜ij = 2−1 − (1 + δij )∂/∂ij

For any f ∈ C∞ (Hm , C), one deﬁnes also the operators Mk f (z) = det(z − z¯)κ−k ∆ det(z − z¯)k−κ+1 f (z) , δk f (z) =

m 1 − det(y)−1 Mk f (z). 4π

Due to H. Maass [Maa], for an integer k and a Dirichlet character ψ modulo N , the diﬀerential operator δk acts on C∞ -Siegel modular forms, δk : ˜ m (N, ψ) → M ˜ m (N, ψ). One deﬁnes then the Shimura diﬀerential operator M k k+2 as the composition (r)

˜ m (N, ψ) → M ˜ m (N, ψ). δk = δk+2r−2 ◦ · · · ◦ δk : M k k+2r In order to describe explicitely the action of δk on Fourier expansions consider, for an integer r ≥ 0 and for a complex number β, the polynomial

3.1 Description of the Shimura diﬀerential operators

97

! Rm (z; r, β) = (−1)mr etr(z) det(z)r+β ∆rm e−tr(z) det(z)−β . According to its deﬁnition the degree of the polynomial Rm (z; r, β) is equal to mr and the term of the highest degree coincides with det(z)r . We have also that for β ∈ Q the polynomial Rm (z; r, β) has rational coeﬃcients. Notice also that for m = 1 one has r r r−t R1 (z, r, β) = z β(β + 1) · · · (β + t − 1). t t=0 M.Courtieu has established in his PhD thesis the following more explicit expression for the function Rm (z; r, β) for arbitrary m (see [Cour], Chapter 3 and Theorem 3.12): Rm (z; r, β) =

r r det(z)r−t t t=0

RL (β)λl1 (z) · . . . · λlt (z),

|L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 +· · ·+lt ≤ mt−t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt−|L|) and with coeﬃcients in the ring Z[1/2]. The polynomials λi (z) are deﬁned by det(t1m + z) =

m

λi (z)tm−i

i=0

(λi (z) is the sum of all diagonal minors of size i × i of the matrix z). Let f ∈ Mm k (N, ψ) be a Siegel modular form whose Fourier expansion is given by f (z) = ξ∈Bm c(ξ)em (ξz). Let us consider the natural representation (0 ≤ r ≤ m) ρr : GLm (C) −→ GL(∧r Cm ) z −→ ρr (z) of the group GLm (C) on the vector space Λr Cm with respect to the basis {ei1 ∧ · · · ∧ eir : i1 < · · · < ir }. m Thus ρr (z) is a matrix of size m r × r composed of the subdeterminants of z of degree r. Put ρr (z) = det(z)ρm−r (tz)−1

(r = 0, 1, · · · , m),

in other words, ρm−r (z) is the matrix representing the action of z on Λm−r Cm with respect to the basis dual to the above basis of Λr Cm . Then the representations ρr and ρr turn out to be polynomial representations so that for each z ∈ Mm (C) the linear operators ρr (z), ρr (z) are well deﬁned.

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3 Arithmetical diﬀerential operators

We prove in this chapter (Theorem 3.14) then that the nearly holomorphic Siegel modular form (r) δk f (z) is given by the following formal power series expansion r r (r) det(ξ)r−t c(ξ) δk f (z) = t t=0 ξ∈Bm

=

RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )q ξ

|L|≤mt−t

ξ∈Bm

=

r r c(ξ) det(ξ)r−t t t=0

c(ξ)Q(R, ξ)q ∈ Q(f )[[q ξ

RL (κ − k − r)QL (R, ξ)q ξ

|L|≤mt−t

Bm

]][Ri,j ]i,j=1,··· ,m ,

ξ∈Bm

where Q(f ) is the subﬁeld of C generated by the Fourier coeﬃcients of f , L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m as above, r r Q(R, ξ) = Q(R, ξ; k, r) = RL (κ − k − r)QL (R, ξ)), det(ξ)r−t t t=0 |L|≤mt−t

and

QL (R, ξ) = tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) ).

For r = 1 this gives: δk f (z) =

c(ξ)

m

ξ∈Bm

(−1)m−l cm−l (k + 1 − κ)tr tρm−l (R) · ρl (ξ) q ξ ,

l=0

where cm−l (k + 1 − κ) =

m−l−1

(k + 1 − κ + j/2) =

j=0

Γm−l (k + 1/2) Γm−l (k − 1)/2)

(l ≤ m − 1),

and in the elliptic modular case m = 1, κ = 1, the only possibilities are t = 0, 1 when c0 (k) = 1, c1 (k) = k, and one obtains again the classical formula k δk ( an q n ) = nan q n − an q n = nan q n − kR an q n 4πy n≥0

n≥1

in which R=

1 , 4πy

n≥1

n≥0

 nan q n =

1 ∂  2πi ∂z

n≥0

n≥1

 an q n  = q

n≥0

 d  dq

 an q n  .

n≥0

Note that the operators δk act on nearly holomorphic forms and they are very diﬀerent from operators studied in [Ch-Eh] and [Eh-Ib] although some polynomials which come up in their action on Fourier expansions have similar properties.

3.2 Nearly holomorphic Siegel modular forms 3.2.1 Algebraic nearly holomorphic Siegel modular forms Let us deﬁne algebraic nearly holomorphic Siegel modular forms over an O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp . Recall that we ﬁx throughout the book two embeddings i∞ : Q → C,

i p : Q → Cp ,

and we often view the ﬁeld Q as a subﬁeld of both C and Cp (the complex and the p–adic numbers) via these embeddings, omitting the symbols i∞ and ip . We assume that A is a closed subring of Cp containing ip (Qp ). Recall some notations and deﬁnitions concerning Siegel modular forms. Let A be a commutative ring with identity, then Mr,s (A) denote the set of all (r × s)-matrices with coeﬃcients in A. For z ∈ Mr (C) put er (z) = e(tr(z)) with e(u) = exp(2iπu) for u ∈ C. We denote by tz ∈ Mr,s (A) the matrix, which is transpose to z ∈ Mr,s (A), and write ξ[η] for tηξη. For an invertible matrix ξ we put ξ = tξ −1 . If ξ is a hermitian matrix then we write ξ ≥ 0 or ξ > 0 according as ξ is non negative or positive deﬁnite. Let Hm denote the Siegel upper half plane on the degree m, Hm = {z ∈ Mm (C) |

t

z = z = x + iy, y > 0},

so that Hm is a complex analytic variety whose dimension is denoted by m = m(m + 1)/2. Let the symbol Am denote the lattice of all half integral symetric matrices in the vector space V = Vm = {y ∈ Mm (R) | ty = y}. This lattice is dual to the latice L = Mm (Z) ∩ V with respect to the pairing given by (u, v) −→ em (uv). Let G = GSpm be the algebraic subgroup of GLm deﬁned by GA = {γ ∈ GLm (A) |

γJm γ = ν(γ)Jm , ν(γ) ∈ A× },

t

for any commutative ring A, where 0m −1m Jm = . 1m 0m The elements of GA are characterized by the conditions bta − atb = dtc − ctd = 0m , dta − ctb = 1m . The multiplier ν deﬁnes a homomorphism ν : GA → A× so that ν(γ)2m = det(γ)2 and ker(ν) is denoted by Spm (A). We also put + + G∞ = GR , G+ ∞ = {γ ∈ G∞ | ν(γ) > 0}, GQ = G∞ ∩ GQ .

The group G+ ∞ acts transitively on the upper half plane Hm by the rule

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3 Arithmetical diﬀerential operators

ab z −→ γ(z) = (az + b)(cz + d)−1 γ = , z ∈ H ∈ G+ m ∞ cd so that the scalar matrices acts trivially, and Hm can be identiﬁed with a homogeneous space of the group Spm (R). Let Km denote the stabilizer of the point i1m ∈ Hm in the group Spm (R), Km = {γ ∈ Spm (R) | γ(i1m ) = i1m }, then there is a bijection Spm (R)/Km Hm and Km = Spm (R) ∩ SO2m . The group Km is a maximal compact subgroup of the Lie group Spm (R) and it can be identiﬁed with the group U(m) of all unitary m × m-matrices via the ab map γ = −→ a + ib. cd For a function f : Hm −→ C of the form f= c(ξ)em (ξz) ξ∈Am

and for an integer k we use the notation of Petersson for the action of γ ∈ GSpm on f : (f |k γ)(z) = det(γ)k−κ det(cz + d)−k f (γ(z)) (where

κ = (m + 1)/2),

we put also for N ∈ N: f |U (N )(z) =

ξ∈Am

f |V (N )(z) =

c(N ξ)em (ξz) =

tu=u∈M

m (Z)

f |k mod N

1m u 0m N 1m

c(ξ)em (N ξz) = f (N z) = N

m(κ−k)

ξ∈Am

f |k

N 1m 0m 0m 1m

√ f |W (N )(z) = det( N z)−k f (−(N z)−1 ) = N m(κ−k/2) f |k W (N ), 0m −1m W (N ) = . N 1m 0m There are the following relations for these operators: (f |W (N ))|W (N ) = (−1)mk f, f |W (N N ) = N mk/2 f |W (N )V (N ). We adopt also the notations c(ξ)em (ξz), Y = {y ∈ Mm (R) | fρ =

t

y = y > 0},

ξ∈Am

Bm = {h ∈ Am | h ≥ 0}, Cm = {h ∈ Am | h > 0}. The bloc notation g = ac db ∈ Spm (Z) deﬁnes the congruence subgroup

, ,

3.2 Nearly holomorphic Siegel modular forms

Γ0m (N ) =

101

" # ab g= ∈ Spm (Z)|c ≡ 0 mod N cd

of the Siegel modular group Γ m = Spm (Z), and its normal subgroup # " ab m Γ1 (N ) = ∈ Spm (Z)det(d) ≡ 1 mod N, c ≡ 0 mod N ⊂ Γ0m (N ). cd A classical Siegel modular form f ∈ Mk (N, ψ) of weight k and character ψ for Γ0m (N ) is a holomorphic function f : Hm → C such that for every ab γ = c d ∈ Γ0m (N ) one has f (γ(z)) = ψ(det d)det(cz + d)k f (z) (for m > 1 the regularity at ∞ is automatically satisﬁed by Koecher). The Fourier expansion of such f uses the semi-group Bm : f (z) = a(ξ)q ξ ∈ C[[q Bm ]] ξ∈Bm

where the symbols q ξ = exp(2πitr(ξz)) =

m i=1

ξii qii

−1 qij ij ⊂ C[[q11 , . . . , qmm ]][qij , qij ]i,j=1,··· ,m 2ξ

i<j

√ are used (with qij = exp(2π( −1zi,j ))); they form a multiplicative semi-group so that one may consider f as a formal q-expansion and one can introduce ˜ k (N, ψ) Siegel modular forms over A as certain elements of A[[q Bm ]]. Let M denote the complex vector space of C∞ -Siegel modular forms of weight k and character ψ for Γ0m (N ). For a non negative integer r and variables q, R = (Ri,j )i,j=1,...,m consider the ring A[[q Bm ]][Ri,j ] (over the complex numbers this notation corresponds to q ξ = exp(2πitr(ξz)), R = (4πIm(z))−1 ). Consider the following A-submodules ' Pr (A) = g = a(ξ, Ri,j )q ξ ∈ A[[q Bm ]][Ri,j ] ξ∈Bm

( a(ξ, Ri,j ) ∈ A[Ri,j ], degR a(ξ, Ri,j ) ≤ r ⊂ A[[q Bm ]][Ri,j ] Consider also for any natural number a the A-modules ( ' −1 Qr,a = g = a(ξ, Ri,j )q a ξ , ξ∈Bm

and put Qr =

3 a≥1

Qr,a .

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3 Arithmetical diﬀerential operators

Deﬁnition 3.1. Let Mm (N, ψ, A) denote the A-submodule of A[[q Bm ]][Ri,j ] r,k generated by all F = a(ξ, R)q ξ such that all a(ξ, R) ∈ A[Ri,j ]i,j=1,...,m ξ∈Bm ≤r

are polynomials of degree

over A ⊂ Cp which belong to

ip (Q)[Ri,j ]i,j=1,...,m and F =

ξ∈Bm

i−1 (a(ξ, R)) p

R=(4πY )−1

m q ξ ∈ Mm r,k (N ), ψ, Q) ⊂ Mr,k (N ), ψ, C)

is a C∞ -complex function on Hm satisfying the following two conditions ab ∀γ = ∈ Γ0m (N ) F |k γ = ψ(det(d))F (3.1) cd ∀γ ∈ Spm (Z) F |k γ ∈ Qr

(3.2)

(a complex nearly holomorphic modular form of type r and weight k over Q in the sense of Shimura, the second condition is necessary only for m = 1). Put Mm r,k (A) =

)

v Mm r,k (N p , ψ; A)

(3.3)

v≥1

(an A-module of nearly holomorphic Siegel modular forms of wight k, Dirichlet character ψ mod N , type r and level N pv (v ≥ 0) with coeﬃcients in A; we view ψ mod N ). 3.2.2 Formal expansions of nearly holomorphic forms. We may view a given cusp form a(ξ; f )q ξ ∈ A[[q Bm ]], f= ξ∈Bm Bm ]][Ri,j ] in the formal q-expansion ring as an element of Mm r,k (A) ⊂ A[[q

A[[q Bm ]][Ri,j ]). 3.2.3 Action of the U -operator. Deﬁne the action of the operator U = U (p) on g= a(ξ, n)Rn q ξ ∈ A[[q Bm ]][Ri,j ], ξ∈Bm ,n≥0

3.2 Nearly holomorphic Siegel modular forms

by

g|U (p) =

103

a(pξ, n)(pR)n q ξ

ξ∈Bm ,n≥0

(over the complex numbers R = (4π)−1 Im(z)−1 and this notation corresponds to 1m u f |U (p)(z) = f |k 0m p1m t u=u∈Mm (Z) mod p

p−κm

tu=u∈M

m (Z)

f ((z + u)/p), mod p

Im ((z + u)/p) = Im(z/p) = Im(z)/p = (4π)−1 (pR)−1 Let f0 = a(ξ, n; f0 )Rn q ξ ∈ A[[q Bm ]][Rij ] denote an eigenfunction ξ∈Bm ,n≥0

of U with the eigenvalue α ∈ A, (U (f0 ) = αf0 ) so that for any ﬁxed ξ = pv · ξ0 we have pv|n| a(pv ξ0 , n, ; f0 ) = αv a(ξ0 , n; f0 ) ∈ A, with |n| = ij nij , (pR)vn = pv|n| Rvn . An element α ∈ A is called a U - characteristic value on M = Mm r,k (A) attached to a cusp form if there exists a cusp form f0 which is an eigenfunction of U with the eigenvalue α ∈ A (U (f0 ) = αf0 ). Deﬁnition 3.2. Let α ∈ A. a) The α-proper submodule M(α) ⊂ M of the operator U is the following A-submodule M(α) = M(α) (A) = Ker(αI − U ) ⊂ M = M(A). b) The α-characteristic (primary) submodule of Mα = Mα (A) ⊂ M of the operator U is the maximal A-submodule on which U − αI is nilpotent. ) Ker(U − αI)r ⊂ M(A). Mα (A) = r≥1

c) For each ﬁxed level N pv put M(α) (N pv ) = M(α) (N pv , ψ; A) = M(α) ∩ M(N pv ), Mα (N pv ) = Mα (N pv , ψ; A) = Mα ∩ M(N pv ). Proposition 3.3. Suppose that α ∈ A× is an invertible element of the algebra A then a) the A-linear map U v takes M(N pv+1 ) to M(N p); b) assume that Mα (N pv ; A) is a locally free A-module of ﬁnite rank then for all v ≥ 1 the A-linear map U is invertible on Mα (N pv ) ⊂ M(N pv ); c) assume that Mα (N pv ; A) is a locally free A-module of ﬁnite rank then for all v ≥ 1 Mα (N pv ; A) = Mα (N p; A) is independent of v;

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3 Arithmetical diﬀerential operators

Proof The statement a) is veriﬁed using a known relation of U v with the trace operator on modular forms of weight k: pv+1 = g|k γ, g|TrN Np γ∈Γ0 (N pv+1 )\Γ0 (N p)

then

v+1

vm(k−m−1)

p 2 g|U v = p g|WN pv+1 TrN WN p , Np 0 −1 is the standard involution. where WM = M 0 v+1 p ˜ m (N pv+1 , ψ) by acts on the space M This operator TrN k Np 1m 0m N pv+1 g|k TrN p (g) = N pu 1m t v u=u∈Mm (Z)

(mod p )

One can write down the action of this operator on Fourier expansions using the operators U (pv ) acting by m(m+1) F |U (pv )(z) = (pv )− 2 F ((z + u)/pv ). tu=u∈M

m (Z)

(mod pv )

One uses the following matrix identity: 1m 0m N pu 1m −1m 0m 1m −u 0m −1m v+1 −1 = (−N p ) = , 0 m pv 1 m N pv+1 1m 0m N p1m 0m which implies the relation: v+1

p TrN Np

(g) = p−

vm(k−m−1) 2

g|W (N pv+1 )U (pv )W (N p).

In order to prove b) notice that the operator U acts on the A-module of ﬁnite rank Mα (N pv ; A) ⊂ M(N pv ; A) and its determinant is in A× hence U is invertible on Mα (N pv ; A). The statement c) then directly follows from a) and b). Deﬁnition 3.4. Suppose that α ∈ A× is an invertible element of the algebra A. a) Deﬁne the α-characteristic projection πα,v : M(N pv ) → Mα (N pv )

(v ≥ 1)

as the canonical projector to the α-characteristic submodule ) Ker(U − αI)r Mα (N pv ) = of the U -operator with the kernel

4

r≥1

r≥1

Im(U − αI)r ;

3.2 Nearly holomorphic Siegel modular forms

105

Proposition 3.5. a) For each ﬁxed level N pv the following diagram is commutativ M(N pv+1 ) −→ Mα (N pv+1 ) πα,v+1    v v U 6 6 U M(N p) −→ Mα (N p)

(3.4)

πα (g) = U −v [πα,1 U v (g)]

(3.5)

πα,1

b) For any g ∈ M put

(note that πα (g) is well deﬁned if v is suﬃciently large so that g ∈ M(N pv+1 ), because of the commutative diagram (3.4)).

3.3 Algebraic diﬀerential operators of Maass and Shimura 3.3.1 Diﬀerential operators on Hm . Let us consider the diﬀerential operator ∆m (the Maass diﬀerential operator) of degree m, acting on complex C ∞ -functions on Hm , deﬁned by the equality: ∆m = det(∂˜ij ),

∂˜ij = 2−1 − (1 + δij )∂/∂ij

(3.6)

For any f ∈ C∞ (Hm , C), one deﬁnes also the operators Mk f (z) = det(z − z¯)κ−k ∆ det(z − z¯)k−κ+1 f (z) , m 1 δk f (z) = − det(y)−1 Mk f (z). 4π Due to H. Maass [Maa], for an integer k and a Dirichlet character ψ modulo N , the diﬀerential operator δk acts on C∞ -Siegel modular forms, δk : ˜ m (N, ψ). One deﬁnes then the Shimura diﬀerential operator ˜ m (N, ψ) → M M k k+2 as the composition (r)

˜ m (N, ψ) → M ˜ m (N, ψ). δk = δk+2r−2 ◦ · · · ◦ δk : M k k+2r In order to describe explicitely the action of δk on Fourier expansions consider, for an integer r ≥ 0 and for a complex number β, the polynomial ! (3.7) Rm (z; r, β) = (−1)mr etr(z) det(z)r+β ∆rm e−tr(z) det(z)−β , with z ∈ V ⊗ C, where the exponentiation is well deﬁned by det(y)β = exp (β log[det(y)]) , for det(y) > 0, y ∈ Y ⊗ C. According to deﬁnition (3.7) the degree of the polynomial Rm (z; r, β) is equal to mr and the term of the highest degree coincides with det(z)n . We have also that for β ∈ Q the polynomial Rm (z; r, β) has rational coeﬃcients. Notice also that for m = 1 one has r r R1 (z, r, β) = β(β + 1) · . . . · (β + t − 1)z r−t . t t=0 M.Courtieu has established in his PhD thesis in 2000 the following more explicit expression for the function Rm (z; r, β) for arbitrary m (see [Cour], Chapter 3)

3.3 Algebraic diﬀerential operators of Maass and Shimura r r Rm (z; r, β) = det(z)r−t t t=0

107

RL (β)λl1 (z) · . . . · λlt (z), (3.8)

|L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 +· · ·+lt ≤ mt−t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt−|L|) and with coeﬃcients in the ring Z[1/2]. The polynomials λi (z) are deﬁned by det(t1m + z) =

m

λi (z)tm−i ,

(3.9)

i=0

In other words λi (z) is the sum of all diagonal minors of size i×i of the matrix z. 3.3.2 The polynomial Rm (z; r, β). We now describe explicitely the polynomial Rm (z; r, β) deﬁned by (3.7). We show that this is a polynomial invariant for the action of conjugation on the space of symetric matrices y ∈ V = {ξ ∈ Mm (R) | tξ = ξ}. Let us consider the natural representation (0 ≤ r ≤ m) ρr : GLm (C) −→ GL(∧r Cm ) z −→ ρr (z) of the group GLm (C) on the vector space Λr Cm with respect to the basis {ei1 ∧ · · · ∧ eir : i1 < · · · < ir }. m Thus ρr (z) is a matrix of size m r × r composed of the subdeterminants of z of degree r. Put ρr (z) = det(z)ρm−r (tz)−1

(r = 0, 1, · · · , m),

in other words, ρm−r (z) is the matrix representing the action of z on Λm−r Cm with respect to the basis dual to the above basis of Λr Cm . Then the representations ρr and ρr turn out to be polynomial representations so that for each z ∈ Mm (C) the linear operators ρr (z), ρr (z) are well deﬁned. We consider ˜ ˜ the diﬀerential operators ρr (∂/∂z) and ρr (∂/∂z) which associate to each C valued function on Hm a certain Mt (C) valued function on Hm with t = m r . In particular, the Maass diﬀerential operator is given by ˜ ˜ ˜ ∆ = ρm (∂/∂z) = ρm (∂/∂z) = det(2−1 (1 + δij )∂/∂z ij ). We deﬁne the invariant polynomials λr (z) = tr(ρr (z)). They are the coeﬃcients of the characteristic polynomial of z

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3 Arithmetical diﬀerential operators

det(t1m − z) =

m

(−1)j λj (z)tm−j .

j=0

The following diﬀerentiation rules are valid (see [Sh8], lemma 9.1): ˜ ˜ tr tρr (∂/∂z)f · ρm−r (∂/∂z)g ,

(3.10)

α ˜ ρr (∂/∂z)det(z) = cr (α)det(z)α−1 ρm−r (z),

(3.11)

α ˜ ρr (∂/∂z)det(z) = cr (α)det(z)α−1 ρm−r (z),

(3.12)

∆(f g) =

m r=0

tr(uz) ˜ ρr (∂/∂z)e = ρr (u)etr(uz) , m ∆ etr(uz) det(z)α = etr(uz) det(z)α−1 cm−t (α)λt (uz),

(3.13) (3.14)

t=0

for α ∈ C with c0 (α) = 1,

cr (α) =

r−1

(α + j/2) =

j=0

Γr (α + (r + 1)/2) (r ≥ 1). Γr (α + (r − 1)/2)

(3.15)

One can extend the formula (3.10) to the case of the action of ∆ on n functions f1 , · · · , fn : Lemma 3.6. Let f1 , · · · , fn : Hm → C be n C∞ -functions, then we get ∆(f1 · · · fn ) = Lt1 Ltn ˜ ˜ (−1)ε(σK,L ) ρt1 (∂/∂z) Kt (f1 ) · · · ρtn (∂/∂z)Ktn (fn ),

|T |=m K,L

(3.16)

1

where T runs over the sets of multiplets T = {0 ≤ t1 , · · · , tn ≤ m} which satisfy |T | = t1 + · · · + tn = m. The second sum is extended over the sets of all multi-indices K and L consisting of strictely ordered subsets of multi-indices Ktj and Ltj such that card(Ktj ) = card(Ltj ) and ∪nj=1 Ktj = ∪nj=1 Ltj = m = {0, · · · , m} are two partitions of m. The permutation σK,L of m is determined by the following rule: for any h ∈ m, take tj such that h = hi ∈ Ktj where i the rank of h in Ktj (with respect to the natural order), then σK,L (hi ) = h i where h i is the element of rank i of Ltj , and denote by ε(σK,L ) the sign of the permutation σK,L . Proof of Lemma 3.6. One proves this lemma by induction on the number n of functions. • If n = 2, this is the formula (3.10) • We assume that (3.16) is true for (n − 1). By hypothesis we have

3.3 Algebraic diﬀerential operators of Maass and Shimura

109

∆(f1 , · · · , fn ) = (3.17) Ltn−1 Lt1 ε(σK,L ) (−1) ρt1 (∂z )Kt (f1 ) · · · ρtn−1 (∂˜z )Kt (fn−1 fn ). 1

|T |=m K,L

n−1

Lt

But the operator ρtn−1 (∂˜z )Ktn−1 coincides with the operator ∆tn−1 (z1 ) (with n−1 respect to the variable z1 = (zij )i∈Ktn−1 ,j∈Ltn−1 ). Thus formula (3.10) gives Lt (3.18) ρtn−1 (∂˜z )Ktn−1 (fn−1 fn ) n−1 L Lt t = (−1)ε(σK,L ) ρtn−1 (∂˜z )K n−1 (fn−1 ) · · · ρtn (∂˜z )Ktn (fn ), t n−1

tn−1 +tn =tn−1

n

with card(Ktn−1 ) = card(Ltn−1 ) = t n−1 , card(Ktn ) = card(Ltn ) = t n and one has Ktn−1 ∪ Ktn = Ktn−1 , Ltn−1 ∪ Ltn = Ltn−1 and (3.17) follows. The following more delicate diﬀerentiation rule holds: Lemma 3.7. Let u = tu be a real symmetric m × m-matrix, α a complex number, r, j two integers such that 0 ≤ r ≤ j ≤ m and z ∈ Hm then one has the following equality: ˜ (λj (z)α ) = cr (α)λj (z)α−1 µ (3.19) ρr (∂) r,j−r (z). m Here µr,j−r (z) is a matrix of size m r × r with the coeﬃcients given by µr,j−r (z)L K =

T ⊃K,L

T \L , T \K

ρj−r (z)

the sum is being taken over all the multi-indices T : 1 ≤ t1 < · · · < tj ≤ m containing the multiindices K : 1 ≤ k1 < · · · < kr ≤ m et L : 1 ≤ l1 < ¯ is used for the the complementary multi-index · · · < lr ≤ m. The notation L in m = {0, · · · , m} of the multi-index L. Recall that the matrix coeﬃcient ρm−r (z)L K is equal (up to a sign) to the determinant of the complementary matrix in z for the determinant ρr (z)L K more precisely one has ¯

ε(σK,L ) ρm−r (z)L ρm−r (z)L ¯, K = (−1) K

where σK,L is the permutation of Sm given by the following rule: for every ¯ σK,L (k¯i ) = ¯li with l¯i ∈ L ¯ and i ki ∈ K, σK,L (ki ) = li and for every k¯i ∈ K, ¯ ¯ denoting the rank of the index ki (respectively li , ki , li ) with respect to the ¯ L) ¯ (recall that ε(σK,L ) denotes the natural order in K (respectively L, K, signature of the permutation σK,L ). 3.3.3 Proof of Lemma 3.7 is deduced from (3.13) as follows. Notice that λj (z) = T ρj (z)TT is the sum ˜ L , one gets non zero of all diagonal minors of size j in z. If one applies ρr (∂) K

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3 Arithmetical diﬀerential operators

terms only for those diagonal minors ρj (z)TT , when the multi-index T contains the multi-indices K et L. The corresponding minor is then the determinant of a submatrix z, so that the computation is similar to that in (3.13) but for a matrix of a lower size. This gives us then ˜ L (λj (z)α ) = cr (α)λj (z)α−1 ρr (∂) K

T \L , T \K

ρj−r (z)

T ⊃K,L

and (3.18) follows. 3.3.4 Action of the Shimura diﬀerential operator on formal Fourier expansions We now apply the above results on Rm (z; r, β) in order to obtain an explicit dscripton of the action of the Shimura diﬀerential operator on formal Fourier expansions of nearly holomorphic Siegel modular forms. First of all we recall that the Shimura diﬀerential operator δk in the Siegel modular case is given as the twisted Maass operator ∆ [Maa], Chapter 19. Deﬁnition 3.8. For any f ∈ C∞ (Hm , C), deﬁne the operators Mk f (z) = det(z − z¯)κ−k ∆ det(z − z¯)k−κ+1 f (z) , δk f (z) =

m 1 − det(y)−1 Mk f (z). 4π

(3.20) (3.21)

Due to H. Maass [Maa], for an integer k and a Dirichlet character ψ modulo N , the Shimura diﬀerential operator acts on C∞ -Siegel modular forms δk : ˜ m (N, ψ) → M ˜ m (N, ψ). One deﬁnes then δ (r) = δk+2r−2 ◦ · · · ◦ δk . M k k+2 k (r) We show now that the Fourier expansion of δk f , where f = Bm c(ξ)em (ξz) ∈ Mm k (N, ψ), is given by the formula (r) c(ξ)det(4πy)−r Rm (4πξy; r, β)em (ξz). δk f = ξ∈Bm (r)

In order to describe the action of δk f one establishes ﬁrst the following Lemma 3.9. Let α be a complex number, r a positive integer, u a real symmetric matrix and z ∈ Hm then one has the following relation det(z)−r ∆ det(z)r+1 ∆r etr(uz) det(z)α = ∆r+1 etr(uz) det(z)α+1 .

3.3 Algebraic diﬀerential operators of Maass and Shimura

111

Proof of Lemma 3.9 Let us use the formula (3.14) giving ∆ etr(uz) det(z)α in order to compare it with the action of the operator det(z)−r ∆ det(z)r+1 · one has det(z)−r ∆ etr(uz) det(z)α+1 · m

·

cm−l1 (α) · . . . · cm−lr (α − r + 1)Pl1 ···lr (u, z) =

l1 ,··· ,lr =0

etr(uz) det(z)α−r · ·

m

cm−l1 (α) · . . . · cm−lr (α − r + 1)cm−lr+1 (α + 1)Pl1 ···lr lr+1 (u, z).

l1 ,··· ,lr+1 =0

The polynomials Pl1 ···lr are deﬁned recursively in such a way that ∆r etr(uz) det(z)α = e

tr(uz)

α−r

det(z)

m

cm−l1 (α) · . . . · cm−lr (α − r + 1)Pl1 ···lr (u, z),

l1 ,··· ,lr =0

hence one has the following equality satisﬁed by Pl1 ···lr (u, z) : Pl1 ···lr (u, z) =

L

L

L

˜ i2 (Pl ···l (u, z)), (−1)ε(σK,L ) ρi1 (u)Kii1 ρm−lr (z)Kllrr ρi2 (∂) 1 r−1 Ki 1

2

i1 +i2 =m−lr K,L

(a polynomial in matrix variables u and z independent of α and symmetric in lj ). The independence of Pl1 ···lr (u, z) on α results from the fact that the terms depending on α factorise out using (3.11). The symmetry in lj results from the theorem of Schwarz on the independence of the mixed partial derivatives of C∞ -functions on the order of diﬀerentiaion. Here the choice of l1 , · · · , lr corresponds to another order of derivation applied to the factor det(z) . Replacing u by −1m and α by −β, and using then the formula deﬁning R(z; r, β), one ﬁnds that this is indeed a polynomial as the factors etr(z) and e−tr(z) cancel, as well as the factors det(z)β and det(z)−β . One obtains then: R(z; r, β) =

m l1 ,··· ,lr =0

cm−l1 (−β) · . . . · cm−lr (−β − r + 1)Pl1 ···lr (−1m , z).

112

3 Arithmetical diﬀerential operators

In particular the polynomials Pl1 ···lr are symmetric in lj , and one can replace the index of summation l1 = lr+1 , l2 = l1 , · · · , lr+1 = lr so that one ob tr(uz) r+1 α+1 e . This gives exactly the announced tains the expression ∆ det(z) (r)

property. In order to compute the Fourier expansion of δk f , one proceeds by induction on r. One computes the action of the Shimura diﬀerential operator on the test functions em (ξz), more precisely one shows that δk (em (ξz)) = det(4πy)−r Rm (4πξy; r, κ − k − r)em (ξz). c(ξ)em (ξz), one obtains then by linearity the In fact if one takes f (z) = (r)

ξ∈Bm (r) δk f ,

resulting Fourier expansion of (r) c(ξ)det(4πy)−r Rm (4πξy; r, κ − k − r)em (ξz), δk f (z) = ξ∈Bm

It remains to prove the following Lemma 3.10. The action of the Shimura diﬀerential operator on the test functions em (ξz) (where ξ ∈ Bm and z ∈ Hm ) is given by the formula δk (em (ξz)) = det(4πy)−r Rm (4πξy; r, κ − k − r)em (ξz). (r)

(3.22)

Proof of Lemma 3.10 : One proves this lemma by induction on r : • If r = 0, it is obvious. • Suppose next that it holds for some r then if one applies the Shimura operator δk+2r one obtains (r+1)

δk

(em (ξz)) = (−1)m det(4πy)−1 det(z − z¯)κ−k−2r · ·∆ det(z − z¯)k+2r+1−κ det(4πy)−r em (ξz) · · Rm (4πξy; r, κ − k − r)] .

Notice that 4πξy = −2iπξ(z − z¯). On the other hand z¯ is aniholomorphic hence it behaves as a constant under the action of ∆, and one can write, by putting u = −2iπξ, β = κ − k − r: z); r, β) = (−1)mr etr(u(z−¯z)) det(z−¯ z )r+β ∆r e−tr(u(z−¯z)) det(z − z¯)−β . Rm (u(z−¯ One replaces this expression in the ﬁrst formula pulling out the factor em (ξ z¯) = etr(−u¯z) which is an antiholomorphic function. One obtains then (r+1)

δk

(em (ξz)) = det(4πy)−(r+1) em (ξz)(−1)m(r+1) etr(u(z−¯z )) det(z − z¯)β ·

3.3 Algebraic diﬀerential operators of Maass and Shimura

113

! ·∆ det(z − z¯)r+1 ∆r e−tr(u(z−¯z)) det(z − z¯)−β . Let us now apply lemma 3.9 which gives (r+1)

δk

(em (ξz)) = det(4πy)−(r+1) em (ξz)(−1)m(r+1) etr(u(z−¯z)) · ·det(z − z¯)r+β ∆r+1 e−tr(u(z−¯z)) det(z − z¯)−β

= det(4πy)−(r+1) Rm (4πξy; r + 1, κ − k − (r + 1))em (ξz), ﬁnishing the proof by induction. Now we can write down the action of the Shimura operator on the Fourier expansions of Siegel modular forms in terms of the polynomial Rm (z; r, β) as follows: Proposition 3.11. Let f ∈ Mm k (N, ψ) be a Siegel modular form whose Fourier expansion is given by f (z) = ξ∈Bm c(ξ)em (ξz), then (r) c(ξ)det(4πy)−r Rm (4πξy; r, κ − k − r)em (ξz). δk f (z) = ξ∈Bm

Let us now use the explicite expressions (3.8) of M.Courtieu for the func(r) tion Rm (z; r, β) for arbitrary m in order to describe the action of δk on Fourier expansions. Theorem 3.12. a) Let f ∈ Mm k (N, ψ) be a Siegel modular form whose Fourier expansion is given by f (z) = ξ∈Bm c(ξ)em (ξz), then r r (r) δk f (z) = c(ξ)det(4πy)−r (3.23) det(2πiξ(¯ z − z))r−t t t=0

ξ∈Bm

RL (κ − k − r)λl1 (2πiξ(¯ z − z)) · . . . · λlt (2πiξ(¯ z − z))em (ξz)

|L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 + · · · + lt ≤ mt − t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt − |L|) and with coeﬃcients in thering Z[1/2] (recall that m−i . the polynomials λi (z) are deﬁned by det(t1m + z) = m i=0 λi (z)t b) In particular, if r = 1, δk f (z) = (−1)m

c(ξ)det(2πi(¯ z − z))−1

ξ∈Bm m

cm−t (k + 1 − κ)λt (−2πiξ(¯ z − z))em (ξz)

t=0

where c0 (α) = 1,

cr (α) =

r−1

(α + j/2) =

j=0

Γr (α + (r + 1)/2) . Γr (α + (r − 1)/2)

(3.24)

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3 Arithmetical diﬀerential operators

Proof of Theorem 3.12 Let us write by (3.22) (r) δk f (z) = c(ξ)det(4πy)−r Rm (4πξy; r, κ − k − r)em (ξz) ξ∈Bm

and substitute in it r r Rm (z; r, β) = det(z)r−t t t=0

RL (β)λl1 (z) · . . . · λlt (z),

|L|≤mt−t

with 4πξy = 2πiξ(¯ z − z), β = κ − k − r: (r) δk f (z)

=

−r

c(ξ)det(4πy)

t=0

ξ∈Bm

r r

t

det(2πiξ(¯ z − z))r−t

RL (κ − k − r)λl1 (2πiξ(¯ z − z)) · . . . · λlt (2πiξ(¯ z − z))em (ξz)

|L|≤mt−t

Note also that by (3.14) m ∆ etr(uz) det(z)α = etr(uz) det(z)α−1 cm−t (α)λt (uz), t=0

for α ∈ C with c0 (α) = 1, cr (α) =

r−1

(α + j/2) =

j=0

Γr (α + (r + 1)/2) Γr (α + (r − 1)/2)

and by deﬁnition (3.7) ! Rm (z; n, β) = (−1)mn etr(z) det(z)n+β ∆nm e−tr(z) det(z)−β , hence

! Rm (z; 1, β) = (−1)m etr(z) det(z)1+β ∆m e−tr(z) det(z)−β = (−1)m etr(z) det(z)1+β e−tr(z) det(z)−β−1

m

cm−t (−β)λt (−z)

t=0

giving δk f (z) =

ξ∈Bm

c(ξ)det(4πy)−1 Rm (4πξy; 1, κ − k − r)em (ξz) =

3.3 Algebraic diﬀerential operators of Maass and Shimura

115

c(ξ)det(2πi(¯ z − z))−1 (−1)m etr(2πiξ(¯z −z)) det(2πiξ(¯ z − z))1+β ·

ξ∈Bm

·e−tr(2πiξ(¯z−z)) det(2πiξ(¯ z −z))−β−1

m

cm−t (k+1−κ)λt (−2πiξ(¯ z −z))em (ξz) =

t=0

(−1)m

c(ξ)det(2πi(¯ z − z))−1

m

cm−t (k + 1 − κ)λt (−2πiξ(¯ z − z))em (ξz).

t=0

ξ∈Bm

3.3.5 Commutation of the Shimura operator with Hecke operators We conclude this section with a general result on commutation of the Shimura diﬀerential operator with Hecke operators (up to a normalization). Recall some basic fact about the action of Hecke algebras (see [An5]). Let q be a prime, q | N , ∆ = ∆m q (N ) = " ab −1 ∩ GL (Z[q ]) γ= ∈ G+ ν(γ)± ∈ Z[q −1 ], c ≡ 0m 2m Q cd

# (mod N )

m be a subgroup in G+ Q containing Γ = Γ0 (N ). The Hecke algebra

L = Lm q (N ) = DQ (Γ, ∆) over Q is then deﬁned as a Q-linear space generated by the double cosets (g) = (Γ gΓ ), g ∈ ∆ of the semigroup ∆ with respect to the subgroup Γ , for which multiplication is deﬁned by the standard rule (see [An5], [Sh1]). We recall the description of the structure of L = Lm q (N ), (q N ): for each j, 1 ≤ j ≤ m let us denote by wj an automorphism of the algebra ±1 ±1 Q[x±1 0 , x1 , · · · , xm ]

deﬁned on its generators by the rule: x0 → x0 xj , xj → x−1 j , xi → xi

(1 ≤ i ≤ m, i = j).

Then the automorphisms wj and the permutation group Σm of the variables xi (1 ≤ i ≤ m) generate together the Weyl group W = Wm , and there is the Satake isomorphism (see (2.1.3) and 2.1.7): ∼

±1 ±1 Wm . Sat : L → Q[x±1 0 , x1 , · · · , xm ]

Recall that we deﬁne Hecke operators using the notation of Petersson and ab Andrianov (2.22): for any g = ∈∆ cd (f |k,ψ g)(z) = (detg)k−(m+1)/2 ψ(deta)det(cz + d)−k f (g(z)).

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3 Arithmetical diﬀerential operators

Theorem 3.13. Let f (z) = c(ξ)em (ξz) ∈ Mm k (N, ψ)

(ξ ∈ Bm ),

ξ∈Bm

and let

(r)

˜ m (N, ψ) → M ˜ m (N, ψ) δk = δk+2r−2 ◦ · · · ◦ δk : M k k+2r be the Shimura diﬀerential operator deﬁned by (3.21). Consider for a ﬁxed prime q an arbitrary element

t(X)

X=

vi (Γ0 gi ) ∈ L0 (Γ0 = Γ0m , L0 = Lm 0,q )

i=1

of the extended Hecke algebra L0 with ν ∗ q i di bi gi = ∈ ∆m 0,q , 0 di and assume X homogeneous (ν1 = ν2 = · · · νt = ν). Then (r) (r) (δk f )|k+2r X = q νrm δk (f |k X).

(3.25)

In particular (r)

(r)

(r)

(r)

m m (δk f )|k+2r Π+ (q) = q rm δk (f |k Π+ (q)),

(3.26)

m m (δk f )|k+2r Π− (q) = q rm δk (f |k Π− (q))

Proof of Theorem 3.13 : We use the known property δk (f |k γ) = (δk f )|k+2 γ for all γ ∈ Sp2m (R) and k ∈ N. Taking into account (δk f )|k+2 gi = det(gi )δk (f |k gi ), one immediately obtains (3.25) from the relation detgi = q νi rm taking into account the homogeneousity of X.

3.4 Arithmetical variables of nearly holomorphic Siegel modular forms and diﬀererntial operators. 3.4.1 Arithmetical nearly holomorphic Siegel modular forms where studied by G.Shimura [Sh9] two diﬀerent descriptions: and they admit a(ξ, Ri,j )q ξ ∈ Q[[q Bm ]][Ri,j ] such that for • as formal power series g = ξ∈Bm

all R = (4πIm(z))−1 and z ∈ Hm the series converges to a C∞ -Siegel modular form of a given weight k and character ψ; • as certain C∞ -Siegel modular forms g taking values in Q at all CMpoints (up to a factor independent of a concrete form) and satisfying certain reciprocity laws at these points. The ﬁrst description motivates our choice of arithmetical variables: q ξ , z − z)−1 . Our task in this Section is to R = (Ri,j ) = (4πIm(z))−1 = 2πi(¯ (r) expliciteltly describe the action of δk in terms of these arithmetical variables. Theorem 3.14. Let f ∈ Mm k (N, ψ) be a Siegel modular form whose Fourier expansion is given by f (z) = ξ∈Bm c(ξ)q ξ , then (r)

δk f (z) =

c(ξ)

ξ∈Bm

RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )q ξ

|L|≤mt−t

=

r r det(ξ)r−t t t=0

ξ∈Bm

r r c(ξ) det(ξ)r−t t t=0

=

RL (κ − k − r)QL (R, ξ)q ξ

|L|≤mt−t

c(ξ)Q(R, ξ)q ξ ∈ Q(f )[[q Bm ]][Ri,j ]i,j=1,··· ,m ,

ξ∈Bm

where Q(f ) is the subﬁeld of C generated by the Fourier coeﬃcients of f , L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m as above, r r Q(R, ξ) = Q(R, ξ; k, r) = RL (κ − k − r)QL (R, ξ)), det(ξ)r−t t t=0 |L|≤mt−t

QL (R, ξ) = tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) ). If r = 1 one has δk f (z) =

ξ∈Bm

where

c(ξ)

m l=0

(−1)m−l cm−l (k + 1 − κ)tr tρm−l (R) · ρl (ξ) q ξ

118

3 Arithmetical diﬀerential operators

cm−l (k + 1 − κ) =

m−l−1

(k + 1 − κ + j/2) =

j=0

Γm−l (k + 1/2) Γm−l (k − 1)/2)

(l ≤ m − 1).

Remark 3.15. In the elliptic modular case m = 1, κ = 1, the only possibilities are t = 0, 1 when c0 (k) = 1, c1 (k) = k, and one obtains again the classical formula δk (

an q n ) =

n≥0

nan q n −

n≥1

k an q n = nan q n − kR an q n 4πy n≥0

n≥1

n≥0

in which

    ∂ 1 1 d  , R= nan q n = an q n  = q  an q n  . 4πy 2πi ∂z dq n≥1

n≥0

n≥0

Proof of Theorem 3.14 Let us change the variable by putting R equal (4πy)−1 = −2πiz − z¯)−1 and let us use the equality (3.14) with u = −2πiξ and the polynomial (3.7) with n = r, β = κ − k − r m 1 det(y)−1 det(z − z¯)κ−k ∆ det(z − z¯)k−κ+1 f (z) δk f (z) = − 4π = det(2πi)−m det(z − z¯)κ−k−1 ∆ det(z − z¯)k−κ+1 f (z) . δk (em (ξz)) = det(4πy)−r em (ξz)R(4πξy; r, κ − k − r) (r)

z − z); r, κ − k − r) = det(2πi(¯ z − z))−r em (ξz)R(2πiξ(¯ We have by Theorem 3.11 (r) δk f (z)

=

−r

c(ξ)det(4πy)

ξ∈Bm

r r det(2πiξ(¯ z − z))r−t · t t=0

(3.27)

RL (κ −k −r)λl1 (2πiξ(¯ z −z))·. . .·λlt (2πiξ(¯ z −z))em (ξz)

|L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 + · · · + lt ≤ mt − t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt − |L|) and with coeﬃcients in the ring Z[1/2] (recall that m the polynomials λj (z) are deﬁned by det(t1m + z) = j=0 λj (z)tm−j . We now express all terms of type λj (2πiξ(¯ z − z)) in terms of our arithz − z)) = λj (ξR−1 )). metical variable R: 2πi(¯ z − z) = R−1 and λj (2πiξ(¯ Viewing symmetric matrices R and ξ as independent variables let us write the deﬁnition

3.4 Arithmetical variables of nearly holomorphic forms

119

det(tR + ξ) = det(R)det(t1m + ξR−1 ) = det(R)det(t1m + ξR−1 ) =

m

det(R)λj (ξR−1 )tm−j = tm det(ξ)det(Rξ −1 + t−1 1m ) =

j=0

tm det(ξ)

m

λj (Rξ −1 )t−m+j =

(3.28)

j=0 m

det(ξ)λj (Rξ −1 )tj

j=0

Let us show that the coeﬃcients of the polynomial det(tR + ξ) admit the following very explicite expression det(tR + ξ) =

m

tr tr tρr (R) · ρm−r (ξ))

(3.29)

r=0

In order to prove 3.29 we use the diﬀerentiation rules (3.10 ) and (3.13 ) ∆(f g) =

m

˜ ˜ tr tρr (∂/∂z)f · ρm−r (∂/∂z)g ,

r=0 tr(uz) ˜ ρr (∂/∂z)e = ρr (u)etr(uz) ,

applied to the functions f = etr(tRz) and g = etr(ξz) : ∆(etr(tR+ξ)z ) = ∆(etr(tRz) · etr(ξz) ) = m

(3.30)

tr(tRz) tr(ξz) ˜ ˜ = tr tρr (∂/∂z)e · ρm−r (∂/∂z)e

r=0 m

tr tρr (tRz)etr(tRz) · ρm−r (ξz)etr(ξz) ) =

r=0 m

tr tρr (tRz) · ρm−r (ξz) etr((tR+ξ)z) ,

r=0

on the other hand tr((tR+ξ)z) ˜ ∆(e(tr(tR+ξ)z) ) = ρm (∂/∂z)e =

ρm ((tR + ξ)z)e

tr((tR+ξ)z)

= det((tR + ξ)z)e

(3.31) tr((tR+ξ)z)

,

and the substutution of z = 1m to (3.28) and (3.31) implies (3.29). It follows from (3.29) and (3.28) that det(R)λj (ξR−1 ) = tr tρm−j (R) · ρj (ξ)

(3.32)

120

3 Arithmetical diﬀerential operators

Let us substitute (3.32) to (3.27) writng 4πy = 2πi(¯ z − z) = R−1 , λj (2πiξ(¯ z − z)) = λj (ξR−1 ) = det(R)−1 tr tρm−j (R) · ρj (ξ)

(r) δk f (z)

=

c(ξ)det(R

−1 −r

ξ∈Bm

)

r r det(ξR−1 )r−t t t=0

RL (κ − k − r)λl1 (ξR−1 ) · . . . · λlt (ξR−1 )em (ξz)

|L|≤mt−t

=

ξ∈Bm

r r c(ξ) det(ξ)r−t det(R)t t t=0

RL (κ − k − r)λl1 (ξR−1 ) · . . . · λlt (ξR−1 )em (ξz)

|L|≤mt−t

r r = c(ξ) (3.33) det(ξ)r−t t t=0 ξ∈Bm RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )em (ξz)

|L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 + · · · + lt ≤ mt − t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt − |L|) and with coeﬃcients in the ring Z[1/2] and the ﬁrst statement of Theorem 3.14 follows. In order to prove the second statement it suﬃces to specialize these formulae to the case of r = 1 using (3.24). 3.4.2 Action on Fourier expansion (r)

Note from (3.33) that the action of δk on the exponential q ξ = em (ξz) written in variables q ξ , R = (Ri,j ) is given by (r) δk (q ξ )

r r = det(ξ)r−t t t=0

(3.34)

RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )q ξ ,

|L|≤mt−t

and one may view ξ also as an independent variable for which the notation ∂ξ = (∂ξ,i,j ), ∂˜ξ = (∂˜ξ,i,j ), ∂˜ξ,i,j = 2−1 (1 + δi,j )∂˜ξ,i,j will be used.

(3.35)

3.4 Arithmetical variables of nearly holomorphic forms

121

3.4.3 Diﬀerentiation on monomials Our next task is to write down the formal action (r)

δk (Ria11,j1 · · · Riann,jn q ξ )

(3.36)

(r)

of δk using the diﬀerentiaion over the formal parameter ξ of the equality (3.34). Let us show ﬁrst that Ria11,j1 · · · Riann,jn q ξ = det(R)a1 +···+an etr(2πiξ(−¯z )) · Dξ (etr(2πiξ(¯z −z)) )

(3.37)

for an appropriate diﬀerential polynomial on ∂˜ξ,i,j which we now specify. Recall that (Ri,j ) = (2πi(¯ z −z))−1 = ρ1 ((2πi(¯ z −z))−1 ) = det(2πi(¯ z −z))−1 ·tρm−1 ((2πi(¯ z −z))) on the other hand t ρm−1 ((2πi(¯ z−

z)))etr(2πiξ(¯z −z)) = tρm−1 ((∂˜ξ ))etr(2πiξ(¯z −z))

hence a1 an Dξ = Dξ,i · · · Dξ,i 1 ,j1 n ,jn

(3.38)

where (Dξ,i,j ) = tρm−1 ((∂˜ξ )). (we used the commutation of ∂˜z,i,j with ∂˜ξ,i,j with respect to the independent formal variables z and ξ). Let us compute (r) (r) δk (Ria11,j1 · · · Riann,jn q ξ ) = δk det(R)a1 +···+an etr(2πiξ(−¯z )) · Dξ (etr(2πiξ(¯z −z)) ) using (3.37), (3.7), Lemma 3.9 and (3.34): r r (r) (3.39) det(ξ)r−t δk (q ξ ) = t t=0 RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )q ξ , |L|≤mt−t (r)

δk (Ria11,j1 · · · Riann,jn q ξ )

det(R)a1 +···+an etr(2πiξ¯z ) · Dξ (etr(2πiξ(z−¯z )) ) (r) = etr(2πi(ξ¯z )) δk det(R)a1 +···+an · Dξ (etr(2πiξ(z−¯z )) ) (r) = etr(2πiξ¯z )) Dξ δk det(R)a1 +···+an · etr(2πiξ(z−¯z )) (r)

= δk

and our computation of (3.39) is reduced to computation of (r) (r) z)−a ·etr(2πi(ξz)) δk det(R)a · etr(2πiξ(z−¯z )) = (−2πi)−am etr(−2πi(ξ¯z )) δk det(z−¯ where a = a1 + · · · + an .

122

3 Arithmetical diﬀerential operators

Lemma 3.16. Let a ≥ 0 be a natural number then the action of the Shimura diﬀerential operator on the functions det(¯ z − z)−a em (ξz) (where ξ ∈ Bm and z ∈ Hm ) is given by the formula (r)

δk

det(z − z¯)−a em (ξz)

= (−2πi)−rm det(z − z¯)−r−a em (ξz)Rm (2πiξ(¯ z − z); r, κ − k − r + a) = (2πi)−rm etr(2πiξ(¯z )) det(z − z¯)κ−k−r ∆r etr(2πiξ(z−¯z )) det(z − z¯)−κ+k+r−a (r)

and δk = δk+2r−2 ◦ · · · ◦ δk , δk f (z) = m 1 det(y)−1 Mk f (z) = (2πi)−m det(z − z¯)κ−k−1 ∆ det(z − z¯)−κ+k+1 f − 4π Mk f (z) = det(z − z¯)κ−k ∆ det(z − z¯)k−κ+1 f (z) , ! Rm (z; r, β) = (−1)mr etr(z) det(z)r+β ∆rm e−tr(z) det(z)−β . Proof of Lemma 3.16 It is very similar to that of Lemma 3.10. One uses induction on r : • If r = 0, it is obviously true. (r+1) (r) • Suppose next that it holds for some r then δk = δk+2r ◦ δk , and if one applies the Shimura operator δk+2r by induction one obtains (r+1) det(z − z¯)−a em (ξz) = (−2πi)−m det(z − z¯)κ−k−2r−1 (3.40) δk ·∆ det(z − z¯)k+2r+1−κ (2πi)−rm det(z − z¯)−a−r em (ξz) · ! ·Rm (2πiξ(¯ z − z); r, κ − k − r + a) (recall that 4πξy = −2iπξ(z − z¯)). On the other hand z¯ is aniholomorphic hence it behaves as a constant under the action of ∆, and it makes it possible to write, by putting u = −2iπξ, β = κ − k − r + a : Rm (u(z − z¯); r, β) =

(−1)mr etr(u(z−¯z)) det(z − z¯)r+β ∆r e−tr(u(z−¯z)) det(z − z¯)−β

(3.41)

(compare with the proof of 3.10). One substituts (3.41) into (3.40) , then the factor em (ξ z¯) = etr(−u¯z) which is an antiholomorphic function can be pulled out. One obtains then

3.4 Arithmetical variables of nearly holomorphic forms (r+1)

δk

det(z − z¯)−a em (ξz)

123

(3.42)

= (−2πi)−m det(z − z¯)κ−k−2r−1 · ·∆ det(z − z¯)k+2r+1−κ (2πi)−rm det(z − z¯)−a−r em (ξz) · ! ·(−1)mr etr(u(z−¯z)) det(z − z¯)r+β ∆r e−tr(u(z−¯z)) det(z − z¯)−β = (2πi)−(r+1)m (−1)m(r+1) det(z − z¯)κ−k−2r−1 em (ξ z¯)) · ! ·∆ det(z − z¯)k+2r+1−κ−a−r+r+β ∆r e−tr(u(z−¯z )) det(z − z¯)−β = (−2πi)−(r+1)m det(z − z¯)κ−k−r−1 em (ξ z¯)) · ! det(z − z¯)−r · ∆ det(z − z¯)r+1 ∆r e−tr(u(z−¯z)) det(z − z¯)−β Let us now apply lemma 3.8 which gives ! det(z − z¯)−r · ∆ det(z − z¯)r+1 ∆r e−tr(u(z−¯z)) det(z − z¯)−β+1 = ∆r+1 e−tr(u(z−¯z)) det(z − z¯)−β+1 and (3.42) transforms to (r+1) δk det(z − z¯)−a em (ξz) =

(3.43) (−2πi)−(r+1)m det(z − z¯)κ−k−r−1 em (ξ z¯)∆r+1 e−tr(u(z−¯z)) det(z − z¯)−β+1 = (−2πi)−(r+1)m det(z − z¯)κ−k−(r+1) em (ξ z¯) · ·∆r+1 e−tr(u(z−¯z )) det(z − z¯)−κ+k+r+1−a

ﬁnishing the proof by induction. Corollary 3.17. Let a ≥ 0 be a natural number then the action of the Shimura diﬀerential operator on the functions det(2πi(¯ z −z))−a em (ξz) (where ξ ∈ Bm and z ∈ Hm ) is given by the formula (r)

δk

det(R)a q ξ = det(R)r+a Rm (ξR−1 ; r, κ − k − r + a)q ξ a ξ

= det(R) q

r r t=0

t

det(ξ)r−t

(3.44)

RL (κ − k − r + a)tr tρm−l1 (R) · ρl1 (ξ) · . . . · tr tρm−lt (R) · ρlt (ξ)

|L|≤mt−t (r)

where δk = δk+2r−2 ◦ · · · ◦ δk , and the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt − |L|) and with coeﬃcients in the ring Z[1/2].

124

3 Arithmetical diﬀerential operators

Proof of Corollary 3.17 Let us substitute into (3.8) 4πξy = 2πiξ(¯ z − z) = ξR−1 as z, β = κ − k − r + a: r r RL (β)λl1 (z) · . . . · λlt (z) Rm (z; r, β) = det(z)r−t t t=0 |L|≤mt−t

giving Rm (ξR−1 ; r, κ − k − r + a) = r r det(ξR−1 )r−t t t=0

RL (κ − k − r + a)λl1 (ξR−1 ) · . . . · λlt (ξR−1 ) =

|L|≤mt−t

r t=0

r det(ξ)r−t det(R)−r t

RL (κ − k − r + a)tr tρm−l1 (R) · ρl1 (ξ) · . . . · tr tρm−lt (R) · ρlt (ξ)

|L|≤mt−t

using again (3.32): λj (ξR−1 ) = det(R)−1 tr tρm−j (R) · ρj (ξ) , and (3.44) follows. In a more general case let us apply Lemma 3.31 to (3.39): (r)

(3.45) δk (Ria11,j1 · · · Riann,jn q ξ ) (r) = etr(2πiξ¯z )) Dξ δk det(R)a1 +···+an · etr(2πiξ(z−¯z )) (r) = etr(2πiξ¯z )) Dξ (−2πi)−am etr(−2πi(ξ¯z )) δk det(z − z¯)−a · etr(2πi(ξz)) (r) = (−2πi)−am Dξ δk det(z − z¯)−a · etr(2πi(ξz)) = (−2πi)−am−rm Dξ det(z − z¯)−r−a em (ξz)Rm (2πiξ(¯ z − z); r, κ − k − r + a) where a = a1 + · · · + an . Let us use again the explicit expression (3.8) for the polynomial Rm (z; r, β) =

r r det(z)r−t t t=0

RL (β)λl1 (z) · . . . · λlt (z),

|L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 + · · · + lt ≤ mt − t. Let us substitute (3.32) to (3.45) writing 4πy = 2πi(¯ z − z) = R−1 ,

3.4 Arithmetical variables of nearly holomorphic forms

125

λj (2πiξ(¯ z − z)) = λj (ξR−1 ) = det(R)−1 tr tρm−j (R) · ρj (ξ) implying directly: (r)

δk (Ria11,j1 · · · Riann,jn q ξ ) (3.46) = (−2πi)−am−rm Dξ det(z − z¯)−r−a q ξ Rm (2πiξ(¯ z − z); r, κ − k − r + a) r r −am−rm −r−a ξ det(2πiξ(¯ z − z))r−t Dξ det(z − z¯) q = (−2πi) t t=0 RL (β)λl1 (2πiξ(¯ z − z)) · . . . · λlt (2πiξ(¯ z − z)) |L|≤mt−t

r r z − z)) q = Dξ det((2πi)(¯ det(2πiξ(¯ z − z))r−t t t=0 RL (β)λl1 (2πiξ(¯ z − z)) · . . . · λlt (2πiξ(¯ z − z))

−r−a ξ

|L|≤mt−t

r r = det(R) Dξ q det(ξ)r−t t t=0 t RL (β)tr ρm−l1 (R) · ρl1 (ξ) · . . . · tr tρm−lt (R) · ρlt (ξ) . a

|L|≤mt−t

ξ

4 Admissible measures for standard L–functions and nearly holomorphic Siegel modular forms

4.1 Congruences between modular forms and p-adic integration 4.1.1 Integration in nearly holomorphic Siegel modular forms Let p be a prime number (we often assume p ≥ 5). The purpose of this chapter is to give a new conceptual construction of admissible measures (in the sense of Amice-V´elu) attached to a standard Lfunction of a Siegel cusp eigenform. For this purpose we use the theory of p-adic integration in spaces of nearly holomorphic Siegel modular forms (in the sense of Shimura, see [Sh9]) over an O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp . Often we simply assume that A = Cp . We view such modular forms as certain formal expansions and we study the action of certain diﬀerential operators on Siegel-Eisenstein distributions and on theta distributions. The important property of these arithmetical diﬀerential operators is their commutation with Hecke operators (under an appropriate normalization). Their action produces natural families of distributions with values in spaces of nearly holomorphic Siegel modular forms. Denote by M = M(A) an A-module (or simply a Cp -vector space) of nearly holomorphic Siegel modular forms. In the present paper we apply to these distributions the canonical projector πα : M → Mα onto the primary (characteristic) A-submodule associated to a non-zero eigenvalue α ∈ A× of the Frobenius operator U = Π + (p) = Up (the Atkin-Lehner operator in the Siegel modular case; Mα is the maximal A-submodule on which U − αI is nilpotent). This operator acts similar to the trace operator lowering the level of modular forms. On the other hand, U is invertible on Mα if α ∈ A× so that one can glue its action on forms of various levels. In this way one obtains the desired distributions with values in a ﬁnite dimensional vector space starting from naturally deﬁned distributions with values in spaces of modular forms (like Siegel-Eisenstein distributions, theta distributions etc.). In order to obtain from them numerically valued distributions interpolating

M. Courtieu and A. Panchishkin: LNM 1471, pp. 127–186, 2004. c Springer-Verlag Berlin Heidelberg 2004

128

4 Admissible measures for standard L–functions

critical values attached to standard L-functions of Siegel modular forms one applies a suitable linear form coming from the Petersson scalar product (using the Andrianov identity of Rankin type for the standard L-function). In previous works [Pa7], [Pa6] a non–Archimedean interpolation of these special values was constructed ﬁxing a non zero Satake p–parameter α0 (p) of the cusp eigenform f . It follows that the normalized critical values D(s, f, χ) of the standard L-function can be explicitely rewritten in terms of certain Cp –valued integrals of admissible p–adic measures (over a proﬁnite group of the type Z× p of p–adic units) provided the character χ is non–trivial. More explicit description of these special values was given by M.Courtieu in his PhD thesis ([Cour], Institut Fourier, 2000) using detailed study of the action of the Shimura diﬀerential operator on Siegel modular forms. In the present chapter we give a conceptual explanation of these p– adic properties satisfyied by the special values of the standard L-function D(s, f, χ), where f is a Siegel cusp form of an even degree m and of weight k > 2m + 2, χ is a varying Dirichlet character. We show that these admissible measures can be lifted to arithmetical nearly holomorphic Siegel modular forms studied by G.Shimura [Sh9]. This lifting is given by a universal sequence Φ± s (χ) of distributions with values in arithmetical nearly holomorphic Siegel modular forms (for critical pairs (s, χ), see Proposition 5.4). It would be interesting to extend these lifting results to Siegel cusp eigenforms of odd degree, using the method of B¨ ocherer-Schmidt [B¨ oSch]. 4.1.2 Arithmetical nearly holomorphic Siegel modular forms They admit two diﬀerent descriptions: a(ξ, Ri,j )q ξ ∈ Q[[q Bm ]][Ri,j ] such that • as formal power series g = ξ∈Bm

for all R = (Ri,j )i,j=1,··· ,m = (4πIm(z))−1 = (2πi(¯ z − z))−1 and z ∈ Hm in the Siegel upper half plane of degree m the series converges to a C∞ -Siegel modular form of a given weight k and character ψ; • as certain C∞ -Siegel modular forms g taking values in Q at all CM-points (up to a factor independnt of a concrete form). Rcall that throughout the paper we ﬁx embeddings i∞ : Q → C,

i p : Q → Cp ,

and we shall often view algebraic numbers (via these embeddings) as both complex and p–adic numbers. 4.1.3 The group For a ﬁxed positive integer N ∈ N consider the proﬁnite group Yv , where Yv = (Z/N pv Z)× . Y = lim ← v

There is a natural projection yp : Y → Z× p.

4.1 Congruences between modular forms and p-adic integration

129

Deﬁnition 4.1. a) For h ∈ N, h ≥ 1 let C h (Y, A) denote the A-module of × locally polynomial functions of degree < h of the variable yp : Y → Z× p → A ; in particular, C 1 (Y, A) ⊂ C loc−const (Y, A) (the A-submodule of locally constant functions). We adopt the notation Φ(U) := Φ(χU ) for the characteristic function χU of an open subset U ⊂ Y . Let also denote C loc−an (Y, A) the A-module of locally analytic functions and C(Y, A) the A-module of continous functions so that C 1 (Y, A) ⊂ C h (Y, A) ⊂ C loc−an (Y, A) ⊂ C(Y, A). b) For a given positive integer h an h-admissible measure on Y with values in M is an A-module homomorphism Φ˜ : C h (Y, A) → M such that for all a ∈ Y and for v → ∞ (yp − ap )j dΦ˜ = o(p−v(j−h) ) for all a+(N pv )

j = 0, 1, · · · , h − 1,

p,M

where ap = yp (a). We adopt the notation (a)v = a+(N pv ) for both an element of Yv and the corresponding open compact subset of Y . 4.1.4 Canonical projection α : We explain in Section 4.3, how to construct an h-admissible measure Φ C h (Y, A) → M(A) out of a sequence of distributions Φj : C 1 (Y, A) → M with values in an A-module M = M(A) of nearly holomorphic modular forms over A (for all j ∈ N with j ≤ h − 1, where A is an O-algebra, and α ∈ A× is a ﬁxed non-zero eigenvalue of the Frobenius operator U = U (p) = Π + (p) acting on g= a(ξ, n)Rn q ξ ∈ A[[q Bm ]][Ri,j ], ξ∈Bm ,n≥0

by g|U (p) =

a(pξ, n)(pR)n q ξ

ξ∈Bm ,n≥0

(over the complex numbers R = (4π)−1 Im(z)−1 and this notation corresponds to u 1 f |k m f |U (p)(z) = = 0m p1m t u=u∈Mm (Z) mod p

130

4 Admissible measures for standard L–functions

p−κm

tu=u∈M

m (Z)

f ((z + u)/p), mod p

Im ((z + u)/p) = Im(z/p) = Im(z)/p = (4π)−1 (pR)−1 (we use the Petersson-Andrianov notation for the action of matrices). Then we consider (for an α ∈ A× ) the canonical projection operator πα : M(A) → M(A)α . We deﬁne an A-linear map α : C h (Y, A) → M Φ on local monomials ypj by

α = πα (Φj ((a)v )) ypj dΦ

(a)v

where Φj : C 1 (Y, A) → M(A) are certain M(A)-valued distributions on Y for all j = 0, 1, · · · , h − 1. 4.1.5 The standard zeta function of a Siegel cusp eigenform In order to describe the main result of the paper let f= a(ξ)em (ξz) ξ>0

be a Siegel cusp form of the even degree m of weight k on the congruence subgroup " # ab ∈ Spm (Z) | c ≡ 0 mod C Γ0 (C) = cd with a Dirichlet character ψ mod C. Here z belongs to the Siegel upper half plane % & Hm = z ∈ GLm (C)|z = tz, Im(z) > 0 and we adopt the standard notation em (z) = exp(2πitr(z)). Suppose that f is an eigenfunction of the global Hecke algebra (m)

L(m) (C) = ⊗qC L(q) (C) with the eigenvalue given by a homomorphism λf : L(m) (C) → C, i.e. (m)

f | X = λf (X)f for all X ∈ L(q) (C). Let α0 (q), α1 (q), · · · , αm (q)

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131

be the (m + 1)–tuple of the Satake q–parameters of λf , which uniquely determine λf . It is known [An5] that the relation α20 (q)α1 (q) . . . αm (q) = ψ(q)m q m(k−(m+1)/2) holds. Recall that the standard L-function of f with a Dirichlet character χ mod N is deﬁned as the Euler product D(s, f, χ) = D(q) (s, f, χ), (4.1) qC

with D(q) (s, f, χ)−1 = m χ(q)ψ(q χ(q)ψ(q)αi (q) χ(q)ψ(q)α−1 i (q) 1− 1− 1− , qs qs qs i=1 (the product is absolutely convergent for Re(s) > 1 + m). Recall the three types of the normalized zeta function: for κ = (m + 1)/2, s ∈ C D∗ (s, f, χ) = (2π)−m(s+k−κ) Γ ((s + δ)/2)

m

Γ (s + k − j)D(s, f, χ),

(4.2)

j=1

D− (s, f, χ) = Γ ((s + δ)/2)−1 D∗ (s, f, χ) = (2π)−m(s+k−κ)

m

(4.3)

Γ (s + k − j)D(s, f, χ),

j=1

D+ (s, f, χ) =

2 iδ Γ (s) cos(π(s − δ)/2) − D (s, f, χ) = (2πi)s

(4.4)

iδ π (1−2s)/2 D∗ (s, f, χ), Γ ((1 + ∆ − s)/2) where δ = 0 or 1 according as ψχ(−1) = (−1)δ with f ∈ Sm k (C, ψ) be an eigenfunction of the global Hecke algebra Lm (C) = ⊗qC Lm (C). Deﬁne q h = [2ordp α0 (p)] + 1, Assume that a(ξ0 ) = 0 for some ξ0 > 0 with det(2ξ0 ) = 1. Moreover, we make the essential assumption (*) that a0 (ξ0 ) = 0

(∗)

for a Fourier coeﬃcient a0 (ξ0 ) of the U -eigenform f0 (z) = ξ>0 a0 (ξ)em (ξz) deﬁned in Section 4.4 and in §1, (1.47), chapter 2 of [Pa6]. The importance of the above non–vanishing condition was pointed out to us by S. B¨ ocherer.

4 Admissible measures for standard L–functions

132

It turns out that it actually may not be satisﬁed by certain Eisenstein series (note that we deal here with cusp forms). However it is not known in general, whether there exist cusp forms violating the non–vanishing condition. Main Theorem. Let f ∈ Skm (N, ψ) be a Siegel cusp eigenform of even degree m and of weight k > 2m + 2 satisfying the above non-vanishing condition (*). Deﬁne h = [2ordp (α0 (p))] + 1 then for any positive integer c > 1, (c, N p) = 1 there exist two Cp -valued h-admissible measures Dc± on Y such that: (i) For all pairs (s, χ) with s ∈ Z, 0 ≤ s ≤ k − m − 2 and a primitive non-trivial Dirichlet character χ ∈ XStors of conductor Cχ = pv one has

m(2s+2k−m)/4

•

χxsp dDc+ = Y

·

GCχ (1m , χ)Cχ (iCχ )m2 /2 α0 (Cχ2 ) q|N

•

G(χψ) ¯

(1 − (χψ) ¯ 2 (c)c−2s−2 )·

¯ 0 (q)q s ) D+ (s + 1, f, χ) (1 − (χψ) ¯ . (1 − (χψ) ¯ 0 (q)q −s−1 ) f, f N m(2−2s+2k−m)/4

χxsp dDc− = Y

s+1 Cχψ ¯

¯ χ GCχ (1m , χ)C (iCχ )m2 /2 α0 (Cχ2 ) ·

¯ 2 (c)c−2s−2 )· (1 − (χ ¯ψ)

D− (−s, f, χ) . f, f N

¯ viewed as elements of the Tate ﬁeld These integrals are algebraic numbers Q ¯ Cp (via the ﬁxed embedding ip : Q → Cp ). (ii) If ordp (α0 (p)) = 0 (i.e. f is p-ordinary), then the measures Dc± are bounded Cp -analytic functions where χ(detξ )em (tξξ /Cχ ), GCχ (1m , χ) = ξ ∈Mm (Z)

(mod Cχ )

denotes the Gauss sum of degree m of χ.

4.2 Algebraic diﬀerential operators and Siegel-Eisenstein distributions 4.2.1 Operatots of Maass and Shimura Consider again the diﬀerential operator ∆m (the Maass diﬀerential operator) of degree m, acting on complex C ∞ -functions on Hm , deﬁned by the equality: ∂˜ij = 2−1 − (1 + δij )∂/∂ij (4.5) ∆m = det(∂˜ij ), For any f ∈ C∞ (Hm , C), one deﬁnes also the operators Mk f (z) = det(z − z¯)κ−k ∆ det(z − z¯)k−κ+1 f (z) , m 1 det(y)−1 Mk f (z). δk f (z) = − 4π Due to H. Maass [Maa], for an integer k and a Dirichlet character ψ modulo N , the diﬀerential operator δk acts on C∞ -Siegel modular forms, δk : ˜ m (N, ψ). Recall also that the Shimura diﬀerential operator ˜ m (N, ψ) → M M k k+2 is the composition (r) δk = δk+2r−2 ◦ · · · ◦ δk . (r)

In order to describe explicitely the action of δk on Fourier expansions consider, for an integer r ≥ 0 and for a complex number β, the polynomial ! Rm (z; r, β) = (−1)mr etr(z) det(z)r+β ∆rm e−tr(z) det(z)−β . (4.6) According to its deﬁnition the degree of the polynomial Rm (z; r, β) is equal to mr and the term of the highest degree coincides with det(z)r . We have also that for β ∈ Q the polynomial Rm (z; r, β) has rational coeﬃcients. Notice also that for m = 1 one has r r r−t R1 (z, r, β) = z β(β + 1) · . . . · (β + t − 1). t t=0 M.Courtieu has established in his PhD thesis the following more explicit expression for the function Rm (z; r, β) for arbitrary m (see [Cour], Chapter 3 and Theorem 3.12: r r Rm (z; r, β) = RL (β)λl1 (z) · . . . · λlt (z), (4.7) det(z)r−t t t=0 |L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 +· · ·+lt ≤ mt−t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt−|L|) and with coeﬃcients in the ring Z[1/2]. The polynomials λi (z) are deﬁned by det(t1m + z) =

m

λi (z)tm−i

i=0

(λi (z) is the sum of all diagonal minors of size i × i of the matrix z).

4 Admissible measures for standard L–functions

134

4.2.2 Formulas for Fourier expansions Let f ∈ Mm k (N, ψ) be a Siegel modular form whose Fourier expansion is given by f (z) = ξ∈Bm c(ξ)em (ξz), then we proved in Chapter 3, (Theorem (r)

3.14) that the nearly holomorphic Siegel modular form δk f (z) is given then by the following formal power series expansion (r) δk f (z)

=

r r c(ξ) det(ξ)r−t t t=0

ξ∈Bm

(4.8)

RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )q ξ =

|L|≤mt−t

ξ∈Bm

r r c(ξ) det(ξ)r−t t t=0

RL (κ − k − r)QL (R, ξ)q ξ =

|L|≤mt−t

c(ξ)Q(R, ξ)q ξ ∈ Q(f )[[q Bm ]][Ri,j ]i,j=1,··· ,m ,

ξ∈Bm

where Q(f ) is the subﬁeld of C generated by the Fourier coeﬃcients of f , L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m as above, r r Q(R, ξ) = Q(R, ξ; k, r) = RL (κ − k − r)QL (R, ξ)), det(ξ)r−t t t=0 |L|≤mt−t

QL (R, ξ) = tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) ). For r = 1 this gives: δk f (z) =

c(ξ)

ξ∈Bm

m

(−1)m−l cm−l (k + 1 − κ)tr tρm−l (R) · ρl (ξ) q ξ ,

l=0

where cm−l (k + 1 − κ) =

m−l−1

(k + 1 − κ + j/2) =

j=0

Γm−l (k + 1/2) Γm−l (k − 1)/2)

(l ≤ m − 1),

4.2.3 Siegel-Eisenstein series. (See also Section 2.2 of Chapter 2). Let us recall the deﬁnition of the SiegelEisenstein series Ek (z, s, χ) which come up in the integral representation of Rankin type for the standard L-function. We introduce some additional notation. Let us ﬁx a positive integer m then a pair of integral matrices c, d ∈ Mm (Z) is called primitive if any matrix with rational coeﬃcients γ ∈ Mm (Q) such that γc, γd ∈ Mm (Z) is integral: γ ∈ Mm (Z). The pair (c, d)

4.2 Algebraic diﬀerential operators and Siegel-Eisenstein distributions

135

is said symmetric if c·td = d·tc. Two primitive symmetric pairs (c1 , d1 ) and (c2 , d2 ) are said equivalent if (c1 , d1 ) = (uc2 , ud2 ) for a matrix u ∈ SLm (Z). Let Dm denote the set of equivalence classes of symmetric couples of coprime matrices. Now let k, N be positive integers, s a complex number and χ a Dirichlet character modulo N such that χ(−1) = (−1)k . For z ∈ Hm the Siegel-Eisenstein series is deﬁned by χ(det(d))det(cz + d)−k−|2s| , (4.9) E(z, s; k, χ, N ) = E(z, s) = det(y)s where the summation is taken over all (c, d) ∈ Dm with the condition c ≡ 0 (mod N ) and we adopt the convenient notation by Deligne and Ribet [De-Ri]: z −k−|2s| = z −k |z|−2s for z ∈ C∗ def

The series (4.2) is absolutely convergent for k + 2Re(s) > m + 1 and it admits a meromorphic analytic continuation over the whole complex s-plane. Put ab j(α, z) = det(cz + d) for α = and z ∈ Hm , then it follows from a cd known description of Dm that χ(det(d)j(α, z)−k−|2s| , (4.10) E(z, s; k, χ, N ) = det(y)s

α∈P ∩Γ \Γ

ab , and P denotes the subgroup of P ⊂ G∞+ , where Γ = Γ0m (N ), α = cd consisting of elements α with the condition c = 0. The series (4.10) deﬁne ˜ k (N, χ) ¯ and if s = 0 and k > m + 1, C∞ -modular forms Ek (z, s, χ) ∈ M m ¯ This series has a Fourier these forms are holomorphic: Ek (z, χ) ∈ Mm k (N, χ). expansion of the type bk (ξ)em (ξz). Ek (z, χ) = ξ∈Bm

Following the works of H. Maass (see [Maa]], chap. 19), the series Ek (z, s, χ) are related to each other for diﬀerent values of k and s via the Shimura diﬀerential operator: Ek+2 (z, s − 1, χ) = (−4π)m

Γm (k + s) δk Ek (z, s, χ). Γm (k + s + 1)

(4.11)

One obtains by iteration of (4.11) r times Ek+2r (z, s − r, χ) = (−4π)mr

Γm (k + s) (r) δ Ek (z, s, χ) Γm (k + s + r) k

(r)

where δk = δk ◦· · ·◦δk+2r−2 . It follows from (4.7) that the series Ek+2r (z, −r, χ) is a nearly holomorphic Siegel modular form with the following Fourier expansion: Γm (k) Ek+2r (z, −r, χ) = det(−y)−r bk (ξ)R(4πξy; r, κ−k −r)em (ξz). Γm (k + r) ξ∈Bm

136

4 Admissible measures for standard L–functions

4.2.4 Normaized Siegel-Eisenstein series In order to obtain arithmetical nearly holomorphic forms whose Fourier coeﬃcients depend nicely on the parameter s = −r ≤ 0 we introduce normalized Siegel-Eisenstein series starting from the series (4.9). Such series where studied in detail by Shimura (voir [Sh8]. Keeping the same notation as above deﬁne (4.12) Ek (z, s, χ) = Ek (−z −1 , s, χ)det(z)−k . (the involuted Siegel-Eisenstein series): Ek (z, s, χ) = Ek (z, s, χ)|k Jm . The series (4.12) admit the following Fourier expansion b(ξ, y, s, χ)em (ξz), (4.13) Ek (z, s, χ) = ξ∈N −1 Am

where the coeﬃcients b(ξ, y, s, χ) are explicitely given in [Sh8] (see also [Pa6] (chap. 2, prop. 3.4.)) in terms of the conﬂuent hypergeometric function. We introduce here other three types of normalized Siegel-Eisenstein series in order to give a precise statement on their holomorphy properties with respect to the variable s, the properties of positivity of matrices ξ by which their Fourier coeﬃcients are indexed, and also algebraic properties of these Fourier coeﬃcients: G∗ (z, s) = G∗ (z, s; k, χ, N ) = N

m(k+s)

Γ(k, s)LN (k + 2s, χ)

(4.14)

[m/2]

LN (2k + 4s − 2i, χ2 )E ∗ (N z, s),

i=1

with E ∗ (N z, s) = E(−(N z)−1 , s)det(N z)−k = N −km/2 E|W (N ),

imk 2−m(k+1) π −m(s+k) ×

"

(4.15)

Γ(k, s) = Γm (k + s)Γ (s + (k − Γm (k + s),

and LN (s, χ) =

m 2

+ µ)/2), if m is even; otherwise.

(1 − χ(q)q −s )−1

q |N

is the Dirichlet series associated to the character χ and µ = ε(m/2 + k), with ε(r) ∈ {0, 1} such that ε(r) ≡ r (mod 2) for any integer r ∈ N. If the character χ2 is non trivial, the function Gk (z, s, χ) is an entire function of the complex variable s (see [Fe], Th. 9.1, p. 49, and [Pa6], Chap. 3, Th. 3.6).

4.2 Algebraic diﬀerential operators and Siegel-Eisenstein distributions

137

Proposition 4.2. Let N > 1 a positive integer such that 2|N then (a) If χ2 = 1 the series Gk (z, s, χ) is an entire function of the variable s. (b) If χ2 is trivial then (b1 ) if either 2k ≥ m and m odd or 2k ≥ m and m is even, but (m/2) + k is odd (i.e. µ = ε((m/2) + k) = 1 ), then the function G∗ (z, s) is an entire function of the variable s; (b2 ) if 2k ≥ m and both numbers m and (m/2) + k are even (i.e. µ = ε(m) = 0) then the function G∗ (z, s) is an entire function of the variable s with the exclusion of a possible simple pole at the point s = (m + 2 − 2k)/4; (b3 ) if m > 2k ≥ 0 then the function G∗ (z, s) is an entire function of the variable s with possible exclusion of simple poles at those points s for which 2s is an integer and [(m − 2k + 3)/2] ≤ 2s ≤ (m + 1 − 2k)/2 ; (b4 ) if k = 0 then the function G∗ (z, s) has a simple pole at the point s = (m + 1)/2 iﬀ χ is trivial, and in this case we have that the function Ress=(m+1)/2 G∗ (z, s; 0, 1, N ) of the variable z is a non-zero constant. Proof of Proposition 4.2 (See [Fe], Th. 9.1, p. 49). If m is odd then we put G+ (z, s, χ) = G− (z, s, χ) = G∗ (z, s, χ). If m is even then we deﬁne G− (z, s) = Γ ((k + 2s − (m/2) + µ)/2)−1 G∗ (z, s), G+ (z, s) =

iµ π κ−k−2s G∗ (z, s) = Γ ((1 − k − 2s + (m/2) + µ)/2)

(4.16) (4.17)

2iµ Γ (k + 2s − (m/2)) cos(π(k + 2s − (m/2) − µ)/2) − G (z, s), (2π)k+2s−(m/2) where µ = ε((m/2) + k), κ = (m + 1)/2. The Fourier expansions of Gk (z, s, χ) and G± k (z, s, χ) are given by Gk (z, s, χ) =

b (ξ, y, s, χ)em (ξz),

ξ∈Am

G± k (z, s, χ) =

b± (ξ, y, s, χ)em (ξz).

ξ∈Am

The Fourier coeﬃcients b (ξ, y, s, χ), b± (ξ, y, s, χ) are explicitely given in [Pa6], (Chap. 2, §3.5) and in Theorem 4.3 below, and we show that for all crtical pairs (s, χ) these series deﬁne arithmetical nearly holomorphic Siegel modular forms.

138

4 Admissible measures for standard L–functions

Next we deﬁne an additional quadratic Dirichlet character χξ depending on ξ ∈ Am and deﬁned only for even rank r. Namely, for ξ = 0 let χξ = χ0 be trivial; for ξ = 0 we know that for some matrix u ∈ GLm (Q) ξ1 0 t uξu = (4.18) with det ξ1 = 0, 0 0 then let χξ denote the quadratic character attached to the quadratic ﬁeld 0 Q( detξ1 )/Q (this deﬁnition does not depend on choice of a matrix u). Theorem 4.3. Let m is even, 2k > m. Then: (a) If 2s is an integer, s ≤ 0, k + 2s ≥ 1 + (m/2) there is the following Fourier expansion b+ (ξ, y, s, χ)em (ξz), (4.19) G+ (z, s) = Am ξ>0

where for s > (m + 2 − 2k)/4 in (4.19) non-zero terms only occur for positive deﬁnite ξ > 0, and for all s from (a) with ξ > 0, ξ ∈ Am the following identity holds b+ (ξ, y, s, χ) = W + (y, ξ, s)L+ N (k + 2s − (m/2), χχξ )M (ξ, χ, k + 2s), with L+ (s, χ) =

2iδ Γ (s) cos(π(s − δ)/2) LN (s, χ) (2π)s

is the normalized Dirichlet L-function, δ = 0 or 1 according to χ(−1) = (−1)δ , the factor M (ξ, χ, k + 2s) is a ﬁnite Euler product Mq (ξ, χ(q)q −k−2s ) M (ξ, χ, k + 2s) = q∈P (ξ)

with Mq (ξ, t) ∈ Z[t], extended over primes q in the set P (ξ) of prime divisors of the number N and of all elementary divisors of the matrix ξ,and the character χξ is given by (4.18). W + (y, ξ, s) = 2−mκ detξ k+2s−κ det(4πy)s R(4πξy; −s; κ − k + s), provided s is an integer, where R(y; n, β) is deﬁned by (4.2.2), and b+ (ξ, y, s, χ) = 0 otherwise (if s ∈ Z). (b)If 2s is an integer with k + 2s ≤ m/2, k + s ≥ κ then there is the following Fourier expansion b− (ξ, y, s, χ)em (ξz), (4.20) G− (z, s) = Am ξ≥0

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139

and for all s from (b) with ξ > 0, ξ ∈ Am the following identity holds b− (ξ, y, s, χ) = W − (y, ξ, s)L− N (k + 2s − (m/2), χχξ )M (ξ, χ, k + 2s), where L− (s, χ) = LN (s, χ), W − (y, ξ, s) = 2−mκ det(4πy)κ−k−s R(4πξy; k+s−κ, −s), provided s + k − κ is an integer, and b− (ξ, y, s, χ) = 0 otherwise. Proof of Theorem 4.3 (See in [Pa6], Chap. 2, Th. 3.8). Let us now write down the action of the Shimura diﬀerential operator on these normailzed Siegel-Eisenstein series. Lemma 4.4. The action of the Shimura diﬀerential operator on Ek (z, s, χ) is given by: δk Ek (z, s, χ) = δk Ek (N z, s, χ) =

Γm (k + s + 1) 1 Ek+2 (z, s − 1, χ), m Γm (k + s) (−4π)

(4.21)

N m Γm (k + s + 1) Ek+2 (N z, s − 1, χ). (−4π)m Γm (k + s)

(4.22)

Proof of Lemma 4.4 We use the known property δk (f |k γ) = (δk f )|k+2 γ for all γ ∈ Sp2m (R) and k ∈ N. Taking into account Ek (z, s, χ) = Ek (z, s, χ)|k Jm , one immediately obtains (4.21) from the relation (4.11) satisﬁed by Ek (z, s, χ). In order to prove (4.22), one changes the variable z = N z. One obtains then δk (z)Ek (N z, s, χ) = det(N 1m )δk (z )Ek (z , s, χ), and (4.22) follows. Corollary 4.5. One has the following relations : δk (Gk (z, s, χ)) = Gk+2 (z, s − 1, χ), + δk (G+ k (z, s, χ)) = Gk+2 (z, s − 1, χ), − δk (G− k (z, s, χ)) = Gk+2 (z, s − 1, χ).

(4.23)

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4 Admissible measures for standard L–functions

Proof of Corollary 4.5 One shows all the three equalities in the same way, so let us consider only the ﬁrst one. One applies the Shimura diﬀerential operator to Gk (z, s, χ). The only factors in the formula (4.14) for Gk (z, s, χ) which are not invariant with respect to the change of variables k → k + 2, s → s − 1, are the following: N m(s+k) imk 2−m(k+1) π −m(s+k) Γm (k + s). These factors come up essentially in the factor Γ(k, s) (see (4.15)). One applies then the Shimura diﬀerential operator to Gk (z, s, χ) using (4.22). A simpliﬁcation of the powers of N , π, 2, and i, and of the factor Γm (k + s) gives δk (Gk (z, s, χ)) = Gk+2 (z, s − 1, χ). 4.2.5 Distributions with values in nearly holomorphic Siegel modular forms. Consider a proﬁnite group Y = lim Yv for example ← v

Yv , where Yv = (Z/N pv Z)× . Y = lim ← v

There is a natural projection yp : Y → Z× p . Let A be a commutative ring, and let M be an A-module. Let Cloc−const (Y, A) denote the A-module of locally constant functions. A distribution on Y with values in M is an A-module homomorphism Φ : Cloc−const (Y, A) → M. We adopt the notations Φ(U) := Φ(χU ) for the characteristic function χU of an open subset U ⊂ Y , and (a)v = a + (N pv ) for both an element of Yv and the corresponding open compact subset of Y . Recall an algebraic version of nearly holomorphic Siegel modular forms over an O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp . We assume that A is equipped with an augmentation eA : A → O, and ﬁx embeddings i∞ : Q → C, ip : Q → Cp . Let us ﬁx another posivite integer N and let denote by g = a(ξ, R)q ξ ∈ ξ∈Bm

Mm r,k (A) an element of m Mm r,k (A) = Mr,k (A; N ) =

)

v Mm r,k (N p ; A)

v≥1

(A-module of nearly holomorphic Siegel modular forms of type r and level N pv (v ≥ 0) with coeﬃcients in A). Let M = Mm r,k (A) and let Φ : loc−const C (Y, A) → M be a distribution on Y with values in M . Assume that

4.2 Algebraic diﬀerential operators and Siegel-Eisenstein distributions

141

A contains (ϕ(N pv ))−1 and all the values of Dirichlet characters χ mod N pv viewed as homomorphisms Y → A× then obviously χ(a)Φ((a)v ), (4.24) Φ(χ) = a mod N pv

1 ϕ(N pv )

Φ((a)v ) =

χ(a)−1 Φ((a)v ).

χ mod N pv

Proposition 4.6 (Normaized Siegel-Eisenstein distributions). a) Let N > 1 a positive integer such that 2|N and let k, s be integers such that − 2k > m and κ − k ≤ s ≤ 0 then there exist distributions Gk,s , G+ k,s , Gk,s on v Y with values in Mm r,k (A) such that for any character χ mod N p Gk,s (χ) = Gk (z, s, χ), G+ k,s (χ)

=

− G+ k (z, s, χ), Gk,s (χ)

(4.25) =

G− k (z, s, χ),

where r = m · max(−s, k − κ + s). b) Suppose that κ − k + 1 ≤ s ≤ 0 then the action of the Shimura diﬀerential − operators on Gk,s , G+ k,s , Gk,s is given by δk (Gk,s (χ)) = Gk+2,s−1 (χ), + δk (G+ k,s (χ)) = Gk+2,s−1 (χ), − δk (G− k,s (χ)) = Gk+2,s−1 (χ).

(4.26)

c) Moreover if t ≥ 0 is an integer such that κ − k + t ≤ s ≤ 0 then the action (t) − of δk on Gk,s , G+ k,s , Gk,s is given by (t)

δk (Gk,s (χ)) = Gk+2t,s−t (χ),

(4.27)

(t)

+ δk (G+ k,s (χ)) = Gk+2t,s−t (χ), − δk (G− k,s (χ)) = Gk+2t,s−t (χ). (t)

Proof of Proposition 4.6 One deduces a) from (4.24) taking into account that for all integrs k, s such that 2k > m and κ − k ≤ s ≤ 0 the Fourier coeﬃcients b (ξ, y, s, χ), b± (ξ, y, s, χ)) are polynomials of the variables Ri,j (, j = 1, . . . , m), where R = (4πIm(z))−1 , of degree ≤ r = m · max(−s, k − κ + s) which belong to Q[Ri,j ]i,j=1,...,m . They are explicitely given in Theorem 4.3 above (4.19), (4.20), and in [Pa6] (Chap. 2, §3.5) . Let us consider the natural representation (0 ≤ r ≤ m) ρr : GLm (C) −→ GL(∧r Cm ) z −→ ρr (z)

142

4 Admissible measures for standard L–functions

of the group GLm (C) on the vector space Λr Cm with respect to the basis {ei1 ∧ · · · ∧ eir : i1 < · · · < ir }. m Thus ρr (z) is a matrix of size m r × r composed of the subdeterminants of z of degree r. Put ρr (z) = det(z)ρm−r (tz)−1

(r = 0, 1, · · · , m),

in other words, ρm−r (z) is the matrix representing the action of z on Λm−r Cm with respect to the basis dual to the above basis of Λr Cm . Then the representations ρr and ρr turn out to be polynomial representations so that for each z ∈ Mm (C) the linear operators ρr (z), ρr (z) are well deﬁned. Then by Lemma 3.10 and (3.34) we have that if r and k − κ − r ≥ 0, then r r −r det(4πy) Rm (4πξy; r, κ − k − r) = (4.28) det(ξ)r−t t t=0 RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) ), |L|≤mt−t

where L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that |L| = l1 + · · · + lt ≤ mt − t, the coeﬃcients RL (β) ∈ Z[1/2][β] are polynomials in β of degree (mt − |L|) and with coeﬃcients in the ring Z[1/2], and the polynomials λi (z) are deﬁned by det(t1m + z) =

m

λi (z)tm−i

i=0

(λi (z) is the sum of all diagonal minors of size i × i of the matrix z). It follows that in (4.19) one has for an integer r = −s ≥ 0: r r det(4πy)s R(4πξy; −s; κ − k + s) = (4.29) det(ξ)r−t t t=0 RL (κ − k − r)tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) ), |L|≤mt−t

and in (4.20) one has for an integer r = k − κ + s ≥ 0: r r κ−k−s R(4πξy; k + s − κ, −s) = (4.30) det(4πy) det(ξ)r −t t t=0 RL (κ − k − r )tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) ). |L|≤mt−t

These are polynomials of degree ≤ m · max(−s, k − κ + s) with rational coeﬃcients , hence the Fourier coeﬃcients b (ξ, y, s, χ), b± (ξ, y, s, χ)) deﬁne for each ξ distributions with values in Q[Ri,j ]i,j=1,...,m (evaluated at χ). In order to deduce b) and c) (by iteration) it suﬃces to use then Corollary 4.5.

4.2 Algebraic diﬀerential operators and Siegel-Eisenstein distributions

143

4.2.6 Convolutions of distributions with values in nearly holomorphic Siegel modular forms. Consider again the proﬁnite group Yv , where Yv = (Z/N pv Z)× . Y = lim ← v

loc−const Let M = Mm (Y, A) → M be two distributions r,k (A) and let Φi : C m on Y with values in Mi = Mri ,ki (A), (i = 1, 2). Assume that A contains (ϕ(N pv ))−1 and all the values of Dirichlet characters χ mod N pv viewed as homomorphisms Y → A× then obviously χ(a)Φi ((a)v ), (4.31) Φi (χ) = a mod N pv

1 ϕ(N pv )

Φi ((a)v ) =

χ mod

χ(a)−1 Φi (χ).

N pv

Let us ﬁx a Dirichlet character ψ on Y with values in A then there exists a unique distribution Φ = Φ1 ∗ψ Φ2 on Y with values in M = Mm r1 +r2 ,k1 +k2 (A) such that for all Dirichlet characters χ on Y with values in A one has Φ(χ) = Φ1 (χ) · Φ2 (ψχ−1 ). One deﬁnes Φ on basic open subsets by Φ((a)v ) = ψ(y)Φ1 (ya)Φ2 (y). (4.32) y∈Yv

for any v such that ψ is deﬁned modN pv . Indeed, Φ((a)v ) =

ψ(y)Φ1 (ya)Φ2 (y) =

y∈Yv

1 ψ(y) v 2 (ϕ(N p )) y∈Yv

χ

mod

χ (ya)−1 Φ1 (χ ) χ

N pv

mod

χ (y)−1 Φ2 (χ )

N pv

hence Φ(χ) =

χ(a)

a∈Yv

χ

mod

ψ(y)Φ1 (ya)Φ2 (y) =

y∈Yv

a∈Yv

χ (ya)−1 Φ1 (χ ) χ

N pv

1 χ(a) ψ(y) v 2 (ϕ(N p ))

mod

χ (y)−1 Φ2 (χ ) = Φ1 (χ) · Φ2 (ψχ−1 ) =

N pv

1 (χχ −1 )(a) (ψχ −1 χ −1 )(y) v 2 (ϕ(N p )) a∈Yv

y∈Yv

y∈Yv

= Φ1 (χ) · Φ2 (ψχ

−1

χ mod N pv

Φ1 (χ )

χ mod N pv

)

because of the orthogonality relations which give χ = χ, χ = ψχ−1 .

Φ2 (χ )

4.3 A general result on admissible measures with values in nearly holomorphic Siegel modular forms. 4.3.1 Proﬁnite group Yv for example Consider again the proﬁnite group Y = lim ← v

Yv , where Yv = (Z/N pv Z)× . Y = lim ← v

There is a natural projection yp : Y → Z× p . Let A be a normed topological ring over Zp with the norm | · | = | · |p,A , and M be a normed A-module with the norm | · |p,M . Deﬁnition 4.7. a) For h ∈ N, h ≥ 1 let Ch (Y, A) denote the A-module of × locally polynomial functions of degree < h of the variable yp : Y → Z× p → A ; in particular, C1 (Y, A) ⊂ Cloc−const (Y, A) (the A-submodule of locally constant functions). We adopt the notation Φ(U) := Φ(χU ) for the characteristic function χU of an open subset U ⊂ Y . Let also denote Cloc−an (Y, A) the A-module of locally analytic functions and C(Y, A) the A-module of continous functions so that C1 (Y, A) ⊂ Ch (Y, A) ⊂ Cloc−an (Y, A) ⊂ C(Y, A). b) For a given positive integer h an h-admissible measure on Y with values in M is an A-module homomorphism : Ch (Y, A) → M Φ such that for ﬁxed a ∈ Y and for v → ∞ (yp − ap )j dΦ = o(p−v(j−h) ) for all a+(N pv )

j = 0, 1, . . . , h − 1,

p,M

where ap = yp (a). We adopt the notation (a)v = a+(N pv ) for both an element of Yv and the corresponding open compact subset of Y . 4.3.2 Measures and sequences of distributions α : Ch (Y, A) → We wish now to construct an h-admissible measure Φ M (A) out of a sequence of distributions r∗

1 Φα j : C (Y, A) → Mr ∗

with values in an A-module M = Mr∗ (A) of nearly holomorphic Siegel modular forms over A of type r∗ (for all j ∈ N with j ≤ h − 1 where r∗ ∈ N be a

4.3 A general result on admissible measures

145

natural number, and A an algebra over O as in Section 1). For this purpose we deﬁne α : Ch (Y, A) → Mr∗ Φ by

α = Φα ((a)v ) = πα (Φj ((a)v ) ypj dΦ j (a)v

where Φj : C (Y, A) → Mr∗ (A) are certain Mr∗ (A)-valued distributions on Y for all j = 0, 1, . . . , h − 1 , and Φα j ((a)v ) their α-characteristic projections given by ! −v πα,1 U v Φj ((a)v ) = πα,v +1 (Φj ((a)v ) Φα j ((a)v ) = U 1

for any v ≥ κv. Note ﬁrst of all that the deﬁnition α = Φα ((a)v ) = U −κv [πα,1 U κv Φj ((a)v )] ypj dΦ j (a)v

α : Ch (Y, A) → Mr∗ (A) is undependent on the choice of of the linear form Φ the level: for any v ≥ κv we have by Proposition 3.5 the following comutative diagram πα,v +1 v +1 Mr∗ (N pv +1 ; A) −→ Mα ; A) r ∗ (N p   6 U v

  Uv 6 πα,1

Mr∗ (N p; A) −→

Mα r ∗ (N p; A)

in which the right vertical arrow is an A-isomorphism by Proposition 1.6 b), and by linear algebra the A-linear endomorphism U commutes with the characteristic projectors πα,v +1 , πα,1 hence ! U −κv [πα,1 U κv Φj ((a)v )] = U −v πα,1 U v Φj ((a)v ) = πα (Φj ((a)v )). Theorem 4.8. Let α ∈ A be an element of A whose absolute value |α|p satisﬁes 0 < |α|p < 1 and put h = [vp (α)] + 1. Suppose that there exists a positive integer κ such that for any (a)v ⊂ Y the following two conditions are satisﬁed: (level) Φj (a)v ∈ Mr∗ (N pκv ), j κv j 0 j−j U ≤ Cp−vj (a) ) Φ ) (growth) (−a j v p j j =0

p

for all j = 0, 1, . . . , κh − 1. α : Ch (Y, A) → Mr∗ with Then there exists an h∗ -admissible measure Φ ∗ h = hκ such that for any ((a)v ) ∈ Y

4 Admissible measures for standard L–functions

146

α = Φα ((a)v ) ypj dΦ j

(a)v

for all j = 0, 1, . . . , h∗ − 1 where −κv Φα [πα,1 U κv Φj ((a)v )] . j ((a)v ) = U

and v πα,v : Mr∗ (N pv ) → Mα r ∗ (N p )

(v ≥ 1)

denotes the α-characteristic projector (canonical projector πα to the α-characteristic submodule of the U -operator ) v Ker(U − αI)n Mα r ∗ (N p ) = n≥1

(with the kernel

7

Im(U − αI)n , see deﬁnition 1.5) (where

n≥1 α U κv Φj ((a)v ) ∈ Mα r ∗ (N p) = Mr ∗ (N p; A)

because of the inclusion U κv−1 (Mr∗ (N pκv ; A)) ⊂ Mr∗ (N p; A) (for all v ≥ 1)). (see 3.3 a)) Proof of Theorem 4.8 We need to check the h∗ -growth condition for the linear form α : Ch∗ (Y, A) → Mr∗ Φ (given by the condition of the theorem 4.8), which says that for ﬁxed a, b ∈ Y and for v → ∞ α (yp − ap )j dΦ = o(p−v(j−hκ) ) (a)v p,Mr∗

for all j = 0, 1, · · · , hκ − 1 where ap = yp (a). α using the binomial formula: Let us develop the deﬁnition of Φ α = (yp − ap )j dΦ (a)v j j (−ap )j−j Φα j ((a)v ) = j

j =0

4.3 A general result on admissible measures

147



α−vκ · αvκ

 j j · U −vκ πα,1 U κv (−ap )j−j Φj (a)v  j j =0

First we notice that all the operators vκ

α

·U

−vκ

−1

= α

U

−vκ

−vκ −vκ n−1 t −1 α−1 Z = I +α Z = t t=0

are uniformely bounded for v → ∞ by a positive constant C1 (where U = (N p; A)) because n does not depend on v αI + Z and Z n = 0 for n = rkA Mr∗ −vκ and the binomial coeﬃcients are all Zp -integers. t On the other hand by the condition (growth) of the theorem (for the distributions Φj ) j κv j j−j U ( Φj ((a)v )) ≤ Cp−vj (−ap ) j j =0 p,Mr∗

for all j = 0, 1, . . . , κh − 1. If we apply to this estimate the previous bounded operators πα,1 U κv we get the following inequality j α (yp − ap ) dΦ ≤ C · C1 |α−vκ |p · p−vj = o(p−v(j−hκ) ) (a)v p,Mr∗

because of the estimate vκ |α−vκ |p = pvp (α) = o(pvhκ )

4.4 The standard L-function of a Siegel cusp eigenform and its critical values. 4.4.1 The standard L function Let f ∈ Mm k (C, ψ) be an eigenfunction of all Hecke operators and let (α0 , α1 , · · · αm ) = (α0,f (q), α1,f (q), · · · , αm,f (q)) ∈ [(C× )m+1 ]Wm

(4.33)

be the Satake q-parameters of the modular form f (for all primes q not dividing the level C). Recall that the standard L-function D(s, f, χ) of f ∈ Mm k (C, ψ) is the $ product D(s, f, χ) = qN D(q) (s, f, χ) with D(q) (s, f, χ) = (1 − χ(q)ψ(q)q −s )−1 Rf,q (χ(q)ψ(q)q −s )−1 , that is D(s, f, χ) = (4.34)

−1 " m χ(q)ψ(q) χ(q)ψ(q)αi (q)−1 χ(q)ψ(q)αi (q) , 1− 1− 1− qs qs qs q i=1 where χ is a Dirichlet character mod M . Analytic properties of the standard zeta functions were studied by A. N. Andrianov and V. L. Kalinin [An-Ka] in the case of even degree m, and S. B¨ ocherer [B¨o2] extended these results to the case of arbitrary degree using a diﬀerent approach (see also [B¨ oSch] where a p-adic version of his construction was developped). For m = 1 and a ∞ normalized cusp eigenform f (z) = n=0 a(n)e(nz) ∈ M1k (C, ψ) we have that D(s, f, χ) = L2,f (s + k − 1, χ), where L2,f (s, χ) = LCM (2s − 2k + 2, χ2 ψ 2 )

∞

χ(n)a(n2 )n−s

n=1

is the symmetric square of the modular form f . Let us recall some basic equalities from 2.2 concerning the integral representation of the standard L-function 4.4.2 Theta series (See in section 2.2.1 (Chapter 2), and in [An2], [An-Ma1], [An-Ma2], [St2]). Let F ∈ 2Cm be an even symmetric positive deﬁnite matrix, and q0 its level (i.e. the smallest positive integer such that q0 F −1 ∈ 2Cm , and χ a Dirichlet character modulo Q (not necesarily primitive). Put ν = 1 or 0 and the theta function was deﬁned by (2.60): (ν) χ(det(ξ))det(ξ)ν em (zF [ξ]/2). (4.35) θ(χ) = θF (z, χ) = ξ∈Mm (Z)

4.4 The standard L-function

149

Proposition 4.9. (a) If γ=

ab cd

∈ Γ0m (q0 Q2 ),

then the following transformation formula (2.61) holds (ν)

(m)

(ν)

θF (γ(z), χ) = χ(det(d))χQ2 F (γ)det(cz + d)(m/2)+ν θF (z, χ),

(4.36)

(m)

where χF (γ) is a root of unity of the eighth degree, and if m is even, then (−1)m/2 det(F ) (m) χF (γ) = . (4.37) det(d) 0m −1m . If χ is primitive (b) Let J(M ) denote for M > 0 the matrix M 1m 0m modulo Q then the action of the involution J(Q2 q0 ) on (2.60) is given by (2.63) (ν) θF (J(Q2 q0 )z, χ) = χ(−1)m (iQ)mν GQ (1m , χ)det(Fˆ )m/2+ν ·

(4.38)

(ν) ·det(−iz)m/2+ν θFˆ (z, χ) ¯

with Fˆ = q0 F −1 and

GQ (ξ, χ) =

χ(det(h))em (tξh/Q)

h∈Mm (Z)modQ

is the Gauss sum of degree m of the character χ.

4.4.3 The Rankin zeta function Let f and g be two holomorphic modular forms of weights k and l on the congruence subgroup Γ0m (N ) ⊂ Γ m , and with Dirichlet characters ψ and ω. More precisely we suppose that k a(ξ)em (ξz) ∈ Sm (N, ψ), (4.39) f (z) = ξ∈Cm

g(z) =

b(ξ)em (ξz) ∈ Mlm (N, ω),

(4.40)

ξ∈Bm

with Bm being the set of half integral non negative matrices of size m × m, and Cm ⊂ Bm the subset of all positive deﬁnite matrices (see section 2.1). Deﬁne an equivalence relation on Bm by ξ1 ∼ ξ2 iﬀ ξ1 = tuξ2 u for some matrix u ∈ SLm (Z). Then the Rankin zeta function (convolution of two Siegel modular forms) was deﬁned in (2.80) as the series [St2]:

150

4 Admissible measures for standard L–functions

L(s, f, g) =

ε(ξ)−1 a(ξ)b(ξ)det(ξ)−s

(s ∈ C),

(4.41)

ξ∈Cm /∼

where ε(ξ) = |Aut(ξ)| is the order of the unit group ' ( Aut(ξ) = η ∈ SLm (Z) | tηξη = ξ[η] = ξ . This series is well deﬁned due to basic properties of Fourier coeﬃcients of Siegel modular forms; Cm / ∼ denotes the orbit space with respect to ∼ and Re(s) is supposed to be large enough: Re(s) ≥ 0. Proposition 4.10 (Integral representation for the Rankin zeta function). (See Proposition 2.8) and [St2], proposition 6). For s with Re(s) 0 there is the following integral representation. (4π)−ms Γm (s)L(s, f, g) = . / f ρ , gE(z, s − k + (m + 1)/2; k − l, ψω, N ) ,

(4.42)

N

where the inner product is deﬁned by (2.14), k ¯ a(ξ)em (ξz) ∈ Sm (N, ψ), f ρ (z) = ξ∈Bm

and Γm (s) denotes the Γ -function of degree m, i.e. Γm (s) = π m(m−1)/4

m−1

Γ (s − (j/2)),

j=0

(see also (2.94)). 4.4.4 The standard zeta function D(s, f, χ) as the Rankin convolution Now let f (z) =

a(ξ)em (ξz) ∈ Skm (N, ψ),

ξ∈Cm

be a cusp form of even degree m which is an eigenfunction of the Hecke algebra Lm q (N ) for q not dividing N , and let (α0 , α1 , · · · , αm ) = (α0 (q), α1 (q), · · · , αm (q))) ∈ [(A× )m+1 ]Wm be the corresponding (m+1)-tuple of the Satake q-parameters of f (see (2.21)). Fix a matrix ξ0 ∈ Cm such that a(ξ0 ) = 0 and consider the primitive quadratic Dirichlet character χξ0 deﬁnied for positive integers d with (d, 2det(2ξ0 )) = 1 by the formula

4.4 The standard L-function

χξ0 (d) =

(−1)m/2 det(2ξ0 ) d

151

,

and when det(2ξ0 ) is odd, then by deﬁnition χξ0 (2) = 1 or −1 according as to which of the two following quadratic forms x1 x2 + x3 x4 + . . . xm−1 xm x1 x2 + · · · + xm−3 xm−2 + x2m−1 + xm−1 xm + x2m our quadratic form ξ0 is equivalent over the ﬁeld F2 of two elements. Assume also that χ mod M is chosen so that (−1)ν = χ(−1). The main result of the A.N.Andrianov’s work [An3] can be stated as a certain identity expressing the standard zeta function D(s, f, χ) as a Rankin zeta function, namely the convolution of the given form f and a theta function with the Dirichlet character χ mod M . The precise statement of the result is given in the following proposition. Proposition 4.11. Under the notation and assumptions as above for the suﬃciently large values of Re(s) the following identity holds a(ξ0 )R(s, f, χ) = 2

−1

(4.43) (s+k−1+ν)/2

det(ξ0 )

L((s + k − 1 +

(ν) ν)/2, f, θ2ξ0 (z; χ)),

where the function R(s, f, χ) is deﬁned by the following equality D(s, f, χ) =



m/2−1

L(s + (m/2), χψχξ0 ) 



(4.44)

L(2s + 2i, χ2 ψ 2 ) R(s, f, χ),

i=0

and it is assumed that the modulus M of the character χ is divisible by all (ν) prime divisors of the number N det(2ξ0 ), where θ2ξ0 (z, χ) is the theta function as in 2.2.1. More explicitly, the right hand side of (2.83) can be represented as the series χ(det(ξ))det(ξ)ν a(ξ0 [ξ])det(ξ0 [ξ])−(s+k−1+ν)/2 = det(ξ0 )(s+k−1+ν)/2 ξ

=

χ(det(ξ))a(ξ0 [ξ])det(ξ)−(s+k−1+ν)/2 ,

(4.45)

ξ

where the summation is taken over the set of equivalence classes ξ ∈ SLm (Z)\M+ m (Z) of the form SLm (Z)ξ. Now we set q0 to be equal to the level of the quadratic form with the matrix 2ξ0 (see 2.2.1). In order to get the integral representation of the standard zeta function we apply to it the result of proposition 2.8. For this purpose we put

152

4 Admissible measures for standard L–functions

in the notation of proposition 2.8 l = (m/2) + ν, ω = χχξ0 with χ being a Dirichlet character modulo M , and take the number N q0 M 2 as N . We note also that both parts of (2.83) and (2.84) converge absolutely for Re(s) > m (see also [An5], p. 133). Proposition 4.12 (integral representation of the standard zeta function). For the notations and assumptions as above about Re(s), the following equality holds: 2a(ξ0 )det(ξ0 )−(s+k−1+ν)/2 (4π)−ms Γm ((s + k − 1 + ν)/2)R(s, f, χ) = . / (ν) = f ρ , θ2ξ0 (z, χ)E(z, (s − k + m + ν)/2) , (4.46) N M 2 q0

where the Siegel-Eisenstein series in the right hand side E(z, s) = E(z, s; k − ν − (m/2), χψχξ0 , N M 2 q0 ) is deﬁned by (2.76). 4.4.5 Algebraic properties of the special values of normalized distributions. Consider only the case of even m. In order to give a precise statements on algebraic properties of the standard zeta functions and of the corresponding distributions, it is convenient to make some additional normalization of these values because these properties look diﬀerent for the integral points to the left and to the right of the critical line Re(s) = 12 (the same eﬀect happens in the case of the Riemann zeta function). Recall the following three types of the normalized zeta function: for κ = m+1 2 , s∈C D∗ (s, f, χ) = (2π)−m(s+k−κ) Γ ((s + δ)/2)

m

Γ (s + k − j)D(s, f, χ), (4.47)

j=1

D− (s, f, χ) = Γ ((s + δ)/2)−1 D∗ (s, f, χ) = (2π)−m(s+k−κ)

m

Γ (s + k − j)D(s, f, χ),

(4.48)

j=1

D+ (s, f, χ) =

2 iδ Γ (s) cos(π(s − δ)/2) − D (s, f, χ) = (2πi)s

(4.49)

iδ π (1−2s)/2 D∗ (s, f, χ), Γ ((1 + δ − s)/2) where δ = 0 or 1 according as ψχ(−1) = (−1)δ with f ∈ Sm k (C, ψ) be an eigenfunction of the global Hecke algebra Lm (C) = ⊗q |C Lm q (C) with the eigenvalue given as a homomorphism λf : Lm (C) → C.

4.4 The standard L-function

153

Now we turn to the normalizations (4.48) and (4.49). Their convenience is illustrated by the following result on algebraicity of the special values of (4.48) and (4.49). Theorem 4.13 (Algebraic properties of the special values of the standard zeta function). Assume that the cusp eigenform f ∈ Sm k (C, ψ) satisﬁes the condition a(ξ0 ) = 1 for some ξ0 ∈ Am , ξ0 > 0. Then a) For all integers s satisfying 1 ≤ s ≤ k − δ − m and s = 1 if the character χ2 ψ 2 is trivial we have that f, f −1 D+ (s, f, χ) ∈ K = Q(f, λf , ψ, χ),

(4.50)

where K = Q(f, λf , ψ, χ) denote the ﬁeld generated by Fourier coeﬃcients of f , by the eigenvalues λf (X) of Hecke operators X on f , and by the values of the characters χ and ψ. Moreover, D+ (s, f, χ) = 0 for s ≡ δ(mod2). b) For all integers s satisfying 1 − k + δ + m ≤ s ≤ 0 we have that f, f −1 D− (s, f, χ) ∈ K, and

(4.51)

D− (s, f, χ) = 0 for s ≡ δ(mod2), s ∈ Z, s ≤ 0.

Proof of Theorem 4.13 (See also [Pa6], Chapter 3). One uses the identity (4.46), which says that for all non-trivial Dirichlet characters χ mod M : 2a(ξ0 )((4π)m det(ξ0 ))−(s+k−1+ν)/2 Γm ((s + k − 1 + ν)/2) D(s, f, χ) = LN (s +

m , ψχξ0 χ) 2

(m/2)−1

LN (2s + 2i, ψ 2 χ2 )) ×

(4.52)

i=0

(ν)

× f ρ (z), θ2ξ0 (z; χ)E(z, (s − k + m + ν)/2) N , E(z, s) = E(z, s; k − (m/2) − ν, ψχξ0 χ, N ) is the Siegel – Eisenstein series of the weight k − (m/2) − ν, the level N = 4q0 CM 2 , and the Dirichlet character ψχξ0 χ is viewed modN . For the convenience of the reader we give here more details on the proof of the algebraicity results (taken from [Pa6], p. 101-110). Proposition 4.14. Le f ∈ Sm k (C, ψ) be a cusp form of weight k ≥ m + 1 where m is even, χ be a Dirichlet character modulo M ≥ 1. Put N = 4qo M 2 C where q0 is the level of a quadratic form with the matrix 2ξ0 such that a(ξ0 ) = 0. Then we have that

4 Admissible measures for standard L–functions

154

∗ a(ξ0 )DN (s, f, χ) = f ρ , K ∗ (z, s; ξ0 , χ) N ,

where K ∗ (z, s; ξ0 , χ) = N −m(2s+m)/4 2m(2k−m+2−κ) i−m(k−(m/2)−ν) × (4.53) (s+k−1+ν)/2 (ν) θ2ξ0 (χ)G∗ (z, (s

×detξ0

− k + ν + m)/2)|W (N ),

∗ (s, f, χ) indicates that all Euler where the subscript N in the notation DN factors corresponding to q, q|N are removed from the Euler product, and the series G∗ (z, s) = G∗ (z, s; k − (m/2) − ν, χχξ0 ψ, N )

being deﬁned by the equality (2.133) of chapter 2. The proof is deduced from the integral representation ((4.46)) rewritten in the form 2a(ξ0 )((4π)m detξ0 )−(s+k−1+ν)/2 Γm ((s + k − 1 + ν)/2) D(s, f, χ) = ˜ s; ξ0 , χ) N , (4.54) f ρ (z), K(z, where ˜ s; ξ0 , χ) = K(z, LN (s +

m , ψχξ0 χ) 2

(ν) ×θ2ξ0 (z; χ)[E(z, (s

(4.55)

(m/2)−1

LN (2s + 2i, ψ 2 χ2 )) ×

i=0

− k + m + ν)/2)|k−(m/2)−ν W (N )]|k−(m/2)−ν W (N ).

In the above equality we used deﬁnition (4.47) of the normalized zeta functions, the deﬁnition of the series G∗ (z, s) and the relation (2.137 ) for the Γ -factors. Now Theorem 4.13 is implied by Proposition 4.14 and Theorem 2.13 of chapter 2 in which we take k to be equal to k − (m/2) − ν. ∗ (s, f, χ) is obtained from the function D∗ (s, f, χ) Note that the function DN if we multiply it by an elementary holomorphic factor; however we do not know in general how to obtain holomorphy properties of the function D∗ (s, f, χ) itself from the theorem, and this interesting question needs a further study. However, under the assumptions of Theorem 4.13 we have that ∗ (s, f, χ), D∗ (s, f, χ) = DN

hence the proof is completed. 4.4.6 Integral representation for the functions D ± (s, f, χ) In order to prove Theorem 4.13 on algebraic properties we need an integral representation for the functions D± (s, f, χ) analogous to that of Proposition 4.14.

4.4 The standard L-function

155

Let χ be a Dirichlet character modulo M and assume that all conditions 2 of Theorem 4.13 are satisﬁed. Put N = 4qM C, then we have the following integral representation ± (s, f, χ) = f ρ , K ± (z, s; ξ0 , χ) N , a(ξ0 )DN

(4.56)

with K ± (z, s; ξ0 , χ) = N −m(2s+m)/4 2m(2k−2−m−κ) i−m(k−(m/2)−ν) × (s+k−1+ν)/2

×detξ0

Hol [θ2ξ0 (χ)|W (N )G± (z, (s − k + ν + m)/2)]|W (N ), (ν)

in which K ± (z, s; ξ0 , χ) ∈ Mm k (N, ψ), the series

G± (z, s) = G± (z, s; k − (m/2) − ν, χχξ0 ψ, N )

are deﬁned by (2.135), (2.136) of Chapter 2 and the symbol Hol denotes the holomorphic projection operator from Theorem 2.16 of chapter 2. The proof of (4.56) is carried out in exactly the same way as that of the proposition 3.9 if we take into account deﬁnitions (4.2) and (4.2) of the functions D± (s, f, χ), the deﬁnition of the series G± (z, χ) and use the relation (3.12) for the Γ -factors. The possibility of applying Hol to the function θ2ξ0 (χ)|W (N )G± (z, (s − k + ν + m)/2) (ν)

(4.57)

by formulas of Theorem 2.147 of chapter 2 is justiﬁed as in the end of Section 2.4 in chapter 2 bearing in mind positivity properties of Fourier expansions of the series G± (z, s) in Theorem 2.147 of chapter 2 and the growth estimates mentioned above. From these estimates it follows that the function (4.57) satisfy the bounded growth condition, and its Fourier expansion contains only terms with positive deﬁnite matrix indices. Remark 4.15. If k > 2m + 2 then for s = k − ν − m the series deﬁning the function G± (z, 0) = G∗ (z, 0; k − (m/2) − ν, χχξ0 ψ, N ) is absolutely convergent so that this function is holomorphic, and we can omit the symbol Hol in the integral representation (4.56): a(ξ0 )Γ + (k − ν − m)DN (k − ν − m, f, χ) = f ρ , K ± (z, k − ν − m; ξ0 , χ) N , with

(4.58)

156

4 Admissible measures for standard L–functions

K ± (z, k − ν − m; ξ0 , χ) =

(4.59)

−m(2k−2ν−m)/4 m(2k−2−m−κ)+m(2k−2ν−m)/2 −m(k−(m/2)−ν)

N 2 (ν) ×θ2ξ0 (χ)[G± (z, 0)|W (N )],

i

×

in which Γ + (s) = (2π)−m(s+k−κ)

m 2iδ Γ (s) cos(π(s − δ)/2) Γ (s + k − j) (2π)s j=1

(4.60)

is the gamma-factor. With these values of k and s the identity of A.N.Andrianov (Proposition 2.84 of Chapter 2 and the equality (4.46) takes the form: a(ξ0 )D+ (s, f, χN ) = a(ξ0 )D+ (s, f, χ) =

(4.61)

(m/2)−1

Γ + (s)LN (s + (m/2), χ χξ0 ψ) ×

LN (2s + 2i, χ2 ψ 2 ) ×

i=0

χN (deth)a(ξ0 [h])deth−(s+k−1) ,

h∈SLm (Z)\M+ m (Z)

where χ be a Dirichlet character modulo N deﬁned by χN (d) = χ(d) for det2ξ0 |N . The series in (4.61 ) is absolutely convergent for Re(s) > 1 + m due to the estimate |a(h)| = O(deth(k/2) ) for the Fourier coeﬃcients. 4.4.7 Action of the group Aut(C) on scalar products of modular forms. Let us ﬁx a positive integer N1 . Recall that the group Aut(C) acts on modular forms a(ξ)em (ξz) ∈ Mm f= k (N1 , ψ) Am ξ≥0

by the following rule: σ fσ = a(ξ)σ em (ξz) ∈ Mm k (N1 , ψ ) (σ ∈ Aut(C)), Am ξ≥0

and this action commutes with the action of the Hecke algebra, see [Sh7], [St2]. Consider the global Hecke algebra L(N1 ) = ⊗qN1 Lm q (N1 ) and suppose that f ∈ M(N1 , ψ) is an eigenfunction of the Hecke algebra L(N1 ) with the eigenvalue given by a homomorphism Λ : L(N1 ) → C, i.e.

4.4 The standard L-function

157

f |X = Λ(X)f for all X ∈ Lm q (N1 ) and all q N1 . Let N1 |N . We deﬁne a Λ-packet of modular forms as the following subspace of Sm k (N, ψ): % & m Hkm (Λ, N, ψ) = f ∈ Sm k (N, ψ)|f |X = Λ(X)f, X ∈ Lq (N1 ), q N , and put Hkm (Λ, ψ) = Sm k (ψ)

=

3 N ≡0( mod N1 )

3

Hkm (Λ, N, ψ),

(4.62)

m N ≡0( mod N1 ) Sk (N, ψ).

We know due to Shimura that the action of Aut(C) commutes with the action of the Hecke algebra [Sh7], therefore for each σ ∈ Aut(C) f ∈ Hkm (Λ, ψ) ⇐⇒ f σ ∈ Hkm (Λσ , ψ σ ),

(4.63)

where Λσ (X) = Λ(X)σ . On the other hand, if we use the normality property of the Hecke operators with respect to the Petersson scalar product and commutativity of the Hecke algebra Λ(N1 ) we see (as in the classical case) that for a certain set of homomorphisms Λ = Λ1 , · · · , Λt there is the decomposition of S(N, ψ) into the orthogonal direct sum of the corresponding Λ-packets: S(N, ψ) =

t 8

Hkm (Λi , N, ψ).

(4.64)

i=1

The following proposition was established by J.Sturm ([St2], Theorem 3). We statehere this result in a form, which is more suitable for our applications. Proposition 4.16. Let m be even then for any integer k with k > 2m + 2, a Dirichlet character ψ mod N , and a homomorphism Λ : L(N ) → C there exists a non zero constant µ(Λ, k, ψ) ∈ C× depending only on Λ, k, ψ such that 2σ 1 ρ f σρ , g σ N f , g N (4.65) = µ(Λ, k, ψ) µ(Λσ , k, ψ σ ) for all f ∈ Hkm (Λ, N, ψ), g ∈ Mm k (N, ψ), σ ∈ Aut(C). Remark 4.17. If we take in equality (4.65) g equal to f ρ then proposition 4.16 implies that f, f N µ(Λ, k, ψ)−1 ∈ Q(Λ, f ), with Q(Λ, f ) being the subﬁeld of C generated by the values of the homomorphism Λ and the Fourier coeﬃcients of f . We give here a proof based on the Andrianov’s identity (4.102) in which the right-hand side has the form: D+ (s, f, χN ) = Γ + (s) R(Λ, q, k)(χ(q)ψ(q)q −s ) (4.66) q|N

158

4 Admissible measures for standard L–functions

where R(Λ, q, k)(t) ∈ Q[Λ(X), X ∈ Lm q (N )] [t] are polynomials satisfying R(Λ, q, k)(0) = 1 which depend only on the Λpacket of the form f and on the numbers q and k. The product in (4.66) converges absolutely for Re(s) > 1 + m. Put s = k − ν − m where ν = 0, 1, k ≡ ν mod 2, take as χN the trivial character modulo N and deﬁne R(Λ, q, k)(ψ(q)q −(k−ν−m) ), (4.67) µ(Λ, k, ψ) = G(ψ)m−1 Γ + (k − ν − m) q2B

with the Γ -factor being deﬁned by (4.93), B = B(Λ, k, ψ) being a positive integer such that the product in (4.67) does not vanish; B exists due to absolute convergence of the product (see the Remark 4.15), so that we can and we will assume that B(Λ, k, ψ)σ = B(Λσ , k, ψ σ ). Now, with the number µ(Λ, k, ψ) already been deﬁned, we prove ﬁrst the proposition 4.16 for the special modular form g = G(ψ)m−1 K + (ξ0 , ψ), where we adopt the notation K + (ξ0 , ψ) = K + (z, k −ν −m; ξ0 , χN ) in order to stress, that the function K + (z, k − ν − m; ξ0 , χN ) depends on ξ0 and on χN (recall that we took χN to be the trivial character modulo N ). Then the following identity holds: [G(ψ)m−1 K + (ξ0 , ψ)]σ = G(ψ σ )m−1 K + (ξ0σ , ψ σ ).

(4.68)

This important fact expresses in a more precise form the result of proposition 2.15 of Chapter 2 on the cyclotomy of the Fourier coeﬃcients of this Siegel modular form. The identity (4.68) will be proved later in Proposition 4.19, and now we deduce from it Proposition 4.16. According to the equalities (4.66) and (4.58), there is the following relation: G(ψ)m−1 f ρ , K + (ξ0 , ψ) N µ(Λ, k, ψ) = a(ξ0 ) R(Λ, q, k)(ψ(q)q −(k−ν−m) ),

(4.69)

q|2B q |det2ξ0

with the ﬁnite Euler product in the right hand side on which the automorphisms σ ∈ Aut(C) act term-by-term. Therefore, from (4.68) and (4.69) it follows that for the functions of the type g = G(ψ)m−1 K + (ξ0 , ψ) satisfy the relation (4.65). In order to deal with the general case we vary ξ0 ∈ Am , ξ0 so that the number N = N (ξ0 ) will now depend on ξ0 , and consider the trace operator Tr(N2 , N1 , ψ) = where elements g(i) = of the right cosets:

a(i) b(i) c(i) d(i)

d

ψ(a(i))F |k g(i), )

i=1

form a complete system of representatives

4.4 The standard L-function

Γ0m (N1 ) =

d )

159

Γ0m (N2 )g(i).

i=1

The important property of this action is that it commutes with the trace operator. This fact is stated more precisely in the following proposition. Proposition 4.18. Let F ∈ Mm k (N2 , ψ) be a Siegel modular form with cyclotomic Fourier coeﬃcients, F (z) = A(ξ)em (ξz), A(ξ) ∈ Qab . ξ≥0

Then for all σ ∈ Aut(C) the following equality holds [f |Tr(N2 , N1 , ψ)]σ = [f σ |Tr(N2 , N1 , ψ σ )]. Proof is given in [St2], lemma 11, and it is based on properties of the action of the restricted adele group GA+ on the graded ring of automorphic forms studied by Shimura [Sh8], where GA+ denotes the subgroup of the adelization % & GA = α ∈ GL2m |t αJm α = ν(α)Jm , ν(α) ∈ GL1 (A) consisting of all elements α ∈ GA for which Archimedean component of ∈ Mm the ν(α) is positive. For x ∈ GA+ and a modular form k = 3 idele $F + m m x M (Γ (M )) the action of x on F is denoted by F . If t ∈ Z an idele k M q q ab whose action on Q by class ﬁeld theory coincides with that of σ ∈ Aut(C), then we have that 1m 0m σ x(t−1 ) F =F with x(t) = ∈ GA+ 0m t1m for the above action on modular forms. Now the proposition is easily deduced from this equality. In view of the strong approximation theorem for the group Sp2m (A) we can choose for each representative g(i) in (4.4.7) elements u(i) ∈ Sp2m (A), h(i) ∈ Sp2m (Z) such that u(i)q ≡ 12m (modN2 ) for all primes q, q|N2 and 1m 0m 1m 0m g(i) = u(i) h(i). 0m t−1 1m 0m t1m −1

Let us take into account that F σ = F x(t

(F σ )|Tr(N2 , N1 , ψ σ

σ−1

)

then we get the equality σ−1 = ψ(a(i))σ F σ |g(i) i

and from the choice of h(i) it follows that

160

4 Admissible measures for standard L–functions

Γ0m (N1 ) =

d )

Γ0m (N2 )h(i),

i=1

(i) b (i) and a(i) ≡ a(i) mod N2 , so that the proposition with h(i) = ac (i) d (i) follows. Now we are able to accomplish the proof of the proposition 4.16. We let the element ξ0 in the equality (4.69) vary, and put N2 = B 2 N det2 2ξ0 , N1 = N. Then f ρ , K + (ξ0 , ψ) N2 = f ρ , K + (ξ0 , ψ)|Tr(N2 , N1 , ψ) N1 ,

and from proposition 4.18 it follows that the equality (4.65) holds for all modular forms g from the set ( ' V = G(ψ)m−1 K + (ξ0 , ψ)|Tr(N2 , N1 , ψ)Am ξ0 > 0, N2 = B 2 det2 2ξ0 , N1 = N . Let

' ( V1 = g1 ∈ Hkm (Λ, N, ψ)g − g1 is orthogonal to some g ∈ V .

In other words, the set V1 consists of those elements in Hkm (Λ, N, ψ) which are orthogonal projections of the special elements g ∈ V as above. One checks as in [St2] that V1 generates the whole Λ-packet Hkm (Λ, N, ψ). Indeed, if a1 (ξ)em (ξz) ∈ Hkm (Λ, N, ψ) f1 = ξ>0

satisﬁes f1 , g1 N = 0 for all g1 ∈ V1 then (3.39) implies that a(ξ0 ) = 0 for all ξ0 ∈ Am , ξ0 > 0 hence f1 ≡ 0. Also, it follows that proposition 4.16 is valid for all g1 ∈ Hkm (Λ, N, ψ). For an arbitrary Siegel modular form g ∈ Mm k (N, ψ) let us write g = g1 + h with g1 ∈ Hkm (Λ, N, ψ) and g1 , h1 N = 0. Now if we combine (4.63) and (4.64) and get the following: σ

g σ = g1σ + hσ1 , g1σ ∈ Hkm (Λσ , N, ψ ), g1σ , hσ1 N = 0.

(4.70)

In order to obtain the above equality we used the important fact about the invariance of the subspace of all Eisenstein series in Mm k (N, ψ) under the action of σ ∈ Aut(C). In turn, this property follows from the general decomposition theorem [Kl3], describing the subspace of all Eisenstein series ( the orthogonal complement to the subspace of cusp forms) in terms of the Klingen – Eisenstein series, and from the invariance properties of these series under the action of complex automorphisms σ ∈ Aut(C), established by M.Harris [Ha2], [Ha3] (see also [B¨o2], [Ku-Mi], [Miz1], [Miz2]). This last fact (stated in a more precise form) comprise the content of the Garrett’s conjecture, which was proved in [Ha3]. However, we do not use this fact any more and therefore will not go into detail of this interesting research. Returning to the equality (4.70) we get

4.4 The standard L-function

161

[ f ρ , g N µ(Λ, k, ψ)−1 ]σ = [ f ρ , g1 N µ(Λ, k, ψ)−1 ]σ = f σρ , g1σ N µ(Λσ , k, ψ σ )−1 = f σρ , g σ N µ(Λσ , k, ψ σ )−1 , To accomplish the proof of proposition 4.16 we need only to check the property (4.68) which will be now stated in the following more general form. Proposition 4.19. Let χ mod N be an arbitrary Dirichlet character, and K + (ξ0 , ψ, χ) = K + (z, k − ν − m; ξ0 , ψ, χ) ∈ Mm k (N, ψ) denote a modular form in the integral representation (3.30). Then for all σ ∈ Aut(C) there is the following relation [G(ψχ)m−1 K + (ξ0 , ψ, χ)]σ = G(ψ σ χσ )m−1 K + (ξ0 , ψ σ , χσ )

(4.71)

Proof From the deﬁnition of the theta series we easily see that (ν)

(ν)

θ2ξ0 (z, χ)σ = θ2ξ0 (z, χσ ). Therefore it suﬃces to check the following property: 1/2

[G(ψχ)m−1 detξ0 G+ (z, 0; χχξ0 ψ, k − (m/2) − ν, N )|W (N )]σ = σ

σ m−1

G(ψ χ )

) ((detξ0 1/2)σ G+ (z, 0; (χχξ0 ψ)σ , k

(4.72)

− (m/2) − ν, N )|W (N ),

(in view of the equality k−(m+1)/2

detξ0

k−(m/2)−1

= detξ0

, m/2 ∈ Z.)

According to formulas in §2.3.1, ( 2.141) of chapter 2, the Fourier coeﬃcient of the series G+ (z, 0; χχξ0 ψ, k − (m/2) − ν, N ) by em (hz) with h ∈ Am , h > 0 has the form (det h)k−ν−m−(1/2) L+ (k − ν − m, χχh χξ0 ψ)M (h, χχξ0 ψ, s), where M (h, χχξ0 ψ, s) =

(4.73)

Mq (h, χχξ0 ψ(q), q −s )

q∈P (h)

is the ﬁnite Euler product (2.132) of Chapter 2, where the product is extended to all primes q in the set P (h) of prime divisors of N and the elementatry divisors of the matrix h, with the property Mq (h, t) ∈ Z[t]. Therefore M (h, χχξ0 ψ, k − ν − (m/2))σ = M (h, (χψ)σ χξ0 , k − ν − (m/2)). Now let us consider the factor L+ (k−ν−m, χχh χξ0 ψ) and recall an elementary result on the special values of the Dirichlet L-functions [La1], [Sh3], [Wa],

162

4 Admissible measures for standard L–functions

[Le2]. Let χ mod N be a Dirichlet character of conductor N0 , and χ0 mod N0 be the corresponding primitive character, G(χ) = G(χ0 ) =

N0

χ0 (x)e(x/N0 )

x=1

be its Gauss sum. We set, for a positive integer r P (r, χ) = G(χ)−1 (2πi)−r L(r, χ). Then for all σ ∈ Aut(C) and χ(−1) = (−1)r we have that P (r, χ)σ = P (r, χσ )

(4.74)

If we apply this property to normalized Dirichlet L-series we see that [G(χ)−1 L+ (r, χ)]σ = G(χσ )−1 L+ (r, χσ ) (r ∈ Z, r > 0)

(4.75)

L− (r, χ)σ = L− (r, χσ ) (r ∈ Z, r ≤ 0)

(4.76)

so that for the values of the “wrong parity” the corresponding values vanish. From the basic property of Gauss sums it follows that G(χ)σ = χσ (v)−1 G(χσ ) f or v ∈ (Z/N Z)× , such that e(1/N )σ = e(v/N ). The last property implies the useful relation: G(ψ)σ G(χ)σ G(ψχ)σ = G(ψ σ χσ ) G(ψ σ )G(χσ )

(4.77)

Let us now apply (4.75) and (4.77) to the coeﬃcients (4.73), then we get [G(ψχ)−1 (detξ0 h)1/2 L+ (k − ν − m, χχh χξ0 ψ]σ = σ

σ −1

G(ψ χ )

((detξ0 h)

(4.78)

) L (k − ν − m, (χψ) χξ0 χh ).

1/2 σ

+

σ

In the equality (4.79) we used the following elementary property of Gauss sums: G(χξ0 χh ) (det(ξ0 h))1/2 ∈ Q ˇ which is due to the fact that χξ0 χh is an even quadratic character (see [Bo-Sa]). To complete the proof of (4.72) and (4.73) we need the following general compatibility property of the action of Aut(C) and of the involution W (N ) (see [St2], lemma 5, and [Sh8]): for a modular form f ∈ Mm k (N, ψ) with cyclotomic Fourier coeﬃcients and for all σ ∈ Aut(C) we have that f σ |W (N ) = ψ(v m )σ (f |W (N ))σ ,

(4.79)

where v ∈ (Z/N Z)× is chosen so that e(1/N )σ = e(v/N ). Now the proof of Proposition 4.16 is ﬁnished.

4.4 The standard L-function

163

4.4.8 Algebraicity properties and Fourier coeﬃcients Now in order to deduce theorem 3.2 on algebraicity of the normalized standard zeta function and the algebraicity property of the normalized distributions we use the already proved algebraicity properties of Fourier coeﬃcients of the functions K ± (z, s; ξ0 , χ)|W (N ) in the corresponding integral representation (3.28) and of the functions F ± (z, χ) in (3.13) and (3.14). We apply these properties in the form given below: let k > 2m + 2, f ∈ Hkm (Λ, N, ψ) ⊂ m Sm k (N, ψ) be a cusp form, an eigenfunction of the Hecke algebra L (N ) with m an eigenvalue given as the homomorphism Λ : L (N ) → C. Let us consider the following linear form Lf : g →

f ρ , g|W (N ) N f, f N

(4.80)

on the vector space Mm k (N, ψ) with g= b(h)em (hz) ∈ Mm k (N, ψ) h≥0

being its arbitrary element. Then there exist positive matrices h1 , h2 , · · · , ht ∈ Am and algebraic numbers α1 , α2 , · · · , αt ∈ Q(f, Λ, ψ) from the ﬁeld Q(f, Λ, ψ) generated by the Fourier coeﬃcients of f and values of the homomorphism Λ and the character ψ such that for all g ∈ Mm k (N, ψ) the linear form is explicitly given by αi b(hi ). (4.81) Lf (g) = i

Indeed, we notice that every Siegel modular form of weight k > 2m is uniquely determined by its Fourier coeﬃcients with positive matrix indices h ∈ Am . This fact is equivalent to saying that for such a weight k there are no singular modular forms (i.e. having only Fourier coeﬃcients with degenerate h ∈ Am , det h = 0), which was established by G.L.Resnikov ([Res],[Ra3]). Then Proposition 4.16 implies that the number f ρ , g|W (N ) N ∈ Q(f, g, Λ, ψ) µ(Λ, k, ψ)−1 belongs to the ﬁeld Q(f, g, Λ, ψ) generated by the Fourier coeﬃcients of the forms f and g, by the values of Λ and ψ. Moreover we have that f, f N µ(Λ, k, ψ)−1 ∈ Q(f, Λ, ψ)× (see the remark after proposition (4.16), and (4.81) follows. In order to prove Theorem 4.13 we use Proposition 4.16 and substitute in it the modular forms K ± (z, s; ξ0 , χ)|W (N ) ∈ Mm k (N ; ψ),

164

4 Admissible measures for standard L–functions

for g. This modular for has cyclotomic Fourier coeﬃcients vanishing for degenerate matrix indices h ∈ Am such that the action of σ ∈ Aut(C) on them is described as in Proposition 4.19 As a consequence, we obtain also the following more explicit description of the action of σ ∈ Aut(C) on the special values in question. Put ˜ + (s, f, χ) = G(ψχ)m−1 D+ (s, f, χ)µ(Λ, k, ψ)−1 , D ˜ − (s, f, χ) = G(ψχ)m D− (s, f, χ)µ(Λ, k, ψ)−1 . D Then under the assumption of Theorem 4.13 for every σ ∈ Aut(C) we have that ˜ ± (s, f, χ)σ = D ˜ ± (s, f σ , χσ ). D

(4.82)

In particular, this relation implies the algebraicity result in Theorem 4.13. Notice that in this situation the constant µ(Λ, k, ψ) depends only on the Λpacket of f and it coincides with that for the original cusp form f as above.

4.5 Algebraic linear forms on arithmetical nearly holomorphic Siegel modular forms 4.5.1 Convolutions of theta distributions and Eisenstein distributions with values in nearly holomorphic Siegel modular forms. Consider again the proﬁnite group Y = lim Yv , where Yv = (Z/N pv Z)× . ← v

loc−const (Y, A) → M be two distributions Let M = Mm r,k (A) and let Φi : C m on Y with values in Mi = Mri ,ki (A), (i = 1, 2). Assume that A contains (ϕ(N pv ))−1 and all the values of Dirichlet characters χ mod N pv viewed as homomorphisms Y → A× then obviously

Φi (χ) =

χ(a)Φi ((a)v ), Φi ((a)v ) =

a mod N pv

1 ϕ(N pv )

χ(a)−1 Φi (χ).

χ mod N pv

Let us ﬁx a Dirichlet character ψ on Y with values in A then there exists a unique distribution Φ = Φ1 ∗ψ Φ2 on Y with values in M = Mm r1 +r2 ,k1 +k2 (A) such that for all Dirichlet characters χ on Y with values in A one has Φ(χ) = Φ1 (χ) · Φ2 (ψχ−1 ) (Φ1 ∗ψ Φ2 is called the twisted convolution of Φ1 and Φ2 . Let us put M = pv , N = 4q0 C and take as Φ1 the theta distribution on Y deﬁned on Dirichlet characters χ mod M by (ν)

Φ1 (χ) = θ2ξˆ (χ)|V (C)

(4.83)

0

and let us take as Φ± 2,s the normalized Siegel-Eisenstein distribution (4.25) deﬁned on Dirichlet characters ω by ± Φ± 2,s (ω) = G (z, (s − k + m + ν)/2); k − (m/2) − ν, ω, N M ) =

(4.84)

±

(r ) ± G± Ψ2,s (ω) (s−k+m+ν)/2),k−(m/2)−ν (ω), N M ) = δ

(we use it for the Dirichlet character ω = ψχξ0 χ ¯ ) where M = M 2 , + Ψ2,s (ω) = G+ m/2+s,0 (ω), − (ω) = G− Ψ2,s m/2+1−s,s−(1/2) (ω)

are two holomorphic Siegel-Eisenstein series. ± Next deﬁne Φ± s = Φ1 ∗ψχξ0 Φ2,s (on basic open subsets one has Φ± s ((a)v ) =

y∈Yv

ψχξ0 (y)Φ1 (ya)Φ± 2,s (y).

(4.85)

166

4 Admissible measures for standard L–functions

for any v such that ψχξ0 is deﬁned modN pv ). We show that for all critical pairs (s, χ) as Theorem 4.13 the function Φ± s (χ) is an arithmetical nearly holomorphic Siegel modular form in m v m v (Γ Mm 1 (N p ), Q), and for an appropriate linear form Lf,α on Mr,k (Γ1 (N p ), Q) r,k ± ¯ the value Lf,α (Φ (χ)) coincides, up to a multiplicative con(deﬁned over Q) s × ± stant in Q , with the corresponding value Ds,χ of the normalized distribution constructed in [Pa6], Chapter 3 (which produces the critical value ¯ D± (s, f, χ))). Let us consider (as in Proposition 3.3) the canonical projection operator m m α πα : Mm r,k (C) → Mr,k (Γ0 (N p), ψ; C)

attached to a non-zero eigenvalue α of U = Π + (p) acting on ) m v Mm Mm r,k (N, p; C) = r,k (Γ0 (N p ); ψ, C) : v≥1

for any g ∈ M = Mm r,k (C) put πα (g) = U −v [πα,1 U v (g)] (πα (g) is well deﬁned if v is suﬃciently large so that g ∈ Mm (N pv+1 )). Proposition 4.20. Let α ∈ Q ⊂ C be a non-zero eigenvalue of U = m Π + (p) on Mm r,k (C) associated with a cusp eigenform f ∈ Sk (Γ0 (N ), ψ) and let f0 = f0,α be a corresponding eigenfunction (f0 |U = αf0 ), let us deﬁne f 0 = f0ρ |WN0 , f0ρ = aξ (f0 )q ξ . ξ

Then a) U ∗ = WN−1 U WN0 , the adjoint operator of U in the hermitian vector 0 space Sr,k (Γ1m (N p), ψ; C) with respect to the Petersson scalar product. b) One has f 0 |U ∗ = α ¯ f 0 , and for all ”good” Hecke operators T ∈ L(N pv ) 0 0 one has T f = λf (T )f . m c) The linear form f,α : g → f 0 , g on Mm r,k (Γ0 (N p), ψ; C) vanishes on Kerπα,1 , where m m m α πα,1 : Mm r,k (Γ0 (N p), ψC) → Mr,k (Γ0 (N p), C)

(the projector onto the α-primary subspace with the kernel Kerπα,0 = Im(U − αI)n ) hence f 0 , g = f 0 , πα,1 (g) = f 0 , πα (g) .

4.5 Algebraic linear forms on modular forms

167

d) If g ∈ Mm (N pv+1 , Q) = Mm (N ppv , Q) and α = 0, one has f 0 , g α = f 0 , πα (g) = α−v f 0 , g | U v

where

g α = πα (g) = (U α )−v πα,1 (g | U v ) ∈ Mm (N p)α

is the α-primary part of g (the image of g under the canonical projection πα ): e) Let us deﬁne Lf,α (g) =

f 0 , α−v g|U v N p f 0 , πα (g) N p = , 0 f , f0 N p f 0 , f0 N p

m hence Lf,α : Mm r,k (N, p; Q) → Q is the linear form Lf,α on Mr,k (N, p; C) which is deﬁned over Q and factorizes through the ﬁnite dimensional projection m α m m πα : Mm r,k (N, p; C) → Mr,k (N, p; C) ⊂ Mr,k (Γ0 (N p); C);

There exists a unique Cp -linear form

f,α

α ∗ ∈ (Mm r,k (N0 , Cp ) ) such that

f,α (ip (πα (g)))

= ip (Lf,α (g)) = ip

f 0 , α−v U v (g) N0 f 0 , f0 N0

(g ∈ Mm r,k (N, p; Q) with coeﬃcients in Q). Proof of Proposition 4.20 a) See [Miy], Th. 4.5.5 for m = 1 (the general case is very similar). ¯ f0ρ | WN p = α ¯f 0. b) By deﬁnition, f 0 |U ∗ = f0ρ | WN p WN−1p U WN p = α c) For any function g1 = (U − αI)n g ∈ Kerπα,1 = Im(U − αI)n one has f 0 , g1 = f 0 , (U − αI)n g = (U ∗ − α ¯ I)f 0 , (U − αI)n−1 g = 0 hence for g1 = g − πα,1 (g) we get f 0 , g = f 0 , πα,1 (g) + (g − g α ) = f 0 , πα,1 (g) + f 0 , g1 = f 0 , πα,0 (g)

d) Let us use directly the equality (U ∗ )v f 0 = α ¯ v f 0 of b): αv · f 0 , g α = (U ∗ )v f 0 , U −v πα,1 (g | U v ) = f 0 , πα,1 (g | U v ) = f 0 , g | U v

by c) because g | U v ∈ Mm (N p). e) Note that Lf,α (f0 ) = 1, f0 ∈ Mm (N p; Q); consider the complex vector space

168

4 Admissible measures for standard L–functions

KerLf,α = f 0 ⊥ = {g ∈ Mm (N p; C) | f 0 , g = 0} which admits a Q-rational basis because it is stable by the action of for all ”good” Hecke operators T ∈ L(N pv ): f 0 , g = 0 =⇒ f 0 , g|T = f 0 |T ∗ , g = 0 and one obtains such a basis by the diagonalisation of the action of all the T (a commutative family of normal operators) and e) follows (see also Section 4.4.7 4.5.2 Evaluation of algebraic linear forms Our next task is to compute the value ± −1 Lf,α (Φ± )) = s (χ)) = Lf,α (Φ1 (χ) · Φ2,s (ψχξ0 χ

(4.86)

Lf,α (θ2ξˆ (χ)|V (C)G± (z, (s − k + m + ν)/2; k − (m/2) − ν, ψχξ0 χ, N M )) = (ν)

0

−1 ))|U 2v N p α−2v f 0 , (Φ1 (χ) · Φ± 2,s (ψχξ0 χ f 0 , f0 N p

α−2v f 0 , (θ2ξˆ (χ)|V (C)G± (z, s−k+m+ν ;k − 2 (ν)

0

m 2

− ν, ψχξ0 χ, N M ))|U 2v N p

f 0 , f0 N p

α−2v f0ρ |WN p , (θ2ξˆ (χ)|V (C)G± (z, s−k+m+ν ;k − 2 (ν)

0

m 2

− ν, ψχξ0 χ, N M ))|U 2v N p

f 0 , f0 N p

where χ mod N0 pv is a Dirichlet character of conductor Cχ = pv . Let us unfold the integral (4.21) using the relation (given in Proposition 3.3) of U 2v with the trace operator 2v+1

p g|U 2v = pvm(k−m−1) g|WN p2v+1 TrN Np

Lf,α (Φ± s (χ)) =

WN p :

(4.87)

mk vm(k−m−1)

(−1)

p α2v

2v+1

p −1 f0ρ , (Φ1 (χ) · Φ± ))|WN p2v+1 TrN 2,s (ψχ Np

f 0 , f

N p

0 N p

mk vm(k−m−1)

p α2v (ν) ρ f0 , (θ2ξˆ (χ)|V (C)G± (z, s−k+m+ν ;k − 2

=

(−1)

0

=

m 2

− ν, ψχξ0 χ, N p2v+1 ))|WN p2v+1 N p2v+1

f 0 , f0 N p

(−1)mk pvm(k−m−1) −1 · f0ρ , Φ1 (χ)|WN p2v+1 Φ± )|WN p2v+1 N p2v+1 2,s (ψχξ0 χ α2v f 0 , f0 N p

4.5 Algebraic linear forms on modular forms

169

Let us compute separately (ν)

Φ1 (χ)|WN p2v+1 = θ2ξˆ (χ)|V (C)WN p2v+1

(4.88)

−1 )|WN p2v+1 = Φ± 2,s (ψχξ0 χ

(4.89)

0

and

m s−k+m+ν ;k − − ν, ψχξ0 χ, N M )|WN p2v+1 2 2 using the action of the main involution on theta series, and using the deﬁnitions (4.16) and (4.17) of the normalized Siegel-Eisenstein series. ˜ M Cχ−2 so that the Now let us put N0 M = 4q0 Cχ2 N with N = N M0m−1 following transformation formula holds: G± (z,

θ2ξˆ (χ)|W (N0 M ) = N m(m+2ν)/4 θ2ξˆ (χ)|W (4q0 Cχ2 )V (N ) = (ν)

(ν)

0

0

GCχ (1m , χ)

˜ (N M0m−1 M )m(m+2ν)/4 −1/2

·det(q0

−m(m+2ν)/2 (iCχ )m2 /2 Cχ

(4.90)

·

ξ0 )m/2+ν θ2ξ0 (χ)|V ¯ (N ) (ν)

Recall the deﬁnition of the normalized Siegel-Eisenstein series of the level N0 M = N p2v+1 : G− (z, s) = Γ ((k + 2s − (m/2) + µ)/2)−1 G∗ (z, s), iµ π κ−k−2s G∗ (z, s) = Γ ((1 − k − 2s + (m/2) + µ)/2)

G+ (z, s) =

2iµ Γ (k + 2s − (m/2)) cos(π(k + 2s − (m/2) − µ)/2) − G (z, s). (2π)k+2s−(m/2) where µ = ε((m/2) + k), and G∗ (z, s) = G∗ (z, s; k, χ, N0 M ) = (N0 M )m(k+s) Γ(k, s)LN0 (k + 2s, χ)

(4.91)

[m/2]

LN0 (2k + 4s − 2i, χ2 )E ∗ (N0 M z, s),

i=1

with E ∗ (N0 M z, s) =

−1

E(−(N0 M z)

i

2

π

−k

, s)det(N0 M z)

Γ(k, s) = mk −m(k+1) −m(s+k)

(4.92)

" ×

−km/2

= (N0 M )

Γm (k + s)Γ (s + (k − Γm (k + s),

m 2

E|W (N0 M ),

+ µ)/2), if m is even; otherwise.

4 Admissible measures for standard L–functions

170

and LN (s, χ) =

(1 − χ(q)q −s )−1

qN

is the Dirichlet series associated to the character χ and µ = ε(m/2 + k), with ε(r) ∈ {0, 1} such that ε(r) ≡ r (mod 2) for any integer r ∈ N. Let us introduce the Gamma-factors Γ ± (k, s) by the equality G± (z, s) = G± (z, s; k, χ, N0 M )

(4.93)

= (N0 M )m(k+s) Γ ± (k, s)

[m/2]

LN0 (k + 2s, χ)

LN0 (2k + 4s − 2i, χ2 )E ∗ (N0 M z, s)

i=1

= (N0 M )m((k/2)+s) Γ ± (k, s)

[m/2]

LN0 (k + 2s, χ)

LN0 (2k + 4s − 2i, χ2 )E|W (N0 M ),

i=1

so that the change of variables s := (s − k + m + ν)/2, k := k − (m/2) − ν in (4.93) gives k + 2s := s + (m/2) = (s + 2m)/2 and m s−k+m+ν ± G ;k − − ν, ψχξ0 χ, N0 M |W (N0 M ) z, 2 2 s−k+m+ν m − ν, ) 2 2

= (N0 M )m((s+2m)/4 Γ ± (k −

[m/2]

LN0 ((2s + m)/2, ψχξ0 χ)

LN0 (2s + m − 2i, χ2 ψ 2 )

(4.94)

i=1

E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχξ0 χ, N0 M ). Let us substitute it to (4.46) with χ replaced by the Dirichlet character χ 2a(ξ0 )(detξ0 )−(s+k−1+ν)/2 (4π)−ms Γm ((s + k − 1 + ν)/2) R(s, f, χ)

(4.95)

(ν)

= f0ρ , θ2ξ0 (z; χ)E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχξ0 χ) N M 2 q0 , where a(ξ0 )R(s, f, χ) = 2−1 det ξ0

(s+k−1+ν)/2

(4.96) (ν)

L((s + k − 1 + ν)/2, f0 , θ2ξ0 (z; χ)).

The function R(s, f, χ) is related to D(s, f, χ) by the following equality

(m/2)−1

D(s, f, χ) = L(s + (m/2), ψχξ0 χ)

i=0

L(2s + 2i, ψ 2χ2 ψ 2 )R(s, f, χ) (4.97)

4.5 Algebraic linear forms on modular forms

171

therefore (4.95) takes the form 2a(ξ0 )(detξ0 )−(s+k−1+ν)/2 (4π)−ms × Γm ((s+ k − 1 + ν)/2)D(s, f, χ) = (4.98)

(m/2)−1

L(s + (m/2), ψχξ0 χ)

L(2s + 2i, ψ 2 χ2 ψ 2 )

i=0 (ν)

f0ρ , θ2ξ0 (z; χ)E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχξ0 χ) N M 2 q0 , We see that the integral in (4.87) transforms to the following −1 )|WN p2v+1 N p2v+1 = f0ρ , Φ1 (χ)|WN p2v+1 Φ± 2,s (ψχ

(4.99)

(ν)

f0ρ , θ2ξˆ (χ)|V (C)WN0 M 0

m s−k+m+ν ;k − − ν, ψχξ0 χ, N0 M )|WN0 M N0 M 2 2 and (4.99) becomes G± (z,

−1 f0ρ , Φ1 (χ)|WN p2v+1 Φ± )|WN p2v+1 N p2v+1 = 2,s (ψχξ0 χ ˜ M )m(m+2ν)/4 (N M0m−1

GCχ (1m , χ) −m(m+2ν)/2 (iCχ )m2 /2 Cχ

(N0 M )m((s+2m)/4 Γ ± (k −

−1/2

det(q0

(4.100) ξ0 )m/2+ν

s−k+m+ν m − ν, ) 2 2

[m/2]

LN0 ((2s + m)/2, ψχξ0 χ)

LN0 (2s + m − 2i, χ2 ψ 2 )

i=1

m s−k+m+ν ;k − − ν, ψχξ0 χ, N0 M ) N0 M 2 2 Let us now compare the value of the linear form (4.87) expliciely given by (4.100), with the integral representation (4.98) for the standard L-function: f0ρ , θ2ξ0 (χ)|V ¯ (N )E(z, (ν)

Lf,α (Φ± s (χ)) = mk vm(k−m−1)

(4.101)

=

(−1) p α2v f 0 , f0 N p

=

(−1)mk pvm(k−m−1) × α2v f 0 , f0 N p

−1 · f0ρ , Φ1 (χ)|WN p2v+1 Φ± )|WN p2v+1 N p2v+1 2,s (ψχξ0 χ

˜ ×(N M0m−1 M )m(m+2ν)/4

GCχ (1m , χ)

−1/2

det(q0 −m(m+2ν)/2 (iCχ )m2 /2 Cχ s−k+m+ν m − ν, )× ×(N0 M )m((s+2m)/4 Γ ± (k − 2 2

ξ0 )m/2+ν ×

4 Admissible measures for standard L–functions

172

[m/2]

×LN0 ((2s + m)/2, ψχξ0 χ)

LN0 (2s + m − 2i, χ2 ψ 2 )

i=1

m s−k+m+ν ;k − − ν, ψχξ0 χ, N0 M ) N0 M 2 2 Let us remove from the Euler products D(s, f, χ) and L(s, χ) all the factors corresponding to the prime divisors of N0 : f0ρ , θ2ξ0 (χ)|V ¯ (N )E(z, (ν)

2a(ξ0 )(detξ0 )−(s+k−1+ν)/2 (4π)−ms × Γm ((s + k − 1 + ν)/2)DN0 (s, f, χ) =

(m/2)−1

LN0 (s + (m/2), ψχξ0 χ)

LN0 (2s + 2i, ψ 2 χ2 ψ 2 )×

(4.102)

i=0 (ν)

× f0ρ , θ2ξ0 (z; χ)E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχξ0 χ) N M 2 q0 . Let χ be a non-trivial Dirichlet character modulo N pv of conductor Cχ such that Cχ |pv ). Consider fhe complex valued distributions Lf,α (Φ± s (χ)) deﬁned for the critical pairs (s, χ) as in Theorem 4.13 by ± −1 Lf,α (Φ± )). s (χ)) = Lf,α (Φ1 (χ) · Φ2,s (ψχξ0 χ

Proposition 4.21. Assume that the cusp eigenform f ∈ Sm k (C, ψ) satisﬁes the condition a(ξ0 ) = 1 for some ξ0 ∈ Am , ξ0 > 0. Then a) For all integers s satisfying 1 ≤ s ≤ k − δ − m and s = 1 if the character χ2 ψ 2 is trivial we have that the value Lf,α (Φ+ s (χ)) ∈ K = Q(f, λf , ψ, χ), where K = Q(f, λf , ψ, χ) denotes the ﬁeld generated by Fourier coeﬃcients of f , by the eigenvalues λf (X) of Hecke operators X on f , and by the values of the characters χ and ψ. Moreover, Lf,α (Φ+ s (χ)) = 0 for s ≡ δ(mod2). b) For all integers s satisfying 1 − k + δ + m ≤ s ≤ 0 we have that Lf,α (Φ− s (χ)) ∈ K, and

Lf,α (Φ− s (χ)) = 0 for s ≡ δ(mod2), s ∈ Z, s ≤ 0.

c) Suppose moreover that N is divisible by all of the prime divisors of 2det(2ξ0 ) then there is the following equality Lf,α (Φ± s (χ)) = t±

GCχ (1m , χ) Cχm(2s+2k−2−m)/4 ± D (s, f, χ), ¯ α0 (Cχ )2 (iCχ )m2 /2

4.5 Algebraic linear forms on modular forms

173

×

where t± is a multiplicative constant in Q , independent of the critical pair (s, χ), ¯ denotes the normalized standard L-function as in Theorem 4.13, D± (s, f, χ) χ(detξ )em (tξξ /Cχ ), GCχ (1m , χ) = ξ ∈Mm (Z)

(mod Cχ )

is the Gauss sum of degree m of χ. Proof of Proposition 4.21 It is directly implied by the equalities (4.101) and (4.102) taking into account Theorem 4.13 on the critical calues of the standard L-function.

4.6 Congruences between nearly holomorphic modular forms and proof of the Main theorem. 4.6.1 Regularized distributions in Siegel modular forms. We now outline the method of the proof of Main Theorem and show the existence of the h–admissible measures attached to the standard zeta functions D(s, f, χ). This proof is based on congruences for the Fourier coeﬃcients of nearly holomorphic Siegel modular forms. Consider again the proﬁnite group Yv , where Yv = (Z/N pv Z)× . Y = lim ← v

and consider fhe complex valued distributions Lf,α (Φ± s (χ)) deﬁned for the critical pairs (s, χ) as in Theorem 4.13 and Proposition 4.21 by ± −1 )). Lf,α (Φ± s (χ)) = Lf,α (Φ1 (χ) · Φ2,s (ψχξ0 χ

(4.103)

The proof is based on a regularization of the distributions Lf,α (Φ+ s (χ)) (for 1 ≤ s ≤ k − ν − m) and Lf,α (Φ− (χ)) (for 1 − k + ν + m ≤ s ≤ 0) (with values s in Q). Recall that by Proposition 4.20 the linear form Lf,α factorizes through the canonical projection πα : ± Lf,α (Φ± s (χ)) = Lf,α (πα (Φs (χ))). ˜ . Consider a positive integer c such that (c, N0 ) = 1 and N0 = 4q0 N M0m−1 One supposes always that m is even and that k > 2m + 2. Hence there exist c− regularized distributions Φc+ s , Φs on Y with values in the module ) m v Mm Mm r,k (Q) = r,k (Γ0 (N p ), ψ; Q) : v≥1

of arithmetical nearly holomorphic Siegel modular forms. These regularized distributions are uniquely determined by their values as follows: s Φc+ ¯ −1 1 − (χψ) (4.104) ¯ 2 (c)c−2s Φ+ s (χ) = Cχψ ¯ G(χψ) s (χ) ¯ 0 (q)q s−1 )(1 − (χψ) (1 − (χψ) ¯ 0 (q)q −s )−1 , q|N0

¯ 2 (c)c2(s−1) Φ− (χ). (χ) = 1 − ( χ ¯ ψ) Φc− s s ¯

Next consider the Cp -linear forms Dc± : Ck−m (Y ; Cp ) → Cp deﬁned on the local monomials xjp for j = 0, 1, · · · , k − m − 1 by: xjp Dc+ = Lf,α (Φc+ ) = Lf,α (πα (Φc+ (4.105) j+1 j+1 )), a+(M)

a+(M)

a+(M)

4.6 Congruences and proof of the Main theorem

xjp Dc− = a+(M)

175

a+(M)

Lf,α (Φc− −j ) =

a+(M)

Lf,α (πα (Φc− −j )).

k−m c± It follows that there exist the Cp -linear maps Φ (Y ; Cp ) → Mm α :C r,k (Cp ) deﬁned on the local monomials xjp for j = 0, 1, · · · , k − m − 1 by: j c± xp Φα = πα (Φc+ (4.106) j+1 )), a+(M)

a+(M)

c± xjp Φ α =

a+(M)

a+(M)

πα (Φc− −j ).

In order to prove that Cp -linear forms Dc± : Ck−m (Y ; Cp ) → Cp deﬁne in fact certain admissible measures we observe that c± ) Dc± = Lf,α (Φ α c± where the linear forms Φ α depend on the choice of α (but not on the choice of f ), and in fact they take values in a ﬁnite dimensional Cp -vector space (the image of the ﬁnite dimensional projection: m α m m πα : Mm r,k (N, p; C) → Mr,k (N, p; C) ⊂ Mr,k (Γ1 (N p); C),

see Proposition 4.20 (d)) hence it suﬀuces to prove the admissibility of the c± linear maps Φ α with values in spaces of arithmetical nearly holomorphic Siegel modular forms. For this purpose we use a general result on admissible measures with values in nearly holomorphic Siegel modular forms given in Theorem 4.8. We have for the positive critical values r r c+ (x − a)r dDc+ = dDj+1 (4.107) (−a)r−j j a+(M) a+(M) j=0 r r 1 c+ (−a)r−j χ−1 (a)Dj+1 (χ) j ϕ(M ) j=0 χmodM r r 1 = χ−1 (a)Lf,α (Φc+ (−a)r−j j+1 (χ)). j ϕ(M ) j=0

=

χmodM

similarly for D

c−

we have

(x − a)r dDc− = a+(M)

=

r r j=0

j

r r c− dD−j (−a)r−j j a+(M) j=0

(−a)r−j

1 c− χ−1 (a)D−j (χ) ϕ(M ) χmodM

r r 1 = χ−1 (a)Lf,α (Φc− (−a)r−j −j (χ)). j ϕ(M ) j=0 χmodM

(4.108)

4 Admissible measures for standard L–functions

176

4.6.2 Suﬃcient conditions for admissibility of measures with values in nearly holomorphic Siegel modular forms. In order to prove the admissibility property Dc± of we need to verify the following growth condition for Dc : r c± p α0 (p) , (4.109) sup (x − ap ) dD = o |M |r−2ord p a∈Y a+(m) p

for r = 0, 1, · · · , k − m − 1, which will be implied by r−2ordp α0 (p) c± , sup (x − ap )r dΦ α = o |M |p a∈Y a+(m)

(4.110)

p

Let us use a general result giving a suﬃcient condition for the admissibility of measures with values in nearly holomorphic Siegel modular forms (given in Theorem 4.8): with κ = 2, h = [2ordp α0 (p)] + 1. Then we need to prove that j 2v j ≤ Cp−vj U (4.111) (−a0p )j−j Φc+ 1+j (a)v ) j j =0 p

and

j 2v j 0 j−j c− U ≤ Cp−vj (a) ) Φ ) (−a v p −j j j =0

(4.112)

p

for all j = 0, 1, · · · , k − m − 1. 4.6.3 Fourier expansions of distributions with values in nearly holomorphic Siegel modular forms. Let us use that fact that (r + )

δm/2+s G+ ¯ h0 ) = G+ ¯ h0 ), k (z, s , χψχ m/2+s (z, 0, χψχ

(4.113)

(r − )

δm/2+1−s G− ¯ h0 ) = G− ¯ h0 ), k (z, s , χψχ m/2+1−s (z, s − 1/2, χψχ

where k = k−m/2−ν, s = (s−k+m+ν)/2, r+ = r− = (k−m−ν−j−1)/2 ∈ N and if j + 1 ≡ δ (mod 2) (otherwise the series vanishes). The Fourier expansions of these series are given by ¯ ξ0 ) = b+ (ξ)em (ξz), (4.114) G+ m/2+s (z, 0, χψχ ξ∈Cm

4.6 Congruences and proof of the Main theorem

G− ¯ψχξ0 ) = m/2+1−s (z, s − 1/2, χ

177

b− (ξ)em (ξz),

ξ∈Cm

where b+ (ξ) = 2−mκ det(ξ)m/2+s−κ L+ N1 (s, ωχξ )M (ξ, ω, m/2 + s),

(4.115)

b− (ξ) = 2−mκ L− N1 (s, ωχξ )M (ξ, ω, m/2 + s), ¯ is a Dirichlet character, ω = ψχξ0 χ M (ξ, ω, m/2 + s) =

Mq (ξ, ω(q)q −m/2−s ),

q∈P (ξ)

is a ﬁnite Euler product such that Mq (ξ, t) ∈ Z[t]. The normalized Dirichlet L-functions are given by δ (1−2s)/2 L+ N1 (s, ωχξ ) = i π

Γ (s + δ)/2) LN (s, ωχξ ), Γ ((1 − s + δ)/2 1

(4.116)

L− N1 (s, ωχξ ) = LN1 (s, ωχξ ). Let us use these Fourier expansions in order to compute the following nearly holmorphic Fourier expansion ± Φ± ¯ s (χ) = Φ1 (χ)Φ2,s (ψχξ0 χ) ± ± (ω) = v ± (R, ξ, s, χ)em (ξz), Φ1 (χ)δ (r ) Ψ2,s

Φ± s (χ)|U (M ) =

=

ξ∈Cm

v ± (M R, M ξ, s, χ)em (ξz)

(4.117)

ξ∈Cm

d± (s, M R, ξ1 , ξ2 )em (ξz)

ξ∈Cm N ξˆ0 [ξ1 ]+ξ2 =M ξ

where M = p2v , d+ (s, M R, ξ1 , ξ2 ) = (q0 N )−m(s−1+k−m−ν)/2 det(ξ0 )(s−1+k−m−ν)/2 (4.118) χ(det(ξ1 ))det(ξ1 )ν det(ξ2 )s−1/2 L+ N1 (s, ωχξ2 ) M (ξ2 , ω, s + m/2)Q(M R, ξ2 ; r+ , β + ), under the condition s ≡ δ (mod 2) (otherwise d+ (s, M Rξ 1 , ξ2 ) = 0). Recall the notation ω = χψχ ¯ ξ0 , β + = (1−s−k+ν +m)/2 and r+ = (k−ν −m−s)/2. − The coeﬃcients d (s, M R, ξ1 , ξ2 ) are given by d− (s, M R, ξ1 , ξ2 ) = (q0 N )−m(s−1+k−m−ν)/2 det(ξ0 )(s−1+k−m−ν)/2 (4.119) χM (det(ξ1 ))det(ξ1 )ν L− N1 (s, ωχξ2 ) M (ξ2 , ω, s + m/2)Q(M R, ξ2 ; r− , β − ), under the condition s ≡ δ (mod 2) (otherwise d− (s, ξ1 , ξ2 ) = 0. Recall also that β − = (s − k + ν + m)/2 and that r− = (k − ν − m + s − 1)/2.

4 Admissible measures for standard L–functions

178

4.6.4 Fourier expansions of regularized distributions. Let us apply the above formulas for the normalized distributions: Φc± v c± (M R, M ξ, s, χ)em (ξz) (4.120) s (χ)|U (M ) = ξ∈Cm

=

dc± (s, M R, ξ1 , ξ2 )em (ξz),

ξ∈Cm N ξˆ0 [ξ1 ]+ξ2 =M ξ

2v

where M = p , dc+ (s, M R, ξ1 , ξ2 ) = (q0 N )−m(s−1+k−m−ν)/2 det(ξ0 )(s−1+k−m−ν)/2 (4.121) χ(det(ξ1 ))det(ξ1 )ν det(ξ2 )s−1/2 Q(M R, ξ2 ; r+ , β + ) M (ξ2 , χψχ ¯ ξ0 , s + m/2)L+ ¯ ξ0 χξ2 ) N1 (s, χψχ   s s−1 ¯ Cχψ (1 − (χψ)0 (q)q )  ¯ (1 − (χψ) ¯ 2 (c)c−2s )  . G(χψ) ¯ (1 − (χψ) ¯ 0 (q)q −s ) q|N0

c−

The coeﬃcients d c−

d

(s, M R, ξ1 , ξ2 ) are given by

(s, M R, ξ1 , ξ2 ) = (q0 N )−m(s−1+k−m−ν)/2 det(ξ0 )(s−1+k−m−ν)/2 (4.122) ¯ 2 (c)c−2s )L− χM (det(ξ1 ))det(ξ1 )ν (1 − (χψ) N1 (s, ωχξ2 ) M (ξ2 , ω, s + m/2)Q(M R, ξ2 ; r− , β − )

under the condition s ≡ δ (mod 2) (otherwise d+ (s, M Rξ 1 , ξ2 ) = 0). Recall the notation ω = χψχ ¯ ξ0 , β + = (1−s−k+ν +m)/2 and r+ = (k−ν −m−s)/2, − β = (s − k + ν + m)/2 and that r− = (k − ν − m + s − 1)/2. 4.6.5 Main congruences for the Fourier expansions of regularized distributions. Let us use the orthogonality relations for Dirichlet characters in order to prove (4.111) and (4.112) using the Fourier expansions (4.120): j j 1 0 j−j −1 c+ χ (a)v (M R, M ξ, 1 + j , χ) (−ap ) j ϕ(M ) j =0

≤ Cp and

χmodM

−vj

p

(4.123)

j j 1 0 j−j −1 c− χ (a)v (M R, M ξ, −j , χ) (−ap ) ϕ(M ) j =0 j χmodM ≤ Cp−vj

for all j = 0, 1, · · · , k − m − 1, M = pv , M = p2v .

p

(4.124)

4.6 Congruences and proof of the Main theorem

179

4.6.6 Kummer congruences and Mazur’s measure. Let S be a ﬁnite set of primes containing p and all the prime divisors of N . Consider the Cp –adic analytic Lie groups × XS = Homcontin (Z× S , Cp ),

where

X = XN = Homcontin (Y, C× p)

(Z/QZ)× Z× S = lim ←− Q

Z× S

The projection homomorphism → Y induces the analytic embedding X → XS . Recall the notion of the h–admissible measures on Z× S and properties of their Mellin transform. These Mellin transform are certain p–adic analytic functions on the Cp –analytic Lie group XS . The canonical Cp –analytic structure on XS is obtained by shifts from the obvious Cp –analytic structure on the subgroup × Homcontin (Z× p , Cp ) ⊂ XS .

We view the elements of ﬁnite order χ ∈ XStors as Dirichlet characters whose conductor C(χ) contain only primes in S, by means of the decomposition × i

∞ × χ : Z× S → Q −→ C ,

where i∞ is the ﬁxed embedding. The characters χ ∈ XStors form a discrete subgroup XStors ⊂ XS . We use also the following natural homomorphism × × xp : Z× S → Zp → Cp ,

xp ∈ XS .

so that all integers k ∈ Z may be viewed as characters of the type xkp : y → y k . Recall that a p–adic measure on Z× S may be viewed as a bounded Cp –linear form µ on the space C(Z× ) of all continuous Cp –valued functions S ϕ → µ(ϕ) = dµ ∈ Cp , ϕ ∈ C(Z× S ), Z× S

which is uniquely determined by its restriction to the subspace C1 (Z× S ) of locally constant functions. We denote by µ(a + (Q)) the value of µ on the characteristic function of the set & % × a + (Q) = x ∈ Z× S | x ≡ a mod Q ⊂ ZS . The Mellin transform Lµ of µ is a bounded analytic function Lµ : XS → Cp , Lµ (χ) = dµ ∈ Cp , χ ∈ XS , Z× S

on XS , which is uniquely determined by its values Lµ (χ) for the characters χ ∈ XStors .

4 Admissible measures for standard L–functions

180

Non–Archimedean integration and admissible measures A more delicate notion of an h–admissible measure was introduced by Amice–Velu and Viˇsik (see [Am-Ve], [Vi1]). Let Ch (Z× S ) denote the space of Cp –valued functions which can be locally represented by polynomials of degree less than a natural number h of the variable xp ∈ XS introduced above. Deﬁnition 4.22. A Cp –linear form µ : Ch (Z× S ) → Cp is called h– admissible measure if for all a ∈ Z× S and for all r = 0, 1, · · · , h − 1 the following growth condition is satisﬁed | sup (xp − ap )r dµ |= o(| Q |r−h ) p a∈Z× S

a+(Q)

We know (essentially due to Amice–Velu and Viˇsik) that each h–admissible measure can be uniquely extended to a linear form on the Cp –space of all locally analytic functions so that one can associate to its Mellin transform Lµ : XS → Cp , Lµ (χ) = χdµ ∈ Cp , χ ∈ XS , Z× S

which is a Cp –analytic function on XS of the type o(log xhp ). Moreover, the measure µ is uniquely determined by the special values of the type Lµ (χxrp ) (χ ∈ XStors , r = 0, · · · , h − 1). Let ω mod such that (A, M0 ) = 1 $ A be a ﬁxed primitive Dirichlet character $ with M0 = q∈S q. Put S = S ∪ S(A), M = q∈S q. Then for any positive integer c with (c, M ) = 1, c > 1 there exist Cp -measures µ+ (c, ω), µ− (c, ω) on Z× S which are uniquely determined by the following conditions: for s ∈ Z, s > 0 Cωχ¯ −1 s + × χ xp dµ (c, ω) = (1 − χω(c)c−s ) ip G(ω χ) ¯ Z× S % & (1 − χω(q)q s−1 )/(1 − χω(q)q −s ) L+ × (4.125) M0 (s, χω), q∈S\S(χ)

and for s ∈ Z, s ≤ 0 i−1 p

Z× S

χ xsp

−

dµ (c, ω)

= (1 − χω(c)cs−1 )L− M0 (s, χ, ω),

(4.126)

2iδ Γ (s) cos(π(s − δ)/2) , (2π)s

(4.127)

where ¯ L+ M0 (s, χω) = LM (s, χω)

4.6 Congruences and proof of the Main theorem

181

L+ ¯ M0 (s, χω) = LM (s, χω) are the normalized Dirichlet L-functions with δ = 0, 1, (−1)δ = χω(−1). The functions (4.125) and (4.126) satisfy the following functional equation LM0 (1 − s, χω) = % & (1 − χω(q)q s−1 )/(1 − χω(q)q −s ) L+ M0 (s, χω).

(4.128)

q∈S\S(χ)

the Recall that by the deﬁnition of the S-adic Mazur measure µc on Z× S distributions (4.125) and (4.126) are given by − x dµ (c, ω) = x−1 ω dµc , Z× S

Z× S

Z× S

dµ+ (c, ω) =

Z×

−1 x x−1 dµc , p ω

S

where x ∈ XS , and XS is regarded as a subgroup of XS . 4.6.7 Reduction of the Main congruence to congruences for partial sums. Recall from (4.2.4) that (r) c(ξ)Q(R, ξ)q ξ ∈ Q(f )[[q Bm ]][Ri,j ]i,j=1,··· ,m , δk f (z) = ξ∈Bm

where Q(f ) is the subﬁeld of C generated by the Fourier coeﬃcients of f , L runs over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m as above, r r Q(R, ξ) = Q(R, ξ; k, r) = RL (κ − k − r)QL (R, ξ), det(ξ)r−t t t=0 |L|≤mt−t

QL (R, ξ) = tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) . Hence combining all the above remarks we deduce that the coeﬃcients ip (v c± (M R, M ξ, s, χM )) are linear combinations with coeﬃcients independent of M , M and s of the following expressions: B+ = χM (det(ξ1 ))det(ξ1 )ν det(ξ2 )s−1/2 det(ξ2 )r

(4.129) +

r+ + t=0

r t

|L|≥t

cL (β + )

182

4 Admissible measures for standard L–functions

r+ +s−1/2 χψ(τ ¯ ) QL (M R, ξ2 ) (q0 N )−m det(ξ0 ) s−1/2 τ

Z× S

r − r

χxsp dµc+ (ω)

−

B− = χ ¯M (det(ξ1 ))det(ξ1 )ν det(ξ2 )r

−

t

t=0

r− QL (M R, ξ2 ) (q0 N )−m det(ξ0 )

cL (β − )

(4.130)

|L|≥t

χx ¯ sp dµc− (ω),

Z× S

with ω = ψχξ0 χξ2 and det(2ξ2 )det(2ξ0 ) = τ a2 for a, τ √∈ Z such that τ is square free. Hence Cχω = τ Cχψ and G(χω) ¯ = χψ(τ ¯ ) τ G(χψ) ¯ and us¯ ¯ ing N ξˆ0 [ξ1 ] + ξ2 = M ξ we show the following congruence det(ξ2 )det(ξ0 ) ≡ (q0 N )m det(ξ1 )2 (mod M ). Moreover m is even, and one deduces then that det(ξ2 )det(ξ0 ) is a square modulo q for all q|N , then we simply obtain (6.28), (6.29) from (6.23) and (6.24). Concider M and M large enough such that M 2 |M and q0m |M and recall that N ∈ Z× S hence on can ﬁnd det(ξ2 )(q0 N )−m det(ξ0 ) ≡ det(ξ1 )2

(mod M ).

Thus the coeﬃcients χM (det(ξ1 )) do not vanish only if det(ξ1 ) ∈ Z× S . We see then that det(ξ2 )(q0 N )−m det(ξ0 ) is a square in Z× and we put S 0 λ = λ+ = τ −1 det(ξ2 )(q0 N )−m det(ξ0 ) ∈ Z× S in the case +, 0 λ = λ− = det(ξ2 )(q0 N )−m det(ξ0 ) ∈ Z× S in the case −. It follows that the expressions B + and B − become B + = ψ(τ )τ k−m−ν−1/2 det(ξ1 )ν χ(λ)λs−1+k−m−ν

cL (β + )QL (M R, ξ2 )

ν

s−1+k−m−ν

B = det(ξ1 ) χ(λ)λ ¯

QL (M R, ξ2 )

Z× S

Z× S

(4.131)

cL (β − )

(4.132)

χxsp dµc+ (ω),

[(k−m−1)/2] − t=0

r t

t=0

|L|≥t −

[(k−m−1)/2] +

r t

|L|≥t

χx ¯ sp dµc− (ω).

In this way the number A± is a linear combination with coeﬃcients independent on M , M and j of type A+ =

(4.133)

t=0

|L|≥t

[(k−m−1)/2]

det(ξ1 )ν λk−m−ν−1−l

QL (M R, ξ2 )

l ± rj j=0

t

4.6 Congruences and proof of the Main theorem

cL (βj± )lj(−aλ−1 )l−j

Z× S

183

1 χ(xa−1 )xj+1 dµc± (ω). ϕ(M ) χmodM

Note that cL (βj+ ) is a polynomial of degree |L| in βj+ hence of degree |L| in j + because βj+ = (−j − k + ν + m)/2 and rtj is a polynomial of degree t in rj+ thus of degree tin j in view of rj+ = (k − m − ν − j − 1)/2. One can therefore write

+ t+|L| (j + n + 1)! rj . µn cL (βj+ ) = t (j + 1)! n=0

Here the coeﬃcients µn are certain ﬁxed rational numbers. Using again the orthogonality relations we see that the sum over j, denoted by C + , takes the form

t+|L|

C+ =

x≡aλ−1 modM n=0

µn

l l (j + n + 1)! j+1 + x dµ (c, .) (−aλ−1 )l−j j (j + 1)! j=0 9 :; < n −n ∂ n+1 −1 l (x − aλ ) x x ∂xn

Therefore (x − aλ−1 )l ≡ 0 (mod M l ) giving the congruence 2νm,k C + ≡ 0 (mod M l−n ) ≡ 0

(mod M l−t−|L| ).

(4.134)

4.6.8 Proof of the Main congruence. Now the expression (6.31) transforms to

t=0

|L|≥t

[(k−m−1)/2]

A+ = det(ξ1 )ν λk−m−ν−1+l

QL (M R, ξ2 )C + ,

(4.135)

where QL (M R, ξ2 ) is a homogeneuos polynomial of degree |L| in the variables M Rij implying the congruence QL (M R, ξ2 ) ≡ 0 We deduce that

(mod M |L| ) ≡ 0 (mod M 2|L| ).

. A+ = o |M |2|L|+l−t−|L| p

(4.136) (4.137)

On the other hand we know from the description of the polynomial r r Q(R, ξ) = Q(R, ξ; k, r) = RL (κ − k − r)QL (R, ξ)), det(ξ)r−t t t=0 |L|≤mt−t

QL (R, ξ) = tr tρm−l1 (R)ρl1 (ξ) · . . . · tr tρm−lt (R)ρlt (ξ) )

4 Admissible measures for standard L–functions

184

that |L| ≥ t so we obtain ﬁnally the congruence A+ = o |M |lp .

(4.138)

This shows that Dc+ is an h-admissible measure by the suﬃcient condition (4.111) (implying (4.109)). ¯ Thus In the case Dc− the proof is similar but one needs to replace χ by χ. − A is also a linear combination with coeﬃcients independent of M , M and j of type

[(k−m−1)/2] −

ν k−m−ν−1

A = det(ξ1 ) λ

t=0

l (−aλ)l−j cL (βj− ) j Z× S(N

0)

QL (M R, ξ2 )

|L|≥t

l − r j

j=0

t

(4.139)

1 −j χ(aλx)x ¯ dµ− (c). ϕ(M ) χmodM

Let us use again the orthogonality for the Dirichlet characters and the fact − that rtj cL (βj− ) is a polynomial of degree t + |L| en j. One represents this polynomial in the form − t+|L| rj (j + 1 + n)! . µn cL (βj− ) = t (j + 1)! n=0

(4.140)

One obtains the following expression for the sum C − over j in the formula giving A− : l l (j + 1 + n)! −j − x dµ (c, .), C = µn (−aλ)l−j j (j + 1)! x−1 ≡aλmodM n=0 j=0 9 :; < n ∂ (aλ − x1 )l xn1 n xn+1 ∂x1 1 (4.141) where x1 = x−1 . Then we use the same argument as for A+ in (4.138) in order to prove that A− = o |M |lp = o |M |l−h . (4.142) p −

t+|L|

It follows from (4.142) that the linear form Dc− is an admissible measure. Theorem 4.23. Let f ∈ Sm k (N, ψ) be a Siegel cusp eigenform of even degree m and of weight k > 2m + 2 satisfying the nonvanishing condition (*) of Main Theorem. For a positive integer c > 1, (c, N p) = 1 there exist two Cp -analytic functions Lc± : XS → Cp such that: (i) For all pairs (s, χ) with s ∈ Z, 0 ≤ s ≤ k − m − 2 and a primitive non-trivial Dirichlet character χ ∈ XStors of conductor Cχ = pv one has

4.6 Congruences and proof of the Main theorem m(2s+2k−m)/4

• Lc+ (χxsp ) =

GCχ (1m , χ)Cχ (iCχ )m2 /2 α0 (Cχ2 ) q|N

s+1 Cχψ ¯

G(χψ) ¯

185

(1 − (χψ) ¯ 2 (c)c−2s−2 )

¯ 0 (q)q s ) D+ (s + 1, f, χ) (1 − (χψ) ¯ . −s−1 (1 − (χψ) ¯ 0 (q)q ) f, f N m(2−2s+2k−m)/4

• Lc− (χxsp ) =

¯ χ GCχ (1m , χ)C (iCχ )m2 /2 α0 (Cχ2 )

¯ 2 (c)c−2s−2 ) (1 − (χ ¯ψ)

D− (−s, f, χ) . f, f N ¯ viewed as elements of the Tate ﬁeld Cp These values are algebraic numbers Q ¯ → Cp ). On the other hand the Cp -analytic (via the ﬁxed embedding ip : Q functions vanish at the following points Lc+ (χxsp ) = Lc− (χxsp ) = 0 if s ≡ 0 (mod 2). (ii) If ordp (α0 (p)) = 0 (i.e. f is p-ordinary), then the holomorphic functions Lc+ et Lc− are bounded Cp -analytic functions. (iii) In the general case the functions Lc+ and Lc− are of type o(log(xhp )) where h = [2ordp (α0 (p))] + 1 and they are Mellin transforms of certain hadmissible measures. (iv) If h ≤ k − m − 1, then the functions Lc± are uniquely determined by (i). 4.6.9 Proof of Theorem 4.23 The assertion (ii) (i.e. the case ordp (α0 (p)) = 0) which was proved in [Pa6], also follows easily from the Main congruence (4.111) and (4.112) taking into account the description of the canonical projection operator in Proposition c± ) let us develop the deﬁnition of Φ α using the 3.3: for α = α0 (p), Φα = πα (Φ binomial formula:

α = (yp − ap )j dΦ

(a)v

j j −vκ · (−ap )j−j Φα j ((a)v ) = α j

j =0



αvκ

 j j · U −vκ πα,1 U κv (−ap )j−j Φj (a)v  j j =0

First we note that all the operators vκ

α

·U

−vκ

−1

= α

U

−vκ

−vκ −vκ n−1 t −1 = I +α Z = α−1 Z t t=0

186

4 Admissible measures for standard L–functions

are uniformely bounded for v → ∞ by a positive constant C1 (where U = αI + Z and Z n = 0 for n = rkA Mr∗ (N p; A)) because n does not depend on v −vκ and the binomial coeﬃcients are all Zp -integers. t On the other hand by the Main congruences (4.111) and (4.112) with κ = 2 c− (for the distributions Φj = Φc+ j+1 or Φj = Φ−j ) j κv j j−j U ( Φj ((a)v )) (−ap ) j j =0

≤ Cp−vj p,Mr∗

for all j = 0, 1, . . . , k−m−1. If we apply to this estimate the previous bounded operators πα,1 U κv we get the following inequality j α (yp − ap ) dΦ ≤ C · C1 |α−vκ |p · p−vj = CC1 p−vj (a)v p,Mr∗

α = πα (Φ0 ) is a because |α−vκ |p = 1. In particular, for j = 0 it implies that Φ bounded measure, and for all j = 0, 1, . . . , k − m − 1 we have moreover that j α dπα (Φj ) = yp dπα (Φ0 ) = ypj dΦ (a)v

(a)v

(a)v

Remark 4.24. We see in this case that all distributions πα (Φj ) are bounded but we do not claim that the distributions Φj are bounded themselves. In the general case h > k − m − 1 the integer j run over {0, · · · , h − 1} and one can extend the values of our functions Lc± by the equality Lc± (χxsp ) = 0 (for all χ ∈ XStors of a conductor divisible by all the prime divisors of N p) for j > k − m − 1 − ν (but keeping the same values for 0 ≤ s ≤ k − m − 1 − ν). ± These are the factors rt which vanish in the above formulas (4.131) for the distributions B ± and the veriﬁcation of the h-admissibility goes without change in this situation. Also, one obtains again the h-admissible measures Dc± with h = [2ordp (α0 (p))] + 1. The functions Lc± coincide therefore with the Mellin transforms of these h-admissible measures. Finally, if h ≤ k − m − 1 then the conditions in (i) uniquely determine the analytic functions Lc± of type o(log(xhp )) by their values following a general property of admissible measures (see [Am-Ve], [Vi1]). In the case h > k − m − 1, there exist many analytic functions Lc± verifying the conditions in (i) which depend on a choice of analytic continuation (interpolation) for the values Lc± (χxjp ) if j > k − m− 1 − ν but one shows in the theorem 4.23 that there exists at least one such continuation (for example the one which was described in the proof of (iii)).

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[Sho] [Si1] [Si2] [St1] [St2]

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Index

S-adic Mazur measure, 34 C ∞ -modular form, 83 h-admissible measure, 9, 37, 129, 132, 144, 185 analytic function, 18 arithmetical Siegel modular forms, 174 Atkin-Lehner operator, 7 canonical projection operator, 7, 130, 146 compatibility criterion, 22 completed group ring, 32 conﬂuent hypergeometric function, 45, 74, 77 convolution of distributions, 32 critical values, 75 equivalent symmetric couples, 65 Fourier expansions, 106, 110, 136 Frobenius element, 58 Frobenius operator, 7 function of bounded growth, 83 function of moderate growth, 84 Gauss sum of degree m, 6, 132, 149 Gauss-Manin connection, 7, 96 growth condition, 37, 146 Hecke Hecke Hecke Hecke

algebra, 53 couple, 56 operators, 7, 53, 95, 127 polynomial, 53, 54

holomorphic projection operator, 45 inertial degree, 14 involuted Siegel-Eisenstein series, 136 Iwasawa algebra, 32 Klingen-Eisenstein series, 71 locally analytic functions, 129, 144 locally constant functions, 9, 129 locally polynomial functions, 9, 129, 144 Maass diﬀerential operator, 75, 96, 106, 133 Mahler’s criterion, 18 measures of bounded growth, 37 Mellin transform, 6, 28, 37, 54 nearly holomorphic Siegel modular form, 7, 95, 96, 99, 102, 110, 127, 135, 144 Newton polygon, 20 non–vanishing condition, 131, 132 non-Archimedean Mellin transform, 31 non-Archimedean zeta function of Kubota-Leopoldt, 34 nonvanishing condition, 184 normaized Siegel-Eisenstein distributions, 141 normalized Siegel-Eisenstein distribution, 165 normalized Siegel-Eisenstein series, 136, 169 normalized standard L-function, 173 notation of Petersson, 100, 115, 130

196

Index

Ramanujan-Petersson conjecture, 54 ramiﬁcation index, 14 Rankin zeta function, 67, 149 Saito-Kurokawa lifting, 54 Satake q-parameters, 53, 131, 148 Satake isomorphism, 51, 52, 57, 115 sequence of distributions, 144 Shimura diﬀerential operator, 140 Shimura diﬀerential operator, 106, 110, 115, 116, 128, 133, 140, 141 Siegel cusp eigenform, 132 Siegel modular forms, 45 Siegel-Eisenstein series, 66, 134, 135 spinor representation, 12 spinor zeta function, 12, 53, 60

standard L-function, 131, 148 standard representation, 12 standard zeta function, 54 sum of eigenvalues, 74 supersingular case, 37 symmetric couple, 65 tame component, 30 Teichm¨ uller representative, 14 totally ramiﬁed extension, 15 trace operator, 7 twisted convolution, 165 unramiﬁed extension, 14 wild component, 30