Automorphic Forms, Representation Theory and Arithmetic

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Automorphic Forms, Representation Theory and Arithmetic

Table of contents :
ON SHIMURA'S CORRESPONDENCE FOR MODULAR FORMS OF HALF-INTEGRAL WEIGHT*......Page 11
The Metaplectic Group......Page 14
Admissible Representations......Page 16
Whittaker Models......Page 18
The Theta-Representations r......Page 20
A Functional Equation of Shimura Type......Page 22
L and -Factors......Page 24
A Local Shimura Correspondence......Page 27
The Metaplectic Group......Page 29
Automorphic Representations of Half-Integral Weight......Page 30
Fourier Expansions......Page 31
Theta-Representations......Page 35
A Shimura-Type Zeta Integral......Page 36
An Euler Product Expansion......Page 39
The Theorem......Page 44
Applications and Concluding Remarks......Page 49
PERIOD INTEGRALS OF COHOMOLOGY CLASSESWHICH ARE REPRESENTED BY EISENSTEIN SERIES......Page 56
The Eisenstein Series......Page 86
Arithmetic Applications......Page 128
Generalities......Page 141
Examples......Page 154
ON P-ADIC REPRESENTATIONS ASSOCIATED WITH Zp-EXTENSIONS......Page 167
Introduction......Page 181
Maass forms......Page 182
Fourier expansions......Page 183
The Mellin Transform......Page 185
The convolution......Page 186
Functional Equations......Page 189
CRYSTALLINE COHOMOLOGY, DIEUDONNÉMODULES, AND JACOBI SUMS......Page 192
ESTIMATES OF COEFFICIENTS OF MODULARFORMS AND GENERALIZED MODULAR RELATIONS......Page 282
A REMARK ON ZETA FUNCTIONS OF ALGEBRAICNUMBER FIELDS1......Page 291
Complex quadratic ground fields......Page 298
L-series considered over Q......Page 306
EISENSTEIN SERIES AND THE RIEMANNZETA-FUNCTION......Page 312
Introduction......Page 342
Statement of the main theorem......Page 346
Eisenstein series and the spectral decomposition.........Page 352
Computation of I(s) for (s) >1.......Page 363
Analytic continuation of I(s),......Page 374

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Automorphic Forms, Representation Theory and Arithmetic

TATA INSTITUTE OF FUNDAMENTAL RESEARCH STUDIES IN MATHEMATICS General Editor : K. G. Ramanathan 1. M. Herv´e : Several Complex Variables 2. M. F. Atiyah and others : Differential Analysis 3. B. Malgrange : Ideals of Differentiable Functions 4. S. S. Abhyankar and others : Algebraic Geometry 5. D. Mumford : Abelian Varieties 6. L. Schwartz : Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures

7. W. L. Baily, Jr., and others : Discrete Subgroups of Lie Groups and Applications to Moduli

8. C. P. RAMANUJAM : A Tribute 9. C. L. Siegel : Advanced Analytic Number Theory 10. S. Gelbart and others : Automorphic Forms, Representation Theory and Arithmetic

Automorphic Forms, Representation Theory and Arithmetic

Papers presented at the Bombay Colloquium 1979, by

GELBART HARDER IWASAWA JACQUET KATZ PIATETSKI–SHAPIRO RAGHAVAN SHINTANI STARK ZAGIER

Published for the TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY

SPRINGER–VERLAG Berlin Heidelberg New York (1981)

© TATA INSTITUTE OF FUNDAMENTAL RESEARCH, 1981

ISBN 3 - 540 - 10697 - 9. Springer Verlag, Berlin - Heidelberg - New York ISBN 0 - 387 - 10697 - 9. Springer Verlag, New York - Heidelberg - Berlin

No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Institute of Fundamental Research, Bombay 400 005 Printed by Spads Phototype Setting Ind., (P.) Ltd. 101 A, Poonam Chambers, Dr. Annie Besand Road, Worli, Bombay 400 018, and Published by H. Goetze Springer-Verlag, Heidelberg, West Germany

© Tata Institute of Fundamental Research, 1969 PRINTER IN INDIA

INTERNATIONAL COLLOQUIUM ON AUTOMORPHIC FORMS REPRESENTATION THEORY AND ARITHMETIC BOMBAY, 8–15 January 1979

REPORT An International Colloquium on Automorphic forms, Representation theory and Arithmetic was held at the Tata Institute of Fundamental Research, Bombay, from 8 to 15 January 1979. The purpose of the Colloquium was to discuss recent achievements in the theory of automorphic forms of one and several variables, representation theory with special reference to the interplay between these and number theory, e.g. arithmetic automorphic forms, Hecke theory, Representation of GL2 and GLn in general, class fields, L-functions, p-adic automorphic forms and p-adic L-functions. The Colloquium was jointly sponsored by the International Mathematical Union and the Tata Institute of Fundamental Research, and was financially supported by them and the Sir Dorabji Tata Trust. An Organizing Committee consisting of Professors P. Deligne, M. Kneser, M.S. Narasimhan, S. Raghavan, M.S. Raghunathan and C.S. Seshadri was in charge of the scientific programme. Professors P. Deligne and M. Kneser acted as representatives of the International Mathematical Union on the Organising Committee. The following mathematicians gave invited addresses at the Colloquium: W. Casselman, P. Deligne, S. Gelbart, G. Harder, K. Iwasawa, H. Jacquet, N.M. Katz, I. Piatetski-Shapiro, S. Raghavan, T. Shintani, H.M. Stark and D. Zagier. Professor R. Howe was unable to attend the Colloquium but has sent a paper for publication in the Proceedings.

6

Report

Professors A. Borel and M. Kneser who accepted our invitation, were unable to attend the Colloquium. The invited lectures were of fifty minutes’ duration. These were followed by discussions. In addition to the programme of invited addresses, there were expository and survey lectures by some invited speakers giving more details of their work. Besides the mathematicians at the Tata Institute, there were also mathematicians from other universities in India who were invitees to the Colloquium. The social programme during the Colloquium included a Tea Party on 8 January; a programme of Western music on 9 January; a programme of Instrumental music on 10 January; a dinner at the Institute to meet the members of the School of Mathematics on 11 January; a performance of classical Indian Dances (Bharata Natyam) on 12 January; a visit to Elephanta on 13 January; a programme of Vocal music on 13 January and a dinner at the Institute on 14 January.

Contents 1

2

ON SHIMURA’S CORRESPONDENCE FOR MODULAR FORMS OF HALF-INTEGRAL WEIGHT∗ 1 The Metaplectic Group . . . . . . . . . . . . . . . . . . 2 Admissible Representations . . . . . . . . . . . . . . . . 3 Whittaker Models . . . . . . . . . . . . . . . . . . . . . 4 The Theta-Representations rχ . . . . . . . . . . . . . . . 5 A Functional Equation of Shimura Type . . . . . . . . . 6 L and ǫ-Factors . . . . . . . . . . . . . . . . . . . . . . 7 A Local Shimura Correspondence . . . . . . . . . . . . 8 The Metaplectic Group . . . . . . . . . . . . . . . . . . 9 Automorphic Representations of Half-Integral Weight . . 10 Fourier Expansions . . . . . . . . . . . . . . . . . . . . 11 Theta-Representations . . . . . . . . . . . . . . . . . . 12 A Shimura-Type Zeta Integral . . . . . . . . . . . . . . 13 An Euler Product Expansion . . . . . . . . . . . . . . . 14 A Generalized Shimura Correspondence . . . . . . . . . 15 The Theorem . . . . . . . . . . . . . . . . . . . . . . . 16 Applications and Concluding Remarks . . . . . . . . . .

1 4 6 8 10 12 14 17 19 20 21 25 26 29 34 34 39

PERIOD INTEGRALS OF COHOMOLOGY CLASSES WHICH ARE REPRESENTED BY EISENSTEIN SERIES 46 2 The Eisenstein Series . . . . . . . . . . . . . . . . . . . 76 4 Arithmetic Applications . . . . . . . . . . . . . . . . . . 118 7

8

CONTENTS

3

WAVE FRONT SETS OF REPRESENTATIONS OF LIE GROUPS 131 1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 131 2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 144

4

ON P-ADIC REPRESENTATIONS ASSOCIATED WITH Z p -EXTENSIONS 157

5

DIRICHLET SERIES FOR THE GROUP GL(N). 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Maass forms . . . . . . . . . . . . . . . . . . . 3 Fourier expansions . . . . . . . . . . . . . . . 4 The Mellin Transform . . . . . . . . . . . . . . 5 The convolution . . . . . . . . . . . . . . . . . 6 Functional Equations . . . . . . . . . . . . . .

6

. . . . . .

. . . . . .

. . . . . .

CRYSTALLINE COHOMOLOGY, DIEUDONNE´ MODULES, AND JACOBI SUMS

. . . . . .

. . . . . .

171 171 172 173 175 176 179

182

7

ESTIMATES OF COEFFICIENTS OF MODULAR FORMS AND GENERALIZED MODULAR RELATIONS 272

8

A REMARK ON ZETA FUNCTIONS OF ALGEBRAIC NUMBER FIELDS1

9

DERIVATIVES OF L-SERIES AT S 1 Introduction . . . . . . . . . . . 2 Complex quadratic ground fields 3 L-series considered over Q . . .

281

=0 288 . . . . . . . . . . . . . 288 . . . . . . . . . . . . . 288 . . . . . . . . . . . . . 296

10 EISENSTEIN SERIES AND THE RIEMANN ZETA-FUNCTION

302

11 EISENSTEIN SERIES AND THE SELBERG TRACE FORMULA I 332 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 332

CONTENTS

1 2 3 4

Statement of the main theorem . . . . . . . . . . Eisenstein series and the spectral decomposition... Computation of I(s) for ℜ(s) > 1. . . . . . . . . Analytic continuation of I(s), . . . . . . . . . . .

9

. . . .

. . . .

. . . .

. . . .

336 342 353 364

ON SHIMURA’S CORRESPONDENCE FOR MODULAR FORMS OF HALF-INTEGRAL WEIGHT∗ By S. Gelbart and I. Piatetski-Shapiro

Introduction 1

G. Shimura has shown how to attach to each holomorphic cusp form of half-integral weight a modular form of even integral weight. More precisely, suppose f (z) is a cusp form of weight k/2, level N, and character χ. Suppose also that f is an eigenfunction of all the Hecke operators 2 N (p2 ), say T (p2 ) f = ω f . If k ≥ 5, then the L-function T k,χ p ∞ X

A(n)n−s =

n=1

Y

p 0. In any case, B0 = B0 × Z2 , and the index of B0 in B is the index of (F x )2 in F x .

On Shimura’s Correspondence for Modular Forms...

7

For any pair of quasi-characters µ1 , µ2 of F x , let µ1 µ2 denote the (genuine) character of B0 /N whose restriction to B0 /N is given by the formula !! a1 0 µ1 µ2 = µ1 (a1 )µ2 (a2 ). 0 a2 The induced representation

7

ρ(µ1 , µ2 ) = Ind(G F , B0 , µ1 µ2 )

(2.3.1)

is admissible and ρ(µ1 , µ2 ) ≈ ρ(ν1 , ν2 ) not only if µ1 = ν2 and µ2 = ν1 , but also if µ2i = νi2 , i = 1, 2.

(2.3.2)

cf. §2 of [Ge PS2] and §5 of [Ge]. Moreover, ρ(µ1 , µ2 ) is irreducible 1 −1 (or all integral points in the real case). In unless µ21 µ−2 2 (x) = |x| or |x| any case, the composition series has length at most 2; cf. [Moen] and [Ge Sa]. 2.4 Classification of Representations If ρ(µ1 , µ2 ) is irreducible, we denote it by π(µ1 , µ2 ) and call it a principal series representation. If ρ(µ1 , µ2 ) is reducible, we let π(µ1 , µ2 ) denote its unique irreducible subrepresentation. In all cases, π(µ1 , µ2 ) defines an infinite-dimensional 1 irreducible admissible representation of G F . If µ21 µ−2 2 (x) = |x| we call π(µ1 , µ2 ) a special representation; it is equivalent to the unique quotient of ρ(µ2 , µ1 ). Suppose (π, V) is any irreducible admissible (genuine) representation of G F . Then π is automatically infinite-dimensional. If it is not of the form π(µ1 , µ2 ) for some pair (µ1 , µ2 ), we say π is supercuspidal. If F is archimedean, no such representations exist. On the other hand, if F is non-archimedean, π is supercuspidal if and only if for every vector v in Vπ , Z π(u)v du = 0

U

8

S. Gelbart and I. Piatetski-Shapiro

for some open compact subgroup U of N ⊂ G F ; cf. [Ge], §5. The construction and analysis of such supercuspidal representations is carried out in [RS] and [Meister]. From [Ge] Section 5, and [Meister], it follows that: 8

2.4.1 An irreducible admissible representation π is class 1 if and only if it is of the form π(µ1 , µ2 ) with µ21 and µ22 unramified and µ21 µ−2 2 (x) , |x|, i.e., π is not special. 2.5 Class 1 Representations Suppose F is non-archimedean and of odd residual characteristic. If π is an admissible representation of G F , recall π is class 1, or spherical, if its restriction to KF contains the identity representation (at least once). If π is also irreducible, it can be shown that π then contains the identity representation exactly once; cf. [Ge] and [Meister]. In particular, suppose 1K denotes the idempotent of the Hecke algebra of G F belonging to the trivial representation of KF , i.e.,    1 if g ∈ K     1K (g) =  −1 if g ∈ K × {−1}     0 if otherwise

Then π class 1 implies π(1K ) has non-zero range, and π class 1 irreducible implies the range is one-dimensional.

3 Whittaker Models Fix once and for all a non-trivial additive character ψ of F. 3.1 Definition Suppose π is an irreducible admissible representation of G F . By a ψ-Whittaker model for π we understand a space W(π, ψ) consisting of continuous functions W(g) on G satisfying the following properties:

On Shimura’s Correspondence for Modular Forms...

3.1.1 W



1 x 01

9

  g = ψ(x)W(g);

3.1.2 If F is non-archimedean, W is locally constant, and if F is archimedean, W is C ∞ ; 3.1.3 The space W(π, ψ) is invariant under the right action of G F , and the resulting representation in W(π, ψ) is equivalent to π. 3.2 In [Ge HPS] we prove that a ψ-Whittaker model always exists. If W(π, ψ) is unique, we say π is distinguished. Note that if π is not genuine, i.e., if π defines an ordinary representation of GL2 (F), then π is always distinguished: this is the celebrated “uniqueness of Whittaker models” result of [Jacquet-Langlands]. In general, if π is genuine (as we are assuming it is), it is not dis- 9 tinguished. To recapture uniqueness, we need to refine our notion of Whittaker model. 3.3 Let ωπ denote the central character of π. This is the genuine character of (F x )2 xZ2 determined by the formula ! a2 0 = ωπ (a2 )I. π (3.3.1) 0 a2 Let Ω(ωπ ) denote the (finite) set of genuine characters of Z whose 2 restriction to Z agrees with ωπ . 3.4 Definition. For each µ in Ω(ωπ ), let W (π, ψ, µ) denote the space of continuous functions W(g) on G F which, in addition to satisfying conditions (3.1.1)-(3.1.3), also satisfy the condition  h i  W z 10 1x g = µ(z)ψ(x)W(g), for z ∈ Z. (3.4.1) In [Ge HPS] we prove that such a Whittaker model is unique. More precisely, there is at most one such model, and for at least one µ in Ω(ωπ ), a (ψ, µ)-Whittaker model always exists.

10

S. Gelbart and I. Piatetski-Shapiro

3.5 Let Ω(π) = Ω(π, ψ) denote the set of µ in Ω(ωπ ) such that W (π, ψ, µ) exists. This set depends on ψ, but its cardinality does not. Indeed if λ ∈ F x , and ψλ denotes the character ψλ (x) = ψ(λx),

(3.5.1)

then W (π, ψ, µ) is mapped isomorphically to W (π, ψλ , µλ ) via the map # ! " λ 0 λ g . (3.5.2) W(g) → W (g) = W 0 1 Here µλ denotes the character  ! !  λ 0 −1 λ 0   µ (z) = µ  z 0 1 0 1 λ

(3.5.3)

with the conjugation carried out in G. The existence of the isomorphism (3.5.2) means that µ ∈ Ω(π, ψ) iff µλ ∈ Ω(π, ψλ ). 3.6 Remark. Ω(π, ψ) is a singleton set if and only if π is distinguished. All possible examples of distinguished π are described in the next Section.

4 The Theta-Representations rχ 10

These representations are indexed by characters of F x and treated in complete detail in [Ge PS2]. We simply recall their definition and basic properties. 4.1 In [Weil] there was constructed a genuine admissible representation of SL2 (F). We call this representation the basic Weil representation and denote it by rψ ; it depends on the non-trivial additive character ψ and splits into two irreducible pieces, one “even”, one “odd”. If χ is an even (resp. odd) character of F x , we can “tensor” χ with ∗ ψ the even (resp. odd) piece of rψ to obtain a representation rχ of G F , the

On Shimura’s Correspondence for Modular Forms...

11

o n  semi-direct product of SL2 (F) with 10 a02 : a ∈ F x . Inducing up to G F produces an irreducible admissible representation which is independent of ψ and denoted rχ . The restriction of rχ to SL2 (F) is the direct sum of a finite number of inequivalent representations, namely λ

{rψ }λ∈Λ , with Λ an index set for the cosets of (F x )2 in F x . 4.2 Each rχ is a distinguished representation of G F . In particular, for each non-trivial character ψ of F, let γ(ψ) denote the eighth root of unity introduced in [Weil], Section 14. Then Ω(rχ , ψ) = {χµψ }, with µψ the projective character of F x defined by µψ (a) =

γ(ψ) γ(ψa )

(4.2.1)

We note that the restriction of µψ to (F x )2 is trivial. Moreover, if ψ has conductor OF , and F is of odd residual characteristic, µψ is also trivial on units. 4.3 When χ is unramified, and F has odd residue characteristic, rχ is class 1. More generally, if χ is an even character, rχ is the unique irreducible subrepresentation of π(χ1/2 | |−1/4 , χ1/2 | |1/4 F F ). 4.4 If χ is an odd character, i.e., χ(−1) = −1, then rχ is super-cuspidal; cf. [Ge]. 4.5 Having observed that each rχ is distinguished, we conjectured that the family {rχ }χ exhausts the irreducible admissible distinguished representations of G. When F is non-archimedean and of odd residue characteristic, the supercuspidal part of this conjecture is established in [Meister]; the nonsupercuspidal part is treated in [Ge PS2].

11

12

S. Gelbart and I. Piatetski-Shapiro

5 A Functional Equation of Shimura Type As always, F is a local field of characteristic not equal to 2 and ψ is a fixed non-trivial character of F. 5.1 Suppose π is any irreducible admissible representation of G F , and χ is any quasi-character of F x . Recall the sets Ω(π, ψ) and Ω(rχ , ψ) introduced in (3.5). In general, Ω(π, ψ) = Ω(ωπ ). However, Ω(rχ , ψ) = {χµψ }. To attach an L-factor to π and χ, we fix some µ in Ω(π, ψ) and introduce the zeta-functions Z Ψ(s, W, Wχ , Φ) = W(g)Wχ (g)| det(g)| s Φ((0.1)g)dg. (5.1.1) N\G

Here W(g) is any element of W (π, ψ, µ), Wχ is any element of W (rχ , ψ−1 , χµψ−1 ), Φ ∈ S (F × F), and s ∈ C. Since W and W x are genuine, and transform contravariantly under N, their product actually defines a function on N\G. Similarly, we define Z e Ψ(s, W, Wχ , Φ) = W(g)Wχ (g)| det g| s ω−1 ∗ (det g)Φ((0, 1)g)dg N\G

(5.1.2)

with ω∗ = µχµψ−1 .

(5.1.3)

Note that ω∗ is an ordinary character of F x whose restriction to (F x )2 is χωπ . 12

5.2 For Re(s) sufficiently large, and g in GL2 (F), the integrals Z s Φ((0., t)g)|t|2s ω∗ (t) dt = f s (g) (5.2.1) | det g| Fx

On Shimura’s Correspondence for Modular Forms...

13

and | det g|

s

ω−1 ∗ (det g)

Z

Fx

Φ((0, t)g)|t|2s ω−1 ∗ (t) dt = h s (g)

(5.2.2)

converge and define elements in the space of the induced representations −1 ρ(s − 1/2, (1/2 − s)ω−1 ∗ ) and ρ(ω∗ (s − 1/2), 1/2 − s) respectively. Cf. e converge. [Ja], 14. Moreover, for such s, the integrals defining Ψ and Ψ Z Ψ(s, W, Wχ , Φ) = W(g)Wχ (g) f (g)dg (5.2.3) NZ\G

and e W, Wχ , Φ) = Ψ(s,

Z

W(g)Wχ (g)h(g)dg

NZ\G

Modifying the methods of [Ja] we obtain : e W, Wχ , Φ) Theorem 5.3. (a) The functions Ψ(s, W, Wχ , Φ) and Ψ(s, extend meromorphically to C; L(s, π, χ) such that for any (b) There exist Euler factors L(s, π, χ) and e W, Wχ , Φ, ψ, and µ, the functions

are entire;

Ψ(s, W, Wχ , Φ) e L(s, π, χ)

and

e W, Wχ , Φ) Ψ(s, e L(s, π, χ)

(c) There is an exponential factor ǫ(s, π, χ, ψ) such that for all W, Wχ and Φ as above,

with

e − s, W, Wχ , Φ) b Ψ(1 Ψ(s, W, Wχ , Φ) , = ǫ(s, π, χ, ψ) e L(s, π, χ) L(1 − s, π, χ) b y) = Φ(x,

"

Φ(u, v)ψ(uy − vx)dudv.

(5.3.1)

14

S. Gelbart and I. Piatetski-Shapiro

5.4 The factor ǫ(s, φ, χ, ψ) might depend on the choice of µ as well as 13 ψ. Therefore, to be precise, we should write ǫ(s, π, χ, ψ, µ) in place of ǫ(s, π, χ, ψ). However, a straightforward computation shows that ǫ(s, π, χ, µλ ) = ωπ (λ−2 )χ−2 (λ)|λ|2−4s ǫ(s, π, χ, ψλ , µ).

(5.4.1)

Also, as we shall see, globally ǫ(s, π, χ, ψ, µ) is easily seen to be independent of both ψ and µ; cf. Remark 13.4. 5.5 If we introduce the “gamma factor” γ(s, π, χ, ψ) =

ǫ(s, π, χ, ψ)e L(1 − s, π, χ) L(s, π, χ)

then the functional equation (5.3.1) takes the simpler form e − s, W, Wχ , Φ) b = γ(s, π, χ, ψ)Ψ(s, W1 , W2 , Φ). Ψ(1

6 L and ǫ-Factors Let π, χ and ψ be as in the last section. In this section we collect together the values of L(s, π, χ), e L(s, π, χ), and ǫ(s, π, χ, ψ) for most representaL we need to analyze the possible tions π. To compute the factors L and e e W, Wχ , Φ). To compute ǫ(s, π, χ, ψ) we poles of Ψ(s, W, Wχ , Φ) and Ψ(s, e explicitly, for judicious choices need to compute the functions Ψ and Ψ of W, Wχ and Φ. Suppose first that F is non-archimedean. 6.1 Suppose π is a supercuspidal. If π is not of the form rν for any quasi-character ν, then L(s, π, χ) = 1 = e L(s, π, χ),

for all χ.

On the other hand, if π = rν , then

L(s, π, χ) = L(2s, χν),

On Shimura’s Correspondence for Modular Forms...

15

and e L(s, π, χ) = L(2s, χ−1 ν−1 )

If χν is unramified, ǫ(s, π, χ, ψ) =

ǫ(2s, χν, ψ)ǫ(2s − 1, χν, ψ)L(1 − 2s, ν−1 χ−1 ) L(2s − 1, νχ)

whereas if χν is ramified

14

ǫ(s, π, χ, ψ) = ǫ(2s, χν, ψ)ǫ(2s − 1, χν, ψ). Here, as throughout, the factors L(s, ω) and ǫ(s, ω, ψ) are the familiar L and ǫ factors attached to each quasi-character ω of F x ; cf. [Jacquet-Langlands, pp. 108-109]. 6.2 Suppose π is of the form π(µ1 , µ2 ) = φ(µ1 , µ2 ). Then L(s, π, χ) = L(2s − 1/2, µ21 χ)L(2x − 1/2, µ22 χ), and −2 e L(s, π, χ) = L(2s − 1/2, µ−2 1 χ)L(2s − 1/2, µ2 χ)

(6.2.1)

If we set s′ = 2s − 1/2, then

ǫ(s, π, χ, ψ) = ǫ(s′ , µ21 χ, ψ)ǫ(s′ , µ22 χ, ψ)

(6.2.2)

In particular, suppose F is class 1, χ(also µ) is trivial on units, ψ has conductor OF , Φ is the characteristic function of OF × OF , and W and Wχ are normalized KF -fixed vectors in W(π, ψ, µ) and W(rχ , ψ−1 ). Then

and

   Ψ(s, W1 , W2 , Φ) = L(s, π, χ)   e W1 , W2 , Φ) e =e Ψ(s, L(s, π, χ) ǫ(s, π, χ, ψ) = 1.

(6.2.3)

16

S. Gelbart and I. Piatetski-Shapiro

6.3 Suppose π is the special representation 1/4 1 π = π(µ1 , µ2 ), with µ21 µ−2 2 (x) = |x|F , and µ1 (x) = ν(x)|x|F

Then

   L(s, π, χ) = L(2s, χν2 ),   e L(s, π, χ) = L(2s, ν−2 χ−1 ),

and-if π(ν2 ) denotes the special representation π(ν2 | |1/2 , ν2 | |−1/2 ) of GL2 (F), ǫ(s, π, χ, ψ) = ǫ(s′ , π(ν2 ) ⊗ χ, ψ). 6.4 If π is of the form rν , with ν(−1) = 1, then    L(s, π, χ) = L(2s − 1, χν)L(2s, χν),   e L(s, π, χ) = L(2s − 1, χ−1 ν−1 )L(2s, χ−1 ν−1 ),

and

ǫ(s, π, χ, ψ) = ǫ(2s − 1, χν, ψ) ∈ (2s, χν, ψ) 15

6.5 Suppose now that F is archimedean. Then each π occurs as the subrepresentation of some ρ(µ1 , µ2 ), with each µi determined up to a character of order 2. Let S (π) denote the unique irreducible admissible representation of GL2 (F) which appears as a subrepresentation of ρ(µ21 , µ22 ). Then

and

   L(s, π, χ) = L(s, S (π) ⊗ χ),   e L(s, π, χ) = L(s, S (π) ⊗ χ−1 ), ǫ(s, π, χ, ψ) = ǫ(s, S (π) ⊗ χ, ψ),

the L and ǫ factors on the right being those of [Jacquet-Langlands].

On Shimura’s Correspondence for Modular Forms...

17

6.6 Stability Given π and ψ, it can be shown that if F is non-archimedean, and χ is sufficiently highly ramified, the corresponding L and ǫ-factors stabilize. More precisely, for all χ sufficiently highly ramified,

and

L(s, π, χ) = 1 = e L(s, π, χ), ǫ(s, π, χ, ψ) = ǫ(s, ωπ χ, ψ) ∈ (s, χ, ψ)

(6.6.1)

ωπ (a) = ωπ (a2 )

(6.6.2)

In (6.6.1), ωπ is the character of F x defined by the equation

7 A Local Shimura Correspondence Suppose π is an irreducible admissible (genuine) representation of G F and ωπ is its central character. 16

7.1 Fixing a non-trivial character ψ of F, we call an irreducible admissible representation π of G F a Shimura image of π if 7.1.1 the central character ωπ of π is such that ωπ (a) = ωπ (a2 ), a ∈ F x ; 7.1.2 for any quasi-character χ of F x ,    L(s, π, χ) = L(s, π ⊗ χ),   e L(s, π, χ) = L(s,e π ⊗ χ−1 ),

and

ǫ(s, π, χ, ψ) = ǫ(s, π ⊗ χ, ψ). 7.2 If the Shimura image of π exists, it is unique, and independent of ψ. We denote it by S (π).

18

S. Gelbart and I. Piatetski-Shapiro

7.3 From Section 6 it follows that S (π) exists whenever π is not a supercuspidal representation (not of the form rν ). Indeed in this case, π = π(µ1 , µ2 ) implies S (π) = π(µ21 , µ22 ). In particular, 1/2 π = rν (ν(−1) = −1) implies (π) = π(ν| |1/2 F , ν| |F ).

On the other hand, as we shall see, if π is supercuspidal (but not of the form rν ) its image S (π) must also be supercuspidal. 7.4 In case F = R, and π corresponds to a discrete series representation of “lowest weight k/2”, S (π) corresponds to a discrete series representation of lowest weight k − 1; cf. [Ge], §4. 7.5 Connections with Shimura’s theory The fact that S takes π(µ1 , µ2 ) to π(µ21 , µ22 ) means (in the non-archimedean unramified situation) that eigenvalues for the Hecke algebras are preserved. See §5.3 of [Ge] for a careful analysis of this phenomenon. Keeping in mind (7.4), it follows that our local Shimura correspondence is consistent with the map defined globally (and classically) in [Shim]. 17

7.6 Summing up, Shimura’s correspondence operates locally as follows: π principal series π(µ1 , µ2 ) special representation π(ν| |1/4 , ν| |−1/4 ) Weil rν (ν(−1) = −1) Weil rν (ν(−1) = 1)

π = S (π) principal series π(µ21 , µ22 ) special rep Sp(ν2 ) special rep Sp(ν) one-dimensional rep ν ◦ det

On Shimura’s Correspondence for Modular Forms...

19

Note all special representations arise as Shimura images (whereas a principal series thus arises if it corresponds to even-or squared-characters of F ∗ ); for the supercuspidal representations, see [Flicker] and [Meister].

Chapter II. Global Theory Throughout this Chapter, F will denote an arbitrary A-field of characteristic not equal to two, A its ring of adeles, and Y ψv ψ= v

a non-trivial character of F\A.

8 The Metaplectic Group For each place v of F we defined in §1 a “local” metaplectic group Gv = G Fv . Roughly speaking, the adelic metaplectic group G A is a product of the local groups Gv . More precisely, recall that if v is non-archimedean and “odd”, Gv splits over Kv = GL2 (OFv ). Thus we can consider the restricted direct product Y e= G Gv (Kv ). v

e by 18 The metaplectic group G A is obtained by taking the quotient of G     Y Y     ee =  . Z2 : ǫv = 1 for all but an even number of v ǫv ∈ Z     v

v

In particular, we can view G A as a group of pairs {(h, ζ) : h ∈ G A , ζ ∈ Z2 }, with multiplication given by (h1 , ζ1 )(h2 , ζ2 ) = (h1 h2 , β(h1 , h2 )ζ1 ζ2 ),

20

S. Gelbart and I. Piatetski-Shapiro

and β a product of the local two-cocycles defining Gv . The fact that the exact sequence 1 → Z2 → G A → G A → 1 splits over the discrete subgroup G F = GL2 (F) is equivalent to the quadratic reciprocity law for F; cf. [Weil].

9 Automorphic Representations of Half-Integral Weight 9.1 Recall that G A is the quotient of

Q v

eA by the subgroup Z ee . Gv = G

9.2 Suppose that for each place v of F we are given an irreducible admissible genuine representation (πv , Vv ) of Gv . Suppose also that for almost every finite v, πv is class 1. Then for almost every v we can choose a Kv -fixed vector ev in Vv and define a restricted tensor product space O Vv (ev ). V= v

eA in V given by The resulting representation of G O π= πv

(9.2.1)

v

ee and defines an irreducible admissible representation of is trivial on Z GA. Conversely, suppose π is an irreducible unitary representation of G A . Following step by step the arguments of §9 of [Jacquet-Langlands] we can show that π must be of the form (9.2.1) with each πv determined uniquely by π.

On Shimura’s Correspondence for Modular Forms...

21

9.3 Let ω denote a character of (A x )2 trivial on (F x )2 . Proceeding as in §10 of [Jacquet-Langlands] we can introduce a space A(ω) of automorphic forms on G A . Each ϕ in A(ω) is a genuine C ∞ function on G F \G A which is “slowly increasing” and transforms under the center 19 (of G A ) according to ω. The group G A acts as expected in A(ω) by right translations. By A0 (ω) we denote the subspace of ω-cuspidal functions, those ϕ in A0 (ω) such that (i) the constant term Z

ϕ0 (g) =

# ! " 1 x g dx = 0 ϕ 0 1

F\A

for each g in GA ; (ii) the integral

Z

|ϕ(g)|2 dg

ZA2 G F \GA

is finite. This space of cusp forms is clearly stable under the action of GA , and each ϕ in A0 (ω) is rapidly decreasing. 9.4 An irreducible admissible representation π of GA is called automorphic (respectively cuspidal) of half-integral weight if it is a constituent of some A(ω) (resp. A0 (ω)).

10 Fourier Expansions Suppose ϕ is an automorphic form on GA , and ψ = Πψv is a fixed nontrivial character of F\A. 10.1 Since

# ! " 1 x g ϕ 0 1

22

S. Gelbart and I. Piatetski-Shapiro

is a C ∞ function on F\A for each fixed g, ϕ(g) admits a Fourier expansion in terms of the characters of F\A. But each non-trivial character ψ of F\A is of the form ψ′ (x) = ψδ (x) = ψ(δx) for some δ ∈ F x . Thus # ! " X 1 x ψδ g = ϕ0 (g) + Wϕ (g)ψ(δx), ϕ 0 1 x

(10.1.1)

(10.1.2)

δ∈F

with ψδ Wϕ (g)

Z

=

! ! 1 x g ψ−1 (δx)dx. ϕ 0 1

(10.1.3)

F\A

20

On the other hand, it is easy to check that ψδ

ψ

ϕ(g) = ϕ0 (g) +

X

Wϕ (g) = Wϕ

! ! δ 0 g . 0 1

(10.1.4)

Thus we also have

δ∈F x

ψ Wϕ

! ! δ 0 g . 0 1

(10.1.5)

In other words—modulo its constant term–ϕ(g) is completely determined by its first Fourier coefficient ! ! Z 1 x ψ g ψ(−x)dx. (10.1.6) Wϕ (g) = ϕ 0 1 F\A

We call this function a ψ-Whittaker function since # ! " 1 x g = ψ(x)W(g), x ∈ A. W 0 1 Now we must refine this notation to bring into play the local theory of §3.

On Shimura’s Correspondence for Modular Forms...

23

10.2 Suppose π = ⊗πv is any automorphic representation of halfintegral weight. Suppose in addition that π actually occurs as a subrepresentation (as opposed to subquotient) of some A(ω), say in the space Vπ . Then ω must be the central character ωπ of π. Now let Ω(ωπ ) denote the set of (genuine) characters of ZF \Z A 2 whose restriction to Z A agrees with ωπ . Then each ϕ in Vπ has a Fourier expansion of the form X X ψδ,µ (10.2.1) Wϕ (g) ϕ(g) = ϕ0 (g) + µ∈Ω(ωπ ) δ∈F x

with ψδ,µ Wϕ (g)

=

Z

2 Z A \Z A

Z

! ! 1 x ϕ Z g ψ−1 (δx)µ−1 (z)dxdz. 0 1

(10.2.2)

F\A

The (ψ, µ) refinement of (10.1.4) is ψδ,µ Wϕ (g)

where

=

ψ,µδ Wϕ

"

# ! δ 0 g 0 1

(10.2.3)

" # # "  δ 0 −1 δ 0   , z ∈ Z A . µ (z) = µ  z 0 1 0 1 δ

Note that for any µ, ψ,µ



z

! ! 1 x g = ψ(x)µ(z)W(g), z ∈ Z A , x ∈ F. 0 1

21

(10.2.4)

10.3 For any µ in Ω(ωπ ), let W (π, ψ, µ) denote a (ψ, µ)-Whittaker space for π (analogous to the local definition (3.1); the crucial property of course is (10.2.4)). Let Ω(π, ψ) denote the set of µ in Ω(ωπ ) such that W (π, ψ, µ) exists; if Ω(π, ψ) is a singleton set we call π distinguished. ψ,µ For π and ϕ as in (10.2), Wϕ (g) is clearly non-zero for at least one µ, and therefore W (π, ψ, µ) exists for at least one µ (ψ being supψ,µ posed fixed). If Wϕ (g) , 0 for exactly one µ in Ω(ωπ ), we say ϕ is

24

S. Gelbart and I. Piatetski-Shapiro

distinguished. We note that π distinguished implies any ϕ in Vπ is distinguished. Of course if π = ⊗πv is any irreducible admissible representation of GA , we might be inclined to call π distinguished if each πv is distinguished in the local sense. Fortunately these notions are compatible. Indeed in [Ge PS2] we prove that an automorphic subrepresentation π of A is distinguished in the above sense if and only if each πv is. If π is a distinguished subrepresentation of A(ωπ ) and ϕ ∈ Vπ , then (10.2.3) implies # ! " X ψ,µ δ 0 g , (10.3.1) ϕ(g) = ϕ0 (g) = Wϕ 0 1 x δ∈F

a familiar GL2 -type Fourier expansion. In particular, if π is cuspidal, ψ,µ the first Fourier coefficient Wϕ (g) completely determines ϕ through the expansion ! X δ 0 g), W ϕ(g) = 0 1 x δ∈F

and we have: Theorem 10.3.2. Every distinguished cuspidal representation of halfintegral weight occurs exactly once in A0 .

22

10.4 Let us explain the classical significance of a distinguished cusp form. Suppose X f (z) = a(n)e2πinz

is a cusp form of weight k/2, and an eigenfunction for all Hecke operators. Since most of these operators act as the zero map, one can’t expect their eigenvalues to relate many of the coefficients a(n). In fact, if T (p2 ) f = ω p f , then ω p serves to relate a(t) only to the coefficients a(tp2 ); in particular, the first Fourier coefficient does not always determine f . In other words, there is more than “one orbit” of coefficients. On the other hand, if f is “distinguished”, i.e., if there is a t such that a(n) = 0 unless n = tm2 for some m, then f is determined by just

On Shimura’s Correspondence for Modular Forms...

25

one coefficient (and the knowledge of the ω p ’s). This is consistent with (10.3.1). Note that in our representation theoretic set-up, our ϕ in Vπ is assumed to be an eigenfunction of the Hecke operators. The fact that ϕ ψ,µ is distinguished means exactly that Wϕ , 0 for exactly “one orbit of characters”. In particular, the relation (10.2.3) implies that if δ < (F x )2 , then ! ! ψδ ,µ ψ,µδ δ 0 ψδ,µ g = 0. Wϕ (g) = Wϕ (g) = Wϕ 0 1 In classical terms, if ϕ corresponds to the form f (z), then XX 2 f (z) = a(δn2 )e2πiδn z =

δ X

n

a(δ0 n2 )e2πiδ0 n

2z

n

For more details, see [Ge PS2] and [Shim]. Examples of distinguished automorphic representations will now be described.

11 Theta-Representations 11.1 Suppose χ =

Q v

χv is any character of F x \A x . Since almost every

χv is unramified, we can define an irreducible admissible representation of GA through the formula rχ = ⊗rχv , where rχv is the local theta-representation described in Section 4.

11.2 In [Ge PS2] we show that rχ occurs in a subspace of χ-automorphic forms on GA . In particular, rχ defines a distinguished automorphic representation of half-integral weight. Our construction 23 χ → rχ

26

S. Gelbart and I. Piatetski-Shapiro

generalizes the classical construction of theta-series associated with Dirichlet characters. To wit, suppose χ : (Z/NZ) x → C is a primitive Dirichlet character, and χ(−1) = 1 say. Then θχ (z) =

∞ X

χ(n)e2πin

2z

n=−∞

defines a “distinguished” modular form of weight 21 , level 4N 2 , and character χ. If χ = Πχv is not totally even, i.e., χv (−1) = −1 for at least one v, then rχ is actually cuspidal. 11.3 In [Ge PS] we conjectured that every distinguished cuspidal representation of half-integral weight is of the form rχ for some χ. In [Ge PS2] we show that this follows from the Shimura correspondence established in this paper; cf. §16,

Chapter III. A Generalized Shimura Correspondence

12 A Shimura-Type Zeta Integral Suppose π = ⊗πv is an automorphic cuspidal representation of halfQ integral weight and χ = χv is a grossencharacter of F. Having inv

troduced L and ǫ factors for each πv and χv , we want to prove that the product Y L(s, πv , χv ) L(s, π, χ) = v

converges in some half-plane, continues to a meromorphic function in C, and satisfies a functional equation of the form   Y  L(s, π, χ) =  ∈ (s, πv , χv ψv ) e L(1 − s, π, χ). v

To do this, we have to introduce a zeta-integral of Shimura type that essentially equals L(s, π, χ).

On Shimura’s Correspondence for Modular Forms...

27

12.1 Let A0 (ωπ ) denote the space of cusp forms which transform under 24 2 Z A according to the character ωπ , and suppose π occurs in the space Vπ in A0 (ωπ ). If ϕ ∈ Vπ then ! ! XX ψ,µδ δ 0 g (12.1.1) Wϕ ϕ(g) = 0 1 x µ δ∈F

Recall that the first summation extends over all characters µ′ of ZF \Z A 2 whose restriction to Z A is ωπ . Now fix O µv in Ω(π, ψ), µ= v

ψ,µ

and fix the embedding Vπ so that Wϕ (g) , 0. Given by character χ = Πχv of F x \A x , let µχ denote the unique element of the singleton set Ω(rχ , ψ−1 ). These µ, µχ determine Whittaker models W (π, ψ, µ) and W (rχ , ψ, µχ ). To define our global analogue of the local zeta functions ψ(s, W, Wχ , Φ) we need first to describe some Eisenstein series on GL2 (A). 12.2 If Φ =

Q v

F s (g) =

Φv is in S (A × A), set

FΦ s (g)

= | det g|

s

Z

Φ((0, t)g)|t|2s ω∗ (t)d x t,

(12.2.1)

Fx

with ω∗ the (ordinary) character of F x \A x given by the formula ω∗ = µµχ = µχµψ − 1; cf. (5.1.3), (5.2.1), and (5.2.2). The integral in (12.2.1) converges for Re(s) ≫ 0 and defines an element F s = Π f s,v

28

S. Gelbart and I. Piatetski-Shapiro

    in the induced space ρ s − 21 , 12 − s ω−1 ∗ . Moreover, the series X

F s (γg ) = E(g, F, s)

γ∈BF \G F

converges for Re(s) ≫ 0, and defines an automorphic form on GL2 , the Eisenstein series E(g, F, s); cf. p. 117 of [Ja] (taking µ1 = α s 21 , 1 µ2 = α 2 −a ω−1 ∗ ). 25

Remark 12.3. E(g, F, s) extends to a meromorphic function in C with functional equation b

E(g, F Φ , s) = E(g, F Φ , 1 − s);

(12.2.1)

b b Rhere, as in the local theory, Φ is the twisted Fourier transform Φ(x, y) = Φ(u, v)ψ(yu − vx)dudv, with du and dv the self-dual measure on A; cf. Prop. 19.3 of [Ja]. We also know that the only poles of E(g, F, s) are simple, and occur for | |2−2s = ω∗ and | |2s = ω−1 ∗ . A A 12.4 Given π, ψ, µ, χ, and F s as above, we define our zeta integral by the equation Z ∗ ϕ(g)θχ (g)E(g, F, s)dg. (12.4.1) ψ (s, ϕ, θχ , F) = ZA2 G F \GA

Here ϕ ∈ Vπ , and θχ (g) =

X

δ∈F x

W

ψ−1 ,µχ

# ! δ 0 g + θ0 (g) 0 1

"

(12.4.2)

belongs to the space of the automorphic distinguished representation rχ . Since the theta-function θχ is also slowly increasing, and since ϕ(g) is a cusp form, the integral in (12.4.1) converges in some right half-plane, and its analytic properties in all of C are reflected by those of E(g, F, s). In particular, we have:

On Shimura’s Correspondence for Modular Forms...

29

Proposition 12.5. For any choice of ϕ, θχ , and F s : (i) the function ψ∗ (s, ϕ, θχ , F) extends to a meromorphic function in C with functional equation b

ψ∗ (1 − s, ϕ, θχ , F Φ ) = ψ∗ (s, ϕ, θχ , F Φ ).

(12.5.1)

(ii) All poles of ψ∗ are simple, with residues proportional to Z (12.5.2) | det g| s0 ϕ(g)θχ (g)dg. ZA2 G F \GA

(iii) ψ∗ is bounded at infinity in vertical strips of finite width. Corollary 12.6. If π is not of the form rν for any grossencharacter ν, then ψ∗ (s, ϕ, θχ , V) is actually entire. Proof. If ψ∗ (s, ϕ, θχ , F) has a pole, the residue (12.5.2) is non-zero for 26 some s0 . In other words, the bilinear form on Vπ × Vrχ defined by (ϕ, θχ ) →

Z

| det g| s0 ϕ(g)θχ (g)dg

ZA2 G F \GA

rχ . Since e rχ ≈ rχ−1 , is not identically zero, and | |As0 ⊗ π is equivalent to e this contradicts our hypothesis. 

13 An Euler Product Expansion 13.1 To relate L(s, π, χ) to ψ∗ (s, ϕ, θχ , F) we need to express ψ∗ as a product of local integrals of the form ψ(s, Wv , Wχv , Φv ). In greater generality, this Euler product decomposition is sketched in [PS]. To treat the explicit case at hand, we assume that the “first Fourier coefficients”

30 ψ,µ

S. Gelbart and I. Piatetski-Shapiro −1 ,µ

Wϕ (g) and W ψ form

χ

of ϕ(g) and θχ (cf. (12.1.1) and (12.4.2)) are of the ψ,µ

Wϕ (g) =

Y

Wv (g)

v

and −1 ,µ



χ

(g) =

Y

Wχv (g),

with Wv ∈ W (πv , ψv , µv ) and Wχv (g) ∈ W (rχv , ψ−1 ). Proposition 13.2. With ϕ, θχ , and F Φ s as above, and Re(s) ≫ 0, Y ψ∗ (s, ϕ, θχ , F Φ ) = Ψ(s, Wv , Wχv , Φv ). v

(Recall the local zeta-functions Ψ are defined by (5.1.1).) Proof. Replacing E by the series defining it, we have Z ϕ(g)θχ (g)E(g, F, s)dg. ψ∗ (s, ϕ, θχ , F Φ ) = ZA2 G F \GA

=

Z

ϕ(g)θχ (g)F Φ s (g)dg

ZA2 BF \GA

Setting B0 = ZN =

 a x 

, we may write X −1 W ψ ,µχ (bg), θχ (g) = θ0 (g) + 0a

B0F \BF

Z

ψ∗ (s, ϕ, θχ , F) =

ϕ(g)θ0 (g)F s (g)dg

ZA2 BF \GA

+

Z

ZA2 B0F \GA

27

−1 ,µχ

ϕ(g)W ψ

(g)F s (g)dg

(13.2.1)

On Shimura’s Correspondence for Modular Forms...

31

We claim now that the first term on the right side of (13.2.1) is zero, i.e., the constant term θ0 (g) contributes nothing to ψ∗ . Indeed θ0 (g)F s (g) is left NA -invariant, and ϕ(g) is a cuspidal. Thus we have Z −1 ∗ ϕ(g)W ψ ,µχ (g)F s (g)dg ψ (s, ϕ, θχ , F) = ZA2 B0F \GA

Z

=

I(g)F s (g)dg

(13.2.2)

BA \GA

where I(g) =

Z

−1 ,µχ

ϕ(bg)W ψ

(bg)ω∗ (b)db

ZA2 B0F \BA

and ω∗ is the character of BA defined by ! 2s a1 x ∗ a1 = ω−1 ω ∗ (a2 ). 0 a2 a2 To continue, we compute  Z  Z  I(g) =  

B0A \BA ZA2 \B0F B0A

=

Z

−1 ,µχ

ω∗ (b)W ψ

(bg)

XX µ

B0A \BA

  Z   

   ′ ψ−1 ,µχ ′ ∗ ′ ′ ϕ(b bg)W (b bg)ω (bb )db  db 

Z

ZA2 \ZA FA



W ψ,µ (bg)

δ

   δ −1 ′ ψ (x)ψ(−x)(µ) (z)µ (z)dxdz db 

But the integral in parenthesis is zero unless δ = 1 and µ′ = µ, in which 28 case it equals 1. Thus we have Z −1 ψ,µ ω∗ (b)Wϕ (bg)W ψ ,µχ (bg)db. I(g) = B0A \BA

32

S. Gelbart and I. Piatetski-Shapiro

Plugging this expression into (13.2.2) gives    Z  Z   ψ,µ ψ−1 ,µχ Φ ∗ Wϕ (bg)W (bg)F s (bg)db dg ψ (s, ϕ, θχ , F) =    BA \GA B0 \BA A

=

Z

−1 ,µχ

ψ,µ

Wϕ (g)W ψ

(g)F s (g)dg

NA ZA \GA −1

So taking into account the infinite product expression for Wϕ , W ψ ,µχ , and F s = Π f s,v , we obtain the desired Euler product expansion for ψ∗ .  Theorem 13.3. Suppose π is any cuspidal representation of half-integral weight. If χ = Πχv is any character of F x \A x set Y L(s, π, χ) = L(s, πv , χv ) v

and e L(s, π, χ) =

Then

Y v

e L(s, πv , χv ).

(i) these infinite products converge in some half-plane Re(s) > s0 ; (ii) L and e L extend meromorphically to all of C, are bounded in vertical strips of finite width, and satisfy the functional equation with

L(s, π, χ) = ǫ(s, π, χ)e L(1 − s, π, χ) ǫ(s, π, χ) =

Y v

∈ (s, πv , χv , ψv );

(iii) the only poles of L(s, π, χ) are simple, and these occur only if π is of the form rv for some character ν of F x \A x .

Proof. For almost every v, πv is of the form π(νv1 , νv2 ), with νvi (x) = |x|ti ,v , and −t0 ≤ ti,v ≤ t0 (independent of v)

On Shimura’s Correspondence for Modular Forms...

33

Therefore, since

29

L(s, π(νv1 , νv2 ))

=

1 ′

1 − q−2t1 ,v−s

!

1 ′

1 − q−2t2,v −s

!

the infinite products in question converge. Now fix a set S outside of which everything is unramified, i.e., if v < S , v is finite and odd, πv and χv are class 1, ψv has conductor OFv , µv is trivial on OFx v , and Wv , Wχv and Φv are chosen so that Ψ(s, Wv , Wχv , Φv ) = L(s, πv , χv ) and ǫ(s, πv , χv , ψv ) = 1; cf. (6.2.3). For v inside S , choose Wv , Wχv and Φv so that Ψ(s, Wv , Wχv , Φv ) = L(s, πv , χv ) modulo a non-vanishing entire factor. Then since ψ∗ (s, ϕ, θχ , F) has the analytic properties asserted in parts (ii) and (iii), so does L(s, πv , χv ). To establish the functional equation, we simply compute (using the local functional equations and (12.5.1)): Y Y L(s, π, χ) = L(s, πv , χv ) L(s, πv , χv ) v∈S

v 2, Y − = (1 − J)Y and since J = J|K is contained in the center of G = Gal(K/Q), Y − is an R-submodule of Y. Let t(Y − ) denote the torsion Λ-submodule of the Λ-module Y − and let Z = Y − /t(Y − ),

Z′ = Z ⊗ Qp. Zp

Then Z is an R-module, and Z ′ an R′ -module. Furthermore, we can prove by using the assumptions of Theorem 2 that there is an exact sequence of R′ -modules Z ′ /ξZ ′ → X ′ → 0. Therefore the proof is now reduced to show that Z ′ /ξZ ′ is cyclic over R′ .

168

Kenkichi Iwasawa

Let

151

R

′−





= (1 − J)R = R (1 − J).

Then we have the following two lemmas: d Lemma 1. Both Z ′ and R′ − are free Λ′ -modules with the same rank 2 and Z ′ /T Z ′ ≃ R′ − /T R′ − as modules over R′ .

Lemma 2. Let A and B be R′ -modules which are free and of the same finite rank over Λ′ , and let A/T A ≃ B/T B as R′ -modules. Then, as modules over R′ , A/pA ≃ B/p B for any non-zero prime ideal p of the principal ideal domain Λ′ . d That Z ′ is a free Λ′ -module of rank , where d = [k : Q], is a known 2 fact in the theory of Z p -extensions. The rest of Lemma 1 can be proved by considering the Galois group of the maximal p-ramified abelian pextension over k. To see the proof of Lemma 2, let us first assume for simplicity that k ∩ Q∞ = Q. In this case, G = Γ × ∆ where ∆ = Gal(k/Q) = Gal(K/Q∞ ), and R = Z p [[G]] is nothing but the group ring of the finite group ∆ over Λ = Z p [[Γ]]: R = Λ[∆]. Hence R′ = Λ[∆] where Λ′ = Λ⊗ Q p is a principal ideal domain. The lemma then follows Zp

152

easily from the results of Swan on the group ring of finite groups over

On P-Adic Representations Associated with Z p -Extensions

169

Dedekind domains9 . The case k ∩ Q∞ , Q can be proved similarly by reducing it to the above mentioned special case. Now, since R′ − = R′ (1 − J) is cyclic over R′ , we see from the above two lemmas that Z ′ /pZ ′ is cyclic over R′ for all p as stated in Lemma 2. As ξ is a non-zero element of the principal ideal domain Λ′ , it follows that Z ′ /ξZ ′ also is cyclic over R′ . This completes the proof of Theorem 2. Finally, we would like to mention here also the following result which can be proved by similar arguments as described above. Namely, changing the notations from the above, let k = an arbitrary (totally) real finite Galois extension over Q, K = k∞ − kQ∞ = the cyclotomic Z p -extension over k, L′ = the maximal unramified abelian p-extension over K in which every p-spot of K is completely decomposed, M = the maximal p-ramified abelian p-extension over K. Then, again, Q ⊆ k ⊆ K ⊆ L′ ⊆ M

and K/Q, L′ /Q, and M/Q are Galois extensions. Let G = Gal(K/Q),

R = Z p [[G]],

R′ = R⊗ Q p . Zp

As in the case discussed above, the Galois groups Gal(M/L′ )

Gal(M/K) and

are modules over R so that Gal(M/K)⊗ Q p and Gal(M/L′ )⊗ Q p are R′ Zp

Zp

modules. Theorem 4. Gal(M/K ′ )× Q p is cyclic over R′ . If in particular λ p (k) = 0, Zp

then Gal(M/K)⊗ Q p also is cyclic over R′ . Zp

The first part of the theorem is proved without any assumption, and the second part without assuming Leopoldt’s conjecture. Hence the theorem might be more useful in some applications than Theorem 2. 9

See Swan [7].

170

Bibliography

Bibliography 153

[1] COATES, J. p-adic L-functions and Iwasawa’s theory, Algebraic Number Fields, Acac. Press, London, 1977, 269–353. [2] GREENBERG, R. On the Iwasawa invariants of totally real number fields, Amer. Jour. Math. 93(1976), 263–284. [3] IWASAWA, K. On p-adic L-functions, Ann. Math. 89(1969), 198– 205. [4] IWASAWA, K. Lectures on p-adic L-functions, Princeton University Press, Princeton, 1972. [5] IWASAWA, K. On Zl -extensions of algebraic number fields, Ann. Math. 98(1973), 246–326. [6] LANG, S. Cyclotomic fields, Springer-Verlag, New YorkHeidelberg-Berlin, 1978. [7] SWAN. R. Induced representations and projective modules, Ann. Math. 71 (1960), 552–578.

DIRICHLET SERIES FOR THE GROUP GL(N). By Herve Jacquet

1 Introduction 155

Suppose ϕ is a modular cusp form with Fourier expansion: X ϕ(z) = an exp(2i πnz).

(1.1)

n≥1

The Mellin transform of ϕ is the integral Z+∞ ϕ(iy)y s−1 dy.

(1.2)

0

If we replace ϕ by its Fourier expansion then we see that (1.2) is equal to Z+∞ X (1.3) an n−s exp(−2πy)y s−1 dy. n≥1

Since

0

Z+∞ exp(−2πy)y s−1 dy = (2π)−s Γ(s). 0

171

(1.4)

172

Herve Jacquet

this integral representation gives the analytic continuation of the series X an n−s , (1.5) n≥1

as a meromorphic function of s in the whole complex plane. Furthermore it shows that the analytic continuation satisfies a simple functional equation. Finally if ϕ is an eigen function of the Hecke operators, then the series (1.5) has an infinite euler product. If ϕ′ is another form then one can also consider the “convolution” of the Dirichlet series attached to ϕ and ϕ′ , namely the Dirichlet series X an a′

n

n≥1

156

ns

(1.6)

It has a simple integral representation and analytic properties similar to that of (1.5). Furthermore, if both ϕ and ϕ′ are eigen functions of the Hecke operators then it has an Euler product. Classically, it is just as easy to pursue the theory for other types of forms: holomorphic forms for congruence sub-groups, Maass forms, Hilbert modular forms... The theory can also be generalied to the groups GL(r) with r > 2. It is still incomplete but, as an introduction, we shall discuss the case of the “Maass forms” for the group Γr = GL(r, Z),

(1.7)

also noted simply Γ. Naturally the discussion of the most general case would entail the use of ad`eles and group representations. This report is based directly on the work of the authors of [J-S-P 1,2,3,]. That work in turn owes much to the results and ideas, published or not, of the authors of [G-K].

2 Maass forms Let ϕ be a function on Gr = GL(r, R),

(2.1)

Dirichlet Series for the Group GL(N)

173

invariant on the left under Γr , on the right under the orthogonal group, and, on both sides, under the center Zr of Gr . The function ϕ will be said to be a cusp form if it satisfies some additional conditions that we now describe. It will be assumed to be C ∞ and an eigen function of the algebra Z of bi-invariant differential differential operators. The corresponding algebra morphism from Z to C will be denoted by λ. We will also assume ϕ cuspidal. This means that for every group of the form   Ir1               V=               0

..

. Ir2

 ∗                 ..    .     Ir s 

the “constant term of ϕ along V”, that is, the integral Z ϕ(ug)du,

(2.2)

(2.3)

Γ∩V\V

vanishes for all g. It is perhaps unnecessary to recall that V ∩ Γ is a discrete cocompact subgroup of V. There is also a condition of growth at infinity which, because we 157 are considering only cuspidal functions, amounts to demand that ϕ be square integrable on the quotient Zr Γ\Gr . Actually, for a given λ, the functions ϕ satisfying the above conditions make up a finite dimensional Hilbert space Vλ . It is invariant under the action of the Hecke algebra; the corresponding algebra of operators on Vλ is diagonalizable and so we may, and do, demand that our forms be eigen vectors of the Hecke algebra.

3 Fourier expansions Let Nr be the group of upper triangular matrices with unit diagonal. For every (r − 1)-tuple of non zero integers (n1 , n2 , . . . , nr−1 ) define a

174

Herve Jacquet

character θn1 ,n2 ,...,nr−1 of Nr by θn1 ,n2 ,...,nr−1 (x) =

Y

exp(2iπn j x j. j+1 ).

(3.1)

ϕ(ug)θ(u)du

(3.2)

1≤ j≤r−1

It is clearly trivial on Nr ∩ Γ. Set ϕn2 ,n2 ,...,nr−1 (g) =

Z

Nr ∩Γ\Nr

where θ stands for θn1 ,n1 ,...,nr−1 . Then ϕ has the following expansion: ϕ(g) =

X

ϕn1 ,n2 ,...,nr−1

"

! # γ 0 g 0 1

(3.3)

where we sum for all (r − 1)-tuples with ni ≥ 1 and γ in a set of representatives for Nr−1 ∩ Γr−1 \Γr−1 . Actually we will need to introduce also, j for 0 ≤ j ≤ r − 1, the subgroup Vr of matrices u ∈ Nr of the form ! 1n− j ∗ . u= 0 ∗ For j = r − 1 this is the group Nr itself. We will set: Z ϕnr− j ,nr− j+1 ,...,nr−1 (g) = ϕ(ug)θ(u)du j

j

Γ∩Vr \Vr

where θ = θn1 ,n2 ,...,nr−1 ; the right hand side does not depend on n1 , n2 , . . . , nr− j−1 which justifies the notation. Then we have the more general expansion: ! # " X γ 0 g (3.4) ϕnr− j+1 ,...,nr−1 (g) = ϕn1 ,n2 ,...,nr−1 0 1j where We sum for all r − j tuples (n1 , n2 , . . . , nr− j ) with ni ≥ 1 and all γ in a set of representatives for Nr− j ∩ Γr− j \Γr− j .

Dirichlet Series for the Group GL(N)

175

It is not simple to explain the ideas involved in these expansions. We 158 will point out however that our assertions are a mere reformulation of the expansions given in [P1] or [Sha]. So far our assertions do not depend on the assumption that ϕ be an eigen function of the Hecke algebra. If this assumption is taken in account, then it is found that ϕn1 ,n2 ,...,nr−1 (g) = an1 ,n2 ,...,nr−1 W(ζg)

(3.5)

where we have denoted by W the function ϕ1,1,...,1 (g) | {z }

(3.6)

diag(n1 n2 . . . nr−1 , n2 . . . nr−1 , . . . , nr−1 , 1).

(3.7)

r−1

and by ζ the diagonal matrix

The constants an1 ,n2 ,...,nr−1 which appear can be computed solely in terms of the homomorphisms of the Hecke algebra into C determined by ϕ. The reader will note that both sides of (3-5) transform on the left under the character θn1 ,n2 ,...,nr−1 of the group Nr . As for W, within a scalar factor, it is determined solely by the morphism λ of Z into C. Again, our assertions are mere reformulation of the results of [C-S], [Sha], [Shi].

4 The Mellin Transform Let us first simplify our notations. For 0 ≤ j ≤ r − 1 we set ϕ j = ϕ1,1,...,1 | {z }

(4.1)

so that ϕ0 = ϕ and ϕr−1 = W. We set also, for 1 ≤ j ≤ r − 1, an1 ,n2 ,...,n j = an1 ,n2 ,...,n j , 1, 1, . . . , 1 . | {z } r− j−1

176

Herve Jacquet

Combining (3.4) [with j = r − 1] with (3.5) we get X

ϕr−2 (g) =

an W

n≥1,ǫ=±1

"

! # nǫ 0 g . 0 1r−1

(4.2)

In view of this formula it is entirely reasonable to define the Mellin 159 transform of ϕ to be the integral Z

ϕ

R× /{±1}

It is equal to

X

−s

an n

n≥1

! a 0 |a| s−1 da. 0 1r−1

(4.3)

! a 0 W |a| s−1 da. 0 1r−1

(4.4)

r−2

Z



If we knew that the integral in (4.4) were a product of Γ -factors–as it should be–then the previous computation would give the analytic continuation of the Dirichlet Series X an n−s . (4.5) n≥1

On the other hand, just as in the case r = 2, the Dirichlet Series has an infinite Euler product: Y X 1 det(1 − p−s X p )−1 , an n−s+ 2 (r−1) = n≥1

p

where X p is a semi-simple conjugacy class in GL(r, C).

5 The convolution The convolution (1.6) also generalizes. Namely let ϕ′ be another cuspform on Gr , with r′ ≤ r. Let us denote with a prime the objects attached to ϕ.

Dirichlet Series for the Group GL(N)

Suppose first r′ ≤ r − 1. Consider the integral ! Z 0 r−1−r′ g ϕ′ (g)| det g| s d× g, ϕ 0 1r−r′

177

(5.1)

Γr′ \Gr′

where d× g is an invariant measure on the quotient Γr′ \Gr′ . Combining (3.4) with (3.5) we have the following expansion: " ! # X γ 0 r−1−r′ ϕ (g) = an1 ,n2 ,...,nr′ W g . (5.2) 0 1r−r′ ′

Replacing ϕr−1−r by this expression in (5.1) we get, after a “few” formal manipulations, X ′ (5.3) an1 ,n2 ,...,nr′ a′n1 ,n2 ,...,nr′ −1 |n1 n22 . . . nrr′ |−s n1 ≥1,n2 ≥1,...,nr ≥1

Z

Nr′ \Gr′

"

g 0 W 0 1r−r′

!#

W ′ [ǫg]| det g| s d× g,

where d× g is now an invariant measure on the quotient Nr′ , \Gr′ , and ǫ 160 is the r′ by r′ diagonal matrix diag(−1, 1, −1, . . .). The multiple series which appears in (5.3) may be regarded as a Dirichlet series in the usual sense. Again if we knew that the integral in (5.3) were a product of Γ-factors, our computations would give the analytic continuation of this series. Just as in the previous case, the series has an Euler product: X 1 ′ (5.4) an1 ,n2 ,...,nr′ a′n1 ,n2 ,...,nr′ −1 |n1 n2 . . . nr′ | s− 2 (r−r ) Y det(1 − p−s X p ⊗ X ′p )−1 . = p

r′ ,

When r = the previous construction needs to be modified. We denote by Φ the Schwartz-function on the space of row matrices with r entries defined by Φ(x) = exp(−πx · t x) (5.5)

178

Herve Jacquet

and we introduce an “Epstein zeta function”: +∞ X Z E(g, s) = Φ(tξg)|t|rs−1 dt| det g| s

(5.6)

ξ∈Z r −{0}−∞

[Here ξg is the product of the row matrix ξ by the square matrix g; t is a scalar]. It can also be written as an “Eisenstein series”: X Z Φ[(0, 0, . . . , 0, t)γg]|t|rs−1 dt | det g| s , (5.7) E(g, s) = ζ(rs) γ∈Γ∩Pr \Γ

where Pr is the standard parabolic subgroup of type (r − 1, 1). Then, instead of (5.2), we have to consider the integral Z ϕ(g)ϕ′ (g)E(g, s)d× g,

(5.8)

Zr Γ\Gr

where d× g is an invariant measure on the quotient Zr Γ\Gr . It turns out to be equal to X −s ζ(rs) an1 ,n2 ,...,nr−1 a′n1 ,n2 ,...,nr−1 |n1 n22 . . . nr−1 (5.9) r−1 | Z W(g)W ′ (ǫg)Φ[(0, 0, . . . , 0, 1)g]| det g| s d× g. Nr \Gr

161

Moreover: ζ(rs)

X

−s an1 ,n2 ,...,nr−1 a′n1 ,n2 ,...,nr−1 |n1 n22 . . . nr−1 r−1 | Y = det(1 − p−s X p ⊗ X ′p )−1 .

(5.10)

p

Remark 5.11. If we take r = 1 then ϕ = ϕ′ = ϕ0 , the constant function equal to one on G1 = R× ; moreover X p = X p′ = 1, and (5.10) reduces to the Euler product for the ζ-function. Similarly, we may regard the theory of §4 as a special case of the theory of §5 where r′ = 1 and ϕ′ = ϕ0 . This remark will be used without further warning.

Dirichlet Series for the Group GL(N)

179

6 Functional Equations We have already pointed out that we do not have enough information on the integrals of (4.4), (5.3), and (5.9). If we assume the missing information then we can address ourselves to the question of the functional equation satisfied by these Euler products. The functional equation should state that the analytic continuation of Y (1 − p−s X p ⊗ X ′p )−1 , p

times the appropriate Γ-factor, is equal to the analytic continuation of Y ′ −1 −1 (1 − p−1+s X −1 p ⊗X p ) , p

times the appropriate Γ-factor. To see this we introduce the function e ϕ(g) = ϕ(t g−1 ).

It is also a Maass cusp form. We denote by a tilda the objects attached to e ϕ. Then:   −1  0   1   t −1 e  , W(g) = W(wr g ), where wr =  −1    . .  . 0 ep = X −1 e an1 ,n2 ,...,nr−1 = anr−1 ,nr−2 ...nt , X p .

If r = r′ our starting point is the functional equation of the Epstein zeta-function: E(g, s) = E(t g−1 , 1 − s); from which we get Z Z ′ e ϕ(g)e ϕ′ (g)E(g, 1 − s)dg. ϕ(g)ϕ (g)E(g, s)dg =

180

Bibliography

The functional equation follows readily. ′ If r′ = r − 1 then ϕr−1−r is just ϕ. Clearly ! ! Z Z 1 g 0 ′ g 0 ′ s− 12 × e e ϕ (g)| det g| 2 −s d× g ϕ ϕ (g)| det g| d g = ϕ 0 1 0 1

and again the functional equation follows readily. However if r′ ≤ r − 2 (which includes the case r′ = 1) we have ′ to take in account a somewhat unexpected relation between e ϕr−r −1 and ′ ϕr−r −1 . Namely    Z 0 0  1r′ ′    ϕr−r −1  x 1r−r′ −1 0 t g−1  dx    0 0 1 ′

is actually a left-translate of e ϕr−r −1 (g); the integral is on the full space of matrices with r′ columns and r − r′ − 1 rows. Rather than trying to explain the details, we refer the reader to [J-S-P1] where the case r′ = 1, r = 3 is discussed.

Bibliography [C-S] Casselman W. and J. Shalika, Unramified Whittaker functions, to appear. [G-K] Gelfand J. M. and D. A. Kazdan, Representations of G1 (n, K) where K is a local field, in Lie groups and their representations, John Wiley & Sons (1975), 95–118. [J-S1] Jacquet H. and J. Shalika, Hecke theory for GL(3), Comp. Math., 29:1 (1974), 75–87. [J-S2] Jacquet H. and J. Shalika, Comparaison des representations automorphes du groupe line aire, C.R. Acad. Sc. Paris. 284 (1977), 741–744. [J-S-P1] Jacquet H., J. Shalika, and J. J. Piatetski-Shaprio, Automorphic forms on GL(3), I and II Annals of Math, 109 (1979).

162

Bibliography

181

[J-S-P2] Jacquet H., J. Shalika, and J. J. Piatetski Shapiro, Facteurs L et ǫ du groupe lineaire, to appear in C.R. Acad. Sci. (1979), Paris. [J-S-P3] Jacquet H., J. Shalika, and J. J. Piatetski-Shapiro, Construc- 163 tions of cusp forms on GL(n), Univ. of Maryland, Lectures Notes in Math. 16(1975). [P1] Piatetski-Shapiro J.J., Euler subgroups, in Lie groups and their representations, John Wiley and Sons (1975), 597–620. [P2] Piatetski-Shapiro J.J., Zeta functions on GL(n), Mimeographed notes, Univ. of Maryland. [Sha] Shalika J., The multiplicity one theorem for GL(n), Annals of Math. 100 (1974), 171–193. [Shi] Shintani T., On an explicit formula for class-1 “Whittaker functions” on GL over p-adic fields, Proc. Japan. Acad. 52 (1976), 180–182.

CRYSTALLINE COHOMOLOGY, DIEUDONNE´ MODULES, AND JACOBI SUMS By Nicholoas M. Katz 165

Introduction 166

Hasse [20] and Hasse-Davenport [21] were the first to realize the connection between exponential sums over finite fields and the theory of zeta and L-functions of algebraic varieties over finite fields. This connection was exploited to Weil; one of the very first applications that Weil gave of the then newly proven “Riemann Hypothesis” for curves over finite fields was the estimation of the absolute value of Kloosterman sums (cf [46]). The basic idea (cf [20]) is that by using the theory of L-functions, one can express the negative of such an exponential sum as the sum of certain of the reciprocal zeroes of the zeta function itself; because the magnitude of these zeroes is given by the “Riemann Hypothesis,” one gets an estimate. In a fixed characteristic p, the estimate one gets in this way for all the fintie fields F pn is best possible. On the other hand, very little is known about the variation with p of the absolute values, even for Kloosterman sums, though in this case there is a conjecture, of Sato-Tate type, which seems inaccessible at present. One case in which the problem of unknown variation with p does not arise is when the expression of the exponential sum as a sum of reciprocal roots of zeta reduces to a sum consisting of a single reciprocal root; then the Riemann Hypothesis tells us the exact magnitude of the 182

Crystalline Cohomology, Dieudonn´e Modules,...

183

exponential sum. Conversely, an elementary argument shows that in a certain sense, this is the only case in which such exact knowledge of the magnitude of exponential sums can arise, and it shows further that a theorem of Hasse-Davenport type always results from such exact knowledge. Examples of exponential sums of this sort are Gauss sums and Jacobi sums. Honda was the first to suggest that the identification of say, Jaboci sums, with reciprocal zeroes of zeta functions could also lead to significant non-archimedean information about Jacobi sums. A few years before his untimely death, Honda conjectured a p-adic limit formula for Jacobi sums in terms of ratios of binomail coefficients ([23]). I gave an over-complicated proof (in a letter to Honda of Nov. 1971) which managed to shed no light whatever on the meaning of the formula. Recently, B. H. Gross and N. Koblitz [14] showed that Honda’s limit formula was really an exact p-adic formula for Jacobi sums in terms of products of values of Morita’s p-adic Γ-function; as such, it constituted the first improvement in this century over Stickelberger’s formula which gave the p-adic valuation and the first non-vanishing p-adic digit in the p-adic 167 expansion of a Jacobi sum! In this paper, I will discuss the cohomological genesis of formulas of the sort discovered by Honda. The basic idea is that the reciprocal zeroes of zeta are the eigenvalues of the Frobenius endomorphism of a suitable cohomology group; if this group, together with the action of Frobenius upon it, can be made sufficiently explicit, one obtains the desired “explicit formulas”. There are two approaches to the question, which differ more in style than in substance. The first and longer is based on Honda’s explicit construction of the Dieudonn´e module of a formal group in terms of “formal de Rham cohomology”. The second, less elementary but more efficient, is grounded in crystalline cohomology, particularly in the theory of the de Rham-Witt complex. I hope the reader will share my belief that there is something to be gained from each of the approaches, and pardon my decision to discuss both of them. I would like to thank B. Dwork for many helpful discussions concerning the original proof of Honda’s conjecture. Whatever I know of

184

Nicholoas M. Katz

the Grothendieck-Mazur-Messing approach to Dieudonn´e theory through exotic Ext’s, I was taught by Bill Messing. I would also like to thank Spencer Bloch for his encouragement when I was trying to understand Honda’s explicit Dieudonn´e theory, and Luc Illusie for gently correcting some extravagent assertions I made at the Colloquium. Finally, I would like to dedicate this paper to the memory of T. Honda.

168

I. Elementary Axiomatics, and the Hasse-Davenport Theorem. Consider a projective, smooth and geometrically connected variety X, say of dimension d, over a finite field Fq . For each integer n ≥ 1, we denote by X(Fqn ) the finite set of points of X with values in Fqn , and by ♯X(Fqn ) the cardinality of this set. The zeta function Z(X/Fq , T ) of X over Fq is the formal power series in T with Q-coefficients defined as   X T n  Z(X/Fq , T ) = exp  ♯X(Fqn ) . n n≥1 Thanks to Deligne [6], we know that this zeta function has a unique expression as a finite alternating product of polynomials Pi (T ) ∈ Z[T ], i = 0, . . . , 2d: Z(X/Fq , T ) =

2d Y

Pi (T )(−1)

i+1

i=0

=

P1 P3 . . . P2d−1 P0 P2 . . . P2s

in which each polynomial Pi (T ) ∈ Z[T ] is of the form Pi (T ) =

deg YPi j=1

(1 − αi, j T )

with αi, j algebraic integers such that |αi, j | =

√ i q

¯ of all algebraic for any archimedean absolute value | | on the field Q numbers. The extreme polynomials P0 , P2d are given explicitly: P0 (T ) = (1 − T ), P2d (T ) = (1 − qd · T )

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185

Despite this apparently “elementary” characterization of the polynomials Pi (T ), their true genesis is cohomological. Let us recall this briefly. For each prime number l different from the characteristic p of Fq , let us denote by Hli (X) the finitely generated Zl -module defined as i (X ⊗ F¯ q , Z/ln Z). Hli (X) = lim Hetale ←−− n

Corresponding to the prime p itself, we denote by W(Fq ) the ring of pi (X) the finitely generated W(F )-module Witt vectors of Fq , and by Hcris q defined as i i Hcris (X) = lim Hcris (X/Wn (Fq )). ←−− n

The Frobenius endomorphism F of X relative to Fq acts, by functoriality, i (X); and on these various cohomology groups Hli (X) for l , p, and Hcris F induces automorphisms of the corresponding vector spaces O O i Hli (X) Ql , Hcris (X) K Zl

W(Fq )

(K denoting the fraction field of W(Fq )). The polynomial Pi (T ) ∈ Z[T ] which occurs in the factorization of the zeta function is then given cohomologically by the formulas Pi (T ) = det(1 − T F Hli (X) ⊗ Ql ) for l , p P (T ) = det(1 − T F H i (X) ⊗ K). i

cris

The resulting formula for zeta as the alternating product of characteristic polynomials of F on the H i , in each of the cohomology theories i (X) ⊗ K, is equivalent, via logarithmic differHli (X) ⊗ Ql for l , p, Hcris entiation, to the identities in those theories X (−1)i trace (F n H i ). for all n ≥ 1. , X(Fqn ) =

169

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By viewing the set X(Fqn ) as the set of fixed points of F n acting on X(F¯ q ), this identity becomes a Lefschetz trace formula # Fix (F n ) =

X

(−1)i trace (F n H i ) all n ≥ 1

for F and its iterates in each of our cohomology theories. If we take as given these Lefschetz trace formulas, then the identification of Pi with det(1 − FT H i ) is equivalent to the assertion: On any of the groups Hli (X) ⊗ Ql with l , p, i Hcris (X) ⊗ K, the eigenvalues of F are alge-

braic integers all of whose archimedean absolute √ values are qi . In fact, there is not a great deal more that is known about the action of F i (X) ⊗ K. It is still not known, for on the Hli (X) ⊗ Ql for l , p, and on Hcris example, whether the action of F on these cohomology groups is always semi-simple when i > 1. (That it is when i = 1 results from the theory of abelian varieties). Suppose that a finite group G operates on X by Fq -automorphisms. Let us choose a number field E big enough that all complex representations of G are realizable over E, and whose residue fields at all p-adic places contain Fq . (For example, the field Q(ζq−1 , ζN ), where N is the l.c.m. of the orders of elements of G, is such an E). We denote by λ an l-adic place of E, l , p, and by P a p-adic place of E. Thus Eλ is a finite extension of Ql , and EP is a finite extension of K. Let M be a finite dimensional E-vector space given with an action of G, say ρ : G → AutE (M). The associated L-function L(X/Fq , ρ, T ) is the formal power series with E-coefficients defined as

170

  X T n 1 X  L(X/Fq , ρ, T ) = exp  · tr(ρ(g−1 ))#Fix(F n g) n #G g∈G n≥1

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187

where Fix (F n g) denotes the finite set of fixed points of F n g acting on X(F¯ q ). We recover the zeta function of X/Fq by taking for ρ the regular representation of G. The usual formalism of zeta and L-functions gives Y Z(X/Fq , T ) = L(X/Fq , ρ, T )deg(ρ) ρ irred

It follows from Deligne’s results that for any representation ρ, we have a unique expression for the corresponding L-function as an alternating product of polynomials Pi,ρ (T ) ∈ E[T ], L(X/Fq , ρ, T ) =

2d Y

i+1

Pi,ρ (T )(−1) ,

i=0

which are of the form Pi,ρ (T ) =

deg Pi,ρ Y j=1

(1 − αi, j,ρ T )

with algebraic integers αi, j,ρ such that |αi, j,ρ | =

√ i q

for any archimedean absolute value | | on the field Q¯ of all algebraic numbers. The cohomological expression of there Pi,ρ is straighforward (cf. [18]). Because the action of G is “defined over Fq ” it commutes with F, and therefore the induced action of G on the cohomology commutes with the action of F. Therefore G, acting by composition, induces automorphisms of the Eλ -vector spaces,l , ρ, O O Eλ ). HomEλ [G] (M Eλ , Hli (X) E

Zl

and of the EP -vector spaces HomEP [G] (M

O E

i EP , Hcris (X)

O

W(Fq )

EP ).

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The polynomials Pi,ρ (T ) ∈ E[T ] are given by the formulas O O Pi,ρ (T ) = det(1 − T F HomEλ [G] (M Eλ , Hli (X) Eλ )) for l , ρ E

Z

l O O i EP )). EP , Hcris (X) Pi,ρ (T ) = det(1 − T F HomEP [G] (M

E

171

W(Fq )

Let us recall the derivation of these formulas. We first vobserve that the characteristic polynomial of F on HomG (M, H i ) ≃ ( M ⊗ H i )G ⊂ v v i divides det(1 − FT H i )dim M , and hence the eigenvalues of F on ⊗ H M HomG (M, H i ) are algebraic integers, all of whose archimedean absolute √ values are qi . So it remains only to verify that the alternating product of those characteristic polynomials is indeed the L-function, i.e. . that Y v i+1 L(X\Fq , ρ, T ) = det(1 − FT ( M ⊗ H i )G )(−1) ,

Equivalently, we must check that

1 X trace ρ(g−1 ) # Fix (F n g) #G X v = (−1)i trace (1 ⊗ F n ( M ⊗ H i )G ) X v 1 X = (−1)i trace (g ⊗ F n g M ⊗ H i ) #G g∈G X v 1 X trace ρ(g) · trace (F n g H i ) = (−1)i #G g∈G X X 1 trace ρ(g−1 ) (−1)i trace (F n g H i ). = #G g∈G To check this last equality, we would like to invoke the Lefschetz trace formula, not for F n , but for F n g, with g an automorphism of finite order which commutes with F; this amounts to invoking the Lefschetz trace formula for Fg on X and on all its “extensions of scalars” X ⊗ Fqn . But an elementary descent argument shows that given an automorphism g of finite order which commutes with F, there is another variety X ′ /Fq

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together with an isomorphism X ⊗ F¯ q ≃ X ′ ⊗ F¯ q under which Fg ⊗ 1 corresponds to F ⊗1. Because this isomorphism also induces isomorphisms of cohomology groups Hli (X)dfn Heti (X ′ ⊗ F¯ q , Zl ) ≃ H i (X ⊗ F¯ q , Zl )dfn Hli (X), H i (X ′ ) ⊗ W(F¯ q ) ≃ H i (X ′ ⊗ F¯ q ) ≃ H i (X ⊗ F¯ q ) ≃ cris

cris

cris

i ≃ Hcris (X) ⊗ W(F¯ q ),

the truth of the Lefschetz formula for Fg on X results from its truth for 172 F on X ′ . Let us now consider in greater detail the case of an irreducible ρ. Then Pi,ρ is a polynomial whose degree is the common multiplicity of ρ i (X) ⊗ E . Decomposing the in any of the Hli (X) ⊗ Eλ , l , ρ, or in Hcris P regular representation leads to the factorization Y Pi (T ) = Pi,ρ (T )deg(ρ) ρ irred

The coarser factorization Pi (T ) =

Y

(Pi,ρ (T )deg(ρ) )

ρirred i (X) ⊗ E , corresponds to the decomposition of Hli (X) ⊗ Eλ , resp. Hcris P into ρ-isotypical components O ρ Hli (X) ⊗ Eλ ≃ Hli (X) × Eλ irredρ

i Hcris (X) ⊗ EP ≃

O

irredρ

i Hcris (X) ⊗ EP



Indeed the corresponding identities, for ρ irreducible, are Pi,ρ (T )deg(ρ) = det(1 − T F (Hli (X) ⊗ Eλ )ρ )l , p P (T )deg(ρ) = det(1 − T F (H i (X) ⊗ E )ρ ). i,ρ

cris

P

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Let us denote by S (X/Fq , ρ, n) the exponential sums used to define the L-function: 1 X S (X/Fq , ρ, n) = tr(ρ(g))# Fix (F n g−1 ). #G g∈G The following lemma gives the cohomological meaning of theorems of Hasse-Davenport type (cf. [20]). Lemma 1.1: Let X/Fq be projective and smooth. Let a finite group G operate on X by Fq -automorphisms, and let p be an irreducible complex representation of G. Fix an N integer i◦ , and denote by H i◦ any None of the i◦ i◦ cohomology groups Hl (X) Zl Eλ with l , p, or Hcris EP . Let (X) W(Fq )

| | be any archimedean absolute value on the filed Q¯ of all algebraic numbers. The following conditions are equivalent: 173

(1) The multiplicity of ρ in H i◦ is one, and the multiplicity of ρ in H i is zero if i , i◦ . (2) For all n ≥ 1, we have and

(−1)i◦ S (X/Fq , ρ, n) = ((−1)i◦ S (X/Fq , ρ, 1))n , √ |S (X/Fq , ρ, 1)| = qi◦

(3) For all n ≥ 1, we have |S (X/Fq , ρ, n)| =

√ i◦ n q

(4) For all n ≥ 1, we have and

|S (X/Fq , ρ, n)| = |S (X/Fq , ρ, 1)|n √ i◦ √ q ≤ |S (X/Fq , ρ, 1)| < q1+i◦

(5) The polynomial Pi◦,ρ (T ) is given by Pi◦,ρ (T ) = 1 − (−1)i◦ S (X/Fq , ρ, 1)T and for i , i◦ , we have Pi,ρ (T ) = 1.

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191

(6) The ρ-isotypical component (H i )ρ = 0 for i , i◦ , (H i◦ )ρ has dimension = deg(ρ), and F operates on (H i◦ )ρ as the scalar (−1)i◦ S (X/Fq , ρ, 1). Proof. This is an easy exercise, using the basic identities: !  Y X Tn  i+1    S (X/F , ρ, n) = L(X/F , ρ, T ) = Pi,ρ (T )(−1) exp q q    n   i   Y  √ i   P (T ) = (1 − αi, j,ρ T ), |αi, j,ρ | = q i,ρ     j       1  i   · dim((H i )ρ).  deg Pi,ρ = multiplicity of ρ in H = deg(ρ)

Suppose, first, that (1) holds, or equivalently that for i , i◦ , Pi,ρ (T ) = √ 1, while Pi◦,ρ is a linear polynomial Pi◦,ρ (T ) = (1 − AT ) with |A| = qi◦ . The cohomological expression for L then becomes !(−1)i◦ ! X Tn 1 exp S (X/Fq , ρ, n) = . n 1 − AT Taking logarithms and equating coefficients, we find (−1)i◦ S (X/Fq , ρ, n) = An

174

for all n ≥ 1.

In particular (2) and (5) hold. The implications (5) ⇒ (1), (6) ⇒ (1) are obvious. Also (5) ⇒ (6), for if Pi◦,ρ is linear, then ρ has multiplicity one in H i◦ , so that (H i◦ )ρ is G-irreducible, and hence F must operate on (H i◦ )ρ as a scalar, which we compute by the formula P (T )deg(ρ) = det(1 − T F (H i◦ )ρ ). i◦,ρ

Clearly we have (2) ⇒ (3) ⇒ (4). We must show that if (4) holds, then exactly one of the Pi,ρ is , 1, and that one is linear. Logarithmically differentiating the cohomological formula for L, we find deg Pi,ρ X X i S (X/Fq , ρ, n) = (−1) (αi, j,ρ )n , i

j=1

|αi, j,ρ | =

√ i q.

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We must show that if (4) holds, then the double sum has only a single term in it. Separating the αi, j,ρ according to the parity of i, we get two disjoint sets of non-zero complex numbers (disjoint because their absolute values are disjoint), to which we apply the following lemma.  Lemma 1.2: Let N ≥ 0 and M ≥ 0 be non-negative integers. Let {Ai } be a family of N not-necessarily distinct elements of C x , and {Bi } a family of M not-necessarily distinct elements of C x . Suppose that for all i, j, Ai , B j . If, for some real number R > 0, we have X X Anj − Bnj = Rn for all n ≥ 1,

then N + M = 1, i.e. either there is just one A and no B’s, or just one B and no A’s.

Proof. Suppose first that either N = 0 or M = 0, say M = 0. Then we have X Ani = Rn . Squaring, we get

X (Ai A j )n = (R2 )n ij

175

whence

Y ij

for n ≥ 1

(1 − Ai A j T ) = (1 − R2 T ),

and hence N = 1. In case both N ≥ 1 and M ≥ 1, squaring leads to X X X X (Ai Ak )n + (B j Bl )n = (R2 )n + (Ai B j )n + (Ai B j )n or equivalently, Q Q (1 − Ai B j T ) (1 − Ai B j T ) 1 =Q Q (1 − R2 T ) (1 − Ai Ak T ) (1 − B j Bl T )

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193

Let Rmax be max(|Ai |, |B j |), and consider the order of pole at T = R−2 max . The numerator’s factors 1 − Ai B j T , 1 − Ai B j T are all non-zero there (for if Ai B j = R2max , by maximality we must have Ai = B j = Rmax , in which case we see, using polar coordinates, that Ai − B j , which is forbidden). In the denominator, each of the terms (1−|Ai |2 T ), (1−|B j |2 T ) with |Ai | = Rmax and |B j | = Rmax vanishes at T = R−2 max . Therefore we may conclude that in fact R = Rmax , and that precisely one among all the Ai and B j has this absolute value. A similar argument shows that Rmin = R.  In a similar but lighter vein, we have the following variant, whose proof is left to the reader. Lemma 1.3. Let X/Fq be projective and smooth. Let a finite group G operate on X by Fq -automorphisms, and let ρ be an irreducible complex representation of G. Denote by H i any of the cohomology groups i (X)⊗ E . The following conditions are Hli (X)⊗Eλ with l , p, or Hcirs P W

Zl

equivalent. (1) For all i, ρ does not occur in H i , i.e. we have (H i )ρ = 0. (2) For all n ≥ 1, we have S (X/Fq , ρ, n) = 0.

II. Gauss and Jacobi Sums as exponential sums, and as eigenvalues 176 of Frobenius We begin by discussing Gauss sums. Let us fix an integer N ≥ 2 prime to p, and a number field E containing the Np’th roots of unity. Given an additive character ψ of F p , i.e. a homomorphism ψ : (F p , +) → E × , we define an additive character ψq of each finite extension Fq by composing ψ with the trace map:

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Given a character of µN , i.e. a homomorphism χ : µN (E) → E × , a p-adic place P of E, with residue field F N(P) , and a finite extension Fq of this residue field, the map “reduction mod P” induces an isomorphism ∼

µN (E) − → µN (F N(P) ) = µN (Fq ) Because Fq× is cyclic, we know that q ≡ 1mod N, and that the map x→x

q−1 N

defines a surjection



Fq× ։ µN (Fq ) = µN (F N(P) ) − → µN (E) We define the character χq of Fq× as the composite

The Gauss sum gq (ψ, χ, P) attached to this situation is defined by the formual X gq (ψ, χ, P) = ψq (x)χq (x) x∈Fq×

177

An elementary computation shows that    q − 1 if ψ, χ both trivial     gq (ψ, χ, P) =  0 if ψ trivial, χ non-trivial     −1 if ψ non-trivial, χ trivial

Crystalline Cohomology, Dieudonn´e Modules,...

while |gq (ψ, χ, P)| =



195

q if ψ, χ both non-trivial

for any archimedean absolute value on E (cf [47]). Now consider the Artin-Schreier curve X/Fq , defined to be the complete non-singular model of the affine smooth geometrically connected curve over Fq with equation T P − T = XN . Set theoretically, X consists of this affine curve plus a single rational point at ∞. The group Fq × µN (Fq ) operates on X/Fq curve by the affine formulas (a, ζ) : (T , X) → (T + a, ζX), fixing the point at ∞. Via the “reduction mod P” isomorphism ∼

µN (E) − → µN (F N(P) ) = µN (Fq ), we may view (ψ, χ) as a character of the group F p × µN (Fq ): (ψ, χ)(a, ζ) = ψ(a)χ(ζ). Thus we may speak of the sums S (X/Fq , (ψ, χ), n) =

1 pN

X

(a,ζ)∈F p ×µN

ψ(a)χ(ζ)♯ Fix (F n · (a, ζ)−1 )

attached to this situation. Lemma 2.1. If χ is non-trivial and ψ is arbitrary, then we have S (X/Fq , (ψ, χ), n) = gqn (ψ, χ, P). Proof. It suffices to treat the case n = 1, for we have S (X/Fqn , (ψ, χ), 1) = S (X/Fq , (ψ, χ), n).

(2.1.1) 178

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Nicholoas M. Katz

We can rewrite S (X/Fq , (ψ, χ), 1) as X

x∈X(Fq)

1 pN

X

ψ(a)χ(ζ)

(a,ζ)s.t. F(x)=(a,ζ)(x)

Given any point x ∈ X(F q ), the set of (a, ζ) ∈ F p × µN which satisfy F(x) = (a, ζ)(x) is either empty or principal homogeneous under the inertia subgroup I x of F p × µN which fixes x; therefore if the restriction of (ψ, χ) to this subgroup is non-trivial, the inner sum above vanishes. Because χ is assumed non-trivial, this vanishing applies to the point at ∞ (for which I x is all of F p × µN ) and to any finite point (T , 0) whose X-coordinate is zero (then I(T ,0) = {0} × µN ). Given a point (T , X) with X , 0, we have F(T , X) = (T q , X q ) and the inertia subgroup I(T ,X) is trival. If there is an element (a, ζ) ∈ F p ×µN satisfying F(T , X) = (T +a, ζX), then it is given by the formulas a = T q − T , ζ = X q−1 Since the point (T , X) is subject to the defining equation T p − T = XN we see that (X N )q−1 = (X q−1 )N = ζ N = 1, hence X N ∈ Fq× , ζ = (X N )

q−1 N

T p − T = X N ∈ Fq× , a = T q − T = traceFq /F p (T p − T ) = traceFq /F p (X N ). For each u ∈ Fq× , the equations (T P − T = u, X N = u) have pN solutions (T , X) over F q , all of which satisfy F(T , X) = (a, ζ)(T , X)

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197

q−1

179

with the same (a, ζ), namely (traceFq /F p (u), u N ), and every point (T , X) which contributes to our sum lies over some u ∈ Fq× . Thus our sum becomes X q−1 ψ(traceFq /F p (u))χ(u N )dfn gq (ψ, χ, P). u∈Fq×

 Corollary 2.2. Let H i denote any of the cohomology groups Hli (X) ⊗ Eλ i (X)⊗ E of the Artin Schreier curve X/F . with l , p, or Hcris P q W

(1) If ψ and χ are both non-trivial, then the eigenspace (H 1 )ψ·χ is one-dimensional, and we have a direct sum decomposition H i = ⊕(H 1 )ψ·χ indexed by the (p−1(N −1) pairs (ψ, χ)) of non-trivial characters. (2) The eigenvalue of F on (H 1 )ψ·χ is −gq (ψ, χ, P), and for each n ≥ 1 we have the Hasse-Davenport formula −gqn (ψ, χ, P) = (−gq (ψ, χ, P))n . (3) The group Fq × µN acts trivially on both H 0 and H 2 . Proof. That the group acts trivially on both H 0 and H 2 follows from the fact that these are one-dimensional spaces on which F always acts as 1 and q respectively. The descent argument shows that for any automorphism of finite order g which commutes with F, Fg also acts as 1 and q on H 0 and H 2 respectively, and hence that g itself acts trivially on H 0 and H 2 . That the multiplicity of (ψ, χ) in H 1 is one when both ψ and χ are non-trivial follows from the lemma of the previous section, given the identity (2.1.1) and the known absolute value of gauss sums; and assertion (2) above is just a repetition of part of that lemma in this particular case. To see that no other characters occurs in H 1 , we recall that the dimension of H 1 is known to be 2g, g = genus of X, and so it suffices

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Nicholoas M. Katz

to verify that 2g = (p − 1)(N − 1). This formula, whose elementary verification we leave to the reader, is in fact valid in any characteristic prime to N(p − 1). (Hing: view T P − T = X N an an N-fold covering of the T -line!)  180

We now turn to the consideration of Jacobi sums. We fix an integer N ≥ 2 prime to p, and a number field E containing the N’th roots of unity. Given a p-adic place P of E, a character χ of µN χ : µN (E) → E × and a finite extension Fq of the residue field F N(P) at P, we obtain the character χq χq : Fq× → E × in the manner explained above. Given two characters χ, χ′ of µN , the Jacobi sum Jq (χ, χ′ , P) is defined by the formula dfn

Jq (χ, χ′ , P) =

X

x∈Fq x,0,1

χq (x)χ′q (1 − x).

An elementary computation (cf [14]) shows that if the product χχ′ is non-trivial, then for any non-trivial additive character ψ of F p , we have the formula gq (ψ, χ, P)gq (ψ, χ′ , P) = Jq (χ, χ′ , P)gq (ψ, χχ′ , P) In particular, from the known absolute values of Gauss sums we obtain |Jq (χ, χ′ , p)| =

√ q

for all archimedean absolute values of E, provided that χ, χ′ , and χχ′ are all non-trivial. Now consider the Fermat curve Y/Fq , defined by the homogeneous equation XN + Y N = ZN

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199

The group µN × µN operates on this curve by the formula (ζ1 , ζ2 ) : (X, Y, Z) → (ζ1 X, ζ2 Y, Z). Viewing (χ, χ′ ) as a character of this group dfn

(χ, χ′ )(ζ1 , ζ2 ) = χ(ζ1 )χ′ (ζ2 ), we may speak of the sums S (Y/Fq , (χ, χ′ )n) attached to this situation. In complete analogy with the situation for the Artin-Schreier curve, 181 we have the following lemma and corollary, whose analogous proofs are left to the reader. Lemma 2.3. If χ and χ′ are non-trivial characters of µN such that χχ′ is also non-trivial, then we have, for all n ≥ 1, S (Y/Fq , (χ, χ′ ), n) = Jqn (χ, χ′ , p).

(2.3.1)

Corollary 2.4. Let H i denote any of the cohomology groups Hli (Y) ⊗ Eλ i (X)⊗ E of the Fermat curve Y/F . with l , p, or Hcris p q W



(1) If χ, χ′ and χχ′ are all non-trivial, then the eigenspace (H 1 )(χ,χ ) is one-dimensional, and we have a direct sum decomposition ′

H 1 = ⊕(H 1 )(χ,χ ) indexed by the (N −1)(N −2) pairs (χ, χ′ ) of non-trivial characters of µN whose product χχ′ is also non-trivial. ′

(2) The eigenvalue of F on (H 1 )(χ,χ ) is −Jq (χ, χ′ , P), and for each integer n ≥ 1 we have the Hasse-Davenport formula −Jqn (χ, χ′ , P) = (−Jq (χ, χ′ , P))n . (3) The group µN × µN operates trivially on both H 0 and H 2 .

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III. The problem of “explicitly” computing Frobenius. We return now to the general setting of a projective, smooth, and geometrically connected variety X/Fq of dimension d. A tantalizing feature of all the cohomology theories that we have been discussing is that when the variety X “lifts” to characteristic zero, then the corresponding cohomology groups H i (X) have an “elementary” description in terms of standard algebro-geometric and topological invariants of the lifting. More precisely, suppose we are given a projective smooth scheme X over W(Fq ), together with an Fq -isomorphism of its special fibre with X. (This is a rather strong notion of what a “lifting” of X should mean, but it is adequate for our purposes, and it avoids certain technical problems related to ramification). Then there is a canonical isomorphism i i Hcris → HDR (X/W(Fq )) i of Hcris with the algebraic de Rham cohomology of the lifting (cf [19], [27]). To discuss Hli (X), we must in addition choose (!) a complex embedding W(Fq ) ֒→ C.

By means of such an embedding, we may “extend scalars” to obtain from X/W a projective smooth complex variety XC , and an associated complex manifold XCan . For each prime number l , p, there is a canonical isomorphism i Hli (X) → Htop (XCan , Z)×Zl , Z

i Htop

where denotes the usual “topological” cohomology. To emphasize the similarity between these two sorts of isomorphisms, recall that by GAGA and the holomorphic Poincar´e lemma, we have a canonical isomorphism i (X/W)⊗C HDR W



/ H i (X/C) DR



/ H i (X an , C) ⊤ C O



182

H⊤i (X an , Z)⊗C Z

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201

Unfortunately, these rather concrete descriptions of the various cohomology groups H i (X) shed little light on their functoriality. In the rather unusual case of an Fq -endomorphism f : X → X which happens to admit a lifting to a W-endomorphism f : X → X, we have the simple formulas   ∗ i ∗ i    f on Hcris (X) = f on HDR (X/W)  an an ∗ i ∗ i    f on Hl (X) = ( fC ) ⊗ 1 on H⊤ (XC , Z)⊗Zl , l , p Z

But for those f which do not lift, we are left somewhat in the dark as to an explicit description of the map f ∗ on cohomology. Suppose for example that a finite group G operates on X by Fq automorphisms, and that this action can be lifted to an action of G on X by W-automorphisms. Then our canonical isomorphisms  ∼ i (X) − i (X/W)   → HDR Hcris  ∼  H i (X) − → H⊤i (XCan , Z) ⊗ Zl for l , p l

are G-equivariant. In particular, we can “explicitly compute” the mul- 183 tiplicities of the various complex irreducible representations ρ of G in the cohomology of X, and we can “explicitly compute” the various isotypical components of the cohomology. If it turns out that a given irreducible representation ρ occurs in a given H i with multiplicity one, then we know a priori that F must operate on the corresponding isotypical component (H i )ρ as a scalar, and we know this even when F itself does not lift. For example, we could recover the isotypical decomposition of H 1 of the Fermat curve Y under the action of µN × µN by lifting the curve and the group action (use the “same” equations) and making an explicit algebro-geometric or topological calculation of the corresponding isotypical decomposition in characteristic zero. In terms of, say, the crystalline cohomology, we obtain an F-stable decomposition ∼



1 1 Hcris (Y) − → HDR (Y/W)(χ,χ ) ;

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1 (Y/W) adapted to this decomposition, the matrix of F in a basis of HDR is the diagonal matrix     . . . O     ′ −Jq (χ, χ , P)    . .  . O

However, it must be borne in mind that the Fermat curve is atypically susceptible to this sort of analysis; it is unusual for a group action, even on a curve, to be liftable to characteristic zero. For example, the action of F p on an Artin-Schreier covering of A1 doesn’t lift to characteristic zero. To get around this non-liftability, we will be led to consider the Washnitzer-Monsky cohomology as well, in Chapter VII.

IV. H1 and abelian varieties; preliminaries. Consider an abelian variety A/Fq , say of dimension g. We denote by End(A) the ring of all Fq endomorphisms of A, and by End(A)0 the opposite ring. As Z-modules, 184 they are free and finitely generated. For each prime l , p, the cohomology group Hl1 (A) is a free Zl -module of rank 2g, and is an End(A)0 1 (A) is a free W-module of rank 2g, module. (It is also the case that Hcris and is an End(A)0 -module, but we will not make use of this fact for the moment). Lemma 4.1. If E is a number field, and λ is a place of E lying over a prime l , p, the natural maps End(A)0 ⊗E

/ End(A)0 ⊗Eλ Z

/ EndZ (H 1 (A))⊗Eλ l

l

O

Zl



Z

EndEλ (Hl1 (A)⊗Eλ ) Zl

are all injective. Proof. The first map is injective simply because E ⊂ Eλ , and because End(A)0 is flat over Z. The second map is obtained from the map End(A)0 ⊗Zl → EndZl (Hl1 (A)) Z

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by tensoring over Zl with the flat Zl -module Eλ . In fact this flatness is irrelevant, for the above map is injective and has Zl -flat cokernel. To see this, recall that (by the Kummer sequence in etale cohomology) we have a canonical isomorphism Hl1 (A) ≃ T l (Pic0 (A))(−1) ≃ Hom(T l (A), Zl ), under which the map considered above is the “opposite” of the map End(A)⊗Zl → EndZl (T l (A)) Z

Our assertion of its injectivity with Zl -flat cokernel is equivalent to the injectivity of (any one of) the maps End(A)/ln End(A) → End(Aln ), and this injectivity follows from the exactness of the sequence ln

0 → Al n → A − →A→0 in the etale topology.



Now consider a projective, smooth and geometrically connected variety X/Fq . Its Albanese variety Alb(X) is an abelian variety over Fq which for our purposes is best viewed as the dual of the Picard va- 185 riety Pic(X), itself defined in terms of the Picard scheme PicX/Fq as (Pic0X/Fq )red . The Kummer sequence in etale cohomology together with the duality of abelian varieties gives isomorphisms for each l , p ∼

Hl1 (X) − → T l (Pic(X))(−1)

(4.1.1)



H 1 (Alb(X)) − → T l (Pic(Alb(X))(−1) = T l (Pic(X))(−1)

(4.1.2)

which combine to give a canonical isomorphism Hl1 (X) ≃ Hl1 (Alb(X)) for

l,p

(4.1.3)

Suppose now that a finite group G operates on X by Fq -automorphisms. Let ρ be an absolutely irreducible representation of G defined

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over a number field E, which occurs in H 1 (X) with multiplicity r. Denote by P1,ρ (T ) = 1 + a1 (ρ)T + · · · + ar (ρ)T r ∈ OE [T ] the reversed characteristic polynomial of F acting on the space HomG (ρ, H 1 (X)) of occurrences of ρ in H 1 ; P1,ρ (T ) = det(1 − T F| HomG (ρ, H 1 (X)). Let us denote by Proj(ρ) ∈ OE [1/♯G][G] the projector Proj(ρ) =

deg(ρ) X tr(ρ(q−1 )) · [g]. ♯G g∈G

By functoriality, G also operates on Alb(X) by Fq -automorphisms, so we may view Proj(ρ), or indeed any element of the OE [1/♯G]-group ring of G, as defining an element of End(Alb(X)) ⊗ OE [1/♯G]. Proposition 4.2. In the above situation, we have the formula (F r + a1 (ρ)F r−1 + · · · + ar (ρ)) · Proj(ρ) = 0

Proj(ρ) · (F r + a1 (ρ)F r−1 + · · · + ar (ρ)) = 0

in End(Alb(X)) ⊗ OE [1/♯G]. (N.B. since F and G commute, these formulas are equivalent).

186

Proof. Since End(Alb(X)) ⊗ OE [1/♯G] is contained in End(Alb(X)) ⊗ E, which is in turn contained in End(Hl1 (Alb(X))⊗Eλ ) for any l , p, it sufZ

fices to verify that F r + a1 (ρ)F r−1 + · · · + ar (ρ) annihilates (H 1 (Alb(X))ρ . But this space is isomorphic to (H 1 (X))ρ , which is in turn isomorphic to ρ ⊗ HomG (ρ, H 1 (X)), with F acting through the second factor, so we need the above polynomial in F to annihilate HomG (ρ, H 1 (X)). This follows from the Cayley-Hamilton theorem. 

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Corollary 4.3. Let D be any contravariant additive functor from the category of abelian varieties over Fq to the category of OE [1/♯G]-modules. For any element m ∈ (D(Alb(X)))ρ , we have F r (m) + a1 (ρ)F r−1 (m) + · · · + ar (ρ) · m = 0 in D(Alb(X)). We will apply this to the functor “Dieudonne module of the formal group of A,” constructed a la Honda. V. Explicit Dieudonn´e Theory a` la Honda; generalities 5.1. Basic Constructions. We being by recalling the notions of formal Lie variety and formal Lie groups. Over any ring R, an n-dimensional formal Lie variety V is a set-valued functor on the category of adic Ralgebras which is isomorphic to the functor. R′ → n-tuples of topologically nilpotent elements of R′ . A system of coordinates X1 , . . . , Xn for V is the choice of such an isomorphism. The coordinate ring A(V) is the R-algebra of all maps of set-functors from V to the “identical functor” R′ 7→ R′ ; in coordinates, A(V) is just the power series ring R[[X1 , . . . , Xn ]]. Although the ideal (X1 , . . . , Xn ) in A(V) is not intrinsic, the adic topology it defines on A(V) is intrinsic, and A(V), viewed as an adic R-algebra, represents the functor V. i (V/R) are the R-modules obThe de Rham cohomology groups HDR tained by taking the cohomology groups of the formal de Rham complex ΩV/R (the separated completion of the “literal” de Rham complex of A(V) as R-algebra); in terms of coordinates X1 , . . . , Xn for V, ΩV/R is the exterior algebra over A(V) on dX1 , . . . , dXn , with exterior differentiation d : Ωi → Ωi+1 given by the customary formulas. A pointed formal Lie variety (V, 0) over R is a formal Lie variety V 187 over R together with a marked point “0” ǫV(R). A formal Lie group G over R is a “group-object” in the category of formal Lie varieties over R. •



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We denote by CFG(R) the additive category of commutative formal Lie groups over R. The “sum” map sum : G × G → G as well as the two projections pr1 , pr2 : G × G → G are morphisms in this category. For G ∈ CFG(R), we define D(G/R) to 1 (G/R) consisting of the primitive elements, be the R-submodule of HDR 1 (G/R) such that i.e. the elements a ∈ HDR 1 sum∗ (a) = pr∗1 (a) + pr∗2 (a) in HDR ((G × G)/R).

Lemma 5.1.1. Over any ring R, the construction G → D(G/R) defines a (contravariant) additive functor from CFG(R) to R-modules. Proof. This is a completely “categorical” result. To begin, let G, G′ ∈ CFG(R), and let f : G′ → G be a homomorphism. Then the diagram G′ × G′ 

sum

/ G′

sum



f×f

G×G

f

/G

commutes, as do the analogous diagrams with “sum” replaced by pr1 or 1 (G/R), we have pr2 . Therefore given any element a ∈ HDR sum∗ ( f ∗ (a)) − pr∗1 ( f ∗ (a)) − pr2 ( f ∗ (a)) =

( f × f )∗ (sum∗ (a) − pr∗1 (a) − pr∗2 (a)).

In particular, if a ∈ D(G/R) then f ∗ (a) ∈ D(G′ /R). Given f1 , f2 homomorphisms G′ → G, let f3 be their sum. Then we have a commutative diagram

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sum

188

as well as a commutative diagram

1 (G/R), we have Therefore for any a ∈ HDR

f3∗ (a) − f1∗ (a) − f2∗ (a) = ( f1 × f2 )∗ (sum∗ (a) − pr∗1 (a) − pr∗2 (a)).

In particular, if a ∈ D(G/R), then f3∗ (a) = f1∗ (a) + f2∗ (a).



For the remainder of this section, we will consider a ring R which is flat over Z, and an ideal I ⊂ R which has divided powers. The flatness means that if we denote by K the Q-algebra R ⊗ Q, then R ⊂ K. That the ideal I ⊂ R has divided powers means that for any integer n ≥ 1, and any element i ∈ I, the element in /n! of K actually lies in I. Given a formal Lie variety V over R, we denote by V ⊗ K the formal Lie variety over K obtained by extension of scalars. In terms of coordinates X1 , . . . , Xn for V, A(V⊗K) is the power-series ring K[[X1 , . . . , Xn ]]. We say that an element of A(V ⊗ K) is integral if it lies in the subring A(V); similarly, an element of the de Rham complex ΩV⊗K/K is said to be integral if it lies in the subcomplex ΩV/R . Lemma 5.1.2. Let (V, 0) be a pointed Lie variety over a Z-flat ring R. Then exterior differentiation induces an isomorphism of R-modules { f ∈ A(V ⊗ K)| f (0) = 0, d f integral} ∼ 1 − → HDR (V/R) { f ∈ A(V)| f (0) = 0}

which is compatible with morphisms of pointed Lie varieties.

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Proof. Because K is a Q-algebra, the formal Poincare lemma gives 0 (V ⊗ K/K) = K, H i (V ⊗ K/K) = 0 for i ≥ 1. Therefore any HDR DR closed one-form on V/R can be written as df with f ∈ A(V ⊗ K), and this f is unique up to a constant. If we normalize f by the condition f (0) = 0, we get the asserted isomorphism.  Key Lemma 5.1.3. Let (V, 0) and (V ′ , 0) be pointed formal Lie varieties over a Z-flat ring R, and let I ⊂ R be an ideal with divided powers. If f1 , f2 are two pointed morphisms V ′ → V such that f1 = f2 mod I, then the induced maps 1 1 f1∗ , f2∗ : HDR (V/R) → HDR (V ′ /R)

are equal. Proof. Let ϕ1 , ϕ2 denote the algebra homomorphisms A(V) → A(V ′ ) corresponding to f1 and f2 . By the previous lemma, we must show that for every element f ∈ A(V ⊗ K) with f (0) = 0 and d f integral, the difference ϕ1 ( f ) − ϕ2 ( f ) lies in A(V ′ ), i.e. is itself integral. (Because f1 and f2 were assumed pointed, this difference automatically has constant term zero). In terms of pointed coordinates X1 , . . . , Xn for V ′ and Y1 , . . . , Ym for V, the maps ϕ1 and ϕ2 are given by substitutions ϕ1 ( f (Y)) = f (ϕ1 (X)) ϕ2 ( f (Y)) = f (ϕ2 (X)) where ϕ1 (X), ϕ2 (X) are m-tuples of series in X = (X1 , . . . , Xn ) without constant term. The hypothesis f1 = f2 mod I means that the componentby-component difference ∆ = ϕ2 (X) − ϕ1 (X) satisfies ∆(0) = 0, ∆ has all coefficients in I. We now compute using Taylor’s formula, and usual multi-index notations: ϕ2 ( f ) − ϕ1 ( f ) = f (ϕ2 (X)) − f (ϕ1 (X))

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= f (ϕ1 (X) + ∆) − f (ϕ1 (X)) X ∆n ∂n ! = f (ϕ1 (X)). n (n)! ∂Y |n|≥1 This last sum is X-adically convergent (because ∆ has no constant term), 190 and its individual terms are integral (because ∆ has coefficients in the divided power ideal I, the terms ∆n /(n)! all have coefficients in I, and hence in R; because d f is integral, all the first partials ∂ f /∂Yi are integral, and a fortiori all the higher partials are integral).  Theorem 5.1.4. Let R be a Z-flat ring, and I ⊂ R a divided power ideal. Let G, G′ be commutative formal Lie groups over R, and denote by G0 , G′0 the commutative formal Lie groups over R0 = R/I obtained by reduction mod I. (1) If f : G′ → G is any morphism of pointed formal Lie varieties whose reduction mod I, f0 : G′0 → G0 , is a group homomor1 (G/R) → H 1 (G ′ /R) maps phism, then the induced map f ∗ : HDR DR ′ D(G/R) to D(G /R). (2) If f1 , f2 , f3 are three maps G′ → G of pointed formal Lie varieties whose reductions mod I are group homomorphisms which satisfy ( f3 )0 = ( f1 )0 + ( f2 )0 in Hom(G′0 , G0 ), then for any element a ∈ D(G/R) we have f1∗ (a) + f2∗ (a) = f3∗ (a). Proof. If f : G′ → G is a pointed map which reduces mod I to a group homomorphism, the diagram G′ × G′ 

sum

/ G′

sum



f×f

G×G

f

/G

commutes mod I, i.e. sum ( f × f ) ≡ f sum mod I.

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1 (G/R) we have, by the previous lemma, and hence for any a ∈ HDR

( f × f )∗ (sum∗ (a)) = sum∗ ( f ∗ (a)) The analogous diagrams with “sum” replaced by pr1 or pr2 commute, hence ( f × f )∗ (pr∗i (a)) = pr∗i ( f ∗ (a)) for i = 1, 2. 191

Combining these, we find ( f × f )∗ (sum∗ (a) − pr∗1 (a) − pr∗2 (a)) =

sum∗ ( f ∗ (a)) − pr∗1 ( f ∗ (a)) − pr∗2 ( f ∗ (a)).

In particular, if a ∈ D(G/R) then f ∗ (a) ∈ D(G′ /R). Similarly, if f1 , f2 and f3 are as in the assertion of the theorem, the diagram sum

commutes mod I, and the diagram

commutes. So again using the preceding lemma, we see that for any 1 (G/R), we have a ∈ HDR f3∗ (a) − f1∗ (a) − f2∗ (a) = ( f1 × f2 )∗ (Sum∗ a) − pr∗1 (a) − pr∗2 (a)). In particular, for a ∈ D(G/R), we obtain the asserted formula f3∗ (a) = f1∗ (a) + f2∗ (a).

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 Let CFG(R; R0 ) denote the additive category whose objects are the commutative formal Lie groups over R, but in which the morphisms are the homomorphisms between their reductions mod I: HomCFG(R,R0 ) (G′ , G) = Hom(G′0 , G0 ). Given a homomorphism f0 : G′0 → G0 , it always lifts to a pointed morphism f : G′ → G of formal Lie varieties (just lift its power-series 192 coefficients one-by-one, and keep the constant terms zero). According to the theorem, the induced map f ∗ : D(G/R) → D(G′ /R) is independent of the choice of pointed lifting f of f0 . So it makes sense to denote the induced map ( f0 )∗ : D(G/R) → D(G′ /R). Theorem 5.1.5. Let R be a Z-flat ring, and I ⊂ R a divided power ideal. Then the construction G 7→ D(G/R), f0 7→ ( f0 )∗ = (any pointed lifting)∗ defines a contravariant additive functor from the category CFG(R; R0 ) to the category of R-modules. Proof. This is just a restatement of the previous theorem.



Remarks. (1) Thanks to Lazard [33], we know that every commutative formal Lie group G0 over R0 lifts to a commutative formal Lie group G over R. If G′ is another lifting of G0 , then the identity endomorphism of G0 is an isomorphism of G′ with G in the category CFG(R; R0 ). Formation of the induced isomorphism ∼ D(G/R) − → D(G′ /R) provides a transitive system of identifications between the D’s of all possible liftings. In this way, it is possible to view the construction G0 7→ D(G/R), where G is some lifting of G0

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as providing a contravariant additive functor from CFG(R0 ) to the category of R-modules. We will not pursue that point of view here. (2) Even without appealing to Lazard, one can proceed in an elementary fashion by observing that any commutative formal Lie group G0 over R0 can certainly be lifted to a formal Lie “monoid with unit” M over R (simply lift the individual coefficients of the group law, and always lift 0 to 0). For a monoid, one can still define 1 (M/R), and one can still D(M/R) as the primitive elements of HDR show exactly as before that the construction G0 → D(M/R), 193

M any monoid lifting of G0

defines a contravariant additive functor from CFG(R0 ) to R-modules. A variant. The reader cannot have failed to notice the purely formal nature of most of our arguments. We might as well have begun with any contravariant functor H from formal Lie varieties over a Z-flat ring R to R-modules for which the key lemma (5.1.3) holds. One such H, which 1 (V/R; I), is defined as H 1 of the subcomplex of the we will denote HDR de Rham complex of V/R “IA(V)′′ → Ω1V/R → Ω2V/R → . . . where “IA(V)” denotes the kernel of reduction mod I: “IA(V)′′ = Ker(A(V) ։ A(V0 )). In terms of coordinates for V, “IA(V)” is the ideal consisting of those series all of whose coefficients lie in I. The analogue of lemma (5.1.2) becomes d

{ f ∈ A(V ⊗ K)| f (0) = 0, dt integral} ∼ 1 −→ HDR (V/R; I). { f ∈ “IA(V)′′ | f (0) = 0}

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This much makes sense for any ideal I ⊂ R. If I has divided powers, then the proof of the key lemma (??) is almost word-for-word the same. (It works because the terms ∆n /(n)! all have coefficients in I.) 1 (G/R; I),” is The corresponding theory, “primitive elements in HDR denoted D1 (G/R). In terms of coordinates X = (X1 , . . . , Xn ) for G, we have the explicit description D1 (G/R) = { f ∈ K[[X]]| f (0) = 0, d f integral, f (X+Y) − f (X) − f (Y) ∈ I[[X, Y]]} G = { f ∈ I[[X]]| f (0) = 0}

as compared with the explicit description

=

D(G/R) = { f ∈ K[[X]]| f (0) = 0, d f integral, f (X+Y) − f (X) − f (Y) integral} G

{ f ∈ R[[X]]| f (0) = 0}

For ease of later reference we summarize the above discussion in a the- 194 orem. Theorem 5.1.6. Let R be a Z-flat ring, and I ⊂ R a divided power ideal. 1 (V/R; I), and theorems (5.1.4) and The key lemma (??) holds for HDR (5.1.5) hold for D1 (G/R). The natural map D1 → D is not an isomorphism, but its kernel and cokernel are visibly killed by I. In the work of Honda and Fontaine, it is D1 rather than D which occurs; in the work of Grothendieck and Mazur-Messing ([17], [35]), it is D which arises more naturally. Let us denote by ωG/R the R-module of translation-invariant, or what is the same, primitive, one-forms on G/R. Because G is commutative, every element w ∈ ωG/R is a closed form, so we have natural maps ωG/R

/ D1 (G/R) 

D(G/R)

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(Notice that in the extreme case I = (0), the map ω → D1 is an isomorphism!) Lemma 5.1.7. Suppose R flat over Z, and I ⊂ R an ideal. We have exact sequences d

0 → HomR-groups (G, Ga ) → − ωG/R → D(G/R)

0 → |HomR/I-groups (G⊗(R/I), (Ga )R/I ) → D1 (G/R) → D(G/R) R

Proof. The first is the special case I = 0 of the second; the second is clear from the explicit description of D1 and D given above.  Corollary 5.1.8. If HomR-groups (G, Ga ) = 0, then the natural maps ωG → D1 (G/R) and

ωG → D(G/R)

are injective. 195

The reader interested in obtaining the limit formula for Jacobi sums conjectured by Honda may skip the rest of this chapter! Others may also be tempted. 5.2 Interpretation via Ext a La Mazur-Messing We denote by Ext(G, Ga ) the group of isomorphism classes of extensions of G by Ga , i.e. of short exact sequences 0 → Ga → E → G → 0 of abelian f.p.p.f. sheaves on (Schemes/R). We denote by Extrigid (G, Ga ) the group of isomorphism classes of “rigidified extensions,” i.e. pairs consisting of an extension of G by Ga together with a splitting of the corresponding extension of Lie algebras:

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Because Lie(G) is a free R-module of rank n = dim(G), any extension of G by Ga admits such a rigidification, which is indeterminate up to an element of Hom(Lie(G), Lie(Ga )) = ωG/R . Passing to isomorphism classes and remembering that the set of splittings of a trivial extension of G by Ga is itself principal homogeneous under Hom(G, Ga ), we obtain a four-term exact sequence (valid over any ring R) d

Hom(G, Ga ) → − ωG → Extrigid (G, Ga ) → Ext(G, Ga ) → 0 Theorem 5.2.1. If R is flat over Z, there is a natural isomorphism ∼

D(G/R) ← − Extrigid (G, G1 ) in terms of which the resulting four term exact sequence 0 → Hom(G, Ga ) → ωG → D(G/R) → Ext(G, Ga ) → 0 is the concatenation of the three-term sequence of (5.1.3) and the map D(G/R) → Ext(G, Ga ) defined by f → the class of the symmetric 2-cocycle ∂ f = f (X+) − f (X) − f (Y) G

Proof. We begin by constructing the isomorphism. Given a rigidified 196 extension

extend scalars from R to K = R ⊗ Q. Because K is a Q-algebra, the Lie functor defines an equivalence of categories between commutative formal Lie groups over K and free finitely generated K-modules. Therefore there is a unique splitting as K-groups

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whose differential is the given splitting S on Lie algebras. At the same time, we may choose a cross section S in the category of pointed f.p.p.f. sheaves over R

The difference f = S − exp(s) is a pointed map from G ⊗ K to (Ga ) ⊗ K, i.e. an element f ∈ A(G ⊗ K), and it satisfied f (0) = 0. We have d f = dS − s, so d f is integral, and the formula f (X+Y) − f (X) − f (Y) = S (X+Y) − S (X) − s(Y), G

G

valid because exp(s) is a homomorphism, shows that f (X+Y) − f (X) − G

f (Y) is integral. Because the initial choice of S is indeterminate up to addition of a pointed map from G to Ga , the class of f = S − exp(s) in D(G/R) is well-defined independently of the choice of S , and it vanishes if and only if exp(s) is itself integral, i.e. if and only if the original rigidified extension is trivial as a rigidified extension. Thus we obtain an injective map Extrigid (G, Ga ) → D(G/R). 197

To see that it is an isomorphism, note that in any case the map D(G/R) → Ext(G, Ga ) defined by f → the class of ∂ f sits in an exact sequence 0 → Hom(G, Ga ) → ωG → D(G/R) → Ext(G, Ga ), which receives the Extrigid exact sequence: 0

/ Hom(G, Ga )



/ D(G/R) O

/ Ext(G, Ga )

0

/ Hom(G, Ga )



? / Extrigid (G, Ga )

/ Ext(G, Ga )

G

The result is now visible.

G

/0



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Given an ideal I ⊂ R, we denote by Ext(G, Ga ; I) the group of isomorphism classes of pairs consisting of an extension of G by Ga together with a splitting of its reduction modulo I. We denote by Extrigid (G, Ga ; I) the group of isomorphism classes of pairs consisting of a rigidified extension and a splitting of the reduction mod I of the underlying extension. Analogously to the previous theorem, we have Theorem 5.2.2. If R is flat over Z, and I ⊂ R an ideal, there is a natural isomorphism ∼ Extrigid (G, Ga ; I) − → D1 (G/R) and a four-term exact sequence ∂

0 → Hom(G, Ga ) → ωG → D1 (G/R) → − Ext(G, Ga ; I) → 0 in which the map ∂, given by f → the class of the symmetric 2-cocycle ∂ f = f (X+Y) − f (X) − f (Y), G

corresponds to the map “forget the rigidification” on Ext’s. 5.3 The Case of p-Divisible Formal Groups Let p be a prime number. A ring R is said to be p-adic if it is complete and separated in its p-adic topology, i.e., if ∼ R− → lim R/pn R. ←−− A commutative formal Lie group G over a p-adic ring R is said to be 198 p-divisible of height h if the map ‘multiplication by p” makes A(G) into a finite locally free module over itself of rank ph . If we denote by Gv the dual of G in the sense of p-divisible groups, it makes sense to speak of the tangent space of Gv at the origin, noted tGv ; it is known that tGv is a locally free R-module of rank h − dim(G), and that there is a canonical isomorphism ∼

Ext(G, Ga ) − → tGv .

(5.3.1)

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Because G is p-divisible and R is p-adic, Hom(G, Ga ) = 0, and the four-term exact sequence becomes a Hodge-like exact sequence 0 → ωG → D(G/R) → tGv → 0

(5.3.2)

Thus we find Theorem 5.3.3. (1) If R is a p-adic ring which is flat over Z, then for a p-divisible commutative formal Lie group G over R, the R-module D(G/R) is locally free of rank h = height (G), and its formation commutes with arbitrary extension of scalars of Z-flat p-adic rings. If an addition I ⊂ R is an ideal which is closed in the p-adic topology, then R/I is again a p-adic ring, G ⊗ (R/I) is still p-divisible, and therefore admits no non-trivial homomorphisms to Ga over R/I. It follows that     D1 (G/R) ⊂ D(G/R) (5.3.4)   I  Ext(G, Ga ; I) → − I Ext(G, Ga ) ≃ I · tGv

and we have a short exact sequence

0 → ωG → D1 (G/R) → I · tGv → 0.

(5.3.5)

5.5 Relation to the Classical Theory Let k be a perfect field of characteristic p > 0, and take R = W(k), I = (p). Let CW denote the k-group-functor “Witt covectors” (in the notations of Fontaine ([13]), with its structure of W(k)-module. According to Fontaine, for any formal Lie variety V over W(k), we obtain a W(k)-linear isomorphism ∼

1 w : CW(A(V ⊗ k)) − → HDR (V/W(k); (p))

199

by defining

  pn  X (e  a ) −a  w(. . . , a−a , . . . , a0 ) = d  n p n≥0

(5.5.1)

(5.5.2)

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219

where e a−n denotes an arbitrary lifting to A(V) of a−n ∈ A(V ⊗ k). Similarly, we can define, following Grothendieck, Mazur-Messing ([35]), a σ-linear isomorphism ∼

1 (V/W(k)) ψ : CW(A(V ⊗ k)) − → HDR

by the formula

  pn+1  X (e  a ) −n  . ψ(. . . , a−n , . . . , a0 ) = d  n+1 p n≥0

(5.5.3)

(5.5.4)

These isomorphisms sit in a commutative diagram

H 1 (V/W(k); (p))

DR ✐✐✐4 ∼✐✐✐✐✐✐ ✐✐✐✐ w ✐✐✐✐

1 pF



CW(A(V ⊗ k)) ❯

❯❯❯❯ ❯❯❯❯∼ ❯❯❯❯ ❯❯❯❯ ψ *

(5.5.5)



1 (V/W(k)). HDR

When G is a commutative formal Lie group over W(k) which is pdivisible, the “classical” Dieudonne module of G0 = G ⊗ k is defined as dfn Homk−gp (G0 , CW) M(G0 ) (5.5.6) the primitive elements in CW(A(G0 )). Combining this definition with the previous isomorphisms, we find a commutative diagram of isomorphisms D p (G/W(k))

❦❦5 ∼❦❦❦❦❦ ❦ ❦ ❦❦ w ❦❦❦

1 pF



M(G0 ) ❙

❙❙❙ ❙❙❙ ∼ ❙❙❙ ψ ❙❙❙❙❙ )



D(G/W(k)).

(5.5.7)

220

5.6 Relation with Abelian Schemes and with the General Theory In this section, we recall without proofs some of the main results and compatibilities of the general D-theory of Grothendieck and MazurMessing. Given an abelian scheme A over an arbitrary ring R, there are canonical isomorphisms  ∼ rigid 1 (A/R)   → HDR Ext (A, Ga ) − (5.6.1)  ∼  Ext(A, Ga ) − → H 1 (A, OA ) = Lie(Av )

0



A

/ Extrigid (A, Ga )



in terms of which the Extrigid -exact sequence “becomes” the Hodge exact sequence:



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0



A

 / H 1 (A/R) DR

 / H 1 (A, OA )

/ Ext(A, Ga )

/0

/0

(5.6.2)

Lie(Av ) Given a p-divisible (Barsotti-Tate) group G = lim Gn over a ring R −−→ in which p is nilpotent, the exact sequence pn

0 → Gn → G −−→ G → 0

(5.6.3)

for any n sufficiently large that pn = 0 in R, leads to a canonical isomorphism ∼

→ Ext(G, Ga ). Lie(Gv ) = Lie(Gvn ) = Hom(Gn , Ga ) −

(5.6.4)

The Extrigid -exact sequence can thus be written 0 → ωG → Extrigid (G, Ga ) → Lie(Gv ) → 0, where ωG is the R-linear dual of Lie(G).

(5.6.5)

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Given an abelian scheme A over a ring R in which p is nilpotent, the exact sequence pn

0 → A pn → A −−→→ A → 0

(5.6.6)

for any n sufficiently large that pn = 0 in R leads to a canonical isomor- 201 phism ∼

→ Ext(A, Ga ). Lie(Av ) = Lie(Avpn ) = Hom(A pn , Ga ) −

(5.6.7)

Therefore the inclusion A p∞ ֒→ A induces an isomorphism ∼

Ext(A, Ga ) − → Ext(A p∞ , Ga )

(5.6.8)

(the identity on Hom(A pn , Ga )!), and consequently we obtain a commutative diagram of isomorphisms



A

A p∞

/ Extrigid (A, Ga )

/ Ext(A, Ga )



0





0

 / Extrigid (A p∞ , Ga )

 / Ext(A p∞ , Ga )

/0

/ 0,

(5.6.9) i.e., an isomorphism ∼

1 HDR (A/R) − → D(A p∞ /R)

(5.6.10)

compatible with the Hodge filtration. For variable B − T groups G over a fixed ring R in which p is nilpotent, the functors ωG , Lie(Gv ), and consequently Extrigid (G, Ga ), are exact functors whose values are locally free R-modules of finite rank; their formation commutes with arbitrary extension of scalars of rings in which p is nilpotent. Following Grothendieck and Mazur-Messing we define D(G/R)

dfn

Extrigid (G, Ga )

when G is a B − T group over a ring R in which p is nilpotent.

(5.6.11)

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When R is a p-adic ring, and G is a B − T group over R, we define   D(G/R) = lim D(G ⊗ (R/pn R)/(R/pn R))   ←−−    n  (5.6.12)  Lie(G) = lim Lie(G ⊗ (R/pn R))   ← − −     ωG = lim ωG ⊗ (R/pn R) ←−−

Thus for variable B − T groups G over a p-adic ring R, the functors ωG , Lie(Gv ) and D(G/R) are all exact functors in locally free R-modules of finite rank, sitting in an exact sequence 0 → ωG → D(G/R) → Lie(Gv ) → 0

(5.6.13)

whose formation commutes with arbitrary extension of scalars of p-adic rings. When A is an abelian scheme over a p-adic ring R, we obtain an isomorphism ∼ 1 HDR (A/R) − → D(A(p∞ )/R), compatible with Hodge filtrations, by passage to the limit. As we have seen in the previous section, this general Extrigid notion of D(G/R) agrees with our more explicit one in the case that both are defined, namely when G is a p-divisible formal group over a Z-flat padic ring R. 5.7 Relation with Cohomology Theorem 5.7.1. Let A be an abelian scheme over the Witt vectors W(k) of an algebraically closed field k of characteristic p > 0. There is a short exact sequence of W-modules β

α

1 b 0 → Het1 (A ⊗ k, Z p ) ⊗ W − → Hcris (A ⊗ k/W) → − D(A/W) →0

which is functorial in A ⊗ k.

Proof. We begin by defining the maps α and β. They will be defined by passage to the limit from maps αn , βn in an exact sequence αn

1 (A ⊗ k/Wn ) 0 → Het1 (A ⊗ k, Z/pn Z) ⊗ Wn −−→ Hcris

(5.7.2)

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βn

b ⊗ Wn /Wn ) → 0. −−→ D(A

of Wn -modules. An element of H 1 (A ⊗ k, Z/pn Z) is (the isomorphism class of) a 1 (A ⊗ k/W ) is (the isoZ/pn Z-torsor over A ⊗ k. An element of Hcris n morphism class of) a rule which assigns to every test situation Y ֒→ Yn 203 consisting an A ⊗ k scheme Y and a divided-power thickening of Y to a Wn -scheme Yn a Ga -torsor on Yn in a way which is compatible with inverse image whenever we have a morphism (Y, Yn ) → (Y ′ , Yn′ ) of such test situations (cf. [35] for more details). Given a Z/pn Z-torsor T on A ⊗ k, we must define for every test situation Y ֒→ Yn , a G-torsor αn (T )(Y,Yn ) on Yn . Because Y is given as an A ⊗ k scheme, we can pull back T to obtain a Z/pn Z-torsor T Y on Y. Because Yn is a Wn -scheme which is a divided-power thickening, its ideal of definition is necessarily a nil-ideal; therefore the etale Y-scheme T Y extends uniquely to an etale Yn -scheme T (Y,Yn ) , and its structure of Z/pn Z-torsor extends uniquely as well. Because Yn is a Wn -scheme, the natural map Z/pn Z → Wn gives rise to a morphism of algebraic groups on Yn αn

(Z/pn Z)Yn −−→ (Ga )Yn ; the required Ga -torsor αn (T )(Y,Yn ) is obtained by “extension of structural groups via αn ” from the Z/pn Z-torsor T (Y,Yn ) . 1 (A ⊗ k/W ). We To define βn , we begin with an element Z of Hcris n rigid b b⊗ must define an element βn (Z) in Ext (A ⊗ Wn , (Ga ) ⊗ Wn ) = D(A Wn /Wn ). Its value on the test object A ⊗ k ֒→ A ⊗ Wn is a Ga -torsor on A ⊗ Wn which is endowed with an integrable connection (cf. [2], [3]), 1 (A ⊗ W /W ). [This interpretation provides i.e., it is an element of HDR n n the canonical isomorphism ∼

1 1 (A ⊗ Wn /Wn ).] Hcris (A ⊗ k/Wn ) − → HDR

Composing with the isomorphism ∼

1 HDR (A ⊗ Wn /Wn ) − → Extrigid (A ⊗ Wn , Ga ⊗ Wn ),

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we obtain an element of Extrigid (A ⊗ Wn , Ga ⊗ Wn ), whose restriction to b ⊗ Wn is the required element βn (Z). the formal group A To see that the map β obtained from these βn by passage to the limit 204 is in fact functorial in A ⊗ k, we first note that it sits in the commutative diagram β



1 (A ⊗ k/W) Hcris

/ D(A/W) b _

inclusion of primitive elements

canonical isom



1 (A/W) HDR

natural map b “restriction to A”

(5.7.3)

 / H 1 (A/W). b DR

What must be shown is that if we are given a second abelian scheme B over W, and a homomorphism f0 : B ⊗ k → A ⊗ k then the diagram 1 (A ⊗ k/W) Hcris

β

( f0 )∗



1 (B ⊗ k/W) Hcris

β

/ D(A/W) b

(any pointed lifting of b f0 )∗

(5.7.4)

 / D(b B/W)

is commutative. But in virtue of the commutativity of the previous diagram (5.7.3), it is enough to show the commutativity of the diagram 1 (A ⊗ k/W) ≃ H 1 (A/W) Hcris DR

restriction

( f0 )∗



1 (B ⊗ k/W) ≃ H 1 (B/W) Hcris DR

restriction

/ H 1 (A/W) b DR

(any pointed lifting of b f0 )∗

 / H 1 (b B/W). DR

(5.7.5)

Crystalline Cohomology, Dieudonn´e Modules,... 205

225

This last commutativity has nothing to do with abelian schemes, nor does it require pointed liftings. It is an instance of the following general fact, whose proof we defer for a moment. General Fact 5.7.6. For any two pointed W-schemes A, B which are both proper and smooth, any pointed map f0 : B ⊗ k → A ⊗ k, and any integer i ≥ 0, we have a commutative diagram 1 (A ⊗ k/W) ∼ H i (A/W) Hcris DR

restriction

( f0 )∗



1 (B ⊗ k/W) ≃ H i (B/W) Hcris DR

restriction

/ H i (A/W) b DR

(any lifting of b f0 )∗

 / H i (b B/W) DR

To conclude the proof of the theorem (!), it remains to see that our marvelously functorial maps α, β really do form an exact sequence. To b is do this, we will use the abelian scheme A over W. Its formal group A p-divisible, and sits in an exact sequence of p-divisible groups over W, bp∞ → A p∞ → E → 0, 0→A

in which E = lim En denote the etale quotient of A p∞ . Because k is alge−−→ braically closed, E is a constant p-divisible group, namely the abstract p-divisible group lim A pn (k) of all p-power torsion points of A(k). −−→ We will identify the exact sequence of the proposition with the exact sequence α′

β′

b → 0, 0 → D(E/W) −−→ D(A p∞ /W) −→ D(A/W)

and we will identify the (αn , βn )-sequence with the exact sequence α′n

β′n

b ⊗ Wn /Wn ) → 0. 0 → D(E ⊗ Wn /Wn ) −−→ D(A p∞ ⊗ Wn /Wn ) −−→ D(A

226

It is clear from the construction of βn that we have a commutative diagram D(A p∞ ⊗ Wn /Wn )

β′n

/ D(A b ⊗ Wn /Wn ) dfn

restriction

/ Extrigid (A b ⊗ Wn , G a ) ✐✐4 ⑤⑤> ✐ ✐ ✐ ✐ restriction ⑤⑤ ✐✐✐✐ ✐✐✐✐ ⑤⑤ ✐ ✐ ⑤ ✐ ✐ ⑤⑤ ⑤⑤ Extrigid (A ⊗ Wn , Ga ) ⑤ ⑤⑤ ⑤⑤ ⑤ ⑤⑤  ⑤⑤ βn ⑤ 1 (A ⊗ W /W ) ⑤⑤ HDR n n ⑤⑤ O ⑤⑤ ⑤ ⑤ ⑤⑤ ⑤⑤

Extrigid (A p∞ ⊗ Wn , Ga )





O



1 (A ⊗ k/W ). Hcris n

To relate the map αn to the D-maps, use the exact sequence pn

0 → En ⊗ Wn → E ⊗ Wn −−→ E ⊗ Wn → 0 to compute D(E ⊗ Wn /Wn )



/ Ext(E ⊗ Wn , Ga )



/ Hom(En ⊗ Wn , Ga )



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(En is constant)



Hom(En (Wn ), Ga (Wn ))

Hom(En (k), Wn ) Hom(En (k), Z/pn Z) ⊗ Wn . Next use the sequence pn

0 → A pn ⊗ Wn → A ⊗ Wn −−→ A ⊗ Wn → 0

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227

to compute

207 ∼

/ Hom(A pn ⊗ Wn , Z/pn Z) O (En = etale quotient of A pn )



Ext(A ⊗ Wn , Z/pn Z)



Hom(En ⊗ Wn , Z/pn Z) Het1 (A ⊗ k, Z/pn Z) o





Hom(En (k), Z/pn Z).

Combining these isomorphisms, and remembering that Ext = Extrigid when either of the arguments is etale, we find a commutative diagram α′n



/ D(A p∞ ⊗ Wn /Wn ) O



D(E ⊗ W O n /Wn ) n Z) ⊗ W Hom(En (k), Z/p n O



D(A ⊗ Wn /Wn ) ∗∗∗

/ Extrigid (A ⊗ Wn , Ga )



Extrigid (A ⊗ WnO , Z/pn Z) ⊗ Wn Het1 (A ⊗ k, Z/pn Z) ⊗ Wn

αn

/ H 1 (A ⊗ k/Wn ) cris

in which the arrow ∗∗∗ is “push-out” along the homomorphism Z/pn Z → Wn → (Ga )Wn .  Corollary 5.7.7. Let A be an abelian scheme over the Witt vectors W(k) of a perfect field k of characteristic p > 0. Then we have a short exact sequence of W(k)-modules  Gal(k/k) 0 → (Het1 (A ⊗ k, Z p ) ⊗ W(k) →

228

Nicholoas M. Katz 1 b Hcris (A ⊗ k/W(k)) → D(A/W(k)) → 0,

in which k denotes an algebraic closure of k, and in which the galois group Gal(k/k) acts simultaneously on Het1 (A ⊗ k, Z p ) and on W(k) by “transport of structure”. 208

Proof. One can obtain this sequence either by passing to Gal(k/k)-invariants in the already-established analogous sequence for A ⊗ W(k), or by repeating the proof given for the proposition. In the latter case, one finds, in the notations of the proof, D(E ⊗ Wn (k)/Wn (k)) ≃ Hom(En ⊗ Wn (k), (Ga )Wn (k) ) ≃ Hom(En (k), Wn (k))Gal(k/k) ≃ Hom(A pn (k), Wn (k))Gal(k/k) Gal(k/k)  = Het1 (A ⊗ k, Z/pn Z) ⊗ Wn (k) and the rest of the proof remains unchanged.



Corollary 5.7.8. Let A be an abelian scheme over the Witt vectors of a perfect field k of characteristic p > 0. The above exact sequence is the Newton-Hodge filtration 1 0 → (slope 0) → Hcris (A ⊗ k/W) → (slope > 0) → 0 1 (A ⊗ k/W)) as an F-crystal. of Hcris

Proof. Since F induces a σ-linear automorphism of (Het1 (A ⊗ k, Z p ) ⊗ W(k))Gal  Gal(k/k) ≃ Hom(T p (A ⊗ k), W(k)) , b it remains only to see that F is topologically nilpotent on D(A/W(k)), b for its p-adic topology. Because D(A/W(k)) is a finitely generated W(k)

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229

1 ( A/W(k)), b b sub-module of HDR the topology induced on D(A/W(k)) by 1 the inverse limit topology on HDR through the isomorphism (cf. lemma 5.8.1. ahead) ∼

1 1 b b ⊗ Wn (k)/Wn (k)) HDR (A/W(k)) − → lim HDR (A ←−−

(5.7.9)

b must be equivalent to the p-adic topology in D(A/W(k)). So it suffices n 1 b to remark that F annihilates HDR (A ⊗ Wn /Wn ) (indeed F n annihilates Ωib for i ≥ 1, since for any pointed lifting of X 7→ X p , F(dX) = A⊗Wn /Wn

d(F(X)) = d(X p + pY) ∈ pΩ1 ) to establish the required topological b nilpotence of F on D(A/W). 

5.8 The Missing Lemmas It remains for us to establish the “general fact” (5.7.7), and to establish the isomorphism (5.7.9). In fact, the two questions are intimately related. We begin with the second.

Lemma 5.8.1. Let R be a Z-flat p-adic ring, and let Rn = R/pn R. For any formal Lie variety V over R, we have isomorphisms ∼

i i HDR (V/R) − → lim HDR (V ⊗ Rn /Rn ). ←−−

Proof. Pick coordinates X1 , . . . , XN for V. Over any ring R, we can define a Z N -grading of the de Rham complex of R[[X1 , . . . , XN ]]/R, by attributing the weight (a1 , . . . , aN ) ∈ Z N to each “monomial”   Y a  Y dX j    Xi i  S any subset of {1, . . . , N}. X j  j∈S Exterior differentiation is homogeneous of degree zero, and the de Rham complex is the product of all its homogeneous graded pieces Ω = ΠΩ (a1 , . . . , aN ). •



Because both cohomology and inverse limits commute with products, we are reduced to proving the lemma homogeneous component by homogeneous component.

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The individual complexes Ω (a1 , . . . , aN ) are quite simple. They vanish except when all ai ≥ 0. The complex Ω (0, . . . , 0) is •



R → 0 → 0 → ... If some ai ≥ 1, and all ai ≥ 0, the complex Ω (a1 , . . . , aN ) is the tensor product complex O  ai  R −→ R . •

i with ai ≥1

What is important for us is that each of these complexes is obtained from a complex of free finitely generated Z-modules (!) by extension of scalars to R. Thus let K denote any complex of free finitely-generated Z p -modules. We must show that for a Z-flat p-adic ring R we have ∼

H i (K ⊗ R) − → lim H i (K ⊗ Rn ). ←−− •

210



The exact sequence of complexes pn

0 → K ⊗ R −−→ K ⊗ R → K ⊗ Rn → 0 •





gives a “universal coefficients” exact sequence 0 → H i (K ⊗ R) ⊗ Rn → H i (K ⊗ Rn ) → pn -Torsion (H i+1 (K ⊗ R)) → 0. •





Passing to the inverse limit over n leads to an exact sequence 0 → lim H i (K ⊗ R) ⊗ Rn → lim H i (K ⊗ Rn ) → T p (H i+1 (K ⊗ R)) → 0. ←−− ←−− •





To see that T p (H i+1 (K ⊗ R)) vanishes, notice that an element of this T p is represented by a system of elements an ∈ K i+1 ⊗ R with d(an ) = 0, pan+1 = an − d(bn ), a0 = 0; because both K i ⊗ R and K i+1 ⊗ R are p-adically complete and separated, we may infer •

an = pan+1 + d(bn ) = p (pan+2 + d(bn+1 )) + d(bn )

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231

= ...   X  pi bn+i  . = d  i≥0

To see that the natural map

H i (K ⊗ R) → lim H i (K ⊗ R) ⊗ Rn ←−− •



is an isomorphism, use the Z-flatness of R and the Z-finite generation of the K i to write ∼

H i (K ⊗ R) ← − H i (K ) ⊗ R = (fin. gen. Z-module) ⊗ R •



= Z n ⊕ (⊕Z/pni ) ⊕

!! prime-to-p ⊗R torsion

= Rn ⊕ (⊕Rni ).  We now turn to the proof of the “general fact”. Lemma 5.8.2. Let k be a perfect field of characteristic p > 0, A and B two proper, smooth pointed W(k)-schemes, f0 : B ⊗ k → A ⊗ k a pointed b a W-lifting of f0 to the formal completions 211 k-morphism and b f :b B→A viewed as functors only on p-adic W-algebras. Then the diagram i (A ⊗ k/W) Hcris



/ H i (A/W) DR

restriction

( f )∗

( f0 )∗



i (B ⊗ k/W) Hcris 0

is commutative.

/ H i (A/W) b DR



/ H i (B/W) DR

restriction

 / H i (b B/W) DR

Proof. If f0 lifted, this would be obvious. But it does lift locally, which is enough for us. More precisely, let U ⊂ A and V ⊂ B be affine open neighborhoods of the marked W-valued points of A and B respectively

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such that f0 maps V ⊗ k to U ⊗ k. Because V is affine and U is smooth over W, we may successively construct a compatible system of Wn -maps fn : V ⊗ Wn → U ⊗ Wn with fn+1 ≡ fn mod pn . The fn induce compatible b ⊗ Wn of formal completions, but these b maps b fn : b B ⊗ Wn → A fn need not be pointed morphisms. b the limit of these b We denote by b f∞ : b B→A fn . (Strictly speaking, b b to f∞ only makes sense as a map of functors when we restrict b B and A the category of p-adic W-algebras). For each n, we have a commutative diagram

Passing to the inverse limit over n, and using the previous lemma to identify the right-hand inverse limits, we obtain a commutative diagram i (A ⊗ k/W) Hcris



i (A/W) HDR

restriction

( f0 )∗



i (B ⊗ k/W) Hcris

212



i (b B/W) HDR

restriction

/ H i (A/W) b DR 

(b f∞ )∗

/ H i (b DR B/W).

To conclude the proof, we need to know that the induced map i i b (b f∞ )∗ : HDR (A/W) → HDR (b B/W)

b ⊗ k, and not on depends only on the underlying map b f0 : b B⊗k → A the particular choice of lifting. In fact this is true for the individual b fn as well!  Lemma 5.8.3. Let R be a p-adic ring. Let V and V ′ be formal Lie varieties over R, and let f1 and f2 be morphisms of functors V ′ → V of the

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233

restrictions of V ′ , V to the category of p-adic R-algebras. If f1 f2 mod p, then for each i, the induced maps i i f1∗ , f2∗ : HDR (V/R) → HDR (V ′ /R)

are equal. Proof. (compare Monsky [39]). In terms of coordinates X1 , . . . , Xn for V ′ , Y1 , . . . Ym for V, the corresponding R-algebra homomorphisms ϕ1 , ϕ2 : R[[Y1 , . . . , Ym ]] → R[[X1 , . . . , Xn ]] are related by ϕ2 (Y) = ϕ1 (Y) + p∆(Y). Introduce a new variable T , and consider the map ϕ : R[[Y1 , . . . , Ym ]] → R[[X1 , . . . , Xn , T ]] ϕ(Y) = ϕ1 (Y) + T · ∆(Y). We have a commutative diagram of algebraic homomorphisms

So it suffices to consider the situation R[[X, T ]]

T →0 T →p

/

/ R[[X]]

and show that these two maps have the same effect on HDR . A form ω on R[[X, T ]] may be written uniquely X X dT ω= an · T n + bn T n T n≥0 n≥1

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with an , bn ’s forms on R[[X]]. This form is closed if and only if d(an ) = 0 for

n ≥ 0,

n · an + d(bn ) = 0 for n ≥ 1.

Its images under T → 0 and T → p are X a0 , an pn n≥0

respectively. Their difference, if ω is closed, is exact, namely   X X pn  · bn  . ω|T =0 − ω|T =p = an pn = d  n n≥1 b≥1



It seems worthwile to point out that this last lemma can be considerably strengthened. Lemma 5.8.4. Let R be a p-adic ring, I ⊂ R a divided power ideal, V and V ′ two formal Lie varieties over R, and f1 , f2 two morphisms of functors V ′ → V of the restrictions of V, V ′ to the category of p-adic R-algebras. If f1 ≡ f2 mod I, then for all i the induced maps i i f1∗ , f2∗ : HDR (V/R) → HDR (V ′ /R)

are equal. Proof. If we had f1 ≡ f2 mod I ′ with I ′ ⊂ I a finitely generated ideal, then we could repeat the proof of the previous lemma, introducing sev214 eral new variables T i , one for each generator of I ′ . In particular, the lemma is true if f1 and f2 are polynomial maps in some coordinate system. But we easily reduce to this situation, for in terms of coordinates X1 , . . . , Xn for V ′ , we have a Z n -graduation of its de Rham complex and a corresponding product decomposition Y i i HDR (V ′ /R) = HDR (V ′ /R)(a1 , . . . , an ). (a1 ,...,an )

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235

Therefore it suffices to show that the composite maps i (V/R) HDR

f1∗

/

f2∗

i (V ′ /R) / HDR

projection

/ H i (V ′ /R)(a1 , . . . , an ) DR

agree, for every (a1 , . . . , an ) ∈ Z n . But for fixed (a1 , . . . , an ), these comP posites depend only on the terms of total degree ≤ ai in the power series formulas for the maps f1 , f2 . Thus we are reduced to the case when f1 and f2 are each polynomial maps.  Remark 5.8.5. If the ideal I is closed, the proof gives the same invarii (V/R; I) defined as the cohomology ance property for the groups HDR of d d “IΩi−1 − ΩiV/R → − Ωi+1 V/R ” → V/R . 5.9 Application to the Cohomology of Curves Throughout this section we work over a mixed-characteristic valuation ring R of residue characteristic p, which is complete for a rank-one (i.e., real-valued) valuation. Let C be a projective smooth curve over R, with geometrically connected fibres of genus g. Its Jacobian J = Pic0 (C/R) is a g-dimensional autodual abelian scheme over R. For each rational point x ∈ C(R), we denote by ϕ x the corresponding Albanese mapping ϕx : C → J given on S -valued points, S any R-scheme, by ϕ x (y) = the class of the invertible sheaf I(y)−1 ⊗ I(x), where I(y) denotes the invertible ideal sheaf of y ∈ C(S ) viewed as a Cartier divisor in C×S . As is well-known (cf. [44], [45]), this morphism 215 R

induces isomorphisms  ∼   H 1 (J, O j ) − → H 1 (C, OC )    ∼  0 1 ) H (J, Ω1J/R ) = ω J − → H 0 (C, ΩC/R     ∼  1 (C/R) H 1 (J/R) − → HDR DR

which are independent of the choice of the rational point x.

(5.9.1)

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bx denote the formal completion of C along x; it is a pointed Let C formal Lie variety of dimension one over R. Because ϕX (0) = 0, ϕ x induces a map of pointed formal Lie varieties bx → J, b b ϕx : C

whence an induced map on cohomology

(b ϕ )∗

b ⊂ H 1 ( J/R) b −−−x−→ H 1 (C b D( J/R) DR DR x /R).

Theorem 5.9.2. The composite map (b ϕ )∗

is injective.

b b −−−x−→ H 1 (C D( J/R) DR x /R)

Corollary 5.9.3. The natural map 1 1 bx /R) H 0 (C, ΩC/R ) → HDR (C

is injective, i.e., a non-zero differential of the first kind cannot be formally exact. Proof. Because Jbis p-divisible, the natural map ω J → D(J/R) is injective. The corollary then follows immediately from the theorem and the commutativity of the diagram b  D( J/R) ∪

ωJ 216



(b ϕ x )∗



/ H 1 (C bx /R) O

(5.9.4)

/ H 0 (C, Ω1 ). C/R

To prove the theorem, we choose an integer n ≥ 2g − 1, and consider the mapping n ϕ(n) x :C → J

Crystalline Cohomology, Dieudonn´e Modules,...

defined by ϕ(n) x (y1 , . . . , yn )

=

n X

237

ϕ x (yi ),

i=1

the summation taking place in J. Passing to formal completions, we obtain b n b b ϕ(n) x : (C x ) → J defined by

b ϕ(n) x (y1 , . . . , yn ) =

In terms of the projections

X

ϕ x (yi ).

bx )n → C bx pbri : (C

onto the various factors, we can rewrite this as b ϕ(n) x =

n X i=1

b ϕ x ◦ pbri ,

b the summation taking place in the abelian group of pointed maps to J. b is defined to consist precisely of the primitive elements Because D( J/R) 1 b b in HDR ( J/R), we have, for any a ∈ D( J/R), ∗ (b ϕ(n) x ) (a) =

n X i=1

(b ϕ x ◦ pbri )∗ (a) =

n X

(b pri )∗ (b ϕ x )∗ (a).

i=1

Therefore the theorem would follow from the injectivity of the map ∗ 1 b b n (b ϕ(n) x ) : D( J/R) → HDR ((C x ) /R).

b b is a flat R-module contained in H 1 ( J/R), it suffices to Because D( J/R) DR show that the kernel of the map ∗ 1 1 b b n (b ϕ(n) x ) : HDR ( J/R) → HDR ((C x ) /R)

consists entirely of torsion elements. In fact, we will show that this kernel is annihilated by n!. To do this, we observe that the map n b ϕ(n) x :C → J

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is obviously invariant under the action of the symmetric group Cn on C n by permutation of the factors. Therefore we can factor it 217

Passing to formal completions, we get a factorization

1 , by showing that We will first show that (b ψ)∗ is injective on HDR the map b ψ has a cross-section. This in turn follows from the global fact that ψ is a Pn−g -bundle over J which is locally trivial on J for the Zariski topology. To see this last point, take a Poincare line bundle C on C × J. Because n ≥ 2g − 1, the Riemann-Roch theorem and standard base-changing results show that the sheaf on J given by (pr2 )∗ (C ⊗ pr∗1 (I −1 (x)⊗n )) is locally free of rank n + 1 − g. The associated projective bundle is naturally isomorphic to ψ. It remains only to show that the kernel of the map 1 bx )/R) → H 1 ((C bx )n /R) (b π)∗ : HDR (Symmn (C DR

bx ) becomes exact is annihilated by n!. But if a one-form ω on Symmn (C n bx ) , say ω = df with f ∈ A((C bx )n ), then when pulled back to (C   X   X σ( f ) σ(ω) = d  n!ω = σ∈Sn

bx ). is exact on Symmn (C

σ∈Sn



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Remark. The fact that for n large the symmetric product Symmn (C) is a projective bundle over J may be used to give a direct proof that C and J have isomorphic H 1 ’s in any of the usual theories (e.g., coherent, Hodge, De Rham, etale, crystalline...). Theorem 5.9.5. Let k be a perfect field of characteristic p > 0, k its algebraic closure, C a projective smooth curve over W(k) with geometrically connected fibre, J = Pic0 (C/W(k)) its jacobian, x ∈ C(W(k)) a rational point of C, and ϕ x : C → J the corresponding Albanese 218 mapping. There is an exact sequence of W-modules

the maps in which are functorial in (C, x) ⊗ k as pointed k-scheme. Proof. The map α is defined exactly as was its abelian variety analogue (cf. 5.7.1); the map β is defined as the composite

By construction, α is functorial in (C, x) ⊗ k. By lemma (5.8.2), β is similarly functorial. To see that the sequence is exact, use the fact that the Albanese map induces isomorphisms on both crystalline (or de Rham!) and etale H 1 ’s, (cf. SGAI, Exp. XI, last page, for the etale case), i.e., we have a commutative diagram



α.



0

b D( J/W(k))







/ (Het1 (J ⊗ k, Z p ) ⊗ W(k))Gal (ϕ,⊗k)∗

/

(Het1 (C ⊗ k, Z p ) ⊗ W(k))(Gal(k/k))

/

1 (J ⊗ k/W(k)) Hcris (ϕ,⊗k)∗

α

/

1 (C ⊗ k/W(k)) Hcris

_

/0

(b ϕ x )∗

β

/

1 (C bx /W(k)). HDR



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Corollary 5.9.6. (1) The kernel of the “formal expansion at a point” map 1 1 bx /W(k)) HDR (C/W(k)) → HDR (C

1 (C/W(k)) ≃ H 1 (C ⊗ k/W(k)) is the “slope-zero” part of the in HDR cris 1 (C ⊗ k/W(k)), i.e., we have a commutative diagram F-crystal Hcris (Gal(k/k))

0

/ (slope 0)

/

/

1 1 / (image of HDR bx /W(k)) (C/W) in HDR (C O

1 HDR (C/W)

O

219

/0



/ (Het1 (C ⊗ k, Z p ) ⊗ W(k))



0

1 (C ⊗ k/W(k)) Hcris

/ (slope > 0)

/ 0.

(2) The image of the “formal expansion at a point” map is the “slope 1 (C ⊗ k/W(k)); this quotient is isomorphic, via the > 0” quotient of Hcris b Albanese map ϕ x , to D( J/W(k)).

VI. Applications to congruences and to Honda’s conjecture. Let C be a projective smooth curve over W(Fq ) with geometrically connected fibres. Let G be a finite group of order prime to p, all of whose absolutely irreducible complex representations are realizable over W(Fq ) (e.g., if the exponent of G divides q − 1, this is automatic). Suppose that G operates on C by W(Fq )-automorphisms. Then G operates also on C ⊗ Fq by Fq -automorphisms. For each absolutely irreducible representation ρ of G, let P1,ρ (T ) ∈ W(Fq )[T ] be the numerator of the associated L-function L(C ⊗ Fq /Fq , G, ρ; T ); P1,ρ (T ) = 1 + a1 (ρ)T + · · · + ar (ρ)T r . 1 )ρ be a differential of the first kind on C which Let ω ∈ H 0 (C, ΩC/W 4 ). Let x ∈ C(W(Fq )) lies in the ρ-isotypical component of H 0 (C, ΩC/W be a rational point on C, and let X be a parameter at x (i.e., X is a bx over coordinate for the one-dimensional pointed formal Lie variety C W(Fq )). Consider the formal expansion of ω around x:

ω=

X n≥1

b(n) · X n

dX X

b(n) ∈ W(Fq ).

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241

We extend the definition of b(n) to rational numbers n > 0 by decreeing that b(n) = 0 unless n is an integer. Theorem 6.1. In the above situation, the coefficients b(n) satisfy the congruences b(n) b(nq) b(nqr ) + a1 (ρ) · + · · · + ar (ρ) ∈ pW(Fq ) n nq nqr for every rational n > 0. Proof. Let J denote the Jacobian of C/W(Fq ), and denote by b ω ∈ ωJ the unique invariant one-form on J which pulls back to give ω under the Albanese mapping ϕ x . The group G operates, by functoriality, on J ∼ 1 ) is G-equivariant. → H 0 (C, ΩC/W and on ω J , and the isomorphism ω J − ρ e lies in (ω J ) . Via the G-equivariant inclusion Therefore ω b ω J ⊂ D(p) ( J/W)

we have

220

ρ b e ∈ (D(p) ( J/W)) ω

Now let F denote the Frobenius endomorphism of J ⊗ Fq relative to Fq . Then both F and the group G act on J ⊗ Fq . By (4.2), we know that (F r + a1 (ρ)F r−1 + · · · + ar (ρ)) · Proj(ρ) = 0 b in End(J ⊗ Fq )⊗W(Fq ). Because D( J/W) is an additive functor of J ⊗ Fq Z

e lies in its ρ-isotypical component, with values in W(Fq )-modules, and ω it follows that e=0 F r (e ω) + a1 (ρ)F r−1 (e ω) + · · · + ar (ρ) · ω

b in D(p) ( J/W). The Albanese map ϕ x : C → J induces a map bx → J, b b ϕx : C

(6.1.1)

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Nicholoas M. Katz

whence a map (b ϕ )∗

x 1 1 bx ; (p)) b b (C D(p) ( J/W) ⊂ HDR ( J/W; (p)) −−−−→ HDR

b 0) ⊗ Fq and (C bx , x) ⊗ Fq . which is functorial in the pointed schemes ( J, So if we denote also by F the q-th power Frobenius endomorphism of bx ⊗ Fq , we have C (b ϕ x )∗ ◦ F = F ◦ (b ϕ x )∗ , whence a relation F r (ω) + a1 (ρ)F r−1 (ω) + · · · + ar (ρ) · ω = 0

(6.1.2)

1 (C bx /W; (p)). in HDR The asserted congruences on the b(n)’s are simply the spelling out bx , a of this relation. Explicitly, in terms of the chosen coordinate X for C b particularly convenient pointed lifting of F on C x ⊗ Fq is provided by

F : X 7→ X q .

221

In terms of the isomorphism ∼ { f ∈ K[[X]]| f (0) = 0, df integral} 1 bx /W; (p)) ← HDR (C − { f ∈ pW[[X]]| f (0) = 0}

the cohomology class of ω is represented by the series X b(n) f (X) = Xn, n n>0 and the cohomology class of F i (ω) is represented by X b(n) i i f (X q ) = X nq . n The relation (??) thus asserts that r

r−1

f (X q ) + a1 (ρ) f (X q ) + · · · + ar (ρ) f (X) is a series whose coefficients all lie in pW(Fq ). The congruence asserted r in the statement of the theorem is precisely that the coefficient of X nq in this series lies in pW(Fq ). 

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243

Remark. In the special case G = {e}, ρ trivial, the polynomial P1,ρ (T ) is the numerator of the zeta function of C ⊗ Fq , and every differential of the 1 ) is ρ-isotypical. The resulting congruences first kind ω ∈ H i (C, ΩC/W on the coefficients of differentials of the first kind were discovered independently by Cartier and by Honda in the case of elliptic curves, and seem by now to be “well-known” for curves of any genus. ([1], [5], [8], [22]). Theorem 6.2. Hypothesis and notation as above, suppose that the polynomial P1,ρ (T ) is linear P1,ρ (T ) = 1 + a1 (ρ)T , i.e., that ρ occurs in H 1 with multiplicity one. Then (1) a1 (ρ) is equal to the exponential sum S (C ⊗ Fq /Fq , ρ, 1) and for every n ≥ 1 we have (−a1 (ρ))n = −S (C ⊗ Fq /Fq , ρ, n). 1 ), then ord p (a1 (ρ)) > 0, i.e., a1 (ρ) is not 222 (2) If ρ occurs in H 0 (C, ΩC/W a unit in W(Fq ). 1 1 )ρ to be non), choose ω ∈ H 0 (C, ΩC/W (3) If ρ occurs in H 0 (C, ΩC/W zero, and such that at least one of coefficients b(n) is a unit in W(Fq ). For any n such that b(n) is a unit, the coefficients b(nq), b(nq2 ), . . . are all non-zero, and we have the limit formulas (in which ρ denotes the contragradient representation)

q · b(nqN ) N→∞ b(nqN+1 )

−S (C ⊗ Fq /Fq , ρ, 1) = −a1 (ρ) = lim −S (C ⊗ Fq /Fq , ρ, 1) = −a1 (ρ) =

−q b(nqN+1 ) . = lim a1 (ρ) N→∞ b(nqN )

Proof. If ρ occurs in H 1 with multiplicity one, then ρ must be a nontrivial representation of G (for if ρ were the trivial representation, G

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would have a one-dimensional space of invariants in H 1 ; but the space of invariants in H 1 of the quotient curve C ⊗ Fq modulo G, so is evendimensional!). Therefore ρ does not occurs in H 0 or H 2 , as both of these are the trivial representation of G. The first assertion now results from (1.1). 1 If ρ also occurs in H 0 (C, ΩC/W ), pick any non-zero ω in 1 H 0 (C, ΩC/W )ρ

and look at its formal expansion around x: X dX ω= b(n)X n . X An elementary “q-expansion principle”-argument (cf. [28]) shows that 1 if all b(n) are divisible by p, then ω is itself divisible by p in H 0 (C, ΩC/W ). So after dividing ω by the highest power of p which divides all b(n), we 1 obtain an element ω ∈ H 0 (C, ΩC/W )ρ which has some coefficient a unit. Consider the congruences satisfied by the b(n): b(n) b(nq) + a1 (ρ) ∈ pW(Fq ). n nq 223

If a1 (ρ) were a unit, we could infer (by induction on the precise power of p dividing n) that for all n ≥ 1,

q b(n) · ∈ W(Fq ). p n

In particular, we would find that

q · ω is formally exact at x, which p

by (5.9.3) is impossible. Given that a1 (ρ) is a non-unit, choose n such that b(n) is a unit. Then ord(b(n)/n) ≤ 0. From the congruences b(nq) b(n) ≡ −a1 (ρ) n nq

mod pW

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245

.. . b(nqN ) b(nqN+1 ) mod pW ≡ −a (ρ) 1 nqN nqN+1 and the fact that ord(a1 (ρ)) > 0, it follows easily by induction on N that ! b(nqN ) ord = ord(b(n)/n) − N ord(a1 (ρ)). nqN Therefore we may divide the congruences, and obtain ! qb(nqN ) + a1 (ρ) ≥ 1 + (N + 1) ord(a1 (ρ)) − ord(b(n)/n) ord b(nqN+1 ) ! ! ! q q b(n) b(nqN+1 ) + ≥ 1 + ord + N ord(a1 (ρ)) − ord . ord a1 (ρ) a1 (ρ) n b(nqN ) Letting N → ∞, we get the asserted limit formulas for −a, (ρ) and for −q/a1 (ρ). By the Riemann Hypothesis for curves over finite fields, we know that −q/a1 (ρ) is the complex conjugate a1 (ρ). Let ρ denote the contragradient representation of ρ; because the definition of the L-series L(C ⊗ Fq /Fq , G, ρ; T ) is purely algebraic, the L-series for ρ is obtained by applying (any) complex conjugation to the coefficients of the L-series for ρ. Therefore a1 (ρ) = a1 (ρ), and ρ also occurs in H 1 with multiplicity 224 one.  Example 6.3. Consider the Fermat curve of degree N over W(Fq ), with q ≡ 1mod N. For each integer 0 ≤ r ≤ N − 1, denote by χr the character of µN given by χr (ζ) = ζ r . We know that under the action of µN × µN (acting as (x, y) → (ζ x, ζ ′ y) in the affine model xN + yN = 1), the characters which occurs in H 1 are precisely χr × χ s 1 ≤ r, s ≤ N − 1, r + s , N,

each with multiplicity one. Those which occur in H 0 (Ω1 ) are precisely the χr × χ s 1 ≤ r, s ≤ N − 1, r + s < N,

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Nicholoas M. Katz

the corresponding eigen-differential ωr,s is given by ωr,s = xr y s

dx . xyN

If we expand ωr,s at the point (x = 0, y = 1), in the parameter x, we obtain s dx −1· N x ! s dx −1 xr+N j N j x

ωr,s = xr (1 − xN ) =

X

(−1) j

X

b(n)xn

j≥0

=

n≥1

225

dx . x

Conveniently, the first non-vanishing coefficient b(r) is 1. The successive coefficients b(rqn ) are given by  s   N − 1  r (qn −1)  n   . N ·  r n b(rq ) = (−1) (q − 1) N

The eigenvalue of F on the χr × χ s -isotypical component of H 1 is the negative of the Jacobi sum Jq (χr , χ s ). There we obtain the limit formulas  − 1   (−1) r n (q − 1) N −Jq (χr , χ s ) = lim   s n→∞  N − 1    r n+1 − 1) N (q r n N (q−1)·q

(−1) −Jq (χN−r , χN−s ) = lim

n→∞

r n N (q−1)·q

  

  

  · 

s N

s N − r n+1 N (q

 − 1   r n N (q − 1) s N

1

− 1)

  

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247

valid for 1 ≤ r, s ≤ N − 1, r + s , N. These formulas are the ones originally conjectured by Honda, and recently interpreted by Gross-Koblitz [14] in terms of Morita’s p-adic gamma function. VII. Application of Gauss sums. In this chapter we will analyze the cohomology of certain Artin-Schreier curves, and then obtain a limit formula for Gauss sums in the style of the preceding section. We fix a prime p, an integer N ≥ 2 prime to p, and consider the smooth affine curve U over Z[1/N(p − 1)] defined by the equation T p − T = XN . It may be compactified to a projective smooth curve C over Z[1/N(p − 1)] with geometrically connected fibres by adding a single “point at infinity”, along which T −1/N is a uniformizing parameter. The group-scheme µN(p−1) operates on U, by ζ : (T , X) → (ζ N T , ζX). This action extends to C, and fixes the point at infinity. A straightforward computation gives the following lemma. Lemma 7.1. (1) The genus of C is 21 (N − 1)(p − 1), and a basis of everywhere holomorphic differentials on C is given by the forms XaT b

dT X N−1

with 0 ≤ a ≤ N − 2, 0 ≤ b ≤ p − 2, and pa + Nb < (p − 1)(N − 1) − 1. ∼

1 (U ⊗ Q/Q) has dimension (N − 1 (C ⊗ Q/Q) − → HDR (2) The space HDR 1)(p − 1), any d basis is given by the cohomology classes of the forms

XqT b

dT X N−1

0 ≤ a ≤ N − 2, 0 ≤ b ≤ p − 2.

1 (C ⊗ Q/Q) are pre(3) The characters of µN(p−1) which occur in HDR cisely those whose restrictions to µN is non-trivial, and each of these occurs with multiplicity one.

226

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Nicholoas M. Katz

In characteristic p, there are new automorphisms. The additive group F p operates on C ⊗ F p by a : (T , X) → (T + a, X). This action does not commute with the action of µN(p−1) . However, the two together define an action of the semi-direct product F p ⋉ µN(p−1) formed via the homomorphism −N

µN(p−1) −−→ µ p−1 ≃ F ×p = Aut(F p ) Explicitly, the multiplication is (a, ζ)(b, ζ1 ) = (a + ζ −N b, ζζ1 ), and the action is (a, ζ) : (T , X) → (ζ N T + ζ N a, ζX). 227

The group F p ⋉ µN(p−1) contains F p × µN as a normal subgroup, acting on C ⊗ F p in the usual manner. Remark. This action of a group of order p(p − 1)N on a curve of genus g = 21 (p − 1)(N − 1) provides a nice example of how “wrong” the characteristic zero estimate 84(g − 1) can become in the presence of wild ramification! Let E be a number field containing the N(p − 1)’st roots of unity, P a p-adic place of E, Fq a finite extension of the residue field F N(P) , of P, G the abstract group F p ⋉ µN(p−1) (Fq ). Let H 1 denote any of the vector 1 (C ⊗ F /W(F )) ⊗ K. spaces Hl1 (C ⊗ Fq )⊗Eλ for l , p, or Hcris q q Zl

By functoriality, the group G operates on H 1 . Because the center of G is µN (Fq ), the decomposition H 1 = ⊗(H 1 )χ of H 1 according to the characters of µN is G-stable.

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249

Proposition 7.2. For each of the N −1 non-trivial E-valued characters χ ∼ of µN (E) − → µN (F N(P) ) = µN (Fq ), the corresponding eigenspace (H 1 )χ is a p − 1 dimensional absolutely irreducible representation of G; the restriction to F p of (H 1 )χ is the augmentation representation of F p ; the restriction to µN(p−1) (Fq ) of (H 1 )χ is the induction, from µN to µN(p−1) , of χ. Proof. All assertions except for the G-irreducibility of (H 1 )χ follow immediately from the preceding lemma, giving the action of µN(p−1) , and from Corollary (2.2), giving the action of F p × µN . The irreducibility follows from these facts together with the fact that in any complex representation of G, the set of characters of F p which occur is stable under the action of µN(p−1) in F p by conjugation; because this action has only the two orbits F ×p and 0, as soon as any one non-trivial character of F p occurs, all non-trivial characters must also occur.  Corollary 7.3. (1) Over any finite extension Fq of F p which contains all the N(p − 1)’st roots of unity (i.e., q ≡ 1mod N(p − 1)), the Frobenius F relative to Fq operates as a scalar on each of the spaces (H 1 )χ , χ a non-trivial character of µN . This scalar is the common value −gq (ψ, χ; P) of the Gauss sums attached to any of the non-trivial additive characters 228 ψ of FP . Proof. Over such an Fq , Frobenius commutes with the action of G on H 1 , so it acts on each (H 1 )χ as a G-morphism. Because (H 1 )χ is Girreducible, this G-morphism must be a scalar, and this scalar is equal to any eigenvalue of F on (H 1 )χ . As we have already seen (2.1), these eigenvalues are precisely the asserted Gauss sums, corresponding to the decomposition of (H 1 )χ under F p .  The common value of these Gauss sums over a sufficiently large Fq is itself a Jacobi sum, in consequence of the fact that universally, i.e., over Z[1/N(p − 1)], the curve C is the quotient of the Fermat curve

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Fermat (N(p − 1)) of degree N(p − 1) by the subgroup H of µN(p−1) × µN(p−1) consisting of all (ζ1 , η2 ) satisfying p−1

ζ1

p

= ζ2

Explicitly, the map is given rationally by the formulas (W, V) on W N(p−1) + V N(p−1) = 1 ↓

(T , X) on T p − T = X N

T = 1/V N , X = W p−1 /V p . Lemma 7.4. Let χ1 be a character of µN(p−1) whose restriction to µN is non-trivial. Under the map ∼

1 HDR (C ⊗ Q/Q) − → H 1 (Fermar (N(p − 1)) ⊗ Q/Q)H

we have ∼

p−1

1 1 HDR → HDR (C ⊗ Q/Q)χ1 − (Fermat (N(p − 1)) ⊗ Q/Q)χ1

−p

×χ1



Proof. That H 1 (C) − → H 1 (Fermat)H in rational cohomology results from the Hochschild-Serre spectral sequence. Since the characters of µN(p−1) (resp of µN(p−1) × µN(p−1) ) occur, if at all, with multiplicity one in H 1 (C) (resp H 1 (Fermar)), it suffices to check that the χ1 -eigenspace p−1 −p of H 1 (C) is mapped to the (χ1 , χ1 )-eigenspace of H 1 (Fermat). This 229 we do by inspection: XaT b

dT = X N−1 =X

a+1−N

T

W p−1 7→ T Vp

b+1 dT

!a+1−N 

Z

 −N b+1

! −NdZ . Z 

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Corollary 7.5. If Fq contains the N(p − 1)’st roots of unity, then for any non-trivial character χ of µN , and extension χ1 of χ to µN(p−1) and any non-trivial additive character ψ of F p , the scalar by which F acts on H 1 (C ⊗ Fq )χ is given by   F|H 1 (C ⊗ Fq )χ = F|H 1 (C ⊗ Fq )ψ×χ = −gq (ψ, χ; P)             p−1 −p F|H 1 (C ⊗ F )χ1 = F|H 1 (Fermat ⊗F )χ1p−1 ×χ−p 1 = −Jq (χ q q 1 , χ1 ; P)

We now turn to the “determination” of the Gauss sum −gq (ψ, χ; P) over an Fq which is merely required to contain the N’th roots of unity. Unless p − 1 and N are relatively prime, such an Fq need not contain the N(p − 1)’st roots of unity! Moreover, the Gauss sum does not in general lie in the Witt vectors W(Fq ), as it does when Fq contains the N(p−1)’st roots of unity! Let π denote any solution of π p−1 = −p. We recall without proof the following standard lemma (cf. [31] or [32]). Lemma 7.6. The fields Q p (ζ p ) and Q p (π) coincide. There is a bijective correspondence primitive p’th roots of 1 ←→ solutions π of π p−1 = −p under which ζ ←→ π if and only if ζ ≡ 1 + πmod π2 . For each solution π of π p−1 = −p, we denote by ψπ : F p → Q p (ζ p )× the unique non-trivial additive character which satisfies ψπ (1) ≡ 1 + πmod π2 .

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If we fix a W(Fq )-valued point x on C, we have the map “formal expansion at x” 1 1 bx ⊗ W(Fq )/W(Fq )). Hcris (C ⊗ Fq /W(Fq )) → HDR (C

If we denote by R the ring

R = W(Fq )[π] which is a free W-module of finite rank (p − 1), we may tensor with R to obtain / H 1 (C bx ⊗ R/R). ✐4 DR ✐ ✐ ✐ ✐✐✐✐ ✐✐✐✐ ✐ ✐ ✐ ✐✐✐✐

1 (C ⊗ F /W(F ))⊗R Hcris q q



W

1 (C ⊗ R/R) HDR

Theorem 7.7. (1) For any W(Fq )-valued point x on C, the “formal expansion” map is injective : 1 1 bx ⊗ W(Fq )/W(Fq )) (C (C ⊗ Fq /W(Fq )) ֒→ HDR Hcris

(2) Let π be any solution of π p−1 = −p, ψπ the corresponding additive character, a an integer 1 ≤ a ≤ N − 1 and χa the corresponding nontrivial character of µN (χa (ζ) = ζ a ). If we take for x the point (T = 0, X = 0) on C, with parameter X, then the image of 1 1 bx ⊗ R/R)⊗Q p (π) (Hcris (C ⊗ Fq /W(Fq )) ⊗ Q p (π))ψw ×χa → HDR (C R

is the one-dimensional Q p (π)-space spanned by the cohomology class of dX X dX exp(−πX N )X a = b(n)X n . X X Corollary 7.8. Notations as above, let f (X) denote the power series f (X) =

X n≥1

b(n)

X n X (−π)n X nN+a = n n! nN + a n≥0

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253

Then the series f (X q ) + gq (ψπ , χa ; P) · f (X) has coefficients with bounded denominators, and we have a limit formula  q·b(aqr )   −g (ψ , χ ; P) = lim b(aq  r+1 )   q π a r→∞ r −1) a (7.8.1)  (q  N   with b(aqr ) = (−π) a r ((q −1) )! N

We first deduce the corollary from the theorem. We know that F has 1 ⊗ Q (π), eigenvalue −gq (ψπ , χa ; P) on the ψπ × χa -eigenspace of Hcris p hence F has the same eigenvalue on the image of this one-dimensional 1 (C bx ⊗ R/R) ⊗ Q p (π). This image is spanned by the eigenspace in HDR cohomology class of df : therefore F +gq (ψπ , χa ; p) annihilates the class of df mod torsion, whence f (X q ) + gq (ψπ , χa ; P) · f (X) has bounded denominators. The final limit formula comes from looking r+1 successively at the coefficients of X aq in the above expression; one has ! b(aqr ) b(aqr+1 ) ord ≥ −A + gq (ψπ , χa ; p) · aqr aqr+1 for some constant A independent of r. An explicit elementary calculation shows that ! b(aqr ) ord → −∞ as r → +∞, aqr and this allows us to “divide” the additive congruence and obtain the asserted limit formula. It remains to prove the theorem. In view of the exact sequence of (5.9.5), the injectivity of 1 1 bx ⊗ W/W) Hcris (C ⊗ Fq /W(Fq )) → HDR (C

1 . 232 is equivalent to the absence of any p-adic unit eigenvalues of F in Hcris

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But these eigenvalues are the Gauss sums X −gq (ψ, χ) ≡ − ψq (x)χq (x). Because ψq (x) ≡ 1(π) for all x, while χq is a non-trivial character of Fqx , we have X −g(ψ, χ) ≡ − χq (x) = 0mod π.

(Alternately, one could observe that each non-trivial character χ of µN 1 has at least one extension χ1 to µN(p−1) which occurs in H 0 (C⊗Q, ΩC⊗Q ); p−1 the eigenvalue of F on this eigenspace is then a non-unit by (??); as F p−1 is a scalar on (H 1 )χ , this scalar is non-unit.) It remains to verify that the image of the ψπ ×χa -eigenspace is indeed spanned by dX exp(−πX N )X a X

This seems to require the full strength of the Washnitzer-Monsky “dagger” cohomology, as follows. Let At denote the “weak completion” of the coordinate ring R[T , X]/(T p − T − X N ) of U ⊗ R. Because U ⊗ Fq is a “special affine variety” with coordinate X, there are unique liftings to At of the actions of F and of the group F p ⊗ µN whose effect on X is given by    F(X) = X q    (a, ζ)(X) = ζX. Thanks to Dwork, we know that the power series in T exp(πT − πT p ) actually lies in R[T ]t , and hence in At for any π satisfying π p−1 = −p. As Monsky pointed out, under the action of F p and At , this series transforms by the character ψπ . It follows that for 1 ≤ a ≤ N − 1 the differential form dX exp(πT − πT p )X a X

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255

transforms by ψπ × χa under the action of F p × µN . Therefore its coho- 233 mology class in dfn

1 HW−M (U ⊗ Fq ; R) ⊗ Q = H 1 (ΩU⊗R/R ⊗ At ) ⊗ Q •

1 lies in the ψπ × χa eigenspace of HW−M . A direct computation ([31], [32]) shows that each of these eigenspaces is one-dimensional, and is spanned by the above-specified form. Furthermore, there is a natural “formal expansion map” attached to any R-valued point x of U; bx ⊗ R/R). H 1 (U ⊗ Fq ; R) → H 1 (U DR

W−M

For the particular choice of point (T = 0, X = 0), the formal expansion map carries dX dX exp(πT − πT p )X a 7→ exp(−πX N )X a . X X 1 To conclude the proof, we need to identify HW M (U ⊗ Fq ; R) ⊗ Q 1 (C ⊗ F /R) ⊗ Q in a way compatible with the formal expansion with Hcris q map and with the action of F and of F p × µN . We will do this with a somewhat ad hoc argument. Because U is the complement of a single point in C, it follows from the theory of residues for both HDR and HW−M that we have isomorphisms ∼



1 1 1 HDR (C ⊗ R/R) ⊗ Q − → HDR (U ⊗ R/R) ⊗ Q − → HW−M (U ⊗ Fq ; R) ⊗ Q.

These sit in a commutative diagram 8

6

2

5 3 1 7

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Nicholoas M. Katz

In this diagram, the maps 2 , 5 and 6 are each compatible with the actions of F and of F p × µN imposed by crystalline and by W − M theory (simply because these actions lift to the U ⊗ Wn ). Therefore the compatibility of the isomorphism 8 with the actions of F and of F p × µN would follow from the injectivity of arrows 2 and 6 . The injectivity of these arrows follows from the commutativity of the diagram and the already noted injectivity of arrow 1 (which is in1 of our jective exactly because F has no p-adic unit eigenvalues in Hcris particular C). A Question 7.8.2. Let U be a smooth affine W-scheme which is the complement of a divisor with normal crossings in a proper and smooth W-scheme. Are the maps 1 1 (U ⊗ Wn /Wn )) ⊗ Q (U/W) ⊗ Q → (lim HDR HDR ←−−

always injective?

7.9 The Gross-Koblitz Formula In this section we will derive the Gross-Koblitz formula from our limit formulas. Morita’s p-adic gamma function is the unique continuous function Γ p : Z p → Z ×p whose values on the strictly positive integers are given by the formula Γ p (1 + n) = (−1)n+1 ·

Y

1≤i≤n p∤n

i=

(−1)n+1 · n! [n/p]!p[n/p]

(7.9.1)

where [ ] denotes “integral part.” Lemma 7.9.2. For any integer n ≥ 0, and any π satisfying π p−1 = −p, we have the identity (−π)n /n! (π)n−p[n/p] . = (−1) · Γ p (1 + n) (−π)[n/p] /[n/p]!

(7.9.3)

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257

Proof. This is just a rearrangement of (7.9.1).



Corollary 7.9.4. Let q = p f with f ≥ 1, π any solution of π p−1 = −p 235 and n ≥ 0 any integer. Let n = n0 + n1 p + · · ·

0 ≤ ni ≤ p − 1

be the p-adic expansion of n. Then we have (−π)n /n! (−1) f · (π)n0 +n1 +···+n f −1 = fQ −1 (−π)[n/q] /[n/q]! Γ p (1 + [n/pi ])

(7.9.5)

i=0

Proof. Simply apply (7.9.3) successively to n, [n/p], . . . [n/p f −1 ].



For a fixed integer i ≥ 0, the map on positive integers n 7→ [n/pi ] extends to a continuous function Z p → Z p which we denote n 7→ [n/pi ] p . In terms of the p-adic “digits” of n, this map is just the i-fold shift: X X n= n j p j 7→ n j+i p j = [n/pi ] (7.9.6) j>0

Lemma 7.9.7. Let 0 < α < 1 be a rational number with a prime-to-p denominator. If p f = 1mod denom (α) for some f ≥ 1, then we have the identity −hp f −1 αi = [−α/pi ] p in Z (7.9.8) for i = 0, 1, . . . , f − 1 (where h i denotes the “fractional part” of a rational number).

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Nicholoas M. Katz

Proof. Write (p f − 1)α = A. Then A is an integer, 0 < A < p f − 1, so we may write its p-adic expansion as A = a0 + a1 p + · · · + a f −1 p f −1 ;

0 ≤ ai ≤ p − 1 ai < p − 1 for some i.

We now extend the definition of an to all n ∈ Z by requiring

236

∀ n ∈ Z.

an = an+ f Then

A p f −i α = p f −i f = p −1 ≡

fP −1

a j p f + j−i

j=0

pf − 1

fP −1

a j+i p j

j=0

pf − 1

mod Z

whence

−hp f −i αi =

But we readily calculate −α =

fP −1

a j+i p j

j=0

1 − pf

=

X

a j+i p j

j≥0

P   a j p j    j≥0  =   pi 

p

X A = a j p j. 1 − pf j≥0 

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Corollary 7.9.9. Let q = p f with f ≥ 1, π any solution of π p−1 = −p, and α any rational number satisfying     0 ≤ α ≤ 1    (q − 1)α ∈ Z. Let

A = (q − 1)α = a0 + a1 p + · · · + a f −1 p f −1 ,

0 ≤ ai ≤ p − 1

be the p-adic expansion of (q − 1)α, and let

237

S ((q − 1)α) = a0 + a1 + · · · + a f −1 be the sum of the p-adic digits of (q − 1)α. Then we have the formula (−π)n /n! (−1) f · (π)S ((q−1)α) = fQ −1 n→−α (−π)[n/q]! Γ p (1 − hpi αi) lim

(7.9.10)

i=0

in which the limit is taken over positive integers n which approach −α p-adically. Proof. Simply combine (7.9.5) and (7.9.8), and use the p-adic continuity of both Γ p and of n → [n/pi ].  Combining this last formula with our limit formula for Gauss sums, we obtain the Gross-Koblitz formulas. Theorem 7.10 (Gross-Koblitz). Let N ≥ 2 prime to p, E a number field containing the N p’th roots of unity, P a p-adic place of E, π ∈ E p a solution of π p−1 = −p, ψπ the corresponding additive character of F p , a an integer 1 ≤ a ≤ N − 1, χa the corresponding character ζ 7→ ζ a of µN , and Fq , q = p f , a finite extension of the residue field F N(P) of E at P. We have the formulas, in EP ,   i Q (−1) f · q · Γ p i − h pNa i −gq (ψπ , χa ; P) =

imod f

a

(π)S ((q−1) N )

(7.10.1)

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Nicholoas M. Katz

−gq (ψπ , χa ; P) = (π)

Y

S ((q−1) Na )

imod f

Γp

! pi a i h N

(7.10.2)

Proof. The sequence nr = (qr − 1)(a/N) tends to −a/N as r grows, and satisfies [nr /q] = nr−1 for r ≥ 1. Therefore the first formula follows from the limit formula (7.8.1) and from the preceding formula (7.9.10) with α = a/N. The second formula is obtained from the first by replacing a by N − a.  238

VIII. Interpretation via the De Rham-Witt Complex. Throughout this chapter, we fix an algebraically closed field k of characteristic p, and a proper smooth connected scheme X over its Witt vectors W = W(k). For each n ≥ 1, we denote by Xn the Wn -scheme X⊗Wn . W

The “second spectral sequence” of de Rham cohomology of Xn /Wn p,q

q

E2 (n) = H p (Xn , HDR (Xn /Wn )) ⇒ H p+q (Xn /Wn ) has an intrinsic interpretation in terms of X ⊗ k as the Leray spectral sequence for the “forget the thickening” map (X ⊗ k/Wn )cris → (X ⊗ k)Zar . As such, it may be rewritten p,q

q

p+q

E2 (n) = H p (X ⊗ k, Hcris (X ⊗ k/Wn )) ⇒ Hcris (X ⊗ k/Wn ). An explicit construction of this spectral sequence may be given in terms of the De Rham-Witt pro-complex on X ⊗ k {Wn Ω }n •

of Deligne and Illusie; it is simply the second spectral sequence of this complex: p,q

E2 (n) = H p (X ⊗ k, H q (Wn Ω )) ⇒ H p+q (X ⊗ k, Wn Ω ). •



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261

It is known that the E2 terms of this spectral sequence are finitely generated Wn (k)-modules. Therefore we may pass to the inverse limit and obtain a spectral sequence p,q

p,q

p+q

E2 = lim E2 (n) ⇒ Hcris (X ⊗ k/W). ←−− n

Let x be a W-valued point of X, and assume X connected. The formal expansion map we have exploited i i i bx /W) Hcris (X ⊗ k/W) ≃ HDR (X/W) → HDR (X

is the composition of the edge-homomorphism

i 0,i Hcris ֒→ E20,i (X/W) ։ E∞

with the natural map i i bx ⊗ Wn /Wn ). (X E20,i = lim H 0 (Xn , HDR (Xn /Wn )) → lim HDR ←−− ←−− n

Lemma 8.1. This map is in fact injective; indeed, the induced maps

as injective.

i i bx ⊗ Wn /Wn ) H 0 (Xn , HDR (Xn /Wn )) → HDR (X

Proof. Because Xn is irreducible, it suffices to show (*) for any closed point y of Xn , and any affine open V ∋ y which is e´ tale over standard affine space A = Spec(Wn [T 1 , . . . , T d ]), the natural map i i b H 0 (V, HDR (Xn /Wn )) → HDR (Vy /Wn ).

is injective. For once (*) is established we argue as follows. Let ξ be a global i which dies formally at x. We must show that section over Xn of HDR for any closed point z in Xn , there is an open set V ∋ z such that ξ dies on V. Let U be an affine open neighborhood of x e´ tale over A, and V an

239

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Nicholoas M. Katz

affine open neighborhood of z e´ tale over A. Because Xn is irreducible, U ∩ V is non-empty. Let y be a closed point of Xn contained in U ∩ V.

Then (*) for U ∋ x shows that ξ dies on U. Therefore ξ dies formally at y. Applying (*) to V ∋ y, we find that ξ dies on V, as required. We now prove (*). Let F : A → A(σ) be any σ-linear map lifting p absolute Frobenius (e.g. T i → T i ). Because V is e´ tale over A, F extends uniquely to a σ-linear map F : V → V (σ) which lifts absolute Frobenius. n Because all iterates of F, especially F n : V → V (σ ) , are homeomorphisms, the functor (F n )∗ is exact. Therefore we have  i (V/W )) = H 0 (V (σn ) , (F ) (H i (V/W )))   H 0 (V, HDR n n n ∗ DR   (F n )∗ H i (V/Wn ) = H i ((F n )∗ (Ω∗ )) V/Wn DR n

But the complex (F n )∗ (ΩV/Wn ) on V (σ ) is a complex of locally free n 240 sheaves of finite rank on V (σ ) , with O-linear differential. For any closed n point y V, the formal stalk at y(σ ) is O bV(σn ),y(σn ) ≃ (F n )∗ Ω (F n )∗ (ΩV/Wn ) O . b •





OV (σn )

Vy /Wn

n)

Therefore the sheaves on V (σ dfn

i (V/Wn )) = H i ((F n )∗ ΩV/Wn ) F i = FiV/Wn = (F n )∗ (HDR •

are coherent, and (by flatness of the completion) their formal stalks are given by bi ) (σn ) = H i (b (F y DR Vy /Wn )

We must show that

n bi )y(σn ) . H ◦ (V (σ ) , F i ) ֒→ (F

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263

For this, it suffices to explicit a finite filtration F i ⊃ Fil1 F i ⊃ . . . n

whose associated graded sheaves are locally free sheaves on V (σ ) ⊗ k. We claim that the filtration induced by the p-adic filtration on Ω V/Wn has this property. To see this, we first reduce to the case V = A, as follows. The diagram •

V



A

Fn

/ V (σn )

 / A(σn )

Fn

is cartesian (because V is e´ tale over A). Therefore we have an isomorphism O ∼ − ((F n )∗ ΩA/Wn ) OV (σn ) . (F n )∗ ΩV/Wn ← •



OA(σn )

Because OV (σn ) is flat over OA(σn ) , this isomorphism is a filtered isomorphism (for the p-adic filtrations of ΩV/Wn and of ΩA/Wn ). By flatness again, this filtered isomorphism induces isomorphisms 241 O j j i i OV (σn ) grFil (FV/W ) ≃ (gr F ) A/W Fil n n •



OA(σn )

j

n

i It remains to show that grFil (FA/W ) is a locally free sheaf on A(σ ) ⊗ k. n n It is certainly a coherent sheaf on A(σ ) (because the p-adic filtration on (F n )∗ ΩA/Wn is OA(σn ) -linear), and it is killed by p; therefore it is a n coherent sheaf on A(σ ) ⊗ k. Because it is coherent, it is locally free on a non-void open set; if we knew that it were translation-invariant, i.e. n isomorphic to all it translates by k-valued points of A(σ ) ⊗ k, we would conclude that it is locally free everywhere. As a sheaf of abelian groups, it is visibly translation-invariant. It’s OA(σn ) ⊗k -module structure is the composite of its natural module-structure •

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Nicholoas M. Katz

over the sheaf of rings ◦ gr◦Fil HDR (A/Wn )

with the σn -linear isomorphism ∼

OA⊗k − → gr◦Fil H ◦ (A/Wn ) f 7→ (F n )∗ ( e f ),

where e f denotes any local section of OA lifting f . To conclude the proof, we must verify that this isomorphism is translation-invariant. For this, it suffices to show that it is independent of the particular choice of F lifting Frobenius which figures in its definition. For this independence, we simply notice that an “intrinsic” description of the same σn -linear isomorphism ∼

OA⊗k − → gr◦Fil H ◦ (A/Wn ) is provided by n f 7→ ( e f )p

where again e f ∈ OA denotes any lifting of f .



Lemma 8.2. The E2i,0 terms of the spectral sequence are given by E2i,0 ≃ Heti (X ⊗ k, Z p ) ⊗ W(k)

Proof. For each integer n ≥ 1, there is an isomorphism (cf. [24], [25]) ∼

0 Wn (OX⊗k ) − → HDR (Xn /Wn )

242

defined by (g0 , . . . , gn−1 ) 7→

n−1 X

pi (e gi ) p

n−i

i=0

where e gi is a local lifting of gi ∈ OX⊗k to O xn (Compare (??)).

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For variable n, these isomorphisms sit in a commutative diagram Wn+r (OX⊗k )



/H

0 DR (Xn+r /Wn+r )

usual projection



reduction mod pn

Wn (OX⊗k ) Fr



Wn (OX⊗k )



/H



0 DR (Xn /Wn ).

Therefore we may calculate 0 (Xn /Wn )) E2i,0 = lim H i (X ⊗ k, HDR ←−− n   \  ∼ r i − → lim  (image of F or H (X ⊗ k, Wn (OX⊗k )) . ←−− r n O ≃ lim(fixed points of F in H i (X ⊗ k, Wn (OX⊗k )) Wn (k) ←−− n Z/p Z

≃ lim Heti (X ⊗ k, Z/pn Z) ⊗ Wn (k). ←−− n

 Consider now the exact sequence of terms of low degree 1 0 → E21,0 → Hcris

d2

(X ⊗ k/W) → E20,1 −−→ E22,0

Lemma 8.3. The map d20,1 : E20,1 → E22,0 vanishes. 1 (X ⊗ k/W) and E 2,0 = H 2 (X ⊗ k, Z ) ⊗ W Proof. Because both Hcris p j et 2 are finitely generated W-modules, we see that E20,1 is a finitely generated W-module. Therefore its inverse limit topology (as lim E20,1 (n)) ←−− is equivalent to its p-adic topology. Because F n annihilates the sheaf 0,1 1 (X ⊗ k/W ), it annihilates its global sections E (n), and hence Hcris n 2 F is topologically nilpotent on E20,1 . But F is an automorphism of the

266

Nicholoas M. Katz

finitely generated W-module E22,0 ; as d2 commutes with F, this forces d20,1 to vanish.  Thus we obtain the following theorem. Theorem 8.4. The exact sequence of terms of low degree 0 → Het1 (X ⊗ k, Z p )⊗

/ E 0,1  2_

/ H 1 (X ⊗ k/W) cris

/0



243





1 (X/W) formal / H 1 ( X bx /W) HDR expansion DR

1 defines the Newton-Hodge filtration on Hcris

1 0 → (slope 0) → Hcris (X ⊗ k/W) → (slope > 0) → 0.

[When X/W is a curve, or an abelian scheme, this exact sequence coincides with the exact sequence ((5.7.2) or (5.9.5)!] Illusie and Raynaud have recently been able to generalize these rei sults to Hcris for all i. Their remarkable result is the following. Theorem 8.5. (Illusie-Raynaud). Let X0 be proper and smooth over an algebraically closed field of characteristic p > 0. The second spectral sequence of the De Rham-Witt complex p,q

p+q

E2 = lim H p (X0 , H q (Wn Ω )) ⇒ Hcris (X0 /W) ←−− •

n

degenerates at E2 after tensoring with Q: p,q

p,q

E2 ×Q ≃ E∞ ⊗Q, dr ⊗ Q = 0 for r ≥ 2, Z

Z

and defines the Newton-Hodge filtration on Hcris (X0 /W) ⊗ Q: p,q

q − 1 < slopes of E2 ⊗ Q ≤ q. Corollary 8.6. If X0 /k lifts to X/W, then for any W-valued point x of X, and any integer i, the image of the formal expansion map i i i bx /W) ⊗ Q Hcris (X ⊗ k/W) ⊗ Q ≃ HDR (X/W) ⊗ Q → HDR (X

i ⊗ Q. is precisely the quotient “slopes > i − 1” of Hcris

Bibliography

267

Bibliography [1] Atkin, A. O. L. and P. H. F. Swinnerton-Dyer, : Modular forms 244 on non-congruence subgroups. Proc. Symposia Pure Math XIX, 1–25, A. M. S. (1971). [2] Berthelot, P. : Cohomologie Cristalline des Sch´emas de Caracteristique p > 0. Springer Lecture Notes in Math 407, SpringerVerlag (1974). [3] Berthelot, P. and A. Ogus, : Notes on Crystalline Cohomology. Princeton University Press (1978). [4] Bloch, S. : Algebraic K-theory and crystalline cohomology. Pub. Math. I. H. E. S. 47, 187–268 (1978). [5] Cartier, P. : Groupes formels, fonctions automorphes et fonctions zeta des courbes elliptiques, Actes, 1970 Congr´es Intern. Math. Tome 2, 291–299 (1971). [6] Deligne, P. : La conjecture de Weil I Pub. Math. I. H. E. S. 43, 273–307 (1974). [7] ———- : Sommes trigonometriques, S. G. A. 4 21 , Cohomologie Etale. Springer Lecture Notes in Math 569, Springer Verlag (1977). [8] Ditters, B. : On the congruences of Atkin and Swinnerton-Dyer. Report 7610, February 1976, Math Inst. Kath. Unis, Nijmegen, Netherlands (preprint). [9] Dwork, B. : On the zeta function of a hypersurface. Pub. Math. I. H. E. S. 12 (1962). [10] ———- : On the zeta function of a hypersurface II. Ann. Math. (2) (80), 227–299 (1964). [11] ———- : Bessel functions as p-adic functions of the argument. Duke Math. J. vol. 41, no. 4, 711–738 (1974).

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[12] M. Boyarsky, : P-adic gamma functions and Dwork cohomology, to appear in T. A. M. S. [13] Fontaine, J.-M. : Groupes p-divisibles sur les corps locaux. Asterisque 47–48, Soc. Math. France (1977). [14] Gross, B. H. and N. Koblitz, : Gauss sums and the p-adic Γfunctions. Ann. Math. vol. 109, no. 3 (1979). [15] Gross, B. H. : On the periods of abelian integrals and a formula of Chowla-Selberg, with an appendix by David Rohrlich. Inv. Math. 45, 193–211 (1978). [16] Grothendieck, A. : Rev´etements Etales et Groupe Fondamental (SGA 1). Springer Lecture Notes in Math. 244, Springer-Verlag (1971). 245

[17] ———- : Groupes de Barsotti-Tate et Cristaux. Actes du Cong. Intern. Math. 1970, tome 1, 431–436 (1971). [17 bis.] ———- : Groupes de Barsotti-Tate et cristaux de Dieudonn´e. Sem. Math. Sup. 45, Presses Univ. de Montr´eal (1970). [18] ———- : Formule de Lefschetz et rationalit´e des fonctions L. Expos´e 279, Seminaire Bourbaki 1964/65. [19] Hartshorne, R. : On the de Rham cohomology of algebraic varieties. Pub. Math. I. H. E. S. 45, 5–99 (1976). [20] Hasse, H. : Theorie der relativ-zyklischen algebraischen Funktionenk¨orp´er, insbesondere bei endlichem Konstantenk¨orp´er. J. Reine Angew. Math. 172, 37–54 (1934). [21] ———- and H. Davenport : Die Nullstellen der Kongruenz zetafunktionen in gewissen zyklischen F¨allen. J. Reine Angew. Math. 172, 151–182. (1934). [22] Honda, T. : On the theory of commutative formal groups. J. Math. Soc. Japan, 22, 213–246 (1970).

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[23] ———- : On the formal structure of the Jacobian variety of the Fermat curve over a p-adic integer ring. Symposia Matematica XI, Istituto Nazionale Di Alta Matematica, 271–284, Academic Press (1973). [24] Illusie, L. : Complex de DeRham-Witt et cohomologie cristalline. to appear. [25] ———- : Complex de DeRham-Witt. Proceedings of the 1978 Journe´es de G´eom´etrie Algebriques de Renn´es, to appear in Asterisque. [26] ———- and M. Raynaud, : work in preparation. [27] Katz, N. : Nilpotent connections and the monodromy theorem. Pub. Math. I. H. E. S. 39, 175–232 (1970). [28] ———- : P-adic properties of modular schemes and modular forms. Proc. 1972 Antwerp Summer School, Springer Lecture Notes in Math 350, 70–189 (1973). [29] ———- and W. Messing, : Some consequences of the Riemann hypothesis for varieties over finite fields. Inv. Math. 23, 73–77 (1974). [30] ———- : Slope filtration of F-crystals. Proceedings of the 1978 Journe´es de G´eom´etrie Alg´ebrique de Rennes, to appear in Asterisque. [31] Koblitz, N. : A short course on some current research in p-adic analysis. Hanoi, 1978, preprint. [32] Lang, S. : Cyclotomic Fields II. Springer Verlag. [33] Lazard, M. : Lois de groupes et analyseurs. Ann. Sci. Ec. Norm. 246 Sup. Paris 72, 299-400 (1955). [34] ———- : Commutative Formal Groupes. Springer Lecture Notes in Math. 443, Springer-Verlag (1975).

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[35] Mazur, B. and W. Messing, : Universal Extensions and OneDimensional Crystalline Cohomology. Springer Lecture Notes in Math. 370, Springer-Verlag (1974). [36] Messing, W. : The Crystals Associated to Barsotti-Tate Groups. Springer Lecture Notes in Math 264, Springer-Verlag (1972). [37] ——— : The universal extension of an Abelian variety by a vector group. Symposia Matematica XI, Istituto Nazionale Di Alta Matematica 358–372, Academic Press (1973). [38] Monsky, P. : P-adic analysis and zeta functions. Lectures at Kyoto University, Kinokuniya Book Store, Tokyo or Brandeis Univ. Math. Dept. (1970). [39] ———- and G. Washnitzer, : Formal Cohomology I. Ann. Math. 88, 181–217 (1968). [40] ———- : One-dimensional formal cohomology, Actes, 1970 Congr´es Intern. Math. Tome 1, 451–456 (1971). [41] Morita, Y. : A p-adic analogue of the Γ-function. J. Fac. Sci. Univ. Tokyo 22, 255-266 (1975). [42] Mumford D. : (1965).

Geometric Invariant Theory. Springer-Verlag

[43] ———- : Abelian Varieties. Oxford Univ. Press (1970). [44] Ida, T. : The first de Rham cohomology group and Dieudonn´e modules. Ann. Sci. Ec. Norm. Sup. Paris, 3i´eme serie, Tome 2, 63–135 (1969). [45] Serre, J.-P. : Groupes Alg´ebrtiques et Corps de Classes. esp. Chapt VII, Hermann (1959). [46] Weil, A. : On some exponential sums. Proc. Nat. Acad. Sci: U.S.A. 34, 204–207 (1948).

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[47] ———- : Number of solutions of equations in finite fields. Bull. A. M. S. 497–508 (1949). [48] ———- : Jacobi sums as Gr¨ossencharaktere. Trans. A. M. S. 73, 487–495 (1952).

ESTIMATES OF COEFFICIENTS OF MODULAR FORMS AND GENERALIZED MODULAR RELATIONS By S. Raghavan 247

We shall be concerned here with two questions, motivated by arithmetic, from the theory of modular forms. The first one deals with the estimation of the magnitude of the Fourier coefficients of Siegel modular forms, while the second pertains to certain generalized modular relations (which may also be called Poisson formulae of Hecke type and) which appear to provide some kind of a link between automorphic forms (of one variable), representation theory and arithmetic.

§Modular forms of degree n Let rm (t) denote the number of ways in which a natural number t can be written as a sum of m squares of integers. We have the well-known Hardy-Ramanujan asymptotic formula [H-R] for m > 4: rm (t) = πm/2 σm (t)t(m/2)−1 /Γ(m/2) + O(tm/4 )

(1)

with σm (t) denoting the ‘singular series’. Arithmetical functions such as rm (t) or, more generally, the number A(S , t) of m-rowed integral columns x with t xS x = t for a given m-rowed integral positive-definite matrix S (where t X = transpose of x) occur as Fourier coefficients of modular forms. While Hardy and Ramanujan used the ‘circle method’ 272

Estimates of Coefficients of Modular Forms...

273

to prove (1), the approach of Hecke [H1] to (1) was via the decomposition of the space of (entire) modular forms into the subspace generated by Eisenstein seris and the subspace of cusp forms, the explicit determination of the Fourier expansion of Eisentein series and the estimation of the Fourier coefficients c(t) of cusp forms of weight k as c(t) = O(tk/2 ). More generally, let A(S , T ) be the number of integral matrices G such that t GS G = T for n-rowed integral T (For any matirx B, let t B denote its transpose and for a square matrix C, let tr(C) and det C denote its trace and determinant respectively). For A(S , T ), we√have, as a ‘genP erating function’, the theta series ϑ(S , Z) = exp(2π −1tr(t GS GZ)) G

where G runs over all (m, n) integral matrices and Z is in the Siegel 248 half-plane ‘Hn ’ of n-rowed complex symmetric matrices Z = (zi j ) with Y = (yi j ) positive definite and yi j = Im zi j ; further, the theta series is a modular form of degree n, weight m/2 and stufe 4 det S . Let Γn (s) denote the principal congruence subgroup of stufe s in the Siegel modular group of degree n and {Γn (s), k} denote the space of modular forms of degree n, weight k and stufe s. Pursuing the approach of Hecke and Petersson and using Siegel’s generalized Farey dissection [S], the following result √was proved in [R]: namely, if k > n + 1 and P a(T ) exp(2π −1tr(T Z)/s) ǫ{Γn (s), k}, there exists a linear f (Z) = T ≥0 √ P b(T ) exp(2π × −1tr(T Z)/s) of Eisenstein secombination g(z) = T ≥0

ries in {Γn (s), k} such that for positive-definite

T , a(T ) = b(T ) + O((min T −1 )n(n+1−2k)/2 (min T )(n+1−k)/2

(2)

(For positive definite R, min R is the first minimum in the sense of Minkowski). Specialising f to be ε(s, z) above, (2) implies the formula: A(s, T ) = λ

Y

α p (S , T )(det T )(m−n−1)/2 + O((det T )(m(2n−1)−2(n

2 −1))/4n

)

p

(3) where m > 2n + 2, = πn(2m−n+1)/4 (det S )−n/2 {Γ(m/2) . . . Γ((m − n + 1)/2)}−1 ,

274

Q p

S. Raghavan

α p (s, T ) is the product (over all primes p) of the p-adic densities

α p (S , T ) of representation of T by S ; further, in (3), T tends to infinity such that for a fixed constant c, min T > c(det T )1/n . From (3), an analogue of a theorem of Tartakowsky resulted for n = 2 [R]: namely, under the conditions above, for larde det T , A(S , T ) , 0 for every matrix in the ‘genus’ of S or for none at all, depending on certain congruence classes to which T belongs. It should be mentioned that, without using Siegel’s generalized Farey dissection, only estimates of the type a(T ) = O((det T )k ) could be derived, in general, earlier; for improving upon (2), it was felt that the decomposition of the space of modular forms of degree n into n + 1 subspaces through Maass’ Poincar´e series should be invoked. Hsia, Kitaoke and Kneser [H-K-K] obtained, using an arithmetic aproach, a very elegant proof of the analogue of Tartakowsky’s theorm for any n > 1 and m > 2n + 3. Quite recently, Kitaoka [KI] gave an analytic proof of the same result in the case when S is an even positive 249 definite m-rowed unimodular matrix with m > 4n+4. By considering for even k > n + r + 2, Zǫ ‘Hn ’ and 0 < r < n, the Eisenstein series E(Z, h) = k (Z, h) which have been studied by Klingen [KL] and which arise En,r by ‘lifting’ a cusp form h in {Γr (1), k} to {Γn (1), k}, Kitaoka has obtained, in the same paper, the estimate a(T , h) = O((det T )k−(n+1)/2 × (det T 1 )!(r+1−k)/2 ) for the Fourier coefficients a(T , h) of E(Z, h) with T = T1 ∗ and r-rowed symmetric T 1 . If f is in {Γn (1), k} with even k > ∗ ∗ 2n + 2 and Φn f = 0 for the Siegel operator Φ, then for the Fourier coefficients a (T ) of f with positive definite T , Kitaoka derived, as a consequence, the estimate a(T ) = O((det T )k−(n+1)/2 (min T )1−k/2 )

(4)

From [C], it can be seen that any f in {Γn (s), k} for k > 2n + 1 is a finite linear combination of Poincar´e series Gk (Z; Γn (s); T ) and their transforms under coset representatives of Γn (1) modulo Γn (s) for non-negative definite T . Following Kitaoka’s method with appropriate

Estimates of Coefficients of Modular Forms...

275

modifications (e.g. of Lemma 7, §2, [KI]), it is not hard to prove the following √ P Theorem . If f (Z) = a(T ) exp(2π −1tr(T Z)/z)ǫ{Γm (s), k} with k > T >0 ! A B 2n + 1 is such that for every M = in Γn (1), the constant term C D in the Fourier expansion of f ((AZ + B)(CZ + D)−1 ) det(CZ + D)−k is 0, then we have a(T ) = O((δT )k−(n+1)/2 (min T )1−k/2 ), for positive definite T. Kitaoka [KI] has conjectured that the above theorem is true even for 2k > 2n + 3. One can also consider the analogues of the theorem above her hermitian and Hilbert-Siegel modular forms.

§Poisson formulae of Hecke type. Arithmetical identities have played a useful role in the estimation of the order or the average order of arithmetical functions. For Ramanujan’s function τ(n), we have an interesting identity X X √ τ(n) exp(−s n) = 236 π23/2 Γ(25/2)s τ(n)(s2 + 16π2 n)−2 5/2 16n 0). 16n 0, there exists in Z , an involution Z 7→ Z × with Z × (s) = Z(κ − s)Γ(s)/Γ(κ − s) and this carries over to a unitary P operator F 7→ WF in F . If ϕ(s) = an n−s is a Dirichlet series 16n 1 and 16 j6r

further, for i , j, αi β j − α j βi is not of the form mα j − nαi with integers m, n > 0. X ( j) Let {ϕ j (s) = an |n|−s ; 1 6 j 6 N} and n,0

{ψ j (s) =

P

n,0

( j)

bn |n|−s ; 1 6 j 6 N} be two sets of N Dirichlet series (each

converging in some right half-plane absolutely) so that if we write ξ j (s) = λ sG(s)ϕ j (s), η j (s) = λ sG(s)ψ j (s) 251

for some fixed λ > 0, then we have the functional equations X ξ j (κ − s) = c jk ηk (s) (1 6 j 6 N)

(1 6 j 6 N)

(6)

16k6N

with real c jk ; we may suppose that (c jk )2 is the identity matrix and also that ξk , η1 have only finitely many poles. Following Igusa [I],

Estimates of Coefficients of Modular Forms...

277

the spaces F , Z may be redefined so that Z consists, for example, only of meromorphic functions Z on the complex plane such that Z/G is entire and PZ is bounded in ‘vertical strips’ (with neighbourhoods of poles removed) for every polynomial P. In the space F , we have a unitary operator W such that for every F in F , (M(WF))(s)/G(s) = (MF)(κ − s)/G(κ − s) for a κ > 0, M being the Mellin transform. Let no ξk have a pole on Re s = κ/2, for simplicity and let u1 , . . . , u p be all the poles of ξk ’s. Then we have a Poisson formula of Hecke type [R-R] given by the following Theorem . For any function F : R+x → C whose Mellin transform MF is such that MF/G is entire and P.MF is bounded in vertical strips (with neighbourhoods of poles removed) for every polynomial P and for ξ1 , . . . , ξN , η1 , . . . ηN satisfying (6), we have X X MF(s) a(k) Residue ξk (s) = (7) n F(|n|/λ) − s=u j G(s) n,0 Re u j 0 in Z or π−s−v × Γ((s + v)/2)Γ((s − v)/2) with v in C. Then for F on Q×A built from F and the various W p0 , we have an adelic analogue of our Poisson formula. Under specialization, a formula of this kind constitutes an important step in the Jacquet-Langlands’ theory, for showing that a global representation of GL2 (QA ) occurs in the space of cusp forms. Further details may be found in [R-R].

Bibliography

279

Bibliography [C] Christian U.: Uber Hilbert-Siegelsche Modulformen and Poincar´esche Reihen, Math. Ann. 148 (1962), 257-307. [H-R] Hardy G. H. and S. Ramanujan: Asymptotic formulae in combinatory analysis, Proc. London Math. Soc., (Ser 2) 17 (1918), 75115. [H1] Hecke E.: Theorie der Eisensteinscher Reihen h¨oherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 199-224; Gesamm Abhand, 461-486. [H2] Hecke E.: Uber die Bestimmung Dirichletscher Reihen durch ihre Funktional-gleichungen, Math. Ann. 112 (1936), 664-699; Gesamm. Abhand., 591-626. [H-K-K] Hsia J. C., Y. Kitaoka and M. Kneser: Representation of positive definite quadratic forms, Jour. reine angew. Math., 301 (1978), 132-141. [I]

Igusa J.-I.: Lectures on forms of higher degree, Tata Institute of Fundamental Research, 1978.

[KI] Kitaoka Y.: Modular forms of degree n and representation by quadratic forms (Preprint). [KL] Klingen H.: Zum Darstellungssatz f¨ur Siegelsche Modulformen, Math. Zeit., 102 (1967), 30-43. [M1] Maass H.: Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktional-gleichungen, Math. Ann., 121 (1949), 141-183. [M2] Maass H.: Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen, Math. Ann., 125 (1953), 233-263.

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[R] Raghavan S.: Modular forms of degree n and representaion by 254 quadratic forms, Annals Math., 70 (1959), 446-477. [R-R] Raghavan S. and S. S. Rangachari: Poisson formulae of Hecke type. V. K. Patodi Memorial Volume; Indian Academy of Sciences (1980), 129-149. [S] Siegel C. L.: On the theory of indefinite quadratic forms, Annals Maths., 45 (1944), 577-622; Gesamm. Abhand. II, 421-466.

A REMARK ON ZETA FUNCTIONS OF ALGEBRAIC NUMBER FIELDS1 By Takuro Shintani

Introduction For a totally real algebraic number field k, it is known that every (partial) 255 zeta function of k is a finite sum of Dirichlet series which are regarded as natural generalizations of the Hurwits zeta function (see [1] and [2]). In this note we show that the similar result holds for arbitrary (not necessarily totally real) algebraic number field. At the time of the Bombay Colloquium (1979), H. M. Stark orally communicated to the author that he has obtained such a result for non-real cubic fields. His oral communication was an initial impetus to the present work. The author wishes to express his gratitude to Stark. Notation. We denote by Z, Q, R and C the ring of rational integers, the field of rational numbers, the field of real numbers and the field of complex numbers respectively. The set of positive real numbers is denoted by R+ . For an algebraic number field k, we denote by E(k) and O(k) the group of units of k and the ring of integers of k respectively. 1 Results presented at the time of the Colloquium were relevant to automorphic forms on unitary groups of order 3. However, later the author found several gaps in the proof of those results. Here, another result obtained after the Colloquium is exposed. 2 Takuro Shintani suddenly passed away on November 14, 1980. Ed.

281

282

Takuro Shintani

1. Let V be an n-dimensional real vector space. For R-linearly independent vectors v1 , v2 , . . . , vt ǫV(1 6 t 6 n), we denote by C(v1 , . . . , vt ) the set of all positive linear combinations of v1 , . . . , vt . We call C(v1 , . . . , vt ) a t-dimensional open simplicial cone with generators v1 , . . . , vt . Note that generators of a given open simplicial cone are unique up to permutations and multiplications by positive scalars. We call a disjoint union of a finite number of open simplicial cones in V a general polyhedral cone. Thus a general polyhedral cone is not necessarily convex. Now assume that V has a Q-structure. Thus, Nan n-dimensional Q-vector sub256 space VQ such that one has V = VQ R is identified in V. An open Q

simplicial cone is said to be Q-rational if, for a suitable choice of generators, all generators are in VQ . A disjoint union of a finite number of Q-rational open simplicial cones is said to be a Q-rational general polyhedral cone. A linear form on V is said to be Q-rational if it is Q-valued on VQ .

Lemma 1. Let C (1) and C (2) be two Q-rational general polyhedral cones. Then C (1) − C (2) is again a Q-rational general polyhedral cone. Proof. If is sufficient to prove the Lemma assuming that both C (1) and C (2) are Q-rational simplicial cones. Let t be the dimension of C (2) . There are n R-linearly independent Q-rational linear forms L1 , . . . Lt ; M1 , . . . , Mn−t on V such that C (2) = {vǫV; La (v) > 0, a = 1, . . . , t,

Mb (v) = 0, b = 1, . . . , n − 1} .

For each b(1 6 b 6 n − t), set n C (1) (b, ±) = vǫC (1) ; M1 (v) = . . . = Mb−1 (v) = 0, ±Mb (v) > 0} .

A Remark on Zeta Functions of Algebraic Number Fields

283

For each a (1 6 a 6 t), set n C (1) (n − t + 1, a) = vǫC (1) ; Mb (v) = 0 for b = 1, . . . , n − t,

L1 (v) > 0, . . . , La−1 (v) > 0, La (v) 6 0} .

Then it is immediate to see that C (1) − C (2) is a disjoint union of sets: C (1) (b, +)(1 6 b 6 n − t), C (1) (b, −)(1 6 b 6 n − t) and C (1) (n − t + 1, a)(1 6 a 6 t). It follows from Lemma 2 of [1] and its corollary that C (1) (b, ±)(1 6 b 6 n − t) and C (1) (n − t + 1, a)(1 6 a 6 t) are all disjoint unions of finite number of Q-rational open simplicial cones.  2. Let k be an algebraic number field of degree n with r1 real and r2 complex infinite primes (n = r1 + 2r2 ). Let x 7→ x(i) (1 6 i 6 n) be n mutually distinct embeddings of k into the field of complex numbers C. We may assume that x(1) , . . . , x(r1 ) are all real and that x(r1 +i) = x−(r1 +r2 +i) (1 6 i 6 r2 ). We embed k into an n-dimensional real vector space 257 V = Rr1 × Cr2 via the map: x 7−→ (x(1) , . . . , x(r1 ) , x(r1 +1) , . . . , x(r1 +r2 ) ). We identify k with an n-dimensional Q-vector subspace of V by means of the embedding. Fix a Q-structure of V by setting VQ = k. Set V+ = Rr+1 × (C)r2 , k+ = V+ ∩ k and E(k)+ = E(k) ∩ k+ . Thus E(k+ ) is the group of totally positive units of k. By componentwise multiplications, the group E(k)+ acts on V+ . Proposition 2. There exists a finite system {C j ; j ∈ J}(|J| < ∞) of open S S simplicial cones with generators all in k+ such that V+ = uC j j∈J u∈E(k)+

(disjoint union).

Proof. For each x ∈ V, we denote by N(x) the “norm” of x given by N(x) = x(1) . . . x(r1 ) |x(r1 +1) . . . x(r1 +r2 ) |2 . Let V+1 be the subset of V+ consisting of all vectors with norm 1: V+1 = {x ∈ V+ ; N(x) = 1} . Note that each vector in V+ is uniquely expressed as a positive scalar multiple of a vector in V+1 : x = {N(x)1/n }{N(x)−1/n .x}.

284

Takuro Shintani

If follows from the Dirichlet unit theorem that the group E(k)+ acts on V+1 properly discontinuously and that E(k)+ /V+1 is compact. Thus, there exists a compact subset F of V+1 such that [ uF. (1) V+1 = u∈E(k)+

Note that the subset of V+1 gives as {N(x)−1/n x; x ∈ k+ } is dense in V+1 . Hence for each X ∈ F, there exists an n-dimensional open simplicial cone C with generators all in k+ such that x ∈ C ∩ V+1 and that C ∩ uC = for any 1 , u ∈ E(k)+ . Thus, there exists a finite system C1 , . . . , C s of n-dimensional open simplicial cones with generators all in k+ such that F=

s [ i=1

(Ci ∩ V+1 )

(2)

and that Ci ∩ uCi = for any 1 , u ∈ E(k)+ (1 6 i 6 s). 258

(3)

If follows from (1) and (2) that V+ =

s [ [

uCi .

i=1 u∈E(k)

Set C1(1) = C1 and set Ci(1) = Ci −

[

uC1 (2 6 i 6 s).

u∈E(k)+

Note that uC1 is disjoint to Ci except for a finite number of u. Hence Lemma 1. implies that Ci(1) is a Q-rational general polyhedral cone. Taking (3) into account, we have V+ =

s [ [

uCi(1) and

i=1 u∈E(k)+

uC1(1)

∩ Ci(1)

= for any u ∈ E(k)+ if i > 2.

A Remark on Zeta Functions of Algebraic Number Fields

285

Now assume that a finite system of Q-rational general polyhedral cones C1(a) , . . . , C (a) s (1 6 a 6 s−2) with the following three properties is given: Ci(a) ⊂ Ci s [ [ uCi(a) , V+ = i=1 u∈E(k)+ (a) uCi ∩ C (a) j = for

(4)(a) (5)(a)

any u ∈ E(k)+ if i 6 a and i , j.

(6)(a)

Then set Ci(a+1) = Ci(a) for i 6 a + 1 and set Ci(a+1) = Ci(a) −

[

(a) uCa+1 for i > a + 2.

u∈E(k)

Then {C1(a+1) , . . . , C (a+1) } is a finite system of Q-rational general polyhes dral cones with properties (4)a+1 , (5)a+1 and (6)a+1 . It is easy to see that {C1(s−1) , . . . , C (s−1) } is a finite system of Qs rational general polyhedral cones such that V+ =

s [ [

uCi(s−1)

(disjoint union).

i=1 u∈E(k)+

 Remark . For totally real fields k, Proposition 2 is obtained in [1] by a different method (cf. Proposition 4 of [1]). 3. We choose and fix a finite system {C j ; j ∈ J(|J| < ∞)} of open simplicial cones with generators all in k+ such that [ [ uC j (disjoint union). (7) V+ = j∈J u∈E(k)+

The existence of such a system is guaranteed by Proposition 2.. For each 259 C j , we denote by t j the dimension of C j and choose and fix generators v j1,..., V jt j of C j so that they are all in O(k)+ = O(k) ∩ k+ .

286

Takuro Shintani

Furthermore, we choose and fix integral ideals a1 , a2 , . . . , ah0 so that they form a complete set of representatives for narrow ideal classes of k. Lef f be an integral ideal of k and let Hk ( f ) be the group of narrow ideal classes modulo f . There is a natural homomorphism from the group Hk ( f ) onto the group of narrow ideal classes of k. Fro each c ∈ Hk ( f ) there uniquely exists an index i(c)(1 6 i(c) 6 h0 ) such that c is mapped to the class represented by f ai(c) . Set n o C 1j = s1 v j1 + s2 v j2 + . . . + st j v jt j ; 0 < s1 , s2 , . . . , st j 6 1 and R(c, C j ) = {x ∈ C 1j ∩ f −1 a−1 i(c) ; (x) f ai(c) ∈ c}. Then R(c, C j ) is finite. Let C be a t-dimensional open simplicial cone with a prescribed system of generators v1 , . . . , vt . For each x ∈ C, we denote by ζ(s, C, x) the Dirichlet series given by X N(x + z1 v1 + . . . + zt vt )−s , (8) ζ(s, C, x) = z

where z = (z1 , . . . , zt ) ranges over the set of all t-tuples of non- negative integers (the notation N is introduced at the beginning of the proof of Proposition 2). Let ζk (s, c) be the zeta functions of k corresponding to the ray class c given by X N(g)−s , (9) ζk (s, c) = g

where g ranges over the set of all integral ideals of k in the ray class c. Proposition 3. The notation and assumptions being as above. X X ζk (s, c) = N(fai(c) )−s ζ(s, C j , x). j∈J x∈R(c,C j )

260

Proof. Let g be an integral ideal in the ray class c. Then g and fai(c) are in the same narrow ideal class of k. Thus, for a suitable w ∈ k+ ,

Bibliography

287

g = fai(c) (w). In view of (7), we may assume that w ∈ C j ∩ k+ for a

suitable j ∈ J. Set w = y1 v j1 + . . . + yt j v jt j . Then y1 , . . . , yt j are all positive rational numbers. Let the integer part of ya be za (a = 1, . . . , t j ). Then x = w − (z1 v j1 + . . . + zt j v jt j ) ∈ C 1j ∩ (fai(c) )−1 . Furthermore (x)fai(c) is in the ray class c. Thus x ∈ R(c, C j ). A simple consideration shows that j, z1 , . . . , zt j and x are uniquely determined by g. On the other hand, for an x ∈ R(c, C j ) and a t j -tuple of non-negative integers z = (z1 , . . . , zt j ), ai(c) f(x + z1 v j1 + . . . + zt j v jt j ) is an integral ideal in the ray class c. We denote by Z+ the set of non-negative integers. We have seen that S t the following map establishes a bijection from the set {R(c, C j ) × Z+j } j∈J

onto the set of integral ideals of k in the ray class c: (x, z) ∈ R(c, C j ) ×

t Z+j

7−→ ai(c) f(x +

tj X

za v ja ).

a=1

Thus Proposition 3. now follow immediately from (9).



Remark. For totally real field k, Proposition 3. is given in the proof of Theorem 1 of [1] (see also [2]).

Bibliography [1]

Shintani, T. On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sec. IA. 23(1976), 393-417.

[2]

Zagier, D. A Kronecker limit formula for real quadratic fields, Math. Ann. 231(1975), 153-184.

DERIVATIVES OF L-SERIES AT S = 0 By H. M. Stark

1 Introduction 261

In 1970, I introduced [5] a rather vague general conjecture on values of Artin L-series at s = 1. Since then the conjecture has been considerably refined, especially for certain types of characters [6, II, III, IV]. It is appropriate to present a paper on this subject here since it was at the Tata Institute that the complex quadratic case was treated in the lectures of Siegel [4] and later work of Ramachandra [3]. It has become clear in recent years that the formulas at s = 0, although equivalent to formulas at s = 1 via the functional equation, are considerably simpler. In this paper, we will concentrate on the case of Artin L-series with first order zeros at s = 0. Included in this category of L-series are the abelian Lseries over complex quadratic ground fields studied by Ramachandra. Since his results have been improved, this is a good place to begin.

2 Complex quadratic ground fields Let k be a complex quadratic field, f an integral ideal of k, f , (1). Suppose G(f) is the ray class group of k(mod f) and let J be a subgroup of G(f) and K the class field corresponding to H = G(f)/J. The characters χ of H are precisely those ray class characters of k(mod f) which are 288

Derivatives of L-Series at S = 0

289

identically 1 on J. We let L(s, χ) denote the L-series corresponding to the primitive version of χ and L(s, χ, f) denote the L-series corresponding to the (possibly imprimitive) character χ(mod f). This is the series that results from L(s, χ) by deleting the p-factors from the Euler product of L(s, χ) for each p/f. Our improvement of Ramachandra’s result is the following theorem which is proved in [6, IV]. Theorem . For each coset c of J in G(f), there is an algebraic integer ε(c) such that the following three properties hold: i). For each character χ of H, L′ (0χ, f) = −

1 X χ(c) log(|ε(c)|2 ) W c∈H

where W is the number of roots of unity in K. ii). The explicit reciprocity law is given by ε(J)N(ρ) ≡ ε(c)(mod p) where p is a prime ideal in c. Further, ε(c)/ε(J)N(p) is a W th power of a number in K and the ε(c) are all associates. iii). If f = pa where p is a prime ideal, then NK/Q (ε(c)) = Nk/Q (p)b where

Wh w and h is the class-number of k, w the number of roots of unity in k. In all other cases, ǫ(c) is a unit. b=

Actually, part iii) is a simple corollary of part i) with χ being the (imprimitive) trivial character of H since by part ii) the ε(c) are the conjugates of ε(J).

262

290

H. M. Stark

√ As an example, suppose k = Q( d) has class-number one and f = pa for a first degree prime ideal p of norm p relatively prime to 6d. Here W = w and for each character χ of G(f), we have −1 X L′ (0, χ, f) = χ(c) log(|ε(c)2 ) W c where ǫ(c) is in the ray class field K(f) of k(mod f). The norm of ǫ(c) from K(f) to Q is p. By our Theorem, ε(c) = εwc ε(c0 ) where c0 is the principal ray class (mod f), εc is in K(f) and is a unit. As we show in [6, IV] by the theory of group determinants as discussed by Siegel [4], the units εc , c , c0 , together with the wth roots of unity generate a subgroup of the unit group of K(f) of index precisely the class-number of K(f). All previous results in this direction have had a much larger index. The situation in this example figured strongly in the work of Coates and Wiles [2]. For the rest of this section, we will suppose that fτ = f andJ τ = J where τ denotes complex conjugation. Thus the field K is normal over Q. We identity H with the Galois group of K/k via our Theorem and 263 now write −1 X L′ (0, χ, f) = χ(h) log(|εh |2 ) W h∈H where ε = ε(J). We let G denote the Galois group of K/Q. We will denote the characters of G by the Greek letter ψ while continuing to denote the characters of H by χ. In particular, if ψ is the character of G induced by χ then for any h in H, ψ(h) = χ(h) + χ(τhτ−1 ), ψ(hτ) = 0. It turns out that ε was constructed so that some power of ε is real. Therefore, −1 |ετhτ | = |εh |

Derivatives of L-Series at S = 0

291

and it follows that L′ (0, ψ, N(f)) = L′ (0, χ, f) −1 X = ψ(g) log(|εg |2 ) 2W g∈G Although different in appearance, this is equivalent to the general conjecture in [6, II] for this case with “fudge constant” −1/(2W). To illustrate some of the possibilities that occur, we will take as an example the case where G is the dihedral group of order 8 with generators√σ, τ and relations σ4 = τ2 = 1, στ = τσ3 . This group arises over Q( −19) with τ being complex conjugation and H = H−19 = {1, σ2 , στ, σ3 τ}, the Klein four group. ❍❍❙❙❙❙ ❧❧ K ❙ ❍❍ στ ❙ ❧❧❧✇✇✇ ❧ ❧ ❍❍ ❙❙❙σ❙❙3 τ ❧ ✇ ❧ ✇ 2 ❧ ❙❙❙ ❧ ❍❍ ✇ σ ❧ ❧ ❙❙❙ ❍ ✇✇ τ ❧❧❧ ✇ ❙❙ ❧ ❧ ❧ σ2 τ

(2) K17

(1) K17

❈❈ ❈❈ ❈❈ ❈❈

K17,−19

① ①① ①① ① ① ①①

(1) K−19

●● ●● ●● ●● ●

k−19 k17 ❍ k−323 ❍❍ ✉✉ ❍❍ ✉ ✉ ❍❍ ✉✉ ❍❍ ❍ ✉✉✉✉ Q

②② ②② ② ② ②②

(2) K−19

√ There is a pair of prime ideals of norm 17 in Q( −19), p(1) 17 = 264 √    √   7 + −19  (1) 7− −19  and p(2)  . For j = 1, 2, there are unique 2 17 = p17 τ = 2 ray class characters χ(i) (mod p(i) 17 ) of order two. They are primitive char(1) (2) (1)τ acters and give rise to ray class fields K−19 and K−19 = K−19 . The composite field K comes from the ray class group (mod 17) modulo a subgroup of index 4 where both χ(1) and χ(2) are defined. Further, G(K/Q) =

292

H. M. Stark

G, the dihedral group of order 8. The product character χ(1) χ(2) of order two is a primitive (mod 17) and corresponds to the class √ character √ field K17,−19 = Q( 17, −19). Of the five possibilities, we see that (1) {1, στ} or {1, σ3 τ} = τ−1 {1, στ}τ must be G(K−19 /k−19 ) and we assume (1) σ has been picked so that G(K−19 /k−19 ) = {1, στ}. This makes H as √ claimed. There are √ two other quadratic subfields of K : k17 = 2Q( 3 17) and k−323 = Q( −323). We see that G(K/k−323 ) = {1, σ, σ , σ } is cyclic while G(K/k17 ) = {1, τ, σ2 , σ2 τ} is the other Klein four group in G. (The real subfield of K is fixed by {1, τ} and is not normal over Q. This is what allows us to decide which group goes to which field.) The remaining two quartic subfields of K are quadratic extensions of (1) (2) k17 : K17 fixed by {1, τ} and K17 fixed by {1, σ2 τ} 1

σ2

στ

σ3 τ

χ1

1

1

1

1

χ(1)

1

-1

1

-1

χ(2)

1

-1

-1

1

χ(1) χ(2)

1

1

-1

-1

Character table of H−19 By our Theorem, there is a number π in K−19 of norm 17 such that L′ (o, χ(1) , p(1) 17 ) =

2 −1 [log(|π|2 ) − log(|πσ |2 )]. 2

(2) There is also such a number in K−19 for L′ (0, χ(2) , p(2) 17 ) but it is just π¯ and the formula is the same. It is no surprise that the formula should (2) (2) give the same answer since L(s, χ(1) , p(1) 17 ) and L(s, χ , p17 ) are the same 265 Dirichlet series. Indeed this series arises in several different ways. From the character table of G (whose characters have been

Derivatives of L-Series at S = 0

293

1

σ2

τ, σ2 τ

στ, σ3 τ

σ, σ3

ψ1

1

1

1

1

1

ψ17

1

1

1

1

ψ−19

1

1

1

1

1

−1

−1

ψ2

2

−2

ψ−323

−1 0

−1 0

−1 1

0

Character table of G given suggestive names), we see that χ(1) and χ(2) both give the same induced character of G, namely ψ2 . But ψ2 also arises as an induced character from G(K/k17 ) and G(K/k−323 ). In particular, there is a primitive ray class character of k17 modulo a prime ideal of norm 19 which (1) corresponds to K17 . It takes the values 1, 1, −1, −1, at 1, τ, σ2 , σ2 τ respectively and also induces ψ2 on G. Further, by our Theorem there is a 1 such that unit E in K−19 π = E2 2 πσ With η = |E|2 = EE τ we have L′ (0, ψ2 ) =

−1 log(|E 2 |2 ) = − log(η) 2

2

2

(1) where η is in K17 . Also E σ = ±1/E so that (EE τ )σ = 1/(EE τ ) and 2 hence, ησ = η−1 . The unit η is precisely what is called for in my conjecture for real quadratic L-series. However, I have proved my conjecture (1) for relative quadratic extensions, such as K17 /k17 without aid of complex multiplication. We return to K/k−19 again and now consider χ(1) and χ(2) as imprim- 266 itive characters (mod 17). According to our Theorem, there is a unit ε of K such that for any of the four characters χ of H,

L′ (0, χ, 17) = −

1X χ(h) log(|εh |2 ). 2 h∈H

294

H. M. Stark

(1) In fact, ǫ is real and so is also in K17 . The question then arises if ε = η. The answer to this question is related to the question as to why we bother with the imprimitive version of L(s, χ(1) ) since χ(1) (p(2) 17 ) = −1 and so

L′ (0, χ(1) , 17) = 2L′ (0, χ(1) ). It turns out that we get new units this way. For instance, since ε is 2 2 2 2 real, εσ2τ = ετσ = εσ = εσ so that εσ is real and 3

3

|εστ | = |ετστ | = |εσ | = |εσ τ | Hence  ε  L (0, χ , 17) = − log 2 εσ = −2 log(η). ′

(1)

2

Of course η/ησ = η2 and so ε = η is still possible. However, σ2 2 ! (εε ) ′ (1) (2) (ε) L (0, χ χ ) = − log NK/k−19 = −2 log(|εεσ2 |),

while by Dirichlet’s class-number formula, L′ (0, χ(1) χ(2) ) = L(0, ψ−323 )L′ (0, ψ17 )

= h(k−323 )h(k17 ) log(ε17 ) = 4 log(ε17 )

2

2

Thus εεσ = ±ε¯ 217 while ηησ = 1 and so ε , η. Since ǫ ε2 = εσ2 εεσ2 2

we also have a confirmation of the fact that ε/εσ is a square in K. 267 Thus far, we have looked at L′ (0, ψ2 ) in three different ways (twice over k−19 and once over k17 ) and found the three different numbers π, η, ǫ

Derivatives of L-Series at S = 0

295

all leading to the same result. But we can also look at L′ (0, ψ2 ) viewed over k−323 . In the table below, the two characters χ′ and χ¯ ′ of order four of H−323 = G(K/k−323 ) induce ψ2 . Here K is actually the Hilbert class field of k−323 . This has the unfortunate consequence that the conductor of L(s, ψ2 ) viewed over k−323 is (1) and our Theorem does not apply directly. However, we may make all four characters of H−323 imprimitive by raising the conductor. It is tempting to use the 1

σ

σ2

σ3

χ′1

1

1

1

1

χ′

1

i

χ′2 χ¯′ = χ′3

1

-1

−1

−i

1

1 −i −1 Character table of H−323

-1 i

unique ideal p′17 of k−323 of norm 17 as our conductor. Since p′17 is in the class of order two, the corresponding Frobenius automorphism of H−323 is σ2 . (Note 17 ramifies from Q to K so we must be very careful in going from H−323 to G with Frobenius automorphisms.) Hence, L′ (0, χ′ , p17 ) = 2L′ (0, χ′ ), (the same is true of χ¯ ′ ) and we are once again evaluating L′ (0, ψ2 , 17) = 2L′ (0, ψ2 ). We see from our Theorem that instead of getting ε again, there is a number π′ , in K such that for all four characters χ of H−323 , 1 X L′ (0, χ, p′17 ) = − χ(h) log(|π′h |2 ) 2 hǫH −323

where

268

NK/Q (π′ ) = 174 Further π′ is not just π2 or even π2 times a unit since (π′ ) = p′17

296

H. M. Stark

(2) (1) 2 so that (π′ )2 = (17) = p(1) 17 p17 while (π ) = p17 . Thus we have found still another number of K. Here again, π′ is real and so L′ (0, χ′ , p′17 ) simplifies to ′ ! π ′ ′ ′ −2 log(η) = L (0, χ , p17 ) = − log 2 . π′σ 2

where π′ /π′σ is real and is a square in K. The difficulty in using conductor (1) is that the trivial character gives ζk−323 (s) whose first derivative at zero is rather horrible. However, for the three non-trivial characters χ of H−323 , one can write all three L′ (0, χ) simultaneously in terms of a nice number given by quotients of Dedekind eta-functions. But this simultaneous expression of all three L-series would appear to require a worse coefficient than −1/2 on the right side of the equation. It does seem possible to express any one of three L′ (0, χ)′ s in a nice manner. For instance, there is a number α in K (non-integral) given by √ η(ω)2 1 + −323 α=3 , , ω= 2 η(ω/9)2 where we have used the eta-function on the right, and L′ (0, χ′ ) = − log(|α|2 ) from which we see that η = NK/K (1) (α) 17

3 L-series considered over Q In this section, K is a normal extension of Q with Galois group G whose characters will again be denoted by ψ. We have seen in the last section that if K has a complex quadratic subfield k such that G(K/k) = H is 269 abelian with conductor f = fτ , (1) and ψ2 is a of G induced by a character of H, then there is an integer ε in K such that −1 X L′ (0, ψ2 , N(f)) = ψ2 (g) log(|εg |2 ). 2W g∈G

Derivatives of L-Series at S = 0

297

P This tempts us to try and relate every L(s, ψ) to ψ(g) log(|εg |2 ). To see the difficulties that we face, let us momentarily return to the dihedral group example of the previous section. We recall that each time we considered L(s, ψ2 ) from a new perspective, we came up with a new number in K related to L′ (0, ψ2 ). From the point of view of characters of G, it is not at all clear why so many different numbers of K should arise or which number we should use. However, for illustrative purposes, let us take the real unit ε in K from the last section which satisfied, L′ (0, ψ2 , 17) =

−1 X ψ2 (g) log(|εg |2 ). 4 g∈G

Further, for ψ = ψ1 , ψ−19 or ψ−323 , X ψ(g) log(|εg |2 ) = 0. g∈G

For ψ = ψ1 , this is because ε is a unit while for ψ = ψ−19 or ψ−323 , it is because ψ(gτ) = −ψ(g) for all g in G which allows a pairing of terms. For ψ = ψ17 , the situation is even more intriguing since L′ (0, ψ17 , 17) = L′ (0, ψ17 ) = log(ε17 ) and so we expect some relation between X L′ (0, ψ17 ) and ψ17 (g) log(|εg |2 ). We found earlier that X X ψ17 (g) log(|εg |2 ) = [ψ17 (g) + ψ−323 (g)] log(|εg |2 ) = −4L′ (0, χ(1) χ(2) )

= −16 log(ε17 ) = −16L′ (0, ψ17 ).

The factor of 16 is rather hard to guess beforehand. Worse still, there are primes p which don’t split in k−19 with ψ17 (p) = −1. For these primes, L(s, χ(1) χ(2) ) has a p-factor (1 − p−2s )−1 and so L(s, ψ17 + ψ−323 , 17p) 270

298

H. M. Stark

has a second order zero at s = 0. This means that we come up with a unit such that X X ψ17 (g) log(|unitg |2 ) = [ψ17 (g) + ψ323 (g)] log(|unitg |2 ) = 0, g∈G

even though L′ (0, ψ17 , 17p) = 2L′ (0, ψ17 ) , 0. Thus it appears that is we wish a common factor such as −1/(2W) in front, we must give up looking simultaneously at all characters ψ of G such that L(s, ψ) has a first order zero at s = 0. For second degree characters, we may still ask if this is possible. Precisely, we ask the following. Question. Suppose that K is a complex normal extension of Q with Galois group G containing W roots of unity. Suppose that f is divisible by the conductor of every irreducible second degree character ψ of G with ψ(τ) = 0 where τ in G represents complex conjugation. Is there an integer π in K such that i). πg is an associate of π for all g in G and some power of π is real. ii). πg /π p is a W th power in K where p is a prime not dividing W f times the discriminant of K and whose associated Frobenius automorphisms are conjugate to g in G. iii). For every irreducible second degree character ψ of G with ψ(τ) = 0, −1 X ψ(g) log(|πg |2 ). L′ (0, ψ, f) = 2W g∈G This question is probably most safely asked when at least one of the characters ψ under consideration is not a character of any quotient group of G. The extra difficulties that arise otherwise can be illustrated by taking√K to be the 36th√degree field generated by the Hilbert class fields of Q( −23) and Q( −31). Also, a study of inertial groups should enable us to replace f by a smaller number in many cases.

Derivatives of L-Series at S = 0

299

Suppose we have a set of n irreducible characters ψ satisfying the hypotheses of our Question such that if ψ is in the set of n characters, so is every algebraic conjugate of ψ. Then we can expect to isolate n pieces of information about units from the numerical values of the L′ (0, ψf). 271 We do this by imitating the orthogonality relations for G. Consider the n-dimensional Z lattice in Cn generated by column vectors of the form vg = (ψ(g)) where ψ runs through the n characters under consideration in some fixed order. The dual lattice consists of those n-dimensional vectors u such that < u, vg > is in Z for all g in G. Without the hypothesis on algebraic conjugates of ψ being present, we needn’t have a lattice and then there may not be any non-zero u such that < u, vg > is in Z for all g. We now have   −1 X 2 < u, L′ (0, ψ, f) > = < u, vg > log πg 2W g∈G = Here εu =

  −1 log |εu |2 . 2W Y

(πg )

g∈G

is a unit since the

πg

are associates and

P g

< u, vg >= 0 by the orthogo-

nality relations. In fact, since πτ = ζπ for a root of unity ζ, Y Y (πg ) , (ζ g ) ετu = g

g

and so up to a root of unity εu is real. It seems likely that π can be chosen so that this root of unity is one (for example, if π itself is real) and εu is positive. We would then expect that < u, L′ (0, ψ, f) >=

−1 log(εu ), W

(1)

where εu is a positive real unit in K. Further, εu is already a W th power in K. To see this, let M be the field of W th roots of unity and H = G(K/M). If χ1 is the trivial character

300

H. M. Stark

of H, then by the definition of M, the induced character χ∗1 is the sum of all the one dimensional characters of G. It follows from the Frobenius reciprocity law that for any of our n characters ψ, the restriction of ψ to H does not contain χ1 . If ρ is a representation of G with character ψ, then for any g in G, X X ρ(gh) = ρ(g) ρ(h) = 0, h∈H

272

and hence

h∈H

X

ψ(gh) = 0.

h∈H

therefore

X

vgh = 0.

h∈H

For each g in G, let pg be chosen according to part ii) of our Question so that πg /π pg is a W th power in K. For any h in H, pgh ≡ pg (mod W) and hence X X pgh < u, vgh >≡ pg < u, vgh >≡ 0(mod W) h∈H

Therefore,

h∈H

Y πg ! Y π pg εu = p g π g∈G g∈G

is a W th power in K as claimed. I have shown numerically in several instances that the Question has an affirmative answer in cases where K is a class field of a real quadratic field [6, III, IV]. Just as this Colloquium was taking place, Ted Chinburg [1] formulated the Conjecture on Artin L-series with first order zeros at s = 0 in terms of (1) and investigated (1) in the case that K is the 48th degree field corresponding to the non-abelian modular form of conductor 133 found by Tate. He found a unit εu in K which is a W th power and which satisfies (1) to 13 decimal places. In fact he found εu by using the numerical values of the L′ (0, ψ) in a manner similar to [6, II] but with a nice improvement in the method that avoids the small searches that I had to make.

Bibliography

301

Bibliography [1]

Chinburg Ted, Stark’s Conjecture for a Tetrahedral Representa- 273 tion, to appear.

[2]

Coates J. and A. Wiles, On the Conjecture of Birch and Swinnerton-Dyer, Inv. Math. 39 (1977), 223-251.

[3]

Ramachandra K., Some aplications of Kronecker’s limits formulas, Ann. of Math. 80 (1964), 104-148.

[4]

Siegel C. L., Lectures on Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1961.

[5]

Stark H. M., Class-number problems in quadratic fields, in Proceedings of the 1970 International congress, Vol. 1, 511-518.

[6]

——-, L-functions at s = 1, I. II, III, IV, Advances in Math. 7 (1971), 301-343; 17 (1975), 60-92; 22 (1976), 64-84;

EISENSTEIN SERIES AND THE RIEMANN ZETA-FUNCTION By D. Zagier1

275

In this paper we will consider the functions E(z, ρ) obtained by setting the complex variable s in the Eisenstein series E(z, s) equal to a zero of the Riemann zeta-function and will show that these functions satisfy a number of remarkable relations. Although many of these relations are consequences of more or less well known identities, the interpretation given here seems to be new and of some interest. In particular, looking at the functions E(z, ρ) leads naturally to the definition of a certain representation of S L2 (R) whose spectrum is related to the set of zeroes of the zeta-function. We recall that the Eisenstein series E(z, s) is defined for z = x + iy ∈ H (upper half-plane) and s ∈ C with Re(s) > 1 by E(z, s) =

X

Im(γz) s =

γ∈Γ∞ /Γ

1 X ys 2 c,dεZ |cz + d|2s

(1)

(c,d)=1

! ) ( 1 n n ∈ Z ⊂ Γ. If we multiply both where Γ = PS L2 (Z), Γ∞ = ± 0 1 1

Supported by the Sonderforschungsbereich “Theoretische Mathematik” at the University of Bonn.

302

Eisenstein Series and the Riemann Zeta-Function

sides of (1) by ζ(2s) =

∞ P

303

r−2s and write m = rc, n = rd, we obtain

r=1

1X y2 , ζ(2s)E(z, s) = 2 m,n |mz + n|2s ′

(2)

P where ′ indicates summation over all (m, n) ∈ Z2 /{(0, 0)}. The function ζ(2s)E(z, s) has better analytic properties than E(z, s); in particular, it has a holomorphic continuation to all s except for a simple pole at s = 1. There is thus an immediate connection between the Eisenstein series at s and the Riemann zeta-function at 2s. This relationship has been made use of by many authors and has several nice consequences, two of which will be mentioned in §1. Our main theme, however, is that there 276 is also a relationship between the Eisenstein series and the zeta function at the same argument. We will give several examples of this in §2. Each takes the form that a certain linear operator on the space of functions on Γ/H, when applied to E(·, s), yields a function of s which is divisible by ζ(s). Then this operator annihilates all the E(·, ρ), and it is natural to look for a space E of functions of Γ/H which contains all the E(·, ρ) and which is annihilated by the operators in question. Such a space is defined in §3. In §4 we show that E is the set of K-fixed vectors of a certain G-invariant subspace V of the space of functions on Γ/G (where G = PS L2 (R), K = PS O(2)). Then V is a representation of G whose spectrum with respect to the Casimir operator contains ρ(1−ρ) discretely with multiplicity (at least) n if ρ is an n-fold zero of ζ(s). In particular, if (as seems very unlikely) one could show that V is unitarizable, i.e. if one could construct a positive definite G-invariant scalar product on V , then the Riemann hypothesis would follow. The paper ends with a discussion of some other representations of G related to V and reformulation in the language of adeles. § 1. We begin by reviewing the most important properties of Eisenstein series. a) Analytic continuation and functional equation.

304

D. Zagier

The function E(z, s) has a meromorphic continuation to all s, the only singularity for Re(s) > 21 being a simple pole at s = 1 whose residue is independent of z: res s=1 E(z, s) =

3 (∀z ∈ H). π

(3)

The modified function E ∗ (z, s) = π−s Γ(s)ζ(2s)E(z, s)

(4)

is regular except for simple poles at s = 0 and s = 1 and satisfies the functional equation E ∗ (z, s) = E ∗ (z, 1 − s). (5) 277

These statements are proved in a way analogous to Riemann’s proof of the analytic continuation and functional equation of ζ(s); we rewrite (2) as Z ′ X 1 −s 1 −s E (z, s) = π Γ(s) Qz (m, n) = (Θz (t) − 1) s−1 dt, 2 2 m,n ∞



(6)

o

where Qz (m, n)(z ∈ H) denotes the quadratic form Qz (m, n) = of discriminant −4 and Θz (t) =

P

m,n∈z

|mz + n|2 y

(7)

e−πtQz (m,n) the corresponding theta-

series; then the Poisson summation formula implies Θz ( 1t ) = tΘz (t) and the functional equation and other properties of E(z, s) follow from this and equation (6). b) “Rankin-Selberg method”. Let F : H → C be a Γ-invariant function which is of rapid decay as y → ∞ (i.e. . F(x + iy) = 0(y−N ) for all N). Let C(F; y) =

Z1 0

F(x + iy)dx

(y > 0)

(8)

Eisenstein Series and the Riemann Zeta-Function

305

be the constant term of its Fourier expansion and I(F; s) =

Z∞

C(F; y)y s−2 dy (Re(s) > 1)

(9)

o

the Mellin transform of C(F; y). From (1) we obtain Z Z I(F; s) = F(z)y s dz = F(z)E(z, s)dz, Γ∞ /H

(10)

Γ/H

dxdy . Therefore the y2 properties of E(z, s) given in a) imply the corresponding properties of I(F; s): it can be meromorphically continued, has a simple pole at s = 1 with Z 3 F(z)dz, (11) res s=1 I(F; s) = π where dz denotes the invariant volume element

Γ/H

and the function I ∗ (F; s) = π−s Γ(s)ζ(2s)I(F; s)

(12)

is regular for s , 0, 1 and satisfies

278

I ∗ (F; s) = I ∗ (F; 1 − s)

(13)

c) Fourier development. The function E ∗ (z, s) defined by (4) has the Fourier expansion E ∗ (z, s) = ζ ∗ (2s)y s + ζ ∗ (2s − 1)y1−s ∞ √ X s−1/2 n σ1−2s (n)K s−1/2 (2πny) cos 2πnx, +2 y

(14)

n=1

where s ζ ∗ (s) = π−s/2 Γ( )ζ(s) 2

(s ∈ C),

(15)

306

D. Zagier

σv (n) =

X

Kv (t) =

Z∞

dv

(n ∈ N, v ∈ C),

d|n

e−t cosh u cosh vu du

(v ∈ C, t > 0).

(16)

o

The expansion (14), which can be derived without difficulty from (2), gives another proof of the statements in a); in particular, the functional equation (5) follows from (14) and the functional equations ζ ∗ (s) = ζ ∗ (1 − s), σv (n) = nv σ−v (n), Kv (t) = K−v (t). Because of the rapid decay of the K-Bessel functions (16), equation (14) also implies the estimates ∂n E(z, s) = O(ymax(σ,1−σ) logn y)(n = 0, 1, 2, . . . , σ = Re(s), (17) ∂sn y = Im(z) → ∞) for the growth of the Eisenstein series and its derivatives. Finally, it follows from (14) or directly from (1) or (2) that the Eisenstein series E(z, s) are eigenfunctions of both the Laplace operator ! 2 ∂2 2 ∂ + ∆=y ∂x2 ∂y2 and the Hecke operators T (n) : F(n) → 279

X

X

ad=n b(mod d) a,d>0

F(

az + b ) d

(n > 0),

namely ∆E(z, s) = s(s − 1)E(z, s),

T (n)E(z, s) = n s σ1−2s (n)E(z, s).

(18)

We now come to the promised applications of the relationship between E(z, s) and ζ(2s). The first (which has been observed by several

Eisenstein Series and the Riemann Zeta-Function

307

authors and greatly generalized by Jacquet and Shalika [3]) is a simple proof of the non-vanishing of ζ(s) on the line Re(s) = 1. Indeed, if ζ(1 + it) = 0, the (14) implies that the function F(z) = E(z, 12 + 21 it) is of rapid decay, does not vanish identically, and has constant term C(F; y) identically equal to 0. But then I(F; s) = 0 for all s, and takings R 1 1 s = − it in (10) we find |F(z)|2 dz = 0, a contradiction. 2 2 Γ/H The second “application” is a direct but striking consequence of the Rankin-Selberg method. Let Cy ⊂ Γ/H be the horocycle Γ∞ /(R + iy); it 1 is a closed curve of (hyperbolic) length . The claim is that, as y → 0, y the curye Cy “fills up” Γ/H in a very uniform way: not only does Cy meet any open set U ⊂ Γ/H for y sufficiently small, but the fraction of Cy contained in U tends to vol(U)/ vol(Γ/H) as y → 0 and in fact length (Cy ∩ U) 1 vol(U) = + 0(y 2 −ε ) (y → 0); length(Cy ) vol(Γ/H) moreover, if the error term in this formula can be replaced by 0(y3/4−ε ) for all U, then the Riemann hypothesis is true! To see this, take F(z) in b) to be the characteristic function χU of U. Then C(F; y) =

length(Cy ∩ U) length (Cy )

and the Mellin transform I(F, s) of this is holomorphic in Re(s) > 21 Θ (where Θ is the supremum of the real parts of the zeroes of ζ(s)) except R vol(U) at s = 1. If for a simple pole of residue κ = 43 F(z)dz = vol(Γ/ H) Γ/H F were sufficiently smooth (say twice differentiable) we could deduce that I(F; σ + it) = 0(t−2 ) on any vertical strip Re(s) = σ > 12 Θ, and 1 the Mellin inversion formula would give C(F; y) = κ + 0(y1− 2 Θ−ε ). For 1 F = χU we can prove only C(F; y) = κ+0(y 2 −ε ); conversely, however, if 280 κ C(F; y) = κ+0(yα ) then I(F; s)− is holomorphic for Re(s) > 1−α, s−1 and if this holds for all F = χU we obtain Θ 6 2(1 − α).

308

D. Zagier

§2. In this section we give examples of special properties of the functions E ∗ (z, ρ) or, more generally, of the functions ∂m F(z) = Fρ,m (z) = m E ∗ (z, s) (0 ≤ m ≤ nρ − 1), (19) ∂s s=ρ where ρ is a non-trivial zero of ζ(s) of order nρ .

Example 1: Let D < 0 be the discriminant of an imaginary quadratic field K. To each positive definite binary quadratic form Q(m, n) =√am2 + bmn + cn2 −b + D ∈ H. The Γof discriminant D we associate the root zQ = 2a equivalence class of Q determines uniquely an ideal class A of K such that the norms of the integral ideals of A are precisely the integers rep2 resented by Q. Also, the form QzQ defined by (7) equals √ Q. There|D| fore (6) gives ! s/2 ′ X 1 |D| Q(m, n)−s π−s Γ(s) E ∗ (zQ , s) = 2 4 m,n ! s/2 w |D| = π−s Γ(s)ζ(A, s), 2 4 where w(= 2, 4 or 6) is the number of roots of unity in K and ζ(A, s) =

−s P a

is the zeta-function of A. (Note that this equation makes sense because E ∗ (zQ , s) depends only on the Γ-equivalence class of zQ and hence of Q.) Thus if Q1 , . . . , Qh(D) are representatives for the equivalence classes of forms of discriminant D, we have ! s/2 h(D) h(D) X X w |D| ∗ π−s Γ(s) ζ(Ai , s) E (zQi , s) = 2 4 i=1 i=1 ! s/2 w |D| π−s Γ(s)ζK (s) = 2 4 ! s/2 w |D| = π−s Γ(s)ζ(s)L(s, D), 2 4

Eisenstein Series and the Riemann Zeta-Function

309

where ζK (s) is the Dedekind zeta-function of K and L(s, D) the L-series 281 ∞ D P n−s . Since the latter is holomorphic, we deduce that the function n=1 n h(D) P ∗ E (zQi , s) is divisible by Γ(s)ζ(s), i.e. that it vanishes with multiplici=1

ity ≥ nρ at a non-trivial zero ρ of ζ(s). A similar statement holds for any negative integer D congruent to 0 or 1 modulo 4 (not necessarily the discriminant of a quadratic field) if we replace ζK (s) in the equation above by the function ζ(s, D) =

h(D) X

X

i=1 (m,n)εZ 2 /ΓQ i Qi (m,n)>0

1 Qi (m, n) s

(20)

where Qi (i = 1, . . . , h(D)) are representatives for the Γ-equivalence classes of binary quadratic forms of discriminant D and ΓQi denotes the stabilizer of Qi in Γ. Again the quotient L(s, D) = ζ(s, D)/ζ(s) is entire ([11], Prop. 3, ii), p. 130). This proves Proposition 1: Each of the functions (19) satisfies h(D) X

F(zQi ) = 0

(21)

i=1

for all integers D < 0, where zQ1 , . . . , zQ j(D) are the points in Γ/H which satisfy a quadratic equation with integral coefficients and discriminant D. Notice how strong condition (21) is: the points satisfying some quadratic equation over Z (“points with complex multiplication”) lie dense in Γ/H, so that it is not at all clear a priori that there exists any non-zero continuous function F : Γ/H → C satisfying eq. (21) for all D < 0. Example 2: This is the analogue of Example 1 for positive discriminants. Let D > 0 be the discriminant of a real quadratic field K and Q1 , . . . , Qh(D)

310

D. Zagier

representatives for the Γ-equivalence classes of quadratic forms of discriminant D. To each Qi we associate, not a point zQi ∈ Γ/H as before, but a closed curve C Qi ⊂ Γ/H as follows: Let wi , w′i ∈ R be the roots of 282 the quadratic equation Qi (x, 1) = ai x2 + bi x + ci = 0 and let Ωi be the semicircle in H with endpoints wi and w′i . The subgroup (

ΓQi = ±

1 2 (t

− bi u) ai u

−ci u 1 (t 2 + bi u)

! ) 2 2 t, u ∈ Z, t − Du = 4

(22)

of Γ, which is isomorphic to {units of K}/{±1} and hence to Z, maps Ωi to itself, and C Qi is the image ΓQi /Ωi of Ωi in Γ/H. On C Qi we have a measure |dQi z|, unique up to a scalar factor, which is invariant under the operation of the group ΓQi ⊗ R obtained by replacing Z by R in (22); if we parametize Ωi by z=

wi ip + w′i ip + 1

(0 < p < ∞),

√ t+u D dp then ΓQi acts by p → (ε = a unit of K) and |dQi z| = .A 2 p theorem of Hecke ([2], p. 201) asserts that the zeta-function of the ideal class Ai of K corresponding to Qi is given by ε2 p

πs ζ(Ai , s) = s 2 D−s/2 Γ( 2 )

Z

E ∗ (z, s)|dQi z|

C Qi

(cf. [10], §3 for a sketch of the proof). Thus h(D) XZ i=1 C

s E ∗ (z, s)|dQi z| = π−s D s/2 Γ( )2 ζK (s), 2

Qi

which again is divisible by ζ(s), and as before we can take for D any positive non-square congruent to 0 or 1 modulo 4 and get a similar identity with ζK (s) replaced by the function (20). Thus we obtain

Eisenstein Series and the Riemann Zeta-Function

311

Proposition 2: Each of the functions (19) satisfies h(D) XZ i=1

F(z)|dQi z| = 0

(23)

C Qi

for all non-square integers D < 0, where Qi (m, n) = ai m2 + bi mn + ci n2 (i = 1, . . . , h(D)) are representatives for the Γ-equivalence classes of 283 binary quadratic forms of discriminant D, C Qi is the image of { z = x + iy ∈ H| ai |z|2 + bi x + ci = 0} in Γ/H, and |dQi z| =



|ai

z2

D ((dx)2 + (dy)2 )1/2 . + bi z + ci |

Example 3: The third example comes from the theory of modular forms. Let f (z) be a cusp form of weight k on S L2 (Z) which is a normalized eigenfunction of the Hecke operators, i.e. f satisfies ! ! az + b a b k ∈ S L2 (Z)) = (cz + d) f (z) (∀z ∈ H, f c d cz + d and has a Fourier development of the form f (z) =

∞ X

an e2πinz

n=1

with a1 = 1 and anm = an am if (n, m) = 1. Define D f (s) by D f (s) =

Y p

(1 −

α2p p−s )(1

1 (Re(s) >> 0), − α p β p p−s )(1 − β2p p−s )

where α p and β p are the roots of X 2 −a p X + pk−1 = 0; it is easily checked that ∞ ζ(2s − 2k + 2) X 2 −s |an | n . D f (s) = ζ(s − k + 1) n=1

312

D. Zagier

Applying the Rankin-Selberg method (10) to the Γ-invariant funcR∞ tion F(z) = yk | f (z)|2 with constant term C(F : yk ) = |an |2 e−4πny , we n=1 R find yk | f (z)|2 E ∗ (z, s)dz = I ∗ (F; s) Γ/H

= π−s Γ(s)ζ(2s) · (4π)−s−k+1 Γ(s + k − 1)

∞ X |an |2 n s+k−1 n=1

= 4−s−k+1 π−2s−k+1 Γ(s)Γ(s + k − 1)ζ(s)D f (s + k − 1). 284

This formula, which was the original application of the RankinSelberg method ([5], [6]), shows that the product ζ(s)D f (s + k − 1) is holomorphic except for a simple pole at s = 1. It was proved by Shimura [7] and also by the author [11] that in fact D f (s) is an entire function of s. Thus the above integral is divisible by Γ(s)Γ(s + k − 1)ζ(s), and we obtain Proposition 3: Each of the functions (19) satisfies Z yk | f (z)|2 F(z)dz = 0 Γ/H

for every normalized Hecke eigenform f of level 1 and weight k. The statement of Proposition 3 remains true if f is allowed to be a non-holomorphic modular form (Maass wave-form); the proof for k = 0 is given in [12] in this volume and the general case is included in the results of [1] or [4]. Finally, we can extend our list of special properties of the functions (19) by observing that each of these functions is an eigenfunction of the Laplace and Hecke operator (eq. (18)) and hence (trivially) has the property that ∆i F(z) and T (n)F(z) satisfy Proposition 1-3 for all i > 0, n > 1. (25) Note that for general functions F : Γ/H → C (not eigenfunctions), eq. (25) expresses a property no contained in Proposition 1 to 3: for

Eisenstein Series and the Riemann Zeta-Function

313

example, eq. (21) for D = −4 says that F(i) = 0, and this does not imply ∆F(i) = 0. § 3. In § 2 we proved that the functions E ∗ (z, ρ), and more generally the functions (19), satisfy a number of special properties. In this section we will both explain and generalize these results by defining in a natural way a space E of functions in Γ/H which contains the functions (19) and has the same special properties. Let D be an integer congruent to 0 or 1 modulo 4. For Φ : R → C a function satisfying certain restrictions (e.g. (27) and (29) below) we define a new function LD Φ : H → C by 1 LD Φ(z) = 2

X

a,b,c∈Z b2 −4ac=D

a|z|2 + bz + c Φ y

!

(z = x + iy ∈ H),

(26)

where the summation extends over all integral binary quadratic forms 285 Q(m, n) = am2 + bmn + cn2 of discriminant D. Since Q and −Q occur together in the sum, we may assume that φ is an even function; the factor 1 2 has then been included in the definition to avoid counting each term twice. The sum (26) converges absolutely for all z ∈ H if we assume that Φ(X) = O(|X|−1−ε )

(|X| → ∞)

(27)

a|z|2 + bx + c is unchanged if y one acts simultaneously on x + iy ∈ H and Q(m, n) = am2 + bmn + cn2 by an element γ ∈ Γ. Hence LD Φ(γz) = LD Φ(z), so LD is an operator from functions on R satisfying (27) to functions on Γ/H. Before going on, we need to know something about the growth of LD Φ in Γ/H. If D is not a perfect square, then a , 0 in (26), so (for Φ even)

for some ε > 0. Moreover, the expression

LD Φ(z) =

∞ X a=1

∞ X

b=−∞ b2 ≡D(mod 4a)

a(x + b/2a)2 − D/4a Φ ay + y

!

314

D. Zagier

=

∞ X a=1

=

  ! 2 − D/4a −1−ε   a(x + b/2a)  O  ay +  y

∞ X

b=−∞ b2 ≡D(mod 4a)  Z∞  ∞ X a=1

 O nD (a) 

ay +

−∞

!−1−ε ax2 y

   dx 

as y = Im(z) → ∞, where n o nD (a) =, b(mod 2a)| b2 ≡ D(mod 4a) . Since the integral is O(a−1−ε y−ε ) and

(28)

∞ nD (a) P converges, we find 1+ε a=1 a

LD Φ(z) = O(y−ε ). 286

If D is a square, then the same argument applies to the terms in (26) with a , 0 and we are left with the sum ∞ bx + c 1 X X Φ( ) 2 2 c=−∞ y b =D

to estimate. If Φ is sufficiently smooth (say twice differentiable), then the inner sum differs by a small amount from the corresponding integral Z∞

−∞

∞ c Φ(X)dX, Φ( )dc = y · y −∞

and this will be small as y → ∞ only if the requirement 2

Φ is C and

Z∞

−∞

R∞

Φ(X)dX vanishes. Thus with

−∞

Φ(X)dX = 0 if D is a square

(29)

as well as (27) we have LD Φ(z) = O(y−ε ) as y → ∞ for all D, and so the scalar product of LD Φ with F in Γ/H converges for any F : Γ/H → C satisfying F(z) = O(y1−ε ) for some ε > 0 (or even F(z) − O(y)). Therefore the definition of E in the following theorem makes sense.

Eisenstein Series and the Riemann Zeta-Function

315

Theorem. For each integer D ∈ Z, D ≡ 0 or 1 (mod 4), let      even functions Φ : R → C  LD  −→ {functions Γ/H → C}   satisfying (27) and (29)  

be the operator defined by (26). Let E be the set of functions F : Γ/H → C such that a) F(z) = O(y1−ε ) for some ε > 0 R P LD Φ(z)F(z)dz = 0 for b) F(z) is orthogonal to Im(LD ), i.e. D

Γ/H

all D ∈ Z and all Φ satisfying (27) and (29).

Then i) E contains the functions (19); ii) E is closed under the action of the Laplace and Hecke operators; iii) Any F ∈ E satisfies (21), (23) (for all D) and (284) (for all f ). Proof. i) The functions (19) satisfy a) because of equation (17), since 0 < Re(ρ) < 1. To prove b), we must show that the integral of any function LD Φ(z) against E ∗ (z, s) is divisible by ζ(s). Consider first the 287 case when D is not a square. Let Φ be any function satisfying (27) and F : Γ/H → C a function which is O(yα ) as y → ∞ for some α 6 1 + ε. If Qi (m, n) = ai m2 + bi mn + ci n2 (i = 1, . . . , h(D)) are representatives for the classes of binary quadratic forms of discriminant D, then any form of discriminant D equals Qi ◦ γ for a unique i and γ ∈ ΓQi /Γ (ΓQi = stabilizer of Qi in Γ). Hence ! h(D) X X ai |γz|2 + bi Re(γz) + ci LD Φ(z) = Φ Im(γz) i=1 γ∈Γ /Γ Qi

and so Z

Γ/H

LD Φ(z)F(z)dz =

h(D) X

Z

i=1 Γ /H Qi

! ai |z|2 + bi x + ci F(z)dz. Φ y

316

D. Zagier

Taking F(z) = ζ(2s)E(z, s) with 1 6 Re(s) < 1 + ε and using (2), we find that the right-hand side of this equations equals h(D) 1X 2 i=1

X

Z

(m,n)∈Z 2 /ΓQi H

  Φ ai |z|2 + bi x + ci

ys dz. |mz + n|2 2s

Since D is not a square, Qi (n, −m) is different from 0 for all (m, n) , (0, 0), so, since Φ is an even function, we can restrict the sum to (m, n) ∈ Z 2 with Qi (n, −m) > 0 if we drop the factor 12 . Then the substitution nz − 12 bi n + ci m introduced in [11], p. 127, maps H to H and z → −mz + ai n − 21 bi m gives Z H

! y ai |z|2 + bi x + ci Φ dz y |mz + n|2s = Qi (n, −m)

−s

Z

! |z|2 − D/4 s y dz. Φ y H

Therefore we have ! Z Z |z|2 − D/4 s y dz ζ(2s) LD Φ(z)E(z, s)dz = ζ(s, D) Φ y Γ/H

288

(30)

H

for 1 < Re(s) < 1 + ε, with ζ(s, D) defined as in (20). Since ζ(2s) and ζ(s, D) have meromorphic continuations to all s and both integrals in (30) converge for 0 < Re(s) < 1 + ε, we deduce that the identity is valid in this larger range; the divisibility of ζ(s; D) by ζ(s) now implies the orthogonality of the functions (19) with LD Φ(z). If D is a square, we would have to treat the terms with Qi (n, −m) = 0 in the above sum separately (as in [11], pp. 127-128). We prefer a different method, whichR in fact works for all D. By the Rankin-Selberg method, we know that LD Φ(z)E(z, s)dz equals the Mellin transform of the constant term of LD Φ, and writing LD Φ(z) as

Eisenstein Series and the Riemann Zeta-Function ∞ X a=1

X

∞ X

b(mod 2a) n=−∞ b2 ≡D(mod 4a)

  a z + Φ 

b 2a

317

 2 + n − D/4a    y

! ∞ 1 X X bx + c + Φ , 2 2 c=−∞ y b =D

we see that this constant term is given by ! Z∞ ∞ X ax2 + ay2 − D/4a dx C(LD Φ; y) = nD (a) Φ y a=1 −∞

  0 if D , m2 ,      Z∞      2   y. Φ(X)dX if D = m > 0, +   −∞    !  ∞   c 1 X    Φ if D = 0,  2 y c=−∞

where nD (a) is defined by (28). The Mellin transform of the first term is ∞  ∞ ∞ ! X nD (a)  Z Z x2 + y2 − D/4 s−2   · y dxdy, Φ as y a=1 P

0 −∞

and since nD (a)a−s = ζ(s, D)/ζ(2s) ([11], Prop. 3, i), p. 130) we recover eq. (30) if D is not a square. The second term vanishes if D = m2 , 0 because of the assumption (29), so eq. (30) remains valid in this case. If D = 0, then, using equation (29) and the Poisson summation formula, we see that the second term in the formula for C(L0 Φ; y) 289 equals Z∞ ∞ ∞ X c 1 1 X ˜ Φ( ) − y Φ(X)dX = y Φ(ny), 2 c=−∞ y 2 n=1 −∞

where ˜ Φ(y) =

Z∞

−∞

Φ(X)e2πiXy dX

318

D. Zagier

is the Fourier transform of Φ. The Mellin transform of this is ζ(s) times ˜ so we obtain the Mellin transform of Φ, ζ(2s)

Z

L0 Φ(z)E(z, s)dz = ζ(s, 0)

Γ/H

Z

Φ

! |z|2 s y dz y

Z∞

s−1 ˜ Φ(y)y dy

H

+ ζ(s)ζ(2s)

(31)

0

for 1 < Re(s) < 1 + ε. Again both sides extend meromorphically to the critical strip and, since ζ(s, 0) = ζ(s)ζ(2s − 1), we again find that the integral on the left is divisible by ζ(s), i.e. that the functions (19) are orthogonal to the image of L0 . This completes the proof of i). We observe that the same calculations as in [12], §4, allow us to perform one of the integrations in the double integral on the right-hand side of (30), obtaining Z

LD Φ(z)E ∗ (z, s)dz

(32)

Γ/H

 Z∞    1  1−s s/2   if D < 0, (2π) |D| Γ(s)ζ(s, D) P s−1 (t)φ(|D| 2 t)dt        1 =  ∞ ! Z   s 2   1 1 s 1−s 1 2   −s s/2  π D Γ ; ; −t Φ(D 2 t)dt if D > 0, ζ(s, D) F ,    2 2 2 2  2 0

! s 1−s 1 2 where P s−1 (t) and F , ; ; −t denote Legendre and hypergeo2 2 2 metric functions, respectively; since both of these functions are invariant under s → 1 − s, we see that (30) is compatible with (and indeed gives another proof of) the functional equation of ζ(s, D) for D , 0 ([11], 290 Prop. 3, ii), p. 130). We can also make the functional equation apparent in the case D = 0 by substituting − 1z for z in the first integral on the

Eisenstein Series and the Riemann Zeta-Function

319

right-hand side of (31) and using the identity Z∞ 0

s 1 1 Γ( 2 ) y s−1 cos 2πXy dy = π 2 −s 1−s |X|−s 2 Γ( 2 )

(0 < Re(s) < 1)

in the second; this gives Z L0 Φ(z)E ∗ (z, s)dz

(33)

Γ/H



= ζ(s)ζ (2 − 2s)

Z∞

Φ(X)X

s−1



dX + ζ(1 − s)ζ (2s)

0

Z∞

Φ(X)X −s dX

0

for 0 < Re(s) < 1, with ζ ∗ (s) as in eq. (15). A calculation similar to the one given here can be found in §2 of Shintani [8]. ii) Since both the Laplace and the Hecke operators are self-adjoint, it P is sufficient to show that the space Im(LD ), or a dense subspace of it, D

is closed under the action of these operators. An elementary calculation shows that ! ! a|z|2 + bx + c a|z|2 + bx + c = Φ1 (34) ∆Φ y y with Φ1 (X) = 2XΦ′ (X) + (X 2 + D)Φ′′ (X).

(35)

Hence ∆LD φ(z) = LD Φ1 (z). If Φ is C ∞ and of rapid decay, then Φ1 also is and satisfies conditions (27) and (29), and since such Φ form a dense subspace the first assertion of ii) is proved. The calculation for the Hecke operators is harder. It suffices to treat the operators T (p) with p prime, since these generate the Hecke algebra. We claim that ! D T (p) ◦ LD = LDp2 ◦ α p + LD + pLD/p2 ◦ β p (36) p

320

D. Zagier

where α p and β p denote the operators

291

α p Φ(X) = Φ(X/p), β p Φ(X) = Φ(pX), ! D is the Legendre symbol, and LD/p2 is to be interpreted as 0 if p2 ∤ p D. To prove this write T (p)LD Φ(z) = LD Φ(pz) + X

=

b2 −4ac=D

+

p X

(

Φ

j=1

=

X

b2 −4ac=Dp

p X j=1

Φ

LD φ

z+ j p

!

ap2 |z|2 + bpx + c py

a|z|2

+ (2a j + bp)x + py

! (a j2

a|z|2 + bx + c n(a, b, c)Φ py 2

+ b jp +

cp2 )

!

!        

with ! X ! p b − 2a j c − b j + a j2 a b n(a, b, c) = ε 2 , , c + , ε a, p p p p2 j=1 (where ε(a, b, c) equals 1 if a, b, c are integral, 0 otherwise). To prove (36) we must show that ! ! ! a b c a b c D ε , , + pε 2 , 2 , 2 n(a, b, c) = 1 + p p p p p p p (a, b, c ∈ Z, b2 − 4ac = Dp2 ). 292

For p odd, this follows from the following table, in which v p1 (m) denotes the exact power of p dividing an integer m.

Eisenstein Series and the Riemann Zeta-Function v p1 (a) v p1 (b) v p′ (c) ε 0 1

>0 >1

>0 >1

>2 >2 >2 >2

>1 1 >2 >2

0 >1 1 >2



p P

a b , ,c p2 p

0 0

j=1

 ε a,

b−2a j c−b j+a j2 , p2 p

1   1 + Dp

1 1 1 1

0 1 0 p



ε



a b c , , p p p

0 1 0 1 1 1



321 ε



a , b, c p2 p2 p2

0 0



0 0 0 1

The proof for p = 2 is similar but there are more cases to be considered. iii) We will show that each of the properties in question is implied by the orthogonality of F with LD Φ for special choices of D and Φ. For (21) we choose Φ(X) = δ(X 2 + D), where δ is the Dirac delta-function. From the identity a|z|2 + bx + c y

!2

+D=

|az2 + bz + c|2 y2

(37)

we see that the support of LD Φ is the set of points in H satisfying some quadratic equation of discriminant D, and an easy calculation shows that Z

Γ/H

h(D) π X F(z)LD Φ(z)dz = √ F(zQi ) 2 |D| i=1

(38)

for any continuous F : Γ/H → C. (Of course, δ(X 2 + D) is not a function, and equation (38) must be interpreted in the sense that it holds in the limit n → ∞ if we choose Φ(X) = δn (X 2 + D) where {δn } is a sequence of smooth, even functions with integral 1 and support tending to {0}.) Hence any F ∈ (Im LD )⊥ satisfies (21). The case D > 0, D not a square, is similar; here we choose Φ(X) = 293 δ(X). so that LD Φ(z) is supported on the semicircles a|z|2 + bx + c = 0

322

D. Zagier

(a, b, c ∈ Z, b2 − 4ac = D), and find Z

Γ/H

h(D) Z 1 X F(z)LD Φ(z)dz = √ F(z)|dQi z|, D i=1

(39)

C Qi

where the equation is to be interpreted in the same way as (38). Thus F ∈ (Im LD )⊥ implies (23). It remains to prove that any F ∈ E satisfies equation (284). We ∞ P follow the proof of the divisibility of |an |2 /n s+k−1 by ζ(s) given in n=1

[11]. Equations (37) and (??) of that paper give the identity r X ai (m) k y | fi (z)|2 ( f , f ) i i i=1

(40)

∞ X (−1)k/2 k−4 k−1 = 2 m (k − 1) Lt2 −4m Φk,t (z) (z ∈ H) π t=−∞

for all integers m > 0, where r = dim S k (S L2 (Z)), fi (z) =

∞ X

ai (m)e2πimz (i = 1, . . . , r)

m=1

are the normalized Hecke eigenforms of weight Z k, ( fi , fi ) = yk | fi (z)|2 dz, Γ/H

and Φk,t (X) = (X − it)−k + (X + it)−k . Thus any function in E is orthogonal to the sum on the left-hand side of (??) and therefore, since the Fourier coefficients ai (m) are linearly independent, to each of the functions yk | fi (z)|2 . Using the computations of [12] and an extension of the RankinSelberg method [13] , it seems to be possible to prove the orthogonality of F ∈ E with | f (z)|2 also for Maass eigenforms (= non-holomorphic cusp forms which are eigenvalues of the Laplace and Hecke operators) of weight 0. 

Eisenstein Series and the Riemann Zeta-Function

323

§4. Let G = PS L2 (R) and K = S O(2)/{±1} its maximal compact subgroup, and identity the symmetric space G/K with H by gK = ! ai + b a b K ↔g·i = . In this section we will construct a represenc d ci + b tation V of G in the space of functions of Γ/G whose space of K-fixed vectors V K is E . Let 294 ! a b/2 a, b, c ∈ R} XR = { b/2 c

be the 3-dimensional vector space of symmetric real 2 × 2 matrices and XZ ⊂ XR the lattice consisting of matrices with a, b, c ∈ Z. The group G acts on XR by g◦ M = gt Mg(g ∈ G, M ∈ XR ), and XZ is stable under the action of the subgroup Γ. For M ∈ XR and g ∈ G, the expression tr(gt Mg) depends only on the right coset gK (since kt = k−1 for k ∈ K), i.e. only on g · i ∈ H. An easy calculation shows that ! a|z|2 + bx + c a b/2 tr(g Mg) = ∈ XR , z = g · i ∈ H). (41) (M = b/2 c y t

a|z|2 + bx + c in the definiy tion of LD comes from and also why this expression is invariant under the simultaneous operation of Γ on the upper half-place (gK → γgK) and on binary quadratic forms (M → (γ−1 )t Mγ−1 ).. Using (41), we can rewrite the definition of LD as X LD Φ(gK) = Φ(tr(gt Mg)). This explains where the strange expression

M∈XZ det M=−D/4

To pass from functions on H to functions on G, we replace the special function M → Φ(tr(M)) by an arbitrary function Φ on the 2-dimensional submanifold ! a b/2 ∈ XR b2 − 4ac = D} XR (D) = { b/2 c

324

D. Zagier

of XR . Thus we extend LD to an operator (still denoted LD ) from the space of nice functions on XR (D) to the space of functions on Γ/G by setting X LD Φ(g) = Φ(gt Mg) (g ∈ G), (42) M∈XZ (D)

where XZ (D) = XR (D) ∩ XZ . Here “nice” means that Φ satisfies the obvious extensions of (27) and (??), i.e. it must be of sufficiently rapid decay in XR (D) and, if D is a square, must be smooth √ !and have zero 1 ε D 0 2 g(g ∈ G, ε = integral along each of the lines lg,ε = gt 1 √ R 2ε D 295 +1) on the ruled surface XR (D). It is clear that LD Φ(g) is left Γ-invariant, since XZ (D) is stable under Γ and the sum (42) is absolutely convergent. Also, the image of LD is stable under the representation π of G given by right translation, since π(g) ◦ LD = LD ◦ π′ (g) (g ∈ G), where π′ (g)Φ(M) = Φ(gt Mg). Hence the space \ V = (Im LD )⊥

(43)

D∈Z

of functions F : Γ/G → C satisfying an appropriate growth condition and such that Z LD Φ(g)F(g)dg = 0 (dg = Haar measure) (44) Γ/G

for all D ∈ Z and all nice functions Φ on XR (D) , is also stable under G. Also, it is clear that V K coincides with the space E defined in §3. In particular, V contains the vectors vρ : g → E ∗ (g · i, ρ) (ρ a non-trivial zero of ζ(s)) and more generally vρ,m · g → Fρ,m (g · i) (Fρ,m as in (19)). 1 Since the various manifolds XR (D) ⊂ XR are disjoint, we can also define V as (Im L )⊥ , where L is the operator from nice functions on XR to functions on Γ/G defined by X L Φ(g) = Φ(g1 Mg). M∈XZ

Eisenstein Series and the Riemann Zeta-Function

325

On the other hand, because the function z → E(z, s) is an eigenfunction of the Laplace operator on H, the representation theory of G tells us that (at least for s < Z) the smallest G-invariant space of functions on Γ/G containing the function g → E(g · i, s) is an irreducible representation isomorphic to the principal series representation P s . (Recall that P s is the representation of G by !right ! translations on the set of functions a b g = |a|2s f (g) and f K ∈ L2 (K)). Thus f : G → C satisfying f −1 0 a V contains the principal series representation Pρ for every non-trivial zero ρ of the Riemann-zeta-function. On the other hand, P s is unitarizable if and only if s(1 − s) > 0, 296 i.e. s ∈ (0, 1) or Re(s) = 21 . Thus the existence of a unitary structure on V would imply the Riemann hypothesis. However, the following argument suggests that it may be unlikely that such a unitary structure can be defined in a natural way. If ρ is a zero of ζ(s) of order n > 1, then the functions Fρ,m (m = 0, . . . , n − 1) belong to V K , and these functions are not eigenfunctions of ∆, through the space they generate is stable under ∆ (for example, differentiating (18) with respect to s we find ∆Fρ,1 = ρ(ρ − 1)Fρ,1 + (2ρ − 1)Fρ,0 . Therefore V contains a G-invariant subspace Vρ,n corresponding to the eigenvalues ρ(1−ρ) which is reducible but is not a direct sum of irreducible represenK = n and V tations (we have dim Vρ,n ρ,n ⊃ Vρ,n−1 ⊃ . . . ⊃ Vρ,1 ⊃ Vρ,0 = {0} with Vρ,m /Vρ,m−1  Pρ ), and such a representation cannot have a unitary structure. Thus the unitarizability of V would imply not only the Riemann hypothesis, but also the simplicity of the zeroes of ζ(s). Since an analogue of V can be defined for any number field or function field (cf. . §5), and since there are examples of such fields whose zeta-functions are known to have multiple zeroes, there cannot be any generally applicable way of putting a unitary structure on V . Of course, ! − 12 b −c We may also identify XR with the Lie algebra iR = { a, b, c ∈ R} of G by 1 a b 2 ! 0 −1 ; then the operation M → g′ Mg of G on XR becomes the adjoint M → M 1 0 representation X → g−1 Xg of G on iR , and V is the set of functions on Γ/G orthogonal P to all functions of the form g → Ψ(Ad(g)X), where Ψ is a nice function on iR . X∈iZ

326

D. Zagier

this does not preclude the possibility that our particular V (for the filed Q) has a unitary structure defined in some special way, and indeed, if the zeros of ζ(s) are simple and lie on the critical line and if (as seems likely) E is spanned by the Vρ , then V is in fact unitarizable, indeed in infinitely many ways, since we are essentially free to choose the norm of vρ . For various reasons, a natural choice seems to be ||vρ || = |ζ ∗ (2ρ)|. Finally, we should mention that the construction of V is closely related to the Weil representation. The functions LD Φ(g) are essentially the Fourier coefficients for a “lifting” operator from functions on Γ/G to autormorphic forms on the metaplectic group, in analogy with the construction of Shintani [9] in the holomorphic case; thus the space V can be interpreted as the kernel of the lifting. §5. The proof of the theorem in § 3 shows that almost every statement of the theorem can be strengthened to a statement about the individual spaces ED = (Im LD )⊥ 297

(D ∈ Z)

rather than just their intersection E . Thus in part iii) of the theorem, to prove that a function F satisfies (21) or (23) we needed only F ∈ ED for T the value of D in question, and it was only for (284) that F ∈ D ED was needed. Similarly in part ii), equation (34) shows that each space ED is stable under the Laplace operator. The same is not true for the Hecke operators, since T (p) maps Im(LD ) to Im(LDp2 ) + Im(LD ) + Im(LD/p2 ) but the intersection of the spaces ED for all D with a common squarefree part is stable under the Hecke algebra. Finally—and most interesting—from equation (30) or (32) we see that ED (D , 0) contains ∂i E ∗ (z, ρ) whenever ρ is a zero of ζ(s, D) (resp. i E ∗ (z, s) for 0 6 i 6 s=ρ ∂s n − 1 if ρ is a zero of multiplicity n). Since ζ(s, D) = ζ(s)L(s, D) and L(s, D) has infinitely many zeroes in the critical strip, this shows that ED contains many more Eisenstein series than just the functions (19). (This conclusion holds also when D is a square; in this case L(s, D) is equal to ζ(s)2 up to an elementary factor and we do not get any new zeroes, but they all occur with twice the multiplicity and so we get twice as many functions as in (19). For D = 0, however, we get only the functions

Eisenstein Series and the Riemann Zeta-Function

327

(19), since the expression on the right-hand side of (31) or (33) is a linear combination of ζ(s)ζ(2s) and ζ(s)ζ(2s − 1) rather than a multiple of ζ(s, 0).) The functions ζ(s, D) for two discriminants D withthe same squarefree part differ by a finite Euler product and have the same non-trivial zeroes ρ. This, together with eq. (36), suggests that the most natural thing to do is to put together the corresponding spaces ED . Thus we let E denote either a quadratic extension of Q, or Q + Q, or Q, and define E (E) =

∞ \

Ed f 2 ,

f =1

where d denotes the discriminant of E, or 1, or 0, respectively. Then the above discussion can be summarized as follows: i) Each of the spaces E (E) is stable under the Laplace and Hecke operators; T ii) E E (E) = E ; ∂i ∗ iii) E (E) contains i E (z, s) for 0 6 i 6 n − 1 if ρ is an n-fold 298 s=ρ ∂s zero of ζE (s), where ζE (s) denotes the Dedekind zeta-function of E if E = Q or E is q quadratic field and ζQ+Q (s) = ζ(s)2 . Of course, we can also define representations VD = (Im LD )⊥ and V (E) = ∩Vd f 2 similarly; then V (E)K = E (E) and V (E) is a representation of PS L2 (R) whose spectrum is related to the zeroes of ζE (s) in the same way as that of V to those of ζ(s). The representations V (E) have a very nice interpretation in the language of adeles; we end the paper by describing this. As motivation, we recall that our starting point for the definition of E was the fact that the zeta-function of a quadratic field E can be written as the integral of E(z, s) over a certain set SE ⊂ Γ/H consisting of a finite number of points if E is imaginary and of a finite number of closed curves if E is real (Proposition 1 and 2). Hence the functions E(z, ρ), ρ a zero of ζE (s), belong to the space of functions whose integral over SE vanishes.

328

D. Zagier

Now let G denote the algebraic group GL(2), Z its center, and A the ring of adeles of Q. Choosing a basis of E over Q gives an embedding of E × in GL(2, Q) and a non-split torus T ⊂ G with T (Q) = E × . There is a projection G(Q)Z(A)/G(A) → Γ/H and under this projection T (Q)Z(A)/T (A) maps to SE . The adelic analogue of Proposition 1 and 2 is the fact that the integral of an Eisenstein series over T (Q)Z(A)/T (A) is a multiple of the zeta-function of E. To prove it, we must recall the definition of the Eisenstein series. Let Φ be a Schwartz-Bruhat function on A2 ; then the Eisenstein series E(g, Φ, s) is defined for g ∈ G(A) and s ∈ C with sufficiently large real part by Z X s Φ[ξzg]| det zg|Q dz, (45) E(g, Φ, s) = 2 Z(Q)/Z(A) ξ∈Q {0}

where | |Q denote the idele norm and dz the Haar measure on Z. (This definition is the analogue of equation (2). The more usual definition of E(g, Φ, s), analogous to eq. (1), is X E(g, Φ, s) = f (γg, Φ, s), γ∈P(Q)/G(Q)

299

where P =

(

∗ ∗ 0 ∗

!)

and

F(G, Φ, s) =

s | det g|Q

Z

Φ[(0, a)g]|a|2s d× a,



which is easily seen to agree with (45); note that f (g, Φ, s) equals ζQ (2s) times an elementary function of s by Tate theory, and that ! ! s a a x g, Φ, s = f (g, Φ, s), f 0 b b

so the analogue of f (g, Φ, s) in the upper half-plane is the function ζ(2s)ℑ(g · i) s .) Identifying Q2 {0} with E × and observing that the Qidele norm of det t(t ∈ T (A)) equals the E-idele norm of t under the

Eisenstein Series and the Riemann Zeta-Function

identification of T (A) with A×E , we find Z Z E(tg, Φ, s)dt =

X

329

s Φ[ξtg]| det tg|Q dt

2 T (Q)/T (A) ξ∈Q {0}

T (Q)Z(A)/T (A)

=

s | det g|Q

s = | det g|Q

Z

E × /A×E

Z

X

Φ[ξtg]|t|Es dt

ξ∈E ×

Φ[eg]|e|Es d× e.

A×E

Since e → Φ[eg] is a Schwartz-Bruhat function on AE , this is precisely the Tate integral for ζE (s). (The computation just given is the basis for Harder’s computations of period integrals in this volume as well as for the generalization of the Selberg trace formula in [4].) In particular, it follows that the integral of ∂i F(g) = i E(g, Φ, s) s=ρ ∂s

over T (Q)Z(A)/T (A)g vanishes if ρ is a zero of ζE (s) of multiplicity > i + 1. The natural adelic definition of V (E) is thus as the space of functions F : G(Q)Z(A)/G(Q) → C satisfying Z F(tg)dt = 0 (∀g ∈ G(A)) (46) T (Q)Z(A)/T (A)

as well as some appropriate growth condition. The space V (E) then contains irreducible principal series representations corresponding to the zeroes of ζE (s). Condition (46) is similar to the condition Z F(ng)dn = 0 (∀g ∈ G(A)) N(Q)/N(A)

defining cusp forms (where N is the unipotent radical of a parabolic 300 subgroup of G), so the space V (E) can be thought of as an analogue of

Bibliography

330

the space L02 (G(Q)Z(A)/G(A)) of cusp forms. Like L02 , it probably has a discrete spectrum. The difference is that, whereas L02 has a unitary structure, the corresponding statement for V (E) would imply the Riemann hypothesis and the simplicity of the zeroes of ζQ (s). We call functions F satisfying (46) toroidal forms (in analogy with the French terminology of “formes paraboliques” for cusp forms) and the V (E) toroidal representations. The calculation given above is unchanged if we replace Q by any global field F and take E to be a quadratic extension of F. In the case where F is the functional field of a curve X over a finite field, there are only finitely many zeroes of ζF (s), their number being equal to the first Betti number of X. Then the K-finite part of our representation V = T E V (E) is a complex vector space of dimension at least (and hopefully exactly) equal to this Betti number, and Barry Mazur pointed out that this space might have a natural interpretation as a complex cohomology group H 1 (X; C). It is not yet clear whether this point of view is tenable. At any rate, however, from conversations with Harder and Deligne it appears that it will at least be possible to show that the dimension of the space in question is finite.

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301

[1]

Gelbart, S. and H. Jacquet,: A relation between automorphic representations of GL(2) and GL(3). Ann. Sc. Ec. Norm. Sup. 11 (1978) 471-542.

[2]

¨ Hecke, E.: Uber die Kroheckersche Grenzformel f¨ur reelle quadratische K¨orper und die Klassenzahl relativ-abelscher K¨orper, Verhandl. d. Naturforschenden Gesell. i. Basel 28, 363-372 (1971). Mathematische Werke, pp. 198-207. Vandenhoeck & Ruprecht, G¨ottingen 1970.

[3]

Jacquet, H. and J. Shalika,: A non-vanishing theorem for zeta functions of GL2 . Invent. Math. 38 (1976) 1-16.

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Jacquet, H. and D. Zagier,: Eisenstein series and the Selberg trace formula II. In preparation.

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Rankin, R.: Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetical functions. I. Proc. Cam. Phil. Soc. 35 (1939) 351-372.

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Selberg, A.: Bemerkungen u¨ ber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43 (1940) 47-50.

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Shimura, G.: On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc. 31 (1975), 79-98.

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[9]

Shintani, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58 (1975) , 83-126.

[10] Zagier, D.: A Kronecker limit formula for real quadratic fields. Math. Ann. 213 (1975), 153-184. [11] Zagier, D.: Modular forms whose Fourier coefficients involve zeta- functions of quadratic fields. In Modular Functions of One Variable VI, Lecture Notes in Mathematics No. 627, Springer, Berlin-Heidelberg-New York 1977, pp. 107-169. [12] Zagier, D.: Eisenstein series and the Selberg trace formula I. This volume, pp. 303-355. [13] Zagier, D.: The Rankin-Selberg method for automorphic functions which are not of rapid decay. In preparation.

EISENSTEIN SERIES AND THE SELBERG TRACE FORMULA I By Don Zagier∗

0 Introduction 303

R The integral K◦ (g, g)E(g, s)dg. Let G = S L2 (R) and Γ be an arithmetic subgroup of G for which Γ/G has finite volume but is not compact. The space L2 (Γ/G) has the spectral decomposition (with respect to the Casimir operator) M M 2 Lcont (Γ/G), L2sp (Γ/G) L2 (Γ/G) = L◦2 (Γ/G) where L◦2 (Γ/G) is the space of cusp forms and is discrete, L2sp (Γ/G) is the 2 discrete part of (L◦2 )⊥ , given by residues of Eisenstein series, and Lcont is the continuous part of the spectrum, given by integrals of Eisenstein series. If ϕ is a function of compact support or of sufficiently rapid decay on G, then convolution with ϕ defines an endomorphism T ϕ of L2 (Γ/G), and the kernel function X K(g, g′ ) = ϕ(g−1 γg′ ) (g, g′ ∈ G) (0.1) γ∈Γ

of T ϕ has a corresponding decomposition as K◦ + Ksp + Kcont , where Ksp and Kcont can be described explicitly using the theory of Eisenstein 332

Eisenstein Series and the Selberg Trace Formula I

333

series. The restriction of T ϕ to L◦2 (Γ/G) is of trace class; its trace is given by Z T r(T ϕ , L◦2 ) =

K◦ (g, g)dg.

(0.2)

Γ/G

The Selberg trace formula is the formula obtained by substituting K(g, g) − K sp (g, g) − Kcont (g, g)

for

K◦ (g, g)

and computing the integral. However, although K◦ (g, g) is of rapid decay in Γ/G, the individual terms K(g, g), K sp (g, g) and Kcont (g, g) are not, so that to carry out the integration one has to either delete small neighbourhoods of the cusps form a fundamental domain or else “truncate” the kernel functions by subtracting off their constant terms in such neighbourhoods, and then to compute the limit as these neighbourhoods shrink to points. This procedure is perhaps somewhat unsatisfactory, both from an aesthetic point of view and because of the analytical difficulties it involves. To get around these difficulties we introduce the integral 304 Z K◦ (g, g)E(g, s)dg, (0.3) I(s) = Γ/G

where E(g, s) (g ∈ G, s ∈ C) denotes an Eisenstein series. The idea of integrating a Γ-invariant function F(g) against an Eisenstein series was introduced by Rankin [5] and Selberg [6], who observed that in the region of absolute convergence of the Eisenstein series this integral equals the Mellin transform of the constant term in the Fourier expansion of F (see §2 for a more precise formulation). Applying this principle to F(g) = K◦ (g, g) we can calculate I(s) for Re(s) > 1 as a Mellin transform, obtaining a representation of I(s) as an infinite series of terms. Each of these terms can be continued meromorphically to Re(s) 6 1; in particular, the contribution of a hyperbolic or elliptic conjugacy class of γ’s in (0.1) is the product of a certain integral transform of ϕ with the Dedekind zeta-functio of the corresponding real or imaginary quadratic field. Since the residue of E(g, s) at s = 1 (resp. the value of E(g, s) at

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Don Zagier

s = 0) is a constant function, we recover the Selberg trace formula by computing res s=1 (I(s)) (resp. . I(0)). This proof of the trace formula is more invariant and in some respects computationally simpler than the proofs involving truncation. It also gives more insight into the origin of the various terms in the trace formula; for instance, the class numbers occurring there now appear as residues of zeta-functions. However, the formula for I(s) has other consequences than the trace formula. The most striking is that I(s) (and in fact each of the infinitely many terms in the final formula for I(s)) is divisible by the Riemann zeta-function, i.e. the quotient I(s)/ζ(s) is an entire function of s. Interpreting this as the statement that the Eisenstein series E(g, ρ) is orthogonal to K◦ (g, g) (in fact, to each of infinitely many functions whose sum equals K◦ (g, g)) whenever ζ(ρ) = 0, one is led to the construction of a representation of G whose spectrum is related to the set of zeros of the Riemann zeta-function (cf. [11] in this volume). On the other hand, the formula for I(s) can be used to get information about cusp forms. The function K◦ (g, g′ ) is a linear combination of terms f j (g) f j (g′ ), where { f j } is an orthogonal basis for L◦2 (Γ/G) and 305 where the coefficients depend on the function ϕ and on the eigenvalues of f j (“Selberg transform”). Moreover, applying the Rankin-Selberg method to the function F(g) = | f j (g)|2 one finds that the integral of this function against E(g, s) equals the “Rankin zeta-function” R f j (s) ∞ P (roughly speaking, the Dirichlet series |an |2 n−s , where the an are n=1

the Fourier coefficients off); indeed, this is the situation for which the Rankin-Selberg method was introduced. Thus I(s) is a linear combination of the functions R f j (s), and so one can get information about the latter from a knowledge of I(s). In particular, using a “multiplicity one” argument one can deduce from the divisibility of I(s) by ζ(s) that in fact each R f j (s) is so divisible (this result had been proved by another method by Shimura [8] for holomorphic cusp forms and by Gelbart and Jacquet [2] in the general case). Other applications of the results proved here might arise by comparing them with the work of Goldfeld [1]. It does not seem impossible that the formula of I(s) can be used to obtain information about the Fourier coefficients of cusp forms.

Eisenstein Series and the Selberg Trace Formula I

335

The idea we have described can be applied in several different situations: 1. By working with an appropriate kernel function, we can isolate the contribution coming from holomorphic cusp forms of a given weight k (discrete series representations in L2 (Γ/G)). This case was treated in [10]. The computation of I(s) here is considerably easier than in the general case because there is no continuous spectrum and only finitely many cusp forms f j are involved. We can therefore represent each Rankin zeta-function R f j (s) as an infinite linear combination of zeta-functions of real and imaginary quadratic fields. Moreover, for certain odd positive values of s the contributions of the hyperbolic conjugacy classes in Γ to I(s) vanish and one is left with an identity expressing R f j (s) as a finite linear combination of special values of zeta-functions of imaginary quadratic extensions of Q. As a corollary of this iden¯ tity one obtains the algebraicity (and behaviour under Gal(Q/Q)) 1 of R f (s)/πk−1 ζ(s) for the values of s in question ([10], ( f j, f j) j Corollary to Theorem 2, p. 115), a result proved independently by Sturm [9] by a different method. 2. The first case involving the continuous spectrum is that of Maass 306 wave forms of weight zero, i.e. cusp forms in L2 (Γ/G/K) = L2 (Γ/H), where K denotes S O(2) and H = G/K the upper half-plane. This is the case treated in the present paper (with Γ = S L2 (Z)). 3. Next, one can replace S L2 (R) and S L2 (Z) by GL2 (2, A) and GL(2, F), respectively, where F is a global field and A the ring of adeles of F. This case, which is the most general one as far as GL(2) is concerned, will be treated in a joint paper with Jacquet [3]. It includes as special cases 1 and 2, as well as their generalizations to holomorphic and non-holomorphic modular forms of

336

Don Zagier

arbitrary weight and level, Hilbert modular forms, and automorphic forms over function fields. 4. Finally, the definition of I(s) makes sense in any context where Eisenstein series can be defined, so it may be possible to apply the method sketched in this introduction to discrete subgroups of algebraic groups other than GL(2).

1 Statement of the main theorem In this section we describe the main result of this paper, namely a formula for I(s) in the critical strip 0 < Re(s) < 1. In order to reduce the amount of notation and preliminaries needed, we will state the formula in terms of a certain holomorphic function h(r); the relationship of h(r) to the function ϕ(g) of the introduction (Selberg transform) is well-known and will be reviewed in § 2. Except at the end of § 5, we will always consider only forms of weight 0 on the full modular group Γ = S L2 (Z)/{±1}. The results for congruence subgroups are similar but messier to state and in any case will be subsumed by the results of [3]. Any continuous Γ-invariant function f : H → C has a Fourier expansion of the form f (z) =

∞ X

2( n = −∞An ( f ; y)e2πinx

(z ∈ H)

(1.1)

(here and in future we use x and y to denote the real and imaginary parts of z ∈ H). We denote by L2 (Γ/H) the Hilbert space of Γ-invariant R functions f : H → C such that ( f , f ) = | f (z)|2 dz is finite dz = dxdy y2 Γ/H

307

and by L◦2 (Γ/H) the subspace of functions with A◦ ( f ; y) ≡ 0. The space L◦2 (Γ/H) is stable under the Laplace operator ! 2 ∂2 2 ∂ + ∆=y ∂x2 ∂y2 and has a basis { f j } j>1 consisting of eigenforms of ∆(see [4], § 5.2).

Eisenstein Series and the Selberg Trace Formula I

We write

! 1 2 ∆ fj = − + rj fj 4

( j = 1, 2, . . .)

337

(1.2)

1 > 0, i.e. r j 4 1 is either real or else pure imaginary of absolute value 6 2 . In fact it is known that the r j are real for S L2 (Z), but the corresponding statement 1 for congruence subgroups is not known and we will use only r2j > − . 4 From (1.2) we find that the nth Fourier coefficient An ( f j , Y) satisfies the second order differential equation ! 2 1 2 d 2 2 2 2 y 2 An ( f j , y) − 4π n y An ( f ; y) = − + r j An ( f j ; y). 4 dy where r j ∈ C. Since ∆ is negative definite, we have r2j +

The only solution of this equation which is bounded as y → ∞ is √ yKir j (2π|n|y), where Kv (z) is the K-Bessel function, defined (for example) by Kv (z) =

Z∞

e−z cosh t cosh vtdt

(v, z ∈ C, Re(z) > 0).

(1.3)

0

Hence the f j have Fourier expansions of the form f j (z) =

∞ X

√ a j (n) yKir j (2π|n|y)e2πinx

(1.4)

n=−∞ n,0

with a j (n) ∈ C. We can choose the f j to be normalized eigenfunctions of the Hecke operators !  1 X X az + b    T (n) : f (z) −→ (n > 0), f    n a,d>0 b(mod d) d  (1.5)   ad=n     T (−1) : f (z) −→ f (−¯z), T (−n) = T (−1)T (n),

338 308

Don Zagier

i.e. f j |T (n) =

a j (n) 1

|n| 2

fj

(n ∈ Z, n , 0)

(1.6)

(then a j (1) = 1, a j (−1) = ±1, and a j (n) is multiplicative). The functions f j chosen in this way are called the Maass eigenforms; they form an orthogonal (but not orthonormal) basis of L◦2 (Γ/H), uniquely determined up to order. For each j we define the Rankin zeta-function R f j (s) by R f j (s) =

∞  s X Γ( 2s )2  s |a j (n)|2 Γ( Γ + ir − ir ) j j 8π s Γ(s) 2 2 |n| s n=−∞

(Re(s) > 1).

n,0

(1.7) We also set R∗f j (s) = π−s Γ(s)ζ(2s)R f j (s) = ζ ∗ (2s)R f j (s), where ζ(s) denotes the Riemann zeta-function and  s ζ ∗ (s) = π−s/2 Γ ζ(s) = ζ ∗ (1 − s). 2

(1.8)

(1.9)

The Rankin-Selberg method implies that R∗f j (s) has a meromorphic continuation to all s, is regular except for simple poles at s = 1 and s = 0 with 1 (1.10) res s=1 R∗f j (s) = ( f j , f j ), 2 and satisfies the functional euqation R∗f j (s) = R∗f j (1 − s)

(1.11)

(the proofs will be recalled in § 2). We will also need the zeta-functions ζ(s, D), where D is an integer congruent to 0 or 1 modulo 4. They are defined for Re(s) > 1 by ζ(s, D) =

XX Q m,n

1 Q(m, n) s

(Re(s) > 1),

(1.12)

Eisenstein Series and the Selberg Trace Formula I

339

where the first summation runs over all S L2 (Z)-equivalence classes of 309 binary quadratic forms Q of discriminant D and the second over all pairs of integers (m, n) ∈ Z2 / Aut(Q) with Q(m, n) > 0, where Aut(Q) is the stabilizer of Q in S L2 (Z). These functions, which were introduced in [10], are related to standard zeta-functions by   ζ(s)ζ(2s − 1) if D = 0      2 ζ(s, D) =  ζ(s) · (finite Dirichlet series ) if D = square , 0 x     ζ √ (s) · (finite Dirichlet series) if D , square, Q( D) (1.13) √ where ζQ( √D) (s) denotes the Dedekind zeta-function of Q( D) (for precise formulas see [10], Proposition 3, p. 130). In particular, ζ(s, D) has a meromorphic continuation in s and ζ(s, D)/ζ(s) is holomorphic except for a simple pole at s = 1 when D is a square. Now let h : R −→ C be a function satisfying   h(r) = h(−r);       1    h(r) has a holomorphic continuation to the strip | Im(r)| < A 2      for some A > 1;      h(r) is of rapid decay in this strip (1.14) (“rapid decay” means O(|r|−N ) for all N). The object of this paper is to ∞ h(r j ) P compute R f j (s). In §§2 and 3 we will show that this function j=1 ( f j , f j ) equals the function I(s) of §0 and compute it in the strip 1 < Re(s) < A by the Rankin-Selberg method; §§4 and 5 give the analytic continuation in s, computation of the residue at s = 1 (Selberg trace formula), ∞ h(r j ) P and generalization to a j (m) R f (s), where the a j (m) are the ( f j, f j) j j=1 Fourier coefficients defined by (1.4). We state here the final result for 0 < Re(s) < 1 and m > 0 in a form which makes the functional equation apparent. Theorem 1: Let h : R −→ C be a function satisfying the conditions (1.14) and m > 1 an integer. Then for s ∈ C with 0 < Re(s) < 1 we have

340

Don Zagier

the identity ∞ X

a j (m)

j=1

310

h(r j ) ∗ R (s) = R(s) + R(1 − s) ( f j, f j) f j

(1.15)

with R(s) = R(s; m, h) given by 1 ∗ 2 ζ (s) 8π

R(s) = −

Z∞

−∞

      X ir  + − 2ir)  a   h(r)dr  ζ ∗ (1 + 2ir)ζ ∗ (1 − 2ir) a,d≥1 d  ζ ∗ (s

2ir)ζ ∗ (s

      1 ζ ∗ (s)ζ ∗ (2s)  X a s/2  is −  h( )  2 ζ ∗ (s + 1) a,d≥1 d  2 +

ad=m ∞ Γ(s)Γ(s − 12 ) X m     ζ(s, t2 4π2 Γ 1+s Γ 2−s t=−∞ 2 2 Z∞ Γ  1−s + ir Γ  1−s − ir 2 2 s−1 2

×

ad=m

− 4m)×

(1.16)

Γ(ir)Γ(−ir)

−∞

! 1−s 3 t2 1−s + ir, − ir; − s; 1 − h(r)dr, ×F 2 2 2 4m where ζ ∗ (s) and ζ(s, t2 − 4m) are defined by equations (1.9) and (1.12) and F(a, b; c; z) denotes the hypergeometric function (defined by analytic continuation if z < 0) and can be expressed in terms of Legendre functions for the special values of the parameters a, b, c occurring in (1.16). For m < 0 there is a similar formula with m replaced by |m| in the first two terms and the function m

s−1 2

1−s 3 t2 1−s F + ir, − ir; − s; 1 − 2 2 2 4m

!

in the third term replaced by a different hypergeometric function.

Eisenstein Series and the Selberg Trace Formula I

341

Corollary. The Rankin zeta-function R∗f j (s) is divisible by ζ ∗ (s) for all j. Proof of the Corollary: Every term on the right-hand side of equation (1.16) (and of the corresponding formula for m < 0) is divisible by ζ ∗ (s); since the series converges absolutely, we deduce that R(s) (and hence, by the functional equation (1.9), also R(1 − s)) is divisible by ζ ∗ (s). Therefore the expression on the left-hand side of equation (1.15), vanishes (with the appropriate multiplicity) at every zero of the Riemann zeta-function, and the linear independence of the eigenvalues a j (m)h(r j )(m ∈ Z − {0}, h satisfying (1.14)) for different j implies that 311 the same holds for each R∗f j (s). A more formal argument is as follows: For z ∈ H define Φ(s, z) =

∞ X j=1

1 f j (z)R∗f j (s); ( f j, f j)

then (1.14) and (1.15) imply the identity ∞ √ X [R(s; m, hmy ) + R(1 − s; m, hmy )]e2πimx , Φ(s, z) = y m=−∞ m,0

where hm,y (r) = Kir (2π|m|y). Therefore Φ(s, z) is divisible by ζ ∗ (s) and the corollary follows because R∗f j (s) equals the scalar product (Φ(s, ·), f j ). As mentioned in the introduction, the above Corollary, which is the analogue of the result for holomorphic forms proved in [8] and [10], is included in the results of Jacquet-Gelbart [2]. We also observe that, up to gamma factors, the quotient R∗f j (s)/ζ ∗ (s) equals ∞ ζ(2s) X |a j (n)|2 . ζ(s) n=1 n s

Using the usual relations among the eigenvalues a j (n) of a Hecke eigenform, we see that this Dirichlet series has the Euler product Y 1 . 2 2 −s −s −s p (1 − α p p )(1 − α p β p p )(1 − β p p )

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Don Zagier

where α p , β p are defined by ∞ X a j (n) n=1

ns

=

Y p

1 (1 − α p

p−s )(1

− β p p−s )

(i.e. α p + β p = a j (p), α p β p = 1). Thus the corollary is the case n = 2 of the conjecture that the “symmetric power L-functions” Ln ( f j , s) =

n YY

p m=0

1 (1 −

−s αmp βn−m p p )

are entire functions of s for all n > 1.

2 Eisenstein series and the spectral decomposition of L2 (Γ/H). 312

In this section we review the definitions and main properties of Eisenstein series, the Rankin-Selberg method, the spectral decomposition formula for L2 (Γ/H), the Selberg transform, and the Selberg kernel function. All of this material is standard and may be skipped by the expert reader. We will try to give at least a rough proof of all of the statements; for a more detailed exposition the reader is referred to Kubota’s book [4]. Eisenstein Series. For z ∈ H and s ∈ C with Re(s) > 1 we set X E(z, s) = Im(γz) s (Re(s) > 1), γ∈Γ∞ /Γ

(2.1)

) ! a b ∈ S L2 (Z) /{±1}  Z is the group of translawhere Γ∞ = 0 d tions in Γ. The series converges absolutely and uniformly and therefore defines a function which is holomorphic in s and real-analytic and Γinvariant with respect to z. Using the 1 : 1 correspondence between Γ∞ /Γ and pairs of relatively prime integers (up to sign) given by ! a b ←→ ±(c, d), Γ∞ c d (

Eisenstein Series and the Selberg Trace Formula I

343

we can rewrite (2.1) as E(z, s) =

1 X ys 2 c,d∈Z |cz + d|2s

(Re z > 1)

(c,d)=1

and hence ζ(2s)E(z, s) =

′ 1 ys X 2 m,n |mz + n|2s

(Re(s) > 1),

(2.2)

P where ′ denotes a summation over all pairs of integers (m, n) , (0, 0). This latter function has better analytic properties than E(z, s), namely: Proposition 1. The function (2.2) can be continued meromorphically to the whole complex s-plane, is holomorphic except for a simple pole at s = 1, and satisfies the functional equation 313 E ∗ (z, s) = E ∗ (z, 1 − s),

(2.3)

E ∗ (z, s) = π−s Γ(s)ζ(2s)E(z, s) = ζ ∗ (2s)E(z, s).

(2.4)

where The residue at s = 1 is independent of z: res s=1 E(z, s) =

3 6 res s=1 E ∗ (z, s) = π 4

(z ∈ H).

(2.5)

We will deduce these properties from the Fourier development of E(z, s), which itself will be needed in the sequel. Separating the terms m = 0 and m , 0 in (2.2) gives ζ(2s)E(z, s) = y s [ζ(2s) +

∞ X

ϕ s (mz)]

(Re(s) > 1),

m=1

where

∞ X

1 ϕ s (z) = |z + n|2s n=−∞

! 1 z ∈ H, Re(s) > . 2

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Don Zagier

The function ϕ s (x+iy) is periodic in x for fixed y and hence has a Fourier ∞ P development a(n, s, y)e2πinx with π=−∞

a(n, s, y) =

Z∞

−∞

e−2πinx dx (x2 + y2 ) s

 1  Γ( 2 Γ(s − 12 )) 1−2s     y (n = 0)    Γ(s)  = ! s− 1 1    π|n| 2 Γ( 2 )    K 1 (2π|n|y)(n , 0)  2 y Γ(s) s− 2

[GR 3.251.2 and /8.432.5]. Hence

ζ(2s)E(z, s) = ζ(2s)y s +

Γ( 12 )Γ(s − 12 ) ζ(2s − 1)y1−s Γ(s)

! s− 1 1 ∞ ∞ π s y 2 X X |n| 2 +2 K s− 1 (2π|n|my)e2πinmx 2 Γ(s) m=1 n=−∞ m n,0

314

or, multiplying both sides by π−s Γ(s), E ∗ (z, s) = ζ ∗ (2s)y s + ζ ∗ (2s − 1)y1−s ∞ √ X τ s− 1 K s− 1 (2π|n|y)e2πinx , +2 y n=−∞ n,0

2

(2.6)

2

where ζ ∗ (s) is defined by (1.9) and τv (n) by X X  a v τv (n) = |n|v d−2v = (n ∈ Z − {0}, v ∈ C). d d|nd>0 ad=|n|

(2.7)

a,d>0

The infinite sum in (2.6) converges absolutely and uniformly for all s and z, so (2.6) implies that E ∗ (z, s) can be continued meromorphically to all s, the only poles being simple poles at s = 0 and s = 1 with residue ± 21 (the poles of ζ ∗ (2s) and ζ ∗ (2s − 1) at s = 21 cancel). Also, it is clear

Eisenstein Series and the Selberg Trace Formula I

345

from (1.3) and the second formula of (2.7) that Kv (z) and τv (n) are even functions of v, so the functional equation of E ∗ (z, s) follows from (2.6) and (1.9). Another consequence of (2.6) is the estimate E(z, s) = O(ymax(σ,1−σ) )

(y → ∞),

(2.8)

where σ = Re(s); this follows because the sum of Bessel functions is exponentially small as y → ∞. The rankin-selberg method. We use this term to designate the general principle that the scalar product of a function f : Γ/H → C with an Eisenstein series equals the Mellin transform of the constant term in the Fourier development of f . More precisely, we have: Proposition 2: Let f (z) be a Γ-invariant function in the upper half-plane which is of sufficiently rapid decay that the scalar product Z ( f , E(., s¯)) = f (z)E(z, s)dz (2.9) Γ/H

converges absolutely for some s with Re(s) > 1. Then for such s ( f , E(., s¯)) =

Z∞

y s−2 A◦ ( f ; y)dy

(2.10)

0

where A◦ ( f , y) is defined by equation (1.1). Proof. Substituting (2.1) into (2.9) we find Z X ( f , E(., s¯)) = f (z) Im(γz) s dz γ∈Γ∞ /Γ

Γ/H

=

Z

f (z) Im(z) s dz

Γ∞ /H

=

Z∞ Z1 0

0

f (x + iy)y s

dxdy y2

315

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Don Zagier

which is equivalent to (2.10). Note that the growth condition on f in the proposition is satisfied if f (z) = O(y−ǫ ) as y → ∞ for some ǫ > 0, for then (2.8) implies that the scalar product (2.9) converges absolutely in the strip −ǫ < Re(s) < 1+ǫ. One of the main applications of Proposition (??) is the one obtained by choosing f (z) = | f j (z)|2 , where f j is a Maass eigenform. (This was the original application made by Ranking [5] and Selberg [6], except that they were looking at holomorphic cusp forms.) From (1.4) we find that the constant term of f is given by A◦ ( f ; y) = y

X n,0

|a j (n)|2 Kir j (2π|n|y)2

(notice that Kir j (2π|n|y) is real by (1.3), since r j is either real or pure imaginary). Hence (2.10) gives Z

2

| f j (z)| E(z, s)dz =

Γ/H

Z∞

y s−1

0

=

X n,0

|a j (n)|2 Kir j (2π|n|y)2 dy

∞ X |a j (n)|2 Z n,0

= R f j (s)

|n| s

y s−1 Kir j (2πy)2 dy

(2.11)

0

(Re(s) > 1)

(the integral is evaluated in [ET 6.8 (45)] and equals the gamma factor in (1.7)). The analytic properties of R f j (s) given in §1 (meromorphic continuation, position of poles, residue formula (1.10), functional equation (1.11)) follow from (2.11) and the corresponding properties of E(z, s). 

316

Spectral decomposition. We now give a rough indication, ignoring analytic problems, of how the Rankin-Selberg method implies the spectral decomposition formula for L2 (Γ/H). This formula states that any

Eisenstein Series and the Selberg Trace Formula I

f ∈ L2 (Γ/H) has an expansion

Z ∞ X ( f , f j) 1 f (z) = f j (z) + ( f j, f j) 4π j=0



−∞

347

!! ! 1 1 f , E ., + ir E z, + ir dr, 2 2 (2.12) L◦2 (Γ/H) and { f◦ } for the space of

where { f j } j≥1 is an orthogonal basis for constant functions (we will choose f j ( j ≥ 1) to be the normalized Maass eigenforms and f◦ (z) ≡ 1). We prove it under the assumption that f is of sufficiently rapid decay, say f (z) = O(y−ǫ ) with ǫ > 0. Let Ψ(s) be the scalar product (2.9). Proposition 2 shows that Ψ(s) is a meromorphic function of s, is regular in 0 < Re(s) < 1 + ǫ except for a simple pole at s = 1 with Z ( f , f◦ ) 3 f◦ , (2.13) f (z)dz = res s=1 Ψ(s) = π ( f◦ , f◦ ) Γ/H

and satisfies the functional equation Ψ(s) =

ζ ∗ (2s − 1) Ψ(1 − s). ζ ∗ (2s)

(2.14)

On the other hand, (2.10) says that Ψ(s) is the Mellin transform of 1 A◦ ( f ; y), so by the Mellin inversion formula y 1 A◦ ( f ; y) = 2πi

C+i∞ Z

Ψ(s)y1−s ds (1 < C < 1 + ǫ).

C−i∞

1 Moving the path of integration from Re(s) = C to Re(s) = and using 2 (2.13) and (2.14) we find Z∞ 1 f , f◦ 1 1 A◦ ( f ; y) = f◦ + Ψ( − ir)y 2 +ir dr f◦ , f◦ 2π 2 −∞

1 ( f , f◦ ) f◦ + = ( f◦ , f◦ ) 4π

Z∞

−∞

1 ζ ∗ (1 − 2ir) 1 −ir 1 y 2 )dr. Ψ( − ir)(y 2 +ir + ∗ 2 ζ (1 + 2ir)

(2.15)

348 317

Don Zagier

ζ ∗ (1 − 2ir) 1 −ir y2 ζ ∗ (1 + 2ir) is the constant term of E(z, 12 + ir), so (2.15) tells us that the Γ-invariant function ! ! Z∞ ( f , f ) 1 1 1 ◦ f˜(z) = f (z) − f◦ (z) − Ψ − ir E z, + ir dr ( f ◦ , f◦ ) 4π 2 2 1

On the other hand, equation (2.6) implies that y 2 +ir +

−∞

has zero constant term. It is also square integrable, because f (z) is and the non-constant terms in the Fourier expansion of E(z, 21 + ir) are expo∞ ( f˜, f j ) P f j (z), and this nentially small. Hence f˜ ∈ L◦2 (Γ/H), so f˜(z) = j=1 ( f j , f j ) proves (2.12) since ( f˜, f j ) = ( f , f j ) for all j ≥ 1.

Selberg transform. As in the introduction, let ϕ be a function on G of sufficiently rapid decay and T ϕ the operator given by convolution with ϕ. Since we are interested only in functions on the upper halfplane !H = G/K (where K = S O(2) and the identification is given by ! ai + b a b K↔ we can assume that ϕ is left and right K-invariant. c d ci + d But the map ! a b K 7−→ a2 + b2 + c2 + d2 − 2 t:K c d gives an isomorphism between K\G/K and [0, ∞) (Cartan decomposition), so we can think of ϕ as a map ϕ : [0, ∞) → C. An easy calculation shows that t(g−1 g′ ) =

|z − z′ |2 yy′

(g, g′ ∈ G),

where z, z′ ∈ H are the images of g and g′ . Therefore T ϕ acts on functions f : H → C by Z T ϕ f (z) = k(z, z′ ) f (z′ )dz′ (z ∈ H), (2.16) H

Eisenstein Series and the Selberg Trace Formula I

where

|z − z′ |2 k(z, z′ ) = ϕ yy′

349 318

!

(z, z′ ∈ H).

The growth condition we want to impose on ϕ is that  1+A  ϕ(x) = O x 2 (x → ∞)

(2.17)

(2.18)

for some A > 1; then (2.16) converges for any f in the vector space   1+A  V = f ; H −→ C| f is continuous, f (z) = O y− 2 . Because k(z, z′ ) = k(gz, gz′ ) for any g ∈ G, the operator T ϕ commutes with the action of G. A general argument (cf. [7], p. 55 or [4], Theorem 1.3.2) then shows that any eigenfunction of the Laplace operator is also an eigenfunction of T ϕ . More precisely, ! 1 2 f ∈ V, ∆ f = − + r f ⇒ T ϕ f = h(r) f , (2.19) 4 where h(r), the Selberg transform of ϕ, is an even function of r, depend1 ing on ϕ but not on f . To compute it, we choose f (z) = y 2 +ir , which A satisfies the conditions is (2.19) if r ∈ C with | Im(r)| < . Then 2 ! Z∞ Z∞ (x − x′ )2 + (y − y′ )2 ′− 32 +ir T ϕ f (z) = y ϕ dx′ dy′ . yy′ −∞

0

√ Making the change of variables x′ = x+ yy′ v in the inner integral gives T ϕ f (z) =

Z∞

′− 32 +ir

y

0

p

yy′ Q

! (y − y′ )2 dy′ , yy′

where the function Q is defined by Q(w) =

Z∞

−∞

ϕ(w + v2 )dv =

319

Z∞ w

ϕ(t)dt √ t−w

(w > 0).

(2.20)

350

Don Zagier

The further change of variables y′ = yeu then gives T ϕ f (z) = y

Z∞

1 2 +ir

−∞

eiru Q(eu − 2 + e−u )du.

Hence, setting g(u) = Q(eu − 2 + e−u ) we have h(r) =

Z∞

−∞

(u ∈ R),

g(u)eiru du (r ∈ C, | Im(r)|
1).

The integral I3 is also quite easy to compute. Since ζ ∗ (1 − 2ir) is nonzero for Im(r) ≥ 0 and since the poles of ζ ∗ (1 + 2ir) and ζ ∗ (1 − 2ir) A at r = 0 cancel, the integrand in K3 (y) is holomorphic in 0 ≤ Im(r) < 2

Eisenstein Series and the Selberg Trace Formula I

357

except for a simple pole of residue 3i  i  1 −1  i  ∗ res y h h i (ζ (1 + 2ir)) = r= 2 ζ ∗ (2) 2 πy 2 i C at . Hence we can move the path of integration to Im(r) = (1 < C < 2 2 A), obtaining iy K3 (y) = 4π

C+i∞ Z

y−s

C−i∞

 is  ζ ∗ (s) ds (1 < C < A). h ζ ∗ (s + 1) 2

The Mellin inversion formula then gives  is  1 ζ ∗ (s) h (1 < Re(s) < A), I3 (s) = − ∗ 2 ζ (s + 1) 2

(3.5)

in agreement with the formula in Theorem (2). We now turn to I2 (s), which is somewhat harder. From (2.23) and (2.20) we have 1 2π

Z∞

h(r)dr = g(0) = Q(0) =

−∞

so

Z∞

1 ϕ(v2 )dv = y

Z∞

−∞

−∞

! u2 ϕ 2 du, y

! Z∞ ! n2 u2 ϕ 2 − ϕ 2 du. K2 (y) = y y n=−∞ ∞ X

−∞

By the Poisson summation formula this equals XZ



n,0−∞

326

! ∞ X u2 2πinu du = 2y ψ(ny), ϕ 2 e y n=1

where ψ(y) =

Z∞

−∞

ϕ(u2 )e2πiuy du.

(3.6)

358

Don Zagier

Since ϕ is smooth, ψ is of rapid decay, so we may interchange summation and integration to get ∞ Z X



I2 (s) = 2

Z∞

ψ(ny)y s−1 dy = 2ζ(s)

n=1 0

ψ(y)y s−1 dy(Re(s) > 1). (3.7)

0

To calculate the integral we begin by substituting the third equation of (2.23) into (3.6). This gives 1 ψ(y) = − π

Z∞

2πiuy

e

−∞

Z∞

Q′ (u2 + v2 )dv du.

−∞

Changing to polar coordinates u + iv = reiθ and using the standard integral representation 1 ·J◦ (x) = 2π

Z2π

eix cos θ dθ

0

of the Bessel function of order 0 [GR 3.915.2] we find ψ(y) = −2

Z∞

J◦ (2πyr)Q′ (r2 )r dr

0

or, making the substitution r = 2 sinh

ψ(y) = −

Z∞

u and using (2.21), 2

u J◦ (4πy sinh )g′ (u)du. 2

0

Using the formula Z∞ 0

J◦ (2ay)y

s−1

! Γ( 2s ) 3 dy = s 0 < Re(s) < , a > 0 2a Γ(1 − 2s ) 2

Eisenstein Series and the Selberg Trace Formula I 327

359

[ET 6.8 (1)] we find Z∞  (2π)−s Γ( 2s ) u −s ′ ψ(y)y dy = − sinh g (u)du (3.8) 2Γ(1 − 2s ) 2 −∞ −∞ ! 3 0 < Re(s) < 2  A  (the integral converges at ∞ because g′ (u) = O e− 2 |u| and at 0 because g′ (u) is an odd function and hence O(u)). Substituting Z∞

s−1

−1 g (u) = 2π ′

Z∞

rh(r) sin ru dr

−∞

and using the Fourier sine transform formula       Z∞ s s     − ir + ir Γ Γ   sin ru   2 2     −   s du = −2 s−1 iΓ(1 − s)     s s  u   Γ 1 − 2 − ir Γ 1 − 2 + ir  sin h 2 0

([ET 2.9 (30)]; the conditions for validity are misstated there) gives     Z∞   Z∞ s iΓ 2s Γ 1−s − ir Γ 2 2  rh(r)dr,  ψ(y)y s−1 dy = − s+2 s+3/2 2 π Γ 1 − 2s − ir 0

−∞

where we have used the fact that h(r) is an even function, and substituting this into (3.7) we obtain the formula stated in the theorem. Since the integral converges for all s with positive real part, the formula is valid for all s with Re(s) > 1 (not just 1 < Re(s) 23 ); we can use the elementary identity           s Γ 2s + ir  Γ 2s + ir Γ 2s − ir r  Γ 2 − ir −    =       s 1+s 2πi  Γ 1 − s − ir + ir Γ 1 − Γ 1−s 2 2 2 Γ 2 Γ(ir)Γ(−ir) to write it in the more elegant form

328

360

Don Zagier

Z∞ Γ  s + ir Γ  s − ir ζ ∗ (s) 2 2 h(r)dr I2 (s) =   s+1 Γ(ir)Γ(−ir) (4π) 2 Γ s+1 2 −∞

(3.9)

(Re(s) > 1).

The proof of (3.9) was rather complicated and required introducing the extraneous function J◦ (x). We indicate a more natural and somewhat simpler derivation which, however, would require more work to justify since it involves non-absolutely convergent integrals. Interchange the order of integration in Z∞

ψ(y)y

s−1

dy =

ϕ(u2 ) cos 2πuyduy s−1 dy.

0 −∞

0

Then the inner integral

Z∞ Z∞

R∞

y s−1 cos 2πuydy converges (conditionally) for

0

0 < Re(s) < 1 (thus in a region of validity disjoint from that of (3.7)!) πs and equals (2π|u|)−s Γ(s) cos there [ET 6.5 (21)]. Using (2.24) we 2 then find Z∞

ψ(y)y s−1 dy =

0

=

Γ(s) cos πs 2 2(2π) s+1

Z∞ 0

x

− s+1 2

Z∞

−∞

 x P− 1 +ir 1 + h(r)r tanh πrdr dx 2 2

for 0 < Re(s) < 1. Interchanging the order of integration again and using the formula Z∞ 0

 x x P− 1 +ir 1 + dx = 2 2       s s Γ + ir Γ − ir Γ 1−s 2 2 2    = 21−s    1+s 1 Γ 2 Γ 2 + ir Γ 21 − ir − s+1 2

(0 < Re(s) < 1)

Eisenstein Series and the Selberg Trace Formula I 329

361

[GR 7.134] we find Z∞

ψ(y)y s−1 dy =

0

  ∞  s  2−s−2 Γ 2s Z  s − ir Γ + ir h(r)r sin hπrdr, Γ   3 2 2 π s+ 2 Γ 1+s 2 −∞

and this now holds whenever Re(s) > 0 (not just 0 < Re(s) < 1) since both sides are holomorphic in that range. Substituting into (3.7) again gives (3.9). To complete the proof of Theorem 2 we must still compute I1 (s) P i.e. the contribution from the main term k(z, γz) of K◦ (z, z). For γ 0 and ∆ < 0. It will also be useful to introduce symmetrization operators S s1 , Sr with S s1 [ f (s)] = f (s) + f (1 − s),

Sr [ f (s)] = f (r) + f (−r)

for any function f . Thus the formula we want to prove can be written " −s # π−s Γ(s) 1 π Γ(s) V(s, t) = S s v(s, t) . (5.3) Γ(s, ∆) γ(s, ∆) If ∆ > 0, then (4.5) and (2.24) give   Z∞ s 1 Γ 2 s r tanh πrh(r) ∆2 V(s, t) = 8π Γ(s) =×

Z1 0

−∞

O¶− 1 +ir 2

! ! s−3 s s 1 dξ ∆/2 2 (1 − ξ) F , ; ; ξ √ dr, 1+ 1−ξ 2 2 2 ξ

u2 . To prove (5.3), where we have made the change of variables ξ = 2 u +1 we must show that the inner integral equals       2 s Γ(S − 12 )Γ 1−s + ir Γ 1−s 2 2 − ir       S s1  s/2 cosh πr (5.4) ∆ Γ 21 Γ 2s Γ 1 − 2s !# 1−s 3 t2 1−s + ir, − ir; − s; 1 − ×F 2 2 2 4

378 346

Don Zagier

(here and from now one we use standard identities for the gamma function without special mention). Using the identity P− 12 +ir

  ! !   1 2 1 1 Γ(2ir)  x 2 −ir F 1+ = Sr   − ir, − ir; 1 − 2ir; −x  (x > 0)  2  1  x 2 2 Γ + ir 2

[EH 3.2 (19)] and expanding the hypergeometric series, we find that the integral in question equals  2  !−n− 21 +ir  ∞ n Γ n + 1 − ir X (−1) ∆ cothπr 2   Sr   2πi n! Γ(n + 1 − 2ir) 4 n=0 ×

Z1

(1 − ξ)

0

From [EH 2.4(2), 2.8(46)] we have Z1 0

s 2 +n−1−ir

 !  s s 1 dξ  F , ; ; ξ √  . 2 2 2 ξ

! s s 1 ξ F , ; ; ξ dξ (1 − ξ) 2 2 2     ! Γ 21 Γ 2s + n − ir s s 1+s  F , ;  = + n − ir; 1 2 2 2 Γ 1+s 2 + n − ir       1 Γ 2s + n − ir Γ 1−s 2 + n − ir Γ 2 , =  2 Γ n + 12 − ir s 2 +n−1−ir

− 21

so our integral equals  ∞  coth πr X (−1)n × Sr  √ 2i π n=0 n!     !−n− 12 +ir  Γ 2s + n − ir Γ 1−s + n − ir  ∆ 2  × Γ(n + 1 − 2ir) 4

Eisenstein Series and the Selberg Trace Formula I

     1  coth πr Γ 2s − ir Γ 1−s − ir ∆ !ir− 2 2  = Sr  √ Γ(1 − 2ir) 4 2i π !# s 1−s 4 ×F − ir, − ir; 1 − 2ir; − 2 2 ∆    s 1−s 1  Γ(s − 2 )Γ 2 − ir ∆ !− 2 1  coth πr   = Sr S s  √ 4 2i π Γ 1+s 2 − ir !# 1−s 3 ∆ 1−s − ir, + ir; − s; − ×F 2 2 2 4

379

(5.5)

(the last formula is [EH 2.10(2)]), and since         coth πr Γ 1−s − ir  cosh πrΓ 1−s + ir Γ 1−s − ir 2 2 2         =  Sr  √ 2i π Γ 1+s Γ 12 Γ 2s Γ 1 − 2s 2 − ir

347

this agrees with (5.4), completing the proof for ∆ > 0. 1 t2 π−s Γ(s) = δ−s/2 , where δ = |∆| = 1 − , so (5.3) is If ∆ < 0, then γ(s, ∆) 2 4 equivalent to   1 δ−s/2 V(s, t) = S s1 δ 2 (s−1) v(1 − s, t) . On the other hand, from (4.4) and (2.24) we have −s/2

π

1 V(s, t) = 2

Z∞

−∞

r tanh πrh(r)

Z∞

P− 1 +ir (1 + 2δ(u2 − 1))P−s (u)dudr. 2

1

Denote the inner integral by I. Then (5.3) will be proved if we show that 348         1 1 Γ 12 − s Γ 2s + ir Γ 2s − ir        I = S s1  δ 2 (s−1)  1+s 2 Γ 21 + ir Γ 21 − ir Γ 2−s 2 Γ 2 !# s 1 s F + ir, − ir; + s; δ . 2 2 2

380

Don Zagier

By [EH 2.10(1)], this is equivalent to         s 1−s 1−s s 1 Γ 2 + ir Γ 2 − ir Γ 2 + ir Γ 2 − ir         I= √ 2−s 2 π Γ 21 + ir Γ 21 − ir Γ 1+s 2 Γ 2 ! 1 s 1 s − ir, + ir; ; 1 − δ . ×δ s− 2 F 2 2 2 To prove this this formula, we being by making the substitution v = u2 − 1 in I and substituting for P− 1 +ir by 2

Z∞

−ax

e

dx Kir (x) √ = x

r

! ! π 1 1 Γ + ir Γ − ir P− 1 +ir (a) 2 2 2 2

0

[GR 6.628. 7]; after an interchange of integration this gives ! ! √ 1 1 2πΓ + ir Γ − ir I 2 2   Z∞ Z∞    √  dv  −x dx −2δxv   e Kir (x) √ . = P−s 1 + v √  e  x 1+v 0

0

By [GR 7.146.2] the inner integral equals (2δx)−3/4 eδx W− 1 , 1 − s (2δx), 4 4 2 where Wλ,µ is Whittaker’s function, and using the Mellin-Barnes integral representation of the latter [GR 9.223] we find that this in turn equals 1+s Γ 2

C+i∞ !−1  ! ! Z 1 1−s s −1 1 s Γ v + Γ − v)Γ( −v × Γ 1− · 2 2πi 2 2 2 C−i∞

1

×(2δx)v− 2 dv, 349

where C is chosen such that − 21 < C < 21 min(σ, 1 − σ). If choose C to satisfy also C > 0 then we may interchange the order of integration again, obtaining ! ! √ 1 s 1+s  1 Γ 1− I 2 πΓ + ir Γ( − ir)Γ 2 2 2 2

Eisenstein Series and the Selberg Trace Formula I

1 = 2πi

381

!  1−s 1 1 s −v × 2v δv− 2 Γ(v + )Γ − v Γ 2 2 2

C+i∞ Z

C−i∞ Z∞

xv−1 e−x Kir (x)dx dv

×

0

Γ( 1 ) = 2 2πi

C+i∞ Z

Γ

C−i∞

!  1−s 1 −v Γ − v Γ(v + ir)Γ(v − ir)δv− 2 dv 2 2

s

[ET 6.8(28)]. The integral is very rapidly convergent (the integrand is 1 O(|v|−3/2 e−2π|v| )), so we may substitute for δv− 2 the binomial expansion   ∞ X Γ 2s − v + n 1 1 1  (1 − δ)n  δv− 2 = δ 2 (s−1) s n! − v Γ n=0 2

and integrate term by term. Using “Barnes’ Lemma” 1 2πi

C+i∞ Z

Γ(α + s)Γ(β + s)Γ(γ − s)Γ(δ − s)ds

C−i∞

=

Γ(α + γ)Γ(α + δ)Γ(β + γ)Γ(β + δ) Γ(α + β + γ + δ)

[GR 6.412] we obtain finally ! ! ! 1 s 1 1+s  1 Γ 1− I = δ 2 (s−1) × 2Γ + ir Γ − ir Γ 2 2 2 2         ∞ Γ s + ir + n Γ s − ir + n Γ 1−s + ir Γ 1−s − ir X 2 2 2 2   (1 − δ)n × 1 Γ 2 + n n! n=0 !−1  ! !  s  1−s s−1 1 s 1−s =Γ + ir Γ − ir δ 2 Γ + ir Γ − ir Γ 2 2 2 2 2 ! s s 1 ×F + ir, − ir; ; 1 − δ . 2 2 2

382

Don Zagier 350

This completes the proof of Proposition 5 and hence of Theorem (1) for m = 1. To calculate the function (5.1) for m > 1 we set K0m (z, z′ )

=

∞ X

a j (m)

j=1

R

Then I m (s) =

h(r j ) f j (z) f j (z′ ). ( f j, f j)

K0m (z, z)E(z, s)dz. On the other hand, from (1.6) we see

Γ/H

K0m (z, z′ )

1

that = m 2 K0 (z, z′ )|T (m), where K0 (z, z′ ) is the kernel function (2.27) and T (m) the Hecke operator (1.5), acting (say) on z′ . Since the constant function and the Eisenstein series E(z, s) are eigen-functions of 1 m 2 T (m) with eigenvalues τ 1 (m) and τ s− 1 (m), respectively (τv (m) as in 2 2 (2.7)), equation (2.31) gives i 3 K0m (z, z′ ) = K m (z, z′ ) − τ 1 (m)h − π 2 2 ! ! Z∞ 1 1 ′ 1 E z, + ir E z , − ir h(r)τir (m)dr, − 4π 2 2 −∞

where √ K (z, z ) = mK(z, z′ )|T (m) = m



! X az′ + b 1 . k z, ′ √ cz + d 2 m a,b,c,d∈Z ad−bc=m

Hence the constant term K m (y) of K0m (z, z) equals

4 P

i=1

Ki m (y), where

K3m and K4m are defined exactly like K3 and K4 but with h(r) replaced 351 by h(r)τir (m) and 1 K1m (y) = √ m

! az + b dz, k z, cz + d ad−bc=m

iy+1 Z X iy

c>0

Eisenstein Series and the Selberg Trace Formula I

1 K2m (y) = √ m



y 2π

383

! az + b dz− k z, d ad=m b=−∞

iy+1 Z ∞ X X iy

a,d>0

Z∞

h(r)τir (m)dr.

−∞

As in §3 we then find I m (s) =

4 P

i=1

Iim (s) for 1 < σ < A, where I3m and I4m

are given by the same formulas as I3 and I4 (equations (3.4) and (3.5)) but with h(r) replaced by τir (m)h(r) and ! ∞ X s−1 t ζ(s, t2 − 4m)V s, √ . ζ(2s)I1m (s) = m 2 m t=−∞ As to I2m , from (2.20) and (2.23) we find K2m (y)

1 = √ m

0



! (x(d − a) − b)2 + (a − d)2 y2 dx ϕ my2 ad=m b=−∞

Z1 X X ∞ a,d>0

Z∞  a ir y X dr h(r) 2π ad=m d a,d>0−∞

! X  a 1 X (a − d)2 −y g log = √ Q m d m ad=m ad=m a,d

 ! ∞  b2 1 X    ϕ   √m my2 + b=−∞      0  √ 1      √m K2 (y m) if =     0 if

if



m∈Z

√ if m < Z √ m∈Z √ m < Z,

384 352

Don Zagier

so

 −s/2   I2 (s) m I2m (s) =    0

√ m ∈ Z, √ if m < Z. if

The analytic continuation to 1 − A < σ < 1 now proceeds as in § 4, the only essential difference being that the terms (4.2) are absent when m is not a square, since I1m then has no summands with t2 − 4m = 0 and I2m vanishes identically. The final formula is that given in Theorem 1. If m < 0 the proof is similar and in fact somewhat easier (since t2 − 4m now always has the same sign and the term I2 is absent), but the calculations with the hypergeometric functions are a little different. Since constant functions and Eisenstein series are invariant under T (−1), the terms I3m (s) and I4m (s) are equal to I3|m| (s) and I4|m| (s), so that first two terms in (1.16) are unchanged except for replacing m by |m|. The term I2m is always zero since m cannot be a square. Finally, for I1m we find I1m (s)

= |m|

s−1 2

∞ X ζ(s, t2 − 4m) V 1 s,t|m|− 2 ζ(2s) t=−∞

(m < 0)

(5.6)

with

V s,t =

Z H

k z,

!

1 y s dz = z¯ + t

Z H

     |z|2 − ∆/4 2  s  2 + t ϕ   y dz,  y2

where now ∆ = t+ 4. This function is easier to compute than V(s, t) since ∆ always has the same sign. Making the same substitutions as in the case ∆ > 0 of Proposition (4) we find that V− (s, t) is given by the same integral (4.5) but with ϕ(∆u2 + t2 ) instead of ϕ(∆u2 + ∆), This integral can then be calculated as in the case ∆ > 0 of Proposition 5, the ! ∆/2 only difference being that the function P− 1 +ir 1 + is replaced by 2 1−ξ ! ∆/2 P− 1 +ir −1 + and we must use 2 1−ξ

Eisenstein Series and the Selberg Trace Formula I

385

! 2 P− 1 +ir −1 + = 2 x   !   1 Γ(2ir) 1 1  − ir, − ir; 1 − 2ir; x  (x > 0) = Sr   x 2 −ir F  2  1  2 2 Γ 2 + ir

! 2 . 353 2 x This has the effect of introducing an extra factor (−1)n in the infinite 4 4 sum and hence of replacing the argument − in (5.4) by + . Using the ∆ ∆ identity      !ir− 12  coth πr Γ 2s − ir Γ 1−s − ir ∆ 2  Sr  √ × Γ(1 − 2ir) 4 2i π !# s 1−s 4 ×F − ir, − ir; 1 − 2ir; 2 2 ∆ ! ! !− 2s  s  1−s cosh2 πr  s 1−s ∆ .× Γ + ir Γ − ir Γ = + ir Γ − ir 2 2 2 2 2 4 π ! 1−s 1−s 1 ∆ ×F − ir, + ir; ; 1 − 2 2 2 4 [EH 3.2(18)] instead of the corresponding formula for P− 1 +ir 1 +

[EH 2.10(3)] and substituting the expression thus obtained for V− (s, t) into (5.6), we find that the last term in (1.16) must be replaced by s−1 ∞ 2 s−4 |m| 2  s 2 X ζ(s, t2 − 4m)× Γ 2 t=−∞ π s+1 Z∞ Γ  s + ir Γ  s − ir Γ  1−s + ir Γ  1−2 − ir 2 2 2 2     × 1 1 Γ 2 + ir Γ 2 − ir Γ(ir)Γ(−ir) −∞ ! 1−s 1 t2 1−s + ir, − ir; ; h(r)dr ×F 2 2 2 4m

if m < 0. This completes the proof of Theorem (1).

386

Don Zagier

Finally, we indicate what happens when Γ is replaced by a congru- 354 ence subgroup Γ1 in the simplest case Γ1 = Γ0 (q)/{±1}, q prime. There are now two cusps and correspondingly two Eisenstein series E1 and E2 , given explicitly by E1 (z) =

X

Im(γz) s ,

E2 (z) =

γ∈Γ∞ \Γ1

X

Im(wγz) s

γ∈w−1 Γ∞ w\Γ1

! 0 −1 ), and formula (2.31) becomes (where w = q 0 K0 (z, z′ ) =



1 4π

X

γ∈Γ1 ∞ 2 Z X

k(z, γz′ ) −

E j (z,

j=1 −∞

i 1 − h vol(Γ1 \H) 2

1 1 + ir)E j (z′ , − ir)h(r)dr, 2 2

where K0 (z, z′ ) is defined as before but with f j now running over all Maass cusp forms of weight 0 on Γ1 (cf. [4]). It is easily checked that 1 qs E(qz, s) − 2s E(z, s), 2s q −1 q −1 qs 1 E2 (z, s) = 2s E(z, s) − 2s E(qz, s), q −1 q −1

E1 (z, s) =

so that Fourier developments R of E1 and E2 can be deduced from (2.6). The calculation of I(s) = K0 (z, z)E1 (z, s)dz (which again can be expressed as

R∞

Γ1 /H

K (y)y s−2 dy, K (y) = constant term of K0 (z, z)) now pro-

0

ceeds as in § 3; the final formula is the same except that I1 (s) is replaced by !! ∞ X t2 − 4 1 −1 1+ ζ(s, t2 − 4)V(s, t), ζ(2s) 2 q q +1 t=−∞

Eisenstein Series and the Selberg Trace Formula I

387

! q−1 ∆ is the Legendre symbol), I3 (s) is multiplied by s+1 (where , q q −1 and the integrand of I4 (s) is multiplied by ! (q + 1)(1 − q−s )(1 − q1−s ) 1 −s + 2q . 1 + q−s (q1+2ir − 1)(q1−2ir − 1) 

Bibliography [1]

Goldfeld, D.: On convolutions of non-holomorphic Eisenstein se- 355 ries. To appear in Advances in Math.

[2]

Gelbart, S. and H. Jacquet,: A relation between automorphic representations of GL(2) and GL(3). Ann. Sc. Ec. Norm. Sup. 11(1978) 471-542.

[3]

Jacquet, H. and D. Zagier: Eisenstein series and the Selberg trace formula II. In preparation.

[4]

Kubota, T.: Elementary Theory of Eisenstein series. Kodansha and John Wiley, Tokyo-New York 1973.

[5]

Rankin, R.: Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetical functions. I. Proc. Camb. Phil. Soc. 35(1939) 351-372.

[6]

Selberg, A.: Bemerkungen u¨ ber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43 (1940) 47-50.

[7]

Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Ind. Math. Soc. 20 (1956) , 47-87.

[8]

Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. 31 (1975), 79-98.

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[9]

Bibliography

Sturm, J.: Special values of zeta-functions, and Eisenstein series of half-integral weight. Amer. J. Math. 102 (1980), 219-240.

[10] Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In Modular Functions of one variable VI, Lecture Notes in Mathematics No. 627, Springer, Berlin-Heidelberg-New York 1977, pp. 107-169. [11] Zagier, D.: Eisenstein series and the Riemann zeta-function. This volume, pp. 275-301. Tables [EH] Erdelyi, A. et al.: Higher Transcendental Functions, Vol. I. McGraw-Hill, New York 1953. [ET] Erdelyi, A. et al.: Tables of Integral Transforms, Vol. I. McGrawHill, New York 1954. [GR] Gradshteyn, I. S. and I. M. Rhyzhik: Table of Integrals, Series, and Products. Academic Press, New York-London 1965.

This book contains the original papers presented at an International Colloquium on Automorphic forms, Representation theory and Arithmetic held at the Tata Institute of Fundamental Research, Bombay in January 1979.