Automorphic Forms, Representation Theory and Arithmetic TATA INSTITUTE of FUNDAMENTAL RESEARCH STUDIES in MATHEMATICS

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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. Herve´ : 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 Mathe- matical 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. Se- shadri was in charge of the scientific programme. Professors P. Deligne and M. Kneser acted as representatives of the International Mathemati- cal Union on the Organising Committee. The following mathematicians gave invited addresses at the Collo- quium: 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 speak- ers 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 per- formance 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 ON SHIMURA’S CORRESPONDENCE FOR MODULAR FORMS OF HALF-INTEGRAL WEIGHT∗ 1 1 The Metaplectic Group ...... 4 2 Admissible Representations ...... 6 3 Whittaker Models ...... 8 4 The Theta-Representations rχ ...... 10 5 A Functional Equation of Shimura Type ...... 12 6 L and ǫ-Factors ...... 14 7 A Local Shimura Correspondence ...... 17 8 The Metaplectic Group ...... 19 9 Automorphic Representations of Half-Integral Weight . . 20 10 Fourier Expansions ...... 21 11 Theta-Representations ...... 25 12 A Shimura-Type Zeta Integral ...... 26 13 An Euler Product Expansion ...... 29 14 A Generalized Shimura Correspondence ...... 34 15 TheTheorem ...... 34 16 Applications and Concluding Remarks ...... 39

2 PERIOD INTEGRALS OF COHOMOLOGY CLASSES WHICH ARE REPRESENTED BY EISENSTEIN SERIES 46 2 TheEisensteinSeries ...... 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 Zp-EXTENSIONS 157

5 DIRICHLET SERIES FOR THE GROUP GL(N). 171 1 Introduction...... 171 2 Maassforms...... 172 3 Fourier expansions ...... 173 4 TheMellinTransform...... 175 5 Theconvolution...... 176 6 Functional Equations ...... 179

6 CRYSTALLINE COHOMOLOGY, DIEUDONNE´ MODULES, AND JACOBI SUMS 182

7 ESTIMATES OF COEFFICIENTS OF MODULAR FORMS AND GENERALIZED MODULAR RELATIONS 272

8 A REMARK ON ZETA FUNCTIONS OF ALGEBRAIC NUMBER FIELDS1 281

9 DERIVATIVES OF L-SERIES AT S = 0 288 1 Introduction...... 288 2 Complex quadratic ground ﬁelds ...... 288 3 L-series considered over Q ...... 296

10 EISENSTEIN SERIES AND THE RIEMANN ZETA-FUNCTION 302

11 EISENSTEIN SERIES AND THE SELBERG TRACE FORMULA I 332 0 Introduction...... 332 CONTENTS 9

1 Statement of the main theorem ...... 336 2 Eisenstein series and the spectral decomposition...... 342 3 Computation of I(s) for (s) > 1...... 353 ℜ 4 Analytic continuation of I(s),...... 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 2 2 T (p ), say T(p ) f = ωp f . If k 5, then the L-function k,χ ≥

∞ s s 2k 2 2s 1 A(n)n− = 1 ωp p− + χ(p)p − − − − Xn=1 pY< ∞ is the Mellin transform of a modular cusp form of weight k 1, level − N/2, and character χ2. for further details, see [Shim] or [Niwa]. Our purpose in this paper is to establish a Shimura correspondence for any (not necessarily holomorphic) cusp form of half-integral weight deﬁned over a global ﬁeld F(not necessarily Q). Our approach is similar to Shimura’s in that we use L-functions. Out point of view is new in that we use the theory of group representations.

1Talk presented by S.G.

1 2 S. Gelbart and I. Piatetski-Shapiro

Roughly speaking, suppose π = πv is an automorphic cuspidal v representation of the metaplectic groupN which doesn’t factor through GL2. Then we introduce an L-factor L(s, πv) for each v and we prove that the L-function

L(s, π) = L(s, πv) Yv

belongs to an automorphic representation of GL2(AF) in the sense of [Jacquet-Langlands]. Since we characterize those π which correspond to cuspidal (as opposed to just automorphic) representations of GL2(AF) we reﬁne as well as generalize Shimura’s results. Let us now describe our correspondence in more detail. Suppose π is an automorphic cuspidal representation of the metaplectic group. Since π is determined by its local components πv, we want to describe its “Shimura image” S (π) in purely local terms. Thus we construct a local correspondence

S : πv πv →

by “squaring” the representation πv; if πv is an induced representation, x this means squaring the characters of Fv which parametrize πv. In general, this process of “squaring” tends to smooth out representations, as we shall now explain. 3 Suppose we consider the theta-representations of the metaplectic group. These representations generalize the classical modular forms of half-integral weight given by the theta-series

∞ 2πin2 θχ(z) = χ(n)e z nX= −∞ where χ is an (even) Dirichlet character of Z. Since these representations arise by pasting together a grossencharacter χ of F with the “even or odd” part of the canonical metaplectic representation constructed in [Weil], we denote these representations by rχ and call them Weil representations. Locally, rχ is supercuspidal when χv( 1) = 1. Almost ev- v − − erywhere, however, χv( 1) = 1, rχ is the class 1 quotient of a reducible − v On Shimura’s Correspondence for Modular Forms... 3 principal series representation at s = 1/2, and the global representation

rχ = rχv Ov is “distinguished” from several different points of view. Most signifi- cantly, these rχ exhaust the automorphic forms of half-integral weight which are determined by just one Fourier coefficient; this is the principal result of [Ge PS2]. Now if πv is an even Weil representation rχ (i.e. χv( 1) = 1), its v − Shimura image will be the one-dimensional representation χv of GL2(Fv), whereas if πv is an “odd” Weil representation, S (πv) will be the special representation Sp(χv); cf. §7. The Shimura correspondence thus takes cuspidal rχ to automorphic representations of GL(2) which almost everywhere are one-dimensional and hence not cuspidal. The main result of this paper, however, guarantees that these representations are the only cuspidal π which map to non-cuspidal automorphic forms of GL(2). This explains the restriction k 5 in [Shim] and ultimately resolves ≥ “Open question (C)” of that paper; cf. §16. We mention also that the cuspidal representations rχ contradict the Ramanujan-Petersson conjecture, in complete analogy to the counter- examples of [Ho PS] for Sp(4). In particular, the L-function we attach to a supercuspidal component rχv can have a pole; cf. §6. Thus these 4 representations rχ distinguish themselves in yet another way, and the regularizing nature of the local correspondence S evidence itself (by “lifting” a supercuspidal representation to a non-supercuspidal one). For a leisurely account of how classical modular forms of half- integral weight can be defined as representations of Weil’s metaplectic group the reader is referred to [Ge]. Most of the results described in the present paper were first announced in [Ge PS]. We note Chapter I is purely local: after describing the local metaplectic group, and the notion of Whittaker models for its irreducible admissible representations, we introduce L and ǫ factors and describe the local Shimura correspondence. In Chapters II and III we piece together these notions to obtain a global correspondence. In the process of doing so, we develop a Jacquet-Langlands theory for the metaplectic 4 S. Gelbart and I. Piatetski-Shapiro

group. Details and related results are to be found in [Ge], [Ge HPS], and [Ge PS2]. The principal contribution of the present paper is the proof of the global Shimura correspondence in Chapter III. It is with pleasure that we acknowledge our indebtedness to G. Shi- mura and R. P. Langlands. Shimura had already suggested the possibility of a representation-theoretic and adelic approach to his results in [Shim]. On the other hand, the concrete suggestions and inspiration of Langlands ﬁrst brought one of us close to the metaplectic group and got this project started. Langlands also suggested how the Selberg trace formula could be used to obtain (and in fact go beyond) our present results; this suggestion has just recently been developed by Flicker, whose results—improvements of our own—will appear in a forthcoming paper [Flicker].

Chapter I. Local Theory

Throughout this Chapter F will denote a local ﬁeld of characteristic not equal to two. By Z2 we shall denote the group of square roots of unity.

1 The Metaplectic Group

2 1.1 Let H (SL2(F).Z2) denote the two-dimensional continuous cohomology group of SL2(F) with coeﬃcients in Z2. From [Weil] and [Moore] 2 it follows that if F , C, H (SL2(F), Z2) = Z2. 5 If F = C, let SL (F) denote the group SL (F) Z . If F , C, let 2 2 × 2 SL2(F) denote the non-trivial central extension of SL2(F) by Z2 deter- 2 mined by the non-trivial element of H (SL2(F), Z2). In all cases, we have an exact sequence of topological groups 1 Z SL (F) SL (F) 1. → 2 → 2 → 2 →

1.2 We want to extend Weil’s metaplectic group to GL2. To do this, we use the fact that any automorphism of SL2(F) lifts uniquely to an automorphism of SL2(F). On Shimura’s Correspondence for Modular Forms... 5

Let D denote the group

a 0 D = : a Fx ( 0 1! ∈ )

Each element of D operates on SL2(F) by conjugation, hence lifts to an automorphism of SL2. If G denotes the resulting semi-direct product of D and SL2(F), we obtain an exact sequences of locally compact groups

1 Z G GL (F) 1. (1.2.1) → 2 → → 2 →

Note G is a non-trivial extension of GL2(F) unless F = C.

1.3 The sequence (1.2.1) splits over the following subgroups of GL2(F):

1 x N = : x F ( 0 1! ∈ ) a 0 D = : a Fx ( 0 1! ∈ ) λ 0 Z2 = : λ (Fx)2 ( 0 λ! ∈ ) and (if F is non-archimedean, of odd residual characteristic, and OF is the ring of integers of F),

K = GL2(OF).

If H is any subgroup of GL2(F), let H denote its full inverse image in GF. If H is such that the sequence (1.2.1) splits over it, then H is the direct product of Z2 with a subgroup of G which we again denote by H.

1.4 The center of GF is

2 Z = Z2 Z . × 2 6 S. Gelbart and I. Piatetski-Shapiro

6 On the other hand, if

α 0 Z = : α Fx , ( 0 α! ∈ )

the group Z is abelian but not central in G. When convenient, we confuse Z with the group Fx, and Z2 with the subgroup (Fx)2.

1.5 If ϕ : G W is any function on G, with values in a vector space → W, we say ϕ is genuine (or doesn’t factor through GL2) if

ϕ(gζ) = ζϕ(g), for all g G, ζ Z . ∈ ∈ 2 Unless speciﬁed otherwise, we henceforth deal only with genuine objects on GF.

2 Admissible Representations

2.1 By modifying the deﬁnitions in [Jacquet-Langlands], we can de- ﬁne, for each local F, the notion of an irreducible admissible representation π of GF.

2.2 If F is archimedean, we shall assume π is actually irreducible unitary, or perhaps the restriction of such a representation to “smooth” vectors. Since GC = GL (C) Z we shall have little to say about the case 2 × 2 when F is complex.

2.3 Induced Representations. Let B denote the Borel subgroup of GL2(F). Although B is not abelian, it contains a convenient subgroup of ﬁnite index which is abelian, and even “splits” in G. Indeed let B0 a1 x denote the subgroup of B consisting of matrices 0 a2 where a1 and a2 have even p-adic order. If F has even residual characteristic we also require that a1 be a square modulo 1 + 4OF. If F is real we simply require that a > 0. In any case, B = B Z , and the index of B in B is the 1 0 0 × 2 0 index of (Fx)2 in Fx. On Shimura’s Correspondence for Modular Forms... 7

x For any pair of quasi-characters µ1, µ2 of F , let µ1µ2 denote the (genuine) character of B0/N whose restriction to B0/N is given by the formula a1 0 µ1µ2 = µ1(a1)µ2(a2). 0 a2!! The induced representation 7

ρ(µ1, µ2) = Ind(GF, B0, µ1µ2) (2.3.1) is admissible and ρ(µ , µ ) ρ(ν , ν ) 1 2 ≈ 1 2 not only if µ1 = ν2 and µ2 = ν1, but also if

2 2 µi = νi , i = 1, 2. (2.3.2) cf. §2 of [Ge PS2] and §5 of [Ge]. Moreover, ρ(µ1, µ2) is irreducible unless µ2µ 2(x) = x 1 or x 1 (or all integral points in the real case). In 1 2− | | | |− 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 2 2 1 irreducible admissible representation of GF. If µ µ (x) = x we call 1 2− | | 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 GF. 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, π(u)v du = 0 Z U 8 S. Gelbart and I. Piatetski-Shapiro

for some open compact subgroup U of N GF; 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(µ , µ ) with µ2 and µ2 unramiﬁed and µ2µ 2(x) , x , 1 2 1 2 1 2− | | 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 GF, 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 GF 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 GF. 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... 9

1 x 3.1.1 W 0 1 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 GF, 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 char- x 2 acter of (F ) xZ2 determined by the formula a2 0 π = ω (a2)I. (3.3.1) 0 a2! π

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 GF which, in addition to satisfying conditions (3.1.1)-(3.1.3), also satisfy the condition

1 x W z 0 1 g = µ(z)ψ(x)W(g), for z Z. (3.4.1) h i ∈ 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 Fx, 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 W(g) Wλ(g) = W g . (3.5.2) → "0 1# !

Here µλ denotes the character

1 λ 0 − λ 0 µλ(z) = µ z (3.5.3)  0 1! 0 1!     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 Fx 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 Fx, we can “tensor” χ with ψ ψ the even (resp. odd) piece of r to obtain a representation rχ of G∗F, the On Shimura’s Correspondence for Modular Forms... 11

1 0 x semi-direct product of SL (F) with 2 : a F . Inducing up to GF 2 0 a ∈ produces an irreducible admissible representationn whicho 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

ψλ r λ Λ, { } ∈ with Λ an index set for the cosets of (Fx)2 in Fx.

4.2 Each rχ is a distinguished representation of GF. 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) = χµ , { ψ } x with µψ the projective character of F deﬁned by γ(ψ) µ (a) = (4.2.1) ψ γ(ψa) x 2 We note that the restriction of µψ to (F ) 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 1/2 1/4 1/2 1/4 irreducible subrepresentation of π(χ − , χ ). | |F | |F 11 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 repre- { } sentations of G. When F is non-archimedean and of odd residue characteristic, the supercuspidal part of this conjecture is established in [Meister]; the non- supercuspidal part is treated in [Ge PS2]. 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 GF, and x χ is any quasi-character of F . 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

s Ψ(s, W, Wχ, Φ) = W(g)Wχ(g) det(g) Φ((0.1)g)dg. (5.1.1) Z | | N G \

Here W(g) is any element of W (π, ψ, µ), Wχ is any element of W 1 S (rχ, ψ− , χµψ 1 ), Φ (F F), and s C. Since W and Wx are − ∈ × ∈ genuine, and transform contravariantly under N, their product actually deﬁnes a function on N G. \ Similarly, we deﬁne

s 1 Ψ(s, W, Wχ, Φ) = W(g)Wχ(g) det g ω− (det g)Φ((0, 1)g)dg Z | | ∗ e N G \ (5.1.2) with

ω = µχµψ 1 . (5.1.3) ∗ − Note that ω is an ordinary character of Fx whose restriction to (Fx)2 is ∗ χωπ.

12 5.2 For Re(s) suﬃciently large, and g in GL2(F), the integrals

s 2s det g Φ((0., t)g) t ω (t) dt = fs(g) (5.2.1) | | Z | | ∗ Fx On Shimura’s Correspondence for Modular Forms... 13 and

s 1 2s 1 det g ω− (det g) Φ((0, t)g) t ω− (t) dt = hs(g) (5.2.2) | | ∗ Z | | ∗ Fx converge and deﬁne elements in the space of the induced representations 1 1 ρ(s 1/2, (1/2 s)ω− ) and ρ(ω− (s 1/2), 1/2 s) respectively. Cf. − − ∗ ∗ − − [Ja], 14. Moreover, for such s, the integrals deﬁning Ψ and Ψ converge. e Ψ(s, W, Wχ, Φ) = W(g)Wχ(g) f (g)dg (5.2.3) Z NZ G \ and

Ψ(s, W, Wχ, Φ) = W(g)Wχ(g)h(g)dg Z NZ G e \ Modifying the methods of [Ja] we obtain :

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

Ψ(s, W, Wχ, Φ) Ψ(s, W, Wχ, Φ) and L(s, π, χ) e L(s, π, χ) are entire; e e

Ψ(1 s, W, Wχ, Φ) Ψ(s, W, Wχ, Φ) − = ǫ(s, π, χ, ψ) , (5.3.1) L(1 s, π, χ) L(s, π, χ) e − b with e Φ(x, y) = Φ(u, v)ψ(uy vx)dudv. " − b 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

λ 2 2 2 4s λ ǫ(s, π, χ, µ ) = ω (λ− )χ− (λ) λ − ǫ(s, π, χ, ψ , µ). (5.4.1) π | | 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”

ǫ(s, π, χ, ψ)L(1 s, π, χ) γ(s, π, χ, ψ) = − L(s, π, χ) e then the functional equation (5.3.1) takes the simpler form

Ψ(1 s, W, Wχ, Φ) = γ(s, π, χ, ψ)Ψ(s, W , W , Φ). − 1 2 e b 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), L(s, π, χ), and ǫ(s, π, χ, ψ) for most representations π. To compute the factors L and L we need to analyze the possible e poles of Ψ(s, W, W , Φ) and Ψ(s, W, W , Φ). To compute ǫ(s, π, χ, ψ) we χ eχ need to compute the functions Ψ and Ψ explicitly, for judicious choices e of W, W and Φ. χ e 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 = L(s, π, χ), for all χ.

On the other hand, if π = rν, thene

L(s, π, χ) = L(2s, χν), On Shimura’s Correspondence for Modular Forms... 15 and 1 1 L(s, π, χ) = L(2s, χ− ν− ) If χν is unramiﬁed,e

ǫ(2s, χν, ψ)ǫ(2s 1, χν, ψ)L(1 2s, ν 1χ 1) ǫ(s, π, χ, ψ) = − − − − 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 Fx; 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, µ2χ)L(2x 1/2, µ2χ), − 1 − 2 and 2 2 L(s, π, χ) = L(2s 1/2, µ− χ)L(2s 1/2, µ− χ) (6.2.1) − 1 − 2 If we setes = 2s 1/2, then ′ − 2 2 ǫ(s, π, χ, ψ) = ǫ(s′, µ1χ, ψ)ǫ(s′, µ2χ, 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 × 1 Wχ are normalized KF-ﬁxed vectors in W(π, ψ, µ) and W(rχ, ψ− ). Then

Ψ(s, W1, W2, Φ) = L(s, π, χ)  (6.2.3) Ψ(s, W1, W2, Φ) = L(s, π, χ)   and e e e ǫ(s, π, χ, ψ) = 1. 16 S. Gelbart and I. Piatetski-Shapiro

6.3 Suppose π is the special representation

2 2 1 1/4 π = π(µ , µ ), with µ µ− (x) = x , and µ (x) = ν(x) x 1 2 1 2 | |F 1 | |F Then L(s, π, χ) = L(2s, χν2),  2 1 L(s, π, χ) = L(2s, ν− χ− ),   and-if π(ν2) denotes thee special representation π(ν2 1/2, ν2 1/2) of | | | |− GL2(F), 2 ǫ(s, π, χ, ψ) = ǫ(s′, π(ν ) 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), −  1 1 1 1 L(s, π, χ) = L(2s 1, χ− ν− )L(2s, χ− ν− ),  −  and e 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 2 2 ρ(µ1, µ2). Then

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

L(s, π, χ) = 1 = L(s, π, χ), and e ǫ(s, π, χ, ψ) = ǫ(s, ωπχ, ψ) (s, χ, ψ) (6.6.1) ∈ x In (6.6.1), ωπ is the character of F deﬁned by the equation

2 ωπ(a) = ωπ(a ) (6.6.2)

7 A Local Shimura Correspondence

Suppose π is an irreducible admissible (genuine) representation of GF 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 GF a Shimura image of π if

7.1.1 the central character ωπ of π is such that

2 x ωπ(a) = ω (a ), a F ; π ∈ 7.1.2 for any quasi-character χ of Fx,

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

2 2 π = π(µ1, µ2) implies S (π) = π(µ1, µ2). In particular,

1/2 1/2 π = rν(ν( 1) = 1) implies (π) = π(ν , ν ). − − | |F | |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 2 2 The fact that S takes π(µ1, µ2) to π(µ1, µ2) 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 π = S (π) principal series principal series 2 2 π(µ1, µ2) π(µ1, µ2) special representation special rep π(ν 1/4, ν 1/4) Sp(ν2) | | | |− Weil rν special rep (ν( 1) = 1) Sp(ν) − − Weil rν one-dimensional rep (ν( 1) = 1) 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

ψ = ψv Yv 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 = GFv . Roughly speaking, the adelic metaplectic group GA 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 G = Gv(Kv). Yv e The metaplectic group GA is obtained by taking the quotient of G by 18 e Ze = ǫv Z : ǫv = 1 for all but an even number of v .  ∈ 2  Yv Yv  e     In particular, we can view GA as a group of pairs (h, ζ): h GA, ζ { ∈ ∈ Z , with multiplication given by 2}

(h1, ζ1)(h2, ζ2) = (h1h2, β(h1, h2)ζ1ζ2), 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 Z GA GA 1 → 2 → → → splits over the discrete subgroup

GF = GL2(F) is equivalent to the quadratic reciprocity law for F; cf. [Weil].

9 Automorphic Representations of Half-Integral Weight

9.1 Recall that GA is the quotient of Gv = GA by the subgroup Ze. v Q e e 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

V = Vv(ev). Ov

The resulting representation of GA in V given by e π = πv (9.2.1) Ov is trivial on Ze and deﬁnes an irreducible admissible representation of G . A e Conversely, suppose π is an irreducible unitary representation of GA. 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 (Ax)2 trivial on (Fx)2. Proceeding as in §10 of [Jacquet-Langlands] we can introduce a space A(ω) of automorphic forms on GA. Each ϕ in A(ω) is a genuine C∞ function on GF GA which is “slowly increasing” and transforms under the center 19 \ (of GA) according to ω. The group GA 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

1 x ϕ0(g) = ϕ g dx = 0 Z "0 1# ! F A \

for each g in GA; (ii) the integral ϕ(g) 2dg Z | | 2 Z GF GA A \ 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 con- stituent of some A(ω) (resp. A0(ω)).

10 Fourier Expansions

Suppose ϕ is an automorphic form on GA, and ψ = Πψv is a ﬁxed non- trivial character of F A. \ 10.1 Since 1 x ϕ g "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 expan- ∞ \ sion in terms of the characters of F A. But each non-trivial character ψ \ of F A is of the form \ δ ψ′(x) = ψ (x) = ψ(δx) (10.1.1)

for some δ Fx. Thus ∈ 1 x ψδ ϕ g = ϕ0(g) + W (g)ψ(δx), (10.1.2) "0 1# ! ϕ δXFx ∈ with ψδ 1 x 1 Wϕ (g) = ϕ g ψ− (δx)dx. (10.1.3) Z 0 1! ! F A \ 20 On the other hand, it is easy to check that

δ δ 0 Wψ (g) = Wψ g . (10.1.4) ϕ ϕ 0 1! ! Thus we also have

ψ δ 0 ϕ(g) = ϕ0(g) + W g . (10.1.5) ϕ 0 1! ! δXFx ∈ In other words—modulo its constant term–ϕ(g) is completely determined by its ﬁrst Fourier coeﬃcient

ψ 1 x Wϕ (g) = ϕ g ψ( x)dx. (10.1.6) Z 0 1! ! − F A \ We call this function a ψ-Whittaker function since

1 x W g = ψ(x)W(g), x A. "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 half- ⊗ integral 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 ZA 2 \ whose restriction to ZA agrees with ωπ. Then each ϕ in Vπ has a Fourier expansion of the form

ψδ,µ ϕ(g) = ϕ0(g) + Wϕ (g) (10.2.1) µ XΩ(ω ) δXFx ∈ π ∈ with

ψδ,µ 1 x 1 1 Wϕ (g) = ϕ Z g ψ− (δx)µ− (z)dxdz. (10.2.2) Z Z 0 1! ! 2 F A Z ZA A\ \ The (ψ, µ) reﬁnement of (10.1.4) is

δ,µ δ δ 0 Wψ (g) = Wψ,µ g (10.2.3) ϕ ϕ "0 1# ! where 1 δ δ 0 − δ 0 µ (z) = µ z , z ZA. "0 1# "0 1# ∈     Note that for any µ,   21

ψ,µ 1 x W z g = ψ(x)µ(z)W(g), z ZA, x F. (10.2.4) ϕ 0 1! ! ∈ ∈

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 sup- ψ,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(ω ) and ϕ V , then π ∈ π (10.2.3) implies

ψ,µ δ 0 ϕ(g) = ϕ0(g) = W g , (10.3.1) ϕ "0 1# ! δXFx ∈

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 δ 0 ϕ(g) = W g), 0 1! δXFx ∈ and we have: Theorem 10.3.2. Every distinguished cuspidal representation of half- integral weight occurs exactly once in A0.

10.4 Let us explain the classical significance of a distinguished cusp form. Suppose f (z) = a(n)e2πinz X 22 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, 2 if T(p ) f = ωp f , then ωp 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 ωp’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 < (Fx)2, then δ,µ δ δ δ 0 Wψ (g) = Wψ ,µ(g) = Wψ,µ g = 0. ϕ ϕ ϕ 0 1! ! In classical terms, if ϕ corresponds to the form f (z), then

2 f (z) = a(δn2)e2πiδn z Xδ Xn 2 2 2πiδ0n z = a(δ0n )e Xn For more details, see [Ge PS2] and [Shim]. Examples of distinguished automorphic representations will now be described.

11 Theta-Representations

x x 11.1 Suppose χ = χv is any character of F A . Since almost every v \ χv is unramiﬁed, weQ can deﬁne an irreducible admissible representation of GA through the formula

rχ = rχ , ⊗ v 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 − ∞ 2πin2z θχ(z) = χ(n)e nX= −∞ 1 2 deﬁnes a “distinguished” modular form of weight 2 , level 4N , 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 half- ⊗ integral weight and χ = χv is a grossencharacter of F. Having in- v troduced L and ǫ factors forQ each πv and χv, we want to prove that the product L(s, π, χ) = L(s, πv, χv) Yv converges in some half-plane, continues to a meromorphic function in C, and satisﬁes a functional equation of the form

L(s, π, χ) = (s, πv, χvψv) L(1 s, π, χ).  ∈  − Yv    e 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 ZA according to the character ωπ, and suppose π occurs in the space Vπ in A (ω ). If ϕ V then 0 π ∈ π δ δ 0 ϕ(g) = Wψ,µ g (12.1.1) ϕ 0 1! ! Xµ δXFx ∈

Recall that the first summation extends over all characters µ′ of ZF ZA 2 \ whose restriction to ZA is ωπ. Now fix µ = µv in Ω(π, ψ), Ov ψ,µ and fix the embedding Vπ so that Wϕ (g) , 0. Given by character

χ = Πχv

x x 1 of F A , let µχ denote the unique element of the singleton set Ω(rχ, ψ ). \ − 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= Φv is in S (A A), set v × Q

Φ s 2s x Fs(g) = Fs (g) = det g Φ((0, t)g) t ω (t)d t, (12.2.1) | | Z | | ∗ Fx with ω the (ordinary) character of Fx Ax 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 ≫

Fs = Π fs,v 28 S. Gelbart and I. Piatetski-Shapiro

1 1 1 in the induced space ρ s 2 , 2 s ω− . Moreover, the series − − ∗ Fs(γg) = E(g, F, s)

γ BXF GF ∈ \

converges for Re(s) 0, and deﬁnes an automorphic form on GL2, ≫ s 1 the Eisenstein series E(g, F, s); cf. p. 117 of [Ja] (taking µ1 = α 2 , 1 a 1 µ2 = α 2 − ω− ). ∗ 25 Remark 12.3. E(g, F, s) extends to a meromorphic function in C with functional equation

E(g, FΦ, s) = E(g, FΦ, 1 s); (12.2.1) b − here, 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. − b b Prop.R 19.3 of [Ja]. We also know that the only poles of E(g, F, s) are 2 2s 2s 1 simple, and occur for A− = ω and A = ω− . | | ∗ | | ∗

12.4 Given π, ψ, µ, χ, and Fs as above, we deﬁne our zeta integral by the equation

ψ∗(s, ϕ, θχ, F) = ϕ(g)θχ(g)E(g, F, s)dg. (12.4.1) Z 2 Z GF GA A \ Here ϕ V , and ∈ π

1 ψ− ,µχ δ 0 θχ(g) = W g + θ0(g) (12.4.2) "0 1# ! δXFx ∈

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 Fs :

(i) the function ψ∗(s, ϕ, θχ, F) extends to a meromorphic function in C with functional equation

Φ Φ ψ∗(1 s, ϕ, θχ, F ) = ψ∗(s, ϕ, θχ, F ). (12.5.1) − b

(ii) All poles of ψ∗ are simple, with residues proportional to

s0 det g ϕ(g)θχ(g)dg. (12.5.2) Z | | 2 Z GF GA A \

(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 s . In other words, the bilinear form on V Vr deﬁned by 0 π × χ

s0 (ϕ, θχ) det g ϕ(g)θχ(g)dg → Z | | 2 Z GF GA A \

s0 is not identically zero, and π is equivalent to rχ. Since rχ r 1 , | |A ⊗ ≈ χ− this contradicts our hypothesis. e e

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 gen- erality, this Euler product decomposition is sketched in [PS]. To treat the explicit case at hand, we assume that the “ﬁrst Fourier coeﬃcients” 30 S. Gelbart and I. Piatetski-Shapiro

1 ψ,µ ψ ,µχ Wϕ (g) and W − of ϕ(g) and θχ (cf. (12.1.1) and (12.4.2)) are of the form ψ,µ Wϕ (g) = Wv(g) Yv and 1 ψ− ,µχ W (g) = Wχv (g), Y 1 with Wv W (πv, ψv, µv) and Wχ (g) W (rχ , ψ ). ∈ v ∈ v − Φ Proposition 13.2. With ϕ, θχ, and F as above, and Re(s) 0, s ≫ Φ ψ∗(s, ϕ, θχ, F ) = Ψ(s, Wv, Wχv , Φv). Yv (Recall the local zeta-functions Ψ are deﬁned by (5.1.1).)

Proof. Replacing E by the series deﬁning it, we have

Φ ψ∗(s, ϕ, θχ, F ) = ϕ(g)θχ(g)E(g, F, s)dg. Z 2 Z GF GA A \

Φ = ϕ(g)θχ(g)F (g)dg Z s 2 Z BF GA A \

0 a x Setting B = ZN = 0 a , we may write

1 ψ− ,µχ θχ(g) = θ0(g) + W (bg), 0 BXBF F \

ψ∗(s, ϕ, θχ, F) = ϕ(g)θ0(g)Fs(g)dg Z 2 Z BF GA A \ 1 ψ− ,µχ + ϕ(g)W (g)Fs(g)dg (13.2.1) Z 2 0 Z B GA A F \ 27 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)Fs(g) is left NA-invariant, and ϕ(g) is a cuspidal. Thus we have

1 ψ− ,µχ ψ∗(s, ϕ, θχ, F) = ϕ(g)W (g)Fs(g)dg Z 2 0 Z B GA A F \

= I(g)Fs(g)dg (13.2.2) Z BA GA \ where ψ 1,µχ I(g) = ϕ(bg)W − (bg)ω∗(b)db Z 2 0 Z B BA A F \ and ω∗ is the character of BA deﬁned by 2s a1 x a1 1 ω∗ = ω− (a2). 0 a2! a2 ∗

To continue, we compute

ψ 1,µχ I(g) =  ϕ(b′bg)W − (b′bg)ω∗(bb′)db′ db Z  Z  0  2 0 0  B BA Z B B  A\  A\ F A     ψ 1,µχ ψ,µ  = ω∗(b)W − (bg) W ′ (bg) Z 0 Xµ Xδ B BA A\

δ 1  ψ (x)ψ( x)(µ)− (z)µ′(z)dxdz db  Z Z −   2  Z ZA FA   A\    But the integral in parenthesis is zero unless δ = 1 and µ′ = µ, in which 28 case it equals 1. Thus we have

1 ψ,µ ψ− ,µχ I(g) = ω∗(b)Wϕ (bg)W (bg)db. Z 0 B BA A\ 32 S. Gelbart and I. Piatetski-Shapiro

Plugging this expression into (13.2.2) gives

ψ,µ 1 ψ (s, ϕ, θ , F) =  W (bg)Wψ− ,µχ(bg)FΦ(bg)db dg ∗ χ  ϕ s  Z  Z  BA GA B0 B  \  A A   \   1  ψ,µ ψ− ,µχ = Wϕ (g)W (g)Fs(g)dg Z NAZA GA \ ψ 1,µχ So taking into account the inﬁnite product expression for Wϕ, W − , and Fs = Π fs,v, we obtain the desired Euler product expansion for ψ∗.

Theorem 13.3. Suppose π is any cuspidal representation of half-integral x x weight. If χ = Πχv is any character of F A set \ L(s, π, χ) = L(s, πv, χv) Yv and L(s, π, χ) = L(s, πv, χv). Yv Then e e

(i) these inﬁnite products converge in some half-plane Re(s) > s0; (ii) L and L extend meromorphically to all of C, are bounded in vertical strips of ﬁnite width, and satisfy the functional equation e L(s, π, χ) = ǫ(s, π, χ)L(1 s, π, χ) − with e ǫ(s, π, χ) = (s, πv, χv, ψv); Yv ∈ (iii) the only poles of L(s, π, χ) are simple, and these occur only if π is x x of the form rv for some character ν of F A . \ 1 2 i t ,v Proof. For almost every v, πv is of the form π(ν , ν ), with ν (x) = x i , v v v | | and t ti v t (independent of v) − 0 ≤ , ≤ 0 On Shimura’s Correspondence for Modular Forms... 33

Therefore, since 29

1 2 1 1 L(s, π(νv, νv)) = 2t ,v s′ 2t2,v s 1 q 1 − ! 1 q− − ′ ! − − − 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 , µ is trivial on Ox , and W , W and Φ are chosen so that v Fv v χv v

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)):

L(s, π, χ) = L(s, πv, χv) L(s, πv, χv) Yv S Yv

~~ǫ(s, πv, χv, ψv)L(1 s, πv, χv) = − Ψ(1 s, Wv, Wχ , Φv) Ψ Φ − v Yv S (1 s, Wev, Wχv , v) Yall v ∈ − e b e 34 S. Gelbart and I. Piatetski-Shapiro~~

~~= (s, πv, χv, ψv) L(1 s, πv, χv) ∈ − Yv S Yall v ∈ e = ǫ(s, π, χ)L(1 s, π, χ), − as was to be shown. e~~

~~30 Remark 13.4. Since L(s, π, χ) doesn’t depend on ψ (or µ), neither does the product ǫ(s, π, χ, ψ) = ǫ(s, πv, χv, ψv). Yv~~

~~14 A Generalized Shimura Correspondence~~

14.1 Suppose π = πv is an irreducible admissible (genuine) repre- ⊗ sentation of GA with central character ω , and π = πv is an irreducible π ⊗ admissible representation of GA. Then we say π is the Shimura image of π, and write π = S (π), if each πv = S (πv), i.e., each πv is the local Shimura image of πv.

~~Example 14.2. Suppose π = rχ, with χ = Πχv a grossencharacter of~~

~~F. Then for each v, S (rχv ) is deﬁned, and for almost every v, S (rχv ) is one-dimensional and class 1. The resulting representation~~

S (π) = S (rχ ) ⊗ v is always automorphic, by the criterion of [Langlands]. Our purpose now is to show that any unitary cuspidal representation of half-integral weight has an automorphic Shimura image, and this image is actually cuspidal if π , rν for any ν.

~~15 The Theorem~~

~~Theorem 15.1. Suppose π = πv is a unitary cuspidal representation of ⊗ half-integral weight. Then:~~

~~(i) S (π) = S (πv) exists; ⊗ On Shimura’s Correspondence for Modular Forms... 35~~

~~(ii) S (π) is automorphic, and is cuspidal if and only if π is not of the x x form rν for any character ν of F A . \~~

~~15.2 Proof. Because of Example 14.2, we may assume π , rν for any ν.~~

15.3 Fixing ψ, let T be the set of places where S (πv) = πν may not be defined. According to Section 7, T is precisely the set of finite places where πv is supercuspidal but not a theta-representation. For almost all v < T, πv is a class 1 representation of the form 1/2 1/4 1/2 1/4 π(µ1, µ2) (possibly of the form π(νv −v , ν v ). Thus S (πv) = 2 2 | | | | < π(µ1, µ2) is class 1 (though possibly one-dimensional) for v T, and we 31 can define T π = πv = S (πv), (15.3.1) Ov

~~T a representation of the restricted product G = Gv. v~~

to ti v to (15.3.3) − ≤ , ≤ To conclude that L(s, πT χ) extends to the L-function of an automorphic ⊗ cuspidal representation of GL (A) we need to know that L(s, πT χ) 2 ⊗ satisﬁes certain analytic properties. In particular, we need to exploit the relation between L(s, πT χ) and the Euler product L(s, π, χ). ⊗ 36 S. Gelbart and I. Piatetski-Shapiro

From Theorem ?? (and our assumption on π) we know that for any x x character χ = Πχv of F A , \ L(s, π, χ) = L(s, πv, χv) Yv and L(s, π, χ) = L(s, πv, χv) Y are entire functions, boundede in verticale strips of ﬁnite width, and such that

L(s, π, χ) = (s, πv, χv, ψv) L(1 s, π, χ). (15.3.4) Y ∈ − On the other hand, we also know from 6.6e that if χv is suﬃciently highly ramiﬁed for v T, ∈ T L(s, π, χ) = L(s, πv, χv) = L(s, π χ), v

and ωπ = Πωπv defines a grossencharacter of F. T Thus we know that for all χ = Πχv highly ramified inside T, L(s, π ⊗ χ) and L(s, πT χ 1) are entire functions, bounded in vertical strips of ⊗ − finite width, and such that e T L(s, π χ) = (s, πv χv) (s, ωπ χv, ψv) (15.3.6) ⊗  ∈ ⊗  ∈ v Yv

~~(i) πv extends to a cuspidal representation π which occurs in A0(ωπ), v~~

~~(ii) there are grossencharacters µ and ν of F such that πv extends a ⊗ quotient π of ρ(µ, ν) (with every component of π inﬁnite-dimensional).~~

~~It remains to show that the v-th component of π equals S (πv) for each v T, and that possibility (ii) can’t occur. ∈ 15.4 In either case, (i) or (ii), we know that~~

~~1 L(s, π χ) = (s, πv, ψv)L(1 s, π χ− ) (15.4.1) ⊗ Yv ∈ − ⊗ e for all grossencharacters χ. Therefore, by (15.3.4) and (15.3.6) we conclude that for all χ,~~

L(s, πv, χv) ǫ(s, πv, χv, ψv) L(1 s, πv, χv) = − (15.4.2) L(s, π χ ) ǫ(s, π χ , ψ ) 1 Yv T v v Yv T v v v L(1 s, πv χv− ) ∈ ⊗ ∈ ⊗ e − ⊗ x Now ﬁx v T and let χv denote an arbitrarye character of F . 0 ∈ 0 Choose χ = Πχv so that its v -th component is χv , but for each v 0 0 ∈ T v , χv is so highly ramiﬁed so that (6.6.1) holds. Then (15.4.2) 33 −{ 0} reads

ǫ(s, πv , χv , ψv )L(1 s, πv , χv ) 0 0 0 − 0 0 (15.4.3) L(s, π , χ ) ve0 v0 1 ǫ(s, πv χv , ψv )L(1 s, πv χ− ) = 0 ⊗ 0 0 − 0 ⊗ v0 L(s, πv χv ) 0 ⊗ 0 e Recall that v T implies πv is supercuspidal and not a theta rep- 0 ∈ 0 resentation. Therefore, by 6.1, L(s, πv , χv ) = 1 = L(1 s, πv , χv ), and 0 0 − 0 0 (15.4.3) implies e 1 L(1 s, πv0 χv− ) ǫ(s, πv , χv , ψv ) − ⊗ 0 = 0 0 0 (15.4.4) L(s, πv χv ) ǫ(s, πv χv , ψv ) e0 ⊗ 0 0 ⊗ 0 0 38 S. Gelbart and I. Piatetski-Shapiro

1 i.e. the quotient L(1 s, πv χ )/L(s, πv χv ) is monomial. Since χv − 0 ⊗ v−0 0 ⊗ 0 0 is arbitrary, it is easy to check this implies πv0 must be super-cuspidal. e (Indeed for other possible πv0 , we could choose χv0 so that the quotient s would be a rational function of q− ). Thus

~~1 L(s, πv χv ) = 1 = L(1 s, πv χ− ) 0 ⊗ 0 − 0 ⊗ v0~~

and (15.4.4) implies πv0 = S (πv0 ). e Remark 15.5. If the set T is non-empty–as we have assumed it is–our Main Theorem is already proved. Indeed in this case, we have just shown that S (πv) still exists for all v. Moreover, we have shown that πv = S (πv) must be supercuspidal for v T. Thus S (π) = S (πv) must ∈ ⊗ be cuspidal automorphic (since possibility (ii) in §15.3 implies πv is a quotient of ρ(µv, νv) for all v). On the other hand, if T is empty, then S (πv) exists a priori, for all v, but π = S (πv) is not a priori cuspidal. To complete the proof in this ⊗ case it remains to note that now

~~L(s, π χ) = L(s, π, χ) ⊗ and 1 L(s, π χ− ) = L(s, π, χ) ⊗ 34 for all grossencharacters χe. Thus, bye the well-known GL2 theory, π must be cuspidal.~~

~~15.6 Corollary (Existence of a Generalized Shimura Correspondence) Given any cuspidal representation π of half-integral weight over F, there exists an automorphic representation~~

~~π = S (π)~~

~~of GL2(AF) such that for any grossencharacter χ of F,~~

~~L(s, π χ) = L(s, π, χ), ⊗~~

~~and S (π) is cuspidal if and only if π is not of the form rν for any ν. On Shimura’s Correspondence for Modular Forms... 39~~

We call S : π π the Shimura map. Its “kernel”—those cuspidal π → which map to non-cuspidal π—consists precisely of those π which come from automorphic forms on GL(1). That a similar situation arises with the lifting of cusp forms from GL(2) to GL(3) (cf. [Ge Ja]) cannot be coincidental.

Remark 15.7. Though we have not written down all the details, it seems likely we can prove that the L and ǫ factors of πv (and their twists by χv) completely determine πv. From this it follows that

~~(i) the Shimura map S : π π is 1-to-1; and →~~

~~(ii) strong multiplicity one holds for A0(ω).~~

~~Apparently similar (and even stronger) results have been obtained by Flicker using the trace formula. Thus we shall not pursue these questions further.~~

Corollary 15.8. (A Weak Ramanujan-Peterson Theorem for G). If π = πv is a unitary cuspidal representation of half-integral weight, and v is ⊗ a complex place, then πv cannot belong to the “outer half” of the com- plementary series for Gv = GL2(C)xZ2; cf. [Ge] Section 4, especially p. 85.

Proof. Suppose v is complex, and πv = πv(µ1, µ2) is as above. Then 35 2 2 S (πv) = π(µ1, µ2) is no longer unitary, contradicting the unitarity of the cuspidal representation S (π) = S (πv) in A0(ωπ). Nv

~~16 Applications and Concluding Remarks~~

~~16.1 In [Ge PS2] we treat the following Corollaries to our Main The- orem 15.1: 40 S. Gelbart and I. Piatetski-Shapiro~~

~~Theorem A. If π is a distinguished cuspidal representation of half- integral weight then~~

~~π = rχ for some grossencharacter χ of F. The classical interpretation of Theorem A is as follows. Suppose~~

~~∞ f (z) = a(n)e2πinz Xn=1~~

~~is a cusp form of weight k/2 which is “distinguished”, i.e, there is a square-free integer t such that a(n) = 0 unless n = tm2 for some m. Then f must be of weight 1/2 or 3/2 and of the form~~

~~∞ ν 2πitn2z f (z) = χ(n)n e = θχ(tz) nX= −∞ for some Dirichlet character χ (with χ( 1) = ( 1)ν). In this form the − − result was ﬁrst established in [Vigneras].~~

Theorem B. Suppose π = πv is a cuspidal representation with the ⊗ property that for at least one place v , π = r with χ an even char- 0 v χv0 v0 x acter of Fv0 . Then π = rχ for some grossencharacter χ of F. Since this theorem results immediately from the existence (and local properties) of the Shimura correspondence it also appears in the recent work of Flicker’s already alluded to.

36 Corollary . Suppose π = πv is a cuspidal representation “of weight ⊗ 1/2”, i.e., for at least one archimedean place v0, πv0 is the “even” piece of the Weil representation. Then there exists a grossencharacter χ of F such that

~~π = rχ. On Shimura’s Correspondence for Modular Forms... 41~~

~~In particular, taking F = Q we obtain an alternate proof of the fact that linear combinations of the theta-series~~

∞ 2πin2tz θχ(tz) = χ(n)e nX= −∞ exhaust the modular forms of weight 1/2 (as χ runs through the set of primitive “even” Dirichlet characters); this result is the principal theorem of [Se-St]. Concluding Remarks.

(i) The Shimura image of any cusp form of weight k/2, k 5, must ≥ again be a cusp form. Indeed it cannot be of the form θχ, and The- orem 15.1 implies that only the θχ’s can map to non-cusp forms. By the same token, if f is a cusp form of weight 3/2, it is mapped to a cuspidal form of weight 2 iﬀ it is orthogonal to the space spanned by θχ’s. This settles the ﬁrst conjecture of problem (C) on p. 478 of [Shim]; the Corollary to Theorem B settles the second.

(ii) In [Flicker] the image of S is characterized and a multiplicity one result is obtained for the full cuspidal spectrum of GA. This resolves question (A) of [Shim], p. 476, and vastly improves our own Theorem 10.3.2.

Appendix We reformulate the “almost everywhere converse theorem” of [Jacquet-Langlands] and [Weil 2]. The hypotheses below are slightly stronger than those of [Jacquet-Langlands], but they are quite tractable and seem to suﬃce for applications; cf. [PS 2] for best possible results.

~~Hypothesis. Suppose we are given: 37~~

~~(i) a non-trivial character ψ = Πψv of F A, and a character x x \ η = ηv of F A ; η \ Q 42 S. Gelbart and I. Piatetski-Shapiro~~

~~(ii) a ﬁnite set of ﬁnite places T, and an irreducible admissible rep- T resentation π = πv of Gv satisfying the following condi- Nv~~

~~T (a) the central character of π is ηv, Nv~~

~~t t ωv < µv(ωv) < ωv − | | | | | | t t ωv < νv(ωv) < ωv − |e | | e | |e |~~

(here t > 0 is a reale numbere independente of v, and ωv is a local uniformizing variable at v); and e (c) for any grossencharacter χ = χv, sufficiently highly ram- v ified inside T, the infinite productsQ

~~L(s, π, χ) = L(s, πv χv) ⊗ Yv~~

~~T T 1 L(s, π , χ) = L(1 s, π , χ− ) (s, πv χv, ψv) − ×  ∈ ⊗  Yv~~

~~T (i) π = πv extends to a cuspidal representation π in A0(η), or Nv~~

~~(ii) there exist grossencharacters µ and ν of F, with µν = η, such that πT extends to an automorphic representation π, with π a quotient of ρ(µ, ν).~~

~~Bibliography~~

[Flicker] FLICKER, Y., “Automorphic forms on covering groups of 38 GL(2)”, Inventiones mathematicae, 57, pp. 119–182 (1980). [Ge] GELBART, S., Weil’s Representation and the Spectrum of the Metaplectic Group, Springer Lecture Notes, No. 530, 1976. [Ge HPS] ———-, R. HOWE, and I.I., PIATETSKI-SHAPIRO, “Uniqueness and Existence of Whittaker Models for the Metaplec- tic Group”, Israel J. Math., 34, pp. 21–37 (1979). [Ge Ja] GELBART, S., and H. JACQUET, “A Relation between Auto- morphic Representations of GL(2) and GL(3)”, Ann. Ecole Nor- male Superieure, 4e serie, t. 11, 1978, p. 471–542. [Ge PS] GELBART, S., and I. I. PIATETSKI-SHAPIRO, “Automor- phic L-functions of half-integral weight”, Proc. N.A.S., U.S.A., Vol. 75, No. 4, pp. 1620–1623, April 1978. [Ge PS2] ————————, “Distinguished Representations and Modular Forms of half-integral weight”, Inventiones Mathemati- cae, 59, pp. 145–188 (1980). [Ge Sa] GELBART, S. and P. J. SALLY, “Intertwining Operators and Automorphic Forms for the Metaplectic Group”, Proc. N.A.S., U.S.A., Vol. 72, No. 4, pp. 1406–1410, April 1975. [Ho] HOWE, R., “θ-series and automorphic forms”, in Proc. Sym. Pure Math., Vol. 33, 1979. [Ho PS] ———-, and I. I. PIATETSKI-SHAPIRO, “A Counterexample to the Generalized Ramanujan Conjecture”, in Proc. Symp. Pure. Math., Vo. 33, A.M.S., 1979. 44 Bibliography

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~~By G. Harder~~

Introduction 41 Our starting point is a very general question. Let Γ be an arithmetic subgroup of a reductive Lie group G . Then the group Γ acts on the symmetric space X = G /K where ∞K G is a maximal compact ∞ ∞ ∞ ⊂ ∞ subgroup. Since X is contractible one knows that the rational cohomology and homology groups of Γ are isomorphic to the (co) homology groups of the quotient Γ X, i.e. \ Hν(Γ, Q) Hν(Γ X, Q) ≃ \ (Comp. [21], 1.6.). In general the quotient space Γ X is not compact. Borel and Serre \ have constructed a natural compactiﬁcation Γ X ֒ Γ X where Γ X is \ → \ \ a manifold with corners and where the inclusion is a homotopy equivalence. (Comp. [3]). In various papers it has been shown that we can construct cohomology classes on Γ X by starting from cohomology classes \ on the boundary. Roughly speaking we associate to a cohomology class ψ on the boundary an Eisenstein series E(ψ, s) which is a diﬀerential form depending on a complex parameters s. For a special value sψ of our

46 Period Integrals of Cohomology Classes... 47 complex parameter this form may become a closed form. This closed form represents a cohomology class and its restriction to the boundary is related to our original class ψ([7], [8] and [18]). We look at this as a procedure to construct cohomology classes on Γ X. \ On the other hand we have another construction which gives us homology classes. To get these homology classes we start from lower dimensional reductive subgroups M ֒ G for which ΓM = Γ M ∞ → ∞ ∩ ∞ is an arithmetic subgroup. If XM is the corresponding symmetric space we get a map ΓM XM Γ X. We even can ﬁnd cases where ΓM XM is \ → \ \ compact and then the fundamental class of ΓM XM gives us a homology \ class on Γ X. Our problem is to ﬁnd situations where the dimension \ of ΓM XM —which is also the dimension of the homology class–equals 42 \ the dimension of an Eisenstein class. If this is the case we can ask for the value of the Eisenstein class on the above homology class which amounts to evaluating the integral

E(ψ, sψ) Z ΓM XM \ This idea of constructing cycles by means of subgroups M ֒ G ∞ → ∞ appears already in [2] and [16]. In this paper we shall not consider the general problem but only a very special example. We take the group G = PGL2(C) and Γ will be a ∞ member of a very specific class of congruence subgroups of PGL2(Z[i]). If γ Γ and if γ is not unipotent then it generates a quadratic field ∈ extension E(γ) in the matrix ring M2(Q(i)) which defines a reductive subgroup in PGL (C). Then the quotient ΓM XM in this case will simply 2 \ be a circle and we shall compute the integrals of Eisenstein classes over these circles. It will turn out that these period integrals are expressible in terms of values of L-functions with Grossencharaktere of type Ao. The results are stated in section ??. Actually we have much more general results. We have a clear picture for those arithmetic groups which come from the group GL2 over an arbitrary algebraic number field. It is planned to write a paper in which we treat this more general situation. But it is clear that this paper 48 G. Harder

will be very long, very diﬃcult to write and certainly also not easy to read. For instance we shall have to use adeles, we have to introduce co- eﬃcient systems and so on. That paper will contain proofs of the results announced in [7] and the results in there have to be generalized. There- fore I made up my mind and decided to write a paper where all this is discussed in a special case. I tried to give many details which will cause some repetition and overlap with older papers and the one planned. But the degree of complexity in the general situation is very high and I think it might be useful to discuss one special case. During the preparation of this paper here I became aware that also the theory of Eisenstein classes which has been announced in [7] has some interesting arithmetic aspects. We shall devote a large part of this 43 paper to recall the theory of these Eisenstein classes and to discuss these arithmetic aspects which also concern values of some L-series. There- fore the title of the paper is not quite appropriate. I want to thank D. Zagier for several discussions and for pointing to me how to compute the period integrals by a method that goes back to E. Hecke. ([10], 200).

1.0 Some Notations. If R is any commutative ring with identity we denote its group of invertible elements by Rx. The field Q[i] will be denoted by F, throughout this paper we consider F as a subfield of C, i.e. we fix an embedding of F into C. The ring of Gaussian integers Z[i] Q(i) will be denoted by O. More general if ⊂ E is any algebraic number field, we denote by OE its ring of algebraic integers. The finite places of F will be denoted by p, q .... The finite places of an extension E/F will be denoted by capital letters P, Q .... We denote by EP the completion at P, by Op Fp the ring of p-adic integers and ⊂ by OE,P = OP the ring of P-adic integers. We drop the index E if it is clear which filed we refer to. = O x = O x We put UP P and UP P. The place of F at infinity will be denoted by and the completion F is canonically identified with C. ∞ ∞ Period Integrals of Cohomology Classes... 49

The ring of adeles of F is denoted by A and by the letter we denote the group ideles of F. If we refer to another filed E we write AE, IE. Elements of adele rings or idele groups will be denoted by underlined latin letters x, a, u, .... If x A then we write ∈ x = (x , ... , xp, ... , xq, ...) ∞ f f i.e. xp, xq are the p, q components. By A (resp I ) we denote the ring (resp. group) of finite adeles (finite ideles) where we drop the component at . Therefore ∞ A = C A f , I = Cx I f × × and for x A we write x f for its finite component, so that we have ∈ x = (x , x f ). ∞ By U f we denote the maximal compact subgroup of units in I f , i.e. 44 f f U = ΠpUp and then U = U U is the maximal compact subgroup ∞ × in I, where U is the circle group. ∞ We start from the group Go/F = PGL2/F. Then Bo/F, Uo/F and To/F will be the standard Borelsubgroup of upper triangular matrices, its unipotent radical and the standard diagonal torus. Sometimes it will be convenient to look at Go/F as a group over Q, this means we put G/Q = RF/Q(Go/F) where RF/Q is the functor of restriction of scalars. ([27], 1.3.). For any group scheme H/A over any ring and any extension A A , → 1 we denote the group of points of H with values in A1 by H(A1).

~~1.1 The Cohomology of Γ and the space Γ X. \ Let us put~~

Γo = PGL2(O) = PGL2(Z[i]) = GL2(O)/Z im 0 where Z = m m Z/4Z . We have Γo PGL (C) and the group 0 i | ∈ ⊂ 2 Γo acts on then three dimensionalo hyperbolic space X = PGL2(C)/K where K is the projective unitary group SU(2)/centre = SO(3). We∞ choose the∞ standard embedding α β SU(2)= αβ C, αα + ββ = 1 SL2(C) ( β α! ∈ ) ⊂ −

~~50 G. Harder~~

~~We choose an ideal a O which has to satisfy one of the following ∈ conditions~~

~~a = ((1 + i)3) (1.1.1) or a is an odd prime where N(a) = p is a prime in Z and p . 1mod 8.~~

This condition (1.1.1) implies that the group W = O x = i, i 1, 1, 1 { − − } injects into the quotient (O/a)x and that i is not a square in (O/a)x. Our main object of study are the congruence subgroups

~~a b a b Γ=Γ(a) = Γo = Id mod a ( c d! ∈ c d! )~~

~~45 this means that Γ is the kernel of the natural homomorphism~~

~~p Γo = PGL (O) PGL (O/a) 2 −→ 2 Lemma 1.1.2. The homomorphism p is surjective.~~

~~Proof. It is very easy to see that the map~~

~~SL (O) SL (O/a) 2 → 2~~

is surjective. The image of SL2(O/a) in PGL2(O/a) is of index 2 and O x O x 2 i 0 the factor group is ( /a) /(( /a) ) . Then we see that p 0 1 < image of SL2(O/a) and this proves the lemma. Let R be any ring in C. We want to assume always that the primes which divide the order of the ﬁnite group PGL2(O/a) are invertible in R. We are interested in the cohomology group Hν(Γ, R) and we can identify

Hν(Γ, R) Hν(Γ X, R) ≃ \ since Γ has no torsion, as one easily checks. First of all we want to summarize some basic facts and definitions of the cohomology theory. If M is a projective R-module on which we Period Integrals of Cohomology Classes... 51 have an action of the finite group G =Γo/Γ= PGL2(O/a) we can define the cohomology groups ν H (Γo, M) 1 We will mainly be concerned with H (Γo, M) and we recall the definition in this case: We write the action of G on M by (g, m) g m and define → · 1 Z (Γ , M) = Φ : Γo M Φ(γ γ ) = Φ(γ ) + γ Φ(γ ) 0 { → | 1 2 1 1 2 } This is the module of 1-cocycles. We have a map

δo 1 M Z (Γo, M) −→ δ : m γ (m γm) →{ → − } 1 1 and H (Γo, M) = Z (Γo, M)/δo(M). There is another way to define these cohomology groups: We look 46 at the projection π : X Γo X → \ and we define a sheaf M on Γo X as follows. For any open set U Γo X \ ⊂ \ we define e 1 m(γu) = γ m(u) and M(U) = m : π− (U) M · ( → m is locally constant. ) e It is well known that under the given assumptions we have ([21], 1.6). ν ν H (Γo X, M) H (Γo, M) \ ≃ Let us look at the special casee where M = R[G] is the group ring of the finite group G. In this case we have two actions of G on M namely by right and left multiplication

~~1 (g , g ): aγγ aγg γg− 1 2 → 1 2 Xγ G X ∈ We deﬁne the cohomology groups~~

~~1 H (Γo, R[G]) 52 G. Harder~~

~~by the module structure given by right multiplication; so if m = aγγ γ G ∈ P∈ R[G] and γ G we have 1 ∈ 1 γ1m = aγγγ1− = aγγ1 γ Xγ Xγ~~

~~The well known Lemma of Shapiro tells us that~~

~~1 1 H (Γ, R) H (Γo, R[G]) (1.1.3) ≃~~

and it is very easy to make this isomorphism explicit. If Φ : Γo R[G] → is a 1-cocycle and if we write Φ(γ) = Φσ(γ)σ. Then the cocycle σ G P∈ relation tells us that Φσ(γ ) + Φσγ (γ ) = Φσ(γ γ ) for all γ , γ 1 1 2 1 2 1 2 ∈ Γ and all σ G. If we restrict Φ to the subgroup Γ all the Φσ are ∈ homomorphisms. It follows from the cocycle relation that for γ Γ, ∈ η Γo and η = ηmod Γ ∈ 1 Φσ(ηγη− ) = Φση(γ)

47 This tells us that Φ1 determines the Φσ for σ , 1 and it is easy to see that Φ : Γ R is the image of the class represented by Φ and the 1 → Shapiro isomorphism (1.1.3). The group G acts on the cohomology groups H1(Γ, R) = H1(Γ X, R) \ where the action is induced by conjugation. On the other hand the action 1 of G by left multiplication induces an action of G on H (Γo, R[G]) and it is not hard to check that (1.1.3) commutes with these actions. This isomorphism (1.1.3) allows us to decompose the cohomology, we have R[G] = Mθ Mθ

where the Mθ are irreducible G G-modules. (Here we use our assump- × tion that 1/ G R). Then we get a decomposition | |∈ 1 1 1 H (Γ, R) = H (Γo, R[G]) = H (Γo, Mθ) Mθ Period Integrals of Cohomology Classes... 53

~~If we assume in addition that R contains enough roots of unity, then the Mθ will be absolutely irreducible and we get~~

Mθ = M Mδ δ ⊗ b where δ runs over the irreducible G-modules and δ is the contragriedient module. Therefore we get b 1 1 H (Γ, R) = H (Γo, Mδ) Mδ M ⊗ δ G ∈ b b and the action of G on the right hand side is trivial on the first factor and the given action on Mδ. b 1.2 The Compactification of Γ X and the Cohomology at Infinity \ It is well known that in this case the space Γ X is not compact. It \ has a finite number of cusps which are in one-to-one correspondence with the Γ-conjugacy classes of Borel subgroups B G/F. ([1]) Borel ⊂ and Serre developed a general theory of compactification of such spaces Γ X. They proved in [3] that we have a homotopy equivalence \ Γ X ֒ Γ X \ → \ where in this special case Γ X is a compact manifold with a boundary. \ The boundary components are in one-to-one correspondence with the Γ- 48 conjugacy classes of Borel subgroups, i.e. they correspond to the cusps. We want to give a precise description of all this in our special situation. Let B be any Borel subgroup defined over the group field F. Let U ⊂ B be its unipotent radical. It follows from the Iwasawa decomposition that the group B(C) acts transitively on X. The positive root defines a homomorphism α : B Gm → and from this we get a homomorphism

~~x α : B(C) Gm(C) = C → 54 G. Harder~~

We put (1) B (C) = b B(C) α(b) C = 1 { ∈ | | | } x where z C = zz for z C . The group | | ∈ B(C) K = B(1)(C) K = KB ∩ ∞ ∩ ∞ ∞ is a one dimensional circle and it is clear that we have a semidirect product B(1)(C) = U(C) KB · ∞ Therefore we have with xo = K G/K ∞ ∈ ∞ (1) (1) X = B (C) xo = U(C) xo X B · · ⊂ (1) and X U(C) C. If we put ΓB = B(C) Γ then we get a homotopy B ≃ ≃ ∩ equivalence (1) ΓB X =ΓB U(C) ֒ ΓB X \ B \ → \ (1) and the Borel-Serre theory gives us that ΓB X is diﬀeomorphic to the \ B boundary component YB of Γ X which corresponds to B ([3]). Since \ ΓB Z Z we get that YB is a product of two circles. ≃ ⊕ Remarks

~~(1) Our congruence condition guarantees that Γ B(C) = Γ U(C) ∩ ∩ since the image of Γ B(C) in B(C)/U(C) has to consist of units ∩ in O.~~

~~49 (2) To give the reader a better feeling for the Borel-Serre compactiﬁ- cation we add a few more comments.~~

~~We mentioned already that B(C) acts transitively on X, we use this fact to deﬁne the function~~

~~x hB : X R → t hB : x = bxo α(b) C → | | Period Integrals of Cohomology Classes... 55~~

We introduce the sets B X (c) = x X hB(x) c { ∈ | ≥ } and the reduction theory tells us ([1], and [5], 1.2.) that for c sufficiently large we have an embedding B ΓB X (c) ֒ Γ X \ → \ and using the geodesic action or the vector field dhB we find B (1) ΓB X (1) =ΓB X [1, ) \ \ B × ∞ The Borel-Serre compactification in this case simply consists of adding in the second factor ∞ B (1) (1) [ ,ΓB X (1) =ΓB X [1, ] ֒ ΓB X [1 \ \ B × ∞ → \ B × ∞ (1) and YB =ΓB X . \ B × {∞} The first part of the paper is devoted to the study of the map 1 1 1 1 H (Γ X, R) ∼ H (Γ X, R) H (∂(Γ X), R) ∼ H (YB, R) \ −→ \ → \ −→ MB where B runs over a set of representatives for the Γ conjugacy classes of Borel subgroups. The group ΓB is free abelian of rank 2 and therefore we have 1 2 H (YB, R) = Hom(ΓB, R) = R If we want to describe the cohomology of the boundary we have to describe the set of cusps or the set of Γ-conjugacy classes of Borel subgroups. This is very simple in this case since O has class number one. Actually we shall do a little bit better. We know that H1(O(Γ X), R) \ is a Γo/Γ= G-module and we give a description of this G-module. Since O has class number one it follows that the group Γo acts tran- 50 sitively on the set of boundary components. It is easy to see (and will also follow from considerations in section 1.3) that the stabilizer of the boundary component YBo is the group im u U+ = Uo W = u O/a, i = imod a · ( 0 1! ∈ )

~~56 G. Harder~~

~~im 0 1 where W = 0 1 m Z . The group U+ acts on H (YBo , R) and it ∈ follows from general principles of representation theory that we have an G-module isomorphism~~

~~1 ∼ G 1 H (∂(Γ X), R) Ind H (YBo , R) \ −→ U+ where the induced module is the space of functions~~

G 1 1 1 Ind H (YB , R) = h : G H (YB , R) h(gu− ) = uh(g) U o o + { → |  for g G and u U+ ∈ ∈   The group G acts on these functions by left translations.  It is easy to decompose this module into irreducible modules. We assume that R contains the (O/a)∗ -roots of unity. The group U+ = | 1 | Uo W and Uo acts trivially on H (YBo , R). Under the action of W we · 1 i 0 have a decomposition H (YBo , R) = Hom(ΓBo , R) = L+ L where 0 1 ⊕ − acts on L+ by multiplication by i and on L by multiplication by i. − − i 0 i 0 1+ = i1+; 1 = i1 0 1! 0 1! − − −

~~We look at the characters φ :(O/a)x S 1 for which φ(i) = i. For each → such character we have a subspace~~

~~1 − Mφ∗ = h : G L+ h(gb ) = φ(b)h(g) { → |  for b Bo and g G ∈ ∈   G G  and Ind L+ and analogously we deﬁne M∗ Ind L . This gives us U+ φ ⊂ U+ − a decomposition~~

~~G 1 1 Ind H (YBo , R) = H (∂(Γ X), R) = (Mφ∗ M∗) (1.2.1) U+ \ ⊕ φ φ:(OM/a)x S 1 φ(i)=→i~~

51 where the M and M are irreducible G-modules. (1.2.2 and [25], φ∗ φ∗ Cor. 4.11.) Here we proﬁt from the fact that φ cannot be a trivial or a quadratic character. Period Integrals of Cohomology Classes... 57

1.2.2 At this point I want to give an idea of one of the main questions of this paper. As we have seen already we can study the restriction map H1(Γ X, R) H1(∂(Γ X), R) \ → \ and we have decomposed the right hand side into irreducible modules (1.2.1). Let us assume that we have selected generators e+ L+ and ∈ e L (We shall see later that we have a rather canonical choice, see − ∈ − 1.6.1) then we can identify Mφ∗ with the induced representation 1 Mφ = ψ : G R ψ(gb− ) = φ(b)ψ(g) { → | } by mapping ψ g ψ(g) g e+ . One knows that Mφ and M are → { → · · } φ irreducible G-modules and they are isomorphic. The operator

Tφ : Mφ M → φ Tφ : ψ Tφψ(g) = ψ(wug) → uXUo ∈ 0 1 with w = 1 0 is a non zero interwining operator ([25], §5). Since there− are no other isomorphisms among these induced representations the decomposition (1.2.1) is isotypical. Let us denote the quotient ﬁeld of R by K. For any φ we pick the 1 isotypical component of Mφ in H (Γ X, K) and get a map \ 1 H (Γ X, K)φ Mφ K M K \ → ⊗ ⊕ φ ⊗ It follows from topological reasons that the image of the restriction 1 map is of multiplicity one (namely the multiplicity of Mφ K M K 2 × ⊗ ⊕ φ⊗ which is two) (comp. [20] 3.4). Therefore the image is of the form (Schur’s lemma)

(ψ, cφTφψ) ψ Mφ K Mφ K M K { | ∈ ⊗ }⊂ ⊗ ⊕ φ ⊗ where cφ K or cφ = in which case the image would be the second 52 ∈ ∞ component. What is the value of cφ? This problem will be attacked by transcendental methods, the theory of Eisenstein series will give us an expression for cφ in terms of values of L-functions. 58 G. Harder

1.2.3 Before I conclude this section I want to translate the questions and assertions 1.2.1 and 1.2.2 in the language of cohomology groups with coeﬃcients. We have the isomorphism (1.1.3) and we put Γo B = Bo(F) Γo. , o ∩ Now we want to give a detailed description of the diﬀerent isomorphisms in the following commutative diagram

~~1 1 1 Sh : H (Γo, R[G]) ∼ / H (Γ, R) H (Γ X, R) ≃ \ res 1 1 ∂Sh : H (Γo B , R[G]) ∼ / H (∂(Γ X), R) , o \ (1.2.3.1)~~

~~ (M M ) φ∗ ⊕ φ∗ φ:(∂/La)x S 1 φ(i)=→i~~

~~In this context it is convenient to identify the group ring R[G] with the ring of R valued functions on G which is denoted by C(G) and~~

~~C(G) ∼ R[G] −→ by f f (σ) σ → · σXG ∈ Now let us assume that~~

~~Φ : Γo C(G) →~~

~~is a 1-cocycle. Then for γ Γo the value Φ(γ) is a function of G and the ∈ value of this function at σ G will be denoted by ∈ Φ(γ)(σ)~~

~~If we restrict Φ to Γ then the map γ Φ(γ)(σ) is a homomorphism for → 53 any σ G and we have seen that Sh is given by ∈ Period Integrals of Cohomology Classes... 59~~

~~[Φ] γ Φ(γ)(1) →{ → } where [Φ] is the class deﬁned by Φ. The class [Φ] deﬁnes a class on the boundary and we get a family of homomorphisms~~

φB : ΓB =Γ B(F) R ∩ → φB(γB) = Φ(γB)(1) where B runs over a set of Γ conjugacy classes of Borel subgroups. If 1 we write ΓB = ηΓB η with η Γo then we get a homomorphism o − ∈ ΓB R o → 1 1 γo φB(ηγoη− ) = Φ(ηγoη− )(1) → But for γo ΓB = Bo(F) Γ we have ∈ o ∩ 1 Φ(ηγoη− ) = ηΦ(γo) = ηΦ(γo) where η is the image of η Γo in G. Therefore we have ∈ 1 Φ(ηγoη− )(1) = Φ(γo)(η) and this tells us that the cocycle Φ deﬁnes a map

~~hΦ : G Hom(ΓB , R) → o hΦ : σ γo (γo)(σ) →{ → }~~

This map is also deﬁned for cocycles on Γo,Bo with values in R[G] and the map [Φ] hφ gives us a direct realisation of ∂Sh and makes → the commutativity of the diagram clear. This means that the study of our restriction map can be reduced to the inverstigation of maps

1 1 H (Γo, M) H (Γo B , M) → , o where M is a projective R-module on which we have an irreducible G- action, i.e. M K is an irreducible G-module. Again we want to as- R sume that R containsN enough roots of unity. 60 G. Harder

~~1 We consider H (Γo,Bo , M). We have 54~~

~~t u x Γo,B = t O , u O =Γo,U W o ( 0 1! | ∈ ∈ ) o ·~~

1 u where Γo U = u O and W is cyclic of order four generated by , o 0 1 | ∈ i 0 n o 0 1 . We always identify W Γo B with its image in ⊂ , o t u x Bo = t (O/a) , u O/a . 0 1 | ∈ ∈ Since we assume that G is invertible in our ring R we see that the | | action of Uo on M is semisimple and it is obvious that

1 Uo H (Γo,Uo , M) = Hom(Γo,Uo , M ) where we have to take into account that MUo = MΓo,Uo . (The notation MUo means of course that we take the invariants). Therefore we can restrict our attention to those modules where MUo , (0). It is well known that in this case M has to be a submodule of an induced module 1 Nχ where χ is a character χ : Bo Bo/Uo S and → →

~~Nχ = f : G R f (bg) = χ(b) f (g) { → | }~~

Uo The module Nχ is easy to compute. We have the Bruhat decomposition 0 1 G = BowUo Bo with w = 1 0 and we put ∪ − bwy χ(b) fo = → b 0  → bwu 0 f = → ∞ b χ(b)  →  Uo  Then Nχ = R fo R f . The group G acts on Nχ by right translations, ⊕ ∞ Uo if we restrict this action to Bo, then Nχ is an invariant subspace and

~~1 b fo = χ(b)− fo, b f = χ(b) f ∞ ∞ Period Integrals of Cohomology Classes... 61~~

~~Since Γo B =Γo U W we have obviously , o , o ·~~

~~1 Uo W H (Γo,Bo , M) = Hom(Γo,Uo , M )~~

The group W acts on Γo,Uo by means of the adjoint action and the O i 0 module Γo,Uo decomposes into two spaces on which 0 1 W acts 55 ⊗ 1 ∈ by the eigenvalues i, i. Then it becomes clear, that H (Γo B , M) , 0 − , o if and only if χ(i) = i. We assume χ(i) = i ans we call this character ± φ again. So φ = χ, then we have that Nφ is irreducible ([25] 4.11.) and M = Nφ. We ﬁnd

~~Uo W W W Hom(Γo,Uo , N ) = Hom(Γo,Uo , R fo) Hom(Γo,Uo , R f ) φ ⊕ ∞ and~~

~~Hom(Γo,Uo , R fo) = Hom(Γo,Uo , R) = Hom(ΓBo , R)~~

~~Hom(Γo,Uo , R f ) = Hom(Γo,Uo , R) = Hom(ΓBo , R) ∞~~

~~But we have to keep in out mind that W acts non trivially on R fo, R f and it acts trivially on R. If we take up our earlier notations we ﬁnd∞~~

~~W Hom(Γo U , R fo) = L+ Hom(ΓB , R) , o ⊂ o W Hom(Γo,Uo , R f ) = L Hom(ΓBo , R) ∞ − ⊂ We constructed an identiﬁcation~~

~~1 H (Γo,Bo , Nφ) = L L = Re+ Re − ⊕ − ⊕ − if we take up the notations in 1.2.2. We look again at our restriction map~~

~~1 1 H (Γo, Nφ) H (Γo,Bo , Nφ) = Re+ Re → ⊕ − and we want to relate this to 1.2.2. 62 G. Harder~~

~~Let us pick the isotypical component R[G]φ in R[G] then we get~~

~~1 H (Γo, R[G]φ)~~

~~1 H (Γ, R)φ / M M = Mφ M φ∗ ⊕ φ∗ ⊕ φ~~

~~On the other hand we realized our given induced representation as a submodule of R[G]φ namely~~

~~Nφ ֒ R[G]φ → 56 Therefore we get a diagram~~

~~1 H (Γo, Nφ) / L+ L = Re+ Re _ ⊕ − _ _ ⊕ − λ λ 1 H (Γo, R[G]φ) / M M = Mφ M φ∗ ⊕ φ∗ ⊕ φ~~

~~and we have to compute the inclusions λ and λ1. To get these inclusions we observe that a generator of L+ is given by the cocycle~~

~~ΓU R fo Nφ o → ⊂ γ e+(λ) fo → · This deﬁnes (1.2.3.1) a map~~

~~h : σ λ e+(λ) fo(σ) →{ → · } h : G Hom(ΓU , R) → o~~

Now we observe that fo Nφ is also an element in Mφ and we see that ∈ λ1 : e+ fo Mφ and λ1 : e f Mφ. The intertwining operator → ∈ → ∞ ∈ Tφ : Mφ Mφ maps Tφ( fo) = N(a) f and we get the proposition. → ∞ Period Integrals of Cohomology Classes... 63

~~Proposition 1.2.4. The image of the restriction map~~

1 H (Γo, Nφ K) Ke+ + Ke ⊗ → − is spanned by the vector (e+, cφN(a)e ) where eφ K . − ∈ ∪ {∞} What is all this good for? If we want to compute explicitely with cocycles it seems to be convenient to work with Γo instead of Γ since it has less generators. We pay for it by introducing coeﬃcients. Later on 1 we shall compute H (Γo, Nφ) in some simple cases and we are then able to compute the number cφ.

~~1.3 Adeles and the Description of the Set of Cusps In the adele group Go(A) = PGL2(A) we have the maximal compact subgroup~~

f K = K K = K PGL2(Op) ∞ · ∞ × pYﬁnite where K = PU(2)(1.1). The ideal a deﬁnes a congruence subgroup 57 ∞ K f (a) K f namely ⊂ f K f (a) = k f K f k 1mod a { ∈ | ≡ } Lemma 1.3.1.. Every element x Go(A) can be written ∈ x = a (y , k f ) · ∞ with k f K f (a). ∈ Proof. We represent x Go(A) by an element x GL (A). If x ∈ ∈ 2 ∈ SL2(A), i.e. det(x) = 1 then the assertion follows from strong approxi- e e mation for SL2. We may modify x by an element z I which we con- e ∈ sider as an element of the center of GL2(A), then det(x) gets multiplied by z2. So the obstruction to get thee determinant x equal to one sits in 2 e I/I . We may modify x by an element in GL2(F) and by an element in f e the inverse image of K (a) in GL2(A). This means that the obstruction against writing x in thee above form lies in

~~I/I2 Fx U f (a) · · 64 G. Harder~~

~~where U f (a) = t U f t 1mod a . Using the fact that F has class { ∈ | ≡ } number one we ﬁnd that this group is equal to~~

~~Cx U f /U f (a) W (U f )2 × · · where W = i Fx. Since nothing is claimed at the inﬁnite place we { } ⊂ may drop the inﬁnite component and the obstruction sits in~~

~~(O/a)x/((O/a)x)2 W = 1. · The lemma is proved. The lemma says simply that~~

~~f Go(F) Go(A)/Go(C) K (a) = 1 . \ × { } Now we consider the double coset decomposition~~

~~f f Bo(F) Go(A )/K (a) \ Let us write f f Go(A ) = Bo(F)ξK (a) [ξ~~

~~We extent ξ to an element ξ′ of Go(A)by(ξ′) = 1. According to our ∞ 58 previous lemma we may write~~

1 f ξ′ = a (a− , k ). · f f 1 with k K (a). Then a Boa = B is a Borel subgroup over F. We ∈ − see that the Γ-conjugacy class of this Borel subgroup depends only on ξ. If we pick a Borel subgroup B/F then we ﬁnd an a Go(F) such 1 ∈ that B = a− Boa. We choose ξ′ = (1, a, ... , a, ...) and therefore we get a bijection between the set of Γ-conjugacy classes of Borel subgroups B/F of G/F and the set of double cosets

~~f f Bo(F) Go(A )/K (a) \ Period Integrals of Cohomology Classes... 65~~

~~We are now able to settle the minor point left open in 1.2 concerning the stabilizer of the boundary component YBo under the action of G. We simply count the number of cusps. We have a map~~

f f f f f Bo(F) Go(A )/K (a) Bo(A ) Go(A )/K (a) \ → \ which is surjective. f f f Since Bo(A ) K = Go(A ) we get · f f f f f Bo(A) Go(A )/K (a) = K /K (a) Bo(A ) = G/Bo \ ∩ The ﬁbers of this map are equal to

B(F) B(A f )/B(A f ) K f (a) \ ∩ where B is any Borel subgroup corresponding to a point in the fiber. Since the unipotent radical has strong approximation we find for these fibers that they are equal to

~~I f /Fx U f (a) = U f /WU f (a) = (O/a)x/W · and that proves that the number of cusps is equal to [G : U+]~~

1.4 Diﬀerential Forms and De Rham Cohomology We should look at Go/F as group over the rationals and therefore we introduce G/Q = RF Q(Go/F). The Lie algebra g = Lie(G/Q) is a Q-vector space and \ we deﬁne g = g R. Then g is the Lie algebra of the real group ∞ ∞ NQ G = G(R) = Go(C). (We shall sometimes denote the group of complex ∞ points of groups over F by the subscript and then we stress the point 59 ∞ of view that they may also be considered as real points of a group over (Q). Now α β g = α, β, γ C ∞ ( γ α! ∈ ) − and the Cartan involution obtained from our given maximal compact group is α β α γ θ : X = tX = − − γ α! →− β α ! − − 66 G. Harder

~~We get the Cartan decomposition~~

g = k + p ∞ ∞ where k = Lie(K ). The real vector space p has the basis ∞ ∞ 1 0 0 1 i 0 H = , E1 = , E2 = 0 1! 1 0! 0 i! − − The group K = PU(2) = S U(2)/ +Id acts on p by the adjoint action. ∞ { } α β If k PU(2) is represented by the matrix β α S U(2) then ∞ ∈ − ∈

ad(k )H = (αα ββ)H 2Re(αβ)E1 2Im(αβ)E2 ∞ − − − 2 2 2 2 ad(k )E1 = 2Re(αβ)H + Re(α β )E1 + Im(α + β )E2 (1.4.1) ∞ − 2 2 2 2 ad(k )E2 = 2Im(αβ)H = Im(α β )E1 + Re(α + β )E2 ∞ − The normalised Killing form on g ∞ 1 X, Y = trace (ad X ad Y) h i 16 · induces a K invariant, positive deﬁnite symmetric quadratic form on p. ∞ With respect to this form our three vectors H, E1, E2 form an orthonormal basis. We shall use this form to identify the space p with its dual space. The projection map

~~π : G G /K = X ∞ → ∞ ∞ deﬁnes an isomorphism~~

~~(dπ)e : p Tx = Tˇx → o o~~

between p and the tangent space of X at the point xo. 60 This allows us to identify the space of diﬀerential p-forms on X with a certain space of ΛpP-valued functions on the group G . To be more ∞ Period Integrals of Cohomology Classes... 67

~~p precise we can identify the space Ω (X) of C∞-p-forms on X and the space of C∞-functions~~

p p p p p 1 C (G , Λ ad, Λ p) = ω : G Λ p ω(g k ) = Λ ad(k− )ω(g ) ∞ { ∞ → | ∞ ∞ ∞ ∞ } ([8], 1.3). We want to make this identification perfect in the sense that we do not distinguish between the p-form and the function on G . The ∞ identification goes as follows: Let ω : G Λpp which satisfies p 1 ∞ → ω(g k ) = Λ ad(k− )ω(g ). If x X and g G satisfies g xo = x ∞ ∞ ∞ ∞ ∈ ∞ ∈ ∞ ∞ then the left translation y g y on X induces an isomorphism of tan- → ∞ gent spaces ∼ dLg : Txo Tx ∞ −→ p If tx Λ Tx then ω considered as a p-form has to have a value on tx ∈ p ω(x)(tx) = ω(g ), Λ dLg 1 (tx) (1.4.2) h ∞ ∞− i This identification is compatible with the action of G from the left on X, so we may divide by Γ and get ∞

~~Ωp(Γ X) = C p(Γ G , Λp ad, Λpp) \ \ ∞ It is important that we can write this also as a space of function on the adele group. Using lemma 1.3.1 we ﬁnd~~

C p(Γ G , Λp ad, Λpp) = \ ∞ p f p p C (Go(F) Go(A)/K (a), Λ ad, Λ p) = \ p ω : Go(F) Go(A) Λ p ω is C∞ in the infinite component \ p → 1 | f  and ω(gk) = Λ ad(k− )ω(g) where k = (k , k ) and   f f ∞ ∞   k K (a)   ∈    1.5 De Rham Cohomology at Infinity : Let B/F be any Borel subgroup of Go/F. This Borel subgroup defines a boundary component YB ∂(Γ X) and we want to describe the cohomology of this boundary ⊂ \ component in terms of differential forms. We still fix our base point xo X. We have seen that the boundary ∈ component associated to B is diffeomorphic to 61 68 G. Harder

~~(1) ΓB X =ΓB U(C)xo =ΓB U(C) \ B \ \ and we have homotopy equivalences~~

(1) ΓB X ֒ ΓB X ֒ ΓB X YB \ B → \ → \ ∪ (1.2 Remark 2). The group B = B(C) acts transitively on X and we put ∞ Kb = B K ∞ ∞ ∩ ∞ Then KB is a circle. This allows us another description of the space of ∞ C -p-forms on ΓB X: ∞ \ p p p p Ω (ΓB X) = C (ΓB B ; Λ ad, Λ p) = \ \ ∞ p ω ω : ΓB B Λ p; ω is C∞ and | \ ∞ → p 1  ω(b k ) = Λ ad(k− )ω(b )   ∞ ∞ B ∞ ∞   for k K .   ∞ ∈ ∞    Under the action of KB we have a canonical decomposition of ∞

~~p = po B p B = po p , ⊕ 1, ⊕ 1 B where po is of dimension 1 and K acts trivially and p1,B is two dimen- ∞ sional irreducible. In the case of B = Bo this decomposition becomes~~

~~p = RH (RE RE ) ⊕ 1 ⊕ 2~~

~~Therefore we get for any 1-form on ΓB X a decomposition \~~

~~ω = ωo,B + ω1,B = ωo + ω1~~

~~It is clear that the ωo component vanishes if we restrict it to the “slices”~~

~~(1) ΓB X ΓB X \ B → \ and this tells us that our decomposition does not depend on the choice of the base point xo. Period Integrals of Cohomology Classes... 69~~

1 Let us assume that ω Ω (ΓB X) is a closed 1-form. Then ω deﬁnes ∈ 1 \ 1 (1) a cohomology class [ω] H (ΓB X; R) = H (ΓB X ; R). We want to ∈ \ \ B compute this class. The group U acts on ΓB X by translations and ∞ \ 62 from this it follows that the cohomology class [ω] is also represented by the form ω(0)(b ) = (u b )du ∞ Z ∞ ∞ ∞ ΓB U \ ∞ where the volume ΓB U is normalized to be equal to 1. If we restrict (0) \ ∞(1) this 1-form ω to ΓB X we get \ (0) (1) (0) (1) ω ΓB X = ω ΓB X \ B \

~~(0) and ω1,B is translation invariant and therefore constant. This means~~

(0) (0) ω (u ) = ω (1) p1,B 1,B ∞ 1,B ∈ (0) This element ω1,B deﬁnes a homomorphism from ΓB into R: Every element γ ΓB can be written in the form γ = exp log γ ∈ where log γ = Id γ Lie(RF/Q(U/F)) and the homomorphism is − ∈ γ log γ, ω(0) (1) = → h 1,B i = log γ, ω(0)(1) h i 1 Since we have H (ΓB, R) = Hom(ΓB, R) we ﬁnd the formula

~~[ω](γ) = log γ, ω(0)(1) = log γ, ω(0) (1) (1.5.1) h i h 1,B i~~

We consider the group B as a real algebraic subgroup of PGL2(C) = ∞ B G(R) where G = RF/Q(Go). The centralizer of K is a real torus T which is of dimension 2 and decomposes into a one∞ dimensional split∞ torus and a one dimensional anisotropic torus. Therefore we have

~~T ∼ Cx = Rx S 1 ∞ −→ × t ∼ (t′ , k(t )) ∞ −→ ∞ ∞ 70 G. Harder~~

~~If ω(1) p B we construct for any complex number s C a form ∈ 1, ∈~~

ωs : ΓB B p1,B C \ ∞ → ⊗ by 1 s 2 + 2 1 ωs(b ) = ωS (u t ) = t ad(k(t )− )ω(1) ∞ ∞ ∞ | ∞|C ∞ where as before z C = zz for z C. | | ∈ 63 Lemma 1.5.2. The 1-form ωs is closed if and only if s = 0. This is an easy computation (see also [8], Lemma 3.1). (1) This lemma allows us to go back and forth from forms on ΓB X \ B to forms on ΓB XB. \ 1.6 The Adelic Description of the Cohomology at the Boundary In the last section we gave a discussion of the de Rham cohomology of an individual boundary component. Now we want to look at all the boundary components and to describe the cohomology in terms of differential forms which depend on adelic variables. We start from our standard Borel subgroup Bo and as in 1.5 we (1) decompose p = po p1 = po,Bo p1,Bo . We define Bo, = b ⊕ ⊕ ∞ { ∞ ∈ Bo, α(b ) = 1 . We introduce the space of maps ∞| | ∞ | } (1) f ω : Uo(A)Bo(F) Bo, Go(A ) p1 C \ ∞ · → ⊗ |  1 f  H =  ω(gk) = ad(k− )ω(g) for k = (k , k )  ∞  ∞ ∞   and k KB,o, k f K f (a)   ∞ ∈ ∞ ∈    We want to show that we have a natural identification 

~~H ∼ H1(∂(Γ X), C) ∞ −→ \ To get this identiﬁcation we start from a computation which is heuristi- cal at the moment, but will also be used later. Let us assume we have a 1-form 1.4~~

~~f ω : Go(F) Go(A)/K (a) p \ → Period Integrals of Cohomology Classes... 71~~

~~We recall the double coset decomposition (1.3)~~

f f Go(A ) = Bo(F) ξK (a) ∪ · where the double cosets are in 1 1 correspondence to the cusps. Let − us pick an element b Bo, and we compute ω(b ξ). As in (1.3) we ∞ ∈ ∞ ∞ write ξ = (1, ξ) = a (a 1, k f ) and get ′ · − 1 ω(b ξ) = ω(b a (a− , 1)) where ∞ ∞ · · 1 b = (b , 1, ... , 1, ...). Then b′ = a− b a B where B is a repre- ∞ ∞ ∞ ∈ ∞ sentative∞ for the Γ-conjugacy class of Borel subgroups corresponding to 64 ξ. Then 1 1 ω(b ξ) = ω(b′ (a− , 1) = ω(b′ a− ) ∞ ∞ · ∞ · 1 where we observe that the adele b′ (a− , 1) is 1 at the ﬁnite components. 1 ∞· We write a− = ba 1 ka 1 with ba 1 B and ka 1 K and ﬁnd − · − − ∈ ∞ − ∈ ∞ 1 ω(b ξ) = ad(k )ω(b b 1 ) a− 1 ′ a− ∞ − ∞

We substitute b′ ba 1 = b′′ and get ∞ − ∞ 1 1 ω(b ) = ad(k 1 ) ω(ab b a ξ) ′′ a− ′′ a− 1 − ∞ · ∞ − · (1) Our forms in H are not deﬁned on all of G but only on Bo, . Therefore we do the∞ following: ∞ ∞ We write f f Go(A ) = Bo(F)ξK (a) [ξ and 1 ξ′ = (1, ξ) = a(a− , k f ) and 1 B = a− Boa and 1 a− = ba ka ba B , ka K · ∈ ∞ ∈ ∞ 72 G. Harder

~~then we put~~

B ω : B p1,B ∞ → B 1 1 ω : b′′ ω(ab′′ ba− 1 a− ξ) ∞ → ∞ − · One checks that B 1 B ω (b′′ k ) = ad(k− )ω (b′′ ) ∞ ∞ ∞ ∞ for k B K = KB and that ∞ ∈ ∞ ∩ ∞ ∞ B B ω (u′′ b′′ ) = ω (b′′ ) ∞ ∞ ∞ 65 Therefore we get for any ω H a collection of diﬀerential forms B 1 (1) ∈ ∞ ω Ω (ΓB X ) which are U invariant and represent cohomology ∈ \ B ∞ classes of the corresponding boundary component at . (1.5.1). This ∞ gives us a map 1 H H (YB, C) ∞ → MB which is obviously an isomorphism and does not depend on any choice. Let us assume that we have a 1-form

ω : G(F) G(A)/K f (a) p \ → which is closed (1.4). So it represents a cohomology class [ω]. We know that the restriction of [ω] to the boundary is given by an element in H , ∞ we want to compute that element. On the adele group Uo(A) we choose a Haar measure du so that the volume Uo(F) Uo(A) becomes equal to \ 1. Then we compute

ω(0)(g) = ω(ug)du Z Up(F) Uo(A) \ (0) (1) f If we restrict ω to Bo, Go(A ) we can project the values to p1 = p1,Bo ∞ · and get (0) (1) f ω1 : Uo(A) Bo(F) Bo, Go(A ) p1 · \ ∞ → which is an element in H . ∞ Period Integrals of Cohomology Classes... 73

Proposition 1.6.1. Under the natural identiﬁcation constructed above (0) the element ω1 corresponds to the restriction of [ω] to the boundary. This follows from 1.5 where we did the corresponding thing for the individual cusps and the computation at the beginning of this section. The normalisation of the measure corresponds exactly to the one in 1.5. We have the decomposition (1.2.1) for the cohomology of the boundary. For the rest of this section we want to analyse our identiﬁcation

H ∼ H1(∂(Γ X); C) ∞ −→ \ from the point of view of (1.2.1). Actually we shall very explicitely associate to any element ψ Mφ∗ or M∗ an element ω(ψ) of H . ∈ φ ∞ We have 66 eiθ 0 KBo = θ Rmod 2π ∞ ( 0 1! ∈ )

~~The group acts on p C = CE E and the vectors 1 ⊗ 1 ⊕ 2 e+ = E i E 1 1 − ⊗ 2 e 1 = E1 + i E2 − ⊗ are eigenvectors with respect to this action:~~

~~iθ iθ e 0 iθ e 0 iθ ad e+1 = e e+1, ad e 1 = e− e 1 0 1! · 0 1! − −~~

The two elements e+1, e 1 deﬁne homomorphisms from ΓBo to R (1.5) − and we shall use them as canonical generators of the two modules L+ and L (1.2.2). Therefore− we have now established the identiﬁcation

~~M = M ; M = M φ∗ φ φ∗ φ in 1.2.2. Now we shall give an explicit formula for the identiﬁcation maps~~

~~1 H (∂(Γ X), C) ∼ (Mφ C Mφ C) H \ −→ ⊗ ⊕ ⊗ → ∞ φ:(OM/a)x S 1 φ(i)=→i 74 G. Harder~~

~~The crucial point is the following simple Lemma 1.6.2. To any φ : (O/a)x S 1 which satisﬁes φ(i) = i 1 there → ± exists exactly one character~~

~~φ : I/FxU f (a) S 1 → for which e φ U f /U f (a) = φ (O/a)x = φ | | any for z Cx e e ∈ 1 z ∓ φ((z, 1, ... ,1)) = . z ! | | e Proof. As in 1.3 we start from~~

~~I/FxU f (a) ∼ Cx x(O/a)x/W −→ 67 Since we have to have~~

~~φ((i, ... , i, ...)) = 1~~

1 and φ(i) = i± we get existencee and uniqueness easily. To any of our characters φ : (O/a)x S 1 for which φ(i) = i 1 we → ± introduce the number ǫ(φ) = 1 such that φ(i) = iǫ(φ). ± We have

~~x Uo(A) Bo(F) Bo(A) To(A)/To(F) = I/F · \ ≃ and therefore we may also look at φ as a character~~

~~1 φ : Bo(F) eBo(A) S \ →~~

which is trivial on Uo(Ae). 1 To any ψ Mφ where φ(i) = i we associate an element ω(, φ, ψ) ∈ ± ∈ H by the formula ∞ ω(b g f , φ, ψ) = ω(b b f k f , φ, ψ) = ∞ ∞ f f 1 φ(b b ) ψ((k )− ) eǫ(φ) ∞ · · e Period Integrals of Cohomology Classes... 75 where we identify K f /K f (a) = G. It’s of course pure routine but we want to check whether this is well deﬁned and the signs are correct. eiθ 0 If b = = h(θ) then we should have ∞ 0 1!

f 1 f ω(h(θ)g , φ, ψ) = ad(h(θ)− ) ω((1, g , φ, ψ) = · 1 f f 1 ad(h(θ)− ) φ(b ) ψ(k )− ) eǫ φ = · · · ( ) f f 1 ǫ(φ)θ φ(b ) ψ(k )− ) e− eǫ φ · · · ( ) and on the othere hand we have

f ǫ(φ) f φ(h(θ) b ) = e− φ(b ) · · so the component at infinitye is ok. To provee that it is well defined we have to write f f f f f f 1 f g = b k = b b (b )− k · · 1 · 1 · and get from the finite places 68

f f f 1 f 1 φ(b b ) ψ(((b )− k )− ) = 1 · 1 f f f 1 f φ(b ) φ(b ) ψ((k )− b ) = e · 1 · · 1 f f F 1 f 1 φ(b ) φ(b ) φ(b )− ψ((k )− ) = e · e 1 · 1 · f f 1 φ(b ) ψ((k )− ). e · e and this proves thate ω(, φ, ψ) is well deﬁned. The map

(Mφ Mφ) C H ⊕ ⊗ → ∞ φM(i)=i which maps ψ Mφ and ψ M to ω(, φ, ψ) and ω(, φ, ψ ) is ∈ ′ ∈ φ ′ equal to the identiﬁcation between H1(∂(Γ X), C) and H if we take \ ∞ (1.2.1) and (1.2.2) into account. One remark concerning the notation. The ψ is always an element in Mφ where φ(i) = i so φ is determined by ψ. But I think it is bet- ± ter always to keep in mind from which space the ψ has been taken, so therefore we keep the φ in the notation. 76 G. Harder 2 The Eisenstein Series

We start from a cohomology class at infinity. We have the identifications 1.2.1 and 1.2.2 and we have seen how to associate to a class ψ Mφ a ∈ map (1) f ω(, φ, ψ): Uo(A) Bo(F) Bo, Go(A ) Ceǫ(φ) p1 C · \ ∞ → ⊂ ⊗ We extend this to a map from G(A) to p C. To get this extension ⊗ (1) x we choose a complex number s C. We have seen that Bo, = Bo, R ∈ ∞ ∞ · (1.5 and 1.5.2) and G = Bo, K . We write an element g G as ∞ ∞ ∞x (1) ∞ ∈ ∞ g = b t k where t (R+) , b B and k K and put ∞ ∞ · ∞ · ∞ ∞ ∈ ∞ ∈ ∞ ∞ ∈ ∞ f f ωs((g , g ), φ, ψ) = ωs((b t k , g ), φ, ψ) = ∞ ∞ ∞ ∞ 1 s 2 + 2 1 f t C ad(k− ) ω((b , g ), φ, ψ) | ∞| · ∞ · ∞ Now we are in the position to define the Eisenstein series. For Re(s) > 1 the series E(g, φ, ψ, s) = ωs(ag, φ, ψ)

a Bo(XF) Go(F) ∈ \ 69 is absolutely and locally uniformly convergent. Moreover it is known that our series has a meromorphic continuation into the entire s-plane ([9], Thm. 7., [13], Chap. 6). We can interpret E(g, φ, ψ, s) as a 1-form on Γ X(1.4) and this 1-form is closed for s = 0. ([8], 4.3). It is well \ known that E(g, φ, ψ) is holomorphic at s = 0. If we want to know the restriction of the Eisenstein class [E(g, φ, ψ,0)] to the boundary we have to compute the constant term (Prop. 1.6.1).

E(ug, φ, ψ, 0)du = E(0)(g, φ, ψ, 0) Z Uo(F) Uo(A) \ We do this by analytic continuation and compute for g = (b , g f ) with b B and Re(s) > 1 ∞ ∞ ∈ ∞ E(ug, φ, ψ, s)du Z Uo(F) Uo(A) \ Period Integrals of Cohomology Classes... 77

This computation has been carried out at several places ([6], 1.6., [11], 6, and [13]). So we recall only the main steps in the computation. We start from the Bruhat decomposition G0(F) = B0(F) 0 1 ∪ B0(F) 1 0 U0(F) and substitute the deﬁnition of the Eisenstein series into the − integral. Then we get two terms

~~ωs(ug, φ, ψ)du + ω(wug, φ, ψ)du Z Z Uo(F) Uo(A) Uo(A) \ 0 1 where w = 1 0 . The ﬁrst integral is constant and therefore we ﬁnd − ~~

ωs(g, φ, ψ) + ωs(wug, φ, ψ)du Z Uo(A) α We have a map Bo(A) I deﬁned by the positive root and for b B (A) −→ ∈ o we deﬁne b = α(b) where x = idelenorm of x I. We write g = | | | | | | ∈ (b , g f ) = (b , b f ) (1, k f ) = b k and get ∞ ∞ · · 1 s 2 + 2 f 1 ωs(g, φ, ψ) = b φ(b) ψ(k )− ) eǫ φ | |C · · · ( ) and e

~~ωs(wubk, φ, ψ)du Z Uo(A)~~

~~1 s 2 2 1 = b − φ(b)− ωs(uk, φ, ψ)du | |C Z e Uo(A) The functions ωS (g, φ, ψ) are product of local functions 70~~

( ) (p) ωs(g, φ, ψ) = ωs∞ (g , φ, ψ) ωs (gp, φ, ψ) ∞ pYﬁnite This is so since they are deﬁned by φ and ψ which are both products of local functions e f ψ(k ) = ψ(kp ) where po = supp (a) o { } 78 G. Harder

~~φ(x) = φ (x ) φp(xp). ∞ ∞ · Yp e e e We have for p ∤ a~~

( ) ( ) ωs∞ (g , φ, ψ) = ωs∞ (b t k , φ, ψ) = ∞ ∞ · ∞ ∞ 1 s 2 + 2 1 t C φ (b ) ad(k− )eǫ(φ) | ∞| · ∞ ∞ · ∞ Therefore the integral decomposese into a product of local integrals. We have to write the measure as a product of local measures and we are in the fortunate case that we can take

du = du dup ∞ Yp O where voldup ( p) = 1 for all p and du = dx dy (Actually there should 1 ∞ be a 2 at (1 + i) and a 2 at inﬁnity but they cancel). For those p which do not divide a (these are all except one) we ﬁnd

2 1+s (p) 1 φp(πp) πp p ωs (wup, φ, ψ)dup = − | | 2 s Z 1 eφp(πp) πp p Uo(Fp) − | | e and this follows from a standard computation ([11], §6). What happens at po where po is the prime dividing a? In this case

~~71 we note that Uo(Fpo ) = Fpo and our integral is a sum~~

~~(po) ωs (wup , kp φ, ψ)dup + Z o o o O po Period Integrals of Cohomology Classes... 79~~

∞ n (po) 1 πp−o ǫpo ωs w kp , φ, ψ dǫp Z 0 1 ! o ! o Xn=1O x po and it is for n > 0 n n 1 0 1 1 πp−o ǫpo πpo ǫp−o 1 1 0 kpo = n − 1 kpo 1 0! 0 1 ! 0 π− ǫp ! πp ǫ− 1! − po o o po − We substitute this into the integrals of the infinite sum. We get that each integral in the infinite sum has value zero since φ2 is not a trivial character. We have only the first term and get

~~(po) ωs (w, up kp , φ, ψ)dup = Z o o o O po~~

~~1 1 1 1 1 ψ(k− u− w) = Tφψ(k− ) N(a) po N(a) po u OXp /po ∈ o where Tφ is the intertwining operator constructed in 1.2.2. At the inﬁnite place we have to compute~~

~~( ) 1 z ωs∞ w , φ, ψ)dxdy Z 0 1!! C where z = x + iy. We introduce polar coordinates and get~~

~~2w ∞ iθ ( ) 1x e ωs∞ w , φ, ψ xdxdθ Z Z 0 1 ! ! 0 0 and we have 1 xeiθ e1θ 0 1 x e iθ 0 = − 0 1 ! 0 1! · 0 1! 0 1! This gives us~~

~~2π ∞ iθ +ǫ(φ)iθ e 0 ( ) 1 x e ad ωs∞ w , φ, ψ xdxdθ Z Z · 0 1!! · 0 1! ! 0 0 80 G. Harder~~

~~Let us write 72~~

~~( ) 1 x ωs∞ w φ, ψ = A(x) eǫ(φ) + B(x) H + C(x) e ǫ(φ) 0 1! ! · · · − Integrating the ﬁrst two terms over θ we ﬁnd zero, so we are left with~~

~~2π ∞ iθ ( ) 1 x e ωs∞ w · , φ, ψ, xdxdθ = Z Z 0 1 !! 0 0~~

~~∞ 1 s 2 + 2 2π  b(x) C(x)xdx e ǫ(φ) Z | |C · −    0    We have to start from the Iwasawa decomposition~~

2 1/2 2 1/2 2 1/2 1 x (1 + x )− x x(1 + x )− (1 x )− w = 2 1/2 − 2 1/2 − − 2 1/2 0 1! 0 (1 + x ) ! · (1 + x )− x(1 + x )− ! − 2 1 Then b(x) = (1 + x )− and a simple computation using (1.4.1) yields C(x) = (1 + x2) 1. − − We get for our integral

∞ 2 1 s 2 1 π  2π (1 + x )− − (1 + x )− xdx e ǫ(φ) = e ǫ(φ) − Z − − s + 1 −    0    Multiplying all this together we ﬁnd for Re(s) > 1 and g = (b , g f ) ∞

~~E(ug, φ, ψ, s)du = Z Uo(F) Uo(A) \ π L(φ2, s) ωs(g, φ, ψ) ω s(g, φ, Tφψ) − s + 1 · 2 · − L(φ e, s + 1) where the L-function is deﬁned as e~~

~~2 2 +s 1 L(φ , s) = (1 φ (π ) π )− − p p | p| pY,po e e Period Integrals of Cohomology Classes... 81~~

~~([12], X/V, §8) Since both sides have meromorphic continuation into the entire s-plane we ﬁnd that the equality holds for all s. Before stating our main result we look at the expression~~

~~π L(φ2, s) − s + 1 L(φ2, s + 1) e s=0 a little bit more closely. The ﬁrst cruciale fact is that L(φ2, 1) , 0 ([12], XV, §4). e So we have to compute 73~~

~~L(φ2, 0) π − 2 L(eφ , 1)~~

~~Now we exploit the functional equation.e Let us assume that a = po is an odd prime and N(po) = p. If we follow the instructions in [12], p. 299 carefully we ﬁnd~~

2 2 1 2 L(φ , 0) =+W(φ ) √p π− L(φ , 1) · and therefore e e e 2 L(φ2, 0) L(φ , 1) π = W(φ2) √p − 2 − · 2 L(eφ , 1) L(eφ , 1) e We apply the formula fore the number W(φ2) givene in [12], p. 300 and 2 2 2 1 2 1 2 get W(φ ) = i τ(φ ) φ (D− ) where τ(φ ) is a Gaussain sum. · √p · (1+i) e ([12],e XIV, §4).e Now ei2 = 1 and φ2(De1) = φ2((1, i , 1, ... ,)) − (1)− 2 where the i/2 stands at the (1 + i)th componente of the idele. Then this e e is φ2(( 2i, 1, 2i, ...)) = ( 1) φ2( 2i) − − − · − where the last 2ieis the residue class of 2i in O/po. Therefore we ﬁnd − − 2 2 L(φ , 0) L(φ , 1) π = τ(φ2) φ2( 2i) − 2 − · − 2 L(eφ , 1) L(eφ , 1) e e e 82 G. Harder

If we have a = (1 + i)3 we find 2 2 L(φ , 0) L(φ , 1) π = W(φ2) 2 = − 2 − · 2 L(eφ , 1) L(eφ , 1) 2 e 2 Le(φ , 1) L(φ , 1) e τ(φ2) = 2 2 − 2 L(eφ , 1) L(eφ , 1) e Now we can state the firste main theoreme of the paper. In the statement we refer to the different identifications made before.

74 Theorem 2.1. For ψ Mφ C the Eisenstein series E(g, φ, ψ, 0) is a ∈ ⊗ closed 1-form and the cohomology class [E(g, φ, ψ,0)] restricted to the boundary is equal to 2 2 2 L(φ , 1) 1 [E(g, φ, ψ,0)] = ψ φ ( 2i)τ(φ ) Tφψ ∂(Γ X) − − · 2 · p \ L(eφ , 1) e e if a = po is prime and N(po) = p and equal to e 2 L(φ , 1) 1 ψ Tφψ − 2 4 L(φe, 1)4 if a = (1 + i)3. e

2.2 Arithmetic Applications In this section we assume that a = po is an odd prime. The theorem gives us the value of the number cφ in 1.2.2, we get with p = N(po) 2 φ2( 2i)τ(φ2) L(φ , 1) cφ = − (2.2.1) − p · 2 e L(eφ , 1) Corollary 2.2.1. We have e 1 cφ = | | √p

~~and in particular cφ , 0, . ∞ Period Integrals of Cohomology Classes... 83~~

~~This is a consequence of the properties of the Gaussian sums. To give another interpretation of the Corollary 2.2.1 we recall that we have a scalar product on Mφ~~

~~ψ, ψ = ψ(g)ψ(g) h i Z G/Bo and the norm of the operator Tφ is obviously √p. So the √p cancels and we ﬁnd that the Corollary says that~~

cφTφ : Mφ C M C ⊗ → φ ⊗ is an unitary operator. I was unable to see this form a topological point of view and shall 75 come back to this kind of questions later1. But we can also reverse the argument. We had the identiﬁcations (1.2.1)

1 H (∂(Γ X), R) ∼ (M∗ M∗) ∼ (Mφ Mφ) \ −→ φ ⊕ φ −→ ⊕ Mφ Mφ φ(i)=i φ(i)=i where the last identiﬁcation has been by means of the elements e+1, e 1 p C (see 1.2.2 and 1.6). Of course we must have cφ R Q − ∈ ⊗ ∈ Z and therefore we get the information that N

~~2 L(φ , 1) τ(φ2) R Q 2 ∈ ⊗ L(eφ , 1) e But we can do better. The cohomologye~~

~~H1(∂(Γ X), R) = H1(∂(Γ X), Z) R \ \ ⊗ and we have an action of the Galois group Gal (K/Q) on the cohomology where K is the ﬁled of fractions of R. But this galois group is also acting~~

~~1Added in Proof: This is actually very easy to see. 84 G. Harder~~

on (M + M ) φ∗ φ∗ Mφ φ(i)=i in an obvious way and the action is compatible with the identification, moreover we see that e+1 and e 1 are both defined over Q(i) and the complex conjugation interchanges− these two homomorphisms. There- fore we can say that also the last identification is compatible with the action of the galois group. The galois group Gal (K/Q) acts on our character φ simply by acting on the values φσ(x) = φ(x)σ

σ and σ Gal(K/Q) maps Mφ into Mφσ. It is clear that T = Tφσ and all ∈ φ this tells us Corollary 2.2.3. We have 2 L(φ , 1) φ2( 2i)τ(φ2) R Q − · L(φ2, 1) ∈ ⊗ e and if σ Gal(K/Q) and φ =eφσ then ∈ 1 2 σ 2 L(φ , 1) 2 2 L(φ1), 1 φ2( 2i)τ(φ2)  = φ ( 2i)τ(φ )  − L(φ2, 1) 1 − 1 · L(φ2, 1)  e  e1  e  76 This follows of coursee from the observation that thee image H1(Γ X, R) H1(∂(Γ X), R) \ → \ has to be invariant under the action of the galois group. This corollary is related to results of Damerell, Shimura and Razar. Damerell’s result is to some extent much stronger since it says that L(φ2, 1) = ω2 α · 1 dx e where ω = and where α is an algebraic number whose de- 3 R0 √x x nominator can be− bounded in terms of our data ([4], II, Thm. 2). But Period Integrals of Cohomology Classes... 85 on the other hand it seems to be so that our information concerning the 2 ratio L(φ , 1)/L(φ2, 1) is much more precise and I do not know whether this can be deduced from his methods. There is also a certain relation to e e the results of Shimura and Razar. Shimura considers Dirichlet L-series corresponding to modular forms ([23])

∞ s ann− = L( f , s) Xn=1 and twists them by Dirichlet characters s s D( f , φ, s) = anφ(n)n− , anψ(n)n− = D( f , ψ, s) X X Then he is able to say something about the values D( f , φ, s) D( f , ψ, s) at special values of s and then his results becomes very similar to ours. But I do not see whether his result implies Corollary ?? or whether it can be obtained from his methods. To conclude this section I want to discuss a few examples very explicitely. We start from the following general remark: The cohomology 1 1 H (Γ X, R) = H (Γo, R[G]) can be computed in principle in an effective \ way once our data-this means a-are given. This will be discussed in the thesis of E. Mendoza ([15]). This means we are also able to compute the 77 number cφ in a given case and this gives an effective way of computing 2 the ratios L(φ , 1)/L(φ2, 1). (Comp. also [23], Intr.). We want to discuss this computation in a couple of cases where we chose a slightly different e 1 method than the one suggested by [15]. We compute H (Γo, Nφ)(1.2.3) by starting from the cochain complex. We look at

Nφ

~~o 1 0 / C (Γo, Nφ) / C (Γo, Nφ) /~~

~~ o 1 0 / C (Γo,Bo , Nφ) / C (Γo,Bo , Nφ) / 86 G. Harder~~

~~We computed~~

~~1 Uo W W H (Γo,Bo , Nφ) = Hom(Γo,Uo , N ) = Hom(Γo,Uo , Rfo Rf ) . φ ⊕ ∞~~

~~Let Φ Hom(Γ , B , Nφ) and let ∈ 0 0 1 1 i 0 0 1 A = , C = , B = 0 1! 0 1! 1 0! − 1 1 Φ : = A a fo + b f 0 1! → ∞~~

1 i 1 Φ : = CAC− c fo + d f . 0 1! → ∞ Such a Φ is invariant under W if and only if c = ia and d = ib. This 1 − means that a cohomology class in H (Γo,Bo , Nφ) is canonically represented by a cocycle

~~Φ : A a fo + b f → ∞ Φ : C 0 → 1 i Φ : +ia fo ib f 0 1! → → ∞~~

~~In 1.6 we introduce e+1, e 1 p1 C and they deﬁne homomorphisms − ∈ ⊗ (1.5)~~

1 1 1   → 2 0 1 e+1 :   1 i  i/2 0 1 →−     1 1 1   → 2 0 1 e 1   − 1 i  i/2   → 0 1     Period Integrals of Cohomology Classes... 87

78 Therefore we have in the notations of 1.2.3, if we put e+ = e+1 and e = e 1 (what we did all the time) that − − Φ= 2ae+ + 2be − 1 1 and cφ = ba N(po) and our problem to compute cφ amounts to: − · − When can we extend the cocycle

Φ : A a fo + b f ; Φ : C 0 → ∞ → to a cocycle on Γo with values im Nφ? The only thing we have to do is we have to give the value Φ(B) Nφ. But we have certain restrictions ∈ for this value. These restrictions come from the relations

~~2 1 3 B = 1 BC C− B (AB) = 1 − which imply~~

~~Φ(B) = CΦ(B), Φ(B) + BΦ(B) = 0 and Φ(AB) + ABΦ(AB) + (AB)2Φ(AB) = 0~~

(These are not all relations, but they are suﬃcient in our special cases) We stick to the case a = po is an odd prime and introduce a basis in Nφ. The basis consists of the functions δu where u O/po = Fp and δ ∈ ∞ and 1 u w 1  · 0 1 →        1 ν δu : w 0 for ν , u  · 0 1 →        1 0   0  0 1 →         1 u w 0    →  0 1 δ :    ∞   1 0  1   →  0 1       88 G. Harder

~~The group acts as follows~~

~~Aδu = δu 1, Aδ = δ ; − ∞ ∞ 2 Bδ = δ , Bδ = δ , Bδ = φ(u )δ 1 (u , 0), o o u u− ∞ ∞ − 1 Cδu = φ(i)− δiu, Cδ = φ(i)δ · ∞ ∞ The cocycles we are looking for are~~

~~Φ : A a  δu + bδ →   ∞ uXFp   ∈  Φ : C 0  → Φ : B ? →~~

79 We write Φ(B) = Σxuδu + x δ and we get from CΦ(B) = Φ(B) ∞ ∞ that xo = x = 0 and x iu = φ(i) xu. The dihedral group generated ∞ − · by B and C acts on B G and we have one orbit of length 2 namely \ p 5 0, , one orbit of length 4 namely 1, i, i, 1 and the other − { ∞} { − − } 8 orbits are of length 8 and those consist of two orbits under C which { } are ﬂipped by B. Therefore we see that the relations CΦ(B) = Φ(B) and p 5 Φ(B)+BΦ(B) = 0 restrict the possible values for Φ(B) to an − + 1 - 8 ! dimensional vector space. Now we look at the relation

~~Φ(AB) + ABΦ(AB) + (AB)2Φ(AB) = 0~~

~~To see what this means it is convenient to look also at the space Nφ. We~~

have a natural pairing Nφ Nφ R and the δu, δ form a dual basis × → ∞ with respect to this pairing. Then this last relation says b b Φ(AB), l = 0 h i for all l N for which ABl = l. This means ∈ φ Φ(A) + A Φ(B), l = 0 h · i Period Integrals of Cohomology Classes... 89 for all such l and this is equivalent to

Φ(A) + Φ(B), Ker(Id BA) = 0 h − i where Id BA : N N . We have a 2-dimensionald space of choices − φ → φ p 5 for Φ(A)d and a − + 1 -dimensional space of choices for Φ(B) and 8 ! p 1 p + 1 p 1 on this − + 3 -dimensional space we get resp. − + 2 8 ! 3 3 ! linear equations if p 1mod 3 (resp p 1mod 3) for the possible ≡ − ≡ cocycles. This gives us some kind of vague feeling the occurence of cohomology is something accidental. But we know that there has to be at least a one dimensional space of solutions. We consider special cases:

~~I. po = (2 i), then we have exactly one character φ with φ(i) = i. − We have the residue classes 0, 1, 2, 3, 4mod 5 80~~

~~Φ(A) = a(δo + δ1 + δ2 + δ3 + δ4) + b δ · ∞ Φ(C) = 0 Φ(B) = x (δ iδ δ + iδ ) · 1 − 2 − 4 3~~

The elements δo + δ1 + δ and δ2 + δ4 δ3 form a basis for ∞ − Ker(Id BA) in the dual module and have to be orthogonal to − b b b b b b Φ(A) + Φ(B) and we get the linear equations d 2a + b + x = 0 a + ( 2i 1)x = 0 − − 1 3 + 4i (1 2i)2 We put a = 1, then x = and b = = − . 2i + 1 −2i + 1 1 + 2i Therefore we have constructed a cocycle representing an Eisen- stein class and this Eisenstein cocycle is

~~(1 2i)2 ΦE : A (δo + δ1 + δ2 + δ3 + δ4) + − δ → (1 2i) · ∞ − 90 G. Harder~~

~~ΦE : C 0 → 1 ΦE : B (δ iδ δ + iδ ) → 2i + 1 1 − 2 − 4 3 This gives in view of 1.2.4.~~

~~(1 2i)2 1 (1 2i)2 1 (1 2i) c = = = φ − − − 2 (1 + 2i) N(po) (1 + 2i) 5 (1 + 2i) If we take (2.2.1) into account we ﬁnd~~

~~2 L(φ , 1) (1 2i)2 (1 2i)2 = φ2( 2i)τ(φ2) 1 − = τ(φ2) 1 − 2 − − − 1 + 2i − − · 1 + 2i L(eφ , 1) e e The deﬁnitione of τ(φ2) is given in [12], and we get~~

~~2 2 + i e 2 2πix 2 + i 1 2i τ(φ ) = φ (x) e 5 = √5 = − √5 2 i · 2 i −1 + 2i − xmodX 5 − e 81 so we end up with~~

~~2 L(φ , 1) (1 2i)3 1 = − 2 (1 + 2i)2 √ L(eφ , 1) 5 e II. po = (3 + 2i), then O/po = Z/13Z and imod po = 5mod 13. Our cocycle has to look like~~

Φ(A) = a δu + bδ , Φ(C) = 0   ∞  O  u X/po   ∈  Φ(B) = x (δ iδs  δ + iδ )+ 1 1 − − 12 8 x (δ iδ δ + iδ φ(4)δ + iφ(9) δ + φ(9)δ iφ(9)δ ) 2 2 − 10 − 11 3 − 6 · 9 7 − 4 The vector Φ(A) + Φ(B) has to be orthogonal to the following vectors in the dual space δ1 + δo δ , δ5 + φ(9)δ3 δ6, δ12 + − ∞ − φ(10)δ + δ , δ φ(4)δ δ and if φ(3) = 1 then also to δ and 7 2 8 − 11 −b 9 b b b b b 10b δ . 4 b b b b b b Period Integrals of Cohomology Classes... 91

(α) φ(3) = 1. Since (Z/13Z)x = 5 3 this ﬁxes φ since we { }×{ } have φ(i) = φ(5) = i. Then its easy to solve the system of linear equations and we ﬁnd the only solution a = 1, x = 1, x = 1 2i, b = 3 + 2i. 2 − 1 − − The Eisenstein cocycle is

ΦE(A) = δu + ( 3 + 2i)δ , ΦE(C) = 0   ∞  O  − uXpo   ∈  ΦE(B) = (1 2i)(φ iδ δ + iδ ) − 1 − 5 − 12 8 (iδ iδ δ + iδ + δ iδ δ iδ ) − 2 − 10 − 11 3 6 − 9 − 7 − 4 ( 3 + 2i) Therefore we ﬁnd c = − and this gives us φ 13 2 L(φ , 1) (3 2i)2 1 = − 2 − (3 + 2i) √ L(bφ , 1) 13 1 e1 (β) φ(3) = ρ = + i √3 where √3 is the positive root. In −2 2 this case our computations gives again one solution and we get

2 2 2 (1 + i ρ )(1 + i ρ) ΦE(A) = iρ − − δ   − (1 i ρ2) ∞ uXO/p − −  ∈  ΦE(C) = 0  3ρ2 ρ2i + 1 ΦE(B) = − − (δ iδ δ + iδ ) 1 i ρ2 1 − 5 − 12 8 − − i ρ iρ2 2 − − − 1 i 1 i ρ2 · − − − (δ + iδ δ + iδ + δ + iδ δ iδ ) 2 10 − 11 3 6 9 − 7 − 4 and we get 82

~~2 2 2 (1 + i ρ )(1 + i ρ) 1 cφ = iρ − − 1 iρ2 · 13 − 92 G. Harder~~

We introduce the Gaussian sum 2π 2 2π 2π G(φ2) = G(φ2) = 2 cos cos 5 cos 3 + cos 2 13 − · 13 − · 13 13 e 2π 2π 2π 2π 2π + 2ρ cos 4 cos 6 cos 6 cos 3 + cos 2 · 13 − · 13 − · 13 − 13 13 ! and get

2 L(φ , 1) (1 + i ρ)3 (1 + i ρ2)2 1 = i − · − L(φ2, 1) (1 i ρ)3 (1 i ρ2) · G(φ2) e − − · − − We want to conclude this section by mentioning an interesting question. One of the consequences of our theory is the non vanishing of cφ. We could now also look at cohomology with torsion coeﬃcients say in the ring R/p = Fq. We have basically the same situation as in characteristic zero if we stay away from some bad characteristics. So we may again ask whether this number cφ , 0, . Of course it is clear that ∞ this question is closely related to the question, what is the prime number decomposition of L(φ, 1)/(L(φ2, 1)? So one might ask for instance 2 whether cφ is always a unit in R, once we inverted the divisors of G . e e | | 3.1 The Period Integrals In the last section we constructed the cohomology classes

[E(g, φ, ψ,0)] H1(Γ X, C). ∈ \ Actually it can be shown that these classes live already in H1(Γ X, K) \ 83 where K is the fraction ﬁeld of R. This will be done in a subsequent paper and we have to use the Hecke algebra and the strong version of the multiplicity one theorem. I ask the reader to accept this fact here, it will be of importance only at the end of this paper. So we have homomorphisms

[E(g, φ, ψ, 0)] : Γ K → 2Added in Proof: Further computations show that this idea is much too naive. Period Integrals of Cohomology Classes... 93 and we may ask for a formula for the value of [E(g, φ, ψ, 0)] on a given element γ Γ. If γ is unipotent, then the answer is given by theorem ∈ 2.1. Therefore we are left with the non unipotent elements γ. In this case the centralizer Tγ of γ is an anisotropic torus over F. The aim of this section is to show that we have a canonical way of associating a cycle zγ to γ and that

~~[E(g, φ, ψ,0)](γ) = E(, φ, ψ, 0) Z zγ~~

~~We want to give an explicit expression for that integral. Our given element γ defines a torus Tγ and this torus defines a quadratic extension E/F, the splitting field of Tγ/F. We have~~

x x Tγ(F) = E /F and it follows from Dirichlet’s theorem on units that Tγ(F) Γ= inﬁnite ∩ cyclic group which contains γ. We shall say that γ is primitive if it generates this inﬁnite cyclic group. The group of complex points of our torus Tγ decomposes in a canonical way x 1 (s) (c) Tγ(C) = R+ xS = Tγ xTγ where S 1 = z C z = 1 and the superscripts s and c stand for spit { ∈ | | | } and compact. We pick an element xγ X = PGL2(C)/K for which (c) ∈ ∞ Tγ xγ = xγ. These elements form a real line in X. We get a map

~~(s) (s) jx : T = Tγ(C)/T X γ γ γ → jx : t txγ γ → and with Γγ =Γ Tγ(F) we get a map ∩ (c) j : Γγ Tγ(C)/T Γ X xγ \ γ → \ 94 G. Harder~~

~~84 and this gives us our one cycle zγ since the left hand side is obviously a circle. Now it is obvious that~~

~~[E(g, φ, ψ,0)](γ) = E(, φ, ψ, 0) Z zγ~~

As I said already we want to have this formula a little bit more explicit. So let us assume that ω is any 1-form on Γ X. We get the identiﬁcation \ x Tγ(C) ∼ C −→ from the selection of a generator in the character group of our torus Tγ C. This selection is unique up to a sign. Therefore we have ×F

~~(s) x Tγ ∼ R+~~

∂ On Rx we have the canonical vector field t which we lift back to a + ∂t (s) (s) vector field Y on Tγ by means of that identification. We have Tγ = (c) (s) Tγ(C)/Tγ and let γ be the image of γ in Tγ . Then

~~γ dt ω = ω(txγ),(d jx (Y)) Z Z h γ i t zγ 1~~

(s) x where we consider Tγ = R+, and d jxγ is of course the derivative of our map jx . We write xγ = gγ xo with gγ PGL (C). We agreed already to γ ∈ 2 look at ω as a function ω : Γ G p \ ∞ → 1 such that ω(g k ) = ad(k− )ω(g ). We apply (1.4.2) and get for the integral ∞ ∞ ∞ ∞ γ dt ω(tgγ), dL 1 d jx (Y) Z h (tgγ)− ◦ γ i t 1 Period Integrals of Cohomology Classes... 95

~~Since our vector ﬁeld is invariant under translations we ﬁnd~~

~~1 dL 1 d jx (Y) = ad(g− )(Y) (tgγ)− · γ γ~~

(s) where we consider Y Lie(Tγ ) Lie(G ) = g . It follows from ∈ ⊂ ∞ ∞ our construction that ad(g 1)(Y) p and the ﬁrst ‘explicit’ form of our γ− ∈ integral is γ 1 dt ω(tgγ), ad(g− )(Y) (3.1.1) Z h γ i t 1

85 3.1.2 Summation over the Classes in the Genus Our final goal is the evaluation of the integrals (3.1.1) for ω = E(g, φ, ψ, 0). But we shall not be able to do this directly since we run into some trouble with class numbers. So we have to discuss these class number problems first. We say that two elements γ, γ Γ are in the 1 ∈ same class if they are conjugate under the action of Γ. If ω is any closed form we shall certainly have ω = ω if γ and γ1 are in the same z z Rγ1 Rγ class. We say that γ, γ1 are in the same genus if they are conjugate under the action of PGL (F) and if we can find an element k f K f (a) 2 ∈ which conjugates γ into γ1. We shall see that the set of classes in a given genus is a finite set which has a natural structure of an abelian group. If γ Γ and if Iγ is the set of classes in the genus of γ then we shall ∈ compute

χ(γ1) E(, φ, ψ, 0) Z γX1 1γ ∈ zγ1 for all characters of Iγ and for this expression we shall write down a rather explicit formula. Now we shall describe the set of classes in a given genus. Let us pick a non unipotent element γ. If we have

~~1 f f 1 f f γ = a− γa with a Go(F), γ = k γ(k )− ; k K (a) 1 ∈ 1 ∈ 96 G. Harder~~

f f then t = a k Tγ(A ). It is very easy to check that the correspondence · ∈ t γ induces a bijection ↔ 1 f f f Iγ ∼ Tγ(F) Tγ(A )/K (a) Tγ(A ) −→ \ ∩ To see this we use 1.3.1 and use the same arguments as in 1.3 for the set of Γ-conjugacy classes of Borel subgroups. This of course deﬁnes the group structure on Iγ. Our next step is to transform the sum

~~χ(γ1) ω Z γX1 Iγ ∈ zγ1~~

~~into an adelic integral over Tγ(A)/Tγ(F). We start from the observation that we can view ω as a function~~

~~ω : Go(F) Go(A) p \ → 1 f f f which satisﬁes ω((gk) = ad(k− )ω(g) of k = (k , k ) and k K (a) ∞ ∞ ∈ (1.4). 86 We have~~

~~f f f f Tγ(F) Tγ(A )/K (a) Tγ(A ) = Tγ(F) Tγ(A)/T(C) (K (a) Tγ(A)) \ ∩ \ · ∩ and we write~~

~~f f Tγ(A) = Tγ(F) Tγ(C) ξ (K (a) Tγ(A )) · · · ∩ [ξ~~

~~f f where ξ Tγ(A ) and where we embed Tγ(A ) ֒ Tγ(A) by putting ∈ → the component at inﬁnity equal to one. Then we use again ?? and write~~

~~f f f ξ = aξk a Go(F), k K (a) ξ ∈ ∈~~

~~and extending this to Tγ(A) we obtain~~

~~1 f ξ = (1, ξ) = aξ(aξ− , kξ ) Period Integrals of Cohomology Classes... 97~~

~~1 The element ξ deﬁnes a class in the genus of γ and γξ = a γaξ Γ is a ξ− ∈ representative of this class. We want to compute~~

~~χ(γξ) ω Xξ Z zγξ~~

~~f f We observe that χ is a character on Tγ(F) Tγ(A)/Tγ(C)(K (a) Tγ(A ) \ ∩ and the above expression becomes~~

~~γξ 1 χ(ξ) ω = χ(ξ) ω(tgγ ), ad(g− )(Y) h ξ γξ i Xξ Z Xξ Z zγγ 1~~

~~1 1 Now we can choose xγξ = aξ− xγ and gγξ = aξ− g. Then we get for our sum~~

~~γξ 1 1 dt χ(ξ) ω(ta− gγ), ad g− ad aξ(Yξ) · Z h ξ γ ◦ i t X 1 In the adele group we have~~

~~1 f aξ (a− , k ) = ξ′ · ξ and therefore~~

~~1 f 1 aξ (a− , 1, ... , 1, ... , 1) = ξ′ (k )− · ξ · 1 We put t′ = aξtaξ− and note that ad aξ(Yξ) = Y then the contribution coming from the class ξ is~~

~~γ 1 1 dt′ ω(aξ− t′gγ), ad(gγ− )(Y) Z h i t′ 1 98 G. Harder~~

~~87 Now we observe that this integration takes place at the component at inﬁnity. We have~~

~~1 1 f 1 (a− , 1, 1, ...) = a− ξ′ (k )− ξ ξ · · and we ﬁnd for the above integral the value~~

1 x ω(tξ), ad(g− )(Y) d t Z h γ i Tγ(F) Tγ(F) Tγ(C) ξ \ · · where the measure dxt on the adele group has to be normalized as follows. We have x x d t = Πd tv dt and dxt = dxt(s) dxt(c); the dxt(s) is the Lebesgue measure on ∞ ∞ × ∞ ∞ t x (s) x (c) R+ = Tγ and d t gives volume one to the circle. At the ﬁnite places we require ∞ vol (T(F ) K f (a)) = 1. dtp p ∩ p At this place we used the fact that γ is primitive. This gives us the ﬁnal formula

~~1 x χ(γ1) ω = ω(tg , ad(g− )(Y)d t (3.1.1.1) Z Z γ γ γX1 Iγ zγ Tγ(F) Tγ(A) ∈ \ where dxt is normalized as above and where g = (g, 1, 1, ... , 1). γ~~

3.1.3 Now we apply this formula in the case where ω = E(g, φ, ψ, 0). What we shall do is to evaluate the right hand side in the case where ω = E(g, φ, ψ, s) with Re(s) > 1. Then the result will be dependent on the diﬀerent choices we made. If we continue analytically and evaluate at s = 0 then the result is intrinsic and we get a formula for the left hand side. Period Integrals of Cohomology Classes... 99

~~We have for Re(s) > 1~~

~~x χ(t) ωs(atg , φ, ψ))d t Z γ a Bo(XF) Go(F) Tγ(F) Tγ(A) \ ∈ \~~

~~The map Bo(F) Tγ(F) Go(F) given by multiplication is easily seen × → to be bijective, so we ﬁnd that the last term is equal to 88~~

~~x χ(t) ωs(ttg , φ, ψ))d t = Z γ t XTγ(F) Tγ(F) Tγ(A) \ ∈~~

~~x χ(t)ωs(tg , φ, ψ)d t Z γ Tγ(A)~~

~~As in section 2 we write ωs(tg, , ) as a product of local functions~~

~~( ) (p) ωS (tg , φ, ψ) = ωs∞ (t gγ, φ , ψ) ωs (tp, φp, ψ) γ ∞ ∞ Yp e e and our integral becomes an inﬁnite product of integrals~~

( ) x (s) x ωs∞ (t gγ, φ , ψ)d t χp(tp)ωp (tp, φp, ψ)d tp ∞ ∞ ∞ Z Yp Z Tγ(C) e e (We should perhaps mention that the ψ enters only in the factor belonging to po). Before we enter into the computation of the individual local terms we want to state the result. Let us look at a finite prime p. We shall call a finite sum ν=r (1/2+s/2)ν aν πp = Gp(s) νX= r | | − where aν K[χ] (i.e. we adjoin the values of χ to the quotient field K ∈ of R) an elementary factor. We shall call such a factor an exponential ( 1 + 1 s)ν factor if it is of the form aν π 2 2 for some ν Z and a , 0. | | ∈ 100 G. Harder

~~If we have any character η of the idele class group of any ﬁeld L/k and if p is a prime of k then we write~~

1 s if η is unramified 1 ηp(πp) πp L(η, s)p = − | | 1 if η is ramified  We know that the character χ above can be identified with a character on our quadratic extension E/F. We have the norm mapping N : IE IF, → so φ N is also a character on the idele class group of E and we put ◦ LE(χ φ N, s) = LE(χ φ N, s)P · ◦ · ◦ YP p e \ e 89 With these conventions we can state: We have for all p

s+1 LE χ φ N, (p) x · ◦ 2 p ωs (tp, φp, ψ)d tp = Gp(φ, ψ, χ, γ, s) (*) 2 Z · LF(φe, s + 1)p e e where Gp(φ, ψ, χ, γ, s) = Gp(s) is an elementary factor.e This elementary factor is equal to one for almost all p. e At the inﬁnite place we shall get an expression s 2 Γ + 1 L χ φ N, s+1 2 E 2 G (s) = G (s) · ◦ ∞ 2 ∞ · Γ(s + 2) ∞ LF(φe, s + 1) ∞ where G (s) is an exponential factor and wheree the Γ-factors are exactly the ones∞ one expects ([12], XIV, §8). We are now ready to enter the long and tedious computations which shall give us (*). We have to be quite careful since we want to get as much information as possible on the local elementary factors. Es- pecially we would like to know whether they can vanish at s = 0 for reasons which become clear later. A last remark concerns our character χ. The formula makes sense for any character on the idele class group which is trivial at inﬁnity. But for our purpose we have only to look at those characters which are trivial f f on Tγ(A ) K (a), since only those enter on the left hand side. So we ∩ shall always assume that Period Integrals of Cohomology Classes... 101

3.1.4 We look, at the ﬁnite places ﬁrst and we begin our computations by discussing an integral representation of our function ωs(gp, φ, ψ). We write the action of GL (F ) on F2 as 2 p p e a b ax + cy (x, y) c d! → bx + dy! and for any locally constant function with compact support

~~Φ : F2 C p → we write 90~~

Lφ(gp, φ, s) = LΦ(gp) = 1 s 2 2 + 2 2 x eΦ[(0, ap)gp] a det gp φ(a det gp)d ap Z | p · |p p x Fp e x where the measure d ap is normalized in such a way that the units get volume 1. One checks very easily that

1 s 2 + 2 t1,p x t1,p t1,p LΦ gp = φ LΦ(gp) 0 t2,p! ! t2,p · t2,p ! · p e This means that our function LΦ transforms under the left action of (p) Bo(Fp) the same way as ωs (gp, φ, ψ) does, so we can expect to ﬁnd a Φ such that LΦ(gp) = ωs(gp, φ, ψ) This is easy: e If p , po then our character φ is unramiﬁed at p, we choose for Φ = (1 φ2(π ) π 1 s) characteristic function of O O . The one − p | p|p− × e p ⊕ p checks easily that e (p) LΦ(gp, φ, s) = ωs (gp, φ, ψ) (3.1.3.1)

~~If p = po then we consider thee space of functions~~

~~2 Φ Φ : O/po O/po C Φ(λ(x, y)) = φ (λ)Φ(x, y) { | ⊕ → | } 102 G. Harder~~

We look at such a function as a function on O O and extend it po ⊕ po outside of O O by zero. Then for g = k K po ⊕ po p p ∈ p LΦ(g , φ, s) = Φ[(0, 1)k ] φ(det(k )) p p · p and therefore if we put e e

ψ(kp) = Φ[(0, 1)kp]φ(det(kp)) (3.1.3.2) we find e LΦ(gp, φ, s) = ωp(gp, φ, ψ) f f 91 Here we identify Mφ = ψ : K R ψ(bk) = φ(b)ψ(k) for all b K { e → | e ∈ ∩ B(A f ) . Now we compute the local integrals by starting from the integral } representation of our functions ωs. We observed that our element γ generates a quadratic extension E/F in the matrix ring M2(F). Then we x x x x have Tγ(F) = E /F and we put Ep = (E Fp) . If p splits in E then x = x x ⊗ Ep EP EP. Our character χ is in a canonical way identified with a × x character χ on IE/I F . We have to compute · (p) x χ(tp)ωs (tp, φ, ψ)d tp = Z x x Ep/Fp e 1 s 2 2 + 2 Φ[(0, ap)tp] a det tp Z Z | p |p x x x Ep/Fp Fp φ(a2 det t ) χ(t )dxa dxt = p p · p p p 1 s e 2 + 2 x Φ[(0, 1)tp] det tp φ(det tp)χ(tp)d tp Z | |p x Ep e We have to spend a moment of thinking about the normalization of the x measures. The measure d tp will be the product measure of the measure x x d ap which gives the volume 1 to the units and the measure d tp which x x gives volume one to Tγ(F ) K E /F . p ∩ p ⊂ p p Let us put I(γ) = (E M (O ))x p p ∩ 2 p Period Integrals of Cohomology Classes... 103 then U(γ) is an open subgroup of the group of units U Ex. If p is p p ⊂ p (γ) O ramified then the image of Up in Tγ( p) is of index one or two, so (γ) voldtp Up ) = 1 or 1/2. Remark . At this point we can derive already the statement (*) very easily, it is almost the definition ([12], XIV, 8). We have to check that t Φ[(0, 1)t ] is a Schwartz-Bruhat function, which is not quite clear p → p if p splits in E. But we said already that we are interested in very specific informations on the local factors so we have to work a little more.

Case I. Let us start with the case where our extension E/F splits at p. At the moment we do not assume p , po. In this case we can ﬁnd two eigenvectors e , e O O such that for t E 1 2 ∈ p ⊕ p ∈ p e1t = t1e1; e2t = t2e2 where t = (t , t ) with respect of the decomposition E = E E . We 92 1 2 p p ⊕ p ﬁnd a constant fp such that

π f p(O O ) O e O e O O p p ⊕ p ⊂ p 1 ⊕ p 2 ⊂ p ⊕ p and we call the embedding regular if we can choose e1, e2 in such a way that O O = O e O e , i.e. f = 0. p ⊕ p p 1 ⊕ p 2 p Let us write

(0, 1) = αe1 + βe2 αβ , 0 f f where α, β E and π p α, π p β O . Any element t E and be ∈ p p p ∈ p p ∈ p written ν ν t = ǫ , ǫ . p  P P YP YP    and we put deg(tp) = (ν, ν′). We have to discuss the question: When do we get (0, t)t O O ? We get a rough picture as follows: We have p ∈ p ⊕ p integer constants Mp, Np and Mp′ , Np′ such that (i) If deg(t ) = (ν, ν ) and ν M then (0, 1)t O O if and only p ′ ≥ p p ∈ p ⊕ p if ν N . ′ ≥ p 104 G. Harder

~~(i ) If deg(t ) = (ν, ν ) and ν M then (0, 1)t O O if and only ′ p ′ ′ ≥ p′ p ∈ p ⊕ p if ν N . ≥ p′~~

(ii) there exists only a ﬁnite number of pairs (ν, ν′) with ν < Mp and ν′ < Mp′ such that we have a number tp with deg(tp) = (ν, ν′) and (0, 1)t O O . p ∈ p ⊕ p So the picture looks like that

93 So if ν M or ν M then (0, 1)t O O depends only on ≥ p ′ ≥ p′ p ∈ p ⊕ p the degree. Let us call the ﬁnite set of points described in (ii) simply S p. If we have a regular embedding the situation becomes nicer. In this case we write (0, 1) = αe + βe with α, β O and if we put +M = 1 2 ∈ p p ord (α), +M = ord (β) then (0, 1)t O O if and only if − p p′ − p p ∈ p ⊕ p we have for deg(t ) = (ν, ν ) that ν M and ν M . We call the p ′ ≥ p ′ ≥ p′ embedding strongly regular if ordp(α) = ordp(β) = 0, this is the nicest case, we have (0, 1)t O O if and only if ν, ν 0. p ∈ p ⊕ p ′ ≥ Now we can evaluate our integral and we write

~~1 s 2 + 2 x Φ[(0, 1)tp] det tp φ(det tp)χ(tp)d tp = Z | |p x Ep e Period Integrals of Cohomology Classes... 105~~

~~∞ 1 s 2 + 2 x Φ[(0, 1)tp] det tp p φ(det tp)χ(tp)d tp = = Z | | ν,νX′ = −∞deg(tp) (ν,ν′) e~~

ν,ν′=+ ∞ 1 + s (ν+ν ) ν+ν ν ν π ( 2 2 ) ′ φ(π ) ′ χ(Π ) χ(Π ) ′ | p| p · p · p′ × ν,νX′= −∞ e Φ[(0, 1)(Πν, Πν′ )ǫ]φ(det(ǫ))χ(ǫ)dxǫ (3.1.3.3) Z p p′ Up

~~Now we should distinguish several cases :~~

~~(α) p , po and χ is unramiﬁed. Then our considerations above tells us that our sum decomposes~~

~~+ + +~~

(νX,ν ) S νXMp νXM νXMp ′ ∈ ≥ ′≥ p′ ≥ Np ν′ M′ N ν M ν′ M′ p 1 p ′ ′ 1 p ≤ ≤ − ≤ ≤ p− ≥ The ﬁrst sum is ﬁnite and in the other sums the value of the inte- x gral is always equal to vold tp (Up). Then these sums can be evaluated easily and we get

G (φ, ψ, χ, γ, s) p × (1 φ(π )2 π 1+s) e − p · | p|p 1 + s 1 + s × Π Π e2 2 Π Π 2 2 1 χ( p)φ(πP) P P 1 χ( P)φ(πP) P P − | | − | | e e where we recall that Π = Π = π and the numerator 94 | P|P | P′ |P′ | P|P stems from the normalization of the characteristic function. If our embedding is regular then we have

1 s ( + )(MP+MP ) MP+MP MP MP G (φ, ψ, χ, γ, s) = π 2 2 ′ φ(π ) ′ χ(Π ) χ(Π ) P | P| P · P P′ so wee get an exponential factor ine this case. It is equal to one in case of a strongly regular embedding. 106 G. Harder

(β) p , po and χ is ramified. Our character φ is unramified in this case and it is clear that χ has to be ramified at P and P since it is e ′ one on I. Therefore most of the integrals disappear and we are left with a finite sum, which gives us the desired elementary factor. In this case we can’t have a regular embedding because of the earlier remark.

~~(γ) P = Po. In this case we have in addition that Φ vanishes on (O O )π and transforms under scalar multiplication as follows P ⊕ P P Φ[λ(x, y)] = φ2(λ)Φ(x, y)~~

~~If we restrict to a speciﬁc degree (νν′) we see that~~

~~ν ν (i) Φ[(0, 1)(Π , Π ′ )ǫ] = 0 if ν and ν′ are large. P P′ = (ii) If ν is large and ǫ (ǫP, ǫP′ )~~

~~ν ν′ 2 ν ν′ Φ[(0, 1)(Π , Π )ǫ] = φ (ǫP )Φ[(0, 1)(Π , Π )] P P′ ′ P P e (ii)′ If ν′ is large then we have~~

~~ν ν′ 2 ν ν′ Φ[(0, 1)(Π , Π )ǫ] = φ (ǫP)Φ[(0, 1)(Π Π )] P P′ P P′ If we decompose our integral accordinge to degree we ﬁnd (??)~~

~~+ +~~

~~ν,νX′ small νXsmall ν largeXν′ small ν′ large where the ﬁrst sum is ﬁnite. Let us consider a term~~

Φ[(0, 1)(Πν , Πν )ǫ]φ(det ǫ) χ(ǫ)dxǫ Z P P′ · UP e 95 where ν is large. We have U = U U , and get P P × P′ ν ν x x Φ[(0, 1)(Π , Π )]φ(ǫP/ǫP )χ(ǫP)χ(ǫP )d ǫPd ǫP Z Z P P′ ′ ′ ′ UP U P′ e Period Integrals of Cohomology Classes... 107

~~Here we should be careful enough to say that of course~~

~~χ(ǫP) = χ(1, ... ,ǫP, 1, ...) = χP(ǫP)~~

~~↑ P-th component = = χ(ǫP′ ) χ(1, ... , 1,ǫP′ , 1, ... 1) χP,(ǫP′ )~~

~~↑ P′-th component~~

~~x and then the condition χ IF = 1 implies χP(ǫ) = χP (ǫ) for ǫ UP F . | ′ ∈ ⊂ P Therefore our integral turns out to be~~

ν ν x x Φ[(0, 1)(Π , Π )]φ(η)χP(η)d ηd ǫ Z Z P P UP U P′ e This integral vanishes if φχ is non trivial on U Fx . But if φχ is P P ⊂ P P trivial on U then the value of the integral is equal to P e e ν ν voldxǫ(UP) Φ[(0, 1)(Π Π )] · P P′ and it is clear that this value does not depend on ν but only on ν′ provided ν is large. A similar assertion holds if ν′ is large. Now we interpret our conditions on χ φ and χ φ to be trivial or non trivial on U resp. U . Of P · P′ · P P′ course we have χ φ is trivial on U if and only if the character χφ N is e P · P ◦ unramified at P and the same holds for P . And we observe that χφ N e ′ e ◦ can be unramified at most one of these place. Therefore we get: e If χφ N is ramified at P and at P then in our decomposition ◦ ′ e + − ν,νX′ small νXlarge νXsmall ν′ small ν′ large the second and third terms contribute zero. Then our integral is given by the first sum which is an elementary factor if ψ takes values in R. If 108 G. Harder

χφ N is unramiﬁed at P then we get a contribution from the second 96 ◦ sum and this gives obviously again e 1 Gp(φ, ψ, χ, γ, s) 1 s 2 + 2 1 χφ N(ΠP) ΠP e − ◦ | ′ |P and this is again the term we want. e The same thing happens at P , if χφ N turns out to be unramiﬁed ′ ◦ at P . ′ e Now we discuss the case of a regular embedding and we want to give explicit expressions for the elementary factor. We write

(0, 1) = αe1 + βe2 and we put ord (α) = M , ord (β) = M . One of these two numbers p − p p − p′ has to be zero, if both are zero then we are in the case of a strongly regular embedding. We recall f f G = PGL2(O/P) = K /K (P) and we recall from 1.2.2. 1 Mφ = ψ : G R ψ(gb− ) = φ(b)ψ(g) = { → | } ψ : K f /K f (P) R ψ(b f k f ) = φ(b f ) ψ(k f ) → | ·  f f f  for all b Bo(A ) K  ∈ e ∩    (1.6.2) Moreover we identiﬁed the two spaces 

~~Φ : O/P /p R Φ[λ(x, y)] = φ2(λ)Φ[(x, y)] { ⊕ → | } and Mφ by (3.1.3.2) e~~

ψ(k f ) = Φ[(0, 1)k f ] φ(det k f ) p · p Now our inﬁnite sum over ν, ν′ specializese to = = + + one term for ν Mp, ν′ Mp′ νX=Mp ν>XMp ν′>Mp′ ν′=Mp′ Period Integrals of Cohomology Classes... 109

~~97 Π Π We choose a uniformizing element πp at P and choose P, P′ to be the projections of π to the components in E F = E E . p ⊗ p P ⊕ p′ Then the ﬁrst term is equal to~~

~~1 s ( + )(Mp+M′ ) Mp+M′ Mp Mp π 2 2 p φ(π ) p χ(Π ) χ(Π ) ′ | p| p P · p′ M M Φ[(0, 1)(Πe p , Π p )ǫ]φ(det(ǫ))χ(ǫ)dxǫ Z P P′ Up e~~

Mp M The vector ξo = (0, 1)(Π ,P , Π ′ ) = α′e1 + β′e2 has both components P′ α , β , 0. We choose an element ko Kp with det(ko) = 1 such that ′ ′ ∈ (0, 1)ko = ξo. Then our integral becomes in view of (3.1.3.2)

~~x χ ψ(koǫ)χ(ǫ)d ǫ = (p 1)P ψ(ko) Z − Up~~

χ Here P is the following projection operator: The basis e1, e2 defines a torus Tγ F GL /F and the image of Tγ F in PGL F is × p ⊂ 2 p × p 2 × p Tγ F . Let T γ be the reduction of Tγmod p. Then this is a split torus × pe e in G defined by the reduction of the basis e1, e2mod p. Our character χ can be considered as a character on the group Tγ because it vanishes by assumption on the center. Then we define e 1 ψ Mφ ψ(gt) = φ(g)χ(t)− Mχ = ∈ | φ  for t T γ ∈   χ χ  and P is the projection from Mφ to M . We get the factor p 1 in front φ − because of the normalization of the measures. The other terms vanish by the same argument as before unless χφ N ◦ is unramified at P or at P . If for instance it is unramified at P we have ′ e to consider

~~M′ ν p′ Φ[(0, 1)(Πp, Π )(ǫp, ǫp′ )] Z p′ × Up x x φ(ǫpǫp′ )χp(ǫp)χp′ (ǫp′ )d ǫpd ǫp′ , e 110 G. Harder~~

~~for ν> Mp. And as before we ﬁnd the value for this integral is~~

~~ν Mp (p 1)Φ[(0, 1)(Πp, Π )] = Φ[e2](p 1) − P′ −~~

~~If we treat the integral the same way we did before in the case ν = Mp, ν = M we get : If k K , such that det(k ) = 1 and (0, 1)k = e then ′ p′ 1 ∈ p 1 1 2 the integral turns out to be~~

~~(p 1)Pχψ(k ) = (p 1)ψ(k ) = Φ[e ]. − 1 − 1 2 e 98 Therefore we ﬁnd: If we put~~

~~1 + s (M +M ) M +M M M W (φ, χ, s) = π ( 2 2 ) P P′ φ(π ) P P′ χ(Π ) P χ(Π ) P′ p | p| p P P′ then the value of the local contributione is equal to~~

χ (p 1)Wp(φ, ψ, s) P ψ(ko) − · if χ φ N is ramiﬁed at P and P′ · ◦ χ 1 + s χ χ Wp(φ, χ, s) (P ψ(ko) + φ(πp)χ(Πp) πp 2 2 (P ψ(ko) P (ko) (p 1) · | | − 1 + s − (1 φe(π )χ(Π ) π 2 2 − p p | p|p if χ φ N is unramiﬁed at P (3.1.3.2) · ◦ e

~~and we get ae corresponding expression in case of non ramiﬁcation at P′.~~

Remark. In the ﬁrst case the expression Wp does depend on the choice of πp but the second factor does so too and the two factors cancel. We did all this since we want to know whether this elementary factor may vanish at s = 0. We ignore the exponential factor and therefore we have to look at the expressions

χ P ψ(ko) if χ φ Nisramiﬁed at P and P′. · ◦ 1 χ 2 χ χ P ψ(ko) + φ(πp)χ(Πp) πp (P ψ(k ) P ψ(ko)) | |p 1 − if χ φ N is unramiﬁede at P. This means that χ considered as a character · ◦ x 1 on (O/P) is equal to φ . The group T γ acts on the projective line e ± Period Integrals of Cohomology Classes... 111

Bo G and we have exactly two ﬁxed points therefore we get an orbit \ decomposition G = Bogo T γ Bo x Bo x · ∪ 1 ∪ 2 Here we have to choose for go and element in G which does not conjugate Tγ into Bo and x1, x2 do conjugate T γ into Bo.

~~We observe that we can for instance choose go simply the reduction of komod p and the reduction of k1 will be x1 or x2mod Bo. Therefore 1 we ﬁnd if χ , φ±~~

χ 1 M = h : G R h(bg t) = φ(b) χ(t)± h Bx = 0, h Bx = 0 φ { → | o · · | 1 · 2 } and we get. 99 χ chi If P ψ , 0 then P ψ(ko) , 0 and we know exactly when our local elementary factor does not vanish. 1 χ χ If χ = φ± then dim Mφ = 2 and we ﬁnd a basis for Mφ by constructing h1 1 h (bg t) = φ(b) χ− (t) 1 o · and h2 , 0 concentrated on Bx1 or Bx2. Then we have

~~h1(ko) , 0, h1(k1) = 0~~

~~h2(ko) = 0, h2(k1) , 0 and again we see that for suitable ψ Mχ the local expression will not ∈ φ be zero, for instance we can choose ψ = λh2 with λ , 0 and then the local factor is , 0.~~

Case II. We assume that E is non split at p. We keep the notation Ep = E F and we choose a uniformizing element Π in E . ⊗ p p p Then we can ﬁnd two constants M < N such that for t = ǫ Πν 1 1 p · p we have

~~(0, 1)t < O O if ν< M p p ⊕ p 1 (0, 1)t O O if ν> N p ∈ p ⊕ p 1 112 G. Harder~~

~~The local contribution is equal to~~

∞ ( 1 + s )ν ν ν det Π 2 2 φ(det Π ) χ(Π ) | P| P · P νX=M1 e Φ[(0, 1)Πν ǫ]φ(det ǫ)χ(ǫ)dxǫ Z P Up e Then it is obvious that this local factor is of the form (*). We simply have to observe that for p , po the integral does not depend on ǫ if ν ν< N1. If p = po then the function Φ[(0, 1)πpǫ] = 0 if ν is large and we get a finite sum. Here we have to take into account that the character φ is ramified at po and that in this case χ φ N is ramified at P. · ◦ Again we discuss the case of a regular embedding more closely

(α) p , po and E/F is unramiﬁed at p. Then we choose of course Π = π and we have U = Ex GL (O ). Then for t = p p p p ∩ 2 p p (0, 1)πνǫ O O if and only if ν 0. We recall that it fol- ∈ p ⊕ p ≥ 100 lows from our assumptions that χ U = 1, only these χ are of | p interest for us. Then the local factor is equal to

~~1 s 2 + 2 1 φ N(πp) χ(πp) det πp p − ◦ · | | = 1 1 φ2(π ) π 1+s e − p | p| e (β) The extension is ramiﬁed at p. We may have two cases, namely~~

(0, 1)Πν ǫ O O if and only if ν 1 P ∈ p ⊕ p ≥− or (0, 1)Πν ǫ O O if and only if ν 0. P ∈ p ⊕ p ≥ If Mo = 1 in the ﬁrst case and Mo = 0 in the second case, we − ﬁnd as local contribution

~~∞ ( 1 + s )ν det Π 2 2 φ(det Π )ν χ(Π )ν | P|p · P · P · νX=Mo e Period Integrals of Cohomology Classes... 113~~

Φ[(0, 1)πνǫ]φ(det ǫ)χ(ǫ)dxǫ Z p Up e The function under the integral sign is constant since φ and χ are x x unramiﬁed. We map Up into Ep/Fp then the image is of index 2. x x e Since we normalized vol (Ep/Fp) = 1 we get as local contribution

~~1 s 1 ( + )Mo det Π 2 2 φ(N(Π ))Mo χ(Π )Mo 2| p|p p · p e 1 φ2(π ) π 1+s − p | p| 1 + s 1 χ(Π )eφ(N(Π )) Π 2 2 − P P | P|P and the elementary factor is exponential. e~~

~~(γ) p , po and the extension is unramiﬁed at p. This is the nicest case the local integral is equal to~~

~~2 χ Φ[(0, 1)ǫ]φ (det ǫ)χ(ǫ)d∗ǫ = (p + 1) P ψ(1) Z · Up e~~

~~Here we observe that the reduction mod p of Tγ is an anisotropic torus T γ in G and P is the projection to the space~~

~~χ 1 M = ψ : G R ψ(bgt) = φ(b)ψ(g)χ(t)− for bǫBo, t T γ φ { → | ∈ }~~

~~Since G = BoT γ this space is of dimension 1 and generated by a function ψ1 which satisﬁes ψ1(1) = 1, the local factor is , 0.~~

~~(δ) p = po and the extension E/F is ramiﬁed at the place p. The 101 group of units injects into GL2(Op) and this induces an injection~~

~~((U /U(2) ֒ GL ((O/p p p → 2 where U(2) = ǫ ǫ 1mod P2 . The units of F mapped into p { | ≡ } p GL2(O/p) ﬁll up the center and we get (2) T γ = Im(U /U G) p p → 114 G. Harder~~

is cyclic of order p, this means that Tγ is the group of O/p-valued points of a Borel subgroup of G. The element πp induces via multiplication an endomorphism of O O . The reduction of p ⊕ p this endomorphism to O/p O/p has obviously a one dimensional ⊕ kernel, which is generated by a vector ξ ǫO O . It is clear that 1 p ⊕ p the reduction of ξ1mod p generates the stabilizer of T γ. This tells us that we have to consider two cases

~~1 δ1)T γ Bo = (0, 1)Π− Op Op ⊂ h i p ∈ ⊕ 1 δ2)T γ 1 Bo = (0, 1)Π− Op Op h i p ∈ ⊕ In the case δ) we get the local contribution~~

1 s ( 2 + 2 ) 1 1 det Π − φ(det Π )− χ(Π )− | P|P P P · 1 x Φ[(0, 1)π− ǫe]φ(det ǫ) χ(ǫ)d ǫ + Z p · Up e + Φ[(0, 1)ǫ]φ(det ǫ)χ(ǫ)dxǫ Z Up e e Now the situation is quite similar to the split case. We choose 1 ko Kp with det(ko) = 1 such that (0, 1)π = (0, 1)ko then we get ∈ P− 1 s π 2 + 2 1 1 χ χ ( ΠP Φ(N(ΠP))− χ(ΠP)− P ψ(ko) + P ψ(1)) 2 | |P

~~e1 The vector (0, 1)Πp− mod p is not stabilized by Bo and therefore the image ko of ko in G is not contained in Bo. We have~~

G = Bo ko T γ Bo · · ∪ χ , and we get that dim Mφ = 1 (resp. 2) if χ 1 (resp χ = 1). In χ both cases we can construct a ψ M such that ψ(ko) , 0 and ∈ φ ψ(1) = 0, and this implies that with this choice the elementary factor is non zero. Period Integrals of Cohomology Classes... 115

~~102 The case δ2) is quite similar. In this case we have also to sum over two terms namely (0, 1)ǫ, (0, 1)Πpǫ. Then we get~~

~~1 p +s/s Pχψ(1) + Π 2 φ(N(Π )) χ(Π )Pχψ(k )) 2 | P|P P · P 1 e where (0, 1)πP = (0, 1)k1. The same argument as above shows the non vanishing if ψ is suitably chosen.~~

~~3.1.5 The Inﬁnite Place We have to evaluate the integral~~

~~( ) x ωs∞ (t gγ, φ)d jxγ (Y))d t Z ∞ ∞ Tγ(C) e We recall that (1.6)~~

( ) ( ) ωs∞ (g , φ) = ωs∞ (b k , φ) = ∞ ∞ ∞ 1 s + 2 + 2 1 b C φ(b )(ad k− ) eeǫ(φ) | ∞| ∞ ∞ · x u where b = 0∞ 1 Bo(C) ande ∞ ∈ 1 + 1 s 1 + 1 s b 2 2 φ(b ) = x 2 2 φ(x ) | ∞|C ∞ | ∞|C · ∞ (s) e (c) x 1 e We recall that Tγ(C) = Tγ xTγ = R+ xS (3.1) and we have selected gγ in such a way that

ω∞s (t gγ, φ), d jxγ (Y) h ∞ i does not depend on the circular variable. (3.1). Now we have of course that E = F E = C E = C C, this ∞ ∞ ⊗ ⊗ ⊕ deﬁnes a split torus in PGL2(C) and this torus is certainly not contained in Bo(C). We can ﬁnd a matrix y u x = Bo(C) 0 1! ∈ such that 1 Tγ(C) = xT1x− 116 G. Harder where 103

a b 2 2 T1 = a, b C, a b , 0 mod center of GL2(C) ( b a! ∈ − ) and one checks that y is unique up to a sign. The maximal compact subgroup of T1 is contained in the maximal compact subgroup K and ∞ therefore we can take gγ = x. Our integral becomes

~~( ) 1 x ωs∞ (xt1, φ), ad(x− ) (Y) d t1 Z h · i T1~~

The generator Y Lie(T (s)) is selected in a canonical way up to a sign ∈ γ since the character module of the torus Tγ has a canonical generator λo up to a sign and dλ(Y) = 1. Therefore we get that

~~1 0 1 ad(x− )(Y) = p ± 1 0! ∈ and we choose y in such a way that~~

~~1 0 1 ad(x− )(Y) = = E1 1 0!~~

~~Observing that the integral does not depend on the circular variable we ﬁnd for the value of the integral~~

~~1 s 2 + 2 ( ) x y φ(y) ωs∞ (t, φ), E1 d t | |C Z h i (s) e T1 e and 0 1 T (s) = exp x x IR = 1 ( 1 0! ∈ )~~

~~cos hx sin hx = t(x) x IR ( sin hx cos nx! ∈ )~~

~~Period Integrals of Cohomology Classes... 117 and the measure was normalized such that we have to compute~~

~~+ 1 s ∞ 2 + 2 ( ) y φ(y) ωs∞ (t(x), φ), E1 dx | |C Z h i e −∞ e 2 2 1 We put h(x) = (cos h x + sin h x) 2 and get~~

~~1 cos hx sin hx h(x) h(x) h(x) t(x) = − ∗ = b(x) k(x) 0 h(x)  sin hx cos hx  ! h(x) h(x) · −    Then the integral is equal to 104~~

~~+ 1 s ∞ 2 + 2 2 2s 1 y φ(y) h(x)− − ad k(x)− eǫ(φ), E1 dx | |C Z h i e −∞ 1 But ad k(x) eǫ φ , E = eǫ φ , ad k(x)E . h − ( ) 1i h ( ) 1i From (1.4.1) we get that~~

~~2 2 eǫ φ , ad k(x)E = h(x)− E , E = h(x)− h ( ) 1i · h 1 1i and we wind up with~~

~~+ 1 s ∞ 2 + 2 2 2 2 s y φ(y) (cos h x + sin h x)− − dx | |C Z e −∞ and this integral turns out to be equal to~~

~~1 1 + s Γ(s/2 + 1)Γ(1/2) y 2 2 φ(y) 2 · | |C · s+1 Γ 2 + 1 e and if we exploit Legendre’s duplication formula ([28], 12.15) we ﬁnd~~

~~s 2 1 + s Γ 2 + 1 2s y 2 2 φ(y) | |C · Γ(s + 2) e for the local contribution. 118 G. Harder~~

~~Now we can evaluate at s = 0. Then the value of the integral is the~~

value of the period of the class [E(φ, ψ, 0)] on the cycle χ(ξ)zγξ and ξ Iγ this is an intrinsic value which does not depend on the diﬀP∈erent choices we made. We get our second main theorem. Theorem 3.1.6. The value of the Eisenstein class [E(φ, ψ,0)] on the cycle

χ(ξ)Zγξ ξXIγ ∈ for a primitive γ Γ is given by ∈ 1 1 LE χφ N, 2 2 y Φ(y) ΠpGp(φ, ψ, χ, γ, 0) ◦ C 2 | | · · LFe(φ , 1) e e 105 We have that almost all local elementary factors Gep(φ, ψ, χ, γ, 0) = 1. If our embedding is regular at all places, then we can choose ψ so that all local factors , 0. Remark. The theorem as it is stated is somewhat weak because we do not say very much about the elementary factors. So we should understand it in connection with our results on these factors which have been obtained in course of the computations. It seems to be diﬃcult to incor- porate these computational results into the statement of the theorem.

~~4 Arithmetic Applications~~

~~In the beginning of 3.1 we stated without proof that the classes~~

~~[E(g, φ, ψ,0)]~~

are cohomology classes in H1(Γ X, K), where K is the ﬁeld of fraction \ of R. We want to accept this fact form now on, in any case we have given several examples in 2.2 in the cases Po = (2 i), Po = (3 + 2i) − where we checked this assumption directly. We abbreviate

~~[E(g, φ, ψ,0)] = ΦE(ψ) Period Integrals of Cohomology Classes... 119 and then we get a homomorphism~~

~~ΦE(ψ): Γ K → Then our main theorem 3.1.6 may also be stated as follows: Let γ Γ be primitive, let Iγ the group of classes in the genus of γ, ∈ 1 for any character χ : Iγ S we consider →~~

~~χ(ξ)γξ Γ/[Γ, Γ] Z[χ] ∈ ⊗ ξXIγ ∈ where (Z[χ] is the ring of integers of the ﬁeld generated by the values of χ). Then~~

~~1 1 LE χ φ N, 2 2 ΦE(ψ)( χ(ξ)γξ) = y φ(y) ΠpGp(φ, ψ, χ, γ, 0) · ◦ C 2 X | | · LF(eφ , 1) e e The ﬁrst consequence of this formula is e~~

~~Corollary 4.1. The number 106~~

~~1 1 LE χ φ N, 2 2 y φ(y) (ΠpGp(φ, ψ, χ, γ,0)) · ◦ C 2 | | · LF(eφ , 1) e e is in K[χ] e We have an action of the Galois group Gal(K(χ)/Q) on the group~~

Hom(Γ, K(χ)) simply given by the action on the group of values. This induces an action of this Galois group on the characters φ, χ and on the functions ψ. Since σ σ it follows from the above theorem, that ΦE(ψ) = Φ(ψ ) we get even information concerning the galois action on the above numbers

~~1 σ 1 LE χ φ N, 2 2 Π · ◦ =  y φ(y)( pGp(φ, ψ, χ, γ,0))  | | · L(φe2, 1)    e e   e  120 G. Harder~~

σ 1 1 L χ N, 2 2 σ σ σ σ ◦ y φ (y)(ΠpGp(φ , ψ , χ , γ,0)) | | L((φσ)2, 1) e e But we can say a little bit more: The Eisenstein classes ΦE(ψ): Γ e → K have of course to satisfy certain integrality conditions. This means that in any given case we find a number d Z, such that ΦE(ψ) takes 1 ∈ its values already in R′ = R χ, d . Then we get of course the same estimates for the denominatorsh of thei right hand side. In any case these questions about the denominators in the Eisenstein classes seem to be very interesting. We discussed already some of the aspects at the end of 2.2. We should certainly expect those primes in the denominator which occur in the torsion of the cohomology of Γ. We should also expect the primes dividing cφ. But we can say that if there are other primes in the denominator of an Eisenstein class, then they will create congruences between the Fourier coefficients of cusp forms and Eisenstein series. We hope to come back to these questions later. 1 107 Of course the main object of interest are the values LE(χ, φ N, ) ◦ 2 themselves. If we want to understand these values we have to get hold of e the local factors, especially we have to prove that they are non zero. We have collected some informations concerning this question, but I do not want to discuss these problems in this paper. Instead of trying to give a general statement I will treat a very special example where one can see how our results can be used to get informations on these special values of L-functions. Before I come to this example I want to say one more word about the relationship of our result to Shimura’s results in [24]. He considers special values of L-functions LE(η, s) where E is a CM field and η a Grossencharakter of type Ao. He proves that for certain special values of s the value of the L-function divided by a suitable power of a period is an algebraic number. Our method here gives some information on the ratios 1 LE χ φ N, · ◦ 2 2 LF(eφ , 1) 2 where the period ω cancels out. Soe our information is weaker to some extent, but we get informations for an infinite number of fields E/Q, Period Integrals of Cohomology Classes... 121 which are not necessarily CM-fields. We get informations on the Galois action and on the denominators. In some cases we get even an effective procedure to compute these ratios, and I want to conclude this paper by describing this procedure and doing the computation in one specific case. We identified the space of R-valued function C(G) with the group ring

~~C(G) ∼ R[G] −→ f f (σ)σ → σXG ∈ This is an isomorphism of G G-modules, the actions on the group ring × are given by~~

~~Lτ : m = aσσ aστσ = a 1 σ → τ− σ σXG σXG σXG ∈ ∈ ∈ 1 Rτ : m = aσσ aσστ− = aστσ → σXG σXG σXG ∈ ∈ ∈~~

~~We consider R[G] as a Γo-module with respect to the action induced by 108 right multiplication and with respect to this action we deﬁned~~

~~1 H (Γo, R[G]) and we have (1.1) 1 1 H (Γo, R[G] = H (Γ, R)~~

~~In C[G] we considered the submodule (1.2.3)~~

~~Nφ = f : G R f (bg) = φ(b) f (g) { → | · } and in some cases we have explicitely computed the cohomology groups (2.2) 1 1 ([H (Γo, Nφ) ֒ H (Γo, R[G → 122 G. Harder~~

~~1 In those case which we considered we found that dimR H (Γo, Nφ) = 1 and we did even better namely we constructed explicitely cocycles~~

~~Φ : Γo Nφ → whose cohomology class generates the cohomology. Here we observe, that such a cocycle Φ : Γo Nφ →~~

is uniquely determined by its class, provided we know that Φ : ΓoBo Uo → Nφ . Therefore we can say that the cocycle Φ is a canonical representative of the given class. Now we got from our construction (1.2.3) that the class Φ which satisﬁed (2.2) 1 1 Φ = δu + b δ 0 1!!   · ∞ uXUo   ∈  has to be equal to the Eisenstein class, i.e.

~~[Φ] = [E(φ, ψo,0)]~~

where we have ψo Mφ and ∈ 1 φ(b)− for g = uwb ψo(g) =  0 for g = b   109 This tells us that the cocycles which we computed in the examples in (2.2) actually are equal to the canonical representatives of a very speciﬁc Eisenstein class.

Remark. This argument of course breaks down if we do not know that 1 H (Γo, Nφ) is of rank 1. In that case we have to separate the Eisenstein class from the other classes by using the action of the Hecke algebra.

~~Now it is clear how we get an “explicit” formula for the value~~

~~[E(φ, ψo,0)](γ) Period Integrals of Cohomology Classes... 123 for γ Γ. Let us assume we have computed the value Φ(γ) of the ∈ representing cocycle on γ. Then~~

~~Φ(γ) = φσ(γ)σ σXG ∈ and according to (1.1) we have~~

~~[E(φ, ψo,0)](γ) = Φ1(γ)~~

~~Now we have a set of generators for Γo namely the matrices~~

1 α u(α) = α O 0 1! ∈ 0 1 i 0 B = and C = 1 0! 0 1! − We know in our examples the value of Φ on each of the generators and if γ = γ1, ... , γt is written as a product of generators then we have

t t Φ(γ) = Rγ1, ... , γν 1Φ(γν) = Φσγ1, ... , γν 1(γν)σ − − Xν=1 Xν=1 σXG ∈ where γ1, ... , γν 1 is the image of γ1, ... , γν 1 in G. We have − − t Φ1(γ) = Φγ1 γν 1(γν) − Xν=1 and if we interpret Φ(γ) R[G] as an R-valued function on G, then we ∈ ﬁnd t Φ1(γ) = [E(φ, ψo,0)](γ) = Φ(γ1)(γ1, ... , γν 1) − Xν=1 We want to generalize this formula slightly, we are interested in 110

~~[E(φ, ψ,0)](γ) for all ψ Mφ. ∈ 124 G. Harder~~

We observe that Mφ is an irreducible G-module with respect to the action induced by multiplication from the left. Therefore it suﬃces to compute these numbers in the special case that ψ = Lσ (φo) for σo G. Then we o ∈ get of course the representing cocycle

~~Lσo Φ(γ) = Φσ(γ)σoσ Xσ and form that we get the formula~~

t 1 [E(φ, Lσo φo,0)](γ) = Φ(γν)σo− γ1, ... , γν 1 (4.1) − Xν=1 if γ = γ1, ... , γt is a presentation of γ as a word in the generators of Γo we gave above. Now we want to evaluate the formula in one special case. We take Po = (2 i) and E/F shall be the ﬁeld of eight roots of unity. − πi If ζ = √i = e 4 then we have

~~OE = OF[ζ]~~

([12], IV, Thm. 3). This field contains the maximal totally real subfield L = Q( √2) and the fundamental unit in L is 1 + √2 = ǫ. We embed E ֒ M (F) by means of the identification → 2 a + bζ (a, b) → and then we have a b E = a, b F M2(F) ( bi a! ∈ ) ⊂

~~Since we have OE = OF[ζ] this embedding is everywhere strongly regular as one checks easily. Now the element η = ǫ3 is a primitive element in the group Γ and it is given by the matrix~~

~~7 5 5i η = − = γ 5 + 5i 7 ! Period Integrals of Cohomology Classes... 125~~

The class number of E is one and starting from this one checks that there x x is only one class in the genus of γ, this is so since (OE/poOE) /(OF/po) 111 global units = 1. This means there is only the trivial character χo = 1 and our main theorem says-

1 LE φ N, 2 [E(φ, ψ,0)](γ) = product of local factors ◦ 2 × LFe(φ , 1) We have to determine the local factors. They are certainlye equal to one at all places except (1 i), po and inﬁnity. So we compute these factors − explicitely at these places. We look at (1+i) ﬁrst. In this case we have the uniformizing element π = (1 ζ). Then 2 − 1 1 + ζ 1 + ζ π 1 = = = 2− 1 ζ (1 ζ)(1 + ζ) 1 i − − − The corresponding matrix is

1 1 1 1 i i 1! − and 1 1 1 1 1 (0, 1) π− = (0, 1) = (i, 1) · 2 1 i i 1! 1 i − − and hence (0, 1)π 1 < O O . We are in case II, β) and ﬁnd the local 2− 2 ⊕ 2 elementary factor 1 G (φ, ψ , χ, γ, 0) = 2 o 2

~~Now we look at the local factore at p = po = (2 i). We are in case II, γ) − and we have to compute~~

~~χo (p + 1)P ψo(1) = ψo(t)~~

~~tXT γ ∈~~

~~3 1 The group T ψ is cyclic of order 6 and generated by η = 2 3 . 126 G. Harder~~

~~Recalling the deﬁnition of ψo and we ﬁnd~~

~~χo Gp (φ, ψo, χo, γ, 0) = 6 P ψo(1) = 2 + i o · And at inﬁnity our toruse is given by 112~~

~~a b T (C) = a, b C center γ bi a ( ! ∈ ) .~~

~~If we choose our matrix ζ 1 0 x = − 0 1! 1 πi then Tγ(C) = xT1x− in our previous notations. We recall that ζ = e 4 . Now we see that the factor at inﬁnity is~~

~~πi e− 4 · Our formula becomes~~

1 πi 1 LE φ N, 2 [E(φ, ψo,)](γ) = e− 4 (2 + i) ◦ 2 · · 2 · LFe(φ , 1) e Now we compute the left hand side by using (4.1).e We have to write down γ in terms of the generators and this is easily done by using the euclidian algorithm.

1 1 i 1 2 2i 1 2 + 2i 1 i 1 γ = − C2 B − − B − B − − B 0 1 ! · · 0 1 ! · 0 1 ! · 0 1 ! Now it is a question of sitting down and to compute the value of Φ(γ), using (4.1) and (2.2). Case I we found 8 + 15i Φ (γ) = 1 1 + 2i Therefore we obtain the formula

~~1 LE φ N, 2 πi 8 + 15i πi i(15 8i) ◦ 4 4 = 2 e− = 2 e− − = L (φ2, 1) · (1 + 2i)(2 + 1) · i(2 i)(2 + i) Fe − e Bibliography 127~~

~~2 πi (4 i) = 2 e− 4 − · 5~~

~~113 Remark 1. This last computation has been done by hand and has not been checked by a numerical computation. But if one believes in the Birch-Swinnerton-Dyer conjecture ([26]) then the value~~

1 LE φ N, ◦ 2! e should have something to do with an order of a Tate-Shafarewic group. Since it is not more than ﬁve minutes ago that I computed the value above I must confess that I am still pleased by the occurrence of the square. Remark 2. For this particular character the value

2 L(φ , 1) (1 2i)3 1 = − 2 (1 + 2i)2 √ L(eφ , 1) 5 has been numerically checked.e In this case I also computed numerically the value 22 1 + 2i L(φ2, 1) = ω2 √1 2i − 5 · (1 2i)2 − − e 1 1 where ω = dx and 2 √x x3 R0 − π arg √1 2i 0 −2h − i But this has to be taken with caution since we have not really proved this. The numerical values are equal up to 8 digits. But if we believe that this value is correct then we ﬁnd

~~1 2 3 1 πi 2 1 + 2i LE φ N, = ω 2 e− 4 (4 i) √1 2i ◦ 2! − · 52 − (1 2i)2 − − e 128 Bibliography Bibliography~~

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~~By Roger Howe1~~

Introduction 117 In the past few years the concept of wave front set [D] has proved fruitful for the theory of distributions and P.D.E. It seems it might also be of use in the representation theory of Lie groups. Its close relative, the singular spectrum of a hyperfunction, has already been discussed in a special context in [K-V], which served as the catalyst for this note. The purpose here is to deﬁne and discuss general properties of wave front sets of representations, and to give some examples. I would like to thank Nolan Wallach for very valuable discussions regarding this paper. Especially, the principle of proof of proposition ?? comes from him. Also I thank Richard Beals for valuable technical discussions.

~~1 Generalities~~

~~Let ρ be a representation of the Lie group G. For convenience we shall assume ρ is unitary, although this is not strictly necessary. Let H be~~

~~1Partially supported by NSF Grant MCS 7610435~~

~~131 132 Roger Howe~~

~~the Hilbert space on which ρ acts, and let J1(H) = J1 be the trace class operators on H. Given T J , put ∈ 1~~

~~trρ(T)(g) = tr(ρ(g)T) g G (1.1) ∈~~

~~where tr is the usual trace functional on J1. Then~~

~~trρ : J Cb(G) (1.2) 1 →~~

where Cb(G) is the space of bounded functions on G, is a norm-decreasing map. The image of trρ is called the space of (continuous) matrix coeﬃcients of ρ. We may also regard trρ(T) as a distribution on G by integration, in the usual fashion

~~trρ(T)( f ) = f (g) trρ(T)dg = tr(ρ( f )T) f C∞(G). (1.3) Z ∈ c G~~

118 Here dg is Haar maasure on G. Since trρ(T) is a distribution on G, we may consider its wave front set WF(trρ(T)). Our basic reference for wave front sets is [D] and we shall recall their basic deﬁnitions and properties as they are needed. For now, recall WF(trρ(T)) is a closed, conical (i.e., closed under positive dilations in the ﬁbers) set in T ∗G, the cotangent bundle of G.

~~Deﬁnition. WFρ is the closure of the union of WF(trρ(T)) as T varies over J1.~~

~~Thus WFρ is also a closed conical set of T ∗G.~~

~~Remark . This is not the same as the wave front set deﬁned in [H1], which is sort of a dual notion to the present one.~~

~~Proposition 1.1. WFρ is invariant under left and right translations of G on T ∗G. Wave Front Sets of Representations of Lie Groups 133~~

~~Proof. Deﬁne as usual left and right translations on functions and distributions:~~

1 Lg( f )(g′) = f (g− g′): Rg( f )(g′) = f (g′g) f Cc∞(G)  ∈ (1.4) Lg(D)(d) = D(L 1 f ); Rg(D)( f ) = D(R 1 f )D D(G).  g− g− ∈  Then we have the well-known relations

~~1 Lg trρ(T) = trρ(Tρ(g)− ); Rg trρ(T) = trρ(ρ(g)T) (1.5)~~

Left and right translations of G also induce in the usual way trans- formations Lg∗ and Rg∗ on T ∗G. By the naturality of the wave front set ([D], proposition 1.3.3.) one has, for a distribution D on G.

~~WF(LgD) = Lg∗(WF(D)) and WF(Rg(D)) = Rg∗(WF(D)). (1.6)~~

The proposition follows directly from the deﬁnition and equations (1.5) and (1.6). Let g be the Lie algebra of G, and let g∗ be the dual of g. Let Ad be the adjoint action of G on g, and let Ad∗ be the contragredient action on g∗. We can identify g∗ with the left invariant exterior 1-forms on G. This leads to an identiﬁcation

T ∗G G g∗ (1.7) ≃ × Thus if ψ C (G), we can regard dψ, the diﬀerential of ψ, as a g - 119 ∈ c∞ ∗ valued function on G. Doing so, we have the following behaviour under right and left translations

~~d(Lgψ) = Lgdψ d(Rgψ) = Ad g(Rgdψ) (1.8)~~

One sees from (1.8) that a bi-invariant set in T ∗(G) is identiﬁed via (1.7) with G X where X g is an Ad∗ G invariant set. Thus we can × ⊆ ∗ associate to WFρ a closed conical Ad∗ G-invariant subset of g∗, to be o o denoted WFρ. The set WFρ then determines WFρ via (1.7). It is conceivable that WFρ could be very uninteresting–it might always be all of g∗ for example. Thus it may be instructive to point out at 134 Roger Howe

the beginning that for irreducible ρ at least, WFρ is limited to a certain characteristic and non-trivial behavior. Let U(g) be the universal enveloping algebra of g. It is well known that there is a canonical linear isomorphism, the symmetrization map

~~σ : U(g) ∼ S (g) P(g∗) (1.9) −→ ≃~~

where S (g) is the symmetric algebra of g, and P(g∗) the polynomial algebra on g∗, the two algebras being identified in the standard way. The symmetrization σ is an intertwining map for the adjoint actions of G on U(g) and on P(g∗). Thus σ restricts to a linear isomorphism between ZU(g), the center of U(g), and IP(g∗), the Ad∗ G invariants in P(g∗). The map σ has a natural interpretation in terms of P.D.E. We can identify each u U(g) to a left invariant differential operator Ru on G. If ∈ Ru has order m, then the leading symbol of Ru, in the sense of P.D.E [D], will be a left-invariant section of S mT(G), the m-th symmetric power of the tangent bundle of G. Thus the symbol of Ru is determined by its value at the identity, which will be an element of S mg Pm(g ). It is ≃ ∗ known and easy to check from the definitions that the symbol of R is just the m-th homogeneous part of σ(u). Let V(g∗) denote the set of common zeroes of the homogeneous elements of positive degree of IP(g∗). We call V the characteristic variety of g∗ (or of G).

~~120 Let ρ be as above a unitary representation of G. Proposition 1.2. Let ρ be irreducible. Then~~

o WF V(g∗) (1.10) ρ ⊆ Proof. Since ρ is irreducible, the action of ZU(g) on the smooth vectors of ρ is by scalars [Se]. Say ρ(z)x = µ(z)x for x a smooth vector and z ZU(g), where µ : ZU(g) C is the inﬁnitesimal character of ρ. ∈ → Thus let x, y be smooth vectors in H, the space of ρ. Let Ex,y be the dyad Ex y(u) = (u, x)y u H (1.11) , ∈ Wave Front Sets of Representations of Lie Groups 135

~~Then trρ(Ex,y)(g) = (ρ(g)y, x) (1.12) It follows by diﬀerentiating (1.5) that~~

~~Rz trρ(Ex,y) = trρ(Ex,φ(z)y) = µ(z) trρ(Ex,y) (1.13)~~

~~Here Rz is as above, the right convolution operator on G corresponding to z. Since every element in J1 is a limit in the trace norm of sums of smooth dyads, and trρ is norm-decreasing, we ﬁnd that~~

~~RZ trρ(T) = µ(z) trρ(T) T J (1.14) ∈ 1~~

That is, the trρ(T) are all eigendistributions for ZU(g). Since as z varies in ZU(g), the symbol in the sense of P.D.E. will vary through all homogeneous elements of IP(g∗), we see that V(g∗) is just the intersection of all the characteristic directions of the Rz, z ZU(g). Hence by [D], ∈ proposition 5.1.1, we have the inclusions WFtrρ (T) G V(g∗) for all o ⊆ × T in J1. By deﬁnition of WFρ, the inclusion (1.10) follows. Remark. We can formulate a relative version of this also. Let N G be ⊆ a normal subgroup. Let ZU(N)G be the Ad G invariants in ZU(N), where N is the Lie algebra of N. The corresponding sub-algebra of P(N∗) G is clearly IP(N∗) , the Ad∗ G invariants in IP(N∗). Let V(N∗; G) be the intersection of the zeroes of the homogeneous elements of positive degree in IP(N∗). Then by the same proof as for the above propositions, we may assert: If ρ is an irreducible representation of G, and ρ H is the | restriction of ρ to N, then 121

~~o WF (ρ H) V(N∗; G) (1.15) | ⊆~~

Next we observe that WFρ behaves very simply under direct sums. If ρ is a representation of G, let nρ, where n is a natural number or , denote the n-fold direct sum of ρ with itself. If ρ and ρ are two ∞ 1 2 representations, recall that ρ and ρ are called quasi-equivalent if ρ 1 2 ∞ 1 and ρ are equivalent. ∞ 2 136 Roger Howe

~~Proposition 1.3.~~

o = o (a) If ρ1 and ρ2 are quasi-equivalent, then WFρ1 WFρ2 . (b) In general WFo(ρ ρ ) = WFo WFo 1 ⊕ 2 ρ1 ∪ ρ2 Proof. To prove (a), it is enough to show that WFo = WFo( ρ); but this ρ ∞ is clear because ρ and ρ have the same matrix coeﬃcients. Similarly, ∞ the general matrix coeﬃcient of ρ ρ is easily seen to have the form 1 ⊕ 2

~~trρ (T ) + trρ (T ), Ti J (Hi) 1 1 2 2 ∈ 1~~

where H1 is the space of ρi. Setting T1 = 0 and letting T2 vary, then vice-versa, we see WFo is contained in WFo(ρ ρ ). On the other ρi 1 ⊕ 2 hand, [D], deﬁnition 1.3.1 assures us of the other inclusion necessary for statement (b). This concludes the proposition. We will now give a technical result oﬀering various descriptions of o WFρ. Recall that if f is a function of a positive real variable t, then f is rapidly decreasing as t if →∞ n sup f (t) t : t 1 = γn( f ) < , all n in Z {| | ≥ } ∞ Let e denote the identity element of G. Let supp(ϕ) denote the support of ϕ C (G). ∈ c∞

~~Theorem 1.4. Let U g be an open set. The following conditions on ⊆ ∗ U are all equivalent~~

(i) U WFo is empty ∩ ρ 122 (ii) For any T in J (H), and every real-valued ψ C (G) such that 1 ∈ ∞ dψ(e) U, there is an open neighborhood V of e such that for ∈ any ϕ C (V) the integral ∈ c∞

~~itψ(g) I(ϕ, ψ, T)(t) = trρ(T)(g)ϕ(g)e dg (1.17) Z G Wave Front Sets of Representations of Lie Groups 137~~

is rapidly decreasing as t . Furthermore, if ψ = ψα and ϕ = →∞ ϕα depend smoothly on a parameter α varying in a neighborhood of 0 in Rk, then for some perhaps smaller neighborhood Y of 0 k in R , the neighborhood V and the quantities γn(I(ϕα, ψα, T)) can be chosen independently of α in Y. (iii) For all T in J , for all ϕ C (G) and for all real-valued ψ 1 ∈ c∞ ∈ C (G) such that dψ(supp ϕ) U, the integral I(ϕ, ψ, T) is rapidly c∞ ⊆ decreasing as t . If ϕ and ψ depend on a parameter α as in → ∞ (ii), then there is uniformity in α as described there. (iv) The same as (iii), but is enough to choose an open neighborhood V of e and choose ϕ C (V). ∈ c∞ (v) The same as (iii), but we have the estimates

~~γn(I(ϕ, ψ, T)) cn(ϕ, ψ) T (1.18) ≤ || ||1~~

for some number cn(ϕ, ψ). If α is an auxiliary parameter as described in (ii), then the numbers cn(ϕα, ψα) may be bounded uniformly on compact sets of α’s. (vi) For ϕ C (G) and real-valued ψ C (G) such that dψ(supp ϕ) ∈ c∞ ∈ ∞ ⊆ U, the norm of the operator ρ(ϕeitφ) is rapidly decreasing as t . If ϕ and ψ depend on a parameter α as in (ii), then → ∞ itψ the quantities γn( ρ(ϕe ) can be bounded uniformly on compact || || sets of α. (vii) Same as (vi), except it is enough to choose a neighborhood V of e and varify (vi) for ϕ C (V). ∈ c∞ Proof. First we will check that statements (ii) though (vii) are equivalent, then we will compare them with (i). It is immediate that (v) implies (iii) and that (iii) implies (iv). Likewise (vi) clearly implies (vii). Also, in view of formula (1.3) and the duality between J1(H) and the space L(H) of all bounded operators on H, we see that (v) and (vi) are equivalent. If ψ C (G), and dψ(e) U, then dψ 1(U) = V is a neighbor- ∈ ∞ ∈ − 1 hood of e in G. If V is as in (iv), then V V will be a neighborhood 123 ∩ 1 that works for (ii). Hence (iv) implies (ii). 138 Roger Howe

Fix ϕ C (G) and ψ C (G). Suppose for any T J , the ∈ c∞ ∈ ∞ ∈ 1 integral I(ϕ, ψ, T)(t) is rapidly decreasing as t . For any n, and → ∞ number a > 0, the set Xa of T such that γn(I(ϕ, ψ, T)) a is convex and ≤ symmetric around 0. Since I(ϕ, ψ, T)(t) is continuous on J1, we see that Xa is also closed. Since Xab = bXa, and Xa = J1 by assumption, we a 0 S≥ see Xa contains a neighborhood of the origin in J1. Thus we see that (iii) implies (v). We will show that (ii) implies (iii) by a partition of unity argument. Observe the identity

~~1 I(ϕ, ψ, T) = I(Lgϕ, Lgψ, Tρ(g)− ). (1.19)~~

This follows from the definition of I(ϕ, ψ, T) and formula (1.5). Suppose that dψ(supp ϕ) U, so that ϕ and ψ satisfy the hypotheses of (iii). By ⊆ formula (1.8), we see d(Lg 1ψ)(e) U if g supp ϕ. Then (ii) tells us − ∈ ∈ that given T J1, there is a neighborhood V = V(Lg 1ψ, Tρ(g)) such ∈ − that if ϕ′ Cc∞(V), then I(ϕ′, Lg 1ψ, Tρ(g)) is rapidly decreasing. From ∈ − (1.19) we can conclude that for g supp ϕ, there is a neighborhood Vg ∈ of g such that if ϕ C (Vg), then I(ϕ , ψ, T) is rapidly decreasing. ′′ ∈ c∞ ′′ We can cover supp ϕ with a finite number of the neighborhoods Vg, and construct a partition of unity subordinate to this cover of supp ϕ. That is, we can find gi such that the Vg cover supp ϕ, and we can find ϕ i i′′ ∈ Cc∞(Vgi ) such that ϕi′′ = 1 on supp ϕ. Then i P

I(ϕ, ψ, T) = I(ϕϕi′′, ψ, T) Xi so I(ϕ, ψ, T) is rapidly decreasing. Clearly we can do this uniformly in some auxiliary parameter α. Thus we see that (ii) implies (iii). A completely analogous, slightly simpler argument shows that (vii) implies (vi). Hence all conditions (ii) through (vii) are equivalent. Finally we observe that by [D], proposition 1.3.2, for any point λ ∈ U, the condition that the point (e, λ) T G not belong to WF(trρ T) for ∈ ∗ T J , is just statement (ii) restricted to those ψ such that dψ(e) = λ ∈ 1 (and with parameter α). Thus we see that (i) implies (ii). Conversely, Wave Front Sets of Representations of Lie Groups 139

(ii) certainly implies that (e, λ) < WF(trρ T) for any T J and any ∈ 1 λ U. Since U is open and WFρ is G-biinvariant, we ﬁnd that also (ii) 124 ∈ implies (i). Thus the theorem is proved. Using theorem 1.4 we can establish a relation between the wave front set of a representation and that of its restriction to a subgroup. Let H G be a Lie subgroup of H, with Lie algebra h. We have the ⊆ restriction map q : g∗ h∗ → Note that q is Ad∗ H-equivariant, and if H is normal in G, then q is Ad∗ G-equivariant.

~~Proposition 1.5. We have the inclusion~~

q(WFo) WFo(ρ H) (1.21) ρ ⊆ \ Proof. Let U be an open subset of h not intersecting WFo(ρ H). Choose ∗ | a small neighborhood V of the identity in G so that in VH there is a smooth cross-section Y to H, so we can write uniquely

~~v = yh v V, y Y, h H. ∈ ∈ ∈ Choose ϕ C (V) and let ψ C (V) be such that dψ q 1(U). We ∈ c∞ ∈ ∞ ⊆ − can compute~~

~~ρ(ϕeitψ) = ϕ(g)eitψ(g)ρ(g)dg (1.22) Z G~~

~~= ϕ(yh)eitψ(yh)ρ(yh)dydh Z Z Y H~~

~~itψ = ρ(y) ρ H(ϕye y) dy Z | Y where we have written~~

~~ϕy(h) = ϕ(yh) ψy(h) = ψ(yh) 140 Roger Howe~~

As y varies, ϕy varies smoothly in Cc∞(H) and ψ varies smoothly in C (H), with dψy(supp ϕy) U. Hence by theorem 1.4, part (vi), the ∞ ⊆ norms of the operators itψy ρ H(ϕye ) | 125 are rapidly decreasing as t , with uniform estimates at least locally →∞ in y. Since ϕy 0 for y outside a compact set we see from (22) that itψ ≡ 1 ρ(ϕe ) is also rapidly decreasing at t , whence q− (U) is disjoint o → ∞ from WFρ, by theorem 4, part (vii). An interesting aspect of proposition 1.5 is that it proceeds in the opposite direction from the standard results ([D] proposition 1.33, see also [?], section 1X.9) concerning restrictions of distributions and wave front sets. This contrast allows us to prove a partial converse to proposition 1.5. Let h⊥ be the kernel of the projection map q of (??). We will say H is crosswise to ρ if h WFo = 0 . When H is crosswise to ρ ⊥ ∩ ρ { } we can, according to [D], proposition 1.3.3, restrict trρ T (or any of its derivatives) to H. The wave front set of (trρ T) H, which will be the same | as the wave front set of trρ H(T), will then be contained in q(WF(trρ T)). | Combining this with proposition 1.5 we may assert.

~~o Proposition 1.6. If H is crosswise to WFρ, then WFo(ρ H) = q(WFo) (1.23) | ρ~~

Note that if H is crosswise to F, the projection q(WFρ) will be closed. In particular if h V(g ) = 0 , then H will be crosswise to all ⊥ ∩ ∗ { } irreducible ρ. Proposition 1.5 also implies a restriction on the wave front set of (outer) tensor products. Let G1 and G2 be two Lie groups, and ρi unitary representations of Gi on spaces Hi. We can form the tensor product representation ρ ρ of G G on H H . 1 ⊗ 2 1 × 2 1 × 2 Proposition 1.7. We have the inclusion

~~WFo(ρ ρ ) WFo WFo g g (1.24) 1 ⊗ 2 ⊆ ρ1 × ρ2 ⊆ 1 × 2 Wave Front Sets of Representations of Lie Groups 141~~

Proof. We have (ρ ρ ) G (dim ρ )ρ . Hence by propositions 1.5 1 ⊗ 2 | 1 ≃ 2 1 and 1.3, we see o o WF (ρ ρ ) WF g∗. 1 ⊗ 2 ⊆ ρ1 × 2 Interchanging G1 and G2, repeating and intersecting gives (24). 126 We remark that the inclusion ?? can be strict. An example of this will be found in part II. For certain representations there is a plausible alternate definition of wave front set. We consider this and compare it with our first notion given above. Recall that there is an antiautomorphism∗ on U(g) defined property that it is 1 on g: − x∗ = x x g. − ∈ If ρ is a unitary representation of G, then

ρ(u∗) = ρ(u)∗ u U(g) (1.26) ∈ where the∗ on the right-hand side indicates the restriction of the adjoint of ρ(u) to the space of smooth vectors of ρ. Thus if u = u∗, then ρ(u) is a symmetric operator, and elements of the form u∗u are mapped to non-negative symmetric operators, and so are sums of such elements.

We call sums ui∗ui in U(g) formally positive. Evidently the formally positive elementsP form a cone in U(g), invariant by∗. In the following discussion we take G to be unimodular for convenience. We will say that ρ is of strong trace class if there is some formally positive element v of U(g) such that ρ(v) (with domain understood to the the smooth vectors of ρ) is essentially self-adjoint, and invertible with trace class inverse. We note irreducible representations are often of strong trace class. If ρ is of strong trace class, then for all ϕ in Cc∞(G), the operator ρ(ϕ) will be trace class, with trace norm satisfying

1 1 ρ(ϕ) ρ(v)− ρ(Rvϕ) ρ(v)− Rvϕ (1.27) || ||1 ≤ || ||1|| || ≤ || ||1|| ||1 where ρ(ϕ) indicates the trace norm on J (H), and v U(g) is a || ||1 1 ∈ formally positive element which makes ρ strongly trace class, and Rv is 142 Roger Howe

the left-invariant operator on G corresponding to v, and ρ(Rvϕ) is the 1 || || usual operator norm of ρ(Rvϕ) and Rvϕ is the L -norm of Rvϕ as a || ||1 function on G. It is clear from (1.27) that the trace linear functional

~~χρ(ϕ) = trρ(ϕ) (1.28)~~

~~127 is a distribution on G. We of course call it the character of ρ. We note that χρ is a conjugation invariant distribution, in the sense that~~

~~χρ(Ad g(ϕ)) = χρ(ϕ) (1.29)~~

where Ad g(ϕ) = LgRg(ϕ). If ρ is of strong trace class, so that its character χρ is well-defined as a distribution, then in the context of this paper, an obvious thing to do is to consider the wave front set WF(χρ). This will be a conjugation invariant set in T ∗G. In particular the intersection of WF(χρ) with the cotangent space at the identity, which is canonically identifiable with g∗, defines a closed, Ad∗ G-invariant, conical set in g∗. Denote this set by o o WF (χρ). It is natural to compare this with our WFρ defined earlier.

~~Theorem 1.8. When ρ is of strong trace class with distributional character χρ, we have o o WF (χρ) = WFρ. (1.30)~~

~~Proof. Let v be a formally positive element of U(g) with respect to which ρ is strongly trace class. Write ρ(v) 1 = T J (H). Then for − ∈ 1 ϕ in Cc∞(G) we have~~

~~χρ(ϕ) = tr(ρ(ϕ)) = tr(Tρ(v)ρ(ϕ))~~

~~= tr(ρ(Lv(ϕ))T) = trρ(T)(Lvϕ) = Lv∗ (trρ(T))(ϕ).~~

~~In other words = χρ Lv∗ (trρ(T)) (1.31) Wave Front Sets of Representations of Lie Groups 143~~

Since action by diﬀerential operators does not increase the wave front set, we see WF(χρ) WF(trρ(T)) WFρ. ⊆ ⊆ Hence, looking at the ﬁbre of T ∗G over the identity of G we see that the left side of (1.30) is contained in the right side. o To prove the reverse inclusion, consider a point p in g WF (χρ). ∗ − Let U be a neighborhood of p with compact closure disjoint from 128 o WF (χρ). Since WF(χρ) is closed in T ∗G, there is a neighborhood V of the identity e in G such that V U T G is disjoint from WF(χρ). × ⊆ ∗ It follows that for ϕ in Cc∞(V) and real-valued ψ in C∞(V) such that dψ itψ (supp ϕ) U, one has that χρ(ϕe ) is rapidly decreasing as t , ⊆ → ∞ with estimates uniform in smooth parametrized families of ϕ’s and ψ’s. 2 Let V1 be a symmetric neighborhood of e such that V1 V. Then if ϕ itψ ⊆ ∈ Cc∞(V1), we see that χρ(Lg(ϕe )) is rapidly decreasing in t, uniformly in g in V1 and in any other auxiliary parameter of interest. Set

~~itψ 1 ϕe = ϕt and ϕt∗(g) = ϕt(g− ) where——indicates complex conjugation. Integrating, we ﬁnd~~

~~1 1 ϕ (g− )χρ(Lgϕ1)dg = χρ(ϕ (g− )(Lgϕt)dg (1.32) Z t Z t G G~~

= χρ(ϕ∗ ϕt) = tr(ρ(ϕ∗ ϕt)) = tr(ρ(ϕt)∗ρ(ϕt)) t ∗ t ∗ is rapidly decreasing as t . Here ϕ ϕt indicates the convolution → ∞ t∗ ∗ of these functions. But the ﬁnal expression in (1.32) is just the Hilbert- Schmidt norm of ρ(ϕt). Since it is rapidly decreasing, the operator norm of ρ(ϕt) is also. Hence criterion (vii) of Theorem 1.4 tells us U is disjoint o from WFρ, and Theorem 1.8 is established. Before concluding this section, let us mention two plausible general properties of wave front sets not established here. First, is it true that WFo(ρ ρ ) (WFo + WFo ) (the here denoting closure) for an 1 ⊗ 2 ⊆ ρ1 ρ2 inner tensor product? Second, is it true that WFo(indG σ) h ? H ⊇ ⊥ 144 Roger Howe 2 Examples

o Here we will show how to compute WFρ for various familiar classes of o groups, and examine the possibilities for WFρ in some interesting cases. A. Abelian Groups. If G is abelian, then G is a homomorphic image of a vector space V, so we may as well assume G = V. Then we may identify V with its 129 Lie algebra. Also the dual vector space V∗ can be identified with V, the Pontrjagin dual of V, by the usual method. Define b α : V∗ V → by b 2πiλ(v) α(λ)(v) = e λ V∗, v V. (2.1) ∈ ∈ Define Fourier transform from L1(V) to C0(V∗) by the usual recipe:

~~2πiλ(v) ϕ(λ) = ϕ(v)e− dv ϕ L1(V), λ V∗ (2.2) Z ∈ ∈ V b Then the inverse Fourier transform is~~

1 2πiλ(v) f − (v) = f (λ)e dv f L1(V∗), v V (2.3) Z ∈ ∈ b V∗ Let ρ be a unitary representation of V on the Hilbert space H. Take T J (H), and consider the matrix coeﬃcient trρ(T). According to ∈ 1 Bochner’s Theorem [R-S], trρ(T)∨ exits as a ﬁnite measure on V∗, positive if T is. Moreover from our formulas (1.5) and (2.2) we can compute that trρ(ρ(ϕ)T) = (ˇϕ) (trρ T) (2.4)

~~where b b b ϕ(v) = ϕˇ( v) (2.5) − We deﬁne supp ρ to be the closure of the union of the supports of the measures (tr(T)) . It is clear from (2.4) that~~

~~ρb(ϕ) = sup (ˇϕ) (λ): λ supp ρ (2.6) | | { ∈ }~~

~~b Wave Front Sets of Representations of Lie Groups 145~~

Given a set S in a vector space U, deﬁne AC(S ), the asymptotic cone of S as follows. Given u in U, if any cone containing a neighborhood of u intersects S in an unbounded set, then u is in AC(S ). In terms of these objects we can give the not unexpected description of WFρ. Proposition 2.1. For a unitary representation ρ of a vector space V, one has WFo = AC(supp ϕ) (2.7) ρ − 130

~~Remark. The minus sign in (2.7) is an artifact of our conventions and could be eliminated by appropriate juggling.~~

Proof. We will apply criterion (vi) of Theorem 1.4, with ψ = 2πλ, λ V . (We will actually use the deﬁnition 1.3.1 of [D] rather than ∈ ∗ proposition 1.3.2 used for Theorem 4). Take ϕ in Cc∞(V). Then one sees from (2.2) that

2πiλ (ϕe )∨ (λ′) = (ˇϕ) (λ′ + tλ) (2.8) Suppose that λ < AC(suppbρ). Then web can choose a small neigh- 0 − borhood U of λ such that the distance between tλ and supp ρ (in any 0 − convenient norm) increases linearly in t. Therefore tλ has a ball around − it of size γt, γ being some constant independent of λ, disjoint from ≥ supp ρ. Since (ϕ ˇ) is rapidly decreasing for ϕ C (V), we see from ∈ c∞ formulas (2.6) and (2.8) that ρ(ϕe2πitλ) decreases rapidly as t . || || → ∞ This shows the leftb side of (2.7) is contained in the right side. The reverse inclusion is equally easy. If λ AC(supp ρ), then no matter how − 0 ∈ small a neighborhood U of λ we choose, the cone on U will intersect 0 − supp ρ in a non-bounded set. This means we can choose t arbitrarily large, and λ in U, such that tλ , is in supp ρ. We may assume for con- 1 − 1 venience that ϕ is positive-deﬁnite, so that ϕ(0) = ϕ . Then we see || ||∞ that ρ(ϕe2πitλ1 ) = ϕ by formulas (2.6) and (2.8), so that U violates || || || ||∞ condition (vi) of Theorem 1.4. Hence the rightb side ofb (2.7) is contained in the left side, and theb proposition is proved. 146 Roger Howe

~~We will use proposition 2.1 to give an example of strict inclusion in proposition 1.7. Let V = R, and let N be the direct sum of the characters~~

~~t e2πin!t n 1 → ≥ Then supp ρ = n!, n Z+ { ∈ } Hence by proposition 2.1, we have~~

o WF (ρ) = AC(supp ρ) = R− = t R, t 0 . − { ∈ ≤ } 131 Consider the tensor product ρ ρ as a representation of R2. Then clearly ⊗ supp(ρ ρ) = (n!, m!) : n, m Z+ . ⊗ { ∈ } It is easy to see that AC(supp(ρ ρ)) consists of the positive x-axis, ⊗ the positive y-axis, and the positive ray of the 45◦ line x = y. Thus WFo(ρ ρ), being the negatives of these 3 rays, is properly contained in ⊗ WFo WFo, which is the whole southwest quadrant. ρ × ρ B: Nilpotent Groups. We will discuss only irreducible representations of general nilpotent groups. Let N be a nilpotent Lie group, assumed to be connected and simply connected for simplicity. Let N be its Lie algebra, and exp : N → N the exponential map. Let ρ be an irreducible representation of N. It is known that ρ is of strong trace class, and according to the orbit theory of Kirillov [K], there is an Ad∗ N orbit O(ρ) = O in N∗, such that

χρ(ϕ) = (ϕO exp) (λ)do(λ) ϕ C∞(N) (2.9) Z ∈ c O b where χρ is the character of ρ, as in (1.27), and is as in (2.2), and do is a properly normalized Ad∗ N invariant measure on O. Given formula (2.9) and theorem 1.8, it is an easy matter tob establish the following result. We omit the details. Wave Front Sets of Representations of Lie Groups 147

~~Proposition 2.2. If ρ is an irreducible representation of N, and O N ⊆ ∗ is the associated orbit, then~~

~~WFo = AC(O). (2.10) ρ −~~

C. Compact Groups. Let K be a compact connected Lie group, and let T K be a maxi- ⊆ mal torus. Let W be the Weyl group of T, the normalizer of T modulo the centralizer of T. Let t and k be the Lie algebras of T and K. If K is semi-simple we can identify k with k∗ via the Killing form. In general, we will suppose given some Ad K-invariant, negative deﬁnite, bilinear form on k allowing us to identify k and k∗. Then we can also identify t and t∗, and may regard t∗ as a subspace of k∗, and we will have

~~Ad∗ K(t∗) = k∗ (2.11)~~

Thus any Ad∗ K invariant set in k∗ is determined by its intersection with 132 t∗, and this intersection will be a Weyl group invariant set. Fix a Weyl + chamber C in t∗, and ﬁx an ordering of the roots of t by letting this chosen Weyl chamber be positive. We have

+ Ad∗ W(C ) = t∗ (2.12) so that an Ad∗ K invariant set in k∗ is determined by its intersection with C+. The irreducible representations of K are described by the celebrated highest weight theory of Cartan and Weyl. Let T be the character group of T. Since T is a quotient of t via the exponential map, we can as b described in paragraph IIA identify T with a lattice in t∗, the so-called lattice of weights. The intersection b T + = T C+ ∩ is called the set of dominant weights.b b The dominant weights parametrize the set K of irreducible unitary representations of K. We recall how. b 148 Roger Howe

~~Let kC be the complexiﬁcation of k. We can write~~

~~kC = tC Lα (2.13) ⊗ Xα~~

where the Lα are the root spaces, that is, the non-trivial eigenspaces of Ad T acting on kC. We parametrize Lα by the character α it deﬁnes, and we regard α as an element of t∗ as explained above. We call a root α positive if (α, c) 0 for all c C+, where (,) is the posited bilinear form ≤ ∈ by means of which we identiﬁed k and k∗. Denote the set of positive roots by +. Put P + N = Lα (2.14) αM+ ∈ P + Then N is a nilpotent subalgebra of kC, and it is known that

+ kC = tC N N− (2.15) ⊕ ⊕ + where N− is the image of N under complex conjugation in kC. Let ρ be a representation of K on a Hilbert space H. Denote by H+ the subspace of H annihilated by all elements of N+. The space H+ is the space of highest weight vectors for ρ. Clearly H is invariant by ρ(T), so it may 133 be decomposed into a direct sum

~~+ + H = Hγ (2.16) Xγ~~

~~where H+ is the eigenspace of T on which T acts by the character γ T. γ ∈ The highest weight theory asserts the following facts: b (i) Each γ is in T +~~

~~b + + + (ii) If ρ is irreducible, then dim H = 1, so that H = Hγ for some well-deﬁned γ.~~

~~(iii) The map from K to T + implied by (ii) is a bijection. b b Wave Front Sets of Representations of Lie Groups 149~~

~~Now consider an arbitrary unitary representation ρ of K. Denote the set of highest weights of ρ by supp ρ. Thus~~

~~supp ρ T + C+ ⊆ ⊆ The following result is very closely akinb to results in [K-V]. Proposition 2.3. For a unitary representation ρ of K, we have~~

WFo C+ = AC(supp ρ), or (2.17) − ρ ∩ o WF = Ad∗ K( AC(supp ρ)) ρ − Proof. By proposition 1.3, it suffices to prove this when ρ is multiplicity free, that is, when ρ contains only one copy of each of its irreducible constituents. Then H+ (the space of ρ being H as usual) will be multiplicity free under the action of T. Let σ denote the representation of T on H+. Then by definition supp σ = supp ρ. Let x and y be two vectors in H+. Consider the matrix coefficient trρ(Ex,y). Since the intersection of the characteristics of the elements of + + N is just t∗, and since trρ(Ex,y) is annihilated by N (acting either on the right or the left), we see by [D], proposition 5.1.1, that the wave-front set of trρ(Ex,y) at the identity of K is contained in t∗. This also implies by [D], proposition 1.3.3, that trρ(Ex,y) restricts to T; this restriction must of course just equal to trσ(Ex,y). One then has again by [D], proposition 1.3.3. o o o WF (trσ(Ex y)) WF (trρ(Ex y)) WF (2.18) , ⊆ , ⊆ ρ where in the first two expressions the o in WFo mean we are looking at the fibre over the identity in K. From (2.18) we immediately have 134

WFoσ WFo (2.19) ⊆ ρ o Since WF is Ad∗ K invariant, we see by proposition 2.1 that the left side of (2.17) contains the right side. On the other hand, since ρ is multiplicity free, it is of strong trace o class, so to compute WFρ, it is enough by Theorem 1.8 to compute 150 Roger Howe

o WF (χρ). Let ∆ be the element in U(k) corresponding to our given bilinear form. Then R∆ is elliptic, and ρ(1 + ∆) is positive definite, and some power of ρ(1+∆) has trace class inverse. Standard and straightforward arguments allow us to find x H+ such that for some sufficiently ∈ large l we have

~~χρ = R(1+∆)l Ad K(trρ(Ex,x))dk (2.20) Z K~~

~~Since R∆ is elliptic, we have~~

~~WF(χρ) = WF  Ad K(trρ(Ex,x))dk (2.21) Z  K    Ad K(WF(trρ(Ex x)))  ⊆ , Hence if we can show~~

o o WF (trρ(Ex x)) WF σ (2.22) , ⊆ we will be done. In fact (2.22) is proven in just the same manner as proposition 1.5. The reasoning is exactly the same as in equation (1.22), except instead of considering simply ρ(ϕeitψ), one looks at the product itψ ρ(ϕe )Ex,x.

D: Semisimple Groups. We come now to the motivating examples of this paper. Let G be a semisimple Lie group with ﬁnite center and with Iwasawa decomposition G = KAN g = k a N (2.23) ⊕ ⊕ One also has the Cartan decomposition

g = k p = k Ad K(a) (2.24) ⊕ ⊕ Wave Front Sets of Representations of Lie Groups 151 where p is the orthogonal complement of k with respect to the Killing 135 form of g. We identify g with g∗ via the Killing form. Thus in what follows we will speak of g when strictly we should say g∗. Let N be the nilpotent set of g. It is well known that N = V(g) is the characteristic variety of g, in the sense of proposition 1.2. It is also known that there are only ﬁnitely many conjugacy classes of nilpotent elements. Thus from proposition 1.2 we have the following result. Proposition 2.4. If ρ is an irreducible unitary representation of G, then

WFo N. (2.25) ρ ⊆ o In particular, there are only finitely many possibilities for WFρ. Let ρ be irreducible, and consider ρ/K. It is a classic result of Harish-Chandra (see [W]) that ρ/K contains each irreducible representation of K a finite number of times, and that in fact ρ/K is of strong trace class. These facts are also reflected in the behavior of wave front sets. From the Cartan decomposition (2.24), noting that p consists of o semisimple elements, we see that k is crosswise to WFρ in the sense of proposition 1.6. Thus we have the following immediate consequence of that result. Proposition 2.5.

~~(a) For irreducible ρ we have~~

~~WFo(ρ/K) = q(WFoρ) (2.26)~~

~~where q is orthogonal projection of g onto k (with kernel p).~~

~~(b) In particular WFo(ρ/K) is the orthogonal projection on K of certain nilpotent orbits in g, and is one of only ﬁnitely many possibilities.~~

Remark . Part (b) of this proposition is very similar to proposition of [K-V]. However, the proof is substantially diﬀerent from the proof in [K-V]. Also, Kashiwara-Vergne do not relate (their version of) WFo(ρ/K) o to an object on G attached intrinsically to ρ. We note that WFρ is a ﬁner 152 Roger Howe

o invariant than WF (ρ/K), as simple examples already on SL3(R) show. (However WF is not ﬁner than ρ/K, when multiplicities are taken into account. It seems to be roughly equivalent to WFo(ρ/K) plus some rough information on multiplicities. See the discussion below of the o analogy with the op-adic case). Furthermore WFρ provides a link be- 136 tween the N-spectrum and the K-spectrum of ρ, empirical observation of which was the original motivation of Kashiwara-Vergne. Indeed applying proposition 1.5 to N, and using proposition 2.5 we arrive at the following fact. Note that the Killing form induces the identiﬁcation

~~N∗ g/(a N) (2.27) ≃ ⊕ Let q′ : g g/(a N) (2.28) → ⊕ be the natural quotient map.~~

~~Proposition 2.6. We have the inclusion~~

o 1 o WF (ρ/K) q(q′− (WF (ρ/N)) (2.29) ⊆ Acutally, this proposition is true with N replaced by any subgroup of G. To illustrate the above results, we offer some observations about the symplectic group Sp2n(R) = Sp. This is the subgroup of GL2n preserv- ing the standard symplectic form on R2n. Similar ideas apply to other classical groups. Each element of sp may be regarded as a linear transformation on R2n in the obvious way, and as such may be assigned its rank, a positive integer. Given an irreducible representation ρ of Sp, we define the singular rank of ρ to be the maximum of the rank of the o elements of WFρ. There are other useful notions of rank for ρ also. Let X be a maximal 2n isotropic subspace of R , and let N1 be the subgroup of Sp that leaves X fixed pointwise. It is well known that N S 2(X), the second sym- 1 ≃ metric power of X. Hence N1∗ is identifiable to the space of symmetric bilinear forms on X, and each of its elements has a well-defined rank. Wave Front Sets of Representations of Lie Groups 153

o We will say a representation ρ of Sp has N1-rank j if WF (ρ/N1) contains elements of rank j, but none of rank greater than j. By means of Proposition 2.1 and a little symplectic geometry, the notion of N1-rank can be made considerably more concrete. It is discussed at more length in [H2]. The singular rank of ρ can vary from zero to 2n 1, while the N - − 1 rank can vary only from zero to n. However, within their common range, they are closely related.

~~Proposition 2.7. Given irreducible ρ with N1 rank less than n, one has 137 the inequality singular rank (ρ) N rank (ρ). (2.30) ≤ 1 Probably (2.30) should be an equality.~~

Proof. By applying proposition 1.5, we reduce the proof to an exercize in symplectic geometry. Let Y be an isotropic subspace of R2n comple- mentary to X. Let N1− be the subalgebra of sp annihilating Y, and let m be the subalgebra of sp stabilizing X and Y. Then

~~sp = N− m N . (2.31) 1 ⊕ ⊕ 1~~

Also the orthogonal complement of N1 in sp with respect to the Killing form is just m N , so we may identify N with N . Thus what we want ⊕ 1 1− 1∗ to show is that in a nilpotent Ad Sp orbit in sp consisting of elements of rank l < n, there are elements whose N1− component in the decomposition (2.31) has rank l also. The rank of the N component of s sp is 1− ∈ easily seen to be rank(s/X) dim(s(X) X) (2.32) − ∩ Hence, reversing the roles of s and X, it will suﬃce, given s of rank l < n to ﬁnd a maximal isotropic X such that (2.32) also equals l. This entails rank(s/X) = l s(X) X = 0. (2.33) ∩ Consider the action of s. It is elementary that

~~im s = (ker s)⊥ 154 Roger Howe~~

~~where ⊥ indicates orthogonal complement for the standard symplectic form on R2n. Hence Z = im s ker s is isotropic, and ∩~~

~~im s + ker s = Z⊥.~~

We can write R2n = Z V V Z ⊕ 1 ⊕ 2 ⊕ where V1 is a complement to Z in im s, ande V2 is a complement to Z in ker s, and Z is an isotropic complement to Z in (V V ) . The 1 ⊕ 2 ⊥ 138 assumption that rank s < n implies dim V < dim V . Hence we may e 1 2 choose an embedding α : V V such that α(v ), α(v ) = v , v , 1 → 2 h 1 1′ i −h 1 1′ i where , denotes the symplectic form on R2. The space U of points h i 1 U = v + α(v): v V 1 { ∈ 1} is then isotropic. Let U be a maximal isotropic subspace of α(V ) V . 2 1 ⊥∩ 2 Then X = U U Z is a maximal isotropic subspace of R2n, and it is 1 ⊕ 2 ⊕ easy to check it satisfies the conditions (2.33). e To illustrate proposition 2.7, consider the two components of the oscillator representation [H3]. It is an easy matter to compute their N1 spectrum, and in particular to see they have N1 rank equal to one, hence by the proposition, singular rank equal to one. (Singular rank zero would imply only a finite number of K-types, hence finite dimen- sionality). There are only two conjugacy classes of rank one nilpotent elements, the transvections

ty : x x, y y x, y R2n → h i ∈ where , denotes the symplectic form, forming one and their negatives h i forming the other. The two holomorphic oscillator representations have the class of transvections for their wave front set, and the antiholomor- phic oscillator representations have the negatives of the transvections for their wave front set. More generally the representation of Sp2n coming from its pairing with Op,q inside Sp2n(p+q) (see [H3]) has N1 rank equal to p + q, with obvious consequences for the wave-front sets of its irreducible components if p + q < n. Bibliography 155

We will conclude the paper with a few remarks. First, the analogy of the present discussion with the results of [H4] and [HC] should be pointed out. In those papers it is shown that for an irreducible representation ρ of a reductive p-adic group G, over a ﬁeld of characteristic zero, the character χρ of ρ has an “asympotic expansion”, valid in a neighborhood of the identity. This expansion expresses χρ as a linear combination of distributions attached in a direct way to nilpotent conjugacy classes in the Lie algebra g of G. From this expansion, one can read oﬀ directly information about the asymptotics of the K-spectrum, or the N-spectrum, where K is a maximal compact subgroup of G, and N a unipotent subgroup. This expansion is thus comparable to the wave- 139 front set, but it is more precise in two ways. First, it permits more precise description of the K – or N – spectra than does the wave front set. Secondly, it attaches to ρ not simply a closed set of nilpotent orbits, but a collection of individual orbits with numbers, which might be thought of as multiplicities, attached. It would clearly be desirable to have an analogue of this expansion for groups over R. It seems that Barbasch and Vogan [BV] have established the existence of such an analogue. A second analogy that might be made is with the characteristic variety of a primitive ideal of a semisimple Lie algebra, as discussed by Borho and Kraft in [B-K]. It would seem the wave front set is the analytical analogue of their construction.

~~Bibliography~~

~~[BV] BARBASCH D. and D. VOGAN, The local structure of characters, preprint.~~

~~[B-K] BORHO W. and H. KRAFT, Uber die Gelfand-Kirillov– Dimension, Math. Ann. 22 (1976), 1–24.~~

~~[D] DUISTERMAAT J. Fourier Integral Operators, Courant Institute of Math. Sci., New York 1973. 156 Bibliography~~

~~[HC] HARISH CHANDRA, The characters of reductive p-adic groups, in Contributions to Algebra, Academic Press, New York (1977), 175-182.~~

~~[H1] HOWE R. On a connection between nilpotent groups and oscilla- tory integrals associated to singularities, Pac. J. Math. 73 (1977), 329–364.~~

~~[H2] HOWE R. On the N-spectrum of representations of semisimple groups, (in preparation.)~~

~~[H3] HOWE R. θ-series and Invariant Theory, to appear Proc. Symp. Pure Math, A.M.S., Providence, R.I., (1979).~~

~~[H4] HOWE R. The Fourier Transform and Germs of Characters, Math. Ann. 208 (1974), 305–322.~~

~~140 [J] JOSEPH A. A. Characteristic Variety for the Primitive Spectrum of a Semisimple Lie Algebra, Springer Lecture Notes in Math. 587, 102–118.~~

~~[K-V] KASHIWARA M. and M. VERGNE, K-types and singular spectrum, Springer Lecture Notes Vol. 728.~~

~~[K] KIRILLOV A.A. Unitary representations of nilpotent Lie groups, Usp. Mat. Nauk 106 (1962), 57–110.~~

~~[R-S] REED M. and B. SIMON, Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self Adjointness.~~

~~[Se] SEGAL I. Hypermaximality of certain operators on Lie groups, P.A.M.S. 3 (1952), 13–15.~~

~~[W] WARNER G. Harmonic Analysis on Semisimple Lie Groups I, Grund, Math. Wiss. 188, Springer Verlag, New York, Berlin, Hei- delberg 1972. ON P-ADIC REPRESENTATIONS ASSOCIATED WITH Zp-EXTENSIONS~~

~~By Kenkichi Iwasawa~~

In the present paper, we shall discuss some results on the p-adic 141 representations of Galois groups, associated with so-called cyclotomic Zp-extensions of finite algebraic number fields. 1. Let p be a prime number which will be fixed throughout the following, and let Zp and Qp denote the ring of p-adic integers and the field of p-adic numbers respectively. A Galois extension K/k is called a Zp- extension if its Galois group is isomorphic to the additive group of the 1 compact ring Zp . Let Ω denote the field of all algebraic numbers, i.e., the algebraic closure of the rational field Q in the field C of all complex numbers, and let W be the group of all pn-th roots of unity in Ω for all n 0. Then the∞ field Q(W ) contains a unique subfield Q which ≥ ∞ ∞ is a Zp-extension over Q. In fact, Q is the unique Zp-extension over Q contained in Ω, and the degree of the∞ extension Q(W )/Q is either p 1 or 2 according as p > 2 or p = 2. For any finite extension∞ ∞ k of Q, − the composite k = kQ is then a Zp-extension over k and it is called ∞ ∞ the cyclotomic Zp-extension over k. For each integer n 0, there then n ≥ exists a unique intermediate field kn with [knK] = p , and

~~k = k0 k1 ... kn ... k = kn. ⊂ ⊂ ⊂ ⊂ ⊂ ∞ [n 0 ≥ 1 For various deﬁnitions and results on Zp-extensions referred to throughout the following, see Iwasawa [5] or Lang [6].~~

~~157 158 Kenkichi Iwasawa~~

~~Let Cn denote the Sylow p-subgroup of the ideal class group of kn. For n m, kn km, there exists a natural homomorphism Cn Cm, and ≤ ⊆ → these homomorphisms deﬁne the direct limit~~

~~C = lim Cn. ∞ −−→ Clearly C is a p-primary abelian group and its Tate module T(C ) is a ∞ ∞ Zp-module. It is known that~~

~~λ T(C ) Zp ∞ ≃~~

~~142 where λ = λp(k) is a non-negative integer, called the λ-invariant of k for the prime number p. Hence~~

~~V = T(C ) Qp ∞ Z⊗p~~

~~is a λ-dimensional vector space over Qp. Let~~

Γ= Gal(k /k) = lim Gal(kn/k) ∞ ←−− so that Γ Zp. Clearly Gal(kn/k) acts on Cn for each n 0 and hence ≃ ≥ Γ acts on C = lim Cn in the natural manner. Therefore Γ acts also ∞ on T(C ) and V.−−→ Thus we have a natural continuous finite dimensional p∞-adic representation of the Galois group Γ = Gal(k /k) on the ∞ λ-dimensional vector space V over Qp. We shall next investigate the properties of the p-adic representation space V for Γ. 2. Let us first consider the special case where p > 2 and where k = Q( √p 1) = the cyclotomic field of p-th roots of unity. Let K = k = kQ = Q(W ). ∞ ∞ ∞ In this case, K/Q is an abelian extension and

~~G = Gal(K/Q) =Γ ∆ ×~~

where Γ = Gal(K/k) Zp and where ∆ = Gal(K/Q ) = Gal(k/Q) is a ≃ ∞ cyclic group of order p 1. Let ∆ denote the character group of ∆; we − b On P-Adic Representations Associated with Zp-Extensions 159 may identify ∆ with Hom(∆, Z×p) where Z×p denotes the multiplicative group of all p-adicb units in Qp. It is well known that ∆ may be identiﬁed also with the group of all Dirichlet characters to the modulus p and that b it is generated by a special character ω called the Teichmuller character for p. A character χ in ∆ is called even or odd according as χ( 1) = 1 − or χ( 1) = 1 respectively. − − b As one sees immediately, in this special case, not only Γ= Gal(K/k) but G = Gal(K/Q) also acts on C , T(C ), and V = T(C ) Qp natu- ∞ ∞ ∞ Z⊗p rally. Hence V is again a p-adic representation space for G. For each χ in ∆, let Vχ = v v V, δ v = χ(δ)v for all δ in ∆ . e { | ∈ · } Since G =Γ ∆, Vχ is then a Γ-subspace of V and 143 × V = Vχ, χ ∆. ⊗χ ∈ b 1+p Let γo denote the element of Γ such that γ0(ζ) = ζ for all ζ in W . γo is a topological generator of Γ; namely, the cyclic subgroup generated∞ by γ0 is dense in Γ. For each χ in ∆, let

~~gχ(X) = the characteristic polynomialb of γ 1 acting on Vχ 0 − and let g(X) = the characteristic polynomial of γ 1 acting on V 0 − = gχ(X). Yχ~~

On the other hand, let Lp(s; χ) denote the p-adic L-function for the Dirichlet character χ in ∆. It is known in the theory of p-adic L-functions2 that for each such χ, there exists a power series ξ (T) in the ring Z [[T]] b χ p of all formal power series in an indeterminate T with coeﬃcients in Zp such that = ξ ((1 + p)s 1) , for χ , 1, s Z , ωχ 1 p Lp(s; χ) − s − 1 s ∈ = ξω((1 + p) 1)/((1 + p) 1), , for χ = 1, s Zp, s , 1. − − − ∈ 2See Iwasawa [4] or Lang [6]. 160 Kenkichi Iwasawa

Since Lp(s; χ) . 0 if χ is even but Lp(s; χ) 0 if χ is odd, ξχ(T) 0 if χ ≡ ≡ is even and ξχ(T) . 0 if χ is odd. By Weierstrass’ preparation theorem, ξχ(T) for add χ can be uniquely written in the form

~~eχ ξχ(T) = ηχ(T)p fχ(T)~~

where ηχ(T) is an invertible power series in the ring Zp[[T]], eχ is a non- 3 negative integer , and fχ(T) is a so-called distinguished polynomial in Zp[T]. The next theorem tells us that there exists a relation between the p-adic representation of Γ = Gal(K/k) on V and the p-adic L-functions Lp(s; χ) for the characters χ in ∆, or, more precisely, between the polynomials g (X) and f (T) deﬁned above. Namely, we have the following χ χ b result4:

~~144 Theorem 1. Let k+ denote the maximal real subﬁeld of the cyclotomic p ﬁeld k = Q( √1) and let h+ be the class number of k+. Assume that h+ is not divisibleq by p. Then~~

~~gχ(X) = 1, Vχ = 0, for all even χ in ∆,~~

~~gχ(X) = fχ(X), for all odd χ in ∆b.~~

The assumption p ∤ h+ in the theorem is knownb as Vandiver’s conjecture. It has been verified by numerical computation for all primes p < 125, 000, and no counter example is yet found. On the other hand, + if we define, following Leopoldt, the p-adic zeta function ζp(s; k ) of the totally real field k+ by

+ + ζp(s; k ) = Lp(s; χ), χ ∆, χ( 1) = 1, Yχ ∈ − b + + then the theorem implies that under the assumption p ∤ h , ζp(s; k ) is essentially equal to the characteristic polynomial g(X) of γ 1 acting 0 − on the representation space V over Qp, up to the change of variables s (1 + p)s. The result is mysteriously analogous to a well known → 3 A recent theorem of B. Ferrero and L. Washington implies that ex = 0 for all odd χ. 4See Iwasawa [3]. On P-Adic Representations Associated with Zp-Extensions 161 theorem of A. Weil which states that a similar relation exists between the zeta function of an algebraic curve defined over a finite field and the characteristic polynomial of the Frobenius endomorphism acting on the p-adic representation space defined by the Jacobian variety of that curve. Now, although the above theorem is proved only for a very special case (and even that under the assumption p ∤ h+), we feel that it is not just an isolated fact for k = Q( √p 1), but is rather a part of a much more general result on teh cyclotomic Zp-extensions over finite algebraic number fields. In fact, Greenberg [2] generalizes Theorem 1 to the case where the ground field k is a certain type of finite abelian extension over the rational field Q, and Coates [1] also discusses such a generalization for an abelian extension k of an arbitrary totally real field. In the following, we shall report some results on cyclotomic Zp-extensions, related to some further generalization of the above Theorem 1. 3. We now assume that p is an odd prime, p > 25, and consider as our ground field a finite algebraic number field k with the following 145 properties: (i) k is a Galois extension of the rational field, (ii) k contains primitive p-th roots of unity so that it is a totally imaginary field, (iii) k also contains a totally real subfield k+ with [k : k+] = 2; namely, k is a number field of C-M type. In general, let J denote the automorphism of the complex field C which maps each complex number α to its complex conjugate α. For simplicity, the restriction of J on any subfield of C, invariant under J, will be denoted again by J. Let ∆= Gal(k/Q) for the field k mentioned above. Then by (ii) and (iii), J is an element in the center of ∆ and J , 1, J2 = 1. As in §1, let K = k = kQ ∞ ∞ 5The case p = 2 can be treated similarly but with some modifications. 162 Kenkichi Iwasawa

denote the cyclotomic Zp-extension over k. Since k contains p-th roots of unity, K = k(W ). Similarly, let K+ = k+ = k+Q be the cyclotomic ∞ + + ∞ ∞ Zp-extension over k . Then K is a totally real subﬁeld of the totally imaginary ﬁeld K with [K; K+] = 2. Clearly K/Q is a Galois extension because both k/Q and Q /Q are Galois extensions. Hence, let ∞

~~G = Gal(K/Q), Γ= Gal(K/k) Zp. ≃ Then we see immediately that Γ is a central subgroup of G and~~

~~∆= Gal(k/Q) = G/Γ.~~

As in the special case of §2, the Galois group G acts on C , T(C ), and ∞ ∞ V = T(C ) Qp so that V provides us with a finite dimensional p-adic ∞ Z⊗p representation space for G = Gal(K/Q). + Theorem 2. Assume that λp(k ) = 0 and that the so-called Leopoldt’s + conjecture holds for all intermediate fields kn , n 0, of the extension + + ≥ K /k . Then V = T(C ) Qp is cyclic over G = Gal(K/Q); namely, ∞ Z⊗p there exists a vector vo in V such that the whole space V is spanned over Qp by the vectors σ v , σ G. · 0 ∈ + Recall that λp(k ) denotes the λ-invariant of the totally real field + + k for the prime p and that Leopoldt’s conjecture for kn states that any + 146 set of units in kn , multiplicatively linearly independent over the ring of rational integers Z, remains multiplicatively linearly independent over Zp when these units are imbedded in the multiplicative group of the + algebra k Qp. We note that both these assumptions are conjectured ⊗Q to be true for any totally real number field k+. Note also that since λ T(C ) Zp, the conclusion of the theorem is equivalent to say that ∞ ≃ there exists an element v0 in T(C ) such that the elements of the form ∞ σ v0, σ G, generate over Zp a submodule of finite index in T(C ). · ∈ ∞ The proof of the theorem will be briefly indicated in the next section. In general, let G be any profinite group and let G = lim Gi with a family of finite groups Gi . The homomorphisms G j ←−−Gi, i j, { } → ≤ On P-Adic Representations Associated with Zp-Extensions 163 which define the inverse limit, induce the homomorphisms Zp[G j] → Zp[Gi] of the group rings of finite groups over Zp, and they in turn define

Zp[[G]] = lim Zp[Gi]. ←−− Zp[[G]] is a compact topological algebra over Zp and it depends only upon G and is independent of the family Gi such that G = lim Gi. { } We apply the above general remark for G = Gal(K/Q) in←−− Theorem 2 and deﬁne R = Zp[[G]], R′ = R Qp. Z⊗p

Let Gn = Gal(kn/Q), Rn = Zp[Gn], n 0. Since G = lim Gn, we then ≥ have −−→ R = lim Rn. ←−− Since Cn is an Rn-module in the obvious manner, C = lim Cn is an ∞ R-module. Hence T(C ) also is an R-module and V = T(C ←−−) Qp is an ∞ ∞ Z⊗p R′-module. We next deﬁne a subset An of Rn by

~~An = α α (1 J)Rn, α Cn = 0 . { | ∈ − · }~~

Note that J = J kn is contained in the center of Gn so that An is a two- | sided ideal of Rn, contained in (1 J)Rn. Furthermore, if n is large − enough and m n, then the homomorphism Rm Rn maps Am into An. ≥ → Therefore A = lim An ←−− is deﬁned and it is a two-sided ideal of R, contained in (1 J)R. Let 147 −

~~A′ = A Qp. Z⊗p~~

~~Clearly A′ is a two-sided ideal of R′ = R Qp, contained in (1 J)R′. Z⊗p − Moreover, it can also be proved that~~

~~A′ = α′ α′ (1 J)R′, α′ V = 0 , { | ∈ − · } 164 Kenkichi Iwasawa~~

~~namely, that A is the annihilator of the R -module V in (1 J)R . Let ′ ′ − ′ d = [k : Q].~~

~~Using Theorem 2, we can then easily prove the following Theorem 3. Let V′ = (1 J)R′/A′. − Under the same assumptions as in Theorem 2, there exist exact sequences of R′-modules~~

~~d V′ V 0, 0 V′ V . → → → →~~

In particular, V′ is a finite dimensional vector space over Qp, and as p- adic representation spaces for G = Gal(K/Q), V and V′ have the same composition factors. At this point, let us consider again the special case where k = Q( √p 1), p > 2; the field Q( √p 1) certainly satisfies the conditions (i), (ii), and (iii) stated at the beginning of this section. In this case, K = k , K+ = k+ , + ∞ ∞ and kn , n 0, are all abelian extensions over Q, and Leopoldt’s conjec- +≥ ture for kn is known to be true by a theorem of Brumer. On the other + + hand, it is easy to deduce λp(k ) = 0 from Vandiver’s conjecture p ∤ h for the class number h+ of k+. Therefore we know by Theorem 2 that under the assumption p ∤ h+, V is cyclic over G = Gal(K/Q), namely,

~~V = R′v0~~

+ with some vector v in V. Now, λp(k ) = 0 also implies V = (1 J)V 0 − so that V = (1 J)R v . Since G = Gal(K/Q) is an abelian group − ′ 0 in this case, both R and R′ are commutative rings. Hence it follows from the above that the map α α v , α (1 J)R , induces an ′ → ′ 0 ′ ∈ − ′ 148 R′-isomorphism V′ = (1 J)R′/A′ ∼ V. − −→ Furthermore, we know in this special case that there are many explicitly described elements in the ideal An of Rn, n 0, called Stickel- ≥ berger operators for kn, and that the p-adic L-functions Lp(s; χ) for χ On P-Adic Representations Associated with Zp-Extensions 165

+ in ∆ = Hom(∆, Ωp) can be constructed by means of such Stickelberger operators6. Thus we obtain a relation between the p-adic representation b space V′ and the p-adic L-functions Lp(s; χ), and hence between V and Lp(s; χ) through the above isomorphism. This is the way how Theorem 1 is proved, and the proof is similar for Greenberg’s generalization. We now consider again the general case where k is any finite algebraic number field satisfying the conditions (i), (ii), and (iii). For each C-M sub-field k′ of k such that k/k′ is abelian, Stickelberger operators for kn/k′ are still defined, and it is proved by Deligne and Ribet that such Stickelberger operators are related to abelian p-adic L-functions for k k+ in much the same way as in the special case mentioned above. ′ ∩ However, it is not known whether such general Stickelberger operators belong to the ideal An and provide us with any essential part of An defined above7. This prevents us from obtaining any nice relation between the p-adic representation space V′ and p-adic L-functions. On the other hand, we can find examples of k/Q, satisfying (i), (ii), (iii) and also the assumptions in Theorem 2, such that the representation spaces V and V′ for G = Gal(K/Q) in Theorem 3 are not isomorphic to each other. Thus we see that the results of Theorems 2, 3 tell us much less on the nature of the p-adic representation space V for G = Gal(K/Q) than Theorem 1 for the special case k = Q( √p 1). Nevertheless, we still feel and hope that those theorems would be of some use in the future investigations to obtain a full generalization of Theorem 1 in §2. We also note in this connection that in such a generalization of Theorem 1, one has certainly to consider p-adic (non-abelian) Artin L-functions. Given any Galois extension L/K of totally real finite algebraic number fields, it is not difficult to define p-adic Artin L-function Lp(s; χ) for each character χ of the Galois group Gal(L/K) so that Lp(s; χ) 149 is related to the classical Artin L-function L(s; χ) in the usual manner and that those Lp(s; χ) share with the classical functions L(s; χ) all essential formal properties such as the formula concerning induced characters. One can even formulate the p-adic Artin conjecture for such

~~6See Iwasawa [4] or Lang [6]. 7See the discussions in Coates [1]. 166 Kenkichi Iwasawa~~

L-functions; the conjecture is not yet verified and, in fact, it is closely related to the above mentioned problem of generalizing Theorem 1. For all these, we refer the reader to forth-coming papers by R. Greenberg and B. Gross, noting here only that Weil’s solution of Artin’s conjecture for L-functions of algebraic curves defined over finite fields is based upon the study of the representations of Galois groups on the spaces similar to V mentioned above. 4. We shall next briefly indicate an outline of the proof of Theorem 28. Following the general definition in §3, let

~~Λ= Zp[[Γ]]~~

for the proﬁnite group Γ = Gal(K/k), and let γ0 be any topological generator of Γ Zp. Let Zp[[T]] denote as in §2 the ring of all formal ≃ power series in T with coeﬃcients in Zp. Then it is known that there is a unique isomorphism of compact algebras over Zp:

~~Λ= Zp[[Γ]] ∼ Zp[[T]] −→ such that γ 1 + T. Hence ﬁxing a topological generator γ , we may 0 → 0 identify Λ= Zp[[Γ]] with Zp[[T]] so that γ0 = 1 + T. Then~~

~~Λ′ = Λ Qp = Zp[[T]] Qp Z⊗p Z⊗p~~

and it is easy to see that Λ′ is a principal ideal domain. One also proves immediately that Λ= Zp[[Γ]] is a central subalgebra of R Zp[[G]] and − that the latter is a free Λ-module of rank d = [k : Q] = [G : Γ]. Hence R′ = R Qp is an algebra over Λ′ = Λ Qp and it is a free module of Z⊗p Z⊗p rank d over the principal ideal domain Λ′. Now, let L denote the maximal unramified abelian p-extension (i.e., Hilbert’s p-class field) over K, and M the maximal p-ramified abelian 150 p-extension over K. Then

~~Q k K L M ⊆ ⊆ ⊆ ⊆ 8Cf. the proof of Theorem ?? in Greenberg [2]. On P-Adic Representations Associated with Zp-Extensions 167 and both L/Q and M/Q are Galois extensions. Let~~

~~X = Gal(L/K), Y = Gal(M/K).~~

These are abelian pro-p-groups and, hence, are Zp-modules in the natural manner. Since G = Gal(K/Q) acts on X and Y in the obvious way, we see that both X and Y are R-modules and, consequently, also Λ-modules. It is know that X is a torsion Λ-module so that there is an element ξ , 0 in Λ such that ξ X = 0. · Let X′ = X Qp. Z⊗p

~~It is clear that X′ is an R′-module. However it is also known in the theory of Zp-extensions that~~

V = T(C ) Qp ∼ X′ ∞ Z⊗p −→ as modules over R′. Hence, in order to prove Theorem 2, we have only to show that X′ is cyclic over R′ under the assumptions of that theorem. Let Y− denote the submodule of all y in Y satisfying (1 + J)y = 0. Since p > 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. Z⊗p~~

~~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′/TZ′ R′−/TR′− ≃ 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/TA B/TB ≃~~

~~as R′-modules. Then, as modules over R′,~~

~~A/pA B/pB ≃~~

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 Zp-extensions. The rest of Lemma 1 can be proved by considering the Galois group of the maximal p-ramified abelian p- extension 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 = × ∞ Zp[[G]] is nothing but the group ring of the finite group ∆ over Λ = Zp[[Γ]]: R = Λ[∆]. Hence R′ = Λ[∆]

where Λ′ = Λ Qp is a principal ideal domain. The lemma then follows Z⊗p 152 easily from the results of Swan on the group ring of ﬁnite groups over On P-Adic Representations Associated with Zp-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 Zp-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 = Zp[[G]], R′ = R Qp. Z⊗p As in the case discussed above, the Galois groups~~

~~Gal(M/K) and Gal(M/L′) are modules over R so that Gal(M/K) Qp and Gal(M/L′) Qp are R′- Z⊗p Z⊗p modules.~~

Theorem 4. Gal(M/K′) Qp is cyclic over R′. If in particular λp(k) = 0, Z×p then Gal(M/K) Qp also is cyclic over R′. Z⊗p 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.

~~9See 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 Univer- sity 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 York- Heidelberg-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:~~

~~ϕ(z) = an exp(2i πnz). (1.1) Xn 1 ≥ The Mellin transform of ϕ is the integral~~

~~+ ∞ s 1 ϕ(iy)y − dy. (1.2) Z 0~~

If we replace ϕ by its Fourier expansion then we see that (1.2) is equal to + ∞ s s 1 ann− exp( 2πy)y − dy. (1.3) − Xn 1 Z ≥ 0 Since + ∞ s 1 s exp( 2πy)y − dy = (2π)− Γ(s). (1.4) Z − 0

~~171 172 Herve Jacquet~~

~~this integral representation gives the analytic continuation of the series~~

s ann− , (1.5) Xn 1 ≥ as a meromorphic function of s in the whole complex plane. Further- more 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 a a n n′ (1.6) ns Xn 1 ≥ 156 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 adeles` 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 =  I  (2.2)  r2     ..   .     0 Ir   s    the “constant term of ϕ along V”, that is, the integral

ϕ(ug)du, (2.3) Z 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 ZrΓ 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) = exp(2iπn j x j. j+1). (3.1) − 1 Yj r 1 ≤ ≤ −~~

~~It is clearly trivial on Nr Γ. Set ∩~~

ϕn2,n2,...,nr 1 (g) = ϕ(ug)θ(u)du (3.2) − Z Nr Γ Nr ∩ \ where θ stands for θn1,n1,...,nr 1 . Then ϕ has the following expansion: − γ 0 ϕ(g) = ϕn1,n2,...,nr 1 g (3.3) − 0 1 X " ! # where we sum for all (r 1)-tuples with ni 1 and γ in a set of repre- − ≥ sentatives for Nr 1 Γr 1 Γr 1. Actually we will need to introduce also, − ∩ − \ − j for 0 j r 1, the subgroup V of matrices u Nr of the form ≤ ≤ − r ∈

~~1n j u = − ∗ . 0 ! ∗~~

~~For j = r 1 this is the group Nr itself. We will set: −~~

ϕnr j,nr j+1,...,nr 1 (g) = ϕ(ug)θ(u)du − − − Z Γ V j V j ∩ r \ r 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:− −

γ 0 ϕnr j+1,...,nr 1 (g) = ϕn1,n2,...,nr 1 g (3.4) − − − 0 1 X " j! # 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) (3.6) r 1 − | {z } and by ζ the diagonal matrix~~

~~diag(n1 n2 ... nr 1, n2 ... nr 1, ... , nr 1, 1). (3.7) − − −~~

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 ≤ ≤ − j ϕ = ϕ1,1,...,1 (4.1) so that ϕ0 = ϕ and ϕr 1 = W. We set| {z also, } for 1 j r 1, − ≤ ≤ −~~

~~an1,n2,...,n j = an1,n2,...,n j , 1, 1, ... , 1 . r j 1 − − | {z } 176 Herve Jacquet~~

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

~~r 2 nǫ 0 ϕ − (g) = anW g . (4.2) " 0 1r 1! # n 1,Xǫ= 1 − ≥ ± In view of this formula it is entirely reasonable to deﬁne the Mellin 159 transform of ϕ to be the integral~~

r 2 a 0 s 1 ϕ − a − da. (4.3) Z 0 1r 1! | | R / 1 − × {± } It is equal to s a 0 s 1 ann− W a − da. (4.4) Z 0 1r 1! | | Xn 1 − ≥ R× 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

~~s ann− . (4.5) Xn 1 ≥ On the other hand, just as in the case r = 2, the Dirichlet Series has an inﬁnite Euler product:~~

~~s+ 1 (r 1) s 1 ann− 2 − = det(1 p− Xp)− , − Xn 1 Yp ≥~~

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

~~5 The convolution~~

~~The convolution (1.6) also generalizes. Namely let ϕ′ be another cusp- form on Gr, with r r. Let us denote with a prime the objects attached ′ ≤ to ϕ. Dirichlet Series for the Group GL(N) 177~~

Suppose ﬁrst r r 1. Consider the integral ′ ≤ − r 1 r g 0 s ϕ − − ′ ϕ′(g) det g d×g, (5.1) Z 0 1r r′ ! | | Γ G − r′ \ r′ where d g is an invariant measure on the quotient Γr Gr . × ′ \ ′ Combining (3.4) with (3.5) we have the following expansion:

~~r 1 r′ γ 0 ϕ − − (g) = an1,n2,...,nr W g . (5.2) ′ " 0 1r r ! # X − ′ r 1 r Replacing ϕ − − ′ by this expression in (5.1) we get, after a “few” formal manipulations,~~

2 r′ s an ,n ,...,n a′ n1n ... n − (5.3) 1 2 r n1,n2,...,nr 1 2 r′ ′ ′− | | n1 1,n2X1,...,nr 1 ≥ ≥ ≥ g 0 s W W′[ǫg] det g d×g, Z " 0 1r r′ !# | | N G − r′ \ r′ 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:

~~s 1 (r r ) a a n n ... n 2 ′ (5.4) n1,n2,...,nr n′ 1,n2,...,nr 1 1 2 r′ − − ′ ′− | | X s 1 = det(1 p− Xp X′p)− . Yp − ⊗~~

When r = 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”:~~

+ ∞ rs 1 s E(g, s) = Φ(tξg) t − dt det g (5.6) | | | | ξ XZr 0 Z ∈ −{ }−∞ [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”:

~~rs 1 s E(g, s) = ζ(rs) Φ[(0, 0, ... , 0, t)γg] t − dt det g , (5.7) Z | | | | γ ΓXPr Γ ∈ ∩ \~~

~~where Pr is the standard parabolic subgroup of type (r 1, 1). − Then, instead of (5.2), we have to consider the integral~~

~~ϕ(g)ϕ′(g)E(g, s)d×g, (5.8) Z ZrΓ Gr \~~

~~where d g is an invariant measure on the quotient ZrΓ Gr. It turns out × \ to be equal to~~

~~2 r 1 s ζ(rs) an1,n2,...,nr 1 an′ 1,n2,...,nr 1 n1n2 ... nr−1 − (5.9) X − − | − | s W(g)W′(ǫg)Φ[(0, 0, ... , 0, 1)g] det g d×g. Z | | Nr Gr \ 161 Moreover:~~

~~2 r 1 s ζ(rs) an1,n2,...,nr 1 an′ 1,n2,...,nr 1 n1n2 ... nr−1 − (5.10) − − | − | X s 1 = det(1 p− Xp X′p)− . Yp − ⊗~~

Remark 5.11. If we take r = 1 then ϕ = ϕ′ = ϕ0, the constant function = = = equal to one on G1 R×; moreover Xp Xp′ 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

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

~~1+s 1 1 1 (1 p− X−p X′−p )− , Yp − ⊗ times the appropriate Γ-factor. To see this we introduce the function~~

~~t 1 ϕ(g) = ϕ( g− ).~~

~~It is also a Maass cusp form.e We denote by a tilda the objects attached to ϕ. Then:~~

~~e 0 1 −  1  = t 1 = W(g) W(wr g− ), where wr  1  ,  −   ..  e . 0     1  an1,n2,...,nr 1 = anr 1,nr 2...nt , Xp = X−p . − − −~~

~~If r = r′ our startinge point is the functionale equation of the Epstein zeta-function: t 1 E(g, s) = E( g− , 1 s); − from which we get~~

~~ϕ(g)ϕ′(g)E(g, s)dg = ϕ(g)ϕ′(g)E(g, 1 s)dg. Z Z − e e 180 Bibliography~~

The functional equation follows readily. 162 If r = r 1 then ϕr 1 r′ is just ϕ. Clearly ′ − − − g 0 s 1 g 0 1 s ϕ ϕ′(g) det g − 2 d×g = ϕ ϕ′(g) det g 2 − d×g Z 0 1! | | Z 0 1! | | and again the functional equation followse readily.e However if r′ r 2 (which includes the case r′ = 1) we have ≤ − r r 1 to take in account a somewhat unexpected relation between ϕ − ′− and r r 1 ϕ − ′− . Namely e

1r′ 0 0 r r′ 1 t 1 ϕ − −  x 1r r′ 1 0 g−  dx Z  0− 0− 1         r r 1   is actually a left-translate of ϕ − ′− (g); the integral is on the full space of matrices with r′ columns and r r′ 1 rows. Rather than trying to e − − 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, Automor- phic forms on GL(3), I and II Annals of Math, 109 (1979). Bibliography 181

~~[J-S-P2] Jacquet H., J. Shalika,andJ.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 Klooster- man 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 Hy- pothesis,” one gets an estimate. In a fixed characteristic p, the estimate one gets in this way for all the fintie fields Fpn 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, DieudonnéModules,... 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 sig- nificant 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 man- aged 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 Dieudonne´ 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 Dieudonne´ 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 Dieudonne´ 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. I. Elementary Axiomatics, and the Hasse-Davenport Theorem. Con- sider 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

T n Z(X/F , T) = exp ♯X(F n ) . q  n q  Xn 1   ≥    168 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:

~~2d ( 1)i+1 P1P3 ... P2d 1 Z(X/F , T) = P (T) = − q i − P P ... P Yi=0 0 2 2s~~

~~in which each polynomial Pi(T) Z[T] is of the form ∈ deg Pi Pi(T) = (1 αi jT) − , Yj=1~~

with αi, j algebraic integers such that i αi j = √q | , | for any archimedean absolute value on the ﬁeld Q¯ of all algebraic | | numbers. The extreme polynomials P0, P2d are given explicitly: d P (T) = (1 T), P d(T) = (1 q T) 0 − 2 − · Crystalline Cohomology, Dieudonn´eModules,... 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, i let us denote by Hl(X) the finitely generated Zl-module defined as

~~i i n H (X) = lim H (X F¯ q, Z/l Z). l etale ⊗ ←−−n~~

Corresponding to the prime p itself, we denote by W(Fq) the ring of p- i Witt vectors of Fq, and by Hcris(X) the ﬁnitely generated W(Fq)-module 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 , i on these various cohomology groups Hl(X) for l p, and Hcris(X); and F induces automorphisms of the corresponding vector spaces

~~i i Hl(X) Ql, Hcris(X) K OZl WO(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 co- homologically by the formulas~~

~~i Pi(T) = det(1 TF H (X) Ql) for l , p − l ⊗ i Pi(T) = det(1 TF H (X) K). − cris ⊗~~

169 The resulting formula for zeta as the alternating product of characteristic polynomials of F on the Hi, in each of the cohomology theories i i H (X) Ql for l , p, H (X) K, is equivalent, via logarithmic diﬀer- l ⊗ cris ⊗ entiation, to the identities in those theories

~~i n i , X(Fqn ) = ( 1) trace (F H ). for all n 1. − ≥ X~~

~~186 Nicholoas M. Katz~~

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

~~# Fix (Fn) = ( 1)i trace (Fn Hi) all n 1 X − ≥~~

~~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 Hi) is equivalent to the assertion: −~~

~~i On any of the groups H (X) Ql with l , p, l ⊗ Hi (X) K, the eigenvalues of F are alge- cris ⊗ 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 i on the H (X) Ql for l , p, and on H (X) K. It is still not known, for l ⊗ cris ⊗ 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

n T 1 1 n L(X/Fq, ρ, T) = exp tr(ρ(g− ))#Fix(F g)  n · #G  Xn 1 Xg G   ≥ ∈  170   Crystalline Cohomology, DieudonnéModules,... 187 where Fix (Fng) denotes the finite set of fixed points of Fng 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

deg(ρ) Z(X/Fq, T) = L(X/Fq, ρ, T) ρYirred 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], , ∈ 2d ( 1)i+1 L(X/Fq, ρ, T) = Pi,ρ(T) − , Yi=0 which are of the form

~~deg Pi,ρ Pi ρ(T) = (1 αi j ρT) , − , , Yj=1 with algebraic integers αi, j,ρ such that~~

i αi j ρ = √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,

~~i HomEλ[G](M Eλ, Hl(X) Eλ). OE OZl and of the EP-vector spaces~~

~~i HomEP[G](M EP, Hcris(X) EP). OE WO(Fq) 188 Nicholoas M. Katz~~

The polynomials Pi ρ(T) E[T] are given by the formulas , ∈ i Pi ρ(T) = det(1 TF HomE G (M Eλ, H (X) Eλ)) for l , ρ , − λ[ ] l OE OZ l i Pi ρ(T) = det(1 TF HomE G (M EP, H (X) EP)). , − P[ ] cris OE WO(Fq)

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

v i G ( 1)i+1 L(X Fq, ρ, T) = det(1 FT (M H ) ) − , \ − ⊗ Y Equivalently, we must check that 1 trace ρ(g 1) # Fix (Fng) #G − X v = ( 1)i trace (1 Fn (M Hi)G) X − ⊗ ⊗ 1 v = ( 1)i trace (g Fng M Hi) − #G ⊗ ⊗ X Xg G ∈ 1 v = ( 1)i trace ρ(g) trace (Fng Hi) − #G · X Xg G ∈ 1 1 i n i = trace ρ(g− ) ( 1) trace (F g H ). #G − Xg G X ∈ To check this last equality, we would like to invoke the Lefschetz trace formula, not for Fn, but for Fng, 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 Crystalline Cohomology, DieudonnéModules,... 189 together with an isomorphism X F¯ q X F¯ q under which Fg 1 cor- ⊗ ≃ ′ ⊗ ⊗ responds to F 1. Because this isomorphism also induces isomorphisms ⊗ of cohomology groups

i dfn i i dfn i H (X) H (X′ F¯ q, Zl) H (X F¯ q, Zl) H (X), l et ⊗ ≃ ⊗ l i i i H (X′) W(F¯ q) H (X′ F¯ q) H (X F¯ q) cris ⊗ ≃ cris ⊗ ≃ cris ⊗ ≃ i H (X) W(F¯ q), ≃ cris ⊗ 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 i in any of the H (X) Eλ, l , ρ, or in H (X) E . Decomposing the l ⊗ cris ⊗ P regular representation leads to the factorization

~~deg(ρ) Pi(T) = Pi,ρ(T) ρYirred~~

~~The coarser factorization~~

~~deg(ρ) Pi(T) = (Pi,ρ(T) ) ρYirred~~

~~i i corresponds to the decomposition of H (X) Eλ, resp. H (X) E , l ⊗ cris ⊗ P into ρ-isotypical components~~

~~i i ρ H (X) Eλ H (X) Eλ l ⊗ ≃ l × irredOρ ρ Hi (X) E Hi (X) E cris ⊗ P ≃ cris ⊗ P irredOρ ~~

~~Indeed the corresponding identities, for ρ irreducible, are~~

~~deg(ρ) i ρ Pi ρ(T) = det(1 TF (H (X) Eλ) )l , p , − l ⊗ deg(ρ) i ρ Pi ρ(T) = det(1 TF (H (X) EP) ). , − cris ⊗~~

~~190 Nicholoas M. Katz~~

Let us denote by S (X/Fq, ρ, n) the exponential sums used to deﬁne the L-function: 1 S (X/F , ρ, n) = tr(ρ(g))# Fix (Fng 1). q #G − Xg 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 i representation of G. Fix an integer i , and denote by H ◦ any one of the i ◦ i cohomology groups H ◦ (X) E with l , p, or H ◦ (X) E . Let l Zl λ cris P N WN(Fq) be any archimedean absolute value on the ﬁled Q¯ of all algebraic | | numbers. The following conditions are equivalent: i i 173 (1) The multiplicity of ρ in H ◦ is one, and the multiplicity of ρ in H is zero if i , i . ◦ (2) For all n 1, we have ≥ i i n ( 1) ◦ S (X/Fq, ρ, n) = (( 1) ◦ S (X/Fq, ρ,1)) , − − i and S (X/Fq, ρ, 1) = √q ◦ | | (3) For all n 1, we have ≥ i n S (X/Fq, ρ, n) = √q ◦ | | (4) For all n 1, we have ≥ n S (X/Fq, ρ, n) = S (X/Fq, ρ, 1) | | | | i 1+i and √q ◦ S (X/Fq, ρ, 1) < √q ◦ ≤ | |

~~(5) The polynomial Pi ,ρ (T) is given by ◦ i Pi ,ρ (T) = 1 ( 1) ◦ S (X/Fq, ρ, 1)T ◦ − − and for i , i , we have Pi,ρ(T) = 1. ◦ Crystalline Cohomology, Dieudonn´eModules,... 191~~

i ρ i ρ (6) The ρ-isotypical component (H ) = 0 for i , i , (H ◦ ) has di- i ρ ◦ mension = deg(ρ), and F operates on (H ◦ ) as the scalar i ( 1) ◦ S (X/Fq, ρ, 1). − Proof. This is an easy exercise, using the basic identities: n T ( 1)i+1 exp S (X/Fq, ρ, n) = L(X/Fq, ρ, T) = Pi,ρ(T) − n !  X Yi   i  Pi,ρ(T) = (1 αi, j,ρT), αi, j,ρ = √q  − | |  Yj   1  deg P = multiplicity of ρ in Hi = dim((Hi)ρ).  i,ρ ρ  deg( ) ·  Suppose, ﬁrst, that (1) holds, or equivalently that for i , i , Pi,ρ(T) = ◦ i 1, while Pi ,ρ is a linear polynomial Pi ,ρ (T) = (1 AT) with A = √q ◦ . ◦ ◦ − | | The cohomological expression for L then becomes

n ( 1)i T 1 − ◦ exp S (X/Fq, ρ, n) = . n ! 1 AT ! X − Taking logarithms and equating coeﬃcients, we ﬁnd 174 i n ( 1) ◦ S (X/Fq, ρ, n) = A for all n 1. − ≥ In particular (2) and (5) hold. The implications (5) (1), (6) (1) are obvious. Also (5) (6), ⇒ ⇒ i ⇒i ρ for if Pi ,ρ is linear, then ρ has multiplicity one in H ◦ , so that (H ◦ ) is ◦ i ρ G-irreducible, and hence F must operate on (H ◦ ) as a scalar, which we compute by the formula deg(ρ) i ρ Pi ,ρ (T) = det(1 TF (H ◦ ) ). ◦ −

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 i n i S (X/Fq, ρ, n) = ( 1) (αi j ρ) , αi j ρ = √q . − , , | , , | Xi Xj=1 192 Nicholoas M. Katz~~

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 ≥ ≥ x { } a family of N not-necessarily distinct elements of C , and Bi a family { } of M not-necessarily distinct elements of Cx. Suppose that for all i, j, Ai , B j. If, for some real number R > 0, we have

~~n n n A j B j = R for all n 1, X − X ≥~~

~~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 n n Ai = R . X Squaring, we get~~

~~n 2 n (AiA j) = (R ) for n 1 ≥ Xi j~~

~~175 whence 2 (1 AiA jT) = (1 R T), − − Yi j and hence N = 1. In case both N 1 and M 1, squaring leads to ≥ ≥ n n 2 n n n (AiAk) + (B jBl) = (R ) + (AiB j) + (AiB j) X X X X or equivalently,~~

~~1 (1 AiB jT) (1 AiB jT) = − − 2 (1 R T) Q(1 AiAkT) Q(1 B jBlT) − − − Q Q Crystalline Cohomology, Dieudonn´eModules,... 193~~

2 Let R be max( Ai , B j ), and consider the order of pole at T = R . max | | | | max− The numerator’s factors 1 AiB jT, 1 AiB jT are all non-zero there 2 − − (for if AiB j = Rmax, by maximality we must have Ai = B j = Rmax, in which case we see, using polar coordinates, that Ai B j, which is 2− 2 forbidden). In the denominator, each of the terms (1 Ai T), (1 B j T) −| 2 | −| | with Ai = R and B j = R vanishes at T = R . Therefore we | | max | | max max− 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 Hi any of the cohomology groups i , i Hl(X) Eλ with l p, or Hcirs(X) EP. The following conditions are ⊗Zl W⊗ equivalent.

~~(1) For all i, ρ does not occur in Hi, i.e. we have (Hi)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 Fp, i.e. a homomorphism

~~ψ :(Fp, +) E×, → we deﬁne an additive character ψq of each ﬁnite extension Fq by composing ψ with the trace map: 194 Nicholoas M. Katz~~

~~Given a character of µN, i.e. a homomorphism~~

~~χ : µN(E) E×, →~~

~~a p-adic place P of E, with residue field FN(P), and a finite extension Fq of this residue field, the map “reduction mod P” induces an isomorphism~~

~~µN(E) ∼ µN(FN P ) = µN(Fq) −→ ( )~~

~~Because Fq× is cyclic, we know that q 1mod N, and that the map q 1 ≡ x x N− deﬁnes a surjection →~~

~~F× ։ µN(Fq) = µN(FN P ) ∼ µN(E) q ( ) −→~~

~~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 gq(ψ, χ, P) = ψq(x)χq(x) xXF ∈ q× 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, DieudonnéModules,... 195 while gq(ψ, χ, P) = √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(FN P ) = µN(Fq), −→ ( ) we may view (ψ, χ) as a character of the group Fp µN(Fq): × (ψ, χ)(a, ζ) = ψ(a)χ(ζ).~~

~~Thus we may speak of the sums~~

~~1 n 1 S (X/Fq,(ψ, χ), n) = ψ(a)χ(ζ)♯ Fix (F (a, ζ)− ) pN · (a,ζ)XFp µN ∈ × 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). (2.1.1)~~

~~Proof. It suﬃces to treat the case n = 1, for we have 178~~

~~S (X/Fqn ,(ψ, χ),1) = S (X/Fq,(ψ, χ), n). 196 Nicholoas M. Katz~~

~~We can rewrite S (X/Fq,(ψ, χ), 1) as 1 ψ(a)χ(ζ) pN x XX(Fq) (aX,ζ)s.t. ∈ F(x)=(a,ζ)(x)~~

Given any point x X(Fq), the set of (a, ζ) Fp µN which satisfy ∈ ∈ × F(x) = (a, ζ)(x) is either empty or principal homogeneous under the inertia subgroup Ix of Fp 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 Ix is all of Fp µN) and to any ﬁnite point (T, 0) whose ∞ × X-coordinate is zero (then I T = 0 µN). ( ,0) { }× Given a point (T, X) with X , 0, we have

F(T, X) = (T q, Xq) and the inertia subgroup I T X is trival. If there is an element (a, ζ) ( , ) ∈ Fp µN satisfying F(T, X) = (T +a, ζX), then it is given by the formulas × q q 1 a = T T, ζ = X − − Since the point (T, X) is subject to the deﬁning equation

~~T p T = XN − we see that~~

q 1 N q 1 q 1 N N N N − (X ) − = (X − ) = ζ = 1, hence X F×, ζ = (X ) N ∈ q p N T T = X F×, − ∈ q q p N a = T T = traceF /F (T T) = traceF /F (X ). − q p − q p For each u F , the equations (T P T = u, XN = u) have pN solutions ∈ q× − (T, X) over Fq, all of which satisfy

~~F(T, X) = (a, ζ)(T, X) Crystalline Cohomology, Dieudonn´eModules,... 197~~

q 1 N− 179 with the same (a, ζ), namely (traceFq/Fp (u), u ), and every point (T, X) which contributes to our sum lies over some u F . Thus our sum ∈ q× becomes q 1 N− dfn ψ(traceFq/Fp (u))χ(u ) gq(ψ, χ, P). uXF ∈ q×

i i Corollary 2.2. Let H denote any of the cohomology groups Hl(X) Eλ i ⊗ with l , p, or H (X) EP of the Artin Schreier curve X/Fq. cris W⊗ 1 ψ χ (1) If ψ and χ are both non-trivial, then the eigenspace (H ) · is one-dimensional, and we have a direct sum decomposition

i 1 ψ χ H = (H ) · ⊕ indexed by the (p 1(N 1) pairs (ψ, χ)) of non-trivial characters. − − 1 ψ χ (2) The eigenvalue of F on (H ) is gq(ψ, χ, P), and for each n 1 · − ≥ we have the Hasse-Davenport formula

n gqn (ψ, χ, P) = ( gq(ψ, χ, P)) . − − 0 2 (3) The group Fq µN acts trivially on both H and H . × Proof. That the group acts trivially on both H0 and H2 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 H0 and H2 respectively, and hence that g itself acts trivially on H0 and H2. That the multiplicity of (ψ, χ) in H1 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 H1, we recall that the dimension of H1 is known to be 2g, g = genus of X, and so it suﬃces 198 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 = XN 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 FN(P) at P, we obtain the character χq χq : F× E× q → 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) = χq(x)χ′ (1 x). q − xXFq x,∈0,1~~

~~An elementary computation (cf [14]) shows that if the product χχ′ is non-trivial, then for any non-trivial additive character ψ of Fp, 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 + YN = ZN Crystalline Cohomology, Dieudonn´eModules,... 199

~~The group µN µN operates on this curve by the formula × (ζ , ζ ):(X, Y, Z) (ζ X, ζ Y, Z). 1 2 → 1 2~~

~~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)~~

~~i i Corollary 2.4. Let H denote any of the cohomology groups Hl(Y) Eλ i ⊗ with l , p, or H (X) Ep of the Fermat curve Y/Fq. cris W⊗~~

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

~~H1 = (H1)(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.~~

~~1 (χ,χ ) (2) The eigenvalue of F on (H ) ′ is Jq(χ, χ , P), and for each − ′ integer n 1 we have the Hasse-Davenport formula ≥ n Jqn (χ, χ′, P) = ( Jq(χ, χ′, P)) . − −~~

~~0 2 (3) The group µN µN operates trivially on both H and H . × 200 Nicholoas M. Katz~~

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 Hi(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 H H (X/W(Fq)) cris → DR i 182 of Hcris with the algebraic de Rham cohomology of the lifting (cf [19], [27]). i To discuss Hl(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 an , complex manifold XC . For each prime number l p, there is a canonical isomorphism i i an H (X) Htop(X , Z) Zl, l → C ×Z i where Htop denotes the usual “topological” cohomology. To emphasize the similarity between these two sorts of isomorphisms, recall that by GAGA and the holomorphic Poincare´ lemma, we have a canonical isomorphism

~~i i i an HDR(X/W) C ∼ / HDR(X/C) ∼ / H (XC , C) W⊗ ⊤ O ∽~~

~~Hi (Xan, Z) C ⊤ ⊗Z Crystalline Cohomology, Dieudonn´eModules,... 201~~

Unfortunately, these rather concrete descriptions of the various cohomology groups Hi(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)  i an i an ,  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

Hi (X) ∼ Hi (X/W) cris −→ DR  i ∼ i an , Hl(X) H (XC , Z) Zl for l p  −→ ⊤ ⊗  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 Hi with multiplicity one, then we know a priori that F must operate on the corresponding isotypical component (Hi)ρ as a scalar, and we know this even when F itself does not lift. For example, we could recover the isotypical decomposition of H1 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

~~H1 (Y) ∼ H1 (Y/W)(χ,χ′); cris −→ DR 202 Nicholoas M. Katz~~

1 in a basis of HDR(Y/W) adapted to this decomposition, the matrix of F 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 1 of Fp on an Artin-Schreier covering of A 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 coho- 1 0 mology group Hl (A) is a free Zl-module of rank 2g, and is an End(A) - 1 module. (It is also the case that Hcris(A) is a free W-module of rank 2g, 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

~~0 0 1 End(A) E / End(A) Eλ / EndZl (Hl (A)) Eλ ⊗Z ⊗Z ⊗Zl O ∽~~

~~1 EndEλ (Hl (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 0 1 End(A) Zl EndZl (Hl (A)) ⊗Z → Crystalline Cohomology, DieudonnéModules,... 203 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

~~1 0 H (A) Tl(Pic (A))( 1) Hom(Tl(A), Zl), l ≃ − ≃ under which the map considered above is the “opposite” of the map~~

~~End(A) Zl EndZl (Tl(A)) ⊗Z →~~

~~Our assertion of its injectivity with Zl-ﬂat cokernel is equivalent to the injectivity of (any one of) the maps~~

~~n End(A)/l End(A) End(Aln ), → and this injectivity follows from the exactness of the sequence~~

~~ln 0 Aln 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 (Pic0 )red. The Kummer sequence in etale cohomology together with X/Fq the duality of abelian varieties gives isomorphisms for each l , p

~~1 H (X) ∼ Tl(Pic(X))( 1) (4.1.1) l −→ − 1 H (Alb(X)) ∼ Tl(Pic(Alb(X))( 1) = Tl(Pic(X))( 1) (4.1.2) −→ − − which combine to give a canonical isomorphism~~

H1(X) H1(Alb(X)) for l , p (4.1.3) l ≃ l Suppose now that a ﬁnite group G operates on X by Fq-automorphisms. Let ρ be an absolutely irreducible representation of G deﬁned 204 Nicholoas M. Katz

~~over a number ﬁeld E, which occurs in H1(X) with multiplicity r. De- note by r P ρ(T) = 1 + a (ρ)T + + ar(ρ)T OE[T] 1, 1 ··· ∈ the reversed characteristic polynomial of F acting on the space~~

~~1 HomG(ρ, H (X))~~

~~of occurrences of ρ in H1;~~

~~1 P ρ(T) = det(1 TF HomG(ρ, H (X)). 1, − |~~

~~Let us denote by Proj(ρ) OE[1/♯G][G] the projector ∈~~

~~deg(ρ) 1 Proj(ρ) = tr(ρ(q− )) [g]. ♯G · Xg 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

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

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

Proof. Since End(Alb(X)) OE[1/♯G] is contained in End(Alb(X)) E, ⊗ 1 ⊗ 186 which is in turn contained in End(H (Alb(X)) Eλ) for any l , p, it suf- l ⊗Z r r 1 1 ρ ﬁces to verify that F +a (ρ)F + +ar(ρ) annihilates (H (Alb(X)) . 1 − ··· But this space is isomorphic to (H1(X))ρ, which is in turn isomorphic 1 to ρ HomG(ρ, H (X)), with F acting through the second factor, so we ⊗ 1 need the above polynomial in F to annihilate HomG(ρ, H (X)). This follows from the Cayley-Hamilton theorem. Crystalline Cohomology, Dieudonn´eModules,... 205

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 ∈ r r 1 F (m) + a (ρ)F − (m) + + ar(ρ) m = 0 1 ··· · 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 Dieudonne´ 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 R- algebras 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 R ; in coordinates, ′ 7→ ′ 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 The de Rham cohomology groups HDR(V/R) are the R-modules obtained by taking the cohomology groups of the formal de Rham com- • plex 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 diﬀerenti- ation 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. 206 Nicholoas M. Katz

~~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~~

pr , pr : G G G 1 2 × → are morphisms in this category. For G CFG(R), we deﬁne D(G/R) to 1 ∈ be the R-submodule of HDR(G/R) consisting of the primitive elements, i.e. the elements a H1 (G/R) such that ∈ DR 1 sum∗(a) = pr∗(a) + pr∗(a) in H ((G G)/R). 1 2 DR ×

~~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 ′ → sum G G / G ′ × ′ ′ f f f × sum G G / G × commutes, as do the analogous diagrams with “sum” replaced by pr1 or pr . Therefore given any element a H1 (G/R), we have 2 ∈ DR

sum∗( f ∗(a)) pr∗( f ∗(a)) pr ( f ∗(a)) = − 1 − 2 ( f f )∗(sum∗(a) pr∗(a) pr∗(a)). × − 1 − 2 In particular, if a D(G/R) then f (a) D(G /R). ∈ ∗ ∈ ′ Given f , f homomorphisms G G, let f be their sum. Then we 1 2 ′ → 3 have a commutative diagram Crystalline Cohomology, Dieudonn´eModules,... 207

~~sum~~

~~188 as well as a commutative diagram~~

Therefore for any a H1 (G/R), we have ∈ DR f ∗(a) f ∗(a) f ∗(a) = ( f f )∗(sum∗(a) pr∗(a) pr∗(a)). 3 − 1 − 2 1 × 2 − 1 − 2 In particular, if a D(G/R), then f (a) = f (a) + f (a). ∈ 3∗ 1∗ 2∗ 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 X , ... , Xn for V, A(V K) is the power-series ring K[[X , ... , Xn]]. 1 ⊗ 1 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 { ∈ ⊗ | } ∼ H1 (V/R) f A(V) f (0) = 0 −→ DR { ∈ | } which is compatible with morphisms of pointed Lie varieties. 208 Nicholoas M. Katz

189 Proof. Because K is a Q-algebra, the formal Poincare lemma gives H0 (V K/K) = K, Hi (V K/K) = 0 for i 1. Therefore any DR ⊗ 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 ⊂ f , f are two pointed morphisms V V such that f = f mod I, then 1 2 ′ → 1 2 the induced maps

~~1 1 f ∗, f ∗ : H (V/R) H (V′/R) 1 2 DR → DR are equal.~~

Proof. Let ϕ , ϕ denote the algebra homomorphisms A(V) A(V ) 1 2 → ′ 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 ϕ ( f ) ϕ ( f ) lies in A(V ), i.e. is itself integral. (Because f 1 − 2 ′ 1 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 component- by-component difference ∆= ϕ (X) ϕ (X) satisfies 2 − 1 ∆(0) = 0, ∆ has all coefficients in I.

~~We now compute using Taylor’s formula, and usual multi-index notations:~~

~~ϕ ( f ) ϕ ( f ) = f (ϕ (X)) f (ϕ (X)) 2 − 1 2 − 1 Crystalline Cohomology, Dieudonn´eModules,... 209~~

= f (ϕ (X) + ∆) f (ϕ (X)) 1 − 1 ∆n ∂n = f (ϕ1(X)). (n)! ∂Yn ! Xn 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, G0′ 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, f : G G , is a group homomor- 0 0′ → 0 phism, then the induced map f : H1 (G/R) H1 (G /R) maps ∗ DR → DR ′ D(G/R) to D(G′/R). (2) If f , f , f are three maps G G of pointed formal Lie varieties 1 2 3 ′ → whose reductions mod I are group homomorphisms which satisfy ( f ) = ( f ) + ( f ) in Hom(G , G ), then for any element a 3 0 1 0 2 0 0′ 0 ∈ 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~~

~~sum G G / G ′ × ′ ′ f f f × sum G G / G × commutes mod I, i.e. sum ( f f ) f sum mod I. × ≡ 210 Nicholoas M. Katz~~

and hence for any a H1 (G/R) we have, by the previous lemma, ∈ DR ( f f )∗(sum∗(a)) = sum∗( f ∗(a)) × The analogous diagrams with “sum” replaced by pr1 or pr2 commute, hence ( f f )∗(pr∗(a)) = pr∗( f ∗(a)) for i = 1, 2. × i i 191 Combining these, we ﬁnd

( f f )∗(sum∗(a) pr∗(a) pr∗(a)) = × − 1 − 2 sum∗( f ∗(a)) pr∗( f ∗(a)) pr∗( f ∗(a)). − 1 − 2 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 a H1 (G/R), we have ∈ DR f ∗(a) f ∗(a) f ∗(a) = ( f f )∗(Sum∗a) pr∗(a) pr∗(a)). 3 − 1 − 2 1 × 2 − 1 − 2 In particular, for a D(G/R), we obtain the asserted formula ∈

~~f3∗(a) = f1∗(a) + f2∗(a). Crystalline Cohomology, Dieudonn´eModules,... 211~~

~~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(G0′ , G0). Given a homomorphism f : G G , it always lifts to a pointed 0 0′ → 0 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~~

( f )∗ : D(G/R) D(G′/R). 0 → Theorem 5.1.5. Let R be a Z-ﬂat ring, and I R a divided power ideal. ⊂ Then the construction G D(G/R), f ( f ) = (any pointed lifting) 7→ 0 7→ 0 ∗ ∗ 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 identiﬁca- −→ ′ tions between the D’s of all possible liftings. In this way, it is possible to view the construction

~~G D(G/R), where G is some lifting of G 0 7→ 0 212 Nicholoas M. Katz~~

~~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 D(M/R) as the primitive elements of HDR(M/R), and one can still show exactly as before that the construction

~~G D(M/R), M any monoid lifting of G 0 → 0~~

~~193 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 1 we will denote HDR(V/R; I), is deﬁned as H of the subcomplex of the de Rham complex of V/R

~~1 2 “IA(V)′′ Ω Ω ... → V/R → V/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~~

~~f A(V K) f (0) = 0, dt integral d { ∈ ⊗ | } ∼ H1 (V/R; I). f “IA(V) f (0) = 0 −→ DR { ∈ ′′| } Crystalline Cohomology, Dieudonn´eModules,... 213~~

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 The corresponding theory, “primitive elements in HDR(G/R; I),” is 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, df 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 ⊂ The key lemma (??) holds for HDR(V/R; I), and theorems (5.1.4) and (5.1.5) hold for D1(G/R). The natural map D D is not an isomorphism, but its kernel and 1 → 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 ω is a closed form, so we have natural maps ∈ G/R

~~ωG/R / D1(G/R)~~

~~ D(G/R) 214 Nicholoas M. Katz~~

~~(Notice that in the extreme case I = (0), the map ω D is an isomor- → 1 phism!) Lemma 5.1.7. Suppose R ﬂat over Z, and I R an ideal. We have exact ⊂ sequences~~

d 0 HomR (G, Ga) ω D(G/R) → -groups −→ 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~~

ω D (G/R) and ω D(G/R) G → 1 G → are injective. The reader interested in obtaining the limit formula for Jacobi sums 195 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 → → → → rigid of abelian f.p.p.f. sheaves on (Schemes/R). We denote by Ext (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: Crystalline Cohomology, Dieudonn´eModules,... 215

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 rigid Hom(G, Ga) ω Ext (G, Ga) Ext(G, Ga) 0 −→ G → → → Theorem 5.2.1. If R is ﬂat over Z, there is a natural isomorphism~~

~~D(G/R) ∼ Extrigid(G, G ) ←− 1 in terms of which the resulting four term exact sequence~~

0 Hom(G, Ga) ω D(G/R) Ext(G, Ga) 0 → → G → → → 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 216 Nicholoas M. Katz

~~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 rigid Ext (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 ex- → → act sequence

~~0 Hom(G, Ga) ω D(G/R) Ext(G, Ga), → → G → → which receives the Extrigid exact sequence:~~

~~0 / Hom(G, Ga) / ω / D(G/R) / Ext(G, Ga) G O~~

~~? rigid 0 / Hom(G, Ga) / ωG / Ext (G, Ga) / Ext(G, Ga) / 0 The result is now visible. Crystalline Cohomology, Dieudonn´eModules,... 217~~

Given an ideal I R, we denote by Ext(G, Ga; I) the group of iso- ⊂ morphism classes of pairs consisting of an extension of G by Ga together rigid with a splitting of its reduction modulo I. We denote by Ext (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 rigid Ext (G, Ga; I) ∼ D (G/R) −→ 1 and a four-term exact sequence

~~∂ 0 Hom(G, Ga) ω D (G/R) Ext(G, Ga; I) 0 → → G → 1 −→ → 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/pnR. −→ ←−− 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 t ; it is known that t is a locally free R-module of rank h dim(G), Gv Gv − and that there is a canonical isomorphism

~~Ext(G, Ga) ∼ t v . (5.3.1) −→ G 218 Nicholoas M. Katz~~

~~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 ω D(G/R) t 0 (5.3.2) → G → → Gv → 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 topol- ⊂ ogy, 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 D (G/R) D(G/R) 1 ⊂  I (5.3.4) Ext(G, Ga; I) I Ext(G, Ga) I t v  −→ ≃ · G  and we have a short exact sequence

~~0 ω D (G/R) I t 0. (5.3.5) → G → 1 → · Gv →~~

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

w : CW(A(V k)) ∼ H1 (V/W(k);(p)) (5.5.1) ⊗ −→ DR 199 by defining pn (a a) w(... , a a, ... , a0) = d − (5.5.2) −  pn  Xn 0   ≥ e    Crystalline Cohomology, DieudonnéModules,... 219 where a n denotes an arbitrary lifting to A(V) of a n A(V k). Sim- − − ∈ ⊗ ilarly, we can define, following Grothendieck, Mazur-Messing ([35]), a σ-lineare isomorphism ψ : CW(A(V k)) ∼ H1 (V/W(k)) (5.5.3) ⊗ −→ DR by the formula

~~pn+1 (a n) ψ(... , a n, ... , a0) = d − . (5.5.4) −  pn+1  Xn 0 e   ≥  These isomorphisms sit in a commutative diagram ~~

1 HDR(V/W(k);(p)) ✐4 ✐✐✐✐ ✐✐✐✐ ✐✐✐✐∼w ✐✐✐✐ 1 F CW(A(V k)) ∼ p (5.5.5) ⊗ ❯❯❯ ❯❯❯❯ ❯❯∼❯❯ ψ ❯❯❯❯ ❯❯* 1 HDR(V/W(k)). When G is a commutative formal Lie group over W(k) which is p- divisible, the “classical” Dieudonne module of G = G k is deﬁned 0 ⊗ as dfn M(G0) Homk gp(G0, CW) − (5.5.6)

~~the primitive elements in CW(A(G0)). Combining this deﬁnition with the previous isomorphisms, we ﬁnd a commutative diagram of isomorphisms~~

~~Dp(G/W(k)) ❦❦5 ❦❦❦ ∼❦❦❦ ❦❦❦ w ❦❦❦ 1 F M(G0) ∼ p (5.5.7) ❙❙❙❙ ❙❙❙ ❙∼❙❙❙ ψ ❙❙❙ ❙) D(G/W(k)). 220 Nicholoas M. Katz~~

200 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 Mazur- Messing. Given an abelian scheme A over an arbitrary ring R, there are canonical isomorphisms

rigid 1 Ext (A, Ga) ∼ H (A/R) −→ DR (5.6.1)  1 v Ext(A, Ga) ∼ H (A, OA) = Lie(A )  −→  in terms of which the Extrigid-exact sequence “becomes” the Hodge exact sequence:

~~rigid 0 / ωA / Ext (A, Ga) / Ext(A, Ga) / 0 ∼ ∼ 1 1 (5.6.2) 0 / ωA / HDR(A/R) / H (A, OA) / 0~~

~~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~~

~~v v Lie(G ) = Lie(G ) = Hom(Gn, Ga) ∼ Ext(G, Ga). (5.6.4) n −→ The Extrigid-exact sequence can thus be written~~

~~rigid v 0 ω Ext (G, Ga) Lie(G ) 0, (5.6.5) → G → → →~~

~~where ωG is the R-linear dual of Lie(G). Crystalline Cohomology, Dieudonn´eModules,... 221~~

Given an abelian scheme A over a ring R in which p is nilpotent, the exact sequence pn 0 Apn A A 0 (5.6.6) → → −−→→ → for any n suﬃciently large that pn = 0 in R leads to a canonical isomor- 201 phism

~~v v Lie(A ) = Lie(A n ) = Hom(Apn , Ga) ∼ Ext(A, Ga). (5.6.7) p −→~~

~~Therefore the inclusion Ap ֒ A induces an isomorphism ∞ →~~

~~Ext(A, Ga) ∼ Ext(Ap , Ga) (5.6.8) −→ ∞~~

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

~~rigid 0 / ωA / Ext (A, Ga) / Ext(A, Ga) / 0 ∼ ∼ rigid 0 / ωA / Ext (Ap∞ , Ga) / Ext(Ap∞ , Ga) / 0, p∞ (5.6.9) i.e., an isomorphism~~

1 H (A/R) ∼ D(Ap /R) (5.6.10) DR −→ ∞ compatible with the Hodge filtration. For variable B T groups G over a fixed ring R in which p is nilpo- − v rigid tent, the functors ωG, Lie(G ), and consequently Ext (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

~~dfn rigid D(G/R) Ext (G, Ga) (5.6.11) when G is a B T group over a ring R in which p is nilpotent. − 222 Nicholoas M. Katz~~

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/pnR)/(R/pnR)) ⊗  ←−−n  Lie(G) = lim Lie(G (R/pnR)) (5.6.12)  ⊗  ←−− n ωG = lim ωG (R/p R)  ⊗  ←−− 202 Thus for variable B T groups G over a p-adic ring R, the functors v − ωG, Lie(G ) and D(G/R) are all exact functors in locally free R-modules of ﬁnite rank, sitting in an exact sequence

0 ω D(G/R) Lie(Gv) 0 (5.6.13) → G → → → 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 H (A/R) ∼ D(A(p∞)/R), DR −→ 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 p- adic 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

1 α 1 β 0 H (A k, Zp) W H (A k/W) D(A/W) 0 → et ⊗ ⊗ −→ cris ⊗ −→ → which is functorial in A k. b ⊗ 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

~~1 n αn 1 0 H (A k, Z/p Z) Wn H (A k/Wn) (5.7.2) → et ⊗ ⊗ −−→ cris ⊗ Crystalline Cohomology, Dieudonn´eModules,... 223~~

βn D(A Wn/Wn) 0. −−→ ⊗ → of Wn-modules. b An element of H1(A k, Z/pnZ) is (the isomorphism class of) a n ⊗ 1 Z/p Z-torsor over A k. An element of H (A k/Wn) is (the iso- ⊗ cris ⊗ 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 , Y ) of such → ′ n′ test situations (cf. [35] for more details). Given a Z/pnZ-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 → n an A k scheme, we can pull back T to obtain a Z/p Z-torsor TY 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

TY extends uniquely to an etale Yn-scheme T(Y,Yn), and its structure of n Z/p Z-torsor extends uniquely as well. Because Yn is a Wn-scheme, the natural map n Z/p Z Wn → gives rise to a morphism of algebraic groups on Yn

n αn (Z/p Z)Y (Ga)Y ; n −−→ n the required Ga-torsor αn(T)(Y,Yn) is obtained by “extension of structural n groups via αn” from the Z/p Z-torsor T(Y,Yn). 1 To deﬁne βn, we begin with an element Z of Hcris(A k/Wn). We rigid ⊗ 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 ⊗ b→ ⊗ b A Wn which is endowed with an integrable connection (cf. [2], [3]), ⊗ 1 i.e., it is an element of H (A Wn/Wn). [This interpretation provides DR ⊗ the canonical isomorphism 1 1 H (A k/Wn) ∼ H (A Wn/Wn).] cris ⊗ −→ DR ⊗ Composing with the isomorphism 1 rigid H (A Wn/Wn) ∼ Ext (A Wn, Ga Wn), DR ⊗ −→ ⊗ ⊗ 224 Nicholoas M. Katz

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

β H1 (A k/W) / D(A/W) cris ⊗ _ b inclusion of primitive ∽ canonical isom (5.7.3) elements 1 natural map 1 HDR(A/W) / HDR(A/W). “restriction to A” b What must be shown is that if web are given a second abelian scheme B over W, and a homomorphism

~~f : B k A k 0 ⊗ → ⊗ then the diagram~~

β H1 (A k/W) / D(A/W) cris ⊗ b (any pointed ( f0)∗ (5.7.4) lifting of f0)∗ b β H1 (B k/W) / D(B/W) cris ⊗ is commutative. b But in virtue of the commutativity of the previous diagram (5.7.3), it is enough to show the commutativity of the diagram

~~restriction H1 (A k/W) H1 (A/W) / H1 (A/W) cris ⊗ ≃ DR DR~~

~~b(any pointed ( f0)∗ (5.7.5) lifting of f0)∗ b restriction H1 (B k/W) H1 (B/W) / H1 (B/W). cris ⊗ ≃ DR DR b Crystalline Cohomology, Dieudonn´eModules,... 225~~

205 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 f : B k A k, and any 0 ⊗ → ⊗ integer i 0, we have a commutative diagram ≥

~~restriction H1 (A k/W) Hi (A/W) / Hi (A/W) cris ⊗ ∼ DR DR~~

b(any lifting ( f0)∗ of f0)∗ b restriction H1 (B k/W) Hi (B/W) / Hi (B/W) cris ⊗ ≃ DR DR b 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 do this, we will use the abelian scheme A over W. Its formal group A is p-divisible, and sits in an exact sequence of p-divisible groups over W, b

0 Ap Ap E 0, → ∞ → ∞ → → b = in which E lim En denote the etale quotient of Ap∞ . Because k is algebraically closed,−−→E is a constant p-divisible group, namely the abstract p-divisible group lim Apn (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′ 0 D(E/W) D(Ap /W) D(A/W) 0, → −−→ ∞ −→ → b and we will identify the (αn, βn)-sequence with the exact sequence~~

~~αn′ βn′ 0 D(E Wn/Wn) D(Ap Wn/Wn) D(A Wn/Wn) 0. → ⊗ −−→ ∞ ⊗ −−→ ⊗ → b 226 Nicholoas M. Katz~~

~~206 It is clear from the construction of βn that we have a commutative diagram~~

~~βn′ D(Ap Wn/Wn) / D(A Wn/Wn) ∞ ⊗ ⊗ b dfn~~

rigid restriction rigid Ext (Ap Wn, Ga) / Ext (A Wn, Ga) ∞ ⊗ ✐4 > ⊗ O ✐✐✐✐ ⑤⑤ restriction✐✐✐✐ ⑤⑤ ∽ ✐✐✐✐ ⑤ b ✐✐✐✐ ⑤⑤ ✐ ⑤⑤ Extrigid(A W , G ) ⑤⑤ n a ⑤⑤ ⊗ ⑤⑤ ⑤⑤ ∽ ⑤ ⑤⑤ βn ⑤⑤ H1 (A W /W ) ⑤⑤ DR n n ⑤⑤ ⊗O ⑤⑤ ⑤⑤ ∽ ⑤⑤ ⑤⑤ 1 H (A k/Wn). cris ⊗ 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) ⊗ ⊗ ⊗~~

~~∽ (En is constant) Hom(En(Wn), Ga(Wn))~~

~~Hom(En(k), Wn)~~

~~n Hom(En(k), Z/p Z) Wn. ⊗ Next use the sequence pn 0 Apn Wn A Wn A Wn 0 → ⊗ → ⊗ −−→ ⊗ → Crystalline Cohomology, Dieudonn´eModules,... 227 to compute 207~~

~~n n Ext(A Wn, Z/p Z) ∼ / Hom(Apn Wn, Z/p Z) ⊗ ⊗O (E = etale quotient of A n ) ∽ n p~~

~~n Hom(En Wn, Z/p Z) ⊗ ∽~~

~~1 n n H (A k, Z/p Z)o ∼ Hom(En(k), Z/p Z). et ⊗ Combining these isomorphisms, and remembering that Ext = Extrigid when either of the arguments is etale, we ﬁnd a commutative diagram~~

~~αn′ D(E Wn/Wn) / D(Ap Wn/Wn) ⊗ O ∞ ⊗O ∽ ∽~~

~~n Hom(En(k), Z/p Z) Wn D(A Wn/Wn) O ⊗ ⊗ ∽~~

~~rigid n rigid Ext (A Wn, Z/p Z) Wn∗∗∗ / Ext (A Wn, Ga) ⊗ O ⊗ ⊗ ∽~~

~~1 n αn 1 H (A k, Z/p Z) Wn / H (A k/Wn) et ⊗ ⊗ cris ⊗ in which the arrow is “push-out” along the homomorphism ∗∗∗ n Z/p Z Wn (Ga)W . → → n~~

~~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~~

~~1 Gal(k/k) 0 (Het(A k, Zp) W(k) → ⊗ ⊗ → 228 Nicholoas M. Katz~~

H1 (A k/W(k)) D(A/W(k)) 0, cris ⊗ → → in which k denotes an algebraic closure ofb k, and in which the galois 1 group Gal(k/k) acts simultaneously on H (A k, Zp) and on W(k) by et ⊗ “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)W k ) ⊗ ≃ ⊗ n( ) Gal(k/k) Hom(En(k), Wn(k)) ≃ Gal(k/k) Hom(Apn (k), Wn(k)) ≃ 1 n Gal(k/k) = Het(A k, Z/p 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~~

~~0 (slope 0) H1 (A k/W) (slope > 0) 0 → → cris ⊗ → → of H1 (A k/W)) as an F-crystal. cris ⊗ Proof. Since F induces a σ-linear automorphism of~~

1 Gal (H (A k, Zp) W(k)) et ⊗ ⊗ Gal(k/k) Hom(T p(A k), W(k)) , ≃ ⊗ it remains only to see that F is topologically nilpotent on D(A/W(k)), for its p-adic topology. Because D(A/W(k)) is a ﬁnitely generated W(k) b b Crystalline Cohomology, Dieudonn´eModules,... 229

~~1 sub-module of HDR(A/W(k)), the topology induced on D(A/W(k)) by the inverse limit topology on H1 through the isomorphism (cf. lemma b DR b 5.8.1. ahead)~~

1 1 H (A/W(k)) ∼ lim H (A Wn(k)/Wn(k)) (5.7.9) DR −→ DR ⊗ ←−− b b must be equivalent to the p-adic topology in D(A/W(k)). So it suﬃces n 1 n to remark that F annihilates H (A Wn/Wn) (indeed F annihilates DR ⊗ b Ωi for i 1, since for any pointed lifting of X X p, F(dX) = A Wn/Wn ≥ b 7→ d(F⊗(X)) = d(X p + pY) pΩ1) to establish the required topological b ∈ nilpotence of F on D(A/W). b 209 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. n Lemma 5.8.1. Let R be a Z-ﬂat p-adic ring, and let Rn = R/p R. For any formal Lie variety V over R, we have isomorphisms

i i H (V/R) ∼ lim H (V Rn/Rn). DR −→ DR ⊗ ←−− Proof. Pick coordinates X1, ... , XN for V. Over any ring R, we can N deﬁne a Z -grading of the de Rham complex of R[[X1, ... , XN]]/R, by N attributing the weight (a , ... , aN) Z to each “monomial” 1 ∈

dX j Xai S any subset of 1, ... , N . i  X  { } Y Yj S j   ∈    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. 230 Nicholoas M. Katz~~

~~• 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 (a , ... , aN) is the tensor ≥ ≥ 1 product complex a R i R . −→ i withOai 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 Zp-modules. We must show that for a Z-flat p-adic ring R we have

~~i • i • H (K R) ∼ lim H (K Rn). ⊗ −→ ⊗ ←−− 210 The exact sequence of complexes~~

~~n • p • • 0 K R K R K Rn 0 → ⊗ −−→ ⊗ → ⊗ → gives a “universal coeﬃcients” exact sequence~~

~~i • i • n i+1 • 0 H (K R) Rn H (K Rn) p -Torsion (H (K R)) 0. → ⊗ ⊗ → ⊗ → ⊗ → Passing to the inverse limit over n leads to an exact sequence~~

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

~~an = pan+1 + d(bn)~~

~~= p (pan+2 + d(bn+1)) + d(bn) Crystalline Cohomology, Dieudonn´eModules,... 231~~

~~= ...~~

~~= i d  p bn+i . Xi 0   ≥    To see that the natural map~~

~~i • i • H (K R) lim H (K R) Rn ⊗ → ⊗ ⊗ ←−− is an isomorphism, use the Z-ﬂatness of R and the Z-ﬁnite generation of the Ki to write~~

~~• • Hi(K R) ∼ Hi(K ) R = (ﬁn. gen. Z-module) R ⊗ ←− ⊗ ⊗ prime-to-p = Zn ( Z/pni ) R ⊕ ⊕ ⊕ torsion !! ⊗ n = R ( Rn ). ⊕ ⊕ i~~

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, f : B k A k a pointed 0 ⊗ → ⊗ k-morphism and f : B A a W-lifting of f to the formal completions 211 → 0 viewed as functors only on p-adic W-algebras. Then the diagram b b b i i restriction i H (A k/W) ∼ / H (A/W) / H (A/W) cris ⊗ DR DR

~~( f0)∗ b( f )∗ i i restriction i H (B k/W) ∼ / H (B/W) / H (B/W) cris 0 ⊗ DR DR is commutative. b~~

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 232 Nicholoas M. Katz

such that f maps V k to U k. Because V is aﬃne and U is smooth 0 ⊗ ⊗ over W, we may successively construct a compatible system of Wn-maps n fn : V Wn U Wn with fn+ fnmod p . The fn induce compatible ⊗ → ⊗ 1 ≡ maps fn : B Wn A Wn of formal completions, but these fn need ⊗ → ⊗ not be pointed morphisms. b b b b We denote by f : B A the limit of these fn. (Strictly speaking, ∞ → f only makes sense as a map of functors when we restrict B and A to ∞ b b b b the category of p-adic W-algebras). b b b 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 i restriction i H (A k/W) ∼ H (A/W) / H (A/W) cris ⊗ DR DR~~

~~( f0)∗ b( f )∗ ∞ i i restriction i b H (B k/W) ∼ H (B/W) / H (B/W). cris ⊗ DR DR b b 212 To conclude the proof, we need to know that the induced map~~

i i ( f )∗ : HDR(A/W) HDR(B/W) ∞ → depends only on theb underlyingb map f : B bk A k, and not on 0 ⊗ → ⊗ the particular choice of lifting. In fact this is true for the individual f as b b b n well! b

Lemma 5.8.3. Let R be a p-adic ring. Let V and V′ be formal Lie varieties over R, and let f and f be morphisms of functors V V of the 1 2 ′ → Crystalline Cohomology, Dieudonn´eModules,... 233 restrictions of V′, V to the category of p-adic R-algebras. If f1 f2mod p, then for each i, the induced maps

~~i i f ∗, f ∗ : H (V/R) H (V′/R) 1 2 DR → DR are equal.~~

~~Proof. (compare Monsky [39]). In terms of coordinates X1, ... , Xn for V′, Y1, ... Ym for V, the corresponding R-algebra homomorphisms~~

~~ϕ , ϕ : R[[Y , ... , Ym]] R[[X , ... , Xn]] 1 2 1 → 1 are related by ϕ2(Y) = ϕ1(Y) + p∆(Y). Introduce a new variable T, and consider the map~~

~~ϕ : R[[Y , ... , Ym]] R[[X , ... , Xn, T]] 1 → 1 ϕ(Y) = ϕ (Y) + T ∆(Y). 1 · We have a commutative diagram of algebraic homomorphisms~~

~~So it suﬃces to consider the situation 213~~

~~T 0 → / R[[X, T]] / R[[X]] T p → and show that these two maps have the same eﬀect on HDR. A form ω on R[[X, T]] may be written uniquely~~

~~n n dT ω = an T + bnT · T Xn 0 Xn 1 ≥ ≥ 234 Nicholoas M. Katz~~

~~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 → → n a0, an p Xn 0 ≥ respectively. Their diﬀerence, if ω is closed, is exact, namely~~

~~n n p ω T= ω T=p = an p = d bn . | 0 − |  n ·  Xb 1 Xn 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 f f mod I, then for all i the induced maps 1 ≡ 2 i i f ∗, f ∗ : H (V/R) H (V′/R) 1 2 DR → DR are equal.

Proof. If we had f f mod I with I I a ﬁnitely generated ideal, 1 ≡ 2 ′ ′ ⊂ then we could repeat the proof of the previous lemma, introducing sev- 214 eral new variables Ti, 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 n X1, ... , Xn for V′, we have a Z -graduation of its de Rham complex and a corresponding product decomposition

~~i i HDR(V′/R) = HDR(V′/R)(a1, ... , an). (a1Y,...,an) Crystalline Cohomology, Dieudonn´eModules,... 235~~

~~Therefore it suﬃces to show that the composite maps~~

f1∗ i / i projection i HDR(V/R) / HDR(V′/R) / HDR(V′/R)(a1, ... , an) f2∗ n agree, for every (a , ... , an) Z . But for ﬁxed (a , ... , an), these com- 1 ∈ 1 posites depend only on the terms of total degree ai in the power ≤ series formulas for the maps f1, f2. Thus we are reducedP 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 invari- i ance property for the groups HDR(V/R; I) deﬁned as the cohomology of i 1 d i d i+1 “IΩ − ” Ω Ω . V/R −→ 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 ϕx the corresponding Albanese mapping ∈ ϕx : C J → given on S -valued points, S any R-scheme, by 1 ϕx(y) = the class of the invertible sheaf I(y)− 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

1 1 H (J, O j) ∼ H (C, OC) −→ H0(J, Ω1 ) = ω ∼ H0(C, Ω1 ) (5.9.1)  J/R J −→ C/R  1 ∼ 1 HDR(J/R) HDR(C/R)  −→ which are independent of the choice of the rational point x. 236 Nicholoas M. Katz

~~Let Cx denote the formal completion of C along x; it is a pointed formal Lie variety of dimension one over R. Because ϕ (0) = 0, ϕ b X x induces a map of pointed formal Lie varieties~~

~~ϕx : Cx J, → whence an induced map on cohomologyb b b~~

~~1 (ϕx)∗ 1 D(J/R) HDR(J/R) HDR(Cx/R). ⊂ −−−−→b b b b Theorem 5.9.2. The composite map~~

~~(ϕx)∗ 1 D(J/R) HDR(Cx/R) −−−−→b is injective. b b Corollary 5.9.3. The natural map~~

~~0 1 1 H (C, Ω ) H (Cx/R) C/R → DR is injective, i.e., a non-zero diﬀerential of theb ﬁrst kind cannot be formally exact.~~

~~Proof. Because J is p-divisible, the natural map ω D(J/R) is injec- J → tive. b The corollary then follows immediately from the theorem and the commutativity of the diagram~~

~~ (ϕx)∗ 1 D(J/R) / H (Cx/R) O b (5.9.4) b∪ b 0 1 ωJ ∼ / H (C, ΩC/R).~~

216 To prove the theorem, we choose an integer n 2g 1, and consider the ≥ − mapping ϕ(n) : Cn J x → Crystalline Cohomology, DieudonnéModules,... 237 defined by n (n) ϕx (y1, ... , yn) = ϕx(yi), Xi=1 the summation taking place in J. Passing to formal completions, we obtain (n) n ϕ :(Cx) J x → defined by b b (n) b ϕx (y1, ... , yn) = ϕx(yi). X In terms of the projectionsb n pr :(Cx) Cx i → onto the various factors, web can rewriteb thisb as n (n) ϕ = ϕx pr , x ◦ i Xi=1 b b b the summation taking place in the abelian group of pointed maps to J. Because D(J/R) is defined to consist precisely of the primitive elements b in H1 (J/R), we have, for any a D(J/R), DR b ∈ b n b n (n) (ϕ )∗(a) = (ϕx pr )∗(a) = (pr )∗(ϕx)∗(a). x ◦ i i Xi=1 Xi=1 b b b b b Therefore the theorem would follow from the injectivity of the map

(n) 1 n (ϕ )∗ : D(J/R) H ((Cx) /R). x → DR b b 1 Because D(J/R) isb a ﬂat R-module contained in HDR(J/R), it suﬃces to show that the kernel of the map b b (n) 1 1 n (ϕ )∗ : H (J/R) H ((Cx) /R) x DR → DR consists entirely ofb torsion elements.b In fact,b we will show that this kernel is annihilated by n!. To do this, we observe that the map

~~ϕ(n) : Cn J x → b 238 Nicholoas M. Katz~~

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

~~Passing to formal completions, we get a factorization~~

1 We will ﬁrst show that (ψ)∗ is injective on HDR, by showing that the map ψ has a cross-section. This in turn follows from the global b fact that ψ is a Pn g-bundle over J which is locally trivial on J for b − 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 C 1 n (pr2) ( pr∗(I− (x)⊗ )) is locally free of rank n + 1 g. The associated ∗ ⊗ 1 − projective bundle is naturally isomorphic to ψ. It remains only to show that the kernel of the map

1 n 1 n (π)∗ : H (Symm (Cx)/R) H ((Cx) /R) DR → DR b bn is annihilatedb by n!. But if a one-form ω on Symm (Cx) becomes exact n n when pulled back to (Cx) , say ω = df with f A((Cx) ), then ∈ b b b = = n!ω σ(ω) d  σ( f ) σXSn σXSn  ∈  ∈    n is exact on Symm (Cx). b Crystalline Cohomology, Dieudonn´eModules,... 239

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 H1’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 ϕx : 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 H1’s, (cf. SGAI, Exp. XI, last page, for the etale case), i.e., we have a commutative diagram

~~1 Gal α. 1 0 / (H (J k, Zp) W(k)) / H (J k/W(k)) / D(J/W(k)) / 0 et ⊗ ⊗ cris ⊗ _ b~~

~~∼ (ϕ, k) ∼ (ϕ, k) (ϕx) ⊗ ∗ ⊗ ∗ ∗ β b 1 (Gal(k/k)) α 1 1 (H (C k, Zp) W(k)) / H (C k/W(k)) / H (Cx/W(k)). et ⊗ ⊗ cris ⊗ DR b 240 Nicholoas M. Katz~~

Corollary 5.9.6. (1) The kernel of the “formal expansion at a point” map 1 1 H (C/W(k)) H (Cx/W(k)) DR → DR in H1 (C/W(k)) H1 (C k/W(k)) is theb “slope-zero” part of the DR ≃ cris ⊗ F-crystal H1 (C k/W(k)), i.e., we have a commutative diagram cris ⊗

~~(Gal(k/k)) 1 1 1 1 0 / (Het(C k, Zp) W(k)) / H (C/W) / (image of H (C/W) in H (Cx/W(k)) / 0 ⊗ ⊗ DR O DR O DR b ∼ ∼~~

0 (slope 0) H1 (C k/W(k)) (slope > 0) 0. / / cris ⊗ / / (2) The image of the “formal expansion at a point” map is the “slope > 0” quotient of H1 (C k/W(k)); this quotient is isomorphic, via the cris ⊗ Albanese map ϕx, to D(J/W(k)).

219 VI. Applications to congruencesb 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 represen- ⊗ tation ρ of G, let P ρ(T) W(Fq)[T] be the numerator of the associated 1, ∈ L-function L(C Fq/Fq, G, ρ; T); ⊗ r P ρ(T) = 1 + a (ρ)T + + ar(ρ)T . 1, 1 ··· 0 1 ρ Let ω H (C, ΩC/W ) be a differential of the first kind on C which ∈ 0 4 lies in the ρ-isotypical component of H (C, Ω ). Let x C(W(Fq)) C/W ∈ be a rational point on C, and let X be a parameter at x (i.e., X is a coordinate for the one-dimensional pointed formal Lie variety Cx over W(F )). Consider the formal expansion of ω around x: q b

~~n dX ω = b(n) X b(n) W(Fq). · X ∈ Xn 1 ≥ Crystalline Cohomology, Dieudonn´eModules,... 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) + a (ρ) + + ar(ρ) pW(Fq) n 1 · nq ··· nqr ∈ for every rational n > 0.~~

Proof. Let J denote the Jacobian of C/W(Fq), and denote by ω ω ∈ J the unique invariant one-form on J which pulls back to give ω under b the Albanese mapping ϕx. The group G operates, by functoriality, on J ∼ 0 1 and on ωJ, and the isomorphism ωJ H (C, ΩC/W ) is G-equivariant. ρ −→ Therefore ω lies in (ωJ) . Via the G-equivariant inclusion e ω D p (J/W) J ⊂ ( ) we have b 220 ρ ω (D p (J/W)) ∈ ( ) Now let F denote the Frobeniuse endomorphismb of J Fq relative to ⊗ Fq. Then both F and the group G act on J Fq. By (4.2), we know that ⊗ r r 1 (F + a (ρ)F − + + ar(ρ)) Proj(ρ) = 0 1 ··· · in End(J Fq) W(Fq). Because D(J/W) is an additive functor of J Fq ⊗ ⊗Z ⊗ with values in W(Fq)-modules, and bω lies in its ρ-isotypical component, it follows that e r r 1 F (ω) + a (ρ)F − (ω) + + ar(ρ) ω = 0 (6.1.1) 1 ··· · in D(p)(J/W). e e e The Albanese map ϕx : C J induces a map b →

~~ϕx : Cx J, → b b b 242 Nicholoas M. Katz~~

~~whence a map~~

1 (ϕx)∗ 1 D(p)(J/W) HDR(J/W;(p)) HDR(Cx;(p)) ⊂ −−−−→b which is functorialb in the pointedb schemes (J, 0) Fbq and (Cx, x) Fq. ⊗ ⊗ So if we denote also by F the q-th power Frobenius endomorphism of b b Cx Fq, we have ⊗ (ϕx)∗ F = F (ϕx)∗, b ◦ ◦ whence a relation b b r r 1 F (ω) + a (ρ)F − (ω) + + ar(ρ) ω = 0 (6.1.2) 1 ··· · 1 in HDR(Cx/W;(p)). The asserted congruences on the b(n)’s are simply the spelling out b of this relation. Explicitly, in terms of the chosen coordinate X for Cx, a particularly convenient pointed lifting of F on Cx Fq is provided by ⊗ b F : X Xq. b 7→ 221 In terms of the isomorphism

1 f K[[X]] f (0) = 0, df integral H (Cx/W;(p)) ∼ { ∈ | } DR ←− f pW[[X]] f (0) = 0 { ∈ | } the cohomologyb class of ω is represented by the series b(n) f (X) = Xn, n Xn>0 and the cohomology class of Fi(ω) is represented by

~~i b(n) i f (Xq ) = Xnq . X n The relation (??) thus asserts that~~

r r 1 q q − f (X ) + a (ρ) f (X ) + + ar(ρ) f (X) 1 ··· 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 Xnq in this series lies in pW(Fq). Crystalline Cohomology, DieudonnéModules,... 243

Remark. In the special case G = e , ρ trivial, the polynomial P ρ(T) is { } 1, the numerator of the zeta function of C Fq, and every differential of the ⊗ first kind ω Hi(C, Ω1 ) is ρ-isotypical. The resulting congruences ∈ C/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 H1 with multiplicity one. Then~~

~~(1) a (ρ) is equal to the exponential sum S (C Fq/Fq, ρ, 1) and for 1 ⊗ every n 1 we have ≥ n ( a (ρ)) = S (C Fq/Fq, ρ, n). − 1 − ⊗~~

0 1 (2) If ρ occurs in H (C, ΩC/W ), then ordp(a1(ρ)) > 0, i.e., a1(ρ) is not 222 a unit in W(Fq). (3) If ρ occurs in H0(C, Ω1 ), choose ω H0(C, Ω1 )ρ to be non- C/W ∈ C/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) S (C Fq/Fq, ρ, 1) = a1(ρ) = lim · − ⊗ − N b(nqN+1) →∞ q b(nqN+1) S (C Fq/Fq, ρ, 1) = a1(ρ) = − = lim . − ⊗ − a (ρ) N b(nqN) 1 →∞~~

~~Proof. If ρ occurs in H1 with multiplicity one, then ρ must be a non- trivial representation of G (for if ρ were the trivial representation, G 244 Nicholoas M. Katz~~

would have a one-dimensional space of invariants in H1; but the space 1 of invariants in H of the quotient curve C Fq modulo G, so is even- ⊗ dimensional!). Therefore ρ does not occurs in H0 or H2, as both of these are the trivial representation of G. The ﬁrst assertion now results from (1.1). 0 1 If ρ also occurs in H (C, ΩC/W ), pick any non-zero ω in

0 1 ρ H (C, ΩC/W ) and look at its formal expansion around x: dX ω = b(n)Xn . X X An elementary “q-expansion principle”-argument (cf. [28]) shows that 0 1 if all b(n) are divisible by p, then ω is itself divisible by p in H (C, ΩC/W ). So after dividing ω by the highest power of p which divides all b(n), we obtain an element ω H0(C, Ω1 )ρ which has some coeﬃcient a unit. ∈ C/W Consider the congruences satisﬁed by the b(n): b(n) b(nq) + a (ρ) pW(Fq). n 1 nq ∈

223 If a1(ρ) were a unit, we could infer (by induction on the precise power of p dividing n) that q b(n) for all n 1, W(Fq). ≥ p · n ∈ q In particular, we would ﬁnd that 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(n) b(nq) a (ρ) mod pW n ≡− 1 nq Crystalline Cohomology, Dieudonn´eModules,... 245~~

. . b(nqN) b(nqN+1) a (ρ) mod pW nqN ≡− 1 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) ord + a1(ρ) 1 + (N + 1) ord(a1(ρ)) ord(b(n)/n) b(nqN+1) ! ≥ − b(nqN+1) q q b(n) + + + ord N 1 ord N ord(a1(ρ)) ord . b(nq ) a1(ρ)! ≥ a1(ρ)! − n ! Letting N , we get the asserted limit formulas for a,(ρ) and for → ∞ − q/a (ρ). By the Riemann Hypothesis for curves over finite fields, we − 1 know that q/a (ρ) is the complex conjugate a (ρ). Let ρ denote the − 1 1 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 1 for ρ. Therefore a1(ρ) = a1(ρ), and ρ also occurs in H 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 r χr(ζ) = ζ .~~

We know that under the action of µN µN (acting as (x, y) (ζx, ζ y) × → ′ in the aﬃne model xN + yN = 1), the characters which occurs in H1 are precisely χr χs 1 r, s N 1, r + s , N, × ≤ ≤ − each with multiplicity one. Those which occur in H0(Ω1) are precisely the χr χs 1 r, s N 1, r + s < N, × ≤ ≤ − 246 Nicholoas M. Katz

~~the corresponding eigen-diﬀerential ωr,s is given by dx ω = xrys . r,s xyN~~

~~If we expand ωr,s at the point (x = 0, y = 1), in the parameter x, we obtain~~

r N s dx ωr s = x (1 x ) 1 , − N − · x s dx = ( 1) j 1 xr+N j − N −j ! x Xj 0 ≥ dx = b(n)xn . x Xn 1 ≥ Conveniently, the first non-vanishing coefficient b(r) is 1. The succes- sive coefficients b(rqn) are given by

s n r (qn 1) N 1 b(rq ) = ( 1) N − − . − ·  r (qn 1)  N −      1 225 The eigenvalue of F on the χr χs-isotypical component of H is the × negative of the Jacobi sum Jq(χr, χs). There we obtain the limit formulas

s 1 r (q 1) qn N ( 1) N − · − −  r (qn 1)  N  Jq(χr, χs) = lim  −  − n s 1  →∞ N −  r n+1  N (q 1)  −    s 1 r (q 1) qn N ( 1) N − · − − ·  r (qn+1 1)  N  Jq(χN r, χN s) = lim  −  − − − n s  1  →∞ N −  r n  N (q 1)  −    Crystalline Cohomology, Dieudonn´eModules,... 247 valid for 1 r, s N 1, r + s , N. These formulas are the ones orig- ≤ ≤ − inally 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 in- 1/N finity”, along which T − is a uniformizing parameter. The group-scheme µN(p 1) operates on U, by − ζ :(T, X) (ζNT, ζX). → This action extends to C, and fixes the point at infinity. 226 A straightforward computation gives the following lemma. Lemma 7.1. (1) The genus of C is 1 (N 1)(p 1), and a basis of 2 − − everywhere holomorphic differentials on C is given by the forms dT XaT b N 1 X − with 0 a N 2, 0 b p 2, and pa + Nb < (p 1)(N 1) 1. ≤ ≤ − ≤ ≤ − − − − (2) The space H1 (C Q/Q) ∼ H1 (U Q/Q) has dimension (N DR ⊗ −→ DR ⊗ − 1)(p 1), any d basis is given by the cohomology classes of the − forms dT XqT b 0 a N 2, 0 b p 2. N 1 X − ≤ ≤ − ≤ ≤ −

1 (3) The characters of µN(p 1) which occur in H (C Q/Q) are pre- − DR ⊗ cisely those whose restrictions to µN is non-trivial, and each of these occurs with multiplicity one. 248 Nicholoas M. Katz

In characteristic p, there are new automorphisms. The additive group Fp operates on C Fp 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−

~~Fp ⋉ µN(p 1) − formed via the homomorphism~~

~~N − µN(p 1) µp 1 F×p = Aut(Fp) − −−→ − ≃ Explicitly, the multiplication is~~

N (a, ζ)(b, ζ1) = (a + ζ− b, ζζ1), and the action is (a, ζ):(T, X) (ζNT + ζNa, ζX). → 227 The group Fp ⋉ µN(p 1) contains Fp µN as a normal subgroup, − × acting on C Fp in the usual manner. ⊗ Remark. This action of a group of order p(p 1)N on a curve of genus − g = 1 (p 1)(N 1) provides a nice example of how “wrong” the char- 2 − − acteristic 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 FN(P), of P, 1 G the abstract group Fp ⋉ µN(p 1)(Fq). Let H denote any of the vector 1 , − 1 spaces Hl (C Fq) Eλ for l p, or Hcris(C Fq/W(Fq)) K. ⊗ ⊗Zl ⊗ ⊗ By functoriality, the group G operates on H1. Because the center of G is µN(Fq), the decomposition H1 = (H1)χ ⊗ 1 of H according to the characters of µN is G-stable. Crystalline Cohomology, DieudonnéModules,... 249

Proposition 7.2. For each of the N 1 non-trivial E-valued characters χ − 1 χ of µN(E) ∼ µN(FN P ) = µN(Fq), the corresponding eigenspace (H ) −→ ( ) is a p 1 dimensional absolutely irreducible representation of G; the − 1 χ restriction to Fp of (H ) is the augmentation representation of Fp; the 1 χ restriction to µN(p 1)(Fq) of (H ) is the induction, from µN to µN(p 1), of χ. − −

Proof. All assertions except for the G-irreducibility of (H1)χ follow immediately from the preceding lemma, giving the action of µN(p 1), and − from Corollary (2.2), giving the action of Fp µN. The irreducibility × follows from these facts together with the fact that in any complex representation of G, the set of characters of Fp which occur is stable under the action of µN(p 1) in Fp by conjugation; because this action has only − the two orbits F×p and 0, as soon as any one non-trivial character of Fp occurs, all non-trivial characters must also occur.

Corollary 7.3. (1) Over any ﬁnite extension Fq of Fp which contains all the N(p 1)’st roots of unity (i.e., q 1mod N(p 1)), the Frobenius − ≡ − 1 χ F relative to Fq operates as a scalar on each of the spaces (H ) , 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 H1, so it acts on each (H1)χ as a G-morphism. Because (H1)χ is G- irreducible, this G-morphism must be a scalar, and this scalar is equal to any eigenvalue of F on (H1)χ. As we have already seen (2.1), these eigenvalues are precisely the asserted Gauss sums, corresponding to the 1 χ decomposition of (H ) under Fp.

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 − 250 Nicholoas M. Katz

~~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 p ζ1 − = ζ2~~

~~Explicitly, the map is given rationally by the formulas~~

~~N(p 1) N(p 1) (W, V) on W − + V − = 1~~

~~↓ (T, X) on T p T = XN − N p 1 p T = 1/V , X = W − /V .~~

~~Lemma 7.4. Let χ1 be a character of µN(p 1) whose restriction to µN is non-trivial. Under the map −~~

~~H1 (C Q/Q) ∼ H1(Fermar (N(p 1)) Q/Q)H DR ⊗ −→ − ⊗ we have~~

~~p 1 p 1 χ1 1 χ − χ− H (C Q/Q) ∼ H (Fermat (N(p 1)) Q/Q) 1 × 1 DR ⊗ −→ DR − ⊗~~

Proof. That H1(C) ∼ H1(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 −1 1 − × − in H (C) (resp H (Fermar)), it suﬃces to check that the χ1-eigenspace 1 p 1 p 1 of H (C) is mapped to the (χ1− , χ1− )-eigenspace of H (Fermat). This 229 we do by inspection:

~~dT XaT b = N 1 X − a+1 N dT W p 1 − b+1 NdZ = Xa+1 NT b+1 − Z N − . − T V p − Z 7→ ! ! Crystalline Cohomology, Dieudonn´eModules,... 251~~

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 Fp, the scalar by which F acts on 1 χ H (C Fq) is given by ⊗ 1 χ 1 ψ χ F H (C Fq) = F H (C Fq) = gq(ψ, χ; P) | ⊗ | ⊗ × −    p 1 p  1 χ 1 χ − χ− p 1 p F H (C F ) 1 = F H (Fermat F ) 1 1 = J (χ − , χ− ; P)  q q × q 1 1  | ⊗ | ⊗ −  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

~~p 1 π − = p. − We recall without proof the following standard lemma (cf. [31] or [32]).~~

~~Lemma 7.6. The ﬁelds Qp(ζp) and Qp(π) coincide. There is a bijective correspondence~~

~~p 1 primitive p’th roots of 1 solutions π of π − = p ←→ − under which ζ π if and only if ←→ ζ 1 + πmod π2. ≡ For each solution π of πp 1 = p, we denote by − −~~

~~ψπ : Fp Qp(ζp)× → the unique non-trivial additive character which satisﬁes 230~~

~~2 ψπ(1) 1 + πmod π . ≡ 252 Nicholoas M. Katz~~

~~Ifweﬁxa W(Fq)-valued point x on C, we have the map “formal expansion at x”~~

~~1 1 H (C Fq/W(Fq)) H (Cx W(Fq)/W(Fq)). cris ⊗ → DR ⊗ If we denote by R the ring b~~

~~R = W(Fq)[π] which is a free W-module of ﬁnite rank (p 1), we may tensor with R − to obtain~~

~~1 1 Hcris(C Fq/W(Fq)) R / HDR(Cx R/R). ⊗ W⊗ ✐4 ⊗ ✐✐✐✐ ✐✐✐✐ b ∽ ✐✐ ✐✐✐✐ ✐✐✐✐ H1 (C R/R) DR ⊗~~

~~Theorem 7.7. (1) For any W(Fq)-valued point x on C, the “formal expansion” map is injective :~~

1 1 ((H (C Fq/W(Fq)) ֒ H (Cx W(Fq)/W(Fq cris ⊗ → DR ⊗ p 1 (2) Let π be any solution of π = p,bψπ the corresponding additive − − character, a an integer 1 a N 1 and χa the corresponding ≤ ≤ −a nontrivial character of µN(χa(ζ) = ζ ). If we take for x the point (T = 0, X = 0) on C, with parameter X, then the image of

1 ψw χa 1 (Hcris(C Fq/W(Fq)) Qp(π)) × HDR(Cx R/R) Qp(π) ⊗ ⊗ → ⊗ ⊗R b is the one-dimensional Qp(π)-space spanned by the cohomology class of dX dX exp( πXN)Xa = b(n)Xn . − X X X Corollary 7.8. Notations as above, let f (X) denote the power series

~~Xn ( π)n XnN+a f (X) = b(n) = − n n! nN + a Xn 1 Xn 0 ≥ ≥ Crystalline Cohomology, Dieudonn´eModules,... 253~~

231 Then the series q f (X ) + gq(ψπ, χa; P) f (X) · has coeﬃcients with bounded denominators, and we have a limit formula q b(aqr) gq(ψπ, χa; P) = lim · + − r b(aqr 1)  →∞(qr 1) a (7.8.1)  r = ( π) − N with b(aq ) ((−qr 1) a )!  − N  We ﬁrst deduce the corollary from the theorem. We know that F has 1 eigenvalue gq(ψπ, χa; P) on the ψπ χa-eigenspace of H Qp(π), − × cris ⊗ hence F has the same eigenvalue on the image of this one-dimensional 1 eigenspace in H (Cx R/R) Qp(π). This image is spanned by the DR ⊗ ⊗ cohomology class of df : therefore F +g (ψ , χ ; p) annihilates the class b q π a of df mod torsion, whence

q f (X ) + gq(ψπ, χa; P) f (X) · has bounded denominators. The ﬁnal limit formula comes from looking r+1 successively at the coeﬃcients of Xaq in the above expression; one has b(aqr) b(aqr+1) ord + gq(ψπ, χa; p) A 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 H (C Fq/W(Fq)) H (Cx W/W) cris ⊗ → DR ⊗ b 1 is equivalent to the absence of any p-adic unit eigenvalues of F in Hcris. 232 254 Nicholoas M. Katz~~

~~But these eigenvalues are the Gauss sums~~

gq(ψ, χ) ψq(x)χq(x). − ≡− X x Because ψq(x) 1(π) for all x, while χq is a non-trivial character of F , ≡ q we have g(ψ, χ) χq(x) = 0mod π. − ≡− X (Alternately, one could observe that each non-trivial character χ of µN 0 1 has at least one extension χ1 to µN(p 1) which occurs in H (C Q, Ω ); − C Q p 1 ⊗ ⊗ the eigenvalue of F − on this eigenspace is then a non-unit by (??); as p 1 1 χ F − is a scalar on (H ) , this scalar is non-unit.) It remains to verify that the image of the ψπ χa-eigenspace is indeed × spanned by dX exp( πXN)Xa − X This seems to require the full strength of the Washnitzer-Monsky “dag- ger” cohomology, as follows. Let At denote the “weak completion” of p N the coordinate ring R[T, X]/(T T X ) of U R. Because U Fq − − ⊗ ⊗ is a “special aﬃne variety” with coordinate X, there are unique liftings t to A of the actions of F and of the group Fp µN whose eﬀect on X is ⊗ given by F(X) = Xq (a, ζ)(X) = ζX.  Thanks to Dwork, we know that the power series in T

exp(πT πT p) − t t p 1 actually lies in R[T] , and hence in A for any π satisfying π − = p. t − As Monsky pointed out, under the action of Fp and A , this series transforms by the character ψπ. It follows that for 1 a N 1 the diﬀer- ≤ ≤ − ential form dX exp(πT πT p)Xa − X Crystalline Cohomology, Dieudonn´eModules,... 255 transforms by ψπ χa under the action of Fp µN. Therefore its coho- 233 × × mology class in

1 dfn 1 • t HW M(U Fq; R) Q = H (ΩU R/R A ) Q − ⊗ ⊗ ⊗ ⊗ ⊗ 1 lies in the ψπ χa eigenspace of H . A direct computation ([31], × W M [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; 1 1 HW M(U Fq; R) HDR(Ux R/R). − ⊗ → ⊗ For the particular choice of point (T = 0, X b= 0), the formal expansion map carries dX dX exp(πT πT p)Xa exp( πXN)Xa . − X 7→ − X 1 To conclude the proof, we need to identify HWM(U Fq; R) Q 1 ⊗ ⊗ with H (C Fq/R) Q in a way compatible with the formal expansion cris ⊗ ⊗ map and with the action of F and of Fp 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~~

~~2 6~~

~~3 1 7 256 Nicholoas M. Katz~~

234 In this diagram, the maps 2 , 5 and 6 are each compatible with the actions of F and of Fp µN imposed by crystalline and by × W M theory (simply because these actions lift to the U Wn). There- − ⊗ fore the compatibility of the isomorphism 8 with the actions of F and of Fp 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 in- 1 jective exactly because F has no p-adic unit eigenvalues in Hcris of our 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 H (U/W) Q (lim H (U Wn/Wn)) Q DR ⊗ → DR ⊗ ⊗ ←−− 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~~

~~Γp : Zp Z× → p whose values on the strictly positive integers are given by the formula~~

~~n+1 n+1 ( 1) n! Γp(1 + n) = ( 1) i = − · (7.9.1) − · [n/p]!p[n/p] 1Yi n ≤p∤≤n~~

~~where [ ] denotes “integral part.” Lemma 7.9.2. For any integer n 0, and any π satisfying πp 1 = p, ≥ − − we have the identity~~

~~( π)n/n! (π)n p[n/p] = ( 1) − . (7.9.3) −[n/p] ( π) /[n/p]! − · Γp(1 + n) − Crystalline Cohomology, Dieudonn´eModules,... 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 πp 1 = p 235 ≥ − − and n 0 any integer. Let ≥~~

~~n = n + n p + 0 ni p 1 0 1 ··· ≤ ≤ − be the p-adic expansion of n. Then we have~~

~~( π)n/n! ( 1) f (π)n0+n1+ +n f 1 − = − · ··· − (7.9.5) ( π)[n/q]/[n/q]! f 1 − i − Γp(1 + [n/p ]) i=0 Q f 1 Proof. Simply apply (7.9.3) successively to n,[n/p], ... [n/p − ].~~

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

~~j j i n = n j p n j+i p = [n/p ] (7.9.6) 7→ X Xj>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 f 1 i p − α = [ α/p ]p in Z (7.9.8) −h i − for i = 0, 1, ... , f 1 (where denotes the “fractional part” of a − h i rational number). 258 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~~

~~f 1 A = a0 + a1 p + + a f 1 p − ; 0 ai p 1 ··· − ≤ ≤ − ai < p 1 for some i. −~~

~~We now extend the deﬁnition of an to all n Z by requiring 236 ∈~~

~~an = an+ f n Z. ∀ ∈ Then~~

~~f 1 − f + j i a j p − f i f i A j=0 p − α = p − = P p f 1 p f 1 − − f 1 − j a j+i p j=0 P mod Z ≡ p f 1 − whence~~

~~f 1 − j a j+i p f i j=0 j p − α = P = a j+i p −h i 1 p f Xj 0 − ≥ j a j p j 0 =  P≥   pi     p   But we readily calculate~~

~~A j α = = a j p . − 1 p f Xj 0 − ≥ Crystalline Cohomology, Dieudonn´eModules,... 259~~

~~Corollary 7.9.9. Let q = p f with f 1, π any solution of πp 1 = p, ≥ − − and α any rational number satisfying~~

~~0 α 1  ≤ ≤ (q 1)α Z.  − ∈  Let ~~

f 1 A = (q 1)α = a0 + a1 p + + a f 1 p − , 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) lim − = − · − (7.9.10) n α ( π)[n/q]! f 1 →− − i − Γp(1 p α ) i=0 − h i Q 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 continu- i ity of both Γp and of n [n/p ]. → 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 Ep a p 1 ∈ solution of π − = p, ψπ the corresponding additive character of Fp, − a a an integer 1 a N 1, χa the corresponding character ζ ζ of ≤ f≤ − 7→ µN, and Fq, q = p , a finite extension of the residue field FN(P) of E at P. We have the formulas, in EP,

~~f pia ( 1) q Γp i N − · · imod f − h i g (ψ , χ ; P) = Q (7.10.1) q π a S ((q 1) a ) − (π) − N 260 Nicholoas M. Katz~~

~~i S ((q 1) a ) p a gq(ψπ, χ ; P) = (π) − N Γp (7.10.2) − a h N i! imodY f~~

r Proof. The sequence nr = (q 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 p q p+q E (n) = H (Xn, H (Xn/Wn)) H (Xn/Wn) 2 DR ⇒ has an intrinsic interpretation in terms of X k as the Leray spectral ⊗ sequence for the “forget the thickening” map~~

~~(X k/Wn) (X k) . ⊗ cris → ⊗ Zar As such, it may be rewritten~~

~~p,q p q p+q E (n) = H (X k, H (X k/Wn)) H (X k/Wn). 2 ⊗ cris ⊗ ⇒ cris ⊗ An explicit construction of this spectral sequence may be given in terms of the De Rham-Witt pro-complex on X k ⊗~~

~~• WnΩ n { } of Deligne and Illusie; it is simply the second spectral sequence of this complex:~~

~~p,q p q • p+q • E (n) = H (X k, H (WnΩ )) H (X k, WnΩ ). 2 ⊗ ⇒ ⊗ Crystalline Cohomology, Dieudonn´eModules,... 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~~

~~E p,q = lim E p,q(n) H p+q(X k/W). 2 2 ⇒ cris ⊗ ←−−n Let x be a W-valued point of X, and assume X connected. The formal expansion map we have exploited~~

~~i i i H (X k/W) H (X/W) H (Xx/W) cris ⊗ ≃ DR → DR is the composition of the edge-homomorphism b~~

~~i ։ 0,i 0,i Hcris(X/W) E ֒ E2 ∞ → with the natural map~~

~~0,i 0 i i E = lim H (Xn, H (Xn/Wn)) lim H (Xx Wn/Wn). 2 DR → DR ⊗ ←−−n ←−− b 239 Lemma 8.1. This map is in fact injective; indeed, the induced maps~~

~~0 i i H (Xn, H (Xn/Wn)) H (Xx Wn/Wn) DR → DR ⊗ as injective. b~~

~~Proof. Because Xn is irreducible, it suﬃces to show~~

(*) for any closed point y of Xn, and any affine open V y which is ∋ etale´ over standard affine space A = Spec(Wn[T1, ... , Td]), the natural map 0 i i H (V, H (Xn/Wn)) H (Vy/Wn). DR → DR is injective. b For once (*) is established we argue as follows. Let ξ be a global H i section over Xn of DR which dies formally at x. We must show that 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 etale´ over A, and V an 262 Nicholoas M. Katz

~~aﬃne open neighborhood of z etale´ 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. Ti T ). Because V is etale´ over A, F extends → i uniquely to a σ-linear map F : V V(σ) which lifts absolute Frobenius. → n Because all iterates of F, especially Fn : V V(σ ), are homeo- → morphisms, the functor (Fn) is exact. Therefore we have ∗ 0 i 0 (σn) i H (V, H (V/Wn)) = H (V ,(Fn) (H (V/Wn))) DR ∗ DR  n H i = H i n Ω (F ) DR(V/Wn) ((F ) ( V∗/W ))  ∗ ∗ n   • n But the complex (Fn) (Ω ) on V(σ ) is a complex of locally free V/Wn ∗ n 240 sheaves of finite rank on V(σ ), with O-linear differential. For any closed n point yV, the formal stalk at y(σ ) is

~~n • O n • (F ) (ΩV/W ) V(σn),y(σn) (F ) Ω . ∗ n ≃ ∗ Vy/Wn O n OV(σ ) b b n Therefore the sheaves on V(σ )~~

F i i dfn n H i H i n • = FV/W = (F ) ( DR(V/Wn)) = ((F ) ΩV/W ) n ∗ ∗ n are coherent, and (by ﬂatness of the completion) their formal stalks are given by F i n i ( )y(σ ) = HDR(Vy/Wn) We must show that b b

~~(σn) i i . (H◦(V , F ) ֒ (F ) (σn → y b Crystalline Cohomology, Dieudonn´eModules,... 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 Fn n V / V(σ )

~~ Fn n A / A(σ ) is cartesian (because V is etale´ over A). Therefore we have an isomorphism • • n ∼ n O n (F ) ΩV/W ((F ) ΩA/W ) V(σ ) . ∗ n ←− ∗ n O n OA(σ )~~

O n O n Because V(σ ) is flat over A(σ ) , this isomorphism is a filtered isomorphism (for the p-adic filtrations of Ω • and of Ω • ). V/Wn A/Wn By flatness again, this filtered isomorphism induces isomorphisms 241

~~j i j i gr (F ) (gr F ) O (σn) Fil V/Wn ≃ Fil A/Wn V O n OA(σ )~~

n It remains to show that gr j (F i ) is a locally free sheaf on A(σ ) k. Fil A/Wn n ⊗ It is certainly a coherent sheaf on A(σ ) (because the p-adic ﬁltration n • on (F ) Ω is O (σn) -linear), and it is killed by p; therefore it is a A/Wn 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 O n A(σ ) k-module structure is the composite of its natural module-structure ⊗ 264 Nicholoas M. Katz

~~over the sheaf of rings~~

~~H grFil◦ DR◦ (A/Wn)~~

~~with the σn-linear isomorphism~~

O ∼ H A k grFil◦ ◦(A/Wn) ⊗ −→ n f (F )∗( f ), 7→ e where f denotes any local section of OA lifting f . To conclude the proof, we must verify that this isomorphism is e 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

~~O ∼ H A k grFil◦ ◦(A/Wn) ⊗ −→ is provided by n f ( f )p 7→ where again f OA denotes any liftinge of f . ∈ e i,0 Lemma 8.2. The E2 terms of the spectral sequence are given by~~

i,0 i E H (X k, Zp) W(k) 2 ≃ et ⊗ ⊗ Proof. For each integer n 1, there is an isomorphism (cf. [24], [25]) ≥ ∼ H 0 Wn(OX k) DR(Xn/Wn) ⊗ −→ 242 deﬁned by n 1 − n i i p − (g0, ... , gn 1) p (gi) − 7→ Xi=0 e where gi is a local lifting of gi OX k to Oxn (Compare (??)). ∈ ⊗ e Crystalline Cohomology, Dieudonn´eModules,... 265

~~For variable n, these isomorphisms sit in a commutative diagram~~

~~0 Wn+r(OX k) ∼ / H (Xn+r/Wn+r) ⊗ DR usual projection n Wn(OX k) reduction mod p ⊗ Fr 0 Wn(OX k) ∼ / H (Xn/Wn). ⊗ DR Therefore we may calculate~~

~~i,0 i 0 E = lim H (X k, H (Xn/Wn)) 2 ⊗ DR ←−−n~~

r i ∼ lim (image of F or H (X k, Wn(OX k)) . −→  ⊗ ⊗  ←−−n \r     i  lim(ﬁxed points of F in H (X k, Wn(OX k)) Wn(k) ≃ ⊗ ⊗ ←−− ZO/pnZ i n lim H (X k, Z/p Z) Wn(k). ≃ et ⊗ ⊗ ←−−n

~~Consider now the exact sequence of terms of low degree~~

d 0 E1,0 H1 (X k/W) E0,1 2 E2,0 → 2 → cris ⊗ → 2 −−→ 2 Lemma 8.3. The map d0,1 : E0,1 E2,0 vanishes. 2 2 → 2 1 2,0 2 Proof. Because both Hcris(X k/W) and E2 = Het(X k, Zp) W j ⊗ 0,1 ⊗ ⊗ are ﬁnitely generated W-modules, we see that E2 is a ﬁnitely gener- 0,1 ated W-module. Therefore its inverse limit topology (as lim E2 (n)) is equivalent to its p-adic topology. Because Fn annihilates←−− the sheaf H 1 0,1 cris(X k/Wn), it annihilates its global sections E2 (n), and hence ⊗ 0,1 F is topologically nilpotent on E2 . But F is an automorphism of the 266 Nicholoas M. Katz

~~2,0 ﬁnitely generated W-module E2 ; as d2 commutes with F, this forces 0,1 d2 to vanish. Thus we obtain the following theorem. 243 Theorem 8.4. The exact sequence of terms of low degree~~

~~1 1 0,1 0 H (X k, Zp) / H (X k/W) / E / 0 → et ⊗ ⊗ cris ⊗ 2_ ∼ 1 formal 1 H (X/W) / H (Xx/W) DR expansion DR~~

1 b defines the Newton-Hodge filtration on Hcris 0 (slope 0) H1 (X k/W) (slope > 0) 0. → → cris ⊗ → → [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 re- i 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 • p+q E = lim H (X , H (WnΩ )) H (X /W) 2 0 ⇒ cris 0 ←−−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 H (X /W) Q: cris 0 ⊗ q 1 < slopes of E p,q Q q. − 2 ⊗ ≤ 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 H (X k/W) Q H (X/W) Q H (Xx/W) Q cris ⊗ ⊗ ≃ DR ⊗ → DR ⊗ is precisely the quotient “slopes > i 1” of Hi Q.b − cris ⊗ Bibliography 267 Bibliography

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~~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:~~

~~m/2 (m/2) 1 m/4 rm(t) = π σm(t)t − /Γ(m/2) + O(t ) (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 tX = 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 tGS G = T for n-rowed integral T (For any matirx B, let tB 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 ‘gen- erating function’, the theta series ϑ(S , Z) = exp(2π √ 1tr(tGSGZ)) G − where G runs over all (m, n) integral matricesP 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 modu- { } lar 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 f (Z) = a(T) exp(2π √ 1tr(TZ)/s) ǫ Γn(s), k , there exists a linear T 0 − { } P≥ combination g(z) = b(T) exp(2π √ 1tr(TZ)/s) of Eisenstein se- T 0 × − P≥ ries in Γn(s), k such that for positive-definite { } 1 n(n+1 2k)/2 (n+1 k)/2 T, a(T) = b(T) + O((min T − ) − (min T) − (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:~~

~~(m n 1)/2 (m(2n 1) 2(n2 1))/4n A(s, T) = λ αp(S , T)(det T) − − + O((det T) − − − ) Yp (3) where m > 2n + 2,~~

~~n(2m n+1)/4 n/2 1 = π − (det S )− Γ(m/2) ... Γ((m n + 1)/2) − , { − } 274 S. Raghavan~~

αp(s, T) is the product (over all primes p) of the p-adic densities p Q αp(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 improv- ing upon (2), it was felt that the decomposition of the space of modular forms of degree n into n + 1 subspaces through Maass’ Poincare´ 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 En,r(Z, h) which have been studied by Klingen [KL] and which arise by ‘lifting’ a cusp form h in Γr(1), k to Γn(1), k , Kitaoka has ob- { } { } k (n+1)/2 tained, in the same paper, the estimate a(T, h) = O((det T) − (r+1 k)/2 × (det T1) − ) for the Fourier coefficients a(T, h) of E(Z, h) with T = T1 ∗ and r-rowed symmetric T1. 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

~~k (n+1)/2 1 k/2 a(T) = O((det T) − (min T) − ) (4)~~

From [C], it can be seen that any f in Γn(s), k for k > 2n + { } 1 is a finite linear combination of Poincare´ 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

Theorem. If f (Z) = a(T) exp(2π √ 1tr(TZ)/z)ǫ Γm(s), k with k > T>0 − { } P A B 2n + 1 is such that for every M = in Γn (1), the constant term C D! 1 k in the Fourier expansion of f ((AZ + B)(CZ + D)− ) det(CZ + D)− is 0, k (n+1)/2 1 k/2 then we have a(T) = O((δT) − (min T) − ), 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~~

~~36 23/2 2 2 25/2 τ(n) exp( s √n) = 2 π Γ(25/2)s τ(n)(s + 16π n)− − 16Xn< 16Xn< ∞ ∞ for s > 0, which looks more involved than the ‘theta-relation’~~

τ(n) exp( ny) = (2π/y)12 τ(n) exp( 4π2n/y) (y > 0). − − 16Xn< 16Xn< ∞ ∞ Such identities (or modular relations as they are referred to in the litera- 250 ture) seem to be included by “Poisson formulae of Hecke type” considered by Igusa [I], which may thus be called generalized modular relations. Let F be the space of complex-valued C∞ functions F on the space x R+ of positive real numbers which behave like Schwartz functions at infinity and which have, as t tends to 0, an asymptotic expansion F(t) r ≈ art which is termwise differentiable (infinitely often). Let Z be r>0 P 276 S. Raghavan

the space of complex-valued functions Z on the complex plane such that Z(s)/Γ(s) is entire in s and further, for every polynomial P, the functions ZP is bounded in any vertical strip s α Re s 6 β with neighbourhoods { | } of 0, 1, 2, ... removed therefrom. The usual Mellin transform F − − 7→ MF established a one-one correspondence between F and Z . On the other hand, for any real κ > 0, there exists in Z , an involution Z Z 7→ × with Z×(s) = Z(κ s)Γ(s)/Γ(κ s) and this carries over to a unitary − − s operator F WF in F . If ϕ(s) = ann− is a Dirichlet series 7→ 16n< (absolutely convergent in a half-plane and)P ∞ of signature λ, κ, γ in the { } sense of Hecke [H2] so that (s κ)ϕ(s) is entire and of ﬁnite genus and − further ξ(s) = (λ/2π)sΓ(s)ϕ(s) = γξ(κ s), then the Poisson formula − established by Igusa in [I] reads:

an(WF)(2πn/λ) = γ anF(2πn/λ) (5) 06Xn< 06Xn< ∞ ∞ κ for every F in F , where ao = γ(λ/2π) Γ(κ). Residue ϕ(s). This includes a result of Yamazaki. m Let G(s) = (Γ(α js + β j)) j with α j > 0, Re β j > 0, m j > 1 and 16 j6r Q further, for i , j, αiβ j α jβi is not of the form mα j nαi with integers − − m, n > 0. ( j) s Let ϕ j(s) = a n − ; 1 6 j 6 N and { n | | } Xn,0 ( j) s ψ j(s) = bn n − ; 1 6 j 6 N be two sets of N Dirichlet series (each { n,0 | | } convergingP in some right half-plane absolutely) so that if we write

~~s s ξ j(s) = λ G(s)ϕ j(s), η j(s) = λ G(s)ψ j(s) (1 6 j 6 N)~~

~~251 for some ﬁxed λ> 0, then we have the functional equations~~

~~ξ j(κ s) = c jkηk(s) (1 6 j 6 N) (6) − 16Xk6N~~

2 with real c jk; we may suppose that (c jk) 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, ... , up 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 : Rx 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

~~(k) MF(s) an F( n /λ) Residue ξk(s) = (7) | | − s=u j G(s) Xn,0 ReXu j<κ/2~~

(k) = ckl b (WF)( n /λ)  n | | − 16X16N Xn,0   (M(WF))(s) Residue η1(s) − s=u j G(s)  Re Xu j<κ/2   Formula (7) generalizes some well-known relations of a similar nature considered, for example, by Maass [M1] in the Hecke theory of non-analytic automorphic forms and by B.C. Berndt. The proof of (7) is on the same lines as in Hecke [H2]; the sum over residues has to be interpreted suitably in terms of the coeﬃcients in the asymptotic expansions of F and WF at 0 and the residue of the Dirichlet series involved and sometimes, it takes a simple form as in (5). A Poisson summation 252 formula for a generalized Fourier transformation due to Kubota can also be treated with arguments similar to those for (7). In the study of non- analytic automorphic forms, Maass [M2] has considered (for Dirichlet series) functional equations in matrix form involving a generalized Γ- function Γ(s; α, β) which is the Mellin transform of the standard Whit- taker function W(y; α, β); in this case again, a general Poisson formula 278 S. Raghavan

x like (7) for pairs (F1, F2) of C∞ functions on R+ with prescribed behaviour at inﬁnity and at 0 can be obtained. Specialising F1(t), F2(t) to be W(ty; α, β), W(ty; β, α) respectively (with y > 0), one gets the corresponding formula in [M2]; in the light of a recent paper of Ranga Rao, it turns out that there are quite a few pairs (F1, F2) for which our Poisson formula holds. In the context of formula (5) proved in the lectures [I], one comes across the natural question as to whether a p-adic analogue of the oper- F F x ator W exists. One may consider, instead of above, the space (Qp) x of complex-valued F on Qp which are locally constant, with

0 for all t with valuation t p large F(t) = 1 | | (8)  2 1/2   aµ1(t) t p + bµ2(t) t p for all t with t p small  | | | | | |    constants a, band quasicharacters µ1, µ2. This is a so-called Kirillov model for irreducible admissible representations πp of GL2(Qp). In this 1 s 1 s 1 case, if L(s, πp) = (1 µ (p)p 2 )(1 µ (p)p 2 ) , then the W- { − 1 − − 2 − }− operator is given again via the Mellin transform M: (M(WF))(1 s) (MF)(s) − = ǫ(s, πp) (9) L(1 s, πp) L(s, πp) − with a certain function ǫ(s, πp) for which ǫ(s, πp). ǫ(1 s, πp) = 1. 0 − Let Wp be the Whittaker function on GL2(Qp) whose Mellin transform x (over Q ) is L(s, πp), for every prime p and further let πp p be such that p { } together with a representation π of GL2(R), the tensor product π pπp ∞ ∞⊗ gives an irreducible unitary representation of GL2(QA) and moreover, s let L(s, πp) be a Dirichlet series an n − converging absolutely in p n,0 | | a rightQ s-half plane, with a functionalP equation s 1 s, involving s (p+1)/2 → − s v 253 L(s, π ) = (2π)− − Γ(s + (p + 1)/2) for p > 0 in Z or π− − Γ((s + ∞ F × v)/2)Γ((s v)/2) with v in C. Then for F on Q×A built from and the −0 various Wp, 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 Poincaresche´ Reihen, Math. Ann. 148 (1962), 257-307.~~

~~[H-R] Hardy G. H. and S. Ramanujan: Asymptotic formulae in combi- natory analysis, Proc. London Math. Soc., (Ser 2) 17 (1918), 75- 115.~~

~~[H1] Hecke E.: Theorie der Eisensteinscher Reihen hoherer¨ 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 fur¨ Siegelsche Modulformen, Math. Zeit., 102 (1967), 30-43.~~

~~[M1] Maass H.: Uber eine neue Art von nichtanalytischen auto- morphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktional-gleichungen, Math. Ann., 121 (1949), 141-183.~~

~~[M2] Maass H.: Die Diﬀerentialgleichungen in der Theorie der ellip- tischen Modulfunktionen, Math. Ann., 125 (1953), 233-263. 280 Bibliography~~

~~[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 commu- nication 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.

1Results 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. 2Takuro 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, ... , vtǫV(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 permu- tations 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, an n-dimensional Q-vector sub- 256 space VQ such that one has V = VQ R is identified in V. An open NQ 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 − (2) C = vǫV; La(v) > 0, a = 1, ... , t, { Mb(v) = 0, b = 1, ... , n 1 . − } For each b(1 6 b 6 n t), set − (1) (1) C (b, ) = vǫC ; M1(v) = ... = Mb 1(v) = 0, ± n − Mb(v) > 0 . ± } A Remark on Zeta Functions of Algebraic Number Fields 283

~~For each a (1 6 a 6 t), set~~

(1) (1) C (n t + 1, a) = vǫC ; Mb(v) = 0 for b = 1, ... , n t, − n − 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 = r + 2r ). Let x x(i) (1 6 i 6 n) be n 1 2 7→ mutually distinct embeddings of k into the field of complex numbers C. (1) (r1) (r1+i) (r1+r2+i) We may assume that x , ... , x are all real and that x = x− (1 6 i 6 r2). We embed k into an n-dimensional real vector space 257 V = Rr1 Cr2 via the map: x (x(1), ... , x(r1), x(r1+1), ... , x(r1+r2)). × 7−→ 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+ = r1 r R (C) 2 , 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 { ∈ } | | ∞ simplicial cones with generators all in k+ such that V+ = uC j j J u E(k)+ (disjoint union). S∈ ∈S~~

Proof. For each x V, we denote by N(x) the “norm” of x given by (1) (r ) ∈(r +1) (r +r ) 2 1 N(x) = x ... x 1 x 1 ... x 1 2 . Let V be the subset of V+ con- | | + sisting of all vectors with norm 1:

~~1 V = x V+; N(x) = 1 . + { ∈ }~~

~~Note that each vector in V+ is uniquely expressed as a positive scalar multiple of a vector in V1 : 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 1 1 on V+ properly discontinuously and that E(k)+/V+ is compact. Thus, 1 there exists a compact subset F of V+ such that~~

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

~~s 1 F = (Ci V ) (2) ∩ + [i=1 and that Ci uCi = for any 1 , u E(k)+(1 6 i 6 s). (3) ∩ ∈ 258 If follows from (1) and (2) that~~

~~s V+ = uCi. [i=1 u[E(k) ∈ (1) Set C1 = C1 and set~~

~~(1) C = Ci uC (2 6 i 6 s). i − 1 u [E(k)+ ∈~~

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

~~s (1) V+ = uCi and [i=1 u [E(k)+ ∈ (1) (1) uC C = for any u E(k)+ if i > 2. 1 ∩ i ∈ A Remark on Zeta Functions of Algebraic Number Fields 285~~

Now assume that a ﬁnite system of Q-rational general polyhedral cones C(a), ... , C(a)(1 6 a 6 s 2) with the following three properties is given: 1 s − (a) C Ci (4) a i ⊂ ( ) s (a) V+ = uCi , (5)(a) [i=1 u [E(k)+ ∈ (a) (a) uC C = for any u E(k)+ if i 6 a and i , j. (6) a i ∩ j ∈ ( ) (a+1) (a) 6 Then set Ci = Ci for i a + 1 and set

C(a+1) = C(a) uC(a) for i > a + 2. i i − a+1 u[E(k) ∈ Then C(a+1), ... , C(a+1) is a ﬁnite system of Q-rational general polyhe- { 1 s } dral cones with properties (4)a+1, (5)a+1 and (6)a+1. (s 1) (s 1) It is easy to see that C − , ... , C − is a ﬁnite system of Q- { 1 s } rational general polyhedral cones such that

~~s (s 1) V+ = uCi − (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~~

V+ = uC j (disjoint union). (7) [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 j ... V jt of C j so that they are all in O(k)+ = O(k) k+. 1, , j ∩ 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

1 6 C j = s1v j1 + s2v j2 + ... + st j v jt j ; 0 < s1, s2, ... , st j 1 n o and 1 1 1 R(c, C j) = x C f − a− ;(x) f ai c c . { ∈ j ∩ i(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 ∈ s ζ(s, C, x) = N(x + z1v1 + ... + ztvt)− , (8) Xz

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 s ζk(s, c) = N(g)− , (9) Xg 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.

~~s ζk(s, c) = N(fai(c))− ζ(s, C j, x). Xj J x RX(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 = y1v 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). 1 1 Then x = w (z v j + ... + zt v jt ) C (fai c ) . − 1 1 j j ∈ j ∩ ( ) − Furthermore (x)fai(c) is in the ray class c. Thus x R(c, C j). A simple consideration shows that j, z , ... , zt ∈ 1 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 + z1v 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 t j the following map establishes a bijection from the set R(c, C j) Z+ j J{ × } onto the set of integral ideals of k in the ray class c: S∈

~~t j t j (x, z) R(c, C j) Z ai c f(x + zav ja). ∈ × + 7−→ ( ) Xa=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 L- series 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). Sup- pose 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,~~

~~1 2 L′(0χ, f) = χ(c) log( ε(c) ) −W | | Xc H ∈ where W is the number of roots of unity in K.~~

~~ii). The explicit reciprocity law is given by 262~~

~~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 Wth power of a number in K and the ε(c) are all associates.~~

~~iii). If f = pa where p is a prime ideal, then~~

~~b NK/Q(ε(c)) = Nk/Q(p)~~

~~where Wh b = 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.~~

~~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). 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 L (0, χ, f) = − χ(c) log( ε(c)2) ′ W Xc | 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) w = εc 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 th by Siegel [4], the units εc, c , c0, together with the w 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 andJτ = 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 h 2 L′(0, χ, f) = − χ(h) log( ε ) W | | hXH ∈ 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,

~~1 ψ(h) = χ(h) + χ(τhτ− ), ψ(hτ) = 0.~~

~~It turns out that ε was constructed so that some power of ε is real. There- fore, 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 = − ψ(g) log( εg 2) 2W | | Xg 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 − 2 3 H = H 19 = 1, σ , στ, σ τ , − { } the Klein four group.

K ❙ ❧❧❧✇ ❍❍❙❙❙ 2 ❧❧ ✇ ❍ ❙❙ 3 σ τ ❧❧❧ ✇✇ ❍❍στ❙❙❙σ τ ❧❧❧ ✇ σ2 ❍❍ ❙❙❙ ❧❧❧ ✇✇ τ ❍ ❙❙❙ ❧❧❧ ✇✇ ❍ ❙❙❙ (2) ❧❧ (1) (1) ❙ (2) K17 K17 K17, 19 K 19 K 19 ❈ ① − ●● − − ❈❈ ①① ●● ②② ❈❈ ①① ●● ②② ❈❈ ①① ●● ②② ❈ ①① ● ②② k k k 17 ❍ 323 19 ❍❍ − ✉✉ − ❍❍ ✉✉ ❍❍ ✉✉ ❍❍ ✉✉ ❍ ✉✉ Q

(1) There is a pair of prime ideals of norm 17 in Q( √ 19), p = 264 − 17 7 + √ 19 (2) (1) 7 √ 19 − and p17 = p17 τ = − 2− . For j = 1, 2, there are unique  2    (i) (i) ray class characters χ (mod p17) 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 character (mod 17) and corresponds to the class field K17, 19 = Q( √17, √ 19). Of the five possibilities, we see that − 3 1 − (1) 1, στ or 1, σ τ = τ− 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 = Q( √17) 2 3 and k 323 = Q( √ 323). We see that G(K/k 323) = 1, σ, σ , σ is − − − { } cyclic while G(K/k ) = 1, τ, σ2, σ2τ is the other Klein four group 17 { } 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 k : K(1) fixed by 1, τ and K(2) fixed by 1, σ2τ 17 17 { } 17 { }

~~1 σ2 στ σ3τ~~

~~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 −~~

~~(1) (1) 1 2 σ2 2 L′(o, χ , p ) = − [log( π ) log( π )]. 17 2 | | − | |~~

(2) (2) (2) There is also such a number in K 19 for L′(0, χ , p17 ) but it is just π¯ and the formula is the same. It is− no surprise that the formula should (1) (1) (2) (2) give the same answer since L(s, χ , p17 ) 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, σ2τ στ, σ3τ σ, σ3~~

ψ1 1 1 1 1 1 ψ 1 1 1 1 1 17 − ψ 19 1 1 1 1 1 − − − ψ 323 1 1 1 1 1 − − − ψ 2 2 0 0 0 2 − 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 prim- − itive ray class character of k17 modulo a prime ideal of norm 19 which corresponds to K(1). It takes the values 1, 1, 1, 1, at 1, τ, σ2, σ2τ re- 17 − − spectively and also induces ψ2 on G. Further, by our Theorem there is a 1 unit E in K 19 such that − π = E2 πσ2 With η = E 2 = EEτ | | we have 1 2 2 L′(0, ψ ) = − log( E ) = log(η) 2 2 | | − (1) σ2 τ σ2 τ where η is in K17 . Also E = 1/E so that (EE ) = 1/(EE ) and σ2 1 ± hence, η = η− . 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. (1) (2) We return to K/k 19 again and now consider χ and χ 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,

~~1 h 2 L′(0, χ, 17) = χ(h) log( ε ). −2 | | hXH ∈ 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)) = 1 and so 17 − (1) (1) L′(0, χ , 17) = 2L′(0, χ ).

~~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~~

~~(1) ε L′(0, χ , 17) = log − σ2 ε = 2 log(η). − 2 Of course η/ησ = η2 and so ε = η is still possible. However,~~

~~(εεσ2)2 L (0, χ(1)χ(2)) = log (ε) ′ − NK/k 19 ! − = 2 log( εεσ2 ), − | | while by Dirichlet’s class-number formula,~~

~~(1) (2) L′(0, χ χ ) = L(0, ψ 323)L′(0, ψ17) − = h(k 323)h(k17) log(ε17) − = 4 log(ε17)~~

~~2 2 Thus εεσ = ε¯2 while ηησ = 1 and so ε , η. Since ± 17 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′ andχ ¯′ 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 di- − rectly. 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

χ1′ 1 1 1 1 χ 1 i 1 i ′ − − 2 χ′ 1 -1 1 -1 χ¯ = χ 3 1 i 1 i ′ ′ − − Character table of H 323 − unique ideal p17′ of k 323 of norm 17 as our conductor. Since p17′ is in the class of order two,− the corresponding Frobenius automorphism of 2 H 323 is σ . (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 ofχ ¯′) 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 h 2 L′(0, χ, p′ ) = χ(h) log( π′ ) 17 −2 | | hǫXH 323 − where 268 4 NK/Q(π′) = 17 2 2 Further π′ is not just π or even π times a unit since

~~(π′) = p17′ 296 H. M. Stark~~

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

~~π′ 2 log(η) = L′(0, χ′, p′ ) = log . − 17 − σ2 ! π′~~

~~σ2 where π′/π′ 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 = − , η(ω/9)2 2 where we have used the eta-function on the right, and

~~2 L′(0, χ′) = log( α ) − | | from which we see that η = N (1) (α) K/K17~~

~~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 τ 269 abelian with conductor f = f , (1) and ψ2 is a of G induced by a character of H, then there is an integer ε in K such that

~~1 g 2 L′(0, ψ , N(f)) = − ψ (g) log( ε ). 2 2W 2 | | Xg G ∈ Derivatives of L-Series at S = 0 297~~

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 momentarilyP 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,

~~1 g 2 L′(0, ψ , 17) = − ψ (g) log( ε ). 2 4 2 | | Xg G ∈ Further, for ψ = ψ1, ψ 19 or ψ 323, − − ψ(g) log( εg 2) = 0. | | Xg 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~~

~~g 2 L′(0, ψ17) and ψ17(g) log( ε ). X | | We found earlier that~~

g 2 g 2 ψ17(g) log( ε ) = [ψ17(g) + ψ 323(g)] log( ε ) | | − | | X X (1) (2) = 4L′(0, χ χ ) − = 16 log(ε ) = 16L′(0, ψ ). − 17 − 17 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, (1) (2) − 2s 1 − L(s, χ χ ) has a p-factor (1 p− )− 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

ψ (g) log( unitg 2) = [ψ (g) + ψ (g)] log( unitg 2) = 0, 17 | | 17 323 | | Xg G X ∈ 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 Ga- lois 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/πp is a Wth 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 g 2 L′(0, ψ, f) = − ψ(g) log( π ). 2W | | Xg 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 2 < u, L (0, ψ, f) > = − < u, v > log πg ′ 2W g Xg G ∈ 1 = − log ε 2 . 2W u | | Here g εu = (π ) Yg G ∈ g is a unit since the π are associates and < u, vg >= 0 by the orthogo- g nality relations. In fact, since πτ = ζπ forP a root of unity ζ,

τ g g εu = (ζ ) (π ) , Yg Yg 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 1 < u, L (0, ψ, f) >= − log(ε ), (1) ′ W u where εu is a positive real unit in K. th Further, εu is already a W power in K. To see this, let M be the th ﬁeld of W 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 χ1∗ 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, ρ(gh) = ρ(g) ρ(h) = 0, hXH hXH ∈ ∈ 272 and hence ψ(gh) = 0. hXH ∈ therefore vgh = 0. hXH ∈ For each g in G, let pg be chosen according to part ii) of our Question g p th so that π /π g is a W power in K. For any h in H, pgh pg(mod W) ≡ and hence

pgh < u, vgh > pg < u, vgh > 0(mod W) ≡ ≡ hXH hXH ∈ ∈ Therefore, πg pg εu = π πpg ! Yg G Yg G ∈ ∈ is a Wth 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 conduc- th tor 133 found by Tate. He found a unit εu in K which is a W 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 Pro- ceedings 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 SL2(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 ∈ 1 ys E(z, s) = Im(γz)s = (1) 2 cz + d 2s γ XΓ /Γ cX,dεZ | | ∈ ∞ (c,d)=1

~~1 n where Γ= PSL2(Z), Γ = n Z Γ. If we multiply both ∞ (± 0 1! ∈ ) ⊂~~

~~1Supported by the Sonderforschungsbereich “Theoretische Mathematik” at the Uni- versity of Bonn.~~

~~302 Eisenstein Series and the Riemann Zeta-Function 303~~

~~∞ 2s sides of (1) by ζ(2s) = r− and write m = rc, n = rd, we obtain r=1 P~~

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

2 where ′ indicates summation over all (m, n) Z / (0, 0) . The func- ∈ { } tion ζ(2Ps)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 = PSL2(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 Eisen- stein series. a) Analytic continuation and functional equation. 304 D. Zagier

The function E(z, s) has a meromorphic continuation to all s, the 1 only singularity for Re(s) > 2 being a simple pole at s = 1 whose residue is independent of z: 3 ress= E(z, s) = ( z H). (3) 1 π ∀ ∈ The modiﬁed function

~~s E∗(z, s) = π− 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) − These statements are proved in a way analogous to Riemann’s proof 277 of the analytic continuation and functional equation of ζ(s); we rewrite (2) as

~~∞ 1 s ′ s 1 s 1 E∗(z, s) = π− Γ(s) Qz(m, n)− = (Θz(t) 1) − dt, (6) 2 2 Z − Xm,n o~~

~~where Qz(m, n)(z H) denotes the quadratic form ∈ mz + n 2 Q (m, n) = | | (7) z y~~

πtQz(m,n) of discriminant 4 and Θz(t) = e− the corresponding theta- − m,n z P∈ 1 series; then the Poisson summation formula implies Θz( t ) = 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 →∞ − 1 C(F; y) = F(x + iy)dx (y > 0) (8) Z 0 Eisenstein Series and the Riemann Zeta-Function 305 be the constant term of its Fourier expansion and

~~∞ s 2 I(F; s) = C(F; y)y − dy (Re(s) > 1) (9) Z o the Mellin transform of C(F; y). From (1) we obtain~~

I(F; s) = F(z)ysdz = F(z)E(z, s)dz, (10) Z Z Γ /H Γ/H ∞ dxdy where dz denotes the invariant volume element . 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 3 ress=1I(F; s) = F(z)dz, (11) π Z Γ/H and the function

~~s I∗(F; s) = π− 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~~

~~s 1 s E∗(z, s) = ζ∗(2s)y + ζ∗(2s 1)y − (14) − ∞ s 1/2 +2 √y n − σ1 2s(n)Ks 1/2(2πny) cos 2πnx, − − Xn=1 where~~

~~s/2 s ζ∗(s) = π− Γ( )ζ(s) (s C), (15) 2 ∈ 306 D. Zagier~~

v σv(n) = d (n N, v C), ∈ ∈ Xd n | ∞ t cosh u Kv(t) = e− cosh vu du (v C, t > 0). (16) Z ∈ 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

v ζ∗(s) = ζ∗(1 s), σv(n) = n 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 ∆= y2 + ∂x2 ∂y2 ! and the Hecke operators az + b T(n): F(n) F( ) (n > 0), → d adX=n b(modX d) a,d>0~~

~~279 namely~~

s ∆E(z, s) = s(s 1)E(z, s), T(n)E(z, s) = n 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, 1 1 if ζ(1 + it) = 0, the (14) implies that the function F(z) = E(z, 2 + 2 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 1 1 s = it in (10) we find F(z) 2dz = 0, a contradiction. 2 − 2 | | Γ/RH 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 → C length ( y U) vol(U) 1 ε ∩ = + 0(y 2 − ) (y 0); length(Cy) vol(Γ/H) →

3/4 ε moreover, if the error term in this formula can be replaced by 0(y − ) 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

~~length(Cy U) C(F; y) = ∩ length (Cy)~~

1 and the Mellin transform I(F, s) of this is holomorphic in Re(s) > 2 Θ (where Θ is the supremum of the real parts of the zeroes of ζ(s)) except vol(U) for a simple pole of residue κ = 3 F(z)dz = at s = 1. If 4 vol(Γ/H) Γ/RH F were suﬃciently smooth (say twice diﬀerentiable) we could deduce 2 1 that I(F; σ + it) = 0(t− ) on any vertical strip Re(s) = σ > 2 Θ, and 1 1 Θ ε the Mellin inversion formula would give C(F; y) = κ + 0(y − 2 − ). For 1 ε F = χU we can prove only C(F; y) = κ+0(y 2 ); 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 of discriminant D we associate the root zQ = − H. The Γ- 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 rep- 2 resented by Q. Also, the form Qz defined by (7) equals Q. There- Q √ D fore (6) gives | | s/2 1 D ′ E (z , s) = | | π sΓ(s) Q(m, n) s ∗ Q 2 4 − − ! Xm,n s/2 w D s = | | π− Γ(s)ζ(A, s), 2 4 ! s where w(= 2, 4 or 6) is the number of roots of unity in K and ζ(A, s) = − a is the zeta-function of A. (Note that this equation makes sense becauseP 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

h(D) s/2 h(D) w D s E∗(zQ , s) = | | π− Γ(s) ζ(Ai, s) i 2 4 ! Xi=1 Xi=1 s/2 w D s = | | π− Γ(s)ζK(s) 2 4 ! s/2 w D s = | | π− 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 s n− . Since the latter is holomorphic, we deduce that the function n=1 n hP(D)

E∗(zQi , s) is divisible by Γ(s)ζ(s), i.e. that it vanishes with multiplic- i=1 ityP 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

h(D) 1 ζ(s, D) = (20) Q (m, n)s Xi=1 X2 i (m,n)εZ /ΓQi Qi(m,n)>0 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)~~

F(zQi ) = 0 (21) Xi=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 zQ Γ/H as before, i ∈ but a closed curve CQi Γ/H as follows: Let wi, wi′ R be the roots of ⊂ 2 ∈ 282 the quadratic equation Qi(x, 1) = ai x + bi x + ci = 0 and let Ωi be the semicircle in H with endpoints wi and wi′. The subgroup

~~1 2 (t biu) ciu 2 2 ΓQi = − 1 − t, u Z, t Du = 4 (22) (± aiu (t + biu)! ∈ − ) 2~~

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

~~wiip + w′ z = i (0 < p < ), ip + 1 ∞~~

2 t + u √D dp then ΓQ acts by p ε p (ε = a unit of K) and dQ z = . A i → 2 | i | 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

~~πs ζ(A , s) = D s/2 E (z, s) d z i s 2 − ∗ Qi Γ( 2 ) Z | | CQi~~

~~(cf. [10], §3 for a sketch of the proof). Thus~~

~~h(D) s s/2 s 2 E∗(z, s) dQ z = π− D Γ( ) ζK(s), | i | 2 Xi=1 Z CQi~~

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) F(z) dQ z = 0 (23) Z | i | Xi=1 CQi~~

2 2 for all non-square integers D < 0, where Qi(m, n) = aim + bimn + cin (i = 1, ... , h(D)) are representatives for the Γ-equivalence classes of 283 binary quadratic forms of discriminant D, CQi is the image of

2 z = x + iy H ai z + bi x + ci = 0 in Γ/H, { ∈ | | | } and √D d z = ((dx)2 + (dy)2)1/2. Qi 2 | | aiz + biz + ci | | Example 3: The third example comes from the theory of modular forms. Let f (z) be a cusp form of weight k on SL2(Z) which is a normalized eigenfunction of the Hecke operators, i.e. f satisﬁes

~~az + b k a b f = (cz + d) f (z) ( z H, SL2(Z)) cz + d ! ∀ ∈ c d! ∈ and has a Fourier development of the form~~

~~∞ 2πinz f (z) = ane Xn=1 with a1 = 1 and anm = anam if (n, m) = 1. Deﬁne D f (s) by~~

1 D (s) = (Re(s) >> 0), f 2 s s 2 s (1 α p )(1 αpβp p )(1 β p ) Yp − p − − − − p − 2 k 1 where αp and βp are the roots of X apX+ p = 0; it is easily checked − − that ζ(2s 2k + 2) ∞ 2 s D f (s) = − an n− . ζ(s k + 1) | | − Xn=1 312 D. Zagier

Applying the Rankin-Selberg method (10) to the Γ-invariant func- k 2 k ∞ 2 4πny tion F(z) = y f (z) with constant term C(F : y ) = an e , we | | | | − n=R1 ﬁnd yk f (z) 2E (z, s)dz = I (F; s) | | ∗ ∗ Γ/RH

2 s s k+1 ∞ an = π− Γ(s)ζ(2s) (4π)− − Γ(s + k 1) | | · − ns+k 1 Xn=1 − s k+1 2s k+1 = 4− − π− − Γ(s)Γ(s + k 1)ζ(s)D f (s + k 1). − − 284 This formula, which was the original application of the Rankin- Selberg 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

~~yk f (z) 2F(z)dz = 0 Z | | 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

~~∆iF(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 LDΦ : H C by → 1 a z 2 + bz + c LDΦ(z) = Φ | | (z = x + iy H), (26) 2 y ! ∈ a,Xb,c Z b2 4ac∈=D − 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 ∈ 1 ε Φ(X) = O( X − − ) ( X ) (27) | | | |→∞ a z 2 + bx + c for some ε> 0. Moreover, the expression | | is unchanged if y one acts simultaneously on x + iy H and Q(m, n) = am2 + bmn + cn2 ∈ by an element γ Γ. Hence LDΦ(γz) = LDΦ(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 LDΦ in Γ/H. If D is not a perfect square, then a , 0 in (26), so (for Φ even)

~~∞ ∞ a(x + b/2a)2 D/4a LDΦ(z) = Φ ay + − y ! Xa=1 bX= b2 D(mod−∞ 4a) ≡ 314 D. Zagier~~

1 ε ∞ ∞ a(x + b/2a)2 D/4a − − = O ay + −  y !  Xa=1 bX=   b2 D(mod−∞ 4a)   ≡   1 ε ∞ ∞ ax2 − − = O nD(a) ay + dx  Z y !  Xa=1    −∞  as y = Im(z) , where  →∞ 2 nD(a) =, b(mod 2a) b D(mod 4a) . (28) n | ≡ o n (a) Since the integral is O(a 1 εy ε) and ∞ D converges, we ﬁnd − − − 1+ε a=1 a P ε LDΦ(z) = O(y− ). 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

~~1 ∞ bx + c Φ( ) 2 y bX2=D cX= −∞ to estimate. If Φ is sufficiently smooth (say twice differentiable), then the inner sum differs by a small amount from the corresponding integral~~

~~∞ c Φ( )dc = y ∞ Φ(X)dX, Z y · −∞ −∞ and this will be small as y only if ∞ Φ(X)dX vanishes. Thus with →∞ R the requirement −∞~~

∞ Φ is C2 and Φ(X)dX = 0 if D is a square (29) Z −∞ ε as well as (27) we have LDΦ(z) = O(y ) as y for all D, and so − → ∞ the scalar product of LDΦ 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 or1(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 → 1 ε a) F(z) = O(y − ) for some ε> 0

b) F(z) is orthogonal to Im(LD), i.e. LDΦ(z)F(z)dz = 0 for D P Γ/RH 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 LDΦ(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. → 2 2 → ∞ If Qi(m, n) = aim + bimn + cin (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 ΓQ /Γ (ΓQ = ◦ ∈ i i stabilizer of Qi in Γ). Hence

~~h(D) 2 ai γz + bi Re(γz) + ci LDΦ(z) = Φ | | Im(γz) ! Xi=1 γ XΓQ /Γ ∈ i and so~~

~~h(D) 2 ai z + bi x + ci LDΦ(z)F(z)dz = Φ | | F(z)dz. y ! Z Xi=1 Z Γ/H ΓQi /H 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) 1 ys Φ a z 2 + b x + c dz. i i i + 2 2 2 Z | | mz n 2s Xi=1 (m,n)XZ /ΓQ ∈ i H | |~~

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) 2 1 ∈ Z with Qi(n, m) > 0 if we drop the factor 2 . Then the substitution 1 − nz bin + cim z − 2 introduced in [11], p. 127, maps H to H and → 1 mz + ain 2 bim gives − −

~~2 z D/4 s ζ(2s) LDΦ(z)E(z, s)dz = ζ(s, D) Φ | | − y dz (30) Z Z y ! Γ/H H~~

for 1 < Re(s) < 1 + ε, with ζ(s, D) deﬁned as in (20). Since ζ(2s) and 288 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 LDΦ(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, which in fact works for all D. By the Rankin-Selberg method, we know that LDΦ(z)E(z, s)dz equals the Mellin transform of the constant term of LR DΦ, and writing LDΦ(z) as Eisenstein Series and the Riemann Zeta-Function 317

b 2 ∞ ∞ a z + + n D/4a Φ 2a −  y  Xa=1 b(modX 2a) nX=   2 −∞   b D(mod 4a)   ≡   1 ∞ bx + c + Φ , 2 y ! bX2=D cX= −∞ we see that this constant term is given by

~~∞ ∞ ax2 + ay2 D/4a C(LDΦ; y) = nD(a) Φ − dx y ! Xa=1 Z −∞ 0 if D , m2,~~

 ∞  2 y. Φ(X)dX if D = m > 0, +  Z   −∞ 1 ∞ c  Φ if D = 0, 2 y!  cX=  −∞ where nD(a) is deﬁned by (28). The Mellin transform of the ﬁrst term is

∞ ∞ 2 2 ∞ nD(a) x + y D/4 Φ − ys 2dxdy,  as  y − = · Z Z ! Xa 1  0   −∞  s  and since nD(a)a− = ζ(s, D)/ζ(2s) ([11], Prop. 3, i), p. 130) we recover eq.P (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(L0Φ; y) 289 equals

~~1 ∞ c 1 ∞ ∞ Φ( ) y Φ(X)dX = y Φ˜ (ny), 2 y − 2 cX= Z Xn=1 −∞ −∞ where ∞ Φ˜ (y) = Φ(X)e2πiXydX Z −∞ 318 D. Zagier~~

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

~~2 z s ζ(2s) L0Φ(z)E(z, s)dz = ζ(s, 0) Φ | | y dz Z Z y ! Γ/H H~~

~~∞ s 1 + ζ(s)ζ(2s) Φ˜ (y)y − dy (31) Z 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

~~LDΦ(z)E∗(z, s)dz (32) Z Γ/H~~

∞ 1 s s/2 1 (2π) − D Γ(s)ζ(s, D) Ps 1(t)φ( D 2 t)dt if D < 0,  | | Z − | |  1 =   2 ∞ 1 s s/2 s s 1 s 1 2 1  π− D Γ ζ(s, D) F , − ; ; t Φ(D 2 t)dt if D > 0, 2 2 Z 2 2 2 − !  0   s 1 s 1 2 where Ps 1(t) and F , − ; ; t denote Legendre and hypergeo- − 2 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 1 for z in the ﬁrst integral on the − z Eisenstein Series and the Riemann Zeta-Function 319 right-hand side of (31) and using the identity

~~∞ Γ s s 1 1 1 s ( 2 ) s y cos 2πXy dy = π 2 X (0 < Re(s) < 1) − − 1 s − Z 2 Γ( − )| | 0 2 in the second; this gives~~

~~L0Φ(z)E∗(z, s)dz (33) Z Γ/H~~

∞ ∞ s 1 s = ζ(s)ζ∗(2 2s) Φ(X)X − dX + ζ(1 s)ζ∗(2s) Φ(X)X− dX − Z − Z 0 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 is suﬃcient to show that the space Im(LD), or a dense subspace of it, D is closed under the action of these operators.P An elementary calculation shows that a z 2 + bx + c a z 2 + bx + c ∆Φ | | = Φ1 | | (34) y ! y ! with 2 Φ1(X) = 2XΦ′(X) + (X + D)Φ′′(X). (35)

Hence ∆LDφ(z) = LDΦ1(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 = L 2 αp + LD + pL 2 βp (36) ◦ Dp ◦ p ! D/p ◦ 320 D. Zagier

~~where αp and βp denote the operators 291~~

~~αpΦ(X) = Φ(X/p), βpΦ(X) = Φ(pX),~~

~~D 2 is the Legendre symbol, and L 2 is to be interpreted as 0 if p ∤ p ! D/p D. To prove this write~~

p z + j T(p)LDΦ(z) = LDΦ(pz) + LDφ p ! Xj=1 ap2 z 2 + bpx + c = Φ | | ( py ! b2 X4ac=D − p a z 2 + (2a j + bp)x + (a j2 + b jp + cp2) + Φ | |  py ! Xj=1  a z 2 + bx + c  = n(a, b, c)Φ | |  py ! b2 4Xac=Dp2 − with

~~p a b b 2a j c b j + a j2 n(a, b, c) = ε , , c + ε a, − , − p2 p ! p p2 ! Xj=1~~

~~(where ε(a, b, c) equals 1 if a, b, c are integral, 0 otherwise). To prove (36) we must show that~~

~~D a b c a b c n(a, b, c) = 1 + ε , , + pε , , p ! p p p! p2 p2 p2 ! (a, b, c Z, b2 4ac = Dp2). ∈ −~~

~~292 For p odd, this follows from the following table, in which vp1 (m) denotes the exact power of p dividing an integer m. Eisenstein Series and the Riemann Zeta-Function 321~~

p a b b 2aj c bj+aj2 a b c a b c v 1 (a) v 1 (b) v (c) ε , , c ε a, − , − ε , , ε , , p p p′ p2 p p p2 p p p p2 p2 p2 j=1 P 0 > 0 > 0 0 1 00 > > D 1 1 1 0 1 + p 1 0 > 2 > 1 0 1 0 00 > 2 1 > 1 1 1 10 > 2 > 2 1 1 0 10 > 2 > 2 > 2 1 p 1 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 LDΦ for special choices of D and Φ. For (21) we choose

~~Φ(X) = δ(X2 + D), where δ is the Dirac delta-function. From the identity~~

a z 2 + bx + c 2 az2 + bz + c 2 | | + D = | | (37) y ! y2 we see that the support of LDΦ is the set of points in H satisfying some quadratic equation of discriminant D, and an easy calculation shows that

h(D) L π F(z) DΦ(z)dz = F(zQi ) (38) Z 2 √ D Xi=1 Γ/H | | for any continuous F : Γ/H C. (Of course, δ(X2 + D) is not a → function, and equation (38) must be interpreted in the sense that it holds 2 in the limit n if we choose Φ(X) = δn(X + 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 2 δ(X). so that LDΦ(z) is supported on the semicircles a z + bx + c = 0 | | 322 D. Zagier

(a, b, c Z, b2 4ac = D), and ﬁnd ∈ − h(D) 1 F(z)LDΦ(z)dz = F(z) dQ z , (39) | i | Z √D Xi=1 Z Γ/H CQi 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 ∈ ∞ 2 s+k 1 follow the proof of the divisibility of an /n − by ζ(s) given in n=1 | | [11]. Equations (37) and (??) of that paperP give the identity r ai(m) k 2 y fi(z) (40) ( f , f ) | | Xi=1 i i k/2 ( 1) k 4 k 1 ∞ = − 2 m (k 1) L 2 Φ (z) (z H) π − − t 4m k,t − t=X − ∈ −∞ for all integers m > 0, where

~~∞ 2πimz r = dim S k(SL2(Z)), fi(z) = ai(m)e (i = 1, ... , r) mX=1 are the normalized Hecke eigenforms of weight~~

~~k 2 k, ( fi, fi) = y fi(z) dz, Z | | Γ/H~~

k k and Φk t(X) = (X it) + (X + it) . Thus any function in E is or- , − − − thogonal to the sum on the left-hand side of (??) and therefore, since the Fourier coeﬃcients ai(m) are linearly independent, to each of the k 2 functions y fi(z) . | | Using the computations of [12] and an extension of the Rankin- Selberg 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 = PSL (R) and K = S O(2)/ 1 its maximal com- 2 {± } pact subgroup, and identity the symmetric space G/K with H by gK = a b ai + b K g i = . In this section we will construct a represen- c 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 XR = a, b, c R { 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 ⊂ t ∈ G acts on XR by g M = g 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 · ∈ 2 t a z + bx + c a b/2 tr(g Mg) = | | (M = XR, z = g i H). (41) y b/2 c ! ∈ · ∈

a z 2 + bx + c This explains where the strange expression | | in the deﬁni- y 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

~~t LDΦ(gK) = Φ(tr(g Mg)).~~

~~MXXZ 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 2 XR(D) = XR b 4ac = 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 t LDΦ(g) = Φ(g Mg) (g G), (42) ∈ M XXZ (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 0 1 ε √D integral along each of the lines l = gt 2 g(g G, ε = g,ε 1 √ 2 ε D R ! ∈ 295 +1) on the ruled surface XR(D). It is clear that LDΦ(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), ◦ ◦ ∈ t where π′(g)Φ(M) = Φ(g Mg). Hence the space

~~V = (Im LD)⊥ (43) D\Z ∈ of functions F : Γ/G C satisfying an appropriate growth condition → and such that~~

~~LDΦ(g)F(g)dg = 0 (dg = Haar measure) (44) Z 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)). , · → , · , 1Since the various manifolds X (D) X are disjoint, we can also define V as R ⊂ R (Im L )⊥, where L is the operator from nice functions on XR to functions on Γ/G defined by L Φ(g) = Φ(g1 Mg). MXX ∈ Z Eisenstein Series and the Riemann Zeta-Function 325

On the other hand, because the function z E(z, s) is an eigenfunc- → tion 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 represen- → · tation isomorphic to the principal series representation Ps. (Recall that Ps is the representation of G by right translations on the set of functions a b C = 2s 2 f : G satisfying f 1 g a f (g) and f K L (K)). Thus → 0 a− ! ! | | ∈ V P contains the principal series representation ρ for every non-trivial zero ρ of the Riemann-zeta-function. On the other hand, Ps is unitarizable if and only if s(1 s) > 0, 296 − i.e. s (0, 1) or Re(s) = 1 . Thus the existence of a unitary structure on ∈ 2 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) be- , − long 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 = ρ(ρ 1)Fρ + (2ρ 1)Fρ . Therefore ,1 − ,1 − ,0 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 represen- − V K V V V V tations (we have dim ρ,n = n and ρ,n ρ,n 1 ... ρ,1 ρ,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,

1 2 b c We may also identify XR with the Lie algebra iR = − − a, b, c R of G by { a 1 b! ∈ } 2 0 1 M M − ; then the operation M g′ Mg of G on XR becomes the adjoint → 1 0 ! → 1 representation X g− Xg of G on i , and V is the set of functions on Γ/G orthogonal → R to all functions of the form g Ψ(Ad(g)X), where Ψ is a nice function on iR. → X i P∈ Z 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 LDΦ(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)⊥ (D Z) ∈ 297 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 ∈ the value of D in question, and it was only for (284) that F ED ∈ D was needed. Similarly in part ii), equation (34) shows that eachT 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. E (z, s) for 0 6 i 6 ∗ ∂si ∗ s=ρ 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;

ii) E E (E) = E ; T ∂i iii) E (E) contains E (z, s) for 0 6 i 6 n 1 if ρ is an n-fold 298 ∂si ∗ − s=ρ zero of ζE(s), where ζE(s) denotes the Dedekind zeta-function of 2 E if E = Q or E is q quadratic ﬁeld and ζQ+Q(s) = ζ(s) .

Of course, we can also define representations VD = (Im LD)⊥ and K V (E) = V 2 similarly; then V (E) = E (E) and V (E) is a repre- ∩ d f sentation of PSL2(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 ∈ E(g, Φ, s) = Φ[ξzg] det zg s dz, (45) Z | |Q ξ XQ2 0 Z(Q)/Z(A) ∈ { }

~~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~~

~~E(g, Φ, s) = f (γg, Φ, s), γ P(XQ)/G(Q) ∈~~

~~299 where P = ∗ ∗ and ( 0 !) ∗~~

~~s 2s F(G, Φ, s) = det g Φ[(0, a)g] a d×a, | |Q Z | | 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~~

~~a x a s f g, Φ, s = f (g, Φ, s), 0 b! ! b~~

so the analogue of f (g, Φ, s) in the upper half-plane is the function ζ(2s) (g i)s.) Identifying Q2 0 with E and observing that the Q- ℑ · { } × idele norm of det t(t T(A)) equals the E-idele norm of t under the ∈ Eisenstein Series and the Riemann Zeta-Function 329 identiﬁcation of T(A) with A×E, we ﬁnd

~~E(tg, Φ, s)dt = Φ[ξtg] det tg s dt Z Z | |Q ξ XQ2 0 T(Q)Z(A)/T(A) T(Q)/T(A) ∈ { } = det g s Φ[ξtg] t s dt | |Q Z | |E ξXE× E×/A×E ∈ s s = det g Φ[eg] e d×e. | |Q Z | |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) = E(g, Φ, s) ∂si s=ρ 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 → F(tg)dt = 0 ( g G(A)) (46) Z ∀ ∈ 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

F(ng)dn = 0 ( g G(A)) Z ∀ ∈ 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 330 Bibliography

2 2 the space L0(G(Q)Z(A)/G(A)) of cusp forms. Like L0, it probably has a 2 discrete spectrum. The difference is that, whereas L0 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 terminol- ogy 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 = E V (E) is a complex vector space of dimension at least (and hopefully Texactly) equal to this Betti number, and Barry Mazur pointed out that this space might have a natural interpretation as a complex cohomology group H1(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.

~~Bibliography~~

~~[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 fur¨ reelle quadratische Korper¨ und die Klassenzahl relativ-abelscher Korper,¨ Verhandl. d. Naturforschenden Gesell. i. Basel 28, 363-372 (1971). Mathematische Werke, pp. 198-207. Vandenhoeck & Ruprecht, Gottingen¨ 1970.

~~301 [3] Jacquet, H. and J. Shalika,: A non-vanishing theorem for zeta functions of GL2. Invent. Math. 38 (1976) 1-16. Bibliography 331~~

~~[4] Jacquet, H. and D. Zagier,: Eisenstein series and the Selberg trace formula II. In preparation.~~

~~[5] 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.~~

~~[6] Selberg, A.: Bemerkungen uber¨ eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43 (1940) 47-50.~~

~~[7] Shimura, G.: On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc. 31 (1975), 79-98.~~

~~[8] Shintani, T.: ON zeta-functions associated with the vector space of quadratic forms J. Fac. Science Univ. Tokyo 22 (1975), 25-65.~~

~~[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 The integral K (g, g)E(g, s)dg. Let G = SL2(R) and Γ be an arithmetic ◦ subgroup ofRG 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)

2 2 2 2 L (Γ/G) = L (Γ/G) Lsp(Γ/G) Lcont(Γ/G), ◦ M M 2 2 where L (Γ/G) is the space of cusp forms and is discrete, Lsp(Γ/G) is the ◦ 2 2 discrete part of (L )⊥, 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 2 on G, then convolution with ϕ deﬁnes an endomorphism Tϕ of L (Γ/G), and the kernel function

~~1 K(g, g′) = ϕ(g− γg′) (g, g′ G) (0.1) ∈ Xγ Γ ∈~~

~~of Tϕ 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~~

2 series. The restriction of Tϕ to L (Γ/G) is of trace class; its trace is given by ◦ 2 Tr(Tϕ, L ) = K (g, g)dg. (0.2) ◦ Z ◦ Γ/G The Selberg trace formula is the formula obtained by substituting

K(g, g) Ksp(g, g) Kcont(g, g) for K (g, g) − − ◦ and computing the integral. However, although K (g, g) is of rapid de- ◦ cay in Γ/G, the individual terms K(g, g), Ksp(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 “trun- cate” 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

I(s) = K (g, g)E(g, s)dg, (0.3) Z ◦ 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 334 Don Zagier

s = 0) is a constant function, we recover the Selberg trace formula by computing ress=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. Inter- preting 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 ◦ 2 of terms f j(g) f j(g′), where f j is an orthogonal basis for L (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 2 method to the function F(g) = f j(g) one finds that the integral of | | this function against E(g, s) equals the “Rankin zeta-function” R f j (s) ∞ 2 s (roughly speaking, the Dirichlet series an n− , where the an are n=1 | | the Fourier coefficients off); indeed, thisP is the situation for which the Rankin-Selberg method was introduced. Thus I(s) is a linear combina-

tion 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 identity one obtains the algebraicity (and behaviour under Gal(Q¯ /Q)) 1 k 1 of R f j (s)/π − ζ(s) for the values of s in question ([10], ( f j, f 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= SL2(Z)).~~

3. Next, one can replace SL2(R) and SL2(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= SL (Z)/ 1 . The results for congruence subgroups are similar but 2 {± } 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 ex- → pansion of the form

∞ 2πinx f (z) = 2( n = An( f ; y)e (z H) (1.1) X −∞ ∈ (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 ∈ functions f : H C such that ( f , f ) = f (z) 2dz is ﬁnite dz = dxdy → | | y2 Γ/RH 307 and by L2(Γ/H) the subspace of functions with A ( f ; y) 0. The space ◦ ◦ ≡ L2(Γ/H) is stable under the Laplace operator ◦ ∂2 ∂2 ∆= y2 + ∂x2 ∂y2 !

~~and has a basis f j j> consisting of eigenforms of ∆(see [4], § 5.2). { } 1 Eisenstein Series and the Selberg Trace Formula I 337~~

~~We write 1 2 ∆ f j = + r f j ( j = 1, 2, ...) (1.2) − 4 j !~~

2 1 where r j C. Since ∆ is negative definite, we have r + > 0, i.e. r j ∈ j 4 6 1 is either real or else pure imaginary of absolute value 2 . In fact it is known that the r j are real for SL2(Z), but the corresponding statement 1 for congruence subgroups is not known and we will use only r2 > . j −4 th From (1.2) we find that the n Fourier coefficient An( f j, Y) satisfies the second order differential equation

~~2 2 d 2 2 2 1 2 y An( f j, y) 4π n y An( f ; y) = + r An( f j; y). dy2 − − 4 j !~~

~~The only solution of this equation which is bounded as y is → ∞ √yKir (2π n y), where Kv(z) is the K-Bessel function, deﬁned (for ex- j | | ample) by~~

~~∞ z cosh t Kv(z) = e− cosh vtdt (v, z C,Re(z) > 0). (1.3) Z ∈ 0~~

~~Hence the f j have Fourier expansions of the form~~

~~∞ 2πinx f j(z) = a j(n) √yKir j (2π n y)e (1.4) nX= | | n,−∞0 with a j(n) C. We can choose the f j to be normalized eigenfunctions ∈ of the Hecke operators~~

~~1 az + b T(n): f (z) f (n > 0),  −→ n d !  aX,d>0 b(modX d) (1.5)  ad=n  T( 1) : f (z) f ( z¯), T( n) = T( 1)T(n),  − −→ − − −  338 Don Zagier~~

308 i.e. a (n) j , f j T(n) = 1 f j (n Z, n 0) (1.6) | n 2 ∈ | | (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 L2(Γ/H), uniquely determined ◦ up to order. For each j we deﬁne the Rankin zeta-function R f j (s) by

~~s 2 2 Γ( ) s s ∞ a j(n) R (s) = 2 Γ + ir Γ( ir ) | | (Re(s) > 1). f j 8πsΓ(s) 2 j 2 j n s − nX= n,−∞0 | | (1.7) We also set~~

~~R (s) = π sΓ(s)ζ(2s)R (s) = ζ (2s)R (s), (1.8) ∗f j − f j ∗ f j where ζ(s) denotes the Riemann zeta-function and~~

s/2 s ζ∗(s) = π− Γ ζ(s) = ζ∗(1 s). (1.9) 2 − The Rankin-Selberg method implies that R (s) has a meromorphic con- ∗f j tinuation to all s, is regular except for simple poles at s = 1 and s = 0 with 1 ress=1R∗ (s) = ( f j, f j), (1.10) f j 2 and satisﬁes the functional euqation

R∗ (s) = R∗ (1 s) (1.11) f j f j − (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 1 ζ(s, D) = (Re(s) > 1), (1.12) Q(m, n)s XQ Xm,n Eisenstein Series and the Selberg Trace Formula I 339 where the first summation runs over all SL2(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 SL2(Z). These functions, which were introduced in [10], are related to standard zeta-functions by ζ(s)ζ(2s 1) if D = 0 − ζ(s, D) = ζ(s)2 (finite Dirichlet series ) if D = square , 0 x  ·  , ζQ( √D)(s) (finite Dirichlet series) if D square,  ·  (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) compute R f j (s). In §§2 and 3 we will show that this function j=1 ( f j, f j) equals theP 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) and generalization to a j(m) R f j (s), where the a j(m) are the j=1 ( f j, f j) Fourier coefficients definedP 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~~

~~∞ h(r j) a j(m) R∗ (s) = R(s) + R(1 s) (1.15) ( f , f ) f j − Xj=1 j j~~

~~310 with R(s) = R(s; m, h) given by~~

∞ ir 1 2 ζ∗(s + 2ir)ζ∗(s 2ir) a R(s) = ζ∗(s) −   h(r)dr −8π Z ζ (1 + 2ir)ζ (1 2ir)  d  ∗ ∗ aX,d 1  −  ≥  −∞ ad=m    1 ζ (s)ζ (2s) a s/2 is ∗ ∗   h( ) − 2 ζ (s + 1)  d  2 ∗ aX,d 1   ≥  ad=m  s 1  1  m −2 Γ(s)Γ(s ) ∞  + − 2 ζ(s, t2 4m) (1.16) 4π2 1+s 2 s − × Γ 2 Γ −2 t=X −∞ 1 s 1 s ∞ Γ − + ir Γ − ir 2 2 − × Z Γ(ir)Γ( ir) − −∞ 1 s 1 s 3 t2 F − + ir, − ir; s; 1 h(r)dr, × 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

~~2 s 1 1 s 1 s 3 t m −2 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 (s) is divisible by ζ (s) for all j. ∗f 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 divisi- − ble 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 (s). A more formal argument is as follows: ∗f j For z H deﬁne ∈ ∞ 1 Φ(s, z) = f j(z)R∗ (s); ( f , f ) f j Xj=1 j j then (1.14) and (1.15) imply the identity

∞ 2πimx Φ(s, z) = √y [R(s; m, hmy) + R(1 s; m, hmy)]e , mX= − 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∗ (s) equals the scalar product (Φ(s, ), f j). 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 (s)/ζ (s) equals ∗f j ∗

~~2 ζ(2s) ∞ a j(n) | | . ζ(s) ns Xn=1~~

Using the usual relations among the eigenvalues a j(n) of a Hecke eigenform, we see that this Dirichlet series has the Euler product 1 . 2 s s 2 s (1 α p )(1 αpβp p )(1 β p ) Yp − p − − − − p − 342 Don Zagier

~~where αp, βp are deﬁned by~~

∞ a (n) 1 j = s s s n (1 αp p )(1 βp p ) Xn=1 Yp − − − − (i.e. αp + βp = a j(p), αpβp = 1). Thus the corollary is the case n = 2 of the conjecture that the “symmetric power L-functions” n 1 L ( f , s) = n j (1 αmβn m p s) Yp mY=0 − 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 Eisen- stein 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 ∈ ∈ E(z, s) = Im(γz)s (Re(s) > 1), (2.1) γ XΓ /Γ ∈ ∞ a b where Γ = SL2(Z) / 1 Z is the group of transla- ∞ ( 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 Γ- invariant 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 Γ (c, d), ∞ c d! ←→ ± Eisenstein Series and the Selberg Trace Formula I 343 we can rewrite (2.1) as

~~1 ys E(z, s) = (Re z > 1) 2 cz + d 2s cX,d Z | | (c,d∈)=1 and hence~~

ys ′ 1 ζ(2s)E(z, s) = (Re(s) > 1), (2.2) 2 mz + n 2s Xm,n | | where ′ denotes a summation over all pairs of integers (m, n) , (0, 0). This latterP 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) − where s E∗(z, s) = π− Γ(s)ζ(2s)E(z, s) = ζ∗(2s)E(z, s). (2.4) The residue at s = 1 is independent of z:~~

6 3 ress= E(z, s) = ress= E∗(z, s) = (z H). (2.5) 1 π 1 4 ∈ 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

~~s ∞ ζ(2s)E(z, s) = y [ζ(2s) + ϕs(mz)] (Re(s) > 1), mX=1 where ∞ 1 1 ϕ (z) = z H,Re(s) > . s z + n 2s 2 nX= ∈ ! −∞ | | 344 Don Zagier~~

The function ϕs(x+iy) is periodic in x for ﬁxed y and hence has a Fourier development ∞ a(n, s, y)e2πinx with π= P−∞ ∞ e 2πinx a(n, s, y) = − dx Z (x2 + y2)s −∞ Γ( 1 Γ(s 1 )) 2 − 2 y1 2s (n = 0)  Γ(s) − =   s 1 1  2 Γ( )  π n − 2 2 | | Ks 1 (2π n y)(n , 0)  y ! Γ(s) 2 | |  −  [GR 3.251.2 and /8.432.5]. Hence Γ 1 Γ 1 s ( 2 ) (s 2 ) 1 s ζ(2s)E(z, s) = ζ(2s)y + − ζ(2s 1)y − Γ(s) − 1 1 s s 2 π y 2 ∞ ∞ n − 2πinmx +2 | | Ks 1 (2π n my)e Γ(s) m ! 2 | | mX=1 nX= − n,−∞0

~~s 314 or, multiplying both sides by π− Γ(s),~~

~~s 1 s E∗(z, s) = ζ∗(2s)y + ζ∗(2s 1)y − (2.6) − ∞ 2πinx + 2 √y τ 1 K 1 (2π n y)e , s 2 s 2 nX= − − | | n,−∞0~~

where ζ∗(s) is deﬁned by (1.9) and τv(n) by v v 2v a τv(n) = n d− = (n Z 0 , v C). (2.7) | | d ∈ −{ } ∈ dXnd>0 adX= n | 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 1 (the poles of ζ (2s) and ζ (2s 1) at s = 1 cancel). Also, it is clear ± 2 ∗ ∗ − 2 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

max(σ,1 σ) E(z, s) = O(y − ) (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

~~( f , E(.,s ¯)) = f (z)E(z, s)dz (2.9) Z Γ/H converges absolutely for some s with Re(s) > 1. Then for such s~~

~~∞ s 2 ( f , E(.,s ¯)) = y − A ( f ; y)dy (2.10) Z ◦ 0 where A ( f , y) is deﬁned by equation (1.1). ◦ Proof. Substituting (2.1) into (2.9) we ﬁnd 315~~

~~( f , E(.,s ¯)) = f (z) Im(γz)sdz Z γ XΓ /Γ Γ/H ∈ ∞ = f (z)Im(z)sdz Z Γ /H ∞ 1 ∞ dxdy = f (x + iy)ys Z Z y2 0 0 346 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 2 by choosing f (z) = f j(z) , 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

~~2 2 A ( f ; y) = y a j(n) Kir j (2π n y) ◦ | | | | Xn,0~~

~~(notice that Kir (2π n y) is real by (1.3), since r j is either real or pure j | | imaginary). Hence (2.10) gives~~

~~∞ 2 s 1 2 2 f j(z) E(z, s)dz = y − a j(n) Kir j (2π n y) dy Z | | Z , | | | | Γ/H 0 Xn 0 a (n) 2 ∞ = j s 1 2 | s | y − Kir j (2πy) dy (2.11) , n Z Xn 0 | | 0~~

~~= R f j (s) (Re(s) > 1)~~

~~(the integral is evaluated in [ET 6.8 (45)] and equals the gamma fac-~~

tor 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).

Spectral decomposition. We now give a rough indication, ignoring an- 316 alytic 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 347

f L2(Γ/H) has an expansion ∈ ∞ ∞ ( f , f j) 1 1 1 f (z) = f j(z) + f , E ., + ir E z, + ir dr, ( f , f ) 4π 2 !! 2 ! Xj=0 j j Z −∞ (2.12) 2 where f j j 1 is an orthogonal basis for L (Γ/H) and f for the space of { } ≥ ◦ { ◦} 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 ◦ ≡ ǫ sufficiently rapid decay, say f (z) = O(y− ) 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 3 ( f , f ) ress=1Ψ(s) = f (z)dz = ◦ f , (2.13) π Z ( f , f ) ◦ Γ/H ◦ ◦ and satisfies the functional equation ζ (2s 1) Ψ(s) = ∗ − Ψ(1 s). (2.14) ζ∗(2s) − 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 ◦ C+i ∞ 1 1 s A ( f ; y) = Ψ(s)y − ds (1 < C < 1 + ǫ). ◦ 2πi Z 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

∞ f , f 1 1 1 +ir A ( f ; y) = ◦ f + Ψ( ir)y 2 dr ◦ f , f ◦ 2π Z 2 − ◦ ◦ −∞ ∞ ( f , f ) 1 1 1 +ir ζ∗(1 2ir) 1 ir = ◦ f + Ψ( ir)(y 2 + − y 2 − )dr. ( f , f ) ◦ 4π Z 2 − ζ∗(1 + 2ir) ◦ ◦ −∞ (2.15) 348 Don Zagier

1 +ir ζ∗(1 2ir) 1 ir 317 On the other hand, equation (2.6) implies that y 2 + − y 2 − ζ∗(1 + 2ir) 1 is the constant term of E(z, 2 + ir), so (2.15) tells us that the Γ-invariant function ( f , f ) 1 ∞ 1 1 f˜(z) = f (z) ◦ f (z) Ψ ir E z, + ir dr − ( f , f ) ◦ − 4π Z 2 − ! 2 ! ◦ ◦ −∞ has zero constant term. It is also square integrable, because f (z) is and 1 the non-constant terms in the Fourier expansion of E(z, 2 + ir) are expo- ˜ 2 ∞ ( f , f j) nentially small. Hence f˜ L (Γ/H), so f˜(z) = f j(z), and this ∈ ◦ j=1 ( f j, f j) P 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 Tϕ the operator given by convolution with ϕ. Since we are interested only in functions on the upper half- plane H = G/K (where K = S O(2) and the identiﬁcation is given by a b ai + b K we can assume that ϕ is left and right K-invariant. c d! ↔ ci + d ! But the map a b t : K K a2 + b2 + c2 + d2 2 c d! 7−→ − gives an isomorphism between K G/K and [0, ) (Cartan decomposi- \ ∞ tion), so we can think of ϕ as a map ϕ : [0, ) C. ∞ → An easy calculation shows that 2 1 z z′ t(g− g′) = | − | (g, g′ G), yy′ ∈

~~where z, z H are the images of g and g . Therefore Tϕ acts on func- ′ ∈ ′ tions f : H C by →~~

~~Tϕ f (z) = k(z, z′) f (z′)dz′ (z H), (2.16) Z ∈ H Eisenstein Series and the Selberg Trace Formula I 349~~

~~where 318 2 z z′ k(z, z′) = ϕ | − | (z, z′ H). (2.17) yy′ ! ∈ The growth condition we want to impose on ϕ is that~~

~~1+A ϕ(x) = O x 2 (x ) (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 Tϕ 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 Tϕ. More precisely,

1 2 f V, ∆ f = + r f Tϕ f = h(r) f , (2.19) ∈ − 4 ! ⇒ where h(r), the Selberg transform of ϕ, is an even function of r, depend- 1 +ir ing on ϕ but not on f . To compute it, we choose f (z) = y 2 , which A satisﬁes the conditions is (2.19) if r C with Im(r) < . Then ∈ | | 2

~~∞ ∞ 2 2 3 +ir (x x′) + (y y′) Tϕ f (z) = y′− 2 ϕ − − dx′ dy′. Z Z yy′ ! 0 −∞~~

~~Making the change of variables x′ = x+ √yy′v in the inner integral gives~~

~~∞ 2 3 +ir (y y′) Tϕ f (z) = y′− 2 yy′Q − dy′, Z yy′ ! 0 p where the function Q is deﬁned by 319~~

~~∞ ∞ ϕ(t)dt Q(w) = ϕ(w + v2)dv = (w > 0). (2.20) Z Z √t w w − −∞ 350 Don Zagier~~

~~u The further change of variables y′ = ye then gives~~

~~∞ 1 +ir iru u u Tϕ f (z) = y 2 e Q(e 2 + e− )du. Z − −∞ Hence, setting~~

u u g(u) = Q(e 2 + e− ) (u R), (2.21) − ∈ we have ∞ A h(r) = g(u)eirudu (r C, Im(r) < ). (2.22) Z ∈ | | 2 −∞ Formulas (2.20)-(2.22) describe the Selberg transform (the notations Q, g, h, due to Selberg, are by now standard and we have retained them). The inverse transform is easily seen to be

∞ 1 iru g(u) = h(r)e du,  2π Z   −∞  1 √w  = 2 (2.23) Q(w) g(2 sinh− ),   ∞  1 2 ϕ(x) = − Q′(x + v )dv.  π Z   −∞  We can also combine these three integrals, obtaining

1 ∞ ∞ sin ru ϕ(x) = rh(r) du dr 2π2 u u 2 x Z Z √e + e− − − 1 + x −∞ cosh− (1 2 ) 1 ∞ x = P 1 +ir(1 + ) r tan hπr h(r)dr, (2.24) 4π Z − 2 2 −∞ Eisenstein Series and the Selberg Trace Formula I 351 where Pv(z)(v C, z C ( , 1]) denotes a Legendre function of the 320 ∈ ∈ − −∞ ﬁrst kind. (For properties of Legendre functions we refer the reader to [EH], Chapter 3; in particular, the integral representation of P 1 just 2 +ir used follows from formulas 3.7 (4) and 3.3.1 (3) there.) The− inversion formula of Mehler and Fock ([EH], p. 175) then gives

~~∞ x A h(r) = 2π P 1 +ir 1 + ϕ(x)dx Im(r) < . (2.25) Z 2 2 | | 2 0 From (2.20) - (2.23) we see easily that the conditions~~

1+A ϕ(x) = O x− 2 , A Q(w) = O(w− 2 ), A u g(u) = O(e− 2 | |), A h(r) holomorphic in Im(r) < | | 2 are equivalent; this also follows from (2.24) and (2.25) since P 1 +ir(x) − 2 1 + Im(r) grows like x 2 as x [EH 3.9.2 (19), (20)]. Thus the growth − | | → ∞ condition (2.18) is equivalent to a holomorphy condition on h, while the condition that ϕ be smooth is equivalent to the requirement that h be of rapid decay. Selberg kernel function. Now suppose that the function f in (2.16) is Γ-invariant. Then Tϕ f is also Γ-invariant and clearly

Tϕ f (z) = K(z, z′) f (z′)dz′ (2.26) Z Γ/H with K(z, z′) = k(z, γz′), (2.27) Xγ Γ ∈ i.e. the action of Tϕ on Γ-invariant functions is given by the kernel function (2.27). We claim that

~~1 A − K(z, z′) = O y′ 2 (z ﬁxed, y′ ) −→ ∞ 352 Don Zagier~~

~~if ϕ satisﬁes (2.18). To see this, write~~

~~K(z, z′) = k(z + n, z′) + k(z + n, γz′). Xn Z γ XΓ /Γ Xn Z ∈ γ∈<Γ∞ ∈ ∞~~

1 A − 321 The ﬁrst term is easily seen to be O(y′ 2 ). In the second term, Im(γz′) is uniformly small as y′ and from this one easily sees that the inner →∞A+1 sum is uniformly O(Im(γz′) 2 ). Therefore the second term is

 A+1  A + 1 A+1 O Im(γ′z) 2 = O E z′, y′ 2   2 ! − ! γ XΓ /Γ   γ∈<Γ∞   ∞    1 A − which by (2.6) is O(y′ 2 ). 2 From (320) is follows that K(z, z′) is in L (Γ/H) with respect to each variable separately and that the scalar product (K( , z′), E( , s)) con- 1 A 1 + A · · verges for − < Re(s) < . Using (2.26) and (2.19) we ﬁnd 2 2

~~(K( , z′), f j) = h(r j) f j(¯z ) ( j 0), (2.29) · ′ ≥ i where r j is given by (1.2) for j 1 and r = , and similarly ≥ ◦ 2 1 1 K( , z′), E , + ir = h(r)E z′, ir · · 2 !! 2 − !~~

~~since ∆E(z, 1 + ir) = 1 + r2 E(z, 1 + ir). Therefore the spectral de- 2 − 4 2 composition formula (2.12) applied to K( , z ) gives · ′~~

∞ ∞ h(r j) 1 1 1 K(z, z′) = f j(z) f j(z )+ E z, + ir E z′, ir h(r)dr. ( f , f ) ′ 4π 2 ! 2 − ! Xj=0 j j Z −∞ We restate this formula as Eisenstein Series and the Selberg Trace Formula I 353

~~Proposition 3: Let h(r) be a function satisfying (1.14) and set~~

∞ h(r j) K (z, z′) = f j(z) f j(z′) (z, z′ H), (2.30) ◦ ( f , f ) ∈ Xj=1 j j 2 where f j is an orthogonal basis of L (Γ/H) satisfying (1.2). Let { } ◦ k(z, z′)(z, z′ H) ∈ be the function deﬁned by (2.17), where ϕ is given by (2.23) or (2.24). Then 3 i K (z, z′) = k(z, γz′) h (2.31) ◦ − π 2 Xγ Γ ∈ 1 ∞ 1 1 E z, + ir E z′, ir h(r)dr. − 4π Z 2 ! 2 − ! −∞ 322 We remark that (2.31) can be proved directly, without recourse to the spectral decomposition formula (2.12): Using the formulas for the Selberg transform and Mellin inversion, one can check directly that the expression on the right-hand side of (2.31) has constant term zero with respect to both variables and hence (using the estimate (320)) is a cusp form; equation (2.29) then implies the desired identity. We leave the details as an exercise for the reader.

~~3 Computation of I(s) for (s) > 1. ℜ Let h(r) be a function satisfying (1.14) and deﬁne~~

~~I(s) = K (z, z)E(z, s)dz, (3.1) Z ◦ ΓH where K (z, z′) is deﬁned by (2.30). Since K (z, z) is of rapid decay, the integral converges◦ for all s(, 1), and from (2.11)◦ we have~~

~~∞ h(r j) I(s) = R (s). (3.2) ( f , f ) f j Xj=1 j j 354 Don Zagier~~

~~The object of this section is to compute I(s) for 1 < Re(s) < A. By the Rankin-Selberg method (eq. (2.10)) we have~~

~~∞ s 2 I(s) = K (y)y − dy (Re(s) > 1), (3.3) Z 0~~

~~where K (y) is the constant term of K (z, z), which we will compute using Proposition (3) above. From (2.6)◦ we ﬁnd that the constant term of E(z, 1 + ir)E(z, 1 ir) equals 2 2 −~~

1 +ir ζ∗(1 2ir) 1 ir 1 ir ζ∗(1 + 2ir) 1 +ir y 2 + − y 2 − y 2 − + y 2 " ζ (1 + 2ir) #" ζ (1 2ir) # ∗ ∗ − 8y ∞ + τ (n)2K (2πny)2. ζ (1 + 2ir)ζ (1 2ir) ir ir ∗ ∗ − Xn=1 323 Of the four terms obtained by multiplying the expressions in square brackets, two are obtained from the other two by replacing r by r and − hence will give the same contribution when integrated against the even function h(r). As to the ﬁrst term in (2.31), we separate the terms with γ Γ and γ < Γ ; the former are their own constant terms since ∈ ∞ n2 ∞ k(z, z + n) = ϕ is independent of x. We thus obtain the decomposi- y2 ! tion 1 4 K (y) = K (x + iy, x + iy)dx = Ki(y) ◦ Z = 0 Xi 1 with

~~1 ∞ K1(y) = k(x + iy, γ(x + iy))dx, Z · Γ 0 Xγ γ<∈Γ ∞ ∞ n2 y ∞ K (y) = ϕ h(r)dr, 2 y2 2π nX= ! − Z −∞ −∞ Eisenstein Series and the Selberg Trace Formula I 355~~

∞ y 2ir ζ∗(1 + 2ir) 3 i K3(y) = y h(r)dr h( ), −2π Z ζ (1 2ir) − π 2 ∗ − −∞ 2y ∞ 1 K4(y) = − π Z ζ∗(1 + 2ir)ζ∗(1 + 2ir)× −∞ ∞ 2 2 τir(n) Kir(2πny) h(r)dr. ×   Xn=1     4 This gives a corresponding decomposition of I(s) as Ii(s) with i=1 P

~~∞ s 2 Ii(s) = Ki(y)y − dy (i = 1, ... , 4). Z 0~~

~~Theorem 2. The integrals Ii(s) converge for 1 < Re(s) < A and are given in that region by the formulas~~

∞ ζ(s, t2 4) z2 + 1 t2 4 2 I (s) = − ϕ | − | | | ys dz, 1 ζ(2s) y2 t=X Z ! −∞ H s 1 s ∞ s iΓ( 2 )Γ( −2 ) Γ( 2 ir) I2(s) = ζ(s) − rh(r)dr, − 2s+1πs+3/2 Z Γ(1 s ir) − 2 − −∞ 1 s π 2 Γ( ) ζ(s) is I (s) = 2 h( ), 3 − s+1 ζ(s + 1) 2 2Γ( 2 ) 1 s s 2 2 π − Γ( ) ζ(s) I (s) = 2 4 − 4Γ(s) ζ(2s)× ∞ Γ( s + ir)Γ( s ir) ζ(s + 2ir)ζ(s 2ir) 2 2 − − h(r)dr. × Z Γ( 1 + ir)Γ( 1 ir) ζ(1 + 2ir)ζ(1 2ir) 2 2 − − −∞ 324 356 Don Zagier

Proof. We begin with I4(s) since it is, despite appearances, the easiest of the four integrals. The very rapid decay of the Bessel functions allows us to interchange the order of the integrations and summation, obtaining

2 ∞ 2 ∞ τir(n) s 1 2 I (s) = y − Kir(2πy) dy 4 −π  ns    × Xn=1 Z     0       h(r)  dr. ×ζ (1 + 2ir)ζ (1 2ir) ∗ ∗ − ζ(s)2 The ﬁrst expression in parentheses equals ζ(s + 2ir)ζ(s 2ir) for ζ(2s) − Re(s) > 1, as one checks by expanding the Dirichlet series as an Euler product. The second expression in parentheses equals

s 2 1 Γ s s 2 Γ + Γ s ir ir 8π Γ(s) 2 2 − (this is the same integral as was used in (2.11)). Putting this together we obtain the formula for I4(s) given in the theorem; it is valid for Re(s) > 1. (The integral converges for all s with Re(s) , 0, 1, as one sees by using Stirling’s formula and standard estimates of ζ(s) and 1 325 ζ(1 + it)− as well as the fact that h(r) is of rapid decay.) Since the gamma factors in the formula are exactly those corresponding to the zeta-functions occurring, we can write the result in the nicer form

2 1 ζ∗(s) I4(s) = (3.4) −4π ζ∗(2s)× ∞ ζ (s + 2ir)ζ (s 2ir) ∗ ∗ − h(r)dr (Re(s) > 1). × Z ζ (1 + 2ir)ζ (1 2ir) ∗ ∗ − −∞ The integral I is also quite easy to compute. Since ζ (1 2ir) is 3 ∗ − 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 K (y) is holomorphic in 0 Im(r) < 3 ≤ 2 Eisenstein Series and the Selberg Trace Formula I 357 except for a simple pole of residue

~~1 1 i 3i i y− h resr= i (ζ∗(1 + 2ir)) = h ζ∗(2) 2 2 πy 2 i C at . Hence we can move the path of integration to Im(r) = (1 < C < 2 2 A), obtaining~~

~~C+i ∞ iy s ζ∗(s) is K3(y) = y− h ds (1 < C < A). 4π Z ζ∗(s + 1) 2 C i − ∞ The Mellin inversion formula then gives~~

~~1 ζ∗(s) is I3(s) = h (1 < Re(s) < A), (3.5) −2 ζ∗(s + 1) 2 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 ∞ ∞ 1 ∞ u2 h(r)dr = g(0) = Q(0) = ϕ(v2)dv = ϕ du, 2π Z Z y Z y2 ! −∞ −∞ −∞ so ∞ n2 ∞ u2 K (y) = ϕ ϕ du. 2 y2 y2 nX= ! − Z ! −∞ −∞ By the Poisson summation formula this equals 326

~~∞ u2 ∞ ϕ e2πinudu = 2y ψ(ny), y2 ! Xn,0 Z Xn=1 −∞ where ∞ ψ(y) = ϕ(u2)e2πiuydu. (3.6) Z −∞ 358 Don Zagier~~

~~Since ϕ is smooth, ψ is of rapid decay, so we may interchange summation and integration to get~~

∞ ∞ ∞ s 1 s 1 I2(s) = 2 ψ(ny)y − dy = 2ζ(s) ψ(y)y − dy(Re(s) > 1). (3.7) = Z Z Xn 1 0 0 To calculate the integral we begin by substituting the third equation of (2.23) into (3.6). This gives

~~∞ ∞ 1 2πiuy 2 2 ψ(y) = e Q′(u + v )dv du. −π Z Z −∞ −∞ Changing to polar coordinates u + iv = reiθ and using the standard integral representation~~

~~2π 1 J (x) = eix cos θdθ · ◦ 2π Z 0 of the Bessel function of order 0 [GR 3.915.2] we ﬁnd~~

~~∞ 2 ψ(y) = 2 J (2πyr)Q′(r )r dr − Z ◦ 0 u or, making the substitution r = 2 sinh and using (2.21), 2~~

~~∞ u ψ(y) = J (4πy sinh )g′(u)du. − Z ◦ 2 0 Using the formula~~

~~∞ Γ( s ) 3 s 1 = 2 J (2ay)y − dy s s 0 < Re(s) < , a > 0 Z ◦ 2a Γ(1 2 ) 2 ! 0 − Eisenstein Series and the Selberg Trace Formula I 359~~

~~327 [ET 6.8 (1)] we ﬁnd~~

~~∞ sΓ s ∞ s s 1 (2π)− ( 2 ) u − ψ(y)y − dy = sinh g′(u)du (3.8) Z − 2Γ(1 s ) Z 2 − 2 −∞ −∞ 3 0 < Re(s) < 2!~~

~~A u (the integral converges at because g (u) = O e 2 and at 0 because ∞ ′ − | | g′(u) is an odd function and hence O(u)). Substituting ~~

~~1 ∞ g′(u) = − rh(r) sin ru dr 2π Z −∞ and using the Fourier sine transform formula~~

∞ Γ s ir Γ s + ir sin ru s 1 2 2 du = 2 − iΓ(1 s) − s  s s  Z sin h u − − Γ 1 ir − Γ 1 + ir  0 2  − 2 − − 2    ([ET 2.9 (30)]; the conditions for validity are misstated there) gives

s 1 s s ∞ iΓ Γ − ∞ Γ ir s 1 2 2 2 − ψ(y)y − dy = rh(r)dr, Z − 2s+2πs+3/2 Z Γ 1 s ir 0 − 2 − −∞ 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 3 for all s with Re(s) > 1 (not just 1 < Re(s) 2 ); we can use the elementary identity

s s s s r Γ ir Γ + ir Γ + ir Γ ir 2 − 2 = 2 2 − 2πi  s s  1 s 1+s Γ 1 ir − Γ 1 + ir  Γ − Γ Γ(ir)Γ( ir)  − 2 − − 2  2 2 −   to write it in the more elegant form 328 360 Don Zagier

~~∞ Γ s + Γ s ζ∗(s) 2 ir 2 ir I2(s) = − h(r)dr (3.9) s+1 s+1 (4π) 2 Γ Z Γ(ir)Γ( ir) 2 − −∞ (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

~~∞ ∞ ∞ s 1 2 s 1 ψ(y)y − dy = ϕ(u ) cos 2πuyduy − dy. Z Z Z 0 0 −∞~~

∞ s 1 Then the inner integral y − cos 2πuydy converges (conditionally) for R0 0 < Re(s) < 1 (thus in a region of validity disjoint from that of (3.7)!) πs and equals (2π u ) sΓ(s) cos there [ET 6.5 (21)]. Using (2.24) we | | − 2 then ﬁnd

∞ s 1 ψ(y)y − dy = Z 0 πs ∞ ∞ Γ(s) cos 2 s+1 x = x 2 P 1 1 + h(r)r tanh πrdr dx s+1 − +ir 2(2π) Z Z − 2 2 0 −∞ for 0 < Re(s) < 1. Interchanging the order of integration again and using the formula

∞ s+1 x 2 x− P 1 +ir 1 + dx = Z − 2 2 0 1 s s s Γ − Γ + ir Γ ir 1 s 2 2 2 = 2 − − (0 < Re(s) < 1) 1+s 1 1 Γ 2 Γ 2 + ir Γ 2 ir − Eisenstein Series and the Selberg Trace Formula I 361

~~329 [GR 7.134] we ﬁnd~~

∞ s 1 ψ(y)y − dy = Z 0 s 2 s ∞ 2− − Γ 2 s s Γ ir Γ + ir h(r)r sin hπrdr, + 3 + s 2 1 s Z 2 − 2 π Γ 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) i.e. the contribution from the main term k(z, γz) of K (z, z). For γ<Γ ◦ each γ Γ denote by [γ] the conjugacy classP∞ of Γ in Γ. Its elements are ∈ of the form σ 1Γσ where σ Γ is well-deﬁned up to left multiplication − ∈ with an element of the stabilizer Γγ of γ in Γ. Hence

~~′ 1 k(z, γz) = k(z, σ− γσz),~~

~~Xγ Γ X[γ] σXΓγ/Γ γ<∈Γ 1∈ σ− Γσ<Γ ∞ ∞~~

where ′ denotes a summation over all non-trivial conjugacy classes [γ] (each suchP class contains at least one element < Γ ) and we have chosen a representative γ for each class. Multiplying ∞σ on the right by an 1 n 1 element Γ does not aﬀect the condition σ− γσ < Γ and ± 0 1! ∈ ∞ ∞ 1 1 replaces k(z, σ− γσz) by k(z + n, σ− γσ(z + n)). Hence

~~′ ∞ 1 k(z, γz) = k(z + n, σ− γσ(z + n))~~

Xγ Γ X[γ] σ ΓXγ/Γ/γ nX= γ<∈Γ ∈1 ∞ −∞ σ− γσ<Γ ∞ ∞ 1 (for this one has to check that σ− Γγσ Γ = 1 , but this follows easily 1 ∩ ∞ { } from σ− γσ < Γ and the fact that the centralizer of any non-trivial ∞ 362 Don Zagier

~~element of Γ is Gamma ). Since the constant term in the Fourier ∞ ∞ expansion of a sum ∞ f (x + n) is the integral ∞ f (x) dx, we ﬁnd n= P−∞ −∞R~~

~~∞ ′ 1 K1(y) = k(x + iy, σ− γσ(x + iy))dx Z X[γ] σ ΓXγ Γ/Γ ∈ 1 \ ∞ σ− γσ<Γ −∞ ∞ 330 and hence~~

′ 1 s I1(s) = k(z, σ− γσz)y dz. Γ Γ Γ Z X[γ] σ Xγ / H ∈ 1 \ ∞ σ− γσ<Γ ∞ a b Now for any element τ = Γ with τ < Γ (i.e. c , 0) we c d! ∈ ∞ have 1 k(z, τz)ysdz = V(s, t) (t = tr(τ)) Z c s H | | where z2 + 1 t2/4 2 V(s, t) = ϕ | − | ysdz. (3.10) Z y2 ! H

~~(To prove this, substitute (2.17) for k(z, z′) and make the change of vari- z a d able z + − .) Hence → c 2c | |~~

∞ 1 ′ 1  I1(s) =   V(s, t), 2 c(σ 1γσ) s  t=X  [[Xγ]] σ ΓXΓ/Γ −  −∞  γ | |   trγ=t ∈ 1 \ ∞   σ− γσ<Γ   ∞    ′ 1 0 where denotes a sum over conjugacy classes in SL2(Z) [[γ]] −(± 0 1!) P1 1 and c(σ− γσ) the element in the lower left-hand corner of σ− γσ (we Eisenstein Series and the Selberg Trace Formula I 363 must work in SL2(Z) rather than Γ in order to have a well-deﬁned trace; 1 notice that Γγ Γ and σ γσ SL (Z) make sense for γ SL (Z), ⊂ − ∈ 2 ∈ 2 σ Γ). Since V(s, t) = V(s, t), we have ∈ −

∞  ′ 1  I1(s) =   V(s, t).  c(σ 1γσ)s  t=X [[Xγ]] σ ΓXΓ/Γ −  −∞  γ  try=t ∈ 1\ ∞   c(σ− γσ)>0    There is a (1:1) correspondence between conjugacy classes [[γ]] of trace t and SL2(Z)-equivalence classes of binary quadratic forms of dis criminant t2 4 given by 331 − a b γ = Q(m, n) = cm2 + (d a)mn bn2. c d! ↔ − − There is also a bijection between Γ/Γ and the set of relatively prime pass of integers (m, n) Z2/ 1 given∞ by mapping an element σ ± ∈ {± } 1 ∈ Γ/Γ to its first column, and under this bijection we have c(σ− γσ) = ∞ Q(m, n) and Γγ = Aut(Q)/ 1 . Hence {± } 1 ζ(s, t2 4) = 1 s − c(σ− γσ) ζ(2s) [[Xγ]] σ Γγ Γ/ΓXc(σ 1γσ)>0 trγ=t ∈ \ ∞ − where ζ(s, t2 4) is defined by (1.12). To complete the proof of the − formula 1 ∞ I (s) = ζ(s, t2 4)V(s, t) (1 < Re(s) < A) (3.11) 1 ζ(2s) t=X − −∞ given in the theorem, it remains only to verify the convergence and justify the various interchanges of summation and integration made. Since the integrals I2, I3 and I4 have already been shown to be convergent for 1 < Re(s) < A (eqs. (3.4), (3.5), (3.7)) and the function K (y) is of rapid decay at infinity, the integral I1(s) is certainly convergent in the same range. By choosing s real and ϕ positive, we see that this convergence is absolute, and this gives an a posteriori proof of the convergence of (3.11) in the range stated and of the validity of the steps leading up to its proof. 364 Don Zagier 4 Analytic continuation of I(s),

In this section we will give the analytic continuation of I(s) to the critical strip 0 < Re(s) < 1 and compute the residue at s = 1 (Selberg trace formula). We will also want to study the functional equations of the various terms in the formula for I(s). From the deﬁnition (3.1) of I(s) it is clear that I(s) is holomorphic for all s , 1 and satisﬁes the functional equation I (s) = I (1 s) where ∗ ∗ − s I∗(s) = π− Γ(s)ζ(2s)I(s) = K0(z, z)E∗(z, s)dz. Z Γ H \ On the other hand, Theorem 2 says that I∗(s) is the sum of the functions s 2 π− Γ(s)ζ(s, t 4)V(s, t) (t Z, t , 2), (4.1) s − ∈ ± π− Γ(s)ζ(s,0)[V(s, 2) + V(s, 2)] + I∗(s), (4.2) − 2 I3∗(s) + I4∗(s) (4.3)

332 for 1 < σ = Re(s) < A, where Ii∗(s) = ζ∗(2s)Ii(s). We will show that each of the functions (4.1)-(4.3) has a meromorphic continuation to the strip 1 A <σ< A with poles at most at 0 and 1 and is invariant under − s 1 s. → − We begin by performing one of the integrations in the double integral (3.10) to write V(s, t) as a simple integral, thus obtaining the analytic continuation and functional equation of V(s, t). Proposition 4. Let ϕ be a function satisfying (2.18), s C, t bR, ∈ ∈ ∆= t2 4. If ∆ , 0, the the integral (3.10) converges for A <σ< 1+A − − and is given by

~~s/2 ∞ ∆ 2 V(s, t) = 2π ϕ ∆ (u 1) P s(u)du 4 Z | | − − 1 ~~

~~( A <σ< 1 + A) (4.4) − if ∆ < 0 and by~~

~~s 2 1 Γ 2 V(s, t) = ∆s/2 (4.5) 2 Γ(s) × Eisenstein Series and the Selberg Trace Formula I 365~~

∞ 2 2 2 1 s s s 1 u ϕ(∆(u + 1))(1 + u ) 2 F , ; ; du( A <σ< 1 + A) × Z 2 2 2 u2 + 1! − −∞ if ∆ > 0, where F(a, b; c; z) and Pv(z) denote hypergeometric and Legen- dre functions, respectively. In particular, V(s, t) satisﬁes the functional equation

π sΓ(s) π 1+sΓ(1 s) − V(s, t) = − − V(1 s, t)(∆ , 0), (4.6) γ(s, ∆) γ(1 s, ∆) − − where s s/2 (2π)− ∆ Γ(s) if ∆ < 0, | | γ(s, ∆) =  2  s s/2 s π− ∆ Γ if ∆ > 0.  2   1 For ∆= 0,V(s, t) converges for 2 <σ< 1 + A and has a meromorphic 333 continuation to 0 < Re(s) < 1 + A given by

~~Γ 1 Γ 1 ∞ ( 2 ) (s 2 ) 2 s 1 V(s, 2) = − ϕ(u )u − ds(0 <σ< 1 + A). (4.7) ± Γ(s) Z 0~~

~~We observe that the functions ζ(s, ∆) deﬁned by (1.12) satisfy the functional equations~~

γ(s, ∆)ζ(s, ∆) = γ(1 s, ∆)ζ(1 s, ∆) − − for ∆ , 0 ([10], Prop. 3, ii), p. 130), so (4.6) tells us that each of the functions (4.1) is invariant under s 1 s. → − Proof. We consider ﬁrst the case ∆ < 0. Mapping the upper half-plane z i √ ∆ 4 to the unit disc bu z − | |\ = reiθ, we ﬁnd → z + i √ ∆ 4 | |\ z2 ∆ 4 2 V(s, t) = ϕ | − \ | ysdz Z Z y2 ! H 366 Don Zagier

~~∆ 4 ∆ r2 1 r2 s rdr dθ = 4 s/2 ϕ | | − , | 4 | " (r2 1)2 ! 1 2r cos θ + r2 ! (1 r2)2 06r61 − − − 06θ62π~~

~~and this is equivalent to (4.4) because~~

~~2π s 1 1 r2 1 + r2 − · dθ = P s (0 6 r < 1, s C) 2π Z 1 2r cos θ + r2 ! − 1 r2 ! ∈ 0 − −~~

~~[EH 3.7(6)]. The functional equation follows since P s(z) = Ps 1(z). If ∆ > 0, then we transform the upper half-plane to− itself by −z z √∆/4 → − = ξ + iη, obtaining z + √∆/4~~

~~ξ2 + η2 ηs dξdη V(s, t) = ∆s/2 ϕ ∆ Z Z η2 ! 1 ξ iη 2s η2 H | − − |~~

∞ ϕ(∆(1 + u2)) ∞ vs 1dv = ∆s/2 − du 2 s/2 s Z (1 + u ) Z 2u 2 0 1 v + v −∞ − √u2+1 334 (u = ξ/η, v = ξ2 + η2). Substituting v = ex we ﬁnd p ∞ s 1 ∞ v − dv s dx s = 2− s Z 2u 2 Z u 0 1 v + v cosh x − √u2+1 −∞ − √u2+1 Γ(s)Γ( 1 ) 1 1 u = 2 F s, s; s + ; 1 + (Re(s) > 0) 2s 1 1 2 2 2 !! 2 − Γ(s + 2 ) √u + 1 π Γ(s) s s 1 u2 = F , ; ; 2s 1 2 2 2 − s+1 2 2 2 u + 1! Γ 2 u π Γ(s) s + 1 s + 1 3 u2 + F , ; ; 2s 2 2 2 √u2 + 1 2 − s 2 2 2 u + 1! Γ 2 Eisenstein Series and the Selberg Trace Formula I 367

~~[EH 2.12 (10), 2.1.5 (28)]. Since the second term is an odd function of u, we ﬁnd the formula~~

∞ 2 2 s π Γ(s) ϕ(∆(1 + u )) s s 1 u V(s, t) = ∆ 2 F , ; ; du, 2s 1 2 2 s/2 2 2 − s+1 Z (1 + u ) 2 2 2 u + 1! Γ 2 −∞ which is equivalent to (4.5); the functional equation follows because

~~2 2 s/2 s s 1 u s 1 s 1 2 (1+u )− F , ; ; = F , − ; ; u [EH2.1.4(22)]. 2 2 2 u2 + 1! 2 2 2 − !~~

~~Finally, if ∆= 0 then the substitution z 1/z gives →− z 4 1 y2 V(s, 2) = ϕ | | ysdz = ϕ dz; ± " y2 ! " y2 ! z 2s H H | |~~

~~1 making the substitution u = y− , t = x/y and using 335~~

∞ Γ( 1 )Γ s 1 2 s 2 2 1 (1 + t )− dt = − Re(s) > Z Γ(s) 2! −∞ [GR 3.251.2] we obtain (4.7). Notice that, since ϕ is assumed to be smooth, the integral in (4.7) has a meromorphic continuation to σ < A + 1 with (at most) simple poles at s = 0, 2, 4, ...; hence V(s, t) − − can also be meromorphically continued to this range and has (at most) 1 1 3 simple poles at s = , , , .... This completes the proof of Proposi- 2 −2 −2 tion (4) except for the various assertions about convergence, which can be checked easily using the asymptotic properties of the Legendre and hypergeometric functions. σ 1 A From (4.5) and (2.18) if follows that V(s, t) grows like t − − as t with s ﬁxed, A <σ< 1 + A. An easy calculation shows that ∈ ∞ − ζ(s, t2 4) = O(tC) for any C > max(1 2σ, 1 σ, 0) as t , and this − − − →∞ implies that the sum (3.11) is absolutely convergent for 1 A <σ< A. − Thus I (s) has a meromorphic continuation to 1 A <σ< A with (at 1 − 368 Don Zagier

most) a double pole at s = 1 (coming from the double pole of ζ(s, 0) = 1 ζ(s)ζ(2s 1)) and simple poles at s = and s = 0. From (3.5) and our − 2 assumptions on h(r) we see that I (s) is meromorphic in A <σ< A, 3 − the only pole in the half-plane σ> 0 being a simple one at s = 1. Thus to obtain a formula for I(s) in the critical strip we must still give the analytic continuations of I2(s) and I4(s). Let J(s) denote the integral in (3.4). As already stated, this integral converges absolutely for all s with σ , 0, 1, because the integrand is of rapid decay as r . However, J(s) is not deﬁned on the lines | |→∞ σ = 0 and σ = 1, because the path of integration passes through a pole of the integrand, so the functions deﬁned by the integral in the three regions σ < 0, 0 <σ< 1 and σ > 1 need not be (and are not) analytic continuations of one another. To obtain the analytic continuation of J(s) (and hence of I4(s)) to 0 <σ< 1, we set

~~ζ∗(s + 2ir)ζ∗(s 2ir) JC(s) = − h(r)dr, Z ζ∗(1 + 2ir)ζ∗(1 2ir) C −~~

336 where C is a deformation of the real axis into the strip 0 < Im(r) < 1 (A 1) which is suﬃciently close to the real axis that all zeroes of the 2 − Riemann zeta-function lie to the left of 1+2iC and ζ(1+2ir) 1 = O( r ǫ) − | | for

r C (see ﬁgure). The integral JC(s) converges for all s C such that ∈ ∈ ζ (s + 2ir) and ζ (s 2ir) remain ﬁnite for all r C, i.e. for s < 1 2iC, ∗ ∗ − ∈ ± 2iC. In particular, JC(s) is holomorphic in the region U bounded by ± 1 + 2iC and 1 2iC. Clearly JC(s) = J(s) for s to the right of 1 2iC, − − Eisenstein Series and the Selberg Trace Formula I 369 but for s in the right half of U we have

ζ∗(2s 1) s 1 J(s) JC(s) = π − h i − (s U,Re(s) > 1) − ζ (2 s)ζ (s) 2 ! ∈ ∗ − ∗ 1 because the integrand has a simple pole (at r = (s 1)) with residue 2 − 1 ζ (2s 1) i ∗ − h( (s 1)) in the region enclosed by R and C. Similarly 2i ζ (2 s)ζ (s) 2 − ∗ − ∗

~~ζ∗(2s 1) s 1 J(s) JC(s) = π − h i − (s U,Re(s) < 1). − − ζ (2 s)ζ (s) 2 ! ∈ ∗ − ∗~~

~~ζ∗(2s 1) s 1 Therefore the function J(s) in 0 <σ< 1 is 2π − h i −2 ζ∗(2 s)ζ∗(s) less than the analytic continuation of the function deﬁned− by J(s) for~~

~~σ > 1. Together with (3.4) this shows that I4∗(s) = ζ∗(2s)I4(s) has an analytic continuation to σ> 0 given by~~

1 2 ζ∗(s) J(s)  − 4π   1 2 1 ζ∗(s)ζ∗(2s 1) s 1  ζ∗(s) J (s) − h i − (s U), I4∗(s) =  C  − 4π − 4 ζ∗(s 1) 2 ! ∈  −  1 1 ζ (s)ζ (2s 1) s 1  ζ (s)2J(s) ∗ ∗ h i (0 <σ< 1),  ∗ − −  − 4π − 2 ζ∗(s 1) 2 !  − (4.8)  where we have used the functional equation ζ (s) = ζ (1 s). Of course, 337 ∗ ∗ − we could use a similar argument to extend past the critical line σ = 0, but since it is obvious that J(s) = J(1 s), we deduce from (4.8) that − I4∗(s) satisﬁes the functional equation

~~1 ζ∗(s)ζ∗(2s 1) s 1 I∗(1 s) = I∗(s) − h i − 4 − 4 − 2 ζ (s 1) 2 ! ∗ − 1 ζ (s)ζ (2s) s + ∗ ∗ h i , 2 ζ∗(s + 1) 2 370 Don Zagier~~

~~and this gives the meromorphic continuation immediately. From (4.8) and (3.5) we ﬁnd~~

1 2 I∗(s) + I∗(s) = ζ∗(s) J(s) (4.9) 3 4 −4π 1 ζ (s)ζ (2s) is ∗ ∗ h (1 <σ< A) 2 ζ∗(s + 1) 2 1 ζ (s)ζ (2s) is 1 ζ (s)ζ (2s 1) s 1  ∗ ∗ h + ∗ ∗ h i <σ<  − − (0 1) − 2 ζ∗(s + 1) 2 2 ζ∗(s 1) 2 !  − 1 ζ∗(s)ζ∗(2s 1) s 1  − h i − (1 A <σ< 0), 2 ζ∗(s 1) 2 ! −  − which proves the invariance of (4.3) under s 1 s. Notice that the → − is ζ∗(s)ζ∗(2s 1) ζ∗(s)ζ∗(2s) s 1 function /ζ∗(s + 1)h resp. − h i has inﬁni- 2 ζ (s 1) −2 ∗ − ! 338 tely many poles in the half-plane σ < 0 (resp.− σ > 1), but drops out of (4.9) before that half-plane is reached. In fact, it is clear from (4.8) and

(4.9) that the function I3∗(s) + I4∗(s) is holomorphic in 1 A <σ< A − 1 except for double poles at s = 0 and s = 1 (the simple poles at s = 2 1 must cancel since I (s) + I (s) is an even function of s ). 3∗ 4∗ − 2 It remains to treat the function (4.2). Using the formulas (2.23) for the Selberg transform we ﬁnd

∞ ∞ ∞ 2 s 1 1 2 2 s 1 ϕ(u )u − du = Q′(u + v )u − dv du Z −u Z Z 0 0 −∞ π ∞ 1 2 s 1 iθ = Q′(r )(r sin θ) − r drdθ(Re = v + iu) −π Z Z 0 0 Γ s ∞ 2 2 s = Q′(r )r dr −Γ( 1 )Γ s+1 Z 2 2 0 s Γ ∞ s 1 2 u − = 2 sinh g′(u)du −2Γ 1 Γ s+1 Z 2 2 2 0 Eisenstein Series and the Selberg Trace Formula I 371 and hence, by (4.7),

s π− Γ(s)ζ(s,0)[V(s, 2) + V(s, 2)] = − 1 (s 1) ∞ s 1 (4π) 2 − u − = ζ∗(s)ζ∗(2s 1) sinh g′(u)du; − Γ s+1 − Z 2 2 0 the integral converges for 1 <σ< 1 + A and hence gives the ana- − lytic continuation of the left-hand side to this strip. On the other hand, formulas (3.7) and (3.8) give

~~1 s ∞ s (4π)− 2 u − I2∗(s) = s ζ∗(s)ζ∗(2s) sinh g′(u)du, −Γ(1 2 ) Z 2 − 0~~

where now the integral converges for A <σ< 2. This shows that 339 − the function (4.2) can be continued to the strip A <σ< 1 + A and is − invariant under s 1 s; equation (3.9) then gives the formula → − s π− Γ(s)ζ(s,0)[V(s, 2) + V(s, 2)] = I∗(1 s) (4.10) − 2 − ∞ 1 s 1 s ζ (s)ζ (2s 1) Γ −2 + ir Γ −2 ir = ∗ ∗ − − h(r)dr (0 <σ< 1) 2 s 2 s (4π) −2 Γ Z Γ(ir) Γ( ir) −2 − −∞ in the critical strip. A similar discussion to that given for the inte- 1 (s 1) π 2 gral I (s) now shows that for σ > 1 we must add − ζ (s)ζ (2s 4∗ s 1 ∗ ∗ − Γ −2 s 1 1)h i −2 to the right-hand side of (4.10) and that near the line σ = 1 we have

s π− Γ(s)ζ(s,0)[V(s, 2) + V(s, 2)] − 1 s 1 s ζ (s)ζ (2s 1) Γ − + ir Γ − ir = ∗ ∗ 2 2 − 2 s − h(r)dr (4.11) − 2 s Γ Γ (4π) 2 Γ − Z (ir) ( ir) 2 C − s 1 1 π −2 s 1 + ζ∗(s)ζ∗(2s 1)h i − (s U). 2 s 1 − 2 ! ∈ Γ −2 372 Don Zagier

Again the analytic continuation to 1 A < σ 6 0 follows using the − functional equation. We have thus proved the analytic continuability and functional equation of each of the functions (4.1) - (4.3) in the strip 1 A <σ< A and − 340 given explicit formulas for these functions in each of the ﬁve regions 1 A <σ< 0, 1 U, 0 <σ< 1, U and 1 <σ< A covering this strip. − − We and this section by using these formulas to compute the residue at s = 1 of the functions in question. From the development 1 1 ζ∗(s) = + (γ log 4π) + O(s 1) (s 1) s 1 2 − − → − and (4.8) we ﬁnd 1 1 γ log 4π I∗(s) = + − + O(1) 4 −4π "(s 1)2 s 1 # − − 2  h(r)dr + (s 1) z(r)h(r)dr + O(s 1)  × Z − Z −  C C     1 1  + h(0) + O(1) 8 s 1 − zeta ′ (1 + 2ir) ζ ′ (1 2ir) as s 1, where z(r) = ∗ + ∗ − . Since z(r) is → ζ∗(1 + 2ir) ζ∗(1 2ir) holomorphic for r near the real line (the poles of the− two terms at r = 0 cancel), we can replace C by R in the two integrals, obtaining κ h(0) I∗(s) = + κ(γ log 4π) + 4 −(s 1)2 − − 8 − ∞ 1 1 z(r)h(r)dr (s 1)− + O(1) −4π Z  −  −∞  1  as s 1, where κ = ∞ h(r)dr. From (3.5) we get → 4π −∞R i ζ∗(s)ζ∗(2s) is 1 h( 2 ) I∗(s) = h = + O(1) (s 1). 3 − 2ζ (s + 1) 2 −2 s 1 → ∗ − Eisenstein Series and the Selberg Trace Formula I 373

~~This takes care of the function (4.3). For (4.2) we use equations (4.11) and (3.9), obtaining (by an argument similar to the one just used for I4∗) 341~~

s π− Γ(s)ζ(s,0)[V(s, 2) + V(s 2)] − 1 1 = + (γ log 4π) + O(s 1) " s 1 2 − − # − 1 1 + (γ log 4π) + O(s 1) × "2(s 1) 2 − − # − 1 1 Γ 1 + log 4π + ′ ( ) (s 1) + O(s 1)2 × "2π 4π Γ 2 ! − − # s 1 Γ Γ h(r)dr − ′ (ir) + ′ ( ir) h(r)dr  Γ Γ × Z − 2 Z − ! C C   2 h(0) 1 +O(s 1) + (s 1)− + O(1) − 8 − κ i = + κ(γ log 8π) (s 1)2 − − ∞ 1 Γ′ 1 1 (1 + ir)h(r)dr + h(0) (s 1)− + O(1) −4π Z Γ 8  −  −∞   and ∞ 1 1 I2∗(s) =  h(r)r tanh πrdr (s 1)− + O(1). 24 Z  −    −∞  Finally, to compute the residue of (4.1) at s = 1 we need the values of 2 V(1, t) and ress= ζ(s, t 4) for t Z, t , 2. From (4.4) and (4.5) we 1 − ∈ ± ﬁnd π ∞ dx  ϕ(x) ( t < 2) 2 Z √x + 4 t2 | |  0 − V(1, t) =   ∞ π dx  ϕ(x) ( t > 2) 2 Z √x + 4 t2 | |  t2 4 −  −  374 Don Zagier

~~(since P0(u) = 1, F(0, b; c; x) = 1). Using the formulas (2.23) for the 342 Selberg transform, we can express this in therms of h(r), obtaining~~

1 ∞ e 2αr π − h(r)dr t = 2 cos α 6 2, 0 6 α 6  2πr 2 Z 1 + e− | | 2  V(1, t) =  −∞  ∞ 1 2iαr  e h(r)dr ( t = 2 cosh α > 2) 4 Z | |   −∞ (4.12)  (we omit the calculation, which is not diﬃcult, since in 5 we will give § a general formula for V(s, t) in terms of h(r)). As to ζ(s, D), we have

2π 1 (D < 0)  √ D Aut(Q)  | | XQ | | ress=1ζ(s, D) =  (4.13)  1  log εQ (D > 0),  √  D XQ   where and Aut(Q) have the same meaning as in (1.12) and, in the Q P second formula, εQ is the fundamental unit for Q (i.e. the larger eigenvalue of M, where M SL (Z) is a matrix with positive trace such that ∈ 2 Aut(Q) = Mn, n Z ). {± ∈ } We have thus given the principal part of each of the functions (4.1) - (4.3) at the pole s = 1. Adding up the expressions obtained, we ﬁnd that the terms in (s 1) 2 cancel and that − − ∞ 3 i i h(r ) = K (z, z) + h dz = 2res = I (s) + h j 0 π 2 s 1 ∗ 2 = Z " # Xj 0 Γ/H 1 ∞ = h(r)r tanh πrdr (4.14) 12 Z −∞ 1 ∞ Γ z(r) + ′ (1 + ir) + log 2 h(r)dr − 2π Z Γ ! −∞ Eisenstein Series and the Selberg Trace Formula I 375

~~1 2 ∞ 2 + h(0) + V(1, t)res = ζ(s, t 4), 2 π s 1 t=X − t2−∞,4 where 343~~

~~ζ ′ (1 + 2ir) ζ ′ (1 2ir) z(r) = ∗ + ∗ − ζ (1 + 2ir) ζ (1 2ir) ∗ ∗ − 1 Γ 1 1 Γ ζ ζ = ′ ( + ir) ′ (ir) + ′ (1 + 2ir) ′ (2ir) 2 Γ 2 − 2 Γ ζ − ζ~~

2 and V(1, t), ress= ζ(s, t 4) are given by equations (4.12) and (4.13), 1 − respectively. Formula (4.14) is the Selberg trace formula. § 5. Complements. In the last section we gave the analytic continuation of I(s) to the strip 1 A <σ< A. To complete the proof of − Theorem 1 we must still

~~1) express V(s, t) in terms of the Selberg transform h(r);~~

~~2) generalize the formula obtained for I(s) to the function~~

m ∞ h(r j) I (s) = a j(m) R f (s) (m Z) (5.1) ( f , f ) j ∈ Xj=1 j j with m , 1 (notations as in 1). In this section we will carry out § these two calculations and also indicate the generalization to congruence subgroups of SL2(Z). The results of 4 show that I (s) equals § ∗

∞ 1 2 ζ∗(s + 2ir)ζ∗(s 2ir) ζ∗(s) − h(r)dr − 4π Z ζ (1 + 2ir)ζ (1 2ir) ∗ ∗ − −∞ 1 ζ (s)ζ (2s) is 1 ζ (s)ζ (2s 1) 1 s ∗ ∗ h ∗ ∗ − h i − − 2 ζ (s + 1) 2 − 2 ζ (s 1) 2 ! ∗ ∗ − ∞ s s ζ (s)ζ (2s) Γ 2 + ir Γ 2 ir + ∗ ∗ − h(r)dr s+1 s+1 (4π) 2 Γ Z Γ(ir)Γ( ir) 2 − −∞ 376 Don Zagier

∞ 1 s 1 s ζ (s)ζ (2s 1) Γ −2 + ir Γ −2 ir = ∗ ∗ − − h(r)dr 2 s 2 s (4π) −2 Γ Z Γ(ir)Γ( ir) −2 − −∞ s ∞ 2 + π− Γ(s) V(s, t)ζ(s, t 4) t=X − t2−∞,4

344 in the critical strip 0 <σ< 1 (cf. equations (4.9) and (4.10). Using the functional equations of V(s, t) and ζ(s, D) we can write this expression as R(s) + R(l s), where − ∞ 1 2 ζ∗(s + 2ir)ζ∗(s 2ir) R(s) = ζ∗(s) − h(r)dr −8π Z ζ (1 + 2ir)ζ (1 2ir) ∗ ∗ − −∞ ζ (s)ζ (2s) is ∗ ∗ h − 2ζ∗(s + 1) 2 Γ(s)Γ(s 1 ) + − 2 ζ(s)ζ(2s 1) s 2 s 1+s − × 2π Γ −2 Γ 2 ∞ 1 s 1 s Γ −2 + ir Γ −2 ir − h(r)dr Z Γ(ir)Γ( ir) − −∞ s ∞ 2 + π− Γ(s) v(s, t)ζ(s, t 4), t=X − t,−∞2 ± v(s, t) being any function such that πs 1Γ(1 s) Γ(s, ∆) V(s, t) = v(s, t) + − − v(1 s, t). (5.2) π sΓ(s) γ(1 s, ∆) − − − (here ∆ = t2 4 as before). Comparing this with (1.16) and observing − that F(a, b; c; 0) = 1, we see that Theorem 1 (for m = 1) will follow from

~~Proposition 5: For t , 2 and 0 < Re(s) < 1 the function V(s, t) ± 345 deﬁned by (3.10) is given by equation (5.2) with Eisenstein Series and the Selberg Trace Formula I 377~~

~~1 ∞ Γ 1 s + ir Γ 1 s ir Γ(s 2 ) −2 −2 v(s, t) = − − 4Γ s+1 Γ 2 s Z Γ(ir)Γ( ir) 2 −2 − −∞ 1 s 1 s 3 t2 F − + ir, − ir; s; 1 h(r)dr. × 2 2 − 2 − − 4 !~~

~~Proof. As in Proposition (4) we must distinguish the cases ∆ > 0 and ∆ < 0. It will also be useful to introduce symmetrization operators S 1 S s , r with~~

~~1 S [ f (s)] = f (s) + f (1 s), Sr[ f (s)] = f (r) + f ( r) s − − for any function f . Thus the formula we want to prove can be written~~

~~π sΓ(s) π sΓ(s) − V(s, t) = S 1 − v(s, t) . (5.3) Γ(s, ∆) s " γ(s, ∆) #~~

~~If ∆ > 0, then (4.5) and (2.24) give~~

~~s ∞ 1 Γ 2 s V(s, t) = ∆ 2 r tanh πrh(r) 8π Γ(s) Z −∞ 1 ∆/2 s 3 s s 1 dξ −2 = O 1 +ir 1 + (1 ξ) F , ; ; ξ dr, × Z ¶− 2 1 ξ ! − 2 2 2 ! √ξ 0 −~~

~~u2 where we have made the change of variables ξ = . To prove (5.3), u2 + 1 we must show that the inner integral equals~~

1 1 s 1 s s Γ(S )Γ − + ir Γ − ir S 1 2 − 2 2 2 − s cosh πr (5.4) ∆s/2 Γ 1 Γ s Γ 1 s  2 2 2  −  1 s 1 s 3 t2 F − + ir, − ir; s; 1 × 2 2 − 2 − − 4 !# 378 Don Zagier

~~346 (here and from now one we use standard identities for the gamma function without special mention). Using the identity~~

2 Γ(2ir) 1 ir 1 1 P 1 + = S x 2 F ir ir ir x x > +ir 1 r  2 − , ; 1 2 ;  ( 0) − 2 x!  1 2 − 2 − − − ! Γ 2 + ir      [EH 3.2 (19)] and expanding the hypergeometric series, we ﬁnd that the integral in question equals

2 1 n 1 n +ir cothπr ∞ ( 1) Γ n + 2 ir ∆ − − 2 Sr  − − 2πi n! Γ(n + 1 2ir) 4 !  Xn=0 −   1 s +n 1 ir s s 1 dξ (1 ξ) 2 − − F , ; ; ξ  . × Z − 2 2 2 ! √ξ  0   From [EH 2.4(2), 2.8(46)] we have

1 s +n 1 ir 1 s s 1 (1 ξ) 2 − − ξ− 2 F , ; ; ξ dξ Z − 2 2 2 ! 0 1 s Γ 2 Γ 2 + n ir s s 1 + s = − F , ; + n ir; 1 Γ 1+s + n ir 2 2 2 − ! 2 − s 1 s 1 Γ 2 + n ir Γ −2 + n ir Γ 2 = − − , 1 2 Γ n + 2 ir − so our integral equals

coth πr ∞ ( 1)n Sr −  i π n! ×  2 √ Xn=0   s 1 s n 1 +ir  Γ + n ir Γ − + n ir ∆ 2 2 − 2 − − − Γ +  × (n 1 2ir) 4 !  −   Eisenstein Series and the Selberg Trace Formula I 379

s 1 s ir 1 coth πr Γ 2 ir Γ −2 ir ∆ − 2 = S − − (5.5) r  Γ  2i √π (1 2ir) 4 !  −  s 1 s 4 F ir, − ir; 1 2ir; × 2 − 2 − − −∆!# 1 1 s s Γ(s )Γ − ir 2 1 coth πr 2 2 ∆ − = SrS − − s  1+s  2i √π Γ ir 4 !  2 − 1 s 1 s 3 ∆ F − ir, − + ir; s; × 2 − 2 2 − − 4 !#

~~(the last formula is [EH 2.10(2)]), and since 347~~

1 s 1 s 1 s coth πr Γ −2 ir cosh πrΓ −2 + ir Γ −2 ir Sr − = −  1+s  1 s s  2i √π Γ ir  Γ Γ Γ 1  2 −  2 2 − 2   this agrees with (5.4), completing the proof for ∆ > 0. π sΓ(s) 1 t2 If ∆ < 0, then − = δ s/2, where δ = ∆ = 1 , so (5.3) is γ(s, ∆) − 2| | − 4 equivalent to

~~1 s/2 S 1 2 (s 1) δ− V(s, t) = s δ − v(1 s, t) . − On the other hand, from (4.4) and (2.24) we have~~

~~∞ ∞ s/2 1 2 π− V(s, t) = r tanh πrh(r) P 1 +ir(1 + 2δ(u 1))P s(u)dudr. 2 Z Z − 2 − − 1 −∞ Denote the inner integral by I. Then (5.3) will be proved if we show that 348~~

Γ 1 s Γ s + ir Γ s ir 1 1 1 (s 1) 2 2 2 I = S δ 2 − − − s 2 1 1 2 s 1+s  Γ + ir Γ ir Γ − Γ  2 2 − 2 2  s s 1 F + ir, ir; + s; δ . 2 2 − 2 !# 380 Don Zagier

~~By [EH 2.10(1)], this is equivalent to~~

s s 1 s 1 s 1 Γ + ir Γ ir Γ − + ir Γ − ir I = 2 2 − 2 2 − 2 √π 1 1 1+s 2 s Γ 2 + ir Γ 2 ir Γ 2 Γ −2 − s 1 s s 1 δ − 2 F ir, + ir; ; 1 δ . × 2 − 2 2 − ! To prove this this formula, we being by making the substitution v = 2 u 1 in I and substituting for P 1 +ir by − − 2

~~∞ ax dx π 1 1 e− Kir(x) = Γ + ir Γ ir P 1 +ir(a) Z √x r2 2 ! 2 − ! − 2 0 [GR 6.628. 7]; after an interchange of integration this gives 1 1 √2πΓ + ir Γ ir I 2 ! 2 − !~~

∞ ∞ 2δxv dv x dx =  e− P s √1 + v  e− Kir(x) . Z Z − √1 + v √x 0  0      3/4 δx By [GR 7.146.2] the inner integral equals (2δx)− e 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

C+i 1 ∞ 1 + s − s 1 1 1 s 1 s Γ Γ 1 − Γ v + Γ v)Γ( − v 2 ! − 2 · 2πi Z 2! 2 − 2 − ! × C i − ∞ v 1 (2δx) − 2 dv, × 349 where C is chosen such that 1 < C < 1 min(σ, 1 σ). If choose C − 2 2 − to satisfy also C > 0 then we may interchange the order of integration again, obtaining 1 1 1 + s s 2 √πΓ + ir Γ( ir)Γ Γ 1 I 2 ! 2 − 2 ! − 2 Eisenstein Series and the Selberg Trace Formula I 381

C+i ∞ 1 v v 1 1 s 1 s = 2 δ − 2 Γ(v + )Γ v Γ − v 2πi Z 2 2 − 2 − ! × C i − ∞ ∞ v 1 x x − e− Kir(x)dx dv × Z 0 C+i Γ 1 ∞ ( 2 ) s 1 s v 1 = Γ v Γ − v Γ(v + ir)Γ(v ir)δ − 2 dv 2πi Z 2 − 2 − ! − C i − ∞ [ET 6.8(28)]. The integral is very rapidly convergent (the integrand is 3/2 2π v v 1 O( v e )), so we may substitute for δ 2 the binomial expansion | |− − | | − Γ s v + n v 1 1 (s 1) ∞ 1 2 n δ − 2 = δ 2 − − (1 δ) n! s − Xn=0 Γ 2 v − and integrate term by term. Using “Barnes’ Lemma”

~~C+i 1 ∞ Γ(α + s)Γ(β + s)Γ(γ s)Γ(δ s)ds 2πi Z − − 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 1 1 + s s 1 (s 1) 2Γ + ir Γ ir Γ Γ 1 I = δ 2 − 2 ! 2 − ! 2 ! − 2 × s s 1 s 1 s ∞ Γ 2 + ir + n Γ 2 ir + n Γ −2 + ir Γ −2 ir − − (1 δ)n × 1 − Xn=0 Γ 2 + n n! 1 1 − s s 1 s 1 s s 1 =Γ Γ + ir Γ ir Γ − + ir Γ − ir δ −2 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~~

~~∞ h(r j) Km(z, z ) = a (m) f (z) f (z ). 0 ′ j ( f , f ) j j ′ Xj=1 j j~~

m m Then I (s) = K0 (z, z)E(z, s)dz. On the other hand, from (1.6) we see Γ/RH m 1 that K (z, z ) = m 2 K (z, z ) T(m), where K (z, z ) is the kernel function 0 ′ 0 ′ | 0 ′ (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 1 (m), respectively (τv(m) as in 2 s 2 (2.7)), equation (2.31) gives −

~~m m 3 i K0 (z, z′) = K (z, z′) τ 1 (m)h − π 2 2 − 1 ∞ 1 1 E z, + ir E z′, ir h(r)τir(m)dr, − 4π Z 2 ! 2 − ! −∞ where~~

m 1 az′ + b K (z, z′) = √mK(z, z′) T(m) = k z, . | cz + d ! 2 √m a,bX,c,d Z ′ ad bc=∈m − 4 K m m K m Hence the constant term (y) of K0 (z, z) equals i (y), where i=1 K m K m K K P 3 and 4 are deﬁned exactly like 3 and 4 but with h(r) replaced 351 by h(r)τir(m) and

~~iy+1 1 az + b K m(y) = k z, dz, 1 cz + d ! √m Z adXbc=m iy −c>0 Eisenstein Series and the Selberg Trace Formula I 383~~

iy+1 1 ∞ az + b K m(y) = k z, dz 2 d ! − √m Z adX=m bX= iy a,d>0 −∞ y ∞ h(r)τir(m)dr. − 2π Z −∞ 4 m m m m As in §3 we then ﬁnd I (s) = Ii (s) for 1 <σ< A, where I3 and I4 i=1 are given by the same formulasP as I3 and I4 (equations (3.4) and (3.5)) but with h(r) replaced by τir(m)h(r) and

~~s 1 ∞ t m −2 2 ζ(2s)I1 (s) = m ζ(s, t 4m)V s, . t=X − √m! −∞ m As to I2 , from (2.20) and (2.23) we ﬁnd~~

1 1 ∞ (x(d a) b)2 + (a d)2y2 K m(y) = ϕ − − − dx 2 my2 ! √m Z adX=m bX= 0 a,d>0 −∞ y ∞ a ir h(r) dr − 2π d adX=m Z a,d>0−∞ 1 (a d)2 a = Q − y g log m ! − d √m adX=m adX=m a,d 1 ∞ b2 ϕ if √m Z +  √m my2 ! ∈  bX=  −∞  0 if √m < Z   1 K2(y √m) if √m Z =  √m ∈   0 if √m < Z,   384 Don Zagier

352 so s/2 m− I2(s) if √m Z, Im(s) = ∈ 2  0 if √m < Z.   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 Im then has no summands with t2 4m = 0 and Im 1 − 2 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 I is absent), but − 2 the calculations with the hypergeometric functions are a little different. Since constant functions and Eisenstein series are invariant under T( 1), m m m m − the terms I3 (s) and I4 (s) are equal to I3| |(s) and I4| |(s), so that first two terms in (1.16) are unchanged except for replacing m by m . The term m | m| I2 is always zero since m cannot be a square. Finally, for I1 we find

~~2 s 1 ∞ ζ(s, t 4m) m −2 I1 (s) = m − V 1 (m < 0) (5.6) ζ(2s) s,t m − 2 | | t=X | | −∞ with~~

2 z 2 ∆/4 1 s 2 s Vs,t = k z, y dz = ϕ  | | − + t  y dz, Z z¯ + t! Z y2   H H     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!~~

Γ(2ir) 1 ir 1 1 = Sr  x 2 − F ir, ir; 1 2ir; x  (x > 0)  1 2 2 − 2 − − ! Γ 2 + ir      2 [EH 3.2(18)] instead of the corresponding formula for P 1 +ir 1 + . 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

s 1 s ir 1 Γ ir Γ − ir ∆ 2 S coth πr 2 − 2 − − r  Γ  2i √π (1 2ir) 4 ! ×  −  s 1 s 4 F ir, − ir; 1 2ir; × 2 − 2 − − ∆!# 2 s cosh πr s s 1 s 1 s ∆ − 2 = Γ + ir Γ ir Γ − + ir Γ − ir . π2 2 2 − 2 ! 2 − ! 4 ! × 1 s 1 s 1 ∆ F − ir, − + ir; ; 1 × 2 − 2 2 − 4 ! [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 4 s 1 2 m −2 s 2 ∞ − | | Γ ζ(s, t2 4m) πs+1 2 t=X − × −∞ ∞ s s 1 s 1 2 Γ 2 + ir Γ 2 ir Γ −2 + ir Γ −2 ir − − × Z Γ 1 + ir Γ 1 ir Γ(ir)Γ( ir) 2 2 − − −∞ 1 s 1 s 1 t2 F − + ir, − ir; ; h(r)dr × 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 Γ in the simplest case Γ = Γ (q)/ 1 , q prime. There 1 1 0 {± } are now two cusps and correspondingly two Eisenstein series E1 and E2, given explicitly by

~~s s E1(z) = Im(γz) , E2(z) = Im(wγz) 1 γ XΓ Γ1 γ w−XΓ w Γ1 ∈ ∞\ ∈ ∞ \~~

~~0 1 (where w = − ), and formula (2.31) becomes q 0 !~~

1 i K0(z, z′) = k(z, γz′) h − vol(Γ1 H) 2 − γXΓ1 ∈ \ 1 2 ∞ 1 1 E j(z, + ir)E j(z′, ir)h(r)dr, −4π 2 2 − Xj=1 Z −∞ 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

qs 1 E (z, s) = E(qz, s) E(z, s), 1 q2s 1 − q2s 1 − − qs 1 E (z, s) = E(z, s) E(qz, s), 2 q2s 1 − q2s 1 − − so that Fourier developments 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 ex- Γ1R/H ∞ s 2 pressed as K (y)y − dy, K (y) = constant term of K0(z, z)) now pro- R0 ceeds as in § 3; the ﬁnal formula is the same except that I1(s) is replaced by 2 1 1 ∞ t 4 2 ζ(2s)− 1 + − ζ(s, t 4)V(s, t), q2 + 1 q t=X !! − −∞ Eisenstein Series and the Selberg Trace Formula I 387

~~∆ q 1 (where is the Legendre symbol), I3(s) is multiplied by − , q ! qs+1 1 − and the integrand of I4(s) is multiplied by~~

~~s 1 s 1 (q + 1)(1 q− )(1 q − ) 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 uber¨ 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 Dirich- let 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. 388 Bibliography

[9] 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. McGraw- Hill, 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.