Modular Forms and Special Cycles on Shimura Curves. (AM-161) [Course Book ed.] 9781400837168

Table of contents :
Contents
Acknowledgments
Chapter 1. Introduction
Bibliography
Chapter 2. Arithmetic intersection theory on stacks
2.1 The one-dimensional case
2.2 Pic(ℳ),CH1ℤ(ℳ)and CH2ℤ(ℳ)
2.3 Green functions
2.4 Pic(A4),CHℤ(ℳ),and CHz(ℳ)
2.5 The pairing CH1(ℳ) x CHℤ (ℳ) — CH2(ℳ)
2.6 Arakelov heights
2.7 The arithmetic adjunction formula
Bibliography
Chapter 3. Cycles on Shimura curves
3.1 Shimura curves
3.2 Uniformization
3.3 The Hodge bundle
3.4 Special endomorphisms
3.5 Green functions
3.6 Special 0-cycles
Bibliography
Chapter 4. An arithmetic theta function
4.1 The structure of arithmetic Chow groups
4.2 The arithmetic theta function
4.3 The vertical component: definite theta functions
4.4 The analytic component: Maass forms
4.5 The Mordell-Weil component
4.6 Borcherds’ generating function
4.7 An intertwining property
Bibliography
Chapter 5. The central derivative of a genus two Eisenstein series
5.1 Genus two Eisenstein series
5.2 Nonsingular Fourier coefficients
5.3 The Siegel-Weil formula
5.4 Singular coefficients
5.5 Eisenstein series of genus one
5.6 Dt
5.7 WT
5.8 The central derivative—the rank one case
5.9 The constant term
Bibliography
Chapter 6. The generating function for 0-cycles
6.1 The case T > 0 with Diff(T, B) = {p} forp ∤ D(B)
6.2 The case T > 0 with Diff(T, B) = {p} forp | D(B)
6.3 The case of nonsingular T with sig(T) = (1,1) or (0,2)
6.4 Singular terms. T of rank 1
6.5 The constant term. T = 0
Bibliography
Chapter 6 Appendix. The case p = 2,p | D(B)
6A. 1 Statement of the result
6A.2 Review of the special cycles Z(j), for q(j) € ℤp \ {0}
6A.3 Configurations
6A.4 Calculations
6A.5 The first nondiagonal case
Bibliography
Chapter 7. An inner product formula
7.1 Statement of the main result
7.2 The case t1t2 is not a square
7.3 A weakly admissible Green function
7.4 A finer decomposition of special cycles
7.5 Application of adjunction
7.6 Contributions forp | D(B)
7.7 Contributions for p ∤ D(B)
7.8 Computation of the discriminant terms
7.9 Comparison for the case t1, t2 > 0, and t1t2 = m2
7.10 The case t1,t2

Citation preview

Annals of Mathematics Studies Number 161

A vertical cycle Z(j) in the case p = 2

The vertical cycle Z(j) in the case p = 2 for the endomorphism j given by (6A.5.2) in Chapter 6: Appendix. Here the half apartment {[Λ] = [Λr ] = [ [e1 , 2r e2 ] ] | mult[Λ] (j) > 0} has been marked, and the multiplicities of components have been indicated.

Modular Forms and Special Cycles on Shimura Curves

Stephen S. Kudla Michael Rapoport Tonghai Yang

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2006

c 2006 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data Kudla, Stephen S., 1950– Modular forms and special cycles on Shimura curves / Stephen S. Kudla, Michael Rapoport, Tonghai Yang. p. cm. — (Annals of mathematics studies ; 161) Includes bibliographical references and index. ISBN-13: 978-0-691-12550-3 (acid-free paper) ISBN-10: 0-691-12550-3 (acid-free paper) ISBN-13: 978-0-691-12551-0 (pbk. : acid-free paper) ISBN-10: 0-691-12551-1 (pbk. : acid-free paper) 1. Arithmetic algebraic geometry. 2. Shimura varieties. I. Rapoport, M., 1948- II. Yang, Tonghai, 1963- III. Title. IV. Annals of mathematics studies ; no. 161. QA242.5.K83 2006 516.30 5—dc22

2005054621

British Library Cataloging-in-Publication Data is available This book has been composed in Times Roman in LATEX The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

Acknowledgments

ix

Chapter 1. Introduction

1 21

Bibliography

Chapter 2. Arithmetic intersection theory on stacks 2.1 2.2 2.3 2.4

The one-dimensional case Pic(M), CH1Z (M), and CH2Z (M) Green functions 1 2 c Z (M), and CH c Z (M) Pc ic(M), CH 1

1

27 30 31 34 2

c (M) × CH c (M) → CH c (M) 2.5 The pairing CH 2.6 Arakelov heights 2.7 The arithmetic adjunction formula Bibliography Chapter 3. Cycles on Shimura curves 3.1 Shimura curves 3.2 Uniformization 3.3 The Hodge bundle 3.4 Special endomorphisms 3.5 Green functions 3.6 Special 0-cycles Bibliography

Chapter 4. An arithmetic theta function 4.1 The structure of arithmetic Chow groups 4.2 The arithmetic theta function 4.3 The vertical component: definite theta functions 4.4 The analytic component: Maass forms 4.5 The Mordell-Weil component 4.6 Borcherds’ generating function 4.7 An intertwining property Bibliography

Chapter 5. The central derivative of a genus two Eisenstein series 5.1 5.2

Genus two Eisenstein series Nonsingular Fourier coefficients

27

36 38 39 43 45 46 47 49 51 56 57 68 71 71 77 79 87 94 96 100 102 105 105 111

vi

CONTENTS

5.3 The Siegel-Weil formula 5.4 Singular coefficients 5.5 Eisenstein series of genus one 5.6 BT 5.7 WT 5.8 The central derivative—the rank one case 5.9 The constant term Bibliography

Chapter 6. The generating function for 0-cycles 6.1 The case T > 0 with Diff(T, B) = {p} for p - D(B) 6.2 The case T > 0 with Diff(T, B) = {p} for p | D(B) 6.3 The case of nonsingular T with sig(T ) = (1, 1) or (0, 2) 6.4 Singular terms, T of rank 1 6.5 The constant term, T = 0 Bibliography

Chapter 6 Appendix. The case p = 2, p | D(B) 6A.1 Statement of the result 6A.2 Review of the special cycles Z(j), for q(j) ∈ Zp \ {0} 6A.3 Configurations 6A.4 Calculations 6A.5 The first nondiagonal case Bibliography

Chapter 7. An inner product formula 7.1 Statement of the main result 7.2 The case t1 t2 is not a square 7.3 A weakly admissible Green function 7.4 A finer decomposition of special cycles 7.5 Application of adjunction 7.6 Contributions for p | D(B) 7.7 Contributions for p - D(B) 7.8 Computation of the discriminant terms 7.9 Comparison for the case t1 , t2 > 0, and t1 t2 = m2 7.10 The case t1 , t2 < 0 with t1 t2 = m2 7.11 The constant terms Bibliography

Chapter 8. On the doubling integral 8.1 The global doubling integral 8.2 Review of Waldspurger’s theory 8.3 An explicit doubling formula 8.4 Local doubling integrals 8.5 Appendix: Coordinates on metaplectic groups Bibliography

121 137 139 140 144 154 161 165 167 169 172 175 177 179 180 181 181 186 188 191 201 204 205 206 208 212 221 225 231 238 245 252 259 262 264 265 266 269 279 285 320 346

CONTENTS

Chapter 9. Central derivatives of L-functions 9.1 The arithmetic theta lift 9.2 The arithmetic inner product formula 9.3 The relation with classical newforms Bibliography

Index

vii 351 351 356 365 369 371

Acknowledgments The authors would like to thank the many people and institutions who provided support over the long period of time during which this project was realized. SK would like to thank J.-B. Bost, J. Bruinier, J. Burgos, M. Harris, J. Kramer, and U. K¨uhn for stimulating mathematical discussions. He would also like to thank the University of Cologne and Bonn University for their hospitality during many visits. The extended research time in Germany, which was essential to the completion of this work, was made possible through the support of a Max-Planck Research Prize from the Max-Planck Society and Alexander von Humboldt Stiftung. In addition, SK profited from stays at the Morningside Center, Beijing, the Fields Institute, Toronto, the University of Wisconsin, Madison, the University of Paris XI, Orsay, and the Newton Institute, Cambridge. He was supported by NSF Grants: DMS-9970506, DMS-0200292 and DMS-0354382. MR acknowledges helpful discussions with T. Zink and with the members of the ARGOS seminar, especially I. Bouw, U. G¨ortz, I. Vollaard, and S. Wewers. He also wishes to express his gratitude to the mathematics department of the University of Maryland for its hospitality on several occasions. TY would like to thank the Max-Planck Institute at Bonn, the Hong Kong University of Science and Technology, The Morningside Center of Mathematics in Beijing, and The National Center for Theoretical Research in Taiwan for providing a wonderful working environment when he visited these institutes at various times. He was supported by NSF Grants DMS-0354353 and DMS-0302043, as well as by a grant from the NSA Mathematical Sciences Program. We thank U. G¨ortz for producing the frontispiece and B. Wehmeyer for her assistance in TeXing several chapters.

Chapter One Introduction In this monograph we study the arithmetic geometry of cycles on an arithmetic surface M associated to a Shimura curve over the field of rational numbers and the modularity of certain generating series constructed from them. We consider two types of generating series, one for divisors and one d 1 (M) and CH d 2 (M), the first and second arithfor 0-cycles, valued in CH metic Chow groups of M, respectively. We prove that the first type is a nonholomorphic elliptic modular form of weight 32 and that the second type is a nonholomorphic Siegel modular form of genus two and weight 32 . In fact we identify the second type of series with the central derivative of an incoherent Siegel-Eisenstein series. We also relate the height pairing of a pair d 1 (M)-valued generating series to the CH d 2 (M)-valued series by an of CH inner product identity. As an application of these results we define an arithmetic theta lift from modular forms of weight 23 to the Mordell-Weil space of M and prove a nonvanishing criterion analogous to that of Waldspurger for the classical theta lift, involving the central derivative of the L-function. We now give some background and a more detailed description of these results. The modular curve Γ \ H, where H = {z ∈ C | Im z > 0} is the upper half plane and Γ = SL2 (Z), is the first nontrivial example of a locally symmetric variety, and of a Shimura variety. It is also the host of the space of modular forms and is the moduli space of elliptic curves. Starting from this last interpretation, we see that the modular curve comes equipped with a set of special divisors, which, like the classical Heegner divisors, are the loci of elliptic curves with extra endomorphisms. More precisely, for t ∈ Z>0 let (1.0.1)

Z(t) = {(E, x) | x ∈ End(E) with tr(x) = 0, x2 = −t · idE },

where E denotes an elliptic curve. The resulting divisor on the modular curve, which we also denote by Z(t), is the set of points where the √ corresponding elliptic curve E admits an action of the order Z[x] = Z[ −t] in √ the imaginary quadratic field kt = Q( −t), i.e., E admits complex multiplication by this order. One may also interpret Z(t) as the set of Γ-orbits in H which contain a fixed point of an element γ ∈ M2 (Z) with tr(γ) = 0 and

2

CHAPTER 1

det(γ) = t. It is a classical fact that the degree of Z(t) is given by deg Z(t) = H(4t), where H(n) is the Hurwitz class number. It is also known that the generating series (1.0.2)

X

deg Z(t) q t =

X

H(4t) q t

is nearly the q-expansion of a modular form. In fact, Zagier [58] showed that the complete series, for τ = u + iv, (1.0.3) Z ∞ X 1 3 1 X 1 2 2 t − 21 H(4t) q + e−4πn vr r− 2 dr · q −n , v E(τ, ) = − + 2 12 t>0 8π 1 n∈Z t>0

t>0

is the q-expansion of the value at s = 21 of a nonholomorphic Eisenstein series E(τ, s) of weight 32 , and hence is a modular form. Generating series of this kind have a long and rich history. They are all modeled on the classical theta series. Recall that if (L, Q) is a positive definite quadratic Z-module of rank n, one associates to it the generating series (1.0.4)

θL (τ ) =

X

q Q(x) = 1 +

∞ X

rL (t) q t .

t=1

x∈L

Here (1.0.5)

rL (t) = |{x ∈ L | Q(x) = t}|,

and we have set, as elsewhere in this book, q = e(τ ) = e2πiτ . It is a classical result going back to the 19th century that θL is the q-expansion of a holomorphic modular form of weight n2 for some congruence subgroup of SL2 (Z). Similarly, Siegel considered generating series of the form (1.0.6)

θr (τ, L) =

X x∈Lr

X

q Q(x) =

rL (T ) q T ,

T ∈Symr (Z)∨

where τ ∈ Hr , and q T = e(tr(T τ )), and (1.0.7)

1 rL (τ ) = |{ x ∈ Lr | Q(x) = ((xi , xj )) = T }|. 2

He showed that they define Siegel modular forms of genus r and weight n2 . Generalizations to indefinite quadratic forms were considered by Hecke and Siegel, and the resulting generating series can be nonholomorphic modular forms. Hirzebruch and Zagier [20] constructed generating series whose

3

INTRODUCTION

coefficients are given by cohomology classes of special curves on HilbertBlumenthal surfaces. They prove that the image under any linear functional of this generating series is an elliptic modular form. For example, they identify the modular form arising via the cup product with the K¨ahler class as an explicitly given Eisenstein series. One can also define special 0-cycles on Hilbert-Blumenthal surfaces and make generating functions for their degrees [25] . These can be shown to be Siegel modular forms of genus two and weight 2. We now turn to the generating series associated to arithmetic cycles on Shimura curves. We exclude the modular curve to avoid problems caused by its noncompactness. It should be pointed out, however, that all our results should have suitable analogues for the modular curve, cf. [57]. We pay, however, a price for assuming compactness. New difficulties arise due to bad reduction and to the absence of natural modular forms. Let B be an indefinite quaternion division algebra over Q, so that (1.0.8)

and

B ⊗Q R ' M2 (R)

Y

D(B) =

def.

p > 1.

B⊗Q Qp division

Let (1.0.9)

V = {x ∈ B | tr(x) = 0},

with quadratic form Q(x) = Nm(x) = −x2 , where tr and Nm denote the reduced trace and norm on B respectively. Then V is a quadratic space over Q of signature type (1, 2). Let (1.0.10)

D = {w ∈ V (C) | (w, w) = 0, (w, w) ¯ < 0}/C× ,

where (x, y) = Q(x+y)−Q(x)−Q(y) is the bilinear form associated to the quadratic form Q. Then D is an open subset of a quadric in P(V (C)) ' P2 , and (B ⊗Q R)× acts on V (R) and D by conjugation. We fix a maximal order OB in B. Since all these maximal orders are conjugate, this is not × really an additional datum. Set Γ = OB . The Shimura curve associated to B is the quotient (1.0.11)

[Γ\D].

Since Γ does not act freely, the quotient here is to be interpreted as an orbifold. Let us fix an isomorphism B ⊗Q R = M2 (R). Then we can also identify (B ⊗Q R)× = GL2 (R) and (B ⊗Q R)× acts on H± = C \ R by fractional linear transformations. We obtain an identification (1.0.12)

D = C \ R, via



z −z 2 7−→ z, 1 −z 

4

CHAPTER 1

equivariant for the action of (B ⊗Q R)× = GL2 (R). The Shimura curve associated to B has a modular interpretation. Namely, consider the moduli problem M which associates to a scheme S over Spec Z the category of pairs (A, ι) where • A is an abelian scheme over S • ι : OB → End(A) is an action of OB on A with characteristic polynomial charpol(x | Lie A) = (T − x)(T − xι ) ∈ OS [T ], for the induced OB -action on the Lie algebra. Here x 7→ xι denotes the main involution on B. If S is a scheme in characteristic zero, then the last condition simply says that A has dimension 2, i.e., that (A, ι) is a fake elliptic curve in the sense of Serre. This moduli problem is representable by an algebraic stack in the sense of Deligne-Mumford, and we denote the representing stack by the same symbol M. We therefore have an isomorphism of orbifolds, (1.0.13)

M(C) = [Γ \ D].

Since B is a division quaternion algebra, M is proper over Spec Z and M(C) is a compact Riemann surface (when we neglect the orbifold aspect). By its very definition, the stack M is an integral model of the orbifold [Γ \ D]. It turns out that M is smooth over Spec Z[D(B)−1 ] but has bad reduction at the prime divisors of D(B). At the primes p with p | D(B), the stack M has semistable reduction and, in fact, admits a p-adic uniformizaˆ In particular, the special fiber Mp tion by the Drinfeld upper half plane Ω. is connected but in general not irreducible. In analogy with the case of the modular curve, we can define special divisors on the Shimura curve by considering complex multiplication points. More precisely, let t ∈ Z>0 and introduce a relative DM-stack Z(t) over M by posing the following moduli problem. To a scheme S the moduli problem Z(t) associates the category of triples (A, ι, x), where • (A, ι) is an object of M(S) • x ∈ End(A, ι) is an endomorphism such that tr(x) = 0, x2 = −t·idA .

An endomorphism as above is called a special endomorphism of (A, ι). The space V (A, ι) of special endomorphisms is equipped with the degree form Q(x) = xι x. Note that for x ∈ V (A, ι) we have Q(x) = −x2 . We denote by the same symbol the image of Z(t) as a cycle in M and use the

5

INTRODUCTION

notation Z(t) = Z(t)C for its complex fiber. Note that Z(t) is a finite set of points on the Shimura curve, corresponding to those √fake elliptic curves which admit complex multiplication by the order Z[ −t]. We form the generating series (1.0.14)

φ1 (τ ) = −vol(M(C)) +

X

deg(Z(t)) q t ∈ C[[q]].

t>0

Here the motivation for the constant term is as follows. Purely formally Z(0) is equal to M with associated cohomology class in degree zero; to obtain a cohomology class in the correct degree, one forms the cup product with the natural K¨ahler class — which comes down to taking (up to sign) the volume of M(C) with respect to the hyperbolic volume element. Proposition 1.0.1. The series φ1 (τ ) is the q-expansion of a holomorphic modular form of weight 3/2 and level Γ0 (4D(B)o ), where D(B)o = D(B) if D(B) is odd and D(B)o = D(B)/2 if D(B) is even.

Just as with the theorem of Hirzebruch and Zagier, this is not proved by checking the functional equations that a modular form has to satisfy. Rather, the theorem is proved by identifying the series φ1 (τ ) with a specific Eisenstein series1 . More precisely, for τ = u + iv ∈ H, set (1.0.15) X 3 1 1 1 (cτ + d)− 2 |cτ + d|−(s− 2 ) ΦB (γ, s), E1 (τ, s, B) = v 2 (s− 2 ) · γ∈Γ0∞ \Γ0

where γ = ac db ∈ Γ0 = SL2 (Z), and ΦB (γ, s) is a certain function depending on B. The Eisenstein series E1 (τ, s, B) is the analogue for the Shimura curve of Zagier’s Eisenstein series (1.0.3). It has a functional equation of the form


(1.0.16)

E1 (τ, s, B) = E1 (τ, −s, B).

Its value at s = 12 is a modular form of weight 32 and we may consider its qexpansion. Proposition 1.0.1 now follows from the following more precise result. Proposition 1.0.2. 1 φ1 (τ ) = E1 (τ, , B), 2

i.e., φ1 is the q-expansion of E1 (τ, 12 , B).

1 Alternatively, φ1 (τ ) can be obtained by calculating the integral over M(C) of a theta function valued in (1, 1) forms; this amounts to a very special case of the results of [33]. The analogous computation in the case of modular curves was done by Funke [11].

6

CHAPTER 1

Proposition 1.0.2 is proved in [38] by calculating the coefficients of both power series explicitly and comparing them term by term. These coefficients turn out to be generalized class numbers. More precisely, for t > 0, the coefficient of q t on either side is equal to (1.0.17)

deg Z(t) = 2δ(d; D(B))H0 (t; D(B)),

where (1.0.18)

δ(d; D) =

Y

(1 − χd (`))

`|D

and (1.0.19)

H0 (t; D) =

h(d) w(d)

X c|n (c,D)=1

c

(1 − χd (`)`−1 ).

Y `|c

Here d denotes√the fundamental discriminant of the imaginary quadratic field kt = Q( −t) and we have written 4t = n2 d; also, h(d) denotes the class number of kt and w(d) the number of roots of unity contained in kt . By χd we denote the quadratic residue character mod d. For t = 0, the identity in Proposition 1.0.2 reduces to the well-known formula for the volume (1.0.20)

vol(M(C)) = ζD(B) (−1),

where in ζD(B) (s) the index means that the Euler factors for p | D(B) have been omitted in the Riemann zeta function. Note that the fact that the generating series φ1 (τ ) is a modular form reveals some surprising and highly nonobvious coherence among the degrees of the various special cycles Z(t). In this book we will establish arithmetic analogues of Propositions 1.0.1 and 1.0.2. In contrast to the above propositions, which are statements about generating series valued in cohomology (just as was the case with the results of Hirzebruch-Zagier), our generating series will have coefficients in the arithmetic Chow groups of Gillet-Soul´e [14], [48], (see also [3]). Let us recall briefly their definition in our case. A divisor on M is an element of the free abelian group generated by the closed irreducible reduced substacks which are, locally for the e´ tale topology, Cartier divisors. A Green function for the divisor Z is a function g on M(C) with logarithmic growth along the complex points of Z = ZC and which satisfies the Green equation of currents on M(C), (1.0.21)

ddc g + δZ = [η],

7

INTRODUCTION

where η is a smooth (1, 1)-form. Let ZˆZ (M) be the group of pairs (Z, g), where g is a Green function for the divisor Z. The first arithmetic Chow d 1 (M) is the factor group of Z ˆZ (M) by the subgroup generated by group CH Z d the Arakelov principal divisors div f associated to rational functions on M. For us it will be more convenient to work instead with the R-linear version d 1 (M). In its definition one replaces Z-linear combinations of divisors by CH R-linear combinations and divides out by the R-subspace generated by the Arakelov principal divisors. Such groups were introduced by Gillet-Soul´e [15]; for the case relevant to us, see [3]. Note that restriction to the generic fiber defines the degree map ∼ d 1 (M) −→ CH1 (MC ) ⊗ R −→ degQ : CH R.

(1.0.22) 2

d (M) is defined in an analogous way, starting with 0-cycles The group CH on M. Since the fibers of M over Spec Z are geometrically connected of dimension 1, the arithmetic degree map yields an isomorphism

(1.0.23)

2

∼

d : CH d (M) −→ R. deg

Finally we mention the Gillet-Soul´e arithmetic intersection pairing, (1.0.24)

1

1

2

d (M) × CH d (M) −→ CH d (M) = R. h , i : CH

It will play the role of the cup product in cohomology in this context. d 1 (M) using the We now define a generating series with coefficients in CH divisors Z(t). For t > 0, we equip the divisor Z(t) with the Green function ˆ v) Ξ(t, v) depending on a parameter v ∈ R>0 , constructed in [24]. Let Z(t, 1 d (M). For t < 0 note that Z(t) = ∅. be the corresponding class in CH However, the function Ξ(t, v) is still defined and is smooth for t < 0, hence ˆ v) it is a Green function for the trivial divisor, and we may define again Z(t, ˆ to be the class of (Z(t), Ξ(t, v)) = (0, Ξ(t, v)). To define Z(0, v), we take our lead from the justification of the absolute term in the generating series (1.0.14). Let ω be the Hodge line bundle on M, i.e., the determinant bundle of the dual of the relative Lie algebra of the universal family (A, ι) over M, (1.0.25)

ω = ∧2 (Lie A)∗ .

The complex fiber of this line bundle comes equipped with a natural metric. This metric is well defined up to scaling.2 We denote by ω ˆ the class of this 2

The normalization of the metric we use differs from the standard normalization.

8

CHAPTER 1 1

c d (M) and metrized line bundle under the natural map from Pic(M) to CH set

(1.0.26)

ˆ v) = −ˆ Z(0, ω − (0, log(v) + c),

where c is a suitable constant. The DM-stack Z(t) is finite and unramified over M. It is finite and flat, i.e., a relative divisor, over Spec Z[D(B)−1 ] but may contain irreducible components of the special fiber Mp when p | D(B). This integral extension of the 0-cycles Z(t) is therefore sometimes different from the extension obtained by flat closure in M. Its nonflatness depends in a subtle way on the p-adic valuation of t. Our definition of Z(t) is a consequence of our insistence on a thoroughly modular treatment of our special cycles, which is essential to our method. We strongly suspect that in fact the closure definition does not lead to (variants of) our main theorems and that therefore our definition is the ‘right one’. We do not know this for sure since the closure definition is hard to work with. We form the generating series, (1.0.27)

φˆ1 =

X

d 1 (M)[[q ±1 ]], ˆ v) q t ∈ CH Z(t,

t∈Z

where the coefficients depend on the parameter v ∈ R>0 via the Green function Ξ(t, v). The first main result of this book, proved in Chapter 4, may now be formulated as follows: Theorem A. For τ = u + iv, φˆ1 (τ ) is a (nonholomorphic) modular form d 1 (M). of weight 23 and level Γ0 (4D(B)o ) with values in CH

To explain the meaning of the statement of the theorem, recall that the d 1 (M) of the arithmetic Chow group splits canonically into a R-version CH d 1 (M, µ), the classical direct sum of a finite-dimensional C-vector space CH Arakelov Chow group with respect to the hyperbolic metric, and the vector space C ∞ (M(C))0 of smooth functions on M(C) orthogonal to the constant functions. Correspondingly, the series φˆ1 is the sum of a series φˆ01 in d 1 (M, µ) and a series φ ˆ∞ in q with coefficients in q with coefficients in CH 1 ∞ C (M(C))0 . The assertion of the theorem should be interpreted as follows. There is a smooth function on H with values in the finite-dimensional d 1 (M, µ) which satisfies the usual transformation law for a vector space CH modular form of weight 32 and of level Γ0 (4D(B)o ) whose q-expansion is equal to φˆ01 , and there is a smooth function on H × M(C) which satisfies the usual transformation law for a modular form of weight 23 and of level

9

INTRODUCTION

Γ0 (4D(B)o ) in the first variable and whose q-expansion in the first variable ˆ0 is equal to φˆ∞ 1 . Obviously, the series φ1 satisfies the above condition if for 1 d (M, µ) → C the series `(φ ˆ1 ) with coefficients in any linear form ` : CH 3 C is a nonholomorphic modular form of weight 2 and level Γ0 (4D(B)o ) in the usual sense. Let us explain briefly what is involved in the proof of Theorem A. The d 1 (M, µ) is encapsulated in the following direct sum decomstructure of CH position (1.0.28)

1

d (M, µ) = MW g ⊕ Rω CH ˆ ⊕ Vert.

Here (1.0.29)

g ' MW(MQ ) := Pic0 (MQ )(Q) ⊗ R MW

is the orthogonal complement to (R ω ˆ ⊕ Vert), and the subspace Vert is spanned by the elements (Y, 0), where Y is an irreducible component of a fiber Mp for some p. Also, MW(MQ ) is the Mordell-Weil group of MQ , tensored with R. By the above remark, we have to prove the modularity of `(φˆ01 ) for linear functionals ` on each of the summands of (1.0.28). g this is done by comparing the restriction to the For the summand MW, generic fiber of our generating series φˆ1 with the generating series considered by Borcherds [2], for which he proved modularity. Proposition 1.0.1 is used to produce divisors of degree 0 in the generic fiber from our special divisors. For the summand R ω ˆ , the modularity follows from the following theorem which is the main result of [38]. Note that this theorem not only gives modularity but even identifies the modular form explicitly. We form the generating series with coefficients in C obtained by cupping with ω ˆ, (1.0.30)

hω ˆ , φ1 i =

X

ˆ v) i q t . hω ˆ , Z(t,

Theorem 1.0.3. The series above coincides with the q-expansion of the derivative at s = 12 of the Eisenstein series (1.0.15), t

1 hω ˆ , φˆ1 i = E10 (τ, , B). 2 Next, consider the pairings of the generating series φˆ1 with the classes (Y, 0) ∈ Vert, where Y is an irreducible component of a fiber with bad reduction Mp , i.e., p | D(B). The corresponding series can be identified with classical theta functions for the positive definite ternary lattice associated to the definite quaternion algebra B (p) with D(B (p) ) = D(B)/p. This

10

CHAPTER 1

is based on the theory of p-adic uniformization and uses the analysis of the special cycles at primes of bad reduction [36]. Finally, for the series φˆ∞ 1 , we show that the coefficients of the spectral expansion of φˆ1 are Maass forms. More precisely, if fλ is an eigenfunction of the Laplacian with eigenvalue λ, then the coefficient of fλ in φˆ1 is up to an explicit scalar the classical theta lift θ(fλ ) to a Maass form of weight 32 and level Γ0 (4D(B)o ). To formulate the second main result of this book, Theorem B, we form a generating series for 0-cycles on M instead of divisors on M. The idea is to impose a pair of special endomorphisms, i.e., ‘twice as much CM’. Let Sym2 (Z)∨ denote the set of half-integral symmetric matrices of size 2, and let T ∈ Sym2 (Z)∨ . We define a relative DM-stack Z(T ) over M by posing the following moduli problem. To a scheme S the moduli problem Z(T ) associates the category of triples (A, ι, x) where • (A, ι) is an object of M(S) • x = [x1 , x2 ] ∈ End(A, ι)2 is a pair of endomorphisms with tr(x1 ) = tr(x2 ) = 0, and 21 (x, x) = T . Here (x, x) = ((xi , xj ))i,j . It is then clear that Z(T ) has empty generic fiber when T is positive definite, since in characteristic 0 a fake elliptic curve cannot support linearly independent complex multiplications. However, perhaps somewhat surprisingly, Z(T ) is not always a 0-divisor on M. To explain the situation, recall from [24] that any T ∈ Sym2 (Z)∨ with det(T ) 6= 0 determines a set of primes Diff(T, B) of odd cardinality. More precisely, let C = (Cp ) be the (incoherent) collection of local quadratic spaces where Cp = Vp for p < ∞ and where C∞ is the positive definite quadratic space of dimension 3. If T ∈ Sym2 (Q) is nonsingular, we let VT be the unique ternary quadratic space over Q with discriminant −1 = discr(V ) which represents T . We denote by BT the unique quaternion algebra over Q such that its trace zero subspace is isometric to VT and define (1.0.31)

Diff(T, B) = { p ≤ ∞ | invp (BT ) 6= inv(Cp ) }.

Note that ∞ ∈ Diff(T, B) if and only if T is not positive definite. If |(Diff(T, B))| > 1 or Diff(T, B) = {∞}, then Z(T ) = ∅. Assume now that Diff(T, B) = {p} with p < ∞. If p - D(B), then Z(T ) is a 0-cycle on M with support in the fiber Mp , as desired. In fact, the cycle is concentrated in the supersingular locus of Mp . If, however, p | D(B), then Z(T ) is (almost always) a vertical divisor concentrated in Mp .

11

INTRODUCTION 2

d (M), Our goal now is to form a generating series with coefficients in CH

(1.0.32)

φˆ2 =

X

ˆ Z(T, v)q T .

T ∈Sym2 (Z)∨

d 2 (M) will in general depend on v ∈ ˆ Here the coefficients Z(T, v) ∈ CH Sym2 (R)>0 . How to define them is evident from the above only in the case when T is positive definite and Diff(T, B) = {p} with p - D(B). In this case we set

(1.0.33)

d 2 (M), ˆ Z(T, v) = (Z(T ), 0) ∈ CH

ˆ independent of v. Then Z(T, v) has image log |Z(T )| ∈ R under the arithmetic degree map (1.0.23). If T ∈ Sym2 (Z)∨ is nonsingular with ˆ |Diff(T, B)| > 1, we set Z(T, v) = 0. In the remaining cases, the definition we give of the coefficients of (1.0.32) is more subtle. If Diff(T, B) = {∞}, ˆ then Z(T, v) does depend on v; its definition is purely archimedean and depends on the rotational invariance of the ∗-product of two of the Green functions in [24], one of the main results of that paper. If Diff(T, B) = {p} ˆ with p | D(B), then the definition of Z(T, v) (which is independent of v) relies on the GL2 (Zp )-invariance of the degenerate intersection numbers on the Drinfeld upper half plane, one of the main results of [36]. Finally, for singular matrices T ∈ Sym2 (Z)∨ ≥0 we are, in effect, imposing only a ‘single CM’, and the naive cycle is a divisor, so that its class lies in the wrong degree; we again use the heuristic principle that was used in the definition of ˆ v) in (1.0.26). In the constant term of (1.0.14) and in the definition of Z(0, these cases we are guided in our definitions by the desire to give a construction that is on the one hand as natural as possible, and on the other hand to obtain the modularity of the generating series. We refer to Chapter 6 for the details. Our second main theorem identifies the generating series (1.0.32) with an explicit (nonholomorphic) Siegel modular form of genus two. Recall that such a modular form admits a q-expansion as a Laurent series in (1.0.34)

q T = e(tr(T τ )), T ∈ Sym2 (Z)∨ ,

and that the coefficients may depend on the imaginary part v ∈ Sym2 (R)>0 of τ = u + iv ∈ H2 . We introduce a Siegel Eisenstein series E2 (τ, s, B) which is incoherent in the sense of [24]. In particular, 0 is the center of symmetry for the functional equation, and E2 (τ, 0, B) = 0. The derivative at s = 0 is a nonholomorphic Siegel modular form of weight 23 .

12

CHAPTER 1

Theorem B. The generating function φˆ2 is a Siegel modular form of genus two and weight 23 of level Γ0 (4D(B)o ) ⊂ Sp2 (Z). More precisely, φˆ2 (τ ) = E20 (τ, 0, B),

i.e., the q-expansion of the Siegel modular form on the right-hand side coincides with the generating series φˆ2 . 2

d cf. (1.0.23). d (M) with R via deg, Here we are identifying implicitly CH Theorem B is proved in Chapter 6 by explicitly comparing the coefficients of ˆ the q-expansion of E20 (τ, 0, B) with the coefficients Z(T, v). This amounts to a series of highly nontrivial identities, one for each T in Sym2 (Z)∨ . Let us explain what is involved. First let T be positive definite with Diff(T, B) = {p} for p - D(B). The calculation of the coefficient of E20 (τ, 0, B) corresponding to T comes down to the determination of derivatives of Whittaker functions or of certain representation densities. This determination is based on the explicit formulas for such densities due to Kitaoka [22] for p 6= 2. For p = 2, corresponding results are given in [55]. The determination of the arithmetic degree of Z(T ) boils down to the problem of determining the length of the formal deformation ring of a 1-dimensional formal group of height 2 with two special endomorphisms. This is a special case of the theorem of Gross and Keating [17]. We point out that for both sides the prime number 2 (‘the number theorist’s nightmare’) complicates matters considerably. Next let T be positive definite with Diff(T, B) = {p} for p | D(B). In this case, the corresponding derivatives of representation densities are determined in [54] for p = 6 2 and in [55] for p = 2. The determination of the corresponding coefficient of φˆ2 depends on the calculation of the intersection product of special cycles on the Drinfeld upper half space. This is done in [36] for p 6= 2. These calculations are completed here for p = 2. Now let T be nonsingular with Diff(T, B) = ∞. Then the calculation of the corresponding coefficients of E20 (τ, 0, B) and of φˆ2 is given in [24] in the case where the signature of T is (1, 1). The remaining case, where the signature is (0, 2), is given here, using the method of [24]. Next, we consider the coefficients corresponding to singular matrices T of rank 1. For such a matrix

(1.0.35)



T =



t1 m ∈ Sym2 (Z)∨ , m t2

with det(T ) = 0 and T 6= 0, we may write t1 = n21 t, t2 = n22 t, and m = n1 n2 t for the relatively prime integers n1 and n2 and t ∈ Z6=0 . The pair n1 , n2 is unique up to simultaneous change in sign, and t is uniquely

13

INTRODUCTION

determined. Also, note that, if t1 = 0, then n1 = 0, n2 = 1, and t = t2 , while if t2 = 0, then n1 = 1, n2 = 0, and t = t1 . Then the comparison between the corresponding singular coefficients of φˆ2 and E20 (τ, 0, B) in this case is based on the following result, proved in Chapter 5. It relates the singular Fourier coefficients of the derivative of the genus two Eisenstein series occurring in Theorem B with the Fourier coefficients of the genus one Eisenstein series occurring in Theorem A. Theorem 1.0.4. (i) Let T ∈ Sym2 (Z)∨ , with associated t ∈ Z= 6 0 as above. Then 1 0 0 E2,T (τ, 0, B) = −E1,t (t−1 tr(T τ ), , B) 2   1 1 det v −1 − · E1,t (t tr(T τ ), , B) · log( −1 ) + log(D(B)) . 2 2 t tr(T v)

(ii) For the constant term

1 1 1 0 0 E2,0 (τ, 0, B) = −E1,0 (i det v, , B) − E1,0 (i det(v), , B) · log D(B). 2 2 2 It is this theorem that motivated our definition of the singular coefficients of the generating series φˆ2 . Just as for Proposition 1.0.1, we see that the modularity of the generating function φˆ2 is not proved directly but rather by identifying it with an explicit modular form. The coherence in our definitions of the generating series φˆ1 and φˆ2 is displayed by the following arithmetic inner product formula, which relates the inner product of the generating series φˆ1 with itself under the GilletSoul´e pairing with the generating series φˆ2 . Let (1.0.36)



H × H −→ H2

τ 0 (τ1 , τ2 ) 7−→ diag(τ1 , τ2 ) = 1 0 τ2

be the natural embedding into the Siegel space of genus two. Theorem C. For τ1 , τ2 ∈ H

hφˆ1 (τ1 ), φˆ1 (τ2 )i = φˆ2 (diag(τ1 , τ2 )).

Explicitly, for any t1 , t2 ∈ Z and v1 , v2 ∈ R>0 , ˆ 1 , v1 ), Z(t ˆ 2 , v2 )i = hZ(t

X (Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 )

ˆ Z(T, diag(v1 , v2 )).



14

CHAPTER 1

Theorem C, which is proved in Chapter 7, is the third main result of this book and provides the arithmetic analogue of Theorem 6.2 in [23], which relates to the cup product of two generating series with values in cohomology. Let us explain what is involved here, first assuming that t1 t2 6= 0. The proof distinguishes two cases. In the first case t1 t2 6∈ Q×,2 . In this case all matrices T occurring in the sum on the right-hand side are automatically nonsingular; at the same time the divisors Z(t1 ) and Z(t2 ) have empty intersection in the generic fiber, so that the Gillet-Soul´e pairing decomposes into a sum of local pairings, one for each prime of Q. Consider the case when ti > 0 for i = 1, 2. Then the key to the formula above is the decomposition of the intersection (fiber product) of the special cycles Z(ti ) according to ‘fundamental matrices’, (1.0.37)

Z(t1 ) ×M Z(t2 ) =

a

Z(T ).

T diag(T )=(t1 ,t2 )

Here Z(T ) appears as the locus of objects ((A, ι), x1 , x2 ) in the fiber product where x = [x1 , x2 ] satisfies 21 (x, x) = T . Note that, by the remarks preceding the statement of Theorem B, the intersection of the Z(ti ) need not be proper since these divisors can have common components in the fibers of bad reduction Mp for p | D(B). Of course, all matrices T occurring in the disjoint sum in (1.0.37) are positive definite. The occurrence in the sum of Theorem C of summands corresponding to matrices T which are not posiˆ v). Similar tive definite is due to the Green functions component of the Z(t, archimedean contributions occur in the cases where one of the ti is negative. In the second case t1 t2 ∈ Q×,2 . In this case, Z(t1 ) and Z(t2 ) intersect in the generic fiber. In addition to the contribution of the nonsingular T to the sum in Theorem C, there is also a contribution√of the two singular matrices T , where T is given by (1.0.35) with m = ± t1 t2 . In this case the GilletSoul´e pairing does not localize. Instead we use the arithmetic adjunction formula from Arakelov theory [10], [40]. To calculate the various terms in this formula we must, among other things, go back to the proof of the Gross-Keating formula and use the fine structure of the deformation locus of a special endomorphism of a p-divisible group of dimension 1 and height 2. We stress that the proof of Theorem C sketched so far has nothing to do with Eisenstein series. However, the modularity of both sides of the identity in Theorem C allows us to deduce from the truth of the statement for all t1 t2 6= 0 first the value of the constant c in (1.0.26) and then the truth of the statement for all (t1 , t2 ). In this way we can also prove our conjecture [38] on the self-intersection of the Hodge line bundle.

15

INTRODUCTION

Theorem 1.0.5. Let ω ˆ 0 be the Hodge line bundle on M metrized with the normalization of Bost [3]. Then 



ζ 0 (−1) 1 1 X p + 1 hˆ ω0 , ω ˆ 0 i = 2 · ζD(B) (−1)  + − log p . ζ(−1) 2 4 p|D(B) p − 1 Formally, this result specializes for D(B) = 1 to the formula of Bost [4] and K¨uhn [39] in the case of the modular curve (note that due to the stacks aspect our quantity is half of theirs). In their case they use the section ∆ of ω ⊗6 to compute the self-intersection of ω ˆ 0 explicitly from its definition. For Shimura curves there is no such natural modular form and our result comes about only indirectly. We note that the general form of this formula is related to formulas given by Maillot and Roessler [42]. The above three theorems are the main results in this book. As an application of these results, we introduce an arithmetic version of the ShimuraWaldspurger correspondence and obtain analogues of results of Waldspurger [53] and of Gross-Kohnen-Zagier [18]. If f is a cusp form of weight 23 for Γ0 (4D(B)o ), we can define the arithmetic theta lift of f by b ) := C · (1.0.38) θ(f

Z

1 d 1 (MB ), f (τ ) φb1 (τ ) v − 2 du dv ∈ CH

Γ0 (4D(B)o )\H

for a constant C given in section 3 of Chapter 9. Of course, this is the analogue of the classical theta lift from modular forms of weight 23 to modular forms of weight 2, but with φb1 (τ ) replacing the classical theta kernel of Niwa [43] and Shintani [47]. By the results discussed above, it follows that (1.0.39) (1.0.40)

b ), 11 i = h f, E1 (τ, 1 ; B) iPet = 0, h θ(f 2 1 b ), ω h θ(f ˆ i = h f, E10 (τ, ; B) iPet = 0, 2

and b ), a(φ) i = h f, θ(φ) iPet = 0, (1.0.41) h θ(f

for all φ ∈ C ∞ (M(C))0 ,

since f is a holomorphic cusp form. Here, for φ ∈ C ∞ (M(C))0 , we denote d 1 (M) and by θ(φ) the corresponding by a(φ) the corresponding class in CH b ) lies in the space of MW g ⊕ Vert0 , Maass cusp form of weight 23 . Thus θ(f where Vert0 is the subspace of Vert orthogonal to ω ˆ. b ), we conIn order to obtain information about the nonvanishing of θ(f

16

CHAPTER 1

b ) i. Using Theorems B and C, we obtain sider the height pairing h φb1 (τ1 ), θ(f b ) i = h f, h φ b1 (τ1 ), φ b1 i i h φb1 (τ1 ), θ(f

(1.0.42)

= h f, φb2 (diag(τ1 , ·)) i = h f, E20 (diag(τ1 , ·), 0; B) i ∂ h f, E2 (diag(τ1 , ·), s; B) i . ∂s s=0 

=



We then consider the integral h f, E2 (diag(τ1 , ·), s; B) i occurring in the last expression. This integral is essentially the doubling integral of PiatetskiShapiro and Rallis [45] (see also [41]), except that we only integrate against one cusp form.

Theorem 1.0.6. Let F be a normalized newform of weight 2 on Γ0 (D(B)) and let f be the good newvector, in the sense defined in section 3 of Chapter 8, corresponding to F under the Shimura-Waldspurger correspondence. Then where

h f, E2 (diag(τ1 , ·), s; B) i = C(s) · L(s + 1, F ) · f (τ1 ),

C(s) = with

3 2π 2

Y

(p + 1)−1 ·



p|D(B)

Cp (s) = (1 − p (F ) p−s ) −

D(B) 2π

s

Γ(s + 1) ·

Y

Cp (s),

p|D(B)

p−1 (1 + p (F ) p−s ) Bp (s). p+1

Here L(s, F ) is the standard Hecke L-function of F , p (F ) is the AtkinLehner sign of F , F |Wp = p (F ) F,

and Bp (s) is a rational function of p−s with Bp (0) = 0

and

Bp0 (0) =

1 p+1 · log(p). 2 p−1

Note that Cp (0) = 2 if p (F ) = −1 and Cp (0) = Cp0 (0) = 0 if p (F ) = 1. As a consequence, we have the following analogue of Rallis’s inner product formula [46], which characterizes the nonvanishing of the arithmetic theta lift.

INTRODUCTION

Corollary 1.0.7. For F with associated f as in Theorem 1.0.6, In particular, and hence b ) 6= 0 θ(f

17

b ) i = C(0) · L0 (1, F ) · f (τ1 ). h φb1 (τ1 ), θ(f

b ), θ(f b ) i = C(0) · L0 (1, F ) · h f, f i, h θ(f

⇐⇒

  p (F ) = −1  L0 (1, F ) 6= 0.

for all p | D(B), and

Let S2new (D(B))(−) be the space of normalized newforms of weight 2 for Γ0 (D(B)) for which all Atkin-Lehner signs are −1. Note that, for F ∈ S2new (D(B))(−) , the root number of L(s, F ) is given by (1, F ) = −

Y

p (F ) = −1.

p|D(B)

Since the vertical part of φb1 (τ ) is a linear combination of theta functions for the anisotropic ternary spaces V (p) , for p | D(B), and since the classical theta lift of a form F with (1, F ) = −1 to such a space vanishes by Waldb ) ∈ MW. g Recall from (1.0.29) spurger’s result [50], [53], it follows that θ(f that this space is isomorphic to MW(MQ ) via the restriction map resQ . Corollary 1.0.8. For each F ∈ S2new (D(B))(−) , let f be the corresponding good newvector of weight 32 . Then
1 ωQ resQ φbB + 1 (τ ) = E1 (τ, ; B) · 2 deg ωQ

X F ∈S2new (D(B))(−)

b ) f (τ ) · resQ θ(f , h f, f i

L0 (1,F )6=0

where ωQ is the restriction of the Hodge bundle to MQ . Next, for each t ∈ Z>0 , write Z(t)(F ) for the component3 of the cycle Z(t) = Z(t)Q in the F -isotypic part CH1 (MQ )(F ) of the Chow group CH1 (MQ ). Note that Z(t)(F ) has zero image in H 2 (MC ) and hence defines a class in MW(MQ ). 3

Here we transfer F to a system of Hecke eigenvalues for the quaternion algebra B via the Jacquet-Langlands correspondence.

18

CHAPTER 1

Theorem 1.0.9. The F -isotypic component of the generating function
resQ φbB 1 (τ ) =

is

Z(t) q t ,

t≥0


resQ φbB 1 (τ ) (F ) =

In particular, where

X

X

Z(t)(F ) q t =

t≥0

Z(t)(F ) =

b ) f (τ ) · resQ θ(f . h f, f i

b ) at (f ) · resQ θ(f , h f, f i

f (τ ) =

X

at (f ) q t

is the Fourier expansion of f . Moreover, for t1 and t2 ∈ Z>0 , the height pairing of the F -components of Z(t1 ) and Z(t2 ) is given by t>0

h Z(t1 )(F ), Z(t2 )(F ) i = C(0) · L0 (1, F ) ·

at1 (f ) · at2 (f ) . h f, f i

This result is the analogue in our case of the result of Gross-KohnenZagier [18], Theorem C, p.503. The restriction to newforms in S2new (D(B)) with all Atkin-Lehner signs equal to −1 is due to the fact that our cycles are invariant under all Atkin-Lehner involutions. To remove this restriction, one should use ‘weighted’ cycles, see section 4 of Chapter 3. In fact, we construct an arithmetic theta lift of automorphic representations σ in the space A00 (G0 ) on the metaplectic extension G0A of SL2 (A). This theta lift, which is only defined for representations corresponding to holomorphic cusp forms of weight 23 , is the analogue of the classical theta lift considered by Waldspurger [50], [51], [53]. We formulate a conjectural analogue of Waldspurger’s nonvanishing criterion and prove it in certain cases as an application of Theorem 1.0.6 and Corollary 1.0.7. For forms F with (1, F ) = +1, Waldspurger proved that the classical theta lift is nonzero if and only if (i) certain local conditions (theta dichotomy) are satisfied at every place, and (ii) L(1, F ) 6= 0. In the arithmetic case, we show that for (certain) forms F of weight 2 with (1, F ) = −1, the arithmetic theta lift is nonzero if and only if (i) the local theta dichotomy conditions are satisfied, and (ii) L0 (1, F ) 6= 0. A more detailed discussion can be found in section 1 of Chapter 9 as well as in [29]. Our construction is similar in spirit

INTRODUCTION

19

to that of [16], where Gross formulates an arithmetic analogue of another result of Waldspurger [52] and shows that, in certain cases, this analogue can be proved using the results of Gross-Zagier [19] and their extension by Zhang [60]. We now mention some previous work on such geometric and arithmeticgeometric generating functions. The classic work of Hirzebruch-Zagier mentioned above inspired much work on modular generating functions valued in cohomology. Kudla and Millson considered modular generating functions for totally geodesic cycles in Riemannian locally symmetric spaces for the classical groups O(p, q), U(p, q), and Sp(p, q) [31], [32], [33]. Such cycles were also considered by Oda [44] and Tong-Wang [49]. In the case of symmetric spaces for O(n, 2), the generating function of Kudla-Millson [33] and Kudla [23] for the cohomology classes of algebraic cycles of codimension r is a Siegel modular form of weight n2 + 1 and genus r. In the case r = n, i.e., for 0-cycles, the generating function was identified in [23] as a special value of an Eisenstein series via the Siegel-Weil formula. A similar relation to Eisenstein series occurs in the work of Gross and Keating [17] for the generating series associated to the graphs of modular correspondences in a product of two modular curves. Borcherds [2] used Borcherds products to construct modular generating series with coefficients in CH1 for divisors on locally symmetric varieties associated to O(n, 2) and proved that they are holomorphic modular forms. We also mention recent related work of Bruinier [5], [6], Bruinier-Funke [8], Funke [11], and Funke-Millson [12], [13]. The results in the arithmetic context are all inspired by the theorem of Gross and Zagier [19]. Part of a generating series for triple arithmetic intersections of curves on the product of two modular curves was implicitly considered in the paper by Gross and Keating [17], where the ‘good nonsingular’ coefficients are determined explicitly, cf. also [1]. For Shimura curves, Kudla [24] considered the generating series obtained from the GilletSoul´e height pairing of special divisors. It was proved that this generating series coincided for ‘good’ nonsingular coefficients with the diagonal pullback of the central derivative of a Siegel Eisenstein series of genus two. The ‘bad’ nonsingular coefficients were determined in [36]. However, the singular coefficients were left out of this comparison. In [37] we considered the 0-dimensional case, where the ambient space is the moduli space of elliptic curves with complex multiplication. In this case we were able to determine the generating series completely and to identify it with the derivative of a special value of an Eisenstein series. Another generating series is obtained in [38] by pairing special divisors on arithmetic models of Shimura curves, equipped with Green functions, with the metrized dualizing line bundle. Again this can be determined completely and identified with a

20

CHAPTER 1

special value of a derivative of an Eisenstein series. A generating series in a higher-dimensional case is constructed by Bruinier, Burgos, and K¨uhn [7]. They consider special divisors on arithmetic models of Hilbert-Blumenthal surfaces whose generic fibers are Hirzebruch-Zagier curves, equip them with (generalized) Green functions [9], and obtain a generating series by taking the pairing with the square of the metrized dualizing line bundle. They identify this series with a special value of an Eisenstein series. Finally we mention partial results in higher-dimensional cases (Hilbert-Blumenthal surfaces, Siegel threefolds) in [34], [35]. This monograph is not self-contained. Rather, we make essential use of our previous papers. We especially need the results in [24] about the particular Green functions we use, as well as the results on Eisenstein series developed there. We also use the results on representation densities from [54], [55]. Furthermore, for the analysis of the situation at the fibers of bad reduction we use the results contained in [36]. These are completed in [38], which is also essential for our arguments in other ways. Finally, we need some facts from [27] in order to apply the results of Borcherds. These papers are not reproduced here. Still, we have given here all the definitions necessary for following our development and have made an effort to direct the reader to the precise reference where he can find the proof of the statement in question. We also have filled in some details in the proof of other results in the literature. Most notable here are our exposition in section 6 of Chapter 3 of the special case of the theorem of Gross and Keating [17] that we use, and the exposition in Chapter 8 of the doubling method of Piatetski-Shapiro and Rallis [45] in the special case relevant to us. In the first instance, we were aided by a project with a similar objective, namely to give an exposition of the general result of Gross and Keating, undertaken by the ARGOS seminar in Bonn [1]. In the second instance, we use precise results about nonarchimedean local Howe duality for the dual pair (SL2 , O(3)) from [30]. We have structured this monograph in the following manner. In Chapter 2 we provide the necessary background from Arakelov geometry. The key point here is to show that the theory of Gillet-Soul´e [14], [3] continues to hold for the DM-stacks of the kind we encounter. We also give a version of the arithmetic adjunction formula. It turns out that among the various versions of it the most naive form, as presented in Lang’s book [40], is just what we need for our application of it in Chapter 7. In Chapter 3 we define the special cycles on Shimura curves and review the known facts about them. Here we also give a proof of the special case of the GrossKeating formula which we need. In Chapter 4 we prove Theorem A, along the lines sketched above. In Chapter 5 we introduce the Eisenstein series of genus one and two which are relevant to us and calculate their Fourier

BIBLIOGRAPHY

21

expansion. In particular, we prove Theorem 1.0.4. In Chapter 6 we define the generating series φˆ2 and prove Theorem B by comparing term by term this series with the Fourier coefficients of the Siegel Eisenstein series of genus two determined in the previous chapter. For the ‘bad nonsingular’ coefficients of φˆ2 , the calculation in the case p = 2 had been left out in [36]. In the appendix to Chapter 6 we complete the calculations for p = 2. Chapter 7 is devoted to the proof of the inner product formula, Theorem C. In Chapter 8 we give an exposition of the doubling method in our case. The point is to determine explicitly all local zeta integrals for the kind of good test functions that we use. The case p = 2 again requires additional efforts. In Chapter 9 we give applications of our results to the arithmetic theta lift and to L-functions and prove Theorems 1.0.6 and 1.0.9 and Corollaries 1.0.7 and 1.0.8 above. This book is the result of a collaboration over many years. The general idea of forming the arithmetic generating series and relating them to modular forms arising from derivatives of Eisenstein series is due to the first author. The other two authors joined the project, each one contributing a different expertise to the undertaking. In the end, we can honestly say that no proper subset of this set of authors would have been able to bring this project to fruition. While the book is thus the product of a joint enterprise, some chapters have a set of principal authors which are as follows: Chapter 2: SK, MR Chapter 4: SK Chapter 5: SK, TY Appendix to Chapter 6: SK, MR Chapter 7: SK, MR Chapter 8: SK, TY The material of this book, as well as its background, has been the subject of several survey papers by us individually: [25], [26], [28], [29], [56], [57]. It should be pointed out, however, that in the intervening time we made progress and that quite a number of question marks which still decorate the announcements of our results in these papers have been removed.

Bibliography

[1] ARGOS (Arithmetische Geometrie Oberseminar), Proceedings of the Bonn seminar 2003/04, forthcoming. [2] R. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J., 97 (1999), 219–233.

22

CHAPTER 1

[3] J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic ´ surfaces, Ann. Sci. Ecole Norm. Sup., 32 (1999), 241–312. [4]

, Lecture, Univ. of Maryland, Nov. 11, 1998.

[5] J. H. Bruinier, Borcherds products and Chern classes of Hirzebruch– Zagier divisors, Invent. Math., 138 (1999), 51–83. [6]

, Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math., 1780, Springer-Verlag, New York, 2002.

[7] J. H. Bruinier, J. I. Burgos Gil, and U. K¨uhn, Borcherds products in the arithmetic intersection theory of Hilbert modular surfaces, preprint, 2003. [8] J. H. Bruinier and J. Funke, On two geometric theta lifts, preprint, 2003. [9] J. I. Burgos Gil, J. Kramer, and U. K¨uhn, Cohomological arithmetic Chow rings, preprint, 2003. [10] G. Faltings, Calculus on arithmetic surfaces, Annals of Math., 119 (1984), 387–424. [11] J. Funke, Heegner divisors and nonholomorphic modular forms, Compositio Math., 133 (2002), 289–321. [12] J. Funke and J. Millson, Cycles in hyperbolic manifolds of noncompact type and Fourier coefficients of Siegel modular forms, Manuscripta Math., 107 (2003), 409–444. [13]

, Cycles with local coefficients for orthogonal groups and vector valued Siegel modular forms, preprint, 2004.

[14] H. Gillet and C. Soul´e, Arithmetic intersection theory, Publ. Math. IHES, 72 (1990), 93–174. [15]

, Arithmetic analogues of standard conjectures, in Proc. Symp. Pure Math., 55, Part 1, 129–140, AMS, Providence, R.I., 1994.

[16] B. H. Gross, Heegner points and representation theory, in Heegner points and Rankin L-Series, Math. Sci. Res. Inst. Publ., 49, Cambridge Univ. Press, Cambridge, 2004.

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[17] B. Gross and K. Keating, On the intersection of modular correspondences, Invent. math., 112 (1993), 225–245. [18] B. H. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of L-functions. II, Math. Annalen, 278 (1987), 497–562. [19] B. H. Gross and D. Zagier, Heegner points and the derivatives of Lseries, Invent. math., 84 (1986), 225–320. [20] F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. math., 36 (1976), 57–113. [21] Y. Kitaoka, A note on local densities of quadratic forms, Nagoya Math. J., 92 (1983), 145–152. [22]

, Arithmetic of Quadratic Forms, Cambridge Tracts in Mathematics, 106, Cambridge Univ. Press, 1993.

[23] S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J., 86 (1997), 39–78. [24]

, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646.

[25]

, Derivatives of Eisenstein series and generating functions for arithmetic cycles, S´eminaire Bourbaki 876, Ast´erisque, 276, 341– 368, Soc. Math. France, Paris, 2002.

[26]

, Derivatives of Eisenstein series and arithmetic geometry, in Proc. Intl. Cong. Mathematicians, Vol II (Beijing, 2002), 173–183, Higher Education Press, Beijing, 2002.

[27]

, Integrals of Borcherds forms, Compositio Math., 137 (2003), 293–349.

[28]

, Special cycles and derivatives of Eisenstein series, in Heegner points and Rankin L-Series, Math. Sci. Res. Inst. Publ., 49, 243– 270, Cambridge Univ. Press, Cambridge, 2004.

[29]

, Modular forms and arithmetic geometry, in Current Developments in Mathematics, 2002, 135–179, International Press, Somerville, MA, 2003.

[30]

f 2 , O(3)), , Notes on the local theta correspondence for (SL preprint, 2005.

24

CHAPTER 1

[31] S. Kudla and J. Millson, The theta correspondence and harmonic forms I, Math. Annalen, 274 (1986), 353–378. [32]

, The theta correspondence and harmonic forms II, Math. Annalen, 277 (1987), 267–314.

[33]

, Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Publ. Math. IHES, 71 (1990), 121–172.

[34] S. Kudla and M. Rapoport, Arithmetic Hirzebruch–Zagier cycles, J. reine angew. Math., 515 (1999), 155–244. [35]

, Cycles on Siegel 3-folds and derivatives of Eisenstein se´ ries, Ann. Sci. Ecole. Norm. Sup., 33 (2000), 695–756

[36]

, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223.

[37] S. Kudla, M. Rapoport, and T. Yang, On the derivative of an Eisenstein series of weight 1, Int. Math. Res. Notices (1999), 347–385. [38]

, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887–951.

[39] U. K¨uhn, Generalized arithmetic intersection numbers, J. reine angew. Math., 534 (2001), 209–236. [40] S. Lang, Introduction to Arakelov Theory, Springer-Verlag, New York, 1988. [41] J.-S. Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. reine angew. Math., 428 (1992), 177–217. [42] V. Maillot and D. Roessler, Conjectures sur les d´eriv´ees logarithmiques des fonctions L d’Artin aux entiers n´egatifs, Math. Res. Lett., 9 (2002), no. 5-6, 715–724. [43] S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J., 56 (1975), 147–161. [44] T. Oda, On modular forms associated to quadratic forms of signature (2, n − 2), Math. Annalen., 231 (1977), 97–144. [45] I. I. Piatetski-Shapiro and S. Rallis, L-functions for classical groups, Lecture Notes in Math., 1254, 1–52, Springer-Verlag, New York, 1987.

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[46] S. Rallis, Injectivity properties of liftings associated to Weil representations, Compositio Math., 52 (1984), 139–169. [47] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J., 58 (1975), 83–126. [48] C. Soul´e, D. Abramovich, J.-F. Burnol, and J. Kramer, Lectures on Arakelov Theory, Cambridge Studies in Advanced Math. 33, Cambridge Univ. Press, Cambridge, 1992. [49] Y. Tong and S. P. Wang, Construction of cohomology of discrete groups, Trans. AMS, 306 (1988), 735–763. [50] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl., 59 (1980), 1–132. [51]

, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl., 60 (1981), 375–484.

[52]

, Sur les valeurs de certaines fonctions L automorphes en leur centre de sym´etrie, Compositio Math., 54 (1985), 173–242.

[53]

, Correspondances de Shimura et quaternions, Forum Math., 3 (1991), 219–307.

[54] T. H. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory, 72 (1998), 309–356. [55]

, Local densities of 2-adic quadratic forms, J. Number Theory, 108 (2004), 287–345.

[56]

, The second term of an Eisenstein series, Proc. Intl. Cong. Chinese Mathematicians, forthcoming.

[57]

, Faltings heights and the derivative of Zagier’s Eisenstein series, in Heegner points and Rankin L-Series, Math. Sci. Res. Inst. Publ., 49, 271–284, Cambridge Univ. Press, Cambridge, 2004.

[58] D. Zagier, Nombres de classes et formes modulaires de poids 3/2, C. R. Acad. Sci. Paris, 281 (1975), 883–886. [59]

, Modular points, modular curves, modular surfaces and modular forms, in Lecture Notes in Math. 1111, 225–248, SpringerVerlag, Berlin, 1985.

[60] Shou-Wu Zhang, Gross–Zagier formula for GL2 , Asian J. Math., 5 (2001), 183–290.

Chapter Two Arithmetic intersection theory on stacks The aim of the present chapter is to outline the (arithmetic) intersection theory on Deligne-Mumford (DM) stacks that will be relevant to us. The stacks M we consider will satisfy the following conditions: • M is regular of dimension 2 and is proper and flat over S = Spec Z, and is a relative complete intersection over Spec Z. Also we assume M to be connected (and later even geometrically connected). • Let M = MC = M ×Spec Z Spec C be the complex fiber of M. Then M is given by an orbifold presentation, M = [Γ \ X], where X is a compact Riemann surface (not necessarily connected) and Γ is a finite group acting on X. 2.1 THE ONE-DIMENSIONAL CASE

As a preparation for later developments we start with the one-dimensional case. First we consider a DM-stack Z which is reduced and proper of relative dimension 1 over an algebraically closed field k. Let L be an invertible sheaf on Z. Before defining the degree of L we recall [10] that if R is an integral domain of dimension 1, with fraction field K, we put for f = ab ∈ K × with a, b ∈ R, (2.1.1)

ordR (f ) = lg(R/a) − lg(R/b).

This is extended in the obvious way to define ordL (s) for an element s ∈ L ⊗R K of a free R-module L of rank one. Now let s be a rational section of L. If x is a closed geometric point of Z ˜Z,x is the strictly local henselian ring of Z in x, we get a direct sum and O decomposition into integral domains according to the formal branches of Z through x, (2.1.2)

˜Z,x = O

M i

Oi .

28

CHAPTER 2

We put (2.1.3)

degx s = Σi ordOi (si ),

where si is the image of s in L ⊗OZ,x Oi . As in [3], VI, 4.3, we put (2.1.4)

deg(L) = deg(Z, L) =

1 · degx (s). |Aut(x)| x∈Z(k) X

If k is not algebraically closed, one defines the degree after extension of ¯ This definition is independent of the choice of s and coincides scalars to k. with the usual definition when Z is a scheme. It satisfies (i) additivity in L: deg(L ⊗ L0 ) = deg(L) + deg(L0 ) (ii) coverings: If f : Z 0 → Z is a finite flat morphism of constant degree, then (2.1.5)

deg(f ∗ L) = deg(f ) · deg(L).

In particular, let π : Z˜ → Z be the normalization of Z. This is the relatively representable morphism such that for any e´ tale presentation X → Z, the resulting morphism X ×Z Z˜ → X is the normalization of X. Then (2.1.6)

˜ π ∗ (L)). deg(Z, L) = deg(Z,

The calculation of the RHS is somewhat easier since if Z = Z˜ is normal, for a rational section s of L we have (2.1.7)

degx (s) = ordx (s).

˜Z,x is a discrete valuation ring and ordx (s) is the (If Z is normal, then O valuation of s.) In Arakelov theory it is more convenient to use the Arakelov degree which is defined as (2.1.8)

d deg(Z, L) = deg(Z, L) · log p,

when Z is of finite type over Fp . Here Z is considered as a stack over Fp . If Γ(Z, O) = Fq and Z is considered as a stack over Fq , then the RHS equals deg(Z, L) · log q. Next we consider the case where Z is a reduced irreducible DM-stack of dimension 1 which is proper and flat over Spec Z. In this case we want to consider metrized line bundles. There are two ways to define the concept of a metrized line bundle on Z. First, one can define a metrized line bundle to be a rule which associates, functorially, to any S-valued point S → Z a

29

ARITHMETIC INTERSECTION THEORY ON STACKS

line bundle LS on S equipped with a C ∞ -metric on the line bundle LS,C on S ×Spec Z Spec C. Second, one can define a metrized line bundle on Z to be a metrized line bundle on an e´ tale presentation X → Z, equipped with a descent datum which respects the metric. c We denote by Pic(Z) the set of isomorphism classes of metrized line bundles on Z. This is an abelian group under the tensor product operation. Let Γ(Z, OZ ) be the ring of regular functions on Z. This may be identified with the ring of regular functions on the coarse moduli space of Z. Then Γ(Z, OZ ) is an order O in a number field K with Γ(Z ⊗Z Q, O) = K. If ˜ O ˜ ) is the ring of integers ν : Z˜ → Z is the normalization of Z, then Γ(Z, Z ˆ OK . We now put for a rational section s of the metrized line bundle L, d ˆ = (2.1.9) deg(Z, L)

X p

1 degx (s) log p − |Aut(x)| 2 

X ¯p ) x∈Z(F

Z

log ksk2 .

Z(C)

Here the integral is defined as Z

(2.1.10)

log ksk2 =

Z(C)

1 · log ks(x)k2 . |Aut(x)| x∈Z(C) X

Let us check that (2.1.9) is independent of the choice of s. This comes down to checking for a function f ∈ K × that (2.1.11) 0=

X p

1 degx (f ) 1 X log p − log |σ(f )|2 . |Aut(x)| 2 σ:K→C |Aut(σ)| 

X ¯p ) x∈Z(F

¯ p ), let [x] be the corresponding geometric point of the coarse For x ∈ Z(F moduli scheme Z = Spec O of Z. Then ˜Z,[x] = (O ˜Z,x )Aut(x)/Aut(¯η) , O

(2.1.12)

where η¯ is any generic geometric point of Z. It follows that (2.1.13)

degx f = |Aut(x)|/|Aut(¯ η )| · deg[x] f.

Inserting this into (2.1.11) we obtain for the right-hand side the expression 1 (2.1.14) · |Aut(¯ η )|

X

X

degx (f ) · log p −

p x∈(Spec O)(F ¯p )

X

2



log |σ(f )|

.

σ

Using the normalization ν : Z˜ → Z we may rewrite this as (2.1.15)

X 1 · |Aut(¯ η )| p

X ¯p ) x ˜∈(Spec OK )(F

ordx˜ (f ) · log p −

X σ



log |σ(f )|2 ,

30

CHAPTER 2

which is zero by the product formula for f ∈ K × . d L) ˆ is again additive in Lˆ and compatible with passThe definition of deg( ing to a finite covering; see (2.1.5). An important line bundle is the relative dualizing sheaf ωZ/S . It is characterized by the fact that its pullback to any e´ tale presentation X → Z is the relative dualizing sheaf ωX/Spec Z . Recall that by Grothendieck duality we have (we identify the sheaves with the Z-modules they define), (2.1.16)

ωX/Z = HomOX (OX , ωX/Z ) = HomZ (OX , Z),

the inverse different of the order Γ(X, OX ) in Γ(X ⊗Z Q, O). In particular we obtain a natural homomorphism (2.1.17)

Γ(X, ωX/Z ) ,→ Γ(X ⊗Z Q, O).

It follows that ωX/Z is equipped with a natural metric k k. For this metric we have for any complex embedding (2.1.18)

σ : Γ(X ⊗Z Q, O) ,→ C

that kσ(1)k = 1. By naturality, this metric descends to a metric on ωZ/S . We define (2.1.19)

d (ω dZ = deg Z/S , k k).

It therefore follows that (2.1.20)

dZ = log |ωZ/S : O| = log |D−1 : O|,

where D−1 = O∗ is the dual module with respect to the trace form of O (inverse of the absolute different of the order O). Indeed the trace map trO/Z defines an element t ∈ O∗ = Γ(Z, ωZ/S ) which goes to 1 ∈ C under every σ. Using the global section t to calculate (2.1.9) for ω ˆ Z/S , we see that the first summand gives log |D−1 : O| while the second summand vanishes. 2.2 Pic(M), CH1Z (M), AND CH2Z (M)

In this section we take up the study of our two-dimensional stack M. Let Pic(M) be the set of isomorphism classes of line bundles on M, an abelian group under the tensor product operation. This is related to the Chow group CH1Z (M) as follows. By a prime divisor on M we mean a closed irreducible reduced substack Z of M which is locally for the e´ tale topology a Cartier divisor. Let ZZ1 (M) be the free abelian group generated by the prime divisors on M. Let f ∈

ARITHMETIC INTERSECTION THEORY ON STACKS

31

Q(M)× be a rational function. In other words, f is the germ of a morphism U → A1 defined on a dense open substack U of M. Equivalently, since M is irreducible, f is an element of the function field of the coarse moduli scheme of M. Then to f there is associated a principal divisor (2.2.1)

div(f ) =

X

ordZ (f ) · Z,

Z

where the sum is over all prime divisors Z of M and where we note that, ˜M,Z , is since M is regular, the strict henselization of the local ring of Z, O a discrete valuation ring so that ordZ (f ) has a meaning. The factor group of ZZ1 (M) by the group of principal divisors is the first Chow group CH1Z (M). The groups Pic(M) and CH1Z (M) are isomorphic. Under this isomorphism, an element L goes to the class of ΣZ ordZ (s) · Z, where s is a meromorphic section of L. Conversely, if Z ∈ Z1Z (M), then its preimage under this isomorphism is O(Z). We denote by Z 1 (M) = ZZ1 (M) ⊗ R the space of real divisors (i.e., the formal sums of prime divisors with coefficients in R), and by CH1 (M) the factor group by the R-subspace generated by the principal divisors. We will also have use for the second Chow group CH2Z (M). By a 0-cycle on M we mean a formal sum ΣP nP P where P ranges over the irreducible reduced closed substacks of M of dimension 0. We denote by ZZ2 (M) the abelian group of 0-cycles on M. We have the homomorphism of abelian groups, L

(2.2.2)

Z

Q(Z)× −→ fZ

7−→

ZZ2 (M) P P⊂Z

degP (f ) · P.

(Here, as in (2.1.3), the degree function degP is the sum of the corresponding degree functions over all formal branches of Z at P.) Then CH2Z (M) is the factor group of ZZ2 (M) by the image of (2.2.2). We also denote by CH2 (M) the R-version of CH2Z (M), i.e. the factor space of Z 2 (M) = ZZ2 (M) ⊗ R by the R-subspace generated by the image of (2.2.2). 2.3 GREEN FUNCTIONS

In this section, we review the theory of Green functions needed to define the arithmetic Chow groups in the next section. Due to the assumptions we made at the beginning of this chapter about the stack M, we can restrict ourselves to the following situation. Let X be a compact Riemann surface (not necessarily connected), and let Γ be a finite group which acts on X by

32

CHAPTER 2

holomorphic automorphisms. We do not assume that the action is effective. The quotient stack M = [Γ\X] then has a presentation (2.3.1)

Γ×X

−→ −→

X,

where one arrow is the projection onto the second factor and the other is the group action. There is a holomorphic projection (2.3.2)

pr : X −→ [Γ\X] = M.

A C ∞ (resp. meromorphic, resp. . . . ) function on M is given by a Γinvariant C ∞ (resp. meromorphic, resp. . . . ) function on X. Similarly, a measure (resp. 2-form) µ on M is given by a Γ-invariant measure (resp. 2-form) on X, and we have (2.3.3)

Z

f ·µ=

Z

−1

f · µ := |Γ|

Z

f · µ.

X

[Γ\X]

M

If z ∈ X is a point, there is a corresponding point, i.e., a closed irreducible substack of dimension 0, (2.3.4)

P = [Γz \z] −→ [Γ\X] = M

of M . Of course, two points z and z 0 in X define the same point of M if and only if they are in the same Γ-orbit. Note that there is an alternative presentation [Γ\pr−1 (P )] = [Γz \z] = P.

(2.3.5)

We define the delta distribution δP of a point P ∈ M. For a function f on M, (2.3.6)

h δP , f iM := |Γ|−1 · h δpr−1 (P ) , f iX = |Γz |−1 · f (z).

A divisor Z on M is an element of the free abelian group on the points of M . Associated to (2.3.7)

X

Z=

nP · P

P ∈X

is a Γ-invariant divisor (2.3.8)

Z˜ =

X

X

nP · z

P ∈M z∈pr−1 (P )

on X. The degree of Z is given by (2.3.9) degM (Z) = h δZ , 1 iM =

X P ∈Z

˜ nP · |ΓP |−1 = |Γ|−1 · degX (Z).

33

ARITHMETIC INTERSECTION THEORY ON STACKS

Definition 2.3.1. A Green function for a divisor Z on M is a Γ-invariant Green function g for the divisor Z˜ on X. In particular, g satisfies the Green equation ddc g + δZ˜ = [ω] of currents on X, where ω is a smooth, Γ-invariant (1, 1)-form on X. To see that this is the correct definition, we check that the corresponding Green equation holds for currents on M . By linearity, we may assume that Z = P is a single point on M , so that Z˜ = pr−1 (P ). Then h ddc g, f iM = |Γ|−1 · h ddc g, f iX (2.3.10)

−1

= |Γ|



·

− h δpr−1 (P ) , f iX +

= −h δx , f iM +

Z



Z

f ·ω

X

f · ω.

M

Thus ddc g + δZ = [ω]

(2.3.11)

as currents on M . Next, if Z1 and Z2 are divisors on M with disjoint supports, then the supports of the Γ-invariant divisors Z˜1 and Z˜2 on X are also disjoint. If g1 and g2 are Green functions for Z1 and Z2 , one may view them as Γ-invariant Green functions for Z˜1 and Z˜2 on X and form their usual star product (2.3.12)

g1 ∗ g2 = g1 δ2 + g2 ω1 ,

where δ2 is the delta current for the divisor Z˜2 on X. This is a Γ-invariant distribution on X, and, as above, we can view it as a distribution on M . Again, we may suppose that Z2 = P2 is a single point and compute h g1 ∗ g2 , f iM : = |Γ|−1 · h g1 ∗ g2 , f iX (2.3.13)

−1

= |Γ|



·

X

Z

g1 (z) f (z) +

f · g2 ω1

X

z∈pr−1 (P2 )

= |Γ|−1 |Γ/Γz | · g1 (z) f (z) +



Z

f · g2 ω1

M

= h g1 δP2 + g2 ω1 , f iM Thus, we may view the formula (2.3.12) for the star product above as an identity of currents on M , where the δ2 on the right side is the delta function

34

CHAPTER 2

of the divisor Z2 . We obtain the symmetry (2.3.14)

h g1 ∗ g2 , 1 iM = h g2 ∗ g1 , 1 iM

by appealing to the corresponding symmetry on X, [1]. When M is not connected, the same identity (2.3.14) holds with 1 replaced by the characteristic function of any connected component. One can check that the notions described so far are intrinsic to M , i.e., do not depend on the particular presentation (2.3.1). For example, suppose that Γ0 ⊂ Γ is a normal subgroup which acts without fixed points on X, let X1 = Γ0 \X be the quotient Riemann surface and let Γ1 = Γ/Γ0 . Then, the orbifolds [Γ\X] and [Γ1 \X1 ] are isomorphic. A Green function g for a divisor Z on M can be given as a Γ-invariant Green function g on X for the divisor pr−1 (Z). But since such a g is then Γ0 -invariant, it may, in turn, be viewed as a Γ1 -invariant Green function on X1 for the Γ1 -invariant divisor pr−1 1 (Z) on X1 . Eventually, we will be in the situation where Γ is a discrete co-compact subgroup of GL2 (R) acting on D = P1 (C) \ P1 (R). Moreover, there will be a normal subgroup Γ0 ⊂ Γ of finite index which acts without fixed points on D. Then, we can work with the orbifold M = [Γ1 \X1 ] = [Γ\D], where X1 = Γ0 \D and Γ1 = Γ/Γ0 . Our Green functions for divisors Z on M will be given as Γ-invariant Green functions on D for the divisor pr−1 (Z). Here, of course, we mean that the corresponding function g on X1 is a Green function for the Γ1 -invariant divisor pr−1 1 (Z) on X1 as discussed above. By the previous remark, the construction is independent of the choice of Γ0 . 1

2

c c Z (M), AND CH c Z (M) 2.4 Pic(M), CH c By Pic(M) we mean, as in the one-dimensional case, the abelian group of isomorphism classes of metrized line bundles on M. Let Z ∈ ZZ1 (M). In the previous section we explained what is meant by a Green function for Z. We denote by ZˆZ1 (M) the group of Arakelov divisors, i.e., of pairs (Z, g) consisting of a divisor Z and a Green function for Z, with componentwise addition. If f ∈ Q(M)× , then f |MC corresponds to a Γ-invariant meromorphic function f˜C on X, and we define the associated principal Arakelov divisor

(2.4.1)

c ) = ( div(f ), − log |f˜C |2 ). div(f

The factor group of ZˆZ1 (M) by the group of principal Arakelov divisors is

d 1 (M). The groups CH d 1 (M) and Pic(M) c the arithmetic Chow group CH Z Z

35

ARITHMETIC INTERSECTION THEORY ON STACKS

are isomorphic. Under this isomorphism, an element Lˆ goes to the class of X

(2.4.2)

ordZ (s) Z, − log ksk2 ,


Z

where s is a meromorphic section of L. Conversely, if (Z, g) ∈ ZˆZ1 (M), then its preimage under this isomorphism is (2.4.3)

(O(Z), k k),

where − log k1k2 = g, with 1 the canonical Γ-invariant section of the pullˆ back of O(Z) to X. d 1 (M) of CH d 1 (M); see [1], 5.5. In We also introduce the R-version CH Z its definition one starts with Zˆ 1 (M), which is the R-vector space of pairs (Z, g), where Z ∈ Z 1 (M) is an R-divisor and g is a Green function for Z, and divides out by the R-subspace generated by the Arakelov principal divisors. As Bost points out [1], whereas CH1 (M) = CH1Z (M) ⊗ R, the 1

1

d (M) cannot be identified with CH d (M) ⊗ R. space CH Z 2 d (M). Let We next turn to CH Z

(2.4.4)

ZˆZ2 (M) = {(Z, g); Z ∈ ZZ2 (M ), g ∈ D1,1 (MC )}.

Here D1,1 (MC ) is the R-vector space of (1, 1)-currents on MC (i.e., the space of Γ-invariant currents of type (1, 1) on X) which are real in the sense that ∗ F∞ (g) = −g

(2.4.5)

where F∞ denotes complex conjugation. Note that Z is ‘in the top degree’ so that there is no Green equation linking Z to g. Then ZˆZ2 (M) is a group ˆ 2 (M) be the subgroup of Zˆ 2 (M) under componentwise addition. Let R Z Z generated by elements of the form (2.4.6)

 X

2



degP (f ) · P, iZ∗ (− log |f |) ,

P⊂Z

where f = fZ ∈ Q(Z)× for some prime divisor Z on M, and by elements ¯ for currents u of type (0, 1) and v of type (1, 0). of the form (0, ∂u + ∂v) 2 Here iZ∗ (− log |f | ) is the current with (2.4.7)

hφ, iZ∗ (− log |f |2 )i = −

1 · φ(x) · log |f (x)|2 . |Aut(x)| x∈Z(C) X

d 2 (M) is the factor group Z ˆ 2 (M)/R ˆ 2 (M). Similarly, we let Then CH Z Z Z Zˆ 2 (M) be the R-vector space of pairs (Z, g) where Z ∈ Z 2 (M) ⊗Z R

36

CHAPTER 2

d 2 (M) be the quotient of Z ˆ 2 (M) by the and where g is as before. We let CH 2 ˆ (M). R-subvector space generated by R Z There is the Arakelov degree map 2

d (M) −→ CH R, Z R d (Z, g) 7−→ deg Z + M(C) g.

(2.4.8) Here for Z =

P

mP P we have put

P

(2.4.9)

dZ = deg

XX p



X

mP

¯p ) x∈P(F

P

1 log p. |Aut(x)| 

The integral M(C) g = |Γ|−1 · X g is to be understood as in the previous section. That this map is well defined follows from Stokes’s theorem for mod¯ For elements of the form (2.4.6), for Z ifications of the form ∂u + ∂v. horizontal, irreducible and reduced, it follows from the product formula, cf. section 1. The degree map obviously factors through the R-arithmetic Chow group, R

R

d : CH d 2 (M) −→ R. deg

(2.4.10)

If M is geometrically irreducible, this last map is an isomorphism. Indeed, in this case the R-vector space D1,1 (MC )/ ( Im ∂ + Im ∂¯ ) has dimension 1. On the other hand, let P ∈ ZZ2 (M) and choose an irreducible horizontal divisor Z ∈ ZZ1 (M) with P ⊂ Z. Now Pic(Z) is finite, hence there exists n ∈ Z and f ∈ Γ(Z ⊗ Q, O)× such that nP = div(f ) as d 2 (M). divisors on Z. But then (nP, 0) ≡ (0, iZ ∗ (log |f |2 ) in CH Z The same argument shows that when M is not geometrically connected, then 2

∼

d (M) −→ Rπ0 (MC ) . CH 1

1

2

c (M) × CH c (M) → CH c (M) 2.5 THE PAIRING CH

Let (2.5.1) (Zb1 (M) × Zb1 (M))o = {(Z1 , g1 ), (Z2 , g2 ); suppZ1Q ∩ suppZ2Q = ∅}

37

ARITHMETIC INTERSECTION THEORY ON STACKS

be the set of pairs of Arakelov divisors with disjoint support on the generic fiber. On this subset we define the intersection pairing by setting (2.5.2)

(Z1 , g1 ).(Z2 , g2 ) = (Z1 .Z2 , g1 ∗ g2 ).

Here the first component is the 0-cycle defined by bilinear extension from the case where Z1 and Z2 are irreducible and reduced. In this case, if Z1 6= Z2 , the definition of Z1 .Z2 is clear (each ‘point’ in the intersection is weighted with the length of the local ring of Z1 ∩ Z2 ). If Z1 = Z2 = Z, then Z is a vertical divisor lying in the special fiber Mp = M ⊗Z Fp for some p. We write (2.5.3)

div(p) = a · Z + R in Z 1 (M),

where R is prime to Z. Then we set 1 Z.Z = − · (R.Z). a We claim that the induced pairing

(2.5.4)

(2.5.5)

2

b 1 (M) × Z b 1 (M))o −→ CH d (M) (Z

is symmetric. This is obvious as far as the symmetry in the first component is concerned. For the second component it follows from the symmetry (2.3.14) which shows that (2.5.6)

mod ( im ∂ + im ∂¯ ).

g1 ∗ g2 ≡ g2 ∗ g1

We now want to show that the pairing above descends to a symmetric pairing (2.5.7)

1

1

2

d (M) × CH d (M) −→ CH d (M). h , i : CH 1

d (M) can be represented by an element Since any pair of two classes in CH 1 1 ˆ ˆ in (Z (M) × Z (M))o the assertion comes down to proving that if

((Z1 , g1 ), (Z2 , g2 )) and

((Z10 , g10 ), (Z20 , g20 )) 1

1

d (M)× CH d (M), in (Zˆ 1 (M)× Zˆ 1 (M))o represent the same element in CH then

(2.5.8)

(Z1 , g1 ).(Z2 , g2 ) = (Z10 , g10 ).(Z20 , g20 ).

This in turn is reduced to the following statement. Let (Z, g) ∈ ZˆZ1 (M) with Z irreducible and reduced. Let f ∈ Q(M)× such that ZQ ∩div(f )Q = ∅. Then (2.5.9)

2

c ).(Z, g) = 0 in CH d (M). div(f

38

CHAPTER 2

If Z is horizontal, then the LHS of (2.5.9) is the image of f |Z under the map (2.4.6) since for the Green function part (2.5.10)

− log |f |2 ∗ g = − log |f |2 · δZ

by the Lelong formula. Therefore, the claim (2.5.9) follows in this case. If Z is vertical, of the form (2.5.3), let div(f ) = mZ + Z 0 , where Z 0 is relatively prime to Z. Then (2.5.11)

div(f a · p−m ) = aZ 0 − mR

is relatively prime to Z. Hence f a · p−m |Z is a nonzero rational function Z and (2.5.12)

div(f a · p−m ).Z = iZ ∗ (div(f a · p−m |Z)) ≡ 0

2

d (M). We are therefore reduced to proving div(p).Z ≡ 0 (recall that in CH d 2 (M) is the R-version of the arithmetic Chow group). But this is exactly CH the content of the definition (2.5.4) of Z.Z. d : CH d 2 (M) → R, Composing (2.5.7) with the arithmetic degree map deg we obtain the Arakelov intersection pairing

(2.5.13)

1

1

d (M) × CH d (M) −→ R. h , i : CH

2.6 ARAKELOV HEIGHTS

Let Lˆ = (L, k k) be a metrized line bundle on M. Let Z ∈ ZZ1 (M) be an irreducible and reduced divisor. We then define the height of Z with respect to L by (2.6.1)

hLˆ(Z) =

 d  ˆ i∗Z (L)), if Z is horizontal, deg(Z,  deg(Z, d i∗Z (L)) if Z is vertical.

We extend hLˆ to all of ZZ1 (M) by linearity. Using this concept we have the following expression for the Arakelov intersection pairing. Let (Z 0 , g 0 ) ∈ c d 1 (M). ZˆZ1 (M) be a representative of Lˆ under the isomorphism Pic(M) = CH Z Then by [1], (5.11), (2.6.2)

h(Z, g), (Z 0 , g 0 )i = hLˆ(Z) +

1 · 2

Z

ˆ g · c1 (L).

[Γ\X]

ˆ = ω 0 is the RHS of the Green equation for (Z 0 , g 0 ), i.e., Here c1 (L) (2.6.3)

ˆ ddc g 0 + δZ˜0 = c1 (L).

ARITHMETIC INTERSECTION THEORY ON STACKS

39

This formula is obvious from our definitions when the supports of Z and Z 0 are disjoint on the generic fiber. The general case follows since all terms c for some f ∈ are unchanged when (Z 0 , g 0 ) is replaced by (Z 0 , g 0 ) + divf × Q(M) . 2.7 THE ARITHMETIC ADJUNCTION FORMULA

In this section, we review the adjunction formula for arithmetic surfaces that we will need. We follow the treatment in [7], which has the advantage of allowing us sufficient flexibility in the choice of the metrics. Let X be a compact Riemann surface with a smooth (1, 1)-form ν. Let ∆ be the diagonal divisor on X × X and let OX×X (∆) be the associated line bundle with its canonical section s∆ with divisor div(s∆ ) = ∆. Definition 2.7.1. A weakly ν-biadmissible Green function g is a function g : X × X − ∆ → R, satisfying the following conditions: (i) g is C ∞ and has a logarithmic singularity along ∆, i.e., on an open neighborhood U × U , g(z1 , z2 ) = − log |ζ1 − ζ2 |2 + smooth with respect to a local coordinate ζ on U , where ζ1 = ζ(z1 ) and ζ2 = ζ(z2 ). (ii) (symmetry) g(z1 , z2 ) = g(z2 , z1 ). (iii) (Green equation) There is a C ∞ function φ : X × X −→ R such that φ(z1 , z2 ) = φ(z2 , z1 ), Z

φ(z1 , z2 ) dν(z2 ) = 1, X

and, for any fixed z1 ∈ X, d2 dc2 g(z1 , z2 ) + δz1 = φ(z1 , z2 ) pr∗2 (ν). Here, in (iii), pr2 is the projection on the second factor of X × X and, as ¯ ¯ usual, dc = (∂ − ∂)/4πi, so that ddc = −∂ ∂/2πi.

40

CHAPTER 2

The definition above generalizes the definition of a ν-biadmissible metric in [8] (bipermise). In [8], φ is required to be the constant 1, and it is explained that in the case of the Arakelov volume form ν, these conditions arise in a very natural way from the relation between X and its Jacobian. Our variant will allow us to make a connection with the peculiar metrics introduced in [6] and is also suitable when X is no longer compact. Note that, for fixed z1 ∈ X, the function z2 7→ g(z1 , z2 ) is a Green function of logarithmic type in the terminology of Gillet and Soul´e [5], [10] for the point z1 ∈ X. A weakly ν-biadmissible Green function g determines a metric || || on OX×X (∆), defined by (2.7.1)

g(z1 , z2 ) = − log ||s∆ (z1 , z2 )||2 .

It therefore also defines a metric on each point bundle (2.7.2)

OX (z) = i∗z OX×X (∆),

where, for a fixed z ∈ X, (2.7.3)

z 0 7→ (z, z 0 ).

iz : X → X × X,

The associated Green function is given by z 0 7→ g(z, z 0 ). For any divisor P Z = i zi , the bundle OX (Z) = ⊗i OX (zi ) gets the tensor product metric. Of course, this metric can depend on the given divisor Z and not just on the isomorphism class of OX (Z) in Pic(X). A metric on the canonical bundle Ω1X is determined by the canonical isomorphism (2.7.4)

Ω1X ' i∗∆ OX×X (−∆).

Note that the Green function on X × X − ∆ associated to the metric on OX×X (−∆) used here is −g. The key point is that, by construction, for these metrics, the residue map (2.7.5)

∼

resz : (Ω1X ⊗ OX (z))|z −→ C

is an isometry for any z ∈ X, where C is given the standard metric with ||1|| = 1. Now suppose that Γ is a finite group of automorphisms of X and let M = [Γ\X], as in section 2.3. Suppose that the weakly ν-biadmissible Green function g and associated function φ, as in Definition 2.7.1, satisfy (2.7.6)

g(γz1 , γz2 ) = g(z1 , z2 )

and

φ(γz1 , γz2 ) = φ(z1 , z2 )

41

ARITHMETIC INTERSECTION THEORY ON STACKS

for all γ ∈ Γ. For a divisor Z = (2.7.7)

gZ (ζ) :=

X

P

P ∈M

nP · P in M , consider the function

nP · g(z, ζ) =

z∈pr−1 (P )

X

nP · e−1 P

P ∈Z

X

g(z, γζ),

γ∈Γ

where, in this last expression, pr(z) = P , and where eP = |Γz |. Then gZ is a Γ-invariant Green function for the divisor Z˜ = pr−1 (Z) in X, and hence, as explained in section 2.3, gZ is a Green function for Z on M . Thus, g defines metrics on all of the bundles OM (Z) on M . Similarly, the diagonal invariance property (2.7.6) implies that the metric on Ω1X determined by the isomorphism (2.7.4) is, in fact, Γ-invariant, and hence defines a metric on Ω1M . We now return to our stack M with ‘uniformization’ M = MC = [Γ\X]. We suppose that X is equipped with a weakly ν-biadmissible Green b M/S function satisfying the diagonal invariance property. We denote by ω the relative dualizing sheaf ωM/S with the metric determined by g. Since we are assuming that M is a relative complete intersection over S, the dualizing sheaf ωM/S is a line bundle on M. bM (Z) Suppose that Z is an irreducible horizontal divisor on M, and let O be the bundle OM (Z) with the metric determined by g. Writing j : Z → M for the closed immersion of Z into M, we have a canonical isomorphism bM (Z) ) = ω b M/S ⊗ O b Z/S , j∗( ω

(2.7.8)

b Z/S is the dualizing sheaf of Z over S, equipped with the metric where ω which makes this an isometry. The first version of the adjunction formula is then

(2.7.9)

hOb

M (Z)

d b (Z) = −hω Z/S ). b M/S (Z) + degZ (ω

As explained in [7], Proposition 5.2, Chapter 4, the second term on the RHS here can be written as (2.7.10)

d (ω deg Z b Z/S ) = dZ +

1 −1 |Γ| 2

X

X

g(z, z 0 ),

P,P 0 ∈Z(C) z∈pr−1 (P ) z 0 ∈pr−1 (P 0 ) z6=z 0

where dZ is the degree of ω Z/S with trivial metric, see (2.1.19) above, and where the second term comes from the use of the metric determined by g,

42

CHAPTER 2

via (2.7.8). Thus, (2.7.11) hOb

M

(Z) = −hω b M/S (Z) + dZ + (Z)

1 −1 |Γ| 2

X

X

g(z, z 0 ).

P,P 0 ∈Z(C) z∈pr−1 (P ) z 0 ∈pr−1 (P 0 ) z6=z 0

Recalling that g is invariant under the diagonal action of Γ, we can rewrite the last term in (2.7.11) as follows: 1 −1 |Γ| 2

X

X

g(z, z 0 )

P,P 0 ∈Z(C) z∈pr−1 (P ) z 0 ∈pr−1 (P 0 ) z6=z 0

(2.7.12)

=

1 −1 |Γ| 2

X

−1 e−1 P eP 0

g(γz, γ 0 z 0 )

X γ, γ 0 ∈Γ

P,P 0 ∈Z(C)

γz6=γ 0 z 0

=

1 2

−1 e−1 P eP 0

X P,P 0 ∈Z(C)

X

g(z, γz 0 ),

γ∈Γ z6=γz 0

where z and z 0 ∈ X are points with pr(z) = P and pr(z 0 ) = P 0 , and eP = |Γz |, eP 0 = |Γz 0 |. If the cycle Z is equipped with the Green function gZ and if (Z, gZ ) ∈ d 1 (M) is the corresponding class, then we also have the arithmetic interCH section pairing (2.6.2), (2.7.13)

h (Z, gZ ), (Z, gZ ) iM = hOb

M

(Z) + (Z)

1 2

Z

φZ · gZ · ν,

M

where 1 2

Z M

φZ · gZ · ν =

X P,P 0 ∈Z(C)

−1 e−1 P eP 0

X 1Z γ∈Γ

2

φ(z, ζ) · g(γz 0 , ζ) dν(ζ).

X

Combining these expressions, we obtain the version of the formula which will be needed later.

Theorem 2.7.2. [Arithmetic Adjunction Formula] Let g be a weakly νbiadmissible Green function on X × X − ∆X for a presentation MC = [Γ\X] of MC , satisfying the diagonal invariance condition (2.7.6). Suppose that Z is an irreducible horizontal divisor on M with Green function bM (Z) be the corresponding metrized line bundle gZ determined by g. Let O

43

BIBLIOGRAPHY

b M/S be the relative dualizing sheaf of M with the metric deterand let ω mined by g. Then (i)

hOb

M (Z)

(Z) = −hω b M/S (Z) + dZ +

1 2

−1 e−1 P eP 0

X P,P 0 ∈Z(C)

where dZ is the discriminant degree of Z; see (2.1.19). (ii)

X

g(z, γz 0 ),

γ∈Γ z6=γz 0

h (Z, gZ ), (Z, gZ ) iM = −hω b M/S (Z) + dZ +

−1 e−1 P eP 0

X P,P 0 ∈Z(C)

X 1 γ∈Γ z6=γz 0

2 Z

+

g(z, γz 0 )

0



φ(z, ζ) · g(γz , ζ) dν(ζ)

M

+

X

e−1 P

1 2

Z

φ(z, ζ) · g(z, ζ) dν(ζ).

M

Here in the last sums, z and z 0 ∈ X are points with pr(z) = P and pr(z 0 ) = P 0 , and the metric on ω M/S depends on the archimedean presentation pr : X −→ [Γ\X] = M . P ∈Z(C)

When we use this formula in Chapter 7, we will, in fact, pass to a presentation M = [Γ\D], where D = H+ ∪ H− and Γ is an infinite discrete group. The expressions in the previous theorem will still be valid for such a presentation provided the, now infinite, sums over Γ are absolutely convergent. Bibliography

[1] J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic ´ surfaces, Ann. Sci. Ecole Norm. Sup., 32 (1999), 241–312. [2] J.-B. Bost, H. Gillet, and C. Soul´e, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., 7 (1994), 903–1027. [3] P. Deligne and M. Rapoport, Les sch´emas de modules de courbes elliptiques, Modular Functions of One Variable II (Proc. Intl. Summer

44

CHAPTER 2

School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. 349, 143–316, Springer-Verlag, Berlin, 1973. [4] G. Faltings, Calculus on arithmetic surfaces, Annals of Math., 119 (1984), 387–424. [5] H. Gillet and C. Soul´e, Arithmetic intersection theory, Publ. Math. IHES, 72 (1990), 93–174. [6] S. Kudla, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646. [7] S. Lang, Introduction to Arakelov Theory, Springer-Verlag, New York, 1988. [8] L. Moret-Bailly, M´etriques permises, S´eminaire sur les Pinceaux Arithm´etiques: La Conjecture de Mordell, Ast´erisque 127, 29–87, Soci´et´e Math´ematique de France, Paris, 1985. [9] J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992. [10] C. Soul´e, D. Abramovich, J.-F. Burnol and J. Kramer, Lectures on Arakelov Theory, Cambridge Studies in Advanced Math. 33, Cambridge Univ. Press, Cambridge, 1992.

Chapter Three Cycles on Shimura curves In this chapter we review the objects which will be our main concern. We introduce the moduli problem attached to a quaternion algebra B over Q solved by the DM-stack M, which has relative dimension 1 over Spec Z, has semistable reduction everywhere, and is smooth outside the ramification locus of B. This stack has a complex uniformization by C \ R = H+ ∪ H− and, for every prime p in the ramification locus of B, a p-adic uniformization by the Drinfeld upper half space. d 1 (M) ˆ v) in CH We then recall from [12] the construction of classes Z(t, for every t ∈ Z and every v ∈ R>0 . This is done in three steps. In a first step we define for every t > 0 a divisor Z(t) on M by imposing a ‘special endomorphism’ of degree t, and we set Z(t) = 0 for t < 0. In a second step we define for every t 6= 0 a Green function for the divisor Z(t) which depends on the positive real parameter v (for t < 0, this is just a ˆ v), we consider the Hodge smooth function on MC ). Finally, to define Z(0, line bundle ω, equipped with a suitable metric depending on v. We then d ˆ v) to be the image of the corresponding class of Pic(M) define Z(0, under 1 d d (M); see Section 2.4. The classes the natural map from Pic(M) to CH ˆ Z(t, v) will be the coefficients of our generating series φˆ1 with coefficients d 1 (M), which will be considered in the next chapter. in CH Finally, we define 0-cycles Z(T ) on M for positive definite ‘good’ semid 2 (M) will be part integral matrices T . The classes of these 0-cycles in CH of our generating series φˆ2 (the remaining coefficients will be defined in Chapter 6). The 0-cycle Z(T ) (if nonempty) is concentrated in the supersingular locus of a fiber Mp for a unique prime number p depending on T . The lengths of all local rings of Z(T ) are identical and are given by the Gross-Keating formula [6]. More precisely, this is the special case of this formula, in which the first entry of the Gross-Keating invariant (a1 , a2 , a3 ) is equal to zero. The proof of this special case is easier than the general case (as is already pointed out in the original paper). Because of the importance of this formula for us, we give an exposition of the proof.

46

CHAPTER 3

3.1 SHIMURA CURVES

Let B be an indefinite quaternion algebra over Q. Let D(B) be the product of primes p for which Bp = B ⊗ Qp is a division algebra. For the moment we allow the case B = M2 (Q), where D(B) = 1, but later on this case will be excluded. Let OB be a maximal order in B. We consider the following moduli problem over Spec Z. The moduli problem M associates to a scheme S the category of pairs (A, ι) where A is an abelian scheme over S and where ι is a ring homomorphism (3.1.1)

ι : OB −→ End A.

The morphisms in this category are the isomorphisms of such objects. We impose the condition on the characteristic polynomial (3.1.2)

char(ι(b) | Lie A)(T ) = (T − b) · (T − bι ), b ∈ OB .

In other words, (A, ι) is special in the sense of Drinfeld. Here b 7→ bι is the main involution of B. If S = C, such an (A, ι) is simply an abelian surface with an action of OB , but if S is a scheme of characteristic p, then the condition (3.1.2) is stronger and excludes some pathologies.

Proposition 3.1.1. (i) The moduli problem M is representable by a DMstack which is flat of relative dimension one over Spec Z. The stack M is regular and in fact is smooth over Spec Z[D(B)−1 ] and semistable over all of Spec Z. If B is a division algebra (i.e., D(B) > 1), then M is proper over Spec Z. Furthermore, M is geometrically connected and so are all its fibers over Spec Z. (ii) Let D(B) > 1. The coarse moduli scheme of M is proper and flat of relative dimension one over Spec Z and is in fact a projective scheme over Spec Z, with all fibers geometrically connected curves. We will not give a proof of this proposition but will give instead some references to the literature. For the representability of M, the main point is to establish the existence and uniqueness of a polarization of (A, ι) of a certain type. This is explained in [2], Chapter 3, Section 3. Once this is done, the construction of M can be performed along familiar lines. One approach is to use the Artin representability theorem for stacks [13], Chapter 10. Another approach is to impose a level structure and use the relative representability over the moduli space of principally polarized abelian varieties of dimension 2 with level structure (cf. [2], Chapter 3). By varying the level structure, one obtains M. That M is smooth over Spec Z[D(B)−1 ] can be shown using deformation theory and the Serre-Tate theorem. In this way one sees that the formal

47

CYCLES ON SHIMURA CURVES

¯ p ) for p 6 | D(B) is isomordeformation space of a point (A, ι) ∈ M(F phic to the formal deformation space of the corresponding p-divisible group (A(p), ι). Now using the idempotents in OB ⊗ Zp ' M2 (Zp ) to write A(p) as the square of a p-divisible group G of dimension 1 and height 2, one sees that this formal deformation space is isomorphic to the formal deformation space of G and hence is formally smooth of relative dimension 1. The semistability of M at primes p dividing D(B) relates to the p-adic uniformization; cf. Corollary 3.2.4 below. If B is a division algebra, the properness of M is checked through the valuative criterion for properness, using the semistable reduction theorem. That the fiber of M over C is connected follows from the complex uniformization; cf. Corollary 3.2.2 below. Then all other fibers are geometrically connected by Zariski’s connectedness theorem. The assertions (ii) about the coarse moduli scheme follow from the corresponding properties of M, except the projectivity of the coarse moduli space, for which one has to use the second approach to the representability of M and the quasi-projectivity of the moduli space of principally polarized abelian varieties. From now on we will assume that B is a division algebra and will call M a Shimura curve, even though M is a stack and not a scheme. 3.2 UNIFORMIZATION

We first recall from [12] the complex uniformization of M. Let (3.2.1)

V = {x ∈ B | tr(x) = 0},

with quadratic form Q(x) = Nm(x) = −x2 . Let H = B × , viewed as an algebraic group over Q. Then H acts on V by conjugation and this induces an isomorphism ∼

(3.2.2)

H −→ GSpin(V, Q),

where GSpin(V, Q) is the spinor similitude group of V . Since B is indefinite, i.e., B ⊗Q R ' M2 (R), the quadratic space VR has signature (1, 2). Let (3.2.3)

D = {w ∈ V (C) | (w, w) = 0, (w, w) ¯ < 0}/C× .

Here (x, y) = tr(xι y) is the bilinear form associated to Q. Then D is an open subset of P1 (C), stable under the action of H(R). If we fix an isomorphism BR ' M2 (R), we have an identification z −z 2 (3.2.4) C\R = H ∪H −→ D, z − 7 → w(z) = 1 −z +

−

∼





mod C× .

48

CHAPTER 3

This identification is equivariant for the action of H(R) ' GL2 (R) by fractional linear transformations on the lefthand side, and by conjugation on the righthand side. × ˆ × in H(Af ). By strong approximation we have Let Γ = OB and K = O B (3.2.5)

H(Q) \ H(A)/K = Γ \ H(R).

The complex uniformization of M is now given by the following statement. Proposition 3.2.1. We have an isomorphism of stacks ∼

[Γ \ D] −→ MC .

Proof. For z ∈ C \ R we obtain a lattice Lz in C2 as the image of OB under the isomorphism (3.2.6)

2

 

λz : BR = M2 (R) −→ C , b 7−→ b ·

z . 1

Then Az = C2 /Lz is an abelian variety with OB -action ιz given by left multiplication. We obtain a map (3.2.7)

D −→ MC , z 7−→ (Az , ιz ).

Two points in D give the same lattice if and only if they are in the same orbit × = Γ. Furthermore, the automorphisms of (Az , ιz ) are given by under OB elements in Γz . Since every (A, ι) over C is isomorphic to some (Az , ιz ), the resulting morphism of stacks [Γ \ D] → MC is an isomorphism. Corollary 3.2.2. MC is connected.

Proof. Γ contains elements b with Nm(b) < 0.

We next turn to p-adic uniformization for p | D(B). We fix such a ˆ the Drinfeld upper half plane. This is an adic forprime and denote by Ω mal scheme over Zp with semistable reduction [2], [16], equipped with an action of PGL2 (Qp ). The name derives from the remarkable property that for any finite extension K of Qp with ring of integers OK we have ˆ K ) = P1 (K) \ P1 (Qp ). We let GL2 (Qp ) act on Ω ˆ × Z by Ω(O (3.2.8)

g : (z, i) 7−→ (gz, i + ordp (det g)).

Let B 0 = B (p) be the definite quaternion algebra over Q whose invariants agree with those of B at all primes ` 6= p, and denote by H 0 the algebraic group over Q with H 0 (Q) = B 0× . We fix isomorphisms (3.2.9)

H 0 (Qp ) ' GL2 (Qp ) and H 0 (Apf ) ' H(Apf ).

49

CYCLES ON SHIMURA CURVES

We write K = K p .Kp with K p ⊂ H(Apf ) and Kp ⊂ H(Qp ). Let K 0p ⊂ H 0 (Apf ) be the image of K p under this last isomorphism and let Γ0 = H 0 (Q) ∩ (H 0 (Qp ).K 0p ). Then Γ0 = OB p−1 tion we have 

×

. By strong approxima-

H 0 (Q) \ H 0 (A)/K 0p = Γ0 \ H 0 (Qp ).

(3.2.10)

¯ p ) of the formal completion ˆ W be the base change to W = W (F Let M ˆ W the base change of Ω ˆ to W . of M along its fiber at p. We denote by Ω The Cherednik-Drinfeld uniformization theorem may now be formulated as follows. ¯ p ), Theorem 3.2.3. There is an isomorphism of formal stacks over W (F h

i

ˆ W ' H 0 (Q)\ (Ω ˆ W × Z) × H 0 (Ap )/K 0p = Γ0 \(Ω ˆ W ×Z) = Γ01 \ Ω ˆW , M f where Γ01 = {g ∈ Γ0 | ordp det g = 0}.

For the proof we refer to [2] and [16]. In these references one also finds a comparison of the descent data from W to Zp of both sides. An informal discussion of this theorem appears in [10]. The principal reason for this ¯ p ) are isogenous to one theorem to hold is that all elements (A, ι) ∈ M(F another. Corollary 3.2.4. M ×Spec

Z

Spec Zp is semistable over Spec Zp .

3.3 THE HODGE BUNDLE

Let (A, ι) be the universal abelian scheme over M. The Hodge line bundle (or Hodge bundle for short) is the following line bundle on M, (3.3.1)

ω = ∗ (Ω2A/M ) = ∧2 Lie(A/M)∗ .

Here  : M → A is the zero section. The Hodge bundle is related to the relative dualizing sheaf ωM/Z by the following proposition; see [12], Proposition 3.2. Proposition 3.3.1. The Hodge bundle ω is isomorphic to the relative dualizing sheaf ωM/Z .

The complex fiber ωC is a line bundle on MC = [Γ \ D]. It may be given by a line bundle on D, equipped with a descent datum with respect to the action of Γ on D as follows ([12], Section 3): The line bundle on D is

50

CHAPTER 3

associated to the trivial vector bundle D × C, and the action of Γ on D is lifted to D × C by the automorphy factor (cz + d)2 , i.e., 



a b γ= : (z, ζ) 7−→ (γ · z, (cz + d)2 · ζ). c d

(3.3.2)

More precisely, the pullback of ωC to D is trivialized by associating to z ∈ C \ R the section αz = D(B)−1 · (2πi)2 · dω1 ∧ dω2 of Ω2Az /C . Here Az = C2 /Lz with Lz = λz (OB ) and w1 , w2 are the coordinates on C2 = BR via λz ; see (3.2.6). It follows that on MC the Hodge bundle is isomorphic to the canonical bundle Ω1MC /C under the map which sends αz to dz. On the Hodge bundle ωC there is a natural metric k knat . If s : z 7→ sz is a section of ωC , then ksk2nat

(3.3.3)

  Z i 2 sz ∧ s¯z . · = 2π Az

We prefer to use the normalized metric defined by k k2 = e−2C · k k2nat .

(3.3.4) Here (3.3.5)

2C = log(4π) + γ,

where γ is Euler’s constant. The reason for this renormalization is explained in the introduction to [12]. Further justification is given in Chapter 7 in connection with the adjunction formula. The metric on Ω1MC resulting under the above isomorphism with ωC is then (3.3.6) 2

2

−2C

kdzk = kαz k = e

  Z i 2 · · αz ∧ α ¯z 2π Az

= e−2C · (2π)−2 · (2π)4 · D(B)−2 · vol(M2 (R)/OB ) · Im(z)2 = e−2C · (2π)2 · Im(z)2 . Here we have used the fact that the pullback under λz of the form dw1 ∧ dw2 ∧ dw ¯ 1 ∧ dw ¯2 on C2 is 4 Im(z)2 times the standard volume form on M2 (R), and that vol(M2 (R)/OB ) = D(B)2 .

51

CYCLES ON SHIMURA CURVES

3.4 SPECIAL ENDOMORPHISMS

Let (A, ι) ∈ M(S). The space of special endomorphisms of (A, ι) is the Z-module (3.4.1)

V (A, ι) = {x ∈ EndS (A, ι) | tr(x) = 0}.

When S is connected, V (A, ι) is a free Z-module of finite rank equipped with the Z-valued quadratic form Q given by −x2 = Q(x) · idA .

(3.4.2)

Proposition 3.4.1. The quadratic form Q is positive-definite, i.e., Q(x) ≥ 0 and Q(x) = 0 only for x = 0.

Proof. We may assume that S is the spectrum of an algebraically closed field. But then it follows from the classification of End(A, ι) ⊗Z Q that any nonscalar x ∈ End(A, ι) ⊗ Q generates an imaginary quadratic field extension k. For x ∈ V (A, ι) ⊗ Q we obtain −x2 = Nmk/Q (x) · idA , which proves the claim.

Definition 3.4.2. Let t 6= 0 be an integer. Let Z(t) be the moduli stack of triples (A, ι, x), where (A, ι) is an object of M and where x ∈ V (A, ι) with Q(x) = t. Then Z(t) is a DM-stack which is relatively representable over M by an unramified finite morphism (rigidity theorem), (3.4.3)

Z(t) −→ M.

If t < 0, then Z(t) = ∅. The degree of the generic fiber Z(t)Q is given by the following formula [12]: Let 4t = n2 d, where −d is the fundamental discriminant of kt . Then (3.4.4)

deg Z(t)Q = 2 · δ(d, D(B)) · H0 (t, D(B)),

where (3.4.5)

δ(d, D) =

Y

(1 − χd (p))

p|D

(zero or a power of 2) and (3.4.6) H0 (t, D) =

X h(c2 d) c|n

w(c2 d)

=

h(d) w(d)

 X c|n (c,D)=1

c·



(1 − χd (`)`−1 ) .

Y `|c

52

CHAPTER 3

Here h(c2 d) is the class number of the order Oc2 d of conductor c in kt , w(c2 d) is the number of units in Oc2 d , and χd is the Dirichlet character for kt . Note that in defining the degree of Z(t)Q over Q, each point η = (A, ι, x) in Z(t)(C) counts with multiplicity 1/|Aut(A, ι, x)|. The proof of (3.4.6) in [12] uses the complex uniformization of the stack Z(t). If (A, √ ι, x) ∈ Z(t)(C), then A is an abelian surface with an action of OB ⊗Z Z[ −t]. If A = Az = C2 /Lz as in (3.2.6), then the action of x on Lie A = BR ' C2 commutes with the action of OB and hence is given by right multiplication by an element jx ∈ OD ∩ V with Nm(jx ) = t. The map given by right multiplication with jx , (3.4.7)

x ˜ = r(jx ) : C2 −→ C2

is holomorphic and preserves D. The point z is fixed by x ˜. Denote by Dx the fixed locus of r(jx ). Let (3.4.8)

L(t) = {x ∈ OB ∩ V | Q(x) = t},

and put (3.4.9)

Dt =

a

Dx .

x∈L(t)

We thus obtain a morphism (3.4.10)

Dt −→ Z(t)C .

One checks, cf. [12], that this induces an isomorphism of stacks over C, (3.4.11)

[Γ \ Dt ] ' Z(t)C ,

compatible with the isomorphism [Γ \ D] ' MC of Proposition 3.2.1. By (9.5) and (9.6) of [12], there is a 2–1 map of orbifolds, (3.4.12)

[ Γ\DZ(t) ] −→ [ Γ\L(t) ].

This arises as follows. For any w ∈ D the negative 2-plane U (w) = (Cw + Cw) ¯ ∩ V has a natural orientation determined by iw ∧ w. ¯ Fix an orientation of V . For x ∈ L(t), with t > 0, let Dx0 = {z0 }, where z0 ∈ Dx is determined by the condition that [x, U (w(z0 ))] is properly oriented. Note 0 = {z 0 ). that D−x ¯0 }, and pr(Dx ) = pr(Dx0 ) ∪ pr(D−x Lemma 3.4.3. Let x ∈ L(t) with Dx0 = {z0 }. Then (i) −x ∈ / Γ · x. (ii) z¯0 ∈ / Γ · z0 .

53

CYCLES ON SHIMURA CURVES

Proof. To prove (i), suppose that γ ∈ Γ is such that γ ·x = γ xγ −1 = −x. It follows that γ and x generate B and hence that γ 2 is central. Thus γ 2 = ±1. The case γ 2 = 1 is excluded, since B is a division algebra. If γ 2 = −1, then B ' (−1, −t) as a cyclic algebra over Q. But this cannot happen, since (−1, −t)∞ = −1 whereas B is indefinite. For (ii), if γ · z0 = z¯0 , then γ reverses the orientation on U (w(z0 )) and hence acts by −1 on x, which is excluded by (i). By (ii) of the lemma, pr(Dx ) consists of two distinct points pr(Dx0 ) and 0 ), while, by (i), the vectors x and −x both contribute the same pair pr(D−x of points to the sum (3.4.13)

Z(t)(C) =

X

X

pr(Dx ) = 2

x∈L(t) mod Γ

pr(Dx0 )

x∈L(t) mod Γ

Thus, (3.4.14)

deg Z(t)Q = 2

X

e−1 x

x∈L(t) mod Γ

so that the computation of deg Z(t)Q is reduced to a counting problem; see [12], Proposition 9.1. Remark 3.4.4. The factor of 2 on the righthand side of (3.4.14) is also due to the fact that the morphism Z(t) → M is not a closed immersion. In fact, Z(t)Q → MQ is of degree 2 over its image ((A, ι, x) and (A, ι, −x) are the two nonisomorphic preimages of (A, ι)); see [12], Remark 9.3). In the sequel we denote by Z(t) both the DM-stack defined above and its image divisor in M. At those places where this notational ambiguity can cause a problem, we have pointed out explicitly which meaning is intended. Proposition 3.4.5. The 0-cycle √ Z(t)C is nonempty if and only if the imaginary quadratic field kt = Q( −t) embeds in B. In this case the stack Z(t) is flat over Spec Z[D(B)−1 ].

Proof. The first assertion is clear by the above description of Z(t)C . Let ¯ p ). The second assertion follows from p - D(B) and let (A, ι) ∈ Mp (F the fact that the locus in the formal deformation space of (A, ι) where a nonzero special endomorphism of (A, ι) deforms is a relative divisor over ¯ p ). Using the idempotents in OB ⊗ Zp ' M2 (Zp ) to write the Spf W (F p-divisible group of A as the square of a p-divisible group G of dimension 1 and height 2, we are reduced to showing that the locus in the formal deformation space of G, where a nonscalar endomorphism of G deforms, is a

54

CHAPTER 3

relative divisor. This is well known; see [17], Proposition 6.1 and also [19], [6]. Just as M has a complex uniformization as well as a p-adic uniformization, so do the special cycles Z(t); see [12], Section 11. We fix p | D(B) and introduce, as in Section 3.2, the definite quaternion algebra B 0 = B (p) , and isomorphisms as in (3.2.9). We fix x ∈ OB 0 with tr(x) = 0 and with x2 = −t. Put (3.4.15)

ˆ p 0 }. I(x) = {gK 0p ∈ H 0 (Apf )/K 0p | g −1 xg ∈ O B

Also put x ˜ = x if ordp (t) = 0, resp. x ˜ = 1 + x if ordp (t) > 0. Then in all cases ordp Nm(x) = 0. Let Hx0 be the stabilizer of x in H 0 and denote by (3.4.16)


x˜

ˆ ˆW × Z C(x) = Ω

the fixed locus of x ˜. Since ordp Nm(x) = 0, this is equal to the product ˆ x˜ × Z. Ω W Then the p-adic uniformization is given as an isomorphism of stacks over W, (3.4.17)

ˆ Z(t)W = Hx0 (Q) \ I(x) × C(x) . 



A closer analysis of the situation for p | D(B) gives the following facts, proved in [11] for p 6= 2 and in [12] for p = 2. Proposition 3.4.6. Let t > 0 and let p | D(B). Then Z(t) has vertical components of the fiber Mp = M ⊗Z Fp if ordp (t) ≥ 2 and the field √ kt = Q( −t) embeds into B 0 = B (p) .

For kt there are therefore the following two alternatives. If kt embeds into B, then it also embeds into B (p) and the cycle Z(t) will, by Proposition 3.4.5, have a nonempty generic fiber and, by Proposition 3.4.6, have vertical components at p, as soon as ordp (t) ≥ 2. In this case p is either inert or ramified in kt . If, on the other hand, p splits in kt , then kt does not embed into B, and the generic fiber of Z(t) is empty. In this case, if kt embeds into B (p) , then all prime divisors of D(B) other than p are non-split in kt and hence the cycle Z(t) has support in the fiber Mp . We therefore see that, while Z(t) is flat over Spec Z[D(B)−1 ], the stack Z(t) is not flat over Spec Z (at least when ordp (t) ≥ 2 for some p | D(B)). This seems to be unavoidable if one insists on a modular definition of Z(t). We will define away another unpleasant feature of our cycles. Namely, by [11], the fiber at p | D(B) of Z(t) contains embedded components when p | t, and it has the ‘wrong dimension’ 0, when ordp (t) = 0 and p splits in

55

CYCLES ON SHIMURA CURVES

kt , and embeds into B 0 = B (p) . We therefore redefine Z(t) by replacing it

by its Cohen-Macauleyfication (denoted Z(t)pure in [11]). By Proposition 3.4.5 this does not change Z(t) over Spec Z[D(B)−1 ], and now Z(t) is pure of dimension 1 (unless it is empty) and without embedded components. We consider Z(t) as a divisor although, strictly speaking, Z(t) is not a closed substack of M; cf. Remark 3.4.4. To form our generating series, we will also need to define Z(0). We take the image of the class of ω under the composition of the isomorphism Pic(M) ' CH1Z (M) and the natural map CH1Z (M) −→ CH1 (M). Remark 3.4.7. In fact, it is possible to slightly refine the definition of the cycle Z(t) as follows. Recall that there is an element δ ∈ OB such that δ 2 = −D(B). For a point (A, ι, x) of√ Z(t), the special endomorphism x defines √ an action on A of the order Z[ −t] of discriminant −4t in kt = Q( −t). If we write 4t = n2 d, √as above, and let n0 be the prime to D(B) part of n, then the action of Z[ −t] extends to an action of the order On2 d 0 of discriminant −n20 d. Note that this order is maximal at each p dividing D(B). Since x commutes with the action of OB , X := ker(ι(δ)) ⊂ A is a finite group scheme of order D(B)2 equipped with an action of (3.4.18)

∼

Y

On2 d /(D(B)) −→

Fp2 ×

0

Y

Fp .

p|D(B) p ramified

p|D(B) p inert

It also carries an action of (3.4.19)

∼

Y

OB /(δ) −→

Fp2 .

p|D(B)

In our situation, the two actions are related by a commutative diagram Q p inert

(3.4.20)

Fp2

(∗) ↓ Q p inert

Fp2

,→

End(X) k

,→ End(X).

There are 2ν = δ(d, D(B)) possibilities, which we call types, for the isomorphism (*), where ν is the number of prime factors of D(B) which are inert in kt . For each type η, we can define a component Z(t, η) of Z(t) by requiring that ker(ι(δ)) be of type η. Note that this construction explains the occurrence of the factor δ(d, D(B)) in the formula (3.5.8) for degC Z(t). This decomposition according to types is analogous to the decomposition of Heegner cycles in [4] and [7]. There, prime divisors of the level N must be split or ramified in kt , and the role of η is played by the parameter r, which

56

CHAPTER 3

determines the isomorphism class of the kernel of the cyclic N -isogeny. In both cases, the group of Atkin-Lehner involutions permutes the components transitively. We will make no further use of this refinement in the present work so that, in effect, we consider only cycles which are invariant under the group of Atkin-Lehner involutions.

3.5 GREEN FUNCTIONS 1

d (M) from the cycles Z(t) introTo obtain arithmetic cycle classes in CH Z duced in Section 3.4, we will equip them with Green functions following [9]. For x ∈ V (R) with Q(x) 6= 0 and z ∈ D set

(3.5.1)

R(x, z) = |(x, w(z))|2 |(w(z), w(z))|−1 = −(prz (x), prz (x)).

where prz (x) is the orthogonal projection of x to the negative 2-plane obtained by intersecting (C·w(z)+C·w(z)) with V (R). Here w(z) ∈ V (C) is any vector with image z in P(V (C)). Then this function for fixed x vanishes precisely when (x, w(z)) = 0, i.e., z ∈ Dx . Let (3.5.2)

∞

Z

β1 (r) =

e−ru u−1 du = −Ei(−r)

1

be the exponential integral. Then (

(3.5.3)

β1 (r) =

− log(r) − γ + O(r) as r → 0, O(e−r ) as r → ∞.

As in [9], we form the functions (3.5.4)

ξ(x, z) = β1 (2πR(x, z))

and (3.5.5)

ϕ(x, z) = [ 2(R(x, z) + 2Q(x)) −

1 ] · e−2πR(x,z) . 2π

Then ξ(x, z) has a logarithmic singularity along Dx and decays exponentially as z goes to the boundary of D. The following result is Proposition 11.1 of [9] (where the functions ξ and ϕ were denoted by ξ 0 and ϕ0 ). Proposition 3.5.1. Let µ = on D = C \ R. Then

i −2 dz 2y

∧ d¯ z be the hyperbolic volume form

ddc ξ(x) + δDx = [ϕ(x)µ]

as currents on D. In particular, ξ(x, ·) is a Green function for Dx .

57

CYCLES ON SHIMURA CURVES

Because of the rapid decay of ξ(x, ·), we can average over lattice points.

Corollary 3.5.2. For v ∈ R>0 let Ξ(t, v) =

X

ξ(v 1/2 x, z).

x∈OB ∩V Q(x)=t

(i) For t > 0, Ξ(t, v) defines a Green function for Z(t). (ii) For t < 0, Ξ(t, v) defines a smooth function on MC = [Γ \ D]. 1

d (M): This allows us to define classes in the arithmetic Chow group CH (

(3.5.6)

ˆ v) = Z(t,

(Z(t), Ξ(t, v)) if t > 0, (0, Ξ(t, v)) if t < 0.

Remark 3.5.3.√ We note that the function Ξ(t, v) vanishes when the quadratic field kt = Q( −t) is not embeddable in B, since then the summation in the definition is empty. d 1 (M). We set, followˆ v) in CH Finally, we have to define the class Z(0, ing [12],

(3.5.7)

ˆ v) = −ˆ Z(0, ω − (0, log(v)) + (0, c).

c Here the first summand is the image of ω ˆ under the natural map from Pic(M) 1 d (M), and c is the real constant determined by the identity into CH (3.5.8)  X p log(p)  ζ 0 (−1) 1 degQ (ˆ ω )·c = h ω ˆ, ω ˆ i−ζD (−1) 2 +1−2C − . 2 ζ(−1) p−1 p|D(B)

In Chapter 7, we will prove that, in fact, c = − log D(B). 3.6 SPECIAL 0-CYCLES

Let (3.6.1)

Sym2 (Z)∨ = {T ∈ Sym2 (Q) | tr(T b) ∈ Z, ∀b ∈ Sym2 (Z)}

(3.6.2)

Z(T ) = {(A, ι; x1 , x2 ); x1 , x2 ∈ V (A, ι); Q(x) = T }.

be the space of semi-integral symmetric matrices. For T ∈ Sym2 (Z)∨ we define

58

CHAPTER 3

Here (A, ι) is an object of M, and for special endomorphisms x1 , x2 ∈ V (A, ι) we set (3.6.3)

1 (x1 , x1 ) (x1 , x2 ) Q(x) = Q(x1 , x2 ) = , 2 (x2 , x1 ) (x2 , x2 ) 



where (x, y) = Q(x + y) − Q(x) − Q(y) is the associated bilinear form. Then Z(T ) is a DM-stack equipped with an unramified morphism (3.6.4)

Z(T ) −→ M.

By Proposition 3.4.1, Z(T ) = ∅ unless T is positive-semidefinite. We next explain when Z(T ) is nonempty of dimension 0 for T with det T 6= 0. We introduce the incoherent collection of quadratic spaces C = (Cp ), where Cp = Vp for p < ∞ and where C∞ is the positivedefinite quadratic space of dimension 3. Note that, since the quaternion algebra B is indefinite, the signature type of V∞ is equal to (1,2), so that indeed the collection C is incoherent, i.e., the product of the Hasse invariants of the local quadratic spaces Cp is equal to −1. Let T ∈ Sym2 (Q) with det(T ) 6= 0. Let VT be the unique ternary quadratic space over Q with discriminant −1 = discr(V) which represents T . Then VT is isometric to the space of trace zero elements of a well-determined quaternion algebra BT over Q. We then define following [9], Definition 5.1, (3.6.5)

Diff(T, B) = { p ≤ ∞ | invp (BT ) 6= inv(Cp ) }.

Note that, since C is an incoherent collection, the cardinality |Diff(T, B)| is a positive odd number. Furthermore, ∞ ∈ Diff(T, B) if and only if the signature of T is equal to (1,2) or (0,2), i.e., if T is not positive-definite.

Theorem 3.6.1. Let T ∈ Sym2 (Z)∨ be nonsingular. (i) If |Diff(T, B)| > 1, then Z(T ) = ∅. (ii) If Diff(T, B) = {p}, with p < ∞, and p 6 | D(B), then Z(T ) is a finite stack in characteristic p with support in the supersingular locus. The lengths of the local rings at all geometric points of Z(T ) are identical and given by the Gross-Keating formula νp (T ) =

if a2 is odd,

 a2 −1 P 2  j   j=0 (a2 + a3 − 4j)p  a  P 22 −1

(a2 + a3 − 4j)pj + 12 (a3 − a2 + 1)p

a2 2

if a2 is even.

Here (0, a2 , a3 ) with 0 ≤ a2 ≤ a3 are the Gross-Keating invariants1 of T˜ = diag(1, T). j=0

1

Z× p .

If p 6= 2, this means that T is GL2 (Zp )-equivalent to diag(ε1 pa2 , ε2 pa3 ) with ε1 , ε2 ∈

CYCLES ON SHIMURA CURVES

59

(iii) If Diff(T, B) = {p}, with p < ∞, and p | D(B), then Z(T ) is concentrated in characteristic p. Furthermore, dim Z(T ) = 1 if p2 |T in Sym2 (Z)∨ . Remark 3.6.2. If p 6= 2, the converse holds in (iii), i.e., if p2 - T , then dim Z(T ) = 0, and in this case the lengths of the local rings at all geometric points of Z(T ) are identical and given by νp (T ) =

if α = 1, β if α = 0 and χ(ε1 ) = −1,  0 if α = 0 and χ(ε1 ) = 1.  β

Here T is GL2 (Zp )-equivalent to diag(ε1 pα , ε2 pβ ) with ε1 , ε2 ∈ Z× p and χ is the quadratic residue character modulo p. If p = 2 in (iii), the situation is more complicated and can be read off from the results in the appendix to Chapter 6. Let us call T ∈ Sym2 (Zp )∨ primitive, if an equation T = AT 0 At with T 0 ∈ Sym2 (Zp )∨ and A ∈ M2 (Zp ) implies A ∈ GL2 (Zp ). Obviously, if T is primitive, then p2 6 | T . If p = 2, then dim Z(T ) = 0 if and only if T is primitive. If dim Z(T ) = 0, then the lengths of the local rings at all geometric points of Z(T ) are the same and can be read off from Theorem 6A.1.1 in the appendix to Chapter 6.

Proof. Let (A, ι; y1 , y2 ) be a point of Z(T ). Then B 0 = End(A, ι) ⊗ Q contains the three linearly independent elements 1, y1 , y2 . Hence (A, ι) is a supersingular point in positive characteristic p. Since T is represented by the space V 0 of trace zero elements of B 0 , it follows that V 0 = VT . On the other hand, B 0 is a definite quaternion algebra whose invariants agree with those of B at all finite primes ` 6= p. It follows that Diff(T, B) = {p}, which proves the first assertion. Now consider (i). Let (A, ι) be a supersingular point and let y = (y1 , y2 ) be a pair of special endomorphisms of (A, ι) with Q(y) = T . We need ˆx /J(y)x . Here O ˆx = O ˆM,x is the formal to determine the length lg O completion of the local ring of x and J(y)x denotes the minimal ideal in ˆx such that y1 and y2 extend to endomorphisms of (A, ι) mod J(y). Here O (A, ι) denotes the universal object over M. We use the Serre-Tate theorem. According to this theorem, the universal deformation space of (Ax , ιx , y) coincides with the universal deformation space of the corresponding p-divisible group (Ax (p), ι) with its special endomorphisms. This p- divisible group is independent of x. More precisely, ¯p let G be the p-divisible formal group of dimension 1 and height 2 over F ˆ (it is unique up to isomorphism). Then there is an isomorphism Ax ' G2

60

CHAPTER 3

compatible with an isomorphism OB ⊗ Zp ' M2 (Zp ). The special endomorphisms in End(Ax (p), ιx ) induced by y1 and y2 can be identified with endomorphisms x1 , x2 ∈ End(G) with tr(xi ) = 0, for i = 1, 2. We therefore obtain isomorphisms (3.6.6)

ˆx ' W [[t]], O ˆx /J(y)x ' W [[t]]/J(x). O

¯ p ) is the ring of Witt vectors, Spf W [[t]] is the base of the Here W = W (F universal deformation space of G, and J(x) is the minimal ideal in W [[t]] such that the special endomorphisms x1 and x2 of G deform to endomorphisms of the universal deformation Γ modulo J(x), (3.6.7)

x1 , x2 : Γ −→ Γ (mod J(x)) .

We therefore have reduced the statement (i) to an assertion about the pdivisible group G, with its pair of special endomorphisms x1 , x2 , given by Theorem 3.6.3 below. For (ii) we refer to [11], Proposition 2.10 and Theorem 6.1 in the case p 6= 2 and to the appendix to Chapter 6 below in the case p = 2. Theorem 3.6.3. Let G be the formal group of dimension 1 and height 2 ¯ p and let Γ be its universal deformation over W [[t]]. Let x1 , x2 be a over F pair of special endomorphisms of G ( tr(xi ) = 0 for i = 1, 2 ) which are linearly independent. Let J(x) be the minimal ideal such that the xi deform to endomorphisms of Γ(mod J(x)). Then lg W [[t]]/J(x) only depends on the GL2 (Zp )-equivalence class of T = 12 (x, x) and is equal to

νp (T ) =

if a2 is odd,

 a2 −1 P 2  j   j=0 (a2 + a3 − 4j)p  a  P 22 −1

(a2 + a3 − 4j)pj + 12 (a3 − a2 + 1)p

a2 2

if a2 is even.

Here (0, a2 , a3 ) with 0 ≤ a2 ≤ a3 are the Gross-Keating invariants of T˜ = diag(1, T). j=0

Theorem 3.6.3 is the special case a1 = 0 of [6], Proposition 5.4. However, this special case is easier to prove and follows from the results in [8], as we now proceed to show. Remark 3.6.4. Theorem 5.1 of [8] is not proved there. Instead, the reader is referred to Keating’s unpublished Harvard thesis. A detailed account of Keating’s result can be found in [1], which also contains a proof of the general result of Gross-Keating. We also point out that Theorem 3.6.3 is overstated as Proposition 14.6 in [9].

CYCLES ON SHIMURA CURVES

Proof. Let L = Zp x1 + Zp x2 ⊂ End(G). It is obvious that (3.6.8)

61

J(L) = J(x),

where J(L) is the minimal ideal in W [[t]] such that (3.6.9)

L ⊂ End(Γ(mod J(L))).

We may therefore choose any basis ϕ, ψ of L and then J(x1 , x2 ) = J(ϕ, ψ) = J(L).

We choose ϕ, ψ, as we may, such that 1, ϕ, ψ form an optimal basis of ˜ = Zp · 1 + Zp x1 + Zp x2 , with v(ϕ) = a2 and v(ψ) = a3 . If p 6= 2, L this simply means that the bilinear form on L is diagonalized by ϕ, ψ and that a2 = v(ϕ) and a3 = v(ψ). Here, as elsewhere, we denote by v the valuation on the quaternion division algebra D = End(G) ⊗Zp Qp .

Lemma 3.6.5. Let k = Qp (ϕ) ⊂ D. Then k/Qp is unramified if a2 is even and is ramified if a2 is odd. The order Zp [ϕ] has conductor ps with s = [a2 /2].

Nm ϕ = ε2 pa2 with ε2 ∈ Z× Proof for p 6= 2. We have tr(ϕ) = 0 and√ p. If a2 is odd, then k = Qp (π) with π = −ε2 p and Ok = Zp [π]. Since ϕ =√ ps · π, the claim follows in this case. If a2 is even, then k = Qp (ξ) with ξ = −ε2 . Then Ok = Zp [ξ]. Since ϕ = ps · ξ, the claim follows. The proof for p = 2 is more complicated and is given at the end of this section. Lemma 3.6.6. ψ ∈ Πa3 OD \ (Ok + Πa3 +1 OD ).

Here Π denotes a uniformizer of OD .

Proof for p 6= 2. In this case ψ anticommutes with k, i.e., conjugation with ψ induces the nontrivial automorphism of k. Since k/Qp is unramified or tamely ramified, we may apply [14], Lemma 2.2. Let v = v(ψ) = a3 . Then [14] tells us that, under the ramification hypothesis made, we have ψ ∈ Πv OD \ (Ok + Πv+1 OD ). The proof for p = 2 is given at the end of this section. Let us now continue the proof of Theorem 3.6.3, assuming (even for p = 2) the validity of Lemmas 3.6.5 and 3.6.6. By Lemma 3.6.5 we may write

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the locus Spf W [[t]]/J(ϕ) where ϕ deforms as a sum of quasi-canonical divisors in Spf W [[t]], (3.6.10)

Spf W [[t]]/J(ϕ) =

s X

Wr (ϕ).

r=0

Recall [5], (cf. also Section 7.7), that the quasi-canonical divisor of level r is a reduced irreducible regular divisor in Spf W [[t]] such that the pullback of the universal p-divisible group on Spf W [[t]] to Wr (ϕ) has the order Or of conductor pr in k (embedded via ϕ into D) as its endomorphism algebra. We have Wr (ϕ) ' Spf Wr (k), where Wr (k) is the ring of integers in a ramified extension Mr of M = Frac W . Now the locus inside Wr (ϕ) to which ψ deforms is defined by an ideal I` in Wr (k) which only depends on the integer ` with ψ ∈ (Or + Π` OD ) \ (Or + Π`+1 OD ). By Lemma 3.6.6, we have ` = a3 . Since a3 ≥ a2 ≥ 2s ≥ 2r − 1, we are in the ‘stable range’ of the formula of [15], Theorem 1.1. Hence (3.6.11)   a3 + 1 (pr−1 − 1)(p + 1) r−1 +p + − r er + 1, lg Wr (k)/Ia3 = p−1 2 where er is the ramification index of Mr over M . Now from the explicit description of Wr (k), as described, e.g., in Section 7.7, we have (3.6.12)

er =

 r   2p

pr

  1

+

pr−1

if k/Qp is ramified, if k/Qp is unramified and r ≥ 1, if k/Qp is unramified and r = 0.

Therefore we have obtained a completely explicit expression for (3.6.13)

lg W [[t]]/J(ϕ, ψ) =

s X

lg Wr (k)/Ia3 .

r=0

Taking into account the first statement of Lemma 3.6.5, which allows us to distinguish the ramified case from the unramified case by the parity of a2 , we get the following values for lg W [[t]]/J(ϕ, ψ): (3.6.14)

s X r=0

(pr−1 − 1)(p + 1) + pr−1 + p−1



!

a3 + 1 − r · 2pr + 1 , 2 

if a2 = 2s + 1 is odd, and (3.6.15)    s  r a3 + 1 X (p − 1)(p + 1) a3 + 1 + + pr−1 + − r (pr + pr−1 ) + 1 , 2 p−1 2 r=1

63

CYCLES ON SHIMURA CURVES

if a2 = 2s is even. An elementary (but tedious) calculation now gives the formula in Theorem 3.6.3. It remains to prove Lemmas 3.6.5 and 3.6.6 for p = 2. For this we use the classification of the possible T˜ and the construction of optimal bases of ˜ in each case; see [18], [3]. L Te’s and Optimal Bases







2 1 (A) T˜ = diag 1, 2β with β ≥ 0 even.2 Then 1 = e1 , ϕ = e2 , 1 2 ψ = e3 is an optimal basis, and GK(T˜) = (0, β + 1, β + 1). (B) T˜ = diag(1, ε2 2β2 , ε3 2β3 ) with 0 ≤ β2 ≤ β3 . Then T˜ is anisotropic if and only if (−1, ε2 ε3 ) = (ε2 , ε3 ) · (2, ε2 )β3 · (2, ε3 )β2 . (1) β2 odd. Then 1 = e1 , ϕ = e2 , ψ = c1 e1 + c2 e2 + e3 for suitable c1 , c2 ∈ Z2 , and GK(T˜) = (0, β2 , β3 + 2). (2) β2 even and β3 ≤ β2 + 1. (a) β2 = β3 . Then 1 = e1 , ϕ = 2β2 /2 e1 +e2 , ψ = 2β2 /2 e1 +e3 , and GK(T˜) = (0, β2 + 1, β3 + 1). (b) β3 = β2 + 1 and ε1 ≡ 1(mod 4). Then 1 = e1 , ϕ = 2β2 /2 e1 + e2 , ψ = 2β2 /2 e1 + e3 , and GK(T˜) = (0, β2 + 1, β3 + 1). (c) β3 = β2 + 1 and ε2 ≡ −1(mod 4). Then 1 = e1 , ϕ = 2β2 /2 e1 + e2 + e3 , ψ = 2β2 /2 e1 + e2 + 2e3 , and GK(T˜) = (0, β2 + 1, β3 + 1). (3) β2 even and β3 ≥ β2 + 2. 2

The parity condition on β comes from the fact that T is anisotropic.

64

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(a) ε2 ≡ −1(mod 4). Then 1 = e1 , ϕ = 2β2 /2 e1 + e2 , ψ = e3 , and GK(T˜) = (0, β2 + 2, β3 ). (b) ε2 ≡ 1(mod 4). Then 1 = e1 , ϕ = 2β2 /2 e1 + e2 , ψ = c1 e1 + c2 e2 + e3 for suitable c1 , c2 ∈ Z2 , and GK(T˜) = (0, β2 + 1, β3 + 1). Proof of Lemma 3.6.5. We distinguish cases.

β+1 . Hence k = Q (π) Case A: Here 2 √ tr(ϕ) = (e1 , e2 ) = 0 and Nm(ϕ) = 2 where π = −2. So k is ramified, Ok = Zp [π] and ϕ = 2β/2 · π, which proves the claim. β2 Case B1: Here = Q2 (π) √ tr ϕ = (e1 , e2 ) = 0 and Nm(ϕ) = ε2 2 . Hence(βk2 −1)/2 · π, with π = −ε2 2. So k is ramified, Ok = Zp [π] and ϕ = 2 which proves the claim.

Case B2a: Here tr(ϕ) = (e1 , 2β2 /2 e1 + e2 ) = 2β2 /2+1 and Nm ϕ = 2β2 (1 + ε2 ). Now by the anisotropy of T˜ we have ε2 ≡ 1(mod 4). Hence k = Q2 [X]/(X 2 − 2X + (1 + ε2 )) is defined by an Eisenstein polynomial. Denoting by π the residue class of X we have Ok = Z2 [π] and ϕ = 2β2 /2 ·π, which proves the claim. Case B2b: This is identical to the previous case. Case B2c: Here tr(ϕ) = 2β2 /2+1 and Nm(ϕ) = 2β2 + ε2 2β2 + ε3 2β3 = 2β2 (2ε3 + 1 + ε2 ). Since ε2 ≡ −1(mod 4), we have k = Q2 [X]/(X 2 − 2X + (2ε3 + 1 + ε2 )), which is defined by an Eisenstein equation. Denoting by π the residue class of X, we have Ok = Z2 [π] and ϕ = 2β2 /2 · π, which proves the claim. Case B3a: Here tr ϕ = 2β2 /2+1 and Nm(ϕ) = 2β2 (1 + ε2 ). Now since T˜ is anisotropic, 1 + ε2 ≡ 4(mod 8). Hence writing 1 + ε2 = 4η we have 2 η ∈ Z× 2 and k = Q2 [X]/(X − X + η). Hence k/Q2 is unramified as asserted and, denoting by ξ the residue class of X, we have Ok = Z2 [ξ] and ϕ = 2β2 /2+1 · ξ, which proves the claim. Case B3b: Here tr(ϕ) and Nm(ϕ) are as in the previous case but this time 1 + ε2 ≡ 2(mod 4). Hence k = Q2 [X]/(X 2 − 2X + (1 + ε2 )) is defined by an Eisenstein polynomial. Denoting by π the residue class of X, we have Ok = Z2 [π] and ϕ = 2β2 /2 · π, which proves the claim. The lemma is now proved in all cases.

65

CYCLES ON SHIMURA CURVES

Proof of Lemma 3.6.6. Again we go through all cases. When p = 2, the endomorphism ψ does not in general anticommute with ϕ (and even if it did, Lemma 2.2 of [14] does not apply when k/Q2 is ramified). Still, the commuting properties of ψ and ϕ can be used to prove Lemma 3.6.6. Case A: Here ϕψ = (eι1 ◦ e2 ) ◦ (eι1 ◦ e3 ) = −eι1 ◦ e2 ◦ (eι3 ◦ e1 ) = −eι1 ◦ (−e3 ◦ eι2 + 2β+1 ) ◦ e1 = eι1 ◦ e3 ◦ eι2 ◦ e1 − 2β+1 = −(eι1 ◦ e3 ) ◦ (eι1 ◦ e2 ) − 2β+1 = −ψϕ − 2β+1 . Here, as usual, x 7→ xι denotes the main involution of D. Hence (3.6.16)

ϕψ + ψϕ = −2β+1 .

Now we can write, following [5], Proposition 4.3, (3.6.17)

D = k ⊕ kj,

where j anticommutes with k and with j 2 ≡ 1 mod 2e−1 , where e is the valuation of the different of k/Q2 , and then (3.6.18)

OD = Ok ⊕ Ok α,

× where α = π 1−e (1 + j) ∈ OD . In the case at hand, the different of k/Qp has valuation 3, hence α = π −2 (1 + j). Writing ψ = a + bα, the assertion of Lemma 3.6.6 comes down to v(b) = a3 . But the LHS of (3.6.16) is equal to (recall ϕ = 2s π),

(3.6.19) 2s (aπ + bπ −1 (1 + j) + aπ + bπ −1 (1 − j)) = 2s+1 · (aπ + bπ −1 ). Noting β = 2s, we get from (3.6.16) the identity (3.6.20)

b = −2s π − aπ 2 .

Comparing valuations of the summands and recalling v(ψ) = β + 1 = a3 , we obtain v(b) = a3 as asserted. Case B1: In this case (3.6.21)

ϕψ − ψϕ = 2 · eι3 ◦ e2 .

× Again D = k + kj and OD = Ok ⊕ Ok α with α = π −2 (1 + j) ∈ OD . s Writing ψ = a + bα, we get for the LHS of (3.6.21), noting ϕ = 2 π,

(3.6.22) 2s (aπ + bπ −1 (1 + j)) − (aπ + bπ −1 (1 − j)) = 2s+1 · bπ −1 j.


66

CHAPTER 3

Hence equation (3.6.21) gives 2s bπ −1 j = eι3 ◦ e2 .

(3.6.23)

Comparing valuations on both sides gives (3.6.24)

2s + v(b) − 1 = β2 + β3 ,

i.e. v(b) = β3 + 2 = a3 , as claimed. Case B2a: In this case ϕψ − ψϕ = 2 · eι3 ◦ e2 .

(3.6.25)

Again D = k + kj and OD = Ok ⊕ Ok α, but this time, since the different has valuation 2, we have α = π −1 (1 + j). Writing ψ = a + bα, we get for the LHS of (3.6.25) and noting ϕ = 2s · π, (3.6.26) 2s (aπ + b(1 + j)) − (aπ + b + bπ ι /π · j) = 2s jb · (2 − 2/π).


Hence equation (3.6.25) gives (3.6.27)

2s jb · (2 − 2/π) = 2 · eι3 ◦ e2 .

Comparing valuations on both sides gives (3.6.28)

2s + v(b) + 1 = β2 + β3 + 2,

i.e., v(b) = β3 + 1 = a3 , as claimed. Case B2b: identical with the previous one. Case B2c: In this case (3.6.29)

ϕψ − ψϕ = 2 · eι3 ◦ e2 .

Again D = k + kj and OD = Ok ⊕ Ok α with α = π −1 (1 + j). Writing ψ = a + bα we obtain by an identical reasoning as in Case B2a that v(b) = β3 + 1 = a3 , as desired. Case B3a: In this case (3.6.30)

ϕψ − ψϕ = 2 · eι3 ◦ e2 .

In this case k/Q2 is unramified. We may choose a uniformizer Π of OD which anticommutes with k and such that Π2 = 2, and then (3.6.31)

OD = Ok ⊕ Ok Π.

67

CYCLES ON SHIMURA CURVES

Now writing ψ = a + bΠ we need to show that v(bΠ) = a3 . Now ϕ = 2s · ξ and the LHS of (3.6.30) is equal to (3.6.32) 2s ((aξ + bξΠ) − (aξ + bξ ι Π)) = 2s b(ξ − ξ ι ) = 2s b(1 − 2ξ) · Π. Hence equation (3.6.30) gives (3.6.33)

2s b · (1 − 2 · ξ ι ) · Π = 2 · eι3 ◦ e2 .

Comparing valuations gives (3.6.34)

2s + v(bΠ) = 2 + β2 + β3 .

Since 2s = β2 + 2, we get v(bΠ) = β3 = a3 , as desired. Case B3b: Again (3.6.35)

ϕψ − ψϕ = 2 · eι3 ◦ e2 .

Here D = k + kj and OD = Ok ⊕ Ok α with α = π −1 (1 + j). Writing ψ = a + bα, we get as in Case B2a (3.6.36)

2β2 /2 · j · b · (2 − 2/π) = 2 · eι3 ◦ e2 .

Hence comparing valuations we get (3.6.37)

β2 + v(b) + 1 = 2 + β2 + β3 ,

i.e., v(b) = β3 + 1 = a3 , as desired. In Chapter 6, when we form the generating series φˆ2 for 0-cycles on M, we will need to associate to every T ∈ Sym2 (Z)∨ and every v ∈ Sym2 (R)>0 an element d 2 (M). ˆ Z(T, v) ∈ CH

When T is positive definite and Diff(T, B) = {p} for some p 6 | D(B), we take the class of (Z(T ), 0) ∈ ZˆZ2 (M), and if T is positive definite 2

d (M). In all other and |Diff(T, B)| > 1, we take the zero class in CH cases, when T is not positive definite, or when T is positive definite and Diff(T, B) = {p} with p | D(B), the definition is less obvious. This issue is dealt with in Chapter 6.

68

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Bibliography

[1] ARGOS (Arithmetische Geometrie Oberseminar), Proceedings of the Bonn seminar 2003/04, forthcoming. [2] J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura, in Courbes Modulaires et Courbes de Shimura, Ast´erisque, 196–197, 45–158, Soc. Math. France, Paris, 1991. [3] I. Bouw, Invariants of Ternary Quadratic Forms, in [1]. [4] B. Gross, Heegner points on X0 (N ), in Modular Forms, Ellis Horwood Ser. Math. and Its Appl.: Statist. Oper. Res., 87–105, Horwood, Chichester, 1984. [5]

, On canonical and quasi-canonical liftings, Invent. math., 84 (1986), 321–326.

[6] B. Gross and K. Keating, On the intersection of modular correspondences, Invent. math., 112 (1993), 225–245. [7] B. H. Gross, and D. Zagier, Heegner points and the derivatives of L-series, Invent. math., 84 (1986), 225–320. [8] K. Keating, Lifting endomorphisms of formal A-modules, Compositio Math., 67 (1988), 211–239. [9] S. Kudla, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646. [10]

, Modular forms and arithmetic geometry, in Current Developments in Mathematics, 2002, 135–179, International Press, Somerville, MA, 2003.

[11] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223. [12] S. Kudla, M. Rapoport, and T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887–951. [13] G. Laumon and L. Moret-Bailly, Champs alg´ebriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 39, Springer-Verlag, Berlin, 2000. [14] M. Rapoport, Deformations of isogenies of formal groups, in [1]. [15] I. Vollaard, Endomorphisms of quasi-canonical lifts, in [1].

BIBLIOGRAPHY

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[16] M. Rapoport and T. Zink, Period Spaces for p-Divisible Groups, Annals of Mathematics Studies, 141, Princeton Univ. Press, Princeton, NJ, 1996. [17] S. Wewers, Canonical and quasi-canonical liftings, in [1] [18] T. H. Yang, Local densities of 2-adic quadratic forms, J. Number Theory, 108 (2004), 287–345. [19] T. Zink, The display of a formal p-divisible group, in Cohomologies p-adiques et applications arithm´etiques, I. Ast´erisque 278, 127–248, Soc. Math. France, Paris, 2002.

Chapter Four An arithmetic theta function In this chapter, we consider the generating function φˆ1 whose coefficients d 1 (M) defined in the previous chapter. ˆ v) ∈ CH are the special divisors Z(t, We prove that φˆ1 is a modular form of weight 23 (Theorem A). In the first section we introduce the decomposition (4.0.1)

1

d (M) = MW g ⊕ Rω CH ˆ ⊕ Vert ⊕ a(A0 (MR )0 )

1

d (M) into its ‘Mordell-Weil’ component, its ‘Hodge bundle’ comof CH ponent, its ‘vertical’ component, and its ‘C ∞ ’-component. We then reduce the proof of Theorem A to an assertion about the various components of φˆ1 with respect to this decomposition. The modularity of the Hodge component follows from [16]. In Section 4.3, we prove the modularity of the vertical component of φˆ1 by identifying it with the theta function of a positive definite ternary form. In Section 4, we prove the modularity of the C ∞ -component of φˆ1 by identifying it with the theta lift of a Maass form. In Sections 4.5 and 4.6, we show that the modularity of the Mordell-Weil component of φˆ1 follows from Borcherds’ theorem [2]. In the last section, we check an intertwining property of the arithmetic theta function which will be crucial in the definition of the arithmetic theta lift in Chapter 9. 4.1 THE STRUCTURE OF ARITHMETIC CHOW GROUPS

Let M be an arithmetic surface over Spec Z, as in Chapter 2. We assume that M is geometrically irreducible. In this section, we review the structure of the arithmetic Chow group of M and set up a convenient decomposition of this group with respect to the arithmetic intersection pairing or ArakelovGillet-Soul´e height pairing. d 1 (M) be the arithmetic Chow group with real coefficients as deLet CH fined in Chapter 2. Recall that this is the quotient of the real vector space Zb 1 (M)R spanned by pairs (Z, g), where Z ∈ Z 1 (M)R is a divisor on M with real coefficients and g is a Green function for Z, by the subspace of c ) = (div(f ), − log |f |2 ) for relations spanned over R by the elements div(f × ˆ (resp. ω(Z)) ˆ for the Green f ∈ Q(M) . We will use the notation g(Z)

72

CHAPTER 4

ˆ These are related by function (resp. (1, 1)-form) associated to a class Z. ˆ + δZ = ω(Z). ˆ ddc g(Z) We will only consider Green functions of C ∞ -regularity, in the terminology d 1 (M) comes of [1], p. 18, as in [5]. As explained in Chapter 1, the group CH equipped with an arithmetic intersection pairing, the Arakelov-Gillet-Soul´e height pairing h , i, which we will also refer to as the height pairing. In d 1 (M) → R obtained addition, there is a geometric degree map degQ : CH as a composition (4.1.1)

resQ

1

deg

d (M) −→ CH1 (MQ ) ⊗ R −→ R, degQ : CH

where CH1 (MQ ) is the usual Chow group of the generic fiber MQ of M. We fix a volume form µ1 on M(C), with vol(M(C), µ1 ) = 1. We also fix a metrized line bundle ω ˆ = (ω, || ||) on M with first Chern form c1 (ˆ ω) = degQ (ω) · µ1 with degQ (ω) > 0, and denote by the same symbol 1

(4.1.2)

d (M) ω ˆ ∈ CH 1

c d (M). Let A0 (MR ) the associated class under the map Pic(M) → CH be the space of C ∞ -functions on M(C) invariant under the archimedean Frobenius F∞ , and let

(4.1.3)

1

d (M), a : A0 (MR ) −→ CH

f 7→ (0, f ),

be the inclusion. Note that this map is R-linear. Let 1

(4.1.4)

d (M). 11 := a(1) ∈ CH

Let A0 (MR )0 be the subspace of f ∈ A0 (MR ) such that Z

(4.1.5)

M(C)

f · c1 (ˆ ω ) = 0.

Let (4.1.6)

Vert = Vert(M) 1

d (M) generated by the classes of the form (Yp , 0), be the subspace of CH c where Yp is an irreducible component of a fiber Mp . The relation div(p) = 2 (Mp , − log(p )) ≡ 0, implies that (Mp , 0) ≡ 2 log(p) 11, so that 11 ∈ Vert. Let

(4.1.7)

Vertp = ⊕i R Yp,i

73

AN ARITHMETIC THETA FUNCTION

be the real vector space with basis the irreducible components Yp,i of the fiber Mp , and let (4.1.8)

Vertp = Vertp /(R · Mp )

and

Vert = Vert/(R · 11).

Then we have a commutative diagram with exact rows and columns R −→ ⊕p R −→ R 1 || ↓ ↓ ↓ R −→ ⊕p Vertp −→ Vert 11 ↓ ↓ ⊕p Vertp −→ Vert

(4.1.9)

where (4.1.10)

X

R = {(λp ) ∈ ⊕p R |

λp log(p) = 0}.

p

The inclusion map on R in the middle line is given by (4.1.11)

(λp ) 7→

X

λp Mp ,

p

while the second arrow in the top line is given by (λp ) 7→ 2 Of course, Vert lies in the kernel of degQ .

P

p λp log(p).

1

d (M) can be written in general as The intersection pairing h ·, · i on CH

(4.1.12)

1 h Zˆ1 , Zˆ2 i = hLˆ1 (Z2 ) + 2

Z M(C)

g(Zˆ2 ) ω(Zˆ1 ),

where hLˆ1 (Z2 ) is the height of the cycle Z2 with respect to the metrized line bundle Lˆ1 corresponding to Zˆ1 ; see Section 2.6 or (5.11) of [1]. The alternative expression h Zˆ1 , Zˆ2 i d 1 · Z2 ) + 1 (4.1.13) = deg(Z

Z

d 1 · Z2 ) + 1 = deg(Z

Z

2

M(C)

g(Zˆ1 ) ∗ g(Zˆ2 )

1 g(Zˆ1 ) δZ2 + 2 M(C) 2

Z M(C)

g(Zˆ2 ) ω(Zˆ1 ),

from [1] (5.8), is valid when the supports |Z1 | and |Z2 | are disjoint on the generic fiber; see Section 2.5. In particular, using (4.1.13), we have (4.1.14)

ˆ 11 i = 1 degQ (Z), ˆ h Z, 2

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CHAPTER 4

so that (4.1.15)

hω ˆ , 11 i =

1 degQ (ω) 2

and

h 11, 11 i = 0.

For f ∈ A0 (MR )0 , we have (4.1.16)

hω ˆ , a(f ) i = h 11, a(f ) i = 0,

since, via (4.1.13), (4.1.17)

hω ˆ , a(f ) i =

1 2

Z M(C)

f · c1 (ˆ ω ) = 0.

Also (4.1.18)

h a(f1 ), a(f2 ) i =

1 2

Z M(C)

f2 · ddc f1 .

Again using (4.1.13), on Vert we have (4.1.19)

hω ˆ , (Yp , 0) i = deg(ω|Yp ) · log(p),

where deg(ω|Yp ) is the degree of the restriction of the line bundle ω to Yp . For fixed p, and denoting by mi the multiplicities of the irreducible components Yi of Mp , (4.1.20)

X

mi · h ω ˆ , (Yp,i , 0) i = h ω ˆ , (Mp , 0) i = degQ ω · log(p),

i

and (4.1.21)

h 11, Vert i = 0.

Let (4.1.22)

MW = MW(M) := Jac(M )(Q) ⊗Z R

be the Mordell-Weil space of the generic fiber MQ . Here Jac(M ) denotes the neutral component of the Picard variety of M . Definition 4.1.1. Let g := MW



Rω ˆ ⊕ Vert ⊕ a(A0 (MR )0 )

⊥

⊂

1

d (M) CH

be the orthogonal complement of R ω ˆ ⊕ Vert ⊕ a(A0 (MR )0 ) with respect to the height pairing. The following result is well known [8], [4], [1].

75

AN ARITHMETIC THETA FUNCTION

Proposition 4.1.2. (i) 1

d (M) = MW g ⊕ Rω CH ˆ ⊕ Vert ⊕ a(A0 (MR )0 ),


and the three summands are orthogonal with respect to the height pairing. (ii) The restriction res Q to the generic fiber induces an isomorphism ∼

g −→ MW res Q : MW

g and the which is an isometry for the Gillet-Soul´e height pairing on MW negative of the Neron-Tate height pairing on MW.

Remark 4.1.3. (i) Note that the finite dimensional real vector space g ⊕ Rω CH1 (M, µ1 ) = MW ˆ ⊕ Vert

(4.1.23)

is the Arakelov Chow group with respect to the normalized volume form µ1 = c1 (ˆ ω )/ deg(ω), except that we have taken real coefficients. Thus we have the decomposition (4.1.24)

1

d (M) = CH1 (M, µ1 ) ⊕ a(A0 (MR )0 ). CH

(ii) If Z is a 0-cycle of degree zero on the generic fiber MQ , let Z be a divisor with rational coefficients on M with Z Q = Z and with Z · Yp = 0 for all irreducible components Yp of closed fibers Mp . Such an extension of Z is unique up to the addition of a finite linear combination of Mp ’s. Let gZ be the µ1 -admissible antiharmonic Green function for Z. Then (4.1.25)

d 1 (M) Zˆ = (Z, gZ ) ∈ CH

lies in (4.1.26)



0

Vert ⊕ a(A (MR )0 )

⊥

There is a real scalar κ ∈ R such that (4.1.27)

hω ˆ , Zˆ + a(κ) i = 0.

Once the extension Z has been chosen, then (4.1.28)

κ = −2 h ω ˆ , Zˆ i.

The resulting class (4.1.29)

g Z˜ := Zˆ + a(κ) ∈ MW

.

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CHAPTER 4

g of the class of Z in MW. It is independent of the is the preimage in MW choice of Z.

We can use the height pairing to write the components in the decomposition (i) in Proposition 4.1.2 more explicitly. For each prime p for which the fiber Mp is not irreducible, choose Yp,i , i = 1, . . . , rp such that the images Y p,i are a basis of Vertp . Recall that ∨ RMp is the radical of the intersection form on Vertp . Let Y p,i be a dual basis ∨ be the preimage for Vertp with respect to the intersection form and let Yp,i ∨ of Y p,i in Vertp such that ∨ hω ˆ , (Yp,i , 0) i = 0.

(4.1.30) For convenience, we write (4.1.31)

1

∨ ∨ d (M), yp,i = (Yp,i , 0) ∈ CH

yp,i = (Yp,i , 0),

and (4.1.32)

ω ˆ 1 = deg(ω)−1 ω ˆ.

Then, if zˆ is any class in Rˆ ω ⊕ Vert, we have the decomposition (4.1.33)

zˆ = degQ (ˆ z) · ω ˆ1 +

rp XX

∨ h zˆ, yp,i i · yp,i + 2 κ(ˆ z ) · 11,

p i=1

where degQ (ˆ z ) = 2 h zˆ, 11 i and (4.1.34)

κ(ˆ z ) = h zˆ, ω ˆ 1 i − degQ (ˆ z) h ω ˆ1, ω ˆ 1 i.

Next consider the archimedean component. Suppose that there is a ‘uniformization’ (4.1.35)

M(C) ' [ Γ\H ],

where H is the upper half plane and Γ ⊂ SL2 (R) is a Fuchsian group. If Γ has elements of finite order, we understand the quotient as an orbifold.1 In addition, we assume that µ1 = vol(M(C), µ)−1 µ where µ = (2π)−1 y −2 dx ∧ dy is the hyperbolic volume form on H. Recall that, for f ∈ A0 (M(C)), (4.1.36) 1

ddc f =

1 ∆f · µ 2

This explains the quotation marks on the term ‘uniformization’.

77

AN ARITHMETIC THETA FUNCTION

where ∆ is the hyperbolic Laplacian. The space A0 (MR ) is spanned by eigenfunctions of the Laplacian fλ satisfying (4.1.37)

∆fλ + λ fλ = 0,

with eigenvalues 0 = λ0 < λ1 ≤ λ2 ≤ . . . and orthonormal with respect to 1 µ. In particular, f0 = vol(M(C), µ)− 2 · 11. For the height pairing, we have, by (4.1.18), (4.1.38)

h a(fλi ), a(fλj ) i = −

1 4

1 fλi · λj fλj µ = − λj δij . 4 M(C)

Z

In summary,

d (M) has a unique decomposition: Proposition 4.1.4. Any zˆ ∈ CH 1

r

zˆ = zˆMW +

p XX degQ (ˆ z) ∨ h zˆ, yp,i i · yp,i ·ω ˆ+ degQ (ˆ ω) p i=1

+ 2 κ(ˆ z ) · 11 − 4 where g and zˆMW ∈ MW.

X

λ−1 h zˆ, a(fλ ) i · a(fλ ).

λ>0

κ(ˆ z ) = h zˆ, ω ˆ 1 i − degQ (ˆ z) h ω ˆ1, ω ˆ 1 i,

4.2 THE ARITHMETIC THETA FUNCTION

In this section, we consider the case of the arithmetic surface M attached to an indefinite division quaternion algebra B over Q, and we define a generd 1 (M). We use the setup and notation of ating function φˆ1 (τ ) valued in CH Chapter 3. In particular, D(B) > 1 is the product of the primes p at which B is ramified and M has good reduction for p - D(B). We let ω ˆ be the Hodge line bundle on M with metric normalized as in Section 3.3. For each t ∈ Z and positive real number v, there is a class (4.2.1)

1

b v) ∈ CH d (M), Z(t,

defined in Section 3.5; see also [16]. For example, if t > 0, then (4.2.2)

b v) = (Z(t), Ξ(t, v)), Z(t,

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CHAPTER 4

where Z(t) is defined in terms of special endomorphisms and Ξ(t, v) is the Green function given in Corollary 3.5.2. On the other hand, for t < 0, (4.2.3)

b v) = (0, Ξ(t, v)), Z(t,

for a smooth function Ξ(t, v). Finally, (4.2.4)

b v) = −ˆ Z(0, ω − a(log(v)) + a(c),

where c is the real constant determined by the identity (4.2.5)  X p log(p)  ζ 0 (−1) 1 degQ (ˆ ω )·c = h ω ˆ, ω ˆ i−ζD (−1) 2 +1−2C − . 2 ζ(−1) p−1 p|D(B) In Chapter 7, we will prove that, in fact, c = − log D(B). We form the generating series (4.2.6)

φb1 :=

X

b v) q t Z(t,

1

d (M)[[q]]. ∈ CH

t

As explained in [12] and [14], we refer to this series as an arithmetic theta function. The main result of this chapter is the following:

Theorem A. The series φb1 is the q-expansion of a nonholomorphic elliptic modular form of weight 23 for the subgroup Γ0 = Γ0 (4D(B)o ) of SL2 (Z) d (M) ⊗R C. valued in CH 1

This means that there is a smooth function φAr of τ ∈ H, valued in the finite dimensional complex vector space CH1 (M, µ1 )C , and a smooth function φan (τ, z) on H × M(C), with Z M(C)

φan (τ, z) dµ(z) = 0,

such that the sum φ(τ ) = φAr (τ ) + φan (τ, z) satisfies the usual transformation law for a modular form of weight 23 for Γ0 (4D(B)o ) and such that the q-expansion of φ(τ ) is the formal generating series φb1 defined above. Of course, the coefficients in the q-expansion of φ(τ ) are functions of v. We will abuse notation and write φb1 (τ ) for both the function φ(τ ) and for the q-expansion φb1 . The proof of Theorem A is based on the application of the decomposition of Proposition 4.1.2 and Proposition 4.1.4 to φb1 . This gives rise to the component functions (4.2.7)

φdeg = degQ (φb1 ) =

X t

degQ (Z(t)) q t ,

79

AN ARITHMETIC THETA FUNCTION

(4.2.8)

h φb1 , ω ˆi=

X

b v), ω h Z(t, ˆ i qt,

t

(4.2.9)

h φb1 , yp,i i =

X

b v), yp,i i q t , h Z(t,

t

(4.2.10)

h φb1 , a(fλ ) i =

X

b v), a(fλ ) i q t , h Z(t,

t

g or, equivalently, in MW, and, finally, the component in MW,

(4.2.11)


φMW = resQ φb1 (τ ) − φdeg (τ ) · ω ˆ 1 ∈ MW ⊗ C.

It will be shown that each of these component functions is modular of weight 3 0 b 2 for Γ , and hence so is φ1 . The modularity of (4.2.7) and (4.2.8) is proved in [16]. More precisely, let E1 (τ, s; D(B)) be the Eisenstein series of weight 32 defined in Section 6 of [16]. Then, Proposition 7.1 of [16] asserts that (4.2.12)

1 φdeg (τ ) = E1 (τ, ; D(B)), 2

and hence gives the modularity of (4.2.7). Here and elsewhere, the meaning of this equation is that the q-expansion of E1 (τ, 21 ; D(B)) is given by φdeg . The main result, Theorem 7.2 of [16], asserts that (4.2.13)

1 h φb1 (τ ), ω ˆ i = E10 (τ, ; D(B)). 2

This gives the modularity of (4.2.8). The vertical, archimedean, and MordellWeil components will be handled in the following three sections. 4.3 THE VERTICAL COMPONENT: DEFINITE THETA FUNCTIONS 1

d (M), In this section, for a prime p | D(B) and a vertical element yp,i ∈ CH b as in (4.1.31), we prove that the component function h φ1 (τ ), yp,i i is a modular form of weight 23 . Our tool will the p-adic uniformization of M and of the special divisors Z(t) recalled in Section 3.2. Recall that B 0 = B (p) denotes the definite quaternion algebra over Q whose local invariants coincide with those of B at all places v 6= p, ∞, and V 0 = {x ∈ B 0 | tr(x) = 0}. For H 0 = (B 0 )× , we have fixed isomorphisms

(4.3.1)

H 0 (Qp ) ' GL2 (Qp ) and H 0 (Apf ) ' H(Apf ).

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CHAPTER 4

b × = K p Kp with K p ⊂ H(Ap ) and Kp ⊂ H(Qp ), and We have K = O B f 0 0 also K 0 = K p Kp0 , where K p = K p under the isomorphism (4.3.1), and ¯ p ) of the formal ˆ W be the base change to W = W (F Kp0 = GL2 (Zp ). Let M ˆ W the completion of M along its special fiber at p. We also denote by Ω base change of the Drinfeld upper half space to W . For convenience of ˆ• = Ω ˆ W × Z, the disjoint sum of copies of Ω ˆW notation, we introduce Ω W parametrized by Z. We then have

ˆ W ' H 0 (Q) \ Ω ˆ •W × H 0 (Ap )/K 0 p . M f

(4.3.2)





From the uniformization (4.3.2), one obtains the following well-known description of the irreducible components over Fp of the special fiber Mp of M. Note that these correspond to the irreducible components of the ˆ W . Recall that γ 0 ∈ H 0 (Q) acts on Ω ˆ • × H(Ap )/K p by special fiber of M W f γ 0 : [ ξ ]i × hK 7→ [ γ ξ ]i+ordp (ν(γ)) × γhK,

(4.3.3)

ˆ W . The components of the special ˆ i of ξ ∈ Ω where [ξ]i is the image in Ω W ˆ p of Ω ˆ W are projective lines P[Λ] indexed by the vertices [Λ] of the fiber Ω building B for PGL2 (Qp ), where [Λ] is the homothety class of a Zp -lattice ˆ • , and, Λ in Q2p . Thus, GL2 (Qp ) acts transitively on the components of Ω W if Λ0 = Z2p is the standard lattice, the stabilizer in GL2 (Qp ) of [ P[Λ0 ] ]0 is Kp0 = GL2 (Zp ). Proposition 4.3.1. There is a bijection 0

0

0

∼

H (Q)\H (Af )/K −→



irreducible components , of Mp 

H 0 (Q)gK 0 7−→ H 0 (Q)-orbit of gp [ P[Λ0 ] ]0 × g p K p .

In particular, the irreducible components of the special fiber Mp are indexed by the double cosets in the decomposition H 0 (Af ) =

a

H 0 (Q)gj K 0 .

j

We will frequently refer to the pair [Λ]i × gK p as the data associated to the component [P[Λ] ]i × gK p . We next recall from [16] the p-adic uniformization of the cycles Z(t); cf. also Section 6.2. Fix t ∈ Z>0 and let Z(t) be the corresponding divisor on M; see Section 3.4. Loosely speaking, Z(t) is the locus where the OB abelian variety is equipped with a special endomorphism of square −t. Let (4.3.4)

C(t) = Z(t) ×Spec

Z

Spec Z(p) ,

81

AN ARITHMETIC THETA FUNCTION

and let d × Cˆ = C(t) Spf Zp Spf W,

(4.3.5)

ˆ W determined by C(t). Let be the cycle in M Vt0 (Q) = { x ∈ V 0 (Q) | Q(x) = −x2 = t }. ˆ Then, (8.17) of [15] gives a p-adic uniformization of C: Cˆ (4.3.6)



H 0 (Q)\

,→

Vt0 (Q)

↓ ˆW M

ˆ • × H(Ap )/K p ×Ω W f



↓ 

∼

H 0 (Q)\

−→



ˆ • × H(Ap )/K p , Ω W f

where the image of Cˆ in the upper right corner is the set of triples (

(4.3.7)

) b p , and (i) g −1 xg ∈ V (Apf ) ∩ O B . (x, (X, ρ), gK ) • p

(ii) for j = j(x), (X, ρ) ∈ Z (j)

ˆ • with the formal moduli space of quasi-isogenies Here we have identified Ω W ρ : X −→ X of special formal OB -modules, where X is the base point and where we have fixed an identification of V 0 (Qp ) with the traceless elements in EndOB (X). Then j(x) is the special endomorphism of the p-divisible ˆ • , as group X, determined by x, and Z • (j) is the corresponding cycle in Ω W defined in [15] (the locus of ρ’s where the isogeny j(x) of X extends to an endomorphism of X). Using the p-adic uniformization just described, we compute the intersection number (C(t), Yp ) for an irreducible component Yp of the special fiber Mp . Proposition 4.3.2. Write t = −εpα , with ε ∈ Z× p and α ∈ Z. Then, for an irreducible component Yp of Mp determined by data [Λ]i × gK p , as above,


C(t), Yp =

X

ϕp (g −1 x) µ[Λ] (x).

0 (Q)

x∈V Q(x)=t

b p and the multiHere ϕp denotes the characteristic function of V (Apf ) ∩ O B plicity µ[Λ] (x) is given as follows: Let j = j(x) be the endomorphism of Q2p associated to x. Let e ≥ f , with e, f ∈ Z be the elementary divisors of the lattice pair (Λ, j(Λ)). Note that

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CHAPTER 4

e + f = α = ordp (det(j)), f ≤ 2 , and that r = 12 (e − f ) = d([Λ], B j ) is the distance from [Λ] to the fixed point set B j of j on B. Then, for α > 0, α

µ[Λ] (x) =

 1−p   

1

  

For α = 0, and p 6= 2, µ[Λ] (x) =

0

if f = 0, if f < 0.

 1 − χp (ε)

For α = 0, and p = 2, µ[Λ] (x) =

if f > 0,

0

 1−p   

1

  

0

if f = 0,

if f < 0.

if f = 0 and (1 + j)(Λ) ⊂ 2Λ,

if f = 0 and (1 + j)(Λ) 6⊂ 2Λ, otherwise.

Here χp (ε) = (ε, p)p where ( , )p is the quadratic Hilbert symbol for Qp . ˆ W corresponding to Yp . Proof. Let Yˆ be the irreducible component of M i p ˆ Fixing the preimage [P[Λ] ] ×gK of Y , we must calculate the total intersecˆ • × H(Ap )/K p , tion number of this curve with the full preimage of Cˆ in Ω W f as described by the incidence set (4.3.7) for the upper right corner in (4.3.6) above. First, for a given (x, (X, ρ), g0 K p ) to contribute, we must have g0 K p = gK p and the height of the quasi-isogeny ρ must be the given i. The first incidence condition becomes ϕp (g −1 x) 6= 0. It remains to calcuˆ W . We write late the intersection number of P[Λ] and Z(j), j = j(x) on Ω h v Z(j) = Z(j) + Z(j) as the sum of horizontal and vertical parts. For p 6= 2, Lemma 4.9 of [15] gives (Z(j)h , P[Λ] ), and Lemma 6.2 of [15] gives (Z(j)v , P[Λ] ), taking into account the relation between the elementary divisors e,f and the distance r, described above; see also Lemma 2.4 of [15]. This yields the result in the case p 6= 2. For p = 2, we use the results of the appendix to Chapter 6. There the cycle Z(j) is described in terms of ‘cycle data’ (S, µ, Z h ); see Section 6A.2 of the appendix to Chapter 6. Here S is a subset of the building (parametrizing the ‘central vertical components’), the integer µ is the common multiplicity of the central vertical components, and Z h denotes the horizontal part of the

83

AN ARITHMETIC THETA FUNCTION

divisor Z(j). Remark (I) right before (6A.4.12) gives (4.3.8) (Z(j)v , P[Λ] ) =

  1−p      1    

0

if 1 ≤ d([Λ], S) ≤ µ − 1, i.e., [Λ] is regular in Z(j), if d([Λ], S) = µ, if d([Λ], S) > µ,

provided µ > 0. If µ > 0 and d([Λ], S) = 0, i.e., [Λ] ∈ S, then (Z(j)v , P[Λ] ) = χ2 (j) − p,

(4.3.9)

where χ2 (j) is equal to 1, −1, or 0, depending on whether the extension of Q2 generated by j is split, unramified, or ramified. Finally, for µ = 0, we have (Z(j)v , P[Λ] ) = 0. As for the intersection number (Z(j)h , P[Λ] ), we have, by a case by case inspection of the cases (1) to (4) in Section A.2 of the appendix, (

(4.3.10)

h

(Z(j) , P[Λ] ) =

1 − χ2 (j) if [Λ] ∈ S, 0 otherwise.

Summing, we obtain

(4.3.11)

  1−p    1

if µ ≥ 1 and d([Λ], S) ≤ µ − 1, if µ ≥ 1 and d([Λ], S) = µ, (Z(j), P[Λ] ) =  1 − χ2 (j) if µ = d([Λ], S) = 0,     0 in all other cases.

We now note, by checking case by case in Section A.2 of the appendix, that (4.3.12)

d([Λ], S) ≤ µ ⇐⇒ d([Λ], B j ) ≤

α ⇐⇒ f ≥ 0. 2

Similarly, d([Λ], S) ≤ µ−1 is equivalent to f > 0. When α > 0, then either µ > 0 and the assertion in the proposition follows from formula (4.3.11), or α = 1 and j 2 = ε · 2 (i.e., j 2 is of type (4) in the terminology of the appendix). In this case, µ = 0, and d([Λ], S) = 0 is equivalent to d([Λ], B j ) = 12 , i.e., (e, f ) = (1, 0), and formula (4.3.11) gives µ[Λ] (x) = 1. Now, in this case, the two vertices in S correspond to the lattices given by the ring of integers O in the ramified extension Q2 (j) and to πO, where π denotes a uniformizer in O. For either of them (1 + j)(Λ) = Λ. Hence the value for µ[Λ] (x) obtained above confirms the statement in the proposition. Suppose that α = 0. Then, when ε is of type (1) or (2), we have µ = 1 and formula (4.3.11) gives the value 1 − p, 1, or 0, depending on whether

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CHAPTER 4

d([Λ], S) = 0, d([Λ], S) = 1, or d([Λ], S) > 1. The first case is characterized by j(Λ) = Λ, (1 + j)Λ ⊂ 2Λ. The second case is characterized by j(Λ) = Λ, (1 + j)Λ 6⊂ 2Λ. This confirms the claim of the proposition in this case. When α = 0 and ε is of type (3), we have µ = 0 and formula (4.3.11) gives 1 if [Λ] ∈ S and 0 otherwise. The first alternative is characterized by j(Λ) = Λ. Again the two lattices with j(Λ) = Λ are given, up to scalar multiples, by the ring of integers O in the ramified extension Q2 (j) and by πO. Hence (1 + j)O = πO so that (1 + j)(Λ) 6⊂ 2Λ, which again confirms the assertion of the proposition in this case. The multiplicity µ[Λ] (x) can be expressed in terms of a Schwartz function on V 0 (Qp ) as follows. Let ϕ[Λ] be the characteristic function of the set {x ∈ V 0 (Qp ) | j(Λ) ⊂ Λ, for j = j(x) }.

(4.3.13)

Note that, with the notation above, (4.3.14)

ϕ[Λ] (x) 6= 0 ⇐⇒ f ≥ 0,

ϕ[Λ] (p−1 x) 6= 0 ⇐⇒ f > 0,

0 = g GL (Z )g −1 on V 0 (Q ) and ϕ[Λ] is invariant under the action of K[Λ] p 2 p p p by conjugation, where Λ = gp Λ0 . Thus, if x ∈ V 0 (Qp ) with Q(x) = t, and excluding the case α = f = 0, we can write

µ[Λ] (x) = ϕ[Λ] (x) − p ϕ[Λ] (p−1 x).

(4.3.15)

− 0 Also, let ϕ+ [Λ] , ϕ[Λ] , and ϕ[Λ] be the characteristic functions of the following sets,

(4.3.16)

XΛ± = { x ∈ V 0 (Qp ) | j(Λ) = Λ, and χp (det(j)) = ±1 } XΛ0 = { x ∈ V 0 (Qp ) | j(Λ) = Λ, (1 + j)(Λ) ⊂ 2Λ }.

These are again Schwartz functions on V 0 (Qp ), invariant under the action 0 on V 0 (Q ) by conjugation. of K[Λ] p Then, we have,

Lemma 4.3.3. (i) For all x ∈ V 0 (Qp ), the function µ[Λ] ∈ S(V 0 (Qp )) is given by µ[Λ] (x) =

 − ϕ[Λ] (x) − p ϕ[Λ] (p−1 x) − ϕ+ [Λ] (x) + ϕ[Λ] (x)

ϕ[Λ] (x) − p ϕ[Λ] (p−1 x) − p ϕ0[Λ]



(ii) If Λ = gp Λ0 , then

µ[Λ] (x) = µ[Λ0 ] (gp−1 x).

if p = 6 2,

if p = 2.

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AN ARITHMETIC THETA FUNCTION

Finally, we can consider the component function of (4.2.9): (4.3.17)

h φb1 (τ ), yp i =

X

b v), yp i q t , h Z(t,

t

= −(ω, Yp ) log(p) + log(p)

X

(Z(t), Yp ) q t

t>0

where, for the constant term, we recall (4.2.4) and (4.1.17). Cancelling the log(p), we define (4.3.18)

φVert (τ ; Yp ) :=

X

(C(t), Yp ) q t ,

t≥0

where we have set C(0) = −ω.

Theorem 4.3.4. Let Yp be the component of Mp corresponding to the double coset H 0 (Q)gK 0 under the bijection of Proposition 4.3.1. Let ϕ0p = µ[Λ0 ] ∈ S(V 0 (Qp ))

be the Schwartz function described in Lemma 4.3.3, and let ϕ0 = ϕ0p ϕp ∈ b p . Then S(V 0 (Af )), where ϕp is the characteristic function of V 0 (Apf ) ∩ O B φVert (τ ; Yp ) = θ(τ, g; ϕ0 ) :=

X

ϕ0 (g −1 x) q Q(x)

x∈V 0 (Q)

is the theta function of weight 23 for the data ϕ0 and g ∈ H 0 (Af ) for the positive definite quadratic space V 0 . Remark 4.3.5. It remains to check that the theta function θ(τ, g; ϕ0 ) is a modular form for Γ0 (4D(B)o ), as claimed in Theorem A. Equivalently, we must determine the compact open subgroup of the metaplectic extension of SL2 (Qp ) which fixes the Schwartz function ϕ0p . This will be done in Section 8.5, where the necessary information about the Weil representation is reviewed. Proof. For t > 0, Proposition 4.3.2 and (ii) of Lemma 4.3.3 give

(4.3.19)

(C(t), Yp ) =

X

ϕ0 (g −1 x).

0 (Q)

x∈V Q(x)=t

On the other hand, Proposition 11.1 of [16] gives (4.3.20)

(ω, Yp ) = − (p − 1) = ϕ0 (0).

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CHAPTER 4

Corollary 4.3.6. The component functions h φb1 (τ ), yp,i i of (4.2.9) are holomorphic theta functions of weight 32 attached to V 0 = V (p) . Remark 4.3.7. (i) The special fiber of M is reduced. Hence we have X

h φb1 (τ ), (Yp,i , 0) i = h φb1 (τ ), Mp i

i

(4.3.21)

= 2 log(p) · h φb1 (τ ), 11 i = log(p) · φdeg (τ ) 1 = log(p) · E(τ, , D(B)). 2

On the other hand, by the Siegel-Weil formula for the anisotropic ternary space V (p) , the sum of the theta functions is the Eisenstein series, (4.3.22)

X i

1 θ(τ, g; ϕ0 ) = E1 (τ, , D(B)). 2

The coincidence of the two formulas is an example of the ‘matching’ identity for the Siegel-Weil formula described in [13]. Up to the factor of log(p), the modular forms h φb1 (τ ), (Yp,i , 0) i are the theta series of weight 23 for the classes, as i varies, in the K 0 -genus for the positive definite ternary space V 0 = V (p) . (ii) Note that the key fact in this proof is that the degree −(p − 1) of the restriction of ω to a line P[Λ] is equal to the intersection number (P[Λ] , C(t)), provided that the vertex [Λ] is regular in C(t) in the sense of the appendix to Chapter 6. This equality seems quite mysterious. (iii) Theorem 4.3.4 carries over to the situation where a level structure away from p is imposed, as in [15]. More precisely, let K p ⊂ H(Apf ) be any × open compact subgroup, but keep Kp = OB , as before. Let K = K p .Kp . p Then we have the moduli space AK over Spec Z(p) as in [15], which again admits a p-adic uniformization. Let ϕp ∈ S(V (Apf )) be the characteristic function of a K p -invariant compact open subset of V (Apf ). (This subset was denoted by ω in [15], but here we avoid this notation because it is already in use.) For t ∈ Z(p),>0 there is a cycle C(t, ϕp ) on AK . Note the slight shift in notation from [15], where the quadratic form had the opposite sign and b p,× , the ω was written in place of ϕp . In the case of M, i.e., when K p = O B b B , and function ϕp is the characteristic function of V (Apf ) ∩ O (4.3.23)

C(t, ϕp ) = Z(t) ×Spec

Z

Spec Z(p) .

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AN ARITHMETIC THETA FUNCTION

We can then form the generating series associated to an irreducible component Yp of (AK )p , as in (4.3.18). Let (4.3.24)

C(0, ϕp ) = −ϕp (0) · ω,

and define the generating function (4.3.25)

φVert (τ, ϕp ; Yp ) :=

X

(C(t, ϕp ), Yp ) q t .

t≥0

Then we can again identify this function with a theta function. Let Yp correspond to the double coset H 0 (Q)gK 0 , and let as before ϕ0p = µ[Λ0 ] ∈ S(V 0 (Qp )) be the Schwartz function described in Lemma 4.3.3, and let ϕ0 = ϕ0p ϕp ∈ S(V 0 (Af )). Then (4.3.26)

φVert (τ, ϕp ; Yp ) = θ(τ, g; ϕ0 ) :=

X

ϕ0 (g −1 x) q Q(x) .

x∈V 0 (Q)

(iv) In his Montreal article [7], Gross gave an analogue of the GrossZagier formula for the central value of the L-function, in the case of root number +1. This should have a natural interpretation here in terms of the geometry of components of the fiber Mp . 4.4 THE ANALYTIC COMPONENT: MAASS FORMS

In this section, we consider the component functions h φb1 (τ ), a(fλ ) i of (4.2.10) associated to eigenfunctions of the Laplacian. In this section we allow an arbitrary level structure K ⊂ (B ⊗ Af )× , although for Theorem A b × is needed. only the case K = O B First we review the definition of the Green function of Section 3.5 and [11], where more details can be found. For H = B × , as in Chapter 3, and for any compact open subgroup K ⊂ H(Af ), the complex points of the associated Shimura curve MK over Q are given by (4.4.1)





MK (C) ' H(Q)\D × H(Af )/K ,

where the right side is, as usual, understood in the sense of stacks or orbifolds. Here D is as in (3.2.3). Recall that, for any K, the irreducible components of MK (C) are indexed by the double cosets (4.4.2)

H(Af ) =

a

H(Q)+ hj K,

j

so that (4.4.3)

MK (C) '

a j

Γj \D+ , 

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CHAPTER 4

+ + where Γj = H(Q)+ ∩ hj Kh−1 j . Here H(Q) = H(Q) ∩ H(R) , where H(R)+ is the identity component of H(R) ' GL2 (R), and D+ is a fixed connected component of D. Thus, in general, MK is not geometrically connected, and the irreducible components are defined over a cyclotomic extension [20], [18]. For t ∈ Q>0 and for a weight function ϕ ∈ S(V (Af ))K , there is a divisor Z(t, ϕ) = Z(t, ϕ; K) on MK , which is rational over Q [11]. We suppress b × , and ϕ is the characthe K to lighten the notation. If t ∈ Z>0 , K = O B b B , then Z(t, ϕ; K) = Z(t)Q coincides with teristic function of V (Af ) ∩ O the generic fiber of the cycle Z(t) defined in Chapter 3. A Green function of logarithmic type for Z(t, ϕ) is constructed as follows; see Section 11 of [11] for more details. Recall that, for z ∈ D and x ∈ V (R),

(4.4.4)

ξ(x, z) = −Ei(−2πR(x, z)),

where Ei is the exponential integral. For a fixed x 6= 0, this function is smooth on D \ Dx and satisfies (4.4.5)

ddc ξ(x, z) + δDx = e2πQ(x) ϕ∗∞ (x, z) · µ,

where, by [13], Proposition 4.10, and [11], (4.4.6)

ϕ∗∞ (x, z) = 4π( R(x, z) + 2Q(x) ) − 1 · ϕ∞ (x, z) 



for (4.4.7)

ϕ∞ (x, z) := e−2πR(x,z) e−2πQ(x) ,

the Gaussian defined by z ∈ D. Here, under an isomorphism D ' P1 (C) \ P1 (R), (4.4.8)

µ=

1 i dz ∧ d¯ z . 2π 2 y 2

For fixed z ∈ D, ϕ∗∞ (·, z) ∈ S(V (R)) is a Schwartz function, and, for fixed x, ξ(x, ·) is a Green function of logarithmic type for the point Dx ⊂ D, in the sense of Gillet-Soul´e. For t > 0, ϕ ∈ S(V (Af ))K , v ∈ R>0 , and [z, h] ∈ D × H(Af ), let (4.4.9)

Ξ(v; t, ϕ)(z, h) =

X

1

ξ(v 2 x, z) ϕ(h−1 x).

x∈V (Q) Q(x)=t

This function is well defined on MK (C) with logarithmic singularities on Z(t, ϕ) and satisfies the Green equation (4.4.10)

ddc Ξ(v; t, ϕ) + δZ(t,ϕ) = Ψ(v; t, ϕ) · µ,

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AN ARITHMETIC THETA FUNCTION

where Ψ(v; t, ϕ)(z, h) · e−2πtv =

(4.4.11)

1

ϕ∗∞ (v 2 x, z) ϕ(h−1 x).

X x∈V (Q) Q(x)=t

In particular, Ξ(v; t, ϕ) is a Green function of logarithmic type for the cycle Z(t, ϕ). Note that the function Ξ(v; t, ϕ) can be defined by the same formula when t < 0 and, in this case, it is a smooth function on MK (C), again satisfying (4.4.10), but without the delta current on the left side. The function Ψ(v; t, ϕ) is defined for all t, with, for example, (4.4.12)

Ψ(v; 0, ϕ) = −ϕ(0).

Let G = SL(2) and let G0R be the metaplectic cover of G(R). Then G0R acts on the space S(V (R)) by the Weil representation ω for the additive character ψ∞ (x) = e(x) = e2πix . For τ = u + iv ∈ H, let 1

1

gτ0

(4.4.13)

uv − 2 1 v− 2

v2

=[

!

, 1] ∈ G0R .

Then (4.4.14) 3

v − 4 (ω(gτ0 )ϕ∗∞ )(x, z) 

=

4πv(w, w) ¯ −1 det







(x, x) (x, w) ¯ − 1 · e−2πvR(x,z) q Q(x) , (w, x) (w, w) ¯

where q = e(τ ) and w = w(z) is as in (3.2.4). Thus, the generating function (4.4.15)

X

3

Ψ(v; t, ϕ)(z, h) · q t = v − 4

t

X

ω(gτ0 )ϕ∗∞ (x, z)ϕ(h−1 x)

x∈V (Q) ∗

=: θ (τ, z, h; ϕ) is the weight 23 theta function associated to ϕ∗∞ (·, z) ⊗ ϕ ∈ S(V (A)). Here the ∗ distinguishes this function from the weight − 12 Siegel theta function (4.4.16)

3

θ(τ, z, h; ϕ) = v − 4

X

ω(gτ0 )ϕ∞ (x, z)ϕ(h−1 x)

x∈V (Q)

=

X

e−2πvR(x,z) q Q(x) ϕ(h−1 x)

x∈V (Q)

defined using the Gaussian ϕ∞ in place of ϕ∗∞ .

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CHAPTER 4

Remark 4.4.1. In fact, this relation of the right side of the Green equation (4.4.10) to the ‘geometric’ theta function θ∗ (τ, z, h; ϕ) was the original motivation for the definition of the Green function Ξ(v; t, ϕ) in [11]. For the moment, we assume that the image of the compact open subgroup ˆ × . This is equivalent to the assumpK under the reduced norm is ν(K) = Z tion that MK is geometrically irreducible, so that X := MK (C) = Γ\D+

(4.4.17)





in (4.4.3), where D+ ' H is one component of D and Γ = H(Q) ∩ H(R)+ K, where H(R)+ ' GL2 (R)+ is the identity component. Let µ1 = vol(X)−1 µ. Recall that there is a unique µ1 -admissible Green function g(t, ϕ) for the cycle Z(t, ϕ) satisfying (4.4.18)

ddc g(t, ϕ) + δZ(t,ϕ) = deg(Z(t, ϕ)) µ1 ,

and Z

(4.4.19)

g(t, ϕ) µ1 = 0. X

The two Green functions, g(t, ϕ) and Ξ(v; t, ϕ) differ by a smooth function on X, which is best described in terms of spectral theory. Let 0 = λ0 < λ1 ≤ λ2 ≤ . . . be the spectrum of the hyperbolic Laplacian on X, with associated smooth eigenfunctions functions fλ satisfying (4.1.37) and orthonormal with respect to µ. For λ > 0, these are the weight 0 Maass forms with respect to Γ. For any smooth function f on X, let (4.4.20)

θ∗ (τ ; ϕ; f ) :=

Z

θ∗ (τ, z; ϕ) f (z) dµ(z)

X

be the associated theta integral. We then can write the spectral expansion of the theta function, as a function of z, as (4.4.21)

θ∗ (τ, z; ϕ) =

X 1 θ∗ (τ ; ϕ; 1) + θ∗ (τ ; ϕ; fλ ) · fλ (z), vol(X) λ>0

where the ‘spectral’ coefficients are the theta lifts of the Maass forms. Taking the Fourier expansion with respect to τ of both sides and recalling (4.4.15), we obtain (4.4.22)

X t

Ψ(v; t, ϕ) · q t =

XX

θt∗ (τ ; ϕ; fλ ) fλ

t λ≥0

where θt∗ (τ ; ϕ; fλ ) is the t-th Fourier coefficient of θ∗ (τ ; ϕ; fλ ). This yields the spectral expansion of the Ψ(v; t, ϕ)’s:

91

AN ARITHMETIC THETA FUNCTION

Proposition 4.4.2. (i) The Fourier expansion of the theta integral of the constant function 1 is θ∗ (τ ; ϕ; 1) = −vol(X) ϕ(0) + Also, for λ > 0,

X

deg(Z(t, ϕ)) q t .

t>0

θ0∗ (τ ; ϕ; fλ ) = 0,

so that the θ∗ (τ ; ϕ; fλ )’s are cuspidal Maass forms of weight 23 . (ii) For t > 0, Ψ(v; t, ϕ)(z) q t = (iii) For t = 0,

X 1 deg(Z(t, ϕ)) q t + θt∗ (τ ; ϕ; fλ ) · fλ (z). vol(X) λ>0

Ψ(v; 0, ϕ) = −ϕ(0)

is a constant. (iv) For t < 0,

Ψ(v; t, ϕ)(z) q t =

X

θt∗ (τ ; ϕ; fλ ) · fλ (z).

denotes the t-th Fourier coefficient of the theta integral of Here the Maass form fλ . λ>0

θt∗ (τ ; ϕ; fλ )

Proof. We only need to prove (i). An easy estimate shows that the theta function θ∗ (τ, z; ϕ) can be integrated termwise on X = Γ\D+ , and (4.4.15) together with a standard Stoke’s Theorem argument [11], pp. 606–608, yields (4.4.23)

Z

Ψ(v; t, ϕ) µ = X

X

1 = deg(Z(t, ϕ)).

P ∈Z(t,ϕ)

Also note that the constant term θ0∗ (τ, z; ϕ) of the whole theta function is the constant function −ϕ(0) on X, so that, for λ > 0, (4.4.24)

θ0∗ (τ ; ϕ; fλ ) = < θ0∗ (τ, ·; ϕ), fλ > = 0.

Remark 4.4.3. In fact, it is possible to give an explicit formula for the Fourier coefficients θt∗ (τ ; ϕ; fλ ). These involve sums of values of fλ over the CM points in Z(t, ϕ) when t > 0 and integrals of fλ over certain geodesics when t < 0; see [9] for the case of modular curves. The spectral expansion of Ψ(v; t, ϕ) can be ‘lifted’ to give the spectral expansion of the Green function Ξ(v; t, ϕ). Here recall (4.1.36): ddc f = 1 2 ∆f · µ.

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CHAPTER 4

Theorem 4.4.4. (i) For t > 0, Ξ(v; t, ϕ) q t = g(t, ϕ) + κ(v; t, ϕ) q t − 2


where (ii) For t < 0,

1 κ(v; t, ϕ) = vol(X)

X 1

λ λ>0

θt∗ (τ ; ϕ; fλ ) · fλ ,

Z

Ξ(v; t, ϕ) µ. X

Ξ(v; t, ϕ) q t = κ(v; t, ϕ) q t − 2

X 1

λ λ>0

θt∗ (τ ; ϕ; fλ ) · fλ .

Proof. For t 6= 0, the function Ξ(v; t, ϕ) − g(t, ϕ) is smooth with 



ddc Ξ(v; t, ϕ) − g(t, ϕ) · q t = Ψ(t, v; ϕ) q t µ − deg(Z(t, ϕ)) q t µ1 (4.4.25)

=

X

θt∗ (τ ; fλ ; ϕ) fλ

λ>0

by (ii) of Proposition 4.4.2. Thus (4.4.26)

Ξ(v; t, ϕ) − g(t, ϕ) = constant − 2

X

λ−1 θt∗ (τ ; fλ ; ϕ) fλ ,

λ>0

and the constant is determined by integrating against µ1 . Let MK be a regular model of MK over Spec Z and let Z(t, ϕ) be a (weighted sum of) divisor(s) on MK with Z(t, ϕ)Q = Z(t, ϕ). Let (4.4.27)

1

b v, ϕ) = (Z(t, ϕ), Ξ(t, v; ϕ)) ∈ CH d (MK )C . Z(t,

Corollary 4.4.5. For λ > 0,

b v), a(fλ ) i · q t = θ ∗ (τ ; ϕ; fλ ). h Z(t, t

Proof. By (4.1.38), we have

b v; ϕ), a(fλ ) i · q t = h a(Ξ(t, v; ϕ) − g(t, ϕ)), a(fλ ) i · q t h Z(t,

(4.4.28)

= −2λ−1 θt∗ (τ ; ϕ; fλ )h a(fλ ), a(fλ ) i = θt∗ (τ ; ϕ; fλ ).

b × , by summing on t ∈ Z, In the case of M as in Section 4.2, i.e., K = O B we obtain

93

AN ARITHMETIC THETA FUNCTION

Corollary 4.4.6. For the generating function φˆ1 (τ ) of Section 4.2, the component h φˆ1 (τ ), a(fλ ) i = θ∗ (τ ; ϕ; fλ ).

is a Maass form of weight 23 .

We now return to the situation at the beginning of this section so that K is an arbitrary compact open subgroup of H(Af ) and the components of MK (C) are described by (4.4.3). For ϕ ∈ S(V (Af ))K , the analogue of (i) of Proposition 4.2 gives (4.4.29)

Z

θ∗ (τ ; ϕ) µ = −vol(MK (C)) ϕ(0) +

MK (C)

X

deg(Z(t, ϕ)) q t

t>0

where θ∗ (τ ; ϕ) is the theta function defined by (4.4.15), and deg(Z(t, ϕ)) is the degree of the weighted 0-cycle, i.e., the sum of its degrees on each of the components of MK (C). The right side of (4.4.29) is an Eisenstein series. We would like to have similar information about the degrees of the weighted 0-cycles Z(t, ϕ) on individual components of MK (C). Note that, by strong approximation, (4.4.30)

π0 (MK (C)) ' Q× \A× /R× + ν(K).

For any function η on π0 (MK (C)), we define a locally constant function η˜ on D × H(Af ) by (4.4.31) η˜(z, h) = η (sgn(z), ν(h)) ,


(sgn(z), ν(h)) ∈ R× × A× f,

where sgn(z) = ±1 if z ∈ D± . The integral of the theta function against η˜ then gives the generating function (4.4.32) Z θ∗ (τ ; ϕ) · η˜ · µ = −volη (MK (C)) ϕ(0) +

MK (C)

X

degη (Z(t, ϕ)) q t .

t>0

for degrees weighted by η.

Proposition 4.4.7. Suppose that η is orthogonal to the constant function on π0 (MK ). Then the generating function for η-degrees is a distinguished cusp form of weight 23 , i.e., lies in the space of cusp forms generated by theta functions for quadratic forms in one variable.

Proof. Let dh be Tamagawa measure on SO(V )(A) ' Z(A)\H(A), where Z is the center of H. Let dzf be the measure on Z(Af ) which gives the b × volume 1, and let dh0 = dz dh be the maximal compact subgroup Z f

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CHAPTER 4

resulting invariant measure on Z(R)\H(A). Write dh0 = dh∞ × dh0f , where the measure dh∞ on SO(V )(R) = Z(R)\H(R) is determined by the identity Z

(4.4.33)

φ(h∞ · z0 ) dh∞ =

Z

φ(z) dµ(z).

Z(R)\H(R)

D

By a very slight modification of the proof of Proposition 4.17 of [13] in our current situation, we obtain

Lemma 4.4.8. For an integrable function φ on D × H(Af ) which is left invariant under H(Q) and right invariant under K, Z H(Q)Z(R)\H(A)

φ(h∞ z0 , hf )dh0 = e−1 K vol(K)

XZ Γj \D+

φ(z, hj ) dµ(z),

where eK = |Z(Q) ∩ K| and vol(K) = vol(K, dh0f ). j

Applying this to the integral in (4.4.32), we obtain (4.4.34) Z Z θ∗ (τ ; ϕ) · η˜ · µ =

MK (C)

H(Q)Z(R)\H(A)

θ(τ, h; ϕ∗∞ ⊗ ϕ) η(ν(h)) dh0 .

We may as well assume that η is a nontrivial character of the component group π0 (MK (C)). The theta function θ is invariant under the center Z(A), since it factors through SO(V )(A). Hence, if η 2 6= 1, the integral is identically zero. If η 2 = 1 with η 6= 1, let kη be the associated quadratic extension of Q. The integral can be unfolded and only terms with x ∈ V (Q) with Q(x) ' kη can give a nonzero contribution. Thus, the theta integral is a distinguished cusp form, and hence, by the results of [6] and [21], lies in the space generated theta functions coming from O(1)’s. 4.5 THE MORDELL-WEIL COMPONENT

In this section, we will prove that the Mordell-Weil component φMW of our generating series given by (4.2.11) is the q-expansion of a modular form valued in MW ⊗ C. The idea is that this series is very closely related to the generating series considered by Borcherds [2], so that his result can be applied. Again, we work in a more general case than is finally needed for the moduli space M. For a compact open subgroup K ⊂ H(Af ), a weight function ϕ ∈ S(V (Af ))K , and t ∈ Q>0 , let Z(t, ϕ; K) be the corresponding weighted 0-cycle on MK . Recall that is it rational over Q, so it defines a class in the Chow group CH1 (MK ) which we also denote by Z(t, ϕ; K). If K 0 ⊂ K is

95

AN ARITHMETIC THETA FUNCTION

an open subgroup, and if pr : MK 0 → MK is the natural projection, then by [10], pr∗ (Z(t, ϕ; K)) = Z(t, ϕ; K 0 ),

(4.5.1)

so that we obtain a class (4.5.2)

Z(t, ϕ) ∈ CH1 (M ) := lim CH1 (MK )C → K

in the direct limit. Let L be the line bundle on D given by the restriction of the bundle O(−1) on P(V (C)), and let LK be the associated bundle on MK . Since the LK ’s are compatible with pullbacks, we have the relation pr∗ (c1 (LK )) = c1 (LK 0 ) on Chern classes and can define classes (4.5.3)

Z(0, ϕ; K) = −ϕ(0) c1 (LK ) ∈ CH1 (MK )C ,

at finite level, and (4.5.4)

Z(0, ϕ) = −ϕ(0)c1 (L) ∈ CH1 (M )C ,

in the direct limit. The following result will be proved in Section 4.6.

Theorem 4.5.1. (Borcherds [2], +) For any ϕ ∈ S(V (Af ))K , the generating series φBor (q, ϕ; K) := Z(0, ϕ; K) +

X

Z(t, ϕ; K) q t ∈ CH1 (MK )C [[ q ]]

is the q-expansion of a holomorphic modular form φBor (τ, ϕ; K) of weight 3 1 2 valued in CH (MK )C . t>0

We can pass to the limit on K and conclude that, for any ϕ ∈ S(V (Af )), the generating series (4.5.5)

φBor (q, ϕ) := Z(0, ϕ) +

X

Z(t, ϕ) q t ∈ CH1 (M )C [[ q ]]

t>0

is also the q-expansion of a holomorphic modular form φBor (τ ) of weight 3 2 . Moreover the map (4.5.6)

φBor (τ ) : S(V (Af )) −→ CH1 (M ),

ϕ 7→ φBor (τ, ϕ)

is equivariant for the natural action of H(Af ) on the two sides. In effect, this equivariance describes the action of the Hecke operators for the Shimura variety M on the generating series.

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CHAPTER 4

The proof of Theorem 4.5.1 will be given in the next section. Here we use it to prove the modularity of the Mordell-Weil component (4.2.11). In b × , MK = MQ is geometrically irreducible. Moreover, by the case K = O B (3.12) of [16] and the remark after (1.15) of [13], we have L = ωQ , for ω the Hodge line bundle, as in Section 3.3. Taking ϕ to be the characteristic b B ), we have function ϕ0 := char(V (Af ) ∩ O 1 deg(φBor (τ, ϕ0 ; K)) = φdeg (τ ) = E1 (τ, ; D(B)). 2 Thus, we have proved the following.

(4.5.7)

Proposition 4.5.2. The generating series φMW (q) ∈ CH1 (MQ )0 [[ q ]] of (4.2.11) is the q-expansion of the holomorphic modular form 1 φ0Bor (τ, ϕ0 ; K) := φBor (τ, ϕ0 ; K) − E1 (τ, ; D(B)) deg(LK )−1 c1 (LK ). 2 4.6 BORCHERDS’ GENERATING FUNCTION

In this section, we explain how to formulate Borcherds’ result [2] in order to obtain the statement of Theorem 4.5.1. A slightly more detailed discussion of some background is given in Section 1 of [13], to which we refer the reader for more information. As in [16] and Section 5.5, let G0A (resp. G0R and G0Af ) be the metaplectic extension of SL2 (A) (resp. SL2 (R) and SL2 (Af )). Let G0Q be the image of SL2 (Q) under the unique splitting homomorphism SL2 (Q) → G0A . Let Γ0 ⊂ G0R be the full inverse image of SL2 (Z) and let K 0 ⊂ G0Af be the full ˆ Thus Γ0 is a central extension of SL2 (Z) by C1 . inverse image of SL2 (Z). 0 0 For each γ ∈ Γ , there is a unique element k 0 ∈ K 0 such that the product γ = γ 0 k 0 ∈ SL2 (Z), identified with a subgroup of G0Q . The map γ 0 7→ k 0 defines a homomorphism from Γ0 to K 0 . For τ = u + iv ∈ H, let gτ0 ∈ G0R 0 ⊂ G0 be the full inverse image of SO(2). be given by (4.4.13), and let K∞ R 0 0 For any γ ∈ GR with projection γ ∈ SL2 (R), we have (4.6.1)

0 0 0 γ 0 gτ0 = gγ(τ ) k∞ (γ , τ ),

0 (γ 0 , τ ) ∈ K 0 . For r ∈ 1 Z, let χ be the character for a unique element k∞ r ∞ 2 0 of K∞ given by (1.26) of [13], and define an automorphy factor

(4.6.2)

0 jr (γ 0 , τ ) = χ−r (k∞ (γ 0 , τ ))) |cτ + d|r ;

see Section 8.5. b and Let ψf be the unique additive character of Af which is trivial on Z such that the additive character ψ = ψ∞ ψf , with ψ∞ (x) = e(x), of A

97

AN ARITHMETIC THETA FUNCTION

is trivial on Q. The group G0Af acts in the space S(V (Af )) by the Weil representation ω associated to ψf . Let L ⊂ V be a lattice on which the quadratic form Q is integral, and let L] = { x ∈ V | (x, L) ⊂ Z } be the dual lattice. Let SL ⊂ S(V (Af )) be the space of functions ϕ with b ] and which are invariant under translation by L. b This space is support in L ] isomorphic to C[L /L] under the map λ 7→ ϕλ where, for λ ∈ L] , ϕλ is the b The Weil representation action of K 0 on characteristic function of λ + L. S(V (Af )) preserves SL and yields a representation ρL of Γ0 on this space via the homomorphism Γ0 → K 0 . Following Borcherds [2], let 1 MF(Γ0 , , ρL ) 2 be the space of SL -valued holomorphic functions f on H such that

(4.6.3)

(4.6.4)

f (γ(τ )) = j 1 (γ 0 , τ ) ρL (γ 0 )f (τ ), 2

for all γ ∈ SL2 (Z), and which have a q-expansion of the form (4.6.5)

f (τ ) =

XX λ

cλ (t) q t ϕλ ,

t

where only a finite number of cλ (t)’s with t < 0 are nonzero. Let 1 MF(Γ0 , , ρL )Z 2

(4.6.6)

be the Z-submodule of f ’s for which cλ (t) ∈ Z for all t ≤ 0. Let ZHeeg(L) be the Q-vector space with generators yλ (t), for λ ∈ L] /L and t ∈ Q>0 with Q(λ) ≡ t mod Z, and an additional element y0 (0). For f ∈ MF(Γ0 , 21 , ρL )Z , let (4.6.7)

rel(f ) = c0 (0) y0 (0) +

XX

cλ (−t) yλ (t) ∈ ZHeeg(L).

λ t>0

Finally, let (4.6.8)

CHeeg(L) =

ZHeeg(L)

. rel(f ) f ∈ MF(Γ0 , 12 , ρL )Z

This group is a kind of formal Chow group for Heegner divisors. In [2], Borcherds proved that the generating series for the yλ (t)’s is a holomorphic modular form, assuming the existence of a basis with rational Fourier coefficients of a certain space of vector valued modular forms. The existence of the required basis was subsequently proved by McGraw [17].

98

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Theorem 4.6.1. (Borcherds) (i) The space CHeeg(L) is finite dimensional over Q. (ii) The generating series φBor (q, L) =

XX

yλ (t) q t ϕ∨ λ

∈ CHeeg(L) ⊗ SL∨ [[ q ]]

is the q-expansion of a holomorphic CHeeg(L) ⊗ SL∨ -valued modular form 0 ∨ φBor (τ, L) of weight 32 and type ρ∨ L for Γ . Here SL is the dual space to SL and {ϕ∨ λ } is the dual basis to {ϕλ }. λ t≥0

We can view φBor (τ, L) as a map

(4.6.9)

φBor (τ, L) : SL −→ CHeeg(L)C ,

where, for ϕ ∈ SL , (4.6.10)

φBor (τ, ϕ; L) := φBor (τ, L)(ϕ) =

X

yϕ (t) q t ,

t≥0

with (4.6.11)

yϕ (t) =

X

yλ (t) < ϕ∨ λ, ϕ > .

λ

This function has the transformation law (4.6.12)

φBor (γ(τ ), ϕ; L) = j 3 (γ 0 , τ ) φBor (τ, ρL (γ 0 )−1 ϕ; L), 2

γ0

Γ0

for all γ ∈ SL2 (Z), where ∈ is any preimage of γ. To relate this formal generating series to one valued in a geometric Chow b × , and let group, we proceed as follows. Let K0 = O B (4.6.13)

KL = { k ∈ K0 | kL = L and k acts trivially in L] /L }.

The key point in Borcherds’ construction is then

Proposition 4.6.2. The map ZHeeg(L) → CH1 (MKL )C defined by sending yλ (t) to Z(t, ϕλ ; KL ) and y0 (0) to −c1 (LKL ) induces a homomorphism CHeeg(L)C −→ CH1 (MKL )C .

Proof. To any f ∈ MF(Γ0 , 21 , ρL )Z , Borcherds constructs a meromorphic section2 Ψ(f ) of LkKL , where k = c0 (0), with explicit divisor (4.6.14)

div(Ψ(f )) =

XX

cλ (−t) Z(t, ϕλ ; KL ).

λ t>0 2 Technically, the transformation law of Ψ(f ) may involve a unitary character. Since this character has finite order [3], and since we are taking Chow groups with Q-coefficients, we can ignore it.

99

AN ARITHMETIC THETA FUNCTION

But then (4.6.15)

rel(f ) 7−→ −c0 (0) c1 (L) + div(Ψ(f )) ≡ 0.

Thus, we have (4.6.16)

SL

φBor (τ,L)

−→

CHeeg(L)C −→ CH1 (MKL )C .

Finally, we pass to the limit over L. Suppose that L2 ⊂ L1 are lattices, as above. Then there is a natural inclusion SL1 ,→ SL2 , compatible with the Γ0 actions ρL1 and ρL2 , since these are, after all, just coming from the action of K 0 on (4.6.17)

S(V (Af )) = lim SL . → L

There is a resulting inclusion 1 1 MF(Γ0 , , ρL1 ) −→ MF(Γ0 , , ρL2 ) 2 2

(4.6.18)

preserving the ‘integral’ elements. It is easily checked that the map (4.6.19)

ZHeeg(L1 ) −→ ZHeeg(L2 ), X

yλ (t) 7→

y0 (0) 7→ y0 (0),

yµ (t),

µ∈L]2 /L2 µ≡λ

mod L1

induces a map pr∗ : CHeeg(L1 ) −→ CHeeg(L2 ).

(4.6.20) We let (4.6.21)

CHeeg := lim CHeeg(L). → L

It is easy to check the following compatibility. Lemma 4.6.3. The diagram SL2

φBor (τ,L2 )

−→

↑ pr∗

↑ SL1

is commutative.

CHeeg(L2 )C −→

φBor (τ,L1 )

−→

CH1 (MKL2 )C ↑ pr∗

CHeeg(L1 )C −→ CH1 (MKL1 )C

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CHAPTER 4

Since the system of subgroups KL is cofinal in the system of all compact ˆ × , we pass to the limit and obtain open subgroups K with K ∩ Z(Af ) ' Z (4.6.22)

φBor (τ ) : S(V (Af )) −→ CHeegC −→ CH1 (M )C ,

where we write φBor (τ ) for the composite map.

Proposition 4.6.4. There is a map φBor (τ ) :

S(V (Af )) −→ CH1 (M )C , ϕ

7−→

φBor (τ, ϕ) = Z(0, ϕ) +

X

Z(t, ϕ) q t

compatible with the H(Af ) actions. Moreover, φBor (τ, ϕ) is a holomorphic modular form of weight 32 . More precisely, for all γ ∈ SL2 (Z), t>0

φBor (γ(τ ), ϕ) = j 3 (γ 0 , τ ) φBor (τ, ρ(γ 0 )−1 ϕ),

where ρ is the representation of Γ0 on S(V (Af )) coming from the action of K 0 and the homomorphism Γ0 → K 0 . 2

Of course, for any compact open subgroup K ⊂ H(Af ), there is a resulting map (4.6.23) φBor (τ ) : S(V (Af ))K −→ CH1 (MK )C , ϕ 7→ φBor (τ, ϕ; K) = Z(0, ϕ; K) +

X

Z(t, ϕ; K) q t

t>0

as claimed in Theorem 4.5.1. 4.7 AN INTERTWINING PROPERTY

In this section, we show that the function on the metaplectic group defined by the Borcherds generating function φBor (τ, ϕ) has the same intertwining property as the usual theta function with respect to the right action of G0Af . We lift the generating function φBor (τ, ϕ) to a function φeBor (g 0 , ϕ) on G0A by setting (4.7.1)

0 0 , i)−1 φBor (g∞ (i), ω(k 0 )ϕ), φeBor (g 0 , ϕ) := j 3 (g∞ 2

0 k 0 , for α ∈ G0 , g 0 ∈ G0 , and k 0 ∈ K 0 . The transformawhere g 0 = α g∞ Q ∞ R tion law of Proposition 4.6.4 implies that this function is well defined; it is

101

AN ARITHMETIC THETA FUNCTION

0 , i) = χ (k 0 ), left G0Q -invariant by construction. In addition, since jr (k∞ −r ∞ we have

(4.7.2)

0 0 φeBor (g 0 k∞ kf0 , ϕ) = χ 3 (k∞ ) φeBor (g 0 , ω(kf0 )ϕ), 2

for kf0 ∈ K 0 . Thus, φeBor has weight 32 . The main result of this section is the following intertwining property. Proposition 4.7.1. For gf0 ∈ G0Af ,

φeBor (g 0 gf0 , ϕ) = φeBor (g 0 , ω(gf0 )ϕ).

Proof. We define another function on G0A by (4.7.3)

e 0 g 0 , ϕ) = j 3 (g 0 , i)−1 φ(g 0 (i), ω(g 0 )ϕ). φ(g ∞ f ∞ ∞ f e

2

This function agrees with φe on G0R K 0 and, by construction, satisfies e 0 g 0 , ϕ) = φ(g e 0 , ω(g 0 )ϕ), φ(g 0 0

(4.7.4)

e

e

for g00 ∈ G0Af . In particular, φe is left invariant under SL2 (Z), and it suffices to prove that it is, in fact, left invariant under all of G0Q .   a 0 δ 0 for δ 0 ∈ G0 ∈ G0Q . Write δ = δ∞ For a ∈ Q× , let δ = ∞ >0 R f a−1 0 0 and δf ∈ GAf . e

Lemma 4.7.2. (i) (ii)

0 Z(t, ω(δf0 )ϕ) = j 3 (δ∞ , τ ) Z(a2 t, ϕ). 2

0 φBor (a2 τ, ω(δf0 )ϕ) = j 3 (δ∞ , τ ) φBor (τ, ϕ).

Proof. Note that ϕ ∈ S(V (Af ))K for some compact open subgroup K ⊂ H(Af ), so (i) can be viewed as an identity between weighted 0-cycles on the generic fiber MK . Recall from Lemma 10.1 of [11], 2

(4.7.5)

Z(t, ϕ; K) =

X

X

j

x∈V (Q) Q(x)=t mod Γj

ϕ(h−1 j x) pr(Dx ),

102

CHAPTER 4

in the notation of Section 4.4 above, especially (4.4.2) and (4.4.3). On the 3 other hand, since |a|Af = a− 2 , we have 0 , τ ) ϕ(ax), ω(δf0 )ϕ(x) = j 3 (δ∞

(4.7.6)

2

which, together with the previous identity, yields (i). Part (ii) is immediate from (i) and the analogous relation for Z(0, ϕ), which is clear from (4.5.4). Thus, we have e 0 , ϕ) = φ(δ e 0 g 0 δ 0 g 0 , ϕ) φ(δg ∞ ∞ f f e

(4.7.7)

e

0 0 = j 3 (δ∞ , τ )−1 j 3 (g∞ , i)−1 φBor (a2 τ, ω(δf0 )ω(gf0 )ϕ) 2

=

2

0 , i)−1 φBor (τ, ω(gf0 )ϕ) j 3 (g∞ 2

e 0 , ϕ). = φ(g e

e Since SL2 (Q) is generated by SL2 (Z) together with the δ’s for a ∈ Q× >0 , φ is left G0Q invariant, and hence coincides with φeBor . This finishes the proof of Proposition 4.7.1. e

Propositions 4.6.4 and 4.7.1 show that φeBor (g 0 , ϕ) behaves just like the classical theta function θ(g 0 , h; ϕ) as far as G0Af × H(Af )-equivariance is concerned. More precisely, let A(G0 ) 3 ,hol be the space of automorphic 2

forms on G0A of weight 32 (for the right action of KR0 ) which are ‘holomorphic’ in the sense that the corresponding functions on H are holomorphic. Then, we have a map (4.7.8) φeBor : S(V (Af )) −→ A(G0 ) 3 ,hol ⊗ CH1 (M ), ϕ 7→ φeBor (g 0 , ϕ), 2

which is G0Af × H(Af )-equivariant. Note that since the action of G0Af × H(Af ) on S(V (Af )) is smooth, any ϕ is fixed by some compact open subgroup K 0 × K, and thus the function φeBor (g 0 , ϕ) takes values in the finite dimensional space CH1 (MK ). Moreover, φeBor (g 0 , ϕ) is right K 0 -invariant, so 0 that the component functions lie in the finite dimensional space A(G0 )K . 3 ,hol 2

Bibliography

[1] J.-B. Bost, Potential theory and Lefschetz theorems for arithmetic ´ surfaces, Ann. Sci. Ecole Norm. Sup., 32 (1999), 241–312.

BIBLIOGRAPHY

103

[2] R. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J., 97 (1999), 219–233. [3]

, Correction to “The Gross-Kohnen-Zagier theorem in higher dimensions,” Duke Math. J., 105 (2000), 183–184.

[4] G. Faltings, Calculus on arithmetic surfaces, Annals of Math., 119 (1984), 387–424. [5] H. Gillet and C. Soul´e, Arithmetic intersection theory, Publ. Math. IHES, 72 (1990), 93–174. [6] S. Gelbart and I. I. Piatetski-Shapiro, Distinguished representations and modular forms of half integral weight, Invent. Math., 59 (1980), 145–188. [7] B. H. Gross, Heights and special values of L-series, in Number Theory (Proc. Montreal Conf., 1985), CMS Conf. Proc. 7, 115–187, AMS, Providence, 1987. [8] P. Hriljac, Heights and arithmetic intersection theory, Amer. J. Math., 107 (1985), 23–38. [9] S. Katok and P. Sarnak, Heegner points, cycles and Maass forms, Israel J. Math., 84 (1993), 193–227. [10] S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J., 86 (1997), 39–78. [11]

, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646.

[12]

, Derivatives of Eisenstein series and generating functions for arithmetic cycles, S´eminaire Bourbaki 876, Ast´erisque, 276, 341– 368, So. Math. de France, Paris, 2002.

[13]

, Integrals of Borcherds forms, Compositio Math., 137 (2003), 293–349.

[14]

, Special cycles and derivatives of Eisenstein series, in Heegner Points and Rankin L-Series, Math. Sci. Res. Inst. Publ., 49, 243– 270, Cambridge Univ. Press, Cambridge, 2004.

[15] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223.

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[16] S. Kudla, M. Rapoport, and T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887–951. [17] W. J. McGraw, Rationality of vector valued modular forms associated to the Weil representation, Math. Annalen, 326 (2003), 105–122. [18] J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in Automorphic Forms, Shimura Varieties and L-functions, Vol I., 283–414, Academic Press, Boston, 1990. [19] S. Rallis and G. Schiffmann, Repr´esentations supercuspidales du groupe m´etaplectique, J. Math. Kyoto Univ., 17 (1977), 567–603. [20] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Princeton Univ. Press, Princeton, N. J., 1971. [21] K. Snitz, The theta correspondence for characters of O(3), Ph. D. Thesis, University of Maryland, August 2003.

Chapter Five The central derivative of a genus two Eisenstein series In this chapter, we study an incoherent Eisenstein series of genus two in detail and, in particular, compute its derivative at s = 0, the central point for the functional equation. This Eisenstein series was first introduced in [4]. In the first few sections, we consider the Fourier coefficients associated to T ∈ Sym2 (Q) with det(T ) 6= 0. These coefficients, which are given by a product of local factors, were studied in [4] and [8]. In Sections 5.4 through 5.8, we deal with the Fourier coefficients for T ’s with rank(T ) = 1. These coefficients, which are not given as a product of local factors, are of global nature and are closely related to the Fourier coefficients of a genus one Eisenstein series. The main point here is to make this relation precise and explicit. The result of this effort, Theorems 5.8.1 and 5.8.7, express the central derivative of the rank one Fourier coefficients of the genus two Eisenstein series in terms of the derivative of a genus one Eisenstein series at a critical noncentral point s = 12 . This genus one Eisenstein series was studied in detail in [9] with this application in mind. In the last section, we prove the analogous relation for the constant terms of the two Eisenstein series. In Chapter 6, we will use these results together with those of Chapter 3 to prove one of the main results in this book—the coincidence of the central derivative of the genus two Eisenstein series computed here with the generating function for 0-cycles on the arithmetic surface attached to a Shimura curve.

5.1 GENUS TWO EISENSTEIN SERIES

In this section, we construct the Eisenstein series of genus two and weight 3 2 attached to an indefinite quaternion algebra B, by specializing the general construction of incoherent Eisenstein series given in part 1 of [4]. For the moment, we allow the case B = M2 (Q). More details can be found in [4]. We begin by fixing some notation and reviewing the structure of the space of Siegel Eisenstein series. Let Sp2 be the rank 2 symplectic group over Q

106

CHAPTER 5

and let GA be the metaplectic extension (5.1.1)

pr

1 −→ C1 −→ GA −→ Sp2 (A) −→ 1.

We write (5.1.2)

P = N M = { n(b)m(a) | b ∈ Sym2 , a ∈ GL2 },

for the Siegel parabolic subgroup of Sp2 , where (5.1.3)





1 b n(b) = , 1



and

m(a) =



a t a−1

.

Let (5.1.4)

Y

K = K∞ ·

K p,

p

where K p = Sp2 (Zp ), for p < ∞, and K ∞ ' U (2) is the standard maximal compact subgroup. Let GR = pr−1 (Sp2 (R)), Gp = pr−1 (Sp2 (Qp )), (5.1.5)

GAf = pr−1 (Sp2 (Af )), PA = pr−1 (P (A)), MA = pr−1 (M (A)), K = pr−1 (K),

K∞ = pr−1 (K ∞ ),

and let GQ (resp. NA ) be the image of Sp2 (Q) (resp. N (A)) in GA under the unique splitting homomorphism. Let ψ be the standard character of A/Q which is unramified and such that ψ∞ (x) = e(x) = e2πix . As explained in Section 8.5.5, there is then an isomorphism of groups, via the Leray coordinates, (5.1.6)

∼

P (A) × C1 −→ PA

(p, z) 7→ [p, z]L,ψ = [p, z]L .

Note that (5.1.7)

PQ := PA ∩ GQ = [P (Q), 1]L .

For a character χ of A× /Q× , we also write χ for the character of PA defined by (5.1.8)

χ([n(b)m(a), z]L ) = z χ(det(a)).

107

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

The character χ on PA depends on the isomorphism (5.1.6) and hence on the choice of ψ, but we suppress this dependence from the notation. For s ∈ C, let I(s, χ) be the global degenerate principal series representation of GA on smooth functions Φ(s) on GA satisfying (5.1.9)

3

Φ([n(b)m(a), z]L g, s) = z χ(det(a)) | det(a)|s+ 2 Φ(g, s).

We also require that Φ(s) be right K∞ -finite, so that I(s, χ) is a (g, K∞ ) × GAf -module. A section Φ(s) is called standard if its restriction to K is independent of s. For such a section Φ(s), the corresponding Siegel Eisenstein series (5.1.10)

E(g, s, Φ) =

X

Φ(γg, s)

γ∈PQ \GQ

converges for Re(s) > 23 . The analytic continuation of E(g, s, Φ) to the whole s-plane is holomorphic at the point s = 0 on the unitary axis and hence there is a (g, K∞ ) × GAf -intertwining map (5.1.11)

E(0) : I(0, χ) −→ A(GA ),

Φ(0) 7→ E(·, 0, Φ),

where A(GA ) is the space of genuine automorphic forms on GA . The representation I(s, χ) is a restricted product (5.1.12)

I(s, χ) = ⊗p≤∞ Ip (s, χp )

of local degenerate principal series representations where χ = ⊗p≤∞ χp . Assume that χ2 = 1 and write (5.1.13)

χ(x) = (x, 2κ)A

for κ ∈ Q× , where ( , )A is the global Hilbert symbol. For any p ≤ ∞, the representation Ip (0, χp ) is unitarizable and is the direct sum (5.1.14)

Ip (0, χp ) = Rp (Vp+ ) ⊕ Rp (Vp− )

of irreducible representations defined as follows. Let Bp± be the quaternion algebra over Qp with invariant invp (Bp± ) = ±1 and let (5.1.15)

Vp± = { x ∈ Bp± | tr(x) = 0 },

with quadratic form Q(x) = −κ · ν(x) = κx2 . Soon we will specialize to the case κ = −1. For the moment, write Vp = Vp± . If p < ∞, the group Gp acts in the Schwartz space S(Vp2 ) by the Weil representation ωVp = ωVp ,ψp . 2 ) be the subspace of the Schwartz space of V 2 If p = ∞, we let S(V∞ ∞ which corresponds to the space of polynomial functions in a Fock model

108

CHAPTER 5

compatible with K∞ and some choice of maximal compact subgroup of 2 ) is a (g, K )-module via the Weil representation O(V∞ ), [2]. Then S(V∞ ∞ ωV∞ = ωV∞ ,ψ∞ . The map (5.1.16) λ = λV : S((Vp± )2 ) −→ Ip (0, χp ), ϕ 7→ (g 7→ ωψ,V (g)ϕ(0)) is Gp -intertwining (resp. (g, K∞ )-intertwining), and the image (5.1.17)

Rp (Vp ) := λVp (S(Vp2 ))

is an irreducible submodule and is isomorphic to the space S(Vp2 )O(Vp ) of O(Vp )-coinvariants [12] (resp. (O(V∞ ), KO(V∞ ) )-coinvariants [6]). In the case p = ∞, we will sometimes write (5.1.18)

− )= R∞ (V∞

(

if κ < 0, if κ > 0,

(

if κ < 0, if κ > 0,

R∞ (3, 0) R∞ (0, 3)

and (5.1.19)

+ R∞ (V∞ )

=

R∞ (1, 2) R∞ (2, 1)

according to the signatures of the quadratic spaces involved. Note that R∞ (p, q) is a cyclic (g, K∞ )-module generated by the vector Φ`∞ (0) with scalar K∞ -type (det)` where ` = 21 (p − q), [6]. Let Φ`∞ (s) be the standard section of I∞ (s, χ∞ ) having this scalar K∞ -type and normalized so that Φ`∞ (e, s) = 1. For a global quaternion algebra B with corresponding ternary quadratic space V B , where the quadratic form is defined by Q(x) = −κ·ν(x) = κx2 , let Π(V B ) = ⊗p Rp (VpB )

(5.1.20)

be the associated irreducible summand of I(0, χ). Similarly, for an incoher ent collection C = {Cp }, Cp = Vp p , i.e., a collection of spaces which differs from the collection {VpB } at an odd number of places, there is an irreducible summand (5.1.21)

Π(C) = ⊗p Rp (Cp ),

and (5.1.22)



I(0, χ) =

B



⊕B Π(V ) ⊕





⊕C Π(C)

is the decomposition of I(0, χ) into irreducibles. The Siegel-Weil formula describes the image of the summands Π(V B ) under the Eisenstein map

109

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

E(0) as spaces of theta functions [4], [7]. The summands Π(C) lie in the kernel of E(0) and occur only as subquotients of the space of automorphic forms on GA [10]. We now specialize to the case of the Eisenstein series of weight 32 whose central derivative will be related to the genus two generating function to be defined in Chapter 6. Fix an indefinite quaternion algebra B, and take κ = −1, so that χ(x) = (x, −2)A and the quadratic form on V B is given by Q(x) = ν(x) = −x2 . Let OB be a maximal order in B and let ϕB ∈ S(V B (Af )2 ) be the characˆ 2 . Then, let ΦB (s) be the teristic function of the set (V (Af ) ∩ (OB ⊗Z Z)) f standard section of If (s, χf ) with B ΦB f (0) = λV (ϕ ).

(5.1.23)

−1

B ) = R (1, 2). Thus, the standard section Φ 2 (s) ⊗ Note that R∞ (V∞ ∞ ∞ B Φf (s) is coherent with −1

Φ∞2 (0) ⊗ ΦB f (0)

(5.1.24)

∈ Π(V B ),

3

2 (s) ⊗ ΦB whereas the standard section Φ∞ f (s) is incoherent with 3

2 Φ∞ (0) ⊗ ΦB f (0)

(5.1.25)

∈ Π(C B ),

B = for the collection C B = {CpB } with CpB = VpB for p < ∞ and with C∞ R∞ (3, 0). This is almost the section we want, but it turns out that a further modification is needed at the primes p | D(B). For any p, let Bp± be the quaternion algebra over Qp with invp (Bp± ) = ±1, and let

(5.1.26)

Vp± = { x ∈ Bp± | tr(x) = 0 }.

For p < ∞, let (5.1.27)

L0p = M2 (Zp ) ∩ Vp+ ,

(5.1.28)

L1p

and (5.1.29)

ra − Lra p = Op ∩ Vp ,





a b ={ ∈ M2 (Zp ) | ordp (c) > 0} ∩ Vp+ , c d

where Opra is the maximal order in the division quaternion algebra Bp− over − 2 Qp . Let ϕ0p , ϕ1p ∈ S((Vp+ )2 ) and ϕra p ∈ S((Vp ) ) be the characteristic

110

CHAPTER 5

2 0 1 functions of (L0p )2 , (L1p )2 , and (Lra p ) respectively. Then, let Φp (s), Φp (s), and Φra p (s) be the standard local sections with

(5.1.30)

Φ0p (0) = λ(ϕ0p ),

(5.1.31) and (5.1.32)

Φ1p (0) = λ(ϕ1p ), ra Φra p (0) = λ(ϕp ).

˜ p (s) is defined as follows [8]: The local modified section Φ ˜ p (s) = Φra (s) + A(s) Φ0 (s) + B(s) Φ1 (s), Φ p p p

(5.1.33)

where Ap (s) and Bp (s) are rational functions of p−s with the property that (5.1.34)

Ap (0) = Bp (0) = 0,

(5.1.35) A0p (0) = −

2 log(p) p2 − 1

and

Bp0 (0) =

1 p+1 · log(p). 2 p−1

˜ p (s) is not a standard section and that Φ ˜ p (0) = Φra Note that Φ p (0). This particular combination of standard sections was originally defined in [8] to match certain intersection numbers of special cycles on the Drinfeld upper half space. It turns out to have good properties for the local doubling integral as well; see Chapter 8. Finally, we define the modified global section (5.1.36)

3

2 ˜ B (s) = Φ∞ ˜ p (s) ⊗ ⊗p-D(B) Φ0p (s) Φ (s) ⊗ ⊗p|D(B) Φ




and the corresponding normalized Eisenstein series (5.1.37)

˜ B ), C D(B) (s) · E(g, s, Φ

where, for any square free positive integer D, (5.1.38)

1 C D (s) = − · (s + 1) c(D) ΛD (2s + 2) 2

with (5.1.39)



ΛD (2s) =

D π

s

Γ(s) ζ(2s)

Y

(1 − p−2s ),

p|D

and (5.1.40)

c(D) = (−1)ord(D)+1

D Y (p + 1)−1 . 2π p|D




THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

111

We often abuse notation and write C B (s) instead of C D(B) (s). Note that, ˜ B ) only in contrast to the situation in [9], the central derivative E 0 (g, 0, Φ depends on the value (5.1.41)

C D (0) = (−1)ord(D)

1 Y (p − 1) 24 p|D

of the normalizing factor, since ˜ B ) = 0. E(g, 0, Φ

(5.1.42)

˜ B (s) is invariant under a suitable compact open subSince the section Φ group of K of GAf (see Section 8.5 for details) and is an eigenfunction for K∞ , E2 (g, s, B) is determined by its restriction to PR . For τ = u+iv ∈ H2 , let 1

(5.1.43)

gτ = [ n(u)m(v 2 ), 1 ]L ∈ PR

and let (5.1.44)

3 ˜ B ). E2 (τ, s, B) := det(v)− 4 · C B (s) E(gτ , s, Φ

Of course, E2 (τ, 0, B) = 0. Our next task is to determine the Fourier expansion (5.1.45)

E20 (τ, 0, B) =

0 E2,T (τ, 0, B)

X T ∈Sym2 (Q)

of the central derivative. 5.2 NONSINGULAR FOURIER COEFFICIENTS

In this section, we review the description obtained in [4] and [8] for the 0 (τ, 0, B) when T ∈ Sym (Q) with det(T ) 6= 0. Fourier coefficients E2,T 2 First there is a product formula ˜ B) = ET (gτ , s, Φ (5.2.1)

Z Sym2 (Q)\Sym2 (A)

Z

=

Sym2 (A)

˜ B ) ψ(−tr(T b)) db E(n(b)gτ , s, Φ

˜ B (w−1 n(b)gτ , s) ψ(−tr(T b)) db Φ 3

2 = WT,∞ (gτ , s, Φ∞ )·

Y p

˜ B ), WT,p (s, Φ p

112

CHAPTER 5

where (5.2.2) 3 2

3

Z

WT,∞ (gτ , s, Φ∞ ) =

Sym2 (R)

−1 2 Φ∞ (w∞ n(b)gτ , s) ψ∞ (−tr(T b)) d∞ b

and (5.2.3)

˜B WT,p (s, Φ p)=

Z Sym2 (Qp )

−1 ˜B Φ p (wp n(b), s) ψp (−tr(T b)) dp b

are local (degenerate) Whittaker integrals. Here, we are writing n(b) for Q [n(b), 1]L = p [n(bp ), 1]L,p , since the splitting homomorphism N (A) → GA is unique. The global measure db is the Tamagawa measure; this measure is self-dual with respect to the pairing [b1 , b2 ] = ψ(tr(b1 b2 )) determined by ψ. The local measures dp b are self-dual with respect to the analogous pairing determined local components ψp of ψ. Also 

(5.2.4)

w=

12



−12

∈ GQ ,

and we choose elements wp = [wp , 1]L ∈ Kp projecting to w in K p and Q with w = p wp in GA ; see Sections 8.5.1 and 8.5.5. Since the local section ˜ p (s) is right invariant under N (Zp ), we have immediately Φ Lemma 5.2.1. For T ∈ Sym2 (Q) with

T ∈ / Sym2 (Z)∨ = { T ∈ Sym2 (Q) | tr(T b) ∈ Z, ∀b ∈ Sym2 (Z) }, ˜ B ) = 0. ET (gτ , s, Φ We next discuss the individual factors in the product on the right side of (5.2.1). For T ∈ Sym2 (Z)∨ with det(T ) 6= 0, let (5.2.5)

S(T, B) = { p | p | 8D(B) det(T ) }.

By Proposition 4.1 of [4], for a finite prime p ∈ / S(B, T ), we have (5.2.6)

−1 ˜B WT,p (s, Φ p ) = ζp (2s + 2) .

For T ∈ Sym2 (Z)∨ with det(T ) 6= 0 and S = S(T, B), we have (5.2.7) 3 Y 2 ˜ B ) = WT,∞ (gτ , s, Φ∞ ˜ B ) · ζ S (2s + 2)−1 . ET (gτ , s, Φ )· WT,p (s, Φ p

p∈S

Note that, since the local Whittaker functions for nonsingular T ’s are entire functions of s, the finite product on the right side of (5.2.7) gives the analytic continuation of the nonsingular Fourier coefficient.

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

113

˜ B ) = 0, at least one of the factors in the product must Since ET (gτ , 0, Φ ˜ B) = vanish at s = 0, and, if more than one factor vanishes, then ET0 (gτ , 0, Φ 0 as well. The vanishing of local factors is controlled by the following principle. Lemma 5.2.2. (i) For any standard local section Φp (s) for which Φp (0) ∈ Rp (Vp ), WT,p (0, Φp ) 6= 0 =⇒ T is represented by Vp .

˜ p (s) defined by (5.1.33), Similarly, for the nonstandard section Φ ˜ p ) 6= 0 =⇒ T is represented by V − . WT,p (0, Φ p

(ii) T ∈ Sym2 (Qp ) with det(T ) = 6 0 is represented by Vp if and only if  = p (T ) χV (det(T )) = p (T ) (− det(T ), −1)p ,

where p (T ) is the Hasse invariant of T .

Note that Vp has Hasse invariant p (Vp ) = (−1, −1)p , so that there is a twist here when p = 2 or ∞, as compared with Lemma 8.2 of [4]. Definition 5.2.3. For T ∈ Sym2 (Zp )∨ with det(T ) 6= 0, let µp (T ) = p (T ) (− det(T ), −1)p . Note that, by (ii) of the previous lemma, µp (T ) = 1 if and only if T is represented by the space Vp+ of trace 0 elements in M2 (Qp ), with quadratic form given by the determinant. If p is odd and T is GL2 (Zp )-equivalent to the diagonal form diag(ε1 pa1 , ε2 pa2 ) with 0 ≤ a1 ≤ a2 and ε1 , ε2 ∈ Z× p, then (5.2.8)

µp (T ) = (ε1 , p)ap2 (ε2 , p)ap1 (−1, p)ap1 a2 ;

cf. Lemma 8.3 of [4]. − Recall that C = C B is the incoherent collection with C∞ = V (3, 0) = V∞ B B B B and Cp = Vp , for p < ∞. Let inv∞ (C ) = −1 and invp (C ) = invp (B) for p < ∞. Then, the set (5.2.9)

Diff(T, B) = { p ≤ ∞ | invp (C B ) 6= µp (T ) }

has odd cardinality, and, by the previous lemma, (5.2.10)

|Diff(T, B)| > 1

=⇒

˜ B ) = 0. ET0 (gτ , 0, Φ

Thus, only T ’s with |Diff(T, B)| = 1 contribute to the central derivative.

114

CHAPTER 5

The following results concerning the values and derivatives of local factors WT,p (s) are collected from [4] for p - 2D(B), from [17] and [8] for p | D(B), p 6= 2, and from [18] for p = 2. For p ≤ ∞, let 3

Cp (V ) = γp (V )2 |D(B)|2p |2|p2 ,

(5.2.11)

where the quantity γp (V ) is a local Weil index defined in (8.5.21). This quantity will frequently appear as a constant of proportionality. Note that Y

(5.2.12)

Cp (V ) = 1.

p≤∞

Theorem 5.2.4. (Kitaoka [3] for p 6= 2, Yang [18], Theorem 5.7) ˜ B (s) = Φ0 (s). Let X = p−s . Suppose that p - D(B), so that Φ p p ∨ (i) If T ∈ / Sym2 (Zp ) , then WT,p (s) = 0. (ii) Suppose that T ∈ Sym2 (Zp )∨ and let (0, a1 , a2 ), with 0 ≤ a1 ≤ a2 , be the Gross-Keating invariants of the matrix diag(1, T ) ∈ Sym3 (Zp )∨ . Then the quantity ˜B WT,p (s, Φ p) Cp (V ) · (1 − p−2s−2 )

is given by a1 −1 2

X

X 2j + µp (T )X a1 +a2 −2j pj


j=0

if a1 is odd, and a1 −1 2

X j=0

X

2j

+ µp (T )X

a1 +a2 −2j




j

p +p

a1 2

X

a1

a2X −a1

(0 X)j

j=0

if a1 is even, where 0 = 0 (T ) is the Gross-Keating -constant [1], p. 236, and the constant Cp (V ) is given by (5.2.11). Note that, when p is odd and T is GL2 (Zp ) equivalent to the matrix diag(ε1 pa1 , ε2 pa2 ), as above, then (0, a1 , a2 ) are the Gross-Keating invariants of diag(1, T ), and 0 = 0 (T ) = (−ε1 , p)p . Corollary 5.2.5. Suppose that p - D(B) and that T ∈ Sym2 (Zp )∨ is as in (ii) of Theorem 5.2.4.

115

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

(i) If µp (T ) = +1, then ˜B WT,p (0, Φ p) Cp (V ) · (1 − p−2 )

 P a12−1 j   2  j=0 p    a1 a = 2 P 2 −1 pj + p 21 j=0     a  a1  P 21 −1 j 2

2

p +p

if a1 is odd,

if a1 is even and 0 = −1,

(a2 − a1 + 1) if a1 is even and 0 = 1.

˜ B ) = 0 and (ii) If µp (T ) = −1, then WT,p (0, Φ p j=0

0 ˜ B ) = Cp (V ) · (1 − p−2 ) log(p) · νp (T ), WT,p (0, Φ p

where νp (T ) =

if a1 is odd,

 a1 −1 P 2   (a1 + a2 − 4j) pj j=0

 P

a1 −1 2

j=0

(a1 + a2 − 4j) pj + 12 (a2 − a1 + 1) p

a1 2

if a1 is even.

ra ˜B Next we consider the case p | D(B) and we recall that Φ p (0) = Φp (0) and that, by (5.1.33), (5.2.13) 0 0 ra 0 0 0 1 ˜B (0, Φ WT,p p ) = WT,p (0, Φp ) + Ap (0) · WT,p (0, Φp ) + Bp (0) · WT,p (0, Φp ),

where the coefficients A0p (0) and Bp0 (0) are given by (5.1.35). When p 6= 2, the following result is a combination of Proposition 8.7 of [4] and Corollary 7.4 of [8]. These results, in turn, rely on the computation of the local densities [11], [17] and their derivatives [17]. In the case p = 2, the densities and their derivatives are computed in [18], and it is shown that the statements of Proposition 8.7 of [4] and Corollary 7.4 of [8] remain valid provided one uses the Gross-Keating invariants of diag(1, T ). Theorem 5.2.6. Suppose that p | D(B) and that T ∈ Sym2 (Zp )∨ is as in (ii) of Theorem 5.2.4. (i) If µp (T ) = −1, then ˜ B ) = WT,p (0, Φra ) = Cp (V ) · 2 (p + 1). WT,p (0, Φ p p

˜ B ) = 0, and (ii) If µp (T ) = 1, then WT,p (0, Φ p

1 0 ˜B WT,p (0, Φ · νp (T ), p ) = Cp (V ) · (p + 1) log(p) · 2

116

CHAPTER 5

where 1 · νp (T ) = a1 + a2 + 1 2

−

 a1 a1  p 2 −1   2 + 2 p   p−1    a (a2 − a1 + 1) p

1 2

if a1 is even and 0 = −1,

a1

2 + 2 pp −−11

   a1 +1     p 2 −1

2

p−1

if a1 is even and 0 = 1, if a1 is odd.

Note that this result is the analogue, for p | D(B), of Corollary 5.2.5. At this point, we omit the analogue of Theorem 5.2.4, i.e., the expressions ˜ B ) for general s. These are rather messy and will be given in for WT,p (s, Φ p Section 5.7 below. Finally, we have the case in which p = ∞, where we use the calculations of [4], which are based on Shimura’s formulas for the confluent hypergeometric functions [15]. Note, however, that in [4], [w−1 , 1]R = [w−1 , −i]L was used as the preimage in GR of w−1 . Thus the values here include an extra factor of i, since we are taking [w−1 , 1]L for preimage of w−1 . Theorem 5.2.7. (i) If T ∈ Sym2 (R)>0 is positive definite, then, for τ = u + iv ∈ H2 , 3 √ 3 2 WT,∞ (gτ , 0, Φ∞ ) = −2 2 (2π)2 det(v) 4 · q T ,

where q T = e(tr(T τ )).

2 (ii) If T ∈ Sym2 (R) has signature (1, 1) or (0, 2), then WT,∞ (gτ , 0, Φ∞ )= 0, and 3

3 √ 3 0 2 WT,∞ (gτ , 0, Φ∞ ) = −2 2 · 2π 2 · det(v) 4 · q T · ν∞ (T, v),

where ν∞ (T, v) is defined as follows. If T has signature (1, 1), then ν∞ (T, v) = − where

1 2

Z

Ei(−4πδ− y −2 |z|2 )

H

× δ+ y −2 (1 + |z|2 )2 −

1

1

1
−πδ+ ( y−2 (1+|z|2 )2 −4 ) e dµ(z), 2π

v 2 T v 2 = t k(θ) · diag(δ+ , −δ− ) · k(θ),

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

117

for k(θ) ∈ SO(2) and δ± ∈ R>0 . Here, for z = x + iy ∈ H, dµ(z) = y −2 dx dy, and Ei is the exponential integral, as in (3.5.2). If T has signature (0, 2), then ν∞ (T, v) = −

1 2

Z

Ei(−4πδ2 y −2 |z|2 )

H

× δ1 y −2 (1 − |z|2 )2 −

where

1
−πδ1 ( y−2 (1−|z|2 )2 +4 ) e dµ(z), 2π

1

1

v 2 T v 2 = −t k(θ) · diag(δ1 , δ2 ) · k(θ),

for k(θ) ∈ SO(2) and δi ∈ R>0 .

Proof. Part (i) and the signature (1, 1) part of (ii) are Proposition 9.5 and Corollary 9.8 in [4], respectively. For the sake of completeness, we give the calculation in the case of signature (0, 2), which is a variant of that given in [4] for signature (1, 1). The manipulations on pp. 585–588 of [4] yield the expression 3

WT,∞ (gτ0 , s, Φ 2 ) β−α 3 √ 1 3 e( 2 ) (2π) = i 2 det(v) 2 (s+ 2 ) η(2 v, T ; α, β) · e(tr(T u)), 2 Γ2 (α) Γ2 (β)

where α = 12 (s + 3) and β = 21 s, and Z

e−2tr(U v) det(U + πT )α−ρ det(U − πT )β−ρ dU,

η(2 v, T ; α, β) = U >−πT U >πT

where ρ = 32 . Writing π T = −t c c for c ∈ GL2 (R)+ , and using Lemma 9.6 of [4], we get η(2 v, T ; α, β) · (π 2 det(T ))−(α+β−ρ) Z

=

e−2 tr(U cv

t c)

det(U − 1)α−ρ det(U + 1)β−ρ dU

U −1>0 U +1>0 2π tr(T v)

Z

=e

0

e−2π tr(U v ) det(U )α−ρ det(U + 2)β−ρ dU

U >0 2(α+β−ρ) 2π tr(T v)

=2

Z

e

U >0

0

e−4π tr(U v ) det(U )α−ρ det(U + 1)β−ρ dU,

118

CHAPTER 5

where v 0 = π −1 cv t c. Writing v 0 = t k(θ) ∆ k(θ), with ∆ = diag(δ1 , δ2 ) and k(θ) ∈ SO(2), we obtain η(2 v, T ; α, β) · (4π 2 det(T ))−(α+β−ρ) Z

= e2π tr(T v)

e−4π tr(U ∆) det(U )α−ρ det(U + 1)β−ρ dU.

U >0

Note that this integral is finite when s = 0. Since s+3 Γ2 (α) Γ2 (β) = π Γ 2 



s+2 Γ 2 



s Γ 2

 

s−1 Γ 2 



has a simple pole at s = 0 with residue −2π 2 , we obtain 3

0 WT,∞ (gτ0 , 0, Φ 2 )

Z √ 3 3 e−4π tr(U ∆) det(U + 1)− 2 dU. = −i 2 det(v) 4 · 2πi · q T · e4π tr(T v) U >0

To put the integral here in a better form, we write 

x z U= z y



and make the substitution x = u, 1

1

z = u 2 (1 + v) 2 w, y = v + (1 + v)w2 = w2 + (1 + w2 )v. Then det(U ) = uv, det(U + 1) = (1 + v)(1 + u + w2 ), ∂(x, y, z) 1 1 ∂(u, v, w) = u 2 (1 + v) 2 ,

119

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

and we have Z

3

e−4π tr(U ∆) det(U + 1)− 2 dU

U >0

Z

Z

Z

e−4π ( δ1 u+δ2 (w

=

2 +(1+w 2 )v)

)

3

(1 + v)− 2

u>0 v>0 w 1

3

1

× (1 + u + w2 )− 2 u 2 (1 + v) 2 du dv dw. Now putting (1 + w2 )u for u, we obtain Z

Z

Z

e−4π ( δ1 (1+w

2 )u+δ

2

(w2 +(1+w2 )v) )

u>0 v>0 w 3

1

× (1 + v)−1 (1 + u)− 2 u 2 du dv dw. The integral with respect to v here is Z

e−4π δ2 (1+w

2 )v

2

(1 + v)−1 dv = −e4π δ2 (1+w ) Ei(−4π δ2 (1 + w2 )).

v>0

Applying integration by parts to the integral with respect to u, we have Z

e−4π δ1 (1+w

2 )u

3

1

(1 + u)− 2 u 2 du

u>0

= −2π

Z

e−4π δ1 (1+w

2 )u

4 δ1 (1 + w2 ) u −

u>0

1 1
−1 u 2 (1 + u)− 2 du. 2π

Combining these facts, we obtain the expression 2π e4π δ2

Z Z w

Ei(−4π δ2 (1 + w2 ))

u>0

1 1
−1 u 2 (1 + u)− 2 du dw. 2π which is the analogue of (9.57) of [4] in the present case. Finally, for z = x + iy ∈ H, we make the substitution given on p. 593 of [4]:

× e−4π δ1 (1+w

2 )u

4 δ1 (1 + w2 ) u −

x w= , y

u=

|z| − |z|−1 2

!2

,

so that 1

1

u− 2 (1 + u)− 2 du dw = 2 y −2 dx dy, 1 −2 y (1 − |z|2 )2 , 4 (1 + w2 ) = y −2 |z|2 ,

(1 + w2 ) u =

120

CHAPTER 5

and the previous expression becomes 4π δ2

Z

2π e

Ei(−4π δ2 y −2 |z|2 )

H

× e−π δ1 y

−2 (1−|z|2 )2

1
−2 y dx dy. 2π

δ1 y −2 (1 − |z|2 )2 −

Returning to the function, this gives 3

0 WT,∞ (gτ0 , 0, Φ 2 ) Z √ 3 = 2 2 · 2π 2 · det(v) 4 · q T · e4π tr(T v) e4π δ2 Ei(−4π δ2 y −2 |z|2 ) H

× e−π δ1 y

−2 (1−|z|2 )2

1
−2 y dx dy 2π

δ1 y −2 (1 − |z|2 )2 −

√ 3 = −2 2 · 2π 2 · det(v) 4 · q T · ν∞ (T, v), where 1 ν∞ (T, v) = − · e−4π δ1 2 × e−π δ1 y

Z

Ei(−4π δ2 y −2 |z|2 )

D −2 (1−|z|2 )2

δ1 y −2 (1 − |z|2 )2 −

1
−2 y dx dy. 2π

Here we note that tr(T v) = −(δ1 + δ2 ). In summary, we obtain the following formulas.

Proposition 5.2.8. Suppose that T ∈ Sym2 (Z)∨ with det(T ) 6= 0. (i) If Diff(T, B) = {p} with p < ∞, then Y √ 0 0 ˜ B ), ˜B (0, Φ E2,T (τ, 0, B) = −C B (0) · 2 2 · 4π 2 · q T · WT,p WT,` (0, Φ p )· ` where

`6=p

0 ˜B WT,p (0, Φ p ) = Cp (V ) · νp (T ) log(p) ·

if p - D(B),

 (1 − p−2 )

if p | D(B).

 1 (p + 1) 2

(ii) If Diff(T, B) = {∞} and sig(T ) = (1, 1) or (0, 2), Y √ 0 ˜ B ). E2,T (τ, 0, B) = −C B (0) · 2 2 · 2π 2 · q T · ν∞ (T, v) · WT,` (0, Φ ` (iii) For all other T ∈ Sym2 (Z)∨ , i.e., if |Diff(T, B)| > 1, `

0 E2,T (τ, 0, B) = 0.

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

121

5.3 THE SIEGEL-WEIL FORMULA

In this section, we suppose that T ∈ Sym2 (Z)∨ with det(T ) 6= 0 and with Diff(T, B) = {p} for p ≤ ∞, and we evaluate the products, Y

(5.3.1)

˜B WT,` (0, Φ ` ),

`6=p

when p < ∞, and Y

(5.3.2)

˜ B ), WT,` (0, Φ `

`

when p = ∞. Note that, both expressions are unchanged if, for each ` | ˜ B by Φra . D(B), we replace the section Φ ` ` 5.3.1 Whittaker functions and orbital integrals

We begin with some general remarks. Suppose that V is any three dimensional anisotropic quadratic space over Q. For ϕ ∈ S(V (A)2 ), let Φϕ (s) ∈ I(s, χV ) be the standard section such that Φϕ (0) = λV (ϕ), where λV : S(V (A)2 ) → I(0, χ) is the map given by (5.1.16). By the Siegel-Weil formula [5], we know that (5.3.3)

E(g, 0, Φϕ ) = 2 I(g, ϕ),

where (5.3.4)

Z

I(g, ϕ) =

O(V )(Q)\O(V )(A)

θ(g, h; ϕ) dh

is the theta integral, with vol(O(V )(Q)\O(V )(A), dh) = 1. In particular, there is an identity of Fourier coefficients: (5.3.5)

ET (g, 0, Φϕ ) = 2 IT (g, ϕ)

for all T ∈ Sym2 (Q). Since the identity (5.3.5) for all nonsingular T ’s was actually one of the key ingredients in the proof of the Siegel-Weil formula in [5] and since we now need a little more information anyway, we return to the proof of such identities given in [13], rather than viewing them as consequences (5.3.3). If det(T ) 6= 0, g ∈ GR , and ϕ is factorizable, then (5.3.6)

ET (g, s, Φϕ ) = WT,∞ (g, s, Φϕ,∞ ) ·

Y p

WT,p (s, Φϕ,p ),

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CHAPTER 5

as in (5.2.1) above. On the other hand, the corresponding Fourier coefficient of the theta integral is given by Z

IT (g, ϕ) =

X

O(V )(Q)\O(V )(A)

ω(g)ϕ(h−1 x) dh

x∈V (Q)2 Q(x)=T

(5.3.7)

Z

=

O(V )(Q)x0 \O(V )(A)

ω(g)ϕ(h−1 x0 ) dh

=

1 2

=

Y 1 OT,p (ϕp ), · OT,∞ (ω(g)ϕ∞ ) · 2 p

Z O(V )(A)

ω(g)ϕ(h−1 x0 ) dh

where x0 ∈ V 2 (Q) with Q(x0 ) = T , and where, in the last step, we have assumed that ϕ is factorizable and written (5.3.8)

Z

OT,p (ϕp ) =

O(V )(Qp )

ϕp (h−1 x0 ) dT,p h,

p ≤ ∞, for the local orbital integral. The measures are normalized as follows ([13], p. 95): Let (5.3.9)

Q : V 2 −→ Sym2 ,

x 7→ Q(x) =

1 ((xi , xj )), 2

be the moment map. VLet α and β be basis vectors for the 1-dimensional V spaces 6 (V 2 )∗ and 3 (Sym2 )∗ , respectively. We also write α and β for the corresponding translation invariant differential forms on V 2 and Sym2 . Let (5.3.10)

2 Vreg = { x ∈ V 2 | det Q(x) 6= 0 },

2 is a subset of the submersive set of the moment map, i.e., and note that Vreg the set of x = [x1 , x2 ] ∈ V 2 such that x1 and x2 span a 2-plane in V .

2 with the following properties. Lemma 5.3.1. There is a 3-form ν on Vreg (i)

α = Q∗ (β) ∧ ν.

(ii) For (h, g) ∈ SO(V ) × GL2 , acting on V 2 by x 7→ hxg −1 , (h, g)∗ ν = ν.

2 , the restriction of ν to ker dQ is nonzero. (iii) For all points x ∈ Vreg x

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

123

2 and s ∈ Sym , we make the usual identifications Proof. For x ∈ Vreg 2 2 ) = V 2 and T (Sym ) = Sym . Then, the differential of Q is given Tx (Vreg s 2 2 by

(5.3.11)

dQx (v) =

Here, v ∈ V 2 , Q(x) = we define the map

1 2

1 1 (x, v) + (v, x). 2 2

2 , ((xi , xj )), (x, v) = ((xi , vj )), etc. For x ∈ Vreg

2 (5.3.12) jx : Sym2 −→ V 2 = Tx (Vreg ),

jx (u) =

1 x · Q(x)−1 · u. 2

Then, (x, jx (u)) = u, and (5.3.13)

dQx ◦ jx (u) = u.

2 ), and note that Let Sx = image(jx ) ⊂ Tx (Vreg

(5.3.14)

2 Tx (Vreg ) = Sx ⊕ ker dQx .

2 as follows. Choose a triple of tangent vectors We define a 3-form ν on Vreg u = [u1 , u2 , u3 ] with ui ∈ Sym2 with β(u) 6= 0. For a triple t = [t1 , t2 , t3 ] of tangent vectors in Tx (V 2 ), let

(5.3.15)

ν(t) = α(jx (u), t) · β(u)−1 .

This quantity is independent of the choice of u, and since the components of jx (u) span Sx , ν(t) depends only on the projection of t onto ker dQx with respect to the decomposition (5.3.14). Property (iii) is clear from the definition. To check (i), it suffices to evaluate both sides on any 6-tuple of tangent vectors whose components span V 2 , e.g., on v = [jx (u), t], where the components of t span ker dQx . Then, Q∗ (β) ∧ ν(v) = β(dQx ◦ jx (u)) · ν(t) (5.3.16)

= β(u) · α(jx (u), t) · β(u)−1 = α(jx (u), t).

To check (ii), note that, if y = (h, g) · x, for (h, g) ∈ SO(V ) × GL2 , then (5.3.17)

(h, g) · jx (u) = jy (t g −1 ug −1 ).

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CHAPTER 5

Then ν((h, g)∗ (t)) = α(jy (u), (h, g)∗ (t)) · β(u)−1 (5.3.18)

= α((h, g)∗ (jx (t gug)), (h, g)∗ (t)) · β(u)−1 = det(g)−3 · α(jx (t gug), t) · β(t gug)−1 det(g)3 = ν(t).

2 with Q(x) = T , then there is an isomorphism If x ∈ Vreg

(5.3.19)

ix : SO(V ) −→ Q−1 (T ),

h 7→ h−1 x,

and by (ii) and (iii) of the previous lemma, ω = ωT = i∗x (ν) is a gauge form on SO(V ). Lemma 5.3.2. The gauge form ω is independent of T .

Proof. Since ixg−1 = (1, g) ◦ ix , the invariance property (ii) yields (5.3.20)

ωT 0 = (ixg−1 )∗ (ν) = i∗x ◦ (1, g)∗ (ν) = i∗x (ν) = ωT ,

where T 0 = t g −1 T g −1 . Over an algebraically closed field, the action of the group GL2 on (Sym2 )reg , the open subset of nonsingular elements of Sym2 , is transitive. The form ω determines the Tamagawa measure dh1 on SO(V )(A). On O(V )(A) = SO(V )(A) × µ2 (A), the measure fixed above is dh = dh1 × dc, where vol(µ2 (A), dc) = 1. On the other hand, ω determines measures dp h1 on the groups SO(V )(Qp ), so that dp h = dp h1 × dcp , with vol(µ2 (Qp ), dcp ) = 1, and a factorization (5.3.21)

dh =

Y

dp h

p≤∞

of the Tamagawa measure. Moreover, by construction, the measure dp h1 coincides, under the isomorphism ix0 , with the measure on Q−1 (T )(Qp ) determined by ν. Our fixed additive character ψ = ⊗p ψp , determines factorizations of the Tamagawa measures on V 2 (A) and on Sym2 (A) as follows. On V 2 (Qp ), we have fixed the measure dp x which is self-dual for the pairing [x, y] = ψp (tr(x, y)), where ( , ) is the bilinear form associated to Q. On the other hand, the gauge form α on V 2 determines a measure (5.3.22)

dα,p x = cp (α, ψ) dp x,

THE CENTRAL DERIVATIVE OF A GENUS TWO EISENSTEIN SERIES

125

for a positive real constant cp (α, ψ). Then, since dα x = dx on V 2 (A), Y

(5.3.23)

cp (α, ψ) = 1.

p≤∞

Similarly, we have fixed the measure dp b on Sym2 (Qp ) which is self-dual with respect to the pairing [b, c] = ψp (tr(bc)). The gauge form β determines a measure (5.3.24)

dβ,p b = cp (β, ψ) dp b,

on Sym2 (Qp ), for a positive real constant cp (β, ψ). Again, on Sym2 (A), dβ b = db and so Y

(5.3.25)

cp (β, ψ) = 1.

p≤∞

Finally, for the Weil representation ωV,ψ = ⊗p ωVp ,ψp , we have, for all ϕ ∈ S(V 2 (A)) and ϕp ∈ S(V 2 (Qp )), (5.3.26)

ωV,ψ (w

−1

Z

)ϕ(x) =

ϕ(y) ψ(−tr(x, y)) dy, V 2 (A)

and (5.3.27) ωVp ,ψp (wp−1 )ϕp (x) = γp (V )2 ·

Z V 2 (Qp )

ϕp (y) ψp (−tr(x, y)) dp y;

see (5.6.3) below and Section 8.5. Here, since we are taking wp−1 = [w−1 , 1]L,p , the constant γp (V )2 is given by (8.5.21). Again, by (5.3.26), Y

(5.3.28)

γp (V ) = 1.

p≤∞

The following result is a very special case of the results of Chapter 4 of [13]. Proposition 5.3.3. Let

Cp (V, α, β, ψ) =

(i) For each p < ∞,

γp (V )2 cp (β, ψ) . cp (α, ψ)

WT,p (0, Φϕ,p ) = Cp (V, α, β, ψ) · OT,p (ϕp ).

(ii) For p = ∞, and g ∈ GR ,

WT,∞ (g, 0, Φϕ,∞ ) = C∞ (V, α, β, ψ) · OT,∞ (ω(g)ϕ∞ ).

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CHAPTER 5

Remark 5.3.4. Note that the constant of proportionality γp (V )2 cp (β, ψ) cp (α, ψ)−1 does not depend on T . Moreover, the Fourier coefficient identity (5.3.5) is an immediate consequence of the combination of this result with (5.3.7) and the fact that Y

(5.3.29)

Cp (V, α, β, ψ) = 1.

p≤∞

Corollary 5.3.5. For any ϕf = Y

Q

p 0, then ∞ ∈ / Diff(T, B). Moreover, |Diff(T, B)| > 1 implies that Z(T ) = φ, so that the only positive definite T ’s of interest are those for which Diff(T, B) = {p} for a prime p. A matrix T ∈ Sym2 (Z)∨ >0 will be called good if Diff(T, B) = {p} with p - D(B). For such a T , Z(T ) is a 0-cycle supported in the fiber Mp , and there is an associated class (6.0.5)

2

b d (M). Z(T, v) = ( Z(T ), 0 ) ∈ CH

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CHAPTER 6

The partial generating series X

(6.0.6)

b Z(T, v) q T

T ∈Sym2 (Z)∨ >0 T good

thus has a natural geometric definition, and our first goal in this chapter is b to complete it by defining terms Z(T, v) for the remaining T ∈ Sym2 (Z)∨ . b d Z(T, We then compute the quantities deg( v)) in all cases and compare them with the Fourier coefficients of the central derivative of the Siegel Eisenstein series E20 (τ, 0; B) computed in Chapter 5. In this way, we prove the following result:

Theorem B. The generating function φˆ2 (τ ) is a Siegel modular form of genus two and weight 23 for Γ0 (4D(B)o ) ⊂ Sp2 (Z), where D(B)o is the odd part of D(B). More precisely φˆ2 (τ ) = E20 (τ, 0; B),

so that for all γ =

φˆ2 (γ(τ )) = sgn(det d) · j 3 (γ, τ ) · φˆ2 (τ ), 2


b

a c d

∈ Γ0 (4D(B)o ).

The transformation law of E2 (τ, s; B) used here is determined in Section 8.5.6. Note that sgn(det d) = (det d, −1)2 and that the automorphy factor j 3 (γ, τ ), which is described explicitly in that appendix, satisfies 2

j 3 (γ, τ )2 = det(cτ + d)3 . 2

To complete the definition of φˆ2 (τ ), we must consider T ’s of the following types: (i) T > 0 with Diff(T, B) = {p} for p | D(B), i.e., T is not good. (ii) T with signature (1, 1) or (0, 2) with Diff(T, B) = {∞}. (iii) T ≥ 0 with det(T ) = 0, but T = 6 0. (iv) T ≤ 0 with det(T ) = 0, but T 6= 0. (v) T = 0.

169

THE GENERATING FUNCTION FOR 0-CYCLES

b b d Z(T, In some cases, we define Z(T, v) by giving the real number deg( v)) directly. On the one hand, it is clear that we want these quantities to coincide with the corresponding Fourier coefficients of E20 (τ, 0; B), since we would like a generating function which is modular. On the other hand, we would like to give a definition which is as natural as possible from the point of view of arithmetic geometry. The main result of Chapter 7, which identifies the inner product h φˆ1 (τ1 ), φˆ1 (τ2 ) i of two genus one generating functions with the restriction to the diagonal φˆ2 (diag(τ1 , τ2 )) gives a further geometric justification for our definitions. The fact that the generating series φˆ2 (τ ) is the q-expansion of a Siegel b modular form implies that the numbers Z(T, v) must satisfy some highly nontrivial identities. For example, for α ∈ GL2 (Z), the transformation law

(6.0.7)

φˆ2 (τ ) = φˆ2 (ατ t α) =

X

b d Z(T, deg( αv t α)) q

t αT α

T

implies that (6.0.8)

b t αT α, v)) = deg( b d Z( d Z(T, deg( αv t α)),

for any T and v. For T ∈ Sym2 (Z)∨ >0,good , this amounts to (6.0.9)

t d d deg(Z( αT α)) = deg(Z(T )).

This identity is immediate from the definition of Z(T ), since the length of the local deformation ring where a pair x of special endomorphisms of (A, ι) deforms is the same as the one where α · x deforms, since this length only depends on the Zp -span of the components x1 and x2 of x. On the other hand, if T ∈ Sym2 (Z)∨ >0 is not good, then the equality above is true for any α ∈ GL2 (Zp ), but this is one of the main results of [5]. 6.1 THE CASE T > 0 WITH Diff(T, B) = {p} FOR p - D(B)

For a positive definite T with Diff(T, B) = {p} for p - D(B), the class b d 2 (M) is defined above, and it remains to comZ(T, v) = (Z(T ), 0) ∈ CH b d Z(T, d pute the real number deg( v)) = deg(Z(T )) and to compare it with the 0 corresponding Fourier coefficient E2,T (τ, 0; B) of E20 (τ, 0; B). d To compute deg(Z(T )) we use the Gross-Keating theorem in the form of Theorem 3.6.1. According to this theorem, the stack Z(T ) has support in the supersingular locus of Mp . Let B (p) be the definite quaternion algebra over Q whose invariants differ from those of B at precisely p and ∞. Let H (p) = B (p),× and (6.1.1)

V (p) = {x ∈ B (p) | tr(x) = 0}.

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CHAPTER 6

We choose a maximal order O(p) in B (p) and an isomorphism B (p) (Apf ) ' ˆ p to OB ⊗ Z ˆ p . We write B(Apf ) which carries O(p) ⊗ Z H (p) (Af ) =

(6.1.2)

a

H (p) (Q)hj K (p) ,

j

where ˆ ×. K (p) = Kp(p) · K p = (O(p) ⊗ Z)

(6.1.3)

ˆ = V (p) (Z ˆ p ) × V (p) (Zp ), with Also let V (p) (Z) ˆ p ) = V (p) (Ap ) ∩ (O(p) ⊗ Z ˆ p) V (p) (Z f and V (p) (Zp ) = V (p) (Qp )∩(O(p) ⊗Zp ). Let ϕ(p) ∈ S((V (p) (Af ))2 ) be the ˆ 2 . Finally, for y ∈ V (p) (Q)2 , we characteristic function of the set ( V (p) (Z)) (p) denote by ey,j the stabilizer of y in the group Γj = H (p) (Q)∩hj K (p) h−1 j . The following proposition gives the number of points in the stack sense of the finite stack Z(T ) (recall that we are assuming that T > 0 with Diff(B, T ) = p is good). Proposition 6.1.1. X

X

¯ p )ss y ∈V (A,ι)2 x∈M(F 1 (y,y)=T 2

X 1 = |Aut(Ax , ιx , y)| j

X

(p) −1 e−1 y,j · ϕ (hj · y)

y ∈V (p) (Q)2 1 (y,y)=T 2

¯ p )ss as a double coset Proof. The proof is based on the description of M(F space, (6.1.4)

¯ p )ss = H (p) (Q)\H (p) (Af )/K (p) . M(F

¯ p )ss by We recall that this is obtained by parametrizing the elements of M(F quasi-isogenies with source a fixed base point (A0 , ι0 ), chosen so that the (p) stabilizer of its Dieudonne module D(A0 ) in H (p) (Qp ) is equal to Kp and the stabilizer of its Tate module Tˆp (A0 ) in H (p) (Apf ) is equal to K p . We also identify V (A0 , ι0 ) ⊗ Q with V (p) (Q). Now if (A, ι) corresponds to the double coset H (p) (Q)hK (p) , then V (A, ι) can be identified with (6.1.5)

ˆ V (A, ι) = {y ∈ V (p) (Q) | h−1 · y ∈ V (p) (Z)}.

171

THE GENERATING FUNCTION FOR 0-CYCLES

Here h · y = hyh−1 . This is an immediate consequence of the fact that (6.1.6) End(A, ι) = {y ∈ B (p) (Q) | yh D(A0 ) ⊂ h D(A0 ), yh Tˆp (A0 ) ⊂ h Tˆp (A0 )} ˆ −1 . = B (p) (Q) ∩ h(O(p) ⊗ Z)h It follows that the left-hand side of the identity in the proposition is equal to (6.1.7)

X

X

j

y ∈V (p) (Q)2

1,

1 (y,y)=T 2 −1 ˆ 2 hj y ∈V (p) (Z) (p) mod Γj

which then leads to the expression of the right-hand side of the proposition. We note that the lengths of the local rings of Z(T ) are all identical and are denoted by νp (T ). They are given by the Gross-Keating formula of d Z(T ): Theorem 3.6.1. We therefore have the following expression for deg

Corollary 6.1.2. Let T be positive definite with Diff(T, B) = {p} for p 6 | D(B). Then d Z(T ) = νp (T ) log p · deg

X

X

e−1 y,j

·ϕ

(p)

(h−1 j



· y) ,

j y ∈V (p) (Q)2 1 2

where νp (T ) is given by Theorem 3.6.1 in terms of the Gross-Keating invariants (0, a1 , a2 ) of T˜ = diag(1, T ) ∈ Sym3 (Zp )∨ : νp (T ) =

(y,y)=T

if a1 is odd,

 a1 −1 P 2   (a1 + a2 − 4j) pj j=0

 P

a1 −1 2

j=0

(a1 + a2 − 4j) pj + 21 (a2 − a1 + 1) p

a1 2

if a1 is even.

Comparing this expression with the corresponding Fourier coefficient of the central derivative of the Eisenstein series computed in Theorem 5.3.8, we have.

Corollary 6.1.3. For T ∈ Sym2 (Z)∨ with det(T ) 6= 0 and Diff(T, B) = {p} with p - D(B), 0 b d Z(T, E2,T (τ, 0, B) = deg( v)) · q T .

172

CHAPTER 6

Proof. Indeed, the stabilizer of y in Γj has twice the size of the stabilizer of y in the group that was denoted Γj in Theorem 5.3.8 and (5.3.54). (p)

Remark 6.1.4. When p 6= 2, this computation and comparison is implicitly done in section 14 of [3]. There, the arithmetic intersection number of a pair b 1 , v1 ), Z(t b 2 , v2 ) ∈ CH d 1 (M) was computed in the case when of classes Z(t t1 t2 is not a square, so that the cycles Z(t1 ) and Z(t2 ) do not meet in the generic fiber. The intersection of such cycles in a fiber Mp , for p - D(B), is a finite union [

Z(T )

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 ) Diff(T,B)={p}

of 0-cycles, and the corresponding contribution to the arithmetic intersecd tion number is the sum of the deg(Z(T ))’s, which are given by the GrossKeating formula.

6.2 THE CASE T > 0 WITH Diff(T, B) = {p} FOR p | D(B)

In this case, the naive cycle Z(T ) defined by imposing a pair of special endomorphisms with fundamental matrix T is supported in the fiber Mp , but when p2 | T , this cycle has components of dimension 1, as described b in Chapter 3. To overcome this difficulty, we define the class Z(T, v) ∈ 2 b d d CH (M) by giving directly the number deg(Z(T, v)). Write (6.2.1)



T =



t1 m ∈ Sym2 (Z)>0 m t2

and recall that Z(T ) is the union of those connected components of the intersection Z(t1 ) ×M Z(t2 ) where the fundamental matrix, as defined in [5], is equal to T . After base change to Zp , we let, by [4], (6.2.2) b d Z(T, d 2 (M), deg( v)) := χ(Z(T ), OZ(t1 ) ⊗L OZ(t2 ) ) · log p ∈ R ' CH where χ is the Euler-Poincar´e characteristic of the derived tensor product of the structure sheaves OZ(t1 ) and OZ(t2 ) ; see [5], Section 4. When p 6= 2, the b d Z(T, v)) was computed in [5], Theorem 8.6. There the result number deg( was expressed as the product of a local multiplicity (which is actually global

173

THE GENERATING FUNCTION FOR 0-CYCLES

on the Drinfeld space) and an orbital integral. The calculation of the multiplicity comes down to a combinatorial problem. The same method works when p = 2, although, as described in the appendix to this chapter, the combinatorics become considerably more elaborate. Also, here we replace the orbital integral by an expression which is more in analogy with the formula obtained in the previous section in the case when p - D(B). To state the result, we introduce, as in the previous section, the twisted quaternion algebra B (p) over Q whose invariants differ from those of B at precisely p and ∞. We again introduce H (p) = B (p),× and V (p) . By strong approximation, we have H (p) (Af ) = H (p) (Q)H (p) (Qp )K p ,

(6.2.3)

ˆ p )× . Let ϕ(p) ∈ S((V (p) (Ap ))2 ) be the characteristic where K p = (O(p) ⊗ Z f ˆ p ))2 . We also set function of the set (V (p) (Z Γ0 = H (p) (Q) ∩ K p .

(6.2.4)

Using this notation, we have

Theorem 6.2.1. Let T ∈ Sym2 (Z)∨ with det(T ) 6= 0 and Diff(T, B) = {p} for p | D(B). Then b d Z(T, deg( v)) = νp (T ) · log p ·

1 2

X

ϕ(p) (y).

y ∈V (p) (Q)2 1 (y,y)=T 2 mod Γ0

Furthermore, the multiplicity νp (T ), defined in (6.2.11), is given as follows in terms of the Gross-Keating invariants (0, a1 , a2 ) and the Gross-Keating -constant 0 = 0 (T˜) , [2], p. 236, of T˜ = diag(1, T ) ∈ Sym3 (Zp )∨ : 1 νp (T ) = a1 + a2 + 1 2

−

 a1 a1  p 2 −1   2 + 2 p   p−1    a

(a2 − a1 + 1) p     a1 +1    p 2 −1  2

1 2

a1 2

+ 2 pp −−11

if a1 is even and 0 = −1, if a1 is even and 0 = 1,

if a1 is odd.

Proof. (Cf. also Section 7.6.) We use the p-adic uniformization of M ⊗ Zp ¯ p ), we have the Drinfeldand of Z(T ) ⊗ Zp . After base changing to W (F Cherednik uniformization of M, p−1

(6.2.5)

¯ p ) = H (p) (Q)\ Ω ˆ • × H (p) (Ap )/K p , M ⊗ W (F f


174

CHAPTER 6

ˆ• = Ω ˆ × Z is the disjoint union of copies of the Drinfeld space where Ω parametrized by Z. The isomorphism depends on the choice of a base point ¯ p ). We also fix an identification of O(p) with End(A0 , ι0 ). (A0 , ι0 ) ∈ M(F Now the analysis of [5], pp. 214–215, gives a uniformization of the formal ¯ p ) as an injection completion of CT = Z(T ) ⊗ W (F (6.2.6)

ˆ • × H (p) (Ap )/K p , CˆT ,→ H (p) (Q)\ V (p) (Q)2T × Ω f


where (6.2.7)

1 V (p) (Q)2T = {y ∈ V (p) (Q)2 | (y, y) = T }. 2

By [5], (8.30), the conditions describing CˆT inside the right-hand side of (6.2.6) are the following: (y, (X, ρ), gK p ) ∈ CˆT if and only if ˆ p ))2 , and (i) g −1 yg ∈ (V (p) (Z (ii) (X, ρ) ∈ Z • (j). Here j is the Zp -span of the images j1 and j2 of y1 and y2 under the injection End(A0 , ι0 ) → End(X) into the endomorphism algebra of the p-divisible group A0 (p) = X2 ; see [5]. Using strong approximation, we can write the right-hand side of (6.2.6) as ˆ• . Γ0 \ V (p) (Q)2T × Ω

(6.2.8)




Taking into account the description of CˆT above, we obtain (6.2.9)

CˆT =

a



Γ0y \Z • (j) . 

ˆ p )2 y ∈V (p) (Q)2T ∩V (p) (Z mod Γ0

The stabilizer Γ0y is the intersection of Γ0 with the center of B (p),× , hence Γ0y = Z[p−1 ]× ' {±1} × Z. The generator of the infinite factor acts by translating the ‘sheets’ of Z • (j) by 2, and the contribution of each sheet to the Euler-Poincar´e characteristic (6.2.2) is the same, see [5]. Denoting by d CˆT ) Z(j) the 0-th sheet, we therefore see that the contribution of y to deg( is equal to (6.2.10)

L e−1 y · 2 · χ(Z(j), OZ(j1 ) ⊗ OZ(j2 ) ) · log p,

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THE GENERATING FUNCTION FOR 0-CYCLES

where ey = 2 is the order of the stabilizer of y in the first factor of Γ0y . Now the quantity (6.2.11)

νp (T ) := 2 · χ(Z(j), OZ(j1 ) ⊗L OZ(j2 ) ) = 2 · (Z(j1 ), Z(j2 ))

only depends on the GL2 (Zp )-equivalence class of T ; see [5], Section 5. Then, as is proved for p 6= 2 in Section 6 of [5], and for p = 2 in the appendix to this chapter, νp (T ) is given by the expression appearing in the statement of the theorem. In any case we obtain e−1 y · νp (T )

X

d deg(Z(T )) = log p ·

ˆ p ))2 y ∈V (p) (Q)2T ∩(V (p) (Z mod Γ0

(6.2.12)

= νp (T ) · log p ·

1 2

X

ϕ(p) (y) ,

y ∈V (p) (Q)2 1 (y,y)=T 2 mod Γ0

as was to be shown. A comparison with Theorem 5.3.8 yields

Corollary 6.2.2. For T ∈ Sym2 (Z)∨ with det(T ) 6= 0 and Diff(T, B) = {p} for p | D(B), 0 b d Z(T, (τ, 0, B) = deg( v)) · q T . E2,T

6.3 THE CASE OF NONSINGULAR T WITH sig(T ) = (1, 1) OR (0, 2)

For T ∈ Sym2 (Z) nonsingular of signature (1, 1) or (0, 2), the cycle Z(T ) is empty, since the quadratic form on any V (A, ι) is positive definite. Thus, b the term Z(T, v) arises as a purely archimedean contribution. For a pair of vectors x = [x1 , x2
] ∈ V (Q)2 with nonsingular matrix of inner products Q(x) = 12 (xi , xj ) , the quantity (6.3.1)

Λ(x) :=

1 2

Z

ξ(x1 ) ∗ ξ(x2 ),

D

where ξ(x1 )∗ξ(x2 ) is the ∗-product of the Green functions ξ(x1 ) and ξ(x2 ), [1], is well defined and depends only on Q(x). In addition, Λ(x), which was denoted by Ht(x)∞ in [3], section 11, has the following invariance property. Theorem 6.3.1. ([3], Theorem 11.6) For k ∈ O(2), Λ(x · k) = Λ(x).

176

CHAPTER 6

For T ∈ Sym2 (Z) of signature (1, 1) or (0, 2) and for v ∈ Sym2 (R)>0 , 1 1 1 choose1 v 2 ∈ GL2 (R) such that v = v 2 · t v 2 , and define (6.3.2)

b d Z(T, deg( v)) :=

2

1

2 e−1 x · Λ(x v )

X

d (M). ∈ R ' CH

x ∈L2 Q(x)=T mod Γ × Here L = OB ∩ V , Γ = OB , and ex is the order of the stabilizer Γx of x in Γ. In fact, ex = 2 for any x with T = Q(x) nonsingular. Note that the invariance property of Theorem 6.3.1 is required to make the right side 1 independent of the choice of v 2 .

b d Z(T, Remark 6.3.2. Notice that deg( v)) = 0 if T is not represented by V , since then the summation in the definition is empty; see also Remark 3.5.3. Recall that T is represented by V if and only if Diff(T, B) = {∞}. For example, if m 6= 0, the matrix 

T =



t1 m m 0

has signature (1, 1) but is not represented by any anisotropic Vp . Thus the b primes dividing D(B) all lie in Diff(T, B) and the corresponding Z(T, v) is zero. Proposition 6.3.3. (i) Suppose that T has signature (1, 1) and write 1

1

v 2 T v 2 = t k(θ) · diag(δ+ , −δ− ) · k(θ)

for k(θ) ∈ SO(2) and δ± ∈ R>0 . Then 1 Λ(x v ) = − 2 1 2

Z

Ei(−4πδ− y

−2



2

|z| ) ·

δ+ y

−2

D

× e−πδ+ [y

where, for z = x + iy ∈ D, dµ(z) = y −2 dx dy. (ii) Suppose that T has signature (0, 2) and write 1

1 (1 + |z| ) − 2π 2 2

−2 (1+|z|2 )2 −4]



· dµ(z),

1

v 2 T v 2 = −t k(θ) · diag(δ1 , δ2 ) · k(θ)

for k(θ) ∈ SO(2) and δ1 , δ2 ∈ R>0 . Then 1

Λ(x v 2 ) = −

1 2

Z

Ei(−4πδ1 y −2 |z|2 ) ·



δ2 y −2 (1 − |z|2 )2 −

D

× e−πδ2 [y 1

1

1

−2 (1−|z|2 )2 +4]

Note that we may choose v 2 ∈ Sym2 (R) with det(v 2 ) > 0.

1 2π



· dµ(z).

177

THE GENERATING FUNCTION FOR 0-CYCLES

Proof. Part (i) is Theorem 6.3.1 together with Lemma 11.7 of [3]. To prove (ii), we write y1 =



p

δ1

1



and

−1

y2 =

p

δ2





1 1

.

Then, writing Ri = 21 (yi , x(z))2 + 2δi , we have 2 R1 = 4 δ1 y −2 |z|2

and

2 R2 = δ2 (y −2 (1 − |z|2 )2 + 4).

Now we use the expression of Lemma 11.5 of [3], taking into account the fact that there is no point evaluation term when both y1 and y2 have negative length.

Corollary 6.3.4. For T of signature (1, 1) or (0, 2), and for any x ∈ V (R)2 with Q(x) = T , 1

Λ(x v 2 ) = ν∞ (T, v),

where ν∞ (T, v) is as in (ii) of Theorem 5.2.7, and b Z(T, v) = ν∞ (T, v) ·

1 2

X

ϕB f (x) .

x ∈V (Q)2 Q(x)=T mod Γ

Comparing this expression with Theorem 5.3.8, we obtain

Corollary 6.3.5. For T of signature (1, 1) or (0, 2),

0 b d Z(T, E2,T (τ, 0, B) = deg( v)) · q T .

6.4 SINGULAR TERMS, T OF RANK 1

We now turn to the case (6.4.1)





t m ∈ Sym2 (Z)∨ T = 1 m t2

with det(T ) = 0. If T 6= 0, we may write t1 = n21 t, t2 = n22 t, and m = n1 n2 t for relatively prime integers n1 and n2 and t ∈ Z6=0 . The pair n1 , n2 is unique up to simultaneous change in sign. Also note that if t1 = 0, then n1 = 0, n2 = 1, and t = t2 . Similarly if t2 = 0, then n1 = 1, n2 = 0, and t = t1 . For the cycle associated to such a singular T , we have simply Lemma 6.4.1. Z(T ) = Z(t).

178

CHAPTER 6

Proof. We fix a choice of the pair n1 and n2 . First suppose that n1 n2 6= 0. Then if an object (A, ι, x) is given, with x = [x1 , x2 ] ∈ V (A, ι)2 , we have Q(n2 x1 − n1 x2 ) = 0, so that n2 x1 = n1 x2 ∈ V (A, ι). Since n1 and n2 −1 are relatively prime, it follows that y := n−1 1 x1 = n2 x2 ∈ V (A, ι). Note that Q(y) = t, so that (A, ι, y) is an object for Z(t). Conversely, for a given (A, ι, y), we have (A, ι, [n1 y, n2 y]) for Z(T ). If, say, t1 = n1 = 0 so that n2 = 1 and t = t2 , and (A, ι, x) is given, we have Q(x1 ) = 0 so that x1 = 0 and (A, ι, x2 ) defines an element of Z(t). Conversely, given (A, ι, y) we can take (A, ι, [0, y]). Of course, if t < 0, then both sides are empty. b d 2 (M) by setting We define Z(T, v) ∈ R ' CH

 b b t−1 tr(T v)) , ω d Z(T, (6.4.2) deg( v)) := − Z(t, ˆ

1 − degQ (Z(t)) 2



det(v)
log −1 + log(D) . t tr(T v) 

As motivation for this definition, note that we are, in some sense, shifting the ‘naive’ class (6.4.3)

1

b t−1 tr(T v)) ∈ CH d (M), Z(t,

which occurs in the wrong degree, by taking its pairing with −ˆ ω . The additional terms involving v are added in order to obtain agreement with the Fourier coefficient of E20 (τ, 0; B). Note that, for all α ∈ GL2 (Z), (6.4.4)

b b t αT α, v)), d Z(T, d Z( deg( αv t α)) = deg(

so that our definition has the invariance which must hold for a Fourier coefficient of a Siegel modular form. Now the results of [6] give explicit expressions for all such singular terms. More precisely, let E1 (τ, s; B) be the modified Eisenstein series of genus one associated to B. We then quote from [6] the following statement: Theorem 6.4.2. For t 6= 0,

and

1 E1,t (τ, ; B) = degQ Z(t) · q t , 2

1 0 b v), ω E1,t (τ, ; B) = h Z(t, ˆ i · qt. 2

179

THE GENERATING FUNCTION FOR 0-CYCLES

Corollary 6.4.3. For T ∈ Sym2 (Z)∨ of rank 1, 1 0 b d Z(T, deg( v)) · q T = −E1,t (t−1 tr(T τ ), ; B) 2 1 1 − E1,t (t−1 tr(T τ ), ; B) · 2 2



det(v)
log −1 + log(D) . t tr(T v) 

Comparing this with Theorem 5.8.7, we have

Corollary 6.4.4. For T ∈ Sym2 (Z)∨ of rank 1,

0 b d Z(T, E2,T (τ, 0; B) = deg( v)) · q T .

6.5 THE CONSTANT TERM, T = 0

We complete the definition of φˆ2 (τ ) by setting b v)) := ω d Z(0, (6.5.1) deg( ˆ, ω ˆ +



1 ω )· degQ (ˆ 2





log det(v)−c+log D ,

2

d (M) ' R, where the constant c is introduced in [6]. We as a class in CH b v) = Zb2 (0, v) to distinguish this constant term will sometimes write Z(0, of the genus two generating function from the constant term Zb1 (0, v) of the genus one generating function. Recall from [6] that

(6.5.2)

1

d (M) Zb1 (0, v) = −ˆ ω − (0, log(v)) + (0, c) ∈ CH

satisfies the identity 1 0 E1,0 (τ, ; B) = h Zb1 (0, v), ω ˆi 2 (6.5.3) 1 = −h ω ˆ, ω ˆ i − degQ ω ˆ· 2 and that this identity determines the constant c. Also, (6.5.4)

log(v) − c




1 E1,0 (τ, ; B) = degQ Zb1 (0, v) 2 = − degQ ω ˆ.

Thus we have (6.5.5) d Zb2 (0, v)) = −E 0 (i det(v), 1 ; B) − 1 · E1,0 (i det(v), 1 ; B) · log(D). deg( 1,0 2 2 2 Comparing this with the result of Theorem 5.9.5, we have the following identity:

180

CHAPTER 6

Theorem 6.5.1. 0 d Zb2 (0, v)). E2,0 (τ, 0; B) = deg(

This concludes the proof of Theorem B above. Bibliography

[1] H. Gillet and C. Soul´e, Arithmetic intersection theory, Publ. Math. IHES, 72 (1990), 93–174. [2] B. Gross and K. Keating, On the intersection of modular correspondences, Invent. math., 112 (1993), 225–245. [3] S. Kudla, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646. [4]

, Derivatives of Eisenstein series and arithmetic geometry, in Proc. Intl. Cong. Mathematicians, Vol II (Beijing, 2002), 173–183, Higher Education Press, Beijing, 2002.

[5] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223. [6] S. Kudla, M. Rapoport, and T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887–951.

Chapter Six: Appendix The case p = 2, p | D(B) In this appendix, we extend the results of [3] concerning intersections of the special cycles on the Drinfeld space to the case p = 2. We will denote by B 0 = M2 (Qp ) the matrix algebra over Qp and by V 0 its subspace of traceless elements.To any j ∈ V 0 there is associated a special cycle on the Drinfeld space. In the appendix to section 11 of [4] the geometry of an individual special cycle Z(j) is described in the case p = 2. What we have to deal with, then, is the relative position of two such special cycles. We remark that elsewhere in this book we used the quadratic form on the space of traceless elements of a quaternion algebra given by the reduced norm, Q(x) = Nm(x) = xι x. On the other hand, in the local context, we used in the appendix to section 11 of [4] the quadratic form given by squaring, q(x) = x2 , which differs from Q by a sign change. This sign change also occurs in [3] when we make the transition from the local results to the global results. It turns out that the main result of this appendix is best expressed in terms of the quadratic form Q; at the same time, to make the relation to the local calculations in our previous papers [3] and [4] easier, in the explicit calculations we use the quadratic form q. The miracle is that, just as in the Gross-Keating formula (Theorem 3.6.1), the same formula for the local intersection index as in the case p 6= 2 continues to hold in the case p = 2 when the result is expressed in terms of the Gross-Keating invariants of the matrix T˜ = diag(1, T ) ∈ Sym3 (Z)∨ .

6A.1 STATEMENT OF THE RESULT

We consider a pair of special endomorphisms j and j 0 ∈ V 0 and assume that they span a 2-dimensional subspace on which the quadratic form is nondegenerate, i.e., such that det(T ) = 6 0, where

(6A.1.1)

T =

Q(j) 1 0 2 (j , j)

!

1 0 2 (j, j ) 0 Q(j )

∈ Sym2 (Q2 ).

182

CHAPTER 6: APPENDIX

Theorem 5.1 of [3], whose proof involves no restriction on the residue characteristic, implies that for intersection numbers (6A.1.2)

(Z(j), Z(j 0 )) = (Z(j)pure , Z(j 0 )pure ).

Moreover, this intersection number depends only on the Z2 -submodule j of V spanned by j and j 0 . Our main result supplements Theorem 6.1 of [3]. To formulate our result we consider the 3-dimensional Zp -lattice given by T˜ = diag(1, T ).

(6A.1.3)

Then Gross and Keating associate to T˜ its GK-invariant GK(T˜) = (0, a2 , a3 ) ∈ Z3

(6A.1.4)

where 0 ≤ a2 ≤ a3 , and furthermore its GK-constant (T˜) ∈ {±1},

(6A.1.5)

provided that a2 ≡ 0 mod 2 and a2 < a3 , [5], appendix B. These terms only depend on the GL2 (Zp )-equivalence class of T . Note that if p 6= 2 and T is GL2 (Zp )-equivalent to diag(ε1 pα , ε2 pβ ) with 0 ≤ α ≤ β with εi ∈ Z× p for i = 1, 2, then (6A.1.6)

GK(T˜) = (0, α, β) and (T˜) = χ(−ε1 ),

where χ is the quadratic residue character modulo p.

Theorem 6A.1.1. The intersection number 1 (Z(j), Z(j 0 )) = ν2 (T ) 2 is equal to a2 + a3 + 1

−

 (a +1)/2 p 2 −1    2   p−1   

(a − a2 + 1) pa2 /2 + 2  3     pa2 /2 − 1   pa2 /2 + 2

p−1

Here GK(T˜) = (0, a2 , a3 ).1

if a2 is odd,

pa2 /2 − 1 p−1

if a2 is even and (T˜) = 1,

if a2 is even and (T˜) = −1.

1 The meaning of the formula in the case where a2 = a3 is even (in which case (T˜) is not defined) is that one takes either of the last two (identical) alternatives.

183

THE CASE p = 2, p | D(B)

If p 6= 2, this formula is exactly the one of Theorem 6.1 of [3] by the remarks preceding the theorem. As a first step of the proof of the above theorem for p = 2, we calculate the Gross-Keating invariants and constants of T˜. Before giving the result, we recall from the appendix to section 11 of [4] the case distinction for q = ε 2α ∈ Z2 , with ε ∈ Z× 2 . We call q of type (1)-(4) according to (1) α even, ε ≡ 1 mod 8 (2) α even, ε ≡ 5 mod 8 (3) α even, ε ≡ 3 mod 4 (4) α odd. The values are now given in Table 2, which is organized as follows: if T is GL2 (Z2 )-equivalent to a diagonal matrix, we write T ∼ diag(ε1 2α , ε2 2β ) with 0 ≤ α ≤ β and call T of type (i):(j) if −ε1 2α is of type (i) and −ε2 2β is of type (j). Table 2. Values for Gross-Keating Invariants Diagonal Cases

GK(T˜)

(1):(*), β ≤ α + 1

(0, α + 1, β + 1)

(1):(*), β ≥ α + 2

(0, α + 2, β)

(2):(*), β ≤ α + 1

(0, α + 1, β + 1)

(2):(*), β ≥ α + 2

(0, α + 2, β)

(3):(*)

(0, α + 1, β + 1)

(4):(*)

(0, α, β + 2)

(T˜)

+1

−1

Other Cases −2α

b 1 2

1 2

b

!

(0, α, α)

b = 0, 1

In Table 2, (*) means that the type of −ε2 2β is arbitrary.

Proof. If T ∼ diag(ε1 2α , ε2 2β ), with α ≤ β, then (6A.1.7)

T˜ ∼ diag(1, ε1 2α , ε2 2β ).

184

CHAPTER 6: APPENDIX

Using the notation of [5], Proposition B.5, we have (6A.1.8)

β1 = 0, β2 = α, β3 = β.

In Cases (1):(*), or (2):(*), one has ε1 ≡ −1 mod 4, and β2 −β1 = α ≡ 0 mod 2. So Proposition B.5 says that the GK-invariant of T˜ = diag(1, T ) is (

(6A.1.9)

GK(T˜) =

(0, α + 1, β + 1) if β ≤ α + 1, (0, α + 2, β) if β ≥ α + 2.

Moreover, when β ≥ α + 2, then by [2], the Gross-Keating constant is equal to (−ε1 , 2)2 as claimed. In Case (3):(*), one has ε1 ≡ 1 mod 4, hence [5], Proposition B.5 gives (6A.1.10)

GK(T˜) = (0, α + 1, β + 1).

The Case (4):(*) is simply [5], Proposition B.5, (1). Finally in the nondiagonal cases, the assertion is [5], Proposition B.4. Taking into account Table 2, it is now an elementary matter to calculate the expression occurring in Theorem 6A.1.1 in all cases. In the reformulation we have passed to the quadratic form q on V 0 which absorbs the minus sign occurring in our definition of the type of a diagonal matrix T in Table 2. Therefore the following statement is equivalent to Theorem 6A.1.1. We will prove Theorem 6A.1.1 by going through all cases in the following theorem. Theorem 6A.1.2. The intersection number

1 (Z(j), Z(j 0 )) = ν2 (T ) 2

depends only on the GL2 (Z2 )-equivalence class of T . The values of 12 ν2 (T ) are the following: Diagonal cases Suppose that T is GL2 (Z2 )-equivalent to the matrix diag(ε1 2α , ε2 2β ). In the following list, the initial pair of numbers indicates the types of q(j) and q(j 0 ), e.g., (1):(3) indicates that q(j) has type (1) and q(j 0 ) has type (3) in the notation introduced above. The comment ‘classical’ indicates that the corresponding configuration, as defined in Section A.3 below, already occurs for p 6= 2. Case (1):(1) with α ≤ β. (

α

α

(β − α − 1)2 2 +1 − 2(2 2 +1 − 1) 1 ν2 (T ) = α + β + 3 − α 2 2 (2 2 +1 − 1)

if α < β, if α = β.

185

THE CASE p = 2, p | D(B)

Case (1):(2).

if α < β, if α = β,

 α α +1  2 + 2 (2 2 +1 − 1)  (β −α α − 1)2

1 ν2 (T ) = α + β + 3 − 2 (2 2 +1 − 1)  2  2 β2 +1 + 2 (2 β2 +1 − 1) Case (1):(3).

 α α (β − α − 1)2 2 +1 + 2(2 2 +1 − 1)

1 ν2 (T ) = α + β + 3 − 2(2 β2 +1 − 1) 2

Case (1):(4) (classical).

if β < α. if α < β

if β ≤ α.

(β − α − 1)2 2 +1 + 2(2 2 +1 − 1), if α < β, 1 ν2 (T ) = α + β + 3 − 2(2 β+1 2 2 − 1), if α > β. α



α

Case (2):(2) with α ≤ β.

(

α

α



α

α

2 2 +1 + 2(2 2 +1 − 1) 1 ν2 (T ) = α + β + 3 − α 2 2(2 2 +1 − 1) Case (2):(3). 2 2 +1 + 2(2 2 +1 − 1) 1 ν2 (T ) = α + β + 3 − 2(2 β2 +1 − 1) 2

Case (2):(4) excluded. Case (3):(3) excluded. Case (3):(4) (classical).

if α < β, if α = β. if α < β, if β ≤ α.

2(2 2 +1 − 1) if α < β, 1 ν2 (T ) = α + β + 3 − β+1 2 2(2 2 − 1) if β < α. (

α

Case (4):(4) (classical), α ≤ β. Additional cases T '

−2α

0 1 2

1 2

α+1 1 ν2 (T ) = α + β + 3 − 2(2 2 − 1). 2

!

0

 2(2 α+1 2 − 1) 1 ν2 (T ) = 2α + 1 − α 2 2 + 2(2 α2 − 1) 2

if α is odd,

if α is even.

186 T ' −2α

CHAPTER 6: APPENDIX

1 1 2

1 2

1

, α even

!

α α 1 ν2 (T ) = 2α + 1 − 2 2 − 2(2 2 − 1). 2

The case when α is odd is excluded.

6A.2 REVIEW OF THE SPECIAL CYCLES Z(j), FOR q(j) ∈ Zp \ {0}

When computing intersection numbers, the divisors Z(j) can be replaced by the corresponding cycles Z(j)pure and these have a simple combinatorial description.

Definition 6A.2.1. A set of cycle data is a triple (S, µ, Z h ), where S ⊂ B is a subset of the building, µ ∈ Z ≥ 0, and Z h is a horizontal divisor on M. Note that, for the computation of the intersection multiplicities of Z h with vertical divisors, it is enough to give a small amount of incidence data about Z h. A set of cycle data (S, µ, Z h ) determines a divisor on M, (6A.2.1)

Z(S, µ, Z h ) =

X

m[Λ] (S, µ) P[Λ] + Z h ,

[Λ]

where (6A.2.2)

m[Λ] (S, µ) = max {µ − d([Λ], S), 0}.

Then Z h is the horizontal part of the divisor Z(S, µ, Z h ), and the first sum is the vertical part of Z(S, µ, Z h ). Conversely, the divisor Z(S, µ, Z h ) determines the set of cycle data (S, µ, Z h ) (except when µ = 0). In the ‘classical’ case, p 6= 2, for q(j) = εpα , Z(j)pure has the following description: (i) α even and χ(ε) = 1. Then (6A.2.3)

S = A = Bj ,

µ=

α , 2

Z h = 0.

Here A denotes an apartment in B. (ii) α even and χ(ε) = −1. Then (6A.2.4)

S = [Λ0 ] = B j ,

µ=

α , 2

Z h = 2 Spf(W).

187

THE CASE p = 2, p | D(B)

Here the horizontal component Z h meets the vertical part only in the central component P[Λ0 ] , in two ordinary special points. (iii) α odd. Then (6A.2.5)

S = [[Λ0 , Λ1 ]],

α−1 , 2

µ=

Z h = Spf(W0 ).

Then B j is the midpoint of the edge [Λ0 , Λ1 ], and Z h meets the vertical part at the superspecial point corresponding to the edge [Λ0 , Λ1 ]. Here we use the notation [[Λ0 , Λ1 ]] for the union of the edge [Λ0 , Λ1 ] and its endpoints. In the case where p = 2, again for q(j) = εpα , Z(j)pure is described in the Appendix to section 11 of [4]:2 (1) α even and ε ≡ 1 mod (8). Then (6A.2.6)

×

S = A = BO ,

µ=

α + 1, 2

Z h = ∅,

where again A is an apartment in B. In this case (6A.2.7)

B j = { x ∈ B | d(x, A) ≤ 1 }.

(2) α even and ε ≡ 5 mod (8). Then (6A.2.8)

×

S = [Λ0 ] = B O ,

µ=

α + 1, 2

Z h = 2 Spf(W),

where Z h meets the vertical part only in the central component P[Λ0 ] in two ordinary special points. Here (6A.2.9)

B j = {x ∈ B | d(x, [Λ0 ]) ≤ 1}.

(3) α even and ε ≡ 3 mod (4). Then (6A.2.10) × S = [[Λ0 , Λ1 ]] = B O = B j ,

µ=

α , 2

µ=

α−1 , 2

Z h = Spf(W0 ),

as in the classical case (iii). (4) α odd. Then ×

(6A.2.11) S = [[Λ0 , Λ1 ]] = B O ,

Z h = Spf(W0 ),

2 We take the occasion to correct an error concerning Case (1) in the Appendix to Section 11 of [4]. It is erroneously asserted on pp. 934–935 that Z2 [¯j]× ( O× and that for the × × associated apartment A one has A = BO . In fact, in this case BO = B j . In the table on p. 934 the header of the last column should be named S(j) and defined to be A in Case (1) × × and BO in Cases (2)–(4). In the rest of the appendix, BO should be replaced everywhere by S(j). Then the statements become correct in all cases.

188

CHAPTER 6: APPENDIX

again as in the classical case (iii). Here (6A.2.12)

B j = midpoint of [Λ0 , Λ1 ].

Note that each of these cycles has the same type of cycle data as one of the classical cases. Also note that S and Z h are unchanged if j is replaced by a scalar multiple. 6A.3 CONFIGURATIONS

In this section, we suppose that a pair j and j 0 is given with q(j) and q(j 0 ) both nonzero and with matrix of inner products T , as in the previous section, and we describe the possible configurations (S, S 0 ), i.e., relative positions, of the sets S and S 0 occuring in the cycle data of Z(j) and Z(j 0 ). In the diagonal cases, these configurations depend only on the types of j and j 0 , rather than on the full matrix T . First suppose that p 6= 2. Then the matrix T is diagonalizable. When it has diagonal form, then jj 0 = −j 0 j, and hence j (resp. j 0 ) preserves 0 the set B j (resp. B j ). From this fact, it is easy to check that the possible configurations are the following (cf. [3]): (i):(i) S and S 0 are apartments which meet at a unique vertex [Λ0 ]. (i):(ii) S is an apartment containing the unique vertex S 0 fixed by j 0 . (i):(iii) S is an apartment containing the edge S 0 whose midpoint is the unique fixed point of j 0 . (ii):(ii) S = S 0 = [Λ0 ] is the common fixed vertex of j and j 0 . (ii):(iii) excluded. (iii):(iii) S = S 0 = [[Λ0 , Λ1 ]] with midpoint the common fixed point of j and j 0 . We will refer to these possibilities as the classical configurations. Proposition 6A.3.1. Suppose that p = 2. Then the possible configurations are the following: When the matrix T is diagonal, (1):(1) S and S 0 are apartments with d(S, S 0 ) = 1. (1):(2) S is an apartment and S 0 is a vertex with d(S, S 0 ) = 1. (1):(3) S is an apartment and S 0 = [[Λ00 , Λ01 ]] with [Λ00 ] ∈ S but [Λ01 ] ∈ / S. (1):(4) S is an apartment and S 0 = [[Λ00 , Λ01 ]] ⊂ S. This coincides with the classical configuration (i):(iii). (2):(2) S = [Λ0 ], S 0 = [Λ00 ], where d([Λ0 ], [Λ00 ]) = 1. 0 (2):(3) S = [Λ0 ] and S 0 = [[Λ0 , Λ01 ]], where [[Λ0 , Λ01 ]] = B j . (2):(4) excluded. (3):(3) excluded.

189

THE CASE p = 2, p | D(B)

(3):(4) S = S 0 = [[Λ0 , Λ1 ]] = B j and B j the midpoint. 0 (4):(4) S = S 0 = [[Λ0 , Λ1 ]] with B j = B j the midpoint. These last two configurations coincide with the classical configuration for (iii):(iii). 0

Finally, if T =

1

−2α

with d(S, S 0 ) = 2.

1 2

1 2

1

, then α is even and S and S 0 are apartments

!

Proof. First note that, when T is diagonal so that j and j 0 anticommute, then conjugation by j preserves O0 and acts by the Galois automorphism and similarly for j 0 and O. In particular, the action of j 0 must preserve both × fixed-point sets B j and B O , and the action of j must preserve both fixed0 0,× point sets B j and B O We now prove selected cases. For (1):(1), j must 0,× act by a nontrivial automorphism of order 2 on the apartment S 0 = B O . It thus has a unique fixed point x0 on the apartment S 0 , which then has × distance ≤ 1 from the apartment S = B O . But this point must have distance exactly 1 from S, since otherwise the intersection of S 0 with the tube of radius 1 around S, i.e., the fixed-point set B j , would contain another point. This confirms the configuration claimed for (1):(1). Case (1):(3) is similar, since j 0 must have a unique fixed point on the apartment S. In Case (1):(2), the fixed points of j 0 all have distance ≤ 1 from S 0 = [Λ0 ]; since the apartment S contains exactly one of them, we must have d(S, S 0 ) = 1. The reasoning for Case (1):(4) is as in the classical situation. In Case (2):(2) we have S = [Λ0 ] and B j = {x | d(x, [Λ0 ]) ≤ 1} and similarly for S 0 = [Λ00 ] 0 0 and B j . It follows that [Λ0 ] ∈ Bj and [Λ00 ] ∈ Bj . On the other hand, we a basis of Λ0 as in (A.10) must have [Λ0 ] 6= [Λ00 ]. To see this, let us choose α ¯ 2 of [4], so that j is given by the matrix j = 2 j with ¯j = −1 2λ 2 1 

(6A.3.1)



with λ ∈ Z× 2.

β

Let j 0 = 2 2 ¯j 0 , where (6A.3.2)





¯j 0 = a b . c −a

The condition that j 0 anticommute with j is given by b = a − λc. If [Λ0 ] = [Λ00 ] then ¯j 0 would have to be a scalar matrix modulo 2. But then 2 | c and 2 | a which contradicts det(¯j 0 ) ∈ Z× 2. × O In Case (2):(3), [Λ0 ] = B has to be fixed by j 0 , which yields the assertion. For the same reason, Case (2):(4) is excluded since then j 0 fixes no vertex. This case could also have been excluded from the fact that j and

190

CHAPTER 6: APPENDIX

j 0 generate the matrix algebra M2 (Qp ) which yields (ε 2α , ε0 2β )2 = 1. For the same reason, Case (3):(3) does not occur. The remaining diagonal cases are obvious. ! 1 12 α For T = −2 , we note that T is representable by the quadratic 1 2 1 space of the traceless matrices in M2 (Q2 ) (with q given by squaring) only α α 2 when α is even. Let j = 2 2 ¯j and j 0 = 2 2 ¯j 0 . Then ¯j 2 = j¯0 = 1. We may 2 choose a basis of Q2 such that 



¯j = 1 0 . 0 −1

(6A.3.3)

Then ¯j 0 = g¯jg −1 for a suitable g ∈ GL2 (Qp ). By the Bruhat decomposition we may assume that g ∈ N A or g ∈ N wN A, where N is the subgroup of unipotent upper triangular  matrices, A the subgroup of diagonal matrices, 0 1 and where w = . The condition is that 1 0 (6A.3.4)

1 = (¯j, ¯j 0 ) = −tr(¯j · ¯j 0 ι ),

where x 7→ xι denotes the main involution on M2 (Q2 ). It is easy to see that g ∈ N A yields no solution. If g ∈ N wN A, we may assume that (6A.3.5)



 

 



1 b1 0 1 1 b2 g= · · . 0 0 1 0 0 1

Then (6A.3.4) is equivalent to b1 b2 = − 34 , and hence we may take !

(6A.3.6)

g=

1 14 . 1 − 43

Now S is the standard apartment A = {[Λr ] | r ∈ Z}, with Λr = [e0 , 2r e1 ], and S 0 = gS. But (6A.3.7)  [[e0 + e1 , 2r e1 ]] if r ≥ 2, r−2 g[Λr ] = [[e0 + e1 , 2 (e0 − 3e1 )]] = [[e − 3e , 24−r e ]] if r < 2. 0 1 1 It is now easy to see that S ∩ S 0 = ∅. It follows that d(S, S 0 ) = 2, the geodesic between S and S 0 being formed by the vertices (6A.3.8) S 3 [e0 , e1 ] = [e0 + e1 , e1 ] ⊃ [e0 + e1 , 2e1 ] ⊃ [e0 + e1 , 22 e1 ] ∈ S 0 .

191

THE CASE p = 2, p | D(B)

6A.4 CALCULATIONS

The principle of the proof of Theorem 6A.1.2 is the same as that of Theorem 6.1 of [3], which is as follows. We write (Z(j), Z(j 0 )) as the sum of two terms, namely of (6A.4.1)

(Z(j)h , Z(j 0 )h ) + (Z(j)h , Z(j 0 )v ) + (Z(j)v , Z(j 0 )h )

and of (Z(j)v , Z(j 0 )v ). For the first term we use the Genestier equations and the information on Z(j)h and Z(j 0 )h encoded in the cycle data of Z(j) and Z(j 0 ). The second term is purely combinatorial and only depends on the first two entries of the cycle data for Z(j) and Z(j 0 ). This allows us to dispose immediately of (Z(j)v , Z(j 0 )v ) in the cases which are termed classical in the list, i.e., (1):(4), (3):(4), and (4):(4). Indeed, these cases have already been dealt with in Section 6 of [3], except that there the multiplicities µ and µ0 of the central components of Z(j)v and Z(j 0 )v were equal to α and β with α ≤ β. Similarly, the calculation of the last two summands of (6A.4.1) is as in Section 6 of [3]. This gives the result claimed in Case (1):(4). For the term (Z(j)h , Z(j 0 )h ) in the Cases (3):(4) and (4):(4) we have to use the Genestier equation — and here the result is different from the case p 6= 2. Case (3):(4). In this case (Z h (j), Z h (j 0 )) = 2 (and not 1, as in the case p 6= 2). Indeed, let us choose coordinates as in (A.14) of the appendix to section 11 of [4] (relative to j) for (6A.4.2)

Λ0 ⊃ Λ1 ⊃ 2Λ0 . 6=

6=

α Then j = p 2 ¯j with

(6A.4.3)

¯j = −1 −2(1 − 2λ) , 1 1 



where ε1 = 4λ − 1 .

Furthermore Z h (j) equals, with µ = −(1 − 2λ)−1 , (6A.4.4) Spec W [X]/(X 2 +2µX−2µ) ' Spec W [T0 , T1 ]/(T0 T1 −2, T0 +2µ−µT1 ) (where T0 7→ X, T1 7→ µ−1 (X + 2µ)). Now j 0 = 2 given terms of the above coordinates as (6A.4.5)



β−1 2

¯j 0 , where ¯j 0 is



¯j 0 = 2a0 2b0 , c −2a0

where b0 and c are units; cf. (A.17) of the Appendix to Section 11 of [3]. Then Z h (j 0 ) is given by the equation b0 T0 − 4a0 − cT1 = 0 in

192

CHAPTER 6: APPENDIX

Spec W [T0 , T1 ]/(T0 T1 − 2). Hence (Z h (j), Z h (j 0 )) equals the length of the Artin ring (6A.4.6)

W [X]/(X 2 + 2µX − 2µ, b0 X − 4a0 −

c (X + 2µ)). µ

˜ is the ring of integers in a ramified Now W [X]/(X 2 + 2µX − 2µ) = W quadratic extension, with the residue class π ˜ of X as uniformizer. Hence the length is equal to the π ˜ -valuation of (6A.4.7)

(b0 −

c )˜ π − 2(c + 2a0 ) = (b0 + c(1 + λ))˜ π − 2(c + 2a0 ). µ

The condition that j anticommute with j 0 is given as −2a0 +b0 −c (1−2λ) = 0. Hence (b0 +c(1+λ))˜ π = 2(b0 −a0 )˜ π , so that the π ˜ -valuation of (6A.4.7) is equal to 2, as claimed. Case (4):(4). In this case (Z h (j), Z h (j 0 )) = 3 (and not 1 as in the case p 6= 2). In this case we may choose standard coordinates as in (A.17) in α−1 the Appendix to section 11 in [4] for Λ0 ⊃ Λ1 ⊃ 2Λ0 . Then j = 2 2 ¯j and 6=

j0 = 2

β−1 2

(6A.4.8)

6=

¯j 0 where 0 0 ¯j = 2a0 2b0 , ¯j 0 = 2a00 2b0 0 , c −2a0 c −2a0









where b0 , c, b00 , c0 are units. Then Z h (j) equals, with µ = c/b0 and ν = 2a0 /c, (6A.4.9) Spec W [X]/(X 2 − 2µνX − 2µ) ' Spec W [T0 , T1 ]/(T0 T1 − 2, b0 T0 − 4a0 − cT1 ) (where T0 7→ X, T1 7→ µ−1 X − 2ν). Furthermore Z h (j 0 ) is given by the equation b00 T0 −4a00 −c0 T1 = 0 in Spec W [T0 , T1 ]/(T0 T1 −p). Denoting by ˜ = W [X]/(X 2 − 2µνX − 2µ), we therefore π ˜ the residue class of X in W have to determine the π ˜ -order of b00 π ˜ − 4a00 − c0 (µ−1 π ˜ − 2ν) (6A.4.10)

= (b00 − c0 /µ)˜ π + 2(νc0 − 2a0 ) = c−1 ((b00 c − b0 c0 )˜ π + 4(a0 c0 − a0 )).

The condition that j and j 0 anticommute is given by 4a0 a00 +(b0 c0 +b00 c) = 0. Hence (b00 c − b0 c0 )˜ π = 2(b00 c + 4a0 a00 )˜ π and the π ˜ -valuation of (6A.4.10) is 3, as claimed.

193

THE CASE p = 2, p | D(B)

We now go through the remaining cases. In the combinatorial problem of calculating (Z(j)v , Z(j 0 )v ), we use the following terminology. By the contribution of a vertex [Λ] we mean the number m[Λ] (S, µ) · (P[Λ] , Z(j 0 )v ).

(6A.4.11)

To calculate (Z(j)v , Z(j 0 )v ), we have to add the contributions of all vertices. It will be useful to note the following elementary results, valid for arbitrary p: (I) A vertex [Λ] of multiplicity ν ≥ 1 in Z(j 0 ) will be called regular in Z(j 0 ) if it has one neighbor with multiplicity ν + 1 and p neighbors with multiplicity ν − 1. Then (P[Λ] , Z(j 0 )v ) = 1 − p. If a vertex [Λ] has multiplicity 0 in Z(j 0 )v and has a unique neighbor with multiplicity 1, then (P[Λ] , Z(j 0 )v ) = 1. (II) We have the following summation formulae, where ν ≤ µ in the second formula, (6A.4.12)

µ+

µ−1 X

(µ − t)(p − 1)pt−1 =

t=1

(6A.4.13)

µ+

ν−1 X

pµ − 1 . p−1

(µ − t)(p − 1)pt−1 − (µ − ν)pν−1 =

t=1

pν − 1 . p−1

Case (1):(1). In this case Z(j)h = Z(j 0 )h = 0. It remains to compute (Z(j)v , Z(j 0 )v ). We are assuming that µ ≤ µ0 . We divide the vertices [Λ] into two groups: Group 1: d([Λ], S 0 ) < d([Λ], S). Each vertex [Λ] with m[Λ] (S, µ) > 0 is regular for Z(j 0 ). Therefore this group contributes (6A.4.14)

−

µ−1 X

(µ − t) 2t−1 = −(2µ − 1) + µ.

t=1

Here we used (I) above. Group 2: d([Λ], S) ⊂ d([Λ], S 0 ). The automorphism j 0 acts on this group, with a unique fixed point [Λ0 ], the vertex on S closest to S 0 . This vertex contributes −µ. The contributions of the other vertices in this group are summed according to the closest vertex in S. Therefore this sum is twice (for the symmetry) (6A.4.15)

−

0 −2 µX

s=1

0

(2min(µ,µ −1−s) − 1) + µ.

194

CHAPTER 6: APPENDIX

Therefore, if µ ≤ µ0 − 1, Group 2 contributes −µ + 2(−

µ−1 X

(2t − 1) − (µ0 − µ − 1) (2µ − 1) + µ)

t=1

= −µ + 2(µ − 1) − 4(2µ−1 − 1) − 2(µ0 − µ − 1) (2µ − 1) + 2µ = µ + 2µ0 − 2(µ0 − µ) 2µ . Therefore, adding the contribution of Group 1, we obtain for µ ≤ µ0 − 1, i.e., α < β, 2(µ + µ0 ) + 1 − (2(µ0 − µ) + 1) 2µ ,

(6A.4.16)

which confirms the formula in this case. If µ = µ0 , the contribution of Group 2 equals (6A.4.17)

−µ + 2(−

µ−2 X

(2s − 1) + µ) = 3µ − 2µ .

s=1

Therefore, adding the contribution of Group 1, we obtain for µ = µ0 , i.e., α = β, 4µ + 1 − 2 · 2µ ,

(6A.4.18)

which confirms the formula in this case. Case (1):(2). In this case Z(j)h = 0 and Z(j 0 )h meets the special fiber in the central component P[Λ00 ] of Z(j 0 ) in two ordinary special points. Therefore, since m[Λ00 ] (S, µ) = µ − 1, (6A.4.19) (Z(j)h , Z(j 0 )h ) + (Z(j)h , Z(j 0 )v ) + (Z(j 0 )h , Z(j)v ) = 2(µ − 1). To calculate (Z(j)v , Z(j 0 )v ), we divide the vertices into two groups: Group 1: d([Λ], [Λ00 ]) < d([Λ], S). The contribution of [Λ00 ] is −3(µ − 1). The other vertices in group 1 contribute  P − µ−2 (µ − 1 − t) 2t t=1 (6A.4.20) t  Pµ0 −1

−

(µ − 1 − t) 2 + (µ − µ0 − 1) 2µ

if µ ≤ µ0 + 1, 0

if µ > µ0 + 1.

Group 2: d([Λ], [Λ00 ]) > d([Λ], S). The automorphism j 0 acts on S with a unique fixed vertex [Λ0 ] characterized by d([Λ0 ], [Λ00 ]) = 1. The contribution of [Λ0 ] is equal to −µ. The contributions of the other vertices in this group are summed according to the t=1

195

THE CASE p = 2, p | D(B)

vertex in S closest to the given vertex. Therefore this sum is twice (for the symmetry) the sum (6A.4.21)

−

0 −2 µX

0

(2min(µ,µ −1−s) − 1) + µ.

s=1

This last sum is equal to (6A.4.22)  P −(µ0 − µ − 1) (2µ − 1) − µ−1 (2s − 1) + µ if µ ≤ µ0 − 1, s=1 − Pµ0 −2 (2s − 1) + µ

if µ > µ0 − 1.

s=1

Using (II) above, this is equal to (6A.4.23)  −(µ0 − µ − 1) (2µ − 1) + (µ − 1) − 2(2µ−1 − 1) + µ if µ ≤ µ0 − 1, (µ0 − 2) − 2(2µ0 −2 − 1) + µ

if µ > µ0 − 1.

The total for (Z(j), Z(j 0 )) is for µ ≤ µ0 − 1, i.e., α < β, equal to 2(µ−1) − 3(µ − 1) −

µ−1 X

(µ − t) 2t−1 − µ

t=2

= −2(µ0 − µ − 1) (2µ − 1) + 2(µ − 1) − 4 (2µ−1 − 1) + 2µ = −(2µ − 1) − 2(µ0 − µ − 1) (2µ − 1) + 4µ − 2(2µ − 1) = 4µ − (3 + 2(µ0 − µ − 1)) (2µ − 1). This equals (6A.4.24)

2(µ + µ0 ) + 1 − (2(µ0 − µ) + 1) 2µ ,

which confirms the given formula in this case (recall µ = For µ = µ0 , i.e., α = β, we obtain

α 0 2 +1, µ

=

β 2 +1).

(6A.4.25) −(2µ − 1) + 2(µ − 2) − 4(2µ−2 − 1) + 2µ = 4µ + 1 − 2 · 2µ , which confirms the formula in this case. For µ = µ0 + 1, i.e., α = β + 1, we obtain 2(µ − 1) − 3(µ − 1) −

µ−1 X

(µ − t) 2t−1

t=2

− µ + 2(µ − 3) − 4(2µ−3 − 1) + 2µ = −(2µ − 1) + 2(µ − 3) − 4(2µ−3 − 1) + 2µ.

196

CHAPTER 6: APPENDIX

This equals 4µ − 1 − 2µ−1 − 2µ ,

(6A.4.26)

which confirms the formula in this case. For µ > µ0 + 1, i.e., α > β, the total for (Z(j), Z(j 0 )) is equal to 0

2(µ − 1) − 3(µ − 1) −

µ X

(µ − t) 2t−1

t=2 0

0

0

+ (µ − µ − 1) 2µ − µ + 2(µ0 − 2) − 4(2µ −2 − 1) + 2µ 0

0

= −(2µ +1 − 1) + 2(µ0 − 2) − 4(2µ −2 − 1) + 2µ. This equals 0

0

2(µ + µ0 ) + 1 − 2µ − 2µ +1 ,

(6A.4.27)

which confirms the formula in this case. Case (1):(3). In this case Z(j)h = 0 and Z(j 0 ) is the spectrum of a ramified quadratic extension of W which passes through the double point pt[Λ00 ],[Λ01 ] . Hence (6A.4.28) (Z(h)h , Z(j 0 )h ) + (Z(j)h , Z(j 0 )v ) + (Z(j 0 )h , Z(j)v ) = 2µ − 1 . To calculate (Z(j)v , Z(j 0 )v ), we first note that the contribution of [Λ00 ] ∈ S is equal to −2µ and the contribution of [Λ01 ] is equal to −2(µ − 1). The remaining vertices are divided into two groups. Group 1: 0 < d([Λ], [Λ01 ]) < d([Λ], [Λ00 ]). These vertices contribute  P − µ−1 (µ − t) 2t−1 t=2 (6A.4.29) t−1  Pµ0

−

t=2 (µ

− t) 2

if µ ≤ µ0 + 1, + (µ − (µ0 + 1)) 2µ

0

if µ > µ0 + 1.

Group 2: 0 < d([Λ], [Λ00 ]) < d([Λ], [Λ01 ]). Here we sum the contributions of the lattices according to the closest vertex on S. We obtain, taking into account the involution of S induced by j0, (6A.4.30)

−2

0 −1 µX

s=1

0

(2min(µ,µ −s) − 1) + 2µ.

197

THE CASE p = 2, p | D(B)

This equals (6A.4.31)  Pµ−1 s −2 (µ0 − µ) (2µ − 1) − 2 s=1 (2 − 1) + 2µ µ0 −1

−2 P

s=1

if µ ≤ µ0 , if µ ≥ µ0 + 1.

(2s − 1) + 2µ

For µ ≤ µ0 , the total for (Z(j), Z(j 0 )) is equal to 2µ − 1 − 2µ − 2(µ − 1) −

µ−1 X

(µ − t) 2t−1 − 2(µ0 − µ)(2µ − 1)

t=2

−2

µ−1 X

(2s − 1) + 2µ.

s=1

This equals 2(µ + µ0 ) + 3 − (2(µ0 − µ) + 3) 2µ ,

(6A.4.32)

which confirms the formula in this case (note that µ = that µ ≤ µ0 ⇔ α < β). If µ = µ0 + 1, i.e., α = β, the total is 2µ − 1 − 2µ − 2(µ − 1) −

µ−1 X

(µ − t) 2t−1 − 2

t=2 µ

α 2

µ−2 X

+ 1, µ0 =

β 2

and

(2s − 1) + 2µ

s=1 µ−2

= −(2 − 1) + 2(µ − 2) − 4(2

− 1) + 2µ.

This equals 4µ + 1 − 2 · 2µ ,

(6A.4.33)

which confirms the formula in this case. If µ > µ0 + 1, i.e., α > β, the total is 0

−2µ + 1 −

µ X

(µ − t) 2

t=2 µ0 +1

= −(2

t−1

µ0

0

+ (µ − (µ + 1)) 2 − 2

s=1 0

µ0 −1

− 1) + 2(µ − 1) − 4(2

− 1) + 2µ.

This equals (6A.4.34)

0 −1 µX

0

2(µ + µ0 ) + 3 − 2 · 2µ +1 ,

which confirms the formula in this case.

(2s − 1) + 2µ

198

CHAPTER 6: APPENDIX

Case (2):(2). In this case S = [Λ0 ] and S 0 = [Λ00 ] with d([Λ0 ], [Λ00 ]) = 1. Now Z(j)h meets Z(j 0 )v in the two ordinary special points of P[Λ0 ] and Z(j 0 )h meets Z(j)v in the two ordinary special points of P[Λ00 ] . Hence (6A.4.35) (Z(j)h , Z(j 0 )h )+(Z(j)h , Z(j 0 )v )+(Z(j)v , Z(j 0 )h ) = 2(µ−1)+2(µ0 −1). To calculate (Z(j)v , Z(j 0 )v ) we divide the vertices into two groups. We may assume µ ≤ µ0 by symmetry. Group 1: 0 ≤ d([Λ], [Λ0 ]) < d([Λ], [Λ00 ]). This group contributes (6A.4.36)  −µ − 2 Pµ−1 (µ − t) 2t−1 if µ ≤ µ0 − 1 t=1 −µ − Pµ−2 (µ − t) 2t + (µ − (µ − 1)) 2µ−1 t=1

if µ = µ0 .

The quantity for µ = µ0 is equal to (6A.4.37) µ−2 µ+

µ−2 X

(µ − t) 2t−1 − (µ − (µ − 1)) 2µ−2 = µ − 2(2µ−1 − 1).


t=1

Hence the contribution of this group is equal to (

(6A.4.38)

µ − 2 (2µ − 1) µ − 2(2µ−1 − 1)

if µ ≤ µ0 − 1, if µ = µ0 .

Group 2: 0 ≤ d([Λ], [Λ00 ] < d([Λ], [Λ0 ]). Here [Λ00 ] contributes −3(µ − 1). The remaining vertices in this group contribute (6A.4.39)

−

µ−1 X

(µ − t) 2t−1 .

t=2

Hence this group contributes (6A.4.40)

−µ + 2 − (2µ − 1).

For the total, we obtain in case µ ≤ µ0 − 1, i.e., α < β, (6A.4.41)

2(µ − 1) + 2(µ0 − 1) + µ − 2(2µ − 1) − µ + 2 − (2µ − 1) = 2(µ + µ0 ) + 1 − 3 · 2µ ,

which yields the formula in this case. When µ = µ0 , i.e., α = β, we get (6A.4.42)

4(µ − 1) + µ − 2(2µ−1 − 1) − µ + 2 − (2µ − 1) = 4µ + 1 − 2 · 2µ ,

199

THE CASE p = 2, p | D(B)

which confirms the formula in this case. Case (2):(3). In this case Z(j)h is unramified and meets the special fiber in two ordinary special points of P[Λ0 ] , and Z(j 0 )h is ramified and meets the special fiber in the double point pt[Λ0 ],[Λ01 ] . Hence (Z(j)h , Z(j 0 )h ) + (Z(j)h , Z(j 0 )v ) + (Z(j)v , Z(j 0 )h ) = 2µ0 + 2µ − 1 = 2(µ + µ0 ) − 1.

(6A.4.43)

To calculate (Z(j)v , Z(j)v ), we note that the contribution of [Λ01 ] is −2(µ− 1) and that the contribution of [Λ0 ] is −2µ. The remaining vertices are divided into two groups. Group 1: 0 < d([Λ], [Λ01 ]) < d([Λ], [Λ0 ]). These vertices contribute t−1 − µ−1 t=2 (µ − t)2 Pµ0 0 − t=2 (µ − t) 2t−1 + (µ − (µ0 + 1)) 2µ

(

(6A.4.44)

if µ − 1 ≤ µ0 , if µ − 1 > µ0 .

P

Group 2: 0 < d([Λ], [Λ0 ]) < d([Λ], [Λ01 ]). These vertices contribute

t−1 −2 µ−1 t=1 (µ − t) 2 0 P −1 0 −2 µt=1 (µ − t) 2t−1 + 2(µ − µ0 ) 2µ −1

(

(6A.4.45)

P

if µ ≤ µ0 , if µ > µ0 .

The total for (Z(j), Z(j 0 )) for µ ≤ µ0 , i.e., α < β, is equal to 2(µ + µ0 ) − 1 − 4µ + 2 −

µ−1 X

(µ − t) 2t−1 − 2

(µ − t) 2t−1

t=1

t=2 0

µ−1 X

µ

µ

= 2(µ + µ ) − (2 − 1) − 2(2 − 1). This equals 2(µ + µ0 ) + 3 − 3 · 2µ ,

(6A.4.46)

which confirms the formula in this case. For µ = µ0 + 1, i.e., α = β, the total is 0

2(µ + µ ) − 1 − 4µ + 2 −

µ−1 X

(µ − t) 2t−1

t=2

−2 0

µ

0 −1 µX

t=1 µ−1

= 2(µ + µ ) − (2 − 1) − 2(2

0

(µ − t) 2t−1 + 2(µ − µ0 )2µ −1 − 1).

200

CHAPTER 6: APPENDIX

This equals 2(µ + µ0 ) + 3 − 2 · 2µ ,

(6A.4.47)

which confirms the formula in this case. For µ − 1 > µ0 , i.e., α > β, the total is 0

0

2(µ + µ ) − 1 − 4µ + 2 −

µ X

(µ − t) 2t−1 + (µ − (µ0 + 1)) 2µ

0

t=2

−2

0 −1 µX

0

(µ − t) 2t−1 + 2(µ − µ0 ) 2µ −1 .

t=1

This equals 0

2(µ + µ0 ) + 3 − 4 · 2µ ,

(6A.4.48)

which confirms the formula in this case. Case T =

1

1 2

!

, α even. In this case Z(j)h = Z(j 0 )h = 0. To 1 calculate (Z(j)v , Z(j 0 )v ), we divide the vertices into three groups. −2α

1 2

Group1: d([Λ], S) = d([Λ], S 0 ). The contribution of these vertices is (6A.4.49)

−(µ − 1) −

µ−2 X

(µ − 1 − t) 2t−1 = −(2µ−1 − 1).

t=1

Group 2: d([Λ], S 0 ) < d([Λ], S). The vertex [Λ00 ] ∈ S 0 closest to S contributes −(µ − 2). The remaining vertices are grouped according to the closest vertex on S 0 . If that vertex has distance t ≥ 1 from [Λ00 ], the contribution is equal to (6A.4.50) −(µ − 2 − t) −

µ−2−t−1 X

(µ − 2 − t − s) 2s−1 = −(2µ−2−t − 1).

s=1

We see that the group 2 contributes (6A.4.51) −(µ − 2) − 2

µ−3 X t=1

(2µ−2−t − 1) = −(µ − 2) − 2

Group 3: d([Λ], S) < d([Λ], S 0 ).

µ−3 X t=1

(2t − 1).

201

THE CASE p = 2, p | D(B)

The vertex [Λ0 ] ∈ S closest to S 0 contributes −µ. The remaining vertices are grouped according to the closest vertex on S. If that vertex on S has distance t with 1 ≤ t ≤ µ − 3 from [Λ0 ], the contribution is equal to (6A.4.52)

−µ −

µ−2−t−1 X

(µ − s) 2s−1 + (µ − (µ − 2 − t)) 2µ−2−t

s=1

= −(2µ−2−t − 1). Finally a vertex on S with distance µ − 2 from [Λ0 ] contributes µ. We see that Group 3 contributes (6A.4.53)

−µ − 2

µ−3 X

(2µ−2−t − 1) + 2µ = µ − 2

t=1

µ−3 X

(2t − 1).

t=1

The total is −(2µ−1 − 1) − (µ − 2) − 2

µ−3 X

(2t − 1) + µ − 2

t=1

µ−3 X

(2t − 1)

t=1

which equals (6A.4.54)

4µ − 1 − 3 · 2µ−1 ,

which confirms the formula in this case. 6A.5 THE FIRST NONDIAGONAL CASE

In this section we consider the case T = −2α

0

1 2

!

. This case is dif0 ferent from the above cases in that the endomorphisms j and j 0 do not anticommute and, while being 6= 0, satisfy q(j) = q(j 0 ) = 0. The case where j ∈ V 0 has q(j) = 0 was excluded from [3], therefore we have to describe first the special cycle Z(j) in M in this case. The method is the same as in [3], therefore we will be brief. Let, thus, j ∈ V 0 be nonzero with q(j) = 0. The proof of Proposition 2.2 of [3] carries over to show that Z(j) ∩ P[Λ] 6= ∅ iff j(Λ) ⊂ Λ, and in fact P[Λ] then occurs in Z(j) with multiplicity (6A.5.1)

1 2

mult[Λ] (j) = max{r ≥ 0 | j(Λ) ⊂ 2r Λ}.

ˆ [Λ] , (see [3], Indeed, this follows from the local equation for Z(j) along Ω   a b equation (3.13)): if j = 2r with a, b, c ∈ Z2 not all simultanec −a ously divisible by 2, then since det(j) = 0, not both b and c are divisible by

202

CHAPTER 6: APPENDIX

2, so that the second factor in (3.13) of [3], which is bT 2 − 2aT − c, is not divisible by 2. Furthermore, the horizontal component Z(j)h is trivial. This can be proved by considering the local equations as in section 3 of [3]; the global argument of Remark 8.2 of [3], that 0 is not represented by a division algebra, can also be applied. We choose a basis e1 , e2 of Q22 such that 

(6A.5.2)

j=



0 1 . 0 0

Then for Λr = [e1 , 2r e2 ] on the standard apartment we have P[Λr ] ∩ Z(j) 6= ∅ iff r ≥ 0 and then mult[Λr ] (j) = r. Conversely, let Λ be such that j(Λ) ⊂ Λ, i.e., P[Λ] ∩ Z(j) 6= ∅. Then Λ ∩ Ker j is generated by a multiple e01 = λe1 of e1 . Furthermore there is e02 ∈ Λ such that j(e02 ) = pr e01 and such that Λ = [e01 , e02 ]. Then r = mult[Λ] (j). In other words, Λ is the r-th vertex on the apartment corresponding to the basis e01 , e02 ; and this apartment shares a half-apartment with the standard apartment. For the multiplicity mult[Λ] (j) there is the formula (6A.5.3)

mult[Λ] (j) = dist([Λ], ∂S) + 1.

Here S = {[Λ] | mult[Λ] (j) > 0} and [Λ] ∈ S. We therefore obtain the picture for Z(j) shown in the frontispiece of this book. ! 1 0 Now let j, j 0 ∈ V 0 with matrix T = −2α 1 2 , i.e., q(j) = q(j 0 ) = 0 2 0 0 0 α and jj + j j = 2 . Let e2 be a generator of ker(j 0 ) and let e1 = j(e2 ). Then e1 is a generator of ker(j). Thus e1 , e2 is a basis of Q22 defining an apartment A with one half-apartment belonging to j (in the sense made clear above) and the opposite half-apartment belonging to j 0 , i.e., 

(6A.5.4)

j=







0 1 0 0 , j0 = α . 0 0 2 0

It is immediate that (6A.5.5)

S ∩ A = {Λr | r > 0}, S 0 ∩ A = {Λr | r < α}.

All vertices in S 0 \ ∂S 0 are regular for Z(j 0 ). Therefore the contribution of a vertex [Λ] which is joined by a geodesic of length t to A and meets there the vertex [Λr ] is equal to (6A.5.6)

   −(r − t)

r−t

  0

if t ≤ r and t ≤ α − r − 1, if t ≤ r and t = α − r, in all other cases.

203

THE CASE p = 2, p | D(B)

Therefore the sum of the contributions of vertices with [Λr ] as the closest vertex on A is equal to (6A.5.7)  Pα−r−1 t−1 + (r − (r − α)) 2r−α−1  if α2 ≤ r < α,  −r − Pt=1 (r − t) 2 t−1 −r − r−1 if 0 < r < α2 , t=1 (r − t) 2   α if r = α. This equals  α−r − 1)   −(2

(6A.5.8)

if α2 ≤ r < α, if 0 < r < α2 , if r = α.

−(2r − 1)

  α

If α is even, the total contribution is equal to −

α−1 X

α −1 2

(2α−s − 1) −

s= α 2

X

(2s − 1) + α

s=1 α −1 2

α 2

=−

X

s

(2 − 1) −

s=1

X

(2s − 1) + α

s=1

α −1 2

= −2

X

α

(2s − 1) − (2 2 − 1) + α.

s=1

This equals α

(6A.5.9)

2α + 3 − 3 · 2 2 ,

which confirms the formula in this case. If α is odd, the total contribution is equal to (6A.5.10) −

α−1 X s= α+1 2

α−1

α−s

(2

− 1) −

2 X

α−1

s

(2 − 1) + α = −2

s=1

s=1

This equals (6A.5.11)

2α + 3 − 2 · 2

which confirms the formula in this case.

2 X

α+1 2

,

(2s − 1) + α.

204

CHAPTER 6: APPENDIX

Bibliography

[1] ARGOS (Arithmetische Geometrie Oberseminar), Proceedings of the Bonn seminar 2003/04, forthcoming. [2] I. Bouw, Invariants of ternary quadratic forms, in [1]. [3] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223. [4] S. Kudla, M. Rapoport, and T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887–951. [5] T. Yang, Local densities of 2-adic quadratic forms, J. Number Theory, 108 (2004), 287–345.

Chapter Seven An inner product formula 1

b1 (τ ) = d (M)-valued generating function φ In Chapter 4 we defined the CH P t b t∈Z Z(t, v) q and showed its modularity. It follows that the height pairing b h φ1 (τ1 ), φb1 (τ2 ) i of two of these generating functions is then a series with coefficients in R which is a modular form of two variables, of weight 23 2

d (M) with R via in each. Here we are identifying, as in Chapter 2, CH the arithmetic degree map. On the other hand, in the previous chapter, we d 2 (M) and defined a generating series for 0-cycles with coefficients in CH proved that it is a Siegel modular form of genus two and weight 32 . This was done by calculating explicitly all coefficients of this generating series and showing that they coincide term by term with the corresponding coefficients of the Siegel Eisenstein series considered in Chapter 5. In the present chapter we prove an inner product formula which asserts that after pulling back the generating series for 0-cycles under the diagonal map, it coincides with the inner product of the generating function for divisors with itself under the height pairing. Recall that there seemed to be some arbitrariness in which the coefficients of the generating series for 0-cycles was defined in Chapter 6, outside the case when the index T ∈ Sym2 (Z)∨ is positive definite and good. Therefore we may view the inner product formula as an additional justification of these definitions, since this formula shows their coherence. It should be stressed that this result is purely geometric, and that formulas for the Fourier coefficients of Eisenstein series do not enter into its proof. In fact, it turns out that we do not need the explicit expressions that we obtained for the coefficients, but we do need the way in which these calculations were organized. When forming the inner product of φb1 (τ ) with itself, there are the pairings of those special divisors Z(t, v) which have no common intersection in the generic fiber. For this part of the intersection product, the claimed formula turns out to be essentially equivalent to the decomposition of the intersection of two such special divisors according to their ‘fundamental matrices’; see [3], (8.24). This part of the formula is therefore essentially tautological. For the special divisors which do have an intersection in the generic fiber, we use the adjunction formula in the form of Section 2.10. For this we construct in Section 7.3 a weakly biadmissible Green function

206

CHAPTER 7

on H × H \ ∆ and show that the induced metric on the sheaf of differentials on M(C) coincides exactly with the metric on ω defined in [4], including the mysterious constant 2C = γ + log(4π). To apply then the adjunction formula, we decompose the cycles into irreducible components according to the conductor of the order generated by the special endomorphism (the ‘type’ of the special endomorphism). To calculate all the terms occurring in the adjunction formula, there are archimedian calculations similar to those in [4]. The calculations at the finite primes lead to Gross’s theory of quasicanonical liftings and the study of their ramification behavior. 7.1 STATEMENT OF THE MAIN RESULT

Let (7.1.1)

φb1 (τ ) =

X

b v) q t Z(t,

t∈Z

be the generating function for divisors on M.1 In Chapter 4, it was proved d 1 (M). that this is a modular form of weight 23 , valued in CH On the other hand, in Chapter 6 we introduced the generating function for 0-cycles (7.1.2)

X

φb2 (τ ) =

b Z(T, v) q T

T ∈Sym2 (Z)∨

and have proved that this is a Siegel modular form2 of weight 23 . Therefore, the pullback of this function under the diagonal map (7.1.3)



H × H −→ H2 ,

(τ1 , τ2 ) 7→



τ1 τ2

is a modular form of two variables, of weight 32 in each of them. In this chapter we will prove that this pullback coincides with the inner product of φb1 with itself under the height pairing.

Theorem C. For τ1 and τ2 ∈ H,

 τ b b b h φ1 (τ1 ), φ1 (τ2 ) i = φ2 ( 1



τ2

).

1

We are using the convention that Z(t) denotes both the moduli stack of triples (A, ι, x) and its image in M, viewed as a divisor. 2

2

c (M) with R via deg. c Recall that, as explained in Chapter 2, we identify CH

207

AN INNER PRODUCT FORMULA

Equivalently, for all t1 , t2 ∈ Z, and v1 , v2 ∈ R>0 , ((?))

b 1 , v1 ), Z(t b 2 , v2 ) i = h Z(t



X

b Z(T,



v1

(Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 )

v2

).

Here (7.1.4) Sym2 (Z)∨ = { T ∈ Sym2 (Q) | tr(T b) ∈ Z,

∀b ∈ Sym2 (Z) }.

To prove the equality ((?)) of Theorem C, we first consider the case t1 t2 6= 0. We write (7.1.5)

Z(ti ) = Z hor (ti ) + Z ver (ti ),

where Z hor (ti ) is the closure in M of the generic fiber Z hor (ti )Q and where Z ver (ti ) is a divisor supported in the fibers Mp for some p | D(B).3 By equipping these cycles with the ‘standard’ Green functions recalled in Section 3.5, we obtain classes (7.1.6)

1

b i , vi ) = Zbhor (ti , vi ) + Zbver (ti ) ∈ CH d (M), Z(t

where (7.1.7)

1

d (M), Zbhor (ti , vi ) = (Z hor (ti ), Ξ(ti , vi )) ∈ CH

and (7.1.8)

1

d (M). Zbver (ti ) = (Z ver (ti ), 0) ∈ CH

Notice that, when ti < 0, then Z(ti ) = φ, and we have 1

b i , vi ) = (0, Ξ(ti , vi )) ∈ CH d (M). Z(t

Such classes might be thought of as being ‘vertical at infinity’. We need to compute b 1 , v1 ), Z(t b 2 , v2 ) i = h Zbhor (t1 , v1 ), Zbhor (t2 , v2 ) i h Z(t

(7.1.9)

+ h Zbhor (t1 , v1 ), Zbver (t2 ) i + h Zbver (t1 ), Zbhor (t2 , v2 ) i + h Zbver (t1 ), Zbver (t2 ) i

3

Here it is essential to interpret the notation Z(ti ) as the image divisor in M.

208

CHAPTER 7

b in terms of the Z(T, v)’s with diag(T ) = (t1 , t2 ). When t1 t2 is not a square, so that only nonsingular T ’s can arise, the required identity comes down essentially to the statement that the intersection can be written as a disjoint union according to the fundamental matrices in the form as, e.g., in [3]. We recall this in Section 7.2. Then, in Sections 7.3–7.9, we handle the cases in which t1 t2 = m2 and ti > 0. In Section 7.10, we consider the case in which t1 t2 = m2 but t1 , t2 < 0. Finally, the cases where t1 t2 = 0 are considered in Section 7.11. Here we rely on arguments involving modularity of our various generating functions combined with the identities ((?)) for t1 t2 6= 0 to conclude that the remaining terms must agree. As a consequence, we obtain the explicit value for hω ˆ, ω ˆ i, which had been conjectured in [4].

Theorem 7.1.1. 

hω ˆ, ω ˆ i = ζD(B) (−1) 2

ζ 0 (−1) 1 + 1 − 2C − ζ(−1) 2

where 2C = log(4π) + γ, for Euler’s constant γ.

X p+1 p|D(B)

p−1



· log(p) ,

7.2 THE CASE t1 t2 IS NOT A SQUARE

When t1 t2 is not a square, the cycles Z(t1 ) and Z(t2 ) can have horizontal components, but these components are disjoint on the generic fiber. Thus, every height pairing on the right-hand side of (7.1.9) can be written as a sum over primes p, p ≤ ∞, of local contributions, h Zbhor (t1 , v1 ), Zbhor (t2 , v2 ) ip , h Zbhor (t1 , v1 ), Zbver (t2 ) ip , etc. For a finite prime p, these contributions come from intersections in the fiber at p. For p = ∞, they come from the Green functions. Theorem 7.2.1. Suppose that t1 t2 is not a square. (i) For a finite prime p with p - D(B),

b 1 , v1 ), Z(t b 2 , v2 ) ip = h Zbhor (t1 , v1 ), Zbhor (t2 , v2 ) ip h Z(t

= Z hor (t1 ), Z hor (t2 ) = Here the quantity

X

b ). Z(T

T >0 diag(T )=(t1 ,t2 ) Diff(T,B)={p}

 v b b Z(T ) = Z(T, 1



v2

)


p

· log p

209

AN INNER PRODUCT FORMULA

is independent of v1 and v2 . (ii) For a finite prime p with p | D(B), X

b 1 , v1 ), Z(t b 2 , v2 ) ip = h Z(t

b ) is independent of v. and, again, Z(T (iii) For p = ∞,

 v b Z(T, 1

X

b 1 , v1 ), Z(t b 2 , v2 ) i∞ = h Z(t

b ). Z(T

T >0 diag(T )=(t1 ,t2 ) Diff(T,B)={p}

T diag(T )=(t1 ,t2 ) Diff(T,B)={∞}



v2

).

The statement in this theorem reflects the decomposition of the intersecb i , vi ) according to ‘fundamental matrices’, as is imtion pairing of the Z(t plicit in [2], Theorem 14.11 (resp. Proposition 12.5) in the case p - D(B) (resp. the case p = ∞), and explicit in [3], Theorem 8.6 in the case p | D(B).

Sketch of Proof. For a fixed prime p < ∞, let B 0 be the definite quaternion algebra with invariants inv` (B 0 ) = inv` (B) for ` 6= p, ∞ and invp (B 0 ) = −invp (B). Let H 0 = (B 0 )× , and let V 0 = {x ∈ B 0 | tr(x) = 0}. For a fixed maximal order OB 0 in B 0 , we fix an isomorphism B 0 (Apf ) ' B(Apf ), ˆ p is identified with OB 0 ⊗Z Z ˆ p . We then have identifications so that OB ⊗Z Z p p p p 0 0 ˆ p )× . H(Af ) = H (Af ) and V (Af ) = V (Af ), and we set K p = (OB ⊗ Z First suppose that p - D(B). In this case, B 0 is ramified at p. Since t1 t2 is not a square, the cycles Z(t1 ) and Z(t2 ) are disjoint on the generic fiber and only meet in Mp at supersingular points. We have (7.2.1)

ˆ 1 , v1 ), Z(t ˆ 2 , v2 )ip = (Z(t1 ), Z(t2 ))p · log (p), hZ(t

where the geometric intersection number can be written as a sum of local intersection numbers (7.2.2)

X

(Z(t1 ), Z(t2 ))p =

(Z(t1 ), Z(t2 ))x .

¯ p )ss x∈Mp (F

¯ ss corresponds to (A, ι), then If x ∈ Mp (F) (7.2.3)

(Z(t1 ), Z(t2 ))x =

X y=(y1 ,y2 )∈V

νp (y). (A,ι)2

Q(yi )=ti

210

CHAPTER 7

Here νp (y) denotes the length of the deformation space of the pair of endomorphisms y = (y1 , y2 ) of (A, ι), and each point is counted with multiplicity e−1 y . But νp (y) = νp (T ) only depends on the fundamental matrix T = Q(y) of (A, ι, y). Therefore, rearranging the sum according to the possible fundamental matrices we obtain, as in Section 6.1, (7.2.4) (Z(t1 ), Z(t2 ))p =

X T ∈Sym2

X (Z)∨

X

¯ p )ss y∈V (A,ι)2 x=(A,ι)∈Mp (F

diag(T )=(t1 ,t2 )

X

=

e−1 y νp (T )

Q(y)=T

|ΛT | · νp (T ),

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 )

with (7.2.5)

|ΛT | =

X

|{y ∈ V (A, ι)2 ; Q(y) = T }|,

¯ p )ss x∈Mp (F

where again the cardinality is to be taken in the stack sense, i.e., (A, ι, y) counts with multiplicity 1/|Aut (A, ι, y)|. Note that since t1 t2 is not a square, the sum (7.2.4) runs only over nonsingular matrices T . Next, we consider the contributions of terms where p | D(B). In this case, we also have to take into account the additional terms coming from vertical components. Let (7.2.6) OB 0 [p−1 ] = OB 0 ⊗Z Z[p−1 ],

(resp. L0 [p−1 ] = L0 ⊗Z Z[p−1 ] ),

and let Γ0 = (OB 0 [p−1 ])× . Note that Γ0 can be identified with an arithmetic subgroup of B 0 (Qp )× ' GL2 (Qp ). Recall that we are assuming that t1 t2 is not a square. Then we can write as in the development leading up to [3], Theorem 8.5, (cf. also the proof of Theorem 6.2.1) b 1 , v1 ), Z(t b 2 , v2 ) ip h Z(t

= h Zbhor (t1 , v1 ), Zbhor (t2 , v2 ) ip + h Zbhor (t1 , v1 ), Z ver (t2 ) ip + h Z ver (t1 ), Zbhor (t2 , v2 ) ip + h Z ver (t1 ), Z ver (t2 ) ip (7.2.7)

=

X

X

T ∈Sym2 (Z)∨ y=[y1 ,y2 ]∈(L0 [p−1 ])2 diag(T )=(t1 ,t2 ) Q(y)=T mod Γ0

e−1 y · νp (T ) log(p),

211

AN INNER PRODUCT FORMULA

=

X

|ΛT | · νp (T ) log(p),

(Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 )

where ey is the order of the stabilizer of y in (7.2.8)

Γ0,1 = { γ ∈ Γ0 | ordp (ν(γ)) = 0 },

and where (7.2.9)

ΛT = Γ0 \{ y ∈ (L0 [p−1 ])2 | Q(y) = T } , 



and (7.2.10)

|ΛT | =

X

e−1 y .

y∈ΛT

The computation is done using the p-adic uniformization of the formal completion of the special fiber Mp . The quantity (7.2.11)

1 νp (T ) = χ(Z(j), OZ(j1 ) ⊗L OZ(j2 ) ), 2

which depends only on the GL2 (Zp )-equivalence class of T ∈ Sym2 (Zp ), b Note that when the matrix T is changed is ‘global’ on the Drinfeld space Ω. by an element of GL2 (Zp ), the geometry of the formal cycle Z(j) changes, although νp (T ) does not. An explicit example is given in Example 5.5 at the end of Section 5 of [3]. This fact will be relevant to us here, since later, in the case in which t1 t2 is a square, we will need to compute the individual terms h Zbhor (t1 , v1 ), Zbhor (t2 , v2 ) ip and h Z ver (t1 ), Z ver (t2 ) ip . Finally, we consider the archimedean part of the height pairing in the case in which t1 t2 is not a square. This part of the pairing is given by the star product of the Green functions. Let D be the union of the upper and the lower half planes and let M = MC = [Γ \ D]. Then, as in the proof of Proposition 12.5 of [2], b 1 , v1 ), Z(t b 2 , v2 ) i∞ = 1 h Z(t

Z

Ξ(t1 , v1 ) ∗ Ξ(t2 , v2 ) 2 M Z 1 1 X X 1 = ξ(v12 x1 ) ∗ ξ(v22 x2 ) 2 [Γ\D] x ∈L x ∈L 1

2

Q(x1 )=t1 Q(x2 )=t2

(7.2.12)

=

X

X

T ∈Sym2 (Z)∨ x=[x1 ,x2 ]∈L2 diag(T )=(t1 ,t2 ) Q(x)=T mod Γ

e−1 x · ν∞ (T, v)

212

CHAPTER 7

X

=

|ΛT | · ν∞ (T, v),

(Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 )

where (7.2.13)

ν∞ (T, v) =

1 2

Z D

1

1

ξ(v12 x1 ) ∗ ξ(v22 x2 ),

and (7.2.14)

ΛT = Γ\{ x ∈ (L)2 | Q(x) = T } . 



As usual, (7.2.15)

|ΛT | =

e−1 x .

X x=[x1 ,x2 ]∈L2 Q(x)=T mod Γ

Summarizing, we have, for any T ∈ Sym2 (Z)∨ with det(T ) 6= 0 and Diff(T, B) = {p}, (7.2.16)

b Z(T, v) =

 |ΛT | · νp (T ) · log(p)

if p < ∞,

|Λ | · ν (T, v) ∞ T

if p = ∞,

where |ΛT | is given by (7.2.5), (7.2.10), and (7.2.15), respectively, νp (T ) is given in (7.2.4) and (7.2.11), and ν∞ (T, v) is given by (7.2.13). This yields the claimed result. 7.3 A WEAKLY ADMISSIBLE GREEN FUNCTION

In this section, we construct a weakly µ-biadmissible Green function on H× H − ∆, as defined in Section 2.7, for the hyperbolic metric µ = y −2 dx ∧ dy on the upper half plane H. This Green function is rapidly decreasing away from the diagonal. The Green functions introduced in [2] (see also Section 3.5) can be recovered by restriction to the ‘slices’ of the form z1 × H. For t > 0, let (7.3.1)

β1 (t) = −Ei(−t) =

Z

∞

e−tr r−1 dr

1

be the exponential integral.

Proposition 7.3.1. For t ∈ R>0 , the function gt0 (z1 , z2 ) = β1 (4πt sinh2 (d(z1 , z2 ))),

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AN INNER PRODUCT FORMULA

where d(z1 , z2 ) is the hyperbolic distance between z1 and z2 ∈ H, defines a weakly µ-biadmissible Green function on H × H − ∆. The associated function φ0t is given by φ0t (z1 , z2 ) = 4t cosh2 (d(z1 , z2 )) − 

For any γ ∈ GL2 (R)+ ,

and

g 0 (γz1 , γz2 ) = g 0 (z1 , z2 )

1  −4πt sinh2 (d(z1 ,z2 )) e . 2π

φ0 (γz1 , γz2 ) = φ0 (z1 , z2 ).

Proof. The symmetry (ii) in Definition 2.7.1 is obvious and condition (i) is immediate from the asymptotics of β1 near zero; see Lemma 7.3.3. For the Green equation, we let R = 2 sinh2 (d(z1 , z2 )) and compute, first on all of H × H − ∆, (7.3.2) ddc gt0 (z1 , z2 ) =

1  ¯ − R−2 ∂R ∧ ∂R ¯ + R−1 ∂ ∂R ¯  e−2πtR − 2πtR−1 ∂R ∧ ∂R 2πi

=

 1  − 2πtR ∂ log(R) ∧ ∂¯ log(R) + ∂ ∂¯ log(R) e−2πtR . 2πi

Since (7.3.3)

cosh(d(z1 , z2 )) =

|z1 − z2 |2 + 1, 2y1 y2

we have (7.3.4)

R=2

|z1 − z2 |2 |z1 − z¯2 |2 · , 2y1 y2 2y1 y2

(7.3.5) log(R) = −2 log(y1 y2 ) − log(2) + log |z1 − z2 |2 + log |z1 − z¯2 |2 , and (7.3.6) ∂ log(R) 

=

iy1−1

1 1 1 1 + dz1 + iy2−1 + + dz2 + z1 − z2 z1 − z¯2 z2 − z1 z2 − z¯1 

= ( ρ−1 + σ1 ) dz1 + (−ρ−1 + σ2 ) dz2 ,





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where, for convenience, we write ρ = z1 − z2 , and (7.3.7)

σ1 = iy1−1 +

1 , z1 − z¯2

σ2 = iy2−1 +

1 . z2 − z¯1

Then (7.3.8)

∂¯ log(R) = (¯ ρ−1 + σ ¯1 ) d¯ z1 + (−¯ ρ−1 + σ ¯2 ) d¯ z2

and (7.3.9) R ∂ log(R) ∧ ∂¯ log(R) 

= 2(2y1 y2 )−2 |z1 − z¯2 |2 |1 + ρσ1 |2 dz1 ∧ d¯ z1 + | − 1 + ρσ2 |2 dz2 ∧ d¯ z2 

+ (1 + ρσ1 )(−1 + ρ¯σ ¯2 )dz1 ∧ d¯ z2 + (−1 + ρσ2 )(1 + ρ¯σ ¯1 )dz2 ∧ d¯ z1 . In addition, (7.3.10) ∂ ∂¯ log(R) = − iy1−2 dx1 ∧ dy1 − iy2−2 dx2 ∧ dy2 + (z1 − z¯2 )−2 dz1 ∧ d¯ z2 + (z2 − z¯1 )−2 dz2 ∧ d¯ z1 . Note that the 2-forms in (7.3.9) and (7.3.10) are both smooth on all of H×H. Now, fixing z1 , we obtain (7.3.11) R ∂2 log(R) ∧ ∂¯2 log(R) = 2 (2y1 y2 )−2 |z1 − z¯2 |2 | − 1 + ρσ2 |2 dz2 ∧ d¯ z2 , and ∂2 ∂¯2 log(R) = −iy2−2 dx2 ∧ dy2 ,

(7.3.12) so that

d2 dc2 gt0 (z1 , z2 ) = ty1−2 |z1 − z¯2 |2 |1 − ρσ2 |2 − 

(7.3.13)



= 4t

1  −2πtR −2 e y2 dx2 ∧ dy2 2π


2 1  −2πtR −2 |z1 − z2 |2 +1 − e y2 dx2 ∧ dy2 2y1 y2 2π

1  −2πtR −2 e y2 dx2 ∧ dy2 . 2π In fact, to finish the proof of (iii) of Definition 2.7.1, we must still check the Green equation for currents, but this will follow from the calculation in [2], Proposition 11.1, once we have identified the restriction of gt0 to the ‘slice’ z1 × H. = 4t cosh2 (d(z1 , z2 )) − 

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AN INNER PRODUCT FORMULA

Remark 7.3.2. Away from the diagonal, (7.3.14)

gt0 (z1 , z2 ) = O( exp(−4πt sinh2 (d(z1 , z2 ))) ),

and hence decays doubly exponentially with respect to the hyperbolic distance. Next, we consider the pullback of the metric on OH×H (∆) determined, on H × H − ∆, by the identity gt0 (z1 , z2 ) = − log ||s∆ ||2 ,

(7.3.15)

where s∆ = z1 − z2 . The point is to determine the behavior of this metric along ∆. Lemma 7.3.3. For the exponential integral β1 of (7.3.1), 

lim



β1 (t) + log(t)

where γ is Euler’s constant. More precisely t↓0

= −γ,

β1 (t) = −γ − log(t) + β10 (t)

where β10 (t) is given by

β10 (t) =

In particular, limt↓0 β10 (t) = 0.

Z 0

t

eu − 1 du. u

It follows from this lemma together with (7.3.4) that (7.3.16) gt0 (z1 , z2 ) = − log |z1 − z2 |2 − γ − log(4πt) + 2 log(2y1 y2 ) − log |z1 − z¯2 |2 + β10 (2πtR). Recalling that ρ = z1 − z2 and using (7.3.6), we compute − log ||1||2 = − log ||ρ−1 s∆ ||2 (7.3.17)

= log |z1 − z2 |2 + gt0 (z1 , z2 ) = log |z1 − z2 |2 − γ − log(2πtR) + smooth = −γ − log(πt) + 2 log(y1 y2 ) − log |z1 − z¯2 |2 + smooth.

Restricting to the diagonal z = z1 = z2 , we get (7.3.18)

− log ||1||2 |∆ = −γ − log(4πt) + 2 log(y).


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CHAPTER 7

Note that the corresponding Chern form is (7.3.19)

−ddc log ||1||2 |∆ = ddc (2 log(y)) = −


1 µ. 2π

Formula (7.3.18) can be interpreted as follows. Let I be the ideal of holomorphic functions vanishing on the diagonal in H × H. Taking s∨ ∆ = (z1 −z2 )−1 , as a section of OH×H (−∆), with divisor −∆, then for any open set U in H × H, there is an isomorphism (7.3.20)

∼

Γ(U, I) −→ Γ(U, OH×H (−∆)),

f 7→ f · s∨ ∆.

Then the canonical isomorphism ∼

i∗∆ (OH×H (−∆)) −→ Ω1H

(7.3.21) is given explicitly by (7.3.22)

∗ f · s∨ ∆ 7→ i∆ (h) · dz,

where (7.3.23)

f (z1 , z2 ) = h(z1 , z2 ) (z1 − z2 ).

1 In particular, ρ · s∨ ∆ 7→ dz in (7.3.21). The metric on ΩH which makes (7.3.21) an isometry is then determined by

(7.3.24)

2 − log ||dz||2 = − log ||ρ · s∨ ∆ || = γ + log(4πt) − 2 log(y),

where the sign has changed compared to (7.3.18), since we are considering OH×H (−∆), rather than OH×H (∆). Suppose that Γ ⊂ GL+ 2 (R) is a discrete group acting properly discontinuously on H with compact quotient Γ\H. Let M = [Γ\H] be the corresponding compact orbifold, and let (7.3.25)

pr : H −→ [Γ\H] = M

be the natural projection. We also suppose that there is a normal subgroup Γ0 ⊂ Γ of finite index which acts without fixed points in H. Thus, there is a compact Riemann surface X = Γ0 \H with a finite group of automorphisms Γ0 = Γ/Γ0 so that M = [Γ0 \X]. Thus, we are in the situation of Sections 2.3 and 2.7. The function (7.3.26)

gt (P, Q) =

X γ∈Γ0

gt0 (z1 , γ z2 ),

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AN INNER PRODUCT FORMULA

for P = pr(z1 ) and Q = pr(z2 ) ∈ X, is then a weakly µ-biadmissible Green function on X × X − ∆X with associated function (7.3.27)

φt (P, Q) =

X

φ0t (z1 , γz2 ).

γ∈Γ0

Since Γ0 is normal in Γ, these functions are invariant with respect to the diagonal action of Γ0 = Γ/Γ0 . In the arithmetic adjunction formula of Section 2.7, we need to compute the metric on Ω1X = i∗∆ OX×X (−∆) determined by the function −gt . Since Γ0 has no elliptic fixed points on H, we have4 (7.3.28)

− log ||dz||2 = γ + log(4πt) − 2 log(y) − ψt (z),

where (7.3.29)

ψt (z) =

X

gt0 (z, γz).

γ∈Γ0 γ6=1

Here, the first three terms on the right-hand side are the contribution for γ = 1, as computed in (7.3.24) above. Since Γ0 is normal in Γ, the function ψt (z) is actually Γ-invariant, as is the quantity − log ||dz||2 + 2 log(y). In particular, ψt defines a function on M . If f is a holomorphic automorphic form of weight 2 with respect to Γ, then the corresponding section f (z) dz of Ω1M has norm (7.3.30)

||f (z)dz||2 = |f (z)|2 y 2 · e−2C−log(t)+ψt (z) ,

where (7.3.31)

2C = γ + log(4π).

Note that the constant C is exactly the quantity occurring in Definition 3.4 of [4], so that its, seemingly mysterious, appearance there is explained in a natural way by (7.3.30). Also note that the Chern form of the metric on Ω1M determined by −gt is (7.3.32)

1 µ − ddc ψt . 2π

Of course, X, ψt , etc. depend on the choice of the subgroup Γ0 . 4

Of course, the reader will not confuse the γ here, which is Euler’s constant, with an element of Γ!

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CHAPTER 7

× Next, we return to the orbifold M = [Γ\D] = M(C), where Γ = OB , + − so that we now have a uniformization by D = H ∪ H ,

(7.3.33)

pr : D −→ [ Γ\D ] = M,

where Γ ⊂ GL2 (R). We want to determine the relation between the Green functions gt0 just constructed and the Green functions introduced in Section 3.5. As in [2], let (7.3.34)

V = {x ∈ M2 (R) | tr(x) = 0},

with quadratic form Q(x) = det(x) = xι x and associated bilinear form (x, y) = tr(xι y), where ι is the main involution on M2 (R).5 The quadratic space (V, Q) has signature (1, 2). Let (7.3.35)

H = {x ∈ V | Q(x) = 1}

be the hyperboloid of two sheets. We embed D = H+ ∪ H− into V by the map (7.3.36)

z = x + iy 7→ x(z) = y

−1



−x |z|2 . −1 x 

Note that Q(x(z)) = 1 and that x(¯ z ) = −x(z). This embedding identifies D with H and is equivariant for the action of GL2 (R): (7.3.37)

g · x(z) = g x(z)g −1 = x(g(z)).

Note that every x ∈ V with t = Q(x) > 0, i.e.,√every vector inside the light cone, can be written uniquely in the form x = t · x(z) for z ∈ D. Remark 7.3.4. Note that this parametrization is different from the one introduced in Section 3.2. Here it is more convenient to identify D with the oriented positive lines in V . Thus we use z 7→ R · x(z) rather than the map z 7→ C× · w(z) of (3.2.4). Given z ∈ D, there is an orthogonal decomposition (7.3.38)

V = R x(z) ⊕ x(z)⊥ , x=

5

(x, x(z)) x(z) + x0 , (x(z), x(z))

We apologize for the awkward notation with x ∈ V and z = x + iy ∈ H. Soon the real part of z will no longer be mentioned, and we will revert to x ∈ V and z ∈ H.

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AN INNER PRODUCT FORMULA

and the restriction of Q to x(z)⊥ is negative definite. In these coordinates, the quantity R(x, z), defined in (3.5.1) can be written as (7.3.39)

1 R(x, z) := −(x0 , x0 ) = (x, x(z))2 − (x, x). 2

Recall that R(x, z) ≥ 0 and that R(x, z) = 0 if and only if x ∈ R x(z). Also, R(−x, z) = R(x, z¯) = R(x, z). Similarly, in these coordinates, the functions ϕ and ξ of (3.5.4) and (3.5.6) become (7.3.40)

1 e−2πR(x,z) , ϕ(x, z) = (x, x(z)) − 2π 

2



and (7.3.41)

ξ(x, z) = β1 (2πR(x, z)),

R(x, z) > 0.

The following result shows that these Green functions are, indeed, the Green functions determined by gt0 via restriction to slices. First, we extend gt0 and φ0t , defined in Proposition 7.3.1, to D × D by setting:

(7.3.42)

gt0 (z1 , z2 ) =

 0 g (z , z ),    t 1 2

if z1 and z2 ∈ H+ ,

  

otherwise,

gt0 (¯ z1 , z¯2 ), if z1 and z2 ∈ H− ,

0,

and similarly for φ0t . These extensions are invariant under the diagonal action of GL2 (R). √ Lemma 7.3.5. If x = ± t x(z0 ), with z0 ∈ H, then, ξ(x, z) = gt0 (z0 , z) + gt0 (¯ z0 , z), ϕ(x, z) = φ0t (z0 , z) + φ0t (¯ z0 , z).

Proof. For z1 and z2 in H = H+ , (7.3.43)

(x(z1 ), x(z2 )) = 2 cosh(d(z1 , z2 )). √ Also note that, if t = Q(x) > 0 and x = ± t · x(z0 ), for z0 ∈ H, then, for z ∈ H, (7.3.44)

R(x, z) = t · 2 sinh2 (d(z0 , z)).

The claimed identities are then immediate.

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CHAPTER 7

√ For x ∈ V with Q(x) = t > 0, and x = ± t x(z0 ), we have Dx = {z0 , z¯0 }. Recall from Propsition 3.5.1 that (7.3.45)

ddc ξ(x) + δDx = ϕ(x) · µ,

as currents on D, and hence that Z

(7.3.46)

ϕ(x, z) dµ(z) = 2. D

The first of these identities completes the proof for gt0 of property (iii) of Definition 2.7.1. Note that ξ(x) is a Green function for the cycle Dx consisting of 2 points. From the definition of the ∗-product and Lemma 7.3.5, we obtain the following fact will be useful in Section 7.5 below.

√ √ Lemma 7.3.6. Write x1 = ± t1 x(z1 ) and x2 = ± t2 x(z2 ) with z1 and z2 ∈ H. Then Z

ξ(x1 ) ∗ ξ(x2 ) =

D

gt01 (z1 , z2 )

Z

gt02 (z2 , ζ) φ0t1 (z1 , ζ) dµ(ζ)

+

+ gt01 (¯ z1 , z¯2 ) +

D

Z D

gt02 (¯ z2 , ζ) φ0t1 (¯ z1 , ζ) dµ(ζ).

For x ∈ V , with Q(x) = t > 0, and z ∈ D, we set (7.3.47)

Ξ(x, z) =

X

ξ(x, γz) = e¯−1 x

X

ξ(x, γz),

¯ γ∈Γ

γ∈Γx \Γ

and (7.3.48)

Φ(x, z) =

X

ϕ(x, γz) = e¯−1 x

γ∈Γx \Γ

X

ϕ(x, γz).

¯ γ∈Γ

¯ = Γ/ΓD , where ΓD is the subgroup of Γ acting trivially on D. Note Here Γ that the function Ξ(t, v) in Corollary 3.5.2 is then obtained by summing Ξ(x, v) over the Γ orbits in the set {x ∈ OB ∩ V | Q(x) = t}.

Lemma 7.3.7. Ξ(x) is a Green function on the stack M for the divisor pr(Dx ); see Section 2.3. Explicitly, ddc Ξ(x) + δpr(Dx ) = Φ(x) · µ.

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AN INNER PRODUCT FORMULA

Proof. In fact, if φ ∈ C ∞ (M ), we have c

h dd Ξ(x), φ iM =

Z

Ξ(x) · ddc φ

M

Z

Ξ(x) · ddc φ

= [Γ\D]

= e−1 x

Z

ξ(x, z) · pr∗ (ddc φ)(z)

D

c ∗ = e−1 x h dd ξ(x), pr φ i

= e−1 x



− h δDx , pr∗ φ i +

= −h δpr(Dx ) , φ iM +

Z

Z

ϕ(x, z) · pr∗ φ(z) · dµ(z)



D

Φ(x) · φ · µ.

M

Here the points in pr(Dx ) on the orbifold M are counted with multiplicity −1 e−1 x = |Γx | . Note, for example, that, if 1M is the constant function on M , then h δpr(Dx ) , 1M iM = 2 e−1 x . 7.4 A FINER DECOMPOSITION OF SPECIAL CYCLES

There is a decomposition of the cycle Z hor (t) via the ‘conductor’, defined as follows. For t ∈ Z>0 ,√write 4t =√n2 d where −d is a fundamental discriminant. Let kt = Q( −t) = Q( −d). Then, on the generic fiber, there is a decomposition (7.4.1)

Z hor (t)Q =

a

Z hor (t : c)Q ,

c|n (c,D(B))=1

where Z hor (t : c)Q is the locus of triples (A, ι, x) in Z hor (t)Q such that, for the embedding φx : kt ,→ End0 (A, ι) determined by x, (7.4.2)

φ−1 x ( Q[x] ∩ End(A, ι) ) = Oc2 d .

Here Oc2 d ⊂ Od is the order of conductor c; see Section 10 of [4]. Recall that the order on the left-hand side of (7.4.2) is maximal at all primes dividing D(B), hence the restriction (c, D(B)) = 1 in the decomposition. Note that, for r ≥ 1, (7.4.3)

Z hor (r2 t : c)Q = Z hor (t : c)Q ,

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CHAPTER 7

since both are the locus of triples (A, ι, ι0 ), where ι0 is an embedding ι0 : Oc2 d ,→ End(A, ι)

(7.4.4)

which is optimal, in the obvious sense. Let6 Z hor (t : c) = Z hor (t : c)Q ,

(7.4.5) Note that, by (7.4.3), (7.4.6)

Z hor (r2 t : c) = Z hor (t : c).

Remark 7.4.1. The conductor condition (7.4.2) does not lead to a good moduli problem over Z(p) , so that we are forced to use the closure in our global definition of Z hor (t : c).

Lemma 7.4.2. Z hor (t : c)Q is irreducible. More precisely, its coarse moduli space is isomorphic to Spec kt,c , where kt,c is the ray class field of kt with × × ˆ× norm subgroup k× t · C · Oc2 d in Ak .

Proof. Let (A, ι, x) be a C-valued point of Z hor (t : c). Then A is of the form Lie(A)/Γ, where Γ is an OB -module from the left and where the order Oc2 d acts from the right on Γ through holomorphic endomorphisms, and is maximal with this property. Now the finite ideles of kt act in the usual way ˆ× transitively on the set of such Γ’s with stabilizer equal to k× t · Oc2 d . Since this action commutes with the action of the Galois group of kt , the action of an element σ ∈ Galkt is given by translation by an element of A× kt ,f . Now the theory of complex multiplication shows that this element is the image of σ under the class field isomorphism. The assertion concerning the coarse moduli space of Z hor (t : c) follows, and this also implies the first claim. t

We have the decomposition (7.4.7)

Z(t) =

[

Z hor (t : c) ∪ Z ver (t).

c|n (c,D(B))=1 1

b v) ∈ CH d (M) corresponding To obtain the decomposition of the class Z(t, to (7.4.7), we must decompose the Green function. As in Section 5 of [4], let L = OB ∩ V ,

(7.4.8) and 6

L(t) = { x ∈ L | Q(x) = t }

Here again, as in the definition of Z hor (t), we use the notation Z hor (t : c)Q to denote the corresponding image divisor in MQ .

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AN INNER PRODUCT FORMULA

(7.4.9)

L(t : c) = { x ∈ L | Q(x) = t, type (x) = c },

where type(x) = c means that (7.4.10)

Q[x] ∩ OB = an order of conductor c in Q[x].

Here Q[x] ' Q[ −Q(x)]. Green functions for Z hor (t) and Z hor (t : c) are given by p

(7.4.11)

X

Ξ(t, v)(z) =

1

ξ(v 2 x, z),

x∈L(t)

and (7.4.12)

X

Ξ(t : c, v)(z) =

1

ξ(v 2 x, z).

x∈L(t:c)

By the constructions explained in Section 7.3, if Q = pr(z), then7 (7.4.13)

X

Ξ(t, v)(z) =

gtv (P, Q),

P ∈Z hor (t)(C)

and (7.4.14)

X

Ξ(t : c, v)(z) =

gtv (P, Q)

P ∈Z hor (t:c)(C)

X

=

gtv (P, Q)

P ∈Z hor (r 2 t:c)(C)

= Ξ(r2 t : c, r−2 v). In the last steps we used (7.4.6). Let (7.4.15)

d 1 (M). Zbhor (t : c, v) = (Z hor (t : c), Ξ(t : c, v)) ∈ CH

Thus we obtain the decomposition (7.4.16)

Zbhor (t, v) =

X

Zbhor (t : c, v).

c|n

Note that, by (7.4.6) and (7.4.14), (7.4.17)

Zbhor (r2 t : c, v) = Zbhor (t : c, r2 v).

b 1 , v1 ), Z(t b 2 , v2 ) i in the case where t1 t2 We now consider the pairing h Z(t is a square. We begin by determining the common components of Z hor (t1 ) and Z hor (t2 ). 7

Here the points in Z hor (t)(C) are counted with their fractional multiplicities.

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CHAPTER 7

Lemma 7.4.3. Write t1 = n21 t and t2 = n22 t with (n1 , n2 ) = 1, and, as above, write 4t = n2 d. Then the greatest common divisor of the divisors8 Z hor (t1 ) and Z hor (t2 ) is equal to Z hor (t) = Proof. We have (7.4.18)

X

Z hor (t : c).

c|n

Z hor (ti ) =

X

Z hor (n2i t : c).

c|ni n (c,D(B))=1

Thus, the common part is the sum of the Z hor (n2i t : c)’s where c divides both n1 n and n2 n, hence c | n. Since, for such c, Z hor (n2i t : c) = Z hor (t : c), this sum is just Z hor (t). b 1 , v1 ), Z(t b 2 , v2 ) i = A + B + C, where Then, we can write h Z(t

(7.4.19)

h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) i,

X

A=

c1 ,c2 c1 |n1 n, c2 |n2 n c1 6=c2

(7.4.20)

B=

h Zbhor (t1 : c, v1 ), Zbhor (t2 : c, v2 ) i,

X c|n

and (7.4.21) C = h Zbhor (t1 ), Zbver (t2 ) i + h Zbver (t1 ), Zbhor (t2 ) i + h Zbver (t1 ), Zbver (t2 ) i. There are, thus, several types of intersections to be computed. First, in the terms in (7.4.19), since c1 6= c2 , the cycles are disjoint on the generic fiber, and hence (7.4.22) h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) i =

X

h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) ip .

p≤∞

We let A = (7.4.23)

p≤∞ Ap ,

P

Ap =

where X

h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) ip .

c1 ,c2 c1 |n1 n, c2 |n2 n c1 6=c2

8

Here the Z hor (ti )’s denote the image divisors on M.

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AN INNER PRODUCT FORMULA

Similarly, let C =

p Cp ,

P

where

(7.4.24) Cp = h Zbhor (t1 , v1 ), Zbver (t2 ) ip + h Zbver (t1 ), Zbhor (t2 , v2 ) ip + h Zbver (t1 ), Zbver (t2 ) ip . Here only primes p with p | D(B) occur. Finally, by (7.4.17), the terms in B have the form (7.4.25) h Zbhor (t1 : c, v1 ), Zbhor (t2 : c, v2 ) i = hZbhor (t : c, n21 v1 ), Zbhor (t : c, n22 v2 )i. These terms are not a sum of local contributions and will be computed in the next section.

7.5 APPLICATION OF ADJUNCTION

In this section, we apply the arithmetic adjunction formula, as formulated in Section 2.7, to compute the arithmetic intersection numbers (7.4.25) of common components of Zbhor (t1 , v1 ) and Zbhor (t2 , v2 ). Recall that, as in (7.3.33), M = M(C) = [Γ\D]. We begin by making explicit what the adjunction formula of Section 2.7 gives for the self-intersection number of an irreducible horizontal divisor Z on M. We write Z(C) =

X

Pi .

i

Taking zi ∈ D with pr(zi ) = Pi , the Green function for Z determined by gv0 , for a parameter v > 0, is given by ΞZ (v) =

X

e−1 zi

i

X

gv0 (zi , γz),

γ∈Γ

with associated function ΦZ (v) =

X i

e−1 zi

X

φ0v (zi , γz).

γ∈Γ

The adjunction formula of Theorem 2.7.2 can be applied here for the infinite presentation M = [Γ\D]. Proposition 7.5.1. Let Z = Z hor be an irreducible, reduced, horizontal divisor on M. For v1 and v2 ∈ R>0 , let Ξ(v1 ) and Ξ(v2 ) be the Green

226

CHAPTER 7

b 1 ) = (Z, ΞZ (v1 )) functions on M for Z with parameters v1 and v2 . Let Z(v 1 b 2 ) = (Z, ΞZ (v2 )) ∈ CH d (M) be the corresponding classes. Then, and Z(v b 1 ), Z(v b 2 ) iM h Z(v

1 degQ (Z) · log(4D(B) v1 ) + dZ 2 X 1 X −1 e−1 e + gv01 (z, γz 0 ) P P0 2 γ∈Γ P,P 0 ∈Z(C)

= −hωˆ (Z) −

z6=γz 0

Z

+ D

e−1 P

X

+

1 2

Z D

φ0v1 (z, ζ) gv02 (γz 0 , ζ) dµ(ζ)



φ0v1 (z, ζ) gv02 (z, ζ) dµ(ζ),

where ω ˆ is the Hodge bundle ω on M with the metric defined by (3.3.4), and where dZ is the discriminant degree of Z. P ∈Z(C)

Proof. The formula of Theorem 2.7.2 involves the height of Z with respect b1 = ω b M/S , where ω M/S is the relative dualizing sheaf and the metric to ω is determined by −gv01 on D × D − ∆D , as in Section 7.3 above. By Proposition 3.3.1, as sheaves ω M/S = ω, where ω is the Hodge bundle on M. The following lemma compares the two metrics on this sheaf. d (M), Lemma 7.5.2. As classes in CH 1




b1 = ω ω ˆ + 0, log(v1 ) + log(4 D(B)) .

b 1 determined by forProof. This follows by comparison of the metric on ω mula (7.3.24) above with that on ω ˆ given in (3.3.4). To compare them, we use the fact that, under the Kodaira-Spencer isomorphism ∼

ω = ∧2 (Ω1A/M ) −→ ω 1 ,

(7.5.1) we have (7.5.2)

2πi f (z) αz 7→ f (z) dz,

where f (z) αz is as in the discussion after (3.13) of [4]. Now the claimed identity is evident. Using the expression of the lemma, we obtain (7.5.3)

1 hω b 1 (Z) = hωˆ (Z) + 2 degQ (Z)

as claimed.





log(v1 ) + log(4D(B))

227

AN INNER PRODUCT FORMULA

It will be important to note that the adjunction formula is not ‘homogeneous’; some of the terms involved are linear in the cycle Z, while others are quadratic. For example, if Z = 2 Z0 , with Z0 irreducible, reduced, and horizontal, then b 1 ), Z(v b 2 ) iM h Z(v

= 4 h Zb0 (v1 ), Zb0 (v2 ) iM = −2 hωˆ (Z) − degQ (Z) · log(4D(B) v1 ) + 2 dZ (7.5.4)

+

X 1

−1 e−1 P eP 0

X P,P 0 ∈Z(C)

γ∈Γ z6=γz 0

2 Z

+ D

X

+

e−1 P

Z

P ∈Z(C)

D

gv01 (z, γz 0 ) φ0v1 (z, ζ) gv02 (γz 0 , ζ) dµ(ζ)



φ0v1 (z, ζ) gv02 (z, ζ) dµ(ζ).

Here, hωˆ (Z) = 2 hωˆ (Z0 ), deg(Z) = 2 deg(Z0 ), and dZ = 2 dZ0 , so that these terms occur multiplied by an additional factor of 2 in the pairing b 1 ), Z(v b 2 ) iM = 4 h Zb0 (v1 ), Zb0 (v2 ) iM . The same is true for the last h Z(v sum, over points of Z(C). On the other hand, the sum over pairs of points of Z(C) amounts to 4 times the corresponding sum over pairs of points of Z0 (C), and hence has no additional factor. Since (A, ι, y) and (A, ι, −y) occur together in Z hor (t : c), all components of Z hor (t) occur with multiplicity 2. Thus, applying (7.5.4) and the irreducibility of Z hor (t : c) , Lemma 7.4.2, we get

Corollary 7.5.3. Let t1 t2 be a square, and define t as in Lemma 4.2. Then, for c | n, h Zbhor (t1 : c, v1 ), Zbhor (t2 : c, v2 ) i = −2 hωˆ (Z hor (t : c)) − degQ (Z hor (t : c)) · log(4D(B) t1 v1 ) + 2 dZ hor (t:c) +

−1 e−1 P eP 0

X P,P 0 ∈Z hor (t:c)(C)

X 1 γ∈Γ z6=γz 0

2 Z

+ D

+

X P ∈Z hor (t:c)(C)

e−1 P

Z D

gt01 v1 (z, γz 0 ) φ0t1 v1 (z, ζ) gt02 v2 (γz 0 , ζ) dµ(ζ)

φ0t1 v1 (z, ζ) gt02 v2 (z, ζ) dµ(ζ).



228

CHAPTER 7

Using Lemma 7.3.6, we obtain the identity (7.5.5) X 1

−1 e−1 P eP 0

X P,P 0 ∈Z hor (t:c)(C)

γ∈Γ z6=γz 0

2

gt01 v1 (z, γz 0 ) Z

+ D

=

−1 e−1 x1 ex2

X

X

X

e−1 x

X

T ∈Sym2 (Z)∨ x=[x1 ,x2 ]∈L2 diag(T )=(t1 ,t2 ) Q(x)=T det(T )6=0 type(x)=(c,c) mod Γ

=

X

X γ∈Γ γDx2 6=Dx1

x1 ∈L(t1 :c) x2 ∈L(t2 :c) mod Γ mod Γ

=

φ0t1 v1 (z, ζ) gt02 v2 (γz 0 , ζ) dµ(ζ)

1 2

1 2

Z D

1

Z D



1

1

ξ(v12 x1 ) ∗ ξ(v22 γx2 ).

1

ξ(v12 x1 ) ∗ ξ(v22 x2 )

|ΛT (c, c)| · ν∞ (T, v),

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 ) det(T )6=0

where ν∞ (T, v) is given by (7.2.13) and (7.5.6)   ΛT (c1 , c2 ) = Γ\{ x = [x1 , x2 ] ∈ L2 | Q(x) = T, type(x) = (c1 , c2 ) } . It is interesting to compare this expression with the ‘nonsingular’ part of the archimedean height pairing from Definition 12.3 of [2]. Recall that, for x ∈ L, the type of x is defined by (7.4.10). We need to evaluate the integrals in the last term of Corollary 7.5.3. Lemma 7.5.4. Z

where

D

φ0v1 (z, ζ) gv02 (z, ζ) dµ(ζ) = log( Z

J(t) =

∞

v1 + v2 ) − J(4π(v1 + v2 )), v2 1

e−tw (w + 1) 2 − 1 w−1 dw. 



Proof. Note that we may as well assume that z ∈ H, so that the contribution of H− is zero. Let ρ = d(z, ζ) be the hyperbolic distance from z to ζ. Then 0

(7.5.7) and

φ0v1 (z, ζ) = 4v1 cosh2 (ρ) − 

1  −4πv1 sinh2 (ρ) e 2π

229

AN INNER PRODUCT FORMULA

gv02 (z, ζ) = β1 (4πv2 sinh2 (ρ)).

(7.5.8)

As in the proof of Proposition 12.1 of [4], (7.5.9) Z D

φ0v1 (z, ζ) gv02 (z, ζ) dµ(ζ) Z

π

∞

Z

=

4v1 cosh2 (ρ) −

0

0

1  −4πv1 sinh2 (ρ) e 2π × β1 (4πv2 sinh2 (ρ)) 2 sinh(ρ) dρ dθ

∞Z ∞

Z

=π 0

1 1  −4π(v1 +v2 r)w −1 e r dr (w + 1)− 2 dw. 2π

4v1 (w + 1) −

1

By integration by parts, we have (7.5.10)

∞

Z

1

e−4π(v1 +v2 r)w (w + 1)− 2 dw

0

Z

= 8π(v1 + v2 r)

∞

1

e−4π(v1 +v2 r)w (w + 1) 2 − 1 dw. 



0

Multiplying this by we obtain (7.5.11) Z Z ∞

0

∞

−(2π)−1

and substituting back into the main integral,

1

− 4πv2 r (w + 1) 2 − 1 + 4πv1 e−4π(v1 +v2 r)w r−1 dr dw 





1

=−

∞

Z

1

e−4π(v1 +v2 )w (w + 1) 2 − 1 w−1 dw 



0

Z

+

∞

v1 (v1 + v2 r)−1 r−1 dr

1

= −J(4π(v1 + v2 )) + log(v1 + v2 ) − log(v2 ). Thus, the last term in Corollary 7.5.3 becomes (7.5.12) e−1 P

X P ∈Z hor (t:c)(C)

= degQ Z

hor

Z D

φ0t1 v1 (z, ζ) gt02 v2 (z, ζ) dµ(ζ)





(t : c) −J(4π(t1 v1 + t2 v2 ))+ log(t1 v1 + t2 v2 )− log(t2 v2 ) .

Thus, we obtain the following pretty result:

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CHAPTER 7

Proposition 7.5.5. h Zbhor (t1 : c, v1 ), Zbhor (t2 : c, v2 ) i = −2 hωˆ (Z hor (t : c)) + 2 dZ hor (t:c) − degQ (Z hor (t : c)) · log(4D(B)) + degQ (Z hor (t : c)) log(t1 v1 + t2 v2 ) − log(t1 v1 ) − log(t2 v2 ) 



− degQ (Z hor (t : c)) J(4π(t1 v1 + t2 v2 )) X

+

|ΛT (c, c)| · ν∞ (T, v).

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 ) det(T )6=0

Summing on c, this yields the following expression for B from (7.4.20).

Corollary 7.5.6. The quantity B= is given by

X

h Zbhor (t1 : c, v1 ), Zbhor (t2 : c, v2 ) i

c|n

B = −2 hωˆ (Z hor (t)) + 2

X

dZ hor (t:c) − degQ (Z(t)) · log(4D(B))

c|n



+ degQ (Z(t)) log(t1 v1 + t2 v2 ) − log(t1 v1 ) − log(t2 v2 ) where



− degQ (Z(t)) J(4π(t1 v1 + t2 v2 )) + B∞ , X

B∞ =

X

|ΛT (c, c)| · ν∞ (T, v).

(Z)∨

c|n T ∈Sym2 diag(T )=(t1 ,t2 ) det(T )6=0

Similarly, when t1 t2 is a square, by the same calculations as in Section 2, we have Proposition 7.5.7. The quantity X

A∞ = is given by

h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) i∞

c1 6=c2

A∞ =

X (Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 )

X c1 6=c2

|ΛT (c1 , c2 )| · ν∞ (T, v).

231

AN INNER PRODUCT FORMULA

Here note that the condition c1 6= c2 does not allow x1 and x2 to be colinear, so that only T ’s with det(T ) 6= 0 contribute to the sum. 7.6 CONTRIBUTIONS FOR p | D(B)

In this section, we consider the contributions of terms Ap and Cp for fixed p where p | D(B). To do this, we use the analysis on pp. 214–215 of [3]. For ¯ p ) of the formal convenience, let Cˆi denote the base change to W = W (F completion of Z(ti ) for i = 1, 2, along its special fiber, and write Cˆ = Cˆ1 ×Aˆ Cˆ2 =

(7.6.1)

a

CˆT ,

T

as in (8.27) and (8.28) of [3]. Here T ranges over all T ∈ Sym2 (Z)∨ ≥0 with diag(T ) = (t1 , t2 ). Now [3], (8.30), gives, as already used in Section 6.2, a description of CˆT as a subset of a quotient space, (7.6.2)





ˆ • × H(Ap )/K p . CˆT ,→ H 0 (Q)\ V 0 (Q)2T × Ω f

By strong approximation, we have H 0 (Af ) = H 0 (Q)H 0 (Qp )K p ,

(7.6.3)

so that, since Γ0 = H 0 (Q) ∩ H 0 (Qp )K p , the quotient on the right side in (7.6.2) can be written as 



ˆ• . Γ0 \ V 0 (Q)2T × Ω

(7.6.4)

Recalling the incidence relations given after (8.30) in [3], which describe the image of CˆT in the right side of (7.6.2), we have (7.6.5)

CˆT

'

X

Γ0y \Z • (j).

]∈(L0 [p−1 ])2

y=[y1 ,y2 Q(y)=T mod Γ0

Here, as in [3], j is the Zp -span of the pair (j1 , j2 ) of endomorphisms associated to y = (y1 , y2 ), of the model p-divisible group X. Recall from [3] that the quantity (7.6.6)

νp (T ) = 2 χ(Z(j), OZ(j1 ) ⊗L OZ(j2 ) ),

232

CHAPTER 7

(see (7.2.11) above) is finite provided det(T ) 6= 0 and gives twice the full intersection number (Z(j1 ), Z(j2 )) of the cycles Z(j1 ) and Z(j2 ) in the ˆ [3], Theorem 6.1. We can decompose this quantity further Drinfeld space Ω, by writing Z(ji ) = Z hor (ji ) + Z ver (ji )

(7.6.7)

for the vertical and horizontal components. Then, let (7.6.8)

G := OZ hor (j1 ) ⊗L OZ hor (j2 ) .

For det(T ) 6= 0, the quantity νphor (T ) := 2 χ(Z(j), G)

(7.6.9)

is finite and gives twice the intersection number (Z hor (j1 ), Z hor (j2 )) of the horizontal parts. Remark 7.6.1. It is asserted here implicitly that the right-hand side of (7.6.9) only depends on the matrix T . To see this, note that if a pair of special endomorphisms (j10 , j20 ) has the same matrix of inner products as (j1 , j2 ), then it follows from Witt’s theorem that there is an element g ∈ GL2 (Qp ) which carries the pair (j1 , j2 ) into the pair (j10 , j20 ). But then g induces an ˆ which carries the pair of special cycles [Z(j1 ), Z(j2 )] automorphism of Ω 0 0 into [Z(j1 ), Z(j2 )], and this automorphism preserves the intersection numbers χ(Z(j), OZ hor (j1 ) ⊗L OZ hor (j2 ) ), resp. χ(Z(j), OZ hor (j10 ) ⊗L OZ hor (j20 ) ). We now consider Ap =

h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) ip .

X c1 6=c2

By the same method as in the case where t1 t2 is not a square (see (7.2.7)), i.e., decomposing according to fundamental matrices, we have, Proposition 7.6.2. Ap =

X

Ap (T ),

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 )

where only T with det(T ) 6= 0 occur, and where Ap (T ) =

X c1 6=c2

|ΛT (c1 , c2 )| · νphor (T ) log(p),

233

AN INNER PRODUCT FORMULA

Here, for any T ∈ Sym2 (Z)∨ , (7.6.10) ΛT (c1 , c2 ) = Γ0 \{ y = [y1 , y2 ] ∈ (L0 [p−1 ])2 

| Q(y) = T, type(y1 , y2 ) = (c1 , c2 ) } . 

For y ∈ L0 [p−1 ], type(y) = c means 1 Q[y] ∩ L0 [p−1 ] = a Z[ ]-order of conductor c in Q[y]. p

(7.6.11)

Note that c is prime to p. In particular, the condition c1 6= c2 implies that y1 and y2 are not colinear, and hence det(T ) 6= 0. Also, the factor 2 occurs in the definition (7.6.9) of νphor (T ) since two ‘sheets’ Z(j) occur in the quotient Γ0y \Z • (j); see [3], (8.37) and (8.40). Next we compute (7.6.12) Cp = h Zbhor (t1 , v1 ), Z ver (t2 ) ip + h Z ver (t1 ), Zbhor (t2 , v2 ) ip + h Z ver (t1 ), Z ver (t2 ) ip . To do this, note that the analysis on pp. 214–215 of [3] leading up to formula (8.30) makes no use of the fact that the given pair of special endomorphisms span a rank 2 module over Zp . Thus, the same argument applies in the present situation. We write Cˆi = Cˆih + Cˆiv

(7.6.13)

for the vertical and horizontal components. Then, using the intersection calculus of Section 4 of [3], we have (7.6.14)

X Cp = (Cˆ1h , Cˆ2v )p + (Cˆ1v , Cˆ2h )p + (Cˆ1v , Cˆ2v )p = χ( CˆT , F ), log(p) T

where (7.6.15)

F = OCˆh ⊗L OCˆv + OCˆv ⊗L OCˆh + OCˆv ⊗L OCˆv . 1

2

1

2

1

2

The quantity (7.6.16)

Cp (T ) := χ( CˆT , F ) · log(p),

is finite for any T with diag(T ) = (t1 , t2 ) and depends only on T ; see Remark 7.6.1. Proposition 7.6.3. For det(T ) 6= 0 with diag(T ) = (t1 , t2 ), Cp (T ) = |ΛT | · (νp (T ) − νphor (T )) log(p),

where |ΛT | is as in (7.2.10).

234

CHAPTER 7

Proof. The condition det(T ) 6= 0 allows us the make essentially the same analysis as on pp. 215–216 of [3]. Using (7.6.5), we have (7.6.17)

Cp (T ) = log(p)

X

χ(Γ0y \ Z • (j), F).

y∈(L0 [p−1 ])2 Q(y)=T mod Γ0

The stabilizer of y in Γ0 is Γ0 ∩ Z 0 (Q), where Z 0 (Q) is the center of Thus Γ0 ∩ Z 0 (Q) ' Z[p−1 ]× ' {±1} × Z, and the generator of the infinite factor of this group acts on Z • (j) by translating the ‘sheet’ by 2. The contribution of y to (7.6.17) is then H 0 (Q).

(7.6.18)

L e−1 y · 2 · χ(Z(j), OCˆ1 ⊗ OCˆ2 ) − χ(Z(j), G)




hor = e−1 y · (νp (T ) − νp (T )).

Here the 2 arises from the two sheets, as in (8.40) of [3], and ey = 2 is the order of the remaining part of the stabilizer of y. Finally, we determine the contribution of the two singular T ’s which occur in the sum (7.6.14). Proposition 7.6.4. For a matrix T ∈ Sym2 (Z)∨ with diag(T ) = (t1 , t2 ) and det(T ) = 0, i.e., for 



t m T = 1 , m t2

m 2 = t 1 t2 ,

write t1 = n1 t and t2 = n2 t with (n1 , n2 ) = 1, and let 4t√= n2 d, where −d is the discriminant of the quadratic extension k = Q( −t). Let k = ordp (n) and let χ = χd (p). √ (i) If p is ramified or inert in Q( −t), then Here,

Cp (T ) =

1 deg Z(t)Q · ν˜p (T ) · log(p). 2

ν˜p (T ) = ordp (t1 t2 )+2(1+χ−ordp (d/4))− √ (ii) If p is split in Q( −t), then

 k −1)   (p+1)(p p−1  2

pk+1 −1 p−1

if p is inert,

if p is ramified.

Cp (T ) = δ(d; D(B)/p) · H0 (t; D(B)) · ν˜p (T ) · log(p),

235

AN INNER PRODUCT FORMULA

with δ(d; D(B)/p) and H0 (t; D(B)) given by (7.6.21) and (7.6.22), and ν˜p (T ) = −2 · (pk − 1).

Proof. Since in this case the components y1 and y2 of y with Q(y) = T , and hence j1 and j2 , are colinear, the analysis is nearly the same as that dealt with in Section 11 of [4] for the cycle given by a single special endomorphism. More precisely, writing t1 = n21 t and t2 = n22 t, with n1 and n2 −1 relatively prime, we have n2 y1 = n1 y2 and we let y = n−1 1 y 1 = n2 y2 , −1 and similarly for the corresponding j1 , j2 we let j = n−1 1 j1 = n2 j2 . There will be three √ cases depending on whether p is inert, ramified, or split in the field k = Q( −t). Let (7.6.19) α = min{ordp (t1 ), ordp (t2 )} and β = max{ordp (t1 ), ordp (t2 )}. In particular, α = ordp (t) = 2k + ordp (d/4) √ and α + β = ordp (t1 t2 ). If p is either inert or ramified in k = Q( −t), then the discussion of the first part of Section 11 of [4] can be applied, so that the intersection number attached to CˆT will be given by (7.6.20) δ(d; D(B)) · H0 (t; D(B)) · (Z hor (j1 ), Z ver (j2 )) 

+ (Z ver (j1 ), Z hor (j2 )) + (Z ver (j1 ), Z ver (j2 )) , 

where the effect of the ‘sheets’ is taken into account; see Lemma 11.4 of [4]. Here (7.6.21)

δ(d; D) =

Y

(1 − χd (`))

`|D

and (7.6.22)

H0 (t; D) =

h(d) w(d)

X c|n (c,D)=1

c

(1 − χd (`)`−1 ).

Y `|c

Recall that by [4], Proposition 9.1, (7.6.23)

deg Z(t)Q = 2 δ(d; D(B)) H0 (t; D(B)).

The sum of intersection numbers on the right-hand side of (7.6.20) can be computed using the results of [3], for p 6= 2, and of the Appendix to Section 11 of [4]. Recall from the appendix to Chapter 6 that, for any j, the vertical cycle Z ver (j) is determined by data (S, µ), where S = S(j) is a subset of the

236

CHAPTER 7

building B and µ = µ(j) ≥ 0, by the formula (7.6.24)

Z ver (j) =

X

mult[Λ] (S, µ) P[Λ] ,

[Λ]

where the multiplicity function is given by (7.6.25)

mult[Λ] (S, µ) = max{µ − d([Λ], S), 0}.

Write q(j) = j 2 =  pα , with α = ordp (q(j)), and let kp = Qp ( q(j)).9 For p inert in kp , we have S = [Λ0 ] and, p

(7.6.26)

µ=

 α

if p = 6 2,

2 α 2

+ 1 if p = 2.

For p ramified in kp , we have S = [[Λ0 , Λ1 ]], the closure of an edge, and

(7.6.27)

µ=

 α−1    2

if p 6= 2,

α 2   α−1 2

if p = 2, and α is even, if p = 2, and α is odd.

For p split in kp , we have S = A, an apartment in B, and (7.6.28)

µ=

 α

if p 6= 2,

2 α 2

+ 1 if p = 2.

Now suppose that j1 and j2 are colinear special endomorphisms, such that {ordp (q(j1 )), ordp (q(j2 ))} = {α, β}, with α ≤ β. The resulting pair of vertical cycles Z ver (j1 ), Z ver (j2 ) is determined by the collection of data (S, µ1 , µ2 ). If p is either inert or ramified, the data (S, µ1 , µ2 ) coincides with one considered in [3] for a pair of anticommuting special endomorphisms j10 and j20 , and so, by the chart (6.15) in [3], (Z ver (j1 ), Z ver (j2 )) =

 µ −1 −(p + 1) pp−1

2−2



9

pµ+1 −1 p−1

if χd (p) = −1, if χd (p) = 0,

Note that, if j arises from a global special endomorphism y with Q(y) = −y 2 = t, then q(j) = −t. The same change of sign occurs in [3] and in the appendix to Chapter 6.

237

AN INNER PRODUCT FORMULA

where µ = min{µ1 , µ2 }. Adding the quantity (7.6.29)

(Z hor (j1 ), Z ver (j2 )) + (Z ver (j1 ), Z hor (j2 )) = 2µ1 + 2µ2 ,

and recalling (7.6.17), (7.6.26), and (7.6.27), we get the claimed value. Finally, we consider the case in which p splits in k, so that there are no horizontal components. The situation is now like the one considered after Lemma 11.4 in [4]. By (11.19) and (11.20) of [4], the intersection number attached to CˆT will be given by (7.6.30)

δ(d; D(B)/p) · H0 (t; D(B))

times the intersection multiplicity of h(y)Z i\Z(j). To compute this multiplicity, recall that the configuration Z(j) in question is now determined by data (S, µ1 , µ2 ), where S = A is an apartment. We use the following easy projection formula: ˆ → ∆\Ω ˆ be the natural projection, where ∆ = Lemma 7.6.5. Let pr : Ω Z ˆ and Z2 ⊂ Ω ˆ be divisors such that the intersections h(y) i. Let Z1 ⊂ ∆\Ω of the supports |Z1 | ∩ |pr∗ (Z2 )| and |pr∗ (Z1 )| ∩ |Z2 | are subsets of the special fibers proper over the base. Then (Z1 , pr∗ (Z2 ))∆\Ωˆ = (pr∗ (Z1 ), Z2 )Ωˆ . ˆ and let Z2 be a fundaNow take Z1 to be the image of Z(j1 ) in ∆\Ω mental domain for the action of ∆ on Z(j2 ). Since ∆ acts by translation by 2 along the fixed apartment, we can take Z2 to be the collection of P[Λ] ’s associated to two neighboring vertices on the apartment and to the set of all vertices joined to them by geodesics running away from the apartment and having a distance at most µ2 from them. The P[Λ] ’s occur with the multiplicity associated to Z(j2 ), i.e., (7.6.31)

mult(P[Λ] ) = mult[Λ] (j2 ) = µ2 − d([Λ], B j ).

We then compute the sum of the intersection numbers of components in Z(j1 ) and Z2 , as in the proof of Theorem 6.1 of [3]. The total is (7.6.32)

−2( pµ − 1),

where µ = min{µ1 , µ2 }, and we get the claimed result from (7.6.28).

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CHAPTER 7

7.7 CONTRIBUTIONS FOR p - D(B)

In this section we compute the quantity (7.7.1)

Ap =

X

h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) ip

c1 ,c2 c1 |n1 n, c2 |n2 n c1 = 6 c2

of (7.4.23) in the case p - D(B). Note that (7.7.2) h Zbhor (t1 : c1 , v1 ), Zbhor (t2 : c2 , v2 ) ip = ( Z hor (t1 : c1 ), Z hor (t2 : c2 ) )p · log(p), where ( , )p is the geometric intersection number at points in the fiber Mp .

√ Lemma 7.7.1. (i) If p - D(B) is not split (resp. split) in kt = Q( −t), ¯ p ) are supersingular (resp. ordinary). then all points of Z hor (t : c)(F hor ¯ p ) corresponds to (A, ι, y), and let (ii) Suppose that x ˜ ∈ Z (t : c)(F ψy : kt ,→ End(A, ι) ⊗Z Q,

√

−t 7→ y.

Then, the order ψy−1 (End(A, ι)) in kt has conductor c0 , where c = c0 ps with p - c0 . (iii) Suppose that p is split in kt . Then the embedding ψy in (ii) is an isomorphism, and V (A, ι) ⊗Z Q ' { x ∈ kt | tr(x) = 0 }

is one-dimensional. For c1 = c01 ps1 and c2 = c02 ps2 with p - c0i , c01 6= c02

=⇒

( Z hor (t1 : c1 ), Z hor (t2 : c2 ) )p = 0.

¯ p ) corresponding to (A, ι) be a point in the image Proof. (i) Let x ∈ M(F hor ¯ p ). Then the action of OB ⊗ Zp ' M2 (Zp ) shows that the of Z (t : c)(F p-divisible group of (A, ι) is isomorphic to X 2 , where X is a p-divisible ˆ m × Qp /Zp group of dimension 1 and height 2. If p splits in kt , then X ∼ G and conversely, which proves (i). To prove (ii), let x ˜ be the specialization of a point x ˜ of Z hor (t : c) with ¯ p and fracvalues in a complete discrete valuation ring with residue field F tion field of characteristic 0. If x ˜ corresponds to (A , ι, y) then the order

239

AN INNER PRODUCT FORMULA

ψy−1 (End(A , ι)) in kt has conductor c. We have the commutative diagram ψy

kt ,→ End(A , ι) ⊗ Q (7.7.3)

∩ ↓

k ψy

kt ,→ End(A, ι) ⊗ Q . Hence it suffices to show that the inclusion End(A , ι) ,→ End(A, ι) induces an isomorphism (7.7.4)

ψy−1 (End(A , ι) ⊗ Z[p−1 ]) = ψy−1 (End(A, ι) ⊗ Z[p−1 ]).

This comes down to showing that any endomorphism α of (A , ι) whose reduction modulo p is divisible by a prime number ` 6= p in End(A, ι) is itself divisible by `. But α is divisible by ` if and only if α annihilates the group scheme A [`]. Since this group scheme is e´ tale, the assertion follows. (iii) The first assertion of (iii) is trivial. To prove the second, suppose that (A, ι) is a point of intersection of Z(t1 : c1 ) and Z(t2 : c2 ) in the fiber Mp for a prime p split in kt . Thus, (A, ι) has special endomorphisms y1 and y2 with Q(y1 ) = t1 and Q(y2 ) = t2 . By (ii), the orders ψy−1 (End(A, ι)) and 1 0 and c0 respectively. On the other hand, ψy−1 (End(A, ι)) have conductors c 1 2 2 by the first part of (iii), ψy2 = ψ±y1 , so that c01 = c02 . Remark 7.7.2. If p is not split in kt , and if (A, ι) corresponds to a point ¯ p )ss , then the space V (A, ι) ⊗ Q has dimension 3, so that there x ∈ M(F can be pairs y = [y1 , y2 ] of special endomorphisms with nonsingular fundamental matrix T = Q(y). The corresponding orders ψy−1 (End(A, ι)) and 1 0 and c0 with c0 6= c0 , where c | n n ψy−1 (End(A, ι)) can have conductors c 1 1 1 2 1 2 2 and c2 | n2 n. In particular, ( Z(t1 : c1 ), Z(t2 : c2 ) )p can be nonzero in this case. By (i) of the lemma, we can write (7.7.5) ( Z(t1 : c1 ), Z(t2 : c2 ) )p =

X

( Z(t1 : c1 ), Z(t2 : c2 ) )x ,

¯ p )• x∈Mp (F

where (7.7.6)

•=

 s.s.

if p is not split in k,

ord

if p is split in k.

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CHAPTER 7

¯ p )• corresponding to (A, ι), we now describe the For a point x ∈ Mp (F local structure of a cycle Z hor (t : c) in the formal neighborhood (M)∧ x of 2. x. Let y ∈ V (A, ι) be a special endomorphism with Q(y) = t = −y √ The corresponding embedding kt = Q( −t) → End(A, ι)Q defines an embedding √ (7.7.7) ψ = ψy : Qp ( −t) −→ End(A(p), ι) ⊗Zp Qp , where A(p) is the p-divisible group of A. Note that the action of the idempotents in OB ⊗ Zp ' Mp (Zp ) gives a decomposition A(p) ' X 2 , where X is the p-divisible group of a supersingular (resp. ordinary) elliptic √ curve ¯ p when p is not split (resp. split) in kt . Since ψ embeds Qp ( −t) over F into End(X)Qp , we may apply the theory of quasicanonical liftings of Gross [1]. In the case in which p is split in kt , Gross’s theory amounts to the clas¯ p , as is pointed sical Serre-Tate theory for an ordinary elliptic curve over F out at the end of [1], so we will use the same terminology in the two cases. For s ∈ Z≥0 , let Ws (ψ) be the quasi-canonical divisor of level s associated to ψ. Recall10 that Ws (ψ) is a reduced, irreducible, regular divisor in the spectrum of the formal completion of M at x, such that the pullback of the universal p-divisible group on M to Ws (ψ) has as its endomorphism algebra the order Zp + ps Okp of conductor ps in kp . For example, when p splits in k, Okp ' Zp ⊕Zp , the order of conductor ps is {(a, b) ∈ Z2p | a ≡ b mod ps }, and (7.7.8)

Ws (ψ) ' Spf W [T ]/Φs (T ),

where Φs (T ) is the cyclotomic polynomial whose roots are the primitive ¯ p ), and let Ms be ps -th roots of 1. Let M be the quotient field of W = W (F the Galois extension of M over which the quasi-canonical liftings of level s are defined. Recall that the extension Ms /M is totally ramified, and, in the case when p is split in kt , Ms is the extension generated by a primitive ps th root of unity. Also note that (7.7.9)

m0 (p) := |M0 : M | =

 2

if p is ramified in kt ,

1

otherwise.

We write (7.7.10)

Ws (ψ) = Spf Ws ,

where Ws is the integral closure of W in Ms . 10

Note that the fact that the canonical morphism from Ws (ψ) to (M)∧ x is a closed immersion follows from [1], Proposition 5.3, (3).

241

AN INNER PRODUCT FORMULA

¯ p ) corresponding to (A, ι, y). Let Lemma 7.7.3. Suppose that x ˜ ∈ Z(t)(F ∧ Z(t)x˜ be the formal completion of Z(t) at x ˜. Then, for 4t = n2 d, as usual, there is an equality of formal schemes ordp (n)

Z(t)∧ x ˜ =

[

Ws (ψ),

where ψ = ψy : kt,p ,→ End(A(p), ι) ⊗Zp Qp is the embedding corresponding to y. s=0

Proposition 7.7.4. Let i : Z(t) → M be the natural morphism, defined by (A, ι, y) 7→ (A, ι). Then i is unramified, and, for x ∈ Mp corresponding to (A, ι), there is an equality of formal divisors11 i∗ Z(t)


∧ x

ordp (n)

X

=

X

Ws (ψ).

y∈V (A,ι) s=0 Q(y)=t

Similarly, if c = c0 ps with p - c0 , then i∗ Z hor (t : c)


∧ x

=

X

Ws (ψ).

y∈V (A,ι) Q(y)=t type(y)=c0

Here, for y ∈ V (A, ι), type(y) = c0 if the order ψy−1 (End(A, ι)) in kt has conductor c0 .
Since the intersection number Ws1 (ψy1 ), Ws2 (ψy2 ) of two distinct quasi-canonical divisors is always finite, we obtain

¯ p )• be a point corresponding to (A, ι). For Corollary 7.7.5. Let x ∈ Mp (F c1 6= c2 , write c1 = c01 ps1 and c2 = c02 ps2 , where p - c01 and p - c02 , as above. Then ( Z hor (t1 : c1 ), Z hor (t2 : c2 ) )x =

X

X

e−1 Ws1 (ψy1 ), Ws2 (ψy2 ) . y


T y=[y1 ,y2 ]∈V (A,ι)2 diag(T )=(t1 ,t2 ) Q(y)=T type(y)=(c01 ,c02 )

11

Note that, elsewhere in this paper, we have been omitting i∗ from the notation, often writing Z(t) for i∗ Z(t).

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CHAPTER 7

Taking the sum on c1 , c2 , and x and regrouping according to the fundamental matrices T , we obtain Corollary 7.7.6. X

Ap = where Ap (T ) =

Ap (T ),

T diag(T )=(t1 ,t2 )

X

X

¯ p )• x∈M(F

c1 ,c2 c1 |n1 n, c2 |n2 n c1 6=c2

e−1 Ws1 (ψy1 ), Ws2 (ψy2 ) · log(p). y

X




y=[y1 ,y2 ]∈V (A,ι)2 Q(y)=T type(y)=(c01 ,c02 )

For p inert or ramified in k, both singular and nonsingular matrices T can make a nonzero contribution to Ap . For nonsingular T ’s, the expression for Ap (T ) given in Corollary 7.7.6 will be just what is needed for the comparison to be made in Section 7.9, so it only remains to determine the contribution of singular T ’s. On the other hand, for p split in kt , only singular T ’s can contribute to Ap , due to (iii) of Lemma 7.7.1. Thus, for the remainder of this section, we consider the quantity Ap (T ) for T ∈ Sym2 (Z)∨ with diag(T ) = (t1 , t2 ) and det(T ) = 0. Note that there are precisely two such T ’s. When det(T ) = 0, a pair of special endomorphisms y ∈ V (A, ι)2 with fundamental matrix Q(y) = T has colinear components y1 and y2 . By (ii) of Lemma 7.7.1, it follows that c01 = type(y1 ) = type(y2 ) = c02 . Since c1 | n1 n and c2 | n2 n, and (n1 , n2 ) = 1, we must have c01 = c02 = c0 with c0 | n. Thus, we obtain, (7.7.11) Ap (T ) =

X

X

X

¯ p )• c0 y=[y1 ,y2 ]∈V (A,ι)2 x∈M(F Q(y)=T type(y)=(c0 ,c0 ) ordp (n1 n) ordp (n2 n)

e−1 y

X

X

s1 =0 s2 =0 s1 6=s2




Ws1 (ψy1 ), Ws2 (ψy2 ) · log(p)

243

AN INNER PRODUCT FORMULA ordp (n1 n) ordp (n2 n)

= |ΛT | ·

X

X




Ws1 (ψ), Ws2 (ψ) · log(p).

s1 =0 s2 =0 s1 6=s2

Here c0 runs over divisors of n which are prime to p D(B), and (7.7.12)

X

|ΛT | =

e−1 y

X

¯ p )• y∈V (A,ι)2 x∈M(F Q(y)=T

X

=

X

e−1 y

¯ p )• y∈V (A,ι) x∈M(F Q(y)=t

=: |Λt |. Here, we have used the bijection given by the map y 7→ [n1 y, n2 y] = y and that ey = ey when y is mapped to y. √ Proposition 7.7.7. (i) For an embedding ψ : Qp ( −t) → End0 (X), and for s = min{s1 , s2 }, (Ws1 (ψ), Ws2 (ψ)) = |Ms : M | = m0 (p) · (ii)

 1 ps−1 (p − χ (p)) d

if s = 0, if s ≥ 1.

ordp (n)

deg Z(t)Q = |Λt | ·

X

|Ms : M |

s=0



= |Λt | · m0 (p) ·

pordp (n) − 1 1 + (p − χd (p)) p−1



,

where |Λt | is given by (7.7.12) and m0 (p) = |M0 : M |, as in (7.7.9).

Proof. The closed embedding of Ws (ψ) into M∧ x is given by a homomorphism of complete W -algebras, (7.7.13)

W [[T ]] −→ Ws , T 7−→ πs

where πs is a uniformizer of Ws , provided that |Ms : M | > 1. In this case, since Ms /M is totally ramified we have (7.7.14)

Ws = W [T ]/(Ps (T )),

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CHAPTER 7

where Ps (T ) is the minimal polynomial of πs over M and is an Eisenstein polynomial of degree |Ms : M |. Now for s1 < s2 , and if |Ms1 : M | > 1, we obtain (Ws1 (ψ), Ws2 (ψ)) = length Ws1 ⊗W [[T ]] Ws2 (7.7.15)

= length (W [T ]/(Ps1 (T ))) ⊗W [T ] (W [T ]/(Ps2 (T ))) = length Ws2 /(Ps1 (πs2 )) = |Ms1 : M |.

The situation is a little different when Ms1 = M . In this case, in (7.7.13), T ∈ W [[T ]] is sent to some element a ∈ W , and the same argument as before works provided |Ms2 : M | > 1. There is one exceptional case which occurs when p = 2 is split in k, s1 = 0 and s2 = 1. In this case, M0 = M1 = M . Then one checks that the maps W [[T ]] → W0 and W [[T ]] → W1 send T to 0 and −2, respectively, and hence W0 ⊗W [[T ]] W1 ' W/2W , so that the formula in the proposition is again true. Part (ii) follows from Lemma 7.7.3 and (7.7.12), since ordp (n)

(7.7.16)

deg Z(t)Q =

X

X

e−1 y

¯ p )• y∈V (A,ι) x∈M(F Q(y)=t

X

|Ms : M |,

s=0

where the inner sum is independent of y.

Corollary 7.7.8. For T ∈ Sym2 (Z)∨ with diag(T ) = (t1 , t2 ) and det(T ) = 0,   Ap (T ) = deg Z(t)Q · ordp (n1 n2 ) + 2ordp (n) log(p)

p−χ − 2 |Λt | · m0 (p) · p−1

where χ = χd (p) and k = ordp (n). Proof. Setting

k = min{ ordp (n1 n), ordp (n2 n) } and ` = max{ ordp (n1 n), ordp (n2 n) },

pk − 1 kp − p−1 k

!

,

245

AN INNER PRODUCT FORMULA

we have k = ordp (n) and ` − k = ordp (n1 n2 ). Then, by (7.7.11) and Proposition 7.7.7, (7.7.17)

Ap (T ) = |Λt | · µp (k, `) log(p),

where (7.7.18)

µp (k, `) =

k ` X X

|Mmin{s1 ,s2 } : M |

s1 =0 s2 =0 s1 6=s2

= (` − k)

k X

|Ms : M | + 2

s=0

k−1 X

|Ms : M | (k − s)

s=0

= [ (` − k) + 2k ]

k X

|Ms : M | − 2

s=0

k X

|Ms : M | s.

s=0

7.8 COMPUTATION OF THE DISCRIMINANT TERMS

In this section we will calculate the discriminant term occurring in Corollary 7.5.3: (7.8.1)

dZ hor (t) =

X

dZ hor (t:c) =

X

log |D(c)−1 : O(c)|.

c|n

c|n

Here O(c) is the ring of regular functions on Z hor (t : c) and D(c)−1 = { α ∈ O(c) ⊗Z Q | tr(α · O(c)) ⊂ Z }.

(7.8.2)

˜ Let Z˜hor (t : c) be the normalization of Z hor (t : c) and let O(c) be its ˜ ring of regular functions. The inclusion O(c) ⊂ O(c) corresponds to the normalization morphism π : Z˜hor (t : c) −→ Z hor (t : c).

(7.8.3)

˜ −1 = {α ∈ O(c) ⊗Z Q | tr(α · O(c)) ˜ ˜ Let D(c) ⊂ Z} be the dual of O(c) with respect to the trace form. Then ˜ ˜ −1 ⊂ D(c)−1 O(c) ⊂ O(c) ⊂ D(c)

(7.8.4)

˜ ˜ −1 |, so that and |O(c) : O(c)| = |D(c)−1 : D(c) (7.8.5)

˜ ˜ −1 : O(c)|. ˜ dZ hor (t:c) = 2 · log |O(c) : O(c)| + log |D(c)

We then write (7.8.6)

dZ hor (t) =

X p

2δp (t) + ∂p (t)




log(p),

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CHAPTER 7

where (7.8.7)

δp (t) =

X

˜ ordp |O(c) : O(c)|,

c

and (7.8.8)

∂p (t) =

X

˜ −1 : O(c)|. ˜ ordp |D(c)

c

To calculate δp (t), recall Serre’s δ-invariant [7] associated to a point x ˜∈ ¯ : c)(Fp ), given by

Z hor (t

(7.8.9)

δx˜ = length ( π∗ OZ˜hor (t:c) /OZ hor (t:c) )x˜ .

Then, δx˜ = 0 unless x ˜ is a singular point of Z hor (t : c). Furthermore (7.8.10)

δp (t) =

X

˜ ordp |O(c) : O(c)| =

X

X

δx˜ .

¯p ) c|n x ˜∈Z hor (t:c)(F

c|n

To calculate δx˜ , we use the following elementary fact.

Lemma 7.8.1. Let (X, x) be the spectrum of a complete regular local ring R of dimension 2. Let C be a reduced divisor on X through x and let C=

n X

Ci

be the decomposition of C into its formal branches through x, i.e., Ci is irreducible, for all i. We assume that Ci is regular at x, ∀i. Let i=1

π : C˜ −→ C

n ` be the normalization morphism, i.e., C˜ = Ci . Let i=1

δx = length (π∗ OC˜ /OC )x

be the Serre invariant of C at x. Then 2δx =

X

(Ci , Cj )x .

Proof. We proceed by induction on n, the case n = 1 being trivial since i6=j

then δ = 0. Let C 0 = morphism factors as (7.8.11)

n−1 P

Ci , so that C = C 0 + Cn . Then the normalization

i=1

0

` π1 π C˜ −→ C 0 Cn −→ C.

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AN INNER PRODUCT FORMULA

Let f 0 = 0, resp. fn = 0, be the equations of C 0 , resp. Cn . Then f 0 and fn are relatively prime elements of R. We have the following diagram of local rings at x with exact rows and columns: 0 x  0 →

OxC 

(7.8.12)

0 →

R x

0 x 

0 x 

→ OC 0 ⊕ x OCn

→

→

→





R⊕ xR

∆ x

→ 0

R x

→ 0







0, f ) → 0 0 → (f 0xfn ) → (f 0 ) ⊕ n x (fn ) → (f x







0

0

0

The first row shows that length ∆ = length (π1∗ (OC 0 tCn )/OC )x . The last column shows that length ∆ = (C 0 , Cn )x . By induction hypothesis, (7.8.13)

length (π∗0 (OC˜ )/OC 0 )x =

X

(Ci , Cj )x .

1≤i 0, we have (7.9.8)Z 1 1 Ξ(t, t−1 tr(T v)) dµ = deg Z(t)Q · J(4π(t1 v1 + t2 v2 )). 2 M(C) 2

We can now begin the comparison. First, comparing Corollary 7.5.6 and Proposition 7.5.7 with (7.2.16) for p = ∞, it is immediate that (7.9.9) A∞ + B∞ X

=

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 ) det(T )6=0 Diff(T,B)={∞}

X

=

 X

|ΛT (c1 , c2 )| +

X



|ΛT (c, c)|

· ν∞ (T, v)

c

c1 6=c2

b Z(T, v),

(Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 ) det(T )6=0 Diff(T,B)={∞}

as required. Next suppose that p | D(B), and that T ∈ Sym2 (Z)∨ , with diag(T ) = (t1 , t2 ), det(T ) 6= 0, and Diff(T, B) = {p}. Then, Propositions 7.6.2 and 7.6.3 together with (ii) of Proposition 7.8.2 yield (7.9.10) Ap (T ) + 2 δp (T ) log(p) + Cp (T ) =

X

|ΛT (c1 , c2 )| · νphor (T ) log(p) +

c1 6=c2

X

|ΛT (c, c)| · νphor (T ) log(p)

c

+ |ΛT | · (νp (T ) − νphor (T ) log(p)


= |ΛT | · νp (T ) log(p) b = Z(T, v).

Finally, suppose that p - D(B), and that T ∈ Sym2 (Z)∨ , with diag(T ) =

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CHAPTER 7

(t1 , t2 ), det(T ) 6= 0 and Diff(T, B) = {p}. Then, by Corollary 7.7.6 (7.9.11) Ap (T ) ordp (n1 n) ordp (n2 n)

=

X

|ΛT (c01 , c02 )|

·

c01 6=c02

X

X

s1 =0

s2 =0




Ws1 (ψ1 ), Ws2 (ψ2 ) · log(p)

ordp (n1 n) ordp (n2 n)

+

X

0

0

|ΛT (c , c )| ·

X

X




Ws1 (ψ1 ), Ws2 (ψ2 ) · log(p),

s1 =0 s2 =0 s1 6=s2

c0

whereas, by (i) of Proposition 7.8.2, (7.9.12) ordp (n)

2 δp (T ) log(p) =

X

|ΛT (c0 , c0 )| ·

X

(Ws (ψ1 ), Ws (ψ2 )) log(p).

s=0

c0

Thus, we have (7.9.13)

b Ap (T ) + 2δp (T ) log(p) = Z(T, v).

Note that identities (7.9.9), (7.9.10), and (7.9.13) account for all terms associated to nonsingular T ’s on the two sides. It remains to compare the global terms and terms associated to the two singular T ’s with diagonal (t1 , t2 ). First, there is the remaining part of B: (7.9.14) X B global = −2 hωˆ (Z hor (t)) + 2 ∂p (t) log(p) p

− degQ (Z(t)) · log(4D(B)) 

+ degQ (Z(t)) log(t1 v1 + t2 v2 ) − log(t1 t2 ) − log(v1 v2 )



− degQ (Z(t)) J(4π(t1 v1 + t2 v2 )). Next, there is the contribution of each of the two singular T ’s. For any p - D(B), Corollary 7.7.8 gives (7.9.15)

  Ap (T ) = deg Z(t)Q · ordp (n1 n2 ) + 2ordp (n) log(p)

p−χ − 2 |Λt | m0 (p) · p−1

pk − 1 kp − p−1 k

!

,

257

AN INNER PRODUCT FORMULA

where χ = χd (p) and k = ordp (n). For p | D(B), recalling Proposition 7.6.4 and the expression for hωˆ (Z ver (t)p ) in Lemma 7.9.1, we have for p inert or ramified in kt , (7.9.16) Cp (T ) = deg Z(t)Q log(p) ·

   12 ordp (t1 t2 ) −

(p+1)(pk −1) 2(p−1)

if p is inert,

  1 ord (t t ) − p 1 2 2

pk+1 −1 p−1

if p is ramified,

+ deg Z(t)Q log(p) · (1 + χ − ordp (d/4)) = −hωˆ (Z ver (t)p ) + deg Z(t)Q

1

2

ordp (t1 t2 ) − k − ordp (d/4) log(p). 

If p is split in kt , then the comparison of Proposition 7.6.4 and Lemma 7.9.1 shows that Cp (T ) = −hωˆ (Z ver (t)p ).

(7.9.17)

Taking into account that there are two singular T ’s and cancelling the terms which the expressions B global + 2 Ap (T ) + 2 Cp (T ) and 2 Z(T, v) have in common, we are left with (7.9.18) 2

X

∂p (t) log(p) − degQ (Z(t)) log(4 t1 t2 )

p

+2

X

deg Z(t)Q

p|D(B)

+2

X 

1

2

ordp (t1 t2 ) − k − ordp (d/4) log(p) 

deg Z(t)Q · ordp (n1 n2 ) + 2 ordp (n) 



p-D(B)

p−χ − 2 |Λt | m0 (p) · p−1

pk − 1 kpk − p−1

! 

,

on the intersection pairing side, and (7.9.19)

− deg Z(t)Q log(t),

on the genus two generating function side. Finally, we use the values for the discriminant terms ∂p (t) and compute the coefficient of each log(p) in (7.9.18).

258

CHAPTER 7

First suppose that p - D(B). We then have terms p−χ (7.9.20) ∂p (t) = |Λt | m0 (p) p−1 

pk − 1 kp −2 p−1 +

(7.9.21)

!

k

1−χ + k p−1



1 deg Z(t)Q · ordp (d), 2

  Ap (T ) = deg Z(t)Q · ordp (n1 n2 ) + 2 ordp (n) log(p)

p−χ − 2 |Λt | m0 (p) · p−1

pk − 1 kp − p−1 k

!

,

and (7.9.22)

−

1 deg Z(t)Q ordp (4t1 t2 ) 2 = − deg Z(t)Q ordp (n1 n2 ) + 2 ordp (n) + ordp (d/2) ], 

whose sum is deg Z(t)Q (7.9.23)

 1 pk+1 − χ pk + χ − 1 ordp (2) − ordp (d) − k|Λt | m0 (p) 2 p−1 



= deg Z(t)Q ordp (2) − 

=−



 1 ordp (d) − ordp (n) 2

1 deg Z(t)Q · ordp (t). 2

Here we have used (ii) of Proposition 7.7.7, (7.9.24)



deg Z(t)Q = |Λt | m0 (p)

pk − 1 1 + (p − χ) p−1



.

Since there are two such terms in (7.9.18), we get the required coincidence with the log(p) part of (7.9.19). Next suppose that p | D(B). We then have terms (7.9.25) (7.9.26) and

1 deg Z(t)Q · ordp (d), 2 1  deg Z(t)Q · ordp (t1 t2 ) − ordp (n) − ordp (d/4) , 2

259

AN INNER PRODUCT FORMULA

(7.9.27)

−

1 deg Z(t)Q · ordp (4t2 t2 ), 2

In this case, the sum is −

1 deg Z(t)Q · ordp (t), 2

so that we again have the claimed agreement. This finishes the proof of the identity ((?)) of Theorem C in the case in which t1 t2 is a square and t1 , t2 > 0.

7.10 THE CASE t1 , t2 < 0 WITH t1 t2 = m2

In this case, the classes b 1 , v1 ) = (0, Ξ(t1 , v1 )), Z(t b 2 , v2 ) = (0, Ξ(t2 , v2 )) ∈ CH d 1 (M) Z(t

only involve the Green functions. More precisely, the cycles Z(ti ) are empty and the functions Ξ(ti , vi ) are smooth on M(C). We then have b 1 , v1 ), Z(t b 2 , v2 ) i = h Z(t

1 2

Z

1 2

Z

=

=

M(C)

Ξ(t1 , v1 ) ∗ Ξ(t2 , v2 ) X

[Γ\D]

=

1

ξ(v12 x1 ) ∗ ξ(v22 x2 )

x1 ∈L x2 ∈L Q(x1 )=t1 Q(x2 )=t2

X T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 )

(7.10.1)

1

X

X

1 2

Z

X

[Γ\D]

1

1

ξ(v12 x1 ) ∗ ξ(v22 x2 )

x∈L2 Q(x)=T

|ΛT | · ν∞ (T, v)

T ∈Sym2 (Z)∨ diag(T )=(t1 ,t2 ) det T 6=0

Z

+ [Γ\D]

X x∈L2 Q(x)=T0

1

1

ξ(v12 x1 ) ∗ ξ(v22 x2 )

260

CHAPTER 7

X

=

b Z(T, v)

(Z)∨

T ∈Sym2 diag(T )=(t1 ,t2 ) det T = 6 0

Z

1

X

+ [Γ\D]

1

ξ(v12 x1 ) ∗ ξ(v22 x2 ),

x∈L2 Q(x)=T0

where, just as in (7.2.12), ν∞ (T, v) is given by (7.2.13) and |ΛT | is given by (7.2.15). Here the matrix T0 is one of the two singular matrices with diag(T0 ) = (t1 , t2 ). It remains to prove the following identity.

Proposition 7.10.1. For a singular matrix T0 with diag(T0 ) = (t1 , t2 ), and for v = diag(v1 , v2 ), b 0 , v) = −h Z(t, b t−1 tr(T v)), ω Z(T ˆi=

1 2

Z

1

X

[Γ\D]

1

ξ(v12 x1 )∗ξ(v22 x2 ).

x∈L2 Q(x)=T0

b 0 , v) is defined as in Section 6.4. Here recall that Z(T

Proof. As in Section 6.4, we write





t m T0 = 1 , m t2 and choose n1 and n2 relatively prime with t1 = n21 t, t2 = n22 t, m = n1 n2 t. Note that we have t−1 tr(T v) = v1 n21 + v2 n22 . Thus, by the results of [4] (see (iii) of Theorem 8.8), (7.10.2) −1 b −h Z(t,t tr(T v)), ω ˆ i = −2 δ(d; D(B)) H0 (t; D(B)) ∞ 1 3 1 −1 2 2 |t| 2 (v1 n21 + v2 n22 )− 2 e−4π|t|(v1 n1 +v2 n2 ) u u− 2 du. 4π 1 Now for any x in the sum in the last expression in the proposition, we can write x1 = n1 y and x2 = n2 y, for a unique vector y ∈ L with Q(y) = t. Using Lemma 11.4 of [2] and unwinding, we have

Z

×

(7.10.3)

1 2

Z [Γ\D]

=

1

X

1

ξ(v12 x1 ) ∗ ξ(v22 x2 )

x∈L2 Q(x)=T0

X y∈L(t) mod Γ

1 4

Z Γy \D

1

1

ξ(v12 n1 y, z) · ϕ(v22 n2 y, z) dµ(z),

261

AN INNER PRODUCT FORMULA

where ϕ is as in (3.5.5). The extra factor of 12 in the second line comes from the stack [Γ\D] in the first line. The integral here can be computed by the method used in the second part of the proof of Proposition 12.1 of [4]. To clean up the notation, we temporarily write v1 for v1 n21 and v2 for v2 n22 . As in (12.9) of [4], we shift y to a standard vector and choose polar coordinates z = reiθ in H. By Lemma 11.5 of [2], we can write the integrand as −Ei(−2πR1 ) · (2 R2 − 4v2 |t| −

1 ) · e−2πR2 , 2π

where 2v1 |t| sin2 (θ)

R1 =

and

R2 =

2v2 |t| . sin2 (θ)

Then, the integral becomes12 2 δy−1

=

1

Z + Γ+ y \D

8 δy−1

1

ξ(v12 y, z) · ϕ(v22 y, z) dµ(z)

log |(y)| ·

π 2

Z 0

4πv1 |t| −Ei − 2 sin (θ)

!

1 4v2 |t| cos2 (θ) − 2 2π sin (θ)

·

4πv2 |t| × exp − 2 sin (θ) =

4 δy−1

log |(y)| ·

∞ Z ∞

Z

e 1

× =

log |(y)| ·

Z

w

−1

!

· (sin2 (θ))−1 dθ



dw

1



4 δy−1

−4πv1 |t|rw

!

∞

1 4v2 |t|(r − 1) − 2π

−4π|t|(v1 w+v2 )

Z

e 1

∞



1

e−4πv2 |t|r (r − 1)− 2 dr

e−4π|t|r(v1 w+v2 )

0



×

1 4v2 |t|r − 2π



1

r− 2 dr w−1 dw.

The integral with respect to r here is −

12

1 3 1 · |t|− 2 (v1 w + v2 )− 2 v1 w, 4π

Note that in [4], dµ(z) =

1 2π

y −2 dx dy, whereas here we omit the

1 ; 2π

see Section 3.5.

262

CHAPTER 7

so we get the expression 1

− 4 δy−1 log |(y)| · |t|− 2 =

−4 δy−1

3 v1 (v1 w + v2 )− 2 dw 4π

e−4π|t|(v1 w+v2 )

1

− 12

log |(y)| · |t|

∞

Z

1 1 (v1 + v2 )− 2 4π

Z

∞

3

e−4π|t|(v1 +v2 )u u− 2 du,

1

where we have used the substitution u = (v1 w + v2 )/(v1 + v2 ) in the last step. Returning to (7.10.3), we obtain X y∈L(t) mod Γ

1

−δy−1 log |(y)| · |t|− 2 ×

1 1 (v1 n21 + v2 n22 )− 2 4π

Z

∞

2

2

3

2

2

3

e−4π|t|(v1 n1 +v2 n2 )u u− 2 du

1

1

= −2 δ(d; D(B)) H0 (t; D(B)) · |t|− 2 ×

1 1 (v1 n21 + v2 n22 )− 2 4π

Z

∞

e−4π|t|(v1 n1 +v2 n2 )u u− 2 du,

1

by Lemma 12.2 of [4]. 7.11 THE CONSTANT TERMS

In this section we prove the remaining cases of identity ((?)) of Theorem C. For clarity, we will write Zb2 (T, v) for a coefficient of the genus two generating function. Theorem 7.11.1. (i) For t1 6= 0,

b 1 , v1 ), Z(0, b v2 ) i = Zb2 (T, v), h Z(t

where T = diag(t1 , 0) and v = diag(v1 , v2 ). (ii)

b v1 ), Z(0, b v2 ) i = Zb2 (0, v), h Z(0,

where v = diag(v1 , v2 ).

Corollary 7.11.2. The constant c = − log(D(B)) and so 

hω ˆ, ω ˆ i = ζD(B) (−1) 2

ζ 0 (−1) 1 + 1 − 2C − ζ(−1) 2

X p+1 p|D(B)

p−1



· log(p) .

263

AN INNER PRODUCT FORMULA

Proof. Recall that, by (5.13) of [4], (7.11.1)

b v) = −ˆ Z(0, ω − (0, log(v) − c),

for the constant c given, as in (0.15) of [4], by the relation (7.11.2)  0 X p log(p)  1 ζ (−1) . deg(ˆ ω )·c = h ω ˆ, ω ˆ i −ζD(B) (−1) 2 +1−2C − 2 ζ(−1) p−1 p|D(B) Therefore, (7.11.3) b 1 , v1 ), Z(0, b v2 ) i = −h Z(t b 1 , v1 ), ω h Z(t ˆi−

1 degQ (Z(t1 ))(log(v2 ) − c) 2

and (7.11.4) b v1 ), Z(0, b v2 ) i = h ω h Z(0, ˆ, ω ˆi+

1 degQ (ˆ ω ) · ( log(v1 v2 ) − 2 c ). 2

On the other hand, in the sum on the right-hand side of ((?)), all contributions of nonsingular matrices T are zero by Remark 6.3.2. Therefore the only matrix T which contributes is the matrix T = diag(t1 , 0). But for T and v as in (i) of the theorem, we have t = t1 , n1 = 1, n2 = 0, and t−1 tr(T v) = v1 , so that, by (7.9.7), (7.11.5)
1 b 1 , v1 ), ω Zb2 (T, v) = −h Z(t ˆ i − degQ (Z(t1 )) · log(v2 ) + log(D(B)) . 2 Finally, for v = diag(v1 , v2 ), (7.11.6) Zb2 (0, v) = h ω ˆ, ω ˆ i+


1 degQ (ˆ ω ) · log(v1 v2 ) − c + log(D(B)) . 2

b 1 , v1 ), with t1 > 0, the function Now, for a fixed cycle Z(t

(7.11.7)

b 1 , v1 ), φ ˆ1 (τ2 ) i = h Z(t

X

b 1 , v1 ), Z(t b 2 , v2 ) i q t2 h Z(t 2

t2 ∈Z

is a modular form of weight 23 , as is the function (7.11.8)

φˆ2 (diag(τ1 , τ2 ))t1 ,

obtained by taking the t1 Fourier coefficient of φˆ2 (diag(τ1 , τ2 )). We have proved that, for any t2 6= 0, (7.11.9)

b 1 , v1 ), φ ˆ1 (τ2 ) it = φˆ2 (diag(τ1 , τ2 ))t ,t , h Z(t 2 1 2

264

CHAPTER 7

and hence, it follows that the constant terms must agree as well (!), i.e., (7.11.10)

b 1 , v1 ), Z(0, b v2 ) i = φ ˆ2 (diag(τ1 , τ2 ))t ,0 = Zb2 (T, v) h Z(t 1

for T = diag(t1 , 0) and v = diag(v1 , v2 ). Comparing (7.11.3) and (7.11.5), we conclude that c = − log(D(B)). Inserting this value into (7.11.4) and (7.11.6), we obtain (ii) of the Theorem. Since deg(ˆ ω ) = vol(M(C)) = −ζD(B) (−1), we obtain the corollary. The proof of Theorem C is now complete. Bibliography

[1] B. Gross, On canonical and quasi-canonical liftings, Invent. math. 84 (1986), 321–326. [2] S. Kudla, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646. [3] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223. [4] S. Kudla, M. Rapoport, and T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math. 140, 887–951. [5] J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992. [6] J.-P. Serre, Corps locaux, Hermann, Paris, 1968. [7]

, Groupes alg´ebriques et corps de classes, Hermann, Paris, 1959.

Chapter Eight On the doubling integral In this chapter, we obtain some information about the doubling zeta integral for the metaplectic cover G0A of SL2 (A). In the standard doubling integral, as considered in [5], one integrates the pullback to Spn (A) × Spn (A) of a Siegel Eisenstein series on Sp2n (A) against a pair of cusp forms f1 and f2 on Spn (A). If f1 and f2 lie in an irreducible cuspidal automorphic representation σ, the result is a product of a partial L-function LS (s + 12 , σ) of σ associated to the standard degree 2n + 1 representation of the L-group L Sp = SO 2n+1 and certain ‘bad’ local zeta integrals at the places in the n finite set S where the data is ramified. If f1 and f2 lie in different irreducible cuspidal automorphic representations, then the global doubling integral vanishes. A similar construction can be made for cusp forms on the metaplectic group [15]. Here, we consider a variant of this procedure, which is closer to what is done in [3] and [2]. The pullback E(ι(g1 , g2 ), s, Φ) of the Siegel Eisenstein series determines a kernel function on Spn (A) × Spn (A), and the associated integral operator on the space of cusp forms preserves each irreducible cuspidal automorphic representation σ. If the Eisenstein series is defined by a factorizable section, the resulting endomorphism of σ ' ⊗p σp is given as a product of endomorphisms of the local components σp , and, when all the data is unramified, the unramified vector in σp is an eigenvector with eigenvalue Lp (s + 12 , σp ) (up to a standard normalizing factor independent of σp ). The difficulty is to determine what happens at the ramified places, including the archimedean places. In this chapter, we restrict ourselves to the case n = 1, and we consider genuine irreducible cuspidal automorphic representations σ of the metaplectic extension G0A of Sp1 (A) = SL2 (A). In this case, the doubling integral represents the degree 2 standard L-function of the representation π = Wald(σ, ψ) associated to σ by the Shimura-Waldspurger lift. Our main result is an explicit construction of ‘good test vectors’, f p ∈ σp and Φp (s) ∈ Ip (s, χp ), the local induced representation, such that f p is an eigenvector of the local zeta integral operator defined by Φp (s). We do this only for unramified principal series and special representations, for p odd, and for mildly ramified principal series and special representations for p = 2. The main local result is Theorem 8.3.1. As a consequence, we can

266

CHAPTER 8

construct certain global eigenfunctions f ∈ σ for the doubling integral operator described above. The resulting explicit doubling formulas are given in Theorem 8.3.2 and Theorem 8.3.3. The latter of these formulas will be used in Chapter 9 to prove, in certain cases, a nonvanishing criterion for the arithmetic theta lift. This chapter is quite long for two reasons. First, in Section 8.2, we have provided a sketch, from our point of view, of Waldspurger’s results [23], [28], which we need in order to best formulate our results. The most important of these are the relations among central signs, local dichotomy invariants, and local root numbers. Second, we have provided a substantial amount of background material concerning coordinates on the metaplectic groups, Weil representations, etc., which is not readily accessible in the literature in the form we need. We hope that the resulting precision will justify the additional space required.

8.1 THE GLOBAL DOUBLING INTEGRAL

As in Sections 5.1 and 5.5, we let G0A be the metaplectic extension of Sp1 (A) = SL2 (A) and GA be the metaplectic extension of Sp2 (A). Let P 0 ⊂ Sp1 (resp. P ⊂ Sp2 ) be the upper triangular Borel subgroup (resp. the Siegel parabolic subgroup), and let PA0 (resp. PA ) be the full inverse image of P 0 (A) (resp. P (A)) in G0A (resp. GA ). Let i0 : Sp1 × Sp1 → Sp2 be the embedding 

(8.1.1)



i0 :





a1



 a1 b1 a b × 2 2 7−→   c1 c1 d1 c2 d2

b1 a2 d1 c2



b2  

.

d2

Also let (8.1.2)

∨

g = Ad



1



−1

· g,

and put (8.1.3)

i(g1 , g2 ) = i0 (g1 , g2∨ ).

Recall that there are two orbits of Sp1 (Q) × Sp1 (Q) on the coset space P (Q)\Sp2 (Q), so that Sp2 (Q) = P (Q) i(Sp1 (Q) × Sp1 (Q)) ∪ P (Q)δ i(Sp1 (Q) × Sp1 (Q)),

267

ON THE DOUBLING INTEGRAL

where 1



(8.1.4)

 1 δ= −1 1

 !  1 −1  = w1 m( ).  1

1 1 Here w1 = i0 (w1 , 1), in the notation of Section 8.5.1. The stabilizers of the chosen orbit representatives are P (Q) ∩ i(Sp1 (Q) × Sp1 (Q)) = i(P 0 (Q) × P 0 (Q)), and δ −1 P (Q)δ ∩ i(Sp1 (Q) × Sp1 (Q)) = i ◦ ∆(Sp1 (Q)), where ∆ is the diagonal embedding. In fact, we have δ i(g, g) = p(g) δ, where d c −c b a −b . p(g) =   a −b −c d 



The map i has a unique lift to a homomorphism i : G0A × G0A −→ GA , whose restriction to C1 × C1 is given by i(z1 , z2 ) = z1 z2−1 . Let ψ be the standard character of A/Q which is unramified and such that ψ∞ (x) = e(x) = e2πix . For ξ ∈ Q× , let ψξ be the character ψξ (x) = ψ(ξx). As explained in Section 8.5.5, there is then a group isomorphism ∼

P (A) × C1 −→ PA ,

(p, z) 7→ [p, z]L,ψ = [p, z]L .

As in Chapter 5, for a character χ of A× /Q× , we also write χ for the character of PA defined by χ([n(b)m(a), z]L ) = z χ(det a), and, for s ∈ C, we let I(s, χ) be the global degenerate principal series representation of GA on smooth functions Φ(s) satisfying (8.1.5)

3

Φ([nm(a), z]L g, s) = z χ(det a) | det a|s+ 2 Φ(g, s),

268

CHAPTER 8

where we require that Φ(s) be K∞ -finite. A section Φ(s) ∈ I(s, χ) is called standard if its restriction to K is independent of s. For a standard section Φ(s) ∈ I(s, χ), the corresponding Siegel Eisenstein series (8.1.6)

X

E(g, s, Φ) =

Φ(γg, s)

γ∈P (Q)\G(Q)

converges for Re(s) > 23 . Let σ ' ⊗p≤∞ σp be an irreducible cuspidal automorphic representation of G0A lying in the space A00 (G0 ) of cusp forms orthogonal to all O(1) theta functions. For a function f ∈ σ and a section Φ(s) ∈ I(s, χ), we consider the global doubling integral defined by (8.1.7)

Z(s, Φ, f )(g10 )

Z

=

Sp1 (Q)\Sp1 (A)

E(i(g10 , g20 ), s, Φ) f (g20 ) dg2 .

Here g20 ∈ G0A is any element with image g2 ∈ Sp1 (A), and we take Tamagama measure dg2 on Sp1 (Q)\Sp1 (A). Note that this is not quite the standard doubling integral of Piatetski-Shapiro and Rallis [5], since we are integrating against only one cusp form, but is of the type considered by B¨ocherer [2] in classical language. Unwinding in the usual way, using the coset information above, we get Z(s, Φ, f )(g10 )

Z

=

Sp1 (A)

Φ(δ i(g10 , g20 ), s) f (g20 ) dg2

Z

+

X

P 0 (Q)\Sp

1 (A)

Φ(i(γ1 g10 , g20 ), s) f (g20 ) dg20 .

γ1 ∈P 0 (Q)\Sp1 (Q)

The second term here vanishes identically, since it can be expressed in terms of the constant term of f . By (i) of Lemma 8.4.1, below, we have Φ(δ i(g10 , g20 ), s) = Φ(δ i(1, (g10 )−1 g20 ), s), so that, for Re(s) > 23 , Z(s, Φ, f )(g10 ) =

Z Sp1 (A)

Φ(δ i(1, g20 ), s) f (g10 g20 ) dg2 .

Thus, the function Z(s, Φ, f ) again lies in the space of σ, and the doubling integral gives the operation of the function g 0 7→ Φ(δ i0 (1, g 0 ), s) on G0A in the representation σ. Recall that I(s, χ) = ⊗p≤∞ Ip (s, χp ),

269

ON THE DOUBLING INTEGRAL

and suppose that Φ(s) = ⊗p≤∞ Φp (s) and that f ' ⊗p≤∞ fp under the isomorphism σ ' ⊗p≤∞ σp . Then, under the latter isomorphism, (8.1.8)

Z(s, Φ, f ) '

O

Zp (s, Φp , fp ),

p≤∞

where (8.1.9)

Z

Zp (s, Φp , fp ) =

Sp1 (Qp )

Φp (δp i(1, g 0 ), s) σp (g 0 )fp dg

∈ σp

is the local doubling integral. Here δp ∈ Gp is an element Q projecting to δ ∈ Sp2 (Qp ), with δp ∈ Kp for almost all p, and such that δ = p δp ∈ GQ . For p < ∞ we choose the measure dgp on SL2 (Qp ) for which vol(SL2 (Zp )) = 1. The measure dg∞ Q on SL2 (R) is then determined by the requirement that the product measure p dgp is Tamagawa measure. See Lemma 8.4.29 for an explicit description of dg∞ . In Section 8.4 below, we will consider the case in which σ has ‘square free’ level and will calculate the local integrals Zp (s, Φp , fp ) ∈ σp explicitly for a certain good choice of fp and Φp (s). To describe the results, it will be helpful to first review Waldspurger’s theory of the Shimura-Shintani correspondence. 8.2 REVIEW OF WALDSPURGER’S THEORY 8.2.1 Local theory

In this section, we review local theta dichotomy as proved by Waldspurger in [28], in particular the important relations among Whittaker models, the central sign, and the representation1 Wald(σ, ψ) and its root number. In addition, we recall the definition of the Waldspurger involution. Since our formulation differs slightly from that given in [28], we explain in some detail how to derive our statements from those of Waldspurger. Let F be a local field and let G0 be the metaplectic extension of Sp1 (F ) = SL2 (F ). As described in the Section 8.5, for a nondegenerate additive ∼ character ψ of F , there is an isomorphism [ , ]R : Sp1 (F ) × C1 −→ G0 giving the Rao coordinates g 0 = [g, z]R of an element g 0 ∈ G0 . We will sometimes write N 0 for the group of F -points of the unipotent radical N 0 of the standard Borel subgroup P 0 ⊂ Sp1 , and we identify this group with its image under the unique splitting homomorphism N 0 → G0 , n 7→ [n, 1]R . The central sign of an irreducible admissible genuine representation σ of G0 is defined as follows ([28], p. 225): 1

This notation was introduced in [7].

270

CHAPTER 8

We have σ([m(−1), 1]R )2 = σ([m(−1), 1]2R ) = σ([1, (−1, −1)F ]R ) = (−1, −1)F , where (·, ·)F is the quadratic Hilbert symbol for F . Since the Weil index γF (−1, ψ) satisfies γF (−1, ψ)2 = (−1, −1)F , there is a sign z(σ, ψ) defined by the relation (8.2.1)

σ([m(−1), 1]R ) = z(σ, ψ) γF (−1, ψ)−1 .

Since (8.2.2)

σ([m(−1), 1]L ) = γF (−1, ψ 1 ) σ([m(−1), 1]R ) 2

= (−1, 2)F γF (−1, ψ) σ([m(−1), 1]R ), in Leray coordinates we have (8.2.3)

z(σ, ψ) = σ([m(−1), 1]L ) χ2 (−1),

where χ2 (x) = (x, 2)F . For an irreducible admissible genuine representation σ of G0 , let (8.2.4)

Fˆ (σ) = { ψ ∈ Fˆ | W (σ, ψ) 6= 0 },

where W (σ, ψ) is the ψ-Whittaker model of σ. For a quadratic space (V, Q) over F and a nondegenerate additive character ψ of F , let ωψ,V be the Weil representation of G0 on the space S(V ); see Section 8.5.3 for the conventions used. We assume that dimF (V ) is odd. Suppose that ξ ∈ F × is such that (i) ψξ ∈ Fˆ (σ), and (ii) there is a vector xξ ∈ V such that Q(xξ ) = ξ. Then, for W ∈ W (σ, ψξ ), ϕ ∈ S(V ), and h ∈ SO(V ), we can consider the integral (8.2.5)

Z

uψ (h; W, ϕ; ξ, V ) =

N 0 \Sp1 (F )

W (g 0 ) ωψ,V (g 0 )ϕ(h−1 xξ ) dg,

as in [28], IV, p. 238, up to a change in notation. Here g 0 ∈ G0 is any element which projects to g ∈ Sp1 (F ). Note that a central element [1, z]R acts by multiplication by z in both σ and the Weil representation so that

271

ON THE DOUBLING INTEGRAL

the integrand is invariant under C1 . Formally, we may view this integral as defining a map2 (8.2.6)

uψ (ξ, V ) : S(V ) −→ W (σ, ψξ )∨ ⊗ C ∞ (SO(V )xξ \SO(V )).

This map is G0 × SO(V )-equivariant. Let (8.2.7)

θψ (σ; ξ, V ) ⊂ C ∞ (SO(V )xξ \SO(V ))

be the subspace spanned by the functions uψ (h; W, ϕ; ξ, V ) as W and ϕ vary; this submodule is stable under the action of SO(V ). For a quaternion algebra B over F , let VB = { x ∈ B | tr(x) = 0 }, with quadratic form3 q(x) = −ν(x) = x2 , and let H B = B × . Recall that the action of H B on VB by conjugation gives an isomorphism H B ' GSpin(VB ). We write VB = V± and H B = H ± , if inv(B) = ±1. Let Pe be the set of (isomorphism classes of) irreducible admissible, genuine unitary representations of G0 which are not of the form θψ (11, Uα ), where θψ (11, Uα ) is the Weil representation ωψ,Uα of G0 on the space of even functions in the Schwartz space S(Uα ), where Uα = F with quadratic form Q(x) = αx2 . Waldspurger proves the following result in [28]. e Suppose that ξ ∈ F × Theorem 8.2.1. (Waldspurger). Assume that σ ∈ P. ˆ with ψξ ∈ F (σ). (i) If V = V+ , then

θψ (σ; ξ, V+ ) 6= 0 ⇐⇒ ψ ∈ Fˆ (σ).

If ψ ∈ Fˆ (σ), then θψ (σ; ξ, V+ ) is an irreducible representation of H + . Moreover, the isomorphism class of θψ (σ; ξ, V+ ) is independent of ξ. 2

We get a C-linear map by composing with complex conjugation ϕ 7→ ϕ ¯ on S(V ). The resulting map S(V ) −→ W (σ, ψξ )∨ ⊗ C ∞ (SO(V )xξ \SO(V )) is equivariant for G0 × SO(V ), where G0 acts on S(V ) by g 7→ ωV,ψ (g ∨ ); see Lemma 8.5.8. Since the representation σ(g ∨ ) is isomorphic to the contragradient σ ∨ (g), [17], Chapitre 4, we obtain an intertwining map S(V ) −→ σ ⊗ C ∞ (SO(V )xξ \SO(V )). Thus, the representation θψ (σ; V ) defined below corresponds to σ under local Howe duality [6]. 3 Here we follow the convention in [28], and hence use the notation VB to distinguish this space from the space V B where the quadratic form Q(x) = ν(x) = −x2 is used.

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CHAPTER 8

(ii) If V = V− , there is an xξ ∈ V− with Q(xξ ) = ξ if and only if ξ ∈ / F ×,2 . Then θψ (σ; ξ, V− ) 6= 0 ⇐⇒ ψ ∈ / Fˆ (σ).

If ψ ∈ / Fˆ (σ), then θψ (σ; ξ, V− ) is an irreducible representation of H − and the isomorphism class of θψ (σ; ξ, V− ) is independent of ξ. In case (i) of the theorem, let θψ (σ, V+ ) ' θψ (σ; ξ, V+ ), and let θψ (σ, V− ) = 0. In case (ii) of the theorem, let θψ (σ, V− ) ' θψ (σ; ξ, V− ),

and let θψ (σ, V+ ) = 0. Define the dichotomy sign to be (8.2.8)

δ(σ, ψ) =

 +1

if θψ (σ, V+ ) 6= 0,

−1

if θψ (σ, V− ) 6= 0.

It follows that, if σ is as in Theorem 8.2.1, then (8.2.9)

δ(σ, ψ) = +1 ⇐⇒ ψ ∈ Fˆ (σ).

For example, if σ is an irreducible principal series representation, then, by [28], Lemma 3, p. 227, Fˆ (σ) = Fˆ , and so δ(σ, ψ) = +1 for all ψ. e take ξ so that ψξ ∈ Fˆ (σ), and let For σ ∈ P, (8.2.10)

Wald(σ, ψ) := θψξ (σ, V+ ) ⊗ χξ ,

where χξ (x) = (x, ξ)F is the quadratic character attached to ξ.4 In [28], Proposition 15, p. 266, Waldspurger proves that this representation is indee Note that the relation pendent of the choice of ξ, for σ ∈ P. (8.2.11)

Wald(σ, ψα ) = Wald(σ, ψ) ⊗ χα

is immediate from the definition. The following result, which is implicit in [28], relates the dichotomy sign, the central sign, and the root number of Wald(σ, ψ). e Theorem 8.2.2. (Waldspurger). For σ ∈ P,

δ(σ, ψ) =  4


1 , Wald(σ, ψ) · z(σ, ψ). 2

This space is called Sψ (T ) in [28], p. 280.

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ON THE DOUBLING INTEGRAL

Proof. First recall that Waldspurger defines Shintani liftings taking irreducible admissible unitary representations of H ± = SO(V± ) to irreducible admissible genuine representations of G0 : π 7→ θ(π, ψ), p. 228 (resp. π 0 7→ θ(π 0 , ψ), p. 235). For the first of these, he proves in [28], Theorem 1, p.249, that the maps π 7→ θ(π, ψ) and σ 7→ θψ (σ, V+ ) define reciprocal bijections:

(8.2.12)

  irreducible, admissible,

unitary, generic rep’s π of PGL(2)



↔

  irreducible, admissible,         unitary, genuine   

rep’s σ of G0 σ 6' θψ (11, Uα ) with ψ ∈ Fˆ (σ)

     



.

     

On the left-hand side here, generic just means infinite dimensional. For the second of these, he proves in [28], Proposition 14, p. 266, that the maps π 0 7→ θ(π 0 , ψ) and σ 7→ θψ (σ, V− ) define reciprocal bijections:

(8.2.13)

irreducible, spherical rep’s π 0 of SO(V− )





↔

   special or supercuspidal      (resp., discrete series)     

rep’s σ of G0 with ψ ∈ / Fˆ (σ)

.

   

On the left-hand side here, spherical means that the representation π 0 occurs in the space C ∞ (Hx− \H − ) for some nonzero x ∈ V − . On the right-hand side here, σ is special or supercuspidal in the nonarchimedean case and is a discrete series representation in the archimedean case. In the nonarchimedean case, π 0 has dimension > 1 if and only if σ is supercuspidal and is not an odd Weil representation. Note that, in fact, all irreducible representations of SO(V− ) are spherical [28], Proposition 18, p. 277. Also recall that for an irreducible admissible unitary generic representation π of PGL(2) = SO(V+ ), (8.2.14)

F (π) := { ξ ∈ F × | U(π, ξ) 6= 0 },

where U(π, ξ) is the ‘hyperboloid model’ of π, [28], p. 226, i.e., a realization of π in the space of functions C ∞ (Hxξ \H) for H = GL2 (F ). The following fundamental relation is then Lemma 6, p. 234 in [28]. Lemma 8.2.3. (i) (ii)

Fˆ (θ(π, ψ)) = { ψξ | ξ ∈ F (π) }.

1 z(θ(π, ψ), ψ) = ( , π). 2

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CHAPTER 8

This second relation says that the central sign of θ(π, ψ) coincides with the root number of π. Suppose that σ lies in the set on the right side of (8.2.12). Then, on the one hand, θψ (σ, V+ ) 6= 0, so that δ(σ, ψ) = +1. On the other hand, σ = θ(π, ψ) for some π on the left side of (8.2.12), and, by (ii) of the fundamental relation, 1 z(σ, ψ) = ( , π). 2

(8.2.15)

Moreover, ψ ∈ Fˆ (σ), so that π = θψ (σ, V+ ) = Wald(σ, ψ),

(8.2.16)

where the first equality is given by the bijection (8.2.12). This gives the identity of Theorem 8.2.2 in this case. Next suppose that σ is as on the right side of (8.2.12), except that ψ ∈ / ˆ F (σ). This condition implies that σ is not an irreducible principal series, and hence lies in the set on the right side of (8.2.13), since the even Weil representations have been excluded. Hence σ = θ(π 0 , ψ) for some π 0 in the set of representations on the left side of (8.2.13). Let π be the representation of SO(V+ ) = PGL(2) associated to π 0 under the Jacquet-Langlands correspondence, and let σ 0 = θ(π, ψ). Here is the picture: G0 W.inv.

(8.2.17) G0

σ

↔ π0

l

l

σ0 ↔

π

SO(V− ) JL SO(V+ ),

where the vertical arrow on the right is the Jacquet-Langlands correspondence and the vertical arrow on the left is, by definition, the Waldspurger involution. On the one hand, since θψ (σ, V− ) 6= 0, we have δ(σ, ψ) = −1. On the other hand, by (3) of Theorem 2, p. 277 of [28] combined with (ii) of Lemma 8.2.3, we have (8.2.18)

1 z(σ, ψ) = −z(σ 0 , ψ) = −( , π). 2

It then remains to relate π and Wald(σ, ψ). First suppose that σ is not an odd Weil representation, so that σ lies in Waldspurger’s set Pe1 . Then Proposition 15, p. 266 of Waldspurger implies that π = Wald(σ, ψ). This proves Theorem 8.2.2 in this case.

275

ON THE DOUBLING INTEGRAL

Finally, we need to consider the case in which σ is an odd Weil representation, i.e., σ = θψ (sgn, Uα ), where Uα = (F, αx2 ). The condition (8.2.19)

ψ∈ / Fˆ (σ) = {ψξ | ξ ∈ α · F ×,2 }

implies that α ∈ / F ×,2 . By Lemma 20, p. 249 of [28], we have (8.2.20)

1

1

θψα (σ, V+ ) = σ(| | 2 , | |− 2 ),

and hence (8.2.21)

1

1

Wald(σ, ψ) = σ(| | 2 , | |− 2 ) ⊗ χα .

Since χα 6= 1, we have (8.2.22)

1 1 1 1 ( , Wald(σ, ψ)) = ( , σ(| | 2 , | |− 2 ) ⊗ χα ) 2 2 = χα (−1),

by the relation given on p. 274 of [28]. On the other hand, the central sign can be computed from the formulas for the Weil representation: (8.2.23)

1

ωψ,U ([m(a), 1]R )ϕ(x) = γF (a, ψ)−1 |a| 2 χα (a) ϕ(xa).

Taking a = −1 and assuming that ϕ is an odd function, we have, via (8.2.1), (8.2.24)

z(σ, ψ) = −χα (−1),

so that the relation of Theorem 8.2.2 holds. Finally, we give a small table of the corresponding representations. Recall that, in [28], for a character µ of F × , the principal series representation is defined on the space of functions B(µ) on G0 satisfying (8.2.25)

f ([nm(a), z]R g 0 ) = z µ(a) γF (a, ψ)−1 |a| f (g 0 ).

Note that this parametrization depends on ψ. Notice also that B(µ) = I(µχ2 ) in our notation for induced representations of G0 ; see (8.4.3) below. If the action of G0 on this space is irreducible, the resulting principal series representation is denoted by π ˜ (µ). In the nonarchimedean case, if 1 µ = χα | | 2 , the action of G0 is not irreducible. The irreducible subrep1 resentation of B(µ) will be the special representation σ ˜ (χα | | 2 ) and the irreducible quotient will be θψ (11, Uα ), the even Weil representation for the one-dimensional quadratic space Uα = (F, αx2 ). In this table, the additive character ψ is fixed and the parameterization of the representations in the first column depends on ψ, as just explained.

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CHAPTER 8

Table 2. The Local Theta Correspondence Wald(σ, ψβ )

σ

π(χβ µ, χβ µ−1 )

π ˜ (µ) 1 2

σ ˜ (χα | | ) 1 2

σ ˜ (χβ | | ) θψ (sgn, Uα ) θψ (sgn, Uβ )

σ(χαβ | | , χαβ | | 1 2

− 21

σ(| | , | |

− 12

σ(χαβ | | , χαβ | | − 21

1 2

σ(| | , | |

)

)

1 2

other s.c. π ˜`±

− 12

1 2

)

s.c.

)

( 12 , Wald(σ, ψβ ))

δ(σ, ψβ )

z(σ, ψβ )

χβ µ(−1)

+1

χβ µ(−1)

χαβ (−1)

+1

χαβ (−1)

−1

−1

1

χαβ (−1)

−1

−χαβ (−1)

−1

+1

−1

...

...

...

±χβ (−1)

±χβ (−1) (−1)`− 2

`− 12

DS2`−1

(−1)

1

Here, in the second and fourth rows, χαβ 6= 1. In the next to last row, ‘s.c.’ stands for supercuspidal representations other than the odd Weil representations. For these, we do not record information about the root number, dichotomy invariant, and central sign, since these quantities depend on the detailed parametrization of such representations [16], and we will not need this information. In the last row, for the archimedean case, ` ≥ 32 , and we suppose that e`+ = HDS` is the holomorphic discrete series repψ(x) = e(x). Here π e`− is the antiholomorphic discrete resentation with lowest weight `, while π series representation with highest weight −`. Note that the signatures of the quadratic spaces in [28] are sig(V+ ) = (2, 1) and sig(V− ) = (0, 3). Also DS2`−1 denotes the discrete series representation of PGL2 (R) of weight 2` − 1. The action of the Waldspurger involution can be seen in the table.5 The pairs of representations which are switched by the involution are (8.2.26)

1

{σ ˜ (χα | | 2 ), θψ (sgn, Uα ) }

and

{π ˜`+ , π ˜`− }.

8.2.2 Global theory

We now review parts of Waldspurger’s global theory. For simplicity, we work over Q. Let A0 (G0 ) be the space of genuine cusp forms on G0A and let A00 (G0 ) be the orthogonal complement in A0 (G0 ) of the subspace spanned by the theta functions coming from one-dimensional quadratic spaces. For an irreducible genuine cuspidal representation σ ' ⊗p≤∞ σp of G0A occuring in A00 (G0 ), the representation (8.2.27) 5

Wald(σ, ψ) = ⊗p≤∞ Wald(σp , ψp )

except for the supercuspidals, of course, whose data we have not recorded.

277

ON THE DOUBLING INTEGRAL

is, in fact, a cuspidal automorphic representation of PGL2 (A). It can also be defined by using global theta integrals; see [23] and [28], p. 280.6 The central signs of the components σp of σ satisfy the product formula, see [28], p. 280 (1), (8.2.28)

Y

z(σp , ψp ) = 1,

p≤∞

and the fibers of the map σ 7→ Wald(σ, ψ) have the following description, see [28], Corollaire 2, p. 286: Proposition 8.2.4. Let Σ be the set of all p ≤ ∞ at which Wald(σp , ψp ) is not an irreducible principal series representation. Then the set of all irreducible cuspidal genuine σ 0 ’s such that Wald(σ 0 , ψ) = Wald(σ, ψ) has cardinality 1 if |Σ| = 0 and 2|Σ|−1 if |Σ| > 0. In the latter case, the representations σ 0 have the form σ 0 ' (⊗p∈Σ0 σpW ) ⊗ (⊗p∈Σ / 0 σp ),

where Σ0 ⊂ Σ is any subset with even cardinality. Here σp 7→ σpW is the Waldspurger involution. Note that, since the Waldspurger involution switches the central sign, the condition on the cardinality of Σ0 is necessary in order to preserve the product formula for the central signs of the local components of σ 0 . Because of the product formula for central signs, Theorem 8.2.2 implies the relation Y Y 1 1 (8.2.29) δ(σp , ψp ) = ( , Wald(σp , ψp )) = ( , Wald(σ, ψ)) 2 2 p≤∞ p≤∞ between the product of the local dichotomy signs and the global root number of Wald(σ, ψ). As usual, for any quaternion algebra B over Q, let VB be the Q-vector space of trace zero elements in B with quadratic form q(x) = −ν(x) = x2 , and let H B = B × ' GSpin(VB ). For g 0 ∈ G0A , h ∈ H B (A) and ϕ ∈ S(VB (A)), there is a theta function (8.2.30)

θ(g 0 , h; ϕ) =

X

ω(g 0 )ϕ(h−1 x),

x∈VB (Q)

where ω = ωVB ,ψ is the Weil representation of G0A in S(VB (A)). For a cusp form f ∈ σ, where σ is as above, and for ϕ ∈ S(VB (A)), the global 6

where Wald(σ, ψ) is denoted by Sψ (T ).

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CHAPTER 8

theta integral is given by Z

(8.2.31)

θ(f , ϕ)(h) =

SL2 (Q)\SL2 (A)

where g 0 the Let

∈ G0A is any element mapping to g ∈ space A0 (H B ) of cusp forms on H B (A) θψ (σ, VB )

(8.2.32)

⊂

f (g 0 ) θ(g 0 , h; ϕ) dg, SL2 (A). This function lies in with trivial central character.

A0 (H B )

be the subspace spanned by the θ(f , ϕ)’s as f varies in σ and ϕ varies in S(VB (A)). This global construction is compatible with the local construction of Section 8.2.1, and we have θψ (σ, VB ) '

(8.2.33)

 ⊗p≤∞ θψ (σp , VB ), p p 0,

so that θψ (σ, VB ) is either an irreducible cuspidal automorphic representation or zero. Now local theta dichotomy comes into play, and (8.2.34)

θψ (σ, VB ) 6= 0

=⇒

invp (B) = δ(σp , ψp ),

∀p ≤ ∞.

Waldspurger’s beautiful result is then the following.

Theorem 8.2.5. (i) If ( 21 , Wald(σ, ψ)) = −1, then θψ (σ, VB ) = 0 for all B. (ii) If ( 21 , Wald(σ, ψ)) = 1, then there is a unique quaternion algebra B over Q such that In this case,

⊗p≤∞ θψp (σp , VpB ) 6= 0.

1 L( , Wald(σ, ψ)) 6= 0. 2 Note: Elsewhere in this chapter and, indeed, throughout the book, we work with the quadratic spaces V B with quadratic form Q(x) = ν(x) = −x2 . As explained in Section 8.5, the associated Weil representation can be written as θψ (σ, VB ) = 6 0

(8.2.35)

⇐⇒

ωV B ,ψ = ωVB ,ψ−1

in both the local and global situations. This implies that (8.2.36)

θψ (σ, V B ) = θψ−1 (σ, VB ).

For convenient later reference, we state the corresponding version of Theorem 8.2.5.

279

ON THE DOUBLING INTEGRAL

Corollary 8.2.6. (i) If ( 12 , Wald(σ, ψ−1 )) = −1, then θψ (σ, V B ) = 0 for all B. (ii) If ( 21 , Wald(σ, ψ−1 )) = 1, then there is a unique quaternion algebra B over Q such that In this case,

⊗p≤∞ θψp (σp , VpB ) 6= 0.

θψ (σ, V B ) 6= 0

⇐⇒

1 L( , Wald(σ, ψ−1 )) = 6 0. 2

In the case ( 12 , Wald(σ, ψ−1 )) = −1, a result analogous to (ii) on the nonvanishing of the arithmetic theta lifting in terms of the central derivative L0 ( 12 , Wald(σ, ψ−1 )) is proved in Chapter 9 in some special cases. 8.3 AN EXPLICIT DOUBLING FORMULA

We now return to the doubling integral of Section 8.1. The additive character ψ is now fixed to be unique unramified character with ψ∞ (x) = e(x). Let χ = χ2κ , where κ = ±1, be a global quadratic character. For convenience, we will write ψpκ for the local component at p of ψκ . Let σ ' ⊗p≤∞ σp be an irreducible genuine cuspidal automorphic representation in the space A00 (G0 ). We will determine good ‘test vectors’ f p ∈ σp and Φp (s) ∈ Ip (s, χ) such that f p is an eigenvector for the operator Z(s, Φp , ·). We limit ourselves to those σp ’s which occur in our arithmetic application, although our method can be extended to all σp ’s. Specifically, we will prove the following result, which collects together from Section 8.4 the results of the calculations of local doubling integrals. Recall that we normalize the Haar measure on SL2 (Qp ) used in the definition of the local doubling integral by vol(SL2 (Zp )) = 1, for p < ∞, and we take −3 da db dθ, for g = n(b)m(a)k ∈ SL (R) with a > 0, so that dg = 12 2 θ π ·a the product of these local measures gives Tamagawa measure on SL2 (A); see Lemma 8.4.29. Also, recall that the parametrization of local components σp is as in Table 2 of Section 8.2 with respect to the local additive character ψp determined by ψ.

0 1 Theorem 8.3.1. For a finite prime p, and for κ ∈ Z× p , let Φp (s), Φp (s), and Φra p (s) ∈ Ip (s, χ2κ ) be the standard sections defined in (8.4.13) below. (i) For p = 6 2, suppose that σp = I(µp ) = π ˜ (χ2 µp ) is an unramified principal series representation. Let f p ∈ σp be the unramified vector. Then

Z(s, Φ0p , f p ) =

L(s + 12 , Wald(σp , ψpκ )) · f p. ζp (2s + 2)

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CHAPTER 8

˜ (χα | | 2 ), with α ∈ Z× (ii) For p 6= 2, suppose that σp = σ p , is an unramified special representation, and let f p = fsp ∈ σp be the Iwahori fixed vector. Then, Z(s, Φ0p , f p ) = 0, 1

Z(s, Φ1p , f p ) = p−2

and

p−1 1 (1 + δ(σp , ψpκ ) p−s )L(s + , Wald(σp , ψpκ ))f p p+1 2

1 κ −2 κ −s Z(s, Φra p , f p ) = −p (1 − δ(σp , ψp ) p )L(s + , Wald(σp , ψp ))f p . 2 Here δ(σp , ψpκ ) is the local dichotomy invariant defined in (8.2.8); see Theorem 8.2.2. (iii) For p = 2, suppose that σp = I(µp ) = π ˜ (χα | |t ), with µp = χ2α | |t , 1 × α ∈ Zp and t 6= ± 2 , is an irreducible principal series representation. Let

f p = fev ∈ σp α be the ‘good newvector’ defined in part (i) of Theorem 8.4.26. Assume that α ≡ κ mod 4. Then (J0 ,χ )

L(s + 12 , Wald(σp , ψpκ )) 1 Z(s, Φ0p , f p ) = δκ √ · · f p, ζp (2s + 2) 2 2

where δκ = 1 if κ ≡ 1 mod 4 and δκ = i if κ ≡ 3 mod 4. 1 (iv) For p = 2, suppose that σp = σ ˜ (χα | | 2 ) with α ∈ Z× p is a special

representation, and let f p = fsp ∈ σp α be the ‘good newvector’ defined in part (ii) of Theorem 8.4.26. Assume that α ≡ κ mod 4. Then, Z(s, Φ0p , f p ) = 0, (J0 ,χ )

Z(s, Φ1p , f p ) = ? p−2 and

p−1 1 (1 + δ(σp , ψpκ ) p−s )L(s + , Wald(σp , ψpκ ))f p p+1 2

1 −2 κ −s κ Z(s, Φra p , f p ) = − ? p (1 − δ(σp , ψp ) p )L(s + , Wald(σp , ψp ))f p , 2

where ? = δκ

1 √ . 2 2

(v) Suppose that p = ∞ and that σ∞ = π ˜`+ is the holomorphic discrete series representation of weight ` ∈ 21 + Z>0 . Let f ∞ ∈ σ∞ be the weight ` vector. Let Φ`∞ (s) ∈ I∞ (s, χ) be the standard normalized section of weight 1 `. Here χ(−1) = χ2κ (−1) = (−1)`− 2 . Then Z(s, Φ`∞ , f ∞ ) = 24 · e(− 41 (`− 12 )) ·

1 2

s+ 12

(s + ` − 12 )

· f ∞.

Returning to the global representation σ ' ⊗p σp , we make the following assumptions about the local components:

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ON THE DOUBLING INTEGRAL

(i) For p = ∞ σ∞ ' π ˜`+ , for some ` ∈

1 2

1

+ Z>0 with κ = (−1)`− 2 .

(ii) For p = 2, σp '

 π ˜ (χα | |t )

1 with α ∈ Z× p and t 6= 2 , or

1 σ ˜ (χα | | 2 )

with α ∈ Z× p.

(iii) For a square free odd integer No , σp '

 π ˜ (χα | |t )

1 with α ∈ Z× p and t 6= 2 , if p - 2No ,

1 σ ˜ (χα | | 2 )

with α ∈ Z× p , if p | No .

Note that in (ii) we are allowing a small amount of ramification. Recall that, by (8.2.1) and (8.2.28), the product over all p of the central signs z(σp , ψp ) must be 1. Note that z(σp , ψp ) is given in the last column of Table 2. Under assumption (iii), z(σp , ψp ) = 1 for p 6= 2, ∞, since χα (−1) = 1 for such p’s. Thus we have (8.3.1)

1=

Y

1

z(σp , ψp ) = (−1)`− 2 · χα,2 (−1),

p≤∞

where χα = χα,2 is the character occurring in σ2 . From this relation and (i), it follows that χα,2 (−1) = κ, and so (8.3.2)

α≡κ

mod 4.

Let N = No if σ2 is an irreducible principal series representation and N = 2No if σ2 is a special representation. The level of Wald(σ, ψκ ) is then N . For p odd, this is clear, while for p = 2, it follows from the fact that Wald(σ2 , ψ2κ ) = Wald(σ2 , ψ2 ) ⊗ χκ =

  σ(χακ | | 21 , χακ | |− 12 ) ακ |

 π(χ

|t , χακ | |−t )

where, by (8.3.2), the character χακ is unramified. Note that δ(σp , ψpκ ) =

 −1

if p | N and χακ,p = 1,

+1

otherwise.

if 2 | N , if 2 - N ,

282

CHAPTER 8

We write N = N + N − , where N± =

Y

p.

p|N δ(σp ,ψpκ )=±1

We then take f ∈ σ, with f ' f ∞ ⊗ (⊗p 1, the intertwining operator is defined by (8.4.28)

Z

M (t)f (g) =

f ([w−1 , 1]L [n(b), 1]L g) db,

Q2

where db is the self-dual measure with respect to ψ. Note that M (t) car0 (J0 ,ξ) . As usual, functions in I(µ) are ries the space I(µt )(J ,ξ) to I(µ−1 t ) determined by their restrictions to K 0 , and the resulting space of functions on K 0 is independent of t. Thus the space of ξ-eigenfunctions for J0 is also independent of t. For simplicity, we assume from now on that ψ ± = 1, i.e., × that ξ = χα is a character of Z× 2 with α ∈ Z2 . Proposition 8.4.16. Let µ = χ2α | |t , and let ξ = χα be a character of J0 with α ∈ Z× 2 . Then the matrix of M (t) with respect to the basis f1 , fw of

303

ON THE DOUBLING INTEGRAL

I(µ)(J ,χα ) is 0

g8 (α)  22t (1 − 2−2t )   

1 4

where Z

g8 (α) =

Here

Z× 2



χα (−1) 

,

g8 (α)  1 − 2−2t

1 u . χ2α (u) ψ( ) du = √ 8 2 2 δα

1 if α ≡ 1 mod 4, i if α ≡ −1 mod 4.

(

δα =

In particular, when t = 12 , the matrix directly above is √   2 2  1 1 δα  ,  √   δα 2 2 δα √ 2 2

and its eigenfunctions are

1 δα fsp = f1 − √ fw ∈ σ ˜ (χα | | 2 ) 2 2

and

with eigenvalues 0 and

δα fev = f1 + √ fw 2

√3 , 2 2 δα

respectively.

Proof. It suffices to compute the values of M (t)f1 and M (t)fw at the points 1 and w. It is easy to check that 

 −wn(b) if ord(b) ≥ 0, −1 −1 w n(b) = = n(−b−1 ) m(b−1 ) n (b−1 ) if ord(b) < 0, 1 b − 

and thus, for b 6∈ Z2 , w−1 n(b) = n(−b−1 )m(b−1 )[n− (b−1 ), 1]L .

304

CHAPTER 8

Here, for convenience, we write w−1 = [w−1 , 1]L , n(b) = [n(b), 1]L , and m(a) = [m(a), 1]L . Then (

f1 (w

−1

0 γ(ψ2b )µ(b)−1 |b|−1

n(b)) =

if b ∈ 12 Z2 , if b ∈ / 21 Z2 ,

and (

fw (w−1 n(b)) =

χα (−1) if b ∈ Z2 , 0 if b ∈ / Z2 .

Therefore Z

M (t)f1 (1) = = =

ord(b)≤−2

f1 (w−1 n(b)) db

∞ Z X

χ2α (b−1 ) |b−1 |t+1 γ(ψ2b ) db ord(b)=−r r=2 Z ∞ X 2−rt χ2α (2r ) × χ2α (u) γ(ψ2r−1 u ) du. Z2 r=2

Recall that, by [19], Proposition A.12, (8.4.29)

γ(ψ2r−1 b ) =

 ψ( b )

8 ψ( b )χ (b) 2 8

if r is even, if r is odd,

so that the integral becomes Z Z× 2

χ2α (u) γ(ψ2r−1 u ) du

R  Z× χ2α (u)ψ( u8 ) du = R 2 u  Z× 2

=

χα (u)ψ( 8 ) du

 g8 (α)

if r is even,

0

if r is odd.

if r is even, if r is odd,

So we have M (t)f1 (1) =

∞ X

2−2nt g8 (α) =

n=1

2−2t · g8 (α), 1 − 2−2t

and Z

M (t)fw (1) = Q2

fw (w−1 n(b)) db = χα (−1).

305

ON THE DOUBLING INTEGRAL

Next, we note that (

w

−1

n(b)w =

[n− (−b), γ(ψ2b )]L if b ∈ 2Z2 , −1 −1 −1 n(−b )m(b )wn(−b ) if b ∈ / 2Z2 ,

so that (

f1 (w

−1

1 0

n(b)w) =

if b ∈ 4Z2 , if b ∈ / 4Z2 ,

and (

fw (w

−1

0 if b ∈ 2Z2 , −1 −1 µ(b )|b| γ(ψ2b ) if b ∈ / 2Z2 .

n(b)w) =

This gives Z

M (t)f1 (w) =

ord(b)≥2

1 db = . 4

Finally, the same calculation as in the beginning gives M (t)fw (w) = = =

∞ Z X

χ2α (b−1 ) |b−1 |t+1 γ(ψ2b ) db

r=0 ord(b)=−r ∞ X 2−rt χα (2)r r=0 X −rt

2

even g8 (α) = . 1 − 2−2t

Z Z× 2

χ2α (b)γ(ψ2r−1 b ) db

χα (2)r g8 (α)

r≥0,

The values of g8 (α) are easy to calculate. It will be useful to record some information about the Whittaker functions of f1 and fw . With the same notation as in Proposition 8.4.16, recall that, for m ∈ Q2 and Re(t) > 1, the Whittaker function attached to f ∈ I(µ) is given by (8.4.30)

Z

Wm (f )(g) =

f ([w−1 , 1]L [n(b), 1]L g) ψ(−mb) db.

Q2

Notice that Wm (f )(m(a)g) = µ(a)−1 |a|Wa2 m (f )(g). We will only need the following values:

306

CHAPTER 8

Proposition 8.4.17. With the same notation as in Proposition 8.4.16, suppose that m ∈ Z2 − 4Z2 . Then (i) Wm (f1 )(w) = χα (−1), Wm (fw )(1) = 41 , and 1 . Wm (fw )(w) = Wm (f1 )(1) + √ 2 2δα

(ii) Wm (f1 )(1) = (iii) Let Then

 −

√1 µ(2)2 2 2δα  √1 µ(2)2 + 1 µ(2)3 ψ( 1−m ) 2 8 2 2δα

√ f

+

= f1 +

if m ≡ α

mod 4,

mod 4.

2µ(2)2 fw . δα

if m ≡ 2, −α

Wm (f + )(1) = 0

if m ≡ 2, −α

mod 4.

We omit the slightly more complicated values in general; these can be computed by the same methods. We remark that part (iii) of the proposition means that f + is in Kohnen’s plus space locally. Additional discussion of this point is given in Section 9.3. Notice that Wm (f + )(w) 6= 0 if m ≡ 2, −α

mod 4,

so that the Fourier coefficients at other cusps will not vanish.

Proof. The same calculation as in the proof of Proposition 8.4.16 gives X

Wm (f1 )(1) =

µ(2)n Im (n, χ2α , ψ),

n≥2

(8.4.31)

and

Wm (f1 )(w) = χα (−1)char(Z2 )(m), 1 Wm (fw )(1) = char(4Z2 )(m), 4 X

Wm (fw )(w) =

µ(2)n Im (n, χ2α , ψ),

n≥0

where (8.4.32)

Z

Im (n, χ2α , ψ) =

Z× 2

γ(ψ21−n b )χ2α (b)ψ(−2n mb) db.

These formulas give a complete description for the ψ-Whittaker functions of f1 and fw if one knows how to compute Im (n, χ2α , ψ).

307

ON THE DOUBLING INTEGRAL

Lemma 8.4.18. Let the notation be as above. Then

Im (n, χ2α , ψ) =

  χ2α (1 − 23−n m) 2√12δ   α  1

2    0

3−n m

ψ( 1−2 8

)

2 | n, 23−n m ∈ 2Z2 , 2 - n, 23−n m ≡ α mod 4, otherwise.

Proof. First we assume that n is even. By (8.4.29), we have b χ2α (b)ψ( (1 − 23−n m)) db 8 1 3−n = χα (1 − 23−n m) √ char(Z× m) 2 )(1 − 2 2 2δα Z

Im (n, χ2α , ψ) =

Z× 2

as claimed. Now we assume that n is odd. Then (8.4.29) implies Im (n, χ2α , ψ) Z

=

=

=

Z× 2

b χα (b)ψ( (1 − 23−n m)) db 8

  1 char(4Z2 )(1 − 23−n m)ψ( 1−23−n m ) 2  1 char(2Z× )(1 2 2

−

8 1−23−n m 3−n 2 m)ψ( ) 8

  1 ψ( 1−23−n m )

if 23−n m ≡ α

0

otherwise.

2

8

if α ≡ 1 if α ≡ −1

mod 4, mod 4,

mod 4,

Now we can complete the proof of Proposition 8.4.17. The first two claims in (i) are parts of (8.4.31). The same formula also gives, together with Lemma 8.4.18, Wm (fw )(w) = Wm (f1 )(1) + Im (0, χ2α , ψ) + µ(2)Im (1, χ2α , ψ) 1 = Wm (f1 )(1) + √ . 2 2δα When m ≡ 2, −α mod 4, one has ( 3−n

ord2 (2

m)

≥ 1 if n ≤ 2, ≤ 0 if n ≥ 4,

308

CHAPTER 8

and 23−n m 6≡ α mod 4. So Lemma 8.4.18 implies Wm (f1 )(1) = µ(2)2 Im (2, χ2α , ψ) µ(2)2 χ2α (1 − 2m) √ 2 2δα µ(2)2 =− √ 2 2δα

=

as claimed. When m ≡ α mod 4, one has 23−n m ≡ α mod 4 iff n = 3. So Lemma 8.4.18 implies that Wm (f1 )(1) = µ(2)2 Im (2, χ2α , ψ) + µ(2)3 Im (3, χ2α , ψ) =

µ(2)2 χ2α (1 − 2m) 1 1−m √ + µ(2)3 ψ( ) 2 8 2 2δα

1 µ(2)2 1−m + µ(2)3 ψ( = √ ). 8 2 2δα 2 This proves (ii). Claim (iii) follows from (i) and (ii) directly. We now return to the local zeta integrals. For κ ∈ Z× 2 , we define lattices and Lra as in (8.4.12). By Proposition 8.5.14, when L = L0 , L1 , or Lra , the standard section ΦL (s) ∈ I(s, χV ) determined by L0 , L1 ,

ΦL (0) = λV (ϕL ⊗ ϕL ) is an eigenfunction of i(J0 × J0 ) with character χκ . Thus, Z(s, ΦL , ·) maps 0 I(µ) into I(µ)(J ,χκ ) . Since the latter space is nonzero only when µ = χ2α | |t , for some α with α ≡ κ mod 4, we assume this condition from now on and are led to compute the zeta integrals, as in (8.4.10), (8.4.33) Z(s, ΦL , fwj )(wi ) Z

= γ(V )

Q× 2

1

(r)

χV µ(a)|a|r+ 2 B(r, a; ωV (wi )ϕL , I(fwj , ϕL )) d× a,

for L = L0 , L1 , Lra , and i, j = 0, 1, where w0 = 1 and w1 = w. Lemma 8.4.19. (i)

ωV (wi )ϕL =

if i = 0,

 ϕL − 21

γ(V )[L# : L]

· ϕL#

if i = 1.

309

ON THE DOUBLING INTEGRAL

(ii)

I(f1 , ϕL ) = vol(J0 )ϕL .

(iii)

1

I(fw , ϕL )(x) = γ(V )−1 [L# : L]− 2 vol(J0 ) · 4 charZ2 (Q(x)) · ϕL# (x)

Proof. Only (iii) needs proof. Since

K 0 ∩ P 0 wJ 0 =

n(c)wJ 0 ,

[ c∈Z/4

we have Z

I(fw , ϕL ) = =

K0

fw (k 0 ) ωV (k 0 )ϕL (x) dk

X Z c∈Z/4

χκ (k0 ) ωV (wk0 )ϕL (x)ψ(c Q(x)) dk0

J0

1

= γ(V )−1 vol(J0 ) [L# : L]− 2 · ϕL# (x)

X

ψ(−c Q(x))

c∈Z/4 1

= γ(V )−1 vol(J0 ) [L# : L]− 2 · ϕL# (x) · 4 charZ2 (Q(x)). Note that charZ2 ◦ Q · ϕL# =

X

ϕy+L .

y∈L] /L Q(y)∈Z2

Lemma 8.4.20. For y ∈ L# − L with Q(y) ∈ / Z2 , and any lattice L0 , B(r, a; ϕL0 , ϕy+L ) = 0.

Proof. Indeed, Z

B(r, a; ϕL0 , ϕy+L ) = Q2

Z

=

Z Vr

(r)

(r)

ϕL0 (x)ϕy+L (−ax) ψ(−b Q(x)) dx db Z

(r) ϕy+L (−ax) lim n→∞ 2−n Z2 0 Lr

ψ(−b Q(x)) db dx.

Now for ax ∈ y + Lr , one has x ∈ a−1 y + a−1 Lr , and thus, since Q(y) ∈ / Z2 , ord(Q(x)) = −2 ord(a) + ord(Q(y)) < −2 ord(a).

310

CHAPTER 8

Thus, Z 2−n Z2

ψ(−b Q(x)) db = 0

when n > −2 ord(a), and hence B(r, a; ϕL0 , ϕy+L ) = 0. Corollary 8.4.21. (i) For L = L0 or Lra , B(r, a; ωV (wi )ϕL , I(fwj , ϕL ))


i,j=0,1



= vol(J0 ) B(r, a; ϕL , ϕL ) ·  

1

4γ(V ) √ #

√γ(V#)

[L :L]

(ii) For L = L1 , and L0 = L or L# ,

−1



[L :L]  . 4 # [L :L]

1

B(r, a; ϕL0 , I(fw , ϕL )) = vol(J0 )·B(r, a; ϕL0 , ϕL# )·4 γ(V )−1 [L# : L]− 2 . Proof. Claim (ii) follows immediately from the previous two lemmas. To prove (i), by the same argument as above, it suffices to prove B(r, a; ϕL1 , ϕL2 ) = B(r, a; ϕL , ϕL ), for Li = L or L# , i = 1, 2. It is easy to check that for y ∈ L# , L = L0 , or Lra , Q(y) ∈ Z2 if and only if y ∈ L. So one has, by the previous lemma and Lemma 8.4.4, B(r, a; ϕL1 , ϕL2 ) =

X

B(r, a; ϕL1 , ϕy+L )

y∈L2 /L

= B(r, a; ϕL1 , ϕL ) = |a|−2r−1 B(r, a−1 ; ϕL , ϕL1 ) = |a|−2r−1 B(r, a−1 ; ϕL , ϕL ) = B(r, a; ϕL , ϕL ). Lemma 8.4.22. where

γ(V ± ) = ±δκ ,

κ−1 δκ = ψ( ) χ2 (κ) = 8

1 if κ ≡ 1 mod 4, i if κ ≡ −1 mod 4.

(

311

ON THE DOUBLING INTEGRAL

Proof. By Lemma 8.5.6,

−1



γ(V ) = χV (−1)γ(ψ1/2 ) γ(det V, ψ1/2 )γ(ψ1/2 )3 (V )

.

Recall that det V = −2κ, and that the matrix of V ± is 2κ diag(1, 1, −1) and −2κ diag(1, 1, 1) respectively. Thus (

(V ) =

(2κ, −1) if V = V + , (−2κ, −1) if V = V − ,

= χ2κ (−1) inv(B), where inv(B) = ±1 is the invariant of the quaternion B associated to V . Then, by a short calculation, inv(B) γ(V ) = δκ . Proposition 8.4.23. (i) For L = L0 , Z(s, ΦL , fwj )(wi )


i,j=0,1

vol(J0 ) L( 21 + s, χV µ)L( 21 + s, χV µ−1 ) = γ(V ) √ ζ(2s + 2) 2

√ ! 2 2 δκ .

1 δκ √ 2

In particular, the eigenfunctions of Z(s, ΦL0 , ·) in I(µ)(J ,χκ ) are

2

0

δκ fev = f1 + √ fw 2

with eigenvalues

γ(V )

and

δκ fsp = f1 − √ fw 2 2

3 vol(J0 ) L( 12 + s, χV µ)L( 21 + s, χV µ−1 ) √ ζ(2s + 2) 2

and 0, respectively. (ii) When L = Lra ,

Z(s, ΦL , fwj )(wi )


i,j=0,1

1 vol(J0 ) L( 12 + s, χV µ)L( 21 + s, χV µ−1 ) 2 = γ(V ) √ δκ −√ 4 ζ(2s) 2 2

√

!

− 2δκ2 . 1

In particular, its eigenfunctions in I(µ)(J ,χκ ) are fev and fsp with eigenvalues 0 and 0

3 vol(J0 ) L( 21 + s, χV µ)L( 12 + s, χV µ−1 ) 1 γ(V ) √ , 4 ζ(2s) 2

respectively.

312

CHAPTER 8

Proof. By the previous corollary and (8.4.10), it suffices to prove that Z Q× 2

1

χV µ(a)|a| 2 +r B(r, a; ϕL , ϕL ) d× a

=

 1 L( 2 + r, χV µ)L( 21 + r, χV µ−1 )    √   2 ζ(2r + 2)

if L = L0 ,

  L( 1 + r, χV µ)L( 21 + r, χV µ−1 )   √  2

if L = Lra .

2 2 ζ(2r)

By Lemma 8.4.4, the integral is equal to L( 12 + r, χV µ)L( 12 + r, χV µ−1 ) · W0 (r, L). ζ(2r + 1) On the other hand, [14] Proposition 13.4 gives (taking into account the different Haar measures used there)  1 ζ(2r + 1)    √

2 ζ(2r + 2) W0 (r, L) =  1 ζ(2r + 1)    √ 2 2 ζ(2r)

if L = L0 , if L = Lra .

Next, we consider the case L = L1 . Lemma 8.4.24.

and

1 W0 (r, L1 ) = √ ζ(2r + 1) 2 2

1 W0 (r, L1,# ) = √ (2 + ζ(2r + 1)). 2 2

Proof. Recall that here the Haar measure on V is the self-dual Haar measure 1 . The same calculation as in so that vol(L1 , dx) = [L1,# : L1 ]−1/2 = 2√ 2

313

ON THE DOUBLING INTEGRAL

[30], Section 4 gives W0 (r, L1,# ) Z

Z

=

L1,# r

Q2

1 = √ 2 2 ×

Z

ψ(−b Q(x)) dx db Z 1 Z 2 2

Q2

ψ(−κbx21 ) dx1 !r

Z

Z ( 21 Z2 )2

√ Z =2 2

ψ(−2κbx2 x3 ) dx2 dx3

min(1, |b

−1

Q

2 √ Z = 2 2

Z22

2 |) · min(1, | |) · b

ψ(−κby1 y2 ) dy1 dy2

Z

ψ(− Z2

db

κb 2 x ) dx db 4 

db + 2

− 32

4Z2

X

2

−n(r+ 21 )

Z× 2

n≥0



Z

b ψ( )χ2 (b)−n+1 db 8



X √ 1 1 1 2−n(r+ 2 )  = 2 2 + 4 8 n≥0, even

1 1 , = √ 2+ 1 − 2−2r−1 2 2 



as claimed. The case W0 (r, L1 ) is a special case of [30], Section 8. Proposition 8.4.25. Let L = L1 and X = 2−s . Then Z(s, ΦL1 , fwj )(wi )

i,j=0,1

vol(J0 )

1 1 L(s + , χV µ)L(s + , χV µ−1 ) 2 2 4 2 √ ! √ 2 2 −1 (2)X − X 2 ) (1 + 2 2 χ µ V δ κ √ . δκ √ (1 + 2 χV µ(2)X − X 2 ) 3 − X2 2

= γ(V ) ×




√

Proof. For convenience, let χ = χV µ. By (8.4.10) and the previous lemmas, the zeta integral matrix, for s = r, is equal to √ 0

γ(V ) vol(J )

I(r, χ; ϕL , ϕL ) δ√ κ I(r, χ; ϕL# , ϕL ) 2 2

2

!

δκ I(r, χ; ϕL , ϕL# ) , 1 2 I(r, χ; ϕL# , ϕL# )

where Z

I(r, χ; ϕL1 , ϕL2 ) =

Q× 2

1

χ(a)|a|r+ 2 B(r, a; ϕL1 , ϕL2 ) d× a.

314

CHAPTER 8

Now Lemma 8.4.4 gives I(r, χ; ϕL , ϕL ) =

L(r + 21 , χ)L(r + 12 , χ−1 ) · W0 (r, L) ζ(2r + 1)

1 1 1 = √ L(r + , χ)L(r + , χ−1 ), 2 2 2 2 and L(r + 21 , χ)L(r + 12 , χ−1 ) · W0 (r, L# ) ζ(2r + 1) 1 1 1 = √ L(r + , χ)L(r + , χ−1 ) · (3 − X 2 ). 2 2 2 2

I(r, χ; ϕL# , ϕL# ) =

Lemma 8.4.4 also gives I(r, χ; ϕL , ϕL# ) 1 = L(r + , χ) · W0 (r, L) 2 1 1 1 + χ−1 (2) 2−r− 2 L(r + , χ−1 ) · ( √ + W0 (r, L)) 2 2 1 1 −1 1 1 = √ L(r + , χ)L(r + , χ ) + χ−1 (2) 2−1 XL(r + , χ−1 ) 2 2 2 2 2 √ 1 1 1 −1 = √ L(r + , χ)L(r + , χ ) · (1 + 2χ(2)−1 X − X 2 ). 2 2 2 2

Switching the roles of L and L# , and of χ and χ−1 , by Lemma 8.4.4, one sees that √ 1 1 1 I(r, χ; ϕL# , ϕL ) = √ L(r + , χ)L(r + , χ−1 )·(1+ 2χ(2)X −X 2 ). 2 2 2 2

Note that the matrix for Z(s, Φ1 , ·) is not diagonal with respect to the 0 basis fev and fsp of I(µ)(J ,χκ ) . 1

Now we consider the case µ = χ2α | | 2 with α ≡ κ mod 4, where the special representation occurs as the unique irreducible submodule of I(µ). 1 In this case, χV µ = χακ | | 2 is unramified. Then matrix in the right hand side of the previous proposition is 2 δ κ √ (1 + χακ (2)X − X 2 ) 2

√ 2 2 δκ (1

!

+ 2χακ (2)X − X 2 ) , 3 − X2

315

ON THE DOUBLING INTEGRAL

and the function δκ fsp = f1 − √ fw 2 2 is an eigenfunction with eigenvalue (1 − χακ (2) X)2 . Thus Z(s, Φ1 , fsp ) = γ(V )

1 vol(J0 ) 1 √ L(s+ , χV µ)L(s+ , χV µ−1 )·(1−χακ (2) 2−s )2 ·fsp . 2 2 4 2 1

As in Section 8.4.4 above, for σ = σ ˜ (µχ2 ) = σ ˜ (χα | | 2 ), we have 1 1 L(s + , χV µ) = L(s + 1, χακ ) = L(s + , Wald(σ, ψκ )). 2 2 On the other hand, 1 1 1 L(s + , χV µ−1 ) = (1 − χακ (2) 2 2 2−s− 2 )−1 = (1 − χακ (2) 2−s )−1 , 2

and δ(σ, ψκ ) = −χ2ακ (2). Also note that, for vol(Sp1 (Z2 )) = 1, we have vol(J0 ) = 61 . Thus Z(s, Φ1 , fsp ) = δκ

1 1 √ L(s + , Wald(σ, ψκ )) · (1 + δ(σ, ψκ ) 2−s ) · fsp . 2 24 2

In summary, we have obtained the following results.

Theorem 8.4.26. (i) For an irreducible principal series representation σ = I(µ) = π ˜ (µχ2 ) with µ = χ2α | |t , t 6= ± 21 and α ≡ κ mod 4, 1 L(s + 12 , Wald(σ, ψκ )) · fev , Z(s, Φ0 , fev ) = δκ √ ζ(2s + 2) 2 2

and Z(s, Φ0 , fsp ) = 0.

(ii) For a special representation σ = σ ˜ (µχ2 ) = σ ˜ (χα | | 2 ) with µ = χ2α | | 2 and α ≡ κ mod 4, Then, Z(s, Φ0 , fsp ) = 0,

1

δα fsp = f1 − √ fw ∈ σ. 2 2

Z(s, Φ1 , fsp ) 1 1 1 = δκ √ · 2−2 (1 + δ(σ, ψκ ) 2−s ) · L(s + , Wald(σ, ψκ )) · fsp , 3 2 2 2

1

316

CHAPTER 8

and

Z(s, Φra , fsp ) 1 1 = −δκ √ · 2−2 (1 − δ(σ, ψκ ) 2−s ) · L(s + , Wald(σ, ψκ )) · fsp . 2 2 2 1 Note that, except for the factor δκ 2√ , these formulas are the same as 2 those given in Corollary 8.4.6 and Corollary 8.4.8 for p odd. As in Corollary 8.4.9, we have

e as in (8.4.14), Corollary 8.4.27. For fsp as in (ii) of Theorem 8.4.26, and Φ e f ) Z(s, Φ, sp

1 1 = −δκ √ 2−2 (1 − δ(σ, ψκ ) 2−s ) − (1 + δ(σ, ψκ ) 2−s ) B(s) 3 2 2 1 × L(s + , Wald(σ, ψκ )) · fsp . 2 



Finally, we consider the zeta integral for the Kohnen plus space section (8.4.34)

f + = (1 + 4χκ (−1)µ(2)2 ) fev + (2 − 4χκ (−1)µ(2)2 ) fsp ,

where α ≡ κ mod 4. By Theorem 8.4.26 (i) for µ = χ2α | |t with t 6= 21 , and α ≡ κ mod 4, (8.4.35) L(s + 12 , Wald(σ, ψκ )) δκ Z(s, Φ0 , f + ) = √ (1 + 4χκ (−1)µ(2)2 ) · fev . ζ(2s + 2) 2 2 8.4.6 The archimedean local integrals

In this section, we deal with the case F = R. We only consider the discrete series representations σ = π ˜`+ of G0 = G0R , where ` ∈ 12 + Z>0 . Recall from Table 2 in Section 8.2 that Wald(σ, ψ) = DS2`−1 , the discrete series representation of PGL2 (R) of weight 2` − 1. Since DS2`−1 ⊗ (sgn) ' DS2`−1 , we see that Wald(σ, ψ−1 ) = DS2`−1 as well. We fix the additive character ψ(x) = e(x) = e2πix . Let (V, Q) be a positive definite quadratic space over R of dimension 2`, 1 1 and note that χV = ((−1)`− 2 , )R = (sgn)`− 2 . The group G0 acts on S(V ) via the Weil representation ωV = ωV,ψ , and the Gaussian ϕ0 = e−2πQ(x) ∈ S(V ) of V is a eigenfunction of weight ` for the action of K 0 , the inverse image of SO(2) ⊂ SL2 (R) = Sp1 (R), i.e., ωV (k 0 )ϕ0 = ξ` (k 0 )ϕ0 .

317

ON THE DOUBLING INTEGRAL

Then f∞ := λV (ϕ0 ) ∈ I(` − 1, χV ) is the lowest weight vector in the unique irreducible submodule of I(`−1, χV ), and this submodule is isomorphic to the discrete series representation π ˜`+ . In what follows, we identify + σ=π ˜` with this submodule. Note that f∞ (1) = 1. Let I(s, χV ) be the degenerate principal series representation of G = GR and let Φ`∞ (s) ∈ I(s, χV ) be the unique standard section12 of weight `, normalized so that Φ`∞ (1, s) = 1. Note that, for k10 and k20 ∈ K 0 , r(i(k10 , k20 ))Φ`∞ (s) = ξ` (k10 ) ξ` (k20 ) · Φ`∞ (s). Thus, as explained in section 8.4.1, the local doubling integral defines a map 0

0

Z(s, Φ`∞ , ·) : σ (K ,ξ` ) −→ σ (K ,ξ` ) , which is given by multiplication by the scalar h∞ (s, `) = Z(s, Φ`∞ , f∞ )(1) =

Z Sp1 (R)

Φ`∞ (δi(1, g 0 ), s) f∞ (g 0 ) dg,

which we now compute using the same ‘interpolation’ method as in the nonarchimedean case. Let Vr = V + Vr,r be the direct sum of V and r copies of standard hyperbolic plane. Note that Vr has signature (2` + r, r). Decompose Vr orthogonally as Vr = Vr+ ⊕ Vr− with Vr+ ⊃ V positive definite and Vr− (r) negative definite. Then the function ϕ0 ∈ S(Vr ) defined by −

+

(r)

ϕ0 (x) = e−2πQ(xr )+2πQ(xr ) , − ± ± where x = x+ r + xr , with xr ∈ Vr , is a Gaussian for Vr , and (r)

(r)

Φ`∞ (s0 + r) = λVr (ϕ0 ⊗ ϕ0 ),

(8.4.36)

where s0 = ` − 23 . By Lemma 8.4.3, writing g 0 = n(b)m(a)k 0 , we have Φ`∞ (δi(1, g 0 ), s0 + r) Z

= γ(V ) Vr

12

(r)

(r)

ϕ0 (x) ωVr (g 0 )ϕ0 (−x) dx

= γ(V ) ξ` (k 0 ) χV (a) |a|r+`

Z

= γ(V ) ξl (k 0 ) χV (a) |a|r+`

Z

Vr

Vr

(r)

(r)

ϕ0 (x) ϕ0 (−ax) ψ(−b Q(x)) dx (r)

p

ϕ0 ( a2 + 1 x) ψ(−b Q(x)) dx.

As usual, we let K be the full inverse image in G of the standard maximal compact subgroup U (2) of Sp2 (R). By weight ` we mean transforming by the character det` under 1 the right action of K. Also note that χV = χ2κ where κ = (−1)`− 2 .

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CHAPTER 8

Therefore h∞ (s0 + r, `) = γ(V ) vol(SO(2))

∞

Z

|a|r+2`−2

0

×

Z

∞

Z

(r)

−∞ Vr

= vol(SO(2))

∞

Z

p

ϕ0 ( a2 + 1 x) ψ(−b Q(x)) dx db d× a

ar+2`−2 (a2 + 1)1−r−` d× a

0

× γ(V )

Z

∞

Z

−∞ Vr

(r)

ϕ0 (x) ψ(−b Q(x)) dx db.

It is well known that Z

∞

ar+2`−2 (a2 + 1)1−r−` d× a =

0

Γ(` + 2r − 1) Γ( 2r ) . 2 Γ(` + r − 1)

On the other hand,13 by [14], Proposition 14.1(iii), Z

∞

Z

γ(V ) −∞ Vr

(r)

ϕ0 (x) ψ(−b Q(x)) dx db = 2π e(− 14 (`− 21 )) ·

21−`−r Γ(` + r − 1) . Γ(` + 2r ) Γ( 2r )

Thus, h∞ (s0 + r, `) = vol(SO(2)) · 2π e(− 41 (`− 12 )) ·

21−`−r . (r + 2` − 2)

In summary,

Proposition 8.4.28. Let the notation be as above. Then with

Z(s, Φ`∞ , f∞ ) = h∞ (s, `) · f∞ ,

h∞ (s, `) = vol(SO(2)) · 2π e(− 41 (`− 12 )) ·

13

1 2

s+ 12

(s + ` − 12 )

.

Here we must be careful since we are working with [w, 1]L rather than [w, 1]R , which was used in [14]. This means that we must multiply the value in [14] by e( 18 ).

319

ON THE DOUBLING INTEGRAL

Recall that Wald(˜ π`+ , ψ) = Wald(˜ π`+ , ψ− ) is the discrete series representation DS2`−1 of PGL2 (R) of weight 2` − 1, so that 1 1 1 1 L(s + , Wald(˜ π`+ , ψ∞ )) = ΓC (s + ` − ) = 2 (2π) 2 −`−s Γ(s + ` − ). 2 2 2 Since

ζ∞ (2s + 2) = π −1−s Γ(s + 1), we have 1

(8.4.37) h∞ (s, `) = vol(SO(2)) · 2`−1 π `− 2 e(− 41 (`− 12 )) 1

`− L(s + 12 , Wald(σ, ψ∞ )) Y2 × · (s + j)−1 . ζ∞ (2s + 2) j=1

Since we are using Tamagawa measure on Sp1 (A) = SL2 (A), we have vol(SL2 (Q)\SL2 (A)) = 1. In the nonarchimedean calculations, we have normalized the measures on SL2 (Qp ) by the condition vol(SL2 (Zp ), dgp ) = 1, for all p. The measure on SL2 (R) is then determined. Lemma 8.4.29. vol(SO(2)) =

12 π.

Proof. Recall that we have used, for g = n(b) m(a) k, with a > 0, dg = a−3 da db dk.

By the remark above, we have 1 = vol(SL2 (Z)\SL2 (R)) = vol(SL2 (Z)\H, =

1 −2 1 y dx dy) vol(SO(2)) 2 2

π · vol(SO(2)), 12

since vol(SL2 (Z)\H,

1 −2 1 y dx dy) = . 2π 6

For the case ` = 32 , this gives:

Corollary 8.4.30. If σ = π ˜+ 3 and f∞ is the lowest weight vector in σ, then, with the notation above, 3 2

2

√

2 i L(s + 21 , Wald(σ, ψ−1 )) · · f∞ s+1 ζ∞ (2s + 2) √ 1 = −12 2 i · s · f∞ . 2 (s + 1)

Z(s, Φ∞ , f∞ ) = −12

320

CHAPTER 8

8.5 APPENDIX: COORDINATES ON METAPLECTIC GROUPS

In this appendix, we collect some information about the metaplectic extension, cocycles, coordinates, Weil representations, etc., which is used in this and other chapters of this book. References for this material include [19], [8], and [9]. However, none of these sources contains all of the facts we need, so we have provided some proofs. 8.5.1 Cocycles and coordinates

Let F be a nonarchimedean local field of characteristic zero with ring of integers O and uniformizer $, and let ψ be an unramified additive character of F .14 We change notation slightly and write G = Spn (F ) = Sp(W ), where W = F 2n with standard basis e1 , . . . , en , f1 , . . . , fn and with symplectic form given by < ei , ej >=< fi , fj >= 0 and < ei , fj >= δij . Let W = X + Y be the complete polarization of W where X (resp. Y ) is the span of the first (resp. last) n standard basis vectors. Let P = PY be the stabilizer of Y in G, and let K = Spn (O) = Spn (F ) ∩ GL2n (O). For 0 ≤ j ≤ n, let 1n−j





(8.5.1)



wj =  

0 1n−j −1j

1j  

 ∈ K,

0

and let w = wn . ˜ be the metaplectic extension Let G ˜ −→ G −→ 1, 1 −→ C1 −→ G ˜ = (G ˜ (2) × C1 )/{±1}, where G ˜ (2) is the unique nontrivial defined by G twofold topological cover of Sp(W ).   a b For each g = ∈ G, there is an operator r(g) on the Schwartz c d space S(F n ) defined by15 (8.5.2) Z
1 1 r(g)ϕ(x) = ψ (xa, xb)+(xb, yc)+ (yc, yd) ϕ(xa+yc) dg (y), 2 2 F n /ker(c)

14

This means that ψ is trivial on O and nontrivial on $−1 O. In the corresponding formula in [9], the notation < , > denotes the symplectic form on W and x ∈ X and y ∈ Y . If we identify these elements with row vectors x = [x0 , 0] and y = [0, y0 ] with x0 and y0 ∈ F n and write g ∈ Sp(W ) as a matrix, as above, then, the term < xb, yc >=< [0, x0 b], [y0 c, 0] >= −(x0 b, y0 c). This accounts for the change in sign in the formula here. 15

321

ON THE DOUBLING INTEGRAL

where, for x, y ∈ F n (row vectors), (x, y) = xt y, and the measure dg (y) on F n /ker(c) is normalized to make this operator unitary [19], [29]. These operators defined a projective representation of G on S(F n ) and (8.5.3)

r(g1 )r(g2 ) = cL (g1 , g2 ) r(g1 g2 ),

where cL is the Leray cocycle, given by [19] and [18]: (8.5.4)

cL (g1 , g2 ) = γ(ψ ◦ q(g1 , g2 )).

Here q(g1 , g2 ) = Leray(Y g1 , Y, Y g2−1 )

(8.5.5)

is the Leray invariant of the triple of isotropic subspaces (Y g1 , Y, Y g2−1 ) andγ(ψ ◦ q) is the Weil index of the character of second degree ψ ◦ q. We then obtain an isomorphism ∼

˜ G × C1 −→ G,

(g, z) 7→ [g, z]L ,

with product [g1 , z1 ]L · [g2 , z2 ]L = [g1 g2 , z1 z2 cL (g1 , g2 )]L .

The cocycle cL is trivial on G × P and P × G but is not trivial on K × K. Thus it must be modified for use in a global situation. Note that (8.5.6) (8.5.7) and (8.5.8)

1

r(m(a))ϕ(x) = | det(a)| 2 ϕ(xa), 1 r(n(b))ϕ(x) = ψ( (x, xb)) ϕ(x), 2 Z

r(w)ϕ(x) =

ψ((x, y)) ϕ(y) dy. Fn

Similarly, r(wj ) is given by the partial Fourier transform with respect to the last j coordinates. Example 8.5.1 In the case of n = 1, if gi = g1 g2 = g3 , then



ai bi ci di

1 cL (g1 , g2 ) = γ(ψ ◦ c1 c2 c3 ). 2

In particular, if c1 c2 c3 = 0, then cL (g1 , g2 ) = 1.



∈ Sp1 (F ) with

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CHAPTER 8

Remark 8.5.2. The dependence of the Leray coordinates on ψ is given explicitly as follows: [g, z]L,ψα = [g, z ξ(g, α, ψ)]L,ψ , where ξ(g, α, ψ) = χα (x(g)) χα (2)j(g) γF (α, ψ)j(g) , for χα (x) = (x, α)F . This can be checked easily using Proposition 4.3 in [9]. Suppose that the residue characteristic of F is odd and let ϕ0 ∈ S(F n ) be the characteristic function of On ⊂ F n . Then the formulas (8.5.6), (8.5.7) and (8.5.8) imply immediately that r(k)ϕ0 = ϕ0

if k = m(a), n(b) or wj .

Since these elements generate K, it follows that ϕ0 is an eigenfunction of r(k) for all k ∈ K, and we define a function λ on K by r(k)ϕ0 = λ(k)−1 ϕ0 .

(8.5.9)

Note that λ(m(a)) = λ(n(b)) = λ(wj ) = 1 and that cL (k1 , k2 ) λ(k1 ) λ(k2 ) λ(k1 k2 )−1 = 1. Thus, for p = n(b)m(a) ∈ P ∩ K, we have λ(p) = λ(n(b)) λ(m(a)) cL (n(b), m(a)) = 1. In fact, λ is bi-invariant under P ∩ K. For example, λ(pk) = λ(p) λ(k) cL (p, k) = λ(k). Thus, for any g ∈ G, writing g = pk with p ∈ P and k ∈ K, we may define (8.5.10)

λ(g) = λ(pk) := λ(k).

This allows us to renormalize coordinates and define an isomorphism ∼ ˜ G × C1 −→ G,

(8.5.11)

(g, z) 7→ [g, z] := [g, z λ(g)]L ,

with cocycle c(g1 , g2 ) = cL (g1 , g2 ) λ(g1 ) λ(g2 ) λ(g1 g2 )−1 . By construction, this cocycle is trivial on K × K, P × P , and P × K, via c(p, k) = cL (p, k) λ(p) λ(k) λ(pk)−1 = λ(k) λ(k)−1 = 1.

323

ON THE DOUBLING INTEGRAL

We will refer to the coordinates given by (8.5.11) as normalized coordinates. Note that, for p ∈ P , [p, 1] = [p, 1]L and [wj , 1] = [wj , 1]L , for example. ˜ under the splitting homoWe write K for the image of K = Spn (O) in G morphism ˜ K −→ G,

k 7→ k = [k, 1] = [k, λ(k)]L .

Remark 8.5.3. We have, for n− (c) ∈ K, i.e., for c ∈ Symn (O), λ(n− (c)) = cL (n− (c), w)−1 .

(8.5.12)

In fact, since n− (c)w = wn(−c), we have λ(n− (c)w) = λ(wn(−c)) = λ(w) = 1, whereas λ(n− (c)w) = λ(n− (c))λ(w) cL (n− (c), w). In the case n = 1, this gives the useful formulas (8.5.13)

λ(n− (c)) = γ(ψ− 1 c ), 2

and (8.5.14)



λ(



 γ(ψ−2cd )

a b )= 1 c d

if c 6= 0 and ord(c) > 0, otherwise.

8.5.2 The lifts of some homomorphisms

We now establish several facts used in the doubling calculation. We continue to work in greater generality than will be needed in this chapter and will use the notation G0 = Spn (F ) and G = Sp2n (F ) for the linear groups ˜ 0 and G ˜ for their metaplectic extensions. This differs slightly from the and G conventions of the rest of the chapter. There is no restriction on the residue characteristic of F . First, recall the embedding i0 : G0 × G0 → G given by the formula (8.1.1). Since the Leray cocycle is compatible with this embedding — this follows from the fact that Y = Y ∩ W1 + Y ∩ W2 for the decomposition ˜0 × G ˜ 0 → G, ˜ W = W1 + W2 used to define i0 — there is a lift i0 of i0 to G given in Leray coordinates by i0 : [g1 , z1 ]L × [g2 , z2 ]L 7→ [i0 (g1 , g2 ), z1 z2 ]L . The operators r(g) are also compatible with i0 , i.e., r(i0 (g1 , g2 )) = r(g1 ) ⊗ r(g2 )

324

CHAPTER 8

on S(F 2n ) ' S(F n ) ⊗ S(F n ). In the case of odd residue characteristic, since ϕ02n = ϕ0n ⊗ ϕ0n , this implies that λ(i0 (k1 , k2 )) = λ(k1 )λ(k2 ), and hence λ(i0 (g1 , g2 )) = λ(g1 )λ(g2 ). Thus, in normalized coordinates, i0 : [g1 , z1 ] × [g2 , z2 ] 7→ [i0 (g1 , g2 ), z1 z2 ], so that i0 is compatible with the splittings homomorphisms of K 0 × K 0 and K. Next, we recall the homomorphism ∨

0

0

: G −→ G ,

∨

g = Ad



1



−1

g

and the twisted embedding i(g1 , g2 ) = i0 (g1 , g2∨ ).

Lemma 8.5.4. The homomorphism ∨ : G0 → G0 has a lift to a homomor˜0 → G ˜ 0 , given in Leray coordinates by phism ∨ : G Proof. We note that

∨

: [g, z]L 7−→ [g ∨ , z −1 ]L .

(8.5.15) r(g ∨ )ϕ(x) ¯
1 1 ψ − (xa, xb) + (xb, yc) − (yc, yd) ϕ(xa ¯ − yc) dg (y) 2 2 F n /ker(−c)

Z

=

Z

= F n /ker(c)


1 1 ¯ + yc) dg (y) ψ − (xa, xb) − (xb, yc) − (yc, yd) ϕ(xa 2 2

= r(g)ϕ(x). This implies that cL (g1∨ , g2∨ ) = cL (g1 , g2 )−1 .

325

ON THE DOUBLING INTEGRAL

Thus, the homomorphism ˜0 × G ˜ 0 −→ G, ˜ i:G

i(g1 , g2 ) = i0 (g1 , g2∨ )

lifts i. In the case of odd residue characteristic, taking ϕ = ϕ0 in (8.5.15), we have r(k ∨ )ϕ0 = λ(k) ϕ0 , so that λ(k ∨ ) = λ(k)−1

λ(g ∨ ) = λ(g)−1 ,

and

and we obtain ∨

˜ 0 −→ G ˜ 0, :G

[g, z] 7−→ [g ∨ , z −1 ],

in normalized coordinates as well. In particular, ∨ and i are compatible with the splitting homomorphisms on K 0 and K 0 × K 0 respectively. We next prove (i) of Lemma 8.4.1. Recall that (8.5.16)

δ i(g, g) = p(g) δ.

Lemma 8.5.5. [δ, 1]L [i(g, g), 1]L = [p(g), 1]L [δ, 1]L .

Proof. On the right side,

[p(g), 1]L [δ, 1]L = [p(g)δ, 1]L , since the cocycle cL is trivial on P × G, whereas on the left side, [δ, 1]L [i(g, g), 1]L = [δ i(g, g), cL (δ, i(g, g))]L . Thus, we must show that the cocycle cL (δ, i(g, g)) is trivial. Note that, by (8.5.16), Leray(Y δ, Y, Y i(g, g)−1 ) = Leray(Y δ, Y i(g, g), Y ), so that, as a function of g the cocycle cL (δ, i(g, g)) is bi-invariant under P 0 . Thus it suffices to compute it for g = wj . To compute the Leray invariant, note that the isotropic 2n-planes Y δ, Y and Y i(g, g)−1 are spanned by the 2n rows of 

−1n 1n 0 0 , 0 0 1 n 1n 



0



1n 0

1n

,

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CHAPTER 8

and 

−t c

ta



tc

ta

,

for 

a=



1n−j

0



,

c=



0 −1j

,

respectively. Thus Y δ ∩ Y is spanned by the rows of


0 0 1n 1n , Y δ ∩ Y i(g, g)−1 is spanned by the rows of −t c

tc

ta

ta




,

and Y ∩ Y i(g, g)−1 is spanned by the rows of 

0 0 0 0

ta

0



0 ta .

Thus R = Y δ ∩ Y + Y ∩ Y i(g, g)−1 + Y δ ∩ Y i(g, g)−1 has rank 2n and q(δ, i(g, g)) = 0, as required. Since any choice of δ in the metaplectic extension of Sp2n (F ) has the form δ = [δ, z]L = [δ, 1]L · [1, z]L and since the element [1, z]L is central, statement (i) of Lemma 8.4.1 is clear. 8.5.3 The Weil representation

˜ is the metaplectic We now return to the notation in Section 8.5.1, so that G extension of Spn (F ), for example. We consider the Weil representation ˜ associated to a quadratic space (V, Q). Here, as before, (ωV , S(V n )) of G ψ has been fixed and η = ψ 1 . In [9], Proposition 4.3, this representation is 2 described in the Rao coordinates which are defined as follows: ∼ ˜ G × C1 −→ G,

(g, z) 7→ [g, z]R ,

where [g, z]R = [g, zβ(g)]L ,

327

ON THE DOUBLING INTEGRAL

for β(g) = γ(x(g), η)−1 γ(η)−j(g) .

(8.5.17)

Here the notation is as in [19] and [8]. Multiplication is then [g1 , z1 ]R · [g2 , z2 ]R = [g1 g2 , z1 z2 cR (g1 , g2 )]R . Note that, since the Rao cocycle cR is valued in {±1}, there is a character ˜ −→ C1 , ζ:G

(8.5.18)

[g, z]R 7→ z 2 .

In Leray coordinates, this character becomes ζ([g, z]L ) = z 2 β(g)−2 .

(8.5.19)

A short calculation using Proposition 4.3 of [9] yields the following formula for the Weil representation in Leray coordinates. Lemma 8.5.6. Let m = dimF (V ). Then

ωV ([g, z]L )ϕ(x) = χV (x(g)) ( z γ(η)j(g) )• γ(η ◦ V )−j(g) · rV (g)ϕ(x),

where

(

•= and

1 0

if m is odd, if m is even,

rV (g)ϕ(x) Z

=

ψ(tr


1 1 (xa, xb) + (xb, yc) + (yc, yd) ) ϕ(xa + yc) dg (y). 2 2

Here, for x, y ∈ V n , (x, y) = ((xi , yj )), and the measure dg (y) on the quotient V n /ker(c) is normalized to make the operator rV (g) unitary. Also, V

n /ker(c)

χV (x) = (x, (−1)

m(m−1) 2

det(V ))F ,

and det(V ) is the determinant of the matrix of the bilinear form on V . In particular (8.5.20)

where

(8.5.21)

ωV ([w, 1]L )ϕ(x) = γ(V )n

Z

ψ(tr(x, y)) ϕ(y) dy, Vn

γ(V ) = χV (−1) γ(η) γ(η ◦ V )−1 ,

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CHAPTER 8

and

γ(η ◦ V ) = γ(det V, η) γ(η)m (V ). Remark 8.5.7. In the formulas of Lemma 8.5.6, we should write ωV,ψ , [g, z]L,ψ and rV,ψ to indicate the dependence on the additive character ψ. The following useful relation is easy to check: (8.5.22)

ωVα ,ψα−1 ([g, z]L,ψ ) = ωV,ψ ([g, z]L,ψ ).

Since, in general, we will work with a fixed additive character ψ, we will suppress it from the notation and write simply ωV . If dim(V ) is odd, we have m

ωV ([n(b)m(a), z]L )ϕ(0) = z χV (det a) | det a| 2 ϕ(0), ˜ so that we obtain a G-intertwining map λV : S(V n ) −→ I(s0 , χV ), where s0 =

m 2

−

n+1 2 .

ϕ 7→ (g 7→ ω(g)ϕ(0),

We note several facts needed in this chapter.

Lemma 8.5.8. ωV ([ g ∨ , z −1 ]L )ϕ(x) = ωV ([ g, z ]L )ϕ(x).

Proof. Note that x(g ∨ ) = (−1)j(g) x(g) and that

rV (g ∨ )ϕ(x) = rV (g)ϕ(x), by the same argument as for r; see (8.5.15). The claimed formula then follows from the easily checked identity χV (−1) γ(η)2 γ(η ◦ V )−2 = 1. Recall that the discriminant of V is given by (8.5.23)

discr(V ) = (−1)

m(m−1) 2

det(V ) ∈ F × /F ×,2 ,

where det(V ) = det((vi , vj )), for any basis {vi } of V . Suppose that V = V1 +V2 is an orthogonal direct sum of quadratic spaces Vi with dimF (Vi ) = mi . Then rV (g) = rV1 (g) ⊗ rV2 (g), γ(ψ ◦ V ) = γ(ψ ◦ V1 ) · γ(ψ ◦ V2 ), discr(V ) = discr(V1 ) · discr(V2 ) · (−1)m1 m2 , and

329

ON THE DOUBLING INTEGRAL 1 m2 . χV (x) = χV1 (x) · χV2 (x) · (x, −1)m F

It follows that

˜ on S(V n ) ' S(V n ) ⊗ Lemma 8.5.9. The Weil representation ωV of G 1 n S(V2 ) is given by ωV (g) = ωV1 (g) ⊗ ωV2 (g) ·

 ζ(g)−1 1

if m1 m2 is odd, otherwise.

The extra factor ζ(g)−1 in the case where both V1 and V2 are odd-dimensional is due to the fact that, in our definition of ωV , [9], we have ‘twisted’ the naturally defined Weil representation by a power of ζ in order to obtain trivial central character in the even-dimensional case and central character [1, z]L 7→ z in the odd-dimensional case. Here we have used the identity (x(g), −1)F z 2 γ(η)2j(g) = ζ([g, z]L ). We finish with two examples. First, suppose that V = V0 = F with quadratic form Q(x) = 21 x2 . Then the operator rV (g) reduces to the operator r(g) on S(F n ). On the other hand, det(V ) = discr(V ) = 1 and γ(η ◦ V ) = γ(η), so that, in the odd residue characteristic case and in normalized coordinates, ωV ([g, z]) = z λ(g) γ(η)j(g) γ(η ◦ V )−j(g) rV (g) = z λ(g) r(g). It follows that the function ϕ0 ∈ S(F n ) is invariant under K, and so g 7→ ωV (g)ϕ0 (0) is the K-fixed vector in the induced representation I(− n2 , χ1 ), where χ1 is the trivial character. Next, suppose that V = Vr,r , where the matrix for the bilinear form is 

1r 1r



.

Then discr(V ) = 1, χV = χ0 = 1, and γ(η ◦ V ) = 1. Thus ωV ([g, z]L ) = rV (g), so the operators rV (g) give a representation of Spn (F ). If ϕ0r is the characteristic function of M2r,n (O) ⊂ V n , then, for ψ unramified, ϕ0r is invariant under the generators of K and hence under all of K. Thus, we again have (8.5.24)

Φ0 (g, r −

n+1 ) = ωV (g)ϕ0r (0), 2

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CHAPTER 8

where Φ0 (g, s) is the K-fixed vector in I(s, χ0 ), the induced representation of Spn (F ). Note that even residue characteristic is allowed here. Taking Vr = V + Vr,r and using Lemma 8.5.9, we have (8.5.25)

ωVr (g) ϕ ⊗ ϕ0r )(0) = Φ(g, s0 + r),

where Φ(s) is the extension to a standard section of the image λV (ϕ) of n+1 0 ϕ ∈ S(V n ) in I(s0 , χV ), with s0 = m 2 − 2 . For V = V0 , ϕ = ϕ , and 0 odd residue characteristic, Φ(s) = Φ (s) is the unique K-fixed vector in I(r − n2 , χ0 ) with Φ0 (1, s) = 1. The following useful fact is easy to check. We refer to Lemma 8.5.14 for the case of even residue characteristic. Lemma 8.5.10. Assume that the residue characteristic of F is odd. Let L ⊂ V be a lattice such that $r L] ⊂ L ⊂ L] , where L] = { x ∈ V | (x, y) ∈ O, ∀y ∈ L }

is the dual lattice and r > 0. Let 



a b Jr = { ∈K|c≡0 c d

mod $r O },

and let Jr ⊂ K be the corresponding subgroup. Then the characteristic function ϕL ∈ S(V n ) of Ln ⊂ V n is an eigenfunction of Jr with ωV (k) ϕL = χV (det(a)) ϕL ,

for all k ∈ Jr .

Finally, for a quaternion algebra B over Qp , we take V = { x ∈ B | tr(x) = 0 } with Q(x) = ν(x) = −x2 . Let ψp be the local component of the global unramified character ψ of A/Q with ψ∞ (x) = e(x). Then χV (a) = (a, −2)p ,

γp (η ◦ V ) = invp (B) ·

 1 e( )    8

if p = 2,

  

otherwise,

e(− 81 ) if p = ∞,

1

and γp (V ) = invp (B) ·

 i   

if p = 2,

  

otherwise.

−i if p = ∞, 1

Note that γp (η) = 1 for p 6= 2, ∞, γ2 (η) = e(− 18 ), and γ∞ (η) = e( 18 ). Recall that γ(η ◦ V )−1 is computed in Lemma 13.3 of [14].

331

ON THE DOUBLING INTEGRAL

8.5.4 The case p = 2

Almost everything in the previous sections is valid for any residue characteristic, except for the results involving the splitting homomorphism over K where p was required to be odd. In this section, we discuss the changes needed when p = 2. Let F be a 2-adic local field with ring of integers O and a uniformizer $. Let ψ be a unramified additive character of F . Let G = Spn (F ), K = Spn (O), and J = {γ ∈ K | c ≡ 0 mod (4)}. ˜ as follows. Let V = F with Q(x) = x2 , We can define a splitting J → G so that (x, y) = 2xy. Then det(V ) = 2, χV (a) = (a, 2)F = χ2 (a), and γ(η ◦ V ) = γ(2, η)γ(η). For L = O, the dual lattice is L] = 12 L, and |L] : L| = |O/2O|. Then the function ϕL = char(L)n ∈ S(F n ) is an eigenvector for the operators ωV ([m(a), 1]L ), ωV ([n(b), 1]L ), and ωV ([n− (c), 1]L ), for c ≡ 0 mod (4). Explicitly, Lemma 8.5.11. For an integer 0 ≤ r ≤ n, let wr be as in (8.5.1). Then ωV ([m(a), 1]L )ϕL = χ2 (det a) ϕL , ωV ([n(b), 1]L )ϕL = ϕL , r

ωV ([wr , 1]L )ϕL = |2| 2 γ(2, η)−r · ϕLr , and

r

ωV ([wr , 1]L )ϕLr = |2|− 2 γ(2, η)−r · ϕL , ωV ([n− (c), 1]L )ϕL = cL (n− (c), w) · ϕL .

Proof. Notice that w = wn and ϕL] = ϕLn . We prove the last formula. Note that n− (c)w = wn(−c). Setting n

∗ = |2| 2 γ(2, η)n , we have ωV ([n− (c), 1]L )ϕL = (∗) · ωV ([n− (c), 1]L ) ωV ([w, 1]L )ϕLn = (∗) cL (n− (c), w) · ωV ([n− (c) w, 1]L )ϕLn = (∗) cL (n− (c), w) · ωV ([w, 1]L ) ωV ([n(−c), 1]L )ϕLn = cL (n− (c), w) · ϕL . Here, in the last step, we have used the fact that, for x ∈ L] , Q(x) ∈ so that ωV ([n(−c), 1]L )ϕL] = ϕL] , since c ≡ 0 mod (4).

1 4

O,

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CHAPTER 8

Every element of J can be written uniquely in the form (8.5.26)

k = n(b)m(a)n− (c),

for a ∈ GLn (O), b ∈ Symn (O), c ∈ 4 Symn (O), and ωV ([k, 1]L )ϕL = λ2 (k)−1 ϕL , with λ2 (k) = χ2 (det a) cL (n− (c), w)−1 .

(8.5.27)

Thus, we obtain a splitting homomorphism ˜ J → G,

(8.5.28)

k 7→ k = [k, λ2 (k)]L ,

˜ be the image of J under this homomorphism. By where k ∈ J. Let J ⊂ G construction, ϕL is invariant under J. Remark 8.5.12. In the case n = 1, 

λ2 (



a b ) = χ2 (a) γ(ψ−2cd ). c d

In Section 4.5.2, we will need some additional cocycle information. For example, note that wn(b)m(a)n− (c)w−1 = n− (−b)m(a−1 )n(−c) ∈ P˜

⇐⇒

b = 0,

and that [w, 1]L [m(a), χ2 (a)]L = [m(a), χ2 (a)]L [w, 1]L . Lemma 8.5.13. (i)

[w, 1]L [n− (c), λ2 (n− (c))]L = [n(−c), 1]L [w, 1]L .

(ii) For k = n(b)m(a)n− (c) ∈ J with a2 = (1 + 2b)(1 − 2c0 ), [n− (2), 1]L [n(b)m(a)n− (c), λ2 (k)]L where

= [n(b(1 + 2b)−1 ) m(a−1 (1 − 2c0 )), ∗]L [n− (2), 1]L ,

∗ = cL (n− (2), k) · λ2 (k) = γ(1 − 2c0 , ψ).

333

ON THE DOUBLING INTEGRAL

Proof. First consider (i). Since the Leray cocycle satisfies cL (P, G) = 1 = cL (G, P ), we have [w, 1]L [n− (c), λ2 (n− (c))]L = [wn− (c), cL (w, n− (c))λ2 (n− (c))]L = [n(−c)w, cL (w, n− (c))λ2 (n− (c))]L = [n(−c), cL (w, n− (c))λ2 (n− (c))]L [w, 1]L = [n(−c), 1]L [w, 1]L , since cL (w, n− (c))λ2 (n− (c)) = 1 by definition of λ2 For (ii), we take k = n(b)m(a)n− (c) ∈ J with (8.5.29)

a2 = (1 + 2b)(1 − 2c0 ).

Then [n− (2), 1]L [n(b)m(a)n− (c), λ2 (k)]L = [n(b(1 + 2b)−1 ) m(a−1 (1 − 2c0 )), ∗]L [n− (2), 1]L , where ∗ = cL (n− (2), k) · λ2 (k). When equation (8.5.29) holds, then 2a + a−1 (1 + 2b)c = 2a−1 (1 + 2b), so that 2 · a−1 c · 2a−1 (1 + 2b) = 4a−2 c(1 + 2b) and cL (n− (2), n(b)m(a)n− (c)) = γ(ψ2c(1+2b) ), where we use the fact that γ(ψα2 β ) = γ(ψβ ). Now ∗ = γ(ψ2c(1+2b) ) · γ(ψ−2c ) = γ(ψ−2c )γ(2c(1 + 2b), ψ)γ(ψ) = γ(ψ−2c )γ(2c, ψ)γ(ψ)(2c, 1 + 2b)γ(1 + 2b, ψ) = (2c0 , 1 − 2c0 )γ(1 + 2b, ψ) = γ(1 − 2c0 , ψ), where (8.5.29) was used in the last two steps. Finally, we consider the Weil representation and record the analogue of Lemma 8.5.10 in the p = 2 case.

334

CHAPTER 8

Proposition 8.5.14. Supppose that (V, Q) is a quadratic space over F and that L ⊂ V is a O-lattice such that, for some r ≥ 0, 2$r L] ⊂ L ⊂ L] ,

where L] is the dual lattice. Let

Jr = { k ∈ K = Spn (O) | c ≡ 0

mod (4$r ) },

and let Jr be the corresponding subgroup of J. Let ϕL ∈ S(V n ) be the characteristic function of Ln ⊂ V n . Then ϕL is an eigenfunction for Jr with eigencharacter χV χ2 , i.e., 

ωV (k)ϕL = χV χ2 (det a) · ϕL ,

k = [k, λ2 (k)],

k=



a b ∈ Jr . c d

Proof. It is enough to check on the generators, (8.5.26), of Jr . We have ωV ([m(a), χ2 (det a)]L ]ϕL = χV χ2 (det a) ϕL , and ωV ([n(b), 1]L )ϕL = ϕL . Since, for x ∈ L] , 4$r Q(x) = (2$r x, x) ∈ O, we have ωV ([n(−c), 1]L )ϕL] = ϕL] for c ≡ 0 mod (4$r ). The same argument as before then yields ωV ([n− (c), λ(n− (c))]L )ϕL = λ(n− (c)) cL (n− (c), w)ϕL = ϕL . 8.5.5 The global metaplectic group

We now turn to the cocycle for the global metaplectic group. Recall that ψ is ˜ A be the our fixed unramified character of A/Q with ψ∞ (x) = e(x). Let G metaplectic cover of G(A) = Spn (A). For p odd, we have the normalized ˜ p , given by (8.5.11), coordinate system for G (8.5.30)

˜ p, Gp × C1 → G

(g, z) 7→ [g, z]p = [g, zλp (g)]L ,

and the associated cocycle c0p (·, ·) is trivial on Kp × Kp . For g1 , g2 ∈ G(A), the global cocycle (8.5.31)

c(g1 , g2 ) = cL (g1,∞ , g2,∞ ) cL (g1,2 , g2,2 )

Y p6=2,∞

c0p (g1,p , g2,p )

335

ON THE DOUBLING INTEGRAL

˜ A ,16 is then well defined and gives a normalized coordinate system for G (8.5.32) Y ˜ A , (g, z) 7→ [g, z] = [g∞ , z]L,∞ [g2 , 1]L,2 G(A) × C1 → G [gp , 1]p . p6=2, ∞

Let 



a b ˆ | c≡0 K0 (4) = {g = ∈ Spn (Z) c d

mod 4} = K0 (4)2 ×

Y

Kp .

p6=2

Then there is a splitting (8.5.33)

˜ A, K0 (4) → G

k 7→ [k, λ2 (k)],

˜ A. where λ2 (k) is given by (8.5.27). Let K0 (4) be the image of K0 (4) in G On the other hand, there is a unique splitting homomorphism of G(Q) into ˜ A , whose image we denote by GQ . In terms of the normalized coordinates, G this splitting has the following description. Lemma 8.5.15. (i) For g ∈ G(Q), λp (g) = 1 for almost all p. In particular, λ(g) := is well defined. Moreover,

Y

λp (g)

p6=2, ∞

c0p (g1 , g2 ) = cL,p (g1 , g2 ) = 1

for almost all p. (ii) For g1 , g2 ∈ G(Q), there is a product formula Y

cL,p (g1 , g2 ) = 1.

˜ A is given by (iii) The splitting homomorphism s : G(Q) → G p≤∞

s(g) = [g, λ(g)−1 ].

Proof. To prove (i), we write g = p1 wr p2 with p1 , p2 ∈ PQ , the Siegel parabolic subgroup of GQ . We have p1 , p2 ∈ Kp ∩ PQ , for almost all primes p. Since λp is bi-invariant under Kp ∩ Pp , we then have λp (g) = λp (p1 wr p2 ) = λp (wr ) = 1. If λp (g1 ) = λp (g2 ) = 1, then c0p (g1 , g2 ) = cL,p (g1 , g2 ). On the other hand, c0p (g1 , g2 ) = 1 whenever g1 and g2 are both in Kp and p = 6 2. This give the 16

˜ p maps to G ˜ A for every p ≤ ∞. Here, in the last expression, we are using the fact that G

336

CHAPTER 8

second statement in (i). To prove (ii), note that the product is well defined by the second statement in (i). Now, for g1 , g2 ∈ GQ , there is a global quadratic form Q = Leray(Y g1 , Y, Y g2−1 ), and the product formula Y

γp (ψ ◦ Q) = 1

p≤∞

for the Weil indices implies (ii). To prove (iii), it suffices to check c(g1 , g2 ) = λ(g1 )λ(g2 )λ(g1 g2 )−1 . By (i), (ii), and the definitions, we have c(g1 , g2 ) =

Y p=2, ∞

cL,p (g1 , g2 )λp (g1 )λp (g2 )λp (g1 g2 )−1

Y

cL,p (g1 , g2 )

p6=2, ∞

= λ(g1 )λ(g2 )λ(g1 g2 )−1

Y

cL,p (gp , gp )

p

= λ(g1 )λ(g2 )λ(g1 g2 )−1 . 8.5.6 The multiplier system for Siegel modular forms of half integral weight

In this subsection, we describe the transformation laws of Siegel modular forms of half integral weight in terms of the metaplectic cocycle. Let K∞ ∼ = U (n) be the standard maximal compact subgroup of G(R) = ˜∞ ∼ ˜ (n) be its preimage in G ˜ R . Then for each half Spn (R) and let K = U 1 ˜ ∞ such that, integer `∈ 2 +Z, there is a unique genuine character ξ` of K

for g =

(8.5.34)

A B −B A

∈ K∞ with A + iB ∈ U (n),

ξ`2 ([k, z]R ) = z 2 det(A + iB)2` .

ξ` ([1, z]R ) = z,

Here [g, z]R are Rao coordinates. Equivalently, in Leray coordinates, we have (8.5.35)

ξ`2 ([k, z]L ) = z 2 β(k)−2 det(A + iB)2` ,

where β is given by (8.5.17). Next we define the automorphy factor. For τ = u + iv ∈ Hn , let 1

gτ = n(u)m(v 2 )

and

gτ0 = [gτ , 1]L .

337

ON THE DOUBLING INTEGRAL

Then, for γ =


b

a c d

∈ G(R), write γ · gτ = gγ(τ ) · k(γ, τ )−1 ,

with k(γ, τ ) =



A B −B A



∈ K∞ ,

det(A + iB) =

cτ + d . |cτ + d|

This implies that 0 0 −1 γ 0 · gτ0 = gγ(τ ) · k (γ, τ ) ,

˜ R. where g 0 = [g, 1]L ∈ G Definition 8.5.16. For γ ∈ Γ0 (4) = G(Q) ∩ K0 (4), and for ` ∈

1 2

+ Z, let

j` (γ, τ ) := λ(γ) λ2 (γ) · ξ` (k 0 (γ, τ )) · | det(cτ + d)|` , where λ(γ) and λ2 (γ) are defined in the previous section. The first part of the next lemma is immediate from the fact that λ(γ), λ2 (γ), and β(k) are all 8-th roots of unity. The second part is a standard classical result. Lemma 8.5.17. (i) For γ ∈ Γ0 (4)

j` (γ, τ )8 = det(cτ + d)8` .

In particular, η` (γ) = j` (γ, τ )j 1 (γ, τ )−2` is independent of τ and is a 2 character of Γ0 (4) of order dividing 8. (ii) When n = 1, j` (γ, τ ) =

and

−1 d

c d

 



η` (γ) =

−1 d

· (cτ + d)` ,

`− 1

2

.

Here d = 1 or i depending on whether d ≡ 1 or −1 mod 4, and 1

z2 =

√

reiθ/2 ,

if z = reiθ ,

−π < θ ≤ π.

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CHAPTER 8

Remark 8.5.18. In the case of classical modular forms of half-integral weight, Shimura [21] uses the automorphy factor ( j 1 (γ, τ ) )` = 2



−1 d

`− 1

2

· j` (γ, τ ).

This is more natural when taking products of such forms. Lemma 8.5.19. For γ =

a b 0d




∈ Γ0 (4), 1

j` (γ, τ ) = sgn(det d)`− 2 .

Proof. Since, for γ = a0 db ∈ Γ0 (4), d ∈ GLn (Z) and det a = det d = ±1, (8.5.27) gives λ2 (γ) = χ2 (det d) = 1 and (8.5.9) gives λp (γ) = 1 for 1 p 6= 2. To compute ξl (k 0 (γ, τ )), let l = (sgn)l− 2 , and let Φl∞ be the eigen˜ ∞ in the induced representation I∞ (l ) with eigencharacter ξl . function of K By evaluating Φ`∞ on the element


γ 0 gτ0 = gγ(τ ) k 0 (γ, τ )−1 , we find that ξl (k 0 (γ, τ )) = l (det a), as claimed. Let N be a positive integer and let χ be a character of Γ0 (4N ). We say that a function f on Hn is of weight l, level 4N , and character χ if (8.5.36)

f (γτ ) = jl (γ, τ )χ(γ)f (τ )

˜ A is of weight l, for every γ ∈ Γ0 (4N ). We say that a function φ on GQ \G level 4N , and character χ, where χ is a character of K0 (4N ), if (8.5.37) and (8.5.38)

φ(gk) = χ(k −1 ) · φ(g) for k ∈ K0 (4N ), ˜ ∞. φ(gk∞ ) = ξl (k∞ ) · φ for k∞ ∈ K

˜ A is of weight l, level 4N , Proposition 8.5.20. If the function φ on GQ \G and character χ, then the function l

f (τ ) := (det v)− 2 · φ(gτ0 ),

τ = u + iv ∈ Hn ,

is of weight l, level 4N , and character χ, where χ is the pullback of χ to Γ0 (4N ).

339

ON THE DOUBLING INTEGRAL

Proof. The argument is just an adelic version of that given by Shintani [22]. We have f (γ(τ )) = det v(γ(τ ))


− 1 ` 2

· φ(gγ(τ ) )

1

= (det v)− 2 ` · det |cτ + d|` · φ(γ 0 gτ0 k 0 (γ, τ ))

(8.5.39)

1

= (det v)− 2 ` · det |cτ + d|` · ξ` (k 0 (γ, τ )) · φ(γ 0 gτ0 ). We write γ 0 = [γ, 1]L,∞ = s(γ) · [1, λ(γ)] · [γ, 1]−1 L,2

(8.5.40)

Y

[γ, 1]−1 p

p6=2, ∞

= s(γ) · [1, λ(γ) λ2 (γ)] · [γ, λ2 (γ)]−1 L,2

Y

[γ, 1]−1 p

p6=2, ∞

= s(γ) · [1, λ(γ) λ2 (γ)] · γ −1 , where γ ∈ K0 (4N ) is the image of γ under the splitting homomorphism. Then, in the last line of (8.5.39), φ(γ 0 gτ0 ) = λ(γ) λ2 (γ) χ(γ) · φ(gτ0 ), as required. This proposition yields the transformation law of the genus two Eisenstein series E2 (τ, s; B) studied in Chapter 5. Proposition 8.5.21. for all γ =

E2 (γ(τ ), s; B) = sgn(det d) · j 3 (γ, τ ) · E2 (τ, s; B),

∈ Γ0 (4D(B)o ), where D(B)o is the odd part of D(B). 2


b

a c d

Note that sgn(det d) = (−1, det d)2 . If c = 0, we have j 3 (γ, τ ) = 2 sgn(det d), so that (8.5.41)

E2 (γ(τ ), s; B) = E2 (τ, s; B),

in this case. Proposition 8.5.21 follows from the eigenproperties of the sections used to define E2 (τ, s; B).

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CHAPTER 8

Proposition 8.5.22. Let χ = χ2κ , with κ = ±1, and let Φ• (s) ∈ In (s, χ), with • = 0, 1, or ra, be the standard sections defined in (8.4.13) and Section 5.1. Let 

Jr = {



a b ∈ Kp | c ≡ 0 c d

mod 4pr },

˜ p under the splitting and let Jr ⊂ K0 (4) be the image of Jr ⊂ K0 (4) in G homomorphism. (i) For p 6= 2, r(k)Φ• (s) = Φ• (s), (ii) For p = 2,

for all k ∈ Jr , where r =

r(k)Φ• (s) = χκ (det d) · Φ• (s),

where k = (k, λ2 (k)) with k =

a b c d




.

 0 1

if • = 0,

if • = 1, ra.

for all k ∈ J0 ,

˜ p intertwining, and the Proof. The map λV : S(V n ) → In (s0 , χV ) is G passage from an element of In (s0 , χV ) to the associated standard section is ˜ p . Thus, by (8.4.13), it suffices to observe compatible with the action of K the following facts, which are a special case of Lemma 8.5.10 and Proposition 8.5.14. Lemma 8.5.23. Let V = V ± , and let L0 , L1 in V + and Lra in V − be the lattices defined in (8.4.12). (i) For p 6= 2, ωV,ψp (k) char(L• ) = char(L• ),

where r = 0 if • = 0 and r = 1 if • = 1 or ra. (ii) For p = 2,

for all k ∈ Jr ,

ωV,ψp (k) char(L• ) = χκ (det d) · char(L• ),

for all k ∈ J0 .

This finishes the proof of Proposition 8.5.22. 8.5.7 The level of the vertical component of φb1 (τ )

In this section, we determine the level of the theta functions associated to the vertical component of the genus one generating function φb1 (τ ) considered in Section 4.3; see Remark 4.3.5. This will complete the proof of Theorem A. We fix a prime p | D(B), and we determine the compact open subgroup

341

ON THE DOUBLING INTEGRAL

of the metaplectic group G0 = G0p which fixes the Schwartz function µ[Λ] defined in Lemma 4.3.3. We slightly simplify the notation of that section by letting V = V 0 (Qp ) = { x ∈ M2 (Qp ) | tr(x) = 0 }, and Λ = Z2p ⊂ Q2p . Then ϕ[Λ] = ϕ0 := char(L),

L = V ∩ M2 (Zp ),

and we have ϕ(x) := µ[Λ] (x) = ϕ0 (x) − p ϕ0 (p−1 x) − ϕ∼ (x), where the function ϕ∼ is defined as follows. If p 6= 2, ϕ∼ (x) = χ(− det(x)) · char(V ∩ GL2 (Zp ))(x), where χ() = (, p)p . If p = 2, ϕ∼ (x) = p · char(V ∩ (1 + p M2 (Zp )))(x). Note that supp(ϕ) ⊂ L and that ϕ is constant on pL cosets. In addition, ϕ is invariant under conjugation by GL2 (Zp ) and under scaling by elements of Z× p. Let 

Kr = { k =



a b ∈ SL2 (Zp ) | ord(c) ≥ r }. c d

There is a splitting homomorphism ∼

Kr −→ Kr ⊂ G0 ,

k 7→ k = [k, λ(k)]L ,

where r = 0, if p 6= 2, and r = 2, if p = 2.

Proposition 8.5.24. (i) For p 6= 2, the function ϕ is invariant under the subgroup K1 . (ii) For p = 2, the function ϕ is an eigenfunction for the subgroup K2 with character 

χ−1



a b : 7→ χ−1 (a) = (a, −1)2 . c d

Proof. If r ≥ 1, any element k ∈ Kr can be written uniquely in the form k = n− (c)m(a)n(b), where, as usual, 



1 n− (c) = , c 1



m(a) =



a a−1



,



1 b n(b) = . 1

342

CHAPTER 8

We write n− (c), m(a), and n(b) for the images of these elements under the splitting homomorphism, where, if p = 2, we suppose that r ≥ 2. Note that χV = χ−2 . Then, for a ∈ Z× p , we have ω(m(a))ϕ(x) = χV χ2 (a) ϕ(ax) = χ−1 (a) ϕ(x), since ϕ is invariant under scaling by units, and ω(n(b))ϕ(x) = ψ(b Q(x)) ϕ(x) = ϕ(x), since supp(ϕ) ⊂ L, Q is Zp -valued on L and ψ is unramified. Next note that we have the relation n− (c) = [w, 1]−1 L n(−c) [w, 1]L , by Lemma 8.5.13, when p = 2 and from the fact that [w, 1]L = w ∈ K0 , in the case p 6= 2. By (8.5.20), we have Z

ωV ([w, 1]L )ϕ(x) = γ(V ) V

b ψ(tr(x, y)) ϕ(y) dy = γ(V ) · ϕ(x).

Thus, it suffices to prove the following result about the support of the Fourier b transform ϕ.

b Lemma 8.5.25. If ϕ(x) = 6 0, then

ord(Q(x)) ≥

if p 6= 2,

 −1

if p = 2.

−2

b ⊂ p−1 L] and that ϕ b is constant on L] cosets. Proof. Note that supp(ϕ) Thus, we may view ϕb as a function on the vector space p−1 L] /L] of dimension 3 over Fp . In addition, this function is invariant under scaling by F× p and under the action of GL2 (Fp ) induced by the conjugation action of GL2 (Zp ) on L] . First consider the case p = 6 2, so that L] = L and we can identify p−1 L/L ' L/pL with the set V (Fp ) of trace zero 2 × 2 matrices over Fp . The conjugation action of GL2 (Fp ) gives the action of SO(V (Fp )) and there are three orbits of GL2 (Fp ) × F× p with orbit representatives 

0,



1 0



,

1



−1



,



1 β

,

where χ(β) = −1. We must show that the function determined by ϕb vanishes on the last two representatives. Of course, setting ϕ0 (x) = ϕ0 (p−1 x) = char(pL)(x)

343

ON THE DOUBLING INTEGRAL

and ϕ00 (x) = ϕ0 (px) = char(p−1 L)(x), we have c0 − p ϕb0 = ϕ0 − p−2 ϕ00 , ϕ

so that it remains to calculate the values of ϕc∼ on the last two representatives. For the first of these, we have ϕc∼ (p−1



1



−1

Z

)= V ∩GL2 (Zp )

= p−3

ψ(−p−1 2y0 ) χ(y02 + y1 y2 ) dy ψ(−p−1 2y0 ) χ(y02 + y1 y2 ).

X y∈V (Fp ) det y6=0

Now the contribution of the set of y where y0 = 0 vanishes, since the character χ of F× p is nontrivial. This leaves the quantity p−3

X

ψ(−p−1 2y0 ) χ(y02 + y1 y2 )

X

y1 ,y2 ∈Fp y0 ∈F× p y02 +y1 y2 6=0

= p−3

X

ψ(−p−1 2y0 ) χ(1 + y1 y2 )

X

y ,y ∈Fp y0 ∈F× p 1 2 y1 y2 6=−1

= −p−3

X

χ(1 + y1 y2 )

y1 ,y2 ∈Fp y1 y2 6=−1

= −p−3

X

X

χ(u)

u∈F× p

1

y1 ,y2 ∈Fp y1 y2 =u−1

= −p−3 (p − 1)

X

χ(u) − p−2

= −p−2 .

u∈F× p

Here, in the last two steps, we note that the number of solutions of y1 y2 = u − 1 is p − 1, if u 6= 1, and 2p − 1, if u = 1. It follows that b ϕ(p

−1



1



−1

) = 0,

344

CHAPTER 8

as claimed. Next we calculate ϕc∼ (p−1



1



Z

)=

β

V ∩GL2 (Zp )

= p−3

ψ(−p−1 (y2 + βy1 )) χ(y02 + y1 y2 ) dy ψ(−p−1 (y2 + βy1 )) χ(y02 + y1 y2 ).

X y∈V (Fp ) det y6=0

The terms where y0 = 0 give p−3 χ(β)

ψ(p−1 (y1 + y2 )) χ(y1 y2 )

X y1 ,y2

=p

−3

ψ(p−1 y1 (1 + y2 )) χ(y2 )

X

χ(β)

= −p−2 χ(−1),

y1 ,y2

where, in the last step, we note that the sum on y1 gives −χ(y2 ), if y2 6= −1, and (p − 1) χ(y2 ), if y2 = −1. Next, the terms with y0 6= 0 give p−3

X

χ(1 + y1 y2 )

y1 ,y2 y1 y2 6=−1

= p−3

X

ψ(−p−1 y0 (y2 + βy1 ))

y0 6=0

X

χ(u)

u∈F× p

X

X

ψ(−p−1 y0 (y2 + βy1 )).

y1 y2 =u−1 y0 6=0

The inner sum is −1, if y2 + βy1 6= 0, and p − 1, if y2 + βy1 = 0. When u = 1, the sum on y1 and y2 then gives −2(p − 1) + p − 1 = −(p − 1). If u = 6 1, then y2 = (u − 1)y1−1 , and we note that the equation (u − 1)y1−1 + βy1 = 0 has two solutions if χ(1 − u) = −1 and no solutions if χ(1 − u) = 1. It follows that, for a fixed u, the sum on y1 and y2 is equal −(p − 1), if χ(1 − u) = 1, and p + 1, if χ(1 − u) = −1. This may be written as 1 − p χ(1 − u), so that the whole expression becomes (8.5.42)

p−3

X

χ(u) − p−2

u6=1

X

χ(u) χ(1 − u) − p−3 (p − 1).

u6=1

It is easy to check that p−2

X u6=1

χ(u) χ(1 − u) = −χ(−1) p−2 ,

345

ON THE DOUBLING INTEGRAL

so that (8.5.42) becomes p−2 (χ(−1) − 1). Thus, ϕc∼ (p−1





1 β

) = −p−2 ,

and b ϕ(p

−1



1



β

) = 0,

as claimed. Remark 8.5.26. For p = 6 2, similar calculations yield the values ϕc∼ (0) = p−2 (p − 1)

and



ϕc∼ (p−1

1



) = 0,

0

and hence b ϕ(0) = p−1 (p − 1)

b −1 ϕ(p

and





1 0

) = p−2 (p − 1).

Next suppose that p = 2. In this case, we can take the following elements as orbit representatives for the action of GL2 (F2 ) on the F2 -vector space L] /2L] : 

0,

1

1 2



0

,

!

− 21

1 2

,

1

!

1 − 12

.

Once again, we need to show that the function on L] /2L] determined by ϕb vanishes on the √ last two elements. Note that the self-dual measure on V gives vol(L) = 1/ 2. Setting ϕ0 = char(2L), we have 1 1 c0 − 2 ϕb0 = √1 char(L] ) − √ ϕ char( L] ). 2 2 4 2 Thus, we must determine ϕc∼ (x) = 2

Z

ψ((x, y)) dy. V ∩(12 +2M2 (Z2 ))

If we write 



y0 2y1 y= , 2y2 −y0

346

CHAPTER 8

with y0 ∈ Z× 2 , we have dy =

(x, y) =

1 √ 4 2

dy0 dy1 dy2 , and

  1   − 2 y0

for x =



   − 12 (y0 + 2y1 + 2y2 )

for x =



1 4



,



.

− 14 1 4 1 2

1 2 1 −4

In either case, we get 1

ϕc∼ (x) = √

2 2

Z

Z Z× 2

(Z2

)2

1 1 = √ ψ(− ) 2 4 2

1 ψ(− y0 ) dy0 dy1 dy2 2

1 =− √ . 4 2

Thus, b ϕ(x) = 0,

as claimed in the lemma. This finishes the proof of Proposition 8.5.24. Remark 8.5.27. Again, we record the other values  ∼ ∼ c c ϕ (0) = ϕ (

0

1 2 )

1 = √ , 4 2

so that  b b ϕ(0) = ϕ(

0

1 2 )

1 = √ . 2 2

Bibliography

[1] E. M. Baruch and Z.Y. Mao, Central value of automorphic Lfunctions, preprint, 2003. ¨ [2] S. B¨ocherer, Uber die Funktionalgleichung automorpher LFunktionen zur Siegelschen Modulgruppe, J. reine angew. Math., 362 (1985), 146–168. [3] P. Garrett, Pullbacks of Eisenstein series; applications, in Automorphic Forms of Several Variables (Taniguchi Symposium, Katata, 1983), Progr. Math., 46, 114–137, Birkh¨auser, Boston, MA, 1984.

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[4] S. Gelbart, Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Math., 530, Springer-Verlag, Berlin, 1976. [5] S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit Constructions of Automorphic L-functions, Lecture Notes in Math., 1254, Springer-Verlag, Berlin, 1987. [6] R. Howe, θ–series and invariant theory, Proc. Symp. Pure Math., 33 (1979), 275–285. [7] R. Howe and I. I. Piatetski-Shapiro, Some examples of automorphic forms on Sp4 , Duke Math. J., 50 (1983), 55–106. [8] S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), 361–401. [9]

, Notes on the local theta correspondence, Lecture Notes of the European School on Group Theory, Schloß Hirschberg, September 1996.

[10]

, Central derivatives of Eisenstein series and height pairings, Annals of Math., 146 (1997), 545–646.

[11]

, Modular forms and arithmetic geometry, in Current Developments in Mathematics, 2002, 135–179, International Press, Somerville, MA, 2003.

[12] S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, forthcoming. [13] S. Kudla and M. Rapoport, Height pairings on Shimura curves and p-adic uniformization, Invent. math., 142 (2000), 153–223. [14] S. Kudla, M. Rapoport, and T. Yang, Derivatives of Eisenstein series and Faltings heights, Compositio Math., 140 (2004), 887–951. [15] J.-S. Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. reine angew. Math., 428 (1992), 177–217. [16] D. Manderscheid, Waldpurger’s involution and types, J. London Math. Soc. (2), 70 (2004), 567–585. [17] C. Moeglin, M.-F. Vigneras, and J.-L. Waldspurger, Correspondances de Howe sur un Corps p-adique, Lecture Notes in Math., 1291, Springer-Verlag, Berlin, 1987.

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[18] P. Perrin, Repr´esentations de Schr¨odinger, indice de Maslov et groupe m´etaplectique, in Noncommutative Harmonic Analysis and Lie Groups (Marseille, 1980), Lecture Notes in Math., 880, 370–407, Springer, Berlin-New York, 1981. [19] R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific J. Math., 157 (1993), 335–370. [20] S. Rallis and G. Schiffmann, Repr´esentations supercuspidales du groupe m´etaplectique, J. Math. Kyoto Univ., 17 (1977), 567–603. [21] G. Shimura, On modular forms of half integral weight, Annals of Math., 97 (1973), 440–481. [22] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J., 58 (1975), 83–126. [23] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl., 59 (1980), 1–132. [24]

, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl., 60 (1981), 375–484.

[25]

, Correspondance de Shimura, in S´eminare de Th´eorie des Nombres (Paris 1979–80), Progr. Math., 12, 357–369, Birkh¨auser, Boston, 1981.

[26]

, Sur les valeurs de certaines fonctions L automorphes en leur centre de sym´etrie, Compositio Math., 54 (1985), 173–242.

[27]

, D´emonstration d’une conjecture de dualit´e de Howe dans le cas p-adique, p 6= 2, in Festschrift in Honor of I. I. Piatetski-Shapiro, part 2, Israel Math. Conf. Proc., 2-3, 267–234, Weizmann Science Press, Jerusalem, 1990.

[28]

, Correspondances de Shimura et quaternions, Forum Math., 3 (1991), 219–307.

[29] A. Weil, Sur certains groupes d’op´erateurs unitaires, Acta Math., 111 (1964), 143–211. [30] T. H. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory, 72 (1998), 309–356. [31]

, Local densities of 2-adic quadratic forms, J. Number Theory, 108 (2004), 287–345.

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[32] D. Zagier, Nombres de classes et formes modulaires de poids 3/2, C. R. Acad. Sci. Paris, 281 (1975), 883–886.

Chapter Nine Central derivatives of L-functions In this chapter, we first use the Borcherds generating function, φBor (τ, ϕ), to define an arithmetic analogue of the classical Shimura-Waldspurger lift described in Section 8.2. We show that this lift, whose target is the MordellWeil group of a Shimura curve over Q, is compatible with the local theta correspondence, and hence there are local obstructions to nonvanishing, just as in the classical case. We then formulate a conjectural analogue of the result of Waldspurger, Theorem 8.2.5, and characterize the nonvanishing of the arithmetic theta lift in terms of theta dichotomy (local obstructions) and the nonvanishing of the central derivative L0 ( 12 , Wald(σ, ψ)). In Section 9.2, we prove this conjecture in certain cases by means of an arithmetic version of Rallis’s inner product formula, obtained by combining the arithmetic inner product formula of Chapter 7, the identity of Chapter 6 relating the genus two generating function and the central derivative E20 (τ, 0, B), and the explicit doubling formula of Chapter 8. In Section 9.3, we explain how the input for our arithmetic theta lift can be described in the classical language of normalized newforms of weight 2. 9.1 THE ARITHMETIC THETA LIFT

We begin by defining the arithmetic theta lift. For an indefinite quaternion algebra B over Q and for any compact open subgroup K ⊂ H B (Af ), where H B = B × ' GSpin(V B ), let (9.1.1)

B B MW(MK ) = Jac(MK )(Q) ⊗Z C

B over Q. be the Mordell-Weil space of the associated Shimura curve MK There is an exact sequence

(9.1.2)

B B B 0 −→ MW(MK ) −→ CH1 (MK ) −→ H 2 (MK ) −→ 0,

B ) is the Betti cohomology with complex coefficients. We can where H 2 (MK pass to the limit over K and let

(9.1.3)

B MW(M B ) = lim MW(MK ). → K

352

CHAPTER 9

The spaces CH1 (M B ) and H 2 (M B ) are defined analogously and there is an exact sequence (9.1.4)

0 −→ MW(M B ) −→ CH1 (M B ) −→ H 2 (M B ) −→ 0.

of admissible H B (Af )-modules, where, for example, B MW(M B )K = MW(MK ).

(9.1.5)

For any ϕ ∈ S(V B (Af )), the associated arithmetic theta function φeBor (·, ϕ), defined in Section 4.7, is a ‘holomorphic’ ‘weight 23 ’ automorphic form on G0A , valued in CH1 (M B ). Recall that this is the lift to G0A of the generating function φBor (τ, ϕ) =

X

Z(t, ϕ) q t

t≥0

for weighted 0-cycles. If ϕ ∈ S(V B (Af ))K for some compact open subgroup K of H B (Af ), then φeBor (·, ϕ) takes values in the finite dimensional B ). Also recall, from Section 4.7, that if g 0 ∈ G0 , then space CH1 (MK 0 Af φeBor (g 0 g00 , ϕ) = φeBor (g 0 , ω(g00 )ϕ),

(9.1.6)

just as for the classical theta function. For any cusp form f ∈ A00 (G0 ), the arithmetic theta lift of f is the class (9.1.7)

ar

Z

θ (f, ϕ) :=

Sp1 (Q)\Sp1 (A)

f (g 0 ) φ˜Bor (g 0 , ϕ) dg,

in CH1 (M B ). In fact, since f is orthogonal to all Eisenstein series and unary theta series, (4.4.29) and Proposition 4.4.7 imply that θar (f, ϕ) maps to zero in H 2 (M B ) in the sequence (9.1.4) and hence θar (f, ϕ) ∈ MW(M B ). Suppose that σ ' σ∞ ⊗ σ0 is a genuine cuspidal automorphic representation of G0A with (9.1.8)

σ∞ ' π ˜+ 3 = HDS 3 , 2

2

the holomorphic discrete series representation of G0R of weight 32 . We write V(σ) ⊂ A00 (G0 ) for the space of σ, and we write V(σ) 3 ,hol ⊂ A00 (G0 ) 3 ,hol 2

2

0 , the full infor the subspace of lowest weight vectors of weight 23 for K∞ 0 verse image of SO(2) in GR . Note that V(σ) 3 ,hol ' σ0 . The arithmetic 2 theta lift of σ,

(9.1.9)

θar (σ, M B ) ⊂ MW(M B ),

353

CENTRAL DERIVATIVES OF L-FUNCTIONS

is the subspace spanned by the elements θar (f, ϕ) as f ∈ σ and ϕ ∈ S(V B (Af )) vary. Note that θar (σ, M B ) is an H B (Af )-invariant subspace. In analogy with (8.2.33), the theory of the local theta correspondence, in Howe’s formulation [5], yields the following information about the space θar (σ, M B ). Write σ0 ' ⊗p σp and recall that, for each p, the maximal σp -isotypic quotient S(σp , VpB ) of S(VpB ) is either zero or S(σp , VpB ) ' σp ⊗ Θ(σp , VpB ), where Θ(σp , VpB ) is a smooth representation of HpB which has a unique irreducible quotient θψ (σp , VpB ); see [10]. Note that, to be consistent with the notation of Chapter 8, we have written θψ (σp , VpB ) = θ(σp , VpB ), even though the additive character ψ is fixed throughout this chapter. Proposition 9.1.1. As a representation of H B (Af ), θar (σ, M B ) '

 ⊗p 1. Then 2

θar (σ, M B ) 6= 0

⇐⇒

1 L0 ( , π) 6= 0. 2

Remark 9.2.11. (i) Note that, when L0 ( 12 , π) 6= 0, π0B ' θar (σ, M B ) ⊂ MW(M B ), B ⊗ π B of where π0B is the finite component of the representation π B ' π∞ 0 H B (A) corresponding to π = Wald(σ, ψ−1 ) under the Jacquet-Langlands correspondence. (ii) In effect, we suppose that σ has square free level and that δp (σp , ψp− ) =

363

CENTRAL DERIVATIVES OF L-FUNCTIONS

−1 for all primes dividing the level. In order to remove these conditions on the level of σ and on the δp (σp , ψp− )’s and to allow D(B) = 1, we would have to extend some of our geometric results about generating functions for arithmetic cycles on M to the case nontrivial level and to the modular curve. Of course, for the application to the central derivative of the L-function, it is possible to work ‘modulo oldforms’, as is done in [4] and [17], so that complete geometric information would not be needed. Some additional remarks are made at the end of Section 9.3. Proof. Let f ∈ σ be the good newvector. By Corollary 9.2.5, we have (9.2.11) 1 L0 ( , Wald(σ, ψ−1 )) 6= 0 2

⇐⇒

b ) 6= 0 θ(f

⇐⇒

θar (f , ϕ0 ) 6= 0 =⇒ θar (σ, M B ) 6= 0.

Thus, it remains to prove that f and ϕ0 are ‘good test vectors’ in the following sense: Proposition 9.2.12. θar (f , ϕ0 ) 6= 0

⇐⇒

θar (σ, M B ) = 6 0.

Proof. Assume that θar (σ, M B ) 6= 0, so that the map jσB of (9.1.16) is an 0 isomorphism. It will then suffice to show that the vector prB σ (ϕ ) ∈ σ0 ⊗ B θ(σ0 , V ) has nonzero pairing with f 0 , the finite component of f . Here prB σ is the natural projection in (9.1.15). This is then a local question. It suffices to show that, for each p, the image of ϕ0p in the quotient S(σp , VpB ) ' σp ⊗ Θ(σp , VpB ) has nonzero pairing with the good newvector f p . First suppose that p 6= 2. Then, if p - D(B), σp is an unramified principal series and f p is the K0 -invariant vector. The vector ϕ0p ∈ S(Vp ) is 0 also K0 -invariant and, as is shown in [10], has nonzero image prB σ (ϕp ) in B B B 0 S(σp , Vp ) ' σp ⊗ θ(σp , Vp ). Thus the pairing of prσ (ϕp ) and f p is nonzero, as required. If p | D(B), then σp is an unramified special representation with δp (σp , ψp− ) = −1 and f p is the J0 -invariant vector. Moreover, 1

the condition δp (σp , ψp− ) = −1 implies that σp ' σ ˜ (χ−1 | | 2 ).5 The map 1

B B prB ˜ (χ−1 | | 2 ) ⊗ 11 σ : S(Vp ) −→ σp ⊗ θ(σp , Vp ) ' σ

is given by λVp : ϕ 7→ (g 7→ ω(g)ϕ(0)). The vector ϕ0p ∈ S(VpB ) is J0 0 invariant and hence its image prB σ (ϕp ), which is nonzero by [10], is also 0 J -invariant and has a nonzero pairing with f p , as required. 5

Here we use the ψ-parametrization in Table 2 in Chapter 8.

364

CHAPTER 9 1

e (χ−1 | | 2 ). By Next suppose that p = 2. If p | D(B), then σp = σ 0 Proposition 8.4.16, the space of χ−1 -eigenvectors for J in σp has dimension 1 and f p = f sp is a basis. On the other hand, by Proposition 8.5.14, the vector ϕ0p is also a χ−1 -eigenvector for J0 and the same argument as for odd p gives the required nonvanishing. e (χ2α | |t ) with Finally suppose that p = 2 with p - D(B), so that σp = π × 1 α ∈ Z2 and t 6= ± 2 . Note that the condition on the global central character, discussed in Section 8.3, implies that α ≡ κ mod 4. Here κ = −1. In any case, the character µ−1 χ2κ is unramified. By Proposition 8.4.15, the space 0 I(µ)(J ,χα ) of χα -eigenvectors for J0 has dimension 2 and is spanned by the functions f1 and fw with support in P J0 and P wJ0 respectively, with f1 (1) = 1 and fw ([w, 1]L ) = 1. The good newvector in σp is f p = f ev = f1 + √i2 fw . The pairing between I(µ−1 ) = I(µ)∨ and I(µ) is given by Z

0

(f, f ) =

K0

f (k 0 ) · f 0 (k 0 ) · ζ(k 0 )−2 dk,

where k 0 ∈ G0 is any element which projects to k ∈ K 0 = SL2 (Zp ), and the factor ζ(k 0 )−2 makes the integrand independent of the choice of k 0 ; see Lemma 8.5.9. Note that I(µ) and I(µ−1 ) can be identified with the same space of functions on K 0 . Using this identification, the restriction of the pairing to the space of χα -eigenvectors for J0 is given by (f1 , f1 ) = vol(J 0 ), (f1 , fw ) = 0 and (fw , fw ) = 4 vol(J 0 ) · ζ([w, 1]L )−2 = −4. Here we use the fact that ζ([w, 1]L ) = (−1, −1)2 γ(ψ 1 )2 = i, 2

by (8.5.19) and (8.4.29). It follows that 1 (f p , f p ) = vol(J 0 ) · 3 = . 2 Using the explicit formula given in [10] for the projection from S(VpB ) to the maximal σp -isotypic quotient, it is easily checked that, when µ−1 χV is unramified, as it is in our case, 0 prB σ (ϕ ) = C · f ev

for a nonzero constant C. It follows that the pairing of this vector with f p = f ev is nonzero. This completes the proof of Theorem 9.2.10.

365

CENTRAL DERIVATIVES OF L-FUNCTIONS

Table 3. Example of Newforms F from Stein’s Tables D(B)

genus of MQ

F

dim of factor

7 ∗ 13

7

91B

1

3 ∗ 41

13

123B

1

7 ∗ 19

9

133C

2

5 ∗ 29

13

145B

2

5 ∗ 31

11

155C

1

5 ∗ 37

13

185C

1

11 ∗ 17

13

187D

2

7 ∗ 37

19

259E

3

2 ∗ 173

14

346B

1

13 ∗ 29

29

377A

1

377D

5

11 ∗ 37

31

407B

4

31 ∗ 43

105

1333A

23

31 ∗ 113

281

3503A

56

43 ∗ 83

287

3569F

61

43 ∗ 89

309

3827A

1

3827B

65

1155N

1

3 ∗ 5 ∗ 7 ∗ 11

41

9.3 THE RELATION WITH CLASSICAL NEWFORMS

In this section, we prove a slight generalization of Proposition 9.2.6. 1 Let l ∈ 32 + Z≥0 be a half integer and let κ = (−1)l− 2 . Let N be a new (N ) be the set of normalized newforms of square free integer and let S2l−1 new (N ), there is a weight 2l − 1 and level N . Associated to each F ∈ S2l−1 cuspidal automorphic representation π = π(F ) ' ⊗p≤∞ πp of PGL2 (A). The following result is essentially that of Kohnen [7], [1], although we allow N to be even.

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CHAPTER 9

new (N ), there is a unique genuine Proposition 9.3.1. (i) For each F ∈ S2l−1 cuspidal automorphic representation σ(F ) = σ ' ⊗p≤∞ σp of G0A with Wald(σ, ψκ ) = π such that σ∞ = HDSl is a holomorphic discrete series and such that, for all p | N , σp is a special representation. Moreover, the dichotomy signs for the local components of σ are given by

δ(σp , ψpκ )

=

  p (F )   

l− 12

(−1)

   1

if p | N ,

if p = ∞,

otherwise,

where, for p | N , p (F ) is the eigenvalue of F for the Atkin-Lehner involution, F | Wp = p (F ) F.

The classical modular f corresponding to the good test vector f in σ, as defined in Chapter 8, has weight l and level 4No , where No = N/2 if N is even and No = N if N is odd. (ii) Conversely, suppose that σ ' ⊗p≤∞ σp is a genuine cuspidal automorphic representation of G0A with σ∞ = HDSl and with finite components σp satisfying the conditions for p - N ,

for p | N ,

σp ' π ˜ (χα | |t ), 1

σp ' σ ˜ (χα | | 2 ),

with t = tp 6=

1 and α = αp ∈ Z× p, 2

with α = αp ∈ Z× p,

for some square free integer N , and with α2 ≡ 1 mod 4. Then the nornew (N ), and malized newform F determined by π = Wald(σ, ψκ ) lies in S2l−1 σ(F ) = σ. Proof. The representation π = π(F ) ' ⊗p≤∞ πp has local components π∞ = DS2l−1 and πp , for p - N , determined by the Hecke eigenvalue ap (F ), where F | Tp = ap (F ) F . For p | N , the local component πp is 1 1 an unramified special representation πp = σ(χα | | 2 , χα | |− 2 ), where α = αp ∈ Z× p and α ≡ 1 mod 4 if p = 2. The quadratic character χαp is determined by the Atkin-Lehner sign p (F ) according to the rule (9.3.1)

χαp (p) = −p (F ).

By a basic result of Waldspurger, the set of genuine cuspidal automorphic representations σ in A00 (G0A ) with Wald(σ, ψκ ) = π is nonempty and has cardinality 1 if N = 1 and 2o(N ) if the number of prime factors o(N ) of N is positive. The possibilities for σ can be described by using the parametrization of local components given in Table 2 in Chapter 8. If Wald(σ, ψκ ) = π,

367

CENTRAL DERIVATIVES OF L-FUNCTIONS

el± , and the local components σp for p - N are irreducible then σ∞ = π principal series which are uniquely determined by the corresponding local components πp = Wald(σp , ψpκ ) of π. For p | N there are two possibilities, 1

ep (χαp κ | | 2 ), θ(sgn, Uαp κ ) }, σp ∈ { σ

related by the Waldspurger involution; see Proposition 8.2.4. Moreover, any choice of σp ’s is allowed, subject to the condition (8.2.28) on the product of the central signs. One possible choice of σp ’s is given by the following. Lemma 9.3.2. The representation σ ' ⊗p≤∞ σp with local components 1 el+ and σp = σ ep (χακ | | 2 ) for all p | N is a genuine cuspidal σ∞ = π automorphic representation of G0A with Wald(σ, ψκ ) = π.

Proof. By the last row of Table 2 in Chapter 8, the central sign of the el+ , ψκ ) = 1. Note that the fact that archimedean component σ∞ is z(π 1 κ = (−1)l− 2 is used here. Then, by the last column of the table, we have the product formula for central signs Y

z(σp , ψκ ) =

p≤∞

Y

χαp (−1) = 1,

p|N

since all of the χαp ’s are unramified. The dichotomy signs for the local components of σ are then given in the fourth column of Table 2. By the local results of Chapter 8, the good test vector f ∈ σ is unique up to a scalar factor. This vector has weight l for 0 , the inverse image of SO(2) ⊂ SL (R) in G0 , and is an eigenvector for K∞ 2 R the group K0 (4No ) ⊂ G0Af with character χκ . The corresponding classical modular form f has weight l and level 4No ; see Proposition 8.5.20. Corollary 9.3.3. (i) The map F 7→ σ(F ) determines a bijection between the new (N ) and the set of genuine cuspidal automorphic representations set S2l−1 σ(F ) = σ ' ⊗p≤∞ σp of G0A with σ∞ = HDSl and with finite components σp satisfying the conditions of (ii) of Proposition 9.3.1. (ii) The representations in the set { σ | Wald(σ, ψκ ) = π(F ), σ∞ = HDSl } have the form σ = σ Σ where, for a set Σ of primes dividing N , with |Σ| even, σpΣ =

 θ(sgn, Uα κ ) p σ(F )

p

if p ∈ Σ,

otherwise.

368

CHAPTER 9

Of course, the second part here is just a restatement of Waldspurger’s result, Proposition 8.2.4. Remark 9.3.4. If we take the classical modular form f associated to the good test vector f in each σ = σ(F ), we obtain a bijection F 7→ f between new (N ) and a set of modular forms of weight l for Γ (4N ). When N is S2l−1 0 o square free and odd, such a bijection was defined by Kohnen [7] where the image is his plus-space, characterized by the support of the Fourier expansion at the cusp at infinity. These two bijections do not agree. The point is that when N is odd, the local component σ(F )2 of the representation σ(F ) is an irreducible principal series representation with a 2-dimensional space π ˜ (χα | |t )K0 (4N ),χκ of newvectors. Our local component δκ f 2 = fev = f1 + √ fw 2 is an eigenfunction for a local zeta operator analyzed in Chapter 8. On the other hand, according to Baruch and Mao [1] or the computation of Whittaker functions given in Proposition 8.4.17, the local component of a function in the Kohnen plus space is, in our notation, √ 2µ(2)2 + fw f = f1 + δα = (1 + 4χκ (−1)µ(2)2 ) fev + (2 − 4χκ (−1)µ(2)2 ) fsp ; see (8.4.34). Of course, the underlying bijection π(F ) 7→ σ(F ) on representations is the same, while the choice of local component at p = 2 can be made according to the particular application at hand. Now take l = 23 , and let F ∈ S2new (N ) be a newform of weight 2 with ( 21 , π(F )) = −1. Since (9.3.2)

−

1 p (F ) = ( , π(F )), 2 p|N Y

there is a unique indefinite quaternion algebra B over Q such that invp (B) = p (F ) for all p | N and invp (B) = 1 otherwise. When D(B) = N , i.e., when all of the Atkin-Lehner signs of F are −1, we take σ = σ(F ), and the results of the previous section describe the nonvanishing of the arithmetic b ), where f ∈ σ(F ) is the good newvector. theta lift θar (α, M B ) and of θ(f In the case in which some Atkin-Lehner signs of F are +1, we can again take σ = σ(F ) and consider the arithmetic theta lift θar (σ, M B ), as in Section 9.1. To extend the results of Section 9.2 to this case, we should define 0 a generating function φbB,N (τ ) for the arithmetic surface MB 0 (N0 ), where 1

369

BIBLIOGRAPHY

N = D(B)N0 , associated to an Eichler order of level N0 in B. This will involve a definition of the special cycles Z(t) with level structure and an 0 extension of the results of this book to the generating function φbB,N (τ ). 1 0 More generally, let B be an indefinite quaternion algebra ramified at a set of primes dividing N , and let Σ = { p | p (F ) = −invp (B 0 ) }. 0

Take σ = σ Σ (F ). Then, the arithmetic theta lift θar (σ, M B ) should be nonzero if and only if L0 (1, F ) 6= 0. The local component of σ Σ at p ∈ Σ is the odd Weil representation θ(sgn, Uαp κ ) where χαp is unramified and χαp (p) = −p (F ). In order to define the analogue of φbB 1 in this case, we need to define cycles Z(t) which are anti-invariant under the Atkin-Lehner involution at the primes p ∈ Σ. This can be done by using a weighted average of the oriented cycles mentioned in the introduction and described in Chapter 3. Again, it should be possible to extend the main results of this book to the generating functions constructed from such cycles. Bibliography

[1] M. Baruch and Z.Y. Mao, Central value of automorphic L-functions, preprint, 2003. [2] B. H. Gross,, Heegner points and representation theory, in Heegner Points and Rankin L-series, Math. Sci. Res. Inst. Publ., 49, 37–65, Cambridge Univ. Press, Cambridge, 2004. [3] B. H. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of L-functions. II, Math. Annalen, 278 (1987), 497–562. [4] B. H. Gross, and D. Zagier, Heegner points and the derivatives of L-series, Invent. math., 84 (1986), 225–320. [5] R. Howe, θ–series and invariant theory, Proc. Symp. Pure Math., 33 (1979), 275–285. [6] R. Howe and I. I. Piatetski-Shapiro, Some examples of automorphic forms on Sp4 , Duke Math. J., 50 (1983), 55–106. [7] W. Kohnen, Newforms of half-integral weight, J. reine angew. Math., 333 (1982) 32–72. [8] S. Kudla, Special cycles and derivatives of Eisenstein series, in Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ., 49, 243– 270, Cambridge Univ. Press, Cambridge, 2004.

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[9]

, Modular forms and arithmetic geometry, in Current Developments in Mathematics, 2002, 135–179, International Press, Somerville, MA, 2003.

[10]

f 2 , O(3)), , Notes on the local theta correspondence for (SL preprint, 2005.

[11] W. Stein, The Modular Form Database, http://modular.ucsd.edu [12] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl., 59 (1980), 1–132. [13]

, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl., 60 (1981), 375–484.

[14]

, Sur les valeurs de certaines fonctions L automorphes en leur centre de sym´etrie, Compositio Math., 54 (1985), 173–242.

[15]

, Correspondances de Shimura et quaternions, Forum Math., 3 (1991), 219–307.

[16] D. Zagier, Modular points, modular curves, modular surfaces and modular forms, in Lecture Notes in Math. 1111, 225–248, SpringerVerlag, Berlin, 1985. [17] Shou-Wu Zhang, Gross–Zagier formula for GL2 , Asian J. of Math., 5 (2001), 183–290.

Index Arakelov Chow group, 8 Arakelov volume form, 40 ARGOS seminar, 20, 60 arithmetic adjunction formula, 14, 42, 225 metric in, 217 not homogeneous, 227 arithmetic Chow group, 6, 71 with real coefficients, 71 arithmetic degree map, 7, 167 arithmetic index theorem, 75 arithmetic inner product formula, 13 arithmetic intersection pairing, 7, 72 projection formula, 237 arithmetic theta function, 77 components of, 78 modularity of, 78 arithmetic theta lift, 351 nonvanishing, 16 vanishing, 355 Atkin-Lehner involutions, 18, 56, 366 Atkin-Lehner signs, 17, 360 automorphy factor, 168, 337 Borcherds generating function, 19, 96, 356 intertwining property, 101 Borcherds products, 19 c, the constant, 8, 57, 78, 179 equal to − log(D(B)), 262 central sign, 269 complex multiplication, 222 configuration, 188, 237 confluent hypergeometric functions, 116, 149 cycle data (S, µ, Z h ), 82, 186, 188 δp (t), Serre’s δ-invariant, 246 dichotomy sign, 272, 355 relation to the root number, 272 Diff(T, B), 58, 113, 167, 176

discriminant term, 245 DM-stack, 46 doubling integral, 16, 265 archimedean case, 316 explicit eigenvectors, 279 explicit global, 282 for the modified Eisenstein series, 284 p = 2, 298 special representations, 293 unramified case, 291 Drinfeld special condition, 46 Drinfeld upper half space, 12, 48 Eisenstein polynomial, 244 Euler-Poincar´e characteristic, 172 exponential integral, Ei, 56 asymptotics of, 215 fibers of the map σ 7→ Wald(σ, ψ), 277 flat closure, 8 flat extension, 54 functional equation, 140 fundamental matrix, 172, 205, 210 γ, Euler’s constant, 50 γ(V ), 327 gauge form, 122, 127, 128 Genestier equations, 191 Gillet-Soul´e arithmetic Chow group, 6 Gillet-Soul´e height pairing, 75 good newvector, 16, 279, 359, 363 good nonsingular T , 167 good test vectors, 265, 363 Green function, 7, 56 Ξ(t, v), 57 ϕ(x, z), 56 ξ(x, z), 56 gt0 , 218 spectral expansion, 91

372 weakly µ-biadmissible on H × H − ∆, 212 weakly ν-biadmissible, 39 Gross Montreal article of, 87 MSRI article of, 356 Gross-Keating formula, 12, 45, 58, 169, 181 Gross-Keating invariants, 45, 58, 114, 145, 173, 181, 183 p = 2, 63 Gross-Kohnen-Zagier Theorem, 15, 18, 362 Gross-Zagier formula, 356 Heegner points, 362 height archimedean contributions to, 176, 211 with respect to a metrized line bundle, 73 Hodge line bundle, 7, 45, 49, 96, 178 metric determined by −gt0 , 226 natural metric, 50 self-intersection number, 14, 57, 208 computed, 262 hyperbolic Laplacian, 90 incoherent collection, 58, 108 Jacquet-Langlands correspondence, 274 Keating’s thesis, 60 Kodaira-Spencer isomorphism, 226 Kohnen plus space, 306, 316, 360, 368 lattices, L0p , L1p , Lra p , 109 Leray cocycle, 321 Leray coordinates, 321 local density functions, Fp (X, T, Lp ), 145 local Howe duality, 271, 353 Lubin-Tate formal group, 250 Maass forms, 87 weight 3/2, 91 metaplectic group cocycles for, 320 in Waldspurger, 287 Leray, 321 coordinates for, 320

INDEX

global, 334 Leray, 321 normalized, 323 Rao, 326 metaplectic group, global, 334 metrized line bundles, 29 moment map, 122 Mordell-Weil space, 74, 94, 351 N´eron-Tate height, 75 normalized Eisenstein series genus one, 139 genus two, 110, 155 normalized newform, 365 weight 2, 360, 368 ‘number theorist’s nightmare’, 12 Oc2 d , order of conductor c, 221 order of conductor c, 221 point bundle, metric on, 40 product formula for central signs, 277, 367 quasi-canonical divisor of level s, Ws (ψ), 240 quasi-canonical lifting, 206, 240 Rallis inner product formula, 359 ramification function, 250 Rao coordinates, 269 relative dualizing sheaf, 30, 43, 49 representation densities, 12, 114, 145 derivatives of, 114 root number, 17 section factorizable, 265 modified global, 110, 155 modified local, 110 standard, 110 Serre’s δ-invariant, δp (t), 246 Shimura-Waldspurger correspondence, 15 Shimura-Waldspurger lift, 265 Siegel Eisenstein series, genus two, 105 Siegel-Weil formula, 86, 108, 121 special cycles and Atkin-Lehner involutions, 55 Cohen-Macauleyfication of, 55

INDEX

decomposition of by conductor, 221 definition of, 51 embedded components, 55 Heegner, 55 vertical components, 79 vertical components of, 54 weighted, 55, 369 with level structure, 369 Z(j), 182 Z(t), 51 0-cycles, 57 Z(T ) for T singular, 177 ˆ v), 57 Z(0, ˆ v), 57 Z(t, b v) for T singular, 178 Z(T, Z hor (t : c), 222 Z hor (t : c)Q , 221 special endomorphisms, 51 Stein’s tables, 362 submersive set, 122, 127 Table 1, the local theta correspondence, 275 Table 2, values of Gross-Keating invariants, 183 Table 3, examples of newforms, 365 transformation law for Siegel modular forms, 336 for the genus two Eisenstein series, 339 b1 (τ ), for the vertical component of φ 340 type (i):(j), 183 uniformization Cherednik-Drinfeld, 49 complex, 47, 76 p-adic, 48, 80, 173 of special cycles, 54, 80 value of h ω ˆ, ω ˆ i, 262 Vert, 72 ‘vertical at infinity’, 207 vertical components, 72 Ws (ψ), quasi-canonical divisor of level s, 240 Wald(σ, ψ), 272 Waldspurger involution, 274, 367 weighted cycles, 18

373 Weil representation in Leray coordinates, 327 Whittaker functions and orbital integrals, 121 derivatives of, 114 local, 112 p = 2, 305 Whittaker integrals, 112, 126, 140 wrong degree, 178