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The Theta Correspondence and Periods of Automorphic Forms

by

Patrick Walls

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2013 by Patrick Walls

Abstract

The Theta Correspondence and Periods of Automorphic Forms

Patrick Walls

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2013

The study of periods of automorphic forms using the theta correspondence and the Weil representation

was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral

weight, periods over tori of modular forms of integral weight and special values of L-functions attached

to these modular forms. In this thesis, we show that there are general relations among periods of auto-

morphic forms on groups related by the theta correspondence. For example, if G is a symplectic group

and H is an orthogonal group over a number field k, these relations are identities equating Fourier

coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods

of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and

follow from the basic properties of theta functions and the Weil representation; further study is required

to show how they compare to the results of Waldspurger. The second part of this thesis shows that,

under some restrictions, the identities alluded to above are the result of a comparison of nonstandard

relative traces formulas. In this comparison, the relative trace formula for H is standard however the

relative trace formula for G is novel in that it involves the trace of an operator built from theta functions.

The final part of this thesis explores some preliminary results on local height pairings of special cycles

on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should

appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built

from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required

to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic

degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods

of automorphic forms over orthogonal subgroups).

ii

Dedication

This thesis is dedicated to my family: Nancy and Darcy, Chris, Liz, Meg and Pam; and Michelle.

Acknowledgements

First and foremost, I would like to thank my advisor, Stephen Kudla, for his support and for generously

sharing his ideas with me.

Brian Smithling and Ying Zong answered all of my questions with tremendous patience while I was

a graduate student; I am grateful for all that I learned from them.

I would like to thank Siddarth Sankaran for our daily discussions about arithmetic geometry and

automorphic forms; sharing an office was certainly a formative experience.

Finally, I would like to thank Ida Bulat and Jemima Merisca for ensuring that issues of funding,

scholarship applications, degree requirements, thesis exams, etc. were always met. Their efforts are deeply

appreciated by the hundreds of graduate students who have studied in the Department of Mathematics

at the University of Toronto.

iii

Contents

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 The Theta Correspondence 8

2.1 The Weil Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Translation and Dual Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3 The Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.4 Intertwining Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.5 The Metaplectic Group and the Weil Representation . . . . . . . . . . . . . . . . . 17

2.2 Dual Reductive Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Symplectic-Orthogonal Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Metaplectic-Orthogonal Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Unitary-Unitary Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 The Theta Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Duality Among Periods 28

3.1 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Orthogonal Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Matching Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Example: Matching Functions for PGL2(Qp) . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Duality Among Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 The Main Theorem: Spectral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Further Directions: Special Values of L-Functions . . . . . . . . . . . . . . . . . . . . . . . 44

iv

4 A Comparison of Relative Trace Formulas 48

4.1 Relative Trace of the Kernel for G(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Relative Trace of the Kernel for H(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Matching Geometric Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Spectral Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Spectral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Height Pairings of Special Cycles 61

5.1 Special Cycles on the p-adic Upper Half Plane . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Height Pairings of Special Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 The Unramified/Unramified Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 The Unramified/Split Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.5 The Split/Unramified Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 The Ramified/Ramified Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 78

v

Chapter 1

Introduction

The study of periods of automorphic forms using the theta correspondence and the Weil representation

was initiated by Waldspurger and his work (see [20], [21], [22] and [23]) relating Fourier coefficients of

modular forms of half-integral weight, periods over tori of modular forms of integral weight and special

values of L-functions attached to these modular forms. In this thesis, we show that there are general

relations among periods of automorphic forms on groups related by the theta correspondence.

This thesis comprises of three separate (but related) results which form Chapter 3, Chapter 4 and

Chapter 5 respectively. The main result is the collection of spectral identities in Theorem 3.6.1, for a

given dual pair (G,H), equating Fourier coefficients of cuspidal automorphic forms on G and periods

of cuspidal automorphic forms on H over orthogonal subgroups. In Theorem 4.5.1, we show, in some

very special cases, that these spectral identities are the result of a comparision of nonstandard relative

trace formulas. Finally, Chapter 5 explores some preliminary results on local height pairings of special

cycles on the p-adic upper half plane. Inspired by the work of the preceding two chapters, these height

pairings should appear as the local factors of arithmetic orbital integrals in a comparison of arithmetic

relative trace formulas. In this introductory section, we give a summary of these results.

1.1 Overview

Let k be a global field, let A be its ring of adeles, let ψ : A/k −→ C× be a nontrivial unitary character and

let (G,H) be a dual reductive pair (in the sense of Howe [6]) defined over k. For example, let G = Spn

be the symplectic group of rank n (consisting of symplectic matrices of size 2n) and let H = OV be

the orthogonal group attached to a space V over k of even dimension m equipped with a nondegenerate

symmetric bilinear form ( , ) and corresponding quadratic form Q(v) = 12 (v, v). The results described

1

Chapter 1. Introduction 2

below apply to symplectic-orthogonal, metaplectic-orthogonal and unitary-unitary dual reductive pairs

as described in Section 2.2, however, for simplicity, we will restrict ourselves in this introduction to the

symplectic-orthogonal dual pair (Spn,OV ). The group G contains a Siegel parabolic subgroup P with

unipotent radical N ⊂ P

N =

n(b) =

1 b

0 1

: b ∈ Symn

where Symn denotes symmetric matrices of size n. For a cuspidal automorphic form fG on G(A) and

symmetric matrix t ∈ Symn(k), the ψt-Fourier coefficient of fG is

Wt(fG) =∫N(k)\N(A)

fG(n)ψt(n) dn

where ψt(n) = ψ(tr tb) for n = n(b).

Let j = (j1, . . . , jn) ∈ V n be an n-tuple of vectors in V and let T be the stabilizer of j in H where H

acts componentwise on V n. The group T is an orthogonal subgroup of H and for a cuspidal automorphic

form fH on H(A), the T -period of fH is

PT (fH) =∫T (k)\T (A)

fH(τ) dτ .

The groups form a dual pair (G,H) in Sp2n and the Weil representation ω = ωψ (relative to ψ) acting

on the Schwartz space S(V nA ) (where VA = V ⊗k A) can be restricted to G(A) × H(A). The elements

ϕ ∈ S(V nA ) define theta functions

θϕ(g, h) =∑

v∈V nω(g)ϕ(h−1v)

and the theta correspondence is built by integrating cuspidal automorphic forms on G(A) against theta

functions to produce automorphic forms on H(A) and vice versa. In particular, if σ is a cuspidal

automorphic representation of G acting on a subspace Vσ ⊂ L2(G(k)\G(A)), the theta lift of fG ∈ Vσ is

θϕ∨fG(h) =

∫G(k)\G(A)

θϕ(g, h) fG(g) dg

and the theta lift of σ is the space of theta lifts θϕ∨fG as ϕ and fG vary and is denoted θψ(σ). Analo-

gously, if π is a cuspidal automorphic representation of H acting on a subspace Vπ ⊂ L2(H(k)\H(A)),


the theta lift of fH ∈ Vπ is

θϕfH(g) =

∫H(k)\H(A)

θϕ(g, h) fH(h) dh

and the theta lift of π is the space of theta lifts θϕfH as ϕ and fH vary and is denoted θψ(π). It is very

important to note that theta lifts are not always cuspidal. In the main theorem, we assume that σ and

π are cuspidal representations of G and H respectively such that θψ(σ) = π (equivalently, θψ(π) = σ).

To state the main result, we need some extra notation: given a Schwartz function ξ on H(A), Rξ is the

Hecke operator

RξfH(h) =

∫H(A)

ξ(x)fH(hx) dx

and, for v ∈ V n, Q[v] = 12 ((vi, vj)) is the symmetric matrix of inner products of the components of v.

Theorem (see Theorem 3.6.1). Let j1, j2 ∈ V n with Q[j1] = t1 and Q[j2] = t2 such that det t1 6= 0

and det t2 6= 0, and let T1 and T2 be the stabilizers in H of j1 and j2 respectively. Let ϕ1, ϕ2 ∈ S(V nA )

and let ξ1 and ξ2 be smooth functions on H(A) such that ξ1 matches ϕ1 relative to j1 and ξ2 matches

ϕ2 relative to j2 as in Proposition 3.3.1. Given cuspidal automorphic representations σ and π of G and

H respectively such that θψ(σ) = π (equivalently θψ(π) = σ), we have

∑FG∈B(σ)

Wt1(θϕ1θϕ∨2 FG)Wt2(FG) =

∑FH∈B(π)

PT1(Rξ1Rξ∨2 FH)PT2(FH) (1.1)

∑FG∈B(σ)

PT1(θϕ∨2 FG) Wt2(FG) =

∑FH∈B(π)

PT1(Rξ∨2 FH) PT2(FH) (1.2)

∑FG∈B(σ)

PT1(θϕ∨1 FG) PT2(θϕ∨2 F

G) =∑

FH∈B(π)

PT1(θϕ∨1 θϕ2FH) PT2(FH) (1.3)

where B(σ) and B(π) are orthonormal bases of Vσ and Vπ respectively.

This result has some special features. First, the relation between the various ϕ, ξ, t and T is explicit

and, in low rank cases, computable. Second, it is well-known that the Fourier coefficient Wt(θϕfH) of

a theta lift of fH is expressible in terms of an orthogonal period PT (fH) however Equation 1.1 allows

one, in principle, to choose the data ϕ1 and ϕ∨2 to select the Fourier coefficients of any fG in σ and

express them in terms of a chosen orthogonal basis of π, say, involving Hecke eigenforms. Third, the

result of Waldspurger [21] (later generalized by Baruch-Mao [1]) is an equality between twisted central

values of L-functions attached to modular forms of integral weight and squares of Fourier coefficients of

modular forms of half-integral weight; the spectral identity Equation 1.1 introduces a formula for a pair

of distinct Fourier coefficients. Finally, by the Siegel-Weil formula, a special value of an L-function is

encoded in the inner product of theta functions θϕ1θϕ∨2 .


Next, we describe the second result of this thesis. When V is anisotropic and n = 1, Equation 1.1

can be interpreted as a spectral identity as a result of a comparison of relative trace formulas. First,

let us introduce the kernel for G(A). Given Schwartz functions ϕ1, ϕ2 ∈ S(VA), the map which takes a

cusp form fG on G(A) to its theta lift θϕ∨2 fG on H(A) and then to the lift θϕ1θϕ∨2 f

G on G(A) again is

an operator on the space of cusp forms. In fact, switching the order of integration shows that it is an

integral operator with kernel function

Kϕ1·ϕ∨2 (g1, g2) =∫H(k)\H(A)

θϕ1(g1, h) θϕ2(g2, h) dh .

Since we have assumed that V is anisotropic, the quotient H(k)\H(A) is compact and this integral

is absolutely convergent. Define the trace of the kernel Kϕ1·ϕ∨2 relative the subgroup N × N and the

characters ψt1 and ψt2 by

JG(ϕ1 · ϕ∨2 ; t1, t2) =∫

[N×N ]

Kϕ1·ϕ∨2 (n1, n2)ψt1(n1)ψt2(n2) dn1dn2

where [N ×N ] = (N ×N)(k)\(N ×N)(A).

The right regular representation of H(A) on L2(H(k)\H(A)) defines an action of the Schwartz space

S(H(A)) by

RξfH(x) =

∫H(A)

ξ(y) fH(xy) dy

which we unwind to find that it is an integral operator with kernel function

Kξ(x, y) =∑

γ∈H(k)

ξ(x−1γy) .

Let j1, j2 ∈ V and let T1 and T2 be the stabilizers in H of j1 and j2 respectively. For each ξ ∈ S(H(A)),

define the trace of the kernel Kξ relative to the subgroup T1 × T2 by

JH(ξ ; T1, T2) =∫

[T1×T2]

Kξ(τ1, τ2) dτ1dτ2

where [T1 × T2] = (T1 × T2)(k)\(T1 × T2)(A).

Both traces JG(ϕ1 ·ϕ∨2 ; t1, t2) and JH(ξ ; T1, T2) have simple geometric expansions whose terms are

sums over the double coset T1\H/T2 and we prove their equality by establishing identities term by term.

Proposition (see Proposition 4.3.2). Let j1, j2 ∈ V be nonzero with Q(j1) = t1 and Q(j2) = t2, and

let T1 and T2 be the stabilizers in H of j1 and j2 respectively. Let ϕ1, ϕ2 ∈ S(VA) and let ξ1 and ξ2 be


smooth functions on H(A) such that ξ1 matches ϕ1 relative to j1 and ξ2 matches ϕ2 relative to j2 as in

Proposition 3.3.1. Then

JG(ϕ1 · ϕ∨2 ; t1, t2) = JH(ξ1 ∗ ξ∨2 ; T1, T2)

where ξ1 ∗ ξ∨2 (h) =∫H(A)

ξ1(x)ξ∨2 (x−1h) dx is the convolution of ξ1 and ξ∨2 for ξ∨2 (x) = ξ2(x−1).

Having established that the traces of the kernels for G and H are equal when the input data is

matching, we express each trace spectrally and proceed to produce equalities of traces at the level of

representations. The first steps are to show that the noncuspidal components of JG(ϕ1 · ϕ∨2 ; t1, t2) and

JH(ξ ; T1, T2) are equal (see Proposition 4.4.1) and then to conclude that the cuspidal components are

equal (see Corollary 4.4.1).

Proposition (see Proposition 4.4.1). For any ϕ1, ϕ2 ∈ S(VA) we have

Keisϕ1·ϕ∨2

(g1, g2) =∫H(k)\H(A)

∑v1∈V

ω(g1)ϕ1(h−11 v1) dh1

∫H(k)\H(A)

∑v2∈V

ω(g2)ϕ2(h−12 v2) dh2

and for ξ1 matching ϕ1 relative to j1 and ξ2 matching ϕ2 relative to j2 as in Proposition 3.3.1 we have

∫H(A)

ξ1 ∗ ξ∨2 (x) dx =∫T1(A)\H(A)

ϕ1(h−11 j1) dh1

∫T2(A)\H(A)

ϕ2(h−12 j2) dh2 .

In particular, JGeis(ϕ1 · ϕ∨2 ; t1, t2) = JHtriv(ξ1 ∗ ξ∨2 ; T1, T2).

Proposition (see Corollary 4.4.1). Let j1, j2 ∈ V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let

T1 and T2 be the stabilizers in H of j1 and j2 respectively. For all ϕ1, ϕ2 ∈ S(VA) and ξ1 matching ϕ1

relative to j1 and ξ2 matching ϕ2 relative to j2 as in Proposition 3.3.1, we have

∑σ∈AG

0

JGσ (ϕ1 · ϕ∨2 ; t1, t2) =∑π∈AH

0

JHπ (ξ1 ∗ ξ∨2 ; T1, T2) (1.4)

where A G0 and A H

0 are the sets of cuspidal automorphic representations of G and H respectively, the

trace along σ ∈ A G0 is

JGσ (ϕ1 · ϕ∨2 ; t1, t2) =∑

FG∈B(σ)

Wt1(θϕ1θϕ∨2 FG)Wt2(FG)

and the trace along π ∈ A H0 is

JHπ (ξ1 ∗ ξ∨2 ; T1, T2) =∑

FH∈B(π)

PT1(Rξ1Rξ∨2 FH)PT2(FH) .


The final step in the comparison of trace formulas is to use a linear independence argument to produce

equalities of traces at the level of representations which correspond by the theta correspondence.

Theorem (see Theorem 4.5.1). Let j1, j2 ∈ V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let

T1 and T2 be the stabilizers in H of j1 and j2 respectively. Let ϕ1, ϕ2 ∈ S(VA) and let ξ1 and ξ2 be

smooth functions on H(A) such that ξ1 matches ϕ1 relative to j1 and ξ2 matches ϕ2 relative to j2 as in

Proposition 3.3.1. Given cuspidal automorphic representations σ and π of G and H respectively such

that θψ(σ) = π (equivalently θψ(π) = σ), we have

∑FG∈B(σ)


∑FH∈B(π)



It should be noted that the method of proving Theorem 3.6.1 was discovered only when the final steps

in the proof of Theorem 4.5.1 were completed. However we have presented the method of Chapter 3

first as the main result since it is more general and to the point.

The final chapter in this thesis explores some preliminary results on height pairings of special cycles

on the p-adic upper half plane following the work of Kudla-Rapoport [12]. Let k = Fp, let W = W (k)

be the Witt ring of k and let B be the quaternion division algebra over Qp with ring of integers OB .

The p-adic upper half plane ΩW is a formal scheme over Spf W and is the moduli space of special formal

OB-modules as in the work of Drinfeld [3] (see also [2]). Fix a special formal OB-module X over k. The

ring of endomorphisms EndOB (X)⊗Q is naturally identified with the algebra M2(Qp) of 2 by 2 matrices

over Qp and we denote by V the subspace of traceless endomorphisms equipped with the quadratic

form q(j) = −det(j). For any j ∈ V , there is a special cycle Z(j) which lives on ΩW (see Chapter 5

for definitions) and the main result of Kudla-Rapoport [12] is a formula for the local height pairings

(Z(j1), Z(j2)) for j1, j2 ∈ V .

Since V can be identified with the space of traceless 2 by 2 matrices over Qp, there is a natural action

of PGL2(Qp) on V by conjugation. The results of Chapter 5 are formulas for the local height pairings

(Z(j1), Z(γj2γ−1)) for fixed j1, j2 ∈ V as a function of γ ∈ PGL2(Qp) and we introduce the notation

O(γ ; j1, j2) = (Z(j1), Z(γj2γ−1)) .

The motivation for studying such height pairings is that they should be the arithmetic analogues of the


local integrals ∫PGL2(Qp)

ϕ1(h−1j1)ϕ2(h−1γj2) dh

appearing in the geometric expansion of JG(ϕ1 ·ϕ∨2 ; t1, t2) as in Proposition 4.1.1. See the introduction

to Chapter 5 for further discussion.

The formula for the pairing O(γ ; j1, j2) depends on the arithmetic of the values q(j1) and q(j2). In

Chapter 5, we consider four of the nine possible cases and we outline one such formula as an example

below. In the statement of Proposition 5.4.1 below, χ is the quadratic residue character, Bj is the set

of points on the Bruhat-Tits tree of PGL2(Qp) fixed under multiplication by j and d(Bj1 ,Bγj2γ−1

) is

the distance from Bj1 to Bγj2γ−1

. Finally, as expected, these formulas are a synthesis of the formulas

given in Theorem 6.1 in [12]

Proposition. Let j1, j2 ∈ V with q(j1) = pαε1 and q(j2) = pβε2 such that 0 ≤ α ≤ β, α and β are

even, χ(ε1) = −1 and χ(ε2) = 1. Let γ ∈ GL2(Qp) and let d = d(Bj1 ,Bγj2γ−1

). Then O(γ ; j1, j2) = 0

if d > α/2 + β/2, O(γ ; j1, j2) = 1 if d = α/2 + β/2, and if d < α/2 + β/2 then O(γ ; j1, j2) is given by

α+ β + 1− 2d−

2p(α/2+β/2−d+1)/2 − 1

p− 1if α

2 + β2 − d is odd and α

2 >β2 − d,

p(α/2+β/2−d)/2 + 2p(α/2+β/2−d)/2 − 1

p− 1if α

2 + β2 − d is even and α

2 >β2 − d,

pα/2 + 2pα/2 − 1p− 1

if α2 ≤

β2 − d.

Chapter 2

The Theta Correspondence

The global theta correspondence is a correspondence between automorphic representations of groups

forming a dual reductive pair in the sense of Howe [6], and the main tool in the construction is the

Weil representation [24]. In Section 2.1, we introduce the Heisenberg group, the Weil representation and

the metaplectic group by focusing on the Fourier transform. There are many introductions to the Weil

representation (for example [9], [16], [19] and the original paper by Weil [24]) therefore our discussion

is informal as we merely outline in Summary 2.1.1 and Summary 2.1.2 the elementary properties of

the Heisenberg group, the Weil representation and the metaplectic group. In Section 2.2, we introduce

the dual pairs and the explicit formulas for the Weil representation used in subsequent chapters in this

thesis. In Section 2.3, we introduce the theta functions and the theta correspondence for these dual

pairs. The results in Chapter 3 and Chapter 4 concern cuspidal automorphic representations which

correspond under the theta correspondence and we conclude with a discussion of the results of Moeglin

[15] and Jiang-Soudry [8] on the irreducibility of cuspidal theta lifts. In general, the theta lift of a

cuspidal representation to another randomly chosen group is almost never cuspidal and so it is worth

noting that the situation where cuspidal representations correspond is quite special. For example, the

results of Rallis [17] show that given a cuspidal representation π of an orthogonal group OV (A) (for a

rational space V of even dimension), there is an integer n such that the theta lift of π to Spn(A) is

cuspidal, the theta lift to Spr(A) is zero for all r < n and the theta lift of π to Spr(A) is not cuspidal

for r > n.

8

Chapter 2. The Theta Correspondence 9

2.1 The Weil Representation

Metaplectic groups and the Weil representation form the basis of the theta correspondence and are

defined by twisting the representations of the Heisenberg groups. There are several introductions to the

Weil representation (see [9], [16], [19] and [24] for example) and so our goal in this section is to introduce

these objects in an informal way while referring to the literature when necessary. Our perspective is

elementary and we take the familiar Fourier transform as our starting point. The abstract Fourier

transform for self-dual locally compact abelian groups and its interaction with translation leads to the

Heisenberg group and we enumerate its properties in Summary 2.1.1. Twisting the natural representation

(ρψ, S) of the Heisenberg group by symplectic automorphisms produces the unitary operators that define

the Weil representation and we describe them in Summary 2.1.2.

2.1.1 The Fourier Transform

The Fourier transform of an integrable function f ∈ L1(R) is the continuous function defined by the

integral

f(y) =∫

Rf(x) e−2πixy dx .

The restriction of the Fourier transform to the Schwartz space S(R) of infinitely differentiable, rapidly

decreasing functions defines an automorphism

F : S(R) −→ S(R) : f 7→ f

such that f (x) = f(−x).

The definition of the Fourier transform can be exteneded to any locally compact abelian group. In

particular, let G be a locally compact abelian group, let G∗ = Homcont(G,R/Z) be the (continuous)

Pontryagin dual of G and denote the natural pairing between G and G∗ by 〈x, x∗〉 for x ∈ G and x∗ ∈ G∗.

Since G is locally compact, there is a Haar measure dx on G unique up to scalars. In particular, dx

is a Borel measure invariant by translation. The abstract Fourier tranform of an integrable function

f ∈ L1(G) is the continuous function on G∗ given by the integral

f(x∗) =∫G

f(x) e−2πi〈x,x∗〉 dx .

Just as in the case of the classical Fourier transform, there are Schwartz spaces S(G) ⊂ L1(G) and


S(G∗) ⊂ L1(G∗) such that the Fourier transform

F : S(G) −→ S(G∗) : f 7→ f

is an isomorphism. The explicit definition of S(G) depends on G however its most important property

is the isomorphism defined by the Fourier transform. Furthermore, there are unique measures on G and

G∗ such that the Fourier inversion formula f (x) = f(−x) holds.

The case G = R is special since R is self-dual in the sense that we have the isomorphism

R ∼−→ R∗ = Homcont(R,R/Z) : y 7→ (x 7→ xy)

in contrast to the case G = R/Z where (R/Z)∗ = Z. Therefore the Fourier transform is an automorphism

of S(R) depending on the identification of R with its dual. There are many other examples of self-dual

groups such as finite-dimensional vector spaces over finite fields or locals fields and the adelic points of

a finite-dimensional vector space over a global field. We make the following definition so that we may

introduce the Heisenberg group and the Weil representation for each of these cases simultaneously.

Definition 2.1.1. Let F and V be one of the following pairs:

1. F is a finite field and V is a finite dimensional F -vector space

2. F is a local field and V is a finite dimensional F -vector space

3. F is the adele ring Ak of a global field k and V is a free F -module of finite rank obtained by

extending scalars to F from a finite dimensional k-vector space

Let ( , ) : V × V → F be a nondegenerate symmetric bilinear form on V which we write as a dot product

(v, w) = v · w, and choose a nonzero unitary additive character

ψ : F −→ C× .

The Fourier transform of a Schwartz function ϕ ∈ S(V ) is

ϕ(w) =∫V

ϕ(v)ψ(v · w) dv

and the map ϕ 7→ ϕ defines an automorphism (depending on the choice ψ) of the Schwartz space S(V ).

The Haar measure dv is chosen to be self-dual relative to ψ so that ϕ(v) = ϕ(−v).


We give some examples of the spaces defined above. In each case, we define a homomorphism from

F to R/Z which we exponentiate to produce a unitary character ψ.

If V is a finite dimensional vector space over a finite field F of characteristic p, then S(V ) is the

space of all complex-valued functions on V and we have the map

Ftr−→ Fp −→ R/Z : x 7→ trx

p

where tr is the trace map from F to Fp and the tilde indicates any lift a ∈ Z of a ∈ Fp (ie. a ≡ a mod p).

If V is a finite dimensional vector space over F = R, then S(V ) is the space of infinitely differentiable,

rapidly decreasing functions on V and we have the quotient map R −→ R/Z. And if F = C, we have

the map C −→ R/Z : z 7→ 2 Re z given by the trace from C to R.

If V is a finite dimensional vector space over a p-adic field F (necessarily of finite degree over Qp), then

S(V ) is the space of locally constant, compactly supported functions on V and we have the compositions

of natural maps

Ftr−→ Qp −→ Qp/Zp → Q/Z → R/Z

where tr is the trace map from F to Qp.

If V is a free module of finite rank over the adele ring F = Ak of a number field k, then V ∼= ⊗′vVv

is isomorphic to a restricted tensor product of finite dimensional vector spaces Vv over the local fields

kv for all the places v of k. The Schwartz space S(V ) is the span of the space of factorizable functions

⊗vϕv where each ϕv is a Schwartz function on S(Vv) (and, for almost all v, ϕv is the characteristic

function of a fixed distinguished lattice Lv) and we have the map given by the product of local maps

kv → R/Z : λv 7→ λv for all places v

Ak −→ R/Z : (λv) 7→∑v|∞

λv −∑v finite

λv .

The choice to take the difference between the infinite places and the finite places ensures that the

resulting map on Ak is trivial on k.

2.1.2 Translation and Dual Translation

Let (V, F, ( , ), ψ, dv) be as in Definition 2.1.1. The space V acts on itself by translation therefore it acts

on functions

τvϕ(x) = ϕ(x+ v) .


Translation interacts with the Fourier transform by the formula

τvϕ(y) =∫V

ϕ(x+ v)ψ(x · y) dx

=∫V

ϕ(x)ψ((x− v) · y) dx

= ψ(v · y)∫V

ϕ(x)ψ(x · y) dx

= ψ(v · y) ϕ(y)

therefore we define the dual translation τ∗v by v ∈ V on functions by

τ∗vϕ(x) = ψ(x · v)ϕ(x) .

By construction, the Fourier transform intertwines the actions τv and τ∗v of V

F τv = τ∗v F .

Translation and dual translation do not commute as we compute

τvτ∗wϕ(x) = ψ((x+ v) · w)ϕ(x+ v) = ψ(v · w)ψ(x · w)ϕ(x+ v)

and

τ∗wτvϕ(x) = ψ(x · w)ϕ(x+ v)

and arrive at the fundamental commutation relation

τvτ∗w = ψ(v · w) τ∗wτv .

Let W = V × V and for each element of W define an operator ρ(v, w) = τ∗wτv on S(V ). This is not

a representation of W as we compute using the relation τvτ∗w = ψ(v · w) τ∗wτv

ρ(v1, w1)ρ(v2, w2) = τ∗w1τv1τ

∗w2τv2

= ψ(v1 · w2)τ∗w1τ∗w2

τv1τv2

= ψ(v1 · w2) ρ(v1 + v2, w1 + w2) .

This relation suggests that the natural actions of translation and dual translation are a representation


of a central extension of W .

2.1.3 The Heisenberg Group

Define the bilinear map on W = V × V

c′ : W ×W −→ F : (v1, w1; v2, w2) 7→ v1 · w2 .

Bilinearity implies that c′ is a 2-cocycle on W and therefore defines a central extension

0 −→ F −→ H ′(W ) −→W −→ 0 .

Define ρ′(v, w, t) = ψ(t)τ∗wτv and compute using the relation τvτ∗w = ψ(v · w) τ∗wτv

ρ′(v1, w1, t1)ρ′(v2, w2, t2) = ψ(t1 + t2)τ∗w1τv1τ

∗w2τv2

= ψ(t1 + t2 + v1 · w2)τ∗w1+w2τv1+v2

= ρ′(w1 + w2, v1 + v2, t1 + t2 + v1 · w2) .

Therefore ρ′ is a representation of H ′(W ) acting on S(V ).

However, our ultimate goal is to have certain automorphisms of W acting on H ′(W ). The automor-

phisms of W which preserve the cocycle define automorphisms of H ′(W ). Therefore, we will adjust the

cocycle by a coboundary so that it is recognizable as a symplectic form on W allowing the symplectic

automorphisms of W to act on this extension.

We must now make the further assumption that the characteristic of F is not 2. Write the bilinear

form c′ as a sum of a symmetric form and a symplectic form

c′(v1, w1; v2, w2) =12

(v1 · w2 + v2 · w1) +12

(v1 · w2 − v2 · w1) .

Define Q(v, w) = 12 v · w and compute the coboundary map

dQ(v1, w1; v2, w2) = Q(v1 + v2, w1 + w2)−Q(v1, w1)−Q(v2, w2)

=12

(v1 · w2 + v2 · w1) .

Therefore

c′ = dQ+ c


where we have defined the symplectic form on W

c(v1, w1; v2, w2) =12

(v1 · w2 − v2 · w1) .

The cocycle c defines a central extension H(W ) called the Heisenberg group

0 −→ F −→ H(W ) −→W −→ 0 .

This extension is isomorphic to the original extension H ′(W ) since the two cocycles are cohomologous

however now the symplectic automorphisms

Sp(W ) = A ∈ GLF (W ) : c(A(v1, w1);A(v2, w2)) = c(v1, w1; v2, w2)

define automorphisms of H(W ) by the formula A(v, w, t) = (A(v, w), t).

Summary 2.1.1. Let (V, F, ( , ), ψ, dv) be as in Definition 2.1.1 and assume 2 6= 0 in F .

1. The Heisenberg group H(W ) attached to V is the central extension of W = V × V defined by the

symplectic form

c(v1, w1; v2, w2) =12

(v1 · w2 − v2 · w1) .

In particular, H(W ) = V × V × F with the group operation

(v1, w1, t1)(v2, w2, t2) = (v1 + v2, w1 + w2, t1 + t2 +12

(v1 · w2 − v2 · w1)) .

2. The symplectic automorphisms of W

Sp(W ) = A ∈ GLF (W ) : c(A(v1, w1);A(v2, w2)) = c(v1, w1; v2, w2)

define automorphisms of H(W ) by the formula A(v, w, t) = (A(v, w), t).

3. The Heisenberg group is equipped with the representation

ρψ : H(W ) −→ U(S) : (v, w, t) 7→ ψ(t+

v · w2

)τ∗wτv

into the group U(S) of unitary automorphisms of the Schwartz space S(V ) arising from the inter-

action of the Fourier transform with translation.


2.1.4 Intertwining Operators

The symplectic automorphisms Sp(W ) of W act on the Heisenberg group H(W ) and therefore on the

functions on H(W ). To obtain a left action on functions, we view the symplectic automorphisms of W

as matrices acting on W by right multiplication

(v, w) ·

a b

c d

= (va+ wc, vb+ wd)

where a, b, c, d ∈ EndF (V ).

Each g ∈ Sp(W ) defines a new representation (ρg, S) where ρg(v, w, t) = ρ((v, w)g, t) (and we write

ρ = ρψ since the character ψ is fixed). Since the central character of the representation (ρg, S) is ψ, the

Stone-Von-Neumann Theorem implies that (ρg, S) ∼= (ρ, S) and so, for all g ∈ Sp(W ), there is a unitary

operator r(g) : S(V )→ S(V ) such that the following diagram commutes

S(V )

r(g)

ρg(v,w,t) // S(V )

r(g)

S(V )

ρ(v,w,t) // S(V )

for all (v, w, t) ∈ H(W ). The operators r(g) are best described via an intermediate space of functions

on the Heisenberg group.

Let X = V × 0 ⊂ W and Y = 0 × V ⊂ W so that W = X + Y is a complete polarization of W .

Let IndH(W )H(Y ) ψ denote the space of functions on the Heisenberg group obtained as the image of the map

ϕ 7→ ρ(v, w, t)ϕ(0) = ψ(t+ 12v · w)ϕ(v) on Schwartz functions S(V ). In other words,

IndH(W )H(Y ) ψ =

Φ(v, w, t) = ψ

(t+

v · w2

)ϕ(v) : ϕ ∈ S(V )

.

Then, by design, (ρ, S) is isomorphic to the right regular representation of H(W ) acting on IndH(W )H(Y ) ψ

and the inverse map is given by restriction to V × 0 × 0 ⊂ H(W ), Φ(v, w, t)|V×0×0 = ϕ(v). The

notation indicates that IndH(W )H(Y ) ψ is the (smooth) induction of the character ψ of the abelian subgroup

H(Y ) = (0, w, t) : w ∈ V, t ∈ F ⊂ H(W ) where ψ(0, w, t) = ψ(t). The essential property of these

functions is the left invariance by Y

Φ((0, w′, 0)(v, w, t)) = Φ(v, w + w′, t− v · w′

2

)


= ψ

(t− v · w′

2+v · (w + w′)

2

)ϕ(v)

= ψ(t+

v · w2

)ϕ(v)

= Φ(v, w, t) .

For g ∈ Sp(W ), we also have the map from the representation (ρg, S) to the right regular representation

of H(W )

ϕ 7→ Φg(v, w, t) = ρg(v, w, t)ϕ(0)

but now these functions are left invariant by Y g−1. As above, we denote by IndH(W )H(Y g−1)ψ the space of

functions Φg(v, w, t) = ρg(v, w, t)ϕ(0). We integrate to get invariance by Y

Φg(v, w, t) =∫

(Y g−1∩Y )\YΦg((0, w′, 0)(v, w, t)) dw′

where dw′ is some measure on the quotient (Y g−1 ∩ Y )\Y . Finally, restricting to V × 0 × 0 ⊂ H(W )

maps the function Φg to S(V ). Since each step was H(W )-equivariant, the result is a H(W )-intertwining

map r(g) from (ρg, S) to (ρ, S)

ϕN

(ρg, S)

r(g) // (ρ, S) Φg|V×0×0

Φgy >>IndH(W )H(Y g−1)ψ

// IndH(W )H(Y ) ψ

OO

ΦgQ

VV

Furthermore, it can be shown (see [19] and [24]) that the measure on (Y g−1 ∩ Y )\Y can be chosen so

that the operator r(g) is unitary. Finally, we can explicitly write the operator r(g) for g =

a b

c d

r(g)ϕ(v) = Φg(v, 0, 0)

=∫

(Y g−1∩Y )\YΦg((0, w, 0)(v, 0, 0)) dw

=∫

ker(c)\YΦ((wc,wd, 0)(va, vb, 0)) dw

=∫

ker(c)\YΦ(va+ wc, vb+ wd,

12

(wc · vb− va · wd)) dw

=∫

ker(c)\Yψ(

12

(wc · vb− va · wd) +12

(va+ wc) · (vb+ wd))ϕ(va+ wc) dw


=∫

ker(c)\Yψ

(wc · vb+

va · vb2

+wc · wd

2

)ϕ(va+ wc) dw

where in the third line we use that fact that since (0, w, 0) · g = (wc,wd, 0) then Y ∩ Y g = ker(c) · g so

Y g−1 ∩ Y = ker(c). In particular, we have the familiar formulas

r

1 b

0 1

ϕ(v) = ψ

(v · vb

2

)ϕ(v) (2.1)

r

a 0

0 ta−1

ϕ(v) = |det(a)|1/2F ϕ(va) (2.2)

r

0 1

−1 0

ϕ(v) =∫V

ϕ(w)ψ(v · w) dw (2.3)

where | · |F is the modulus character of F and dw is the measure on V that is self-dual relative to ψ.

2.1.5 The Metaplectic Group and the Weil Representation

In the previous section, we found a map from the symplectic group to the group of unitary automorphisms

of S(V )

r : Sp(W ) −→ U(S) : g 7→ r(g)

however this is not a representation of Sp(W ). The cohomology class [Cψ] ∈ H2(Sp(W ),C×1 ) (where

C×1 is the group of complex numbers of norm 1) defined by the 2-cocyle

r(g1)r(g2) = Cψ(g1, g2)r(g1g2)

defines a central extension

1 −→ C×1 −→ Mp(W ) −→ Sp(W ) −→ 1

called the metaplectic group Mp(W ) attached to Sp(W ). The end result is a representation of the

metaplectic group

ωψ : Mp(W ) −→ U(S) : (g, z) 7→ z r(g)

called the Weil representation.

Summary 2.1.2. Let (V, F, ( , ), ψ, dv) be as in Definition 2.1.1 (assume 2 6= 0 in F ), let H(W ) be the

Heisenberg group attached to V and let (ρψ, S) be the representation of H(W ) defined in Summary 2.1.1.


1. The action of the symplectic automorphisms Sp(W ) on H(W ) defines representations (ρgψ, S) of

H(W ) where

ρgψ(v, w, t) = ρψ((v, w)g, t) .

2. Each representation (ρgψ, S) is isomorphic to (ρψ, S) and the intertwining operator

r(g) : (ρgψ, S) −→ (ρψ, S)

is given by the unitary operator

r(g)ϕ(v) =∫

ker(c)\Yψ

(wc · vb+

va · vb2

+wc · wd

2

)ϕ(va+ wc) dcw

where g =

a b

c d

and dcw is the unique Haar measure on ker(c)\Y such that r(g) is unitary.

3. The operators r(g) define the 2-cocyle Cψ on Sp(W ) with values in C×1 = z ∈ C : |z| = 1 by

r(g1)r(g2) = Cψ(g1, g2)r(g1g2)

and gives the central extension

1 −→ C×1 −→ Mp(W ) −→ Sp(W ) −→ 1

where Mp(W ) is the metaplectic group. In particular, Mp(W ) = Sp(W ) × C×1 with the operation

(g1, z1)(g2, z2) = (g1g2, z1z2Cψ(g1, g2)) and is equipped with the representation

ωψ : Mp(W ) −→ U(S) : (g, z) 7→ z r(g)

called the Weil representation.

The explicit form of the metaplectic group depends on the choice of the 2-cocycle representing the

cohomology class in H2(Sp(W ),C×1 ) corresponding to the extension Mp(W ) of Sp(W ). In particular,

the cocycle Cψ can be altered by a coboundary to produce an equivalent form of the metaplectic group.

In the next section, we introduce the dual pairs and the explicit formulas for the Weil representation

(based on the cocycle introduced by Rao [19] which is not the cocycle in Summary 2.1.2) used in the

remainder of this thesis.


2.2 Dual Reductive Pairs

Throughout this section, let k be a global field with A = Ak its ring of adeles and fix once and for all an

additive unitary class character ψ : Ak/k −→ C×. We will introduce the dual pairs and explicit formulas

for the Weil representation (based on the cocycle introduced by Rao [19] which is not the cocycle in

Summary 2.1.2) used in the remainder of this thesis.

2.2.1 Symplectic-Orthogonal Dual Pairs

Let V be a k-vector space of even dimension m equipped with a nondegenerate symmetric bilinear form

( , ). Let Q(v) = 12 (v, v) be the corresponding quadratic form and let OV be the k-algebraic group of

automorphisms of V that preserve the symmetric form

OV = h ∈ GL(V ) : (hv, hw) = (v, w) for all v, w ∈ V .

Let W be the standard k-vector space of dimension 2n equipped with the nondegenerate skew-

symmetric bilinear form 〈 , 〉 defined by the matrix

J =

0 1n

−1n 0

where 1n is the identity matrix of size n. Let Spn = Sp(W ) be the k-algebraic group of automorphisms

of W that preserve the symplectic form. In particular, we view W as the space of row vectors of size 2n

and the automorphisms of W as matrices acting by right multiplication therefore

Spn =g ∈ GL2n : g J tg = J

.

The Siegel parabolic subgroup P of Spn has a decomposition P = MN where the Levi component M is

M =

m(a) =

a 0

0 ta−1

: a ∈ GLn

and the unipotent radical N is

N =

n(b) =

1n b

0 1n

: b ∈ Symn


where Symn denotes symmetric matrices of size n.

Let W = V ⊗k W and let 〈〈 , 〉〉 be the symplectic form on W defined by

〈〈v1 ⊗ w1, v2 ⊗ w2〉〉 =12

(v1, v2)〈w1, w2〉 .

Let Sp(W) be the k-algebraic group of automorphisms of W that preserve the symplectic form. Define

right actions of h ∈ OV and g ∈ Spn on W by

(v ⊗ w) · h = h−1v ⊗ w and (v ⊗ w) · g = v ⊗ wg .

It is clear that the algebraic groups OV and Spn are subgroups of Sp(W) such that each is the other’s

centralizer. The standard polarization W = kn ⊕ kn defines a polarization W = V n ⊕ V n where

V n = v = (v1, . . . , vn) : v1, . . . , vn ∈ V

and so v · h = h−1v where

h−1v = (h−1v1, . . . , h−1vn) .

By construction, the symplectic form is

〈〈v1 + w1,v2 + w2〉〉 =12(v1,w2)− (v2,w1)

where v1 + w1,v2 + w2 ∈ V n ⊕ V n. Extending scalars to A, we have the self-dual group V nA equipped

with the nondegenerate symmetric bilinear form (v,w) =∑i(vi, wi) therefore we proceed as in the

previous section to define the Heisenberg group H(WA), the metaplectic group Mp(WA) and the Weil

representation.

We will take the form of metaplectic group described by the cocyle defined by Rao [19]. In particular,

the metaplectic group Mp(WA) is a central extension of the group of adelic points Sp(WA) of the

symplectic group Sp(W) and fits into the exact sequence

1 −→ C×1 −→ Mp(WA) −→ Sp(WA) −→ 1

defined by the cocycle described in [19]. The assumption that m is even implies that the cocycle

is cohomologically trivial when restricted to Spn(A) therefore the extension splits over Spn(A). The


extension is also split over OV (A) therefore there is an injective homomorphism

Spn(A)×OV (A) −→ Mp(WA)

(normalized to produced the formulas below) and so we view Spn(A) and OV (A) as commuting subgroups

of Mp(WA) and we call (Spn(A),OV (A)) a symplectic-orthogonal dual pair.

Let ω = ωψ be the Weil representation of Mp(WA) (relative to ψ) acting on the space S(V nA ) of

Schwartz functions on V nA . Restricted to the symplectic-orthogonal dual pair by the chosen embedding

above, the Weil representation satisfies

ω(h)ϕ(v) = ϕ(h−1v) h ∈ OV (A)

ω(m(a))ϕ(v) = χV (det a) |det a|m2

A ϕ(va) m(a) ∈M(A)

ω(n(b))ϕ(v) = ψ(tr bQ[v])ϕ(v) n(b) ∈ N(A)

(2.4)

where

χV (x) =(x, (−1)

m(m−1)2 detV

)A

is the character of the quadratic space (V,Q) and

Q[v] =12

((vi, vj))

where ((vi, vj)) is the symmetric matrix of inner products of the components of v = (v1, . . . , vn).

2.2.2 Metaplectic-Orthogonal Dual Pairs

Let V be a k-vector space of odd dimension m equipped with a nondegenerate symmetric bilinear form

( , ). Let Q(v) = 12 (v, v) be the corresponding quadratic form and let OV be the k-algebraic group of

automorphisms of V that preserve the symmetric form.

Let W be the standard symplectic space over k of dimension 2n and let W = V ⊗kW be the symplectic

space over k defined analogous to the construction in the previous section. Note again that in this case

OV and Spn are subgroups of Sp(W) such that each is the other’s centralizer. The metaplectic group

Mp(WA) is a central extension

1 −→ C×1 −→ Mp(WA) −→ Sp(WA) −→ 1

defined by the cocycle described in [19] however the assumption that m is odd implies that the cocycle


is not cohomologically trivial when restricted to Spn(A) and it defines the central extension Mpn(A) of

Spn(A)

1 −→ C×1 −→ Mpn(A) −→ Spn(A) −→ 1 .

This extension splits over the Siegel parabolic subgroup P (A) as well as the group of rational points

Spn(k) and we view each as a subgroup of Mpn(A) by fixed embeddings.

The extension Mp(WA) of Sp(WA) is still split over OV (A) therefore there is an injective homomor-

phism

Mpn(A)×OV (A) −→ Mp(WA)

(normalized to produced the formulas below) and so we view Mpn(A) and OV (A) as commuting sub-

groups of Mp(WA) and we call (Mpn(A),OV (A)) a metaplectic-orthogonal dual pair.

Let ω = ωψ be the Weil representation of Mp(WA) (relative to ψ) acting on the space S(V nA ) of

Schwartz functions on V nA . Restricted to the metaplectic-orthogonal dual pair by the chosen embedding

above, the Weil representation satisfies

ω(h)ϕ(v) = ϕ(h−1v) h ∈ OV (A)

ω(m(a))ϕ(v) = χψV (det a) |det a|m2

A ϕ(va) m(a) ∈M(A)


ω(z)ϕ(v) = z ϕ(v) z ∈ C×1 ⊂ Mpn(A)

(2.5)

where χψV (x) = χV (x) γ(x, ψ)−1 such that γ(x, ψ) = γ(xψ)/γ(ψ) is the Weil index (see [24] or [9]).

2.2.3 Unitary-Unitary Dual Pairs

Let K be a quadratic extension of k, let AK be the ring of adeles of K and continue to let A denote

the ring of adeles of k. Let V be a K-vector space of dimension m equipped with a nondegenerate

hermitian form ( , ). In particular, we have (v, w) = (w, u)σ and (λu, µv) = λµσ(u, v) for λ, µ ∈ K

and the nontrivial element σ ∈ Gal(K/k). Let Q(v) = 12 (v, v) and let UV be the k-algebraic group of

K-linear automorphisms of V that preserve the hermitian form

UV = h ∈ GLK(V ) : (hv, hw) = (v, w) for all v, w ∈ V .

Let W be the standard K-vector space of dimension 2n equipped with the nondegenerate skew-


hermitian form 〈 , 〉 defined by the matrix

J =

0 1n

−1n 0

where 1n is the identity matrix of size n. Let Un,n be the k-algebraic group of K-linear automorphisms

of W that preserve the skew-hermitian form. In particular, we view W as the space of row vectors of

size 2n and the automorphisms of W as matrices acting by right multiplication therefore

Un,n =g ∈ GL2n/K : g J tgσ = J

.

The Siegel parabolic subgroup P of Un,n has a decomposition P = MN where the Levi component M

is

M =

m(a) =

a 0

0 (ta−1)σ

: a ∈ GLn/K

and the unipotent radical N is

N =

n(b) =

1n b

0 1n

: b ∈ Hermn

where Hermn denotes hermitian matrices of size n.

Define W = V ⊗KW as a k-vector space and let 〈〈 , 〉〉 be the symplectic k-bilinear form on W defined

by

〈〈v1 ⊗ w1, v2 ⊗ w2〉〉 =12

trK/k (v1, v2) 〈w1, w2〉σ .

Let Sp(W) be the k-algebraic group of k-linear automorphisms of W that preserve the symplectic form.

Define right actions of h ∈ UV and g ∈ Un,n on W by

(v ⊗ w) · h = h−1v ⊗ w and (v ⊗ w) · g = v ⊗ wg .

It is clear that the algebraic groups UV and Un,n are subgroups of Sp(W) such that each is the other’s

centralizer. The standard polarization W = Kn ⊕Kn defines a polarization W = V n ⊕ V n where

V n = v = (v1, . . . , vn) : v1, . . . , vn ∈ V


and so v · h = h−1v where

h−1v = (h−1v1, . . . , h−1vn) .

The metaplectic group Mp(WA) is a central extension of the group of adelic points Sp(WA) of the

symplectic group Sp(W) and fits into the exact sequence

1 −→ C×1 −→ Mp(WA) −→ Sp(WA) −→ 1

defined by the cocycle described in [19]. Let χ : A×K/K× −→ C× be a character such that χ|A = εmK/k

where εK/k is the character attached to the extension K/k by class field theory. Then χ defines a splitting

of the metaplectic extension over Un,n(A)×UV (A) (see [4] for the local construction) therefore there is

an injective homomorphism

Un,n(A)×UV (A) −→ Mp(WA)

(depending on χ) and so we view Un,n(A) and UV (A) as commuting subgroups of Mp(WA) and we call

(Un,n(A),UV (A)) a unitary-unitary dual pair.

Let ω = ωχ,ψ be the Weil representation of Mp(WA) (relative to ψ and χ, see [4]) acting on the space

S(V nA ) of Schwartz functions1 on V nA . Restricted to the unitary-unitary dual pair, the Weil representation

satisfiesω(h)ϕ(v) = ϕ(h−1v) h ∈ UV (A)

ω(m(a))ϕ(v) = χ(det a) |det a|m2

AK ϕ(va) m(a) ∈M(A)


(2.6)

and recall the groups Un,n and UV are defined over k by restriction of scalars and so m(a) ∈ M(A) is

given by a ∈ GLn(AK) and n(b) ∈ N(A) is given by b ∈ Hermn(AK).

2.3 The Theta Correspondence

Let (G(A), H(A)) be a dual reductive pair as in the previous section with the Weil representation ω

acting on S(V nA ). In particular, we have

(G(A), H(A)) =

(Spn(A),OV (A)) symplectic-orthogonal

(Mpn(A),OV (A)) metaplectic-orthogonal

(Un,n(A),UV (A)) unitary-unitary

1Note that V is defined over K and VA = V ⊗k A ∼= (V ⊗K K)⊗k A ∼= V ⊗K AK = VAK .


and we adopt the convention that G(k) = Spn(k) in the metaplectic case G(A) = Mpn(A). Each

Schwartz function ϕ ∈ S(V nA ) defines a theta function

θϕ(g, h) =∑

v∈V nω(g)ϕ(h−1v) (2.7)

which is an automorphic form on G(A)×H(A). In particular, it is left-invariant by G(k)×H(k) and is

of moderate growth. The map

ω −→ A (G×H) : ϕ 7→ θϕ

is a G(A) ×H(A)-equivariant map from the Weil representation to the space of automorphic forms on

G(A) × H(A). The theta functions act as kernel functions which provide a method for transferring

automorphic forms on one group to forms on the other.

Let σ be an irreducible cuspidal automorphic representation of G(A) acting on a subspace Vσ of

L2(G(k)\G(A)). The theta lift of fG ∈ Vσ is the function

θϕ∨fG(h) =

∫G(k)\G(A)

θϕ(g, h) fG(g) dg (2.8)

and the space of functions obtained as ϕ and fG vary is the theta lift of σ

θψ(σ) = θϕ∨fG : ϕ ∈ S(V nA ) and fG ∈ σ . (2.9)

The measure dg is chosen to be the Tamagawa measure of G(A) when G is either symplectic or unitary

and, in the metaplectic case, dg is the product of the unit measure on C×1 and Tamagawa measure on

Spn(A). The cuspidality of σ ensures the absolute convergence of the integrals. Furthermore, the map

ω∨ ⊗ σ −→ θψ(σ) : ϕ∨ ⊗ fG 7→ θϕ∨fG

is a G(A) × H(A)-equivariant map where the action of G(A) on ω∨ ⊗ σ is diagonal and the action of

G(A) on θψ(σ) is trivial.

Let π be an irreducible cuspidal automorphic representation of H(A) acting on a subspace Vπ of

L2(H(k)\H(A)). The theta lift of fH ∈ Vπ is the function

θϕfH(g) =

∫H(k)\H(A)

θϕ(g, h) fH(h) dh (2.10)


and the space of functions obtained as ϕ and fH vary is the theta lift of π

θψ(π) = θϕfH : ϕ ∈ S(V nA ) and fH ∈ π . (2.11)

The measure dh is chosen to be the Tamagawa measure of H(A). Again, the map

ω ⊗ π −→ θψ(π) : ϕ⊗ fH 7→ θϕfH

is a G(A)×H(A)-equivariant map where the action of H(A) on ω⊗π is diagonal and the action of H(A)

on θψ(π) is trivial.

Basic questions concerning theta lifts are as follows:

1. When is a theta lift nonzero?

2. When is a theta lift cuspidal?

These questions are quite deep. Solutions to the first question in some cases involve the nonvanishing of

special values of L-functions as in the work of Waldspurger [20] and the Rallis inner product formula as

in the work of Rallis [18]. The answer to the second question appears in the work of Moeglin [15] and

Jiang-Soudry [8] and relies heavily on the regularized Siegel-Weil formula as proved by Rallis-Kudla [11]

and Ichino [7]. We summarize the results on irreduciblity in the symplectic-orthogonal and metaplectic-

orthogonal cases.

Theorem 2.3.1 (Moeglin [15]). Let (Spn(A),OV (A)) be a symplectic-orthogonal dual pair.

1. Let σ be an irreducible cuspidal automorphic representation of Spn(A) and suppose θψ(σ) con-

tains cuspidal automorphic forms of OV (A). Then θψ(σ) is an irreducible cuspidal automorphic

representation of OV (A) and θψ(θψ(σ)) = σ.

2. Let π be an irreducible cuspidal automorphic representation of OV (A) and suppose θψ(π) con-

tains cuspidal automorphic forms of Spn(A). Then θψ(π) is an irreducible cuspidal automorphic

representation of Spn(A) and θψ(θψ(π)) = π.

Theorem 2.3.2 (Jiang-Soudry [8]). Let (Mpn(A),OV (A)) be a metaplectic-orthogonal dual pair.

1. Let σ be an irreducible genuine cuspidal automorphic representation of Mpn(A) and suppose θψ(σ)

contains cuspidal automorphic forms of OV (A). Then θψ(σ) is an irreducible cuspidal automorphic

representation of OV (A) and θψ(θψ(σ)) = σ.


2. Let π be an irreducible cuspidal automorphic representation of OV (A) and suppose θψ(π) contains

cuspidal automorphic forms of Mpn(A). Then θψ(π) is an irreducible genuine cuspidal automorphic

representation of Mpn(A) and θψ(θψ(π)) = π.

Chapter 3

Duality Among Periods

The main result of this thesis is the collection of spectral identities in Theorem 3.6.1 relating Fourier

coefficients and orthogonal periods of cuspidal automorphic forms for a given dual pair (G,H). The

essential calculation is contained in the main duality Proposition 3.5.1 where we show that the Fourier

coefficient of a theta lift Wt(θϕfH) is equal to the period of a Hecke translate PT (RξfH) when the

functions ϕ and ξ are matching as in Proposition 3.3.1. This calculation is inspired by the work of Wald-

spurger [20] for the dual pair (SL2(A),PGL2(A)). Regarding both operations Wt(θϕfH) and PT (RξfH)

as linear functionals on a given cuspidal automorphic representation π of H(A), we show in Proposi-

tion 3.1.1 and Proposition 3.2.1 that these functionals are given by integrating against certain elements

of π. The equality of these functions yield the spectral identities in Theorem 3.6.1 by computing their

various periods.

The method for proving Theorem 3.6.1 contained in this chapter was discovered while attempting the

comparison of relative trace formulas appearing in Chapter 4. In fact, it was only when the comparison

was complete that the more direct (and more general) argument below was formulated. The comparison

of relative trace formulas is included in the next chapter because it is an interesting calculation that

further illuminates the resulting spectral identities.

The spectral identities Theorem 3.6.1 are quite formal and follow from the simplest properties of the

Weil representation and the theta correspondence. The real potential of these identities lies in applying

them in conjunction with the Siegel-Weil formula to produce formulas equating Fourier coefficients,

orthogonal periods and special values of L-functions. In Proposition 3.4.2, we take the first steps in this

direction for PGL2(Qp) (which is the local special orthogonal group, for almost every prime p, when the

rational quadratic space V has dimension 3) by explicitly computing the local functions ξp matching

28

Chapter 3. Duality Among Periods 29

the characteristic function of the standard lattice ϕLp relative to various j. In Section 3.7, we roughly

sketch a method for producing formulas for special values of L-functions in terms of orthogonal periods

using the Siegel-Weil formula following a result of Waldspurger appearing in [20] and [23].

Let k be a global field and let A be the ring of adeles of k. Let (G(A), H(A)) be a dual pair of the

type described in Section 2.2. In particular, we have

(G(A), H(A)) =

(Spn(A),OV (A)) symplectic-orthogonal

(Mpn(A),OV (A)) metaplectic-orthogonal

(Un,n(A),UV (A)) unitary-unitary

and we adopt the convention that G(k) = Spn(k) in the metaplectic case G(A) = Mpn(A). In each case

there is the Siegel parabolic subgroup P (A) = M(A)N(A) ⊂ G(A). In our attempt to treat each case

simultaneously, we let

Sym∗n =

Symn symplectic-orthogonal

Symn metaplectic-orthogonal

Hermn unitary-unitary

and in each case we have Sym∗n ∼= N via b 7→ n(b).

In the orthogonal case, the quadratic space V defining OV is a k-vector space and we have the space

of Schwartz functions S(V nA ). In the unitary case, the hermitian space V defining UV is a K-vector

space where K is a quadratic extension of k. Since V ⊗K AK ∼= V ⊗k Ak, we again write the space of

Schwartz functions as S(V nA ) where as always A is the adele ring of k.

The set of irreducible cuspidal automorphic representations of G(A) (resp. H(A)) is denoted A G0

(resp. A H0 ). Given σ ∈ A G

0 (resp. π ∈ A H0 ), let Vσ (resp. Vπ) denote the subspace of L2(G(k)\G(A))

(resp. L2(H(k)\H(A))) on which the representation acts. The inner product of square-integrable func-

tions on G(k)\G(A) (resp. H(k)\H(A)) is denoted by 〈fG1 , fG2 〉G (resp. 〈fH1 , fH2 〉H).

3.1 Fourier Coefficients

For t ∈ Sym∗n(k), the ψt-Fourier coefficient of an automorphic form fG on G(A) is

Wt(fG) =∫N(k)\N(A)

fG(n)ψt(n) dn (3.1)

where ψt(n) = ψ(tr tb) for n = n(b). The goal of this chapter is to relate the Fourier coefficients of theta

lifts to orthogonal periods as in Proposition 3.5.1. Our first step is to compute the spectral expansion


of the operation fH 7→Wt(θϕfH).

Proposition 3.1.1. Let σ and π be cuspidal automorphic representations of G and H respectively such

that θψ(σ) = π. For all t ∈ Sym∗n(k) and ϕ ∈ S(V nA ), the linear functional on Vπ ⊂ L2(H(k)\H(A))

defined by fH 7→Wt(θϕfH) is given by the inner product with the function

θϕ∨,σ,t(h) =∑

FG∈B(σ)

θϕ∨FG(h) Wt(FG) (3.2)

where B(σ) is an orthonormal basis of Vσ.

Proof. Using the basis B(σ) we write

θϕfH(g) =

∑FG∈B(σ)

⟨θϕf

H , FG⟩GFG(g) .

We find the adjoint property⟨θϕf

H , FG⟩G

=⟨fH , θϕ∨F

G⟩H

by computing

⟨θϕf

H , FG⟩G

=∫G(k)\G(A)

(∫H(k)\H(A)

θϕ(g, h) fH(h) dh

)FG(g) dg

=∫H(k)\H(A)

fH(h)

(∫G(k)\G(A)

θϕ(g, h)FG(g) dg

)dh .

Therefore, we compute the ψt-Fourier coefficient

Wt(θϕfH) =∑

FG∈B(σ)

⟨θϕf

H , FG⟩GWt(FG)

=∑

FG∈B(σ)

⟨fH , θϕ∨F

G⟩HWt(FG)

=

⟨fH ,

∑FG∈B(σ)

θϕ∨FG Wt(FG)

⟩H

.

3.2 Orthogonal Periods

Let j ∈ V n and let T be the stabilizer of j in H where H acts componentwise on V n. The period over

T of a cuspidal automorphic form fH on H(A) is

PT (fH) =∫T (k)\T (A)

fH(τ) dτ . (3.3)


We call these integrals orthogonal periods because the subgroup T is the orthogonal/unitary group

attached to the subspace v ∈ V : v ⊥ ji for i = 1, . . . , n ⊂ V which is orthogonal to each of the

components of j = (j1, . . . , jn).

Given a Schwartz function ξ ∈ S(H(A)), the Hecke operator Rξ is the convolution operator

RξfH(h) =

∫H(A)

ξ(x) fH(hx) dx .

Looking ahead to Proposition 3.5.1, the orthogonal periods PT (RξfH) appear when the integrals defining

Wt(θϕfH) are unfolded.

Proposition 3.2.1. Let π ∈ A H0 . For all ξ ∈ S(H(A)), the linear functional on Vπ ⊂ L2(H(k)\H(A))

defined by fH 7→ PT (RξfH) is given by the inner product with the function

Rξ∨,π,T (h) =∑

FH∈B(π)

Rξ∨FH(h) PT (FH) (3.4)

where B(π) is an orthonormal basis of Vπ and ξ∨(h) = ξ(h−1).

Proof. Using the basis B(π) we write

RξfH(h) =

∑FH∈B(π)

⟨Rξf

H , FH⟩HFH(h) .

We find the adjoint property⟨Rξf

H , FH⟩H

=⟨fH , Rξ∨F

H⟩H

by computing

⟨Rξf

H , FH⟩H

=∫H(k)\H(A)

(∫H(A)

ξ(x) fH(hx) dx

)FH(h) dh

=∫H(A)

ξ(x)

(∫H(k)\H(A)

fH(hx)FH(h) dh

)dx

=∫H(k)\H(A)

fH(h)

(∫H(A)

ξ(x−1)FH(hx) dx

)dh .

Therefore, we compute the period over T

PT (RξfH) =∑

FH∈B(π)

⟨Rξf

H , FH⟩HPT (FH)

=∑

FH∈B(π)

⟨fH , Rξ∨F

H⟩HPT (FH)


=

⟨fH ,

∑FH∈B(π)

Rξ∨FH PT (FH)

⟩H

.

3.3 Matching Functions

The next proposition introduces a notion of matching between the Schwartz functions ϕ ∈ S(V nA ) defining

the theta functions θϕ and the Schwartz functions ξ ∈ S(H(A)) defining the Hecke operators Rξ. This

is the main input into Proposition 3.5.1 which shows that Wt(θϕfH) and PT (RξfH) are equal exactly

when the functions ϕ and ξ are matching.

Proposition 3.3.1. Let j ∈ V n with Q[j] = t such that det t 6= 0, and let T be the stabilizer of j in H.

For all ϕ ∈ S(V nA ) there is a smooth function ξ on H(A) such that

ϕ(h−1j) =∫T (A)

ξ(τh) dτ (3.5)

and ξv = ξ|H(kv) is compactly supported for all finite places v and rapidly decreasing for all infinite places

v. In this case, we say ξ matches ϕ relative to j.

Proof. We may assume that ϕ = ⊗v ϕv is factorizable therefore we will prove the analogous equality for

each place v. The group H(A) can be written as the restricted product∏′vH(kv) with respect to the

compact open subgroups Kov = Aut(Lv) where Lv = L⊗O Ov for a fixed global O-lattice L ⊂ V . Here

O is the ring of integers of k and Ov is the ring of integers of the competion kv. We will show that ξv is

the characteristic function of Kov for almost every place v.

Suppose v is a finite place. If Vv is anisotropic, the group H(kv) is compact and so

ξv(h) =1

volT (kv)ϕv(h−1j)

is a smooth function of compact support which matches ϕv relative to j. Note that, since det t 6= 0, the

group H(kv) acts transitively on the set Ωt = v ∈ V nv : Q[v] = t (which is a closed subset of V nv ) and

the map h 7→ h−1j is a homeomorphism between T (kv)\H(kv) and Ωt.

Suppose Vv is isotropic. Let ϕv be a locally constant compactly supported function and let Kv be

a compact open subgroup of H(kv) such that ϕv(kv) = ϕv(v) for all k ∈ Kv and v ∈ V nv . Use the


notation CS to denote the characteristic function of a set S and write

ϕv(h−1j) =∑

γ ∈ H(kv)/Kv

γ−1j ∈ suppϕv

ϕv(γ−1j)CγKv (h)

=∑

γ ∈ T (kv)\H(kv)/Kv

γ−1j ∈ suppϕv

ϕv(γ−1j)∑

τ∈(T (kv)∩γKvγ−1)\T (kv)

CγKv (τh) .

We claim that the outer sum is finite. Since det t 6= 0, the group H(kv) acts transitively on the set

Ωt = v ∈ V nv : Q[v] = t and therefore the map γ 7→ γ−1j is a homeomorphism between T (kv)\H(kv)

and Ωt. Since ϕv has compact support, its restriction to the closed subset Ωt has compact support and

therefore the support of ϕv(h−1j) in T (kv)\H(kv) is also compact. Finally, since T (kv)\H(kv)/Kv is

discrete, the set γ ∈ T (kv)\H(kv)/Kv : γ−1j ∈ suppϕv is finite.

We make the observation

∫T (kv)

CγKv (τh) dτ = vol(T (kv) ∩ γKvγ−1)

∑τ∈(T (kv)∩γKvγ−1)\T (kv)

CγKv (τh)

for all γ, h ∈ H(kv). Therefore

ξv(h) =∑

γ ∈ T (kv)\H(kv)/Kv

γ−1j ∈ suppϕv

ϕv(γ−1j)vol(T (kv) ∩ γKvγ−1)

CγKv (h)

is a locally constant compactly supported function which matches ϕv relative to j.

We claim that, for almost all v, the sum above has a single term and ξv(h) = CK0v(h). For almost

all v, we are in the following situation:

1. ϕv = ϕLv ⊗ · · · ⊗ ϕLv is n copies of the characteristic function ϕLv of the lattice Lv = L⊗O Ov

2. j1, . . . , jn ∈ Lv

3. detQ[j] ∈ O×v

4. Q is Ov-valued on Lv

The symmetric matrix Q[j] represents the quadratic form Q restricted to Λj = spanOvj1, . . . , jn

relative to the basis j1, . . . , jn. Since detQ[j] ∈ O×v , the lattice Λj is regular therefore Lv = Λj ⊕ Λ′ is


an orthogonal direct sum of Ov-lattices where

Λ′ = v ∈ Lv : (v, w) = 0 for all w ∈ Λj .

Suppose γ ∈ H(kv) such that ji ∈ γLv for each i = 1, . . . , n. Then Λj is a regular subspace of γLv

therefore we have the orthogonal direct sum γLv = Λj ⊕ Λ′′ of Ov-lattices where

Λ′′ = γv ∈ γLv : (γv, w) = 0 for all w ∈ Λj .

Since Λj ⊕ Λ′ and Λj ⊕ Λ′′ are isometric, we must have that Λ′ and Λ′′ are isometric by the Witt

cancellation property for local rings. In other words, there is some δ ∈ T (kv) (note that T (kv) is the

orthogonal group of spankvj1, . . . , jn⊥) such that δΛ′′ = Λ′. Finally, δγLv = Lv and so δγ ∈ Ko

v .

Therefore, in this most unramified case, the sum above has a single term and ξv(h) = CKov(h) is the

characteristic function of Kov . Here we have used the fact that vol(T (kv) ∩Ko

v) = 1 for almost all v.

Suppose v is an infinite place. If Vv is anisotropic, the group H(kv) is compact and so

ξv(h) =1

volT (kv)ϕv(h−1j)

is a smooth compactly supported function which matches ϕv relative to j.

Suppose Vv is isotropic. We follow the construction in [5, Lemma 1.10, p.92]. Let us simplify the

notation and let H = H(kv) which we view as a real Lie group and let T = T (kv) which we view a Lie

subgroup of H. Define a map f 7→ f on the space Cc(H) of continuous compactly supported functions

by

f(h) =∫T

f(τh) dτ .

We will proceed as in [5] to show that the map f 7→ f is a linear map of Cc(H) onto Cc(T\H).

The map h 7→ h−1j defines a diffeomorphism from T\H to Ωt = v ∈ V nv : Q[v] = t. First suppose

that ϕ = ϕv is a smooth function of compact support on V nv which we view by restriction as a smooth

function of compact support on T\H with support in a compact set C ⊂ T\H. Let C ′ be a compact

subset of H such that π(C ′) = C for the projection map π : H → T\H. Let CT be a compact subset of

T of positive measure and put C = CT · C ′. Then π(C ′) = C. Select f ∈ Cc(G) such that f ≥ 0 on G


and f > 0 on C ′. Then f > 0 on C and the function

ξ(h) =

f(h)

ϕ(h−1j)f(h)

if π(h) ∈ C

0 if π(h) 6∈ C

has compact support and ξ = ϕ.

If ϕ ∈ S(V nv ) then let ϕi be a sequence of smooth compactly supported functions such that ϕi → ϕ

uniformly. Then we can choose a sequence fi of smooth compactly supported functions and define the

sequence of functions ξi as in the construction above so that ξi = ϕi. There is much flexibility in the

choice of the functions fi and so we can choose them such that the sequence ξi converges uniformly

to a smooth, rapidly decreasing function ξ on H which matches ϕ relative to j.

3.4 Example: Matching Functions for PGL2(Qp)

In this section, we specialize to the case dimV = 3 and explicitly compute some local matching functions

ϕp and ξp following Proposition 3.3.1. After this work was completed, we found similar calculations in

[14]. Note that below we restrict to Qp while the local field considered in the local calculations in [14] is

arbitrary. Furthermore, the value Q(j) for the calculations in [14] (when ϕp is the characteristic function

of the standard lattice in Vp, see [14, Lemma 3.1 p.599]) is assumed to be either a unit or a prime while

below we allow the value Q(j) to have any p-adic order.

Let Vp be the space of traceless 2 by 2 matrices over Qp equipped with the quadratic form Q(x) =

−det(x). In this case, we have O(Vp) = SO(Vp)× 〈±1〉 and SO(Vp) = PGL2(Qp) acting on Vp by conju-

gation. Let Lp be the lattice of integral matrices and let Kp ⊂ SO(Vp) denote the proper automorphisms

of Lp. Let j ∈ Vp and let Q(j) = pαε with α ∈ Z and ε ∈ Z×p . There are three possibilities for j:

(unramified) if α is even and ε is a nonsquare, then there is some h ∈ O(Vp) such that

h(j) = pα/2

0 ε

1 0

(ramified) if α is odd, then there is some h ∈ O(Vp) such that

h(j) = p(α−1)/2

0 pε

1 0


(split) if α is even and ε is a square, then there is some h ∈ O(Vp) such that

h(j) = pα/2√ε

1 0

0 −1

To compute matching functions we need to have representatives for the double coset space Tp\SO(Vp)/Kp

where Tp is the stabilizer of j.

Proposition 3.4.1. Let j ∈ Vp and let Tp be the stabilizer of j in SO(Vp).

(i) A set of coset representatives of SO(Vp)/Kp is given by

pd u

0 1

: d ≥ 0 and 0 ≤ u < pd

∪1 0

u pd

: d > 0, 0 ≤ u < pd and p | u

.

(ii) If j =

0 ε

1 0

where ε ∈ Z×p is a nonsquare, then Tp\SO(Vp)/Kp =⋃d≥0

Tp

pd 0

0 1

Kp.

(iii) If j =

0 pε

1 0

where ε ∈ Z×p , then Tp\SO(Vp)/Kp =⋃d≥1

Tp

pd 0

0 1

Kp.

(iv) If j =

1 0

0 −1

, then Tp\SO(Vp)/Kp =⋃d≥0

Tp

pd 1

0 1

Kp.

Proof. The Iwasawa decomposition states that SO(Vp) = BKp where B are the upper triangular matrices

and so we need only consider elements of B. Let g ∈ B and choose a matrix representative that has

integral entries with at least one entry a unit. There are three possibilities:

Case 1: Write g =

pdε1 b

0 ε2

for some d ≥ 0, ε1, ε2 ∈ Z×p and b ∈ Zp. There is a unique u ∈ Z

and x ∈ Zp such that 0 ≤ u < pd and bε−12 = u+ pdx. Modify the element g by multiplying on the right

by elements of Kp

pdε1 b

0 ε2

ε−1

1 0

0 ε−12

1 −x

0 1

=

pd bε−12

0 1

1 −x

0 1

=

pd u

0 1

.

Case 2: Write g =

ε1 b

0 pdε2

for some d ≥ 0, ε1, ε2 ∈ Z×p and b ∈ Zp. Modify the element g by


multiplying on the right by elements of Kp

ε1 b

0 pdε2

ε−1

1 0

0 ε−12

1 −bε−1

2

0 1

=

1 bε−12

0 pd

1 −bε−1

2

0 1

=

1 0

0 pd

.

Case 3: Write g =

pαε1 u

0 pβε2

for some α, β ≥ 0 and ε1, ε2, u ∈ Z×p . There is a unique u ∈ Z and

x ∈ Zp such that 0 ≤ u < pα+β and pβu−1ε2 = u+ pα+βx. Modify the element g by multiplying on the

right by elements of Kp

pαε1 u

0 pβε2

ε−1

1 0

0 ε−12

1 0

−pαu−1ε2 1

0 −uε−1

2

u−1ε2 0

1 0

−x 1

=

pα uε−12

0 pβ

1 0

−pαu−1ε2 1

0 −uε−1

2

u−1ε2 0

1 0

−x 1

=

0 uε−12

−pα+βu−1ε2 pβ

0 −uε−1

2

u−1ε2 0

1 0

−x 1

=

1 0

pβu−1ε2 pα+β

1 0

−x 1

=

1 0

u pα+β

.

If p - u then u ∈ Z×p and

pα+β u−1

−u 0

∈ Kp. There is a unique u′ ∈ Z and x ∈ Zp such that

0 ≤ u′ < pα+β and u−1 = u′ + pα+βx. Compute

1 0

u pα+β

pα+β u−1

−u 0

1 −x

0 1

=

pα+β u−1

0 1

1 −x

0 1

=

pα+β u′

0 1

.


Therefore, we have the coset representatives in (i).

Given the element j in (ii), we have

Tp =

a bε

b a

∈ SO(Vp)

.

Modify the coset representatives obtained above by multiplying on the left by an element of Tp and on

the right by an element of Kp

1 −u

−uε−1 1

pd u

0 1

1 0

pduε−1(1− u2ε−1)−1 (1− u2ε−1)−1

=

pd 0

−pduε−1 1− u2ε−1

1 0

pduε−1(1− u2ε−1)−1 (1− u2ε−1)−1

=

pd 0

0 1

We also compute 0 ε

1 0

1 0

u pd

0 1

ε−1 0

=

pd uε

0 1

which can be modified further as above to produce

pd 0

0 1

. Therefore we obtain the representatives

in (ii). The other two statements are proved analogously.

Proposition 3.4.2. Let ϕp be the characteristic function of Lp, let j ∈ Vp, let Tp be the stabilizer of j

in SO(Vp) and let Q(j) = pαε with α ∈ Z and ε ∈ Z×p . We write CS for the characteristic function of a

set S.

(i) If α < 0, then ϕp(h−1j) = 0 for all h ∈ SO(Vp).

(ii) If j = pα/2

0 ε

1 0

with α ≥ 0 even and ε ∈ Z×p a nonsquare, then

ξp(h) =1

volTp

CKp(h) +α/2∑d=1

(pd + pd−1)C[pd]Kp(h)

, [pd] =

pd 0

0 1

, (3.6)

matches ϕp relative to j.


(iii) If j = p(α−1)/2

0 pε

1 0

with α > 0 odd, then

ξp(h) =1

volTp

(α+1)/2∑d=1

2pd−1 C[pd]Kp(h) , [pd] =

pd 0

0 1

, (3.7)


(iv) If j = pα/2√ε

1 0

0 −1

with α ≥ 0 even and ε ∈ Z×p a square, then

ξp(h) =1

vol(Tp ∩Kp)

CKp(h) +α/2∑d=1

C[pd,1]Kp(h)

, [pd, 1] =

pd 1

0 1

, (3.8)


Proof. We have ordpQ(v) ≥ 0 for all v ∈ Lp and Q(h−1j) = Q(j) for all h ∈ SO(Vp) therefore if

ordpQ(j) < 0 then h−1j 6∈ Lp for all h ∈ SO(Vp) and so ϕp(h−1j) = 0.

In the proof of Proposition 3.3.1, we saw that the function

ξp(h) =∑

γ ∈ Tp\SO(Vp)/Kp

γ−1j ∈ Lp

1vol(Tp ∩ γKpγ−1)

CγKp(h) (3.9)

is a smooth function of compact support that matches ϕp relative to j.

In the first case, if j = pα/2

0 ε

1 0

then:

• Tp =

a bε

b a

∈ PGL2(Qp)

and Tp\SO(Vp)/Kp =⋃d≥0

Tp[pd]Kp where [pd] =

pd 0

0 1

• For d > 0,

1vol(Tp ∩ [pd]Kp[p−d])

=

[Tp : Tp ∩ [pd]Kp[p−d]

]volTp

=pd + pd−1

volTp

• [p−d] · j = pα/2−d

0 ε

p2d 0

, therefore [p−d] · j ∈ Lp if d ≤ α/2

To prove the second point, note that Tp ⊂ Kp fixes the base point Kp ∈ SO(Vp)/Kp in the Bruhat-Tits

tree of PGL2(Qp) and acts transitively on the pd + pd−1 points at a distance d from the base point while


the subgroup Tp ∩ [pd]Kp[p−d] is the stabilizer in Tp of a point at a distance d from the base point.

Therefore

ξp(h) =1

volTp

CKp(h) +α/2∑d=1

(pd + pd−1)C[pd]Kp(h)

.

If j = p(α−1)/2

0 pε

1 0

, then

• Tp =

a bpε

b a

∈ PGL2(Qp)

and Tp\SO(Vp)/Kp =⋃d≥1

Tp[pd]Kp where [pd] =

pd 0

0 1

• For d > 0,1

vol(Tp ∩ [pd]Kp[p−d])=

[Tp : Tp ∩ [pd]Kp[p−d]

]volTp

=2pd−1

volTp

• [p−d] · j = p(α+1)/2−d

0 ε

p2d−1 0

, therefore [p−d] · j ∈ Lp if d ≤ (α+ 1)/2

To prove the second point, note that Tp fixes the midpoint between Kp and [p]Kp in the tree SO(Vp)/Kp

and acts transitively on the 2pd−1 points at a distance d− 1/2 from that midpoint while the subgroup

Tp∩[pd]Kp[p−d] is the stabilizer in Tp of the point [pd]Kp which is at a distance d−1/2 from the midpoint

between Kp and [p]Kp. Therefore

ξp(h) =1

volTp

(α+1)/2∑d=1

2pd−1 C[pd]Kp(h) .

Finally, if j = pα/2√ε

1 0

0 −1

, then

• Tp =

a 0

0 b

∈ PGL2(Qp)

and Tp\SO(Vp)/Kp =⋃d≥0 Tp[p

d, 1]Kp where [pd, 1] =

pd 1

0 1

.

• For d ≥ 0,1

vol(Tp ∩ [pd, 1]Kp[pd, 1]−1)=

1vol(Tp ∩Kp)

• [pd, 1]−1 · j = pα/2−d√ε

pd 2

0 −pd

therefore [pd, 1]−1 · j ∈ Lp if d ≤ α/2

To prove the second point, note that Tp acts on the set of vertices [pd]Kp : d ∈ Z in the tree SO(Vp)/Kp

and [pd, 1]Kp[pd, 1]−1 is the stabilizer in SO(Vp) of the point [pd, 1]Kp. Therefore, if τ ∈ Tp stabilizes


[pd, 1]Kp it must fix Kp also therefore Tp ∩ [pd, 1]Kp[pd, 1]−1 = Tp ∩Kp. Therefore

ξp(h) =1

vol(Tp ∩Kp)

CKp(h) +α/2∑d=1

C[pd,1]Kp(h)

, [pd, 1] =

pd 1

0 1

.

Notice that when d = 0, we have

1 1

0 1

Kp = Kp.

3.5 Duality Among Periods

The next proposition gives the main duality between Fourier coefficients and orthogonal periods. It is a

simple calculation using the notion of matching described in Proposition 3.3.1. During the preparation

of this thesis, we found a version of this proposition for the dual pair (SL2, PB×) (where SL2 is the

metaplectic cover of SL2 and B is a quaternion algebra over k) appearing in a paper by Mao [14]. An

immediate consequence, stated in the following corollary, is that the functions θϕ∨,σ,t(h) and Rξ∨,π,T (h)

found in Proposition 3.1.1 and Proposition 3.2.1 respectively are equal when the functions ϕ and ξ are

matching relative to j. Looking ahead, it is the equality of these functions which yields the spectral

identities in Theorem 3.6.1 in the next section.

Proposition 3.5.1. Let j ∈ V n with Q[j] = t such that det t 6= 0, and let T be the stabilizer of j

in H. Let ϕ ∈ S(V nA ) and let ξ be a smooth function on H(A) which matches ϕ relative to j as in

Proposition 3.3.1. Then, for all fH ∈ L2(H(k)\H(A)),

Wt(θϕfH) = PT (RξfH) . (3.10)

Proof. We simply compute the left hand side by changing the order of integration

Wt(θϕfH) =∫N(k)\N(A)

∫H(k)\H(A)

∑v∈V n

ω(n(b))ϕ(h−1v) fH(h)ψt(b) db

=∫H(k)\H(A)

∫N(k)\N(A)

( ∑v∈V n

ϕ(h−1v)ψ(b(Q[v]− t)) db

)fH(h) dh

=∫H(k)\H(A)

∑v ∈ V n

Q[v] = t

ϕ(h−1v) fH(h) dh .


Since det t 6= 0, the sum is a single H(k)-orbit therefore we continue

Wt(θϕfH) =∫H(k)\H(A)

∑γ∈T (k)\H(k)

ϕ(h−1γ−1j) fH(h) dh

=∫T (k)\H(A)

ϕ(h−1j) fH(h) dh

=∫T (A)\H(A)

ϕ(h−1j)∫T (k)\T (A)

fH(τh) dτ dh .

The function ξ matches ϕ relative to j as in Proposition 3.3.1 therefore

Wt(θϕfH) =∫T (A)\H(A)

(∫T (A)

ξ(τ ′h) dτ ′) ∫

T (k)\T (A)

fH(τh) dτ dh

=∫H(A)

∫T (k)\T (A)

ξ(h)fH(τh) dτ dh

=∫T (k)\T (A)

∫H(A)

ξ(h)fH(τh) dh dτ

and we conclude Wt(θϕfH) = PT (RξfH).

Corollary 3.5.1. Let j ∈ V n with Q[j] = t such that det t 6= 0, and let T be the stabilizer of j in H. Let

ϕ ∈ S(V nA ) and let ξ be a smooth function on H(A) which matches ϕ relative to j as in Proposition 3.3.1.

Given cuspidal automorphic representations σ and π of G and H respectively such that θψ(σ) = π, we

have ∑FG∈B(σ)

θϕ∨FG(h) Wt(FG) =

∑FH∈B(π)

Rξ∨FH(h) PT (FH) (3.11)


Proof. Proposition 3.5.1 says that the linear functionals fH 7→ Wt(θϕfH) and fH 7→ PT (RξfH) are

equal therefore the functions in Proposition 3.1.1 and Proposition 3.2.1 (which are clearly elements of

Vπ) are equal.

3.6 The Main Theorem: Spectral Identities

Equipped with Corollary 3.5.1, we are now prepared to prove our main result: a collection of spectral

identities relating Fourier coefficients and orthogonal periods of cuspidal automorphic forms for a given

dual pair (G,H). As the analysis in the previous sections show, these period relations are manifestations

of the main duality Wt(θϕfH) = PT (RξfH) in Proposition 3.5.1. These identities are quite formal and, in

the next section, we explore the possibility of proving formulas involving Fourier coefficients, orthogonal


periods and, via the Siegel-Weil formula, special values of L-functions.

Theorem 3.6.1. Let j1, j2 ∈ V n with Q[j1] = t1 and Q[j2] = t2 such that det t1 6= 0 and det t2 6= 0,

and let T1 and T2 be the stabilizers in H of j1 and j2 respectively. Let ϕ1, ϕ2 ∈ S(V nA ) and let ξ1 and

ξ2 be smooth functions on H(A) such that ξ1 matches ϕ1 relative to j1 and ξ2 matches ϕ2 relative to j2

as in Proposition 3.3.1. Given cuspidal automorphic representations of G and H respectively such that

θψ(σ) = π, we have

∑FG∈B(σ)


∑FH∈B(π)


∑FG∈B(σ)


∑FH∈B(π)

PT1(Rξ∨2 FH) PT2(FH) (3.13)

∑FG∈B(σ)

PT1(θϕ∨1 FG) PT2(θϕ∨2 F

G) =∑

FH∈B(π)

PT1(θϕ∨1 θϕ2FH) PT2(FH) (3.14)


Proof. Write Equation 3.11 with the data j2, t2, T2, ϕ2 and ξ2

∑FG∈B(σ)

θϕ∨2 FG(h) Wt2(FG) =

∑FH∈B(π)

Rξ∨2 FH(h) PT2(FH)

and then apply the linear functional fH 7→Wt1(θϕ1fH) to each side

∑FG∈B(σ)

Wt1(θϕ1θϕ∨2 FG) Wt2(FG) =

∑FH∈B(π)

Wt1(θϕ1Rξ∨2 FH) PT2(FH) .

Using the equalityWt1(θϕ1Rξ∨2 FH) = PT1(Rξ1Rξ∨2 F

H) as in Proposition 3.5.1, we arrive at Equation 3.12

∑FG∈B(σ)

Wt1(θϕ1θϕ∨2 FG) Wt2(FG) =

∑FH∈B(π)

PT1(Rξ1Rξ∨2 FH) PT2(FH) .

For the second equation, write Equation 3.11 with the data j2, t2, T2, ϕ2 and ξ2 and then apply the

linear functional fH 7→ PT1(fH) to each side to obtain Equation 3.13

∑FG∈B(σ)


∑FH∈B(π)

PT1(Rξ∨2 FH) PT2(FH) .


For the third equation, write f ∈ Vσ in terms of the basis B(σ)

f(g) =∑

FG∈B(σ)

〈f, FG〉G FG(g)

therefore

PT2(θϕ∨2 f) =

⟨f,

∑FG∈B(σ)

PT2(θϕ∨2 FG) FG

⟩G

.

Alternatively, write θϕ∨2 f in terms of the basis B(π)

θϕ∨2 f(h) =∑

FH∈B(π)

〈θϕ∨2 f, FH〉H FH(h)

and compute PT2(θϕ∨2 f)

PT2(θϕ∨2 f) =∑

FH∈B(π)

⟨θϕ∨2 f, F

H⟩HPT2(FH) =

⟨f,

∑FH∈B(π)

PT2(FH) θϕ2FH

⟩H

therefore ∑FG∈B(σ)

PT2(θϕ∨2 FG)FG(g) =

∑FH∈B(π)

PT2(FH) θϕ2FH(g) .

Apply the linear functional fG 7→ PT1(θϕ∨1 fG) and we arrive at Equation 3.14

∑FG∈B(σ)

PT1(θϕ∨1 FG)PT2(θϕ∨2 F

G) =∑

FH∈B(π)

PT1(θϕ∨1 θϕ2FH)PT2(FH) .

3.7 Further Directions: Special Values of L-Functions

We conclude this chapter by roughly sketching a method for producing formulas for special values of

L-functions in terms of orthogonal periods using the Siegel-Weil formula. Consider the second spectral

identity in Theorem 3.6.1

∑FG∈B(σ)


∑FH∈B(π)

PT1(Rξ∨2 FH) PT2(FH)

applied to the dual pair (SL2(A),OV (A)). Here SL2(A) is the metaplectic cover of SL2(A) and OV

is the orthogonal group of the vector space V of traceless elements of a quaternion division algebra


B over k equipped with the norm form. During the course of Waldspurger’s deep study of the theta

correspondence for such dual pairs in [20], [21], [22] and [23], Waldspurger proves the following theorem

which links the orthogonal periods of theta lifts to the special values of L-functions.

Theorem 3.7.1. (Waldspurger, [23, Lemma 45 p.293]) Let σ be a cuspidal automorphic representation

of SL2(A) and let π be an automorphic representation of OV (A) such that θψ(σ) = π. Let Waldψ(σ) be

the Jacquet-Langlands transfer of π to PGL2(A). Let j ∈ V be nonzero with Nm(j) = t, let T be the

stabilizer of j in OV , let ϕ ∈ S(VA) and let f ∈ Vσ. There is a finite set S of places v such that

PT (θϕ∨f) =LS(Waldψ(σ), 1/2)

LS(χt, 1)

∏v∈S

Zv(f, ϕ, j, 1/2)

where χt is the quadratic character attached to the extension k(√t),

Zv(f, ϕ, j, s) =∫N(kv)\SL2(kv)

W fv (g)ω(g)ϕv(j)|a(g)|s−1/2

v dg

are local zeta functions and ∫[N ]

f(ng)ψt(n) dn =∏v

W fv (g)

is the decomposition of the ψt-Whittaker function of f into local ψt,v-Whittaker functions.

Proof. Consider the orthogonal decomposition V = V ′ ⊕ V ′′ where V ′ = span j and V ′′ = (span j)⊥.

Note that O(V ′′) = T . Suppose ϕ(aj +w) = ϕ′(aj)ϕ′′(w) for all aj +w ∈ V ′⊕ V ′′ with ϕ′ ∈ S(V ′A) and

ϕ′′ ∈ S(V ′′A ). Now we compute

PT (θϕ∨f) =∫

[T ]

∫[SL2]

∑v∈V

ω(g)ϕ(τ−1v) f(g) dg dτ

=∫

[SL2]

∑aj∈V ′

ω(g)ϕ′(aj)∫

[T ]

∑w∈V ′′

ω(g)ϕ′′(τ−1w) dτ f(g) dg

=∫

[SL2]

∑aj∈V ′

ω(g)ϕ′(aj)E(g, 0,Φϕ′′) f(g) dg .

The last line uses the Siegel-Weil formula (which Waldspurger proves in [20, Proposition 31 p.119] for

the dual reductive pair (SL2(A), T (A)) (see also [10] and [18])

∫[T ]

∑w∈V ′′

ω(g)ϕ′′(τ−1w) dτ = E(g, 0,Φϕ′′)


where the Eisenstein series is given by the analytic continuation of the absolutely convergent sum

E(g, s,Φϕ′′) =∑

γ∈P (k)\SL2(k)

ω(γg)ϕ′′(0) |a(γg)|s , Re s > 1 .

Note that we are following the standard conventions in [10] so that there is a slight shift in the variable

s as compared to [20]. To unfold this integral further, introduce the integral for s ∈ C with Re s > 1

PT (θϕ∨f ; s) =∫

[SL2]

∑aj∈V ′

ω(g)ϕ′(aj)E(g, s,Φϕ′′) f(g) dg

=∫

[SL2]

∑aj∈V ′

ω(g)ϕ′(aj)∑

γ∈P (k)\SL2(k)

ω(γg)ϕ′′(0) |a(γg)|s f(g) dg

=∫P (k)\SL2(A)

∑aj∈V ′

ω(g)ϕ′(aj)ω(γg)ϕ′′(0) |a(g)|s f(g) dg

=∫P (k)\SL2(A)

∑aj∈V ′

ω(g)ϕ(aj) |a(g)|s f(g) dg .

Since ω(g)ϕ(aj) = ω(m(a)g)ϕ(j) for all a ∈ k, we continue

PT (θϕ∨f ; s) =∫P (k)\SL2(A)

∑m(a)∈M(k)

ω(m(a)g)ϕ(j) |a(g)|s f(g) dg

+∫P (k)\SL2(A)

ω(g)ϕ(0) |a(g)|s f(g) dg

where the second term vanishes since f is cuspidal. Finally,

PT (θϕ∨f ; s) =∫P (k)\SL2(A)

∑m(a)∈M(k)

ω(m(a)g)ϕ(j) |a(g)|s f(g) dg

=∫N(k)\SL2(A)

ω(g)ϕ(j) |a(g)|s f(g) dg

=∫N(A)\SL2(A)

ω(g)ϕ(j) |a(g)|s∫

[N ]

f(ng)ψt(n) dn dg .

The ψt-Whittaker function of f can written as a product of local Whittaker functions

∫[N ]

f(ng)ψt(n) dn =∏v

W fv (g)

therefore the value PT (θϕ∨f) is the value at s = 0 of the analytic continuation of

PT (θϕ∨f ; s) =∏v

∫N(kv)\SL2(kv)

W fv (g)ω(g)ϕ(j) |a(g)|s dg .


The set S is given by the set of places v such that for all v 6∈ S the functions ϕv and W fv are unramified

and t is an integral unit. Finally, Waldspurger does the calculation for v 6∈ S in [20, Lemma 61 p.120]

(see also [20, Lemma 50 p.89])

∫N(kv)\SL2(kv)

W fv (g)ω(g)ϕ(j) |a(g)|s dg

∣∣∣∣∣s=0

=Lv(Waldψ(σ), 1/2)

Lv(χt, 1).

Let us return to the second spectral identity in Theorem 3.6.1. If there exists a sufficient theory of

newforms for SL2(A), then it is possible in principle to choose fG ∈ Vσ and ϕ ∈ S(VA) such that the

left hand side of Equation 3.13 will reduce to a single term. Furthermore, if there exists a theory of

newforms for OV (A), then it is possible in principle that the right hand side of Equation 3.13 will reduce

to a single term. Finally, we suppose that this newform fG ∈ Vσ yields the exact equality

PT (θϕ∨fG) =L(Waldψ(σ), 1/2)

L(χt, 1)

With these assumptions and the result of Waldspurger, we would like to make the following conjecture.

Conjecture 3.7.1. Let σ and π be cuspidal automorphic representations of SL2(A) and OV (A) respec-

tively such that θψ(σ) = π. There exist automorphic forms fG ∈ Vσ and fH ∈ Vπ and data j, t, T, ϕ, ξ

such that Equation 3.13 reduces to

L(Waldψ(σ), 1/2)L(χt, 1)

Wt(fG)〈fG, fG〉G

?=1√t

∣∣PT (fH)∣∣2

〈fH , fH〉H. (3.15)

The factor 1/√t appearing on the right is due to the operator Rξ∨ acting on fH based on the

calculations in Proposition 3.4.2.

Chapter 4

A Comparison of Relative Trace

Formulas

The goal of this chapter is to show, in some very special cases, that the spectral identity Equation 3.12

appearing in Theorem 3.6.1 is the result of a comparison of nonstandard relative trace formulas. One

trace formula is standard and given by the trace of the operator Rξ1Rξ∨2 for H relative to T1 × T2. The

other trace formula is novel and is given by the trace of the operator θϕ1θϕ∨2 for G relative to N × N

and the characters ψt1 and ψt2 (see Theorem 3.6.1 for notation). The comparison requires the following

strong assumptions:

• n = 1, therefore G(A) is either Sp1(A) = SL2(A), its metaplectic cover Mp1(A) or U1,1(A).

• V is anisotropic.

The first assumption implies that the Siegel parabolic subgroup is the only proper parabolic subgroup

of G(A) and this simplifies the computation of the noncuspidal part of θϕ1θϕ∨2 in Proposition 4.4.1. It

is likely that the techniques in the work of Rallis [17] can be used to compute the noncuspidal part of

θϕ1θϕ∨2 in the general situation by calculating the constant terms along all parabolic subgroups.

In general, given a dual reductive pair (G(A), H(A)), the theta lift of a cuspidal automorphic form on

G(A) is not always (in fact, almost never) a cuspidal automorphic form on H(A) as shown in the results

of Rallis [17], Moeglin [15] and Jiang-Soudry [8]. The second assumption ensures that the theta lift of a

cuspidal representation of G(A) is a cuspidal representation of H(A) therefore the operator θϕ1θϕ∨2 on

cusp forms is well-defined.

48

Chapter 4. A Comparison of Relative Trace Formulas 49

The second assumption also implies that G(k)\G(A) is compact therefore the inner product of theta

functions θϕ1θϕ∨2 is absolutely convergent and defines an integral operator with kernel Kϕ (where ϕ =

ϕ1 · ϕ∨2 ) on the space of cusp forms L2cusp(G(k)\G(A)). The action of Hecke operators Rξ1Rξ∨2 unfolds

in the standard way to show that it is an integral operator with kernel Kξ (where ξ = ξ1 ∗ ξ∨2 ). The

comparison then follows the standard procedure:

1. Compute the geometric expansions of the traces of Kϕ and Kξ (Proposition 4.1.1 and Proposi-

tion 4.2.1).

2. Find a natural bijection between the terms, the orbital integrals, appearing in the geometric

expansions.

3. Find a relation between the data ϕ1, ϕ2 and ξ1, ξ2 such that the orbital integrals matching by the

bijection above are equal and conclude that the traces are equal when the input data is matching

(Proposition 4.3.1 and Proposition 4.3.2).

4. Compute the spectral expansions of the traces and show that the noncuspidal parts of each are

equal (Proposition 4.4.1).

5. Use a linear independence argument to conclude that the trace of Kϕ along a cuspidal repre-

sentation σ is equal to the trace of Kξ along π = θψ(σ) to obtain the desired spectral identity

(Theorem 4.5.1).

We retain the notation from the previous section for the dual pair (G(A), H(A)). In particular, there

is the Siegel parabolic subgroup P (A) = M(A)N(A) ⊂ G(A) and, in the special case n = 1, we have

k = N via b 7→ n(b). Finally, for any algebraic group G defined over k, we commonly denote the adelic

quotient G (k)\G (A) as [G ].

4.1 Relative Trace of the Kernel for G(A)

Given ϕ1, ϕ2 ∈ S(VA), the map which takes a cusp form fG on G(A) to its theta lift θϕ∨2 fG on H(A) and

then to the lift θϕ1θϕ∨2 fG on G(A) again is an operator on the space of cusp forms. In fact, switching

the order of integration shows that it is an integral operator

∫H(k)\H(A)

θϕ1(g1, h)

(∫G(k)\G(A)

θϕ2(g2, h) fG(g2) dg2

)dh

=∫G(k)\G(A)

(∫H(k)\H(A)

θϕ1(g1, h) θϕ2(g2, h) dh

)fG(g2) dg2


and we let Kϕ1·ϕ∨2 denote the kernel function

Kϕ1·ϕ∨2 (g1, g2) =∫H(k)\H(A)

θϕ1(g1, h) θϕ2(g2, h) dh . (4.1)

For each pair ϕ1, ϕ2 ∈ S(VA), define the trace of the kernel Kϕ1·ϕ∨2 relative the subgroup N ×N and

the characters ψt1 and ψt2 by

JG(ϕ1 · ϕ∨2 ; t1, t2) =∫

[N×N ]

Kϕ1·ϕ∨2 (n1, n2)ψt1(n1)ψt2(n2) dn1dn2 . (4.2)

Proposition 4.1.1. Let j1, j2 ∈ V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let T1 and T2 be the

stabilizers in H of j1 and j2 respectively. Then

JG(ϕ1 · ϕ∨2 ; t1, t2) =∑

γ∈T1(k)\H(k)/T2(k)

vol[H(j1,γj2)

] ∫H(j1,γj2)(A)\H(A)

ϕ1(h−1j1)ϕ2(h−1γj2) dh (4.3)

where H(j1,γj2) = T1 ∩ γT2γ−1 is the stabilizer in H of the pair (j1, γj2).

Proof. Interchanging the order of integration, we compute the integral Equation 4.2

JG(ϕ1 · ϕ∨2 ; t1, t2)

=∫

[N×N ]

∫H(k)\H(A)

∑(v1,v2)∈V 2

ω(n1)ϕ1(h−1v1)ω(n2)ϕ2(h−1v2) dh

ψt1(n1)ψt2(n2) dn1dn2

=∫H(k)\H(A)

∑(v1,v2)∈V 2

(∫[N×N ]

ψ(n1(Q(v1)− t1) + n2(t2 −Q(v2))) dn1dn2

)ϕ1(h−1v1)ϕ2(h−1v2) dh

=∫H(k)\H(A)

∑(v1, v2) ∈ V 2

Q(v1) = t1

Q(v2) = t2

ϕ1(h−1v1)ϕ2(h−1v2) dh .

The group H(k) acts transitively on the sets v ∈ V : Q(v) = t1 and v ∈ V : Q(v) = t2 therefore

JG(ϕ1 · ϕ∨2 ; t1, t2) =∫H(k)\H(A)

∑γ∈T1(k)\H(k)/T2(k)

∑g∈H(j1,γj2)(k)\H(k)

ϕ1((gh)−1j1)ϕ2((gh)−1γj2) dh .

Therefore we obtain Equation 4.3

JG(ϕ1 · ϕ∨2 ; t1, t2) =∑

γ∈T1(k)\H(k)/T2(k)

vol[H(j1,γj2)

] ∫H(j1,γj2)(A)\H(A)

ϕ1(h−1j1)ϕ2(h−1γj2) dh .


4.2 Relative Trace of the Kernel for H(A)

The right regular representation of H(A) on L2(H(k)\H(A)) defines an action of the space S(H(A)) of

smooth, rapidly decreasing functions on H(A) by

RξfH(x) =

∫H(A)

ξ(y) fH(xy) dy (4.4)

which we unwind to find that it is an integral operator

RξfH(x) =

∫H(A)

ξ(y) fH(xy) dy

=∫H(k)\H(A)

∑γ∈H(k)

ξ(x−1γy)

fH(y) dy

with kernel function

Kξ(x, y) =∑

γ∈H(k)

ξ(x−1γy) . (4.5)

Let j1, j2 ∈ V and let T1 and T2 be the stabilizers inH of j1 and j2 respectively. For each ξ ∈ S(H(A)),

define the trace of the kernel Kξ relative to the subgroup T1 × T2 by

JH(ξ ; T1, T2) =∫

[T1×T2]

Kξ(τ1, τ2) dτ1dτ2 . (4.6)


stabilizers in H of j1 and j2 respectively. Then

JH(ξ ; T1, T2) =∑

γ∈T1(k)\H(k)/T2(k)

vol [(T1 × T2)γ ]∫

(T1×T2)γ(A)\(T1×T2)(A)

ξ(τ−11 γτ2) dτ1dτ2 (4.7)

where (T1 × T2)γ = (τ1, τ2) ∈ T1 × T2 : τ1γτ−12 = γ.

Proof. We unfold the integral Equation 4.6 in the usual manner

JH(ξ ; T1, T2) =∫

[T1×T2]

Kξ(τ1, τ2) dτ1 dτ2

=∫

[T1×T2]

∑γ∈T1(k)\H(k)/T2(k)

∑(s1,s2)∈(T1×T2)γ(k)\(T1×T2)(k)

ξ(τ−11 s−1

1 γs2τ2) dτ1 dτ2 .


Therefore we obtain Equation 4.7

JH(ξ ; T1, T2) =∑

γ∈T1(k)\H(k)/T2(k)

vol [(T1 × T2)γ ]∫

(T1×T2)γ(A)\(T1×T2)(A)

ξ(τ−11 γτ2) dτ1 dτ2 .

4.3 Matching Geometric Expansions

The geometric expansions Equation 4.3 and Equation 4.7 are both sums over the double coset space

T1(k)\H(k)/T2(k) therefore there is a clear bijection between the terms. The following proposition

shows that the corresponding orbital integrals are equal when the functions ϕ1, ϕ2 match ξ1, ξ2 as in

Proposition 3.3.1. As a consequence, which we state in a subsequent proposition, the relative traces

JG(ϕ1 · ϕ∨2 ; t1, t2) and JH(ξ1 ∗ ξ∨2 ; T1, T2) are equal when all the data is matching.


stabilizers in H of j1 and j2 respectively. Let ϕ1, ϕ2 ∈ S(VA) and let ξ1 and ξ2 be smooth functions on

H(A) such that ξ1 matches ϕ1 relative to j1 and ξ2 matches ϕ2 relative to j2 as in Proposition 3.3.1.

Then ξ1 ∗ ξ∨2 is a smooth function on H(A), compactly supported at finite places and rapidly decreasing

at infinite places, such that

∫H(j1,γj2)(A)\H(A)

ϕ1(h−1j1)ϕ2(h−1γj2) dh =∫

(T1×T2)γ(A)\(T1×T2)(A)

ξ1 ∗ ξ∨2 (τ−11 γτ2) dτ1 dτ2 (4.8)

for all γ ∈ T1(k)\H(k)/T2(k).

Proof. We know that ξ1 and ξ2 are smooth functions on H(A), compactly supported at finite places and

rapidly decreasing at infinite places, such that

ϕ1(h−1j1) =∫T1(A)

ξ1(τ1h) dτ1 and ϕ2(h−1j2) =∫T2(A)

ξ2(τ2h) dτ2

and we have

(ξ1 ∗ ξ∨2 )(x) =∫H(A)

ξ1(h)ξ∨2 (h−1x) dh =∫H(A)

ξ1(h)ξ2(x−1h) dh .

Note that ξ1 ∗ ξ∨2 is a smooth function on H(A), compactly supported at finite places and rapidly

decreasing at infinite places, since it is a convolution of such functions. We compute

∫(T1×T2)γ(A)\(T1×T2)(A)

ξ1 ∗ ξ∨2 (τ−11 γτ2) dτ1 dτ2


=∫

(T1×T2)γ(A)\(T1×T2)(A)

∫H(A)

ξ1(h)ξ2(τ−12 γ−1τ1h) dh dτ1dτ2

=∫

(T1×T2)γ(A)\(T1×T2)(A)

∫H(A)

ξ1(τ−11 h)ξ2(τ−1

2 γ−1h) dh dτ1dτ2

=∫

(T1×T2)γ(A)\(T1×T2)(A)

∫H(j1,γj2)(A)\H(A)

∫H(j1,γj2)(A)

ξ1(τ−11 τ0h)ξ2(τ−1

2 γ−1τ0h) dτ0 dh dτ1dτ2 .

Note that for all γ ∈ T1(k)\H(k)/T2(k) there is the isomorphism

H(j1,γj2)∼−→ (T1 × T2)γ : τ0 7→ (τ0, γ−1τ0γ) .

We continue

∫(T1×T2)γ(A)\(T1×T2)(A)

ξ1 ∗ ξ∨2 (τ−11 γτ2) dτ1 dτ2

=∫

(T1×T2)γ(A)\(T1×T2)(A)

∫H(j1,γj2)(A)\H(A)

∫(T1×T2)γ(A)

ξ1((τ−10 τ1)−1h)

× ξ2((γ−1τ−10 γτ2)−1γ−1h) dτ0 dh dτ1dτ2 .

=∫

(T1×T2)(A)

∫H(j1,γj2)(A)\H(A)

ξ1(h)ξ2(τ−12 γ−1τ1h) dh dτ1dτ2

=∫H(j1,γj2)(A)\H(A)

∫T1(A)

ξ1(τ−11 h) dτ1

∫T2(A)

ξ2(τ−12 γ−1h) dτ2 dh

=∫H(j1,γj2)(A)\H(A)

ϕ1(h−1j1)ϕ2(h−1γj2) dh .




Then

JG(ϕ1 · ϕ∨2 ; t1, t2) = JH(ξ1 ∗ ξ∨2 ; T1, T2) . (4.9)

Proof. For all γ ∈ T1(k)\H(k)/T2(k) we have (T1 × T2)γ ∼= H(j1,γj2) therefore vol [(T1 × T2)γ ] =

vol[H(j1,γj2)

]. Comparing Proposition 4.1.1 and Proposition 4.2.1, we see that both JG(ϕ1 · ϕ∨2 ; t1, t2)

and JH(ξ1 ∗ ξ∨2 ;T1, T2) can be written as sums over the set T1(k)\H(k)/T2(k). The terms in each sum

are exactly the terms on each side of the equation Equation 4.8.


4.4 Spectral Decompositions

The kernel function Kϕ1·ϕ∨2 is an integral operator on the space of cuspidal automorphic forms on

G(A) however it is not an operator on the entire space of square-integrable automorphic functions

L2(G(k)\G(A)). Nevertheless, we may consider its cuspidal part

Kcuspϕ1·ϕ∨2

(g1, g2) =∑σ∈AG

0

∑FG∈B(σ)

θϕ1θϕ∨2 FG(g1)FG(g2) (4.10)

where A G0 is the set of irreducible cuspidal automorphic representations of G(A) and B(σ) is an or-

thonormal basis of the space Vσ on which σ acts. The orthogonal complement of the space of cusps

forms is called the Eisenstein spectrum and we define

Keisϕ1·ϕ∨2

(g1, g2) = Kϕ1·ϕ∨2 (g1, g2)−Kcuspϕ1·ϕ∨2

(g1, g2) (4.11)

therefore the relative trace of Kϕ1·ϕ∨2 is the corresponding sum of integrals

JG(ϕ1 · ϕ∨2 ; t1, t2) = JGcusp(ϕ1 · ϕ∨2 ; t1, t2) + JGeis(ϕ1 · ϕ∨2 ; t1, t2) . (4.12)

In particular, the cuspidal part of the spectral expansion of the relative trace of Kϕ1·ϕ∨2 is

JGcusp(ϕ1 · ϕ∨2 ; t1, t2) =∑σ∈AG

0

JGσ (ϕ1 · ϕ∨2 ; t1, t2) (4.13)

where the spectral expansion along σ is

JGσ (ϕ1 · ϕ∨2 ; t1, t2) =∑

FG∈B(σ)

Wt1(θϕ1θϕ∨2 FG)Wt2(FG) . (4.14)

Similarly, we may write the kernel function Kξ1∗ξ∨2 as

Kξ1∗ξ∨2 (h1, h2) =∑π∈AH

0

∑FH∈B(π)

Rξ1∗ξ∨2 FH(h1)FH(h2) +

∫H(A)

ξ1 ∗ ξ∨2 (x) dx (4.15)

where A H0 is the set of nontrivial irreducible automorphic representations π of H(A), B(π) is an or-

thonormal basis of Vπ and the integral is the action of Rξ1∗ξ∨2 on the trivial representation. The relative


trace of Kξ1∗ξ∨2 is the sum

JH(ξ1 ∗ ξ∨2 ; T1, T2) =∑π∈AH

0

JHπ (ξ1 ∗ ξ∨2 ; T1, T2) + JHtriv(ξ1 ∗ ξ∨2 ; T1, T2) (4.16)

where

JHtriv(ξ1 ∗ ξ∨2 ; T1, T2) = vol [T1 × T2]∫H(A)

ξ1 ∗ ξ∨2 (x) dx (4.17)

is the relative trace of Kξ1∗ξ∨2 along the trivial representation and

JHπ (ξ1 ∗ ξ∨2 ; T1, T2) =∑

FH∈B(π)

PT1(Rξ1Rξ∨2 FH)PT2(FH) . (4.18)

Proposition 4.4.1. For any ϕ1, ϕ2 ∈ S(VA) we have

Keisϕ1·ϕ∨2

(g1, g2) =∫H(k)\H(A)

∑v1∈V

ω(g1)ϕ1(h−11 v1) dh1

∫H(k)\H(A)

∑v2∈V

ω(g2)ϕ2(h−12 v2) dh2

and for ξ1 matching ϕ1 relative to j1 and ξ2 matching ϕ2 relative to j2 as in Proposition 3.3.1 we have

∫H(A)

ξ1 ∗ ξ∨2 (x) dx =∫T1(A)\H(A)

ϕ1(h−11 j1) dh1 ·

∫T2(A)\H(A)

ϕ2(h−12 j2) dh2 .

In particular, JGeis(ϕ1 · ϕ∨2 ; t1, t2) = JHtriv(ξ1 ∗ ξ∨2 ; T1, T2).

Proof. The Eisenstein spectrum of G(A) has a dense subspace consisting of Poincare series

F(g) =∑

γ∈P (k)\G(k)

F (γg)

where F ∈ C∞c (P (k)N(A)\G(A)). Indeed, for f ∈ L2(G(k)\G(A)) we compute

∫G(k)\G(A)

f(g)∑

γ∈P (k)\G(k)

F (γg) dg =∫P (k)\G(A)

f(g)F (g) dg

=∫P (k)N(A)\G(A)

(∫N(k)\N(A)

f(ng)

)F (g) dg

therefore f is cuspidal if and only if f is orthogonal to all Poincare series. We compute for any Poincare


series F

(θϕ1θϕ∨2 F)(g1)

=∫G(k)\G(A)

∫H(k)\H(A)

∑(v1,v2)∈V 2

ω(g1)ϕ1(h−1v1)ω(g2)ϕ2(h−1v2) dhF(g2) dg2

=∫H(k)\H(A)

∑v1∈V

ω(g1)ϕ1(h−1v1)∫G(k)\G(A)

∑v2∈V

ω(g2)ϕ2(h−1v2)∑

γ∈P (k)\G(k)

F (γg2) dg2 dh

=∫H(k)\H(A)

∑v1∈V

ω(g1)ϕ1(h−1v1)∫P (k)N(A)\G(A)

∫[N ]

∑v2∈V

ω(ng2)ϕ2(h−1v2) dnF (g2) dg2 dh

=∫H(k)\H(A)

∑v1∈V

ω(g1)ϕ1(h−1v1) dh∫P (k)N(A)\G(A)

ω(g2)ϕ2(0)F (g2) dg2 .

Here we have used the fact that V is anisotropic therefore if Q(v) = 0 then v = 0. We also compute

∫G(k)\G(A)

∫H(k)\H(A)

∑v∈V

ω(g2)ϕ2(h−1v) dhF(g2) dg2

=∫G(k)\G(A)

∫H(k)\H(A)

∑v∈V

ω(g2)ϕ2(h−1v) dh∑

γ∈P (k)\G(k)

F (γg2) dg2

=∫P (k)N(A)\G(A)

∫H(k)\H(A)

∫[N ]

∑v∈V

ω(ng2)ϕ2(h−1v) dn dhF (g2) dg2

=∫P (k)N(A)\G(A)

ω(g2)ϕ2(0)F (g2) dg2 .

We conclude that there is an equality of kernel functions on the Eisenstein spectrum of G(A)

Keisϕ1·ϕ∨2

(g1, g2) =∫H(k)\H(A)

∑v1∈V

ω(g1)ϕ1(h−11 v1) dh1

∫H(k)\H(A)

∑v2∈V

ω(g2)ϕ2(h−12 v2) dh2

and therefore

Jeis(ϕ1 · ϕ∨2 ; t1, t2) =∫H(k)\H(A)

∫[N ]

∑v1∈V

ω(n1)ϕ1(h−11 v1)ψt1(n1) dn1 dh1

×∫H(k)\H(A)

∫[N ]

∑v2∈V

ω(n2)ϕ2(h−12 v2)ψt2(n2) dn2 dh2

=∫H(k)\H(A)

∑v1 ∈ V

Q(v1) = t1

ϕ1(h−11 v1) dh1 ·

∫H(k)\H(A)

∑v2 ∈ V

Q(v2) = t2

ϕ2(h−12 v2) dh2

= vol [T1]∫T1(A)\H(A)

ϕ1(h−11 j1) dh1 · vol [T2]

∫T2(A)\H(A)

ϕ2(h−12 j2) dh2 .


Now we compute JHtriv(ξ1 ∗ ξ∨2 ; T1, T2)

JHtriv(ξ1 ∗ ξ∨2 ; T1, T2) = vol [T1] vol [T2]∫H(A)

ξ1 ∗ ξ∨2 (h) dh

= vol [T1] vol [T2]∫H(A)

ξ1(x) dx∫H(A)

ξ2(h) dh

= vol [T1]∫T1(A)\H(A)

ϕ1(x−1j1) dx · vol [T2]∫T2(A)\H(A)

ϕ2(h−1j2) dh

and we conclude Jeis(ϕ1 · ϕ∨2 ; t1, t2) = JHtriv(ξ1 ∗ ξ∨2 ; T1, T2).

Corollary 4.4.1. Let j1, j2 ∈ V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let T1 and T2 be the

stabilizers in H of j1 and j2 respectively. For all ϕ1, ϕ2 ∈ S(VA) and ξ1 matching ϕ1 relative to j1 and

ξ2 matching ϕ2 relative to j2 as in Proposition 3.3.1, we have

∑σ∈AG

0

JGσ (ϕ1 · ϕ∨2 ; t1, t2) =∑π∈AH

0

JHπ (ξ1 ∗ ξ∨2 ; T1, T2) . (4.19)

Proof. The equality Equation 4.19 follows from Proposition 4.3.2, Equation 4.12, Equation 4.13, Equa-

tion 4.16 and Proposition 4.4.1.

4.5 Spectral Identities

The last step in the comparison of trace formulas is to obtain an equality on the level of representations

from Equation 4.19. For representations σ and π corresponding by the theta correspondence, the key is

to write both JGσ (ϕ1 ·ϕ∨2 ; t1, t2) and JHπ (ξ1 ∗ ξ∨2 ; T1, T2) as linear functionals in the variable ϕ1 ∈ S(VA)

and see that each is defined by an element in the representation π. Then Equation 4.19 is an equality of

orthogonal sums from which we deduce the spectral identity Equation 3.12 (and the stronger statement

of Corollary 3.5.1).

Theorem 4.5.1. Let j1, j2 ∈ V be nonzero with Q(j1) = t1 and Q(j2) = t2, and let T1 and T2 be the



Given cuspidal automorphic representations σ and π of G and H respectively such that θψ(σ) = π, we

have ∑FG∈B(σ)


∑FH∈B(π)




Proof. First, note that

Wt(θϕfH) =∫N(k)\N(A)

(∫H(k)\H(A)

∑v∈V

ω(n)ϕ(h−1v) fH(h) dh

)ψt(n) dn

=∫H(k)\H(A)

∑v∈V

(∫N(k)\N(A)

ψ(n(Q(v)− t)) dn

)ϕ(h−1v) fH(h) dh

=∫H(k)\H(A)

∑v ∈ V

Q(v) = t

ϕ(h−1v) fH(h) dh

=

⟨ ∑v ∈ V

Q(v) = t

ϕ(h−1v) , fH(h)

⟩

and so we compute

JGσ (ϕ1 · ϕ∨2 ; t1, t2) =∑

FG∈B(σ)

Wt1(θϕ1θϕ∨2 FG)Wt2(FG)

=∑

FG∈B(σ)

⟨ ∑v ∈ V

Q(v) = t1

ϕ1(h−1v) , θϕ∨2 FG(h)

⟩Wt2(FG)

=

⟨ ∑v ∈ V

Q(v) = t1

ϕ1(h−1v) ,∑

FG∈B(σ)

θϕ∨2 FG(h)Wt2(FG)

⟩.

Therefore, with all other data fixed, the map ϕ1 7→ JGσ (ϕ1 · ϕ∨2 ; t1, t2) is a linear functional given by

sending ϕ1 to the subspace

Θ0t1 =

∑v ∈ V

Q(v) = t1

ϕ(h−1v) : ϕ ∈ S(VA)

⊆ L2(H(k)\H(A))

and then taking the inner product with the function

∑FG∈B(σ)

θϕ∨2 FG(h)Wt2(FG) . (4.21)


Recall Proposition 3.5.1 which states

PT1(Rξ1Rξ∨2 FH) = Wt1(θϕ1Rξ∨2 F

H)

when ξ1 matches ϕ1 relative to j1 therefore we compute

JHπ (ξ1 ∗ ξ∨2 ; T1, T2) =∑

FH∈B(π)

PT1(Rξ1Rξ∨2 FH)PT2(FH)

=∑

FH∈B(π)

Wt1(θϕ1Rξ∨2 FH)PT2(FH)

=∑

FH∈B(π)

⟨ ∑v ∈ V

Q(v) = t1

ϕ1(h−1v) , Rξ∨2 FH(h)

⟩PT2(FH)

=

⟨ ∑v ∈ V

Q(v) = t1

ϕ1(h−1v) ,∑

FH∈B(π)

Rξ∨2 FH(h)PT2(FH)

⟩.

Therefore, with all other data fixed, the map ϕ1 7→ JHπ (ξ1 ∗ ξ∨2 ; T1, T2) is a linear functional given by

sending ϕ1 to the subspace Θ0t1 and then taking the inner product with the function

∑FH∈B(π)

Rξ∨2 FH(h)PT2(FH) . (4.22)

The closure Θt1 of the subspace Θ0t1 is a H(A)-invariant subspace of L2(H(k)\H(A)) and by the calcula-

tions above we see that it contains all the representations π such that θψ(π) is nonzero and has nonzero

ψt1 -Fourier coefficients. We write Equation 4.19 as

∑σ ∈ AG

0

0 6= θψ(σ) ⊂ Θ∨t1

JGσ (ϕ1 · ϕ∨2 ; t1, t2) =∑

π ∈ AH0

π ⊂ Θ∨t1

JHπ (ξ1 ∗ ξ∨2 ; T1, T2)

where we view each side as an orthogonal sum of elements in the dual Θ∨t1 of Θt1 defined by the functions

in Equation 4.21 and Equation 4.22. Therefore

JGσ (ϕ1 · ϕ∨2 ; t1, t2) = JHπ (ξ1 ∗ ξ∨2 ;T1, T2)

for all σ and π corresponding by the theta correspondence.


In fact, we have proved the stronger statement of Corollary 3.5.1: Let j ∈ V be nonzero with Q(j) = t

and let T be the stabilizer of j in H. Let ϕ ∈ S(VA) and let ξ be a smooth function on H(A) which

matches ϕ relative to j as in Proposition 3.3.1. Given cuspidal automorphic representations σ and π of

G and H respectively such that θψ(σ) = π, we have

∑FG∈B(σ)

θϕ∨FG(h)Wt(FG) =

∑FH∈B(π)

Rξ∨FH(h)PT (FH) .

Chapter 5

Height Pairings of Special Cycles

The goal of this chapter is to explicitly calculate height pairings of special cycles on the p-adic upper

half plane using the results of Kudla and Rapoport [12]. These local height pairings are the arithmetic

analogues of the local orbital integrals appearing in Proposition 4.1.1. In particular, in most cases the

group H(j1,γj2) is equal to 〈±1〉 (see Proposition 4.1.1 for notation) and the integrals in the geometric

expansion of JG(ϕ1 · ϕ∨2 ; t1, t2) are determined by the local integrals

∫H(Qp)

ϕ1,p(h−1j1)ϕ2,p(h−1γj2) dh .

In the arithmetic setting, there is a quadratic space V over Qp of special endomorphisms of a fixed

p-divisible group X over Spec Fp and, given j1, j2 ∈ V , there are special cycles Z(j1) and Z(j2) which

are formal schemes over Spf W and live on the p-adic upper plane ΩW over Spf W (where W is the Witt

ring of Fp). The results of Kudla and Rapoport [12, Theorem 6.1, p.198] compute explicit formulas

for the local height pairing (Z(j1), Z(j2)). The space V can be identified with the space of traceless 2

by 2 matrices over Qp and therefore possesses an action of PGL2(Qp). The results below compute the

local height pairing (Z(j1), Z(γj2)), the arithmetic local orbital integrals, as a function of γ ∈ PGL2(Qp)

where γj2 = γj2γ−1 and we introduce the notation

O(γ ; j1, j2) def= (Z(j1), Z(γj2)) , γj2 = γj2γ−1 . (5.1)

These calculations are the first steps towards a comparison of arithmetic relative trace formulas

analogous to the comparison in Chapter 4. In that comparison there was the operator on cusp forms

given by the inner product of theta functions θϕ1θϕ∨2 (see Equation 4.1). In the arithmetic setting we

61

Chapter 5. Height Pairings of Special Cycles 62

can consider the global height pairing⟨φ1(τ), φ1(τ ′)

⟩(which is a function on two copies of the complex

upper half plane) of the arithmetic theta functions φ1(τ) defined by Kudla, Rapoport and Yang [13, p.8].

These are generating series with coefficients in the first arithmetic Chow group CH1(M) attached to an

integral model M of a Shimura curve. In the setting of an arithmetic relative trace formula, we would

like to consider the global height pairing⟨φ1(τ), φ1(τ ′)

⟩as a kernel function on a certain space of cusp

forms of half-integral weight and compute its trace relative to the standard unipotent subgroup N ×N

with characters ψt1 and ψt2 . This amounts to computing the t1, t2-Fourier coefficient of⟨φ1(τ), φ1(τ ′)

⟩which can be written as a sum involving local height pairings in the fibers of M over Spec Z [13, p.206-

212] and, in the fibers of bad reduction, the local height pairings involve the sum of pairings of special

cycles (Z(j1), Z(γj2)) where j1 and j2 are fixed and have length t1 and t2 respectively and γ varies in

PGL2(Qp). It is with this application in mind that we proceed to calculate the arithmetic local orbital

integrals O(γ ; j1, j2) = (Z(j1), Z(γj2)) in various cases.

There are two methods for calculating the height pairing (Z(j1), Z(γj2)). The first method is alge-

braic: Kudla and Rapoport show [12, Theorem 5.1, p.188] that the pairing (Z(j1), Z(j2)) depends only

on the Zp-span of the pair j1, j2 in V . Therefore, given j1,γj2 and assuming that j1, j2 are orthogonal

in V , one could find a new basis, consisting of orthogonal vectors, for the Zp-span of the pair j1, γj2

and then use the formulas in [12, Theorem 6.1, p.198]. Finding an orthogonal basis of the Zp-span of

the pair j1, γj2 is equivalent to diagonalizing the matrix of inner products of j1, γj2 therefore this for-

mulation would compute the local height pairing (Z(j1), Z(γj2)) as a function of the value of the inner

product (j1, γj2) (where the inner product on V is defined by the quadratic form q = −det). The second

method is geometric: one could use the explicit description of the (divisors attached to the) cycles Z(j1)

and Z(γj2) and use the combinatorial method in the proof of [12, Theorem 6.1, p.198] to compute the

height pairing in terms of the position of Z(j1) relative to Z(γj2) in ΩW . We have chosen the geometric

approach.

Finally, there are three kinds of special cycles Z(j) depending on the arithmetic of q(j) = pαε where

α ≥ 0 and ε ∈ Z×p :

(i) if α is even and ε is a nonsquare, then Z(j) is called unramified,

(ii) if α is even and ε is a square, then Z(j) is called split,

(iii) if α is odd, then Z(j) is called ramified.

In the first section, we recall the description of (the divisors Z(j)pure attached to) the special cycles Z(j)

in terms of the Bruhat-Tits tree of PGL2(Qp). We also recall the preliminary results of [12, Section


6, p.198] which reduce the height pairing computations to combinatorial arguments on the tree. Since

there are 3 possibilities for both Z(j1) and Z(j2), there are 9 cases to consider. In the remaining sec-

tions, we find explicit formulas for the height pairings (Z(j1), Z(γj2)) in 4 cases: unramified/unramified,

unramified/split, split/unramified and ramified/ramified. The remaining cases can be found by a similar

analysis.

5.1 Special Cycles on the p-adic Upper Half Plane

The p-adic upper half plane Ωp is a formal scheme over Spf Zp [2, 3] and the irreducible components of

its special fiber Ωp ×Spf Zp Spec Fp are projective lines P[Λ] over Spec Fp indexed by the vertices [Λ] on

the Bruhat-Tits tree B of PGL2(Qp). Here [Λ] is a homothety class of lattices in the standard vector

space over Qp of dimension 2. Vertices [Λ] and [Λ′] are connected by an edge in the tree if there are

representatives Λ and Λ′ for the classes such that pΛ ( Λ′ ( Λ. Therefore each vertex [Λ] is connected

to p + 1 other vertices and each irreducible component P[Λ] meets p + 1 other irreducible components

transversely at the Fp-rational points of P[Λ].

Let W = W (k) be the Witt ring of k = Fp, let B be the quaternion division algebra over Qp

and let OB denote its maximal order of integral elements. It is a theorem of Drinfeld that the base

change ΩW = Ωp ×Spf Zp Spf W to the Witt ring W is the moduli space of special formal OB-modules

[2, 3]. Recall, a special formal OB-module X over a W -scheme S is a p-divisible formal group over S

of dimension 2 and height 4 with an action ι : OB → EndS(X) satisfying the “special” condition [2, 3].

Fix a special formal OB-module X over Spec k and let (Nilp) be the category of schemes over SpecW

on which p is locally nilpotent. Let M be the functor which attaches to a scheme S in (Nilp) the set of

isomorphism classes of pairs (X, ρ) where X is a special formal OB-module over S and

ρ : X×Spec k S −→ X ×S S

is a quasi-isogeny of height 0. Here, we write S = S×SpecW Spec k. Drinfeld’s theorem states that there

is an isomorphism ΩW ∼= M of functors on (Nilp). In other words, the formal scheme ΩW is a p-adic

uniformization of M .

The moduli problem M is studied via the theory of Dieudonne modules therefore, as an exercise, let

us construct the Dieudonne module of a special formal OB-module over Spec k. The main theorem of

Dieudonne theory states that the category of p-divisible groups over Spec k is equivalent to the category

of triples (M,F ,V ) where M is a free W -module of finite rank, F : M → M is a σ-linear map and


V : M → M is a σ−1-linear map such that FV = V F = p (where σ is the Frobenius automorphism

of W ). We can identify OB with the noncommutative ring Zp2 [Π] where Zp2 is the ring of integers

of the unramified quadratic extension of Qp and Π satisfies the relations Π2 = p and Πa = σ(a)Π for

all a ∈ Zp2 . Therefore we let M = W⊕4 and we can define maps F : M → M , V : M → M and

ι : OB → End(M,F ,V ) by choosing the data

F =

0 0 p 0

0 0 0 1

1 0 0 0

0 p 0 0

σ V =

0 0 p 0

0 0 0 1

1 0 0 0

0 p 0 0

σ−1

ι(Π) =

0 0 p 0

0 0 0 1

1 0 0 0

0 p 0 0

ι(a) =

a 0 0 0

0 a 0 0

0 0 σ(a) 0

0 0 0 σ(a)

for all a ∈ Zp2 .

Then (M,F ,V ) together with ι is the Dieudonne module of a special formal OB-module over Spec k

and it is easy to see that its OB-linear isogenies are

EndOB (M,F ,V )⊗Zp Qp = M2(Qp)

where j ∈M2(Qp) defines

j 0

0 j′

∈ EndOB (M,F ,V )⊗Zp Qp , where j′ =

p 0

0 1

−1

j

p 0

0 1

.

Returning to the p-adic upper half plane, we can choose X to be the special formal OB-module over

Spec k whose Dieudonne module is the tuple (M,F ,V , ι) defined above. We define the space of special

endomorphisms of X as the quadratic space

V = j ∈ EndOB (X)⊗Zp Qp : tr j = 0 (5.2)

where, as above, we have identified EndOB (X) ⊗Zp Qp with the space of 2 by 2 matrices over Qp and

we equip V with the quadratic form q(j) = −det(j). Finally, the main objects of this chapter are the


special cycles attached to special endomorphisms and we recall the definition.

Definition 5.1.1. ([12, p.167]) Let j ∈ V be a special endomorphism with q(j) 6= 0. The special cycle

Z(j) associated to j is the closed formal subscheme of M consisting of all points (X, ρ) such that the

quasi-isogeny ρ j ρ−1 of X ×S S lifts to an endomorphism of X.

5.2 Height Pairings of Special Cycles

The main result of [12] is the following theorem which computes explicit formulas for the height pairings

of special cycles.

Theorem 5.2.1. (Kudla-Rapoport, [12, Theorem 6.1, p.198]) Let j and j′ be special endomorphisms

such that their Zp-span j = Zpj + Zpj′ is of rank 2 and nondegenerate for the quadratic form. Let

T =

q(j) 12 (j, j′)

12 (j′, j) q(j′)

,

and suppose that T is GL2(Zp)-equivalent to diag(pαε1, pβε2), where 0 ≤ α ≤ β. Then

(Z(j), Z(j′)) = α+ β + 1−

pα/2 + 2pα/2 − 1p− 1

if α is even and χ(ε1) = −1

(β − α+ 1)pα/2 + 2pα/2 − 1p− 1

if α is even and χ(ε1) = 1

2p(α+1)/2 − 1

p− 1if α is odd

where χ is the quadratic residue character of Z×p .

The goal of this chapter is calculate the pairings (Z(j1), Z(γj2)) in various cases as functions of

γ ∈ PGL2(Qp). Obviously, the formulas we obtain are variants of the ones above. To compute these

pairings, we recall the following results.

Proposition 5.2.1. ([12, Lemma 4.2 p.184 and Lemma 4.3 p.185]) Let Z(j1) and Z(j2) be special

cycles on ΩW . There are closed formal subschemes Z(j1)pure ⊂ Z(j1) and Z(j2)pure ⊂ Z(j2) such that

Z(j1)pure and Z(j2)pure are divisors on ΩW and

(Z(j1), Z(j2)) = (Z(j1)pure, Z(j2)pure) .

This result reduces the computation of the height pairing (Z(j1), Z(j2)) to the computation of


the pairing (Z(j1)pure, Z(j2)pure) and the following result gives a complete description of the divisors

Z(j)pure.

Proposition 5.2.2. ([12, Proposition 4.5 p.186]) Let q(j) = pαε such that α ≥ 0 and ε ∈ Z×p and, for

a vertex [Λ] in the Bruhat-Tits tree B of PGL2(Qp), define

mult[Λ](j) = maxα/2− d([Λ],Bj), 0

where Bj is the set of fixed points on B by the action of j and d([Λ],Bj) denotes the distance between

[Λ] and Bj. The set of fixed points Bj is:

(i) if α is even and χ(ε) = −1, then there is a unique vertex [Λ] such that Bj = [Λ],

(ii) if α is even and χ(ε) = 1, then Bj = [Λi] : i ∈ Z is an infinite line of adjacent vertices,

(iii) if α is odd, then there is a unique edge ∆ such that Bj = midpoint of ∆,

and we have the equality of divisors of ΩW

Z(j)pure = Z(j)h + Z(j)v

where the vertical part of Z(j) is

Z(j)v =∑[Λ]

mult[Λ](j) · P[Λ]

and the horizontal part of Z(j) is:

(i) if α is even and χ(ε) = −1, then Z(j)h is the disjoint sum of two divisors each projecting isomor-

phically to Spf W and meeting the special fibre in an ordinary special point of P[Λ] where [Λ] is the

unique vertex fixed by j.

(ii) if α is even and χ(ε) = 1, then Z(j)h = 0,

(iii) if α is odd, then Z(j)h is the formal spectrum of the ring of integers in a ramified quadratic

extension of WQ which meets the special fibre in pt∆, where ∆ is the edge containing the unique

fixed point of j.

We now have a complete description of the divisors Z(j)pure as sums of irreducible components with

multiplicities. The next result gives us all the tools we need to compute the pairings (Z(j1)pure, Z(γj2)pure)

as γ varies in PGL2(Qp).


Proposition 5.2.3. ([12, Lemma 4.9 p.187 and Lemma 6.2 p.199]) Let j ∈ V with q(j) = pβε, let

[Λ] ∈ B and let r = d([Λ],Bj). Then

(P[Λ], Z(j)h) =

2 if β is even, χ(ε) = −1 and r = 0

1 if β is odd and r = 1/2

0 otherwise

(P[Λ], Z(j)v) =

1− p if 1 ≤ r ≤ β/2− 1

χ(ε)− p if r = 0 and β is even

−p if r = 1/2 and β is odd

1 if r = β/2

0 otherwise

5.3 The Unramified/Unramified Case

The first case we consider is when both cycles are unramified. This result is slightly incomplete since it is

possible for the horizontal components Z(j1)h and Z(γj2)h to meet nontrivially and we do not compute

the contribution of (Z(j1)h, Z(γj2)h) in this case. However, when the distance between Bj1 and Bγj2 is

positive, these horizontal components do not meet.

Proposition 5.3.1. Let j1, j2 ∈ V with q(j1) = pαε1 and q(j2) = pβε2 such that 0 ≤ α ≤ β, α and β

are even and χ(ε1) = χ(ε2) = −1. Let γ ∈ GL2(Qp) and let d = d(Bj1 ,Bγj2). Then O(γ ; j1, j2) = 0 if

d > α/2 + β/2, O(γ ; j1, j2) = 1 if d = α/2 + β/2, and if d < α/2 + β/2 then O(γ ; j1, j2) is given by

α+ β + 1− 2d−

2p(α/2+β/2−d+1)/2 − 1

p− 1if α


2 >β2 − d,

p(α/2+β/2−d)/2 + 2p(α/2+β/2−d)/2 − 1

p− 1if α


2 >β2 − d,

pα/2 + 2pα/2 − 1p− 1

+ (Z(j1)h, Z(γj2)h) if α2 ≤

β2 − d.

Proof. The set of fixed points Bj1 is a single vertex and the vertical component of the cycle Z(j1) is the

sum

Z(j1)v =∑[Λ]

d([Λ],Bj1 ) < α/2

(α/2− d([Λ],Bj1)

)· P[Λ]


which is the ball of radius α/2 centered at Bj1 and similarly

Z(γj2)v =∑[Λ]

d([Λ],Bγj2 ) < β/2

(β/2− d([Λ],B

γj2))· P[Λ] .

Our proof will calculate the intersection pairing beginning with the case where the distance d between

Bj1 and Bγj2 is so large that the cycles Z(j1) and Z(γj2) do not meet and then we proceed as d goes to

zero.

First of all, the extreme case in which the cycles meet only at a superspecial point in a boundary

component corresponds to d = α/2 + β/2− 1 and so (Z(j1), Z(γj2)) = 0 if d ≥ α/2 + β/2.

Consider the case in which Z(γj2) does not contain the component P[Λ] at Bj1 , that is, α/2+β/2−d >

0 and β/2 − 1 < d. Let [Λ1], [Λ2], . . . , [Λ`] denote the ` = α/2 + β/2 − d vertices on the geodesic from

Bj1 to Bγj2 such that each component P[Λi] for i = 1, . . . , `− 1 is contained in Z(j1) and [Λ`] is on the

boundary of Z(γj2). The indices indicate that [Λi] is at a distance d([Λi],Bj1) = α/2− i from Bj1 . The

multiplicity of the component P[Λ`] of Z(j1) is mult[Λ`](j1) = α/2 + β/2 − d and (P[Λ`], Z(γj2)v) = 1

according to Proposition 5.2.3 therefore its contribution is

(mult[Λ`](j1) · P[Λ`], Z(γj2)v) = α/2 + β/2− d . (5.3)

Proposition 5.2.3 implies that (P[Λi], Z(γj2)v) = 1−p for i = 1, . . . , `−1 and the sum of their contributions

is`−1∑i=1

mult[Λi](j1) · (P[Λi], Z(γj2)v) = (1− p)`−1∑i=1

i = (1− p)`(`− 1)2

. (5.4)

We will sum the contribution of all the other components of Z(j1)v according to their distance r

from the geodesic connecting Bj1 and Bγj2 . For a fixed distance r, there are (p − 1)pr−1 components

of Z(j1)v with multiplicity 1 at a distance r from the geodesic. They are connected to the geodesic

via [Λr+1]. There are (p − 1)pr−1 components of Z(j1)v at a distance r from the geodesic and on the

boundary of Z(γj2) (and therefore at a distance β/2 from Bγj2) and they have multiplicity `− 2r. They

are connected to the geodesic via [Λ`−r]. Therefore, analogous to the computation above, starting with

those components with multiplicity m = 1 and summing up to those on the boundary of Z(γj2) with

multiplicity m = ` − 2r, we find the total contribution of the components at a distance r from the


geodesic

pr−1(p− 1)(`− 2r) + pr−1(p− 1)`−2r−1∑m=1

m(1− p) =(`− 2r + (1− p) (`− 2r)(`− 2r − 1)

2

)(p− 1)pr−1 .

(5.5)

The maximum distance from the geodesic of a component of Z(j1) contributing to the height pairing

is `/2 − 1 if ` is even (corresponding to ` − r − 1 = r + 1) and (` − 1)/2 if ` is odd (corresponding to

`− r = r + 1). Therefore, if ` = α/2 + β/2− d is even, the contribution of the components of Z(j1)v at

a distance r from the geodesic is

`/2−1∑r=1

(`− 2r + (1− p) (`− 2r)(`− 2r − 1)

2

)(p− 1)pr−1 (5.6)

and then summing Equation 5.3, Equation 5.4 and Equation 5.6 we arrive at

(Z(j1), Z(γj2)) = α+ β + 1− 2d− p(α/2+β/2−d)/2 − 2p(α/2+β/2−d)/2 − 1

p− 1. (5.7)

If ` = α/2 + β/2 − d is odd, the contribution of the components of Z(j1)v at a distance r from the

geodesic is(`−1)/2∑r=1

(`+ 1− 2r + (1− p) (`− 2r)(`− 2r − 1)

2

)(p− 1)pr−1 (5.8)

and then summing Equation 5.3, Equation 5.4 and Equation 5.8 we arrive at

(Z(j1), Z(γj2)) = α+ β + 1− 2d− 2p(α/2+β/2−d+1)/2 − 1

p− 1. (5.9)

Consider the case in which Z(γj2) contains the component P[Λ] at Bj1 but Z(j1) does not contain

the component at Bγj2 , that is, α/2 − 1 < d ≤ β/2 − 1. Furthermore, the distance from Bj1 to the

boundary of Z(γj2) in the direction away from Bγj2 is β/2 − d and we consider the case where Z(j1)

contains that part of the boundary, that is, α/2 > β/2−d. The contribution of the components of Z(j1)

along the p branches emanating from Bj1 away from the geodesic connecting Bj1 and Bγj2 is given by

(1− p)α/2 + (1− p)β/2−d−1∑r=1

(α/2− r)pr + (α/2− β/2 + d)pβ/2−d (5.10)

where the first term is the contribution of the component P[Λ] at Bj1 , the second term is the contribution

of the components of Z(j1)v contained in Z(γj2) and the last term is the contribution of the components

on the boundary of Z(γj2). There are α/2− 1 components of Z(j1) along the geodesic connecting Bj1


and Bγj2 and their contribution is

α/2−1∑i=1

(1− p)(α/2− i) . (5.11)

Label the vertices along the geodesic connecting Bj1 and Bγj2 which are contained in Z(j1) by [Λ1], [Λ2], . . . , [Λα/2−1].

The indices indicate that [Λi] is at a distance d([Λi],Bj1) = α/2− i from Bj1 . The distance from [Λi] to

the boundary of Z(γj2) in the direction away from the geodesic is α/2 + β/2− d− i. The distance from

[Λi] to the furthest component of Z(j1) (where its multiplicity is 1 and the distance to Bj1 is α/2− 1)

is i− 1. Therefore, if i− 1 < `− i (where ` = α/2 + β/2− d) then the contribution of the components

of Z(j1)v on the p− 1 branches emanating from [Λi] is

i−1∑r=1

(1− p)(p− 1)pr−1(i− r) , i < (`+ 1)/2. (5.12)

If i− 1 ≥ `− i then the contribution of the components of Z(j1)v on the p− 1 branches emanating from

[Λi] is`−i−1∑r=1

(1− p)(p− 1)pr−1(i− r) + (p− 1)p`−i−1(2i− `) , i ≥ (`+ 1)/2. (5.13)

Summing (Z(j1)h, Z(γj2)) = 2(β/2 − d) with Equation 5.10, Equation 5.11, Equation 5.12 and Equa-

tion 5.13 when ` is odd

(Z(j1), Z(γj2)) = β − 2d+ (1− p)α/2 + (1− p)β/2−d−1∑r=1

(α/2− r)pr + (α/2− β/2 + d)pβ/2−d

+α/2−1∑i=1

(1− p)(α/2− i)

+(`−1)/2∑i=1

i−1∑r=1

(1− p)(p− 1)pr−1(i− r)

+α/2−1∑

i=(`+1)/2

`−i−1∑r=1

(1− p)(p− 1)pr−1(i− r) + (p− 1)p`−i−1(2i− `)

= α+ β + 1− 2d− 2p(α/2+β/2−d+1)/2 − 1

p− 1.

Summing (Z(j1)h, Z(γj2)) = 2(β/2 − d) with Equation 5.10, Equation 5.11, Equation 5.12 and Equa-

tion 5.13 when ` is even

(Z(j1), Z(γj2)) = β − 2d+ (1− p)α/2 + (1− p)β/2−d−1∑r=1

(α/2− r)pr + (α/2− β/2 + d)pβ/2−d


+α/2−1∑i=1

(1− p)(α/2− i)

+`/2∑i=1

i−1∑r=1

(1− p)(p− 1)pr−1(i− r)

+α/2−1∑i=`/2+1

`−i−1∑r=1

(1− p)(p− 1)pr−1(i− r) + (p− 1)p`−i−1(2i− `)

= α+ β + 1− 2d− p(α/2+β/2−d)/2 − p(α/2+β/2−d)/2 − 1p− 1

.

Consider the case in which Z(γj2) contains the component P[Λ] at Bj1 but Z(j1) does not contain

the component at Bγj2 , that is, α/2 − 1 < d ≤ β/2 − 1. However, now consider the case where Z(j1)

does not contain the boundary of Z(γj2), that is, α/2 ≤ β/2− d. Then Z(j1) is completely contained in

Z(γj2) therefore

(Z(j1), Z(γj2)) =2(β/2− d) + (1− p)α/2 + (p+ 1)(1− p)α/2−1∑r=1

(α/2− r)pr−1

= α+ β + 1− 2d− pα/2 − 2pα/2 − 1p− 1

.

The only cases left to consider are those in which Z(j1) contains the component at Bγj2 . The only

differences are the contribution of the component of Z(j1) at Bγj2 is −(1 + p)(α/2 − d) according to

Proposition 5.2.3 instead of (1− p)(α/2− d) and there is a contribution of 2(α/2− d) from Z(j1) with

the horizontal part Z(γj2)h according to Proposition 5.2.3. The changes cancel each other and we are

finished.

5.4 The Unramified/Split Case


are even, χ(ε1) = −1 and χ(ε2) = 1. Let γ ∈ GL2(Qp) and let d = d(Bj1 ,Bγj2). Then O(γ ; j1, j2) = 0



α+ β + 1− 2d−

2p(α/2+β/2−d+1)/2 − 1

p− 1if α


2 >β2 − d,

p(α/2+β/2−d)/2 + 2p(α/2+β/2−d)/2 − 1

p− 1if α


2 >β2 − d,

pα/2 + 2pα/2 − 1p− 1

if α2 ≤

β2 − d.

Proof. The proof is identical to the unramified/unramified situtation in Proposition 5.3.1 except at the

final step when we consider the cases when Z(j1) meets Bγj2 . However, in this case, Z(γj2)h = 0

according to Proposition 5.2.2 and the value of the pairing of the components of Z(j1) at the vertices

along Bγj2 remains 1− p according to Proposition 5.2.3.

5.5 The Split/Unramified Case


are even, χ(ε1) = 1 and χ(ε2) = −1. Let γ ∈ GL2(Qp) and let d = d(Bj1 ,Bγj2). Then O(γ ; j1, j2) = 0


α+ β + 1− 2d−

2p(α/2+β/2−d+1)/2 − 1

p− 1if α


2 >β2 − d,

p(α/2+β/2−d)/2 + 2p(α/2+β/2−d)/2 − 1

p− 1if α


2 >β2 − d,

(β − α+ 1− 2d)pα/2 + 2pα/2 − 1p− 1

if α2 ≤

β2 − d.

Proof. Our first calculations are identical to the unramified/unramified case. The case in which the

cycles meet only at a superspecial point in a boundary component corresponds to d = α/2 + β/2 − 1

and so (Z(j1), Z(γj2)) = 0 if α/2 + β/2 ≤ d.

The case in which Z(γj2) does not meet the centre of Z(j1), that is, α/2+β/2−d > 0 and β/2−1 < d,

proceeds by the same calculations as in Proposition 5.3.1. If ` = α/2 + β/2− d is even, then


p− 1


and, if ` is odd, then

(Z(j1), Z(γj2)) = α+ β + 1− 2d− 2p(α/2+β/2−d+1)/2 − 1

p− 1.

Consider the case in which Z(γj2) meets Bj1 but Z(j1) does not meet Bγj2 , that is, α/2− 1 < d ≤

β/2−1. Furthermore, we restrict to the case where α/2 > β/2−d. Again, the calculations are the same

as in Proposition 5.3.1. If ` = α/2 + β/2− d is odd, then

(Z(j1), Z(γj2)) = α+ β + 1− 2d− 2p(α/2+β/2−d+1)/2 − 1

p− 1(5.14)

and, if ` is even, then


p− 1. (5.15)

Continue with the case α/2 − 1 < d ≤ β/2 − 1 but now with α/2 ≤ β/2 − d. Let [Λ] be the vertex

on Bj1 nearest to Bγj2 . The components of Z(j1) along the branches emanating from [Λ] in the p − 1

directions away from the fixed apartment Bj1 are all contained in Z(γj2). Their contribution, including

the component P[Λ], is

(1− p)α/2 + (p− 1)(1− p)α/2−1∑r=1

(α/2− r)pr−1 (5.16)

The contribution of the components in Z(j1) along the 2 branches emanating from [Λ] along the fixed

apartment Bj1 is

α(1− p)(β/2− d− 1) + α+ 2(1− p)(p− 1)β/2−d−α/2∑

s=1

α/2−1∑r=1

(α/2− r)pr−1

+ 2β/2−d−1∑

s=β/2−d−α/2+1

(1− p)(p− 1)β/2−d−s−1∑

r=1

(α/2− r)pr−1 + (α/2− β/2 + d+ s)(p− 1)pβ/2−d−s−1

.

(5.17)

The first term is the contribution of the components of Z(j1) on Bj1 (that is, those with multiplicity

α/2), the second term is the contribution of the branches emanating from a vertex on Bj1 and which

are completely contained in Z(γj2) and the last term is the contribution of the branches emanating

from a vertex on Bj1 but which are only partially contained in Z(γj2) (hence the truncated sums up to


β/2− d− s− 1 as opposed to α/2− 1). Finally, summing Equation 5.16 and Equation 5.17, we arrive at

(Z(j1), Z(γj2)) = α+ β + 1− 2d− (β − α+ 1− 2d)pα/2 − 2pα/2 − 1p− 1

. (5.18)

The only cases left to consider are those in which Z(j1) meets Bγj2 . The only differences are the

contribution of the component of Z(j1) at Bγj2 is −(1+p)(α/2−d) according to Proposition 5.2.3 instead

of (1− p)(α/2− d) and there is a contribution of 2(α/2− d) from Z(j1) with the horizontal component

Z(γj2)h according to Proposition 5.2.3. The changes cancel each other and we are finished.

5.6 The Ramified/Ramified Case

Proposition 5.6.1. Let j1, j2 ∈ V with q(j1) = pαε1 and q(j2) = pβε2 such that 0 ≤ α ≤ β and α

and β are odd. Let γ ∈ GL2(Qp) and let d = d(Bj1 ,Bγj2). Then O(γ ; j1, j2) = 0 if d > α/2 + β/2,

O(γ ; j1, j2) = 1 if d = α/2 + β/2, and if d < α/2 + β/2 then O(γ ; j1, j2) is given by

α+ β + 1− 2d−

2p(α/2+β/2−d+1)/2 − 1

p− 1if α


2 >β2 − d,

p(α/2+β/2−d)/2 + 2p(α/2+β/2−d)/2 − 1

p− 1if α


2 >β2 − d,

2p(α+1)/2 − 1

p− 1if α

2 ≤β2 − d.

Proof. The calculations for this case closely resemble those of the unramified/unramified case. The

extreme case in which the cycles meet only at a superspecial point in a boundary component corresponds

to d = α/2 + β/2− 1 and so (Z(j1), Z(γj2)) = 0 if α/2 + β/2 ≤ d.

Consider the case in which Z(γj2) does not meet Z(j1)h, that is, α/2+β/2−d > 0 and (β−1)/2 < d.

The calculation here is the same as in Proposition 5.3.1 where ` = α/2 + β/2 − d is multiplicity of the

component of Z(j1) which is at the boundary of Z(γj2) and on the geodesic connecting Bj1 and Bγj2 .

If ` is even we have


p− 1.


If ` is odd we have

(Z(j1), Z(γj2)) = α+ β + 1− 2d− 2p(α/2+β/2−d+1)/2 − 1

p− 1.

If d = (β−1)/2, then the component of Z(j1) on the geodesic from Bj1 to Bγj2 at a distance of β/2 from

Bγj2 contributes ` = (α−1)/2 as opposed to `+1. However, there is a contribution (Z(j1)h, Z(γj2)) = 1

and so the result is unchanged.

Consider the case in which Z(γj2) meets Z(j1)h but Z(j1) does not reach Z(γj2)h, that is, (α−1)/2 <

d < (β − 1)/2. The difference of the multiplicities of the components of each cycle at a distance of 1/2

from Bj1 and d− 1/2 from Bγj2 is ` = (α− 1)/2− ((β − 1)/2− d+ 1), that is, ` = α/2− β/2 + d− 1.

The horizontal divisors yields

(Z(j1)h, Z(γj2)) = (β − 1)/2− d+ (β − 1)/2− d+ 1 = β − 2d .

If ` is odd and (β − 1)/2− d < (α− 1)/2 we have (Z(j1)v, Z(γj2)) =

(1− p) (α− 1)2

+ p(1− p)(β−1)/2−d−1∑

r=1

((α− 1)/2− r)pr−1 + ((α− 1)/2− (β − 1)/2 + d)p(β−1)/2−d

+(`+1)/2∑s=1

(1− p)((α− 1)/2 + 1− s) + (1− p)(p− 1)(β−1)/2−d+s−1∑

r=1

((α− 1)/2 + 1− s− r)pr−1

+((α− 1)/2 + 1− s− ((β − 1)/2− d+ s))(p− 1)p(β−1)/2−d+s−1

+(α−1)/2∑

s=(`+1)/2+1

(1− p)((α− 1)/2 + 1− s) + (1− p)(p− 1)(α−1)/2−s∑

r=1

((α− 1)/2 + 1− s− r)pr−1

= α+ 1− 2p(α/2+β/2−d+1)/2 − 1

p− 1.

The first line is the contribution of the components of Z(j1) along the branches emanating from Bj1 in

the direction away from Bγj2 and the second line is the contribution of the components of Z(j1) along

the branches emanating from Bj1 in the direction towards Bγj2 . Combining with the contribution from


the horizontal divisor we have

(Z(j1), Z(γj2)) = α+ β + 1− 2d− 2p(α/2+β/2−d+1)/2 − 1

p− 1.

If ` is even and (β − 1)/2− d < (α− 1)/2 we have

(1− p) (α− 1)2

+ p(1− p)(β−1)/2−d−1∑

r=1

((α− 1)/2− r)pr−1 + ((α− 1)/2− (β − 1)/2 + d)p(β−1)/2−d

+`/2∑s=1

(1− p)((α− 1)/2 + 1− s) + (1− p)(p− 1)(β−1)/2−d+s−1∑

r=1

((α− 1)/2 + 1− s− r)pr−1

+((α− 1)/2 + 1− s− ((β − 1)/2− d+ s))(p− 1)p(β−1)/2−d+s−1

+(α−1)/2∑s=`/2+1

(1− p)((α− 1)/2 + 1− s) + (1− p)(p− 1)(α−1)/2−s∑

r=1

((α− 1)/2 + 1− s− r)pr−1

= α+ 1− p(α/2+β/2−d)/2 − 2p(α/2+β/2−d)/2 − 1

p− 1.

Combining with the contribution from the horizontal divisor we have


p− 1.

Consider the case (α− 1)/2 ≤ (β − 1)/2− d. Then

(Z(j1), Z(γj2)) =(β − 1)/2− d+ (β − 1)/2− d+ 1 + 2(1− p)(α− 1)/2 + 2(1− p)(α−1)/2∑r=1

((α− 1)/2− r)pr

= α+ β + 1− 2d− 2p(α+1)/2 − 1

p− 1.

The only cases left to consider are those in which Z(j1) meets Z(γj2)h. The only differences are the

contribution of the components of Z(j1) at a distance of 1/2 from Bγj2 are −p((α − 1)/2− d + 1) and

−p((α− 1)/2− d) instead of (1− p)((α− 1)/2− d+ 1) and (1− p)((α− 1)/2− d) respectively however


there are contributions of (α − 1)/2 − d + 1 and (α − 1)/2 − d from Z(j1) with Z(γj2)h. The changes

cancel each other and we are finished.

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