Rankin Triple Products and Quantum Chaos

Thomas C. Watson

September 2003

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Contents

1 Introduction 5

2 Automorphic Forms 7 2.1 OrdersandUnitsofQuaternionAlgebras ...... 7 2.2 Automorphic Forms on Quaternion Groups ...... 8 2.3 HeckeOperators ...... 11 2.3.1 DoubleCosets ...... 11 2.3.2 Non-ArchimedeanOperators ...... 13 2.3.3 ArchimedeanOperators ...... 16 2.4 AutomorphicRepresentations ...... 17 2.5 WhittakerModels ...... 21

3 Theta Lifting 27 3.1 BasicSetup ...... 27 3.1.1 Groups ...... 27 3.1.2 Measures ...... 28 3.1.3 Dual-Pairs...... 29 3.2 Jacquet–LanglandsCorrespondence...... 30 3.2.1 Shimizu’sThetaLift ...... 30 3.2.2 Adjoint of Shimizu’s Lift ...... 35 3.3 LocalData ...... 36

4 L-functions 39 4.1 LocalLanglandsCorrespondence ...... 39 4.1.1 AdjointParameters ...... 41 4.1.2 TripleProductParameters ...... 44 4.2 ZetaIntegrals...... 45 4.2.1 Rankin–Selberg...... 45 4.2.2 Garrett/Rallis–Piateski-Shapiro ...... 48

5 Conclusion 59 5.1 TripleProductIdentities...... 59 5.2 ApplicationstoQuantumChaos ...... 60 5.2.1 Proofs...... 63

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4 CONTENTS

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Chapter 1

Introduction

In this paper, we demonstrate the chaotic nature of some archetypical quantum dynamical systems, using machinery from analytic number theory. Consider the quantized geodesic flow on a finite-volume hyperbolic surface Γ\H, with Γ ⊂ SL2R consisting of the norm-1 units of an Eichler order in an indefinite quaternion algebra B over Q. Such Γ generalize the congruence subgroups of SL2Z and are co-compact whenever B is ramified. For Γ = SL2Z, we prove that high-energy bound eigenstates obey the Random Wave conjecture of Berry / Hejhal for third moments. In fact we show that the third moment of − 1 +ǫ a wave’s amplitude distribution decays like E 12 . In the more general case of maximal orders, we reduce an optimal quantitative version of the Quantum Unique Ergodicity conjecture of Rudnick–Sarnak to the Lindelöf Hypothesis, itself a consequence of the Riemann Hypothesis, for particular families of automorphic L-functions. Furthermore, our analysis shows that any lowering of the exponent in the Phragmen–Lindelöf convexity bound implies QUE. In the moment problem as well, a decisive role is played by ‘convexity-breaking.’ That is to say, the maximum non-trivial exponents precisely agree when translated between physical and arithmetical formulations for both of these problems. (Those readers primarily interested in the physical side of this correspondence are ad- vised to skip ahead to the final chapter.) We accomplish this translation by proving identities expressing triple-cor- relation integrals of eigenforms in terms of central values of the corresponding Rankin triple-product L-functions. Very general forms of such identities were proved by Harris–Kudla, and certain (more precise) explicit classical versions of these were given by Gross–Kudla and Böcherer–Schulze-Pillot, for definite quaternion algebras. In using the Harris–Kudla method to prove our own classical identities, we have to solve two main problems. The first problem is to explicitly compute the adjoint of Shimizu’s theta lift, which realizes the Jacquet–Langlands correspondence by transferring automor- × 2 phic forms from GL2 to GO(B), the latter being nearly the same as (B ) . As is well known, theta liftings from metaplectic to orthogonal groups are gen- erally more difficult to characterize than lifts in the opposite direction, which

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 6 CHAPTER 1. INTRODUCTION can be evaluated directly in terms of Whittaker functions. Since B× and hence GO(B) have ‘multiplicity-one’—as Jacquet–Langlands proved with the trace- formula—we are able to determine the adjoint of Shimizu’s lift by duality, from explicit knowledge of Shimizu’s lift itself. It is, however, necessary to generalize Shimizu’s original calculations, since he only considered averages of lifts over isotypic bases of forms, which allowed him to employ Godement’s theory of spherical functions. In order to deal with individual ramified forms, we re- place this argument by explicit calculations involving Hecke operators. Thus we determine the Shimizu lifts of oldforms and newforms of square-free level, with any (possibly imprimitive) nebentypus. As a byproduct of these calculations, we obtain explicit formulas for all relevant GL2 Whittaker functions. These play an important role in our second main problem: evaluation of Garrett / Rallis–Piateski-Shapiro local zeta integrals in terms of the standard functorial triple-product L-factors. Our contribution lies in the calculation of archimedean zeta integrals with various types of ramifica- tion. In all of these ramified cases, the canonical choices of local data for theta lifting are apparently useful only for evaluating central values of the associated zeta integrals. This was observed by Gross–Kudla in the non-archimedean case, and it appears to be a significant unexplained feature of the Harris–Kudla method. Finally, we re-prove (by elementary brute-force computation) the important result of Rallis–Piateski-Shapiro on unramified non-archimedean zeta integrals, in anticipation of future generalizations to the many ramified non- archimedean cases. Acknowledgements. This work is the product of my doctoral research at Princeton University, directed by Peter Sarnak. I am very grateful to him for explaining this fascinating problem to me, and for his generous help and encouragement, which cannot be overstated. I also thank Henryk Iwaniec and Goro Shimura for teaching me analytic and algebraic number theory, as well as Michael Boshernitzan and Yakov Sinai for teaching me ergodic theory. I gratefully acknowledge receiving funding through an NSF Graduate Fellowship and a Charlotte E. Proctor Honorific Fellowship. Finally, I thank my friends and family for their love and support.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Chapter 2

Automorphic Forms

2.1 Orders and Units of Quaternion Algebras

Let B be a quaternion algebra over Q. Then Bv = B ⊗Q Qv is a quaternion algebra over Qv, for each place v of Q. B is called ramified at v if Bv is a division algebra. We will always assume that B is unramified at ∞, i.e. B is indefinite. Then the reduced discriminant dB of B over Q equals the product of the finite even number of finite places p where B is ramified. The 2 × 2 matrix algebra M = M2 over Q is totally unramified, and hence dM = 1. We denote by Dv the unique division quaternion algebra over Qv, while Mv is the unique split (i.e. non-division) quaternion algebra. We will sometimes use the coordinate α α functions α on α = 11 12 ∈ M , and also the notation ij α α v 21 22

[Xij ] = {α ∈ Mv ; αij ∈ Xij ⊂ Qv}.

ι Write the canonical anti-involution ι of B or Bv as α 7→ α , noting its compatibility with ﬁeld extensions, and deﬁne the reduced trace and norm by ι ι τ(α)= α + α and ν(α)= αα . On M and Mv, ι is given explicitly as ι a b d −b = , c d −c a so τ and ν correspond to the usual trace and determinant. Dp has the valuation ordp ◦ ν, and its valuation ring Rp is also the unique maximal order of Dp. Choose a uniformizer ̟p ; all bilateral Rp ideals in Dp n n are principal and of the form ̟p Rp = Rp̟p , n ∈ Z. Every maximal order of Mp is conjugate to M2(Zp). We will be interested −n n −1 n Zp p Zp p p in those of the form n = M2(Zp) for p Zp Zp 1 1 n ∈ Z, although the set of all maximal orders is larger, having the structure of a (p + 1)-homogeneous tree for a natural notion of adjacency. It is a theorem of Hijikata that every Eichler order, an intersection of two maximal orders, is

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 8 CHAPTER 2. AUTOMORPHIC FORMS

Z Z conjugate to p p for some n ≥ 0. All Eichler orders containing the pnZ Z p p Z p−n˙ Z previous one are of the form p p for 0 ≤ n˙ ≤ n¨ ≤ n. pn¨ Z Z p p Orders O of B exist and have the property that Op := O⊗Z Zp is a maximal ′ ′ order in Bp for almost all p. Furthermore, for any other order O of B, Op = Op for almost all p. Conversely, given a choice over all p of local orders (O˜p) which ′ satisfies O˜p = Op for a.a. p, then O := {α ∈ B ; αp ∈ O˜p ∀p} is an order in B ′ ˜ with p-adic completions Op = Op. The adele ring BA is defined as the restricted direct product of all Bv relative × to Ofin := p Op, for any order O of B, and the idele group BA as the restricted direct product of all B× relative to O× . B is contained diagonally in B , and Q v fin A we will write B = BQ ⊂ BA to emphasize this inculsion. By the previous paragraph, orders of B correspond 1-1 with compact open subgroups of Bfin. Maximal orders in B are characterized as orders O s.t. every Op is maximal. Choose a maximal order for each B, denoted O(dB ). Throughout the rest of the paper, for p ∤ dB we will identify Bp with Mp in such a way that Op(dB)= M2(Zp). Furthermore, in the case dB = 1 we identify B with M: O(1) = M2(Z). + Now for N,˙ N¨ ∈ Q relatively prime to dB and s.t. N = N˙ N¨ ∈ Z, we define the order O(dB , N~ ) of B by

Rp p | dB,

Op(dB , N~ ) = Z N˙ Z  p p p ∤ d .  ¨ B " NZp Zp #

−1  Then O(dB , N,˙ N˙ ) is a maximal order, and

−1 −1 O(dB , N~ ) = O(dB , N,˙ N˙ ) ∩ O(dB , N¨ , N¨) is an Eichler order. It has reduced discriminant dB N and is called of level N. Using strong approximation and Hijikata’s result, it is easy to show that any Eichler order of level N in B is conjugate to O(dB , (1,N)). Note that the only ′ ′ ′ Eichler orders containing O(dB , N~ ) are the O(dB , N~ ) s.t. N˙ | N˙ and N¨ | N¨. Throughout the rest of the paper, for any X possessing a character ν, we will use the notations

(a) X∗ = {β ∈ X∗ ; ν(β)= a}, (A) X∗ = {β ∈ X∗ ; ν(β) ∈ A},

± × ± (R ) [a] (aZp ) X∞ = X∞ , Xp = Xp .

2.2 Automorphic Forms on Quaternion Groups

~ × In this section, fix O = O(dB , N). Strong approximation for Ofin is proved in × × + × × × [26]: BA = BQ B∞Ofin since B is indefinite and ν(Ofin)= Zfin. Because of this,

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.2. AUTOMORPHIC FORMS ON QUATERNION GROUPS 9

(1) × + × and since O = BQ ∩ B∞Oﬁn, there are homeomorphisms

× × × (1) + BQ \BA /Ofin ≃ O \B∞, × + × × (1) + BQ R \BA /Ofin ≃ O \PGL2 (R). χ Given a primitive Dirichlet character χ mod N , factor χ = p χp and extend its domain to Z× by means of the Chinese remainder theorem, fin Q χ × χ × (Z/N Z) ≃ p|N χ (Zp/N Zp) × χ = Qp|N χ Zp /(1 + N Zp). × Q× + × Using strong approximation, A = Q R Zfin , we define the grossen-character × χ χ˜ byχ ˜(qrz)= χ(z). Note thatχ ˜∞(−1) = χ(−1) andχ ˜p(pZp )= χ(p) if p ∤ N . Since O has level N, the map rp : Op → Zp/NZp defined by rp(β) = [β11] for p ∤ dB , is a ring homomorphism, and hence so is r : Ofin → Z/NZ, r(β) = χ × × (rp(βp))p|N . This permits us, for N | N, to defineχ ˜ on Ofin and χ on O by composition with r. (Note χ(β)= χ˜(β) for β ∈ O×.) Sinceχ ˜ is well-defined on × × × × × Qfin ∩ Ofin = Zfin, it extends to a character of QfinOfin. × As usual, B∞ = GL2(R) acts on CrR by linear fractional transformations, az + b a b z 7→ β|z = , β = , cz + d c d 2 2 2 2 + preserving the metric ds = (dx + dy )/y . Furthermore, B∞ acts on weight-k + k Maass forms (k ∈ Z) on H = R + ı R by ψ 7→ ψ|β,

k cz+d −k ψ|β(z) = |cz+d| ψ(β|z). Consider the space Ak(dB , N,χ~ ) of smooth weight-k Maass forms ψ on H which have moderate growth and satisfy the automorphy condition k (1) ~ ψ|γ∞ = χ(γ) ψ, γ ∈ O (dB , N). Since −1 ∈ O(1), such ψ must vanish identically unless χ(−1) = (−1)k, so we assume this from now on. Let Ck(dB , N,χ~ ) ⊂ Ak(dB, N,χ~ ) denote the cuspidal subspace, equipped with the Petersson inner-product

dx dy hψ1, ψ2i := ψ1(z) ψ2(z) y2 . (1) ZO \H 2 × × In the adelic setting, we deﬁne L (BQ \BA , χ˜) as usual to consist of automorphic functions Ψ with central characterχ ˜,

× × Ψ(γzβ) =χ ˜(z) Ψ(β), γ ∈ BQ , z ∈ A , and ﬁnite norm under the inner-product

1 × hΨ1, Ψ2i := 2 Ψ1(β) Ψ2(β) d β. P B×\P B× Z Q A

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 10 CHAPTER 2. AUTOMORPHIC FORMS

The Tamagawa measure d×β is deﬁned in §3.1.2. Note these two inner-products × are normalized diﬀerently. We denote the right regular action of α ∈ BA by

2 × × α|Ψ(β) = Ψ(βα), Ψ ∈ L (BQ \BA , χ˜),

2 × × and deﬁne L0(BQ \BA , χ˜) as the closed invariant subspace of cuspidal Ψ. Now let × ~ Op (dB , N) v = p, Kv(dB , N~ ) = (SO(2, R) v = ∞,

+ × and extendχ ˜∞ (dependent on k) to R K∞ from R by setting

cos θ sin θ χ˜ (rκ ) = eikθ, κ = . ∞ θ θ − sin θ cos θ ˜ ~ 2 × × We deﬁne Ck(dB , N, χ˜) to consist of all smooth Ψ ∈ L0(BQ \BA , χ˜) satisfying

κv|Ψ =χ ˜v(κv)Ψ, κv ∈ Kv(dB, N~ ).

× × + × (Note this is consistent with the inclusion of the center A ⊂ BQ R Oﬁn.) Then ∼ we have an isomorphism Ck(dB , N,χ~ ) −→ C˜k(dB , N,~ χ˜), given by

+ k Ψ(γβ κﬁn) =χ ˜(κﬁn) ψ| +(ı). ∞ β∞

′ χ ′ If O(dB , N~ ) ⊃ O(dB , N~ ) and N | N , there is an inclusion

′ .( ~Ck(dB , N~ ,χ) ֒→ Ck(dB , N,χ

Any form in the image of such a map with pN ′ | N is called p-old, as are linear combinations of such forms, and any form orthogonal to all p-old forms is called p-new. Likewise at the archimedean place, we have raising and lowering operators,

± ~ ~ Rk : Ck(dB, N,χ) −→ Ck±2(dB , N,χ), + z−z ∂ k − z−z ∂ k Rk = 2 ∂z + 2 , Rk = 2 ∂z + 2 .

± ′ A weight-k form ψ is called ∞-old if ψ = Rk∓2 ψ for ±k ≥ 0, and ∞-new if it is ± ∓ orthogonal to all ∞-old forms. Since Rk∓2 and Rk are adjoints, ψ is ∞-new iﬀ ∓ |k| − 2 Rk ψ = 0 for ±k ≥ 0, and this is equivalent to y ψ(z) being a holomorphic / anti-holomorphic function of z. All forms with k =0 are ∞-old. For the rest of this paper we assume that N is square-free, and factoring it ♭ ♯ χ ♭ into N = N N N , we deﬁne Sk(dB , N,χ,N~ ) ⊂ Ck(dB , N,χ~ ) as the subspace of forms which are p-old for p | N ♭, p-new for p | N ♯, and ∞-new if |k| > 0. We will also assume that |k|= 6 1.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.3. HECKE OPERATORS 11

2.3 Hecke Operators

χ In this section, ﬁx O = O(dB , N~ ) and χ primitive with conductor N | N, and assume that N = N ♭N ♯N χ is square-free. For p | N ♭N ♯, set

˙ ˙ N¨ ¨ N˙ ¨ Op = Op(dB , (N, p )), Op = Op(dB, ( p , N)).

2.3.1 Double Cosets

× × × For α ∈ Bp , define the double coset Tp(α) = Op α Op . If p | dB , every Tp(α) n n × × n n is equal to some Tp(̟p ) = ̟p Rp = Rp ̟p for n ∈ Z, and Tp(̟ ) ⊂ Op iff × 2 n ≥ 0. Now for p ∤ dB and ~a ∈ (Qp ) , define

a˙ a˙ T (~a) = T , R (~a) = T . p p a¨ p p a¨ N˙ Since all a ∈ Q× and α = normalize O×, p N~ N¨ p × × ~ × × Tp(a) = aOp = Op a, Rp(N) = αN~ Op = Op αN~ .

~n If p ∤ dB N, every Tp(α) is equal to Tp(p ) for somen ˙ ≥ n¨ (set n =n ˙ − n¨), ~n and Tp(p ) ⊂ Op iﬀn ¨ ≥ 0. The parametersn, ˙ n¨ uniquely solve

n¨ n˙ +¨n × α ∈ p (Op r pOp), ν(α) ∈ p Zp .

Furthermore, these double cosets are partitioned by left cosets: (For right cosets, apply ι ; for lower triangular representatives, conjugate by Rp(N~ ).)

n pm+¨n Nxp˙ n¨ T (p~n) = O×. p pn−m+¨n p m=0 x mod pm [ p∤x if[ 0

We prove both well-known assertions of the previous paragraph at the same ~n time. Note that Tp(p ) contains the above union of left cosets, and they are disjoint since

−1 ′ pm+¨n Nxp˙ n¨ pm +¨n Nx˙ ′pn¨ ′ pn−m+¨n pn−m +¨n m′−m ˙ ′ −m −m′ p N(x p − xp ) × = ′ ∈/ O pm−m p unless m = m′, x ≡ x′ mod pm. Now it suﬃces to show that the union of these left cosets over alln ˙ ≥ n¨ ≥ 0 has additive volume (see §3.1.2) equal to that of

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 12 CHAPTER 2. AUTOMORPHIC FORMS

× × × × 2 Op ∩ Bp , since for any α ∈ Op ∩ Bp , vol(αOp )/vol(Op )= |α|Bp = |ν(α)|p > 0. Finally,

(pZ ) Z× N˙ Z× p pZ Np˙ Z O r O× = p p ∪ p p ∪ . . . p p N¨Z× Z× N¨Z× Z× p p p p pZ Np˙ Z pZ Np˙ Z ∪ p p ∪ . . . ∪ p p , so Np¨ Z Z× Np¨ Z pZ p p p p

3 2 1 × −1 p−1 −2 p−1 −3 p−1 −4 vol(Op ) = 1 − p p +4p p +4p p + p = 1 − p−1 1 − p−2 , and

× × × ζp(1)ζp(2) = vol(Op)/vol(Op ) ≥ vol(Op ∩ Bp )/vol(Op ) ≥ 1 + (p − 1) 1+ p + · · · + pn˙ −n¨−2 + pn˙ −n¨ p−2˙n−2¨n n˙ ≥n¨≥0 X = 1 + (p − 1) p−n−1 p−4¨n n>0 ! X nX¨≥0 1+ p−2 1 = = ζ (1)ζ (2). 1 − p−1 (1 − p−4) p p × ~n ~n ~ If p | N, every Tp(α) (with α ∈ Bp ) is equal to either Tp(p ) or Rp(p N) for somen, ˙ n¨ ∈ Z, and Tp(α) ⊂ Op iﬀn, ˙ n¨ ≥ 0. We distinguish among these m as follows: First, m = min{n,˙ n¨} satisﬁes α ∈ p (Op r pOp). Then the type of double coset and sign of n =n ˙ − n¨ are determined by the partition

T,+ T,0 T,− pZ N˙ Z Z× N˙ Z Z× N˙ Z O r pO = p p ∪ p p ∪ p p p p N¨Z Z× N¨Z Z× N¨Z pZ p p p p p p R,+ R,0 R,− pZ Np˙ Z pZ N˙ Z× pZ N˙ Z× ∪ p p ∪ p p ∪ p p . N¨Z× pZ N¨Z× pZ Np¨ Z pZ p p p p p p Lastly, the remaining parameter max{n,˙ n¨} is recovered from

pn˙ +¨n Z× α ∈ T (p~n), ν(α) ∈ p p p1+n ˙ +¨n Z× α ∈ R (p~nN~ ). p p

These Tp(α), taking α diagonal or anti-diagonal, have left coset representatives as follows: (For right cosets, apply ι and conjugate by Rp(N~ ).)

1 Nx˙ α n ≥ 0, 1 T (α) = α O×, α =  p x p x 1 x mod p|n|  α n ≤ 0. [  Nx¨ 1  

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.3. HECKE OPERATORS 13

As in the case p ∤ dBN, we prove all of these claims together by showing that × the left cosets above (withn, ˙ n¨ ≥ 0) completely partition Op ∩ Bp . The double cosets listed above are disjoint for unequal types / parameters by the preceding × characterization, and the left cosets αxOp listed within the same double coset −1 × ′ |n| are disjoint, since αx αx′ ∈/ Op unless x ≡ x mod p . Z× N˙ Z Finally, O× = p p has vol(O×)= p−1(1 − p−1)2, and p N¨Z Z× p p p 2 × × × ζp(1) = vol(Op)/vol(Op ) ≥ vol(Op ∩ Bp )/vol(Op ) ≥ p−4¨n +2 pn˙ −n¨−2˙n−2¨n nX¨≥0 n>˙ Xn¨≥0 + p−4¨n−2 +2 pn˙ −n¨−2˙n−2¨n−2 nX¨≥0 n>˙ Xn¨≥0 = 1+ p−2 1+2 p−n p−4¨n n>0 ! n¨≥0 X X 1+ p−1 1 = 1+ p−2 = ζ (1)2. 1 − p−1 (1 − p−4) p ♭ ♯ ˙ × ¨× ˙ ¨ In the subcase p | N N , we write double cosets of Op , Op as Tp(α), Tp(α). It is easy to check by the disjointness / volume argument that

˙ ˙ N¨ Tp(1) = Tp(1) ∪ Rp(N, p ), ¨ N˙ ¨ Tp(1) = Tp(1) ∪ Rp( p , N).

2.3.2 Non-Archimedean Operators

× × The space S(Bp ) of C-valued Schwarz functions on Bp forms an algebra under × convolution (using the measure dp β deﬁned in §3.1.2),

′ −1 ′ × (φp ∗ φp)(α) := φp(αβ ) φp(β) dp β, × ZBp 2 × × and it acts on L0(BQ \BA , χ˜) via the right regular representation,

× φp|Ψ := β|Ψ φp(β) dp β. × ZBp ⋆ ♭ ♯ χ We will deﬁne and examine subquotient algebras Hp = Hp, Hp, Hp, Hp , in the ♭ ♯ χ ♭ respective cases p ∤ N, p | N , p | N , p | N , acting on S˜k(dB, N,~ χ,N˜ ). × For any p, deﬁne Hp to be the subalgebra of S(Bp ) generated by all

1 × φTp(α) := × × ITp(α), α ∈ Bp . vol (Op )

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 14 CHAPTER 2. AUTOMORPHIC FORMS

We abuse notation by writing Tp(α) in place of φTp(α) and omitting ∗’s when × the meaning is clear. For any a ∈ Qp , let

[a] T = × [a] × Tp(α) ∈ Hp. p α∈Op \Op /Op

χ P ♭ If p ∤ N , Hp acts on S˜k(dB , N,~ χ,N˜ ) non-trivially. The element ∨ −1 Tp(α) := Tp(α ) ∈ Hp acts as the adjoint of Tp(α), and Tp(p) acts as the scalar χ(p). n [pn] Tp(̟p) n ≥ 0, If p | dB then Hp ≃ C Tp(̟p) and Tp = ( 0 n< 0. −1 If p ∤ dBN, it is well known (see [36]) that Hp ≃ C Tp(p, 1),Tp(p),Tp(p) , [pn] ~n Tp = Tp(p ), n˙ +¨n=n n˙ ≥Xn¨≥0 ∞ [pn] n 2 −1 Hp[[X]] ∋ Tp X = Tp(1) − Tp(p, 1)X + pTp(p)X . n=0 X For p | N, we begin by computing multiplication rules in Hp :

2 Rp(N~ ) = Tp(p), n n Tp(~a) = Tp(~a ) if n ≥ 0,

Tp(~a) Tp(p) = Tp(~ap) = Tp(p) Tp(~a),

Tp(~a) Rp(N~ ) = Rp(~aN~ ) = Rp(N~ ) Tp(¨a, a˙ ), ~ Tp(p, 1) Tp(1,p) = pTp(p) + (p − 1) Tp(p, 1) Rp(N),

Tp(1,p) Tp(p, 1) = pTp(p) + (p − 1) Tp(1,p) Rp(N~ ). Also, [pn] ~n ~n ~ Tp = Tp(p )+ Rp(p N). n˙ +¨n=n 1+n ˙ +¨n=n n,˙Xn¨≥0 n,˙Xn¨≥0 ♭ ~ Now define Hp as the centralizer of Rp(N) in Hp. Shimizu proved in [35] (see ♭ also [28]) that Hp is a maximal commutative subalgebra of Hp, that it contains [pn] all Tp , and that ∞ ~ [pn] n Tp(1) + p Rp(N)X Tp X = . [p] ~ 2 n=0 Tp(1) − Tp X + p Rp(N)X + pTp(p)X X ♯ To define Hp, we first note that ˙ 1 ˙ N¨ Tp(1) = p+1 Tp(1) + Rp(N, p ) , ¨ 1 N˙ ¨ Tp(1) = p+1 Tp(1) + Rp( p , N),

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.3. HECKE OPERATORS 15 and compute the relations ˙ p ~ ˙ Tp(1,p) Tp(1) = p+1 Tp(1,p)+ Rp(N) = Tp(1) Tp(1,p), ¨ p ~ ¨ Tp(p, 1) Tp(1) = p+1 Tp(p, 1)+ Rp(N) = Tp(1) Tp(p, 1). ♯ ♭ If p | N , the elements T˙p(1), T¨p(1) of Hp annihilate S˜k(dB , N,~ χ,N˜ ), while Tp(1,p), Tp(p, 1) are invertible, as we will see in §2.4. Such a representation of Hp factors through the quotient ♯ ~ Hp := Hp Tp(1,p)+ Rp(N) ~ ~ = HpTp(p, 1)+ Rp(N) ≃ C Rp(N) , ~ n [pn] −Rp(N) n ≥ 0, ♯ Tp ≡ in Hp. ( 0 n< 0, χ × ′ Now suppose p | N . For β ∈ Bp , δ = det β, m ∈ {1, 2}, m = 3 − m, we deﬁne

χ˜p(βmm) nm ≤ nm′ , χ m,m 1 φ (n ,n ) (β) = × × IT (p(n1,n2))(β) Tp(p 1 2 ) vol (Op ) p (χ˜p(δ/βm′m′ ) nm ≥ nm′ , χ m,m′ φ (n ,n ) (β)= 0 , Tp(p 1 2 )

′ χ˜p(βmm′ ) nm ≤ nm′ , χ m,m 1 φ (n ,n ) (β) = × × I (n1,n2) ~ (β) Rp(p 1 2 N~ ) vol (Op ) Rp(p N) (χ˜p(−δ/βm′m) nm ≥ nm′ , χ m,m φ (n ,n ) (β)= 0 . Rp(p 1 2 N~ ) These cases were treated somewhat diﬀerently by Miyake in [26, 27]. It is straightforward to check, using the partition in §2.3.1, that χ ~m χ ~m × φ (κβ ˙ κ¨) = χ˜p(κ ˙ m˙ m˙ κ¨m¨ m¨ ) φ (β), κ,˙ κ¨ ∈ O . Tp(α) Tp(α) p χ × Any function φ ∈ S(Bp ) satisfying this transformation rule and supported on χ a double-coset Tp(α) ∈ Hp is determined by it’s value φ (β) on any β ∈ Tp(α). By taking β to be diagonal or anti-diagonal and considering the left and right × χ actions of diagonal κ ∈ Op , we see that in the diagonal case,m ˙ 6=m ¨ ⇒ φ = 0, and in the anti-diagonal case,m ˙ =m ¨ ⇒ φχ = 0. Thus, any such φχ is proportional to the corresponding φχ ~m listed above, which are normalized to Tp(α) have multiplicative values, φχ m˙ 1,m¨ 1 (α ) · φχ m˙ 2,m¨ 2 (α ) = φχ m˙ 1,m¨ 2 (α α ), Tp(α1) 1 Tp(α2) 2 Tp(α1α2) 1 2 × for diagonal or anti-diagonal αj ∈ Bp , whenever neither side vanishes. Now it χ follows from convolving supports in Hp that the φ convolve according to:

′ ′ φχ m,m ∗ φχm ,m = φχ m,m, Rp(N~ ) Rp(N~ ) Tp(p) χ m,m ∗n χ m,m φ = φ n for n ≥ 0, Tp(~a) Tp(~a )

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 16 CHAPTER 2. AUTOMORPHIC FORMS

φχ m,m ∗ φχ m,m = φχ m,m = φχ m,m ∗ φχ m,m, Tp(~a) Tp(p) Tp(~ap) Tp(p) Tp(~a) ′ ′ ′ ′ ′ φχ m,m ∗ φχ m,m = φχ m,m = φχ m,m ∗ φχm ,m , Tp(~a) Rp(N~ ) Rp(~aN~ ) Rp(N~ ) Tp(¨a,a˙ ) φχ m,m ∗ φχ m,m = p φχ m,m = φχ m,m ∗ φχ m,m . Tp(p,1) Tp(1,p) Tp(p) Tp(1,p) Tp(p,1)

χ Deﬁne Hp to be the algebra of ﬁnite C-linear combinations of the functions

T χ(~a) := φχ 1,1 + φχ 2,2 , ~a ∈ (Q×)2 ; p Tp(~a) Tp(~a) p χ χ χ =⇒ Hp ≃ C Tp (p, 1),Tp (1,p) .

Also deﬁne χ ~ χ 1,2 χ 2,1 Rp (~aN) = φ + φ . Rp(~aN~ ) Rp(~aN~ )

χ χ χ × All of the functions φ satisfy φ (zβ)= χ˜p(z) φ (β) for z ∈ Zp , and hence act 2 × × non-trivially on L0(BQ \BA , χ˜). χ Given η ∈ p|N χ (Z/2Z), deﬁne

Q χ χ η χ ~ ηp Ψ = Rp (N) Ψ, and χ pY|N ηχ χ ˜ ~ χ ~ ηp ˜ ~ Ck (dB, N, χ˜) = Rp (N) Ck(dB, N, χ˜), χ pY|N 2 × × which consists of Ψ ∈ L0(BQ \BA , χ˜) satisfying

χ χ˜p(κ11)Ψ ηp =0, × χ κ|Ψ = χ κ ∈ Op , p | N , (χ˜p(κ22)Ψ ηp =1,

χ χ χ plus the usual Kv condition for v ∤ N . The action of Tp (~a) ∈ Hp stabilizes χ ˜η ~ χ ∨ χ −1 each Ck (dB, N, χ˜) and is adjoint to the action of Tp (~a) := Tp (~a ), while χ Tp (p) acts as the identity.

2.3.3 Archimedean Operators ⋆ ∆ Let H∞ denote the semigroup generated by T∞,

∆ 2 ∂2 ∂2 ∂2 T∞ Ψ = −y ∂x2 + ∂y2 − y ∂x∂θ σzκθ|Ψ, 1 T − Ψ = ǫ |Ψ, ǫ = ∈ B× . ∞ ∞ ∞ −1 ∞ It is easy to check that these commute and

∆ ˜ηχ ~ ˜ηχ ~ T∞ : Ck (dB , N, χ˜) −→ Ck (dB , N, χ˜), − ˜ηχ ~ ˜ηχ ~ T∞ : Ck (dB , N, χ˜) −→ C−k(dB , N, χ˜).

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.4. AUTOMORPHIC REPRESENTATIONS 17

χ For η ∈ v(Z/2Z), η = (ηp)p|N χ , deﬁne

χ Q η − η∞ η Ψ = (T∞) Ψ , ˜η ~ ˜ηχ ~ Ck (dB , N, χ˜) = C(−1)η∞ k(dB, N, χ˜).

Now consider the C-valued coordinates,

1 i X(β) = 2 (a + d)+ 2 (b − c), a b β = ∈ B∞, Y (β) = 1 (a − d)+ i (b + c), c d 2 2 with which we write the (2, 2)-form ν and a positive-deﬁnite majorant P :

1 ι ν(β) = 2 tr(ββ ) = XX − Y Y = ad − bc, 1 t 1 2 2 2 2 P (β) := 2 tr(ββ ) = XX + Y Y = 2 (a + b + c + d ).

ret/2 It is straightforward to compute, writing B+ ∋ β = κ κ , that ∞ θ˙ re−t/2 θ¨ i(+θ˙+θ¨) t 2 X(β) = e r cosh 2 , ν(β) = r , i(−θ˙+θ¨) t 2 Y (β) = e r sinh 2 , P (β) = r cosh t.

Now deﬁne φk,ξ , φk,ξ ∈ S(R+\B+ ) by X∞ Y∞ ∞

k k,ξ 1 φ (β) = 1 X(β)/ν(β) 2 e (iξP (β)/ν(β)) , X∞ π 1 t |k| zk k> 0, = π χ˜(κθ˙)χ˜(κθ¨) (cosh 2 ) e(iξ cosh t), zk = 1 k =0, k  k,ξ 1 |k| φ (β) = 1 Y (β)/ν(β) 2 e (iξP (β)/ν(β)) , z k< 0. Y∞ π 1 t |k|  = π χ˜(κθ˙)χ˜(κθ¨) (sinh 2 ) e(iξ cosh t), 

∆ These give rise to Hecke operators commuting with T∞ ,

k,ξ η ~ η˜ ~ 0 ∗ = X, T∗∞ : Ck (dB , N, χ˜) −→ Ck (dB , N, χ˜), η˜∞ = η∞ + (1 ∗ = Y,

k,ξ k,ξ × χ χ T∗∞ Ψ = β|Ψ φ∗∞ (β) d∞β, η˜ = η . + + ZR \B∞ 2.4 Automorphic Representations

˜ ~ ♭ 2 × × Given Ψ ∈ Sk(dB, N, χ,N˜ ), deﬁne π ⊂ L0(BQ \BA , χ˜) as the subspace of × KA(dB )-ﬁnite vectors in the closure of the span of {β|Ψ ; β ∈ BA }. This is the × cuspidal automorphic representation of BA attached to Ψ. If π is irreducible, then π = ⊗˜ vπv and Ψ = ⊗˜ vΨv factor as restricted tensor products over all

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 18 CHAPTER 2. AUTOMORPHIC FORMS

places v, relative to a choice of spherical unit vectors Ψv at almost all places. In this section, we recall the classiﬁcation of pre-unitary irreducible admissible × representations of Bv having square-free conductor [17, 6, 4], and explicitly compute the actions of Hecke operators on these models. As a consequence, we have

♭ ⋆ Lemma 1. The eigenforms in S˜k(dB , N,~ χ,N˜ ) of H comprise an orthogonal basis, and their tuples of eigenvalues have multiplicity one. The corresponding ♭ automorphic representations are each shared by 2#{p|N } basis elements, and these have the same eigenvalues for all v ∤ N ♭.

Proof: This follows from multiplicity-one for automorphic representations × of BA (proved by Jacquet–Langlands using the trace-formula) and Casselman’s theorem, plus the local calculations below for p | N ♭. s˙p s¨p For p ∤ dB N, πp is equivalent to the unramiﬁed continuous series π(| |p , | |p ) with central characterχ ˜p. Since πp is pre-unitary, either

2πi s˙p = −s¨p − logp(χ(p)) ∈ iR/ log(p) Z, or

s˙p = −s¨p = σp + itp , 1 i π 2π σp ∈ (0, 2 ), tp ∈ 2 logp(χ(p)) + log(p) Z/ log(p) Z.

× Casselman’s theorem implies that the right Op -invariant vector Ψp is unique 0 s˙p s¨p up to scaling in πp, and hence corresponds to Vp ∈ π(| |p , | |p ) of the form

a˙ ∗ 1 V 0 κ = |a˙ |s˙p |a¨|s¨p a˙ 2 V 0(1), κ ∈ O×. p a¨ p p a¨ p p p

1 −s˙p −s¨p In terms of the parameters, Ψp has Tp(p, 1) eigenvalue λp = p 2 p + p n −s˙p−s¨p [p ] and Tp(p) eigenvalue χ(p)= p , and hence by the analysis in §2.3.2, Tp eigenvalue

n −ns˙p −(n−1)s ˙p−s¨p −ns¨p λpn = p 2 p + p + · · · + p

−(n+1)˙sp −(n+1)¨sp n p − p = p 2 . p−s˙p − p−s¨p ♭ If p | N , πp is again equivalent to an unramiﬁed continuous series, exactly × as above. However by Casselman’s theorem, the subspace of right Op -invariant ˙ × ˙ 0 ¨× vectors is spanned by a right Op -invariant vector Vp and a right Op -invariant ¨ 0 vector Vp , each unique up to scaling. We relate their normalizations by deﬁning

1 V¨ 0 = V˙ 0 = R (N~ ) V˙ 0, p p p p p ˙ 0 ˙ 0 ~ ¨ 0 =⇒ χ(p) Vp = Tp(p) Vp = Rp(N) Vp .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.4. AUTOMORPHIC REPRESENTATIONS 19

Using coset representatives, we see that

˙ 0 ˙ 0 ¨ 0 Tp(p, 1) Vp = λp Vp − Vp , ¨ 0 ˙ 0 Tp(p, 1) Vp = pχ(p) Vp .

We have now determined the actions of Rp(N~ ) and Tp(p, 1), and hence also

−1 Tp(1,p) = Rp(N~ ) Tp(p, 1) Rp(N~ ), [p] ~ Tp = Tp(p, 1)+ Tp(1,p)+ Rp(N).

˙ 0 ¨ 0 With respect to the basis Vp , Vp , these operators are represented by the matrices h i λ pχ(p) −χ(p) T (p, 1) ∼ p ,T (1,p) ∼ , p −1 p p λ p λ pχ(p) χ(p) T [p] ∼ p , R (N~ ) ∼ . p p λ p 1 p The latter two are normal and commute, and so have orthogonal eigenvectors

♭ ˙ 0 ¨ 0 2 Vp := Vp + εp Vp , εp = χ(p), [p] ♭ ♭ ~ ♭ ♭ Tp Vp = (λp + pεp)Vp , Rp(N) Vp = εp Vp .

Then by the considerations of §2.3.2,

[pn] ♭ ♭ Tp Vp = (λpn + pεp λpn−1 )Vp .

˙ ¨ ˙ 0 ¨ 0 The idempotents Tp(1), Tp(1) act as orthogonal projectors to C Vp , C Vp , and ˙ ¨ 0 λp ˙ 0 using the above calculations we see that Tp(1) Vp = p+1 Vp , so

¨ 0 ˙ 0 λp ˙ 0 ¨ 0 ♭ ♭ εp λp hVp , Vp i = p+1 = χ(p) hVp , Vp i, hVp , Vp i = 2 1+ p+1 .

1 i π 2π s˙p = −s¨p = 2 + itp , tp ∈ 2 logp(χ(p)) + log(p) Z/ log(p) Z.

This is an irreducible invariant subspace of IndGp (| |s˙p , | |s¨p ), consisting of V s.t. Tp p p p

1 V dx = 0. p x 1 ZQp × ♯ For an Op -invariant vector Vp , this is equivalent to

1 |N¨| V ♯(1) + |N˙ | V ♯ = 0, p p p p 1

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 20 CHAPTER 2. AUTOMORPHIC FORMS and so a˙ ∗ V ♯ κ = |a˙ a¨|itp a˙ V ♯(1), κ ∈ O×, p a¨ p a¨ p p p ¨ ♯ ∗ a˙ itp a˙ N ♯ Vp κ = −|a˙ a¨|p ˙ Vp (1). a¨ a¨N p ♯ Casselman’s theorem implies that Ψp corresponds to Vp of this form, and thus ~ −itp 2 has Rp(N) eigenvalue εp = −p , Tp(p) eigenvalue εp = χ(p), both Tp(p, 1) −1 and Tp(1,p) = Rp(N~ ) Tp(p, 1) Rp(N~ ) eigenvalues −εp, and hence ﬁnally by n n [p ] [p ] −nit the calculations of §2.3.2, Tp eigenvalues λp = p p . χ itp −itp 2π For p | N , πp ≃ π(˜χp| |p , | |p ), tp ∈ R/ log(p) Z, is a pre-unitary continu- χ ous series, and the new vector Vp corresponding to Ψp has the form,

1 a˙ ∗ +itp V χ κ =χ ˜ (˙aκ) a˙ 2 V χ(1), κ ∈ O×, p a¨ p a¨ p p p ∗ a˙ V χ κ = 0 . p a¨ n χ n −nitp χ n Therefore Ψp has Tp (p , 1) eigenvalues p 2 , all Tp (p ) eigenvalues 1, and n χ n +nitp hence Tp (1,p ) eigenvalues p 2 . × × For p | dB , πp is an irreducible representation of Dp , trivial on Rp , with unramiﬁed central characterχ ˜p. It is easy to see from this that πp is one- itp dimensional and given by the character |ν(·)|p , with i π 2π tp ∈ 2 logp(χ(p)) + log(p) Z/ log(p) Z. [pn] Ψp must be a multiple of this character, and therefore have Tp eigenvalues n [p ] −nitp JL λp = p . The representation πp of GL2(Qp) associated to πp by Jacquet– 1 1 2 +itp − 2 +itp Langlands is the special representation σ(| |p , | |p ). For v = ∞ and k = 0, we haveχ ˜∞ = 1, and π∞ is equivalent to the pre- δ∞ s∞ δ∞ −s∞ 0 unitary continuous series π(sgn | |∞ , sgn | |∞ ). V∞ corresponding to Ψ∞ has the form:

1 a˙ ∗ +s∞ V 0 κ = sgnδ∞ a˙ a˙ 2 V 0 (1), κ ∈ SO(2, R). ∞ a¨ a¨ a¨ ∞ ∞ − δ∞ ∆ 1 2 Thus Ψ∞ has T∞ eigenvalue (−1) , T∞ eigenvalue λ∞ = ( 4 − s∞) > 0, and T 0,ξ = T 0,ξ eigenvalues X∞ Y∞

λ0,ξ = (V 0 (β)/V 0 (1)) φ0,ξ (β) d× β ∞ ∞ ∞ X∞ ∞ + + ZR \B∞ 2π 1 2 +s∞ 1 2 2 −1 −2 = y π exp(−πξ(y + x + 1)y ) y dx dy dθ + ZR ZR Z0 − 1 −1 s −1 = 2 ξ 2 exp(−πξ(y + y )) y ∞ dy R+ 1 Z − 2 = 4 ξ Ks∞ (2πξ).

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.5. WHITTAKER MODELS 21

We are not considering |k| = 1, but we mention that in this case π∞ is 1+δ∞ s∞ δ∞ −s∞ equivalent to the continuous series π(sgn | |∞ , sgn | |∞ ). k s∞ −s∞ If |k| ≥ 2, π∞ is equivalent to the discrete series σ(sgn | |∞ , | |∞ ) for |k|−1 s∞ = 2 . Upon restriction to SL2(R), this representation decomposes as the direct sum of the weight-|k| holomorphic and anti-holomorphic discrete series, k and the vector V∞ corresponding to Ψ∞ is a lowest-weight vector in either the k ﬁrst or second of these, depending on whether k > 0 or k < 0. Thus V∞ has the form, a˙ ∗ |k| V k κ =χ ˜ (˙aκ) a˙ 2 V k (1), κ ∈ SO(2, R), ∞ a¨ ∞ a¨ ∞ ∞

∆ |k|(2−| k |) − ±k ∓k and Ψ∞ has T∞ eigenvalue λ∞ = 4 , while T∞V∞ = V∞ . We deﬁne T k,ξ Ψ = λk,ξ Ψ ,T −T k,ξΨ = λk,ξ Ψ . X∞ ∞ X∞ ∞ ∞ Y∞ ∞ Y∞ ∞ Our calculation is similar to that for k = 0, but we now use the mean value and conformal mapping properties of harmonic functions. These eigenvalues can also be computed by application of Selberg’s lemma on invariant integral operators.

2π k k,ξ |k| 1 y+1−ix y2+x2+1 −2 λ = y 2 exp −πξ y dx dy dθ X∞ π 2y1/2 y R R+ 0 Z Z Z 1−iz k −2 = 2 2 exp(−2πξ cosh dist(i,z)) y dx dy + ZR ZR y2+x2+1 −2 = 2exp −πξ y y dx dy + ZR ZR − 1 −1 = 4 ξ 2 K 1 (2πξ) = 2 ξ e(iξ), − 2

2π k k,ξ |k| 1 y−1−ix y2+x2+1 −2 λ = y 2 exp −πξ y dx dy dθ Y∞ π 2y1/2 y R R+ 0 Z Z Z −1−iz k −2 = 2 2 exp(−2πξ cosh dist(i,z)) y dx dy = 0. + ZR ZR 2.5 Whittaker Models

× In this section, we restrict our attention to B = GL2 = G (i.e. dB = 1) and O(1, (1,M)), M square-free. Let e∞(x) = e(x) = exp(2πix) for x ∈ R, −1 and deﬁne the character ep of Qp, with kernel Zp, on x ∈ Z[p ] ⊂ Qp by ep(x)=e(−x). Thus eA = v ev is trivial on Q and unramiﬁed. 2 Any F ∈ L (GQ\GA, χ˜) has the Fourier–Whittaker series expansion, 0 Q ξ F (g) = F g , 1 × ξX∈Q b 1 x F (g) := F g e (x) dx. 1 A ZQ\A b

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 22 CHAPTER 2. AUTOMORPHIC FORMS

We will see in §4.2.1 that

2 a −1 hF, F i = F |a|A da. × 1 ZA

b Deﬁne the global Whittaker model W = W(π) ≃ π as the image of π under F 7→ F . Then W = ⊗˜ vWv, where all Wv ∈ Wv ≃ πv are smooth functions on

GQv such that b 1 x W g = e (x) W (g), x ∈ Q . v 1 v v v We will explicitly compute those Wv having the same right Kv types as forms in ♭ S˜k(1, (1,M), χ,˜ M ). By the uniqueness of Wv proved in [17], it suffices for us to ⋆ ⋆ write down candidates Wv and check that they are Hv eigenvectors. Of course we could solve for these functions directly, by working backwards through the calculations below, but we find it easier to ‘guess’ them based on results of §3.2. An important feature of the Whittaker models is that they endow forms on GL2 with an arithmetic normalization distinct from the spectral one. An ⋆ ⋆ eigenform F is called Hecke normalized if F = v Wv . The Wv have been ⋆ ⋆ scaled so that Wp (1) = 1 and W∞ has prescribed asymptotics in the cusp. We ˇ ⋆ ∨ Q denote by Wv ∈ Wv the Whittaker functionsb of F , −1 Wˇ ⋆(g) = W ⋆ (ǫg), ǫ = . v v 1 0 For p ∤ M, we define Wp by

a˙ s˙ s¨ s¨ s˙ 1 W 0 = |ap˙ | |a¨| −|ap˙ | |a¨| a˙ 2 I a˙ , p a¨ |p|s˙ −|p|s¨ a¨ Zp a¨ and check that a ap ax a T (p, 1) W 0 = W 0 + W 0 p p 1 p 1 p p x modX p |ap2|s˙ −|ap2|s¨ 1 2 = ep(ax) |p|s˙ −|p|s¨ |ap| IZp (ap) x modX p s˙ s¨ s¨ s˙ 1 |ap| |p| −|ap| |p| a 2 a + |p|s˙ −|p|s¨ p IZp p p−s˙ |ap|s˙ −p−s¨ |ap|s¨ 1 1 2 2 = p−s˙ −p−s¨p p |a| IZp (a) p−s¨ |ap|s˙ −p−s˙ |ap|s¨ 1 1 2 2 + p−s˙ −p−s¨ p |a| IZp (a) a = λ W 0 . p p 1 Recall that the case of p | M ♭ is essentially the same as the previous one, but ˙ 0 we must consider a two-dimensional space of oldvectors. Taking Wp as above,

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.5. WHITTAKER MODELS 23

˙ 0 we have already shown Wp ∈ Wp , and we simply deﬁne

1 W¨ 0 = W˙ 0 = R (1,p) W˙ 0, p p p p p ♭ ˙ 0 ¨ 0 2 Wp = Wp + εp W p , εp = χ(p). For p | M ♯, we deﬁne

a˙ W ♯ = |a˙ a¨|itp a˙ I a˙ , p a¨ p a¨ p Zp a¨ a˙ W ♯ = −|a˙ a¨|itp ap˙ I ap˙ , p a¨ p a¨ p Zp a¨ and check a a R (1,p)[W ♯] = W ♯ p p 1 p p a = −p−itp W ♯ , p 1 a a R (1,p)[W ♯] = p−2itp W ♯ p p p p 1 a = −p−itp W ♯ , p p

a ap ax T (p, 1)[W ♯] = W ♯ p p 1 p 1 x modX p 1+itp = ep(ax) |ap|p IZp (ap) x modX p 1+itp = |ap|p p IZp (a) a = p−itp W ♯ , p 1

a a T (p, 1)[W ♯] = W ♯ p p 1 p p x x modX p a 1 a ap − 1 = W ♯ + W ♯ x x p p p 1 1 p x xX6≡0 1+itp a 1+itp = −p |ap|p IZp (a)+ ep( x ) |ap|p IZp (ap) xX6≡0 1+itp 1+itp = −p |ap|p IZp (a)+ |ap|p p IZp (a) − IZp (ap) a = −|ap|1+itp I (ap) = p−itp W ♯ . p Zp p 1

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 24 CHAPTER 2. AUTOMORPHIC FORMS

For p | M χ, we deﬁne

1 a˙ +itp W χ =χ ˜ (˙a) a˙ 2 I a˙ , p a¨ p a¨ p Zp a¨ 1 a˙ 2 −itp W χ = G χ˜ (¨ a) ap˙ 2 I ap˙ , p a¨ p p a¨ p Zp a¨ Gp = χ˜p( x)ep (x) dx, 1 Z× Z p p and check that a ap ax T χ(p, 1)[W χ] = W χ χ˜ (p) p p 1 p 1 p x modX p 1 2 +itp = ep(ax)˜χp(a) |ap|p IZp (ap) x modX p 1 +itp 1 −it χ a =χ ˜ (a) |ap| 2 p I (a) = p 2 p W , p p Zp p 1

a a T χ(p, 1)[W χ] = W χ χ˜ (p) p p 1 p p x p x modX p a 1 a ap − 1 = W χ 1 + W χ x x 1 p p p p 1 1 p x p xX6≡0 1 1 2 −itp a a 2 +itp = Gp |ap|p IZp (a)+ ep( x )˜χp(− x ) |ap|p IZp (ap) xX6≡0 1 1 2 −itp −itp χ a = Gp |ap|p IZ (a)+ Gp I × (ap) = p 2 W . p Zp p 1 For v = ∞, recall that the classical Whittaker function wκ,µ uniquely solves

1 κ 1 − µ2 w′′ (y)+ − + + 4 w (y) = 0, κ,µ 4 y y2 κ,µ κ − y wκ,µ(y) ∼ y e 2 as y → ∞. In particular,

µ+ 1 − y ∞ y 2 e 2 −yt µ−κ− 1 µ+κ− 1 wκ,µ(y) = e t 2 (1 + t) 2 dt, Γ(µ − κ + 1 ) 2 Z0 1 k y y 2 y − w0,µ(y) = Kµ( ), w k k−1 (y) = y 2 e 2 . π 2 2 , 2

0 If k = 0, deﬁne W∞ ∈ W∞ by a˙ W 0 = sgnδ∞ a˙ w a˙ . ∞ a¨ a¨ 0,s∞ a¨

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 2.5. WHITTAKER MODELS 25

This has moderate growth and satisﬁes

− 0 δ∞ 0 T∞[W∞] = (−1) W∞ , ∆ 0 1 2 0 T∞[W∞] = ( 4 − s∞) W∞ .

k If |k|≥ 2, deﬁne W∞ ∈ W∞ by

k a˙ a˙ a˙ W∞ = IkR+ a¨ w |k| |k|−1 a¨ , a¨ 2 , 2 which has moderate growth and satisﬁes

∓ k Rk [W∞]= 0, ±k ≥ 2 , ∆ k |k|(2−|k|) k T∞[W∞] = 4 W∞ .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 26 CHAPTER 2. AUTOMORPHIC FORMS

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Chapter 3

Theta Lifting

3.1 Basic Setup 3.1.1 Groups 3 Following [8], let G = GL2 , H = GSp6 , G = G ∩ H as linear algebraic groups over Q. The similitude character of H and its restriction to G will be denoted ν (and distinguished by context from the reduced norm of B). Also consider B, B×, H′ = GO(B, ν), G′ = H′3 ∩ GO(B3, ⊕3ν) as linear algebraic groups over Q, again denoting the similitude characters by ν. Write H˘ ′ for the connected component of the identity in H′, and deﬁne ρ : B× × B× → H˘ ′ by ρ(α, ˙ α¨)(β) =αβ ˙ α¨−1. There is an exact sequence

diag ρ 1 −→ Z(B×) −−−→ (B× × B×) −→ H˘ ′ −→ 1.

′ We may view the anti-involution ι as an element of order two in HQ, ι(β) = βι. It and H˘ ′ generate H′ = H˘ ′ ⋊ hιi as a semi-direct product, ρ(α, ˙ α¨) ι = α¨ α˙ ιρ ν(¨α) , ν(˙α) . Note that ∼ v(Z/2Z) −→ hιiA, δv Q (δv) 7−→ (ιv ). To compute inner-products on H′, we will use the parametrization

ρ : PB× × PB× −→∼ P H˘ ′.

We will also need Shimizu’s parametrization,

B× ⋊ PB× −→∼ H˘ ′, B(δ) × PB× −−→1-1 H˘ ′(δ), (β,˙ β¨) 7−→ ρ(β˙β,¨ β¨).

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 28 CHAPTER 3. THETA LIFTING

3.1.2 Measures We now describe the canonical Tamagawa measures on each of these groups, as in [42, 40]. Let dx = dxv be the Haar measure on Qv such that the Fourier transform is self-dual (vol(R/Z) = vol(Zp) = 1):

Fϕ(y) = ϕ(y)ev(xy) dx, ϕ ∈ S(Qv), ZQv = F −1ϕ(−y).

× ζv (1) × + Z × × Then deﬁne d xv = dxv, so that vol (R /e ) = vol (Z )=1. |x|v p ∗ Identify Bv with its algebraic dual Bv using the non-degenerate symmetric bilinear form hα, βi = tr(αβι) = αβι + βαι. Now choose the Haar measure dα = dαv on Bv so that the ‘ι-twisted’ Fourier transform is self-dual:

Fϕ(β) = ϕ(α)ev (hα, βi) dα, ϕ ∈ S(Bv), ZBv = F −1ϕ(−β).

1 If v ∤ dB , dα = ij dαij , while for p | dB , vol(Rp) = p . It follows from ~ calculating vol(B∞Q/O(dB , N)) = dBN that vol(BA/BQ)=1. × ζv (1) As before, deﬁne d αv = dαv. It is well known and follows from the |α|v calculations in §2.3 that

−1 (p − 1) p | dB, × × ~ −1 −1 vol (Op (dB , N)) = ζp(2) (p + 1) p | N,  1 p ∤ dBN.

× × ×  On B∞, d α∞ = d a1 d a2 dx dθ in terms of the coordinates a 1 x α = 1 κ . ∞ a 1 θ 2 × × We will also write d αv for the measure on PBv compatible with the exact sequence × × × 1 −→ Qv −→ Bv −→ PBv −→ 1, (1) (1) and d αv for the measure on Bv compatible with

(1) × ν × 1 −→ Bv −→ Bv −→ Qv −→ 1. To be explicit, × × ~ × × ~ vol (P Op (dB, N)) = vol (Op (dB , N)),

× × × while d α∞ = d a dx dθ in the PB∞ coordinates a 1 x α = κ , 0 ≤ θ<π. ∞ 1 1 θ

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 3.1. BASIC SETUP 29

Also (1) (1) ~ × × ~ vol (Op (dB , N)) = vol (Op (dB, N)),

(1) × (1) while d α∞ = d a dx dθ in the B∞ coordinates

a 1 x α = κ , 0 ≤ θ <π. ∞ a−1 1 θ × × × (1) (1) (1) It is well known (see [40]) that vol (PBQ \PBA ) = 2 and vol (BQ \BA )=1. × (1) × The diﬀerence between d α∞ and d α∞ may be seen as follows: Lift d α∞ to + × × 1 × the double cover R \B∞ of PB∞. Then 2 d α∞ is compatible with

(1) + × ν + × 1 −→ B∞ −→ R \B∞ −→ R \R −→ 1.

(1) The Tamagawa measures on G = GL2, P G = PGL2, G = SL2 are special cases of those above, for dB = 1. Now deﬁne dgv on Gv to be compatible with

(1) 3 ν × 1 −→ (Gv ) −→ Gv −→ Qv −→ 1.

Explicitly, × ~ 3 × × ~ 3 vol(Op (1, N) ∩ Gp) = vol (Op (1, N)) , × × and dg∞ = d a j (d aj dxj dθj ) in the G∞ coordinates Q a a 1 x g = j j κ , 0 ≤ θ < π. ∞j 1 a−1 1 θj j j

Also deﬁne dgv on P Gv to be compatible with

× 1 −→ Qv −→ Gv −→ P Gv −→ 1.

Then × ~ 3 × × ~ 3 vol(P (Op (1, N) ∩ Gp)) = vol (Op (1, N)) , × and dg∞ =2 j (d aj dxj dθj ) in the P G∞ coordinates Q ǫ a 1 x g = j j κ , ∞j 1 a−1 1 θj j 2 ǫ =1, a1a2a3 > 0, 0 ≤ θj < π.

It is easy to check that vol(P GQ\P GA)=2.

3.1.3 Dual-Pairs

n Extend the deﬁnition of h· , ·i to Bv as an orthogonal direct sum, and deﬁne F, n dαv on Bv as before. We may write Fj for the Fourier transform in βj alone,

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 30 CHAPTER 3. THETA LIFTING

n n β = (βj ) ∈ Bv . The Weil representation ω of Sp2n(Qv) on S(Bv ) is uniquely determined by: [17, 34]

1 U ω n ϕ(β) = e ( 1 hUβ,βi) ϕ(β), U = U t ∈ M (Q ), 1 v 2 n v n At ω ϕ(β) = | det A|2 ϕ(Aβ), A ∈ GL (Q ), A−1 v n v 1n nI ω ϕ(β) = (−1) v|dB Fϕ(β). −1 n Now consider the similitude dual-pairs

R(X, Y ) = {(x, y) ∈ X × Y ; ν(x)= ν(y)}, (n,X,Y ) = (1, G, H′), (3, G, G′), (3,H,H′).

We extend the deﬁnition of ω to each of these as follows: Let L denote the n unitary left regular representation of Yv on S(Bv ),

′ ′ −1 ′ −1 L(h ) ϕ(β) = |ν(h )|v ϕ (h ) β , ′ ′ −3 ′ −1 L(g ) ϕ(β) = |ν(g )|v ϕ(hj ) βj , ′ ′ −3 ′ −1 L(h ) ϕ(β) = |ν(h )|v ϕ (h ) βj . 1 If ν(x, y)= δ, set α = n ∈ X , x(1) = xα−1, (1)x = α−1x. Then δ 1 v n ω(x, y) = ω(x(1)) L(y) = L(y) ω((1)x)

n deﬁnes a representation of R(X, Y )v on S(Bv ). z 1 Note ω(z,z)=1 for z ∈ Q×, since ω n = L(z)−1. v z−1 1 n

3.2 Jacquet–Langlands Correspondence

In this section we explicitly compute the Shimizu theta lift and it’s adjoint, which realize the Jacquet–Langlands correspondence.

3.2.1 Shimizu’s Theta Lift

˙ ¨ ˘′ 2 ˘ ′ ˘ ′ For Ψ, Ψ as in §2.2 and η ∈ v(Z/2Z), deﬁne F ∈ L0(HQ\HA, χ˜) by Q F˘′(ρ(β,˙ β¨)) = Ψ˙ η(β˙) Ψ¨ η(β¨),

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 3.2. JACQUET–LANGLANDS CORRESPONDENCE 31

with matching local dataϕ ˘ ∈ S(BA) as in §3.3. Shimizu’s theta lift is deﬁned (δ) in [34]: For g ∈ GA ,

˘ ′ ˘′ 1 ′ ˘′ ′ ′ Θϕ˘(F )(g) = 2 ω(g,h )˘ϕ(α) F (h ) dh ˘ ′(1) ˘ ′(δ) HQ \HA Z αX∈BQ = 1 ω(g,ρ(β˙β,¨ β¨))ϕ ˘(α) 2 × × (1) (δ) P BQ \P BA BQ \BA Z Z αX∈BQ · Ψ˙ η(β˙β¨)Ψ¨ η(β¨) d(1)β˙ d×β¨

= 1 |δ|−1ω((1)g)˘ϕ((β˙β¨)−1αβ¨) 2 × × (1) (δ) A P BQ \P BA BQ \BA Z Z αX∈BQ · Ψ˙ η(β˙β¨)Ψ¨ η(β¨) d(1)β˙ d×β.¨

This expression is readily converted into a Fourier series, following [34]. First, (1) (1) × × decompose the sum over BQ into left BQ orbits. Note that BQ \BQ ≃ Q by (0) ν, and if BQ is a division algebra, BQ = {0}. In the unramiﬁed case B = M , the remaining elements are those α ∈ MQ of rank 1, and every such can be QQ written as α = γ−1α,˜ for representatives of unique classesα ˜ ∈ P , 0 0 γ ∈ NQ\SL2(Q). ˘ ′ ˘′ The contribution to Θϕ˘(F ) from α = 0 is proportional to

Ψ˙ η(β˙β¨) d(1)β˙ Ψ¨ η(β¨) d×β¨ = 0, P B×\P B× B(1) \B(δ) Z Q A Z Q A ! sinceπ ˙ is not one-dimensional, by the argument in [34]. For B = M (which is (0) not considered in [34], but see [36]), the contribution from all α ∈ MQ r {0} is

1 2 (1) (δ) PGQ\PGA G \G Z Z Q A α˜ γ∈N \G(1) X XQ Q −1 (1) ˙ ¨ −1 ¨ ˙ η ˙ ¨ ¨ η ¨ (1) ˙ ×¨ |δ|A ω( g)˘ϕ((γββ) α˜β) Ψ (ββ)Ψ (β) d β d β

= 1 2 (δ) PGQ\PGA NQ\GA Xα˜ Z Z −1 (1) ˙ ¨ −1 ¨ ˙ η ˙ ¨ ¨ η ¨ (1) ˙ ×¨ |δ|A ω( g)˘ϕ((ββ) α˜β) Ψ (ββ)Ψ (β) d β d β

= 1 |δ|−1ω((1)g)˘ϕ((β˙β¨)−1α˜β¨) 2 (δ) A α˜ PGQ\PGA NA\GA X Z Z · Ψ˙ η(nβ˙β¨) dn Ψ¨ η(β¨) d(1)β˙ d×β¨ = 0, ZNQ\NA !

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 32 CHAPTER 3. THETA LIFTING sinceπ ˙ is cuspidal. Therefore in all cases under consideration,

˘ ′ ˘′ 1 Θϕ˘(F )(g) = 2 P B×\P B× B(1)\B(δ) Z Q A Z Q A α˜∈B(1)\B× γ∈B(1) XQ Q XQ ω(g,ρ(γβ˙β,¨ β¨))ϕ ˘(˜α) Ψ˙ η(β˙β¨)Ψ¨ η(β¨) d(1)β˙ d×β¨

1 = 2 P B×\P B× B(δ) (1) × Z Q A Z A α˜∈BXQ \BQ ω(g,ρ(β˙β,¨ β¨))ϕ ˘(˜α) Ψ˙ η(β˙β¨)Ψ¨ η(β¨) d(1)β˙ d×β¨

1 ν(˜α) ι = 2 ω , ρ(˜α , 1) ◦ P B×\P B× B(δ) 1 (1) × Z Q A Z A α˜∈BXQ \BQ ω(g,ρ(β˙β,¨ β¨))ϕ ˘(1) Ψ˙ η(β˙β¨)Ψ¨ η(β¨) d(1)β˙ d×β¨ ξ = Θ˘\′ (F˘′) g , where ϕ˘ 1 × ξX∈Q

˘\′ ˘′ 1 ˙ ¨ ¨ Θϕ˘(F )(g) := 2 ω(g,ρ(ββ, β))ϕ ˘(1) P B×\P B× B(δ) Z Q A Z A ·Ψ˙ η(β˙β¨)Ψ¨ η(β¨) d(1)β˙ d×β¨

= 1 ω(g,ρ(β¨β,˙ β¨))ϕ ˘(1) 2 × × (δ) ZP BQ \P BA ZBA ·Ψ˙ η(β¨β˙)Ψ¨ η(β¨) d(1)β˙ d×β¨

1 ˜ η ¨ η × = 2 Ψ (g,β)Ψ (β) d β, P B×\P B× Z Q A

Ψ˜ η(g,β) := ω(g,ρ(α, 1))˘ϕ(1) Ψ˙ η(βα) d(1)α. B(δ) Z A At this point our proof diverges from [34]. Using the calculations in §3.3, we realize the last integral as the action of Hecke operators on Ψ˙ η and show Ψ˜ η(g,β) = F (g) Ψ˙ η(β),

⋆ where F is the unique Hecke normalized Hb eigenform ˜ ♭ JL F ∈ S|k|(1, (1, dBN), χ,N˜ ) ∩ π (π ˙ ) s.t. ♭ Rp(1, dBN) F =ε ˙p F for all p | N , and so ˘ ′ ˘′ ˙ η ¨ η ˙ ¨ Θϕ˘(F ) = hΨ , Ψ i F = hΨ, Ψi F.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 3.2. JACQUET–LANGLANDS CORRESPONDENCE 33

It is clear from the deﬁnition of ω that 1 x Ψ˜ η g,β = e (x) Ψ˜ η(g,β), x ∈ A, 1 A and since ω(z,ρ(z, 1)) = 1,

Ψ˜ η(zg,β) =χ ˜(z) Ψ˜ η(g,β), z ∈ A×.

η Furthermore, the right Kv-types of Ψ˜ (g,β) are determined by Lemma 3 in §3.3. The remainder of our calculation consists identifying the Whittaker functions ♭ ♭ ♯ ♯ χ χ from §2.5 (with M = dB N, M = N , M = dB N , M = N ). a If p ∤ d N, it suﬃces to consider g = : B 1 ι ω (g,ρ(α, 1))ϕ ˘p(1) = |a|p ϕ˘p(α ) = |a|p ϕp(α),

˜ η ˙ η (1) Ψp (g,β) = |a|p ϕp(α) Ψp(βα) dp α (a) ZBp η × = |a|p ϕp(α) Ψ˙ (βα) d α [a] p p ZBp [a] ˙ η = |a|p Tp Ψp(β)

1 s˙p s¨p 2 |ap|p −|ap|p ˙ η = |a|p s˙ s¨ IZ (a) Ψ (β) |p| p −|p| p p p p p 0 ˙ η = Wp (g) Ψp(β).

˜ η [a] ˙ η Ψp(g,β) = |a|p Tp Ψp(β)

1 s˙p s¨p 2 |ap|p −|ap|p ˙ η = |a|p s˙ s¨ IZ (a) Ψ (β) |p| p −|p| p p p p p 1 s˙p s¨p a 2 |a|p −|a|p a ~ ˙ η + s˙ s¨ IZ Rp(N)Ψ (β) p p |p| p −|p| p p p p p p ˙ 0 ¨ 0 ˙ η = Wp (g)+ ε˙ p W p (g) Ψp(β),

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 34 CHAPTER 3. THETA LIFTING

′ ι ω(g ,ρ(α, 1))˘ϕp(1) = |a|p Fϕ˘p(α ) = |a|p Fϕp(α),

˜ η ′ ˙ η × Ψp(g ,β) = |a|p Fϕp(α) Ψp(βα) dp α [a] ZBp ~ −1 [ap] ˙ η = |ap|p Rp(N) Tp Ψp(β)

1 2 s˙p 2 s¨p 2 |ap |p −|ap |p ~ −1 ˙ η = |ap|p s˙ s¨ IZ (ap) Rp(N) Ψ (β) |p| p −|p| p p p p p 1 s˙p s¨p 2 |ap|p −|ap|p ˙ η +|a|p s˙ s¨ IZ (a) Ψ (β) |p| p −|p| p p p p p ˙ 0 ′ ¨ 0 ′ ˙ η = Wp (g )+ ε˙p Wp (g ) Ψp(β).

If p | N ♯,

ι ω(g,ρ(α, 1))˘ϕp(1) = |a|p ϕ˘p(α ) = |a|p ϕp(α),

˜ η [a] ˙ η Ψp(g,β) = |a|p Tp Ψp(β) 1+ıtp ˙ η = |a|p IZp (a) Ψp(β) ♯ ˙ η = Wp (g) Ψp(β),

′ ι ω(g ,ρ(α, 1))˘ϕp(1) = |a|p Fϕ˘p(α ) = |a|p Fϕp(α),

˜ η ′ ~ −1 [ap] ˙ η Ψp(g ,β) = |ap|p Rp(N) Tp Ψp(β) ıtp ˙ η = −|a|p |ap|p IZp (ap) Ψp(β) ♯ ′ ˙ η = Wp (g ) Ψp(β).

If p | N χ,

ι 1−ηp ω(g,ρ(α, 1))˘ϕp(1) = |a|p ϕ˘p(α ) = |a|p ϕp(ι α),

˜ η χ ˙ ηp Ψp(g,β) = |a|p χ˜p(a) IZp (a) Tp (a, 1) Ψp (β) 1 +ıt 2 p ˙ η =χ ˜p(a) |a|p IZp(a) Ψp(β) χ ˙ η = Wp (g) Ψp(β),

′ ι 1−ηp ω(g ,ρ(α, 1))˘ϕp(1) = |a|p Fϕ˘p(α ) = |a|p Fϕp(ι α),

˜ η ′ 2 χ −1 ˙ ηp Ψp(g ,β) = |ap |p Gp χ˜p(−1) IZp (ap) Tp (p ,ap) Ψp (β), 1 −ıt 2 2 p ˙ η = Gp χ˜p(−1) |ap |p IZp (ap) Ψp(β) χ ′ ˙ η = Wp (g ) Ψp(β).

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 3.2. JACQUET–LANGLANDS CORRESPONDENCE 35

If p | dB,

ι ω(g,ρ(α, 1))˘ϕp(1) = |a|p ϕ˘p(α ) = |a|p ϕp(α),

˜ η [a] ˙ η Ψp(g,β) = |a|p Tp Ψp(β) 1+ıtp ˙ η = |a|p IZp (a) Ψp(β) ♯ ˙ η = Wp (g) Ψp(β),

′ ι ω(g ,ρ(α, 1))˘ϕp(1) = −|a|p Fϕ˘p(α ) = −|a|p Fϕp(α),

˜ η ′ −1 [ap] ˙ η Ψp(g ,β) = −|ap|p Tp(̟) Tp Ψp(β) ıtp ˙ η = −|a|p |ap|p IZp (ap) Ψp(β) ♯ ′ ˙ η = Wp (g ) Ψp(β).

±a If v = ∞, we consider g± = , a ∈ R+ : 1 ± ι ω(g ,ρ(α, 1))˘ϕ∞(1) = a ϕ˘∞(α ),

η + 1+ |k| −k,a˘ ι η (1) Ψ˜ (g ,β) = a 2 φ (α )Ψ (βα) d α ∞ (a) X∞ ∞ ∞ ZB∞ 1+ |k| k,a˘ η 1 × = a 2 φ (α)Ψ (βα) d α X∞ ∞ 2 ∞ + + ZR \B∞ 1 1+ |k| k,a˘ η = a 2 T Ψ˙ (β) 2 X∞ ∞ |k| + ˙ η = W∞ (g ) Ψ∞(β),

η − 1 1+ |k| − k,a˘ η Ψ˜ (g ,β) = a 2 T T Ψ (β) ∞ 2 ∞ Y∞ ∞ |k| − ˙ η = W∞ (g ) Ψ∞(β).

3.2.2 Adjoint of Shimizu’s Lift

˙ ¨ ′ ′(δ) Let Ψ = Ψ= Ψ, F be as in §3.2, and for h ∈ HA deﬁne

′ ′ Θϕ F (h ) = ω(g,h )ϕ(α) F (g) dg. G(1)\G(δ) Z Q A α∈BQ X Theorem 1.

2 kF k ′ 2 ′ ′ Θϕ F = kΨk2 F ∈ L0(HQ\HA, χ˜), F′(ρ(β,˙ β¨)ιη ) = Ψη(β˙) Ψη(β¨).

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 36 CHAPTER 3. THETA LIFTING

Proof: Since the theta kernel is smooth, automorphic, and has moderate growth, it follows from Lemma 3 that

η Θϕ F (ρ(β,˙ β¨)ι ) = Θϕ˘ F (ρ(β,˙ β¨)) ˜η ~ ˜η ~ ∈ Ak(dB, N, χ˜) × Ak(dB , N, χ˜).

Furthermore by Lemma 4, for p ∤ dB N, [p] ˙ [p] ¨[p] ∨ Θϕ˘ Tp F = Tp Θϕ˘ F = Tp Θϕ˘ F , × and hence by strong multiplicity-one on BA , ∨ Θϕ˘ F ◦ ρ ∈ π × π = π × π.

Thus by Lemma 1, Θϕ˘ F is determined up to scaling. We compute its normalization using the adjoint identity below (compare with the similitude see-saw identity in [9]). Lemma 2. ˘ ′ ˘′ ˘′ Θϕ˘ F , F = F , Θϕ˘ F . PG P H˘ ′ Proof: D E D E

1 1 ′ ˘′ ′ (1) ′ × LHS = 2 2 ϑ(g,h , ϕ˘)F (h )F (g) d h d g G A×\G ˘ ′(1) \H˘ ′(δ) Z Q A ZHQ A 1 1 ′ ˘′ ′ (1) ′ × = 2 2 ϑ(g,h , ϕ˘)F (h )F (g) d h d g G R+\G H˘ ′(1) \H˘ ′(δ) Z Q A Z Q A 1 ′ ˘′ ′ (1) ′ (1) × = 2 ϑ(g,h , ϕ˘)F (h )F (g) d h d g d δ, Q×R+\A× G(1)\G(δ) ˘ ′(1) \H˘ ′(δ) Z Z Q A ZHQ A RHS = the same symmetric expression.

3.3 Local Data

With the notation of §3.2, deﬁne ϕ, ϕ˘ ∈ S(BA) by

χ 1 1 p ∤ N , ϕp(β) = × × IOp (β) χ vol (Op ) χ˜ (β ) I × (β ) p | N , ( p 11 Zp 11 1 k ϕ∞(β) = π X (β) e(iP (β)),

ηv ηv ϕ˘v(β) = ω(1, ιv )ϕv(β) = ϕv(ιv β), i.e. 1 p ∤ N χ, 1 χ × ϕ˘p(β) = × × IOp (β) χ˜p(β11) IZ (β11) p | N , ηp =0, vol (Op )  p χ χ˜ (β ) I × (β ) p | N , η =1,  p 22 Zp 22 p ˘ 1 k  ˘ η∞ ϕ˘∞(β) = π X (β) e(iP (β)), k = (−1) k.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 3.3. LOCAL DATA 37

Now we compute the Fourier transforms of these functions in each case, and × relate them to Hecke operators for Bv . a b N˙ If p ∤ d N χ, writing β = , α = , B c d N~ N¨ 1 ϕ˘p(β) = × × IZp (a) I ˙ (b) I ¨ (c) IZp (d), vol (Op ) NZp NZp [pn] ϕ˘p × = Tp , Bp n≥0 1 ˙ 1 ¨ 1 Fϕ˘p (β) = P × × IZp (d) |N|pI Z (c) |N|pI Z (b) IZp (a), vol (Op ) N˙ p N¨ p 1 ¨ ˙ = × × |N|p IZp (a) IZp (Nb) IZp (Nc) IZp (d), vol (Op )

ϕ˘p(β) p ∤ dBN, = 1 ♭ ♯ ( p ϕ˘p(αN~ β) p | N N ,

ϕ˘p × p ∤ dB N, Bp Fϕ˘p × = Bp 1 ~ −1 ♭ ♯ ( p R p(N) ϕ˘p B× p | N N . p χ If p | N , (note ιp and F commute) 1 × ϕp(β) = × × χ˜p(a)I (a) I ˙ (b) I ¨ (c) IZp (d) , vol (Op ) Zp NZp NZp χ 2,2 ϕp × = (˜χp ◦ ν) · φ n , Bp n≥0 Tp(1,p ) 1 1 × Fϕp(β) = P × × 2 Gp χ˜p(−d) I 1 (d) I 1 N¨Z (c) I 1 N˙ Z (b) IZp (a) , vol (Op ) p p Zp p p p p 1 χ 2,2 Fϕp × = 2 Gp χ˜p(−1) φ n −1 . Bp p n≥0 Tp(p ,p ) ∗ P −1 If p | d B, Rp := {α ∈ Dp ; ∀β ∈ Rp , tr(αβ) ∈ Zp} = ̟p Rp , 1 ϕ˘p = × IR , vol (Rp) p [pn] ϕ˘p × = Tp , Bp n≥0 1 1 1 1 −ω(J)˘ϕp = F ϕ˘p = P × I(R∗)ι = × I −1 , vol (Rp) p p vol (Rp) p ̟p Rp 1 −1 Fϕ˘p × = Tp(̟p) ϕ˘p × . Bp p Bp If v = ∞,

|k| −k,ν˘ 2 ϕ˘∞ + = ν · φ , B∞ X∞ |k| −k,ν˘ − 2 T ϕ˘∞ − = ν · φ , ∞ B∞ Y∞

0 1 Dω = 2πı (XX − Y Y )· , 0 0 0 0 −Dω = 1 ∂ ∂ − ∂ ∂ , 1 0 2πı ∂X ∂X ∂Y ∂Y Dω(J)ϕ ˘∞ = ı|k| ϕ˘∞, ı|k|θ =⇒ ω(κθ)ϕ ˘∞ = e ϕ˘∞.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 38 CHAPTER 3. THETA LIFTING

Lemma 3. For κ ∈ Kp(1, (1, dBN)) and κ,˙ κ¨ ∈ Kp(dB, N~ ) such that det κ = ν(κ ˙ )/ν(¨κ) , (Note χ˜p has diﬀerent deﬁnitions on these two groups.)

ηp ηp ω(κ,ρ(κ, ˙ κ¨))ϕ ˘p =χ ˜p(κ) χ˜p(ιp κ˙ )˜χp(ιp κ¨)ϕ ˘p , ı(|k|θ−k˘θ˙+k˘θ¨) ω(κθ,ρ(κθ˙,κθ¨))ϕ ˘∞ = e ϕ˘∞ .

Proof: First consider diagonal κ ∈ Kp(1, (1, dBN)) : −1 ω(κ,ρ(κ, ˙ κ¨))ϕ ˘p(β) =ϕ ˘p(κ11κ˙ βκ¨)

ηp ηp =χ ˜p(κ) χ˜p(ιp κ˙ )˜χp(ιp κ¨)ϕ ˘p . Then it suﬃces to check 1 x ω ϕ˘ =ϕ ˘ for x ∈ Z , 1 p p p 1 x ω Fϕ˘ = Fϕ˘ for x ∈ d NZ . 1 p p B p The archimedean statement follows from previous calculations. 1 Lemma 4. For p ∤ d N, α = , B p −1 (1) −1 (1) ω(ακα )ϕp(β) dp κ = ω(α, ρ(κ ˙ α, 1))ϕp(β) dp κ˙ (1) ~ ZSL2(Zp) ZKp (dB ,N) −1 (1) = ω(α, ρ(1, κα¨ ))ϕp(β) dp κ¨ . (1) ~ ZKp (dB ,N) −1 (1) 1 Proof: Since ω(ακα ) ϕp = ϕp for κ ∈ Kp (1, (p, p )), the κ integral re- (1) (1) duces to an average over the Kp (1, (1, 1)) Kp (1, (p, 1)) coset representatives, 1 x 1 (x mod p), , 1 −1 and hence evaluates to ′ 1 2 ϕp(β) = p I (pZp) (β)+ p I ~ (β) . p+1 Op (dB,N~ ) p Op(dB ,N) Now theκ ˙ andκ ¨ integrals evaluate as:

−1 (1) p [p] p ϕp(α κβ˙ ) dp κ˙ = p+1 Tp ∗ IOp (β), (1) ~ ZKp (dB ,N) −1 (1) p [p] p ϕp(βκα¨ ) dp κ¨ = p+1 IOp ∗ Tp (β). (1) ~ ZKp (dB ,N) It follows from the calculations in §2.3.2 (extending by continuity to Bv) that [p] I × = I ∗ T (1) − T + pT (p) , Op Op p p p [p] [p] I ∗ T = I −I × + p I = T ∗ I Op p Op Op pOp p Op p+1 ′ Z = I (p p) + p IpOp = ϕp . Op p

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Chapter 4

L-functions

In this chapter we compute special values of adjoint and triple product L- functions. First we describe the relevant Langlands parameters (admissible representations of Weil or Weil-Deligne groups), which determine canonical L and ε factors. Then we compute special values of the global L-functions using the Rankin–Selberg and Garrett / Rallis–Piateski-Shapiro integral representations.

4.1 Local Langlands Correspondence

Let WF denote the Weil group of the local ﬁeld F (as in [39, 22, 32]) and rF the × ∼ ab × local class-ﬁeld theory isomorphism rF : F −→ WF . If χ is a character of F , r r denote by χ the corresponding character of WF . In particular, write kk = | |F . × Recall that WC = C is abelian and rC is the identity (so kzk = |z|C), and

× × 2 −1 WR = C ∪  C ,  = −1, z = z,

c (1) has closed commutator subgroup WR = C , with rR being deﬁned by

+ 1 (1) (1) R ∋ r 7→ r 2 C , −1 7→  C (so kzk = kzk = |z|C).

As described in [21] (with a normalization error that we have corrected), all 1 1 one-dimensional representations of WR are isomorphic to ̺ = (s,δ)R , for some s ∈ C, δ ∈{0, 1},

̺1(z) = kzks, ̺1() = (−1)δ,

2 2 and all irreducible two-dimensional semi-simple representations to ̺ = (s,l)R , for some s ∈ C, l ∈ Z+,

r2seilθ (−1)l ̺2(reiθ) = , ̺2() = . r2se−ilθ 1 39

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 40 CHAPTER 4. L-FUNCTIONS

Allowing l ∈ Z, we have

2 1 1 2 2 (s, 0)R ≃ (s, 0)R ⊕ (s, 1)R , (s,l)R ≃ (s, −l)R , and

1 1 1 (s1,δ1)R ⊗ (s2,δ2)R ≃ (s1 + s2,δ)R , δ ≡ δ1 + δ2 mod 2, 2 2 2 2 (s1,l1)R ⊗ (s2,l2)R ≃ (s1 + s2,l1 + l2)R ⊕ (s1 + s2,l1 − l2)R , 1 2 2 (s1,δ1)R ⊗ (s2,l2)R ≃ (s1 + s2,l2)R .

The standard local factors are deﬁned as follows:

s − 2 s ζR(s) = π Γ( 2 ), ζC(s) = ζR(s) ζR(s + 1) = 2(2π)−s Γ(s),

1 L(s,̺ ) = ζR(s + s∞ + δ∞), 1 C(t,̺ ) = (1+ |ıt + s∞|), 1 δ∞ ε(s,̺ , e∞) = ı ,

2 l∞ L(s,̺ ) = ζC(s + s∞ + 2 ), 2 l∞ 2 C(t,̺ ) = (1+ |ıt + s∞ + 2 |) , 2 l∞+1 ε(s,̺ , e∞) = ı .

W′ Now let Qp denote the Weil-Deligne group of Qp. We implicitly identify the W W′ characters of Qp and Qp . Recall from [39, 22] the indecomposable admissible spn W′ representation of Qp on C{e0,..., en−1} deﬁned by

n i n ei+1 i

Note (spn)∨ ≃kk1−n ⊗ spn. Since spn ≃ Symn−1sp2, it follows that

n−1 spm ⊗ spn ≃ kki ⊗ spm+n−2i−1, m ≥ n, i=0 M ⊗2sp2 ≃ kk1 ⊕ sp3, ⊗3sp2 ≃ ⊕2 kk1 ⊗ sp2 ⊕ sp4.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.1. LOCAL LANGLANDS CORRESPONDENCE 41

The standard local factors we need are deﬁned as:

−s −1 ζp(s) = 1 − p , ̺ = kksp ⊗ spn,

L(s,̺) = ζp(s + sp + n − 1), C(̺) = pn−1, n−2 ζp(−s − sp − i) −s−s − n−2 n−1 ε(s,̺, e ) = = − p p 2 . p ζ (s + s + i) i=0 p p Y × k Also consider a characterχ ˜p of Qp with ker(˜χp)=1+ p Zp for k > 0. It has these corresponding standard local factors:

′ r ̺ =χ ˜p , L(s,̺′) = 1, C(̺′) = pk,

′ −ks −1 ε(s,̺ , ep) = p χ˜p (x)ep(x) dx. −k × Zp Zp

4.1.1 Adjoint Parameters

Consider a representation πv of GL2(Qv) as in §2.4 . Below we list the corresponding Langlands parameters ̺ and their local factors, plus those of compo- sitions with the adjoint representation Ad : GL2(C) → GL3(C),

Ad̺ = Ad ◦ ̺ ≃ ̺ ⊗ ̺∨ ⊖ 1.

Archimedean

δ∞ s∞ δ∞ −s∞ If k = 0, then π∞ ≃ π(sgn | |∞ , sgn | |∞ ), δ∞ ∈{0, 1}, corresponds to

0 1 1 ̺ ≃ (s∞,δ∞)R ⊕ (−s∞,δ∞)R , 0 L(s,̺ ) = ζR(s + s∞ + δ∞) ζR(s − s∞ + δ∞), 0 C(t,̺ ) = (1+ |ıt + s∞|)(1+ |ıt − s∞|), 0 δ∞ ε(s,̺ , e∞) = (−1) ,

0 1 1 1 Ad̺ ≃ (2s∞, 0)R ⊕ (0, 0)R ⊕ (−2s∞, 0)R , 0 L(s, Ad̺ ) = ζR(s +2s∞) ζR(s) ζR(s − 2s∞), 0 C(t, Ad̺ ) = (1+ |ıt +2s∞|)(1+ |ıt|)(1+ |ıt − 2s∞|), 0 ε(s, Ad̺ , e∞) = 1.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 42 CHAPTER 4. L-FUNCTIONS

k s∞ −s∞ k−1 If k ≥ 2, then π∞ ≃ σ(sgn | |∞ , | |∞ ), s∞ = 2 , corresponds to

k 2 ̺ ≃ (0, k − 1)R , k L(s,̺ ) = ζC(s + s∞), k 2 C(t,̺ ) = (1+ |ıt + s∞|) , k k ε(s,̺ , e∞) = ı ,

k 2 1 Ad̺ ≃ (0, 2k − 2)R ⊕ (0, 1)R , k L(s, Ad̺ ) = ζC(s +2s∞) ζR(s + 1), k 2 C(t, Ad̺ ) = (1+ |ıt +2s∞|) (1 + |ıt|), k k ε(s, Ad̺ , e∞) = (−1) .

Non-Archimedean

♯ χ s˙p s¨p If p ∤ M M , then πp ≃ π(| |p , | |p ) corresponds to

̺0 ≃ kks˙p ⊕kks¨p , 0 L(s,̺ ) = ζp(s +s ˙p) ζp(s +¨sp), C(̺0) = 1, 0 ε(s,̺ , ep) = 1,

Ad̺0 ≃ kks˙p−s¨p ⊕kk0 ⊕kks¨p−s˙p , 0 L(s, Ad̺ ) = ζp(s +s ˙p − s¨p) ζp(s) ζp(s +¨sp − s˙p), C(Ad̺0) = 1, 0 ε(s, Ad̺ , ep) = 1.

♯ − 1 +ıt 2 ̺ ≃ kk 2 p ⊗ sp , ♯ 1 Lp(s,̺ ) = ζp(s + 2 + ıtp), C(̺♯) = p , 1 ♯ −ıtp −s+ εp(s,̺ , ep) = (−p )p 2 ,

Ad̺♯ ≃ kk−1 ⊗ sp3, ♯ L(s, Ad̺ ) = ζp(s + 1), C(Ad̺♯) = p2, ♯ −2s+1 ε(s, Ad̺ , ep) = p .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.1. LOCAL LANGLANDS CORRESPONDENCE 43

χ r ıtp −ıtp ̺ ≃ χ˜p kk ⊕kk , χ L(s,̺ ) = ζp(s − ıtp), C(̺χ) = p,

1 1 χ −ıtp− −s+ ε(s,̺ , ep) = p 2 χ˜p(x)ep(x) dx p 2 , 1 Z× Z p p

χ r 2ıtp 0 r −2ıtp Ad̺ ≃ χ˜p kk ⊕kk ⊕ χ˜p kk χ L(s, Ad̺ ) = ζp(s), C(Ad̺χ) = p2, χ −2s+1 ε(s, Ad̺ , ep) =χ ˜p(−1) p .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 44 CHAPTER 4. L-FUNCTIONS

4.1.2 Triple Product Parameters Archimedean

If k1 = k2 = k3 = 0, let δ ≡ δ1 + δ2 + δ3 (mod 2), δ ∈{0, 1}, so that

0 0 0 1 ̺ := ̺1 ⊗ ̺2 ⊗ ̺3 ≃ (±s1 ± s2 ± s3,δ)R , 3 M{±} L(s,̺) = ζR(s ± s1 ± s2 ± s3 + δ), 3 {±}Y C(t,̺) = (1 + |ıt ± s1 ± s2 ± s3|), 3 {±}Y ε(s,̺, e∞) = 1.

If k = |k1| = |k2|≥ 2 and k3 = 0,

k k 0 2 2 ̺ := ̺1 ⊗ ̺2 ⊗ ̺3 ≃ (s3, 2k − 2)R ⊕ (s3, 0)R 2 2 ⊕(−s3, 2k − 2)R ⊕ (−s3, 0)R , L(s,̺) = ζC(s + s3 + k − 1) ζC(s + s3)

·ζC(s − s3 + k − 1) ζC(s − s3), 2 2 C(t,̺) = (1+ |ıt + s3 + k − 1|) (1 + |ıt + s3|) 2 2 ·(1 + |ıt − s3 + k − 1|) (1 + |ıt − s3|) ,

ε(s,̺, e∞) = 1 .

If |k1| = |k2| + |k3| > |k2| ≥ |k3|≥ 2,

|k1| |k2| |k3| 2 2 ̺ := ̺1 ⊗ ̺2 ⊗ ̺3 ≃ (0, 2|k1|− 3)R ⊕ (0, 2|k2|− 1)R 2 2 ⊕(0, 2|k3|− 1)R ⊕ (0, 1)R , 3 1 L(s,̺) = ζC(s + |k1|− 2 ) ζC(s + |k2|− 2 ) 1 1 ·ζC(s + |k3|− 2 ) ζC(s + 2 ) , 3 2 1 2 C(t,̺) = (1+ |ıt + |k1|− 2 |) (1 + |ıt + |k2|− 2 |) 1 2 1 2 ·(1 + |ıt + |k3|− 2 |) (1 + |ıt + 2 |) , ε(s,̺, e∞) = 1 .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 45

Non-Archimedean

η1 η2 0 0 s1 +s2 ̺1 ⊗ ̺2 ≃ kk , 2 η∈{·M,··}

η1 η2 η3 0 0 0 s1 +s2 +s3 ̺1 ⊗ ̺2 ⊗ ̺3 ≃ kk , 3 η∈{·M,··} 0 0 0 η1 η2 η3 Lp(s,̺1 ⊗ ̺2 ⊗ ̺3) = ζp(s + s1 + s2 + s3 ) , 3 η∈{·Y,··} 0 0 0 C(̺1 ⊗ ̺2 ⊗ ̺3) = 1, 0 0 0 εp(s,̺1 ⊗ ̺2 ⊗ ̺3, ep) = 1,

♯ η1 η2 1 0 0 s1 +s2 − 2 +ıt3 ̺1 ⊗ ̺2 ⊗ ̺3 ≃ kk ⊗ sp2, 2 η∈{·M,··}

♯ ♯ ıt1+ıt2 −1+ıt1+ıt2 3 ̺1 ⊗ ̺2 ≃ kk ⊕ kk ⊗ sp ,

♯ ♯ ♯ 1 2 − 2 +ı(t1+t2+t3) 2 ̺1 ⊗ ̺2 ⊗ ̺3 ≃ ⊕ kk ⊗ sp − 3 +ı(t +t +t ) 4 ⊕ kk 2 1 2 3 ⊗ sp , ♯ ♯ ♯ 1 2 L(s,̺1 ⊗ ̺2 ⊗ ̺3) = ζp(s+ 2 + ı(t1 + t2 + t3)) 3 ·ζp(s + 2 + ı(t1 + t2 + t3)) , ♯ ♯ ♯ 5 C(̺1 ⊗ ̺2 ⊗ ̺3) = p , ♯ ♯ ♯ 5 −5ı(t1+t2+t3) −5s+ 2 ε(s,̺1 ⊗ ̺2 ⊗ ̺3, ep) = (−p )p ,

♯ ♯ 0 ıt1+ıt2+˙s3 −1+ıt1+ıt2+˙s3 3 ̺1 ⊗ ̺2 ⊗ ̺3 ≃ kk ⊕ kk ⊗ sp ıt +ıt +¨s −1+ıt +ıt +¨s 3 ⊕kk 1 2 3 ⊕ kk 1 2 3 ⊗sp . 4.2 Zeta Integrals 4.2.1 Rankin–Selberg ♭ ⋆ Lemma 5. Let F ∈ S˜k(1, (1,M), χ,˜ M ) be a Hecke normalized H eigenform, with Langlands parameters ̺ = (̺v). Then

2 a −1 v cv ∗ hF, F i = F |a|A da = ∗ L (1, Ad̺), × 1 ζ (2) ZA Q

b 1 p ∤ M, 1 k =0, p ♯ χ c∞ = cp = p | M M , 2−|k|−1 k 6=0,  p+1 (  εpλp ♭ 2 1+ p+1 p | M . 

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 46 CHAPTER 4. L-FUNCTIONS

♭ In terms of the corresponding classical Hecke form f ∈ Sk(1, (1,M),χ,M ),

2 dxdy 1+εpλp ∗ |f(z)| y2 = 2c∞M 2 1+ p L (1, Ad̺). Γ (M)\H Z 0 p|M ♭ Y ♭ Proof: Suppose M = 1 for now, and deﬁne the P GA Eisenstein series:

E(s,g,ξ) = ξ(s,γg), γ∈PX TQ\PGQ a x s 1 ξv s, κ = |a| IPK (1,(1,M))(κ), 1 v volv v κ ∈ PKv(1, (1, 1)), volv = volv PKv(1, (1,M)) .

Now consider the Rankin–Selberg zeta integral,

Z(s, F × F , ξ) = |F (g)|2 E(s,g,ξ) dg ZPGQ\PGA = |F (g)|2 ξ(s,g) dg ZP TQ\PGA 2 a x s−1 × = F |a|A d a dx × × 1 ZQ \A ZQ\A 2 ξa s−1 × = F |a|A d a × × 1 ZQ \A ξ∈Q× X b = Zv(s, Fv × F v), v Y b b 2 a s−1 × Zv(s, Fv × F v) = Fv |a|v dv a. × 1 ZQv

b b b These local factors are easy to compute: If k = 0, 0 ˇ 0 2 −1 Z∞(s, Wp × Wp ) = ζR(s +2s∞) ζR(s) ζR(s − 2s∞) ζR(2s) −1 = L∞(s,̺∞ ⊗ ̺∞) ζR(2s) . If |k|≥ 2, k ˇ k −(s+|k|−1) Z∞(s, Wp × Wp ) = (4π) Γ(s + |k|− 1) −|k|−1 −1 = 2 ζC(s + |k|− 1) ζC(s) ζR(2s) −|k|−1 −1 = 2 L∞(s,̺∞ ⊗ ̺∞) ζR(2s) . If p ∤ M, 0 ˇ 0 2 −1 Zp(s, Wp × Wp ) = ζp(s +s ˙p − s¨p) ζp(s) ζp(s +¨sp − s˙p) ζp(2s) −1 = Lp(s,̺p ⊗ ̺p) ζp(2s) .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 47

If p | M ♯, ♯ ˇ ♯ Zp(s, Wp × Wp ) = ζp(s + 1) −1 = Lp(s,̺p ⊗ ̺p) ζp(s) . If p | M χ, χ ˇ χ Zp(s, Wp × Wp ) = ζp(s) −1 = Lp(s,̺p ⊗ ̺p) ζp(s) . By unfolding the integral of an incomplete Eisenstein series as above, one can show that res E(s,g,ξ)= 1 . Therefore the Petersson norm of F is given by s=1 2

1 2 hF, F i = 2 |F (g)| dg ZPGQ\PGA = res Z(s, F × F , ξ) s=1 = lim Z(s, F × F , ξ) ζ∗(s)−1 s→1 −1 = Zv(1, Fv × F v) ζv(1) v Y b 2 b a −1 = F |a|A da. × 1 ZA

It is easy to check the value of each bc , deﬁned by v cv Zv(1, Fv × F v) = Lv(1,̺ ⊗ ̺). ζv(2) The restriction M ♭ = 1 isb removedb using the calculation from §2.4,

♭ ♭ εpλp ˙ 0 ˙ 0 hVp , Vp i = 2 1+ p+1 hVp , Vp i. Now we consider Eisenstein series which are Hecke eigenforms. For any ǫ choice of signs ǫ ∈ p|M {±1}, deﬁne E(s,g,ξ ) using Rp = Rp(1, (1,M)), Q ǫ ξ (s,g) = p|M (1 + ǫpRp) ξ(s,g). Then by Hejhal’s calculation of the Q scattering matrix [10], η(s, ξǫ) E(s,g,ξǫ) = η(1 − s, ξǫ) E(1 − s,g,ξǫ), ǫ s ∗ η(s, ξ ) := p|M (1 + ǫpp ) ζ (2s). Finally, we make the spectral renormalization Q [10, 14],

− 1 #{p|M}+ 1 2 2 2 E1(s,g,ξǫ) = E(s,g,ξǫ), so that vol(Γ0(M)\H) ∞ 2 ∞ 1 1 ǫ dt dxdy 2 dt h(t) E ( 2 + ıt,σz, ξ ) π y2 = |h(t)| π . ZΓ0(M)\H Z0 Z0

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 48 CHAPTER 4. L-FUNCTIONS

⋆ Lemma 6. Let ψ ∈ Sk(1, (1,M), χ, 1) be a classical Hecke normalized H eigenform, with Langlands parameters ̺ = (̺v). Then

2 1 ǫ dxdy |ψ(z)| E (s, σz, ξ ) y2 ZΓ0(M)\H 2 dxdy |ψ(z)| y2 ZΓ0(M)\H 1 #{p|M}− 3 ∗ 2 2 2 L (s,̺ ⊗ ̺) = 1+ǫp . p|M 2 M 1−s L∗(1, Ad̺) η(s, ξε) Q ♭ 2 2 Proof: Let F ,f be as in Lemma 5, with M = 1. Since Rp |F | = |F | for all p | M, b b ǫ 1+ǫp #{p|M} Z(s, F × F , ξ ) = p|M 2 2 Z(s, F × F , ξ) c L∗(s,̺ ⊗ ̺) = Q 1+ǫp 2#{p|M} ∞ , p|M 2 M −s η(s, ξǫ)

Q1 1 ǫ 2 #{p|M}+ 2 2 1 ǫ dxdy Z(s, F × F , ξ ) = 2 |f(z)| E (s, σz, ξ ) y2 . ZΓ0(M)\H The result now follows by dividing out the similar formula of Lemma 5. Since the new formula is self-normalizing, we only require ψ ∈ C×f to be an eigenform. Note that the identity of Lemma 6 is consistent with the functional equations of E1(s,g,ξǫ) and L∗(s,̺ ⊗ ̺). For the purpose of comparison with Theorem 3, we observe that the Langlands parameters of the unitary Eisenstein series 1 1 ǫ E ıt −ıt E ( 2 + ıt,g,ξ ) are given by ̺v = kk ⊕kk , and that

2 2 1 1 ǫ dxdy |ψ(z)| E ( 2 + ıt,σz, ξ ) y2 Γ0(M)\H Z 2

2 dxdy |ψ(z)| y2 ZΓ0(M)\H ! #{p|M}−3 ∗ 1 E E 2 L ( 2 ,̺⊗̺⊗̺ ) = Q ∗ ∗ , p|M p 2 L (1,Ad̺)2 res L (s,Ad̺E ) M s=1 Q 1+ǫ 1 −1 1 −1 E p − 2 −ıt − 2 +ıt Qp = 2 1+ p 1+ p . 4.2.2 Garrett / Rallis–Piateski-Shapiro In this section we explicitly calculate zeta integrals for the Rankin triple L- ♯ function, as defined in [33]. Let Fj ∈ πj ∩ S˜k(1, (1, dBN ), 1, 1) be three Hecke ⋆ normalized Hv eigenforms as in Theorem 1, with matching data ϕj as defined in §3.3. Consider F = F1 × F2 × F3 as an automorphic form on P GA and 3 0 ϕ = ϕ1 × ϕ2 × ϕ3 ∈ S(BA). We distinguish totally unramified data as ϕ .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 49

Following [33, 8], we consider Φv(s,h) satisfying A B Φ (s,th) = | det AD−1|s+1 Φ (s,h), h ∈ H , t = ∈ T . v v v v 0 D v Note the Iwasawa decomposition Hv = TvKv, where

K∞ = U(3),Kp = GSp6(Zp). ′ ′ ′ Now for h ∈ Hv, choose any h ∈ Hv s.t. ν(h )= ν(h) and deﬁne ′ Φv(0,h) = ω(h,h )ϕv(0). 0 Since Φv(0,h) is constant on Kv,

0 s Φv (0,h) Φv(s,h) = 0 Φv(0,h). Φv (0,1) The local zeta integral is deﬁned as in [33, 3, 7] by

Zv(s, F v, ϕv) = Φv(s,γ0g) F v(g) dg, × 0 G ZQv Nv \ v where b b

x1 0 13 x2 Nv =   ; x1 + x2 + x3 =0 , x3    0 1    3 3   1 1 1 −1 0 0    0 1 0 −1 1 0  0 0 1 −1 0 1  H(1) ∋ γ = . Q 0  111 000     0 0 0 −1 1 0     0 0 0 −1 0 1    3   Lemma 7. For any ϕv ∈ S(Bv ),

Iv|d ω(γ0)ϕv(0, 0, 0) = (−1) B ϕv(β,β,β) dβ. ZBv −1 Proof: Write γ0 = J1 EJ1F G, −1 0 1 1 0 1 0 E = 3 , J = 2 2 ,  0  1  −1 0   03 13   0 1     2 2     1 1 1  −1 1 0   13 1 1 F =   , G = . 1  1     0 1   −1 1   3 3       −1 0 1     

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 50 CHAPTER 4. L-FUNCTIONS

Then

′ ϕv(β1,β2,β3) = ω(F G)ϕv(β1,β2,β3) = ev(−ν(β1)+ ν(β2)+ ν(β3))

· ϕv(β1,β1 + β2,β1 + β3),

−1 ′ ω(γ0)ϕv(0, 0, 0) = ω(J1 EJ1)ϕv(0, 0, 0) −1 ′ = (F1 ev(−ν(α1)) ·F1ϕv)(0, 0, 0)

Iv|d ′ = (−1) B ev(ν(β)) ϕv(β, 0, 0) dβ ZBv Iv|d = (−1) B ϕv(β,β,β) dβ. ZBv By the calculations of Piateski-Shapiro–Rallis, Ikeda, Gross–Kudla, plus our own which follow, we have

Theorem 2. Suppose k = |k1| ≥ |k2| ≥ |k3|≥ 0, k1+k2+k3 =0 (⇒ k2k3 ≥ 0), ♯ ♭ χ and for some ﬁxed square-free N, all Nj = N and all Nj = Nj =1. Then

C Wˇ ⋆ ϕ v 1 Zv(0, v , v) = 2 Lv( 2 , π1 × π2 × π3), ζv(2)

1 p ∤ dBN, ε∞+1 2 k =0, p (p+εp) C∞ = Cp = (εp − 1) 3 p | dB, 2−2k−2 k ≥ 2,  (p+1) (  p (p+εp) (εp + 1) (p+1)3 p | N.  Proof: We go through each case separately. Note that although we only describe central values above, they are deﬁned by analytic continuation of Zv(s).

Archimedean Recall that

ı|kj |θ ı|kj |θ κθ|Fj = e Fj , ω(κθ)ϕj = e ϕj .

Deﬁning

ǫa x a−1 ǫ g = j 3 j , h′ = ρ , 1 , j a−1 1 j × ǫ ∈ {±1}, aj ∈ R , x ∈ R, we have

k ′ 1 |kj |+2 j x 2 ω(gj,h )ϕj (β) = π3 |aj | Xǫ (β) e( 3 ν(β)+ ıaj P (β)),

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 51

where X1 = X, X−1 = Y . Then

Φ(0,δg) = ω(g)ϕ(β,β,β) dβ ZM2(R)

1 |k1|+2 |k2|+2 |k3|+2 2k 2 2 = π3 |a1| |a2| |a3| |Xǫ| e z |X| − z |Y | dβ, ZM2(R) 2 2 2 where z = x + ı(a1 + a2 + a3)= x + ıy,

1 |k1|+2 |k2|+2 |k3|+2 2k 2 2 = π3 |a1| |a2| |a3| F |Xǫ| e z |X| − z |Y | (0) 1 −2k |k1|+2 |k2|+2h |k3|+2 i = π3 (2πı) |a1| |a2| |a3| k · ∂ ∂ F e z |X|2 − z |Y |2 (0) ∂Xǫ ∂Xǫ 1 h −2k |k1|+2 |k2|+2 |k3|i+2 −2 = π3 (2πı) |a1| |a2| |a3| |z| k · ∂ ∂ e − 1 |X|2 + 1 |Y |2 (0) ∂Xǫ ∂Xǫ z z 1 h |k1|+2 |k2|+2 |k3|+2 i ǫ −k −2 = π3 k! |a1| |a2| |a3| (2πǫz /ı) |z| ,

Φ(s,δg)

1 2s+|k1|+2 2s+|k2|+2 2s+|k3|+2 ǫ −k −2s−2 = π3 k! |a1| |a2| |a3| (2πǫz /ı) |z| . We used the formulas

t t − 1 t −1 t exp(−πx Qx +2πiy Bx) dx = (det Q) 2 exp(−πy BQ B y), n ZR ∂ ∂ n 2 n ∂z ∂z exp(a|z| ) (0) = n! a . h i

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 52 CHAPTER 4. L-FUNCTIONS

In the case of all Fj being even Maass forms, the diﬃcult archimedean zeta integral was evaluated by Ikeda [12]. It is trivial to extend his result to arbitrary combinations of parity. Recall that

{0, 1} ∋ δ ≡ δ1 + δ2 + δ3 mod 2,

y = y1 + y2 + y3.

Then

ˇ 0 Z∞(s, W∞, ϕ∞)

Q>0 1 2s −2−2s = π3 |a1a2a3| |z| (R×)3 R [0,π)3 {±X1} Z Z Z δj 2 × · j ǫ w0,sj (4πaj ) e(x)2 d aj dx dθj 4 Q −2−2s 2s 2 × = (1 − δ)2 |z| e(x) j aj w0,sj (4πaj ) d aj dx + 3 Z(R ) ZR Q 1 4 −2−2s s+ 2 × = (1 − δ)2 |z| e(x) j yj Ksj (2πyj ) d yj dx + 3 Z(R ) ZR 5 s+1 1 Q 1 (1 − δ)2 π −s− s+ 2 × 2 1 = y Ks+ (2πy) j yj Ksj (2πyj ) d yj Γ(s + 1) + 3 2 Z(R ) −s Q (1 − δ) π s 1 s1 s2 s3 = Γ( 2 + 4 ± 2 ± 2 ± 2 ) Γ(s + 1)Γ(2s + 1) 3 {±}Y (1 − δ) 1 = ζR(s + 2 ± s1 ± s2 ± s3) ζR(2s + 2) ζR(4s + 2) 3 {±}Y 1 L∞(s + , π1 × π2 × π3) = (1 − δ) 2 . ζR(2s + 2) ζR(4s + 2)

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 53

Our calculation in the case |k3| ≥ 2 is very similar to that of Gross–Kudla for all-equal positive weights [7].

ˇ |k1| ˇ |k2| ˇ |k3| Z∞(s, W∞ × W∞ × W∞ , ϕ∞)

2s+2|k1| 2s+2|k2| 2s+2|k3| = 4 IR− (ǫ) a1 a2 a3 (R+)3 {±X1} Z ǫ −k −2s−2 × · k! (2πǫz /ı) |z| e(z)2 d aj dx ZR −k s+|k1| s+|k2| s+|k3| = (2πı) k! y1 y2 y3 + 3 Z(R ) −k −2s−2 × · (z) |z| e(z) d yj dx ZR s+1 s+k+1 −k (−2πı) (2πı) s+|k1| s+|k2| s+|k3| = (2πı) k! y1 y2 y3 Γ(s + 1)Γ(s + k + 1) + 3 Z(R ) ∞ −4π(y1+y2+y3)(1+t) s s+k × · e t (1 + t) d yj dt Z0 (2π)2s+2 k! Γ(s + |k |) Γ(s + |k |) Γ(s + |k |) = 1 2 3 Γ(s + 1)Γ(s + k + 1) (4π)3s+2k ∞ · ts(1 + t)−2s−k dt Z0 k! Γ(s + |k |) Γ(s + |k |) Γ(s + |k |) Γ(s + 1)Γ(s + k − 1) = 1 2 3 24s+4k−2 πs+2k−2 Γ(s + 1)Γ(s + k + 1) Γ(2s + k) 2−2k−2 k! 24 (2π)−4s−2k = −3s−2 (s + k) (2s + 1)k π Γ(s + k − 1)Γ(s + 1)Γ(s + |k |) Γ(s + |k |) · 2 3 Γ(s + 1)Γ(2s + 1) −2k−2 1 2 k! L∞(s + , π1 × π2 × π3) = 2 , (s + k) (2s + 1)k ζR(2s + 2)ζR(4s + 2)

2−2k−2 ˇ |k1| ˇ |k2| ˇ |k3| ϕ 1 .Z∞(0, W∞ × W∞ × W∞ , ∞) = 2 L∞( 2 , π1 × π2 × π3) ζR(2) It is easy to check that our evaluation of the above integral is consistent with 1 Ikeda’s result for the overlapping parameter values kj = 0, sj = − 2 .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 54 CHAPTER 4. L-FUNCTIONS

Finally, in the case |k2|≥ 2, |k3| = 0, we have

ˇ k ˇ k ˇ 0 Z∞(s, W∞ × W∞ × W∞, ϕ∞)

−k s+k s+k s 2πy3 = (2πı) k! y1 y2 y3 w0,s3 (4πy3)e + 3 Z(R ) −k −2s−2 × · (z) |z| e(z) d yj dx ZR s+1 s+k+1 −k (−2πı) (2πı) s+k s+k s = (2πı) k! y1 y2 y3 w0,s3 (4πy3) Γ(s + 1)Γ(s + k + 1) + 3 Z(R ) ∞ −4π(y1+y2+y3)(1+t)+2πy3 s s+k × · e t (1 + t) d yj dt 0 Z 2s+2 2 (2π) k! Γ(s + k) s = 2s+2k y3 w0,s3 (4πy3) Γ(s + 1)Γ(s + k + 1) (4π) R+ ∞ Z −2πy3(1+2t) s −s−k × · e t (1 + t) d y3 dt Z0 k! Γ(s + k) Γ(s + 1) = 2−2s−4k+2 π−2k+2 Γ(s +1)(s + k) (4π)s

k s+ 2 −1 × · (4πy3) w−s− k , 1−k (4πy3) w0,s3 (4πy3) d y3 + 2 2 ZR k! Γ(s + k) = 2−4s−4k+2 π−s−2k+2 (s + k) 1 1 Γ(s + 2 ± s3) Γ(s + k − 2 ± s3) Γ(∓2s3) · 1 1 1 Γ( 2 ∓ s3)Γ(2s + k + 2 ± s3) {±}X s + 1 ± s , s + k − 1 ± s , 1 ± s ; · F 2 3 2 3 2 3 . 3 2 1 ± 2s , 2s + k + 1 ± s ; 1 3 2 3

3F2 denotes a generalized hypergeometric series. We can simplify this further at s = 0 using the three-term relation,

a, b, c; Γ(1 − a) Γ(d) Γ(e) Γ(c − b) F = 3 2 d, e; 1 Γ(d − b) Γ(e − b)Γ(1 + b − a) Γ(c) bX↔c b, 1+ b − d, 1+ b − e; · F . 3 2 1+ b − c, 1+ b − a; 1

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 55

ˇ k ˇ k ˇ 0 Z∞(0, W∞ × W∞ × W∞, ϕ∞)

2 1 1 Γ(k) Γ( ± s3) Γ(k − ± s3) Γ(∓2s3) = 2 2 24k−2 π2k−2 1 1 1 Γ( 2 ∓ s3) Γ(k + 2 ± s3) {±}X 1 ± s , k − 1 ± s , 1 ± s ; · F 2 3 2 3 2 3 3 2 1 ± 2s , k + 1 ± s ; 1 3 2 3 Γ(k) 1 3 1 1 = 4k−2 2k−2 Γ( 2 ∓ s3) Γ(−k + 2 ∓ s3) Γ( 2 ± s3) Γ(k − 2 ± s3) 2 π 1 {±}X Γ(k) Γ(∓2s3) · 1 3 1 1 Γ( 2 ∓ s3) Γ(−k + 2 ∓ s3) Γ( 2 ∓ s3) Γ(k + 2 ± s3) 1 ± s , k − 1 ± s , 1 ± s ; · F 2 3 2 3 2 3 3 2 1 ± 2s , k + 1 ± s ; 1 3 2 3 Γ(k) (−1)k−1 π2 1 = 4k−2 2k−2 2 lim 2 π cos (πs3) f→2−k Γ(f)

Γ(k)Γ(1)Γ(f) Γ(∓2s3) 1 1 1 1 1 Γ( 2 ∓ s3) Γ(f − 2 ∓ s3) Γ(k + 2 ± s3) Γ( 2 ∓ s3) {±}X 1 ± s , 1 ± s , 3 ± s − f; · F 2 3 2 3 2 3 3 2 1 ± 2s , k + 1 ± s ; 1 3 2 3 k−1 2 1 1 Γ(k) (−1) π 1 −k +1, 2 + s3, 2 − s3; = lim 3F2 24k−2 π2k−2 cos2(πs ) f→2−k Γ(f) 1, f; 1 3 k−1 2 ∞ 1 1 Γ(k) (−1) π (−k + 1)n ( 2 + s3)n ( 2 − s3)n = 4k−2 2k−2 2 2 π cos (πs3) n! (1)n Γ(2 − k + n) n=Xk−1 k−1 2 k−1 1 1 Γ(k) (−1) π (−1) Γ(k) Γ(k − + s3) Γ(k − − s3) = 2 2 4k−2 2k−2 2 2 1 1 2 π cos (πs3) Γ(k) Γ( 2 + s3) Γ( 2 − s3) −4k+2 −2k+2 1 1 1 1 = 2 π Γ(k − 2 + s3) Γ(k − 2 − s3) Γ( 2 + s3) Γ( 2 − s3)

−2k−2 2 1 = 2 L∞( 2 , π1 × π2 × π3). ζR(2)

1 This is consistent with the previous case if we set s3 = − 2 . It would be 1 interesting to directly factor the function Z∞(s), for either k =0 or s3 = − 2 , by comparing with the previous cases.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 56 CHAPTER 4. L-FUNCTIONS

Non-Archimedean

First consider p ∤ dBN. A more illuminating treatment was given in [33], however our calculation is elementary and generalizes in an obvious way to the ramiﬁed cases. Note that the restriction p 6= 3 is unnecessary. a a x aa Writing α = , g = , g = j for j = 2, 3, 1 1 1 j a−1 j and choosing h′ = ρ (αι, 1), we compute

Φp(0,γ0g) = ω(g)ϕp(β,β,β) dβ M Z 2,Qp 3 2 2 = |a a2a3|p ep(xν(β)) ϕp1(αβ) ϕp2(αa2β) ϕp3(αa3β) dβ M Z 2,Qp 3 3 2 2 = ζp(2) |a a2a3|p ep(x(β11β22 − β12β21)) 4 Z(Qp) · I (aaβ˜ ) I (aaβ˜ ) I (˜aβ ) I (˜aβ ) dβ Zp 11 N˙ Zp 12 N¨ Zp 21 Zp 22

( |a˜|p := max{1, |a2|p, |a3|p} )

3 3 2 2 −1 = ζp(2) |a a2a3|p |aa˜|p Iaa˜Zp (xβ22) IZp (ãβ22) dβ22 ZQp · |N˙ | |aa˜|−1 I (xβ ) I (ãβ ) dβ p p aaÑ¨Zp 21 N¨Zp 21 21 ZQp 3 3 2 2 −2 2 = ζp(2) |a a2a3|p |z|p ( |z|p := max{|aa˜ |p, |x|p} ),

3 3 2 2 s+1 −2s−2 Φp(s,γ0g) = ζp(2) |a a2a3|p |z|p .

It is straightforward to check that for |a|p ≤ 1,

λ−1 1−pλ−1 2 λ |z|p ep(x) dx = 1−p−λ (1 − |paa˜ |p ). Q Z p ˇ 0 Then Zp(s, Wp , ϕp)

|ap|−s˙1 −|ap|−s¨1 1 3 2 2 s+1 −2s−2 2 = |a a2a3| |z| ep(x) |p|−s˙1 −|p|−s¨1 |a| IZp (a) (Q×)3 Q Z p Z p |aa p|−s˙2 |a |s¨2 −|aa p|−s¨2 |a |s˙2 1 2 2 2 2 2 2 2 · |p|−s˙2 −|p|−s¨2 |aa2| IZp (aa2) |aa p|−s˙3 |a |s¨3 −|aa p|−s¨3 |a |s˙3 1 3 3 3 3 2 2 2 · |p|−s˙3 −|p|−s¨3 |aa3| IZp (aa3) 3 2 2 −1 × × × · |a a2a3| dx d a d a2 d a3 1−p−2s−2 2 −2s−1 = 2s+1 (1 − |paa˜ | ) − 1 − 1 1−p |a|≤1 |a |≤|a| 2 |a |≤|a| 2 Z Z 2 Z 3 −s˙1 −s¨1 −s˙2 s¨2 −s¨2 s˙2 |ap| −|ap| |aa2p| |a2| −|aa2p| |a2| · |p|−s˙1 −|p|−s¨1 |p|−s˙2 −|p|−s¨2 |aa p|−s˙3 |a |s¨3 −|aa p|−s¨3 |a |s˙3 1 3 3 3 3 3 2 2 s+ 2 × × × · |p|−s˙3 −|p|−s¨3 |a a2a3| d a d a2 d a3 .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 4.2. ZETA INTEGRALS 57

−k −kj Now write |a|p = p , |aj |p = p , so

ˇ 0 Zp(s, Wp , ϕp) = S(k, k2, k3, min{k, k +2k2, k +2k3}), k≥0 k ≥− k k ≥− k X 2X 2 3X 2 −2s−2 1 1−p −(3k+2k2+2k3)(s+ 2 ) S(k, k2, k3,l) := 1−p2s+1 p

(l+1)(2s+1) p(k+1)s ˙1 −p(k+1)¨s1 · 1 − p ps˙1 −ps¨1 p(k+k2+1)s ˙2−k2s¨2 −p(k+k2+1)¨s2−k2s˙2 · ps˙2 −ps¨2 p(k+k3+1)s ˙3−k3s¨3 −p(k+k3+1)¨s3−k3s˙3 · ps˙3 −ps¨3 . We can make the sums independent:

ˇ 0 Zp(s, Wp , ϕp) = µ=0,1 k′≥0 k′ ≥0 k′ ≥0 X X X2 X3 ′ ′ ′ ′ ′ ′ ′ ′ S(2k + µ, k2 − k , k3 − k , 2 min{k , k2, k3} + µ) = η=0,1 l≥0 µ=0,1 k′′≥0 k′′≥0 k′′≥0 X X X X X2 X3 η ′′ ′′ ′′ ′′ ′′ (−1) S(2(k + l + η)+ µ, k2 − k , k3 − k , 2l + µ).

After multiplying out the summand as a polynomial and rearranging the expo- ′′ ′′ ′′ nents (as linear combinations of {1,l,k , k2 , k3 }), we recognize Zp to be a finite sum of products of geometric series. This evaluates to a complicated rational expression, which we factor using Mathematica. It is necessary to impose the constraint on central characters, j (s ˙j +¨sj ) = 0, e.g. by substitution fors ˙1. Then P L (s + 1 , π × π × π ) ˇ 0 p 2 1 2 3 Zp(s, Wp , ϕp) = . ζp(2s + 2) ζp(4s + 2) ♯ In the cases p | dB N , the difficult ramified zeta integrals were calculated by Gross–Kudla [7]. We have quoted their results, taking into account our differing normalizations of measures and local data.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 58 CHAPTER 4. L-FUNCTIONS

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Chapter 5

Conclusion

5.1 Triple Product Identities ~ Let ψj ∈ Skj (dB , N) be three Hecke-eigen newforms of the same square-free level N, with k1 + k2 + k3 = 0, and let ̺j denote the corresponding Langlands parameters. Deﬁne εv = j εjv for the eigenvalues εjv as in §2.4: −Q T∞ ψj = εj∞ ψj , kj =0, [p] Tp ψj = −εjp ψj , p | dBN.

(1) Writing X = O (dB, N~ )\H, we have Theorem 3. 2 dxdy ψ (z) ψ (z) ψ (z) 2 1 2 3 y #{p|dBN}−3 ∗ 1 2 L ( ,̺1 ⊗ ̺2 ⊗ ̺3) ZX = Q 2 , v v (d N)2 L∗(1, Ad̺ ) |ψ (z)|2 dx dy B j j j j y2 Q ZX Q Q 1+ε∞ 1−εp 2 k1 = k2 = k3 =0, 2 p | dB , 1+εp Q∞ =  1 |k1| = |k2| > |k3| =0, Qp =  2 p | N,  2 |k1| > |k2| ≥ |k3| > 0,  1 p ∤ dB N.

 × ′ Proof: Deﬁne the corresponding adelic forms Ψj on BA , Fj on GA, Fj on ′ HA, and the local data ϕj ∈ S(BA), as in §2.2, §3.2. As Harris–Kudla showed, it follows from the see-saw identity,

′ 1, Θϕ(F ) ′ = Θϕ(1), F , P HA P GA

′ 1 and the Siegel–Weil formula, Θϕ(1) = 2 E(0, h, Φ), that

′ ′ Θϕ(F )(h ) dh = Z(0, F , ϕ). P H′ \P H′ Z Q A 59

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 60 CHAPTER 5. CONCLUSION

Then by Theorem 1,

2 × Ψ1(β)Ψ2(β)Ψ3(β) d β P B×\P B× Z(0, F , ϕ) Z Q A = , Ψj, Ψj × Fj , Fj j P B j PGA A and by Theorem 2 andQ Lemma 5, Q

2 dx dy ψ1(z) ψ2(z) ψ3(z) y2 ∗ ∗ 1 ζ (2) C L ( ,̺1 ⊗ ̺2 ⊗ ̺3) ZX = v v 2 . 4 vol(X) c L∗(1, Ad̺ ) |ψ (z)| dx dy j,v jv j j j j y2 Q ZX Q Q Q Noting that vol(X) = 2ζ∗(2) (p − 1) (p + 1), p|dB p|N it is straightforward to compute theQ constants. Q

5.2 Applications to Quantum Chaos

Consider the geodesic flow on a finite volume hyperbolic surface X = Γ\H. This is a classical Hamiltonian dynamical system with phase space T ∗X, the ∗ ∗ 2 cotangent bundle of X, and Hamiltonian H(vx)= |vx| , the Riemannian norm squared. We restrict our attention to S∗X, the constant energy submanifold of unit cotangent vectors, which is invariant under the flow. Under the homeo- morphic identification

∗ ∼ + S X −→ Γ\PGL2 (R), 1 (dy) 7−→ Γ , ı 1 the ﬂow map for time t is given explicitly by right multiplication,

et T (Γg) = Γg . t 1 The normalized Liouville measure on S∗X is ergodic for the geodesic flow and has entropy equal to 1. Furthermore, Ornstein–Weiss proved this system is measurably isomorphic to a Bernoulli flow. Topologically, the geodesic flow is less simple. There are no stable periodic orbits, but since the flow is Anosov, unstable periodic orbits form a dense subset. Each of these closed geodesics obviously carries an ergodic measure, but there are also ergodic measures with supports having any Hausdorff dimension between 1 and 3. The standard quantization of the classical geodesic flow has state space L2(X) and Hamiltonian H˜ = −∆, a pseudo-differential operator with principal symbol H. For a particle described by the normalized quantum state

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 5.2. APPLICATIONS TO QUANTUM CHAOS 61

ψ ∈ L2(X), measuring its position is equivalent to sampling an X-valued random variable deﬁned by the probability distribution

2 dx dy dµ(z) = |ψ(z)| y2 . kψkL2 =1. Similarly, measuring the particle’s energy is equivalent to sampling a Spec(H˜ )- valued random variable, deﬁned according to the spectral expansion of ψ. If X is compact, then L2(X) has an orthonormal basis of eigenforms,

Hψ˜ j = λj ψj , 0= λ0 < λ1 ≤ λ2 ≤ ... , in which case the corresponding spectral measure is simply

2 |hψ, ψj i| δλj . j X ˜ 1 For non-compact X, H also has absolutely continuous spectrum equal to 4 , ∞ . All of the corresponding (non-square-integrable) eigenspaces are spanned by unitary Eisenstein series and have dimension equal to the number of cusps. The special arithmetic X we will consider are distinguished by their numer- ous symmetries, giving rise to Hecke operators. Since H⋆ is commutative, it represents a collection of simultaneously observable quantities. Furthermore, ˜ ⋆ ⋆ since H ∈ H∞ , we may assume that the ψj described above are in fact H - eigenforms. As we saw in Lemma 1 and the discussion before Lemma 6, H⋆ completely decomposes L2(X) with multiplicity 1. (Note that for these X, the residual spectrum consists of {0}, i.e. all ψj with j > 0 are cuspidal.) Quantum chaos is concerned with the behavior of eigenstates in quantiza- tions of classically chaotic systems, particularly under semi-classical / high- energy limits [31]. An important general result due to Colin-de-Verdiere / Shnirelman / Zelditch is that

N 1 2 f dµ − f dµ −→ 0 as N → ∞, f ∈ C∞(X). N j 0 c j=1 X X X Z Z This is called quantum ergodicity, and it implies that almost all high-energy states are nearly equidistributed. The only assumption on X is ergodicity of the Liouville measure under geodesic ﬂow, one of the mildest chaotic properties ensured by negative curvature. However, quantum ergodicity does not exclude the possibility that there may be other weak limits of the µj along subsequences, a phenomenon known as scarring. Scarring has been observed numerically in some related systems, such as the Bunimovich Stadium billiard. In the special cases of congruence hyperbolic surfaces (and 3-manifolds), Rudnick–Sarnak ruled out strong scarring along closed geodesics [29], leading them to conjecture that all high-energy states become equidistributed. This is called quantum unique ergodicity (QUE). In [24], Luo–Sarnak formulated this conjecture quan- titatively, predicting the rate of equidistribution:

1 − 4 +ǫ ∞ f dµj − f dµ0 ≪f,ǫ λj , f ∈ Cc (X). ZX ZX

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 62 CHAPTER 5. CONCLUSION

They proved this, in the case X = SL2Z\H, on average over µj and for the individual measures associated to unitary Eisenstein series. We now reduce their quantitative conjecture for individual µj to appropriate Lindel¨of Hypotheses, using triple product identities. To quantify the smoothness of f, we use the Sobolev norms

r 2 f L2,r = (1 − ∆) f L2 , r ∈ R. 2,r ∞ Let L (X) denote the Hilbert space completion of Cc (X) under kkL2,r . (1) 2 Theorem 4. Fix X = O (dB )\H and let the ψj ∈ L (X) be as before, with ′ Langlands parameters ̺j . Suppose that for some 0 ≤ a ≤ 1 and all j, j ≥ 1,

a 1 4 +ǫ L( 2 ,̺j ⊗ ̺j ⊗ ̺j′ ) ≪X,ǫ C∞(0,̺j ⊗ ̺j ⊗ ̺j′ ) , and also a 1 4 +ǫ L( 2 ,̺j ⊗ ̺j ⊗ ̺E) ≪X,ǫ C∞(0,̺j ⊗ ̺j ⊗ ̺E) if dB =1. a+1 1 Then for 0 ≤ r ≤ 2 and λj ≥ 4 ,

a−r 2 1 2 +ǫ |ψj | − vol(X) L2,−r ≪X,ǫ λj . a+1 Corollary 1. Under the same hypotheses, set r = 2 . Then

a−1 4 +ǫ f dµj − f dµ0 ≪X,ǫ λj kfkL2,r . (5.1) ZX ZX For a = 1 , these hypotheses are Phragmen–Lindel¨of convexity bounds, while a variant of (5.1) follows trivially from the Sobolev inequality,

kfkL∞ ≪X,ǫ kfkL2,1+ǫ . Any value a < 1 (‘convexity breaking’) implies the QUE conjecture. Further- more, the Grand Riemann Hypothesis implies a = 0, a generalization of Lin- del¨of’s Hypothesis, and this is best-possible. It follows from our proof that the 1 corresponding exponent − 4 of λj in (5.1) can’t be lowered. Our methods also apply to weight-k eigenforms ψ, giving the same result, although more smoothness is required from f. We will present these details later. Corollary 2. Under the same hypotheses, set r =0. Then

a 4 +ǫ kψjkL4 ≪X,ǫ λj . (5.2)

1 The exponent 4 of λj in (5.2) corresponding to a = 1 is not the best which 1 is presently known for general surfaces ( 16 is proved in [30]), and it is not the best that can be done using Theorem 3. In joint work with Sarnak, we have 1 obtained the exponent 24 unconditionally, and we hope to improve this further. Another related problem in quantum chaos concerns the amplitude distribution of high-energy eigenstates. Viewing (X, µ0) as a probability space, each ψj becomes an R-valued random variable with cumulative distribution function

−1 fj (t) = µ0(ψj (−∞,t]).

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 5.2. APPLICATIONS TO QUANTUM CHAOS 63

Berry / Hejhal’s random wave conjecture asserts that these fj converge in a − 1 suitable sense to the normal distribution N (0, vol(X) 2 ) as λj → ∞. It follows from the proof of QE that the truth of the random wave conjecture for fourth moments alone would imply QUE. Regarding third moments, this conjecture says 1 3 dx dy vol(X) ψj (z) y2 −→ 0 as λj → ∞. ZX We now prove a stronger form of this (unconditionally) for the full modular group. 2 Theorem 5. Fix X = SL2Z\H and let the ψj ∈ L (X) be deﬁned as before. Then 1 3 dx dy − 12 +ǫ ψj (z) y2 ≪ǫ λj . ZX 5.2.1 Proofs 1 2 Lemma 8. (Weyl’s Law) Writing λj = 4 − sj , sj = σj + ıtj , vol(X) 2 1+ǫ 1 = 4π T + OX,ǫ (1 + T ) , 0≤t 0 X 5 2 −r 2 1 1 2 dt + ( 4 + t ) |ψj | , E ( · , 2 + ıt) π , + ZR where the integral only appears if dB = 1. First we analyze the individual terms, using Theorem 3 and Lemma 6. We may assume εj′∞ = 1, or else the 1 corresponding term vanishes by symmetry, and also λj ≥ 4 . Then 1 1 1 2 1 L∞( 2 ,̺j ⊗ ̺j ⊗ ̺j′ ) = ζR( 2 +2sj + sj′ ) ζR( 2 + sj′ ) ζR( 2 − 2sj + sj′ ) 1 1 2 1 · ζR( 2 +2sj − sj′ ) ζR( 2 − sj′ ) ζR( 2 − 2sj − sj′ ), 2 4 2 C∞(0,̺j ⊗ ̺j ⊗ ̺j′ ) = (1+ |2sj + sj′ |) (1 + |sj′ |) (1 + |2sj − sj′ |) ,

1 E 1 1 2 1 L∞( 2 ,̺j ⊗ ̺j ⊗ ̺ ) = ζR( 2 +2sj + ıt) ζR( 2 + ıt) ζR( 2 − 2sj + ıt) 1 1 2 1 · ζR( 2 +2sj − ıt) ζR( 2 − ıt) ζR( 2 − 2sj − ıt), E 2 4 2 C∞(0,̺j ⊗ ̺j ⊗ ̺ ) = (1+ |2s∞ + ıt|) (1 + |ıt|) (1 + |2s∞ − ıt|) .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 64 CHAPTER 5. CONCLUSION

Recall Stirling’s approximation: For s = σ + ıt and | arg(s)|≤ π − ǫ < π,

−1 1 s− 1 −s Γ(s) = (1+ Oǫ(|s| )) (2π) 2 s 2 e , −1 1 σ− 1 −σ−t arg(s) |Γ(s)| = (1+ Oǫ(|s| )) (2π) 2 |s| 2 e .

We will write X ≍ Y in place of X ≪ Y ≪ X. Then

π − 2 (|2tj +tj′ |+2|tj′ |+|2tj −tj′ |) 1 e ′ L∞( 2 ,̺j ⊗ ̺j ⊗ ̺j ) ≍ 1 , C(0,̺j ⊗ ̺j ⊗ ̺j′ ) 4

−π|t ′ | L∞(1, Ad̺j′ ) ≍ e j , 1 π − 2 (|2tj +tj′ |−2|2tj |+|2tj −tj′ |) L∞( 2 ,̺j ⊗ ̺j ⊗ ̺j′ ) e 2 ≍ 1 , L∞(1, Ad̺j ) L∞(1, Ad̺j′ ) C(0,̺j ⊗ ̺j ⊗ ̺j′ ) 4

π − 2 (|2tj +t|+2|t|+|2tj −t|) 1 E e L∞( ,̺j ⊗ ̺j ⊗ ̺ ) ≍ , 2 E 1 C(0,̺j ⊗ ̺j ⊗ ̺ ) 4 E −π|t| res L∞(z, Ad̺ ) ≍ e , z=1 1 E − π (|2t +t|−2|2t |+|2t −t|) L∞( ,̺j ⊗ ̺j ⊗ ̺ ) e 2 j j j 2 ≍ . 2 E E 1 L∞(1, Ad̺j) res L∞(z, Ad̺ ) C(0,̺j ⊗ ̺j ⊗ ̺ ) 4 z=1 Note that |a + b|− 2|a| + |a − b| = 2max{0, |b| − |a|}. Finally, recall the estimate of Hoﬀstein–Lockhart [11],

−1 ǫ L(1, Ad̺j′ ) ≪X,ǫ (1 + |tj′ |) , ǫ> 0.

We divide the discrete spectrum contributions into

I= {0 ≤ tj′

Then

a−1+ǫ a−1−2r+ǫ ≪X,ǫ (1 + tj ) (1 + tj′ ) I I max{a−1,2a−2r}+ǫ X ≪X,ǫ (1 + tj ) X 2a−2r+ǫ ≪X,ǫ (1 + tj ) ,

3 (a−1)−2r+ǫ 1 (a−1)+ǫ ≪X,ǫ (1 + tj ) 2 (1 + |2tj − tj′ |) 2 II II 2a−2r+ǫ X ≪X,ǫ (1 + tj ) , X

ǫ 2a−2−2r+ǫ −π(t ′ −2tj ) ≪X,ǫ (1 + tj ) (1 + tj′ ) e j III III −(π−ǫ)tj X ≪X,ǫ e X 2a−2r ≪X (1 + tj ) .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 5.2. APPLICATIONS TO QUANTUM CHAOS 65

Similarly, we divide up the continuous spectrum:

tj tj a−1+ǫ a−1−2r+ǫ ≪X,ǫ (1 + tj ) (1 + t) dt Z0 Z0 max{a−1,2a−2r−1}+ǫ ≪X,ǫ (1 + tj ) 2a−2r+ǫ ≪X,ǫ (1 + tj ) ,

3tj 3tj 3 (a−1)−2r+ǫ 1 (a−1)+ǫ ≪X,ǫ (1 + tj ) 2 (1 + |2tj − t|) 2 dt Ztj Ztj 2a−2r−1+ǫ ≪X,ǫ (1 + tj ) ,

∞ ∞ ǫ 2a−2−2r+ǫ −π(t−2tj ) ≪X,ǫ (1 + tj ) (1 + t) e dt Z3tj Z3tj −(π−ǫ)tj ≪X,ǫ e 2a−2r ≪X (1 + tj ) .

Proof of Theorem 5: First note that

3 3 2 Lv(s, ⊗ ̺j ) = Lv(s, Sym ̺j ) Lv(s,̺j ) .

1 Then for λj ≥ 4 ,

1 3 L∞( 2 , ⊗ ̺j ) −2 ≍ (1 + |tj |) , 1 3 L∞( 2 , Ad̺j ) 3 4 C∞(0, Sym ̺j ) ≍ (1 + |tj |) .

3 To prove the convexity bound for L(s, Sym ̺j ), we ﬁrst note that it is entire and has ﬁnite order by the work of Kim–Shahidi and Gelbart–Shahidi [19, 5]. Furthermore, using the converse theorem of Cogdell–Piateski-Shapiro [2], Kim– 3 Shahidi proved that Sym ̺j is cuspidal on GL4 [20]. Recent results of Molteni [25] then imply

3 3 ǫ L(1 + ıt, Sym ̺j ) ≪ǫ C∞(t, Sym ̺j ) , 3 3 1 +ǫ L(0 + ıt, Sym ̺j ) ≪ǫ C∞(t, Sym ̺j ) 2 , and hence by the Phragmen–Lindel¨of / Hadamard three-circles method,

1 1 3 3 4 +ǫ 1+ǫ L( 2 , Sym ̺j ) ≪ǫ C(0, Sym ̺j ) ≪ (1 + |tj |) . Now to prove that the third moment tends to zero, we only need subconvexity 1 for L( 2 ,̺j ). The ﬁrst such estimate was proved conditionally by Iwaniec in [14]; his result is made unconditional at a small cost to the exponent by the same technique as in [15]. The current record bound is due to Ivic and subsequently Jutila [13, 18]: 1 1 3 +ǫ L( 2 ,̺j ) ≪ǫ (1 + |tj |) .

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 66 CHAPTER 5. CONCLUSION

Therefore 2 1 3 dx dy − 3 +ǫ ψj (z) y2 ≪ǫ (1 + |tj |) . ZX

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals Bibliography

[1] S. B¨ocherer, R. Schulze-Pillot, On the central critical value of the triple product L-function, Number theory (Paris, 1993–1994), London Math. Soc. Lecture Note Ser. 235, Cambridge Univ. Press, Cambridge (1996), 1–46.

[2] J. Cogdell, I. Piatetski-Shapiro, A converse theorem for GL4, Math. Res. Lett. 3 (1996), 67–76.

[3] P. Garrett, M. Harris, Special values of triple product L-functions, Amer. J. Math. 115 (1993), 161–240.

[4] S. Gelbart, Automorphic Forms on Adele Groups, Annals of Mathematics Studies 83, Princeton Univ. Press (1975)

[5] S. Gelbart, F. Shahidi, Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc. 14 (2001), 79–107.

[6] R. Godement, Notes on Jacequet–Langlands Theory, IAS mimeograph (1970)

[7] B. Gross and S. Kudla, Heights and the central critical values of triple product L-functions, Compositio Math. 81 (1992), 143–209.

[8] M. Harris and S. Kudla, The central critical value of a triple product L- function, Annals of Math. 133 (1991), 605–702

[9] M. Harris and S. Kudla, Arithmetic automorphic forms for the non- holomorphic discrete series of GSp(2), Duke Math. J. 66 (1992), 59–121.

[10] D.A. Hejhal, The Selberg Trace Formula for PSL(2,R), Lecture Notes in Math. 1001 (1983)

[11] J. Hoﬀstein, P. Lockhart, Coeﬃcients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), 161–181.

[12] T. Ikeda, On the gamma factor of the triple L-function, I, Duke Math. J. 97 (1999), 301–318.

[13] A. Iv´ıc, On sums of Hecke series in short intervals, J. Th´eor. Nombres Bordeaux 13 (2001), no. 2, 453–468.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals 68 BIBLIOGRAPHY

[14] H. Iwaniec, The spectral growth of automorphic L-functions, J. Reine Angew. Math. 428 (1992), 139–159. [15] H. Iwaniec, P. Sarnak, L∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), 301–320. [16] H. Iwaniec, P. Sarnak, Perspectives on the analytic theory of L-functions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. (2000) Special Volume, Part II, 705–741. [17] H. Jacquet, R. Langlands, Automorphic forms on GL(2), Lecture Notes in Math. 114, Springer (1970) [18] M. Jutila, The fourth moment of central values of Hecke series, Number theory (Turku, 1999), de Gruyter, Berlin (2001), 167–177.

[19] H. Kim, F. Shahidi, Symmetric cube L-functions for GL2 are entire, Ann. of Math. 150 (1999), 645–662.

[20] H. Kim, F. Shahidi, Functorial products for GL2 × GL3 and functorial symmetric cube for GL2, C. R. Acad. Sci. Paris Sr. I Math. 331 (2000), 599–604. [21] A. Knapp, Local Langlands correspondence: the Archimedean case, Mo- tives (Seattle, 1991), Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., Providence, RI (1994), 393–410. [22] S. Kudla, The local Langlands correspondence: the non-Archimedean case, Motives (Seattle, 1991), Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., Providence, RI (1994), 365–391. [23] G. Lione and M. Vergne, The Weil representation, Maslov index, and theta series, Birkhauser, Boston (1980)

2 [24] W. Luo, P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2(Z)\H , Inst. Hautes Etudes´ Sci. Publ. Math. 81 (1995), 207–237. [25] G. Molteni, Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product, Duke Math. J. 111 (2002), no. 1, 133–158.

[26] T. Miyake, On automorphic forms on GL2 and Hecke operators, Ann. of Math. 94 (1971), 174–189. [27] T. Miyake, Modular Forms, Springer–Verlag, Berlin (1989)

[28] A. Pizer, Hecke operators for Γ0(N), J. Algebra 83 (1983), 39–64. [29] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195–213. [30] A. Seeger, C. Sogge, Bounds for eigenfunctions of diﬀerential operators, Indiana Univ. Math. J. 38 (1989), 669–682.

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals BIBLIOGRAPHY 69

[31] P. Sarnak, Arithmetic quantum chaos, The Schur lectures (Tel Aviv, 1992), Israel Math. Conf. Proc. 8 (1995), 183–236. [32] F. Shahidi, Local coeﬃcients as Artin factors for real groups, Duke Math. J. 52 (1985), 973–1007. [33] I. Piatetski-Shapiro and S. Rallis, Rankin triple L-functions, Comp. Math. 64 (1987), 31–115.

[34] H. Shimizu, Theta series and automorphic forms on GL2, J. Math. Soc. Japan 24 (1972), 638–683. [35] H. Shimizu, On zeta functions of quaternion algebras, Ann. of Math. 81 (1965), 166–193. [36] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Func- tions, Princeton University Press, Princeton (1971) [37] G. Shimura, On certain zeta functions attached to two Hilbert modular forms. II. The case of automorphic forms on a quaternion algebra, Ann. of Math. 114 (1981), 569–607. [38] D. Ornstein, B. Weiss, Geodesic flows are Bernoullian, Israel J. Math. 14 (1973), 184–198. [39] J. Tate, Number theoretic background, Automorphic forms, representations and L-functions (Corvallis, 1977), Proc. Sympos. Pure Math. 33, Part 2, Amer. Math. Soc., Providence, R.I. (1979), 3–26. [40] M-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math. 800, Springer, Berlin (1980) [41] A. Weil, Basic Number Theory, Die Grundlehren der mathematischen Wis- senschaften, Band 144 Springer–Verlag, New York (1967) [42] A. Weil, Adeles and Algebraic Groups, Progress in Mathematics 23, Birkhuser, Boston (1982)

Feb 19 2009 7:55:20 PST Vers. 1 - Sub. to Annals