On Critical Values of L-Functions for Holomorphic Forms on GSp(4) X GL(2) Citation Saha, Abhishek (2009) On Critical Values of L-Functions for Holomorphic Forms on GSp(4) X GL(2). Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5HWD-TH76. https://resolver.caltech.edu/CaltechETD:etd-05222009-162600 Abstract Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight ℓ. We derive an explicit integral representation for the degree eight L-function L(s, F x g). As an application, we prove a reciprocity law --- predicted by Deligne's conjecture --- for the critical special values L(m, F x g) where m ∈ Z, 2 ≤ m ≤ ℓ/2 -1. The proof of our integral representation has two major components: the generalization of an earlier integral representation due to Furusawa and a ``pullback formula" relating the complicated Eisenstein series of Furusawa with a simpler one on a higher rank group. The critical value result follows from our integral representation using rationality results of Garrett and Harris and the theory of nearly holomorphic forms due to Shimura. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: automorphic forms; critical values; Deligne's conjecture; integral representation; L-functions; siegel modular Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2009. Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2005-2006. Thesis Availability: Public (worldwide access) Research Advisor(s): Ramakrishnan, Dinakar Thesis Committee: Ramakrishnan, Dinakar (chair) Balasubramaniam, Baskar Mantovan, Elena Flach, Matthias Defense Date: 11 May 2009 Non-Caltech Author Email: abhishek.saha (AT) gmail.com Record Number: CaltechETD:etd-05222009-162600 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05222009-162600 DOI: 10.7907/5HWD-TH76 ORCID: Author ORCID Saha, Abhishek 0000-0002-2867-0386 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1964 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 29 May 2009 Last Modified: 26 Nov 2019 20:23 Thesis Files Preview PDF (thesis.pdf) - Final Version See Usage Policy. 614kB Repository Staff Only: item control page

On critical values of L-functions for holomorphic forms on GSp(4) × GL(2) Thesis by Abhishek Saha In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2009 (Defended May 11, 2009) ii c 2009 Abhishek Saha All Rights Reserved iii To my Parents iv Acknowledgements I would like to express my deepest gratitude to my advisor, Prof. Dinakar Ramakrishnan, for the opportunities he has given me. Without his enthusiastic support, helpful instructions and constant guidance, this work would not have been possible. I am thankful to the Department of Mathematics at Caltech for providing an excellent work environment and the staff and faculty here for their helpfulness and professionalism. I have made many wonderful friends at Caltech over the last five years and I will treasure the great times we spent together. I am grateful to them for their friendship, good cheer, encouragement and for contributing to my growth as an individual. Finally, I would like to thank my parents, who have been a tireless source of love and support to me throughout. This thesis is dedicated to them. v Abstract Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight `. We derive an explicit integral representation for the degree eight Lfunction L(s, F ×g). As an application, we prove a reciprocity law — predicted by Deligne’s conjecture — for the critical special values L(m, F × g) where m ∈ Z, 2 ≤ m ≤ 2` − 1. The proof of our integral representation has two major components: the generalization of an earlier integral representation due to Furusawa and a “pullback formula” relating the complicated Eisenstein series of Furusawa with a simpler one on a higher rank group. The critical value result follows from our integral representation using rationality results of Garrett and Harris and the theory of nearly holomorphic forms due to Shimura. vi Contents Acknowledgements iv Abstract v 1 Introduction 1 1.1 Overview and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Extending the Furusawa Integral Representation 10 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 The Rankin-Selberg integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Strategy for computing the p-adic integral . . . . . . . . . . . . . . . . . . . 17 2.4 The evaluation of the local Bessel functions in the Steinberg case . . . . . . 39 2.5 The case unramified πp , Steinberg σp . . . . . . . . . . . . . . . . . . . . . 47 2.6 The case Steinberg πp , Steinberg σp . . . . . . . . . . . . . . . . . . . . . . 54 2.7 The case Steinberg πp , unramified σp . . . . . . . . . . . . . . . . . . . . . 59 2.8 The global integral and some results . . . . . . . . . . . . . . . . . . . . . . 62 3 The Pullback Formula and the Second Integral Representation 79 3.1 Eisenstein series on GU (3, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Statement of the pullback formula . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 The local integral and the unramified calculation . . . . . . . . . . . . . . . 86 3.4 The local integral for the ramified and infinite places . . . . . . . . . . . . . 99 3.5 Proof of the Pullback formula . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.6 The integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 vii 4 Rationality of Eisenstein series and Deligne’s conjecture on critical Lvalues 124 4.1 Eisenstein series on Hermitian domains . . . . . . . . . . . . . . . . . . . . . 124 4.2 The integral representation in classical terms . . . . . . . . . . . . . . . . . 129 4.3 Nearly holomorphic Eisenstein series . . . . . . . . . . . . . . . . . . . . . . 131 4.4 Holomorphic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.5 Deligne’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Bibliography 140 1 Chapter 1 Introduction 1.1 Overview and main results If M is an arithmetic or geometric object, then we can often associate a very interesting invariant to it. This invariant is a complex analytic function known as the L-function for M and is denoted L(s, M). The simplest example of an L-function is the Riemann Zeta function defined by ζ(s) = ∞ X 1 n=1 n = s Y (1 − p−s )−1 p for Re(s) > 1 and by analytic continuation elsewhere. This function encodes rich number theoretic information about the distribution of primes. For instance, the fact that ζ(s) has no zeroes on the line Re(s) = 1 is equivalent to the prime number theorem. One can also associate L-functions to Dirichlet characters, number fields, elliptic curves and modular forms. The case of modular forms is particularly interesting because they, in a sense, include all the other examples mentioned so far. The classical modular forms are holomorphic functions that live on the upper-half plane and satisfy certain symmetries. These functions and their generalizations arise naturally from fundamental investigations in several mathematical areas. Indeed, there is a quote attributed to Martin Eichler saying that there are five basic operations in arithmetic: addition, subtraction, multiplication, division and modular forms. The theory of automorphic representations provides a natural setting in which to study modular forms and their generalizations. The fascinating work of Langlands associates L-functions to automorphic representations and use these to express deep relationships be- 2 tween number theory, geometry and representation theory. One of the tools that has been successfully used to study L-functions and their special values is the method of integral representations; this is sometimes called the Rankin-Selberg method after Rankin and Selberg’s fundamental work in this direction. Often, sharper and more explicit results are obtained when one restricts attention to holomorphic forms. The papers [GH93], [GK92], [HK91] treating the triple-product L-function are good examples, and in fact, provided an inspiration for this thesis. A conjecture of Deligne [Del73] predicts that certain special values of such families of L-functions are algebraic numbers up to multiplication by certain period integrals. Indeed, let L(s, M) be an arithmetically defined (or motivic) L-function associated to an arithmetic object M. Then Deligne’s conjectures assert that for certain critical points m, (a) L(m, M) is the product of a suitable transcendental number Ω and an algebraic number A(m, M), (b) If σ is an automorphism of C, then A(m, M)σ = A(m, Mσ ). To give a simple example, Deligne’s conjecture for the Riemann zeta function is just the fact that ζ(m)/π m is a rational number for all positive even integers m. In this thesis, we prove a key special case of Deligne’s conjecture when M corresponds to the product F × g where F is a Siegel modular form of genus two and g a classical modular form; see Theorem 3.6.1 below. As is often the case for such problems, the key ingredient in our proof is the interpretation of the transcendental factor as the period arising from a certain integral representation of Rankin Selberg type. Fix odd, squarefree integers M, N . Let F be a genus two Siegel newform of level M and g an elliptic newform of level N ; see § 2.8 for the precise definitions of these terms. One can associate a degree eight L-function L(s, F × g) to the pair (F, g). We assume that F and g have the same even integral weight ` and have trivial central characters. We also make the following assumption about F : Suppose F (Z) = X a(S)e(tr(SZ)) S>0 is the Fourier expansion; then we assume that 3  a(T ) 6= 0 for some T =  a b 2  b 2 (1.1.1) c √ such that −d = b2 − 4ac is the discriminant of the imaginary quadratic field L = Q( −d), and all primes dividing M N are inert in L. Given a Hecke character Λ of L, we define in §3.1 a Siegel Eisenstein series EΥ] (g, s) on GU (3, 3; L)(A). Let R denote the subgroup of GSp(4) × GU (1, 1; L) consisting of elements h = (h1 , h2 ) such that h1 ∈ GSp(4), h2 ∈ GU (1, 1; L) and h1 , h2 have the same multiplier. We define in §3.2 an embedding ι : R ,→ GU (3, 3; L). Let Φ, Ψ denote the adelizations of F, g respectively. We can extend the definition of Ψ to GU (1, 1; L)(A) by defining Ψ(ag) = Ψ(g) for all a ∈ L× (A), g ∈ GL(2)(A). Our integral representation is as follows. Theorem 3.6.1. We have Z 1 EΥ] (ι(g1 , g2 ), s)Φ(g1 )Ψ(g2 )Λ−1 (det g2 )dg = A(s)L(3s + , F × g) 2 g∈Z(A)R(Q)\R(A) where g = (g1 , g2 ), Λ is a suitable Hecke character of L and A(s) is an explicit normalizing factor, defined in §3.6. The first step towards proving Theorem 3.6.1 is achieved in §2 where we extend an integral representation due to Furusawa. Let π, σ be the automorphic representations corresponding to F, g respectively. Furusawa [Fur93] defined a certain integral of Rankin Selberg type and proved that it factorizes as a product of local zeta integrals Zp (s). He computed the local zeta integrals only in the case when the local representations πp , σp are both unramified and showed that they equal the local L-functions up to a normalizing factor. Thus, he was able to find an integral representation for L(s, F × g) in the full level case, i.e., when M = 1, N = 1. In §2 we compute, under certain conditions, the local integrals Zp (s) when the local representations are either Steinberg or unramified. This allows us to generalize the Furusawa integral representation to our case; see Theorem 2.8.7 for the precise statement. The generalization is not straightforward. The explicit evaluation of the local zeta integral involves several steps. First of all, we need to perform certain technical volume and double-coset computations. These computations — easy in the unramified case — are tedious and challenging for the remaining cases and are carried out in §2.3. Secondly, it is 4 necessary to suitably choose the sections of the Eisenstein series at the bad places to insure that the local zeta integrals do not vanish. Thirdly, and perhaps most crucially, the local computations require an explicit knowledge of the local Whittaker and Bessel functions. The formulae for the Whittaker model are well known in all cases; the same, however, is not true for the Bessel model. In fact, the only case where the local Bessel model for a finite place was computed before this work was when πp is unramified [BFF97, Sug85]. However, that does not suffice for the cases when we have πp Steinberg. As a preparation for the calculations in these cases, we find in §2.4, an explicit formula for the Bessel function for πp when it is Steinberg. We believe this is of independent interest. The second and final step towards proving Theorem 3.6.1 is achieved in §3. We prove a certain pullback formula (Theorem 3.2.1) that expresses our earlier Eisenstein series as the inner product of the cusp form and the pullback of the simpler higher-rank Siegel Eisenstein series EΥ] . Formulas in this spirit were first proved in a classical setting by Shimura [Shi97]. Unfortunately, Shimura only considers certain special types of Eisenstein series in his work which does not include ours (except in the full level case M = 1, N = 1). Furthermore his methods are classical and cannot be easily modified to deal with our case. The complicated sections at the ramified places and the need for precise factors make the adelic language the right choice for our purposes. We provide a complete proof of the pullback formula for our Eisenstein series which explicitly gives the precise factors at the ramified places needed by us. Combining the results of §2 and §3, we deduce Theorem 3.6.1. It seems appropriate to mention here that the referee of our paper [Sah09] has indicated that it was well-known to some experts that one could use such a pullback formula to rewrite the Furusawa integral representation. From Theorem 3.6.1, we easily conclude that L(s, F × g) is a meromorphic function whose only possible pole on the right of the critical line Re(s) = 21 is simple and at s = 1. Moreover, with the aid of basic techniques and results due to Garrett, Harris and Shimura, we prove the following Theorem. Theorem 4.5.1. Suppose that the Fourier coefficients of F and g are totally real and 5 algebraic and that ` ≥ 6. For a positive integer k, 1 ≤ k ≤ 2` − 2, define A(F, g; k) = L( 2` − k, F × g) . π 5`−4k−4 hF, F ihg, gi Then we have, (a) A(F, g; k) is algebraic (b) For an automorphism σ of C, A(F, g; k)σ = A(F σ , g σ ; k). We remark here that the completely unramified case M = 1, N = 1 of the above theorem was already known by the works of Heim [Hei99] and Böcherer–Heim [BH06], who used a very different integral representation from the one in this thesis. Also, just the algebraicity part of the above Theorem (i.e., part (a)) has been proved for the right-most critical value (corresponding to k = 1) in various settings earlier by Furusawa [Fur93], Pitale–Schmidt [PS09] and the author [Sah09]. To relate Theorem 4.5.1 to the conjecture of Deligne for motivic L-functions mentioned at the beginning of this introduction, we note that Yoshida [Yos01] has shown that the set of all critical points for L(s, F ×g) is {m : 2− 2` ≤ m ≤ 2` −1, m ∈ Z}. In particular, the critical points are always non-central (since the weight ` is even) and so the L-value is expected to be non-zero. Assuming the existence of a motive attached to F (this seems to be now known for our cases by the work of Weissauer [Wei05]) and the truth of Deligne’s conjecture for the standard degree 5 L-function of F , Yoshida also computes the corresponding motivic periods. According to his calculations, the relevant period for the point m is precisely the quantity π 4m+3`−4 hF, F ihg, gi that appears in our theorem above (once we substitute m = 2` − k). We note here that Yoshida only deals with the full level case; however, as the periods remain the same (up to a rational number) for higher level, his results remain applicable to our case. Thus, Theorem 4.5.1 is compatible with (and implied by) Deligne’s conjecture, and furthermore, it covers all the critical values to the right of Re(s) = 21 except for the L-value at the point 1. The proof for the critical values to the left of Re(s) = 12 would follow from the expected functional equation. Extending our result to L(1, F × g) is intimately connected to proving the analyticity of the L-function at that point (see Corollary 3.6.2). 6 These questions, related to analyticity and the functional equation are also of interest for other applications and will be considered in a future paper. We also note that the integral representation (Theorem 3.6.1) is of interest for several other applications. For instance, we hope that this integral representation will pave the way to certain new results involving stability, hybrid subconvexity, and non-vanishing results for the L-function under consideration following the methods of [MR]. We are also hopeful that we can prove results related to non-negativity of the central value L( 12 , F ). These results appear to be new for holomorphic Siegel modular forms. For example, the nonnegativity result is known in the case of generic automorphic representations by Lapid and Rallis [LR02]; however, automorphic representations associated to Siegel modular forms are never generic. Another interesting application of the integral representation would be to the construction of p-adic L-functions. We intend to address these questions elsewhere. We expect most of the results of this thesis to hold for arbitrary totally real base fields. It would be particularly interesting to work out the special value results when the HilbertSiegel modular forms have different weights for each Archimedean place. This case will be considered in a future work. 1.2 Notation • The symbols Z, Z≥0 , Q, R, C, Zp and Qp have the usual meanings. A denotes the ring of adeles of Q, Af the finite adeles. For a complex number z, e(z) denotes e2πiz . • For a matrix M we denote its transpose by M t . Denote by Jn the 2n by 2n matrix given by  0 Jn =  −In In 0  .   0 1 0 0     1 0 0 0 . We use J to denote J2 and let s1 denote the matrix    0 0 0 1   0 0 1 0 7 • For a positive integer n define the group GSp(2n) by GSp(2n, R) = {g ∈ GL2n (R)|g t Jn g = µn (g)Jn , µn (g) ∈ R× } for any commutative ring R. Define Sp(2n) to be the subgroup of GSp(2n) consisting of elements g1 ∈ GSp(2n) with µn (g1 ) = 1. • For a quadratic extension L of Q define GU (n, n) = GU (n, n; L) by GU (n, n)(Q) = {g ∈ GL2n (L)|(g)t Jn g = µn (g)Jn , µn (g) ∈ Q× } where g denotes the conjugate of g. e = GU (3, 3), H e 1 = U (3, 3), H = GSp(6), H1 = Sp(6), G e = GU (2, 2), G e1 = • Let H U (2, 2), G = GSp(4), G1 = Sp(4), Fe = GU (1, 1), Fe1 = U (1, 1). • Define e n = {Z ∈ Mn (C)|i(Z − Z) is positive definite}, H Hn = {Z ∈ Mn (C)|Z = Z t , i(Z − Z) is positive definite}.   A B e n define  ∈ GU (n, n)(R), Z ∈ H For g =  C D J(g, Z) = CZ + D. The same definition works for g ∈ GSp(2n)(R), Z ∈ Hn . • For a commutative ring R we denote by I(2n, R) the Borel subgroup of GSp(2n, R)  A B  where A is lowerconsisting of the set of matrices that look like  t −1 0 λ(A ) × triangular and λ ∈ R . 8 • For a quadratic extension L of Q and v be a finite place of Q, define Lv = L ⊗Q Qv . ZL denotes the ring of integers of L and ZL,v its v-closure in Lv .For a prime p, let Z× L,p denote the group of units in ZL,p . √ If p is inert in L, the elements of Z× L,p are of the form a + b −d with a, b ∈ Zp and such that at least one of a and b is a unit. Let Γ0L,p be the subgroup of Z× L,p consisting of the elements with p|b. • For a positive integer N the subgroups Γ0 (N ) and Γ0 (N ) of SL2 (Z) are defined by  Γ0 (N ) = {A ∈ SL2 (Z) | A ≡  ∗ ∗   0 ∗  ∗ 0  Γ0 (N ) = {A ∈ SL2 (Z) | A ≡  ∗ ∗ (mod N )},  (mod N )}. For p a finite place of Q, their local analogues Γ0,p (resp. Γ0p ) are defined by   ∗ ∗  Γ0,p = {A ∈ GL2 (Zp ) | A ≡  0 ∗ (mod p)},   ∗ 0  Γ0p = {A ∈ GL2 (Zp ) | A ≡  ∗ ∗ (mod p)}. • The local Iwahori subgroup Ip is defined to be the subgroup of Kp = G(Zp ) consisting of those elements of Kp that when reduced mod p lie in the Borel subgroup of G(Fp ). Precisely,  ∗ 0 ∗ ∗      ∗ ∗ ∗ ∗   Ip = {A ∈ Kp | A ≡   0 0 ∗ ∗    0 0 0 ∗ (mod p)} e denote the subgroup of G e × Fe consisting of elements h = (h1 , h2 ) such that • Let R e h2 ∈ Fe and µ2 (h1 ) = µ1 (h2 ). Let R denote the subgroup of R e consisting of h1 ∈ G, those (h1 , h2 ) where h1 ∈ G. e For a fixed element g ∈ G(A), let Fe1 [g](A) denote the subset of Fe(A) consisting of all 9 elements h2 such that µ2 (g) = µ1 (h2 ). 10 Chapter 2 Extending the Furusawa Integral Representation 2.1 Preliminaries Bessel models We recall the definition of the Bessel model of Novodvorsky and Piatetski-Shapiro [NPS73] following the exposition of Furusawa [Fur93]. Let S ∈ M2 (Q) matrix. We let disc(S) = −4det(S) andput d =  be a symmetric  a b/2 b/2 c  then we define the element ξ = ξS =  . −disc(S). If S =  b/2 c −a −b/2 √ Let L denote the subfield Q( −d) of C. We always identify Q(ξ) with L via p Q(ξ) 3 x + yξ 7→ x + y ( − d) ∈ L, x, y ∈ Q. 2 (2.1.1) We define a subgroup T = TS of GL2 by T (Q) = {g ∈ GL2 (Q)|g t Sg = det(g)S}. (2.1.2) The center of T is denoted by ZT . It is not hard to verify that T (Q) = Q(ξ)× and ZT (Q) = Q× . We identify T (Q) with L× via (2.1.1)). 11 We can consider T as a subgroup of G via   g 0  ∈ G. T 3g→ 7  0 det(g).(g −1 )t (2.1.3) Let us denote by U the subgroup of G defined by  U = {u(X) =  12 X 0 12   |X t = X}. Let B be the subgroup of G defined by R = T U . Let ψ be a non trivial character of A/Q. We define the character θ = θS on U (A) by θ(u(X)) = ψ(tr(S(X))). Let Λ be a character of T (A)/T (Q) such that Λ|ZT (A× ) = 1. Via (2.1.1) we can think of Λ as a character of L× (A)/L× such that Λ|A× = 1. Denote by Λ⊗θ the character of B(A) defined by (Λ⊗θ)(tu) = Λ(t)θ(u) for t ∈ T (A) and u ∈ U (A). Let π be an automorphic cuspidal representation of G(A) with trivial central character and Vπ be its space of automorphic forms. Then for Φ ∈ Vπ , we define a function BΦ on G(A) by Z BΦ (h) = (Λ ⊗ θ)(r)−1 Φ(rh)dr. (2.1.4) B(A)/B(Q)ZG (A) e The C - vector space of function on G(A) spanned by {BΦ |Φ ∈ Vπ } is called the global Bessel space of type (S, Λ, ψ) for π. We say that π has a global Bessel model of type (S, Λ, ψ), if the global Bessel space has positive dimension, that is if there exists Φ ∈ Vπ such that BΦ 6= 0. In §2.1–§2.7 , we assume that: There exists S, Λ, ψ such that π has a global Bessel model of type (S, Λ, ψ). (2.1.5) Eisenstein series We briefly recall the definition of the Eisenstein series used by Furusawa in [Fur93]. Let e consisting of the elements in G e that look P be the maximal parabolic subgroup of G  ∗   0 like  0  0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 12  ∗   ∗ . We have the Levi decomposition P = M N with M = M (1) M (2) where  ∗  ∗ the groups M, N, M (1) , M (2) are as defined in [Fur93]. Precisely,   a        0 (1) M (Q) =     0       0   1        0 M (2) (Q) =     0       0 0 0 α 0 0 λ γ 0 0 0 1 0    0         0 × |a∈L ≃ L× .    0       1 0 a−1 0 0   0           β α β α β  |  ∈ GU (1, 1)(Q), λ = µ1     0 γ δ γ δ        δ (2.1.6)  (2.1.7) ≃ GU (1, 1)(Q).   1        0 N (Q) =     0       0 x 1 0 0  1 0 0   0 0 0   1 0 0  0 −x 1 0 a 1 y 0 1 0 0   y         0  | a ∈ Q, x, y ∈ L .    0       1  (2.1.8) We also write  a   0 m(1) (a) =   0  0 0 0 1 0  0   0 ,  0  1 0 a−1 0  0 1 0 0 0      0 α 0 β  α β (2)   .  m =  0 0 λ 0  γ δ   0 γ 0 δ   Let σ be an irreducible automorphic cuspidal representation of GL2 (A) with central char- 13 acter ωσ . Let χ0 be a character of L× (A)/L× such that χ0 | A× = ωσ . Finally, let χ be a character of L× (A)/L× = M1 (A)M1 (Q) defined by χ(a) = Λ(a)−1 χ0 (a)−1 . (2.1.9) Π(m1 m2 ) = χ(m1 )(χ0 ⊗ σ)(m2 ), m1 ∈ M1 (A), m2 ∈ M2 (A) (2.1.10) Then defining we extend σ to an automorphic representation Π of M (A). We regard Π as a representation of P (A) by extending it trivially on N (A). Let δP denote the modulus character of P . If p = m1 m2 n ∈ P (A) with mi ∈ Mi (A)(i = 1, 2) and n ∈ N (A), δP (p) = |NL/Q (m1 )|3 · |µ1 (m2 )|−3 , (2.1.11) where || denoted the modulus function on A. e Then for s ∈ C, we form the family of induced automorphic representations of G(A) e G(A) I(Π, s) = IndP (A) (Π ⊗ δPs ) (2.1.12) where the induction is normalized. Let f (g, s) be an entire section in I(Π, s) viewed cone cretely as a complex-valued function on G(A) which is left N (A)-invariant and such that e for each fixed g ∈ G(A), the function m 7→ f (mg, s) is a cusp form on M (A) for the automorphic representation Π ⊗ δPs . Finally we form the Eisenstein series E(g, s) = E(g, s; f ) by E(g, s) = X f (γg, s) (2.1.13) e γ∈P (Q)\G(Q) e for g ∈ G(A). This series converges absolutely (and uniformly in compact subsets) for Re(s) > 1/2, has a meromorphic extension to the entire plane and satisfies a functional equation (see [Lan76, Fur93] ). 14 2.2 The Rankin-Selberg integral The global integral The main object of study in this chapter is the following global integral of Rankin-Selberg type Z E(g, s, f )Φ(g)dg, Z(s) = Z(s, f, Φ) = (2.2.1) G(Q)ZG (A)\G(A) where Φ ∈ Vπ and f ∈ I(Π, s). Z(s) converges absolutely away from the poles of the Eisenstein series. e Let Θ = ΘS be the following element of G(Q)  1 0 0 0    √   α 1 0 0  b + −d   Θ=  where α = 2c  0 0 1 −α   0 0 0 1 The ‘basic identity’ proved by Furusawa in [Fur93] is that Z Z(s) = Wf (Θh, s)BΦ (h)dh (2.2.2) B(A)\G(A) e where for g ∈ G(A) we have  1 0 0 0         0 1 0 x      Wf (g, s) = f   g, s ψ(cx)dx, A/Q 0 0 1 0      0 0 0 1 Z (2.2.3) and BΦ is the Bessel model of type (S, Λ, ψ) defined in § 2.1. The local integral In this section v refers to any place of Q. Let π = ⊗v πv and σ = ⊗v σv . Now suppose that Φ and f are factorizable functions with Φ = ⊗v Φv and f (·, s) = ⊗v fv (·, s). 15 By the uniqueness of the Whittaker and the Bessel models, we have Wf (g, s) = Y Wf,v (gv , s) (2.2.4) BΦ,v (hv , s) (2.2.5) v BΦ (h) = Y v e for g = (gv ) ∈ G(A) and h = (hv ) ∈ G(A) and local Whittaker and Bessel functions Wf,v , BΦ,v respectively. Henceforth we write Wv = Wf,v , Bv = BΦ,v when no confusion can arise. Therefore our global integral breaks up as a product of local integrals Z(s) = Y Zv (s) (2.2.6) v where Z Wv (Θg, s)Bv (g)dg. Zv (s) = Zv (s, Wv , Bv ) = B(Qv )\G(Qv ) The unramified case The local integral is evaluated in [Fur93] in the unramified case. We recall the result here. Suppose that the characters ωπ , ωσ , χ0 are trivial. Now let q be a finite prime of Q such that (a) The local components πq , σq and Λq are all unramified. (b) The conductor of ψq is Zq .   a b/2  ∈ M2 (Zq ) with c ∈ Z× (c) S =  q . b/2 c (d) −d = b2 − 4ac generates the discriminant of Lq /Qq . Since σq is spherical, it is the spherical principal series representation induced from unramified characters αq , βq of Q× q . Suppose M0 is the maximal torus (the group of diagonal matrices) inside G and P0 the Borel subgroup containing M0 as Levi component. πq is a spherical principal series repreM (Q ) sentation, so there exists an unramified character γq of M0 (Qq ) such that πq = IndP00(Qqq) γq 16 (i) , (where we extend γq to P0 trivially). We define characters γq (i = 1, 2, 3, 4) of Q× q by  x 0 0 0   0 (1) γq (x) = γq   0  0  1   0 γq(3) (x) = γq   0  0 x 0 0 1 0 0 0 0 1 0 0 x 0 0    0 ,  0  1  0   0 ,  0  x  x   0 γq(2) (x) = γq   0  0  1   0 γq(4) (x) = γq   0  0 0 0 0 1 0 0 1 0 0 0 0 x 0 0 x 0 0    0 ,  0  x  0   0 .  0  1 fq − spherical vector in Iq (Πq , s) and Φq be Now let fq ( , s) be the unique normalized K the unique normalized Kq − spherical vector in πq . Let Wq , Bq be the coresponding vectors in the local Whittaker and Bessel spaces. The following result is proved in [Fur93] Theorem 2.2.1 (Furusawa). Let ρ(Λq ) denote the Weil representation of GL2 (Qq ) corresponding to Λq . Then we have Zq (s, Wq , Bq ) = L(3s + 12 , πq × σq ) L(6s + 1, 1)L(3s + 1, σq × ρ(Λq )) where, L(s, πq × σq ) = 4  Y −1 (1 − γq(i) αq (q)q −s )(1 − βq(i) αq (q)q −s ) , i=1 L(s, 1) = (1 − q −s )−1 , = L(s, σq × ρ(Λq ))   (1 − αq2 (q)q −2s )−1 (1 − βq2 (q)q −2s )−1               (1 − αq (q)Λq (q1 )q −s )−1 (1 − βq (q)Λq (q1 )q −s )−1           (1 − αq (q)Λq (q1 )q −s )−1 (1 − βq (q)Λq (q1 )q −s )−1       ·(1 − αq (q)Λ−1 (q1 )q −s )−1 (1 − βq (q)Λ−1 (q1 )q −s )−1 q q if q is inert in L, if q is ramified in L, if q splits in L, 17 where q1 ∈ Zq ⊗Q L is any element with NL/Q (q1 ) ∈ qZ× q . 2.3 Strategy for computing the p-adic integral Throughout this section we fix an odd prime p in Q such that p is inert in L. Moreover, we assume that S ∈ M2 (Zp ). The fact that p is inert in L implies that if w, z are elements of Zp then w + zξ ∈ (T (Qp ) ∩ Kp ) if and only if at least one of w, z is an unit. Moreover the additional assumption S ∈ M2 (Zp ) forces that a, c are units in Zp . An explicit set of coset representatives Recall the Iwahori subgroup Ip . It will be useful to describe a set of coset representatives of Kp /Ip . But first some definitions. Let Y be the set {0, 1, .., p − 1}. Let V = Y ∪ {∞} where ∞ is just a convenient formal symbol.  1   0 For x = (n, q, r) ∈ Z3p , let Ux ∈ U (Qp ) be the matrix   0  0   1 y 0 0     0 1 0 0   ∈ Kp . For y ∈ Zp define Zy =   0 0 1 0   0 0 −y 1   0 1 0 0     1 0 0 0  ∈ Kp .  Also define Z∞ =   0 0 0 1   0 0 1 0 0 n q    r .  0 1 0  0 0 1 1 q In particular, the definitions Ux , Zy make sense for x ∈ Y 3 , y ∈ V . Now we define the following three classes of matrices. We call them matrices of class A, class B and class D, respectively. 18 (a) For x = (n, q, r) ∈ Y 3 , y ∈ V , let Ayx = Ux JZy . (b) For x = (n, q, r) ∈ Y 3 with q 2 − nr ≡ 0 (mod p) and y ∈ V , let Bxy = JUx JZy .     0 1   −λ 0           −1    1 0 0 λ        Zy if λ 6= 0, ∞,      −1   0 1 λ 0              0 λ −1 0                    0 0 0 1           −1 0 0 0   y (c) For λ, y ∈ V , let Dλ = if λ = 0,   Zy        0 1 0 0            0 0 1 0                   −1 0 0 0             0 0 0 1      if λ = ∞.   Zy        0 0 1 0            0 1 0 0 Let S be the set obtained obtained by taking the union of the class A, class B and class S S D matrices, precisely S = {Ayx } y∈V {Bxy }y∈V,x=(n,q,r)∈Y 3 {Dλy }λ∈V . Clearly S has x∈Y 3 q 2 −nr≡0 (mod p) 3 2 2 cardinality p (p + 1) + p (p + 1) + (p + 1) = (p + 1)2 (p2 + 1). y∈V Lemma 2.3.1. S is a complete set of coset representatives for Kp /Ip . Proof. Let us first verify that S has the right cardinality. Clearly the cardinality of Kp /Ip is the same as the cardinality of G(Fp )/I(4, Fp ). By [Kim98, Theorem 3.2], |G(Fp )| = p4 (p − 1)3 (p + 1)2 (p2 + 1). On the other hand I(4) has the Levi-decomposition    12 X   I(4) =  0 v.(g −1 )t 0 12 g 0 19 with g upper-triangular, X symmetric and v ∈ GL(1). So |I(4, Fp )| = p4 (p − 1)3 . Thus |G(Fp )/I(4, Fp )| = (p + 1)2 (p2 + 1) which is the same as the cardinality of S. So it is enough to show that no two matrices in S lie in the same coset. For a 2 × 2 matrix H with coefficients in Zp , we may reduce H mod p and consider the Fp -rank of the resulting matrix;  we denote this quantity by rp (H). It is easy to see that  if the matrix A =  A1 A2 A3 A4  varies in a fixed coset of Kp /Ip , the pair (rp (A1 ), rp (A3 )) remains constant. Observe now that if A is of class A, then rp (A3 ) = 2; for A of class B, rp (A3 ) < 2 and rp (A1 ) = 2; while for A of class D we have rp (A3 ) < 2, rp (A1 ) < 2. This proves that elements of S of different classes cannot lie in the same coset. Now we consider distinct elements of S of the same class, and show that they too must lie in different cosets. For x1 = (n1 , q1 , r1 ), x2 = (n2 , q2 , r2 ) ∈ Y 3 , y1 , y2 ∈ Y , consider the elements Ayx11 , Ayx22 , Bxy11 , Bxy22 of S. We have (Ayx11 )−1 Ayx22 = (Bxy11 )−1 Bxy22 =  y2 − y1 1 0 0       0 1 0 0  .    −n2 + n1 −n2 y2 − q2 + n1 y2 + q1 1 0   y1 (n1 − n2 ) + q1 − q2 y1 y2 (n1 − n2 ) + (y1 + y2 )(q1 − q2 ) − r2 + r1 y1 − y2 1 So if the above matrix belongs to Ip , we must have y1 = y2 , n1 = n2 . That leads to q1 = q2 , and finally by looking at the bottom row we conclude r1 = r2 . This covers the case of class A and class B matrices in S whose y-component is not equal to ∞. y1 −1 ∞ Now (Ayx11 )−1 A∞ x2 = (Bx1 ) Bx2 =  −y1 1 0 0       1 0 0 0      q1 − q2 n1 − n2 0 1   q1 y1 + r1 − y1 q2 − r2 n1 y1 + q1 − y1 n2 − q2 1 y1 (2.3.1) 20 which cannot belong to Ip . −1 ∞ ∞ −1 ∞ Also (A∞ x1 ) Ax2 = (Bx1 ) Bx2 =  1 0    0 1   −r2 + r1 −q2 + q1  −q2 + q1 −n2 + n1  0 0   0 0   1 0  0 1 (2.3.2) and if the above matrix lies in Ip we must have x1 = x2 . Thus we have completed the proof for class A and class B matrices. To complete the proof of the lemma we need to show that no two class D matrices are in the same coset. The calculations for that case are similar to those above and are therefore omitted. Reducing the integral to a sum By [Fur93, p. 201]) we have the following disjoint union G(Qp ) = a B(Qp ) · h(l, m) · Kp (2.3.3) l∈Z 0≤m∈Z where  p2m+l    0 h(l, m) =    0  0 0 0 0    0 .  1 0  0 pm pm+l 0 0 0 We wish to compute Z Zp (s) = Wp (Θh, s)Bp (h)dh. (2.3.4) B(Qp )\G(Qp ) By (2.3.3) and (2.3.4) we have Zp (s) = X Z l∈Z,m≥0 B(Qp )\B(Qp )h(l,m)Kp Wp (Θh, s)Bp (h)dh. (2.3.5) 21 For m ≥ 0 we define the subset Tm of S by 0 ∞ 0 ∞ 0 ∞ Tm = {B(1,0,0) , B(1,0,0) , B(0,0,1) , B(0,0,1) , B(0,0,0) , B(0,0,0) , A0(0,0,0) , A∞ (0,0,0) } if m > 0, 0 ∞ 0 T0 = {B(1,0,0) , B(1,0,0) , B(0,0,0) , A0(0,0,0) }. 0 ∞ Also, we use the notation t1 = B(1,0,0) , t2 = B(1,0,0) , ..., t8 = A∞ (0,0,0) . Thus Tm = {ti |1 ≤ i ≤ 8} if m > 0 and T0 = {t1 , t2 , t5 , t7 }. Proposition 2.3.2. Let l ∈ Z, m ≥ 0. Then we have B(Qp )\B(Qp )h(l, m)Kp = a B(Qp )\B(Qp )h(l, m)tIp . t∈Tm Proof. Define two elements f and g in Kp to be (l, m)-equivalent if there exists r ∈ B(Qp ) and k ∈ Ip such that rh(l, m)f k = h(l, m)g. Furthermore observe that if two elements of Kp are congruent mod p then they are in the same Ip -coset and therefore are trivially (l, m)-equivalent. The proposition can be restated as saying that any s ∈ S is (l, m)-equivalent to exactly one of the elements t with t ∈ Tm . This will follow from the following nine claims which we prove later below. Claim 1. Any class A matrix in S by left-multiplying by an appropriate element of U (Zp ) can be made congruent mod p to Ay(0,0,0) for some y ∈ V . Claim 2. If m > 0 all the Ay(0,0,0) , y ∈ V \ {0} are (l, m)-equivalent. In the case m = 0 all the Ay(0,0,0) , y ∈ V are (l, 0)-equivalent. Claim 3. Any class B matrix in S by left-multiplying by an appropriate element of U (Zp ) can be made congruent mod p to one of the matrices y y −λ ∞ B(1,λ,λ 2 ) , B(1,λ,λ2 ) B(0,0,1) , B(0,0,0) , where λ ∈ Y, y ∈ V . 22 y Claim 4. The matrices B(1,λ,λ 2 ) , λ ∈ Y, y ∈ {−λ, ∞} are all (l, m)-equivalent to one of the y matrices B(1,0,0) , y ∈ {0, ∞}. y Claim 5. The matrices B(0,0,1) , y ∈ V by left-multiplying by an appropriate element of y U (Zp ) can be made equal to one of the matrices B(0,0,1) with y ∈ {0, ∞}. y y Claim 6. The matrices B(0,0,0) , y ∈ V are (l, m)-equivalent to one of the matrices B(0,0,0) with y ∈ {0, ∞}. In the case m = 0 these two matrices are also equivalent. 0 ∞ Claim 7. The matrices B(1,0,0) , B(0,0,1) are (l, 0)-equivalent and the matrices ∞ 0 B(1,0,0) , B(0,0,1) are also (l, 0)-equivalent. Claim 8. Any class D matrix Dλy by left-multiplying by an appropriate element of U (Zp ) can be made equal to a class B matrix. Claim 9. No two elements of Tm are (l, m)-equivalent for any m ≥ 0. Indeed claims 1, 2 imply that any class A matrix is (l, m)-equivalent to one of t7 , t8 (and when m = 0, t7 alone suffices). On the other hand claims 3, 4, 5, 6, 7 tell us that any class B matrix is (l, m)-equivalent to one of the ti , 1 ≤ i ≤ 6 (and that just t1 , t2 , t5 suffice if m = 0). Also claim 8 says that any class D matrix is also (l, m)-equivalent to one of the above. Since the class A, class B and class D matrix exhaust S, this shows that any element of S is (l, m)-equivalent to some element of Tm ; in other words we do have the union stated in Proposition 2.3.2. Finally claim 9 completes the argument by implying that the union is indeed disjoint. We now prove each of the above claims. The proofs are just computations, we simply multiply by suitable elements of R to get the results we desire. Proof of claim 1. This follows from the fact that U−x Ayx ≡ JZy (mod p) and JZy = Ay(0,0,0) . Proof of claim 2. We first deal with the case m = 0. For y ∈ V, y 6= 0 let j = (− ay + b 2 ) + ξ ∈ (T (Qp ) ∩ Kp ) (here and elsewhere we interpret 1/∞ = 0). Consider the element 23 (A0(0,0,0) )−1 h(l, 0)−1 jh(l, 0)Ay(0,0,0) . By direct calculation this equals  − ay 0 0    −c    0  0 −cy 2 +a−yb y 0 0 − cy +a−yb y 0 0 0 2    0    c   − ay if y 6= ∞ and equals  a 0 0 0       b −c 0 0      0 0 c b    0 0 0 −a if y = ∞. Both of these matrices lie in Ip and this proves the claim for m = 0. Now consider m > 0. For y ∈ V, y 6= 0, ∞, let j = cy + pm ξ ∈ (T (Qp ) ∩ Kp ). Consider −1 h(l, m)−1 j h(l, m)Ay the element (A∞ (0,0,0) ) (0,0,0) , which by direct calculation equals  −c  m  cy − bp2    0  0 bpm 2 ypm b 2 cy − 2 + p2m a 0 0 −p2m a − cy 2 + yp2 b 0 bpm 2 0 0      m  cy − bp2   c 0 m and this lies in Ip . Thus Ay0,0,0 is (l, m)-equivalent to A∞ 0,0,0 and this completes the proof of the claim. Proof of claim 3. Before proving this claim let us make a small remark. If λ ∈ Y is such y that λ2 does not belong to Y one may ask what we mean by the notation B(1,λ,λ 2 ) ; in such a case, we understand λ2 to refer to the unique element in Y that is congruent to λ2 mod p. This convention will govern any such situation. y We now begin proving the claim. Given a class B matrix B(n,q,r) with n 6= 0 we must have q ≡ nλ, r ≡ nλ2 (mod p) for some appropriate λ ∈ Y . 1 First assume that y 6= −λ. If y 6= ∞ put s = λ−y+n(λ+y) , t = − n(y+λ) , u = 0 and check n(y+λ) 24 y ∞ that (Bn,q,r )−1 Us,t,u B1,λ,λ 2 is congruent mod p to  0 1 − n(y+λ) 1 n(y+λ) 0 0 1 − y+λ 0 0 −λ − y   λ+n(y+λ)  n(y+λ)    0  0  λ+n(y+λ) n(y+λ)   1  − n(y+λ) . λ+n(y+λ)   y+λ  n(y + λ) ∞ −1 ∞ If y = ∞ put s = n−1 n , t = 0, u = 0 and check that (Bn,q,r ) Us,t,u B1,λ,λ2 is congruent to  1   (n−1)λ − n    0  0 0 0 1 n 0 0 0  0    .  1 (n − 1)λ  0 n n−1 n Both of these matrices belong to Ip . −λ −λ −1 Now suppose that y = −λ. Put s = n−1 n , t = 0, u = 0 and observe that (B(n,q,r) ) U(s,t,u) B(1,λ,λ2 ) is congruent mod p to  0 n−1 n  0   0  0 1 0 0 n 0 0 1 n  0   0   0  1 which belongs to Ip also. Finally assume that n = 0. So q = 0 as well. If r = 0 there is nothing to prove. So supy y −1 pose r 6= 0. If y 6= ∞ put s = 0, t = y(r−1) , u = r−1 r r and observe that (B(0,0,r) ) U(s,t,u) B(0,0,1) equals  2) 1 0 − (r−1)y r   0 1r 0   0 0 1  0 0 0 0    0   0  r y y −1 which belongs to Ip . If y = ∞ put s = 0, t = 0, u = r−1 r and observe that (B(0,0,r) ) U(s,t,u) B(0,0,1) 25 equals  1 r 0 r−1 r  0 1   0 0  0 0 0 r 0  0   0   0  1 which belongs to Ip . Thus the claim is proved. Proof of claim 4. First suppose that y = ∞. If m > 0, put j = c/λ + pm ξ ∈ (T (Qp ) ∩ ∞ ∞ Kp ). By direct calculation we verify that (B(1,0,0) )−1 h(l, m)−1 jh(l, m)B(1,λ,λ 2 ) is congruent mod p to  c λ  c   0  0 0 0 c λ 0 0 c λ 0 0 0    0   −c  c λ and this belongs to Ip . + ξ ∈ (T (Qp ) ∩ Kp ) and check that if we let n ∈ Y \ 0 be the If m = 0, put j = (2c−bλ) 2λ 2 element congruent mod p to aλ −bλ+c and y ∈ Y \ 0 be the element congruent mod p to c y c ∞ − aλ then (B(n,0,0) )−1 h(l, 0)−1 jh(l, 0)B(1,λ,λ 2 ) is congruent  c(aλ2 −bλ+c) aλ2        c−bλ λ 0 0 0 −a 0 0 0 0 a 0 0 0 mod p to c−bλ λ c(aλ2 −bλ+c) − aλ2         y ∞ which lies in Ip . Hence B(1,λ,λ 2 ) is (l, 0)-equivalent to B(n,0,0) and by the proof of Claim 3 ∞ it follows that it is (l, 0)-equivalent to B(1,0,0) . Now assume that y = −λ. If λ = 0 there is nothing to prove so assume λ 6= 0. If m > 0, put j = c + pm λξ ∈ (T (Qp ) ∩ Kp ). By a direct calculation we see that −λ −λ 0 (B(1,0,0) )−1 h(l, m)−1 jh(l, m)B(1,λ,λ 2 ) is congruent to cI4 (mod p) and thus B(1,λ,λ2 ) is (l, m)0 + ξ ∈ (T (Qp ) ∩ Kp ) and check that if we let equivalent to B(1,0,0) . If m = 0, put j = (2c−bλ) 2λ 2 n ∈ Y \ 0 be the element that is congruent mod p to aλ −bλ+c then, c 26 −λ 0 (B(n,0,0) )−1 h(l, 0)−1 jh(l, 0)B(1,λ,λ 2 ) is congruent mod p to  c  λ  −a    0  0 0 0 0 aλ2 −bλ+c λ 0 0 aλ2 −bλ+c λ 0 0    0   a  c λ −λ 0 which lies in Ip . Hence B(1,λ,λ 2 ) is (l, 0)-equivalent to B(n,0,0) and by Claim 3 it follows that 0 it is (l, 0)-equivalent to B(1,0,0) . Proof of claim 5. If y = ∞ there is nothing to prove. So assume y ∈ Y . Put s = 0, t = y 0 −y, u = 0 and observe that (B(0,0,1) )−1 U(s,t,u) B(0,0,1) equals  1   0   0  0 y2 0 1 −y 0 1 0 0 −y    0    0   1 which is in Ip . Proof of claim 6. First consider the case m > 0. Take y ∈ V \{0, ∞} and let j = c/y+pm ξ ∈ (T (Qp ) ∩ Kp ). By direct calculation we verify that y 0 (B(0,0,0) )−1 h(l, m)−1 jh(l, m)B(0,0,0) y 0 is (l, m)-equivalent to B(0,0,0) is congruent mod p to yc I4 and so B(0,0,0) . Now let m = 0. Take y ∈ V \ {0, ∞} and let j = c/y + b/2 + ξ ∈ (T (Qp ) ∩ Kp ). By y 0 direct calculation we verify that (B(0,0,0) )−1 h(l, 0)−1 jh(l, 0)B(0,0,0) equals  ay 2 +by+c y  0 0 0 −a c y 0 0 0 0 c y       0 which lies in Ip . 0 0 a ay 2 +by+c y         27 Finally, if we take j = b/2 + ξ ∈ (T (Qp ) ∩ Kp ) we can verify that ∞ 0 (B(0,0,0) )−1 h(l, 0)−1 jh(l, 0)B(0,0,0)  −a 0    b =   0  0 0 0    0   0 −c b   0 0 a c 0 which lies in Ip . Proof of claim 7. Putting j = 2b + ξ and s = 1 − ac , t = 0, u = 0 we verify that ∞ 0 (B(0,0,1) )−1 h(l, 0)−1 jU(s,t,u) h(l, 0)B(1,0,0)  −c   bc a =  0  0  0 c−a 0 c b(a−c) a 0 −a 0 0   0   b  a which lies in Ip . Putting j = − 2b + ξ and u = 1 − ac , t = 0, s = 0 we verify that ∞ 0 (B(0,0,1) )−1 h(l, 0)−1 jU(s,t,u) h(l, 0)B(1,0,0)  −a − ba c    0 =   0  0 0 a 0 0 −c 0 −b  b(a−c) c   c − a   0   c which lies in Ip . Proof of claim 8. Suppose y 6= ∞, λ 6= 0, ∞. Put s = 1 − 2λy, t = y, u = 0 and check that 28 ∞ (Dλy )−1 U(s,t,u) B(1,λ,λ 2 ) is congruent mod p to  −1   λ   0  0  y λ 0 1 1 − 2λ 0 1 2 0 0 1−2λy 2λ   − 12  . 1   − 2λ  − 12 Now suppose y 6= ∞ and put s = 1, t = −y, u = 0 and check that ∞ (D0y )−1 U(s,t,u) B(1,0,0)  1   0 =  0  0 0 0 0    −1 .  0 −1 0   0 0 −1 1 0 y −1 0 ) U(s,t,u) B(0,0,1) equals Next put s = 0, t = −y, u = 1 and check that (D∞  1 0 0 0      0 1 0 −1  .   0 0 −1 0    0 0 0 −1 −λ Now let λ 6= 0, ∞. Put s = 1, t = 0, u = 0 and check that (Dλ∞ )−1 U(s,t,u) B(1,λ,λ 2 ) is congruent mod p to  1 0 1 −1 − 2λ   1 0 −1 2λ   0 0 − 12  0 0 0    0  .  0   1 2 0 Next put s = 1, t = 0, u = 0 and check that (D0∞ )−1 U(s,t,u) B(1,0,0) is congruent mod p 29 to  1 0 −1 0      0 1 0 0 .    0 0 −1 0    0 0 0 −1 ∞ )−1 U ∞ Finally put s = 0, t = 0, u = 1 and check that (D∞ (s,t,u) B(0,0,1) is congruent mod p to  1 0 −1 0      0 1 0 0 .    0 0 −1 0    0 0 0 −1 Proof of claim 9. Suppose two elements f and g in Tm are (l, m)-equivalent. Then there exists r ∈ B(Qp ) and k ∈ Ip such that rh(l, m)f k = h(l, m)g. Denote r0 = h(l, m)−1 rh(l, m) 0 f k. Then r 0 is upper-triangular and belongs to K . Writing f, g in block form so that p  g = r   f1 f2 g g , g =  1 2  we conclude as in the proof of Lemma 2.3.1 that the Fp -rank f = f3 f4 g3 g4 of f3 equals the Fp -rank of g3 . For class A matrices the Fp -rank of the corresponding 2 × 2 block is 2 and for class B matrices it is less than 2. So it is not possible that one of f, g is class A and the other class B. Thus we can assume that f and g are in the same class. We first deal with the case m > 0. Continuing with the generalities, let r = tu with T ∈ Qp , u ∈ U (Qp ). Put t0 = h(l, m)−1 th(l, m), u0 = h(l, m)−1 uh(l, m). Thus r0 = t0 u0 and this forces t0 ∈ (T (Qp )∩Kp ), u0 ∈ U (Zp ). We must then have t = x+zpm ξ 0 with x ∈ Z× p , z ∈ Zp . Let u = U(s,t,u) . Let us first consider the class A case. We can check that (A0(0,0,0) )−1 t0 u0 A∞ (0,0,0) is con- 30 gruent mod p to  0 x 0 0       x −zc 0 0     −tx − zcu −sx − zct zc x   −ux −tx x 0 and so can never belong to Ip because x is a unit. We now consider the class B case. Suppose for (n1 , q1 , r1 ), (n2 , q2 , r2 ) we compute  0 0 = )−1 t0 u0 B(n (B(n 2 ,q2 ,r2 ) 1 ,q1 ,r1 ) A B C D   and prove that C is never congruent to 0 (mod p). It will follow then that for any y1 , y2 ∈ V , y1 y2 B(n and B(n are not (l, m)-equivalent because the introduction of the new terms 1 ,q1 ,r1 ) 2 ,q2 ,r2 ) Zy1 , Zy2 cannot affect C. 0 0 (B(1,0,0) )−1 t0 u0 B(0,0,0) is congruent mod p to   x zc sx + zct tx + zcu     0 x tx ux   .   x zc sx + x + zct tx + zcu   0 0 −zc x 0 0 (B(1,0,0) )−1 t0 u0 B(0,0,1) is congruent mod p to   x zc − tx − zcu sx + zct tx + zcu     0 x − ux tx ux   .   x zc − tx − zcu sx + x + zct tx + zcu   0 −x −zc x 0 0 (B(0,0,0) )−1 t0 u0 B(0,0,1) is congruent mod p to   x zc − tx − zcu sx + zct tx + zcu     0 x − ux tx ux   .   0  0 x 0   0 −x −zc x 31 Each of these three matrices have this property because x is a unit. 0 ∞ Now consider (B(1,0,0) )−1 t0 u0 B(1,0,0) . This is congruent mod p to  zc x − sx − zct tx + zcu   x   zc  0 −tx −sx − zct zc sx + zct       tx + zcu sx + x + zct  x −zc ux tx which cannot belong to Ip because x is a unit. 0 ∞ Next consider (B(0,0,0) )−1 t0 u0 B(0,0,0) . This is congruent mod p to   zc x tx + zcu sx + zct     x 0 ux tx      0 0 0 x    0 0 x −zc which cannot belong to Ip for the same reason. 0 ∞ Finally consider (B(0,0,1) )−1 t0 u0 B(0,0,1) . This is congruent mod p to  −tx + zc − zcu    x − ux    0  −ux  x tx + zcu sx + zct   0 ux tx    0 0 x   0 ux + x tx − zc which can again not belong to Ip . Thus we have completed the proof of the claim for m > 0. For m = 0 we can only say that t0 = x + zξ with atleast one of x, z an unit. 0 0 (B(1,0,0) )−1 t0 u0 B(0,0,0) =  x + 12 zb zc sx + 12 zb + zct tx + 12 zb + zcu      1 1 1  −za x − 2 zb tx − 2 zb − zas ux − 2 zb − zat      1 1 1 x + 12 zb zc sx + x − 2 zb + 2 szb + zct tx + 2 zb + zcu + za   1 0 0 −zc x + 2 zb 32 which if in Ip implies p|z which in turn implies p|x, a contradiction. ∞ 0 (B(1,0,0) )−1 t0 u0 B(0,0,0) =  −za   x + 12 zb    0  x + 12 zb x − 12 zb −zas + tx − 12 tzb −zat + ux − 12 uzb  zc sx + 21 szb + zct tx + 12 tzb + zcu 0 −zc x + 21 zb        zc sx + x + zct + 12 szb − 21 zb tx + 12 tzb + zcu + za which cannot belong to Ip for the same reason. ∞ 0 Put G = z(ct + 12 sb − 12 b). (B(1,0,0) )−1 t0 u0 B(1,0,0) =  za(s − 1) − tx + 12 tzb x − 12 zb −zas + tx − 21 tzb        x(1 − s) + G zc sx + 21 szb + zct zc 0 −zc 1 2 zb − G − sx zc x(s + 1) + G −zat + ux − 12 uzb       1  x + 2 zb  1 tx + 2 tzb + zcu + za tx + 12 tzb + zcu which cannot belong to Ip for the same reason. This completes the proof of the final claim. In which we calculate a certain volume For any t ∈ Kp we define the volume Itl,m as follows. Itl,m = vol(B(Qp )\B(Qp )h(l, m)tIp ). (2.3.6) In this subsection we shall explicitly compute the volume Itl,m . By Proposition 2.3.2, it is enough to do this for t ∈ Tm . The next two propositions state the results and the rest of the section is devoted to proving them. 3l+4m l,m p Proposition 2.3.3. Let m > 0. Let Ml,m denote (p+1)(p for 2 +1) . Then the quantities Iti 33 1 ≤ i ≤ 8 are as follows. Itl,m = pMl,m 1 Itl,m = Ml,m 5 Itl,m = p2 Ml,m 2 Itl,m = pMl,m 6 Itl,m = pMl,m 3 Itl,m = p2 Ml,m 7 Itl,m = Ml,m 4 Itl,m = p3 Ml,m 8 Proposition 2.3.4. For m = 0 the quantities Itl,m are as follows. p3l+1 (p + 1)(p2 + 1) p3l+2 Itl,m = 2 (p + 1)(p2 + 1) Itl,m = 1 p3l (p + 1)(p2 + 1) p3l+3 Itl,m = 7 (p + 1)(p2 + 1) Itl,m = 5 Remark. That the volume Itl,m is finite can be viewed either as a corollary of the above propositions, or as a consequence of the fact that vol(B(Qp )\B(Qp )h(l, m)Kp ) is finite [Fur93, §3]. For each t ∈ Tm define the subgroup Gt of Kp by Gt = t−1 U (Zp )GL2 (Zp )t ∩ Ip   12 M  with where U (Zp ) is the subgroup of Kp consisting of matrices that look like  0 12 M =M t ∈ M2 (Zp ), and  GL2 (Zp ) (more generally GL2 (Qp )) is embedded in G(Qp ) via g 7→  g 0 . 0 det(g) · (g −1 )t Also let G1t = tGt t−1 be the corresponding subgroup of U (Zp )GL2 (Zp ). And finally, define Ht = {x ∈ GL2 (Zp ) | ∃y ∈ U (Zp ) such that yx ∈ G1t }. It is easy to see that Ht = U (Zp )G1t ∩ GL2 (Zp ), thus Ht is a subgroup of GL2 (Zp ). (2.3.7) 34 Lemma 2.3.5. We have a disjoint union B(Qp )\B(Qp )h(l, m)tIp = a B(Qp )\B(Qp )h(l, m)G1t ty. y∈Gt \Ip Proof. Since tIp = S y∈Gt \Ip tGt y = S 1 y∈Gt \Ip Gt ty, the only thing to prove is that the union in the statement of the lemma is indeed disjoint. So suppose that y1 , y2 are two coset representatives of Gt \Ip and rh(l, m)g1 ty1 = h(l, m)g2 ty2 with g1 , g2 ∈ G1t , r ∈ B(Qp ).  This means ty2 y1−1 t−1 is an element of Kp that is of the form  Hence ty2 y1−1 t−1 ∈ U (Zp )GL2 (Zp ). Thus y2 y1−1  A B 0 det(A) · (A−1 )t ∈ t−1 U (Zp )GL2 (Zp )t ∩ Ip . = Gt which completes the proof. By the above lemma it follows that Itl,m = Z Z dg · 3(l+m) =p dt (2.3.8) B(Qp )\B(Qp )h(l,m)G1t Gt \Ip −1 [Kp : Ip ] where we have normalized [GL2 (Zp )U (Zp ) : G1t ] Z dt B(Qp )\B(Qp )h(0,m)G1t R U (Zp )GL2 (Zp )\Kp dx = 1. On the other hand, B(Qp )\B(Qp )h(0, m)G1t = B(Qp )\B(Qp )h(0, m)U (Zp )G1t = B(Qp )\B(Qp )h(0, m)(U (Zp )G1t ∩ GL2 (Zp )) = T (Qp )\T (Qp )h(m)Ht   pm 0 . where h(m) =  0 1 For each t ∈ Tm let us define At = [GL2 (Zp )U (Zp ) : G1t ] (2.3.9) 35 and Z Vt,m = dt. T (Qp )\T (Qp )h(m)Ht We use the same normalization of Haar measures as in [Fur93], namely we have Z dt = 1. T (Qp )\T (Qp )h(m)GL2 (Zp ) We summarize the computations above in the form of a lemma. Lemma 2.3.6. Let m ≥ 0. For each t ∈ Tm we have Itl,m = p3(l+m) · At · Vt,m . (p + 1)2 (p2 + 1) Proof. This follows from equation (2.3.9). By exactly the same arguments as in [Fur93, p. 202-203], we see that t Vt,m = [GL2 (Zp ) : Ht ]−1 [T (Zp ) : Om ] (2.3.10) t = T (Q ) ∩ h(m)H h(m)−1 . where Om p t Let Γ0p (resp. Γ0,p ) be the subgroup of GL2 (Zp ) consisting of matrices that become lower-triangular (resp. upper-triangular) when reduced mod p. Lemma 2.3.7. (a) We have Hti = Γ0p for i = 1, 2, 5, 8 and Hti = Γ0,p for i = 3, 4, 6, 7. (b) The quantities Ati = [U (Zp )GL2 (Zp ) : G1ti ] are as follows: At1 = p(p + 1) At5 = p + 1 At2 = p2 (p + 1) At6 = p + 1 At3 = p2 (p + 1) At7 = p3 (p + 1) At4 = p(p + 1) At8 = p3 (p + 1) Proof. We will prove this directly using (2.3.7) and the definition of Ati . First observe that the cardinality of U (Fp )GL2 (Fp ) is p3 ·(p2 −p)(p2 −1) = p4 (p−1)2 (p+ 1). Recall also that the images of Γ0p and Γ0,p have cardinality p(p − 1)2 in GL2 (Fp ). 36 Suppose  1   0 U =  0  0 0 n q   a b 0 0        c d 0 r 0 ,G =  .    0 0 d −c 0 1 0    0 0 1 0 0 −b a 1 q We have  a − nd + qb nd − qb b −nc + qa        c − qd + rb d qd − rb −qc + ra −1 . t1 U Gt1 =    a − nd + qb − d b nd − qb + d −nc + qa − c   b 0 −b a By inspection, this belongs to Ip if and only if b ≡ 0 (mod p), n ≡ ad − 1 (mod p). So 4 2 (p+1) Ht1 = Γ0p and At1 = p (p−1) = p(p + 1). p(p−1)2 p2   d c − qd + rb ra − qc qd − rb       b a − nd + qb qa − nc nd − qb −1  . t2 U Gt2 =   0  b a −b   b a − nd + qb − d qa − c − nc nd − qb + d By inspection, this belongs to Ip if and only if b ≡ 0 (mod p), n ≡ ad − 1 (mod p), q ≡ dc . 4 2 (p+1) So Ht2 = Γ0p and At2 = p (p−1) = p2 (p + 1). p(p−1)2 p   a b + nc − qa nd − qb −nc + qa     c d + qc − ra qd − rb ra − qc  −1  . t3 U Gt3 =   0  c d −c   c d + qc − ra − a qd − rb − b ra − qc + a By inspection, this belongs to Ip if and only if c ≡ 0 (mod p), r ≡ ad − 1 (mod p), q ≡ ab . So 4 2 (p+1) = p2 (p + 1). Ht3 = Γ0,p and At3 = p (p−1) p(p−1)2 p  d + qc − ra    b + nc − qa  t−1 4 U Gt4 =  d + qc − ra − a  c c ra − qc qd − rb    a qa − nc nd − qb  .  c ra − qc + a qd − rb − b  0 −c d 37 By inspection, this belongs to Ip if and only if c ≡ 0 (mod p), r ≡ ad − 1 (mod p). So 4 2 (p+1) Ht4 = Γ0,p and At4 = p (p−1) = p(p + 1). p(p−1)2 p2  a b nd − qb −nc + qa      c d qd − rb −qc + ra  −1 .  t5 U Gt5 =   0 0 d −c    0 0 −b a  By inspection, this belongs to Ip if and only if b ≡ 0 (mod p). So Ht5 = Γ0p and At5 = p4 (p−1)2 (p+1) = (p + 1). p(p−1)2 p3  d c ra − qc qd − rb        b a qa − nc nd − qb  . U Gt = t−1 6 6   0 0 a −b    0 0 −c d By inspection, this belongs to Ip if and only if c ≡ 0 (mod p). So Ht6 = Γ0,p and At6 = p4 (p−1)2 (p+1) = (p + 1). p(p−1)2 p3  d −c 0 0       −b a 0 0 −1  . t7 U Gt7 =   qb − nd nc − qa a b    rb − qd qc − ra c d By inspection, this belongs to Ip if and only if c ≡ 0 (mod p), n ≡ 0 (mod p), q ≡ 0 4 2 (p+1) = p3 (p + 1). (mod p), r ≡ 0 (mod p). So Ht7 = Γ0,p and At7 = p (p−1) p(p−1)2  a −b 0 0        −c d 0 0 −1 . t8 U Gt8 =     qc − ra rb − qd d c    nc − qa qb − nd b a By inspection, this belongs to Ip if and only if b ≡ 0 (mod p), n ≡ 0 (mod p), q ≡ 0 4 2 (p+1) (mod p), r ≡ 0 (mod p). So Ht8 = Γ0p and At8 = p (p−1) = p3 (p + 1). p(p−1)2 38 Let t be such that Ht = Γ0p . Then by working through the definitions, we see that t Om = x + pm+1 yξ0 , x, y ∈ Zp . (2.3.11) On the other hand if t is such that Ht = Γ0,p , then we see that t Om = x + pm yξ0 , x, y ∈ Zp . (2.3.12) Lemma 2.3.8. Let m > 0. Then we have Vti ,m = pm for i = 1, 2, 5, 8 and Vti ,m = pm−1 for i = 3, 4, 6, 7. Proof. This follows from (2.3.10), (2.3.11), (2.3.12), Lemma 2.3.7 and [Fur93, Lemma 3.5.3] Proof of Proposition 2.3.3. The proof is a consequence of Lemma 2.3.6, Lemma 2.3.7 and Lemma 2.3.8. Let us now look at the case m = 0. In this case T0 = {t1 , t2 , t5 , t7 }. The groups Hti and the quantities [GL2 (Zp ) : Hti ]−1 have already been calculated. On the other hand we now have O0ti = x + pyξ0 , x, y ∈ Zp . (2.3.13) for each ti ∈ T0 . Proof of Proposition 2.3.4. We have already calculated each Ati . Also by (2.3.10),(2.3.13) and Lemma 2.3.7 we conclude that each Vti ,0 = 1. Now the result follows as before, from Lemma 2.3.6. Simplification of the local zeta integral Recall the definition of the key local integral Zp (s) from §2.2. In (2.3.5) we reduced this integral to an useful sum. Now suppose that Wp and Bp are right Ip -invariant. Then 39 Proposition 2.3.2 allows us to further simplify that expression as follows. Zp (s) = X X Wp (Θh(l, m)t, s) · Bp (h(l, m)t) · Itl,m (2.3.14) l∈Z,m≥0 t∈Tm Note that in the above formula we mildly abuse notation and use Θ to really mean its e p ). We will continue to do this in the future for notational economy. natural inclusion in G(Q Remark. The importance of §2.3, where we calculated Itl,m for each t ∈ Tm , is that we can now use the formula (2.3.14) to evaluate the local zeta integral whenever the local functions Wp and Bp can be explicitly determined. 2.4 The evaluation of the local Bessel functions in the Steinberg case Background Because automorphic representations of GSp(4) are not necessarily generic, the Whittaker model is not always useful for studying L-functions. For many problems, the Bessel model is a good substitute. Explicit evaluation of local zeta integrals then often reduces to explicit evaluation of certain local Bessel functions. Formulas for the Bessel functions have been established in the following cases. [Sug85] unramified representations of GSp4 (Qp ) [BFF97] unramified representations (the Casselman-Shalika like formula) [Niw91] class-one representations on Sp4 (R) [Miy00] large discrete series and PJ -principal series of Sp4 (R) [Ish02] principal series of Sp4 (R) In this section we give an explicit formula for the Bessel function for an unramified quadratic twist of the Steinberg representation of GSp4 (Qp ). By [Sch05] this is precisely the representation corresponding to a local newform for the Iwahori subgroup. Throughout this section we let p be an odd prime that is inert in  L. We suppose  that the a b/2  ∈ M2 (Zp ). local component (ωπ )p is trivial, the conductor of ψp is Zp and S =  b/2 c Because p is inert, Lp is a quadratic extension of Qp and we may write elements of Lp 40 √ √ in the form a + b −d with a, b ∈ Qp ; then ZL,p = a + b −d where a, b ∈ Zp . We identify √ Lp with T (Qp ) and ξ with −d/2. Then T (Zp ) = Z× L,p consists of elements of the form √ a + b −d where a, b are elements of Zp not both divisible by p. √ We assume that Λp is trivial on the elements of T (Zp ) of the form a + b −d with a, b ∈ Zp , p | b, p - a. Further, we assume that Λp is not trivial on the full group T (Zp ), that is, it is not unramified. Finally, assume that the local representation πp is an unramified twist of the Steinberg representation. This is representation IVa in [Sch05, Table 1]. The space of πp contains a unique normalized vector that is fixed by the Iwahori subgroup Ip . We can think of this vector as the normalized local newform for this representation. Bessel functions Let B be the space of locally constant functions ϕ on G(Qp ) satisfying ϕ(tuh) = Λp (t)θp (u)ϕ(h), for t ∈ T (Qp ), u ∈ U (Qp ), h ∈ G(Qp ). Then by Novodvorsky and Piatetski-Shapiro [NPS73], there exists a unique subspace B(πp ) of B such that the right regular representation of G(Qp ) on B(πp ) is isomorphic to πp . Let Bp be the unique Ip -fixed vector in B(πp ) such that Bp (14 ) = 1. Therefore Bp (tuhk) = Λp (t)θp (u)ϕ(h), (2.4.1) where t ∈ T (Qp ), u ∈ U (Qp ), h ∈ G(Qp ), k ∈ Kp . Our goal is to explicitly compute Bp . By Proposition 2.3.2 and (2.4.1) it is enough to compute the values Bp (h(l, m)ti ) for l ∈ Z, m ∈ Z≥0 , ti ∈ Tm . Let us fix some notation. Recall the matrices ti which were defined in §2.3. Also we will frequently use other notation from §2.3. We now define 41 al,m 0 = Bp (h(l, m)t7 ), al,m ∞ = Bp (h(l, m)t8 ), bl,m 0 = Bp (h(l, m)t2 ), 1 l,m b0 = Bp (h(l, m)t1 ), bl,m ∞ = Bp (h(l, m)t3 ), 1 l,m b∞ = Bp (h(l, m)t4 ), cl,m 0 = Bp (h(l, m)t5 ), cl,m ∞ = Bp (h(l, m)t6 ). Lemma 2.4.1. Let m ≥ 0, y ∈ {0, ∞}. The following equations hold: (a) al,m y =0 if l < −1. l,0 l,m 1 l,0 (b) 1 bl,m 0 = b0 = b∞ = b∞ = 0 l,m (c) 1 bl,m ∞ = b∞ = 0 (d) cl,m y =0 if l < 0. if l < −1. if l < 0. Proof. First note that U(0,0,p) ti ≡ ti (mod p), hence they are in the same coset of Kp /Ip . Hence Bp (h(l, m)ti ) = Bp (h(l, m)U(0,0,p) ti ) = Bp (U(0,0,pl+1 ) h(l, m)ti ) = ψp (pl+1 c)Bp (h(l, m)ti ). Since the conductor of ψp is Zp and c is a unit, it follows that Bp (h(l, m)ti ) = 0 for l < −1. This completes the proof of (a) and (c). Next, observe that cl,m y = Bp (h(l, m)Zy ) = Bp (h(l, m)U(0,0,1) Zy ) = Bp (U(0,0,pl ) h(l, m)Zy ) = ψp (pl c)Bp (h(l, m)Zy ). It follows that Bp (h(l, m)Zy ) = 0 for l < 0. This completes the proof of (d). 42 Next, we have Bp (h(l, m)JU(1,0,0) JZy ) = Bp (h(l, m)JU(1,0,0) JU0,0,1 Zy ) = Bp (h(l, m)U0,0,1 JU(1,0,0) JZy ) = ψp (pl c)Bp (h(l, m)JU(1,0,0) JZy ). l,m It follows that 1 bl,m 0 = b0 = 0 for l < 0. Finally, Bp (h(l, 0)JU(0,0,1) JZy ) = Bp (h(l, 0)JU(0,0,1) JU1,0,0 Zy ) = Bp (h(l, 0)U1,0,0 JU(0,0,1) JZy ) = ψp (pl a)Bp (h(l, 0)JU(0,0,1) JZy ). l,0 It follows that 1 bl,0 ∞ = b∞ = 0 for l < 0. This completes the proof of (b). By our normalization, we have c0,0 0 = 1. From Proposition 2.3.2, proof of Claim 6, it √ b+ −d follows that c0,0 ). ∞ = Λp ( 2 To get more information, we have to use the fact that the local Iwahori-Hecke algebra acts on Bp in a precise manner. Hecke operators and the results Henceforth we always assume that l ≥ −1, m ≥ 0. In particular, all equations that are stated without qualification will be understood to hold in the above range. We know that πp is either StGSp(4) or ξ0 StGSp(4) where ξ0 is the non-trivial unramified quadratic character. Put wp = −1 in the former case and wp = 1 in the latter. Put  0 0 0 1       0 0 1 0 . ηp =     0 p 0 0   p 0 0 0 43 Also, for y ∈ V , define the matrices Ry as follows: If y ∈ Y , Ry = (U(y,0,0) )t , and  0 0    0 −1 R∞ =   −1 0  0 0 −1 0 0 0 0    0 .  0  1 Let t ∈ G(Qp ). By [Sch05], we know the following: X Bp (tZy ) = 0, (2.4.2) Bp (tηp ) = wp Bp (t), (2.4.3) X (2.4.4) y∈V Bp (tRy ) = 0. y∈V (2.4.2) and Proposition 2.3.2 immediately imply l,m al,m 0 + pa∞ = 0, for m > 0 (2.4.5) 1 l,m pbl,m y + by = 0, for y ∈ {0, ∞} (2.4.6) l,m pcl,m 0 + c∞ = 0, for m > 0 (2.4.7) Next we act upon by ηp . Check that  1 0 0 0      0 1 0 0 ∞ −1 0 .  (h(l + 1, m)B(0,0,0) ) h(l, m)A(0,0,0) ηp =   0 0 −1 0    0 0 0 −1 44 So we have 0 al,m 0 = Bp (h(l, m)A(0,0,0) ) = wp Bp (h(l, m)A0(0,0,0) ηp ) ∞ ). = wp Bp (h(l + 1, m)B(0,0,0) Thus l+1,m al,m . 0 = wp c∞ (2.4.8) We also have  1 0 0 0        0 1 0 0 0 −1 ∞ . (h(l + 1, m)B(0,0,0) ) h(l, m)A(0,0,0) ηp =    0 0 −1 0    0 0 0 −1 So similarly, we conclude l+1,m . al,m ∞ = wp c0 (2.4.9) Next, check that 1 1 (h(l, m)B(1,0,0) ηp )−1 h(l − 1, m + 1)U(−1/p,0,0) D∞ = (Z 1 )t ∈ Ip . Hence 1 1 Bp (h(l, m)B(1,0,0) ) = wp Bp (h(l − 1, m + 1)D∞ ). (Note that both sides are zero if l = −1, m = 0). 1 1 By the proof of Proposition 2.3.2, Bp (h(l, m)B(1,0,0) ) = bl,m 0 and Bp (h(l−1, m+1)D∞ ) = l−1,m+1 ψp (pl−1 c)b∞ . Thus we have proved l−1 bl,m c)bl−1,m+1 . ∞ 0 = wp ψp (p (2.4.10) At this point we pause and note that on account of (2.4.5)–(2.4.10) it is enough to l,m compute the quantities bl,m ∞ , a0 , l ≥ −1, m ≥ 0, l + m 6= −1. Of course, we already know √ that a−1,0 = wp Λp ( b+ 2 −d ). 0 45 Next, we use (2.4.4). 0 . For each x ∈ Y , we can check that A00,0,0 Rx = A0−x,0,0 . Furthermore, A00,0,0 R∞ = D∞ 0 = ψ (pl c)bl,m . Assuming l + m ≥ 0 we have Bp (h(l, m)A0−x,0,0 ) = al,m and Bp (h(l, m)D∞ ∞ p 0 So using (2.4.4) we conclude l l,m pal,m 0 = −ψp (p c)b∞ , (2.4.11) for l + m ≥ 0. ∞ ∞ However we can do more. Check that for x ∈ Y , A∞ (0,0,0) Rx = A(0,0,−x) and A(0,0,0) R∞ ≡ l,m l,m 0 D00 (mod p). If l ≥ 0 we have Bp (h(l, m)A∞ (0,0,−x) = a∞ and Bp (h(l, m)D0 ) = b0 . So again using (2.4.4) we have l,m pal,m ∞ = −b0 , (2.4.12) for l ≥ 0. So (2.4.5), (2.4.10) and (2.4.12) imply that for l ≥ 0, m > 0 l,m l−1 c)bl−1,m+1 . bl,m ∞ ∞ = −pb0 = −pwp ψp (p (2.4.13) 0 0 0 R∞ ≡ D0∞ (mod p) and for x ∈ Y , B0,0,0 Rx = B−x,0,0 . Assuming Now observe that B0,0,0 l + m 6= −1 we have Bp (h(l, m)D0∞ ) = 1 bl,m and for x ∈ y, x 6= 0, B (h(l, m)B 0 p −x,0,0 ) = 0 1 bl,m . Hence using (2.4.4) 0 1 l,m cl,m 0 = −p b0 (2.4.14) So by equations (2.4.6) and (2.4.9) we have, 2 l l,m+1 al,m ∞ = p ψp (p c)b∞ (2.4.15) The above equation, along with our normalization tells us that −1,1 b∞ = 1 c ψp (− )wp . 2 p p (2.4.16) Also, using (2.4.12),(2.4.13) and (2.4.15) we get bl,m+1 = ∞ for l ≥ 0, m > 0. 1 l,m b p4 ∞ (2.4.17) 46 (2.4.13), (2.4.17) and (2.4.16) imply : (−pwp )l bl,m ∞ = − 4l+4m+1 if l ≥ 0, m ≥ 1 p b−1,m = ∞ 1 p4m−2 (2.4.18) c ψp (− )wp if m ≥ 1. p (2.4.19) √ b+ −d 1 l,0 In the case m = 0, Proposition 2.3.2, proof of Claim 7, tells us that 1 bl,0 ) b0 ∞ = Λp ( 2 which implies b+ bl,0 ∞ = Λp ( √ 2 −d b+ l−1 )bl,0 c)Λp ( 0 = wp ψp (p √ 2 −d l−1,1 )b∞ (2.4.20) EquationS (2.4.18)–(2.4.20), along with the earlier equations that specify the interdependence of various quantities, determine all the values Bp (h(l, m)ti ). For convenience, we compactly state the facts proven above as two propositions. We only state it for l ≥ 0 since that is the only case needed for our later applications. The values for l = −1 can be easily gleaned from these and the above equations. Proposition 2.4.2. Let l ≥ 0, m > 0. Put M = (−pwp )l p−4(l+m) . Then the following hold: (a) Bp (h(l, m)t1 ) = M · −1 p , (b) Bp (h(l, m)t2 ) = M · p12 , (c) Bp (h(l, m)t3 ) = M · −1 p , (d) Bp (h(l, m)t4 ) = M (e) Bp (h(l, m)t5 ) = M (f ) Bp (h(l, m)t6 ) = M · (−p), (g) Bp (h(l, m)t7 ) = M · p12 , (h) Bp (h(l, m)t8 ) = M · −1 . p3 Proposition 2.4.3. Let l ≥ 0. Put M = (−pwp )l p−4l . Then the following hold: (a) Bp (h(l, 0)t1 ) = M · −1 p , (b) Bp (h(l, 0)t2 ) = M · p12 , 47 (c) Bp (h(l, 0)t5 ) = M, √ −Λp ( b+ 2 −d ) . (d) Bp (h(l, m)t7 ) = M · p3 2.5 The case unramified πp , Steinberg σp Assumptions Suppose that the characters ωπ , ωσ , χ0 are trivial. Let p 6= 2 be a finite prime of Q such that √ (a) p is inert in L = Q( −d). (b) The local components Λp and πp are unramified. (c) σp is the Steinberg representation (or its twist by the unramified quadratic character). (d) The conductor of ψp is Zp   a b/2  ∈ M2 (Zp ). (e) S =  b/2 c (f) −d = b2 − 4ac generates the discriminant of Lp /Qp . Remark: σp is concretely realized as (possibly the unramified quadratic twist of) the special representation on the locally constant functions of Bp \GL2 (Qp ) modulo the constant functions (where Bp is the standard Borel subgroup consisting of upper-triangular matrices). It corresponds to the local newform for the Iwahori subgroup Γ0 (p) of GL2 (Qp ). Description of Bp and Wp For any choice of local Whittaker and Bessel functions Wp and Bp we can define the local zeta integral Zp (s) by (2.2.6). We now fix such a choice. As in the unramified case from §2.1, we let Bp be the unique normalized Kp -vector in the local Bessel space. Sugano[Sug85] has computed the function Bp explicitly. 48 fp be the subgroup of K fp defined by We now define Wp . Let U  ∗ 0 ∗ ∗      ∗ ∗ ∗ ∗ f f   Up = z ∈ Kp | z ≡   ∗ 0 ∗ ∗    0 0 0 ∗  (mod p) . fp -fixed vectors. Now let Wp be the unique U fp -fixed It is not hard to see that I(Πp , s) has U vector in the local Whittaker space with the following properties: • Wp (e, s) = 1, fp • Wp (g, s) = 0 if g does not belong to P (Qp )U Concretely we have the following description of Wp ( s). We know that σp = Sp ⊗ τ where Sp denotes the special (Steinberg) representation and τ is a (possibly trivial) unramified quadratic character. We put ap = τ (p), thus ap = ±1 is the eigenvalue of the local Hecke operator T (p). Let Wp0 be the unique function on GL2 (Qp ) such that Wp0 (gk) = Wp0 (g), for g ∈ GL2 (Qp ), k ∈ Γ0,p ,  Wp0 ( 1 x (2.5.1)   g) = ψp (−cx)Wp0 (g), for g ∈ GL2 (Qp ), x ∈ Qp , (2.5.2)    τ (a)|a| if |a|p ≤ 1, a 0 0  Wp =  0 0 1 otherwise. (2.5.3)     −p−1 τ (a)|a| if |a|p ≤ p, a 0 0 1 0      Wp =  0 0 1 −1 0 otherwise. (2.5.4) 0 1   We extend Wp0 to a function on GU (1, 1)(Qp ) by Wp0 (ag) = Wp0 (g), for a ∈ L× p , g ∈ GL2 (Qp ). 49 e p ) such that Then, Wp ( s) is the unique function on G(Q fp , Wp (mnk, s) = Wp (m, s), for m ∈ M (Qp ), n ∈ N (QP ), k ∈ U (2.5.5) fp , Wp (e) = 1 and Wp (g, s) = 0 if g ∈ / P (Qp )U (2.5.6)    0 1 0 0 0         0 1 0 0 0 a1 0 b1    , s     Wp     0 0 a−1 0 0 0 c1 0       0 d1 0 e1 0 0 0 1   a1 b1 3(s+1/2)  · Wp0  = NL/Q (a) · c−1 1 p d1 e1 (2.5.7) and  a 0 0     a b a b  1 1  ∈ GU (1, 1)(Qp ), c1 = µ1  1 1 . for a ∈ Q× p, d1 e1 d1 e1   a b  ∈ GU (1, 1) we let Let us use the following notation: For  c d  1     0 a b ) =  m(2) (  0 c d  0  where β = µ1 ( a b c d 0 0 0    b   0 β 0  c 0 d a 0  ). The results (i) For i = 1, 2, 3, 4, define the characters γp of Q× p as in §2.1. We now state and prove the main theorem of this section. 50 Theorem 2.5.1. Let the functions Bp , Wp be as defined in §2.5. Then we have Zp (s, Wp , Bp ) = L(3s + 12 , πp × σp ) 1 · p2 + 1 L(3s + 1, σp × ρ(Λp )) where, L(s, πp × σp ) = 4 Y (1 − γp(i) (p)ap p−1/2 p−s )−1 , i=1 and L(s, σp × ρ(Λp )) = (1 − p−2s−1 )−1 . Before we begin the proof, we need a lemma. Lemma 2.5.2. We have the following formulae for Wp (Θh(l, m)ti , s) where ti ∈ Tm .     p−6ms−3ls−3m−5l/2 alp    (a) If m > 0 then Wp (Θh(l, m)ti , s) = p−6ms−3ls−3m−5l/2 alp · −1 p      0 (b) Wp (Θh(l, 0)ti , s) =   p−3ls−5l/2 alp if i ∈ {1, 5}  0 if i ∈ {2, 7} if i ∈ {1, 5} if i ∈ {3, 7} otherwise Proof. We have  1 0 0 0     m  p α 1 0 0    Θh(l, m) = h(l, m)   m  0 0 1 −p α   0 0 0 1 (2.5.8) First consider the case m > 0. f if i ∈ {2, 4, 6, 8}. We claim that Θh(l, m)ti ∈ / P (Qp )U p  1 0 0 0     m p α 1 0 0   . Using (2.6.8), it suffices to prove that Θm ti ∈  / Put Θm =   m  0 0 1 −p α   0 0 0 1 51 fp . So take a typical element element P (Zp )U  a ax at + axy ay   0 P =  0  0 m my − βx 0 λ/a γ γy − δx    β  ∈ P (Zp )  0  δ (2.5.9)     m β m β .  ∈ GU (1, 1)(Zp ) with λ = µ1  where all the variables lie in ZL and  γ δ γ δ We have   m m ax a + axp α − at − axy −(at + axy)αp + ay at + axy     m m m mp α − my + βx −(my − βx)(α)p + β my − βx   P Θm t2 =   −1 m −1 −1 0 −λ(a) −αp λ(a) λ(a)    m m −(γy − δx)αp + δ γy − δx γ γp α − γy + δx fp would imply that p | λ, a contradiction. which, if it were an element of U We have   ax + (at + axy)αpm − ay a + axpm α −(at + axy)αpm + ay at + axy     m m  m + (my − βx)αpm − β mp α −(my − βx)αp + β my − βx   P Θm t4 =   m −1 m −1 −1  αp λ(a) 0 −αp λ(a) λ(a)    m m m γ + (γy − δx)αp − δ γp α −(γy − δx)αp + δ γy − δx fp would imply that p | λ, a contradiction. which, if it were an element of U We have   ax a + axpm α −(at + axy)αpm + ay at + axy     m mpm α −(my − βx)αpm + β my − βx   P Θm t6 =   0 0 −αpm λ(a)−1 λ(a)−1    m m γ γp α −(γy − δx)αp + δ γy − δx fp would imply that p | γ, p | δ, a contradiction. which, if it were an element of U 52 Finally we have  at + axy)αpm − ay −at − axy ax a + axpm α   (my − βx)αpm − β −my + βx P Θm t8 =    λαpm (a)−1 −λ(a)−1  (γy − δx)αpm − δ −γy + δx m 0 γ    mpm α     0  γpm α fp would imply that p | γ, p | δ, a contradiction. This which, if it were an element of U completes the proof of the claim. For the remaining ti (i.e., i ∈ {1, 3, 5, 7}) we have the following decompositions: Θh(l, m)t1  p2m+l 0    0 =   0  0 0 1 0 0 p−2m−l 0 0    0 −1 0 0 0         m m+l −p α −1 0 0 (2) p 0 0    m ( )     m m  1 0 0 p 0 −1 p α    0 0 0 −1 1 Θh(l, m)t3 =  p2m+l    0    0  0 0 0 1 0 0 p−2m−l 0 0    0 −1 0 0 0        m  m+l   0 −p α −1 0 0  p 0  m(2) (  )     m m m m   0 0 −p α −1 p α −p p    m 1 −p α 0 0 −1 Θh(l, m)t5  p2m+l 0    0 =   0  0 1 0 0 0 p−2m−l 0 0   0 1       m p α 0 (2) pm+l 0  m ( )    0 pm  0 0   0 1 0 0 0    0    0 1 pm α   0 0 1 1 0 53 Θh(l, m)t7  p2m+l 0    0 =   0  0 0 1 0 0 p−2m−l 0 0   0 0 0 1       0 (2) 0 −pm+l  0 1 0  m ( )    −1 pm α 0 −pm 0 0   1 0 0 pm α 0    0   0  1 Part (a) of the lemma now follows from the above decompositions and equations (2.5.1)(2.5.7). Let us now look at m = 0. Once again, let P be the matrix defined in (2.5.9). The same fp . As for t7 , proof as above for t2 shows that P Θm t2 ∈ /U   −at − axy (at + axy)α − ay a + axα ax     −my + βx (my − βx)α − β mα m  . P Θm t7 =    −λ(a)−1 λα(a)−1 0 0   −γy + δx (γy − δx)α − δ γα γ fp then we have p | γα which implies p | γ. But that immediately If the above matrix lies in U implies, by looking at the bottom left entry, that p | δx, hence (by looking at the second entry of the bottom row) p | δ. Thus p | γ, p | δ, a contradiction. fp if i ∈ {2, 7}. For t1 and t5 we have the above decompositions, Thus Θh(l, 0)ti ∈ / P (Qp )U from which part (b) follows via the equations (2.5.1)-(2.5.7). Proof of Theorem 2.5.1. By (2.3.14) we have Zp (s, Wp , Bp ) = X l≥0,m≥0 Bp (h(l, m)) X Wp (Θh(l, m)ti , s) · Itl,m i (2.5.10) ti ∈Tm We first look at the terms corresponding to m > 0. From Lemma 2.5.2 and ProposiP tion 2.3.3 we have ti ∈Tm Wp (Θh(l, m)ti , s) · Itl,m = 0. So only terms corresponding to i m = 0 contribute. From Proposition 2.3.4 and Lemma 2.5.2 we have X ti ∈T0 Wp (Θh(l, 0)ti , s) · Itl,0 = i 1 p2 + 1 · p−3ls+l/2 alp . 54 Hence (2.6.9) reduces to Zp (s, Wp , Bp ) = X 1 · Bp (h(l, 0))p−3ls+l/2 alp . p2 + 1 l≥0 Define C(y) = l l≥0 Bp (h(l, 0))y . We are interested in the quantity P Zp (s, Wp , Bp ) = 1 C(ap p−3s+1/2 ). p2 + 1 (2.5.11) Sugano, in [Sug85, p. 544], has computed C(y) explicitly. His results imply that C(y) = 2 where H(y) = 1 − yp4 , Q(y) = H(y) Q(y) (i) −3/2 y). i=1 (1 − γp (p)p Q4 Plugging in these values in (2.5.11) we get the desired result. 2.6 The case Steinberg πp , Steinberg σp Assumptions Suppose that the characters ωπ , ωσ , χ0 are trivial. Let p 6= 2 be a finite prime of Q such that √ (a) p is inert in L = Q( −d). (b) Λp is not trivial on T (Zp ); however it is trivial on T (Zp ) ∩ Γ0p . (c) πp is the Steinberg representation (or its twist by the unique non-trivial unramified quadratic character). (d) σp is the Steinberg representation (or its twist by the unique non-trivial unramified quadratic character). (e) The conductor of ψp is Zp .   a b/2  ∈ M2 (Zp ). (f) S =  b/2 c 55 (g) −d = b2 − 4ac generates the discriminant of Lp /Qp . Remark. πp corresponds to a local newform for the Iwahori subgroup Ip (see [Sch05]). Also, as in the previous section, σp corresponds to the local newform for the Iwahori subgroup Γ0 (p) of GL2 (Qp ). Description of Bp and Wp Let Φp be the unique normalized local newform for the Iwahori subgroup Ip , as defined by Schmidt [Sch05]. Let wp be the local Atkin-Lehner eigenvalue for πp ; this equals −1 when πp is the Steinberg representation and equals 1 when πp is the unramified quadratic twist of the Steinberg representation. We let Bp be the normalized vector that corresponds to Φp in the Bessel space. §2.4 was devoted to the computation of the values Bp (h(l, m)t) for l, m ∈ Z, m ≥ 0, t ∈ Tm . Because p is inert, Lp is a quadratic extension of Qp and we may write elements of Lp in √ √ the form a + b −d with a, b ∈ Qp ; then ZL,p = a + b −d where a, b ∈ Zp . We also identify √ Lp with T (Qp ) and ξ with −d/2. We now define Wp . By Assumption (b) above, we have √ Λp is trivial on the elements of T (Qp ) of the form a + b −d with a, b ∈ Zp , p | b, p - a. Take e p → G(F e p ) and define I 0 = r−1 (I(4, Fp )). the canonical map r : K   p 0 1 0 0     1 0 0 0  . Let s1 denote the matrix   0 0 0 1   0 0 1 0 Let Wp ( , s) be the unique vector in I(Πp , s) with the following properties: • Wp (1, s) = 1, • Wp (s1 , s) = 1, • Wp (gk, s) = Wp (g, s) is k ∈ Ip0 , • Wp (g, s) = 0 if g does not belong to P (Qp )Ip0 t P (Qp )s1 Ip0 Concretely we have the following description of Wp ( , s) : 56 We know that σp = Sp ⊗ τ where Sp denotes the special (Steinberg) representation and τ is a (possibly trivial) unramified quadratic character. We put ap = τ (p), thus ap = ±1 is the eigenvalue of the local Hecke operator T (p). Let Wp0 be the unique function on GL2 (Qp ) such that Wp0 (gk) = Wp0 (g), for g ∈ GL2 (Qp ), k ∈ Γ0,p ,  1 x (2.6.1)   g) = ψp (−cx)Wp0 (g), for g ∈ GL2 (Qp ), x ∈ Qp , (2.6.2)    τ (a)|a| if |a|p ≤ 1, a 0 0  = Wp  0 0 1 otherwise (2.6.3)     −p−1 τ (a)|a| if |a|p ≤ p, a 0 0 1 0    = Wp  0 0 1 −1 0 otherwise (2.6.4) Wp0 ( 0 1   We extend Wp0 to a function on GU (1, 1)(Qp ) by Wp0 (ag) = Wp0 (g), for a ∈ L× p , g ∈ GL2 (Qp ). e p ) such that Then, Wp ( s) is the unique function on G(Q Wp (mnuk, s) = Wp (mu, s), for m ∈ M (Qp ), n ∈ N (QP ), u ∈ {1, s1 }, k ∈ Ip0 , Wp (t) = 0 if t ∈ / P (Qp )Ip0 t P (Qp )s1 Ip0     0 1 0 0 0          0 1 0 0 0 a1 0 b1        Wp  u, s    0 0 a−1 0 0 0 c1 0       0 d1 0 e1 0 0 0 1   a1 b1 3(s+1/2) , = NL/Q (a) · c−1 · Λp (a)Wp0  1 p d1 e1 a 0 (2.6.5) (2.6.6) 0 (2.6.7) 57   for a ∈ Q× p , u ∈ {1, s1 },   a1 b1  ∈ GU (1, 1)(Qp ), c1 = µ1  . e1 d1 e1 a1 b1 d1  The results We now state and prove the main theorem of this section. Theorem 2.6.1. Let the functions Bp , Wp be as defined in subsection* 2.6. Then we have Zp (s, Wp , Bp ) = 1 p−6s−3 1−p · L(3s + , πp × σp ) · p2 + 1 1 − ap wp p−3s−3/2 2 where L(s, πp × σp ) = (1 + ap wp p−1 p−s )−1 (1 + ap wp p−2 p−s )−1 . Before we begin the proof, we need a lemma. Lemma 2.6.2. We have  the following formulae for Wp (Θh(l, m)ti , s) where ti ∈ Tm .    if i = 3, 4, m > 0 p−6ms−3ls−3m−5l/2 alp · −1  p   Wp (Θh(l, m)ti , s) = p−6ms−3ls−3m−5l/2 alp if i = 5, 6, m > 0      0 otherwise. Proof. We have  1 0 0 0     m  p α 1 0 0    Θh(l, m) = h(l, m)   m  0 0 1 −p α   0 0 0 1 (2.6.8) Put Kp0 = r−1 (G(Fp )). Thus Θh(l, m)ti ∈ P (Qp )Kp0 when m > 0 and Θh(l, m)ti ∈ P (Qp )ΘKp0 when m = 0. A direct computation shows that P (Qp )Kp0 and P (Qp )ΘKp0 are disjoint; the fact that P (Qp )Ip0 ⊂ P (Qp )Kp0 then implies that Wp (Θh(l, m)ti , s) = 0 for m = 0 . From now on we assume m > 0. We can check that Θh(l, m)ti ∈ / P (Qp )Ip0 t P (Qp )s1 Ip0 if i ∈ {1, 2, 7, 8}. For the remaining ti (i.e. i ∈ {3, 4, 5, 6}) we have the decompositions: 58 Θh(l, m)t3 =  p2m+l    0    0  0 0 0 1 0    0 −1 0 0 0         m −p α 0 (2) pm+l 0 −1 0 0   m (  )     0 −pm pm  0 −pm α −1 pm α    1 −pm α 0 0 −1 0 p−2m−l 0 0 0 0 1 0 Θh(l, m)t4 =  p2m+l    0    0  0   0 1 pm α 0        0 0 (2) pm+l 0 1 0  m ( )s1     0 0 −pm pm pm α 1   0 −pm α 1 pm α 0 p−2m−l 0 0  0   0   0  1 Θh(l, m)t5  p2m+l 0    0 =   0  0 0 1 0 0 p−2m−l 0 0   0 1       pm α 0 (2) pm+l 0  m ( )    0 0 pm  0   0 1 0 0 0    1 0 0    m 0 1 p α  0 0 1 Θh(l, m)t6  p2m+l    0 =   0  0 0 1 0 0 0 p−2m−l 0 0   0 1 pm α 0       0 0 (2) pm+l 0 1 0  m ( )s1    0 0 0 pm 0 1   1 0 0 −pm α 0    0   0  1 The lemma now follows from the above decompositions and equations (2.6.1)-(2.6.7). Proof of Theorem 2.6.1. By (2.3.14) we have Zp (s, Wp , Bp ) = X X l≥0,m≥0 ti ∈Tm Bp (h(l, m)ti )Wp (Θh(l, m)ti , s) · Itl,m i (2.6.9) 59 From Proposition 2.3.3, Proposition 2.4.2 and Lemma 2.6.2 we have X Bp (h(l, m)ti )Wp (Θh(l, m)ti , s) · Itl,m = i i∈{3,4,5,6} (1 − p)(−ap wp p−3s−5/2 )l (p−6s−3 )m . p2 + 1 Hence (2.6.9) implies Zp (s, Wp , Bp ) = 1 1 (1 − p)p−6s−3 · · −6s−3 −3s−1/2 −2 p2 + 1 1 − p 1 + ap wp p p This completes the proof. 2.7 The case Steinberg πp , unramified σp Assumptions Suppose that the characters ωπ , ωσ , χ0 are trivial. Let p 6= 2 be a finite prime of Q such that √ (a) p is inert in L = Q( −d). (b) Λp is not trivial on T (Zp ); however it is trivial on T (Zp ) ∩ Γ0p . (c) πp is the Steinberg representation (or its twist by the unique non-trivial unramified quadratic character) while σp is unramified. (d) The conductor of ψp is Zp .   a b/2  ∈ M2 (Zp ). (e) S =  b/2 c (f) −d = b2 − 4ac generates the discriminant of Lp /Qp . Remark. πp corresponds to a local newform for the Iwahori subgroup Ip (see [Sch05]). Description of Bp and Wp Let Φp be the unique normalized local newform for the Iwahori subgroup Ip , as defined by Schmidt [Sch05]. Let wp be the local Atkin-Lehner eigenvalue for πp ; this equals −1 when 60 πp is the Steinberg representation and equals 1 when πp is the unramified quadratic twist of the Steinberg representation. We let Bp be the normalized vector that corresponds to Φp in the Bessel space. Section 2.4 was devoted to the computation of the values Bp (h(l, m)t) for l, m ∈ Z, m ≥ 0, t ∈ Tm . e p → G(F e p ) and define Ip0 = r−1 (I(4, Fp )). We now define Wp . Take the canonical map r : K Let Wp ( , s) be the unique vector in I(Πp , s) with the following properties: • Wp (Θ, s) = 1, • Wp (1, s) = 1, • Wp (gk, s) = Wp (g, s) is k ∈ Ip0 , • Wp (g, s) = 0 if g does not belong to P (Qp )ΘIp0 t P (Qp )Ip0 Concretely we have the following description of Wp ( , s). Suppose σp is the principal series representation induced from the unramified characters 0 α, β of Q× p . Let Wp be the unique function on GL2 (Qp ) such that Wp0 (gk) = Wp0 (g), for g ∈ GL2 (Qp ), k ∈ GL2 (Zp ),  Wp0 ( 1 x 0 1  Wp0  (2.7.1)   g) = ψp (−cx)Wp0 (g), for g ∈ GL2 (Qp ), x ∈ Qp , a 0 0 b  =  1   a 2 · α(ap)β(b)−α(b)β(ap) if  0 otherwise b p α(p)−β(p) a b p ≤ 1, (2.7.2) (2.7.3) We extend Wp0 to a function on GU (1, 1)(Qp ) by Wp0 (ag) = Wp0 (g), for a ∈ L× p , g ∈ GL2 (Qp ). e p ) such that Then, Wp ( s) is the unique function on G(Q Wp (mnuk, s) = Wp (mu, s), for m ∈ M (Qp ), n ∈ N (QP ), u ∈ {1, Θ}, k ∈ Ip0 , (2.7.4) 61 Wp (t) = 0 if t ∈ / P (Qp )ΘIp0 t P (Qp )Ip0    0 1 0 0 0        1 0 0 0 a1 0 b1   u, s       0 a−1 0 0 0 c1 0     0 d1 0 e1 0 0 1   a b 1 1 3(s+1/2) , = NL/Q (a) · c−1 · Λp (a−1 )Wp0  1 p d1 e1  a    0  Wp    0  0   for a ∈ Q× p , u ∈ {1, Θ}, 0 0 (2.7.6)   a1 b1 .  ∈ GU (1, 1)(Qp ), c1 = µ1  d1 e1 e1 a1 b1 d1 (2.7.5)  The results We now state and prove the main theorem of this section. Theorem 2.7.1. Let the functions Bp , Wp be as defined in subsection* 2.7. Then we have Zp (s, Wp , Bp ) = 1 1 · L(3s + , πp × σp ), 2 (p + 1)(p + 1) 2 where L(s, πp × σp ) = (1 + wp p−3/2 α(p)p−s )−1 (1 + wp p−3/2 β(p)p−s )−1 . Before we begin the proof, we need a lemma. Lemma 2.7.2. Let ti ∈Tm , l ≥ 0. We have   l+1 −β(p)l+1    p−3ls−2l α(p)α(p)−β(p)      l+1 l+1 Wp (Θh(l, m)ti , s) = p−6ms−3ls−3m−5l/2 α(p) −β(p) α(p)−β(p)      0 if m = 0, i = 5 if m > 0, i = 3, 5 otherwise Proof. By the proof of Lemma 2.6.2 we have Θh(l, m)ti ∈ / P (Qp )ΘIp0 if m > 0. As for the case m = 0, we can check that Θh(l, 0)ti ∈ / P (Qp )ΘIp0 if i ∈ {1, 2, 7}. On the other hand, again by the proof of Lemma 2.6.2, we have Θh(l, m)ti ∈ P (Qp )Ip0 if and only if m > 0 and i ∈ {3, 5}. The lemma now follows immediately from (2.7.1) - (2.7.6). 62 Proof of Theorem 2.7.1. We have Zp (s, Wp , Bp ) = X Wp (Θh(l, 0)t5 , s)Bp (h(l, 0)t5 ) · Itl,0 5 l≥0 X + X Wp (Θh(l, m)ti , s)Bp (h(l, m)ti ) · Itl,m i (2.7.7) l≥0,m>0 i∈{3,5} Using Proposition 2.4.3 , Proposition 2.3.4 and Lemma 2.7.2 we have X Wp (Θh(l, m)ti , s)Bp (h(l, m)ti ) · Itl,m =0 i i∈{3,5} and hence Zp (s, Wp , Bp ) = X Wp (Θh(l, 0)t5 , s)Bp (h(l, 0)t5 ) · Itl,0 5 l≥0 X 1 = p−3ls−2l 2 (p + 1)(p + 1) l≥0  α(p)l+1 − β(p)l+1 α(p) − β(p)  (−pwp )l p−l 1 1 = L(3s + , πp × σp ). 2 (p + 1)(p + 1) 2 This completes the proof of the theorem. Remark. We might equally well have chosen Wp to be the simpler vector supported only on Θ (rather than on Θ and 1). The only reason we include 1 in the support of the section is because this definition will be necessary for §3. 2.8 The global integral and some results Classical Siegel modular forms and newforms for the minimal congruence subgroup For M a positive integer define the following global parahoric subgroups. 63  Z    Z B(M ) := Sp(4, Z) ∩   M Z  MZ  Z    Z U1 (M ) := Sp(4, Z) ∩   M Z  MZ  Z    Z U2 (M ) := Sp(4, Z) ∩    Z  MZ  Z    Z U0 (M ) := Sp(4, Q) ∩   M Z  MZ MZ  Z   Z Z ,  Z Z  MZ Z  Z Z   Z Z ,  Z Z  Z Z  Z Z   Z Z ,  Z Z  MZ Z MZ Z MZ Z MZ MZ Z Z MZ MZ MZ Z MZ Z  Z   M −1 Z .  MZ Z Z   MZ MZ Z Z Z When M = 1 each of the above groups is simply Sp(4, Z). For M > 1, the groups are all distinct. If Γ0 is equal to one of the above groups, or (more generally) is any congruence subgroup, we define Sk (Γ0 ) to be the space of Siegel cusp forms of degree 2 and weight k with respect to the group Γ0 . More let H2 = {Z ∈ M2 (C)|Z = Z t , i(Z − Z) is positive definite}. For any  precisely,  A B  ∈ G let J(g, Z) = CZ + D. Then f ∈ Sk (Γ0 ) if it is a holomorphic function g= C D on H2 , satisfies f (γZ) = det(J(γ, Z))k f (Z) for γ ∈ Γ0 , Z ∈ H2 and disappears at the cusps. We know that f has a Fourier expansion f (Z) = X S>0 a(S, F )e(tr(SZ)), 64 where e(z) = exp(2πiz) and S runs through all symmetric semi-integral positive-definite matrices of size two. Now let M be a square-free positive integer. For any decomposition M = M1 M2 into coprime integers we define, following Schmidt [Sch05], the subspace of oldforms Sk (B(M ))old to be the sum of the spaces Sk (B(M1 ) ∩ U0 (M2 )) + Sk (B(M1 ) ∩ U1 (M2 )) + Sk (B(M1 ) ∩ U2 (M2 )). For each prime p not dividing M there is the local Hecke algebra Hp of operators on Sk (B(M )) and for each prime q dividing M we have the Atkin-Lehner involution ηq also acting on Sk (B(M )). For details, the reader may refer to [Sch05]. By a newform for the minimal congruence subgroup B(M ), we mean an element f ∈ Sk (B(M )) with the following properties (a) f lies in the orthogonal complement of the space Sk (B(M ))old . (b) f is an eigenform for the local Hecke algebras Hp for all primes p not dividing M . (c) f is an eigenform for the Atkin-Lehner involutions ηq for all primes q dividing M . Remark. By [Sch05], if we assume the hypothesis that a nice L-function theory for GSp(4) exists, (b) and (c) above follow from (a) and the assumption that f is an eigenform for the local Hecke algebras at almost all primes. Description of our newforms Let M be an odd square-free positive integer and F (Z) = X a(T )e(tr(T Z)) T >0 be a Siegel newform for B(M ) of even weight l. Let N be an odd square-free positive integer and g be a normalized newform of weight l for Γ0 (N ). g has a Fourier expansion g(z) = ∞ X n=1 b(n)e(nz) 65 with b(1) = 1. It is then well known that the b(n) are all totally real algebraic numbers. We make the following assumption:  a(T ) 6= 0 for some T =  a b 2  b 2 (2.8.1) c √ such that −d = b2 − 4ac is the discriminant of the imaginary quadratic field Q( −d), and √ all primes dividing M N are inert in Q( −d). We define a function Φ = ΦF on G(A) by Φ(γh∞ k0 ) = µ2 (h∞ )l det(J(h∞ , iI2 ))−l F (h∞ (i)) where γ ∈ G(Q), h∞ ∈ G(R)+ and k0 ∈ ( Y Kp ) · ( p-M Y Ip ). p|M Because we do not have strong multiplicity one for G we can only say that the representation of G(A) generated by Φ is a multiple of an irreducible representation π. However that is enough for our purposes. We know that π = ⊗πv where     holomorphic discrete series    πv = unramified spherical principal series      ξv StGSp(4) where ξv unramified, ξ 2 = 1 v if v = ∞, if v finite , v - M, if v | M. Next, we define a function Ψ on GL2 (A) by l Ψ(γ0 mk0 ) = (det m) 2 (γi + δ)−l g(m(i))  where γ0 ∈ GL2 (Q), m =  α β γ δ   ∈ GL+ 2 (R), and k0 ∈ Y p-N GL2 (Zp ) Y p|N Γ0,p 66 Let σ be the automorphic representation of GL2 (A) generated by Ψ. We know that σ = ⊗σv where     holomorphic discrete series    σv = unramified spherical principal series      ξStGL(2) where ξv unramified, ξ 2 = 1 v if v = ∞, if v finite , v - N, if v | N. Description of our Bessel model In order to use our results from the previous sections, we need to associate a Bessel model to π (or more accurately, we associate it to π e). This involves making a choice of (S, Λ, ψ). This subsection is devoted to doing that. Q Let ψ = v ψv be a character of A such that • The conductor of ψp is Zp for all (finite) primes p, • ψ∞ (x) = e(−x), for x ∈ R, • ψ|Q = 1. √ Put L = Q( −d). where d is the integer defined in (2.8.1). First we deal with the case M = 1. In this case, our choice of S and Λ is identical to [Fur93]. To recall, put h(−d) T (A) = a tj T (Q)T (R)(Πp<∞ T (Zp )) (2.8.2) j=1 where tj ∈ Q p<∞ T (Qp ) and h(−d) is the class number of L. Write tj = γj mj κj , where γj ∈ GL2 (Q), mj ∈ GL+ 2 (R), and κj ∈ ((Πp<∞ GL2 (Zp )). Choose       d/4 0         0 1  S=     (1 + d)/4 1/2            1/2 1 if d ≡ 0 (mod 4) if d ≡ 3 (mod 4) 67 Let Sj = det(γj )−1 γjt Sγj . Then, any primitive semi-integral two by two positive definite matrix with discriminant equal to −d is SL2 (Z)-equivalent to some Sj . So, by our assumption, we can choose Λ a character of T (A)/T (Q)T (R)((Πp<∞ T (Zp )) such that h(−d) X Λ(tj )a(Sj ) 6= 0. j=1 Thus, we have specified a choice of S and Λ for M = 1. In the rest of this subsection, unless otherwise mentioned, assume M > 1. √ Suppose p is a prime dividing M . We can identify Lp with elements a + b −d with × a, b ∈ Qp . Let Z× L,p denote the units in the ring of integers of Lp . The elements of ZL,p are √ of the form a + b −d with a, b ∈ Zp and such that at least one of a and b is a unit. Let Γ0L,p × 0 be the subgroup of Z× L,p consisting of the elements with p|b. The group ZL,p /ΓL,p is clearly √ cyclic of order p + 1. Moreover, the elements {(−b + −d)/2} where b is a positive integer 0 satisfying {1 ≤ b ≤ 2p : b = d (mod 2)} are distinct in Z× L,p /ΓL,p . Note that d = 0 or 3 (mod 4) and hence b = d (mod 2) implies that 4 divides b2 + d. So we have the lemma: Lemma 2.8.1. There exists an integer b such that 4 divides b2 + d and (−b + √ −d)/2 is 0 a generator of the group Z× L,p /ΓL,p for each p|M . Proof. By the comments above, we can choose, for each prime pi dividing M , an integer bi √ 0 such that bi ≡ d (mod 2) and (−bi + −d)/2 is a generator of the group Z× L,pi /ΓL,pi . Now, using the Chinese Remainder theorem, choose b satisfying b ≡ bi (mod 2pi ) for each i . Now we define  S= b2 +d  4 b 2  b 2 . 1 As in section 2.1 we define the matrix ξ = ξS and the group T = TS . We have T (Q) ≃ L× . We write T (Zp ) for T (Qp ) ∩ GL2 (Zp ). Let h(−d) T (A) = a tj T (Q)T (R)(Πp<∞ T (Zp ) (2.8.3) p<∞ T (QP ) and h(−d) is the class number of L. For each p|M put Γ0L,p = j=1 where tj ∈ Q T (Zp ) ∩ Γ0p . Note that under the isomorphism T (Zp ) ≃ Z× L,p sending x + yξ 7→ x + y our two definitions for Γ0L,p agree, so there is no ambiguity. √ −d 2 , 68 Let M = p1 p2 ...pr be its decomposition into distinct primes. For each 1 ≤ i ≤ r we (p ) choose coset representatives uki i ∈ T (Zpi ) such that T (Zpi ) = pa i +1 (p ) uki i Γ0L,pi . ki =1 We write an r-tuple (k1 , .., kr ) in short as e k. Let X denote the Cartesian product of the r sets Xi = {x : 1 ≤ x ≤ pi }. For e k ∈ X, define uek = r Y (p ) uki i . i=1 Then it is easy to see that as e k varies over X the elements uek form a set of coset representatives of Πp|M T (Zp )/Πp|M Γ0L,p . Also note that |X| = |SL2 (Z)/Γ0 (M )| = Πp1 |M (pi + 1). We denote the quantity Πp1 |M (pi + 1) by g(M ). Let T (Z) denote the (finite) group of units in the ring of integers ZL of L. Let t(d) denote the cardinality of the group T (Z)/{±1}. We know that,     3    t(d) = 2      1 if d = 3 if d = 4 otherwise. × × Let TM be the image of T (Z) in Πp|M T (Zp ). Then TM ∩ Πp|M Γ0L,p = {±1}. Choose a set of elements r1 , r2 , ..rt(d) in T (Z) such that they form distinct representatives in T (Z)/{±1}. × . We have Let ri denote the image of ri in TM t(d) × TM Πp|M Γ0L,p = a ri (Πp|M Γ0L,p ). (2.8.4) i=1 Finally, choose x1 , x2 , ..., xg(M )/t(d) in Πp|M T (Zp ) such that we have the disjoint coset decomposition: g(M )/t(d) Πp|M T (Zp ) = a i=1 × xi TM Πp|M Γ0L,p (2.8.5) 69 This immediately gives us the fundamental coset decomposition: a T (A) = tj xk T (Q)T (R)(Πp-M T (Zp ))(Πpi |M Γ0L,pi ) (2.8.6) 1≤j≤h(−d) 1≤k≤g(M )/t(d) Also from (2.8.4) and (2.8.5) we immediately get another coset decomposition: Πp|M T (Zp ) = a xi rj Πp|M Γ0L,p (2.8.7) 1≤i≤g(M )/t(d) 1≤j≤t(d) But we know that an alternate set of coset representatives in the above equation is given by the elements uek . It follows that for any 1 ≤ i ≤ g(M )/t(d), 1 ≤ j ≤ t(d), there exists a unique e k ∈ X such that ue−1 xi rj ∈ Πp|M Γ0L,p . This correspondence is bijective. k Write tj xk = γj,k mj,k κj,k , where γj,k ∈ GL2 (Q), mj,k ∈ GL+ 2 (R), and κj,k ∈ (Πp<∞,p-M GL2 (Zp )· Πp|M Γ0p . Also, by (γj,k )f we denote the finite part of γj,k , that is, (γj,k )f = γj,k mj,k . −1 Lemma 2.8.2. For each j, the elements γj,1 rl γj,k form a system of representatives of SL2 (Z)/Γ0 (M ) as l, k vary over 1 ≤ l ≤ t(d), 1 ≤ k ≤ g(M )/t(d). Proof. Fix j. Let 1 ≤ l2 ≤ t(d), 1 ≤ k2 ≤ g(M )/t(d). We have −1 −1 −1 −1 γj,k r rl γj,k = mj,k2 κj,k2 x−1 k2 rl2 rl xk (mj,k κj,k ) . 2 l2
−1 −1 Therefore γj,k r rl γj,k ∈ GL+ 2 (R)Πq<∞ GL2 (Zq ) ∩ GL2 (Q) = SL2 (Z). Moreover, if it 2 l2 −1 0 belongs to Γ0 (M ) then we must have x−1 k2 r l2 r l xk ∈ Πp|M Γp and by (2.8.7) this can happen −1 only if l = l2 , k = k2 . Now the lemma follows because the size of the set γj,1 rl γj,k equals the cardinality of SL2 (Z)/Γ0 (M ). t Sγ . So, looking at S and S + Let Sj,k = det(γj,k )−1 γj,k j,k j,k as elements of GL2 (R) we −1 t have Sj,k = det(mj,k ) (m−1 j,k ) Smj,k . Lemma 2.8.3. There exists j, k, 1 ≤ j ≤ h(−d), 1 ≤ k ≤ g(M )/t(d) such that a(Sj,k ) 6= 0. Proof. By assumption (2.8.1), a(T ) 6= 0 for some primitive semi-integral positive definite matrix T with discriminant equal to −d. By [Fur93, p.209] there exists j such that T is SL2 (Z)-equivalent to Sj,1 . This means there is R ∈ SL2 (Z) such that T = Rt Sj,1 R. By 70 −1 Lemma 2.8.2, we can find k, l such that R = γj,1 rl γj,k g where g ∈ Γ0 (M ). This gives us −1 t −1 t T = g t γj,k rlt (γj,1 ) Sj,1 γj,1 rl γj,k g t = det(γj,k )−1 g t γj,k rlt Srl γj,k g t = det(γj,k )−1 g t γj,k Sγj,k g = g t Sj,k g Hence 0 6= a(T ) = a(g t Sj,k g) = a(Sj,k ), using the fact that the image of g t in Sp4 (Z) falls in B(M ) and F is a modular form for B(M ). Proposition 2.8.4. There exists a character Λ of T (A)/(T (Q)T (R)Πp<∞,p-M T (Zp )·Πp|M Γ0L,p ) such that X Λ(tj xk )−1 a(Sj,k ) 6= 0. 1≤j≤h(−d) 1≤k≤g(M )/t(d) Moreover for any such Λ we have Λp non-trivial on T (Zp ) for each prime p|M . Proof. By Lemma 2.8.3 we can find Sj,k such that a(Sj,k ) 6= 0. Hence using (2.8.6) we know that a character Λ satisfying the condition listed in the proposition exists. Let Λ be such a character and pi a fixed prime dividing M . We will show that Λpi is not the trivial character on T (Zpi ). For any 1 ≤ j ≤ h(−d) and e k ∈ X we can write tj uek = γj,ek mj,ek κj,ek , where γj,ek ∈ 0 GL2 (Q), mj,ek ∈ GL+ 2 (R) and κj,k ∈ (Πp<∞,p-M GL2 (Zp ) · Πp|M Γp . We put Sj,ek = det(γj,ek )−1 γ t e Sγj,k j,k Suppose Λpi is trivial on T (Zpi ). We claim that X Λ(tj uek )−1 a(Sj,ek ) = 0. (2.8.8) 1≤j≤h(−d) e k∈X Suppose we fix k1 , k2 , .., ki−1 , ki+1 , ..kr . For 1 ≤ y ≤ pi + 1, let e k y ∈ X be the r-tuple obtained by putting ki = y. Then, by essentially the same argument as in Lemma 2.8.2 we see that γ −1 γ ky form a set of representatives of Γ0 (M/pi )/Γ0 (M ). In particular, this j,e k1 j,e P implies, by [Sch05, 3.3.3], that y a(Sj,eky ) = 0, and therefore, because Λpi is trivial on 71 T (Zpi ), we must have P ) y Λ(tj ue ky −1 a(S ) = 0. It follows, by breaking up j,e ky X Λ(tj uek )−1 a(Sj,ek ) 1≤j≤h(−d) e k∈X into quantities as above, (2.8.8) follows. Given 1 ≤ k ≤ g(M )/t(d), 1 ≤ l ≤ t(d), let e k(k, l) be the unique element in X such that ue−1 xk rl ∈ Πp|M Γ0L,p . k(k,l) (2.8.9) Such an element exists by our comment after (2.8.7). Suppose we write rl = rl rl,f rl,∞ where rl,f ∈ Πp-M T (Zp ) and rl,∞ ∈ T (R) Then, using (2.8.9) we have −1 tj uek(k,l) = rl tj xk rl,∞ k with k ∈ (Πp<∞,p-M GL2 (Zp ) · Πp|M Γ0p . In other words we can take γj,ek(k,l) = rl γj,k . But then a(Sj,ek(k,l) ) = a(Sj,k ). Also from (2.8.9) it is clear that Λ−1 (tj uek(k,l) ) = Λ−1 (tj xk ). On the other hand if we let k, l vary over all elements in the range 1 ≤ k ≤ g(M )/t(d), 1 ≤ l ≤ t(d), the corresponding e k(k, l) vary over all e k ∈ X. As a result we conclude that X Λ(tj uek )−1 a(Sj,ek ) = t(d) X Λ(tj xk )−1 a(Sj,k ). (2.8.10) 1≤j≤h(−d) 1≤k≤g(M )/t(d) 1≤j≤h(−d) e k∈X But we have already shown that if Λpi is trivial on T (Zpi ) then X Λ(tj uek )−1 a(Sj,ek ) = 0. 1≤j≤h(−d) e k∈X The proof follows. Consider now the global Bessel space of type (S, Λ, ψ) for π e. We shall prove that this space is non zero. 72 For that, we consider Z BΦ (h) = (Λ ⊗ θ)(r)−1 Φ(rh)dr (2.8.11) ZG (A)B(Q)\B(A) where θ is defined as in Section 2.1 and Φ(h) = Φ(h). We will show that this function is non-zero. In fact, we shall explicitly evaluate BΦ (g∞ ) for g∞ ∈ G(R)+ . Proposition 2.8.5. Let g∞ ∈ G(R)+ and define BΦ (g∞ ) as in (2.8.11). The following hold: (a) If M = 1 we have BΦ (g∞ ) = det(J(g∞ , i))−l µ2 (g∞ )l e(−tr(S · g∞ (i)) X Λ(tj )−1 a(Sj ) 1≤j≤h(−d) (b) If M > 1 we have BΦ (g∞ ) = 1 det(J(g∞ , i))−l µ2 (g∞ )l e(−tr(S ·g∞ (i)) g(M ) X Λ(tj xk )−1 a(Sj,k ) 1≤j≤h(−d) 1≤k≤g(M )/t(d) Remark. This is a mild generalization of [Sug85, (1-26)]. We present a proof below. But first, we need some preliminary results. For any f a function on H2 and g∞ ∈ G(R)+ define (f |g∞ )(Z) = f (g∞ (Z))µ2 (g∞ )l det(J(g∞ , i))−l . Let M2Sym denote the space of symmetric two by two matrices. We shall think of M2Sym as a subgroup of G via x 7→ u(x). Also, for any continuous function f on G(Q)\G(A) define Z Cf (g) = M2Sym (Q)\M2Sym (A) f (u(X)g)ψ(tr(SX))−1 dX. The following lemma is the content of [Sug85, (1-19)]. However it is not proved there, so for convenience we include a proof here. Lemma 2.8.6. Let g∞ ∈ G(R)+ , gf ∈ GL2 (Af ). We consider gf as an element of G(Af ) 73   g 0 . Then via g → 7  0 det(g) · (g −1 )t CΦ (g∞ gf ) = det(J(g∞ , i))−l µ2 (g∞ )l a(gf , S)e(−tr(S · g∞ (i))) where a(gf , T ) is the (T )’th Fourier coefficient of F |gR , i.e., F |gR (Z) = X a(gf , T )e(−tr T Z), T and gR is defined by the equation gf = gQ gR gK with gQ ∈ G(Q), gR ∈ G(R)+ , gK ∈ Q Q p<∞,p-M Kp · p|M Ip . Proof. Put Up = Q p<∞,p-M Kp · Q p|M Ip ⊂ G(Af ). Define Φf by Φf (g) = Φ(ggf ). Then −1 . From that it follows that Φf is left invariant by G(Q) and right invariant by gQ Up gQ Q −1 . Also note that CΦ (g∞ gf ) = CΦf (g∞ ) and Φf (gg∞ ) = Φf (g∞ ) if g ∈ gQ ( q<∞ U (Zq ))gQ Φf (g∞ ) = µ2 (g∞ )l det(J(g∞ , i))−l (F |gR )(g∞ (i)). Finally, by approximation, we have ! M2Sym (A) = M2Sym (Q) + det(gQ )−1 gQ M2Sym (R) Y t . M2Sym (Zq ) gQ q<∞ Therefore Z CΦf (g∞ ) = M2Sym (Q)\M2Sym (A) Φf (u(X)g∞ )ψ(tr(SX))−1 dX Z = t \M Sym (R) det(gQ )−1 gQ M2Sym (Z)gQ 2 = µ2 (g∞ )l det(J(g∞ , i))−l X Φf (u(X)g∞ )e(tr(SX))dX a(gf , T )e(-tr(T · g∞ (i))) T ! Z · M2Sym (Z)\M2Sym (R) e(tr(T + S) · X)dX = det(J(g∞ , i))−l µ2 (g∞ )l a(gf , S)e(−tr(S · g∞ (i))) 74 Proof of Proposition 2.8.5. The case M = 1 is proved in [Sug85]. So we assume M > 1. Note that Z BΦ (g) = Z(A)T (Q)\T (A) CΦ (tg)Λ−1 (t)dt. Hence, using (2.8.6) and the fact that CΦ is right invariant by Πp<∞,p-M T (Zp ) · Πp|M Γ0L,p we have −1 0 BΦ (g∞ ) = [SL2 (Z) : Γ (M )] X Λ −1 j,k Z (tj xk ) ZT (R)\T (R) CΦ (tj xk t∞ g∞ )dt∞ . (2.8.12) Our Haar measure is normalized so that the compact set ZT (R)\T (R) has volume 1. We henceforth write R∗ instead of ZT (R) for simplicity. We have, Z R∗ \T (R) CΦ (tj xk t∞ g∞ )dt∞ Z CΦ (t∞ g∞ tj xk )dt∞ = R∗ \T (R) Z = CΦ (t∞ g∞ (γj,k )f )dt∞ R∗ \T (R) Z = det(J(t∞ g∞ , i))−l µ2 (t∞ g∞ )l a((γj,k )f , S)e(−tr(S · t∞ g∞ (i)))dt∞ ∗ R \T (R) Z l −l = det(J(g∞ , i)) µ2 (g∞ ) a((γj,k )f , S)( e(−tr(S · g∞ (i)))dt∞ ) R∗ \T (R) = det(J(g∞ , i))−l µ2 (g∞ )l a((γj,k )f , S)e(−tr(S · g∞ (i))) (2.8.13) Let us compute a((γj,k )f , S). We have F |mj,k (Z) = X a(T )e(−tr T · (mj,k (Z))) T >0 = X t a(T )e(−tr det(m−1 j,k ) · ((mj,k ) T mj,k ) · Z). T >0 −1 t So, the S’th Fourier coefficient corresponds to T = det(mj,k ) (m−1 j,k ) Smj,k = Sj,k . Thus a((γj,k )f , S) = a(Sj,k ). (2.8.14) Putting together (2.8.12),(2.8.13) and (2.8.14), we have the proof of the proposition. 75 Description of the Eisenstein series e fv This section describes the Eisenstein series on G(A). For each finite place v, recall that K e v ) and is defined by is the maximal compact subgroup of G(Q fv = G(Q e v ) ∩ GL4 (ZL,v ). K Let us now define g e K ∞ = {g ∈ G(R)|µ2 (g) = 1, g < iI2 >= iI2 }. Equivalently g K ∞ = U (2, 2; R) ∩ U (4, R). We define ρl (k∞ ) = det(k∞ )l/2 det(J(k∞ , i))−l .  A B  g  By [Ich07, p. 5], any matrix k∞ in K ∞ can be written in the form k∞ = λ  −B A where λ ∈ C, |λ| = 1, and A + iB, A − iB ∈ U (2; R) with det(A + iB) = det(A − iB). Then, ρl (k∞ ) = det(A − iB)−l Note that if k∞ has all real entries, i.e. k∞ ∈ Sp(4, R) ∩ O(4, R), then ρl (k∞ ) = det(J(k∞ , i))−l . Extend Ψ to GU (1, 1; L)(A) by Ψ(ag) = Ψ(g) for a ∈ L× (A), g ∈ GL2 (A). We define subsets S1 , S2 , S3 of the (finite) primes by • S1 is the set of primes that divide M but not N (2.8.15) 76 • S2 is the set of primes that divide gcd(M, N ) • S3 is the set of primes that divide N but not M √ and put S = S1 t S2 t S3 . Let L = Q( −d), Λ be as in §2.8. Note that all primes in S are odd and inert in L. e f ) by Now define the compact open subgroup U G of G(A e UG = e Y e Y KpG e Ip0 (2.8.16) p∈S1 ∪S2 p∈S3 p∈S / Y UpG Define s+ 1 fΛ (g, s) = δP 2 (m1 m2 )Λ(m1 )−1 Ψ(m2 )ρl (k∞ ) e if g = m1 m2 ne kk ∈ G(A) (2.8.17) G , k ∈ U G and where mi ∈ M (i) (A) (i = 1, 2), n ∈ N (A), k = k∞ k0 with k∞ ∈ K∞ 0 Q Q e G e / S1 t S2 , kp ∈ {1, s1 } for p ∈ S2 and k = p kp ∈ p Kp is such that kp = 1 if p ∈ e e kp ∈ {1, Θ} for p ∈ S1 . Put fΛ (g, s) = 0 if g is not of the form above. e We define the Eisenstein series EΨ,Λ (g, s) on G(A) by X EΨ,Λ (g, s) = fΛ (γg, s). (2.8.18) e γ∈P (Q)\G(Q) The global integral The global integral for our consideration is Z Z(s) = EΨ,Λ (g, s)Φ(g)dg. ZG (A)G(Q)\G(A) Then, by (2.2.6), Theorem 2.2.1, Theorem 2.5.1, Theorem 2.6.1 and Theorem 2.7.1 we have Z(s) = Y L(3s + 12 , π × σ) Qf Z∞ (s) p−6s−3 · · g(M/f )PM N 1 − ap wp p−3s−3/2 ζM N (6s + 1)L(3s + 1, σ × ρ(Λ)) p|f (2.8.19) 77 where f denotes gcd(M, N ) and Y L(s, π × σ) = L(s, πq × σq ) q<∞ Y L(s, σ × ρ(Λ))) = L(s, σq × ρ(Λq )), q<∞,q-M Y ζA (s) = (1 − p−s )−1 , p-A p prime Y PA = (r2 + 1), r|A r prime , QA = Y (1 − r), r|A r prime and Z Z∞ (s) = B(R)\G(R) WfΛ (Θg, s)BΦ (g)dg (2.8.20) As for the explicit computation of Z∞ , Furusawa’s calculation in [Fur93], mutatis mutandis, works for us. The only real point of difference is the choice of S. Furusawa chooses     d    4 0   ,      0 1  S=   1+d 1    2  4   ,      1 1 2 if d ≡ 0 (mod 4), if d ≡ 3 (mod 4). He computes Z∞ (s) for the case d ≡ 0 (mod 4) and uses it to deduce the other case via a simple change of variables, using  1+d  4 1 2  1 2 1  1 = − 12 t   d 0 0 1  4  1 0 1 − 12  0 . 1 78 In our case we have,   S= b2 +d  4 b 2 b 2 1 t    d 0 1 0 1 0  4   = b b 1 0 1 1 2 2  and so a similar change of variables works. Define a(Λ) = a(F, Λ) by  P    1≤j≤h(−d) Λ(tj )a(Sj ) a(Λ) = 1 P  −1  1≤j≤h(−d) Λ(tj xk ) a(Sj,k )  g(M ) if M = 1 if M > 1. 1≤k≤g(M )/t(d) Then we have (cf. [Fur93, p. 214]) 3 3 l Z∞ (s) = πa(Λ)(4π)−3s− 2 l+ 2 d−3s− 2 · Γ(3s + 32 l − 23 ) . 6s + l − 1 Henceforth we simply write L(s, F × g) for L(s, π × σ). We can summarize our computations in the following theorem. Theorem 2.8.7 (The integral representation). Let F and EΨ,Λ be as defined previously. Then Z 1 EΨ,Λ (g, s)Φ(g)dg = C(s) · L(3s + , F × g) 2 ZG (A)G(Q)\G(A) where C(s) = 3 3 l Y A(f )πa(Λ)(4π)−3s− 2 l+ 2 d−3s− 2 Γ(3s + 32 l − 32 ) p−6s−3 g(M/f )PM N (6s + l − 1)ζM N (6s + 1)L(3s + 1, σ × ρ(Λ)) 1 − ap wp p−3s−3/2 p|f with f = gcd(M, N ). Remark. Note that l 1 π 4−2l a(F, Λ) C( − ) = × (an algebraic number). 6 2 ζ(l − 2)L( l−1 2 , σ × ρ(Λ)) 79 Chapter 3 The Pullback Formula and the Second Integral Representation 3.1 Eisenstein series on GU (3, 3) e with Let PHe = MHe NHe be the Siegel parabolic of H,       A 0  |A ∈ GL3 (L), v ∈ Q× , MHe (Q) := m(A, v) =    0 v · (A−1 )t       1 b  |b ∈ M3 (L), bt = b . NHe (Q) := n(b) =    0 1 For s ∈ C, we form the induced representation e H(A) I(Λ, s) = ⊗v Iv (Λv , s) = IndP (A) (Λk · k3s ) e H e consisting of smooth functions Ξ on H(A) such that 1 1 Ξ(nm(A, v)g, s) = |v|−9(s+ 2 ) |NL/Q (det A)|3(s+ 2 ) Λ(det A)Ξ(g, s) (3.1.1) e for n ∈ NHe (A), m(A, v) ∈ MHe (A), g ∈ H(A). Finally, given such a section Ξ, we form the Eisenstein series EΞ (h, s) by EΞ (h, s) = X e γ∈PH e (Q)\H(Q) Ξ(γh, s) (3.1.2) 80 for Re(s) large, and defined elsewhere by meromorphic continuation. Some compact subgroups fp of (respecFor each finite place p of Q, define the maximal compact subgroups KpH , KpF , K e e e p ), Fe(Qp ), G(Q e p ) by tively) H(Q e e p ) ∩ GL6 (ZL,p ), KpH = H(Q e KpF = Fe(Qp ) ∩ GL2 (ZL,p ), fp = G(Q e p ) ∩ GL4 (ZL,p ). K Let UpH be the subgroup of KpH defined by e e  ∗   ∗   ∗  e e H H Up = z ∈ Kp | z ≡   ∗   0  ∗ ∗ ∗ ∗ ∗    ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗   0 0 0 ∗ ∗  0 0 0 0 ∗ ∗ (mod p) . e e p ) be the canonical map and define the subgroup Let r : KpH → H(F Ip0H = r−1 I(6, Fp ). e Also, put e H e K∞ = {g ∈ H(R)|µ 3 (g) = 1, g < iI3 >= iI3 }, g e K ∞ = {g ∈ G(R)|µ2 (g) = 1, g < iI2 >= iI2 } and Fe K∞ = {g ∈ Fe(R)|µ1 (g) = 1, g < i >= i}. H F By [Ich07,  p.5], any matrix k∞ in K∞ (resp. K∞ ) can be written in the form k∞ = A B  where λ ∈ C, |λ| = 1, and A + iB, A − iB lie in U (3; R) (resp. U (1; R)) with λ −B A e e 81 det(A + iB) = det(A − iB). For a positive even integer `, define ρ` (k∞ ) = det(A − iB)−` . (3.1.3) Note that an alternate definition for ρ` (k∞ ) is simply ρ` (k∞ ) = det(k∞ )`/2 det(J(k∞ , i))−` . Also note that if k∞ has all real entries, then ρ` (k∞ ) = det(J(k∞ , i))−` . A particular choice of section e 1 (Z). For any place v let Qv (resp. Ωv ) Fix an element Q ∈ H1 (Z) and an element Ω ∈ H e v ). denote the natural inclusion of Q (resp. Ω) into H(Q We impose the following condition on Ω for all primes p ∈ S2 : 0 If nm(A, v) ∈ PHe (Qp ) ∩ Ωp Ip0H Ω−1 p , then det(A) ∈ ΓL,p . e We next define, for each place v, a particular section Υv (s) ∈ Iv (Λv , s). e v ) such that Recall that Iv (Λv , s) consists of smooth functions Ξ on H(Q −9(s+ 21 ) Ξ(nm(A, t)g, s) = |t|v 3(s+ 12 ) |NL/Q (det A)|v Λv (det A)Ξ(g, s) (3.1.4) e v ). for n ∈ NHe (Qv ), m(A, t) ∈ MHe (Qv ), g ∈ H(Q • Clearly Ip (Λp , s) has a KpH fixed vector whenever Λp is unramified. e For all finite places p ∈ / S, choose Υp to be the unique KpH fixed vector with e Υp (1, s) = 1. (3.1.5) • For all finite places p ∈ S3 , choose Υp to be the unique UpH fixed vector with e Υp (Qp , s) = 1 (3.1.6) 82 and Υp (t, s) = 0 if t ∈ / PHe (Qp )Qp UpH . e • Suppose p ∈ S2 . Choose Υp to be the unique Ip0H fixed vector with e Υp (Qp , s) = 1 (3.1.7) and Υp (t, s) = 0 if t ∈ / PHe (Qp )Qp Ip0H . e • Let p ∈ S1 . Choose Υp to be the unique Ip0H fixed vector with e Υp (Ω, s) = 1, Υp (Qp , s) = 1 (3.1.8) and Υp (t, s) = 0 if t ∈ / PHe (Qp )ΩIp0H t PHe (Qp )Qp Ip0H . Note that such a vector exists by our assumption e e on Ω. • Finally choose Υ∞ to be the unique vector in I∞ (Λ∞ , s) such that Υ∞ (k∞ , s) = ρ` (k∞ ) H. for k∞ ∈ K∞ e e H(A) Let Υ be the factorizable section in IndP (A) (Λk · k3s ). defined by e H Υ(s) = (⊗v Υv (s)). As explained in (3.1.2), this gives rise to Eisenstein series EΥ (g, s). (3.1.9) 83 Also, for each place v, we define the local section Υ]v by Υ]v (g, s) = Υv (gQv , s). Define Υ] (s) = ⊗v Υ]v (s). Note that Υ] is right invariant by Q e −1 Q e −1 Q e 0H H H p∈S / Kp . p∈S1 tS2 QIp Q p∈S3 QUp Q p<∞ So, by (3.1.2), we construct an Eisenstein series EΥ] (g, s). Note that EΥ] (g, s) = EΥ (gQ, s) 3.2 (3.1.10) Statement of the pullback formula We henceforth fix Q to equal the following matrix:  0 1 0 0 0   1   0 Q=  0   0  0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0    0   −1 .  −1   0  0 An important embedding   g1 0 e ,→ H. e Let (g1 , g2 ) ∈ R(A), e . We now define an embedding ι : R and put h =  0 g2 −1 e Then we define ι(g1 , g2 ) to equal the element p0 hp0 ∈ H(A) where p0 ∈ GL6 (Q) is defined by   0 0 0      1 0 0 0 0 0      0 0 0 0 1 0  . p0 =    0 0 −1 1 0 0     −1 0 0 0 0 1   0 0 1 0 0 0 0 1 0 (3.2.1) 84 An essential feature of this embedding is the following. Suppose g1 = m1 (a)m2 (b)n ∈ P (A), g2 = b where  a 0 0   0 1 0 m1 (a) =   0 0 a−1  0 0 0  0   0  ∈ M (1) (A),  0  1   α β  ∈ Fe(A), n ∈ N (A), b =  γ δ and  1   0 m2 (b) =   0  0  where λ = µ1  α β γ δ 0 0 0    α 0 β  ∈ M (2) (A),  0 λ 0  γ 0 δ  . Then ι(g1 , g2 ) ∈ PHe (A). (3.2.2) e It is this key fact that enables us to pass from Klingen Eisenstein series on G(A) to Siegel e Eisenstein series on H(A). Henceforth, we fix Ω = ι(Θ, 1)Q. The Pullback formula e For an element g ∈ G(A), let Fe1 [g](A) denote the subset of Fe(A) consisting of all elements h2 such that µ2 (g) = µ1 (h2 ). 85 We will compute the integral Z EΥ] (ι(g, h), s)Ψ(h)Λ−1 (det h)dh. E(g, s) = (3.2.3) f1 (Q)\Fe1 [g](A) F Define ζ S (s) = Y (1 − p−s )−1 , p∈S / LS (s, χ−D ) = Y (1 − (χ−D )p (p)p−s )−1 p∈S / gcd(p,D)=1 where χ−D denotes the character of A× associated to L. Also define PS = Y (p2 + 1). p∈S Also, let ρ(Λ) denote the representation of GL2 (A) obtained from Λ by automorphic induction. Hence, for a prime q ∈ / S, we have: = L(s, σq × ρ(Λq ))   (1 − α2 (q)q −2s )−1 (1 − β 2 (q)q −2s )−1               (1 − α(q)Λq (q1 )q −s )−1 (1 − β(q)Λq (q1 )q −s )−1           (1 − α(q)Λq (q1 )q −s )−1 (1 − β(q)Λq (q1 )q −s )−1       ·(1 − α(q)Λ−1 (q1 )q −s )−1 (1 − β(q)Λ−1 (q1 )q −s )−1 q q if q is inert in L, if q is ramified in L, if q splits in L, where q1 ∈ Zq ⊗Q L is any element with NL/Q (q1 ) ∈ qZ× q . Also for a prime p ∈ S3 , put L(s, σp × ρ(Λp )) = (1 − p−2s−1 )−1 . Put L(s, σ × ρ(Λ)) = Y L(s, σq × ρ(Λq )). q-M Now define B(s) = B∞ (s)L(3s + 1, σ × ρ(Λ)) g(M )2 PS3 LS (6s + 2, χ−D )ζ S (6s + 3) (3.2.4) 86 where g(M ) = Y (p + 1) p|M and B∞ (s) = (−1)`/2 2−6s−1 π . 6s + ` − 1 Then the pullback formula says: e Theorem 3.2.1 (Pullback formula). For g ∈ G(A) define E(g, s) as above and EΨ,Λ (g, s) as in (2.8.18). Then we have E(g, s) = B(s)EΨ,Λ (g, s). We will prove the Pullback formula in §3.5 using the machinery developed in the next two sections. 3.3 The local integral and the unramified calculation Definitions We retain the notations and definitions of the previous section. Furthermore, for any prime p, we define the following compact subgroups of Fe(Qp ):  • ΓF0,p = {A ∈ KpF | A ≡  e ∗ ∗ e 0 ∗   (mod p)} • Let rp : KpF → GU (1, 1)(Fp ) be the canonical map and let Kp0F = rp−1 (GL2 (Fp )). e e Define F Γ00,p = Kp0F ∩ ΓF0,p . e e e Some useful properties e First, we note some properties of the section Υ] . Fix (g1 , g2 ) ∈ R(A). fp , k2 ∈ KpFe with µ2 (k1 ) = µ1 (k2 ). • Let p be a prime not dividing M N and k1 ∈ K Then, note that Q−1 ι(k1 , k2 )Q ∈ KpH . e 87 Because Υ]p is KpH -fixed, it follows that e Υ] (ι(g1 k1 , g2 k2 ), s) = Υ] (ι(g1 , g2 ), s), (3.3.1) fp , k2 ∈ ΓF with µ2 (k1 ) = µ1 (k2 ) then check that • Let p|N, p - M . If k1 ∈ U 0,p e Q−1 ι(k1 , k2 )Q ∈ UpH . e (3.3.2) Because Υ]p is UpH -fixed, it follows that e Υ] (ι(g1 k1 , g2 k2 ), s) = Υ] (ι(g1 , g2 ), s), (3.3.3) F with µ (k ) = µ (k ) then check • Let p be a prime dividing M . If k1 ∈ Ip0 , k2 ∈ Γ00,p 2 1 1 2 e that Q−1 ι(k1 , k2 )Q ∈ Ip0H . e (3.3.4) Because Υ]p is Ip0H -fixed, it follows that e Υ] (ι(g1 k1 , g2 k2 ), s) = Υ] (ι(g1 , g2 ), s), (3.3.5) F g • Finally, let k1 ∈ K ∞ , k2 ∈ K∞ with µ2 (k1 ) = µ1 (k2 ). Check that e H Q−1 ι(k1 , k2 )Q ∈ K∞ . (3.3.6) Υ] (g1 k1 , g2 k2 , s) = ρ` (k1 )ρ` (k2 )−1 Υ] (g1 , g2 , s). (3.3.7) e Hence we have The key local zeta integral Let ψ = Q v ψv be a character of A such that • The conductor of ψp is Zp for all (finite) primes p, • ψ∞ (x) = e(x), for x ∈ R, • ψ|Q = 1. 88 Let WΨ be the Whittaker model for Ψ. It is a function on Fe(A) defined by  Z WΨ (g) = Ψ  Q\A 1 x 0 1    g  ψ(−x)dx. We have the Fourier expansion    λ 0  g Ψ(g) = WΨ  0 1 λ∈Q× X (3.3.8) By the uniqueness of Whittaker models, we have a factorization WΨ = ⊗v WΨ,v . fv , define the local zeta integral Now, for each place v, and elements gv ∈ Fe(Qv ), kv ∈ K Z Υ]v (ι(kv , hv ), s)WΨ,v (gv hv )Λ−1 v (det hv )dhv , Zv (gv , kv , s) = (3.3.9) Fe1 (Qv ) The evaluation of this local integral at each place v lies at the heart of our proof of the pullback formula. First of all, by (3.2.2) and the properties proved earlier, observe that it is enough to fv )\K fv /Uv , evaluate the integral for kv lying in a fixed set of representatives of (P (Qv ) ∩ K where    fv  K    fv Uv = U      I 0 v if v ∈ /S if v ∈ S3 if v ∈ S1 t S2 For 1 ≤ i ≤ 5, define the matrices si ∈ G(Q) as follows:      0 0 0 1 0 0 0 0 0 1           0 0 1 0 0 0 0 0 1 0  , s3 =   , s2 =  s1 =        0 −1 0 0 −1 0 0 0 0 1      0 −1 −1 0 0 0 0 0 1 0 1 0    0 1 ,  0 0  0 0 89  0   −1 s4 =   0  1 −1 0 0 0 0 0    0 ,  0 −1  −1 0 0  0 −1   −1 s5 =   0  0 0 1 0 0 0    0 .  0 −1  −1 0 0 e p ) as follows: Define the set Y∞ = {1} and for a (finite) prime p, define the set Yp ⊂ G(Q • Yp = {1} if p - M N . • Yp = {1, s1 , s2 } if p|N, p - M . • Yp = {1, s1 , s2 , s3 , Θ, Θs2 , Θs4 , Θs5 } if p|M . e p ). This Remark. In the above definition, we consider the si and Θ as elements of G(Q e v ) for all places v. makes Yv a subset of G(Z fv )\K fv /Uv at all places v. Lemma 3.3.1. Yv is a set of representatives for (P (Qv ) ∩ K Proof. For v infinite or v a prime not dividing M N , this is obvious. Now let p be a prime dividing N but not M . If W denotes the eight element Weyl group, then W is generated fp )\K fp /I G where I G denotes the by s1 and s2 . W is a set of representatives for (P (Qp ) ∩ K p p e e e p ). Since U fp is larger than I G , there is some collapsing, as expected. Iwahori subgroup of G(Q p e By explicit computation we find that {1, s1 , s2 } do form a set of distinct representatives. The case when p|M is also proved similarly by explicit computation. For brevity, we do not include the details here. The rest of this section and the next will be devoted to evaluating at each place v the integral Zv (gv , kv , s) for every kv ∈ Yv , gv ∈ Fe(Qv ). The local integral at unramified places In the rest of this section, q will denote a prime that does not divide M N . Hence, both Λq and σq are unramified. In particular, σq is a spherical principal series representation induced from unramified characters α, β of Q× q . By abuse of notation we use q to also denote its inclusion in Q× q . Thus q is an uniformizer in our local field. 90 Let ρ(Λ) denote the representation of GL2 (A) obtained from Λ by automorphic induction. Hence we have: = L(s, σq × ρ(Λq ))   (1 − α2 (q)q −2s )−1 (1 − β 2 (q)q −2s )−1               (1 − α(q)Λq (q1 )q −s )−1 (1 − β(q)Λq (q1 )q −s )−1           (1 − α(q)Λq (q1 )q −s )−1 (1 − β(q)Λq (q1 )q −s )−1       ·(1 − α(q)Λ−1 (q1 )q −s )−1 (1 − β(q)Λ−1 (q1 )q −s )−1 q q if q is inert in L, if q is ramified in L, if q splits in L, where q1 ∈ Zq ⊗Q L is any element with NL/Q (q1 ) ∈ qZ× q .   (1 − χ(q)q −s ) if χ is unramified at q, × For a character χ of Qq define L(s, χ) =  1 otherwise. Proposition 3.3.2. Let q be a prime such that q - M N . Let 1 denote the trivial character and χ−D denote the Hecke character associated to the quadratic extension L/Q. Then, we have Zq (gq , 1, s) = WΨ,q (gq ) · L(3s + 1, σq × ρ(Λq )) . L(6s + 2, (χ−D )q )L(6s + 3, 1) Proof. Let KqF1 denote the maximal compact subgroup of Fe1 (Qq ) defined by e KqF1 = Fe1 (Qq ) ∩ GL2 (ZL,q ). e e Note that for g ∈ Fe1 (Qq ), k1 , k2 ∈ KqF1 , we have using (3.1.1), (3.3.1) Υ]q (ι(1, k1 gk2 ), s) = Υ]q (ι(m2 (k1 )m2 (k1 )−1 , k1 gk2 ), s) = Υ]q (ι(m2 (k1 )−1 , gk2 ), s) = Υ]q (ι(1, g), s) In other words Υ]q (ι(1, g), s) only depends on the double coset KqF1 gKqF1 . e e There are three distinct cases: q can be inert, split or ramified in L. We consider each of these cases separately. Case 1. q is inert in L. 91 In this case, Lq is a quadratic extension of Qq . We may write elements of Lq in the √ √ form a + b −d with a, b ∈ Qq ; then ZL,q = a + b −d where a, b ∈ Zq . Also note that Λq is trivial. We know (Cartan decomposition) that Fe1 (Qq ) = G KqF1 An KqF1 e e n≥0  where An =  qn 0 0 q −n   . So (3.3.9) gives us Zq (gq , 1, s) = X Υ]q (ι(1, An ), s) Z e F e F Kq 1 An Kq 1 n≥0 WΨ,q (gq hq )dhq . (3.3.10) Given an element k ∈ KqF1 we can find l ∈ Z× L,q such that kl ∈ GL2 (Zq ). It follows that e if GL2 (Zq )An GL2 (Zq ) = G ai GL2 (Zq ), i where ai ∈ SL2 (Zq ) then KqF1 An KqF1 = e e G ai KqF1 . e i The importance of this observation is that we can use the theory of Hecke operators for R GL2 to evaluate e e WΨ,q (gq hq )dhq . F F 1 1 Kq An Kq Recall that classically T (q k ) denotes the Hecke operator corresponding to the set GL2 (Zq )Sk GL2 (Zq ) where Sk comprises of the matrices of size 2 with entries in Zq whose determinant generates the ideal (q k ). Also observe that   qn 0  GL2 (Zq )An GL2 (Zq ) GL2 (Zq )S2n GL2 (Zq ) =  0 qn   G q 0   GL2 (Zq )S2n−2 GL2 (Zq ). 0 q 92 So we have Z e F WΨ,q (gq hq )dhq = e F X WΨ,q (gq ai ) (3.3.11) = (β2n − β2n−2 )WΨ,q (gq ) (3.3.12) Kq 1 An Kq 1 i where βk is the eigenvalue corresponding to Ψ for the Hecke operator T (q k ). We put βk = 0 if k < 0. Using [Bum97, Propostion 4.6.4] we have βk = q k/2 (α(q)k+1 − β(q)k+1 ) α(q) − β(q) (3.3.13) for k ≥ 0. On the other hand, using (3.2.1) we see that ι(1, An ) is the matrix  1    0    0 C=   0    0  1 − qn 0 0 0 0 1 0 0 0 0 q −n 0 0 0 q −n − 1 1 0 0 0 0 1 0 0 0 0 0    0   0   0   0  n q We can write C = P K where  qn   0   0 P =  0   0  0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 q −n 0 0 q −n  0 1   0 0   0 0  ∈ P e (Qq )  H 0 0   1 0  0 1 93 and  1    0    0 K=   0    0  1 − qn 0 0 1 0 0 1 0 1 − qn 0 0 0 0  0 −1   0 0 0   −1 0 0  .  n q 0 0   0 1 0  0 0 0 (3.3.14) qn So, by (3.1.1) we have Υ]q (ι(1, An ), s) = q −6n(s+1/2) Υ]q (K, s) (3.3.15) Also K ∈ KqH , hence Υ]q (K, s) = 1. e So, by (3.3.10),(3.3.12),(3.3.13),(3.3.15), we have X q n (α(q)2n+1 − β(q)2n+1 ) Zq (gq , 1, s) = WΨ,q (gq ) q −6n(s+1/2) α(q) − β(q) n≥0 − X n≥1 q −6n(s+1/2) q n−1 (α(q)2n−1 − β(q)2n−1 )  α(q) − β(q) (1 − q −6s−3 )(1 + q −6s−2 ) (1 − α(q)2 q −6s−2 )(1 − β(q)2 q −6s−2 ) L(3s + 1, σq × ρ(Λq )) = WΨ,q (gq ) · L(6s + 2, χ−D )L(6s + 3, 1) = WΨ,q (gq ) Case 2. q is split in L. We can identify Lq with Qq ⊕ Qq with Qq embedded diagonally as t 7→ (t, t). For g ∈ GLn (Qq ) denote g ∗ = Jn−1 (g t )−1 Jn . Note that for n = 2, g ∗ = detg g . Now there is a natural isomorphism of GLn (Qq ) into U (n, n)(Qq ) given by g 7→ (g, g ∗ ). Thus specializing to the n = 2 case, g 7→ (g, detg g ) takes GL2 (Qq ) isomorphically onto Fe1 (Qq ).   q m+k 0 . Define Am,k to be the image of  0 qm   m+k −m (q ,q ) 0 . Thus Am,k =  m −m−k 0 (q , q ) 94 The Cartan decomposition gives us Fe1 (Qq ) = G KqF1 Am,k KqF1 . e e k≥0 m∈Z Let q1 denote the element (q, 1) ∈ Lq . So NL/Q (q1 ) = q. For brevity, let us denote Λq (q1 ) by λ. Note that for any integer m, Λq (q m , q −m ) = λ2m . Now, using (3.3.9), we have Zq (gq , 1, s) = X Υ]q (ι(1, Am,k ), s)λ−4m−2k k≥0 m∈Z Z e F e F Kq 1 Am,k Kq 1 WΨ,q (gq hq )dhq . (3.3.16) Using the above conventions, and the notation of the inert case, we have   q −m 0  GL2 (Zq )Am,k GL2 (Zq ) GL2 (Zq )Sk GL2 (Zq ) =  0 q −m   G q 0   GL2 (Zq )Sk−2 GL2 (Zq ). 0 q So, we have Z e F e F Kq 1 Am,k Kq 1 WΨ,q (gq hq )dhq = (βk − βk−2 )WΨ,q (gq ) where we put βk = 0 if k < 0. Now ι(1, Am,k ) is the matrix C where        C=       1 0 0 0 0 0 0 1 0 0 0 0 0 qm 0 0 0 0 qm − 1 1 0 0 0 0 0 1 1 − q m+k 0 0 0 0 q m+k    0    0  .  0    0   (3.3.17) 95 [Note that by C we actually mean the pair (C, C ∗ ). This convention will be used throughout our treatment of the split case; thus the letters P, K etc. are really a shorthand for (P, P ∗ ), (K, K ∗ ), etc.] First we consider the case m ≥ 0. We can write C = P K where  q m+k    0    0 P =   0    0  0 0 0 0 1 0 0 0 q m −q m 0 0 1 0 0 0 0 0 0  0 1   0 0   0 0   0 0   1 0  0 1 and        K=       −1 1 0 0 0 0 0 1 0 0 0 0 0 qm 1 0 0 0 qm − 1 1 0 0 0 0 0 1 1 − q m+k 0 0 0 0 q m+k    0    0  .  0    0   Since P ∈ PHe (Qq ) we have, using (3.1.1) Υ]q (ι(1, Am,k ), s) = λ2m+k q −3(2m+k)(s+1/2) Υ]q (K, s) (3.3.18) Since K ∈ KqH , Υ]q (K, s) = 1. e Thus when m ≥ 0 we have Υ]q (ι(1, Am,k ), s) = λ2m+k q −(6m+3k)(s+1/2) . (3.3.19) Now suppose 0 ≥ m ≥ −k. For convenience we temporarily put n = −m. So 0 ≤ n ≤ k. We can write C = P K 96 where  q k−n    0    0 P =   0    0  0 0 0 0 1 0 0  0 1   0 0   0 0   0 0   1 0  0 1 q −n 0 0 q −n 0 0 0 0 0 0 0 1 and        K=       −1 1 0 0 0 0 0 1 0 0 0 0 0 1 −1 0 0 0 1 − qn qn 0 0 0 0 0 1 1 − q k−n 0 0 0 0 q k−n    0    0  .  0    0   Since P ∈ PHe (Qq ) we have, using (3.1.1) Υ]q (ι(1, Am,k ), s) = λ−2n+k q −3k(s+1/2) Υ]q (K, s) (3.3.20) = λ2m+k q −3k(s+1/2) Υ]q (K, s) As before Υ]q (K, s) = 1. So, when −k ≤ m ≤ 0 we have Υ]q (ι(1, Am,k ), s) = λ2m+k q −3k(s+1/2) . (3.3.21) Finally, consider the case m ≤ −k. For convenience we again put n = −m. So 0 ≤ k ≤ n. We can write C = P K where  1   0   0 P =  0   0  0 0 0 0 1 0 0 0 0 1 q −n 0 0 0 q −n 0 0    0    0    0    0   0 0 0 1 0 0 0 0 0 q k−n 97 and        K=       1 0 0 0 0 1 0 0 0 1 0 0 1 − qn 0 0 0 q n−k − 1 0 0 0 0    0 0   −1 0 0   n q 0 0   0 1 0  0 0 1 0 Since P ∈ PHe (Qq ) we have, using (3.1.1) Υ]q (ι(1, Am,k ), s) = λ−2n+k q −3(2n−k)(s+1/2) Υ]q (K, s) 2m+k −3(−2m−k)(s+1/2) =λ q (3.3.22) Υ]q (K, s) So, when m ≤ −k we have Υ]q (ι(1, Am,k ), s) = λ2m+k q (6m+3k)(s+1/2) . (3.3.23) Substituting (3.3.13),(3.3.17),(3.3.19),(3.3.21),(3.3.23) into (3.3.16) we obtain Zq (gq , 1, s) = WΨ,q (gq ) + ∞ X X ∞ (βk − βk−2 ) λ−2m−k q (−6m−3k)(s+1/2) k=0 0 X m=1 λ −2m−k −3k(s+1/2) q + −k−1 X λ −2m−k (6m+3k)(s+1/2)  q m=−∞ WΨ,q (gq )(1 − q −6s−3 )(1 − q −6s−2 ) = (1 − α(q)λq −3s−1 )(1 − β(q)λq −3s−1 )(1 − α(q)λ−1 q −3s−1 )(1 − β(q)λ−1 q −3s−1 ) m=−k = WΨ,q (gq ) · L(3s + 1, σq × ρ(Λq )) L(6s + 2, χ−D )L(6s + 3, 1) Case 3. q is ramified in L. √ We largely revert to the notation of the inert case. Write elements of Lq as a + b −d √ √ with a, b ∈ Qq ; so ZL,q = a+b −d with a, b ∈ Zq . Also let q1 = −d; thus NL/Q (q1 ) ∈ qZ× q . Put λ = Λq (q1 ). We have λ2 = 1. 98 The Cartan decomposition takes the form Fe1 (Qq ) = G KqF1 An KqF1 e e n≥0  where An = qn  1 0  0 q1−n  . So (3.3.9) gives us Zq (gq , 1, s) = X Υ]q (ι(1, An ), s) n≥0 Z e F e F Kq 1 An Kq 1 WΨ,q (gq hq )dhq . (3.3.24) Now,     −n n 0 q1 q 0 e e  KqFe1   KqFe1 KqF1 An KqF1 =  0 q1−n 0 1 . So, by the same argument as in the inert case, we have, Z e F e F Kq 1 An Kq 1 WΨ,q (gq hq )dhq = (βn − βn−2 )WΨ,q (gq ) (3.3.25) where, of course, we put βn = 0 for negative n. Now ι(1, An ) is the same matrix as in the inert case with q replaced by q1 . So the same choice of P and K work. Thus, by (3.1.1) we have Υ]q (ι(1, An ), s) = λn q −3n(s+1/2) (3.3.26) Substituting (3.3.13),(3.3.25), (3.3.26) in (3.3.24) we have X q n/2 (α(q)n+1 − β(q)n+1 ) Zq (gq , 1, s) = WΨ,q (gq ) λn q −3n(s+1/2) α(q) − β(q) n≥0 − X n≥2 n −3n(s+1/2) q λ q n/2−1 (α(q)n−1 − β(q)n−1 ) α(q) − β(q) (1 − q −6s−3 ) (1 − α(q)λq −3s−1 )(1 − β(q)λ q −3s−1 ) L(3s + 1, σq × ρ(Λq )) = WΨ,q (gq ) · L(6s + 2, χ−D )L(6s + 3, 1) = WΨ,q (gq )  99 (Note that L(s, χ−D ) = 0 in this case) This completes the proof. 3.4 The local integral for the ramified and infinite places The local integral for primes in S3 Let r be a prime dividing N but not M . Note that r is inert by our assumptions. In this section we will prove the following proposition. Proposition 3.4.1.  We have
  21 WΨ,r (gr ) · L 3s + 1, σr × ρ(Λr ) r +1 Zr (gr , kr , s) =  0 if kr = 1 if kr = s1 or s2 . where the local L-function L(s, σr × ρ(Λr )) is defined by L(s, σr × ρ(Λr )) = (1 − r−2s−1 )−1 . Proof. Recall that σ is the irreducible automorphic representation of GL2 (A) generated by e Let σr be the local component of σ at the place r. We know that σr = Sp ⊗ τ where Ψ. Sp denotes the special (Steinberg) representation and τ is a (possibly trivial) unramified quadratic character. We put ar = τ (r), thus ar = ±1 is the eigenvalue of the local Hecke operator T (r). We first deal with the case kr = 1. Let ΓF0,r1 denote the compact open subgroup of e Fe1 (Qr ) defined by e e ΓF0,r1 = ΓF0,r ∩ Fe1 (Qr ). Note that for g ∈ Fe1 (Qr ), k1 , k2 ∈ ΓF0,r1 , we have using (3.1.1), (3.3.3) e Υ]r (ι(1, k1 gk2 ), s) = Υ]r (ι(m2 (k1 )m2 (k1 )−1 , k1 gk2 ), s) = Υ]r (ι(m2 (k1 )−1 , gk2 ), s) (3.4.1) = Υ]r (ι(1, g), s) In other words Υ]r (ι(1, g), s) only depends on the double coset ΓF0,r1 gΓF0,r1 . e e Because r is inert in L, Lr is a quadratic extension of Qr . We may write elements of Lr 100 √ √ in the form a + b −d with a, b ∈ Qr ; then ZL,r = a + b −d where a, b ∈ Zr . Also note that Λr is trivial. We know (Bruhat-Cartan decomposition) that e e Fe1 ∪ ΓF0,r1 wΓF0,r1 Fe1 (Qr ) = Γ0,r G e e ΓF0,r1 An ΓF0,r1 ∪ ∪ G G ΓF0,r1 wAn ΓF0,r1 G ∪ e e where An =  rn 0 0 r−n  (3.4.2) ΓF0,r1 wAn wΓF0,r1 . e e n>0 n>0  e e n>0 n>0 ∪ ΓF0,r1 An wΓF0,r1  0 1   and w =  . So (3.3.9) gives us −1 0 Zr (gr , 1, s) = Υ]r (ι(1, 1), s) Z + Υ]r (ι(1, w), s) Z + X + WΨ,r (gr hr )dhr e F e F 1 wΓ 1 Γ0,r 0,r Υ]r (ι(1, An ), s) n>0 X e F 1 Γ0,r Z X Υ]r (ι(1, An w), s) Z Υ]r (ι(1, wAn ), s) Z n>0 + X e F WΨ,r (gr hr )dhr (3.4.3) e F WΨ,r (gr hr )dhr e F e F e F 1 A wΓ 1 Γ0,r n 0,r 1 wA Γ 1 Γ0,r n 0,r Υ]r (ι(1, wAn w), s) n>0 e F 1A Γ 1 Γ0,r n 0,r n>0 + WΨ,r (gr hr )dhr WΨ,r (gr hr )dhr Z e F e F 1 wA wΓ 1 Γ0,r n 0,r WΨ,r (gr hr )dhr . Now WΨ,r is an eigenvector for the Iwahori-Hecke algebra, hence each of the integrals in (3.4.3) evaluates to a constant multiple of WΨ,r (gr ). Thus for some function A(s) (not depending on gr ) we have Zr (gr , 1, s) = A(s)WΨ,r (gr ). Since WΨ,r (1) = 1 it follows that Zr (gr , 1, s) = Zr (1, 1, s)WΨ,r (gr ) (3.4.4) 101 Given an element k ∈ ΓF0,r1 we can find l ∈ Z× L,q such that kl ∈ Γ0,r . It follows that if e Γ0,r An Γ0,r = G ai Γ0,r , i where ai ∈ SL2 (Zq ) then ΓF0,r1 An ΓF0,r1 = e e G ai ΓF0,r1 . e i  From [Miy89, Lemma 4.5.6], we may choose ai =  rn mr−n r−n   where 0 ≤ m < r2n . Using 0 the formula in [GK92, Lemma 2.1], we have WΨ,r (ai ) = r−2n and hence X WΨ,r (a) = 1 (3.4.5) e e e F 1 A ΓF1 /ΓF1 a∈Γ0,r n 0,r 0,r Also, from [Miy89] we have ΓF0,r1 wAn wΓF0,r1 = e e G bi ΓF0,r1 . e i  r−n 0  . Using the formula in [GK92, Lemma 2.1], and doing some where bi =  −mr1−n rn simple manipulations, we have X WΨ,r (b) = 1 (3.4.6) e e e F 1 wA wΓF1 /ΓF1 b∈Γ0,r n 0,r 0,r Next, we check that the quantities Υ]r (ι(1, An w), s), Υ]r (ι(1, wAn ), s), are both equal to e r ) does not belong to 0. Indeed Υ]r (ι(1, A), s) = 0 whenever ι(1, A) as an element of H(Q PHe (Qr )QUrH Q−1 . Let K be the matrix defined in (3.3.14) with q replaced by r. It suffices to e prove that the quantities Kι(m(w), 1)Q, Kι(1, w) · Q do not belong to (PHe (Qr ) ∩ KrH )QUrH . e e We check this by taking a generic element P of (PHe (Qr )∩KrH ) and showing that Q−1 P K0 ∈ / e UrH where K0 is one of the above quantities. That is a simple computation and is omitted. e 102 On the other hand, putting  rn   0   0 P =  0   0  0 0 0 0 1 0 0 0 1 r−n 0 0 r−n 0 0 0 0 0 0 0 1    0 0   0 0  ∈ P e (Qr )  H 0 0   1 0  0 1 we can check that Q−1 P −1 · (1, An )Q ∈ UrH , e hence Υ]r (ι(1, An ), s) = r−6n(s+1/2) (3.4.7) Also, putting  0   0   1 P =  0   0  0 rn 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0    0    −n r   ∈ P e (Qr )  H −n r    0   0 we can check that Q−1 P −1 · ι(w, An w)Q ∈ UrH , e hence Υ]r (ι(1, wAn w), s) = Υ]r (ι(w, An w), s) = r−6n(s+1/2) (3.4.8) 103 So, using (3.4.5), (3.4.6), (3.4.7) and (3.4.14), Zr (1, 1, s) = Υ]r (ι(1, 1), s) Z e F 1 Γ0,r dhr + Υ]r (ι(1, An ), s) Z n>0 e F e F 1A Γ 1 Γ0,r n 0,r WΨ,r (hr )dhr  Z + X e F e F 1A Γ 1 Γ0,r n 0,r WΨ,r (hr )dhr = [KrF : ΓF0,r1 ]−1 1 + 2 e e X Υ]r (ι(1, An ), s)
n>0 = 1 (1 + 2 r+1 X r−6n(s+1/2) ) n>0 1 1 + r−6s−3 = r + 1 1 − r−6s−3 whence (3.4.4) implies Zr (gr , 1, s) = 1 + r−6s−3 1 WΨ,r (gr ) · . r+1 1 − r−6s−3 (3.4.9) Finally, we deal with the case when kr = s1 or s2 . The key observation is that if k ∈ KrF1 e then for i = 1, 2, f s−1 i m2 (k)si ∈ Ur . By the same argument as in (3.4.1), it follows that Υ]r (si , g, s) only depends on the double P e e coset KrF1 gΓF0,r1 . So, if we can show that for all h ∈ Fe1 (Qr ) we have e e e WΨ,r (gr a) = F F F 1 1 1 a∈Kr hΓ0,r /Γ0,r 0, it would follow that Zr (gr , si , s) = 0. If we define X W (gr ) = WΨ,r (gr a) e e e F F 1 /ΓF1 a∈Kr 1 hΓ0,r 0,r then W (gr k) = W (gr ) for all k ∈ KrF1 ; in other words W is a vector in the Whittaker space e that is right KrF1 invariant. But the only such vector is the 0 vector and this completes the e proof. The local integral for primes in S2 Proposition 3.4.2. Let p be a prime dividing gcd(M, N ) and kp ∈ Yp . We have 104 Zp (gp , kp , s) =    WΨ,p (g2p ) if kp = 1 or kp = s1  0 otherwise . (p+1) Proof. Recall that σ is the irreducible automorphic representation of GL2 (A) generated by Ψ. Let σp be the local component of σ at the place p. We know that σp = Sp ⊗ τ where Sp denotes the special (Steinberg) representation and τ is a (possibly trivial) unramified quadratic character. We put ap = τ (p), thus ap = ±1 is the eigenvalue of the local Hecke operator T (p). Fe1 denote the compact open subgroup of Fe1 (Qp ) defined by Let Γ00,p Fe1 Fe Γ00,p = Γ00,p ∩ Fe1 (Qp ). F1 F1 We first consider the case kp = 1. Note that for g ∈ Fe1 (Qp ), k1 ∈ Γ00,p , k2 ∈ Γ00,p , we e e have using (3.1.1), (3.3.5), Υ]p (ι(1, k1 gk2 ), s) = Υ]p (ι(m2 (k1 )m2 (k1 )−1 , k1 gk2 ), s) = Υ]p (ι(m2 (k1 )−1 , gk2 ), s) (3.4.10) = Υ]p (ι(1, g), s) F1 F1 In other words Υ]p (ι(1, g), s) only depends on the double coset Γ00,p gΓ00,p . e e Because p is inert in L, Lp is a quadratic extension of Qp . We may write elements of √ √ Lp in the form a + b −d with a, b ∈ Qp ; then ZL,p = a + b −d where a, b ∈ Zp . Also note that Λp is not trivial. 0 Fix a set U of representatives of Z× L,p /ΓL,p . For definiteness we may take U = {1} ∪ {b +  l 0 √ −d : b ∈ Z, 0 ≤ b < p}  e1 × e   . We know that given g ∈ ΓF0,p there exists l ∈ ZL,p such that For l ∈ L× p put l = −1 0 l F1 ge l ∈ Γ00,p . From this fact and the Bruhat-Cartan decomposition (3.4.2), it follows that e 105 Fe1 (Qp ) = G F1 e 0F1 lΓ0,p Γ00,p e G ∪ e G Fe1 Fe1 An e lΓ00,p Γ00,p G ∪ G Fe1 Fe1 An we lΓ00,p Γ00,p (3.4.11) n>0 l∈U n>0 l∈U ∪ e e l∈U l∈U ∪ F1 e 0F1 wlΓ0,p Γ00,p Fe1 Fe1 wAne lΓ00,p Γ00,p G ∪ Fe1 Fe1 . wAn we lΓ00,p Γ00,p n>0 l∈U n>0 l∈U     rn 0 0 1  and w =  . where as before An =  0 r−n −1 0 Now, in the proof of Proposition 3.4.1 we saw that the elements ι(1, An w), ι(1, wAn ) of e p ) do not belong to P e (Qp )QU He Q−1 . In particular therefore, the elements ι(1, An we H(Q l), p H e p ) cannot belong to P e (Qp )QI 0He Q−1 . ι(1, wAne l) of H(Q p H So (3.3.9) gives us Zp (gp , 1, s) = X ] e Λ−2 p (l)Υp (ι(1, l), s) l∈U + XX Z e 0F e lΓ0,p1 WΨ,p (gp hp )dhp ] e Λ−2 p (l)Υp (ι(1, An l), s) Z n>0 l∈U + XX e e 0F 0F Γ0,p1 Ane lΓ0,p1 ] e Λ−2 p (l)Υp (ι(1, wAn w l), s) WΨ,p (gp hp )dhp (3.4.12) Z e 0F e 0F Γ0,p1 wAn we lΓ0,p1 n>0 l∈U WΨ,p (gp hp )dhp . If we choose ai , bi as in the proof of Proposition 3.4.1 then we have Fe1 Fe1 Γ00,p An e lΓ00,p = G F1 ai Γ00,p , e i Fe1 Fe1 Γ00,p wAn we lΓ00,p = G F1 bi Γ00,p . e i Hence, by the same argument as in the proof of that proposition, we have Z Z WΨ,p (gp hp )dhp = e 0F e 0F Γ0,p1 Ane lΓ0,p1 F1 −1 WΨ,p (gp hp )dhp = [KpF : Γ00,p ] . e e e 0F 0F Γ0,p1 wAn we lΓ0,p1 1 It is easy to check that the last quantity is equal to (p+1) 2. e 106 So we have WΨ,p (gp ) Zp (gp , 1, s) = (p + 1)2 + XX X ] e Λ−2 p (l)Υp (ι(1, l), s) l∈U ] e Λ−2 p (l)Υp (ι(1, An l), s) + n>0 l∈U XX  (3.4.13) . ] e Λ−2 p (l)Υp (ι(1, wAn w l), s) n>0 l∈U e We can check that for n > 0, ι(1, Ane l) does not belong to PHe (Qp )QIp0H Q−1 , hence Υ]p (ι(1, Ane l), s) = 0. We can also check that for l 6= 1, l ∈ U, (1, e l) does not belong to e PHe (Qp )QIp0H Q−1 , hence Υ]p (ι(1, e l), s) = 0. Also, putting  0   0   1 P =  0   0  0 pn l −1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0    0    p−n l  ∈ P e (Qp )  H p−n l   0   0 we can check that e Q−1 P −1 ι(w, An we l)Q ∈ Ip0H , hence Υ]p (ι(1, wAn we l), s) = Λp (l)p−6n(s+1/2) (3.4.14) ] −1 −6n(s+1/2) and hence for all n > 0 we e Thus we have Λ−2 p (l)Υp (ι(1, wAn w l), s) = Λp (l)p have X ] e Λ−2 p (l)Υp (ι(1, wAn w l), s) = 0. l∈U So we conclude that Zp (gp , 1, s) = WΨ,p (gp ) . (p + 1)2 Next, we deal with the case kp = s1 . F1 0 If k ∈ Γ00,p then s−1 1 m2 (k)s1 ∈ Ip . So, by the same argument as before, we know that e F1 F1 Υ]p (ι(s1 , g), s) depends only on the double coset Γ00,p gΓ00,p . e e Also, by explicit computation, we check that ι(s1 , An we l), ι(s1 , wAne l) do not belong to e e PHe (Qp )QIp0H Q−1 for any n ≥ 0. Moreover, the quantity ι(s1 , Ane l) belongs to PHe (Qp )QIp0H Q−1 107 if and only if n = 0, l = 1. On the other hand, for n > 0, the quantity ι(s1 , wAn we l) does belong to PHe (Qp )QIp0H Q−1 . These last two facts are reflected in the following equations. e ι(s1 , 1) = P QIQ−1 , (3.4.15) where  0 1 0 0 0 0   1   0 P =  0   0  0    0 0 0 0 0   0 1 0 0 0  ∈ P e (Qp ),  H 0 0 0 1 0   0 0 1 0 0  0 0 0 0 1 and  1 0 0 0   0 1 0 0   1 −1 1 0 I=  0 0 0 1   0 0 0 0  0 0 0 0 0 0 1 0 0 0 0    0   0  ∈ Ip0He .  −1   1  1 ι(s1 , w−1 An we l) = P QIQ−1 where  0   1   0 P =  0   0  1 0 0 0 0 0 0 0 0 −pn l −1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0    0    0   ∈ P e (Qp ),  H 0    0   n −p l (3.4.16) 108 and  1 0 0 0 0  0     0 1 0 0 0 0      −n −1  −p l −1 −1 0 0 0    ∈ Ip0He . I=  −1  0 0 0 1 0 −pn l       0 0 0 0 1 −1    0 0 0 0 0 −1 So, we have   XX WΨ,p (gp ) −2 ] e 1+ Λp (l)Υp (ι(s1 , wAn wl), s) . Zp (gp , s1 , s) = (p + 1)2 (3.4.17) n>0 l∈U But from (3.4.16) we see that Υ]p (ι(s1 , wAn we l), s) = Λp (l)p−6n(s+1/2) and hence X ] e Λ−2 p (l)Υp (ι(s1 , wAn w l), s) = 0. l∈U This completes the proof that Zp (gp , s1 , s) = WΨ,p (gp ) . (p + 1)2   0 1 Fe1 Fe1 . Next, we consider kp = s2 . Let Γ00, = J1 Γ00,p J1 where J1 =  p −1 0 F1 0 If k ∈ Γ00, then s−1 p 2 m2 (k)s2 ∈ Ip . So, by the same argument as before, we know that e F1 F1 Υ]p (ι(s2 , g), s) depends only on the double coset Γ00, gΓ00,p . p e e Now, the Bruhat-Cartan decomposition (3.4.11) continues to hold when we replace the F1 F1 left Γ00,p in each term by Γ00, . So, to prove that Zp (gp , s2 , s) = 0 it is enough to prove p e e that each of the elements ι(s2 , Ane l), ι(s2 , An we l), ι(s2 , wAne l), ι(s2 , wAn we l) cannot belong to PHe (Qp )QIp0H Q−1 for any n ≥ 0. This we do by an explicit computation. The details are e omitted. F1 0 Next, take kp = s3 . Once again, we check that if k ∈ Γ00, then s−1 p 3 m2 (k)s3 ∈ Ip . On the e other hand, an explicit computation again shows that the elements ι(s3 , Ane l), ι(s3 , An we l), e ι(s3 , wAne l), ι(s3 , wAn we l) cannot belong to PHe (Qp )QIp0H Q−1 . So by exactly the same argu- ment as the previous case, Zp (gp , s3 , s) = 0. 109 Next consider the case kp = Θ. Define  1 ∗ Fe1 Fe1  |A≡ = {A ∈ Γ00,p Γ01,p 0 1  (mod p)}. F1 1 , there exthen Θ−1 m2 (k)Θ ∈ Ip0 . We know that given g ∈ ΓF0,p We can check that if k ∈ Γ01,p e e 0F1 e ists l ∈ Z× L,p such that g l ∈ Γ1,p . Thus, the Bruhat-Cartan decomposition (3.4.11) continues e F1 F1 . An explicit computation again in each term by Γ01,p to hold when we replace the left Γ00,p e e e shows that the elements ι(Θ, Ane l), ι(Θ, An we l), ι(Θ, wAne l) never belong to PHe (Qp )QIp0H Q−1 . e On the other hand, if n > 0, then ι(Θ, wAn we l) does belong to PHe (Qp )QIp0H Q−1 . Indeed, ι(Θ, w−1 An we l) = P QIQ−1 (3.4.18) where  1   0   0 P =  0   0  α 0 0 0 0 1 0 0 0 0 0 0 0 0 −pn l −1 0 0 1 0 0 0 0 −α 1 0 0 0 0 0 0 −p−n l         ∈ P e (Qp ),  H      and  1 0 0 0 0 0      0 1 0 0 0 0        −αpn l−1 −1 − pn l−1 −1 0 0 0  ∈ Ip0He .  I=  −1 n  0 0 0 1 0 −p l α     −1  n  0 0 0 0 1 −1 + p l    0 0 0 0 0 −1 So, we have WΨ,p (gp ) Zp (gp , Θ, s) = (p + 1)2 XX n>0 l∈U ] e Λ−2 p (l)Υp (ι(Θ, wAn w l), s)  . (3.4.19) 110 But from (3.4.18) we see that Υ]p (ι(Θ, wAn we l), s) = Λp (l)p−6n(s+1/2) and hence X ] e Λ−2 p (l)Υp (ι(Θ, wAn w l), s) = 0. l∈U This completes the proof that Zp (gp , Θ, s) = 0. Next consider the case kp = Θs2 . Define the subgroup  1 e Fe1 |A≡ Γ0pF1 = {A ∈ Γ00,p 0 0 1   (mod p)}. F1 We can check that if k ∈ Γ0pF1 then (Θs2 )−1 m2 (k)Θs2 ∈ Ip0 . We know that given g ∈ Γ01,p ,   1 x e . So, to there exists x ∈ Z/pZ such that gu(x) ∈ Γ0pF1 , where u(x) is the matrix  0 1 prove that Zp (gp , Θs2 , s) = 0, it is enough to check that the elements ι(Θs2 , u(x)Ane l), ι(Θs2 , u(x)An we l), ι(Θs e e do not belong to PHe (Qp )QIp0H Q−1 . This can be done by an explicit computation (omitted e for brevity). Next, consider the case kp = Θs4 . As before, we can check that if k ∈ Γ0pF1 then e (Θs4 )−1 m2 (k)Θs4 ∈ Ip0 . To prove that Zp (gp , Θs4 , s) = 0, it is enough to check that the elements ι(Θs4 , u(x)Ane l), ι(Θs4 , u(x)An we l), ι(Θs4 , u(x)wAne l), ι(Θs4 , u(x)wAn we l) do not belong to PHe (Qp )QIp0H Q−1 . This is done by an explicit computation, which we omit. e F1 Finally, we consider the case kp = Θs5 . We can check that if k ∈ Γ01,p then (Θs5 )−1 m2 (k)Θs5 ∈ e Ip0 . To prove that Zp (gp , Θs5 , s) = 0, it is enough to check that the elements ι(Θs5 , Ane l), e ι(Θs5 , An we l), ι(Θs5 , wAne l), ι(Θs5 , wAn we l) do not belong to PHe (Qp )QIp0H Q−1 . This is done by an explicit computation, which we omit. This completes the proof of the theorem. The local integral for primes in S1 In this subsection, we prove the following proposition. Proposition 3.4.3. Let p be a prime dividing M but not N and kp ∈ Yp . We have 111 Zp (gp , kp , s) =    WΨ,p (g2p ) if kp = 1 or kp = Θ  0 otherwise . (p+1) Proof. Recall that σ is the irreducible automorphic representation of GL2 (A) generated by Ψ. Let σp be the local component of σ at the place p. We also let α, β be the unramified characters of Q× p from which σp is induced. F1 F1 be as defined in the previous subsection. , Γ01,p Let Γ00,p e e We first consider the case kp = 1. As in the previous case, Υ]p (ι(1, g), s) only depends F1 F1 . gΓ00,p on the double coset Γ00,p e e By explicit computation we check that, ι(1, Ane l), ι(1, An we l), ι(1, wAne l), ι(1, wAn we l) do not belong to PHe (Qp )ΩIp0H Q−1 . So, by the results of the previous subsection, and e by (3.4.11), we have Zp (gp , 1, s) = 0. Next, consider the case kp = s1 . Again, by explicit computation, we check that for n > 0, ι(1, Ane l), ι(1, An we l), ι(1, wAne l), ι(1, wAn we l) do not belong to PHe (Qp )ΩIp0H Q−1 . e Furthermore ι(1, we l) does not belong to PHe (Qp )ΩIp0H Q−1 and ι(1, e l) belongs only when e l 6= 1. This last fact is reflected by the following equation: Let l satisfy the equation l2 + b1 l + c1 = 0 and let α = (l − r)/2 for some r ∈ Z. Then we have ι(1, e l) = P ΩIQ−1 (3.4.20) where  0 − 2c   1 2lc   0 0 P =  0 0   0 0  0 0 0 0 0 0 0 0 l c 0 0 0 − cb − 1l − 2c 0 1 0 0 0 0 0    0   0  ∈ P e (Qp ),  H 0   0  l 112 and  − 2c   b+r  c    0 I=   0    0  0 0 0 0 0 1 0 0 0 −1 1 0 0 0 0 − 2c 0 0 0 1 0 0 0 0 − b+r 2 0    0    0   ∈ Ip0He . b+r   2   1   1 Hence we have  Zp (gp , s1 , s) =  X  WΨ,p (gp )  −2 ] e  . Λ (l)Υ (ι(s , l), s) + 1 1 p p   (p + 1)2 (3.4.21) l∈U l6=1 where the 1 comes from the results of the previous subsection. P Noting that Υ]p (ι(s1 , e l), s) = Λp (l) and that l∈U Λ−1 p (l) = −1, l6=1 we get Zp (gp , s1 , s) = − WΨ,p (gp ) WΨ,p (gp ) + = 0. (p + 1)2 (p + 1)2 F1 Next, we consider kp = s2 . Let Γ00, be as in the previous subsection. p e By the argument there, we know that Υ]p (ι(s2 , g), s) depends only on the double coset F1 F1 Γ00, gΓ00,p . p e e To prove that Zp (gp , s2 , s) = 0 it is enough to prove that each of the elements ι(s2 , Ane l), e ι(s2 , An we l), ι(s2 , wAne l), ι(s2 , wAn we l) cannot belong to PHe (Qp )ΩIp0H Q−1 for any n ≥ 0. This we do by an explicit computation. The details are omitted. Next, take kp = s3 . Once again, an explicit computation shows that the elements e ι(s3 , Ane l), ι(s3 , An we l), ι(s3 , wAne l), ι(s3 , wAn we l) cannot belong to PHe (Qp )ΩIp0H Q−1 . So by exactly the same argument as the previous case, Zp (gp , s3 , s) = 0. Next, consider the case kp = Θ. By explicit calculation, we check that for n > 0 the e elements ι(Θ, An we l), ι(Θ, wAne l), ι(Θ, wAn we l) do not belong to PHe (Qp )ΩIp0H Q−1 .Also check e e that ι(Θ, we l) ∈ / PHe (Qp )ΩIp0H Q−1 . Also, provided l 6= 1, we have ι(Θ, we l) ∈ / PHe (Qp )ΩIp0H Q−1 . Thus, the only term that contributes is ι(Θ, 1). 113 So by the same argument as before, we have Zp (gp , Θ, s) = Υ]p (ι(Θ, 1), s) Z e 0F Γ0,p1 WΨ,p (gp hp )dhp (3.4.22) WΨ,p (gp ) = (p + 1)2 Next consider the case kp = Θs2 . For x ∈ Z, let u(x) be as in the previous subsection. As before, to prove that Zp (gp , Θs2 , s) = 0, it is enough to check that the elements ι(Θs2 , u(x)Ane l), ι(Θs2 , u(x)An we l), ι(Θs2 , u(x)wAne l), ι(Θ, u(x)wAn we l) do not belong to PHe (Qp )ΩIp0H Q−1 . This can be done by an explicit computation (omitted for brevity). e Next, consider the case kp = Θs4 . To prove that Zp (gp , Θs4 , s) = 0, it is enough to check that the elements ι(Θs4 , u(x)Ane l), ι(Θs4 , u(x)An we l), ι(Θs4 , u(x)wAne l), ι(Θs4 , u(x)wAn we l) do not belong to PHe (Qp )ΩIp0H Q−1 . This is done by an explicit computation, which we omit. e Finally, we consider the case kp = Θs5 . To prove that Zp (gp , Θs5 , s) = 0, it is enough to check that the elements ι(Θs5 , Ane l),ι(Θs5 , An we l), ι(Θs5 , wAne l), ι(Θs5 , wAn we l) do not belong to PHe (Qp )ΩIp0H Q−1 . This is done by an explicit computation, which we omit. e The local integral at infinity Proposition 3.4.4. We have Z∞ (g∞ , 1, s) = B∞ (s)WΨ,∞ (g∞ ), `/2 −6s−1 π 2 where B∞ (s) = (−1)6s+`−1 . Fe is the maximal compact subgroup of F e1 (R). Furthermore, note that Proof. Note that K∞ any element h of Fe1 (R) can be written in the form    1 x b 0  k h= −1 0 1 0 b  F . Let us henceforth denote u(x) =  where x ∈ R, b ∈ R+ , k ∈ K∞ 1 x e 0 1   b 0  , t(b) =  . 0 b−1 F has volume 1. Also, note that Λ We normalize our Haar measures such that K∞ ∞ is trivial e 114 Fe , g, h ∈ F e1 (R) we have and for k ∈ K∞ Υ]∞ (ι(1, hk), s)WΨ,∞ (ghk) = Υ]∞ (ι(1, h), s)WΨ,∞ (gh). Hence we have Z ∞Z ∞ Z∞ (g∞ , 1, s) = 0 −∞ Υ]∞ (ι(1, u(x)t(b)), s)WΨ,∞ (g∞ u(x)t(b))b−3 dxdb (3.4.23) where dx, db are the usual Lebesgue measures. H = K H ∩ H(R). To calculate Υ] (ι(1, u(x)t(b)), s) we need to write the Iwasawa Let K∞ ∞ ∞ e decomposition of ι(1, u(x)t(b)). However, finding an explicit decomposition is not really necessary. Indeed, we know that there exists some decomposition  ι(1, u(x)t(b)).Q =   A X 1 (At )−1 K H , A ∈ GL (R) and that with K ∈ K∞ 3 Υ]∞ (ι(1, u(x)t(b)), s) = | det(A)|6(s+1/2) det(J(K, i))−` . Now, let Abx = ι(1, u(x)t(b)).Q. By explicit computation, we see that  0   1   0 b Ax =   0   0  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 b 0 0 0    0   1 −b  1 −b   0   x −b By (3.4.24) we have det(J(Abx , i)) = det(A)−1 det(J(K, i)). (3.4.24) 115 2 Since det(J(Abx , i)) = x−i(bb +1) we have Υ]∞ (ι(1, u(x)t(b)), s) = | det(A)|6(s+1/2) det(A)−` b` (x − i(b2 + 1))−` . (3.4.25) On the other hand, we have (Abx )(i) = (AAt i + XAt ). By explicit computation, we see that  (Abx )(i) = 1 4 2 b + 2b + x2 + 1  −x b4 + b2 + x2 0 −x 0 b4 + 2b2 + x2 + 1 0 −x 0 b2 + 1     0 b2 + 1   + 0 0  b2 + 1 0 0    i       x b . From this we get det(A) = √b4 +1+2b 2 +x2 Therefore, we have Υ]∞ (ι(1, u(x)t(b)), s) = b6s+3 (b4 + 1 + 2b2 + x2 )−3(s+1/2)+`/2 (x − i(b2 + 1))−` . (3.4.26) On the other hand, we know that 2 WΨ,∞ (u(x)t(b)) = e2πix e−2πb b` We will prove the proposition only for g∞ = 1, the calculations in the general case are similar. We need to evaluate the integral Z ∞Z ∞ 0 2 b6s+` (x − i(b2 + 1))−3(s+1/2)−`/2 (x + i(b2 + 1))−3(s+1/2)+`/2 e−2πix e−2πb dxdb −∞ (3.4.27) 116 Putting b2 = y , the above integral becomes 1 2 Z ∞Z ∞ `−1 1 ` 1 ` y 3s+ 2 (x − i(y + 1))−3(s+ 2 )− 2 (x + i(y + 1))−3(s+ 2 )+ 2 e2πix e−2πy dxdy (3.4.28) −∞ 0 Applying [GK92, (6.11)] to the inner integral, (3.4.28) becomes (−1)`/2 (2π)6s+3 2Γ(3s + 23 + 2` )Γ(3s + 23 − 2` ) times Z ∞ 1 ` 1 ` e−2π(1+2t) (t + 1)3s+ 2 + 2 t3s+ 2 − 2 Z ∞ 0 Now,  `−1 y 3s+ 2 e−4πy(1+t) dy dt. (3.4.29) 0 R∞ 0 `−1 y 3s+ 2 e−4πy(1+t) dy evaluates to ` 1 1 2−6s−`−1 π −3s− 2 − 2 Γ(3s + `/2 + ). 2 Using this, and the formula Z ∞ ` 1 3 e−2π(1+2t) t3s+ 2 − 2 = 2−6s+`−3 e−2π π −3s+`/2− 2 Γ(3s + 0 3 ` − ) 2 2 we see that (3.4.28) simplifies to (−1)l/2 2−6s−1 π WΨ,∞ (1). 6s + ` − 1 3.5 Proof of the Pullback formula In this section, we will prove Theorem 3.2.1. Recall the definition of E(g, s) from §3.2. Our main step in computing E(g, s) will be the evaluation of the following integral: Z ΥΨ (g, s) = Υ] (ι(g, h), s)Ψ(h)Λ−1 (det h)dh (3.5.1) Fe1 [g](A) By [Shi97], we know that the integral above converges absolutely and uniformly on 117 compact sets for Re(s) large. We are going to evaluate the above integral for such s. Q f e f Note that G(A) = P (A) v K v . Moreover if k ∈ Kv , we may write   λ 0  k 0 k = m 2  0 1 where λ = µ2 (k), so that µ2 (k 0 ) = 1. For any p ∈ S3 we have, by the Bruhat decomposition, fp t (P (Qp ) ∩ K fp )s1 U fp t (P (Qp ) ∩ K fp )s2 U fp . fp = (P (Qp ) ∩ K fp )U K Also, for p|M , we have, by Lemma 3.3.1, fp = K a fp )sIp0 . (P (Qp ) ∩ K s∈Yp e f ) in (2.8.16). Recall that we defined the compact subgroup U G of G(A Q f So write g = m1 (a)m2 (b)nk where k ∈ v Kv , µ2 (k) = 1 and further write k = e k∞ kram kur where e G g k∞ ∈ K ∞ , kur ∈ U and kram = Q v (kram )v , with     {1}    (kram )v ∈ {1, s1 , s2 }      {1, s1 , s2 , s3 , Θ, Θs2 , Θs4 , Θs5 } if v ∈ /S if v ∈ S3 if v ∈ S1 t S2 . 118 Therefore we have Z Υ] (ι(m1 (a)m2 (b)nk, b(b−1 h)), s)Ψ(h)Λ−1 (det h)dh ΥΨ (g, s) = Fe1 [m2 (b)](A) = ρ` (k∞ ) Z × Υ] (ι(m1 (a)m2 (b)nkram , b(b−1 h)), s)Ψ(h)Λ−1 (det h)dh Fe1 [m2 (b)](A) (using properties from §3.3) = Λ(a)kNL/Q (a) · µ2 (b)−1 |3(s+1/2) ρ` (k∞ ) Z Υ] (ι(kram , h), s)Ψ(bh)Λ−1 (det h)dh × Fe1 (A) (using (3.1.1)). We write Z Ub (kram , s) = Υ] (ι(kram , h), s)Ψ(bh)Λ−1 (det h)dh. Fe1 (A) Thus we have ΥΨ (g, s) = Λ(a)|NL/Q (a) · µ2 (b)−1 |3(s+1/2) ρl (k∞ ) × Ub (kram , s) (3.5.2) Recall the Whittaker expansion    λ 0  g Ψ(g) = WΨ  0 1 × λ∈Q (3.5.3)    λ 0  b, kram , s Ub (kram , s) = Z  0 1 λ∈Q× (3.5.4) X Therefore X Q f where for g ∈ Fe(A), k ∈ v K v , µ2 (k) = 1, we define Z Z(g, k, s) = Υ] (ι(k, h), s)WΨ (gh)Λ−1 (det h)dh. Fe1 (A) Note that the uniqueness of the Whittaker function implies Z(g, k, s) = Y v Zv (gv , kv , s), 119 where the local zeta integral Zv (gv , kv , s) is defined as in (3.3.9). So, by the results of the previous two sections, we have Z(g, kram , s) = where we define   B(s)WΨ (g) if (kram )v ∈ Yv0 for all places v  0 otherwise     {1}    Yv0 = {1, s1 }      {1, Θ} (3.5.5) if v ∈ / S1 t S2 if v ∈ S2 if v ∈ S1 t S2 . From (3.5.2),(3.5.3),(3.5.4),(3.5.5) we conclude that ΥΨ (g, s) = B(s)fΛ (g, s) (3.5.6) where fΛ (g, s) is defined as in §2.8. We are now in a position to prove the Pullback formula. Proof of Theorem 3.2.1. Recall the definition of B(s) from (3.2.4). Also recall that we defined Z EΥ] (ι(g, h), s)Ψ(h)Λ−1 (det h)dh. E(g, s) = (3.5.7) f1 (Q)\Fe1 [g](A) F The pullback formula states that E(g, s) = B(s)EΨ,Λ (g, s). e e By abuse of notation, we use R(Q) to denote its image in H(Q). First, we recall e e R(Q)|=2. We take the identity element as one of the from [Shi97] that |PHe (Q)\H(Q)/ double coset representatives, and denote the other one by τ . Thus e e e H(Q) = PHe (Q)R(Q) t PHe (Q)τ R(Q). Let us denote by R1 , R2 the corresponding sets of coset representatives, i.e. R1 ⊂ e e R(Q), R2 ⊂ R(Q) and 120 e PHe (Q)R(Q) = G PHe (Q)s s∈R1 and e PHe (Q)τ R(Q) = G PHe (Q)s. s∈R2 Recall that we defined X EΥ] (h, s) = Υ] (γh, s) e γ∈PH e (Q)\H(Q) 1 (h, s) + E 2 (h, s) where for Re(s) large. We can write EΥ] (h, s) = EΥ ] Υ] X 1 EΥ ] (h, s) = Υ] (γh, s) γ∈R1 and X 2 EΥ ] (h, s) = Υ] (γh, s). γ∈R2 Now, by [Shi97, 22.9] the orbit of τ is ’negligible’ for our integral, that is for all g, Z f1 (Q)\Fe1 [g](A) F 2 −1 EΥ (det h)dh = 0. ] (ι(g, h), s)Ψ(h)Λ It follows that Z E(g, s) = f1 (Q)\Fe1 [g](A) F 1 −1 EΥ (det h)dh. ] (g, h, s)Ψ(h)Λ (3.5.8) On the other hand, by [Shi97, 2.7] we can take R1 to be the following set: e R1 = {ι(m2 (ξ)β, 1) : ξ ∈ Fe1 (Q), β ∈ P (Q)\G(Q)} For Re(s) large, we therefore have 1 EΥ ] (ι(g, h), s) = X ξ∈Fe1 (Q) e β∈P (Q)\G(Q) Υ] (ι((m2 (ξ)βg, h), s). (3.5.9) 121 Substituting in (3.5.8) we have Z X E(g, s) = f1 (Q)\Fe1 [g](A) F ξ∈Fe1 (Q) e β∈P (Q)\G(Q) Z X = f1 (Q)\Fe1 [g](A) F = X Υ] (ι(βg, h), s)Ψ(h)Λ−1 (det h)dh Fe1 [g](A) e β∈P (Q)\G(Q) = Υ] (ι(βg, ξ −1 h), s)Ψ(ξ −1 h)Λ−1 (det ξ −1 h)dh ξ∈Fe1 (Q) e β∈P (Q)\G(Q) Z X Υ] (ι(m2 (ξ)βg, h), s)Ψ(h)Λ−1 (det h)dh ΥΨ (βg, s) e β∈P (Q)\G(Q) X = B(s) fΛ (βg, s) e β∈P (Q)\G(Q) = B(s)EΨ,Λ (g, s) Thus Z −1 e EΥ] (g, h, s)Ψ(h)Λ (det h)dh = B(s)EΨ,Λ (g, s) (3.5.10) f1 (Q)\Fe1 [g](A) F for Re(s) large (so that all sums and integrals converge nicely and our manipulations are valid). However, EΥ] (g, h, s) is slowly increasing away from its poles, while Ψ(h) is rapidly decreasing. Thus the left side above converges absolutely for s ∈ C away from the poles of the Eisenstein series. Hence (3.5.10) holds as an identity of meromorphic functions. 3.6 The integral representation The following result was proved in the previous chapter: Z 1 EΨ,Λ (g, s)Φ(g)dg = C(s) · L(3s + , F × g) 2 ZG (A)G(Q)\G(A) 122 where C(s) = 3 3 ` Y Qf πa(Λ)(4π)−3s− 2 `+ 2 d−3s− 2 Γ(3s + 23 l − 32 ) p−6s−3 g(M/f )PM N (6s + ` − 1)ζ M N (6s + 1)L(3s + 1, σ × ρ(Λ)) 1 − ap wp p−3s−3/2 p|f where f = gcd(M, N ), Y QA = (1 − r), r|A r prime and g(A), PA , ζ A are as defined earlier. Recall the definition of B(s) from (3.2.4) and let A(s) = B(s)C(s). e consisting of elements h = (h1 , h2 ) such that h1 ∈ Let R denote the subgroup of R G, h2 ∈ Fe and µ2 (h1 ) = µ1 (h2 ). The above Theorem, along with our pullback formula, implies the following result. Theorem 3.6.1. We have Z 1 EΥ] (ι(g1 , g2 ), s)Φ(g1 )Ψ(g2 )Λ−1 (det g2 )dg = A(s)L(3s + , F × g) 2 g∈Z(A)R(Q)\R(A) where g = (g1 , g2 ). This new integral representation has a great advantage over the previous one: the Eisenstein series EΥ] (g, s) is much simpler than EΨ,Λ (g, s) (even though it lives on a higher rank group). This is because it is because it is induced from a character of the Siegel parabolic. Thus, it is more suitable for applications, especially with regard to special value results. Corollary 3.6.2. L(s, F × g) can be continued to a meromorphic function on the entire complex plane. It’s only possible pole to the right of the critical line Re(s)= 12 is at s = 1. Proof. The integral representation of Theorem 3.6.1 immediately proves the meromorphic continuation. Furthermore by [Ich04], we know that the only possible poles of the Eisenstein 123 series EΥ] (g, s) to the right of s = 0 are at s = 16 and s = 21 . However, as we remark in the proof of Proposition 4.1.3, the pole at s = 12 is impossible. So the only possible pole of the Eisenstein series in that half plane is at s = 16 which corresponds to a pole of the L-functions at s = 1. 124 Chapter 4 Rationality of Eisenstein series and Deligne’s conjecture on critical L-values 4.1 Eisenstein series on Hermitian domains Let e + (R) = {g ∈ G(R) e G : µ2 (g) > 0}. e + (R), Fe+ (R) similarly. Define the groups G+ (R), H e n . We define the ‘standard Also recall the definitions of the symmetric domains Hn , H e2 × H e 1 into H e 3 by embedding’ of H  (Z1 , Z2 ) 7→   Z1 Z2 . e2 × H e 1 and its image in H e3 We use the same notation (Z1 , Z2 ) to denote an element of H under the above embedding. Note that this embedding restricts to an embedding of H2 ×H1 into H3 . e2 × H e 1 into H e 3 by We also define another embedding u of H u(Z1 , Z2 ) = (Z1 , −Z2 ). Clearly this embedding also restricts to an embedding of H2 × H1 into H3 . Furthermore, the following is true, as can be verified by an easy calculation: 125 e 1 (R), g2 ∈ Fe1 (R), such that g1 (i) = Z1 , g2 (i) = Z2 . In the event that (Z1 , Z2 ) ∈ Let g1 ∈ G H2 × H1 we may even take g1 ∈ G1 (R), g2 ∈ SL2 (R). Then u(Z1 , Z2 ) = (Q−1 ι(g1 , g2 )Q)i. e 3 . Recall Now, let us interpret the Eisenstein series on GU (3, 3) as a function on H e H(Q ) e the definitions of the sections Υv (s) ∈ IndP (Qv v ) (Λv k · k3s v ). Also, for Z ∈ Hn , we set e H b = i (Z t − Z). Z 2 e + (R). Then Lemma 4.1.1. Let g∞ ∈ H 3(s+1/2)−`/2 Υ∞ (g∞ , s) = det(g∞ )`/2 det(J(g∞ , i))−` det(g\ ∞ (i)) H. Proof. Let us write g∞ = m(A, v)nk∞ where m(A, v) ∈ M (A), n ∈ N (A) and k ∈ K∞ e Then, (3.1.1) and (3.1.9) tells us that Υ∞ (g∞ , s) = v −9(s+1/2) | det A|6s+3 det(k∞ )`/2 det(J(k∞ , i))−` . On the other hand, we can verify that t −1 g\ ∞ (i) = v AA and therefore −3 2 det(g\ ∞ (i)) = v | det A| . Also we see that t J(g∞ , i) = v(A )−1 J(k∞ , i) which implies det(J(g∞ , i)) = v 3 det(A)−1 det(J(k∞ , i)). Finally det(g∞ ) = v 3 det(k∞ ) det(A) det(A)−1 . Putting the above equations together, we get the statement of the lemma. e f ) be fixed. Then the function Σ on H e + (R) defined Corollary 4.1.2. Let s ∈ C, uf ∈ H(A 126 by Σ(g∞ ) = det(g∞ )−`/2 det(J(g∞ , i))` EΥ (uf g∞ , s ` + − 1/2) 3 6 depends only on g∞ (i). Proof. We have X EΥ (uf g∞ , s) = Υ∞ (γ∞ g∞ , s)Υf (γf uf , s). e γ∈PH(Q) \H(Q) e So, by the above lemma, Σ(g∞ ) = X [ s det(γ)`/2 det(J(γ, Z))−` det(γ(Z)) (4.1.1) e γ∈PH(Q) \H(Q) e where Z = g∞ (i). Now,consider the coset decomposition Fe(A) = h G  Fe(Q)Fe+ (R)   ti i=1 t∗i  U Fe (4.1.2) F Γ00,p . (4.1.3) −1 where ti ∈ Fe(Af ), t∗i = ti , and UF = e Y p∈S / e Y KpF p∈S3 Y ΓF0,p e e p∈S1 tS2 We note here that the constant h comes up because the class number of L may not be 0F to Z× is not surjective. In particular, note that if 1 and because the det map from Γ0,p L,p e M = 1, we have h = h(−d), the class number of L. Also, we note that by the Cebotarev density theorem, we may choose ti such that (NL/Q ti ) = qi−1 where qi corresponds to an ideal of Z that splits in L. In particular gcd(qi , M N ) = 1. 127 Now, let  ti Γi = SL2 (Z) ∩   t∗i   K Fe   t−1 i (t∗i )−1  Fe(R) = Γ0 (M ) ∩ Γ0 (N qi ) . Also, we define the congruence subgroup ΓM,N of Sp4 (Z) by ΓM,N = B(M ) ∩ U2 (N ). Recall the definition of U G from (2.8.16). Let us define the compact open subgroup U G e of G(Af ) by U G = U G ∩ G(A). e (4.1.4) Observe that ΓM,N = U G Sp4 (R) ∩ Sp4 (Q). Next, put  ti si =   t∗i  and e 1 (Af ). ri = ι(1, si ) ∈ H e 3 , define the Eisenstein series E i (Z; s) by For Z ∈ H Υ i EΥ (Z; s) = det(g∞ )−`/2 det(J(g∞ , i))` EΥ (Q−1 ri Qg∞ , s/3 + `/6 − 1/2), (4.1.5) e + (R) is such that g∞ (i) = Z. We note that E i (Z, s) is well defined by where g∞ ∈ H Υ Corollary 4.1.2. i (Z , Z ; 0) for Z ∈ H , Z ∈ H . Now, consider the function EΥ 1 2 1 2 2 1 i (Z , Z ; 0) is a modular form of weight ` for Proposition 4.1.3. Assume ` ≥ 6. Then EΥ 1 2 i (Z , Z ; s ) (which is not holomorphic ΓM,N × Γi . Furthermore, for any s0 , the function EΥ 1 2 0 i (Z , Z ; 0) under the action of Γ in Z1 , Z2 unless s0 = 0) transforms just like EΥ 1 2 M,N × Γi . 128 Proof. We know that EΥ (g, s) converges absolutely and uniformly for s > 21 . So if ` > 6, it i (Z; 0) is holomorphic. Furthermore, the case ` = 6 corresponds to the point follows that EΥ s = 21 of EΥ (g, s). From the general theory of Eisenstein series, we know that the residue of H at s = 1 must be a constant function. However, because E (g, s) EΥ (g, s) restricted to K∞ Υ 2 e H with non-trivial eigencharacter, this residue must be zero. Hence is an eigenfunction of K∞ e i (Z; 0) is a holomorphic function of Z even for ` = 6. EΥ Let A ∈ ΓM,N , B ∈ Γi . It suffices to show that i i EΥ (AZ1 , BZ2 ; s0 ) = det(J(A, Z1 ))` det(J(B, Z2 ))` EΥ (Z1 , Z2 ; s0 ).     −b  denote ge =   . Let g1 ∈ G1 (R), g2 ∈ SL2 (R) such that g1 i = For g =  c d −c d Z1 , g2 i = Z2 . Put s1 = s0 /3 + `/6 − 1/2. We have a b a i i EΥ (AZ1 , BZ2 ; s0 ) = EΥ (u(AZ1 , −BZ2 ; s0 ) i e ge2 )Q)i; s0 ) = EΥ ((Q−1 ι(Ag1 , B e ge2 )Q, i))` EΥ (Q−1 ri ι(Ag1 , B e ge2 )Q, s1 ) = det(J(Q−1 ι(Ag1 , B e ge2 )Q, i))` EΥ] (ι(Ag1 , si B e ge2 ), s1 ) = det(J(Q−1 ι(Ag1 , B F e Now, because s−1 i Bsi ∈ U we have e e ge2 ), s1 ) = EΥ] ((g1 , si ge2 ); s1 ). EΥ] (ι(Ag1 , si B On the other hand, we can check that e ge2 )Q, i))` = det(J(A, Z1 ))` det(J(B, Z2 ))` det(J(g1 , i))` det(J(g2 , i))l . det(J(Q−1 (Ag1 , B Putting everything together, we see that i i EΥ (AZ1 , BZ2 ; s0 ) = det(J(A, Z1 ))` det(J(B, Z2 ))` EΥ (Z1 , Z2 ; s0 ) as required. 129 4.2 The integral representation in classical terms Henceforth, we assume ` ≥ 6. Recall the definitions of the compact open subgroups U G , U F e from (4.1.4), (4.1.3) respectively. Let us define U R ⊂ R(Af ) to be subgroup consisting of R = K × K F . Note elements (g, h) with g ∈ U G , h ∈ U F and µ2 (g) = µ1 (h). Also put K∞ ∞ ∞ e e R is a compact subgroup of R(A). that K R K∞ Also, define VM,N = [Sp4 (Z) : ΓM,N ][K F : U F ], where K F = e e e Q Fe p<∞ Kp . We now rephrase Theorem 3.6.1 in classical terms. Theorem 4.2.1. For any k, we have X Λ−2 (ti ) Z Z Γi \H1 i = VM,N A( ΓM,N \H2 i EΥ (Z1 , −Z2 ; 1 − k)F (Z1 )g(qi Z2 ) det(Y1 )` det(Y2 )` dZ1 dZ2 ` − 1 − 2k ` )L( − k, F × g) 6 2 where for i = 1, 2, we define the invariant measure dZi on H3−i by 1 dZi = (det Yi )i−4 )dXi dYi 2 where Zi = Xi + iYi . Proof. By Theorem 3.6.1, it suffices to prove that for g = (g1 , g2 ), ` − 1 − 2k )Φ(g1 )Ψ(g2 )Λ−1 (det g2 )dg (4.2.1) EΥ] (ι(g1 , g2 ), 6 Z(A)R(Q)\R(A) Z Z X i = Λ−2 (ti ) EΥ (Z1 , −Z2 ; 1 − k)F (Z1 )g(qi Z2 ) det(Y1 )` det(Y2 )` dZ1 dZ2 Z VM,N i Γi \H1 ΓM,N \H2 (4.2.2) R . Also, we Now, the quantity inside the integral in (4.2.1) is right invariant by U R K∞ R is equal to (V −1 (recall that we normalize the volume note that the volume of U R K∞ M,N ) of the maximal compact subgroup to equal 1). Hence we see that (4.2.1) equals 130 Z R Z(A)R(Q)\R(A)/U R K∞ EΥ] (ι(g1 , g2 ), ` − 1 − 2k )Φ(g1 )Ψ(g2 )Λ−1 (det g2 )dg 6 (4.2.3) Now, by strong approximation for Sp4 (A) and (4.1.2) we know that R Z(A)R(Q)\R(A)/U R K∞ = h a  (ΓM,N \Sp4 (R)/K∞ ) ×  i=1 ti 0 0 t∗i   (Γi \SL2 (R)/SO(2)) .  ti Suppose g ∈ Sp4 (R), h ∈ SL2 (R). Also, put si =   t∗i , ri = ι(1, si ), g(i) = Z1 , h(i) = Z2 . We have EΥ] (ι(g, si h), ` − 1 − 2k ` − 1 − 2k ) = EΥ (Q−1 ri QQ−1 ι(g, h)Q, ) 6 6 i = det(J(Q−1 ι(g, h)Q, i))−` EΥ (Z1 , −Z2 ; 1 − k) On the other hand Φ(g) = F (Z1 )det(J(g, i))−` and Ψ(si h) = g(qi Z2 ) det(J(h, i))−` . The result now follows from the observations det(J(Q−1 ι(g, h)Q, i)) = det(J(g, i))det(J(h, i)), | det(J(g, i))|2 = det(Y1 ), | det(J(h, i))|2 = det(Y2 ). and the fact that the Haar measure dg equals dZ1 dZ2 under the above equivalence. Let us take a closer look at the quantity A( `−1−2k ) that appears in the statement of 6 the above theorem in the case when k is an integer, 1 ≤ k ≤ 2` − 2. Write a ∼ b if a/b is rational. From the definition of A(s), it is clear that √ ` − 1 − 2k π 4+k−2` a(Λ) d A( )∼ . 6 L(` + 1 − 2k, χ−d )ζ(` − 2k)ζ(` + 2 − 2k) But it is well known that L(`+1−2k,χ−d ) ζ(`−2k) √ , π`−2k and ζ(`+2−2k) are all rational numbers. π `+2−2k π `+1−2k d It 131 follows that A( ` − 1 − 2k ) ∼ π 7k+1−5` a(Λ). 6 (4.2.4) Rationality of holomorphic Eisenstein series Suppose f1 , f2 are modular forms of weight ` for some congruence subgroup Γ of Sp2n (Z) containing {±1}. We define the Petersson inner product 1 hf1 , f2 i = V (Γ)−1 2 Z f1 (Z)f2 (Z)(det Y )`−n−1 dXdY Γ\Hn where V (Γ) = [Sp2n (Z) : Γ]. Note that these definitions are independent of our choice for Γ. i in order to show the dependence on Λ, ` and a(F, Λ) for i for EΥ We henceforth use EΛ,` a(Λ) to show the dependence on F . By a result of M. Harris [Har97, Lemma 3.3.5.3], we know how Aut(C) acts on the i (Z; 0). In particular he proves the following result. Fourier coefficients of EΛ,` i (Z; 0) lie in Qab . Furthermore, Proposition 4.2.2 (Harris). The Fourier coefficients of EΛ,` if σ ∈ Gal(Qab /Q), then i (Z; 0)σ = EΛi σ ,` (Z; 0) EΛ,` i (Z; 0)σ is obtained by letting σ act on the Fourier coefficients of E i (Z; 0). where EΛ,` Λ,` 4.3 Nearly holomorphic Eisenstein series i (Z; s) from (4.1.5). The section Υ defining Recall the definition of the Eisenstein series EΥ this Eisenstein series depends on the Hecke character Λ as well as on the integer `. Hencei (Z; s) to denote E i (Z; s). Moreover, for forth, to make this dependence explicit, we use EΛ,` Υ i (Z; s) to denote the Eisenstein series that is any other positive even integer k, we use EΛ,k defined similarly except that the integer ` has been replaced by k everywhere. In particular, i (Z; 0) is a holomorphic Eisenstein series (of weight k), whenever k ≥ 6. we know that EΛ,k e n uniquely as Z = X + iY where X, Y are Hermitian and Y is We can write any Z ∈ H positive definite. We can also write any Z ∈ Hn uniquely as Z = X + iY where X, Y are symmetric and Y is positive definite. These decompositions are compatible with each other e n. in the obvious sense under the inclusion Hn ⊂ H 132 We briefly recall Shimura’s theory of differential operators and nearly holomorphic functions. A thorough exposition of this material can be found in his book [Shi00]. e n . For a non negative integer q, we let N q (H) Let H temporarily stand for Hn or H denote the space of all polynomials of degree ≤ q in the entries of Y −1 with holomorphic functions on H as coefficients. e n ). For a Suppose Γ is a congruence subgroup of Sp2n (if H = Hn ) or U (n, n) (if H = H positive integer k, we let Nkq (H, Γ) stand for the space of functions f ∈ N q (H) satisfying f (γZ) = det(J(γ, Z))k f (Z) for all γ ∈ Γ, Z ∈ H, with the standard additional (holomorphy at cusps) condition on the e 1 . It is well-known that N q (H, Γ) is finite dimensional. In Fourier expansion if H = H1 = H k particular, if q = 0, then Nkq (H, Γ) is simply the corresponding space of weight k modular forms. e n and N = (n2 + n)/2 if H = Hn . We let N = n2 if H = H Whenever convergent, the Petersson inner product for nearly holomorphic forms is defined exactly as before. Any f ∈ Nqt (H, Γ) has a Fourier expansion [Shi00, p. 117] as follows: f (Z) = X QT ((2πY )−1 )e2πiT rT Z T ∈L where L is a suitable lattice and for each T , QT is a polynomial in N variables and of degree ≤ t. For an automorphism σ of C we define f σ (Z) = X QσT ((2πY [σ] )−1 )e2πiT rT Z T ∈L where QσT is obtained by letting σ act on the coefficients of QT and Y [σ] =   Y t e n and if H = H  Y otherwise √ √ σ −d = − −d We say that f ∈ Nqt (H, Γ; Q) if f ∈ Nqt (H, Γ) and f σ = f for all σ ∈ Aut(C/Q). We will occasionally omit the weight q and the congruence subgroup Γ when we do not wish to 133 specify those. In particular, we write Nqt (H; Q) to denote S t Γ Nq (H, Γ; Q) where the union is taken over all congruence subgroups Γ. Now, from (4.1.1), it is easy to see that for a positive integer k (assume k ≤ 2` − 2 to i (Z; 1 − k) ∈ N 3(k−1) (H e 3 ). Then, exactly the same proof ensure convergence) we have EΛ,` as Proposition 4.1.3 tells us that the restriction of this function to H2 × H1 is a nearly holomorphic modular form with respect to the appropriate subgroups. More precisely, we have 2(k−1) i EΛ,` (Z1 , Z2 ; 1 − k) ∈ N` (k−1) (H2 , ΓM,N ) ⊗ N` (H1 , Γi ). (4.3.1) e 3 ) we can only say that f (Z1 , Z2 ) ∈ We remark here that for a general f ∈ N 3(k−1) (H P N λ1 (H2 ) ⊗ N λ2 (H1 ) where the sum should be extended over all (λ1 , λ2 ) with λ1 + λ2 = 3(k − 1). However, in this case, we know by (4.1.1) the exact nature of the polynomial of degree 3(k − 1); thus we can conclude that λ1 = 2(k − 1), λ2 = k − 1. To prove the desired algebraicity result for critical L-values, we will need to know arithmeticity properties for the nearly holomorphic modular forms in (4.3.1). That is the substance of the next proposition. Proposition 4.3.1. Let ` ≥ 6 and let k be an integer satisfying 1 ≤ k ≤ 2` − 2. Then the i (Z , Z ; 1 − k) on H × H belongs to function EΛ,` 1 2 2 1   2(k−1) (k−1) (H2 , ΓM,N ; Q) ⊗ N` π 3(k−1) N` (H1 , Γi ; Q) . Furthermore, for an automorphism σ of C, we have i (Z1 , Z2 ; 1 − k))σ = π −3(k−1) EΛi σ ,` (Z1 , Z2 ; 1 − k). (π −3(k−1) EΛ,` i (Z , Z ; 1 − k) Proof. Since we already know (4.3.1) and since the Fourier coefficients of EΛ,` 1 2 i (Z; 1 − k), it is enough to prove that are just sums of those of EΛ,` i (π −3(k−1) EΛ,` (Z; 1 − k))σ = π −3(k−1) EΛi σ ,` (Z; 1 − k). (4.3.2) For positive integers p, q, we have the (modified) Maass-Shimura differential operator ∆pq e 3 . This operator is that acts on the space of nearly holomorphic forms of weight q on H 134 defined in [Shi00, p.146]. By [Shi00, Theorem 14.12], we know that e 3 ; Q). e 3 ; Q) ⊂ π 3p N t+3p (H ∆pq Nqt (H q+2p However, more is true; in fact ((πi)−3p ∆pq f )σ = (πi)−3p ∆pq (f σ ) (4.3.3) e 3 ). This easily follows from [Shi00, p. 118] since the Maass-Shimura whenever f ∈ Nqt (H operators are special cases of the operators considered there and the projection map is Aut(C)-equivariant. An alternative way to directly see (4.3.3) is to observe that the action of the Maass-Shimura operator on the Fourier coefficients of a nearly holomorphic form can be explicitly computed and observed to satisfy the desired property. The details in the symplectic case was worked out by Panchishkin [Pan05, Theorem 3.7]; the calculations in the unitary case are very similar. 0 i e 3 ; Q). So, we can apply (4.3.3) when t = (H (Z; 0) ∈ N`+2−2k We know that EΛ,`+2−2k i (Z; 0). 0, p = k − 1, q = ` + 2 − 2k, f = EΛ,`+2−2k Moreover, by the result of Harris stated above, i (Z; 0)σ = EΛi σ ,`+2−2k (Z; 0). EΛ,`+2−2k So, (4.3.2) will follow if we know that k−1 i i ∆`−2(k−1) EΛ,`+2−2k (Z; 0) = c · i3(k−1) · EΛ,` (Z; 1 − k) (4.3.4) for some rational number c (The superscript i should not be confused with the quantity √ i = −1 that appears above!). But (4.3.4) is precisely the content of Shimura’s calculations in [Shi00, (17.27)]. We remark here that the Eisenstein series Shimura considers has different sections than ours at the finite places dividing M N ; however that does not make a difference because the differential operator only depends on the archimedean section. In particular, we apply [Shi00, Theorem 12.13] to each term of the definition of our Eisenstein series using (4.1.1) and p ` observe that (4.3.4) follows with c = 2−3(k−1) ck−1 `−2(k−1) ( 2 − k + 1) where cq (s) is defined as 135 in [Shi00, (17.20)]. 4.4 Holomorphic projection Shimura observed [Shi00, p. 123] that for q > n + t, there exists a holomorphic projection operator A on Nqt (Hn ). For a nearly holomorphic form f ∈ Nqt (Hn ), Af is a modular form of weight q (i.e. an element of Nq0 (Hn )). For any cusp form g of weight q on Hn , < f, g >=< Af, g > . More precisely, by the proof of [Shi00, Theorem 15.3], we can write f = Af + Lq f 0 where Lq is a rational polynomial of certain differential operators and f 0 is a certain nearly holomorphic form. The differential operators which are used to define Lq are Aut(C)equivariant by [Shi00, Theorem 14.12]. Thus, for an automorphism σ of C, we have f σ = (Af )σ + Lq (f 0σ ). So we can conclude that A(f σ ) = (Af )σ . Furthermore because the space of modular forms is a direct sum of the space of Eisenstein series and the space of cusp forms, there exists an orthogonal projection from the space of modular forms on Hn to the space of cusp forms on Hn . Because the space of Eisenstein series is preserved under automorphisms of C, this cuspidal projection is also Aut(C)-equivariant. From the above comments we conclude the existence of a projection map Acusp from Nqt1 (H2 , Γ2 )⊗Nqt2 (H1 , Γ1 ) to Sq (H2 , Γ2 )⊗Sq (H1 , Γ1 ) for q > 2+ti and congruence subgroups Γ2 ⊂ Sp4 , Γ1 ⊂ SL2 . This projection map satisfies, for any E(Z1 , Z2 ) ∈ Nqt1 (H2 , Γ2 ) ⊗ Nqt2 (H1 , Γ1 ), F (1) ∈ Sq (H2 , Γ2 ), g (1) ∈ Sq (H1 , Γ1 ), the following properties: (a) hhAcusp E(Z1 , Z2 ), F (1) (Z1 )i, g (1) (Z2 )i = hh(E(Z1 , Z2 ), F (1) (Z1 )i, g (1) (Z2 )i, 136 (b) (Acusp E)σ = Acusp (Eσ ). i (Z , Z ; 1− In particular, everything above can be applied to the case when E(Z1 , Z2 ) = π −3(k−1) EΛ,` 1 2 k). We use gi (z) to denote the cusp form g(qi z) on Γ0 (N qi ). We can rewrite Theorem 4.2.1 as follows. X i Λ−2 (ti )hhEΛ,` (Z1 , Z2 ; 1 − k), F (Z1 )i, gi (Z2 )i i = VM,N ` − 1 − 2k ` A( )L( − k, F × g) V (Γi )V (ΓM,N ) 6 2 Note that we have used the fact that gi has real Fourier coefficients. Together with (4.2.4) the above equation implies that X i 4.5 ` i (Z1 , Z2 ; 1 − k), F (Z1 )i, gi (Z2 )i ∼ π 7k+1−5` a(F, Λ)L( − k, F × g). (4.4.1) Λ−2 (ti )hhEΛ,` 2 Deligne’s conjecture Motives and periods Let L(s, M) be the L-function associated to a motive M over Q. Suppose M has coefficients in an algebraic number field E; then L(s, M) takes values in E ⊗Q C. Note that E sits naturally inside E ⊗Q C. Let d be the rank of M and d± the dimensions of the ± eigenspace of the Betti realization of M. Deligne defined the motivic periods c± (M) and conjectured that for all “critical points” m, L(m, M) ∈E (2πi)md c (M) where  = (−1)m . Now, let F , g have algebraic Fourier coefficients. Assuming the existence of motives MF , Mg attached to F, g respectively, Yoshida computed the critical points for MF ⊗ Mg . He also computed the motivic periods c± (MF ⊗ Mg ) under the assumption that Deligne’s conjecture holds for the degree 5 L-function for F . We note here that Yoshida only deals 137 with the full level case; however as the periods remain the same (up to a rational number) for higher level, his results remain applicable to our case. Yoshida’s computations [Yos01, Theorem 13] show that Deligne’s conjecture implies the following reciprocity law:  L(m, F × g) 4m+3`−4 π hF, F ihg, gi α = L(m, F α × g α ) π 4m+3`−4 hF α , F α ihg α , g α i (4.5.1) for all 2 − 2` ≤ m ≤ 2` − 1, α ∈ Aut(C). In the next subsection we prove the above statement for all the critical points m to the right of Re(s) = 12 except for the point 1. The proof for the critical values to the left of Re(s) = 21 would follow from the expected functional equation. The proof that L(1, F × g) behaves nicely under the action of Aut(C) would probably require further work because we do not know that this quantity is even finite (see Corollary 3.6.2). Thus, the problem of extending our result to the remaining critical values is closely related to questions of analyticity and the functional equation for the L-function. These questions are also of interest for other applications, such as transfer to GL(4) and will be considered in a future paper. We also note that the integral representation (Theorem 3.6.1) is of interest for several other applications. Indeed, we hope that this integral representation will pave the way to stability, hybrid subconvexity, non-vanishing, non-negativity and p-adic results for the L-function under consideration. We intend to deal with these questions elsewhere. The main result Theorem 4.5.1. Let ` ≥ 6. Further, assume that F has totally real algebraic Fourier coefficients and define A(F, g; k) = L( 2` − k, F × g) . π 5`−4k−4 hF, F ihg, gi Then, we have: (a) A(F, g; k) ∈ Q, (b) For all α ∈ Aut(C), A(F, g; k)α = A(F α , g α ; k). Proof. Let U be the least common multiple of M, N and all the qi . Let Γ1 be the principal congruence subgroup of Sp4 (Z) of level U and Γ2 the principal congruence subgroup of 138 SL2 (Z) of level U . For each i, we can write i Acusp (π −3(k−1) EΛ,` (Z1 , Z2 ; 1 − k)) = X F1r (Z1 )f1r (Z2 ) (4.5.2) r where F1r (resp. f1r ) is a cusp form for Γ1 (resp. Γ2 ); all of weight `. Then X i hf1r , gi ihF1r , F i = π −3(k−1) hhEΛ,` (Z1 , Z2 ; 1 − k), F (Z1 )i, gi (Z2 )i. (4.5.3) r We also have X h(f1r )α , giα ih(F1r )α , F α i = π −3(k−1) hhEΛi α ,` (Z1 , Z2 ; 1 − k), F α (Z1 ), giα (Z2 )i (4.5.4) r using Proposition 4.3.1 and the properties of holomorphic projection stated above. By (4.4.1) we know that A(F, g; k) = W · (a(F, Λ)) −1 · X Λ −2 r r r hf1 , gi ihF1 , F i P (ti ) (4.5.5) hF, F ihg, gi i for some rational number W . Making α act on both sides of the above equation we get α A(F, g; k) = W · (a(F α , Λα ))−1 · X α −2 (Λ ) α r r r hf1 , gi ihF1 , F i P (ti ) hF, F ihg, gi i . (4.5.6) We also note that hg, gi = hgi , gi i. Now by a result of Garrett [Gar92, p. 460], we know that for each r,  hf1r , gi ihF1r , F i hF, F ihg, gi α  = h(f1r )α , giα ih(F1r )α , F α i hF α , F α ihg α , g α i  . so we have α A(F, g; k) = W · (a(F α , Λα ; k))−1 · X α −2 (Λ ) i r α α r α α  r h(f1 ) , gi ih(F1 ) , F i . hF α , F α ihg α , g α i P (ti ) Using (4.5.5) for F α , g α , Λα , we conclude that A(F, g; k)α = A(F α , g α ; k). (4.5.7) 139 Remark. 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