Luận án tiến sĩ lifting from sl2 to gspin1 4

Lifting from SL(2) to GSpin(1,4) DISSERTATION Presented in Partial Fulfillment of the Requirement for the Degree Doctor of Philosophy in the Graduate School of The Ohio State Univesity By Ameya Pitale ***** The Ohio State Univesity 2006 Dissertation Committee: Approved by Professor Steve Rallis, Adviser Professor James Cogdell Advisor Professor Cary Rader Graduate Program in Mathematics UMI Number: 3226302 UMI Microform 3226302 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.

Box 1346 Ann Arbor, MI 48106-1346 ABSTRACT In this thesis we construct liftings of automorphic forms from the metaplectic group SL f 2 to GSpin(1, 4) using the Maaß Converse Theorem. In order to prove the non-vanishing of the lift we derive Waldspurger’s formula for Fourier coefficients of half integer weight Maaß forms. We analyze the automorphic representation of the adelic spin group obtained from the lift and show that it is CAP to the Saito- Kurokawa lift from SL f 2 to GSp4 (A). ii Dedicated to Swapna iii ACKNOWLEDGMENTS I would like to thank my thesis advisor Professor Steve Rallis for introducing me to this field of Mathematics and this problem in particular.

His guidance, patience and generosity were invaluable to me. I would like to thank James Cogdell and Cary Rader for humoring my endless questions and giving me so much of their time and help. For their comments and suggestions I am grateful to Dinakar Ramakrishnan, Tamotsu Ikeda, Winfred Kohnen, Dihua Jiang, Steve Kudla, William Duke, Peter Sarnak and Eitan Sayag. iv VITA September 16, 1977.

Born - Nagpur, India 2000. MSc Mathematics, Indian Institute of Technology, Kanpur, India 2000 − present. Graduate Teaching Assistant, The Ohio State University PUBLICATIONS Pitale A., “Lifting from SL f 2 to GSpin(1, 4)”, Intern. 63, Pg 3919 - 3966 FIELDS OF STUDY Mathematics, Number Theory v TABLE OF CONTENTS Page Abstract.

Maaß Converse Theorem .1 Clifford algebras and Vahlen matrices. Definition and Automorphy of the Lift .1 Maaß forms on SL2 of half-integral weight .3 Proof of Theorem 3.1 Theta Function and Eisenstein Series .2 Rankin Integral Formula .3 The functional equation. Non-Vanishing of the Lift .1 Criteria for non-vanishing .2 Waldspurger’s formula for Maaß forms .3 Non-vanishing of special values of L−functions .1 Hecke Algebra Hp for p odd .2 Hecke operator Tp .3 Hecke operator Tp2. Automorphic representation corresponding to F.

102 vii CHAPTER 1 INTRODUCTION In [16], Ikeda defines a lifting of automorphic forms from SL f 2 , the metaplectic cover of SL2 , to the symplectic group GSp4n for all n. Ikeda proves his result using the theory of Fourier Jacobi forms. For n = 1 he shows that his definition agrees with the classical Saito-Kurokawa lift. In this thesis we present the lifting of automorphic forms from SL f 2 to the spin group GSpin(1, 4) using the Maaß Converse Theorem [25].

The reason we choose to study these lifts is the following observation : For all primes p 6= 2 we have that GSp4 (Qp ) ' GSpin(1, 4)(Qp ) (l. This relation between GSp4 and GSpin(1, 4) suggests the possibility of developing liftings for the spin group analogous to Ikeda’s liftings for the symplectic group. Another motivation for considering spin groups is to give applications for the Maaß Converse Theorem. As far as we know this theorem has been applied only once by Duke [6] to show that a non-cuspidal theta series is automorphic for the group GSpin(1, 5).

In this paper we obtain a family of cuspidal automorphic forms for GSpin(1, 4) using the converse theorem. 1 To define the lifting we start with a weight 1/2 Maaß Hecke eigenform f on the complex upper half plane with respect to Γ0 (4). The candidate function F on a symmetric space of GSpin(1, 4) is defined in terms of a Fourier expansion with Fourier coefficients related to the Fourier coefficients of f as in the formula (3. The Maaß Converse Theorem 2.5 states that the function F is cuspidal automorphic with respect to the integer points of GSpin(1, 4) if and only if a family of Dirichlet series is “nice” (“nice” means analytic continuation, bounded on vertical strips and functional equation).

To obtain the “nice” properties of the Dirichlet series we use the method of Duke and Imamoḡlu in [7] to rewrite the Dirichlet series in terms of a Rankin triple integral of the function f, a certain Theta function and an Eisenstein series with respect to Γ0 (4). Then essentially the “nice” properties of the Eisenstein series give us the corresponding “nice” properties of the Dirichlet series. Another interesting result of the thesis is related to the non-vanishing of the lift. In our case, the non- vanishing of the lift F is equivalent to the non-vanishing of certain negative Fourier coefficients of f.

It seems that it is not possible to get non-vanishing of specifically negative Fourier coefficients of f using elementary methods as in [19]. To achieve this we use the work of Baruch and Mao [2] to prove a Waldspurger type (or Kohnen- Zagier type) formula which relates the square of the negative Fourier coefficient of f to special values of twisted L−functions (Theorem 4. Then we get the non- vanishing of the lift F by the results of Friedberg and Hoffstein [10] on non-vanishing of special values of twisted L−functions. 2 We also analyze the adelic automorphic representation πF of the group GSpin(1, 4)(A) obtained from the automorphic function F.

To do this, we first show that if f is a Hecke eigenfunction then so is F and explicitly calculate the eigenvalues of F in terms of the eigenvalues of f. Since the p−adic component (p an odd prime) of πF is an irreducible unramified representation of GSpin(1, 4)(Qp ) (w GSp4 (Qp )) we know that it is the unique spherical constituent of the representation obtained by induction from an unramified character on the Borel subgroup.5, we obtain the explicit description of this character in terms of the eigenvalue of f using the Hecke theory for the spin group. Using the precise information about the local representations, we show that the representation πF is a CAP representation. Let us remind the reader that if G is a reductive algebraic group and P a parabolic subgroup then an irreducible cuspidal automorphic representation π of G(A) is called Cuspidal Associated to Parabolic (CAP) subgroup P if there is an irreducible cuspidal automorphic representation σ of the Levi subgroup of P such that π is nearly equivalent to an irreducible com- ponent of IndG P (σ).

The notion of CAP representations was first introduced by I. Piatetski-Shapiro [28] for the group Sp4. The CAP representations are very interest- ing because they provided counter-examples to the earlier versions of the generalized Ramanujan conjecture. CAP representations on Sp4 have been extensively studied by Piatetski-Shapiro in [28] and Soudry in [35], [36].

In [11] and [29], families of CAP representations on the split exceptional group of type G2 have been constructed. In [12] the authors give a criterion for an irreducible cuspidal automorphic representa- tion of a split group of type D4 to be a CAP representation. In these papers the method used to construct CAP representations is theta lifting of various types. 3 The representation πF constructed here is interesting in this context because it is CAP to a representation of GSp4 (A) instead of GSpin(1, 4)(A) (Theorem 6.

If one considers Langlands’ philosophy then it is natural to extend the notion of CAP representations to the situation where the group G is replaced by two groups G1 , G2 which satisfy G1,ν w G2,ν for almost all ν which is the case in our present setting. 4 CHAPTER 2 Maaß Converse Theorem Maaß stated his converse theorem in terms of Clifford algebras and Vahlen matri- ces. In this chapter we give the definition and basic properties of Vahlen matrices and illustrate their relation to the spin group. We then define automorphic functions and state the Maaß Converse Theorem.

We give a brief sketch of the proof of the converse theorem. In this chapter we perform all the calculations for the case of Spin(1, k + 2) for k ≥ 0. We will restrict to the case of k = 2 from Chapter 3. The main reference for this chapter is Maaß [25].

For details on Vahlen’s group of matrices we refer the reader to [8] and [9].1 Clifford algebras and Vahlen matrices Let E be a vector space over R of dimension k. Let Q : E → R be a nondegenerate quadratic form and {i1 , · · · , ik } a basis of E orthogonal with respect to Q. Define i2ν = Q(iν ), iν iµ = −iµ iν for 1 ≤ µ, ν ≤ k and µ 6= ν. The Clifford algebra Ck (Q) is generated by {i1 , · · · , ik } with relations as above.

Hence the 2k elements iν1 · · · iνp where 1 ≤ ν1 < ν2 < · · · < νp ≤ k, 0 ≤ p ≤ k form a basis of Ck (Q) over the reals 5 (For p = 0 we mean the element 1). Denote by Ck (Q)× the invertible elements of Ck (Q). Note that Ck−1 (Q) is embedded in Ck (Q) by setting the coefficient of all the above basis elements containing ik to be equal to zero. Ck (Q) consists of elements of the form k X X a = a0 + aν1 ···νp iν1 iν2 · · · iνp p=1 1≤ν1 <···<νp ≤k with aν1 ···νp , a0 ∈ R.

We define the following maps on Ck (Q) : k X X 0 a 7→ a := a0 + (−1)p aν1 ···νp iν1 iν2 · · · iνp , p=1 1≤ν1 <···<νp ≤k k X p(p+1) X a 7→ a := a0 + (−1) 2 aν1 ···νp iν1 iν2 · · · iνp , p=1 1≤ν1 <···<νp ≤k k X p(p−1) X ∗ 0 a 7→ a := a = a0 + (−1) 2 aν1 ···νp iν1 iν2 · · · iνp. p=1 1≤ν1 <···<νp ≤k These maps satisfy x = −x, x0 = −x, x∗ = x for all x ∈ Ri1 ⊕ · · · ⊕ Rik and (uv) = v u, (uv)0 = u0 v 0 , (uv)∗ = v ∗ u∗ for all u, v ∈ Ck (Q). Let us denote Re(x0 + · · · + xk ik ) := x0. Let Vk (Q) ⊂ Ck (Q) be the (k + 1)−dimensional vector space over reals with basis {1, i1 , i2 , · · · , ik }.

In [25], Maaß calls the elements of Tk (Q) as transformers. 6 The Vahlen group of matrices is defined by    α β  SVk (Q) := {  ∈ M2 (Ck (Q)) : α, β, γ, δ ∈ Tk (Q) ∪ {0}, γ δ αδ ∗ − βγ ∗ = 1, αβ ∗ , δγ ∗ ∈ Vk (Q)}. For the proof we refer the reader to [8] Pg 377. We will now give the relationship between SVk (Q) and the Spin group.

Then define Spink (Q) := {s ∈ Ck+ (Q)|sEs∗ ⊂ E, ss∗ = 1}.2) Let E0 be a 3−dimensional vector space over reals with basis {f0 , f1 , f2 }. Let Q0 be the quadratic form on E0 defined by Q0 (y0 f0 + y1 f1 + y2 f2 ) = y02 − y12 − y22. Equip + Ẽ := E0 ⊕ E with the quadratic form Q̃ := Q0 ⊥ Q.