luận án tiến sĩ lifting from sl(2) to gspin(1,4)

In this thesis we construct liftings of automorphic forms from the metaplecticgroup fSL2 to GSpin1, 4 using the Maaß Converse Theorem.. In order to provethe non-vanishing of the lift we

Lifting from SL(2) to GSpin(1,4)

* * * * *

The Ohio State Univesity

2006

Professor Steve Rallis, Adviser

UMI Number: 3226302

3226302 2006

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In this thesis we construct liftings of automorphic forms from the metaplecticgroup fSL2 to GSpin(1, 4) using the Maaß Converse Theorem In order to provethe non-vanishing of the lift we derive Waldspurger’s formula for Fourier coefficients

of half integer weight Maaß forms We analyze the automorphic representation ofthe adelic spin group obtained from the lift and show that it is CAP to the Saito-Kurokawa lift from fSL2 to GSp4(A)

Dedicated to Swapna

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, patienceand generosity were invaluable to me I would like to thank James Cogdell and CaryRader for humoring my endless questions and giving me so much of their time andhelp For their comments and suggestions I am grateful to Dinakar Ramakrishnan,Tamotsu Ikeda, Winfred Kohnen, Dihua Jiang, Steve Kudla, William Duke, PeterSarnak and Eitan Sayag

September 16, 1977 Born - Nagpur, India

2000 MSc Mathematics, Indian Institute of

Technology, Kanpur, India

2000 − present Graduate Teaching Assistant, The Ohio

TABLE OF CONTENTS

Page

Abstract ii

Dedication iii

Acknowledgments iv

Vita v

Chapters: 1 Introduction 1

2 Maaß Converse Theorem 5

2.1 Clifford algebras and Vahlen matrices 5

2.2 Automorphic functions 9

3 Definition and Automorphy of the Lift 14

3.1 Maaß forms on SL2 of half-integral weight 14

3.2 Definition of A(β) 19

3.3 Proof of Theorem 3.3 22

3.3.1 Theta Function and Eisenstein Series 25

3.3.2 Rankin Integral Formula 29

3.3.3 The functional equation 31

4 Non-Vanishing of the Lift 36

4.1 Criteria for non-vanishing 36

4.2 Waldspurger’s formula for Maaß forms 38

4.3 Non-vanishing of special values of L−functions 44

5 Hecke Theory 47

5.1 Hecke Algebra Hp for p odd 48

5.2 Hecke operator Tp 63

5.3 Hecke operator Tp2 78

6 Automorphic representation corresponding to F 89

6.1 Unramified Calculation 91

6.2 CAP representation 99

Bibliography 102

CHAPTER 1

INTRODUCTION

In [16], Ikeda defines a lifting of automorphic forms from fSL2, the metaplecticcover of SL2, to the symplectic group GSp4n for all n Ikeda proves his result usingthe theory of Fourier Jacobi forms For n = 1 he shows that his definition agrees withthe classical Saito-Kurokawa lift In [7], Duke and Imamo¯glu prove the automorphy

of the classical Saito-Kurokawa lift from holomorphic half integer weight forms usingthe converse theorem for Sp4 due to Imai [17]

In this thesis we present the lifting of automorphic forms from fSL2 to the spingroup 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 thatGSp4(Qp) ' GSpin(1, 4)(Qp) (l.c Proposition 6.3) This relation between GSp4 andGSpin(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 spingroups is to give applications for the Maaß Converse Theorem As far as we know thistheorem has been applied only once by Duke [6] to show that a non-cuspidal thetaseries is automorphic for the group GSpin(1, 5) In this paper we obtain a family ofcuspidal automorphic forms for GSpin(1, 4) using the converse theorem

To define the lifting we start with a weight 1/2 Maaß Hecke eigenform f onthe 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 withFourier coefficients related to the Fourier coefficients of f as in the formula (3.9).The Maaß Converse Theorem 2.5 states that the function F is cuspidal automorphicwith respect to the integer points of GSpin(1, 4) if and only if a family of Dirichletseries is “nice” (“nice” means analytic continuation, bounded on vertical strips andfunctional equation) To obtain the “nice” properties of the Dirichlet series we usethe method of Duke and Imamo¯glu in [7] to rewrite the Dirichlet series in terms of aRankin triple integral of the function f, a certain Theta function and an Eisensteinseries with respect to Γ0(4) Then essentially the “nice” properties of the Eisensteinseries give us the corresponding “nice” properties of the Dirichlet series

Another interesting result of the thesis is related to the non-vanishing of thelift In [16] it is shown that the non-vanishing of the Ikeda lift is equivalent to thenon-vanishing of certain Fourier coefficients of the holomorphic half-integer weightform, which follows from a straightforward calculation in [19] In our case, the non-vanishing of the lift F is equivalent to the non-vanishing of certain negative Fouriercoefficients of f It seems that it is not possible to get non-vanishing of specificallynegative 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 Zagier type) formula which relates the square of the negative Fourier coefficient of

Kohnen-f to special values oKohnen-f twisted L−Kohnen-functions (Theorem 4.3) Then we get the vanishing of the lift F by the results of Friedberg and Hoffstein [10] on non-vanishing

non-of special values non-of twisted L−functions

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 showthat 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 byinduction from an unramified character on the Borel subgroup In Theorem 6.5, weobtain the explicit description of this character in terms of the eigenvalue of f usingthe Hecke theory for the spin group

Using the precise information about the local representations, we show that therepresentation πF is a CAP representation Let us remind the reader that if G is areductive algebraic group and P a parabolic subgroup then an irreducible cuspidalautomorphic 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 ponent of IndG

com-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 generalizedRamanujan 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 CAPrepresentations 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 themethod used to construct CAP representations is theta lifting of various types

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 6.7) Ifone considers Langlands’ philosophy then it is natural to extend the notion of CAPrepresentations to the situation where the group G is replaced by two groups G1, G2which satisfy G1,ν w G2,ν for almost all ν which is the case in our present setting

CHAPTER 2

Maaß Converse Theorem

Maaß stated his converse theorem in terms of Clifford algebras and Vahlen ces In this chapter we give the definition and basic properties of Vahlen matrices andillustrate their relation to the spin group We then define automorphic functions andstate the Maaß Converse Theorem We give a brief sketch of the proof of the conversetheorem 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 referencefor this chapter is Maaß [25] For details on Vahlen’s group of matrices we refer thereader to [8] and [9]

matri-2.1 Clifford algebras and Vahlen matrices

Let E be a vector space over R of dimension k Let Q : E → R be a nondegeneratequadratic form and {i1, · · · , ik} a basis of E orthogonal with respect to Q Define

i2

ν = Q(iν), iνiµ = −iµiν 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· · · iνpwhere 1 ≤ ν1 < ν2 < · · · < νp ≤ k, 0 ≤ p ≤ k form a basis of Ck(Q) over the reals

(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 theabove basis elements containing ik to be equal to zero Ck(Q) consists of elements ofthe form

The Vahlen group of matrices is defined by

For the proof we refer the reader to [8] Pg 377

We will now give the relationship between SVk(Q) and the Spin group Let Ck+(Q)

be the subalgebra of Ck(Q) with basis {iν1· · · iνp : 1 ≤ ν1 < · · · < νp ≤ k, p ≡ 0(mod 2)} Then define

Ck+3+ ( ˜Q) From [8] Pg 372 we get that the map ψ : M2(Ck(Q)) → Ck+3+ ( ˜Q) given by

Proposition 2.2 The R−algebra isomorphism ψ defined above restricts to a groupisomorphism

ψ : SVk(Q) → Spink+3( ˜Q)

For the proof, we again refer the reader to [8] Pg 382

Let us now fix Q = Qk(x1, · · · , xk) := −x2

1− · · · − x2

k on Rk and consider Ck(Qk).Define the upper half space as

∂

∂xj) = x

2 k+1

2.2 Automorphic functions

Let us now define the automorphic functions for SVk We fix the quadratic form

Q = Qk given in the previous section and drop Q from the notation whenever there

where T is a fixed lattice in Vk

Definition 2.4 (Automorphic Function) A complex-valued C∞ function F on Hk+1

is called automorphic with respect to ΓT if F satisfies the following conditions :

k+1 2 k+1log(xk+1), if r = 0

Here S is the lattice in Vk dual to the lattice T defined by S = {β ∈ Vk :Re(βT ) ⊂ Z} For β = β0 + β1i1 + · · · + βkik we define the norm by the formula

|β|2 = β2

0 + β2

1 + · · · + β2

k Also, Kir is the classical K−Bessel function satisfying

Kir(y) → 0 as y → ∞ If u0(xk+1) = 0 then we call F a cuspidal automorphicfunction In the next chapter we will make a choice of the lattice T such that ΓT hasonly one cusp, hence the notation is justified

For every integer l fix a basis {Pl,ν} of spherical harmonic polynomials of degree

l in k + 1 variables In [25], Maaß proves the following Converse Theorem

Theorem 2.5 (Maaß Converse Theorem) The following two statements are lent

equiva-1 F (x) = u0(xk+1) + P

β∈S β6=0

A(β)x

k+1 2 k+1Kir(2π|β|xk+1)e2πiRe(βx) is an automorphicfunction with respect to ΓT

2 For all l, Pl,ν, the functions ξ(s, Pl,ν) := π−2sΓ(s + ir2)Γ(s − ir2) P

β∈S β6=0

A(β)P l,ν (β) (|β| 2 ) s

satisfy

(a) s − k+1+ir2
s − k+1−ir2
ξ(s, P0,ν) and ξ(s, Pl,ν), l > 0, have analyitic tinuation to the complex plane,

con-(b) ξ(s, Pl,ν) are bounded on vertical strips and

(c) ξ(s, Pl,ν) satisfy the functional equation

ξ(k + 1

2 + l − s, Pl,ν) = (−1)

lξ(s, Pl,ν0 ) (2.8)

where Pl,ν0 (β) := Pl,ν(β0) for β ∈ Vk

The residues of ξ(s, P0,ν) at s = k+1+ir2 and k+1−ir2 determine the constants a1 and

a2 in the constant term u0(xk+1) Hence, if we want F to be a cuspidal automorphicfunction then we require ξ(s, P0,ν) to have analytic continuation to C

We give a sketch of the proof here Note that the automorphy of a function Fgiven by the Fourier expansion (2.7) is equivalent to the condition F (−x−1) = F (x),i.e g(x) := F (−x−1) − F (x) is identically zero Since F is an eigenfunction of

Ωk+1 and x → −x−1 preserves Ωk+1 we conclude that g is also an eigenfunction of

Ωk+1 The main ingredient of the proof of Theorem 2.5 is the following Propositionproved in [25] which gives a criterion for an eigenfunction of the operator Ωk+1 to beidentically zero

Proposition 2.6 Let g(x) be twice continuously differentiable function on Hk+1 isfying Ωk+1g + (r2+(k+1)4 2)g = 0 for some r ∈ R Since Ωk+1 is an elliptic operator,g(x) can be extended to a complex neighbourhood U about any point in Hk+1 suchthat g(x) represents a regular analytic function in complex variables x0, x1, · · · , xk+1

sat-on U g(x) vanishes identically in Hk+1 if and only if for any set of complex numbers

a0, a1, · · · , ak with a0 + a1 + · · · + ak2 = 0 and any real xk+1 > 0, the conditions

are satisfied for l = 0, 1, 2, · · ·

For the proof of the above Proposition we refer the reader to [25] Observe that

in the case of a holomorphic function f on the complex upper half plane modularitywith respect to SL2(Z) is equivalent to checking the periodicity (f (z +1) = f (z)) andthe condition f (−z−1) = (−z)kf (z) only for z = iy (The second condition is a direct

consequence of the holomorphy of f.) In our case, the above proposition replaces thiscondition Note that l = 0 corresponds to g(ik+1xk+1) = 0.

Fix complex numbers a0, a1, · · · , akwith a0 +a1 +· · ·+ak2 = 0 and real xk+1 > 0

To prove Theorem 2.5 using Proposition 2.6 we need to show that for all l = 0, 1, 2, · · ·

A(β)Pl(β)yk+12 +lKir(2π|β|y) (2.10)

where y ∈ R, y > 0 and Pl is a spherical harmonic polynomial of degree l in k + 1variables Define Pl0(β0+ β1i1+ · · · + βkik) = Pl(β0− β1i1− · · · − βkik) i.e Pl0(β) =

β∈S, β6=0

A(β)Pl,ν(β)(|β|2)s Using Mellin inversion formula, we now get that Fl(y, Pl,ν) = (−1)lFl(1y, Pl,ν0 ) for every

l and Pl,ν if and only if for every l, Pl,ν the ξ(s, Pl,ν) satisfy the following conditions :

s − k+1+ir2
s −k+1−ir2
ξ(s, P0.ν) and ξ(s, Pl,ν), l > 0 have analyitic continuation

to the complex plane, ξ(s, Pl,ν) is bounded on vertical strips and ξ(s, Pl,ν) satisfy thefunctional equation ξ(k+12 + l − s, Pl,ν) = (−1)lξ(s, Pl,ν0 ) 

CHAPTER 3

Definition and Automorphy of the Lift

Let us restrict ourselves to the case k = 2 This corresponds to the groupSpin(1, 4) In this chapter we first give some details about half-integral weight Maaßforms in section 3.1 These will be the input data used to define a candidate for acuspidal automorphic function F (x) on H3 We do this by giving an explicit formula(3.9) for the Fourier coefficients A(β) of F in section 3.2 In section 3.3 we prove theautomorphy of the candidate function using the Maaß Converse Theorem 2.5 Themain idea of the proof is to get a Rankin integral formula for the Dirichlet serieswhich is achieved in Proposition 3.9

3.1 Maaß forms on SL2 of half-integral weight

Automorphic forms for the metaplectic group fSL2 can be realised as half integerweight forms on the upper half plane with respect to the group Γ0(4) := {

3 f vanishes at the cusps of Γ0(4) (The cusps of Γ0(4) are given by ∞, 0 and 12).

f has the Fourier expansion

f (z) =X

n∈Z n6=0

c(n)Wsign(n)(t+1/2)

where λ = 14 + (r/2)2 and Wν,µ(y) is the classical Whittaker function The numbers{c(n)} are called the Fourier coeffficients of f For more details on half-integral weightMaaß forms we refer the reader to [21] and [18]

Define the plus space

S+

t+1(4) := {f ∈ St+1

2 | c(n) = 0 whenever (−1)tn ≡ 2, 3 (mod 4)}

This is analogous to the Kohnen plus space for holomorphic half integral weight

modular forms introduced in [19] For t = 0, S+1

2

(4) is same as the plus space defined

in [18] where it is shown to be nonempty It then follows from Lemma 3.1 below that

is true for c(n) since np2, n/p2 ≡ n (mod 4) This means that the operators T (p2)

us the freedom to move f ∈ S1/2+ (4) to functions in St+1/2+ (4) for t ∈ Z This will becrucial in the proof of the non-vanishing result (Theorem 4.5) of the next chapter

The following recurrence relations ([26] Pg 302) for Whittaker functions Wν,µ(y)

Using (3.5) with ν = (t+1/2)2 , µ = ir2 and y replaced with 4πny we get

2

c(n) for n < 0 We get (3.6)

Proof It is clear that ΓT ⊂ SV2(Z) To show the other inclusion we will first showthat given a matrix M =

∈ SV2(Z) we can find an element g ∈ ΓT such

that gM is upper triangular By multiplying on the left by the matrix

1 ⇒ |α − uγ|2 < |γ|2 Now consider

α = ±1, ±i1, ±i2 or ± i1i2 and δ = α0 One can check easily that 

Remark : Let us note here that if we consider Spin(1, n) with n > 4 then theabove Proposition is not true, i.e., ΓT (with T = Z + Zi1+ · · · + Zin−2) is not equal

to the integer points of Spin(1, n), n > 4 In fact, the group ΓT does not have finitecovolume and is not arithmetic So, in a sense, Spin(1, 4) is the only interestingcase to apply the Maaß Converse Theorem For all higher values of n the conversetheorem is an analog of Hecke’s converse theorem for triangle subgroups in SL2 [15].Let f ∈ S+1(4) be a non-zero Hecke eigenform with Fourier coefficients {c(n)}.Let β = β0 + β1i1 + β2i2 ∈ V2(Z) and write gcd(β0, β1, β2) = 2ud, where u ≥ 0and d is odd, positive

The main result of this section is

Theorem 3.3 With A(β) as above,

β∈V 2 (Z), β6=0

A(β)x33Kir(2π|β|x3)e2πiRe(βx)

is a cuspidal automorphic function on H3 with respect to SV2(Z)

1 We observe that the definition of A(β) above is similar to the Saito-Kurokawalift for Sp4 In the Sp4 case, one starts with g(z) =

∞

P

n=1

b(n)e2πnz, a holomorphiccusp form with respect to Γ0(4) of weight k + 12 with b(n) = 0 if (−1)kn ≡ 2, 3(mod 4) The Saito-Kurokawa lift is then given by G(Z) = P

The above formula differs from (3.9) in the sense that in (3.9) we have to treatthe prime 2 dividing gcd(β0, β1, β2) separately

2 Note that A(β) satisfies |A(β)| < C0|β|k 0 for some constants C0, k0 independent

of β which ensures the convergence of F (x) above We can see this as follows :Since f ∈ S+1(4) there exists positive C1, k1 ∈ R such that |c(n)| < C1|n|k 1 forall nonzero integers n This gives us

β∈V 2 (Z)

A(β)P l,ν (β) (|β| 2 ) s =(−1)l X0

to formulate the Rankin integral Let us set D(s, Pl,ν) := X0

β∈V 2 (Z)

A(β)P l,ν (β) (|β| 2 ) s Observethat from (3.10) we can conclude that D(s, Pl,ν) converges absolutely in some righthalf plane Re(s) ≫ 0

Pl,ν(β)

Proof As in (3.9), for β = β0+ β1i1+ β2i2 ∈ V2(Z) we write gcd(β0, β1, β2) = 2udwith u ≥ 0 and d > 0 odd Set ν2(β) := u.

Now rearranging terms we get

c−|β|n22



n−1/2Pl,ν(β)

Notice that the third summation is nonempty only when v is divisible by n2 Hence

we can replace v by mn2 to get

Substituting (3.12) in (3.11), replacing s by s +2l +14 and multiplying numeratorand denominator by 2s we then have

of the theta function and Eisenstein series

3.3.1 Theta Function and Eisenstein Series

Here we have identified the lattice V2(Z) defined in Section 3.2 with Z3 By [33]

Pg 456, Θl,ν(z) is a holomorphic modular form of weight l + 32 for the discrete group

Γ0(4) i.e

Θl,ν(γz) =d−1c

d

(cz + d)12

The following transformation laws for Θl,ν(z) will be used in Section 3.3.3 where

we obtain the functional equation of ξ(s, Pl,ν) using the Rankin integral formula

h (mod 2)

e2πitkh2



Θl,ν(z; k)

Use the fact that P

|β| 2 =m β≡0 (mod 2)

The Eisenstein series satisfies

3.3.2 Rankin Integral Formula

Define the integral

(cz + d)12

Here we have used (3.25) Now using (3.1), (3.19) we get

(cz + d)1

m∈Z m6=0

Comparing (3.8) with (3.16) we get

ana-at all points where the Eisenstein series is defined, the integral I(s) converges since f

is a cusp form and Θl,ν and the Eisenstein series have moderate growth At the poles

of the Eisenstein series its residue is a constant and hence the residue of I(s) is given

by a constant multiple of R

Γ 0 (4)\Hf (z)Θl,ν(z)yl+22 − 1

4 dxdy

y 2 This integral is zero since f

is a non-holomorphic cusp form and Θl,ν is a holomorphic modular form Also, I(s)

is bounded on vertical strips since the same is true of the Eisenstein series

3.3.3 The functional equation

Observe that the functional equation from the Maaß Converse Theorem ξ(32+ l −

s, Pl,ν) = (−1)lξ(s, Pl,ν0 ) is equivalent to ξ(s +2l+14, Pl,ν) = ξ(1 − s +2l+14, Pl,ν0 ) Thisimplies that we have to show the functional equation

23s− 2s I(s) = 23(1−s)− 2(1−s) I(1 − s) (3.30)since the term 2s + 21−s in the denominator of (3.29) is already invariant under

s → 1 − s (Note that I(s) is unchanged if Pl,ν is replaced by Pl,ν0 since the same istrue of Θl,ν.)