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

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 Professor Cary Rader Advisor 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.O Box 1346 Ann Arbor, MI 48106-1346 ABSTRACT In this thesis we construct liftings of automorphic forms from the metaplectic group SL2 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 SaitoKurokawa lift from SL2 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 2000 2000 − present Born - Nagpur, India MSc Mathematics, Indian Institute of Technology, Kanpur, India Graduate Teaching Assistant, The Ohio State University PUBLICATIONS Pitale A., “Lifting from SL2 to GSpin(1, 4)”, Intern Math Res Not 2005, No 63, Pg 3919 - 3966 FIELDS OF STUDY Mathematics, Number Theory v TABLE OF CONTENTS Page Abstract ii Dedication iii Acknowledgments iv Vita v Chapters: Introduction Maaß Converse Theorem 2.1 2.2 Clifford algebras and Vahlen matrices Automorphic functions Definition and Automorphy of the Lift 14 3.1 Maaß forms on SL2 of half-integral weight 3.2 Definition of A(β) 3.3 Proof of Theorem 3.3 3.3.1 Theta Function and Eisenstein Series 3.3.2 Rankin Integral Formula 3.3.3 The functional equation 14 19 22 25 29 31 Non-Vanishing of the Lift 36 4.1 4.2 Criteria for non-vanishing Waldspurger’s formula for Maaß forms vi 36 38 4.3 Non-vanishing of special values of L−functions 44 Hecke Theory 47 5.1 5.2 5.3 Hecke Algebra Hp for p odd Hecke operator Tp Hecke operator Tp2 48 63 78 Automorphic representation corresponding to F 89 6.1 6.2 Unramified Calculation CAP representation 91 99 Bibliography 102 vii CHAPTER INTRODUCTION In [16], Ikeda defines a lifting of automorphic forms from SL2 , 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 = he shows that his definition agrees with the classical Saito-Kurokawa lift In [7], Duke and Imamo¯lu prove the automorphy g of the classical Saito-Kurokawa lift from holomorphic half integer weight forms using the converse theorem for Sp4 due to Imai [17] In this thesis we present the lifting of automorphic forms from SL2 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 = we have that GSp4 (Qp ) GSpin(1, 4)(Qp ) (l.c Proposition 6.3) 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 6.1 Unramified Calculation For the rest of the chapter p is an odd prime We have shown in Chapter that the classical Hecke algebra Hp is generated by double cosets ΓM Γ where M ∈ GSpin(1, 4)+ (Z[p−1 ]) =: G+ (Z[p−1 ]) Here Z[p−1 ] is the set of rational numbers with only powers of p in the denominator In Theorem 5.8 we gave the four algebraically independent generators of this algebra H(Gp , Kp ) is the convolution algebra of Kp −biinvariant compactly supported functions on Gp This is generated by the characteristic functions of double cosets Kp gKp with g ∈ Gp Proposition 6.1 With notations as above Hp H(Gp , Kp ) (6.2) Proof We have the following bijections from the natural inclusions Γ\G+ (Z[p−1 ])/Γ G(Z)\G(Z[p−1 ])/G(Z) Kp \Gp /Kp     The first bijection is obtained from the fact that G(Z) = Γ ∪   Γ and −1     G(Z[p−1 ]) = G+ (Z[p−1 ]) ∪   × G+ (Z[p−1 ]) To get the second one, note −1 that we get injectivity from the fact that G(Z[p−1 ]) ∩ Kp = G(Z) We get surjectivity from the isomorphism Gp G(Q)Kp G(Z[p−1 ])Kp This tells us that the two Hecke algebras are isomorphic as vector spaces We get similar bijections for single cosets This is used to check that the classical double coset multiplication coincides with the convolution product on the p−adic Hecke algebra 91 Hence H(Gp , Kp ) is generated by the functions     ˆ    α  1 φ1 := V ol(Kp ) char(Kp   Kp ), φ2 := V ol(Kp ) char(Kp   Kp ), p pˆ α     −1  p   p  1 φ3 := V ol(Kp ) char(Kp   Kp ), φ4 := V ol(Kp ) char(Kp   Kp ) −1 p p Since the representation πp is unramified we have a unique (up to a scalar) Kp −fixed unramified vector Fp ∈ Vπp Now Proposition 6.1, Theorem 5.12 and Theorem 5.15 give us the following proposition 0 Proposition 6.2 With notations as above we have φ1 Fp = (p3/2 λp + p(p + 1))Fp , 0 0 0 φ2 Fp = ((p + 1)p3/2 λp + (p2 − 1))Fp , φ3 Fp = Fp and φ4 Fp = Fp Since the φi generate the Hecke algebra freely Proposition 6.2 gives us an algebra homomorphism of H(Gp , Kp ) Our next step is to find the unramified character which corresponds to this homomorphism For this it is convenient to work in the setting    A B  of the symplectic group Define GSp4 (Qp ) := {M =   ∈ M at4 (Qp ) : C D At D − B t C = µ(M )I2 , µ(M ) ∈ Q× , B t D = Dt B, At C = C t A} p Proposition 6.3 Let p be an odd prime Then with notations as above we have Gp GSp4 (Qp ) and Kp GSp4 (Zp ) Proof Fix r, s ∈ Zp such that r2 + s2 = −1 (We can always choose r, s as above √ √ since p does not divide the discriminant of Zp [ −1] and hence every unit in Zp [ −1] is a norm.) For α = α0 +α1 i1 +α2 i2 +α3 i1 i2 ∈ C2 (Qp ) define ψp : C2 (Qp ) → M at2 (Qp ) by the formula 92    α0 − α1 r − α2 s −α3 − α1 s + α2 r  ψp (α) :=   α3 − α1 s + α2 r α0 + α1 r + α2 s ψp satisfies the following properties : ψp is an isomorphism of Qp −algebras det(ψp (α)) = |α|2 ψp (α ) = p t ψp (α)−1 , ψp (α∗ ) = t ψp (α) and ψp (¯ ) = p ψp (α)−1 α Extend this map to ψp : Gp → GSp4 (Qp )      ψp (α) ψp (β)   α β  M =  → ψp (γ) ψp (δ) γ δ This is well-defined since according to the definition of Gp we have αδ ∗ − βγ ∗ = µ(M ), αγ ∗ and βδ ∗ are vectors which gives us the precise conditions of the definition of GSp4 (Qp ) above Hence we get the isomorphism Gp GSp4 (Qp ) One can also check that ψp maps Kp onto GSp4 (Zp ) From now on, we will use the notation Gp for GSp4 (Qp ) and Kp for GSp4 (Zp ) Following Asgari-Schmidt [1], Gp = BKp where B is the Borel subgroup Let N :=   t −1 X   D {  : D upper triangular with on the diagonal, X symmetric } be the D    a1      a2    : a0 , a1 , a2 ∈ Q× } be unipotent radical and A := {a =  p   −1   a1 a0     −1 a2 a0 the torus so that B = N A Then δB (a) = |a−3 a2 a4 |p is the modular function coming 93 from the Haar measure Given unramified characters χ0 , χ1 , χ2 on Q× (trivial on Z× ) p p define the character χ on A by χ(a) := χ0 (a0 )χ1 (a1 )χ2 (a2 ) Extend χ to a character G of B = N A by setting it to be trivial on N Define I(χ) := IndBp (χ) = {f : Gp → C : 1/2 locally constant function such that f (nag) = δB (a)χ(a)f (g) for n ∈ N, a ∈ A, g ∈ Gp } Gp acts on this space by right translation and we get the normalized induced representation I(χ) We will now find the unramified character χ such that πp is isomorphic to the unique spherical constituent πχ of I(χ) The strategy is to apply the generators of the Hecke algebra H(Gp , Kp ) to the unramified vector by convolution and evaluate at identity The values will be polynomials in χ0 (p), χ1 (p), χ2 (p) and then from Proposition 6.2 we get a relation between the character values and the eigenvalues of our lift F We solve these equations to get the character χ Note that χ is unique upto the action of the Weyl group W The Weyl group of Gp is of order and is generated by the matrices      0 w1 :=    0   0      0        , w2 :=     −1      0        0   0  , w3 :=     0 0      −1  0          0 The Weyl group acts on the character χ by the formula χw (a) := χ(w−1 aw) If χ(a) = χ0 (a0 )χ1 (a1 )χ2 (a2 ) then we have χw1 (a) = χ0 (a0 )χ2 (a1 )χ1 (a2 ), χw2 (a) = (χ0 χ1 )(a0 )χ−1 (a1 )χ2 (a2 ) χw3 (a) = (χ0 χ2 )(a0 )χ1 (a1 )χ−1 (a2 ) 94 0 Let Fp be the unramified vector in the space of πχ satisfying Fp (1) = Then 1/2 0 Fp (nak) = δB (a)χ(a) where n ∈ N, a ∈ A, k ∈ Kp Any φ ∈ H(Gp , Kp ) acts on FP by convolution according to the formula φ(hg)Fp (g)dg for h ∈ Gp (φ ∗ Fp )(h) := Gp We will use Proposition 6.2 and the action of H(Gp , Kp ) on Fp above to get equations satisfied by χ For this we need the following right coset decomposition : Proposition 6.4            Kp =  Kp     p     p     b1 b2    p       p b2 b     Kp        bi (mod p)     i=1,2,3       ∗ ∗ ∗ ∗   p       p−1  ∗ ∗   −b p ∗ ∗   Kp       b=0  p b        p p p      Kp           Kp (6.3)     In the third and fourth term there are exactly p choices for the upper right hand corner 95         p−1  −b p α   ψp (ˆ )    Kp Kp   Kp =   b=0  p bp  pψp (ˆ ) α     p      ∗ ∗  ∗ ∗   p  p            p−1  −bp p2 ∗ ∗  p ∗ ∗     Kp   Kp        b=0     p b            p     p       p         Kp     p2  ∗ ∗    p ∗ ∗   Kp   p   p p In the third and fourth term there are exactly p3 choices for the upper right hand corner and in the fifth term there are exactly p2 − choices Proof The proposition follows from Proposition 5.3, Proposition 5.6 and Proposition 6.3 Now let φ = char(Kp M Kp ) ∈ H(Gp , Kp ) with Kp M Kp = Mi Kp Let Mi = ni i Then φ(g)Fp (g)dg = (φ ∗ FP )(e) = Gp FP (g)dg = i MK i p Kp M Kp 1/2 Fp (ni ) = V ol(Kp ) = V ol(Kp ) Fp (g)dg i δB (ai )χ(ai ) (6.4) i      Apply (6.4) with φ = φ1 = V ol(Kp ) char(Kp    p   96  p      Kp ) Then by the single     coset decomposition obtained in (6.3) and Proposition 6.2 we get p3/2 λp + p(p + 1) = p3/2 χ0 (p) [χ1 (p)χ2 (p) + + χ2 (p) + χ1 (p)] (6.5)    ψp (α)  Now take φ = φ2 = V ol(Kp ) char(Kp   Kp ) in (6.4) Then by the pψp (α ) single coset decomposition obtained in (6.4) and Proposition 6.2 we get (p + 1)p3/2 λp + (p2 − 1) = p2 χ2 (p) χ2 (p) + χ1 (p) + χ2 (p)χ2 (p) + χ1 (p)χ2 (p) +(p2 − 1)χ2 (p)χ1 (p)χ2 (p)    p      p    Kp ) in (6.4) to get Finally take φ = φ3 = V ol(Kp ) char(Kp      p     p = χ2 (p)χ1 (p)χ2 (p) (6.6) (6.7) This tells us that πχ has trivial central character (6.7) implies χ0 (p) = (χ1 (p)χ2 (p))−1/2 Using this we get p3/2 λp + p(p + 1) = p3/2 ((χ1 (p)χ2 (p))1/2 + + (p + 1)p3/2 λp = p2 χ1 (p) χ2 (p) 1/2 + (χ1 (p)χ2 (p))1/2 χ2 (p) χ1 (p) 1/2 ) 1 + + χ1 (p) + χ2 (p) χ1 (p) χ2 (p) (6.8) (6.9) From the earlier remark on the Weyl group of Gp it is clear that the above equations are not changed if we replace the character χ with χw for any w ∈ W, the Weyl group Hence the solutions to the above equations will be in the same Weyl group orbit as expected 97 Theorem 6.5 Up to the action of the Weyl group the character χ is given by χ1 (p) = p1/2 λ2 − p λp + , χ2 (p) = p1/2 λ2 − p λp − , χ0 (p) = p−1/2 Proof Denote a = (χ1 (p)χ2 (p))1/2 + (χ1 (p)χ2 (p))−1/2 and b = χ2 (p) χ1 (p) 1/2 χ1 (p) χ2 (p) (6.10) 1/2 + Then we have χ1 (p) + χ2 (p) + χ1 (p)−1 + χ2 (p)−1 = ab Hence from (6.8) and (6.9) we get p3/2 λp + p(p + 1) = p3/2 (a + b) (p + 1)p3/2 λp = p2 ab Hence we get the equation p2 ab − p3/2 (a + b) + p(p + 1) = p+1 p2 a − p3/2 p+1 b − p−1/2 (p + 1) = If a = p−1/2 (p + 1) = p1/2 + p−1/2 then b = λp and this implies (χ1 (p)χ2 (p))1/2 = p±1/2 √ 1/2 λp ± λ2 −4 p χ1 (p) = This gives us four solutions and χ2 (p) χ1 (p) = p±1/2 χ2 (p) = p λ2 − p λp ± λ2 − p λ ±1/2 p If b = p−1/2 (p + 1) = p1/2 + p−1/2 then a = λp and this implies (χ1 (p)χ2 (p))1/2 = √ 1/2 λp ± λ2 −4 p and χ1 (p) = p±1/2 This gives us four more solutions χ2 (p) χ1 (p) = p±1/2 χ2 (p) = p 1/2 λ2 − p λp ± λ2 − p λp ± From the comments before Proposition 6.4 regarding the Weyl group action on characters, one can check that the choices for the character χ obtained above are in the same Weyl group orbit This completes the proof of Theorem 6.5 98 Since χ1 χ2 = | |−1 it follows from [30] Lemma (3.2) that the induced representation I(χ) obtained above is not irreducible From the classification of automorphic representations of GSp(4) given in [31] we can conclude that the representation πχ is a representation of type IIb 6.2 CAP representation Definition 6.6 Let G1 and G2 be two groups such that G1,ν G2,ν for almost all places ν and P2 be a parabolic subgroup of G2 Then we will call an irreducible cuspidal automorphic representation π of G1 a CAP representation associated to P2 if there is an irreducible cuspidal automorphic representation σ of M2 , the Levi component of P2 , such that πν πν for almost all places ν, where π is an irreducible component of IndG2 (σ) P To define the normalized induction, extend (σ, Vσ ) to a representation of P2 = M2 N2 by setting it to be trivial on the unipotent radical N2 and let δP2 be the modular function obtained from the Haar measure Then 1/2 IndG2 (σ) := {f : G2 → Vσ |f smooth, f (pg) = δP2 (p)σ(p)f (g) for p ∈ P2 , g ∈ G2 } P (6.11) Let G1 := GSpin(1, 4) and G2 := GSp4 Then from Proposition 6.3 we have G1,p G2,p for every odd prime p Let   B   g P := {  : g ∈ GL2 , B symmetric matrix , µ the similitude } µt g −1 be the Siegel parabolic subgroup of G2 We will now construct an irreducible cuspidal + automorphic representation σ of the Levi subgroup of P Consider f ∈ S1/2 (Γ0 (4)) 99 which is a Hecke eigenform with eigenvalue λp for every odd prime p Let h be the weight Maaß form with respect to SL2 (Z) associated to f by the Shimura correspondence given in [18] If we define the Hecke operator T (p) on weight Maaß forms by p−1 h (T (p)h)(z) := j=0 z+j p + h(pz) then from [18], Pg 199 and 223 we know that T (p)h = p1/2 λp Let σ be the irreducible cuspidal automorphic representation of GL2 (A) obtained from h using the decomposition of GL2 given by : GL2 (A) GL2 (Q)GL+ (R) p GL2 (Zp ) Write σ ⊗p σp ˆ ˆ ˆ I(η1 , η2 ) := {f : GL2 (Qp ) → C | f smooth, f (bg) =    a1  ˆ δB2 (b)1/2 η(b)f (g)∀ b ∈ B2 , g ∈ GL2 } Here B2 := {b =  } is the stana3 a2    a1  −1 dard lower-triangular Borel subgroup of GL2 , δB2   = |a1 a2 |p is the a2    a1  modular function obtained from the Haar measure on GL2 and η   := a3 a2 η1 (a1 )η2 (a2 ) where η1 and η2 are unramified unitary characters of Q∗ (We consider p where σp is given by σp the lower triangular Borel subgroup here instead of the upper triangular since we have used the notations of [1] for our calculations regarding the symplectic group.) Following calculations similar to (6.4), (6.5), (6.6) we get η1 (p) + η2 (p) = λp and η1 (p)η2 (p) = Theorem 6.7 Let πF be the representation of G1 from the previous section Then πF is CAP to an irreducible component of IndG2 (η0 × σ × |det|−1/2 ) where σ is as P above and η0 (µ) := |µ|1/2 is an unramified character that acts on the similitude 100 G Proof We claim that for every odd prime p we have IndB2,p (χ) G IndP 2,p (η0 × σp × |det|−1/2 ) with χ as in Theorem 6.5 We can then take π to be the irreducible G automorphic constituent of IndP (η0 ×σ ×|det|−1/2 ) such that the p−adic component G of π is the spherical constituent of IndP 2,p (η0 ×σp ×|det|−1/2 ) It follows from Lemma of [24] that we can always find a π with the above property Then we have πF,p πp for every odd prime p and hence we get the result of the theorem We get the claim above essentially by transitivity of induction Define the map G G L : IndP 2,p (η0 × σp × |det|−1/2 ) −→ IndB2,p (χ) by (Lf )(g) := (f (g))(I2 ) Here f : G G2,p → I(η1 , η2 ) is a function in IndP 2,p (η0 × σp × |det|−1/2 ) and I2 is the identity matrix in GL2 We have to show that this map is well-defined and an isomorphism To show that L is well-defined we have to prove that for any b ∈ B, Lf satisfies   ∗   s 1/2 (Lf )(bg) = δB (b)χ(b)(Lf )(g) Write b =   where s ∈ GL2 (Qp ) is lower µt s−1 triangular with s = an, a = diag(a1 , a2 ) and n in the unipotent radical of GL2 Since b ∈ P and s ∈ B2 we have 1/2 (Lf )(bg) = δP (b)η0 (µ)|det(s)|−1/2 (σp (s)f (g))(I2 ) 1/2 = δP (b)η0 (µ)|det(s)|−1/2 f (g)(s) 1/2 1/2 = δP (b)η0 (µ)|det(s)|−1/2 δB2 (s)η(s)f (g)(I2 ) Now we have 1/2 1/2 δP (b)η0 (µ)|det(s)|−1/2 δB2 (s)η(s) = |µ−3 a3 a3 |1/2 |µ|1/2 |a−1 a2 |1/2 |a1 a2 |−1/2 η1 (a1 )η2 (a2 ) 1/2 This implies that (Lf )(bg) = δB (b)χ0 (µ)χ1 (a1 )χ2 (a2 )(Lf )(g) since by Theorem 6.5 we have p1/2 η1 (p) = χ1 (p), p1/2 η2 (p) = χ2 (p) and χ0 (µ) = |µ|1/2 101 G It is easy to see that L is injective If f1 and f2 are two functions in IndP 2,p (η0 × σ × |det|−1/2 ) then, by definition, Lf1 = Lf2 implies that f1 (g)(I2 ) = f2 (g)(I2 ) for all g Now applying the right translation σ(s) for s ∈ GL2 we see that f1 (g)(s) = f2 (g)(s) and hence f1 = f2 G ˜ To show that L is an isomorphism we define the inverse map L : IndB2,p (χ) −→    s  G ˜ IndP 2,p (η0 × σp × |det|−1/2 ) by (Lf )(g)(s) := |det(s)|−3/2 f (  g) One can t −1 s ˜ show that L is well-defined using a similar calculation as above It is clear from the ˜ definition that L ◦ L is the identity map We note that from [32] Lemma 2.2 and Theorem 6.7 above we can conclude that the representation πF,p is precisely the local Saito-Kurokawa lift of σ We want to point out that Definition 6.6 is not the same as the definition of CAP representation found in the literature in the sense that we allow two different groups G1 and G2 satisfying G1,ν G2,ν for almost all places ν instead of considering just one group To the best of our knowledge this is the first example where such CAP representations are constructed One can explain why we get CAP representation involving two different groups if we consider Langlands functoriality The two groups GSpin(1, 4) and GSp4 are inner forms of each other Hence they have the same L−groups Langlands functoriality tells us that corresponding to the identity L−homomorphism we should get a lifting of automorphic representations from the inner form GSpin(1, 4) to the split group GSp4 Locally, when GSpin(1, 4)ν GSp4,ν the lifting is given by an isomorphism which is the content of Theorem 6.7 Hence one can say that Theorem 6.7 is a special case of the Langlands functoriality expected in this situation 102 BIBLIOGRAPHY [1] Asgari M., Schmidt R Siegel modular forms and representations Manuscripta Math 104 (2001), p 173-200 [2] Baruch E., Mao Z Central value of automorphic L−functions Preprint [3] Bump D Automorphic forms and representations Cambridge studies in Advanced Mathematics, 55 Cambridge University Press, Cambridge, 1977 [4] Bump D., Cogdell J., de Shalit E., Gaitsgory D., Kowalski E., Kudla S.S An Introduction to the Langlands Program Lectures presented at Hebrew University of Jerusalem, Jerusalem, March 12 - 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We analyze the automorphic representation of the adelic spin group obtained from the lift and show that it is CAP to the SaitoKurokawa lift from SL2 to GSp4 (A) ii Dedicated to Swapna iii ACKNOWLEDGMENTS... ABSTRACT In this thesis we construct liftings of automorphic forms from the metaplectic group SL2 to GSpin(1, 4) using the Maaß Converse Theorem In order to prove the non-vanishing of the lift... Saito-Kurokawa lift In [7], Duke and Imamo¯lu prove the automorphy g of the classical Saito-Kurokawa lift from holomorphic half integer weight forms using the converse theorem for Sp4 due to