TTWISTING OF PARAMODULAR VECTORS
JENNIFER JOHNSON-LEUNGBROOKS ROBERTS
Abstract.
Let F be a non-archimedean local field of characteristic zero, let ( π, V ) be an irre-ducible, admissible representation of GSp(4 , F ) with trivial central character, and let χ be a qua-dratic character of F × with conductor c ( χ ) >
1. We define a twisting operator T χ from paramodularvectors for π of level n to paramodular vectors for χ ⊗ π of level max( n + 2 c ( χ ) , c ( χ )), and provethat this operator has properties analogous to the well-known GL(2) twisting operator. Introduction
Let k and M be positive integers, and let χ be a quadratic Dirichlet character mod C . If f ∈ S k (Γ ( M )) is a cusp form of weight k with respect to Γ ( M ) with Fourier expansion f ( z ) = ∞ (cid:88) m =1 a ( m ) e πimz , then the twist f χ of f by χ is the element of S k (Γ ( M C )) with Fourier expansion f χ ( z ) = ∞ (cid:88) m =1 χ ( m ) a ( m ) e πimz . See, for example, Proposition 3.64 of [S]. In fact, twisting of cusp forms is a local operation whencusp forms are identified as automorphic forms on the adeles of GL(2) over Q .Let F be a nonarchimedean local field of characteristic zero with ring of integers o and maximalideal p , let ( π, V ) be a smooth representation of GL(2 , F ) for which the center of GL(2 , F ) actstrivially, and let χ be a quadratic character of F × . For n a non-negative integer, we let V ( n ) and V ( n, χ ) be the spaces of v ∈ V such that π ( k ) v = v and π ( k ) v = χ (det( k )) v , respectively, for k ∈ Γ ( p n ); here Γ ( p n ) is the subgroup of GL(2 , o ) of elements which are upper triangular mod p n . For v ∈ V , define the χ -twist T χ ( v ) of v as in (2). The main result about GL(2) twisting issummarized by the following known theorem. See section 2 for further definitions and section 3 fora proof. Theorem (
GL(2) twisting).
Let ( π, V ) be a smooth representation of GL(2 , F ) for which thecenter of GL(2 , F ) acts trivially, and let χ be a quadratic character of F × with conductor c ( χ ) > .Let n be a non-negative integer and define N = max( n, c ( χ )) . If v ∈ V ( n ) , then T χ ( v ) ∈ V ( N, χ ) .Moreover, assume that π is generic, irreducible and admissible with Whittaker model W ( π, ψ ) . Let W ∈ V ( n ) . The χ -twisted zeta integral (3) of T χ ( W ) is Z ( s, T χ ( W ) , χ ) = (1 − q − ) G ( χ, − c ( χ )) W (1) . For n ≥ N π , the image of T χ : V ( n ) → V ( N, χ ) is spanned by the non-zero vector T χ ( β (cid:48) n − N π W π ) ,where W π is a newform for π . a r X i v : . [ m a t h . N T ] J u l JOHNSON-LEUNG AND ROBERTS
The goal of this paper is to construct an analog of quadratic twisting for paramodular vectors inrepresentations of GSp(4 , F ) with trivial central character. Let ( π, V ) be a smooth representationof GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially. Let V ( n ) and V ( n, χ ) be the spaces of v ∈ V such that π ( k ) v = v and π ( k ) v = χ ( λ ( k )) v , respectively, for k in the paramodular subgroupK( p n ) of GSp(4 , F ) of level p n . For v ∈ V , we define the χ -twist T χ ( v ) of v as in (9). Our mainresult is the following theorem. We refer to section 2 for more definitions and section 4 for theproof. Main Theorem.
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially, and let χ be a quadratic character of F × with conductor c ( χ ) > . Let n be a non-negative integer and define N = max( n +2 c ( χ ) , c ( χ )) . If v ∈ V ( n ) , then T χ ( v ) ∈ V ( N, χ ) .Moreover, assume that π is generic, irreducible and admissible with Whittaker model W ( π, ψ c ,c ) where c , c ∈ o × . If W ∈ V ( n ) , then the χ -twisted zeta integral (7) of T χ ( W ) is Z ( s, T χ ( W ) , χ ) = ( q − q c ( χ ) χ ( c ) G ( χ, − c ( χ )) W (1) . For n ≥ N π , the image of T χ : V ( n ) → V ( N, χ ) is spanned by the non-zero vector T χ ( θ (cid:48) n − N π W π ) ,where W π is a newform for π . In another work we will consider the application of the paramodular twisting operator T χ toSiegel modular forms and the resulting Fourier coefficients. One reason that Siegel paramodularforms of degree 2 are of interest is their conjectural connection to abelian surfaces over Q . This isdiscussed in [BK]; see also [PY].We note that the integer N in the Main Theorem is optimal in the following sense. We mayidentify the space V ( n, χ ) with the space V χ ⊗ π ( n ) of K( p n ) fixed vectors in the twisted represen-tation χ ⊗ π . Then there exist generic, irreducible, and admissible representations π such that N = N χ ⊗ π . For example, if π is a type I representation χ × χ (cid:111) σ with χ , χ and σ unramified,then N χ ⊗ π = 4 c ( χ ) = max(0 + 2 c ( χ ) , c ( χ )). Further, suppose that π is a type X represen-tation π (cid:111) σ with π having trivial central character, σ unramified, and 2 c ( χ ) < N π . Then N χ ⊗ π = N π + 2 c ( χ ) = max( N π + 2 c ( χ ) , c ( χ )). It is interesting to observe, as in this last example,that N χ ⊗ π > N π no matter how large N π is.2. Notation and preliminaries
In this paper F is a nonarchimedean local field of characteristic zero, with ring of integers o andgenerator (cid:36) of the maximal ideal p of o . We fix a non-trivial continuous character ψ of ( F, +)such that ψ ( o ) = 1 but ψ ( p − ) (cid:54) = 1. We let q be the number of elements of o / p and use theabsolute value on F such that | (cid:36) | = q − . We use the Haar measure on the additive group F thatassigns o measure 1 and the Haar measure on the multiplicative group F × that assigns o × measure1 − q − . Throughout the paper χ is a quadratic character of F × with conductor c ( χ ), i.e., c ( χ ) isthe smallest non-negative integer n such that χ (1 + p n ) = 1, where we take 1 + p = o × .If n is a non-negative integer, then we let Γ ( p n ) be the subgroup of GL(2 , o ) of elements whichare upper triangular mod p n ; we will also write Γ ( p n ) for the analogous subgroup of SL(2 , o )when there is no risk of confusion. Let ( π, V ) be a smooth representation of GL(2 , F ) for whichthe center of GL(2 , F ) acts trivially, and let n be a non-negative integer. The subspace V ( n )consists of the vectors in V fixed by Γ ( p n ) and V ( n, χ ) is the subspace of vectors v ∈ V such that π ( k ) v = χ (det( k )) v for k ∈ Γ ( p n ). We define the level raising operators β, β (cid:48) : V ( n ) → V ( n + 1)and β, β (cid:48) : V ( n, χ ) → V ( n + 1 , χ ) by β ( v ) = π ([ (cid:36) ]) v and β (cid:48) v = v . If π is generic, irreducibleand admissible, then V ( n ) is non-zero for some n ; we let N π be the smallest such n . The space V ( N π ) is one-dimensional; if W π is a non-zero element of V ( N π ) so that V ( N π ) = C · W π , then WISTING OF PARAMODULAR VECTORS 3 we refer to W π as a newform . The space V ( n ) for n ≥ N π is spanned by the vectors β (cid:48) i β j W π where i and j are non-negative integers with i + j = n − N π . If W π is viewed as an element ofthe Whittaker model W ( π, ψ ) of π , then W π (1) (cid:54) = 0. As usual, the elements W of W ( π, ψ ) satisfy W ([ x ] g ) = ψ ( x ) W ( g ) for x ∈ F and g ∈ GL(2 , F ). See [C] and [D].The theory of paramodular newforms is developed in [RS], and we will use the notation of [RS]concerning GSp(4 , F ). We recall some necessary definitions and results. In particular, GSp(4 , F )is the subgroup of g ∈ GL(4 , F ) such that t g − − g = λ ( g ) − − for some λ ( g ) ∈ F × . If n is a non-negative integer, we let Kl( p n ) (respectively K( p n )) be thesubgroup of k ∈ GSp(4 , F ) such that λ ( k ) ∈ o × and k ∈ o o o op n o o op n o o op n p n p n o (resp. k ∈ o o o p − n p n o o op n o o op n p n p n o ) . The group Kl( p n ) is called the Klingen congruence subgroup of level p n and K( p n ) is called theparamodular subgroup of level p n . For a, b, c, d ∈ F × , we setdiag( a, b, c, d ) = a b c d . This element is in GSp(4 , F ) if and only if ad = bc . Let ( π, V ) be a smooth representation ofGSp(4 , F ) such that the center of GSp(4 , F ) acts trivially. If n is a non-negative integer, then V Kl ( n ) and V ( n ) are the subspaces of vectors fixed by the Klingen congruence subgroup Kl( p n ),and paramodular subgroup K( p n ), respectively; additionally, we let V Kl ( n, χ ) and V ( n, χ ) be thesubspaces of vectors v in V such that π ( k ) v = χ ( λ ( k )) v for k ∈ Kl( p n ) and k ∈ K( p n ), respectively.Also, we define η = (cid:36) − (cid:36) , τ = (cid:36) − (cid:36) , t n = − (cid:36) − n (cid:36) n . (1)Sometimes we will write η and τ for π ( η ) and π ( τ ), respectively. We define the level raisingoperators η : V ( n ) → V ( n + 2) and θ, θ (cid:48) : V ( n ) → V ( n + 1) as in [RS]. Let ( π, V ) be an irreducible,admissible representation of GSp(4 , F ) with trivial central character. If V ( n ) is non-zero for somenon-negative integer n then we say that π is paramodular and let N π be the smallest such integer.It is known that if π is paramodular, then V ( N π ) is one-dimensional; if W π is a non-zero element of V ( N π ) so that V ( N π ) = C · W π , then we refer to W π as a newform . The space V ( n ) for n ≥ N π isspanned by the vectors θ (cid:48) i θ j η k W π where i, j and k are non-negative integers with i + j +2 k = n − N π .It is known that if π is generic, then π is paramodular; in general, all paramodular, irreducible,admissible representations of GSp(4 , F ) with trivial central character have been classified. If π is ageneric, irreducible, admissible representation of GSp(4 , F ) with trivial central character then welet W ( π, ψ c ,c ) be the Whittaker model of π with respect to the character ψ c ,c of the unipotent JOHNSON-LEUNG AND ROBERTS radical of the Borel subgroup of GSp(4 , F ) with c , c ∈ o × . If W π is viewed as an element of theWhittaker model W ( π, ψ c ,c ) of π , then W π (1) (cid:54) = 0.We will refer to the following basic lemma. Lemma 2.1.
Let X be a complex vector space. Let f : o × → X be a locally constant function. Let b ∈ o and let t be a positive integer. We have (cid:90) o × f ( u (1 + bu − (cid:36) t )) du = (cid:90) o × f ( u ) du. Proof.
Let n be a positive integer such that f ( x + p n ) = f ( x ) for x ∈ o × . We have (cid:90) o × f ( u (1 + bu − (cid:36) t )) du = q − n (cid:88) u ∈ o × / (1+ p n ) f ( u (1 + bu − (cid:36) t )) . and similarly (cid:90) o × f ( u ) du = q − n (cid:88) u ∈ o × / (1+ p n ) f ( u ) . The lemma now follows from the fact that the function o × / (1 + p n ) → o × / (1 + p n ) defined by u (cid:55)→ u (1 + bu − (cid:36) t ) is a well-defined bijection. (cid:3) The following lemma about Gauss sums is well-known.
Lemma 2.2.
Let χ be a character of o × with conductor c ( χ ) , and let k be an integer. Define G ( χ, k ) = (cid:90) o × χ ( u ) ψ ( u(cid:36) k ) du. If χ is ramified, then G ( χ, k ) is non-zero if and only if k = − c ( χ ) . Twist in genus 1
Let ( π, V ) be a smooth representation of GL(2 , F ) for which the center of GL(2 , F ) acts trivially,let χ be a quadratic character of o × with conductor c ( χ ), and let n be a non-negative integer. For v ∈ V ( n ) we define T χ ( v ) = (cid:90) o × χ ( b ) π ( (cid:20) b(cid:36) − c ( χ ) (cid:21) ) v db. (2)If χ is unramified, then T χ ( v ) = (1 − q − ) v for v ∈ V ( n ). Thus, we will usually assume that χ is ramified. Assume further that π is generic, irreducible and admissible with Whittaker model W ( π, ψ ). For W ∈ W ( π, ψ ) we define the χ -twisted zeta integral of W as Z ( s, W, χ ) = (cid:90) F × W ( (cid:20) t (cid:21) ) | t | s − / χ ( t ) d × t. (3) Theorem 3.1.
Let ( π, V ) be a smooth representation of GL(2 , F ) for which the center of GL(2 , F ) acts trivially, let χ be a quadratic character of o × with conductor c ( χ ) > , and let n be a non-negative integer. Let N = max( n, c ( χ )) . If v ∈ V ( n ) , then π ( k ) T χ ( v ) = χ (det( k )) T χ ( v ) for k ∈ Γ ( p N ) . Moreover, assume that π is generic, irreducible and admissible with Whittaker model W ( π, ψ ) . Let W ∈ V ( n ) . The χ -twisted zeta integral of T χ ( W ) is Z ( s, T χ ( W ) , χ ) = (1 − q − ) G ( χ, − c ( χ )) W (1) . WISTING OF PARAMODULAR VECTORS 5
For n ≥ N π , the image of T χ : V ( n ) → V ( N, χ ) is spanned by the non-zero vector T χ ( β (cid:48) n − N π W π ) .Proof. The group Γ ( p N ) is generated by the elements contained in the sets (cid:20) o × o × (cid:21) , (cid:20) o (cid:21) , (cid:20) p N (cid:21) . It is easy to verify that π ( k ) T χ ( v ) = χ (det( k )) T χ ( v ) for generators k of the first two types. Let y ∈ p N . Noting that N − c ( χ ) ≥ c ( χ ) > N ≥ n , we have π ( (cid:20) y (cid:21) ) (cid:90) o × χ ( b ) π ( (cid:20) b(cid:36) − c ( χ ) (cid:21) ) v db = (cid:90) o × χ ( b ) π ( (cid:20) (1 + (cid:36) − c ( χ ) yb ) − b(cid:36) − c ( χ ) (cid:36) − c ( χ ) yb (cid:21) ) π ( (cid:20) (cid:36) − c ( χ ) yb ) − y (cid:21) ) v db = (cid:90) o × χ ( b ) π ( (cid:20) (cid:36) − c ( χ ) yb ) − b(cid:36) − c ( χ ) (cid:21) ) π ( (cid:20) (1 + (cid:36) − c ( χ ) yb ) − (cid:36) − c ( χ ) yb (cid:21) ) v db = (cid:90) o × χ ( b ) π ( (cid:20) (cid:36) − c ( χ ) yb ) − b(cid:36) − c ( χ ) (cid:21) ) v db = (cid:90) o × χ ((1 + (cid:36) − c ( χ ) yb ) − b ) π ( (cid:20) (cid:36) − c ( χ ) yb ) − b(cid:36) − c ( χ ) (cid:21) ) v db = (cid:90) o × χ ( b ) π ( (cid:20) b(cid:36) − c ( χ ) (cid:21) ) v db = T χ ( v ) . For the penultimate equality we applied Lemma 2.1. Assume now that π is generic, irreducible andadmissible as in the statement of the theorem. Then: Z ( s, T χ ( W ) , χ ) = (cid:90) F × T χ ( W )( (cid:20) t (cid:21) ) | t | s − / χ ( t ) d × t = (cid:90) F × (cid:90) o × χ ( b ) W ( (cid:20) t (cid:21)(cid:20) b(cid:36) − c ( χ ) (cid:21) ) | t | s − / χ ( t ) db d × t = (cid:90) F × ( (cid:90) o × χ ( b ) ψ ( tb(cid:36) − c ( χ ) ) db ) W ( (cid:20) t (cid:21) ) | t | s − / χ ( t ) d × t = (cid:90) o × ( (cid:90) o × χ ( b ) ψ ( tb(cid:36) − c ( χ ) ) db ) W ( (cid:20) t (cid:21) ) χ ( t ) d × t = (cid:90) o × χ ( t ) G ( χ, − c ( χ )) W ( (cid:20) t (cid:21) ) χ ( t ) d × t = (1 − q − ) G ( χ, − c ( χ )) W (1) . Here, we have used Lemma 2.2. (cid:3)
JOHNSON-LEUNG AND ROBERTS Twist in genus 2
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) actstrivially. Let χ be a quadratic character. For v ∈ V we define v χ = (cid:90) o × (cid:90) o × (cid:90) p − c ( χ ) χ ( ab ) π ( − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) τ c ( χ ) v dz da db. (4)Evidently, if χ is unramified, then v χ = (1 − q − ) v . Lemma 4.1.
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially. Let χ be a quadratic character with c ( χ ) > . Let n be a non-negative integer. Let v ∈ V Kl ( n ) . We have π ( k ) v χ = χ ( λ ( k )) v χ for the subgroup of k ∈ GSp(4 , F ) such that λ ( k ) ∈ o × and k ∈ o o o p − c ( χ ) o o op c ( χ ) o oo . Proof.
The subgroup in the statement of the lemma is generated by the elements of the formdiag( w w w, w w, w w, w ) for w, w , w ∈ o × , and elements of the subgroups p − c ( χ ) , o o , o o , o , p c ( χ ) . Using v ∈ V Kl ( n ), the definition of v χ and Lemma 2.1 one can verify that π ( k ) v χ = χ ( λ ( k )) v χ foreach type of generator k . As an illustration, let x ∈ o . Then π ( − x x ) v χ = (cid:90) o × (cid:90) o × (cid:90) p − c ( χ ) χ ( ab ) π ( − a(cid:36) − c ( χ ) − x b(cid:36) − c ( χ ) z − xb(cid:36) − c ( χ ) b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) + x ) τ c ( χ ) v dz da db = (cid:90) o × (cid:90) o × (cid:90) p − c ( χ ) χ ( ab ) π ( − a (1 + a − x(cid:36) c ( χ ) ) (cid:36) − c ( χ ) b(cid:36) − c ( χ ) z b(cid:36) − c ( χ ) a (1 + a − x(cid:36) c ( χ ) ) (cid:36) − c ( χ ) ) τ c ( χ ) v dz da db = v χ . This completes the proof. (cid:3)
Lemma 4.2.
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially. Let χ be a quadratic character and let n be a non-negative integer. Let v ∈ V Kl ( n ) WISTING OF PARAMODULAR VECTORS 7 and define v χ as in (4) . Then v χ is invariant under the subgroup GSp(4 , F ) ∩ p max( n + c ( χ ) , c ( χ )) p max( n + c ( χ ) , c ( χ )) p max( n +2 c ( χ ) , c ( χ )) p max( n + c ( χ ) , c ( χ )) p max( n + c ( χ ) , c ( χ )) . Proof.
This is clear if χ is unramified; assume that c ( χ ) >
0. Let a, b ∈ o × and c ∈ o , and set g = − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) c(cid:36) − c ( χ ) b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) τ c ( χ ) . Let L be an integer and y ∈ o . We have the following identities: y(cid:36) L − y(cid:36) L g = g ay(cid:36) L − c ( χ ) − a y(cid:36) L − c ( χ ) ( ab + c(cid:36) c ( χ ) ) y(cid:36) L − c ( χ ) acy(cid:36) L − c ( χ ) y(cid:36) L + c ( χ ) − ay(cid:36) L − c ( χ ) by(cid:36) L ( ab + c(cid:36) c ( χ ) ) y(cid:36) L − c ( χ ) ay(cid:36) L − c ( χ ) a y(cid:36) L − c ( χ ) − y(cid:36) L + c ( χ ) − ay(cid:36) L − c ( χ ) , y(cid:36) L g = g − cy(cid:36) L − c ( χ ) acy(cid:36) L − c ( χ ) − bcy(cid:36) L − c ( χ ) − c y(cid:36) L − c ( χ ) − by(cid:36) L − c ( χ ) aby(cid:36) L − c ( χ ) − b y(cid:36) L − c ( χ ) − bcy(cid:36) L − c ( χ ) − ay(cid:36) L − c ( χ ) a y(cid:36) L − c ( χ ) − aby(cid:36) L − c ( χ ) − acy(cid:36) L − c ( χ ) y(cid:36) L − ay(cid:36) L − c ( χ ) by(cid:36) L − c ( χ ) cy(cid:36) L − c ( χ ) . These identities prove that v χ is invariant under the groupGSp(4 , F ) ∩ p max( n + c ( χ ) , c ( χ )) p max( n +2 c ( χ ) , c ( χ )) p max( n + c ( χ ) , c ( χ )) . To prove the remaining invariance, set L = max( n + c ( χ ) , c ( χ )). A calculation shows that y(cid:36) L y(cid:36) L g = g − byc + ab y (cid:36) L − c ( χ ) ) u − (cid:36) L − c ( χ ) abyu − (cid:36) L − c ( χ ) − abyu − (cid:36) L − c ( χ ) k JOHNSON-LEUNG AND ROBERTS for some k ∈ Kl( p n ) with u = 1 + by(cid:36) L − c ( χ ) . Therefore, π ( y(cid:36) L y(cid:36) L ) v χ = q c ( χ ) (cid:90) o × (cid:90) o × (cid:90) o χ ( ab ) π ( − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) c(cid:36) − c ( χ ) b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) τ c ( χ ) π ( − byc + ab y (cid:36) L − c ( χ ) ) u − (cid:36) L − c ( χ ) abyu − (cid:36) L − c ( χ ) − abyu − (cid:36) L − c ( χ ) ) v dz da db = q c ( χ ) (cid:90) o × (cid:90) o × (cid:90) o χ ( ab ) π ( − a (1 − byu − (cid:36) L − c ( χ ) ) (cid:36) − c ( χ ) b(cid:36) − c ( χ ) c (cid:48) (cid:36) − c ( χ ) b(cid:36) − c ( χ ) a (1 − byu − (cid:36) L − c ( χ ) ) (cid:36) − c ( χ ) ) τ c ( χ ) v dz da db, where c (cid:48) = c (1 − byu − (cid:36) L − c ( χ ) ) − ab yu − (1 + byu − (cid:36) L − c ( χ ) ) (cid:36) L − c ( χ ) . Changing variables, weobtain v χ . (cid:3) Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) actstrivially, let χ be a quadratic character, and let n be a non-negative integer. For v ∈ V Kl ( n ) wedefine T Kl χ ( v ) = (cid:90) o π ( x ) v χ dx + (cid:90) p π ( − y ) v χ dy. (5)Here, v χ as in (4). If χ is unramified, then T Kl χ ( v ) = (1 + q − )(1 − q − ) v . Lemma 4.3.
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially, let χ be a quadratic character, and let n be a non-negative integer. Let v ∈ V Kl ( n ) .Then π ( k ) T Kl χ ( v ) = χ ( λ ( k )) T Kl χ ( v ) (6) for k in Kl( p N ) where N = max( n + 2 c ( χ ) , c ( χ )) . Moreover, π ( k ) T Kl χ ( v ) = T Kl χ ( v ) for k ∈ GSp(4 , F ) such that k ∈ p N − c ( χ ) p N − c ( χ ) p N − c ( χ ) p N − c ( χ ) . WISTING OF PARAMODULAR VECTORS 9
Proof.
The group Kl( p N ) is generated by its elements contained in the sets p N p N p N p N p N , o o o o o , o × o oo o o × , and there is a disjoint decompositionSL(2 , o ) = (cid:71) x ∈ o / p c ( χ ) (cid:20) x (cid:21) Γ ( p c ( χ ) ) (cid:116) (cid:71) y ∈ p / p c ( χ ) (cid:20) − (cid:21) (cid:20) y (cid:21) Γ ( p c ( χ ) ) . The lemma follows from these two facts, Lemma 4.1, and Lemma 4.2. (cid:3)
Let ( π, V ) be a generic, irreducible, admissible representation of GSp(4 , F ) with trivial centralcharacter with Whittaker model W ( π, ψ c ,c ); we take c , c ∈ o × . Let χ be a quadratic characterof F × . If W ∈ W ( π, ψ c ,c ) we define the zeta integral of W twisted by χ to be Z ( s, W, χ ) = (cid:90) F × (cid:90) F W ( t tz ) | t | s − / χ ( t ) dz d × t. (7)This is the same as the zeta integral of W in the twist χ ⊗ π of π by χ . See [RS]. Lemma 4.4.
Let ( π, V ) be a generic, irreducible, admissible representation of GSp(4 , F ) with trivialcentral character with Whittaker model W ( π, ψ c ,c ) ; we take c , c ∈ o × . Let χ be a quadraticcharacter of F × such that c ( χ ) > . Let n be a non-negative integer. Let W ∈ V Kl ( n ) , and define T Kl χ ( W ) as in (5) . We have Z ( s, T Kl χ ( W ) , χ ) = (1 − q − ) q c ( χ ) χ ( c ) G ( χ, − c ( χ )) W (1) . (8) In particular, if W is the newform of π , then T Kl χ ( W ) (cid:54) = 0 .Proof. To begin, we note that by Lemma 4.1.1 of [RS] we have Z ( s, T Kl χ ( W ) , χ ) = (cid:90) F × T Kl χ ( W )( t t ) | t | s − / χ ( t ) d × t. Therefore, the first part of Z ( s, T Kl χ ( W ) , χ ) is: (cid:90) F × (cid:90) o (cid:90) o × (cid:90) o × (cid:90) p − c ( χ ) W ( t t x − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) χ ( ab ) dz da db dx d × t = q c ( χ ) (cid:90) F × (cid:90) o (cid:90) o × (cid:90) o × W ( t t x − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) χ ( ab ) da db dx d × t = q c ( χ ) (cid:90) F × (cid:90) o (cid:90) o × (cid:90) o × ψ ( c ( − a(cid:36) − c ( χ ) − bx(cid:36) − c ( χ ) )) W ( t t x (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) χ ( ab ) da db dx d × t = q c ( χ ) ( (cid:90) o × χ ( a ) ψ ( − c a(cid:36) − c ( χ ) ) da ) (cid:90) o ( (cid:90) o × χ ( b ) ψ ( − c bx(cid:36) − c ( χ ) ) db ) (cid:90) F × W ( t t x (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) d × t dx. By Lemma 2.2 the integral in the b variable is zero unless v ( x ) = c ( χ ). Continuing,= q c ( χ ) ( (cid:90) o × χ ( a ) ψ ( − c a(cid:36) − c ( χ ) ) da ) (cid:90) o × ( (cid:90) o × χ ( b ) ψ ( − c bx(cid:36) − c ( χ ) ) db ) (cid:90) F × W ( t t x(cid:36) c ( χ ) (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) d × t dx = q c ( χ ) G ( χ, − c ( χ )) (cid:90) o × χ ( x ) (cid:90) F × W ( t t x − (cid:36) − c ( χ ) − x − (cid:36) − c ( χ ) − x(cid:36) c ( χ ) − x − (cid:36) − c ( χ ) (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) d × t dx WISTING OF PARAMODULAR VECTORS 11 = q c ( χ ) G ( χ, − c ( χ )) (cid:90) o × χ ( x ) (cid:90) F × ψ ( c tx − (cid:36) − c ( χ ) ) W ( t t ) | t | s − / χ ( t ) d × t dx = q c ( χ ) G ( χ, − c ( χ )) (cid:90) F × ( (cid:90) o × χ ( x ) ψ ( c tx(cid:36) − c ( χ ) ) dx ) W ( t t ) | t | s − / χ ( t ) d × t. Again, by Lemma 2.2 the integral in the x variable is zero unless v ( t ) = 0. Thus, our quantity is:= q c ( χ ) G ( χ, − c ( χ )) (cid:90) o × ( (cid:90) o × χ ( x ) ψ ( c tx(cid:36) − c ( χ ) ) dx ) W ( t t ) | t | s − / χ ( t ) d × t = q c ( χ ) G ( χ, − c ( χ )) (cid:90) o × χ ( t )( (cid:90) o × χ ( x ) ψ ( c x(cid:36) − c ( χ ) ) dx ) W (1) χ ( t ) d × t = q c ( χ ) χ ( c ) G ( χ, − c ( χ )) (cid:90) o × W (1) d × t = (1 − q − ) q c ( χ ) χ ( c ) G ( χ, − c ( χ )) W (1) . Finally, we prove that the second part of Z ( s, T Kl χ ( W ) , χ ) is zero: (cid:90) F × (cid:90) p (cid:90) o × (cid:90) o × (cid:90) p − c ( χ ) W ( t t − y − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) χ ( ab ) dz da db dy d × t = q c ( χ ) (cid:90) F × (cid:90) p (cid:90) o × (cid:90) o × ψ ( − c ty ) W ( t t − − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) χ ( ab ) da db dy d × t = q c ( χ ) (cid:90) F × (cid:90) p (cid:90) o × (cid:90) o × ψ ( − c ty ) ψ ( c b(cid:36) − c ( χ ) ) W ( t t − (cid:36) − c ( χ ) (cid:36) c ( χ ) ) | t | s − / χ ( t ) χ ( ab ) da db dy d × t = 0 . The last equality holds because χ is ramified by assumption. (cid:3) Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) actstrivially, let χ be a quadratic character, and let n be a non-negative integer. Define N = max( n +2 c ( χ ) , c ( χ )). For v ∈ V Kl ( n ) we define T χ ( v ) = q (cid:90) o π ( z(cid:36) − N ) T Kl χ ( v ) dz + π ( t N ) (cid:90) o π ( z(cid:36) − N +1 ) T Kl χ ( v ) dz. (9)Here, T Kl χ ( v ) is defined as in (5). Explicitly, q − c ( χ ) T χ ( v )= q (cid:90) o (cid:90) o (cid:90) o × (cid:90) o × χ ( ab ) π ( x − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z(cid:36) − N b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) τ c ( χ ) v da db dx dz (10)+ q (cid:90) o (cid:90) p (cid:90) o × (cid:90) o × χ ( ab ) π ( − y − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z(cid:36) − N b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) τ c ( χ ) v da db dy dz (11)+ (cid:90) o (cid:90) o (cid:90) o × (cid:90) o × χ ( ab ) π ( t N x − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z(cid:36) − N +1 b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) τ c ( χ ) v da db dx dz (12) WISTING OF PARAMODULAR VECTORS 13 + (cid:90) o (cid:90) p (cid:90) o × (cid:90) o × π ( t N − y − a(cid:36) − c ( χ ) b(cid:36) − c ( χ ) z(cid:36) − N +1 b(cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) τ c ( χ ) v da db dy dz. (13) Lemma 4.5.
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially and let χ be a quadratic character. Let n be a non-negative integer and define N =max( n + 2 c ( χ ) , c ( χ )) . Let v ∈ V Kl ( n ) . i) We have π ( k ) T χ ( v ) = χ ( λ ( k )) T χ ( v ) for k ∈ K( p N ) . ii) Assume that c ( χ ) > . If T χ ( v ) is invariant under under the elements r (cid:36) − r (cid:36) − r (cid:36) − − r (cid:36) − (14) for r , r ∈ o , then T χ ( v ) = 0 .Proof. i) Fix a Haar measure for the group GSp(4 , F ). By Lemma 3.3.1 of [RS] there is a disjointdecompositionK( p N ) = (cid:71) z ∈ o / p N z(cid:36) − N Kl( p N ) (cid:116) (cid:71) z ∈ o / p N − t N z(cid:36) − N +1 Kl( p N ) . Here, the second disjoint union is not present if N = 0. Therefore, by (6), (cid:90) K( p N ) χ ( λ ( k )) π ( k ) T Kl χ ( v ) dk = vol(Kl( p N )) (cid:88) z ∈ o / p N π ( z(cid:36) − N ) T Kl χ ( v )+ vol(Kl( p N )) (cid:88) z ∈ o / p N − π ( t N z(cid:36) − N +1 ) T Kl χ ( v )= vol(Kl( p N )) q N (cid:90) o π ( z(cid:36) − N ) T Kl χ ( v ) dz + vol(Kl( p N )) q N − (cid:90) o π ( t N z(cid:36) − N +1 ) T Kl χ ( v ) dz. This is a positive multiple of T χ ( v ), and thus implies the desired transformation rule. ii) Assume that T χ ( v ) is invariant under the elements in (14). Then T χ ( v ) = (cid:90) o (cid:90) o π ( r (cid:36) − r (cid:36) − r (cid:36) − − r (cid:36) − ) T χ ( v ) dr dr = q (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − r (cid:36) − ) T Kl χ ( v ) dr dr dz + π ( t N ) (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) N − − r (cid:36) N − − r (cid:36) N − − r (cid:36) N − z(cid:36) − N +1 ) T Kl χ ( v ) dz dr dr . (15)We claim that the first summand of (15) is zero. Now (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − r (cid:36) − ) T Kl χ ( v ) dr dr dz = (cid:90) o (cid:90) o (cid:90) o (cid:90) o π ( x r + xr ) (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − ( r + xr ) (cid:36) − ) v χ dr dr dx dz + (cid:90) p (cid:90) o (cid:90) o (cid:90) o π ( − y r y − r ) (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − ( r y − r ) (cid:36) − ) v χ dr dr dy dz = (cid:90) o (cid:90) o (cid:90) o (cid:90) o π ( x r (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − r (cid:36) − ) v χ dr dr dx dz + (cid:90) p (cid:90) o (cid:90) o (cid:90) o π ( − y r (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − r (cid:36) − ) v χ dr dr dy dz. WISTING OF PARAMODULAR VECTORS 15
Moreover, (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) − r (cid:36) − z(cid:36) − N r (cid:36) − − r (cid:36) − ) v χ dr dr dz = q c ( χ ) (cid:90) o × (cid:90) o × (cid:90) o (cid:90) o χ ( ab ) π ( − au (cid:36) − c ( χ ) bu (cid:36) − c ( χ ) z(cid:36) − N bu (cid:36) − c ( χ ) au (cid:36) − c ( χ ) ) v dr dr dz da db with u = 1 − r a − (cid:36) c ( χ ) − and u = 1 + b − r (cid:36) c ( χ ) − . Assume first c ( χ ) = 1. Then this integral is: (cid:90) o × (cid:90) o × (cid:90) o (cid:90) o χ ( ab ) π ( − ( a − r ) (cid:36) − c ( χ ) bu (cid:36) − c ( χ ) z(cid:36) − N bu (cid:36) − c ( χ ) a − r ) (cid:36) − c ( χ ) ) v dr dr dz da db (cid:90) o × (cid:90) o × (cid:90) o (cid:90) o χ ( ab ) π ( r (cid:36) − c ( χ ) bu (cid:36) − c ( χ ) z(cid:36) − N bu (cid:36) − c ( χ ) − r (cid:36) − c ( χ ) ) v dr dr dz da db = 0 . Assume that c ( χ ) >
1. Changing variables in r and then in a , this integral is: (cid:90) o × (cid:90) o × (cid:90) o (cid:90) o χ ( ab ) π ( − a (1 + r (cid:36) c ( χ ) − ) (cid:36) − c ( χ ) bu (cid:36) − c ( χ ) z(cid:36) − N bu (cid:36) − c ( χ ) a (1 + r (cid:36) c ( χ ) − ) (cid:36) − c ( χ ) ) v dr dr dz da db = (cid:90) o × (cid:90) o × (cid:90) o (cid:90) o χ ( a (1 + r (cid:36) c ( χ ) − ) b ) π ( − a(cid:36) − c ( χ ) bu (cid:36) − c ( χ ) z(cid:36) − N bu (cid:36) − c ( χ ) a(cid:36) − c ( χ ) ) v dr dr dz da db = 0 . This proves that the first summand of (15) is zero, as claimed. We now have: T χ ( v ) = π ( t N ) (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) N − − r (cid:36) N − − r (cid:36) N − − r (cid:36) N − z(cid:36) − N +1 ) T Kl χ ( v ) dz dr dr . Applying π ( t N ) − to both sides and using the invariance of T χ ( v ) under t N ∈ K( p N ) from i), wesee that T χ ( v ) is: (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) N − − r (cid:36) N − − r (cid:36) N − − r (cid:36) N − z(cid:36) − N +1 ) T Kl χ ( v ) dz dr dr = (cid:90) o (cid:90) o (cid:90) o π ( z(cid:36) − N +1 r z r zr (cid:36) N − r z − r (cid:36) N − − r z − r (cid:36) N − − r (cid:36) N − ) T Kl χ ( v ) dz dr dr = (cid:90) o π ( z(cid:36) − N +1 ) T Kl χ ( v ) dz where we have used the invariance properties of T Kl χ ( v ) from Lemma 4.3. By assumption, T χ ( v ) isinvariant under the elements of the form (14); integrating again over these elements we have T χ ( v ) = (cid:90) o (cid:90) o (cid:90) o π ( r (cid:36) − r (cid:36) − z(cid:36) − N +1 r (cid:36) − − r (cid:36) − ) T Kl χ ( v ) dr dr dz. This integral is zero by an argument analogous to the one above proving that the first term of (15)is zero. The proof is complete. (cid:3)
Theorem 4.6.
Let ( π, V ) be a smooth representation of GSp(4 , F ) for which the center of GSp(4 , F ) acts trivially, and let χ be a quadratic character of F × with conductor c ( χ ) > . Let n be a non-negative integer and define N = max( n + 2 c ( χ ) , c ( χ )) . If v ∈ V ( n ) , then T χ ( v ) ∈ V ( N, χ ) .Moreover, assume that π is generic, irreducible and admissible with Whittaker model W ( π, ψ c ,c ) where c , c ∈ o × . If W ∈ V ( n ) , then the χ -twisted zeta integral (7) of T χ ( W ) is Z ( s, T χ ( W ) , χ ) = ( q − q c ( χ ) χ ( c ) G ( χ, − c ( χ )) W (1) . (16) For n ≥ N π , the image of T χ : V ( n ) → V ( N, χ ) is spanned by the non-zero vector T χ ( θ (cid:48) n − N π W π ) ,where W π is a newform for π .Proof. The first assertion was proven in i) of Lemma 4.5. Assume now that π is generic andirreducible. We work in the Whittaker model W ( π, ψ c ,c ) with c , c ∈ o × . By Lemma 4.1.1 of[RS] we have Z ( s, T χ ( v ) , χ ) = (cid:90) F × T χ ( v )( t t ) | t | s − / χ ( t ) d × t. By the definition of T χ ( v ), this is WISTING OF PARAMODULAR VECTORS 17 q (cid:90) F × (cid:90) o T Kl χ ( v )( t t z(cid:36) − N ) | t | s − / χ ( t ) dz d × t + (cid:90) F × (cid:90) o T Kl χ ( v )( t t t N z(cid:36) − N +1 ) | t | s − / χ ( t ) dz d × t. We assert that the second summand is zero; it will suffice to prove that the integrand is zero. Let t ∈ F × and z ∈ o . Let x ∈ o . Then ψ ( c x(cid:36) − ) T Kl χ ( v )( t t t N z(cid:36) − N +1 )= T Kl χ ( v )( x(cid:36) − − x(cid:36) − t t t N z(cid:36) − N +1 )= T Kl χ ( v )( t t t N z(cid:36) − N +1 − x(cid:36) N − − x(cid:36) N − xz x z(cid:36) N − − xz )= T Kl χ ( v )( t t t N z(cid:36) − N +1 ) , where the last equality follows from the invariance properties of Lemma 4.3. Since ψ ( p − ) (cid:54) = 1, thisimplies that the integrand is zero. The first summand is q (cid:90) F × (cid:90) o T Kl χ ( v )( t t z(cid:36) − N ) | t | s − / χ ( t ) dz d × t = q (cid:90) F × (cid:90) o T Kl χ ( v )( t t ) | t | s − / χ ( t ) dz d × t = q (cid:90) F × T Kl χ ( v )( t t ) | t | s − / χ ( t ) d × t = qZ ( s, T Kl χ ( v ) , χ ) . The formula (16) follows now from (8). To prove the final assertion, we note first by Theorem 7.5.7of [RS] that the space V ( n ) is spanned by the vectors θ (cid:48) i θ j η k W π with i + j + 2 k = n − N π . Theformula (3.7) of [RS] implies that Z ( s, T χ ( θ (cid:48) n − N π W π ) , χ ) = ( q − q c ( χ ) χ ( c ) G ( χ, − c ( χ )) ( θ (cid:48) n − N π W π )(1)= ( q − q c ( χ )+ n − N π χ ( c ) G ( χ, − c ( χ )) W π (1) , and this is non-zero. To complete the proof, it will suffice to prove that T χ ( θ (cid:48) i θ j η k W π ) = 0 if j > k >
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