aa r X i v : . [ m a t h . N T ] N ov WALDSPURGER FORMULA OVER FUNCTION FIELDS
CHIH-YUN CHUANG AND FU-TSUN WEI
Abstract.
In this paper, we derive a function field version of the Waldspurger formulafor the central critical values of the Rankin-Selberg L -functions. This formula states thatthe central critical L -values in question can be expressed as the “ratio” of the global toricperiod integral to the product of the local toric period integrals. Consequently, this resultprovides a necessary and sufficient criterion for the non-vanishing of these central critical L -values, and supports the Gross-Prasad conjecture for SO(3) over function fields.
Introduction
In 1985, Waldspurger [15] established a fundamental formula for the central critical valueof the Rankin-Selberg L -function associated to an automorphic cuspidal representation of GL over a given number field F convolved with a Hecke character on the idele class groupof a quadratic field extension over F . This formula asserts that “global toric period integrals”can be written as the central critical L -value in question multiplying the product of “localtoric period integrals.” From this result, these critical L -values now have been studied exten-sively over number fields and lead to plenty of arithmetic consequences (cf. [2], [3], and [22]).The main purpose of this paper is to derive a function field analogue of Walspurger’s formula.Let k be a global function field with odd characteristic, and denote the adele ring of k by k A . Let D be a quaternion algebra over k , and K be a separable quadratic algebra over k with an embedding ι : K ֒ → D . We put D A and K A to be the adelization of D and K ,respectively. Let Π D be an infinite dimensional automorphic representation of D × A (cuspidal if D is the matrix algebra) with a unitary central character η . Given a unitary Hecke character χ : K × \ K × A → C × , suppose η · χ (cid:12)(cid:12) k × A = 1 . Let P D χ ∈ Hom K × A (Π D , χ − ) be the global toricperiod integral: P D χ ( f ) := Z K × k × A \ K × A f (cid:0) ι ( a ) (cid:1) χ ( a ) d × a, ∀ f ∈ Π D . The measure d × a chosen here is the Tamagawa measure (cf. Section 1.2). This then givesus a linear functional P D χ : Π D ⊗ e Π D → C (where e Π D is the contragredient representation of Π D ) defined by: P D χ ( f ⊗ ˜ f ) := P D χ ( f ) · P D χ − ( ˜ f ) , ∀ f ⊗ ˜ f ∈ Π D ⊗ e Π D . On the other hand, write Π D = ⊗ v Π D v , and e Π D = ⊗ v e Π D v . We may assume that theidentification between Π D (resp. e Π D ) and ⊗ v Π D v (resp. ⊗ e Π D v ) satisfies the following equality: h· , ·i D Pet = 2 L (1 , Π , Ad ) ζ k (2) · Y v h· , ·i D v : Π D × e Π D → C , Mathematics Subject Classification.
Key words and phrases.
Function field, Automorphic form on GL , Rankin-Selberg L -function.This work was supported by grants from Ministry of Science and Technology, Taiwan. where: • the pairing h· , ·i D Pet is induced from the Petersson inner product (with respect to theTamagawa measure, i.e. the total volume of D × k × A \D × A is , cf. Section 1.2). • for each place v of k , h· , ·i D v is the natural duality pairing between Π D v and e Π D v . • Π is the automorphic cuspidal representation of GL ( k A ) correspoding to Π D via theJacquet-Langlands correspondence. • L ( s, Π , Ad ) is the adjoint L -function of Π . • ζ k ( s ) is the Dedekind-Weil zeta function of k .Write χ = ⊗ v χ v . Then for each v , the local toric period integral P D χ,v : Π D v ⊗ e Π D v → C isgiven by: P D χ,v ( f v ⊗ ˜ f v ) := ∗ · Z K × v /k × v h Π D v (cid:0) ι ( a v ) (cid:1) f v , ˜ f v i v χ v ( a v ) d × a v . Here d × a v is the Tamagawa measure on K × v /k × v (chosen in Section 1.2), and ∗ is a productof “local L -factors” so that P D χ,v ( f v ⊗ ˜ f v ) = 1 when v is “good” (cf. Lemma 5.1). These localtoric period integrals induce another linear functional P D χ := ⊗ P D χ,v : Π D ⊗ e Π D → C . Wenow state the main theorem of this paper as follows (cf. Theorem 5.2): Theorem 0.1.
Under the above assumptions, we have P D χ = L ( 12 , Π × χ ) · P D χ , where L ( s, Π × χ ) is the Rankin-Selberg L -function associated to Π and χ . We remark that L ( s, Π × χ ) can be identified with L ( s, Π K ⊗ χ ) , the L -function of Π K twisted by χ , where Π K is Jacquet’s lifting of Π to GL ( K A ) (cf. [7, Theorem 20.6]).Let ς K be the quadratic Hecke character of K/k and put ς K,v := ς K (cid:12)(cid:12) k × v . From the workof Tunnell [13] and Waldspurger [15, Lemme 10], the local toric period integral P D χ,v is nottrivial if and only if ǫ v (Π × χ ) = η v ( − ς K,v ( − ǫ v ( D ) . ( ⋆ ) Here ǫ v (Π × χ ) is the local root number of L ( s, Π × χ ) at v and ǫ v ( D ) is the Hasse invariantof D at v . This leads us to the following consequence. Corollary 0.2.
Suppose Q v ǫ v (Π × χ ) = 1 . Let D be the unique (up to isomorphism)quaternion algebra over k so that the equality ( ⋆ ) holds for every place v of k . Then thenon-vanishing of L (1 / , Π × χ ) is equivalent to the existance of an automorphic form f ∈ Π D so that P D χ ( f ) = Z K × k × A \ K × A f (cid:0) ι ( a ) (cid:1) χ ( a ) d × a = 0 . In particular, via the isomorphism
PGL ∼ = SO(3) , Corollary 0.2 supports the Gross-Prasadconjecture for the SO(3) case over function fields (cf. [6]).The proof of Theorem 0.1 basically follows Waldspurger’s approach in [15] for the numberfield case. Suppose first that K is a quadratic field over k . Let ( V D , Q V D ) be the quadraticspace ( D , Nr D /k ) , where Nr D /k is the reduced norm from D to k . Given φ ∈ Π and a Schwartzfunction ϕ ∈ S ( V D ( k A )) , suppose φ and ϕ are both pure tensors. From the Rankin-Selberg ALDSPURGER FORMULA OVER FUNCTION FIELDS 3 method, we have (cf. Corollary 3.3 (2)) L (2 s, ς K ) · Z ( s ; φ, ϕ ) = L ( s, Π × χ ) · Y v Z ov ( s ; φ v , ϕ v ) , (0.1)where the zeta integral Z ( s ; φ, ϕ ) (resp. Z ov ( s ; φ v , ϕ v ) ) is defined in the beginning of Section 3.2(resp. Corollary 3.3 (2)). Applying the Siegel-Weil formula in Theorem 3.1 and the seesawidentity (cf. the diagram (4.2)), we may connect L (1 , ς K ) · Z (1 / φ, ϕ ) with a global toricperiod integral T ( φ, ϕ ) (cf. the equation (4.3) and Proposition 4.3). On the other hand, thelocal zeta integral Z ov (1 / φ v , ϕ v ) can be rewritten as a local toric period integral T v ( φ v , ϕ v ) (cf. Proposition 4.1). The global (resp. local) Shimizu correspondence in Theorem 2.2 (resp.Section 2.3.1) then enables us to connect T (resp. T v ) with P D χ (resp. P D χ,v ), which completesthe proof. Note that in our approach, we always take the original Schwartz functions (i.e.functions in S ( V ( k A )) , cf. Section 2), instead of using the “extended ones” (i.e. functions in S ( V ( k A ) × k × A ) as in [15, Section 3]. This simplifies the arguments.One ingredient of the above proof is to decompose the global Shimizu correspondence asthe tensor product of local ones (cf. Section 2.3.1 and Appendix A). To achieve this, we needto verify the Siegel-Weil formula for the dual pair ( f SL , O ( D o )) , where f SL is the metaplecticcover of SL , and D o consists of all the pure quaternions in D (cf. Appendix B).When K = k × k , the existance of the embedding ι : K ֒ → D forces that D = Mat . Wemay write χ = χ × χ where χ i are unitary Hecke characters on k × \ k × A . In this case we have L ( s, Π × χ ) = L ( s, Π ⊗ χ ) · L ( s, Π ⊗ χ ) . Note that the assumption η · χ (cid:12)(cid:12) k × A = 1 says that Π ⊗ χ = e Π ⊗ χ − . The global (resp. local)toric period integrals can then be easily identified with the product of the special values ofthe global (resp. local) zeta integrals of forms in Π ⊗ χ and e Π ⊗ χ − at s = 1 / . ThereforeTheorem 0.1 follows immediately (cf. Appendix C).Identifying e Π D with the space { ¯ f : f ∈ Π D } via the Petersson inner product on Π D , weput k f k D Pet := h f, ¯ f i D Pet (resp. k f v k D v := h f v , ¯ f v i D v ). For non-zero pure tensors φ = ⊗ v φ v ∈ Π and f = ⊗ v f v ∈ Π D , from Theorem 0.1 we obtain that | P D χ ( f ) | k f k D Pet = L (1 / , Π × χ ) k φ k Mat Pet · Y v α v ( φ v , f v ) , (0.2)where α v ( φ v , f v ):= (cid:18) L v (1 , Π , Ad ) ζ v (2) k φ v k Mat v (cid:19) · L v (1 , ς K ) L v (1 / , Π × χ ) Z K v /k × v h Π D v (cid:0) ι ( a v ) (cid:1) f v , ¯ f v i v k f v k D v χ v ( a v ) d × a v ! . Taking suitable φ and f , it is possible to calculate the local quantities α v ( φ v , f v ) in concreteterms. Therefore the equality (0.2) leads us to an explicit formula of L (1 / , Π × χ ) . This willbe studied in a subsequent paper.The content of this paper is given as follows. In Section 1, we first set up basic notationsused throughout this paper, and fix all the Haar measures in the paper to be the Tamagawameasures. In Section 2, we recall needed properties of theta series associated to quadraticfields and quaternion algebras, and state the Shimizu correspondence in the version usedhere. In Section 3, we apply the Rankin-Selberg method to show the equation (0.1). InSection 4, we first rewrite Z ov (1 / φ v , ϕ v ) in terms of the local toric period integral T v ( φ v , ϕ v ) CHIH-YUN CHUANG AND FU-TSUN WEI associated to φ v and ϕ v in Section 4.1. Applying the seesaw identity, the special value L (1 , ς K ) · Z (1 / φ, ϕ ) equals to the global toric period integral T ( φ, ϕ ) associated to φ and ϕ in Section 4.2. We thereby arrive at the main theorem in Section 5 by applying the globaland local Shimizu correspondence. In Appendix A, we recall the decomposition of the globalShimizu correspondence into the tensor product of local ones. In Appendix B, we verify theSiegel-Weil formula for the dual pair ( f SL , O ( D o )) , where f SL is the metaplectic cover of SL ,and D o consists of all the pure quaternions in a division quaternion algebra D . The case when K = k × k for Theorem 0.1 is proven in Appendix C.1. Prelimilaries
Basic settings.
Give a ring R , the multiplicative group of R is denoted by R × . By S ) for each set S , we mean the cardinality of S .Let k be a global function field with finite constant field F q . Throughout this paper, wealways assume q to be odd . For each place v of k , let k v be the completion of k at v , and O v be the valuation ring in k v . Choose a uniformizer ̟ v once and for all. Set F v := O v /̟ v O v ,the residue field at v , and put q v := F v ) . The valuation on k v is denoted by ord v , and wenormalize the absolute value | · | v on k v by | a v | v := q − ord v ( a v ) v for every a v ∈ k v .Let k A be the ring of adeles of k , i.e. k A = Q ′ v k v , the restricted direct product of k v withrespect to O v . The maximal compact subring of k A is denoted by O A . The group of ideles of k is k × A , with the maximal compact subgroup O × A . For a = ( a v ) v ∈ k × A , we put | a | A := Q v | a v | v .Finally, fix a non-trivial additive character ψ : k A → C × which is trivial on k . For eachplace v of k , put ψ v := ψ (cid:12)(cid:12) k v . Let δ v be the “conductor” of ψ v , i.e. ψ v is trivial on ̟ − δ v v O v but not trivial on ̟ − δ v − v O v . Then P v δ v · deg v = 2 g k − , where g k is the genus of k .1.2. Tamagawa measures.
For each place v of k , choose the self-dual Haar measure dx v on k v with respect to the fixed additive character ψ v , i.e. vol ( O v , dx v ) = q − δ v / v . The Haarmeasure dx = Q v dx v on k A is then self-dual with respect to ψ , and vol ( k \ k A , dx ) = 1 . Forthe multiplicative group k × v , we take the Haar measure d × x v := ζ v (1) · dx v | x v | v , where ζ v ( s ) = (1 − q − sv ) − is the local zeta function of k at v . Then vol ( O × v , d × x v ) = q − δ v / v .This gives us a Haar measure d × x = Q v d × x v on k × A .Given a separable quadratic algebra K over k , let T K/k and N K/k be the trace and normfrom K to k , respectively. Put K v := K ⊗ k k v . The Haar measures on K v and K × v are chosenas above for each place v of k (with respect to the character ψ v ◦ T K/k ). This induces aHaar measure d × h v on K × v /k × v , and one has vol ( O × K v /O × v , d × h v ) = q − (ord v ( d K )+ δ v ) / v , where d K ∈ Div( k ) is the discriminant divisor of K over k . Let K A := K ⊗ k k A . We then take theHaar measure on K × A /k × A to be d × h := Q v d × h v . Let ς K be the quadratic character of K/k ,i.e. ς K : k × \ k × A → C × is the character with the kernel precisely equal to k × · N K/k ( K × A ) .When K is a field, one has vol ( K × \ K × A /k × A , d × h ) = 2 · L (1 , ς K ) . ALDSPURGER FORMULA OVER FUNCTION FIELDS 5
By Hilbert’s theorem 90, we may identify K × /k × with K := { a ∈ K × | N K/k ( a ) = 1 } .Thus the chosen Haar measure d × h on K × A /k × A can be identified with a Haar measure d × h on K A . In particular, for each place v of k , we havevol ( O K v , d × h v ) = (ord v ( d K ) + 1) · q − (ord v ( d K )+ δ v ) / v . Given a quaternion algebra D over k , let Tr D /k and Nr D /k be the reduced trace and normfrom D to k , respectively. Put D v := D ⊗ k k v for each place v of k . The Haar measure db v on D v for each v is taken to be self-dual with respect to ψ v ◦ Tr D /k . For the multiplicativegroup D × v , we choose d × ˜ b v := ζ v (1) · db v | Nr D /k ( b v ) | v . Globally, put D A := D ⊗ k k A . We choose the Haar measure d × b on D × A satisfying that foreach maximal compact open subgroup K = Q v K v ⊂ D × A , one hasvol ( K , d × b ) := Y v vol ( K v , d × ˜ b v ) . Via the exact sequence → D → D × → k × → the chosen Haar measures d × b on D × A and d × x on k × A determine a Haar measure d × b on D A .Moreover, it is known that (cf. [21, Theorem 3.3.1])vol ( D × k × A \D × A , d × b ) = 2 and vol ( D \D A , d × b ) = 1 . Theta series
Weil representation.
Let ( V, Q V ) be a non-degenerate quadratic space over k witheven dimension (then dim k V ≤ ). Set h x, y i V := Q V ( x + y ) − Q V ( x ) − Q V ( y ) , ∀ x, y ∈ V, the bilinear form associated to Q V . Given an arbitrary k -algebra R , set V ( R ) := V ⊗ k R . Forour purpose, the (local) Weil representation ω Vv of (cid:0) SL × O( V ) (cid:1) ( k v ) on the Schwartz space S ( V ( k v )) is chosen with respect to ψ v for every place v of k . We denote by ω V := ⊗ v ω Vv the(global) Weil representation of (cid:0) SL × O( V ) (cid:1) ( k A ) on the Schwartz space S ( V ( k A )) .Let GO( V ) be the orthogonal similitude group of V over k . Put [GL × GO( V )] := { ( g, h ) ∈ GL × GO( V ) | det( g ) = ν ( h ) } . Here ν ( h ) is the factor of similitude for h ∈ GO( V ) . We extend ω V to a representation(still denoted by ω V ) of [GL × GO( V )]( k A ) on S ( V ( k A )) by the following: for every pair ( g, h ) ∈ [GL × GO( V )]( k A ) and ϕ ∈ S ( V ( k A )) , set (cid:0) ω V ( g, h ) ϕ (cid:1) ( x ) := | det( g ) | − A · (cid:0) ω V ( (cid:18) g ) − (cid:19) g ) ϕ (cid:1) ( h − x ) , ∀ x ∈ V ( k A ) . Given ( g, h ) ∈ [GL × GO( V )]( k A ) and ϕ ∈ S ( V ( k A )) , let θ V ( g, h ; ϕ ) := X x ∈ V ( k ) (cid:0) ω V ( g, h ) ϕ (cid:1) ( x ) . For every ϕ ∈ S ( V ( k A )) , the theta series θ V ( · , · ; ϕ ) is invariant by [GL × GO( V )]( k ) via leftmultiplications. CHIH-YUN CHUANG AND FU-TSUN WEI
Quadratic theta series.
Let K be a quadratic field extension of k . Given γ ∈ k × ,let ( V ( γ ) , Q ( γ ) ) := ( K, γ · N K/k ) , where N K/k is the norm form on
K/k . Then one has
GO( V ( γ ) ) ∼ = K × ⋊ h τ K i , where τ K ( x ) := ¯ x for every x ∈ K = V ( γ ) ( k ) . We may identify K := { h ∈ K | N K/k ( h ) = 1 } with the special orthogonal group SO( V ( γ ) ) .Let GL + K be the image of natural projection of [GL × GO( V ( γ ) )] into GL . Given a unitaryHecke character χ on K × \ K × A and ϕ ∈ S ( V ( γ ) ( k A )) , set θ ( γ ) χ ( g ; ϕ ) := Z K \ K A θ V ( γ ) ( g, rh g ; ϕ ) χ ( rh g ) dr, ∀ g ∈ GL + K ( k A ) . Here h g ∈ K × A is chosen so that N K/k ( h g ) = det( g ) . Then θ ( γ ) χ ( · ; ϕ ) is invariant under GL + K ( k ) by left multiplications, and has a central character equal to ς K · χ (cid:12)(cid:12) k × A , where ς K isthe quadratic Hecke character of K/k . When γ = 1 , we will denote by θ K ( · , · ; ϕ ) and θ Kχ ( · ; ϕ ) the quadratic theta series θ V (1) ( · , · ; ϕ ) and θ (1) χ ( · ; ϕ ) , respectively.2.2.1. Whittaker functions.
Given γ ∈ k × , the Whittaker function (with respect to ψ ) at-tached to θ ( γ ) χ ( · ; ϕ ) for ϕ ∈ S ( V γ ( k A )) is: W ( γ ) χ ( g ; ϕ ) := Z k \ k A θ ( γ ) χ (cid:18)(cid:18) n (cid:19) g ; ϕ (cid:19) ψ ( n ) dn. Then W ( γ ) χ (cid:18)(cid:18) n (cid:19) g ; ϕ (cid:19) = ψ ( n ) · W ( γ ) χ ( g ; ϕ ) , ∀ g ∈ GL + K ( k A ) and n ∈ k A . It is straightforward that:
Lemma 2.1.
Suppose ϕ = ⊗ v ϕ v ∈ S ( V γ ( k A )) is a pure tensor. Then W ( γ ) χ ( · ; ϕ ) is fac-torizable. More precisely, for g = ( g v ) v ∈ GL + K ( A k ) , choose h g = ( h g,v ) v ∈ K × A so that det( g ) = N K/k ( h g ) . One has W ( γ ) χ ( g ; ϕ ) = Q v W ( γ ) χ,v ( g v ; ϕ v ) , where W ( γ ) χ,v ( g v ; ϕ v ) := Z K v (cid:0) ω V ( γ ) v ( g v , r v h g,v ) ϕ v (cid:1) (1) · χ v ( r v h g,v ) dr v . Quaternionic theta series.
Let D be a quaternion algebra over k , and denote by Nr D /k (resp. Tr D /k ) the reduced norm (resp. trace) on D /k . Let ( V D , Q V D ) := ( D , Nr D /k ) .Then we have the following exact sequence: −→ k × −→ ( D × × D × ) ⋊ h τ D i −→ GO( V D ) −→ . Here: • k × embeds into D × × D × diagonally; • every pair ( b , b ) ∈ D × × D × is sent to [ b , b ] := ( x b xb − , x ∈ D ) ∈ GO( V D ); • τ D ( x ) := ¯ x = Tr D /k ( x ) − x for every x ∈ D .Let Π D be an infinite dimensional automorphic representation of D × A which is cuspidal if D = Mat . Suppose the central character of Π D is unitary. Let Π be the automorphic cuspidalrepresentation of GL ( k A ) corresponding to Π D via the Jacquet-Langlands correspondence.Given ϕ ∈ S ( V D ( k A )) and φ ∈ Π , for b , b ∈ D × A we set θ D ( b , b ; φ, ϕ ) := Z SL ( k ) \ SL ( k A ) φ (cid:0) g α ( b b − ) (cid:1) · θ V D (cid:0) g α ( b b − ) , [ b , b ]; ϕ (cid:1) dg . ALDSPURGER FORMULA OVER FUNCTION FIELDS 7
Here α ( b ) := (cid:18) D /k ( b ) (cid:19) for every b ∈ D × A , and dg is the Tamagawa measure on SL ( k A ) (cf. Section 1.2). It is clear that θ D ( · , · ; φ, ϕ ) is invariant by D × × D × via left multiplications.Put Θ D (Π) := (cid:8) θ D ( · , · ; φ, ϕ ) | φ ∈ Π , ϕ ∈ S ( V D ( k A )) (cid:9) . The Shimizu correspondence says (cf. [12, Theorem 1]):
Theorem 2.2.
Given an infinite dimensional automorphic representation Π D of D × A (cusp-idal if D = Mat ), suppose the central character of Π D is unitary. Then Θ D (Π) = { f ⊗ ¯ f : D × A × D × A → C | f , f ∈ Π D } C − span . (2.1) Here f ⊗ ¯ f ( b, b ′ ) := f ( b ) · ¯ f ( b ′ ) for every b, b ′ ∈ D × A . Consequently, let e Π D be the contra-gredient representation of Π D . Identifying e Π D with the space { ¯ f | f ∈ Π D } via the Peterssoninner product, the equality (2.1) induces an isomorphism Sh : Θ D (Π) ∼ = Π D ⊗ e Π D . Local Shimizu correspondence.
We may identify Π with ⊗ v Π v naturally via the Whit-taker model of Π (with respect to ψ ). Let v be a place of k . For φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) ,put θ D ,ov ( b v , b ′ v ; φ v , ϕ v ):= ζ v (2) L v (1 , Π , Ad ) · Z U( k v ) \ SL ( k v ) W φ v (cid:0) g v α ( b v b ′− v ) (cid:1) · (cid:0) ω D v ( g v α ( b v b ′− v ) , [ b v , b ′ v ]) ϕ v (cid:1) (1) dg v . Here W φ v is the Whittaker function of φ v (with respect to ψ v ), the map α is defined in theabove of Theorem 2.2, and U ⊂ SL is the standard unipotent subgroup. Observe that when v is “good” we have θ D ,ov ( b , b ′ v ; φ v , ϕ v ) = 1 (cf. Theorem A.3 (1)). Moreover, for pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ v φ v ∈ S ( V D ( k A )) we have (cf. Theorem A.3) Z D × k × A \D × A θ D ( bb , bb ; φ, ϕ ) d × b (2.2) = 2 L (1 , Π , Ad ) ζ k (2) · Y v θ D ,ov ( b ,v , b ,v ; φ v , ϕ v ) , ∀ b , b ∈ D × A . Put Θ D v (Π v ) := { θ D ,ov ( · , · ; φ v , ϕ v ) | φ v ∈ Π v , ϕ v ∈ S ( V D ( k v )) } . Then the above equality implies that (cf. Proposition A.4) Θ D v (Π v ) = { f v ⊗ ˜ f v : D × v × D × v → C | f v ∈ Π D v , ˜ f v ∈ e Π D v } C − span . Here f v ⊗ ˜ f v is viewed as a matrix coefficient: f v ⊗ ˜ f v ( b v , b ′ v ) := h Π D v ( b v ) f v , e Π D v ( b ′ v ) ˜ f v i D v , ∀ b v , b ′ v ∈ D × v , where h· , ·i D v : Π D v × e Π D v → C is the natural duality pairing. Consequently, we have anisomorphism Sh v : Θ D v (Π v ) ∼ = Π D v ⊗ e Π D v . Remark . Let h· , ·i D Pet : Π D × e Π D → C be the Petersson pairing. The equality (2.2),together with Sh and Sh v , provide us a way to indentify Π D (resp. e Π D ) with ⊗ v Π D v (resp. ⊗ v e Π D v ) so that for pure tensors f = ⊗ v f v ∈ Π D and ˜ f = ⊗ v ˜ f v ∈ e Π D , we have h f, ˜ f i D Pet = 2 L (1 , Π , Ad ) ζ k (2) · Y v h f v , ˜ f v i D v . CHIH-YUN CHUANG AND FU-TSUN WEI Zeta integrals and Rankin-Selberg method
Siegel Eisenstein series.
Let K be a quadratic field over k . Fix γ ∈ k × . Recall thatwe put ( V ( γ ) , Q ( γ ) ) = ( K, γ · N K/k ) . Given ϕ ∈ S ( V ( γ ) ( k A )) , the Siegel section associated to ϕ is defined by Φ ϕ ( g, s ) := | a | s A | b | s A · ς K ( b ) · (cid:0) ω V ( γ ) ( κ ) ϕ (cid:1) (0) for every g = (cid:18) a n b (cid:19) κ ∈ GL ( k A ) with a, b ∈ k × A , n ∈ k A , κ ∈ SL ( O A ) , and s ∈ C . Here ς K is the quadratic character of K/k . The Siegel Eisenstein series associated to ϕ is E ( g, s, ϕ ) := X γ ∈ B( k ) \ GL ( k ) Φ ϕ ( γg, s ) , ∀ g ∈ GL ( k A ) , which converges absolutely for Re( s ) > . It is known that E ( g, s, ϕ ) has meromorphic con-tinuation to the whole complex s -plane and satisfies a functional equation with the symmetrybetween s and − s . Note that E ( g, s, ϕ ) is always holomorphic at the central critical point s = 1 / , and the following formula holds (cf. [17, Theorem 0.1]): Theorem 3.1. (The Siegel-Weil formula)
Fix γ ∈ k × . Given ϕ ∈ S ( V ( γ ) ( k A )) , one has E ( g, , ϕ ) = 1 L (1 , ς K ) · θ ( γ ) K ( g, ϕ ) , ∀ g ∈ GL + K ( k A ) , where K is the principal character on K × A . Zeta integrals.
Let D be a quaternion algebra over k . Given a quadratic field extension K over k with an embedding K ֒ → D , we write D = K + Kj where j = γ ∈ k × and jb = ¯ bj for every b ∈ K . Set ( V D , Q V D ) := ( D , Nr D /k ) . Then ( V D , Q V D ) = ( V (1) , Q (1) ) ⊕ ( V ( − γ ) , Q ( − γ ) ) . Let Π be an automorphic cuspidal representation of GL ( k A ) with a unitary central char-acter denoted by η . Given a Hecke character χ : K × \ K × A → C × , suppose that χ is unitaryand η · χ (cid:12)(cid:12) k × A = 1 . For φ ∈ Π and ϕ ∈ S ( V D ( k A )) , we are interested in the following (global)zeta integral: writing ϕ = P i ϕ ,i ⊕ ϕ ,i with ϕ ,i ∈ S ( V (1) ( k A )) and ϕ ,i ∈ S ( V ( − γ ) ( k A )) , weset Z ( s ; φ, ϕ ) := X i Z Z( k A ) GL + K ( k ) \ GL + K ( k A ) φ ( g ) θ Kχ ( g, ϕ ,i ) E ( g, s ; ϕ ,i ) dg. Here Z is the center of GL , and dg is the Tamagawa measure on GL ( k A ) restricting to GL + K ( k A ) (cf. Section 1.2). This integral is a meromorphic function on the complex s -plane.Moreover, one asserts: Proposition 3.2.
Given pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ v ϕ v ∈ S ( V D ( k A )) , one has Z ( s ; φ, ϕ ) = Y v Z v ( s ; φ v , ϕ v ) , where Z v ( s ; φ v , ϕ v ) is equal to Z K × v Z SL ( O v ) W φ v (cid:18)(cid:18) N K/k ( h ) 00 1 (cid:19) κ v (cid:19) · (cid:16) ω D v ( κ v ) ϕ v (cid:17) (¯ h ) dκ v ! χ v ( h ) | N K/k ( h ) | s − v d × h ; and W φ v is the local Whittaker function associated to f v (with respect to ψ v ). ALDSPURGER FORMULA OVER FUNCTION FIELDS 9
Proof.
Without loss of generality, assume ϕ = ϕ ⊕ ϕ . Let B + K := B ∩ GL + K = Z · T + K · U ,where T := (cid:18) ∗
00 1 (cid:19) , T + K := T ∩ GL + K , and U := (cid:18) ∗ (cid:19) . Put GL + K ( O A ) := GL ( O A ) ∩ GL + K ( k A ) . From the Iwasawa decomposition GL + K ( k A ) = B + K ( k A ) · GL + K ( O A ) , we write the zeta integral Z ( s ; φ, ϕ ) as Z ( s ; φ, ϕ )= Z Z( O A ) \ GL + K ( O A ) Z T + K ( k A ) W φ ( tκ ) W Kχ ( tκ ; ϕ ) (cid:0) ω ( − γ ) ( κ ) ϕ (cid:1) (0) | t | s − A d × tdκ, where for every κ ∈ GL ( O A ) , we put κ := (cid:18) det( κ ) −
00 1 (cid:19) κ ∈ SL ( O A ) . Note that for eachplace v of k , we have the following exact sequence: / / {± } / / SL ( O v ) / / Z( O v ) \ GL ( O v ) det / / O × v ( O × v ) / / . Therefore when ϕ and f are pure tensors, one has Z ( s ; φ, ϕ ) = Y v Z ′ v ( s ; φ v , ϕ v ) , where Z ′ v ( s ; φ v , ϕ v ) := Z SL ( O v ) Z T + K ( k v ) W φ v ( t v κ v ) W Kχ,v ( tκ v ; ϕ ,v ) (cid:0) ω ( − γ ) v ( κ v ) ϕ ,v (cid:1) (0) | t | s − v d × t v dκ v . By Lemma 2.1, the local zeta integral Z ′ v ( s ; φ v , ϕ v ) becomes Z ′ v ( s ; φ v , ϕ v )= Z SL ( O v ) Z T + K ( k v ) W φ v ( t v κ v ) · Z K v (cid:0) ω Kv ( κ v ) ϕ ,v (cid:1) ( r v h t,v ) χ v ( r v h t,v ) dr v ! (cid:0) ω ( − γ ) v ( κ v ) ϕ ,v (cid:1) (0) | t | s − v d × t v dκ v = Z v ( s ; φ v , ϕ v ) . (cid:3) The following results are straightforward.
Corollary 3.3. (1)
Suppose v is “good”, i.e. the conductor of ψ v is trivial, Π v is an unramifiedprincipal series, φ v ∈ Π v is spherical with W φ v (1) = 1 , v is unramified in K , χ v is unramified, ord v ( γ ) = 0 , and ϕ = ϕ ⊕ ϕ with ϕ = ϕ = O Kv . We have Z v ( s ; φ v , ϕ v ) = L v ( s, Π × χ ) L v (2 s, ς K ) . (2) Given φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) , put Z ov ( s ; φ v , ϕ v ) := L v (2 s, ς K ) L v ( s, Π × χ ) · Z v ( s ; φ v , ϕ v ) . Then Z ov ( s ; φ v , ϕ v ) = 1 for all but finitely many v , and Z ( s ; φ, ϕ ) = L ( s, Π × χ ) L (2 s, ς K ) · Y v Z ov ( s ; φ v , ϕ v ) for every pure tensors φ ∈ Π and ϕ ∈ S ( V D ( k A )) . (3) The (local) zeta integral Z ov ( s ; φ v , ϕ v ) always converges at s = 1 / . Central critical values of zeta integrals
Let D , K , Π , η , and χ be as in the above section. For pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ ϕ v ∈ S ( V D ( k A )) , we shall express Z (1 / φ, ϕ ) (resp. Z v (1 / φ v , ϕ v ) ) in terms of global(resp. local) “toric period integrals” of the pair ( φ, ϕ ) (resp. ( φ v , ϕ v )).4.1. Local case.
We may rewrite Z v (1 / φ v , ϕ v ) for φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) as follows: Proposition 4.1.
Given φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) , we have Z v ( 12 ; φ v , ϕ v ) = L v (1 , Π , Ad ) ζ v (2) · Z K × v /k × v θ D ,ov ( h v , φ v , ϕ v ) χ v ( h v ) d × h v . Here θ D ,ov ( · , · ; φ v , ϕ v ) is defined in Section 2.3.1 .Proof.
Given h v ∈ K × v and g v ∈ SL ( k v ) , one has ω D v (cid:0) g v α ( h v ) , [ h v , (cid:1) ϕ v (1) = | N K/k ( h v ) | − v · ω D v (cid:0) α ( h v ) − g v α ( h v ) (cid:1) ( h − v ) . From the Iwasawa decomposition: SL ( k v ) = B ( k v ) · (cid:16) α ( h v ) SL ( O v ) α ( h v ) − (cid:17) , we may write dg v = | N K/k ( h v ) | v · d L b v · d R κ v . Thus L v (1 , Π , Ad ) ζ v (2) · θ D ,ov ( h v , φ v , ϕ v )= Z SL ( O v ) Z k × v W φ v (cid:18)(cid:18) a v a − v (cid:19) α ( h v ) κ v (cid:19) (cid:18) ω D v ( (cid:18) a v a − v (cid:19) κ v ) ϕ v (cid:19) ( h − v ) d × a v | a | v dκ v = Z k × v Z SL ( O v ) W φ v (cid:18)(cid:18) N K/k ( a v h v ) 00 1 (cid:19) κ v (cid:19) (cid:0) ω D v ( κ v ) ϕ v (cid:1) ( a v h v ) dκ v ! χ v ( a v ) d × a v Therefore the result follows immediately. (cid:3)
Let Π D = ⊗ v Π D v be, if exists, the automorphic representation of D × A corresponding to Π via the Jacquet-Langlands correspondence. For φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) , we may view θ D ,ov ( · , · ; φ v , ϕ v ) as a matrix coefficient of Π D v ⊗ e Π D v (cf. Proposition A.4). Define the localtoric period integral of the pair ( φ v , ϕ v ) by T v ( φ v , ϕ v ) := L v (1 , ς K ) L v (1 , Π , Ad ) L v ( , Π × χ ) ζ v (2) · Z K × v /k × v θ D ,ov ( h v , φ v , ϕ v ) χ v ( h v ) d × h v . (4.1)Then the above proposition says Z ov ( 12 ; φ v , ϕ v ) = T v ( φ v , ϕ v ) . ALDSPURGER FORMULA OVER FUNCTION FIELDS 11
Global case.
Put [GO( V (1) ) × GO( V ( − γ ) )] := { ( h , h ) ∈ GO( V (1) ) × GO( V ( − γ ) ) | ν ( h ) = ν ( h ) } , which is viewed as a subgroup of GO( V D ) . Note that GO( V ( a ) ) ∼ = K × ⋊ h τ K i for every a ∈ k × ,and we have the following exact sequence −→ k × −→ ( D × × D × ) ⋊ h τ D i −→ GO( V D ) −→ . Here k × embeds into D × × D × diagonally, and every pair ( b , b ) ∈ D × × D × is sent to ( x b xb − , x ∈ V = D ) ∈ GO( V D ) . Let [ K × × K × ] := { ( h , h ) ∈ K × × K × | N K/k ( h ) = N K/k ( h ) } = K × × K × ∩ [GO( V (1) ) × GO( V ( − γ ) )] . Define ι : [ K × × K × ] ֒ → D × × D × by sending ( h , h ) to ( h h ′ , h ′ ) ∈ ( D × × D × ) /k × , where h ′ ∈ K × such that h ′ /h ′ = h /h . Then the following diagram commutes: [ K × × K × ] (cid:31) (cid:127) ι / / (cid:127) _ (cid:15) (cid:15) D × × D × (cid:15) (cid:15) [GO( V (1) ) × GO( V ( − γ ) )] (cid:31) (cid:127) / / GO( V D ) . Suppose ϕ = ϕ ⊕ ϕ ∈ S ( V D ( k A )) , where ϕ ∈ S ( V (1) ( k A )) and ϕ ∈ S ( V ( − γ ) ( k A )) . InSection 2.1 we put θ Kχ ( g ; ϕ ) = Z K \ K A θ V (1) ( g, rh g ; ϕ ) χ ( rh g ) dr, ∀ g ∈ GL + K ( k A ) . The Siegel-Weil formula in Theorem 3.1 says E ( g,
12 ; ϕ ) = 1 L (1 , ς K ) · Z K \ K A θ V ( − γ ) ( g, rh g ; ϕ ) dr. Note that the following lemma is straightforward.
Lemma 4.2.
Given g ∈ GL + K ( k A ) and h , h ∈ K × A with det( g ) = N K/k ( h ) = N K/k ( h ) ,one has θ V (1) ( g, h ; ϕ ) · θ V ( γ ) ( g, h ; ϕ ) = θ V D (cid:0) g, [ h h ′ , h ′ ]; ϕ ⊕ ϕ (cid:1) . Here h ′ ∈ K × A is chosen so that h ′ /h ′ = h /h , and [ h h ′ , h ′ ] ∈ ( D × A × D × A ) /k × A is consideredas an element in GO( V D )( k A ) . Applying the “seesaw identity” (cf. [10]) with respect to the following diagram GL + K (cid:127) _ diagonal (cid:15) (cid:15) ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ [GO( V (1) ) × GO( V ( − γ ) )] (cid:127) _ (cid:15) (cid:15) [GL + K × GL + K ] ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ GO( V D ) + K , (4.2)where [GL + K × GL + K ] (resp. [GO( V (1) ) × GO( V ( − γ ) )] ) is the subgroup of GL + K × GL + K (resp. GO( V (1) ) × GO( V ( − γ ) ) ) consisting of all pairs ( g , g ) where g and g have the same deter-minants (resp. the factor of similitudes), we then obtain that: Proposition 4.3.
Given φ ∈ Π , and ϕ ∈ S ( V D ( k A )) , we have Z ( 12 ; φ, ϕ ) = 1 L (1 , ς K ) · Z K × k × A \ K × A Z K × k × A \ K × A θ D ( h , h ; φ, ϕ ) · χ ( h h − ) dh dh . Proof.
The above discussion says that Z ( 12 ; φ, ϕ ) = 1 L (1 , ς K ) · Z Z( k A ) GL + K ( k ) \ GL + K ( k A ) φ ( g ) · Z K \ K A Z K \ K A θ V (1) ( g, r h g ; ϕ ) θ V ( − γ ) ( g, r h g ; ϕ ) χ ( r h g ) dr dr ! dg = 1 L (1 , ς K ) · Z SL ( k ) \ SL ( k A ) φ (cid:0) g α (N K/k ( h )) (cid:1) · Z K × k × A \ K × A Z K × k × A \ K × A θ D ( g α (N K/k ( h )) , [ hh ′ , h ′ ]; ϕ ) χ ( h ) dhdh ′ ! dg = 1 L (1 , ς K ) · Z K × k × A \ K × A Z K × k × A \ K × A θ D ( h , h ; φ, ϕ ) · χ ( h h − ) dh dh . (cid:3) For each pair ( φ, ϕ ) with φ ∈ Π and ϕ ∈ S ( V D ( k A )) , define the global toric period integralby: T ( φ, ϕ ) := Z K × k × A \ K × A Z K × k × A \ K × A θ D ( h , h ; φ, ϕ ) · χ ( h h − ) dh dh . (4.3)Then by Corollary 3.3, Proposition 4.1 and 4.3, we arrive at: Corollary 4.4.
Given pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ v ϕ v ∈ S ( V D ( k A )) , we have T ( φ, ϕ ) = L ( 12 , Π × χ ) · Y v T v ( φ v , ϕ v ) . Waldspurger formula
Let Π be an automorphic cuspidal representation of GL ( k A ) with a unitary central charac-ter η . For a quaternion algebra D over k , let Π D be, if exists, the automorphic representationof D × A corresponding to Π via the Jacquet-Langlands correspondence. Let K be a separablequadratic algebra over k together with an embedding ι : K ֒ → D . Given a unitary Heckecharacter χ : K × \ K × A → C × , suppose η · χ (cid:12)(cid:12) k × A = 1 . For each f ∈ Π D , put P D χ ( f ) := Z K × k × A \ K × A f (cid:0) ι ( h ) (cid:1) χ ( h ) d × h. This induces a linear functional P D χ : Π D ⊗ e Π D → C defined by P D χ ( f ⊗ ˜ f ) := P D χ ( f ) · P D χ − ( ˜ f ) , ∀ f ⊗ ˜ f ∈ Π D ⊗ e Π D . On the other hand, write Π D = ⊗ v Π D v and e Π D = ⊗ v e Π D v . For each place v of k , let h· , ·i v : Π D v × e Π D v → C be the natural duality pairing. We assume that the identificationbetween Π D (resp. e Π D ) and ⊗ v Π D v (resp. ⊗ v e Π D v ) satisfies: h· , ·i Pet = 2 L (1 , Π , Ad ) ζ k (2) · Y v h· , ·i v , (5.1) ALDSPURGER FORMULA OVER FUNCTION FIELDS 13 where h· , ·i Pet : Π D × e Π D → C is the pairing induced from the Petersson inner product on Π D . The local toric period integral P D χ,v : Π D v ⊗ e Π D v → C is defined by: P D χ,v ( f v ⊗ ˜ f v ) := L v (1 , ς K ) L v (1 , Π , Ad ) L v (1 / , Π × χ ) ζ v (2) · Z K × v /k × v h Π D v (cid:0) ι ( h v ) (cid:1) f v , ˜ f v i v · χ v ( h v ) d × h v . Lemma 5.1.
Suppose v is “good,” i.e. the additive character ψ v has trivial conductor, thequaternion algebra D splits at v , the local representation Π D v = Π v is an unramified principalseries, the place v is unramified in K , the character χ v is unramified. Take f v ∈ Π D v and ˜ f v ∈ e Π D v to be spherical and invariant by ι ( O K v ) with h f v , ˜ f v i v = 1 . Then P D χ,v ( f v ⊗ ˜ f v ) = 1 . Proof.
Suppose v is inert in K . Then the choices of f v and ˜ f v satisfy P D χ,v ( f v ⊗ ˜ f v ) = L v (1 , ς K ) L v (1 , Π , Ad ) L v (1 / , Π × χ ) ζ v (2) . It is straightforward that the right hand side of the above equality equals to under theabove assumptions on v .Suppose v splits in K , i.e. K v = k v × k v . Write χ v = χ v, × χ v, on k × v × k × v . Then L v ( 12 , Π × χ ) = L v ( 12 , Π v ⊗ χ v, ) · L v ( 12 , Π v ⊗ χ v, ) = L v ( 12 , Π v ⊗ χ v, ) · L v ( 12 , e Π v ⊗ χ − v, ) . The last equality follows from the assumption η v · χ v (cid:12)(cid:12) k × v = 1 , where η v is the central characterof Π v . The pairing h· , ·i v can be realized by h f v , ˜ f v i v := ζ v (2) ζ v (1) L v (1 , Π , Ad ) · Z k v W f v (cid:18) a v
00 1 (cid:19) W ′ ˜ f v (cid:18) a v
00 1 (cid:19) d × a v , ∀ f v ∈ Π D v , ˜ f v ∈ Π D v , where W f v (resp. W ′ ˜ f v ) is the Whittaker function of f v (resp. ˜ f v ) with respect to ψ v (resp. ψ v ). We may assume the embedding ι : K v → D v satisfies ι ( a v , a ′ v ) = (cid:18) a v a ′ v (cid:19) ∈ Mat ( k v ) = D v , ∀ ( a v , a ′ v ) ∈ k v × k v . Then P D χ,v ( f v ⊗ ˜ f v ) = (cid:18) L v (1 / , Π v ⊗ χ v, ) Z k × v W f v (cid:18) a v
00 1 (cid:19) χ v, ( a v ) d × a v (cid:19) · L v (1 / , e Π v ⊗ χ − v, ) Z k × v W ′ ˜ f v (cid:18) a v
00 1 (cid:19) χ − v, ( a v ) d × a v ! . Therefore when f v and ˜ f v are spherical and invariant by ι ( O K v ) with h f v , ˜ f v i v = 1 , we get P D χ,v ( f v ⊗ ˜ f v ) = 1 . (cid:3) Set P D χ := ⊗ v P D χ,v : Π D ⊗ e Π D → C . We finally arrive at: Theorem 5.2.
The linear functionals P D χ and P D χ on Π D ⊗ e Π D satisfy P D χ = L ( 12 , Π × χ ) · P D χ . Proof.
The case when K = k × k is proven in Appendix C. Suppose K is a quadratic fieldover k . Take pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ v ϕ v ∈ S ( V D ( k A )) . Applying the globaland local Shimizu correspondence (cf. Theorem 2.2 and Section 2.3.1), Corollary 4.4 implies P D χ (cid:0) Sh ( θ D ( · , · ; φ, ϕ )) (cid:1) = T ( φ, ϕ )= L ( 12 , Π × χ ) · Y v T v ( φ v , ϕ v )= L ( 12 , Π × χ ) · Y v P D χ,v (cid:0) Sh v ( θ D ,o ( · , · ; φ v , ϕ v )) (cid:1) = L ( 12 , Π × χ ) · P D χ (cid:0) θ D ( · , · ; φ, ϕ ) (cid:1) . Therefore the result holds. (cid:3)
Non-vanishing criterion.
For each place v of k , we have: Lemma 5.3. (cf. [13])
The space
Hom K × v (Π D v , χ − v ) is at most one dimensional. Moreover, Hom K × v (Π D v , χ − v ) = 0 if and only if ǫ v (Π v × χ v ) = χ v ( − ς K,v ( − · ǫ v ( D ) . (5.2) Here ǫ v (Π v × χ v ) is the local root number of L v ( s, Π × χ ) , and ǫ v ( D ) is the Hasse invariantof D at v . It is clear that P D χ,v lies in Hom K × v (Π D v , χ − v ) ⊗ Hom K × v ( e Π D v , χ v ) . Moreover, followingWaldspurger [15, Lemme 10] one gets Lemma 5.4. P D χ,v is a generator of the C -vector space Hom K × v (Π D v , χ − v ) ⊗ Hom K × v ( e Π D v , χ v ) . Consequently, P D χ generates the space Hom K × A (Π D , χ − ) ⊗ Hom K × A ( e Π D , χ ) , in which P D χ lies. Therefore Theorem 5.2 implies: Corollary 5.5.
Let Π be an automorphic cuspidal representation of GL ( k A ) with a unitarycentral character η . Given a separable quadratic algebra K over k and a unitary Heckecharacter χ : K × \ K × A → C × with η · χ (cid:12)(cid:12) k × A = 1 , assume Q v ǫ v (Π v × χ v ) = 1 . Let D be thequaternion algebra over k satisfying (5.2) for every place v of k , and Π D be the automorphicrepresentation of D × A corresponding to Π via the Jacquet-Langlands correspondence. Choosean embedding ι : K ֒ → D . Then the non-vanishing of the central critical value L (1 / , Π × χ ) is equivalent to the existence of f ∈ Π D so that P D χ ( f ) = Z K × k × A \ K × A f (cid:0) ι ( h ) (cid:1) χ ( h ) d × h = 0 . Appendix A. Local Shimizu correspondences
Recall the Shimizu correspondence stated in Theorem 2.2:
Theorem A.1.
Given an infinite dimensional automorphic representation Π D of D × A whichis cuspidal if D = Mat , suppose the central character of Π D is unitary. Then Θ D (Π) = { f ⊗ ¯ f : D × A × D × A → C | f , f ∈ Π D } C − span . Here f ⊗ ¯ f is viewed as the function (cid:0) ( b, b ′ ) f ( b ) · ¯ f ( b ′ ) (cid:1) . Consequently, let e Π D be thecontragredient representation of Π D . Identifying e Π D with the space { ¯ f | f ∈ Π D } via thePetersson inner product, the equality (2.1) induces an isomorphism Sh : Θ D (Π) ∼ = Π D ⊗ e Π D . ALDSPURGER FORMULA OVER FUNCTION FIELDS 15
Recall that for φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) , we define θ D ,ov ( b v , b ′ v ; φ v , ϕ v ) for b v , b ′ v ∈ D × v in Section 2.3.1 by θ D ,ov ( b v , b ′ v ; φ v , ϕ v )= ζ v (2) L v (1 , Π , Ad) · Z U( k v ) \ SL ( k v ) W φ v (cid:0) g v α ( b v b ′− v ) (cid:1) · (cid:0) ω D v ( g v α ( b v b ′− v ) , [ b v , b ′ v ]) ϕ v (cid:1) (1) dg v . Lemma A.2.
Suppose v is “good,” i.e. ψ v has trivial conductor, the representation Π v isan unramified principle series, the vector φ v ∈ Π v is spherical with W φ v (cid:18) (cid:19) = 1 , thequaternion algebra D v = Mat ( k v ) , and the Schwartz function ϕ v = Mat ( O v ) . One has θ D ,ov ( b v , b ′ v ; φ v , ϕ v ) = 1 , ∀ b v , b ′ v ∈ GL ( O v ) . Proof.
From the Iwasawa decomposition SL ( k v ) = B ( k v ) · SL ( O v ) , the above assumptionsimply that for b v , b ′ v ∈ GL ( O v ) , we have θ D ,ov ( b v , b ′ v ; φ v , ϕ v ) = ζ v (2) L v (1 , Π , Ad) · Z k × v W φ v (cid:18) a v a − v (cid:19) O v ( a v ) d × a v = 1 (cid:3) The aim of this section is to show:
Theorem A.3.
Given pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ v ϕ v ∈ S ( V D ( k A )) , we have Z D × k × A \D × A θ D ( bb , bb ; φ, ϕ ) d × b = 2 L (1 , Π , Ad ) ζ k (2) · Y v θ D ,ov ( b ,v , b ,v ; φ v , ϕ v ) , ∀ b , b ∈ D × A . The proof of the above theorem is given in Section A.1 when D = Mat , and in Section A.2when D is division.Via the Petersson pairing h· , ·i D Pet : Π D × e Π D , the representation Π D ⊗ e Π D is isomorphic tothe space of the matrix coefficients of Π D ⊗ e Π D : f ⊗ ˜ f ←→ m f ⊗ ˜ f ∀ f ⊗ ˜ f ∈ Π D × e Π D , where m f ⊗ ˜ f ( b, b ′ ) := h Π D ( b ) f, e Π D ( b ′ ) ˜ f i Pet , ∀ b, b ′ ∈ A ×D . On the other hand, for each place v of k , we may also identify Π D v ⊗ e Π D v with the space ofmatrix coefficients, i.e. for f v ∈ Π D v and ˜ f v ∈ e Π D v , the matrix coefficient m f v ⊗ ˜ f v associated to m f v ⊗ ˜ f v is defined by m f v ⊗ ˜ f v ( b v , b ′ v ) := h Π D v ( b v ) f v , e Π D v ( b ′ v ) ˜ f v i D v , ∀ b v , b ′ v ∈ D × v . Here h· , ·i D v : Π D v × e Π D v → C is the natural duality pairing. Put Θ D v (Π v ) := { θ D ,ov ( · , · ; φ v , ϕ v ) | φ v ∈ Π v , ϕ v ∈ S ( V D ( k v )) } . Proposition A.4.
We have the following equality: Θ D v (Π v ) = { m f v ⊗ ˜ f v | f v ∈ Π D v , ˜ f v ∈ e Π D v } C − span . This induces an isomorphism Sh v : Θ D v (Π v ) ∼ = Π D v ⊗ e Π D v . Proof.
Pick φ o = ⊗ v φ ov ∈ Π and ϕ o = ⊗ v ϕ ov ∈ S ( V D ( k A )) so that C ( φ o , ϕ o ) := Z D × k × A \D × A θ D ( b, b ; φ o , ϕ o ) db = 0 . Then for each place v of k , the space of matrix coefficients of Π D v can be generated by m v ( φ v , ϕ v ) for φ v ∈ Π v and ϕ v ∈ S ( V D ( k v )) , where m v ( φ v , ϕ v ) is defined by: m v ( φ v , ϕ v )( b v , b ′ v ) := Z D × k × A \D × A θ D ( bb v , bb ′ v ; φ, ϕ ) db, ∀ b v , b ′ v ∈ D × v , where φ = φ v ⊗ v = v φ ov ∈ Π and ϕ = ϕ v ⊗ v = v ϕ ov ∈ S ( V D ( k A )) . By Theorem A.3 we mayassume that the chosen φ o and ϕ o satisfy C ( φ o , ϕ o ) = θ D ,ov (1 , φ ov , ϕ ov ) . Then m v ( φ v , ϕ v )( b v , b ′ v ) = R D × k × A \D × A θ D ( bb v , bb ′ v ; φ, ϕ ) dbC ( φ o , ϕ o ) · C ( φ o , ϕ o )= θ D ,ov ( b v , b ′ v ; φ v , ϕ v ) θ D ,ov (1 , φ ov , ϕ ov ) · C ( φ o , ϕ o ) (by Theorem A.3) = θ D ,ov ( b v , b ′ v ; φ v , ϕ v ) . (cid:3) A.1.
Proof of Theorem A.3 when D = Mat . Given φ ∈ Π and ϕ ∈ S ( V D ( k A )) , considerthe Whittaker function associated to φ and ϕ : W φ,ϕ ( b , b ):= Z k \ k A Z k \ k A θ D (cid:18)(cid:18) u (cid:19) b , (cid:18) u (cid:19) b ; φ, ϕ (cid:19) ψ ( u − u ) du du , ∀ b , b ∈ GL ( k A ) . Then:
Proposition A.5.
When φ and ϕ are both pure tensors, we have W φ,ϕ = Y v W φ,ϕ,v . Here for b , b ∈ GL ( k v ) , let W φ,ϕ,v ( b , b ) := Z U( k v ) \ SL ( k v ) W φ v ( g α ( b b − )) · (cid:16) ω D v ( g α ( b b − ) , [ b , b ]) ϕ v (cid:17) ∼ (cid:18) (cid:19) dg , and ( ϕ v ) ∼ (cid:18) a bc d (cid:19) := Z k v ϕ v (cid:18) a b ′ c d (cid:19) ψ v ( bb ′ ) db ′ for ϕ v ∈ S ( V D ( k v )) .Proof. Let V := (cid:26)(cid:18) ∗ ∗ (cid:19)(cid:27) ⊂ V D , V := (cid:26)(cid:18) ∗∗ (cid:19)(cid:27) ⊂ V D , and Q i := Q D (cid:12)(cid:12) V i for i = 1 , .Then ( V D , Q D ) = ( V , Q ) ⊕ ( V , Q ) . For ϕ ∈ S ( V ( k A )) and g ∈ SL ( k A ) , it is observedthat (cid:0) ω V ( g ) ϕ (cid:1) ∼ (cid:18) bc (cid:19) = ϕ ∼ (cid:18) b ′ c ′ (cid:19) , where ( c ′ , b ′ ) = ( c, b ) · g . Thus for ϕ ∈ S ( V D ( k A )) , by Poisson summation formula we maywrite θ V D ( g α ( b b − ) , [ b , b ]; ϕ ) = θ V D ( g α ( b b − ) , [ b , b ]; ϕ ) + θ V D ( g α ( b b − ) , [ b , b ]; ϕ ) , ALDSPURGER FORMULA OVER FUNCTION FIELDS 17 where θ V D ( g α ( b b − ) , [ b , b ]; ϕ ) := X γ ∈ U( k ) \ SL ( k ) X a,d ∈ k (cid:0) ω D ( γg α ( b b − ) , [ b , b ]) ϕ (cid:1) ∼ (cid:18) a d (cid:19) , and θ V D ( g α ( b b − ) , [ b , b ]; ϕ ) := X a,d ∈ k (cid:0) ω D ( g α ( b b − ) , [ b , b ]) ϕ (cid:1) ∼ (cid:18) a d (cid:19) . Since for u , u ∈ k A one has (cid:16) ω D (1 , (cid:20)(cid:18) u (cid:19) , (cid:18) u (cid:19)(cid:21) ) ϕ (cid:17) ∼ (cid:18) a bc d (cid:19) = ψ (cid:0) b ( − au + du + cu u ) (cid:1) · ϕ ∼ (cid:18) a bc d (cid:19) , we get Z k \ k A Z k \ k A θ V D (cid:18) g α ( b b − ) , (cid:20)(cid:18) u (cid:19) b , (cid:18) u (cid:19) b (cid:21) ; ϕ (cid:19) ψ ( u − u ) du du = X γ ∈ U( k ) \ SL ( k ) (cid:0) ω D ( γg α ( b b − ) , [ b , b ]) ϕ (cid:1) ∼ (cid:18) (cid:19) . Therefore W φ,ϕ ( b , b ) = Z U( k ) \ SL ( k A ) φ ( g α ( b b − )) · (cid:0) ω D ( γg α ( b b − ) , [ b , b ]) ϕ (cid:1) ∼ (cid:18) (cid:19) dg = Z U( k A ) \ SL ( k A ) W φ ( g α ( b b − )) · (cid:0) ω D ( γg α ( b b − ) , [ b , b ]) ϕ (cid:1) ∼ (cid:18) (cid:19) dg = Y v W φ,ϕ,v ( b ,v , b ,v ) . (cid:3) The space consisting of all the W f,ϕ,v is actually the Whittaker model of Π v ⊗ e Π v . More-over, the following straightforward lemma connects the local Whittaker function W φ,ϕ,v with θ D ,ov ( · , · ; φ v , ϕ v ) : Lemma A.6.
For b , b ∈ GL ( k v ) , one gets ζ v (2) L v (1 , Π , Ad ) · Z k × v W f,ϕ,v (cid:18)(cid:18) a v
00 1 (cid:19) b , (cid:18) a v
00 1 (cid:19) b (cid:19) d × a v = ζ v (1) · θ D ,ov ( b , b ; φ v , ϕ v ) . Recall that for pure tensors f , f ∈ Π D = Π , we have h f , f i Mat Pet = 2 · L (1 , Π , Ad ) ζ k (2) · h f ,v , ¯ f ,v i Mat v , where h f ,v , ¯ f ,v i Mat v := ζ v (2) ζ v (1) L v (1 , Π , Ad ) · Z k × v W f ,v (cid:18) a v
00 1 (cid:19) W f ,v (cid:18) a v
00 1 (cid:19) d × a v , and h f ,v , ¯ f ,v i v = 1 when v is “good.” Therefore by Proposition A.5 and Lemma A.6, theShimizu correspondence in Theorem 2.2 implies that: Proposition A.7.
Theorem A.3 holds when D = Mat . A.2.
Proof of Theorem A.3 when D is division. Given g ∈ SL ( k A ) , we set I D ( g , s, ϕ ) := X γ ∈ B ( k ) \ SL ( k ) | a ( γg ) | s − A · X x ∈ k (cid:0) ω D ( γg ) ϕ (cid:1) ( x ) , ∀ ϕ ∈ S ( V D ( k A )) . Here a ( g ) = a ∈ k × A is chosen so that g can be written as g = (cid:18) a ∗ a − (cid:19) κ with κ ∈ SL ( O A ) . This series converges absolutely when
Re( s ) > / , and has meromorphic continuation to thewhole complex s -plane. Given pure tensors φ = ⊗ v φ v ∈ Π and ϕ = ⊗ v ϕ v ∈ S ( V D ( k A )) , onehas J D ( s ; φ, ϕ ) := Z SL ( k ) \ SL ( k A ) φ ( g ) I D ( g , s, ϕ ) dg = Z U( k ) \ SL ( k A ) φ ( g ) | a ( g ) | s − A · (cid:0) ω D ( g ) ϕ (cid:1) (1) dg = Y v J D v ( s ; φ v , ϕ v ) , where J D v ( s ; φ v , ϕ v ) := Z U( k v ) \ SL ( k v ) W φ v ( g ) (cid:0) ω D v ( g ) ϕ v (cid:1) (1) | a ( g ) | s − v dg . It is clear that:
Lemma A.8.
The local integral J D v ( s ; φ v , ϕ v ) always converges when Re( s ) ≥ . Moreover,when v is “good,” one has J D v ( s ; φ v , ϕ v ) = L v ( s, Π , Ad ) ζ v (2 s ) . In particular, we obtain that J D ( s ; φ, ϕ ) = L ( s, Π , Ad ) ζ k (2 s ) · Y v J D ,ov ( s ; φ v , ϕ v ) , where J D ,ov ( s ; φ v , ϕ v ) := ζ v (2 s ) L v ( s, Π , Ad ) · J D v ( s ; φ v , ϕ v ) . For b , b ∈ D × v , one has θ D ,ov ( b , b ; φ v , ϕ v ) = J D ,ov (1; φ ′ v , ϕ ′ v ) , where φ ′ v := Π v ( α ( b b − )) φ v and ϕ ′ v := ω D v ( α ( b b − ) , [ b , b ]) ϕ v . Thus we obtain that: Proposition A.9.
Theorem A.3 holds when D is division.Proof. From the equality (B.1) in
Remark
B.2, we get that for b , b ∈ D × A , Z D × k × A \D × A θ D ( bb , bb ; φ, ϕ ) db = 2 J D (1; φ ′ , ϕ ′ )= 2 L (1 , Π , Ad ) ζ k (2) · Y v J D ,o (1; φ ′ v , ϕ ′ v )= 2 L (1 , Π , Ad ) ζ k (2) · Y v θ D ,ov ( b ,v , b ,v ; φ v , ϕ v ) . (cid:3) ALDSPURGER FORMULA OVER FUNCTION FIELDS 19
Note that the equality (B.1) follows from a “metaplectic type” Siegel-Weil formula, whichis verified in the next section.
Appendix B. Siegel-Weil formula for the metaplectic Eisenstein series
B.1.
Metaplectic groups.
For each place v , the Kubota -cocycle σ ′ v is defined by (cf. [9,Section 3]): σ ′ v ( g , g ) := (cid:18) x ( g g ) x ( g ) , x ( g g ) x ( g ) (cid:19) v , ∀ g , g ∈ SL ( k v ) . Here x (cid:18) a bc d (cid:19) := ( c, if c = 0 , d, if c = 0 ;and ( · , · ) v is the Hilbert quadratic symbol at v . Define a map s v : SL ( k v ) → {± } by setting s v (cid:18) a bc d (cid:19) := ( ( c, d ) v , if ord v ( c ) is odd and d = 0 , , otherwise.Let σ v be the -cocycle defined by σ v ( g , g ) := σ ′ v ( g , g ) s v ( g ) s v ( g ) s v ( g g ) − , ∀ g , g ∈ SL ( k v ) . It is known that (cf. [4, Section 2.3]) σ v ( κ , κ ) = 1 ∀ κ , κ ∈ SL ( O v ) . Hence σ v induces acentral extension f SL ( k v ) of SL ( k v ) by {± } which splits on the subgroup SL ( O v ) . Moreprecisely, the extension f SL ( k v ) is identified with SL ( k v ) × {± } (as sets) with the followinggroup law: ( g , ξ ) · ( g , ξ ) = (cid:0) g g , ξ ξ σ v ( g , g ) (cid:1) . Globally, we define a -cocycle σ on SL ( k A ) by setting σ := ⊗ v σ v , and let f SL ( k A ) be thecorresponding central extension of SL ( k A ) by {± } . The section SL ( k A ) −→ f SL ( k A ) κ ( κ, becomes a group homomorphism when restricting to SL ( O A ) , which embeds SL ( O A ) into f SL ( k A ) as a subgroup. Moreover, for every γ ∈ SL ( k ) , the value s ( γ ) := Q v s v ( γ ) iswell-defined, and the embedding SL ( k ) −→ f SL ( k A ) γ (cid:0) γ, s ( γ ) (cid:1) preserves the group law. Thus we may view SL ( k ) as a discrete subgroup of f SL ( k A ) .B.2. Weil representation.
Let D be a division quaternion algebra over k . We first write D as V ⊕ V , where V = k and V := { b ∈ D : Tr D /k ( b ) = 0 } . Put Q V i := Nr D /k | V i .Then the quadratic space ( V i , Q V i ) is anisotropic with dimension i , and SO ( V ) ∼ = D × /k × .Let ω V i = ⊗ v ω V i v be the Weil representation of the metaplectic group g SL ( k A ) × O( V i ) on theSchwartz space S ( V i ( k A )) , where for each place v , ω V i v is defined as follows (cf. [4, Section (1) ω V i v ( h ) φ ( x ) := φ ( h − x ) , h ∈ O( V i );(2) ω V i v (1 , ξ ) φ ( x ) := ξ · φ ( x ) , ξ ∈ {± } ;(3) ω V i v (cid:18)(cid:18) u (cid:19) , (cid:19) φ ( x ) = ψ v ( uQ V i ( x )) φ ( x ) , u ∈ k v ;(4) ω V i v (cid:18)(cid:18) a v a − v (cid:19) , (cid:19) φ ( x ) = | a v | i v ( a v , a v ) v ε V i v ( a v ) ε V i v (1) · φ ( a v x ) , a v ∈ k × v ; (5) ω V i v (cid:18)(cid:18) − (cid:19) , (cid:19) φ ( x ) = ε V i v (1) · b φ ( x ) . Here: • ε V i v ( a v ) := Z L v ψ v ( a v Q V i ( x )) d a v x, ∀ a v ∈ k × v , where L v is a sufficiently large O v -lattice in V i ( k v ) , and the Haar measure d a v x isself-dual with respect to the pairing ( x, y ) ψ v ( a v · Tr D /k ( x ¯ y )) , ∀ x, y ∈ V i ( k v ); • b φ ( x ) is the Fourier transform of φ : b φ ( x ) := Z V i ( k v ) φ ( y ) ψ v (Tr D /k ( x ¯ y )) dy. For a = ( a v ) v ∈ k × A , we put ε V i ( a ) := Q v ε V i v ( a v ) .B.3. Siegel-Eisenstein series.
Recall that B denotes the standard Borel subgroup of SL ,and let e B be the preimage of B in f SL ( k A ) .For each φ ∈ S ( V ( k A )) , the Siegel section associated to φ is defined by: Φ φ (˜ g, s ) = ξ · ε V ( a ) ε V (1) · | a | s + k A · (cid:0) ω V ( κ ) φ (cid:1) (0) , for ˜ g ∈ f SL ( k A ) , where a ∈ k × A and κ ∈ SL ( O A ) so that ˜ g = (cid:18)(cid:18) a ∗ a − (cid:19) , ξ (cid:19) · κ . The Siegel-Eisensteinseries associated to ϕ is then defined by E (˜ g, s, φ ) := X γ ∈ B ( k ) \ SL ( k ) Φ φ ( γ ˜ g, s ) , which converges absolutely for Re( s ) > / and has meromorphic continuation to the wholecomplex s -plane. In this section, we shall verify that: Theorem B.1.
Given φ ∈ S ( V ( k A )) , we have E (˜ g, , φ ) = 12 · I (˜ g, φ ) , ∀ ˜ g ∈ f SL ( k A ) , where I (˜ g, φ ) := Z D × k × A / D × A (cid:16) X x ∈ V ( k ) ω V (˜ g, b ) φ ( x ) (cid:17) db. ALDSPURGER FORMULA OVER FUNCTION FIELDS 21
Remark
B.2 . Write ( V D , Q V D ) = ( V , Q V ) ⊕ ( V , Q V ) . Given ϕ ∈ S ( V D ( k A )) , suppose ϕ = φ ⊕ φ where φ ∈ S ( V ( k A )) and φ ∈ S ( V ( k A )) . Then I D ( g, s, ϕ ) = θ k (( g, , φ ) · E (( g, , s, φ ) , ∀ g ∈ SL ( k A ) , where θ k (( g, , φ ) := P x ∈ k (cid:0) ω V ( g ) φ (cid:1) ( x ) . On the other hand, θ k (( g, , φ ) · I (( g, , φ ) = Z D × k × A / D × A θ V D ( g, [ b, b ]; ϕ ) db. Therefore I D ( g, , ϕ ) = 12 Z D × k × A / D × A θ V D ( g, [ b, b ]; ϕ ) db, ∀ g ∈ SL ( k A ) and ϕ ∈ S ( V D ( k A )) . In particular, for φ ∈ Π we have J D (1; φ, ϕ ) = Z SL ( k ) \ SL ( k A ) φ ( g ) I D ( g, , ϕ ) dg = 12 · Z D × k × A \D × A θ D ( b, b ; φ, ϕ ) db. (B.1) Proof of
Theorem B.1 . The proof of the above theorem basically follows the approach in [16]for the even dimensional case. Here we recall the strategy as follows: For each β ∈ k , the β -th Fourier coefficient of a given metaplectic form f is: f ∗ β (˜ g ) := Z k \ k A f (cid:18)(cid:16) (cid:18) u (cid:19) , (cid:17) ˜ g (cid:19) ψ ( − βu ) du. Given φ ∈ S ( V ( k A )) , we verify that:(1) The equality holds for the “constant terms,” i.e. E ∗ (˜ g, , φ ) = (cid:0) ω V (˜ g ) φ (cid:1) (0) = 12 · I , ∗ (˜ g, φ ) . In particular, this says that E (˜ g, , φ ) − / · I (˜ g, φ ) is a cusp form on f SL ( k A ) .(2) It is straightforward that E ( · , s, φ ) is orthogonal to all the cuspidal metaplectic formson f SL ( k A ) with respect to the Petersson inner product.(3) The theta integral I ( · , φ ) is orthogonal to all the cuspidal metaplectic forms on f SL ( k A ) . Indeed, adapting the proof of [16, Theorem A.4], there exists a constant C so that I , ∗ β (˜ g, φ ) = C · E ∗ β (˜ g, , φ ) , ∀ β = 0 . In particular, the function I ′ ( · , φ ) := I ( · , φ ) − C · E ( · , , φ ) satisfies I ′ (cid:18)(cid:16) (cid:18) u (cid:19) , (cid:17) ˜ g, φ (cid:19) = I ′ (˜ g, φ ) , ∀ u ∈ k A . Thus I ′ ( · , φ ) is orthogonal to all the cuspidal metaplectic forms, and so is I ( · , φ ) by (2) .Since a cusp form orthogonal to itself must be zero, we get E (˜ g, , φ ) = 1 / · I (˜ g, φ ) for every ˜ g ∈ f SL ( k A ) . (cid:3) Appendix C. The case when K = k × k When K = k × k , we have L ( s, ς K ) = ζ k ( s ) . Moreover, the existance of an embedding ι : K ֒ → D implies that D = Mat ( k ) , and Π D = Π . We may write the given Hecke character χ as χ × χ , where χ i is a unitary Hecke character on k × \ k × A for i = 1 , . The assumption η · χ (cid:12)(cid:12) k × A = 1 says that Π ⊗ χ = e Π ⊗ χ − . Thus we have L ( s, Π × χ ) = L ( s, Π ⊗ χ ) · L ( s, Π ⊗ χ ) .Without lose of generality, suppose ι ( a, b ) = (cid:18) a b (cid:19) for every a, b ∈ k . Then for f ∈ Π and ˜ f ∈ e Π , P D χ ( f ⊗ ˜ f ) = Z ( 12 ; f, χ ) · Z ( 12 ; ˜ f , χ − ) , where Z ( s ; f, χ ) := Z k × \ k × A f (cid:18) y
00 1 (cid:19) χ ( y ) | y | s − / A d × y and Z ( s ; ˜ f , χ − ) := Z k × \ k × A ˜ f (cid:18) y
00 1 (cid:19) χ − ( y ) | y | s − / A d × y are entire functions on the complex s -plane. Note that for a pure tensor f = ⊗ v f v ∈ Π , onehas (cf. [1, the equality (5.31) in Chapter 3 and Proposition 3.5.3]) Z ( s ; f, χ ) = L ( s, Π ⊗ χ ) · Y v Z ov ( s ; f v , χ ,v ) , (C.1)where for each place v of k , put Z ov ( s ; f v , χ ,v ) := 1 L v ( s, Π ⊗ χ ) · Z k × v W f v (cid:18) y v
00 1 (cid:19) χ ,v ( y v ) | y v | s − / v d × y v . Here W f v is the Whittaker function associated to f v with respect to the chosen ψ v . Similarly,for a pure tensor ˜ f = ⊗ ˜ f v ∈ e Π we have Z ( s ; ˜ f , χ − ) = L ( s, e Π ⊗ χ − ) · Y v Z ov ( s ; ˜ f v , χ − ,v ) , (C.2)where Z ov ( s ; ˜ f v , χ − ,v ) = 1 L v ( s, e Π ⊗ χ − ) · Z k × v W ′ ˜ f v (cid:18) y v
00 1 (cid:19) χ − ,v ( y v ) | y v | s − / v d × y v . Here we take W ′ ˜ f v to be the Whittaker function associated to ˜ f v with respect to ψ v . Note thatthe validity of Ramanujan bounds for Π and e Π implies that Z ov ( s ; f v , χ ,v ) and Z ov ( s ; ˜ f v , χ − ,v ) both converge absolutely at s = 1 / .Recall that we may choose h· , ·i Mat v : Π v × e Π v → C by: h f v , ˜ f v i Mat v := ζ v (2) ζ v (1) L v (1 , Π , Ad ) · Z k × v W f v (cid:18) y
00 1 (cid:19) W ′ ˜ f v (cid:18) y
00 1 (cid:19) d × y. Indeed, by the Rankin-Selberg method these local pairings satisfy: h· , ·i Mat Pet = 2 L (1 , Π , Ad ) ζ k (2) · Y v h· , ·i Mat v . Therefore we get P D χ,v ( f v ⊗ ˜ f v ) = Z ov ( 12 ; f v , χ ,v ) · Z ov ( 12 ; ˜ f v , χ − ,v ) . From the equation (C.1) and (C.2), we arrive at:
ALDSPURGER FORMULA OVER FUNCTION FIELDS 23
Theorem C.1.
Theorem 0.1 holds for the case when K = k × k .Acknowledgments. The authors are grateful to Jing Yu for his steady encouragements. Thiswork was completed while the second author was visiting Institute for Mathematical Sciencesat National University of Singapore. He would like to thank Professor Wee Teck Gan for theinvitation, and the institute for kind hospitality and wonderful working conditions.
References [1] Bump, D.,
Automorphic forms and representations,
Cambridge studies in advanced mathematics 55,(1996).[2] Chida, M. & Hsieh, M.-L.,
Special values of anticyclotomic L-functions for modular forms,
J. reineangew. Math. (2016) (DOI:10.1515/crelle-2015-0072)[3] Chida, M. & Hsieh, M.-L.,
On the anticyclotomic Iwasawa main conjecture for modular forms,
Compo-sitio Mathematica 151 no. 5 (2015) 863-897.[4] Gelbart, S. S.,
Weil’s representation and the spectrum of the metaplectic group,
Lecture Notes in Math-ematics 530, Springer 1976.[5] Gross, B. H.,
Heights and the special values of L -series, CMS Conference Proceedings, H. Kisilevsky, J.Labute, Eds., 7 (1987) 116-187.[6] Ichino, A. & Ikeda, T,
On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture,
Geom. Funct. Anal., to appear.[7] Jacquet, H.,
Automorphic forms on
GL(2) part II,
Lecture Notes in Mathematics 278, Springer 1972.[8] Jacquet, H. & Piatetski-Shapiro, I.I. & Shalika, J.,
Rankin-Selberg convolutions,
American Journal ofMathematics 105 (1983) 367-464.[9] Kubota, T.,
On automorphic functions and the reciprocity law in a number field,
Lectures in Math. 21,Kyoto University 1969.[10] Kudla S.,
Seesaw dual reductive pairs, in Automorphic forms of several variables, Taniguchi Symposium(Katata 1983) (Birkhäuser, Basel 1984), 244-267.[11] Papikian, M.,
On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves,
Journal of Number Theory 115 (2005) 249-283.[12] Shimizu, H.,
Theta series and automorphic forms on GL , J. Math. Soc. Japan Vol. 24 No. 4 (1972)638-683.[13] Tunnell, J.,
Local ǫ -factors and characters of GL(2), Amer. J. Math. 105 no. 6 (1983) 1277-1307.[14] Waldspurger, J-L.,
Sur les coefficients de Fourier des formes modulaires de poids demi-entier,
J. Math.pures et appl. 60 (1981) 375-484.[15] Waldspurger, J-L.,
Sur les valeurs de certaines fonctions L automorphices en leur centre de symétrie, Compositio Math., 54 (2)(1985) 173-242.[16] Wei, F.-T.,
On metaplectic forms over function fields,
Mathemetische Annalen Volume 355 Issue 1 (2013)235-258 (DOI 10.1007/s00208-012-0785-1).[17] Wei, F.-T.,
On the Siegel-Weil formula over function fields,
The Asian Journal of Mathematics Volume19 Number 3 (2015) 487-526.[18] Wei, F.-T. & Yu, J.,
Theta series and function field analogue of Gross formula,
Documenta Math. 16(2011) 723-765.[19] Weil, A.,
Sur certains groupes d’opérateurs unitaires,
Acta math., 111 (1964) 143-211.[20] Weil, A.,
Dirichlet Series and Automorphic Forms,
Lecture notes in mathematics 189, Springer 1971.[21] Weil, A.,
Adeles and algebraic groups,
Progress in Mathematics vol. 23, 1982.[22] Yuan X. & Zhang, S.-W. & Zhang, W.,
The Gross-Zagier Formula on Shimura Curves,
Annals ofMathematics Studies, Princeton University Press, 2012
Department of Mathematics, National Taiwan University, Taiwan
E-mail address : [email protected] Department of Mathematics, National Central University, Taiwan
E-mail address ::