Rankin-Selberg method for Jacobi forms of integral weight and of half-integral weight on symplectic groups
aa r X i v : . [ m a t h . N T ] A ug RANKIN-SELBERG METHOD FOR JACOBI FORMS OF INTEGRALWEIGHT AND OF HALF-INTEGRAL WEIGHT ON SYMPLECTICGROUPS.
SHUICHI HAYASHIDA
Abstract.
In this article we show analytic properties of certain Rankin-Selberg typeDirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integralweight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of sum-mation of these Dirichlet series are like the ones of the Koecher-Maass series. Themeromorphic continuations and functional equations of these Dirichlet series are ob-tained. Moreover, an identity between the Petersson norms of Jacobi forms with respectto linear isomorphism between Jacobi forms of integral weight and half-integral weightis also obtained. Introduction ∞ X m =1 h f m , g m i m s , where f m and g m are m -th Fourier-Jacobi coefficients of two Siegel cusp forms of degreetwo F and G , respectively, and where h f m , g m i is the Petersson inner product of Jacobicusp forms f m and g m . They showed a meromorphic continuation and the functionalequation of this Dirichlet series. This result has been generalized to Siegel cusp formsof arbitral degree n by Yamazaki [Ya 90]. It means that F and G can be replaced bySiegel cusp forms of arbitral degree n and f m and g m are Fourier-Jacobi coefficients of F and G of not only integer index, but also matrix index m . If m runs over matrices ofa fixed size, the range of summation and the denominators of Yamazaki Dirichlet seriesare the same to the ones of the Koecher-Maass Dirichlet series.In this paper we start with two Jacobi cusp forms φ M and ψ M instead of two Siegelcusp forms F and G . Here φ M and ψ M are Jacobi cusp forms of index M and where M is a half-integral symmetric matrix. The Fourier-Jacobi coefficients φ N and ψ N of Date : August 27, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Siegel modular forms, Jacobi forms, Dirichlet series, Half-integral weight. φ M and ψ M , respectively, are also Jacobi cusp forms. We will show a meromorphiccontinuation and the functional equation of similar Dirichlet series as above.The importance of this result is that one can apply it to obtain also similar resultfor Jacobi cusp forms (which includes Siegel cusp forms) of half-integral weight in ageneralized plus space. This result is a generalization of the result for the Rankin-Selbergtype Dirichlet series of Siegel cusp forms of half-integral weight obtained by Katsuradaand Kawamura in [K-K 15, Corollary to Proposition 3.1]. It means that in [K-K 15] theytreated Fourier coefficients of Siegel cusp forms of half-integral weight to construct theRankin-Selberg type Dirichlet series. In this paper we shall treat Jacobi cusp forms ofhalf-integral weight of arbitral degree instead of Siegel cusp forms of half-integral weightand we also shall treat
Fourier-Jacobi coefficients instead of Fourier coefficients.1.2. To be more precise, let L ∗ n be the set of all half-integral symmetric matrices of size n and L + n be the subset of all positive-definite matrices in L ∗ n .We fix a matrix M ∈ L + r . We set L + t,r ( M ) := (cid:26) N = (cid:18) N R t R M (cid:19) ∈ L + t + r | N ∈ L + t , R ∈ Z ( t,r ) (cid:27) and we put B t,r ( Z ) := (cid:26) γ = (cid:18) γ γ r (cid:19) ∈ GL ( t + r, Z ) | γ ∈ GL t ( Z ) , γ ∈ Z ( t,r ) (cid:27) . The group B t,r ( Z ) acts on L + t,r ( M ) by γ · N := N [ t γ ] for γ ∈ B t,r ( Z ) and N ∈ L + t,r ( M ).Here we put X [ Y ] := t Y XY for matrices X and Y of suitable sizes.We set ǫ ( M ) := { γ ∈ GL ( r, Z ) | γ · M = M} . For
N ∈ L + t,r ( M ) we set ǫ t,r ( N ) := { γ ∈ B t,r ( Z ) | γ · N = N } . The symbol Sp( n, R ) denotes the real symplectic group of size 2 n . We put Γ n :=Sp( n, Z ) = Sp( n, R ) ∩ Z (2 n, n ) . The symbol H n denotes the Siegel upper half space ofsize n .Let φ M and ψ M be Jacobi cusp forms of weight k of index M on Γ n (cf. [Zi 89,Definition 1.3]). In the case r = 0 we regard φ M and ψ M as Siegel cusp forms of weight k with respect to Γ n . Let 0 < t ≤ n and we take Fourier-Jacobi expansions of φ M and ψ M : φ M ( τ, z ) e ( M ω ) = X N ∈ L + t,r ( M ) φ N ( τ ′ , z ′ ) e ( N ω ′ ) ,ψ M ( τ, z ) e ( M ω ) = X N ∈ L + t,r ( M ) ψ N ( τ ′ , z ′ ) e ( N ω ′ ) , (1.1) ANKIN-SELBERG METHOD FOR JACOBI FORMS 3 where (cid:18) τ z t z ω (cid:19) = (cid:18) τ ′ z ′ t z ′ ω ′ (cid:19) , τ ∈ H n , ω ∈ H r , z ∈ C ( n,r ) , τ ′ ∈ H n − t , ω ′ ∈ H t + r and z ′ ∈ C ( n − t,t + r ) . We have the fact that φ N and ψ N are Jacobi cusp forms of weight k ofindex N on Γ n − t .The aim of this article is to obtain analytic properties of the Dirichlet series D t ( φ M , ψ M ; s ) := X N ∈ B t,r ( Z ) \ L + t,r ( M ) h φ N , ψ N i| ǫ t,r ( N ) | det( N ) s , (1.2)where we denote by h φ N , ψ N i the Petersson inner product of φ N and ψ N (see § s ) (see Lemma 2.2). If t = n , then φ N and ψ N are Fourier coefficients of φ M and ψ M , respectively, and we set h φ N , ψ N i = φ N ψ N in this case. We remark D ( φ M , ψ M ) = 12 X N ∈ Z > h φ N , ψ N iN s if t = 1 and r = 0.Remark that if r = 0 and if n ≥
2, then φ M and ψ M are Siegel cusp forms. Andanalytic properties of the Dirichlet series D t ( φ M , ψ M ; s ) in the case r = 0 have beenshown by Maass [Ma 73] (for t = n = 2), by Kohnen-Skoruppa [K-S 89, Theorem 1] (for t = 1 and n = 2), by Kalinin [Kal 84] (for t = n ≥
2) and by Yamazaki [Ya 90] (for t ≥ n ≥ r = 1, then meromorphic continuations, functional equationsand residues of the above Dirichlet series have been shown in the papers by Kohnen-Zagier [K-Z 81, p.189-191] (for t = n = 1 and M = 1), by Katsurada-Kawamura[K-K 15,Proposition 3.1] (for t = n ≥ M = 1) and by Imamo¯glu-Martin[I-M 03, Theorem2(d) and Proposition 1(b)] (for t = n = 1 and for arbitral integer M ).The main result in the present article is that the Dirichlet series D t ( φ M , ψ M ; s ) (for1 ≤ t ≤ n and for arbitral index M ) has a meromorphic continuation to the wholecomplex plane and has a functional equation (see Theorem 2.5). The residue at s = k − r is also determined (cf. Theorem 2.5). Such properties are shown by using Rankin-Selberg method. The properties of certain real analytic Siegel-Eisenstein series, whichare necessarily to prove Theorem 2.5, have been shown by Kalinin [Kal 77] (for t = n )and by Yamazaki [Ya 90] (for 1 ≤ t < n ). Hence the main issue in this paper is toobtain an integral expression of the Dirichlet series D t ( φ M , ψ M ; s ) by using such Siegel-Eisenstein series. To obtain the integral expression, we shall refine a method treatedby Katsurada and Kawamura in [K-K 15, Proposition 3.1]. It means that we takevector valued modular forms through theta decompositions of two Jacobi forms andtake the inner product of these two vector valued modular forms. By coupling this innerproduct with a certain Siegel-Eisenstein series we obtain the integral expression of theDirichlet series D t ( φ M , ψ M ; s ) (cf. Proposition 2.3). To show the integral expression ofthe Dirichlet series, we use the compatibility between the theta decomposition and theFourier-Jacobi expansion of Jacobi forms. S. HAYASHIDA half-integral weight of certain indices . Such Jacobi forms of half-integral weight belong to theso-called plus-space, which is a generalization of Kohnen plus-space of elliptic modularforms of half-integral weight. A generalization of Kohnen plus-space for Jacobi formshas been introduced in [Ha 18]. Let M = (cid:18) N R t R (cid:19) be a matrix in L + r − , (1). Weput M = 4 N − R t R . We assume that k is an even integer. It is shown in [Ib 92] (for r = 1) and in [Ha 18] (for r >
1) that there exists a linear isomorphism between thespace J ( n ) k, M of Jacobi forms of weight k of index M on Γ n and the space of Jacobi forms J ( n )+ k − , M , where J ( n )+ k − , M is a certain subspace of Jacobi forms of weight k − of index M on Γ n (see § J ( n ) k, M and J ( n )+ k − , M corresponds each other.We remark that if r = 1, then M = ∅ and J ( n )+ k − , M is the plus-space of Siegel modularforms of weight k − introduced by Kohnen [Ko 80] (for n = 1) and by Ibukiyama [Ib 92](for n > φ M and ψ M be Jacobi cusp forms in J ( n )+ k − , M . We construct a Dirichletseries D t ( φ M , ψ M ; s ) in the same manner as in the case of integral weight. Then, by usingthe linear isomorphism between J ( n ) k, M and J ( n )+ k − , M , we obtain a meromorphic continuation,a functional equation and residues of D t ( φ M , ψ M ; s ) (see Theorem 3.3).Acknowledgement:This work was supported by JSPS KAKENHI Grant Number 80597766.2. Dirichlet series of Jacobi forms of integral weight
We denote by H n the Siegel upper half space of size n . For any ring R , we denoteby R ( l,m ) the set of all matrices of size l × m with the entries in R . The symbol 0 ( l,m ) denotes the zero matrix in C ( l,m ) . We denote by δ i,j the Kronecker delta. It means that δ i,j = 1 if i = j and 0 otherwise. By abuse of language we put det( M ) = det(4 M ) = 1,if the size of the matrix M is 0.Let k be an integer and let M ∈ L + r . We denote by J ( n ) k, M (resp. J ( n ) cuspk, M ) the spaceof Jacobi forms (resp. Jacobi cusp forms) of weight k of index M on Γ n . (See thedefinition [Zi 89, Definition 1.3]).For φ , ψ ∈ J ( n ) cuspk, M , the Petersson inner product is defined by h φ, ψ i := Z F n,r φ ( τ, z ) ψ ( τ, z ) e − πT r ( M v − [ y ]) det( v ) k − n − r − du dv dx dy, ANKIN-SELBERG METHOD FOR JACOBI FORMS 5 where F n,r := Γ Jn,r \ ( H n × C ( n,r ) ), τ = u + iv , z = x + iy , du = Q i ≤ j u i,j , dv = Q i ≤ j v i,j , dx = Q i,j x i,j and dy = Q i,j y i,j . Here we putΓ Jn,r := A B ∗∗ r ∗ ∗ C D ∗ r ∈ Γ n + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) A BC D (cid:19) ∈ Γ n , and the group Γ Jn,r acts on H n × C ( n,r ) in the usual manner (cf. [Zi 89, p.193]). For thesake of simplicity we put J (0) cuspk, M := C and for φ , ψ ∈ C , we set h φ, ψ i := φψ . Lemma 2.1.
Let φ N be the N -th Fourier-Jacobi coefficient of φ M defined in (1.1).Then, there exists a constant C ′ which does not depend on the choice of N such that | φ N ( τ ′ , z ′ ) e ( i N v ′− [ y ′ ])(det v ′ ) k | < C ′ det( N ) k for any ( τ ′ , z ′ ) ∈ H n − t × C ( n − t,t ) , and where v ′ = Im ( τ ′ ) and y ′ = Im ( z ′ ) .Proof. Since φ M is a Jacobi cusp form, there exists a constant C φ which depends onlyon φ M such that | φ M ( τ, z ) det( v ) k exp ( − π Tr( M v − [ y ])) | < C φ for any ( τ, z ) ∈ H n × C ( n,r ) , and where v = Im( τ ) and y = Im( z ). On the other hand,we have φ N ( τ ′ , z ′ ) e ( i N T ′ )= Z Sym r ( Z ) \ Sym r ( R ) Z R ( t,r ) Z Sym t ( Z ) \ Sym t ( R ) φ M (cid:0) τ + (cid:0) X ′ (cid:1) , z + (cid:0) X ′ (cid:1)(cid:1) × e ( M ( ω + X ′ )) e ( −N (cid:16) X ′ X ′ t X ′ X ′ (cid:17) ) dX ′ dX ′ dX ′ , where ( τ z t z ω ) = (cid:0) τ ′ z ′ t z ′ ω ′ (cid:1) ∈ H n + r , τ ∈ H n , ω ∈ H r , z ∈ M n,r ( C ), τ ′ ∈ H n − t , ω ′ ∈ H t + r , and z ′ ∈ M n − t,t + r ( C ).We decompose the matrix ( τ z t z ω ) as (cid:18) τ z t z ω (cid:19) = τ ′ z ′ z ′ t z ′ τ z t z ′ t z ω = (cid:18) τ ′ z ′ t z ′ ω ′ (cid:19) ∈ H n + r with τ = (cid:18) τ ′ z ′ t z ′ τ (cid:19) , z = (cid:18) z ′ z (cid:19) , z ′ = (cid:0) z ′ z ′ (cid:1) , ω ′ = (cid:18) τ z t z ω (cid:19) ,z ′ ∈ C ( n − t,t ) , z ′ ∈ C ( n − t,r ) , z ∈ C ( t,r ) , and τ ∈ H t . S. HAYASHIDA
We write v ′ y ′ y ′ t y ′ v y t y ′ t y T := Im τ ′ z ′ z ′ t z ′ τ z t z ′ t z ω ,v ′ = Im( τ ′ ), v = Im( τ ), T = Im( ω ) and T ′ := Im( ω ′ ) = (cid:18) v y t y T (cid:19) .Furthermore, we write y ′ := Im( z ′ ) = Im (cid:0) z ′ z ′ (cid:1) = (cid:0) y ′ y ′ (cid:1) , then we have | φ N ( τ ′ , z ′ ) e ( i N v ′− [ y ′ ])(det v ′ ) k | = (cid:12)(cid:12)(cid:12)(cid:12) e ( − i N T ′ ) Z Sym r ( Z ) \ Sym r ( R ) Z R ( t,r ) Z Sym t ( Z ) \ Sym t ( R ) φ M (cid:0) τ + (cid:0) X ′ (cid:1) , z + (cid:0) X ′ (cid:1)(cid:1) × e ( M ( ω + X ′ )) e ( −N (cid:16) X ′ X ′ t X ′ X ′ (cid:17) ) dX ′ dX ′ dX ′ e ( i N v ′− [ y ′ ])(det v ′ ) k (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) e ( − i N T ′ ) Z Sym r ( Z ) \ Sym r ( R ) Z R ( t,r ) Z Sym t ( Z ) \ Sym t ( R ) φ M (cid:0) τ + (cid:0) X ′ (cid:1) , z + (cid:0) X ′ (cid:1)(cid:1) × det(Im( τ + (cid:0) X ′ (cid:1) )) k e ( i M (Im( τ + (cid:0) X ′ (cid:1) ) − )[Im( z + (cid:0) X ′ (cid:1) )]) × e ( M ( ω + X ′ )) e ( −N (cid:16) X ′ X ′ t X ′ X ′ (cid:17) ) dX ′ dX ′ dX ′ e ( i N v ′− [ y ′ ])(det v ′ ) k × det( v ) − k e ( − i M v − [ y ]) (cid:12)(cid:12)(cid:12) < C φ e ( − i N T ′ ) e ( i M T ) e ( i N v ′− [ y ′ ]) det( v ′ ) k det( v ) − k e ( − i M v − [ y ])= C φ e ( − i N ( T ′ − v ′− [ y ′ ])) e ( i M ( T − v − [ y ])) det( v ′ ) k det( v ) − k . We now write N := (cid:18) N R t R M (cid:19) and put∆ := v − v ′− [ y ′ ] , η := y − t y ′ v ′− y ′ . Then, by a straightforward calculation we haveTr( N ( T ′ − v ′− [ y ′ ])) − Tr( M ( T − v − [ y ])) = Tr (cid:18) N (cid:18) ∆ η t η ∆ − [ η ] (cid:19)(cid:19) . Since N = (cid:18) N − M − [ t R ] 00 M (cid:19) (cid:20)(cid:18) t M − t R r (cid:19)(cid:21) and since (cid:18) ∆ η t η ∆ − [ η ] (cid:19) = (cid:18) ∆ 00 0 (cid:19) (cid:20)(cid:18) t ∆ − η r (cid:19)(cid:21) , ANKIN-SELBERG METHOD FOR JACOBI FORMS 7 there exists (cid:0) v y (cid:1) ∈ Sym + t × R ( t,r ) which satisfies N (cid:18) ∆ η t η ∆ − [ η ] (cid:19) = (cid:18) t
00 0 (cid:19) . By astraightforward calculation, such (cid:0) v y (cid:1) is given by v = v ′− [ y ′ ] + ( N − M − [ t R ]) − ,y = −
12 ( N − M − [ t R ]) − M − t R + t y ′ v ′− y ′ . With this (cid:0) v y (cid:1) we have | φ N ( τ ′ , z ′ ) e ( i N v ′− [ y ′ ])(det v ′ ) k | < C φ e ( − it ) det( v ′ ) k det( v ) − k = C φ e ( − it ) det( v − v ′− [ y ′ ]) − k = C φ e ( − it ) det( N − M − [ t R ]) k = C φ e ( − it ) det( M ) − k det( N ) k . Hence we conclude this lemma. ⊓⊔ Let φ M and ψ M be Jacobi cusp forms of weight k of index M on Γ n . Let D t ( φ M , ψ M ; s )be the Dirichlet series defined in (1 . Lemma 2.2.
The Dirichlet series D t ( φ M , ψ M ; s ) converges absolutely with sufficientlarge Re ( s ) .Proof. By virtue of Lemma 2.1 we have h φ N , ψ N i < C det( N ) k with a certain positive number C which does not depend on the choice of N . Hence, itis enough to show that the series X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) | det( N ) s . converges absolutely for sufficiently large Re ( s ).It is known that the series X T ∈ GL ( t, Z ) \ L + t | ǫ ( T ) | det( T ) s is absolutely convergent for Re ( s ) > t +12 (cf. [Shi 75]).There exists a natural map B t,r ( Z ) \ L + t,r ( M ) → GL ( t, Z ) \ L + t given by N = (cid:18) N R t R M (cid:19) → lN − l M − [ t R ] , S. HAYASHIDA where 2 l is the smallest positive integer which satisfies 2 l (2 M ) − Z ( r, ⊂ Z ( r, . This mapmay not be injective, but if two matrices N i = (cid:18) N ,i R ,i t R ,i M (cid:19) ∈ L + t,r ( M ) ( i = 1 , N , − M − [ t R , ] = 4 N , − M − [ t R , ]and R , − R , ∈ Z ( t,r ) (2 M ), then N and N belong to the same equivalent class in B t,r ( Z ) \ L + t,r ( M ). Therefore, for a fixed representative T in GL ( t, Z ) \ L + t , there exist atmost det(2 M ) t representatives in B t,r ( Z ) \ L + t,r ( M ) which map to T .For N = (cid:18) N R t R M (cid:19) , we remark the identitydet( N ) = (4 l ) − t det( M ) det(4 lN − l M − [ t R ]) . For any real number s , we have1 | ǫ t,r ( N ) | det( N ) s ≤ N ) s = (4 l ) ts det( M ) − s | ǫ ( N ′ ) || ǫ ( N ′ ) | det( N ′ ) s , where we put N ′ = 4 lN − l M − [ t R ].It is not difficult to show that there exists a constant c such that | ǫ ( N ′ ) | < det( N ′ ) c for any N ′ ∈ L + t . Therefore, for sufficiently large real number s , we have X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) | det( N ) s < (4 l ) ts rt det( M ) − s + t X N ′ ∈ GL ( t, Z ) \ L + t | ǫ ( N ′ ) | det( N ′ ) s − c . Thus we conclude this lemma. ⊓⊔ For 0 ≤ t ≤ n , we put P n − t,t := (cid:26)(cid:18) ∗ ∗ ( t, n − t ) ∗ (cid:19) ∈ Γ n (cid:27) = ( ∗ ( n − t,t ) ∗ ∗∗ ∗ ∗ ∗∗ ( n − t,t ) ∗ ∗ ( t,n − t ) ( t,t ) ( t,n − t ) ∗ ! ∈ Γ n ) . For τ ∈ H n and for s ∈ C , we set E ( n ) t ( s ; τ ) := X γ ∈ P n − t,t \ Γ n (cid:18) det(Im( γ · τ ))det(Im(( γ · τ ) )) (cid:19) s , where ( γ · τ ) is the left upper part of γ · τ of size ( n − t ) × ( n − t ). The series E ( n ) t ( s ; τ )converges absolutely for Re( s ) > n − t − (see [Ya 90, p.42-43]).For R ∈ Z ( n,r ) we put ϑ M ,R ( τ, z ) := X p ∈ Z ( n,r ) p ≡ R mod Z ( n,r ) (2 M ) e (cid:18) p M − t pτ + p t z (cid:19) . (2.1)We remark that ϑ M ,R is defined for R modulo Z ( n,r ) (2 M ). ANKIN-SELBERG METHOD FOR JACOBI FORMS 9
We take the following decompositions with theta series: φ M ( τ, z ) = X R mod Z ( n,r ) (2 M ) f R ( τ ) ϑ M ,R ( τ, z ) ,ψ M ( τ, z ) = X R mod Z ( n,r ) (2 M ) g R ( τ ) ϑ M ,R ( τ, z ) . (2.2)We call it theta decomposition. We also take the Fourier expansions of φ M and ψ M : φ M ( τ, z ) = X N,R C φ ( N, R ) e ( N τ + R t z ) ,ψ M ( τ, z ) = X N,R C ψ ( N, R ) e ( N τ + R t z ) , where in the above summations N ∈ L + n and R ∈ Z ( n,r ) run over matrices which satisfy4 N − M − [ t R ] >
0. Then we have f R ( τ ) = X N C φ ( N, R ) e (cid:18)
14 (4 N − M − [ t R ]) τ (cid:19) ,g R ( τ ) = X N C ψ ( N, R ) e (cid:18)
14 (4 N − M − [ t R ]) τ (cid:19) , where in the above summations N ∈ L + n runs over matrices which satisfy the condition4 N − M − [ t R ] > Proposition 2.3.
We have an integral expression of D t ( φ M , ψ M ; s ) as follows. If Re ( s ) is sufficiently large, then we obtain (1 + δ t,n ) − π − t ( t − t ( s + k − n + t − r − ) t Y j =1 Γ (cid:18) s + k − n + t − r − j (cid:19) − × Z Γ n \ H n X R mod Z ( n,r ) (2 M ) f R ( τ ) g R ( τ )(det( Im ( τ ))) k − r E ( n ) t ( s ; τ ) dτ = det(2 M ) n − t + s + k − n + t − r − ( n − t ) r − (2 t + r )( s + k − n + t − r − ) × D t ( φ M , ψ M ; s + k − n + ( t − r − / . We will show this proposition in § ξ ( s ) := π − s Γ (cid:16) s (cid:17) ζ ( s ) and set E ( n ) t ( s ; τ ) := t Y i =1 ξ (2 s + 1 − i ) [ t/ Y i =1 ξ (4 s − n + 2 t − i ) E ( n ) t ( s ; τ ) . The following theorem has been shown by Kalinin [Kal 77] for t = n and by Ya-mazaki [Ya 90] for 1 ≤ t < n . Theorem 2.4 ([Kal 77], [Ya 90]) . The function E ( n ) t ( s ; τ ) has a meromorphic contin-uation to the whole complex plane as the function of s and holomorphic for Re ( s ) > (2 n − t + 1) / . Moreover, E ( n ) t ( s ; τ ) satisfies the functional equation E ( n ) t ( s ; τ ) = E ( n ) t (cid:18) n − t + 12 − s ; τ (cid:19) . It has a simple pole at s = n − ( t − / with the residue δ t,n t Y j =2 ξ ( j ) [ t/ Y j =1 ξ (2 n − t + 2 j + 1) when n > and with the residue / when n = t = 1 . (cf. [Kal 77, Theorem 2] for t = n > , [Ya 90, Theorem 2.2] for ≤ t < n ).Moreover, if t = n , then the function ξ (2 s ) [ n/ Y i =1 ξ (4 s − i ) E ( n ) n ( s ; τ ) has a meromorphiccontinuation to the whole complex plane in s except the possible poles of finite order at s = j/ for integers j (0 ≤ j ≤ n + 2) .If t = 1 , then E ( n )1 ( s ; τ ) has a meromorphic continuation to the whole complex planein s except the poles at s = n and with residues and − , respectively. It is remarked in [Ya 90] that if t ≥ n − t + 2, then we can simplify the gammafactor of E ( n ) t by virtue of the cancellation of the above functional equation. It meansthat it is possible to take n − t +1 Y i =1 ξ (2 s + 1 − i ) [ t/ Y i =1 ξ (4 s − n + 2 t − i ) E ( n ) t ( s ; τ ) as thechoice of the definition of E ( n ) t in this case. The residue of E ( n ) t in the theorem will bechanged if we change the gamma factor.We put D t ( φ M , ψ M ; s ):= (4 π ) − ts (det M ) s t Y j =1 (cid:18) Γ (cid:18) s − j − (cid:19) ξ (2 s − k + 2 n + r + 2 − t − j ) (cid:19) × [ t/ Y j =1 ξ (4 s − k + 2 n + 2 r + 2 − j ) × D t ( φ M , ψ M ; s ) . We remark that if r = 0, then we regard det( M ) as 1. ANKIN-SELBERG METHOD FOR JACOBI FORMS 11
By virtue of Proposition 2.3 the function D t ( φ M , ψ M ; s ) equals to Z Γ n \ H n X R mod Z ( n,r ) (2 M ) f R ( τ ) g R ( τ )(det(Im τ )) k − r E ( n ) t ( s − k + n − ( t − r − / τ ) dτ times the constant π − t ( t − det(4 M ) − n − t (1 + δ t,n ) − .Thus, due to Theorem 2.4 we have the following. Theorem 2.5.
The function D t ( φ M , ψ M ; s ) has a meromorphic continuation to thewhole complex plane and holomorphic for Re ( s ) > k − r . It has a simple pole at s = k − r with the residue (1 + δ ,r ) − π − t ( t − det(4 M ) t h φ M , ψ M i t Y j =2 ξ ( j ) [ t/ Y j =1 ξ (2 n − t + 2 j + 1) when n > and with the residue (1 + δ ,r ) − det(4 M ) h φ M , ψ M i when n = t = 1 .It satisfies a functional equation D t ( φ M , ψ M ; s ) = D t (cid:18) φ M , ψ M ; 2 k − n − r + t − − s (cid:19) . Moreover, if t = 1 , then D ( φ M , ψ M ; s ) has a meromorphic continuation to the wholecomplex plane and holomorphic except for simple poles at s = k − r and s = k − r − n .The residue at s = k − r is (1 + δ ,n ) − (1 + δ ,r ) − det(4 M ) h φ M , ψ M i . The case r = 0 has been shown in [Ya 90] . We remark that if n = r = t = 1, then the above residue coincides with Proposi-tion 1 (a) in [I-M 03]. However, Proposition 1 (a) in [I-M 03] should readRes s = k − / D F,G ( s , s ) = π k + Γ (cid:18) k − (cid:19) − ζ (2) − L ( F, G, s + k − . After we shall explain some similar results of Theorem 2.5 for Jacobi cusp forms of half-integral weight in Section 3, we will prove Proposition 2.3 in Section 4.3.
Rankin-Selberg method for the plus space of Jacobi forms
In this section we shall explain the half-integral weight case. In this section we assumethat k is an even integer. We assume r ≥
1. Let M = (cid:18) M L t L (cid:19) ∈ L + r with M ∈ L + r − and L ∈ M r − , ( Z ). If r ≥
2, we set M := 4 M − L t L. If r = 1, then M = 1 and we regard M = ∅ as the empty set and we put det( M ) = 1by abuse of notation.We set Γ ( n )0 (4) := (cid:26)(cid:18) A BC D (cid:19) ∈ Γ n (cid:12)(cid:12)(cid:12)(cid:12) C ∈ Z ( n,n ) (cid:27) .Let J ( n )+ k − , M be the plus-space of Jacobi forms of weight k − of index M on Γ ( n )0 (4)which is a generalization of generalized plus-space of Siegel modular forms of weight k − to Jacobi forms. The space J ( n )+ k − , M is defined as follows. Let φ be a Jacobi formof weight k − of index M on Γ ( n )0 (4). The reader is referred to [Ha 18] for the precisedefinition of Jacobi forms of half-integral weight. We take the Fourier expansion φ ( τ, z ) = X N ′ ,R ′ C φ ( N ′ , R ′ ) e ( N ′ τ + R ′ t z )for ( τ, z ) ∈ H n × C ( n,r − , where N ′ and R ′ run over L ∗ n and Z ( n,r − , respectively, suchthat 4 N ′ − R ′ M − t R ′ ≥
0. Then φ belongs to J ( n )+ k − , M if and only if C φ ( N ′ , R ′ ) = 0 unless (cid:18) N ′ R ′ t R ′ M (cid:19) ≡ λ t λ mod 4with some λ ∈ Z ( n + r − , .If r = 1, then the space J ( n )+ k − , M coincides with the generalized plus-space of Siegel mod-ular forms. There exists a linear isomorphism map ι M from J ( n ) k, M to J ( n )+ k − , M (cf. [E-Z 85](for r = n = 1), [Ib 92] (for r = 1, n > r > n ≥ ι M isgiven as follows.Let φ M ∈ J ( n ) k, M be a Jacobi form. We denote by C φ M ( ∗ , ∗ ) the Fourier coefficients of φ M . For τ ∈ H n and for z = ( z , z ) ∈ C ( n,r ) ( z ∈ C ( n,r − , z ∈ C ( n, ), we take thetheta decomposition φ M ( τ, z ) = X R ∈ Z ( n, R mod 2 Z ( n, f R, M ( τ, z ) ϑ ,L,R ( τ, z , z ) , where f R, M ( τ, z ) = X N ∈ L ∗ n ,N ∈ Z ( n,r − C φ M ( N , (cid:0) N R (cid:1) ) × e (( N − R t R ) τ + ( N − R t L ) t z )and the function ϑ ,L,R will be denoted in (4.3) (cf. [Ha 18, Lemma 4.1]). We put ι M ( φ M )( τ, z ) = X R ∈ Z ( n, / (2 Z ( n, ) f R, M (4 τ, z ) . ANKIN-SELBERG METHOD FOR JACOBI FORMS 13
For the sake of simplicity we write φ M = ι M ( φ M ). Then φ M belongs to J ( n )+ k − , M (cf. [Ha 18,Proposition 4.4]). If φ M is a Jacobi cusp form, then φ M is also a Jacobi cusp form. If r = 1, then φ M is a Siegel modular form (cf. [E-Z 85], [Ib 92]).Let φ and ψ be Jacobi cusp forms of weight k − of index S ∈ L + r on Γ ( n )0 (4). ThePetersson inner product is defined by h φ, ψ i := h Γ n : Γ ( n )0 (4) i − Z F n,r, φ ( τ, z ) ψ ( τ, z ) e − πT r ( Sv − [ y ]) det( v ) k − n − r − du dv dx dy, where F n,r, := Γ Jn,r (4) \ ( H n × C ( n,r ) ), τ = u + iv , z = x + iy , du = Q i ≤ j u i,j , dv = Q i ≤ j v i,j , dx = Q i,j x i,j , dy = Q i,j y i,j and h Γ n : Γ ( n )0 (4) i denotes the index of Γ ( n )0 (4) in Γ n . Herewe put Γ Jn,r (4) := A B ∗∗ r ∗ ∗ C D ∗ r ∈ Γ n + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) A BC D (cid:19) ∈ Γ ( n )0 (4) . Lemma 3.1.
Let φ M and ψ M be Jacobi cusp forms in J ( n ) cuspk, M . We put φ M = ι M ( φ M ) and ψ M = ι M ( ψ M ) . As for the Petersson inner product we obtain the identity h φ M , ψ M i = (1 + δ ,r ) − n ( k − h φ M , ψ M i . We remark that, in the case of r = 1 and n = 1 , this identity has been obtained by com-bining Kohnen and Zagier [K-Z 81, p.p. 189–191] and Eichler-Zagier [E-Z 85, Theorem5.3] . Remark that the denominator of RHS of [E-Z 85, Theorem 5.3] is neither √ m nor √ m but √ m . We remark that, in the case of r = 1 and n > , the above identityhas been obtained by Katsurada and Kawamura [K-K 08, p. 2051 (6)] . Remark that (2 k − n − in [K-K 08, p. 2051 (6)] should read (2 k − n − − .Proof. We recall the symbol P ,n = { ( ∗ ∗ ( n,n ) ∗ ) ∈ Γ n } . We set E ( n ) n, ( s ; τ ) := X γ ∈ P ,n \ Γ ( n )0 (4) det(Im( γ · τ )) s . We take the theta decompositions φ M ( τ, z ) = X R mod Z ( n,r − (2 M ) ˜ f R ( τ ) ϑ M ,R ( τ, z ) ,ψ M ( τ, z ) = X R mod Z ( n,r − (2 M ) ˜ g R ( τ ) ϑ M ,R ( τ, z ) . We put I n ( φ M , ψ M ; s ) := Z Γ ( n )0 (4) \ H n X R mod Z ( n,r − (2 M ) ˜ f R ( τ )˜ g R ( τ )(det(Im τ )) k − − r − E ( n ) n, ( s ; τ ) d τ. where we put τ = u + √− v and dτ := det( v ) − n − du dv and du = Q l ≤ m u l,m , dv = Q l ≤ m v l,m . Here u = ( u l,m ) and v = ( v l,m ). We put I n ( φ M , ψ M ; s ) := Z Γ n \ H n X R mod Z ( n,r ) (2 M ) f R ( τ ) g R ( τ ) det(Im( τ )) k − r E ( n ) n ( s ; τ ) dτ, where f R and g R are denoted in (2.2) through the decompositions of φ M and ψ M withthe theta series.We will show this lemma by comparing the residue of I n ( φ M , ψ M ; s ) with the one of I n ( φ M , ψ M ; s ) at s = n +12 . Let φ M ( τ, z ) = X N ′ R ′ t R ′ M ! ∈ L + n,r − ( M ) C φ M ( N ′ , R ′ ) e ( N ′ τ + R ′ t z )and φ M ( τ, z ) = X N R t R M ! ∈ L + n,r − ( M ) A φ M ( N , R ) e ( N τ + R t z )be the Fourier expansions of φ M and φ M , respectively. We similarly denote by C ψ M ( N ′ , R ′ )and A ψ M ( N , R ) the Fourier coefficients of ψ M and ψ M , respectively. We remark thatif (cid:18) N ′ R ′ t R ′ M (cid:19) = 4 (cid:18) N R , t R , M (cid:19) − (cid:18) R , L (cid:19) t (cid:18) R , L (cid:19) (3.1)and if (cid:18) N R t R M (cid:19) = N R , R , t R , M L t R , t L , (3.2)then C φ M ( N ′ , R ′ ) = A φ M ( N , R ) and C ψ M ( N ′ , R ′ ) = A ψ M ( N , R ).By the similar argument of the proof of the identity (4.2) in Proposition 4.1 whichwill be appeared in §
4, we have I n ( φ M , ψ M ; s )= 2 π n ( n − n Y i =1 Γ( s + k − − r − − n + 12 − i −
12 ) × X N ∈ L + n,r − ( M ) /B n,r − ( Z ) | ǫ n,r − ( N ) | C φ M ( N ′ , R ′ ) C ψ M ( N ′ , R ′ ) × det( π (4 N ′ − M − [ t R ′ ])) − k + + r − − s + n +12 , ANKIN-SELBERG METHOD FOR JACOBI FORMS 15 where in the summation we set N = (cid:18) N ′ R ′ t R ′ M (cid:19) . We remark that if the identities(3.1) and (3.2) hold, thendet( π (4 N ′ − M − [ t R ′ ])) = 2 n det( π (4 N − M − [ t R ]))and | ǫ n,r − ( N ) | = | ǫ n,r ( N ) | , where we put N = (cid:18) N R t R M (cid:19) . The map (cid:18) N R t R M (cid:19) ∈ L + n,r ( M ) /B n,r ( Z ) (cid:18) N ′ R ′ t R ′ M (cid:19) ∈ L + n,r − ( M ) /B n,r ( Z ) given by the identities (3.1)and (3.2) is bijective. Therefore, by using the identity (4.2) which will be appearedin §
4, we have I n ( φ M , ψ M ; s )= 2 π n ( n − n Y i =1 Γ( s + k − − r − − n + 12 − i −
12 ) × X N ∈ L + n,r ( M ) /B n,r ( Z ) | ǫ n,r ( N ) | A φ M ( N , R ) A ψ M ( N , R ) × n ( − k + r − s + n +12 ) det( π (4 N − M − [ t R ])) − k + + r − − s + n +12 = 2 n ( − k + r − s + n +12 ) I n ( φ M , ψ M ; s ) . (3.3)We put γ n ( s ) := n Y i =1 ξ (2 s + 1 − i ) [ n/ Y i =1 ξ (4 s − i ) , where ξ ( s ) = π − s Γ (cid:16) s (cid:17) ζ ( s ) is the symbol denoted before Theorem 2.4, and we put E ( n ) n, ( s ; τ ) := γ n ( s ) E ( n ) n, ( s ; τ ) . Then E ( n ) n, ( s ; τ ) has a meromorphic continuation to the whole complex plane in s (cf. [Kal 77,Theorem 1]). Moreover, E ( n ) n, ( s ; τ ) has a simple pole at s = ( n + 1) / s = n +12 E ( n ) n, ( s ; τ ) = 1[Γ n : Γ ( n )0 (4)] Res s = n +12 E ( n ) n ( s ; τ )= 1[Γ n : Γ ( n )0 (4)] n Y j =2 ξ ( j ) [ n/ Y j =1 ξ (2 j + 1) . Therefore the residue of γ n ( s ) I n ( φ M , ψ M ; s ) at s = n +12 is1[Γ n : Γ ( n )0 (4)] n Y j =2 ξ ( j ) [ n/ Y j =1 ξ (2 j + 1) × Z Γ ( n )0 (4) \ H n X R mod Z ( n,r − (2 M ) ˜ f R ( τ )˜ g R ( τ )(det(Im τ )) k − − r − d τ = 21 + δ ,r − det(4 M ) n n Y j =2 ξ ( j ) [ n/ Y j =1 ξ (2 j + 1) h φ M , ψ M i . On the other hand the residue of γ n ( s ) I n ( φ M , ψ M ) at s = n +12 is2 det(4 M ) n n Y j =2 ξ ( j ) [ n/ Y j =1 ξ (2 j + 1) h φ M , ψ M i . We remark the identity det(4 M ) = 2 r − det(4 M ). Thus, by virtue of the identity (3.3),we have the lemma. ⊓⊔ Let φ M , ψ M ∈ J ( n ) cuspk − , M be Jacobi cusp forms of weight k − with the index M ∈ L + r − on Γ ( n )0 (4). We remark that if r = 1, then φ M and ψ M are Siegel cusp forms of weight k − . For any natural number t (1 ≤ t ≤ n ), we take the Fourier-Jacobi expansions φ M ( τ, z ) e ( M ω ) = X N ∈ L + t,r − ( M ) φ N ( τ ′ , z ′ ) e ( N ω ′ ) ,ψ M ( τ, z ) e ( M ω ) = X N ∈ L + t,r − ( M ) ψ N ( τ ′ , z ′ ) e ( N ω ′ ) . For complex number s which real part is sufficient large, we set D t ( φ M , ψ M ; s ) := X N ∈ B t,r − ( Z ) \ L + t,r − ( M ) h φ N , ψ N i| ǫ t,r − ( N ) | det( N ) s . Lemma 3.2.
We assume that φ M and ψ M belong to the plus space J ( n )+ cuspk − , M . Let φ M and ψ M ∈ J ( n ) cuspk, M be Jacobi cusp forms which satisfy φ M = ι M ( φ M ) and ψ M = ι M ( ψ M ) .Then, for any t (1 ≤ t ≤ n ) , we have D t ( φ M , ψ M ; s ) = (1 + δ ,r )2 − k − n − t ) − r + t − s D t ( φ M , ψ M ; s ) . In the case of r = 1, M = 1 and t = n this lemma has been shown in [K-K 15]. Proof.
Assume N is a matrix in L + t,r ( M ). Let φ N and ψ N be the N -th Fourier-Jacobicoefficients of φ M and ψ M , respectively. We put φ N = ι N ( φ N ) and ψ N = ι N ( ψ N ). Then ANKIN-SELBERG METHOD FOR JACOBI FORMS 17 φ N and φ N are N -th Fourier-Jacobi coefficients of φ M and ψ M , respectively. We remarkthat φ N and ψ N belong to J ( n − t )+ cuspk − , N and remark that φ N and ψ N belong to J ( n − t )+ cuspk, N .By virtue of Lemma 3.1, we have h φ N , ψ N i = (1 + δ ,r )2 − n − t )( k − h φ N , ψ N i . We havealso | ǫ t,r − ( N ) | = | ǫ t,r ( N ) | and det N = 2 r + t − det N . Thus we conclude the lemma. ⊓⊔ We set D t ( φ M , ψ M ; s ):= π − ts (det M ) s t Y j =1 (cid:18) Γ (cid:18) s − j − (cid:19) ξ (2 s − k + 2 n + r + 2 − t − j ) (cid:19) × [ t/ Y j =1 ξ (4 s − k + 2 n + 2 r + 2 − j ) D t ( φ M , ψ M ; s ) . Then D t ( φ M , ψ M ; s ) = (1 + δ ,r )2 − k − n − t ) D t ( φ M , ψ M ; s ) . Due to Theorem 2.5 we have the followings.
Theorem 3.3.
The function D t ( φ M , ψ M ; s ) has a meromorphic continuation to the wholecomplex plane and holomorphic for Re ( s ) > k − r . It has a simple pole at s = k − r with the residue (1 + δ ,r ) − tk − t π − t ( t − det( M ) t h φ M , ψ M i t Y j =2 ξ ( j ) [ t/ Y j =1 ξ (2 n − t + 2 j + 1) when n > and with the residue k − det( M ) h φ M , ψ M i when n = t = 1 .It satisfies the functional equation D t ( φ M , ψ M ; s ) = D t (cid:18) φ M , ψ M ; 2 k − n − r + t − − s (cid:19) . Moreover, if t = 1 , then D ( φ M , ψ M ; s ) has a meromorphic continuation to the wholecomplex plane and holomorphic except for simple poles at s = k − r and k − r − n . Theresidue at s = k − r is (1 + δ ,n ) − (1 + δ ,r ) − k − det( M ) h φ M , ψ M i . In particular, if r = 1 and M = 1, then the space J ( n ) cuspk, of Jacobi cusp formsof degree n is linearly isomorphic to the generalized plus-space S + k − (Γ ( n )0 (4)) as Heckealgebra modules. Here S + k − (Γ ( n )0 (4)) is a certain subspace of Siegel cusp forms of weight k − of degree n (see [Ib 92] for the definition and the isomorphism). We have thefollowing. Corollary 3.4.
Let F , G ∈ S + k − (Γ ( n )0 (4)) . The function D t ( F, G ; s ) has a meromorphiccontinuation to the whole complex plane and holomorphic for Re ( s ) > k − . It has asimple pole at s = k − with the residue tk − t − π − t ( t − h F, G i t Y j =2 ξ ( j ) [ t/ Y j =1 ξ (2 n − t + 2 j + 1) when n > and with the residue k − h F, G i when n = t = 1 .It satisfies the functional equation D t ( F, G ; s ) = D t (cid:18) F, G ; 2 k − n + t − − s (cid:19) . Moreover, if t = 1 , then D ( F, G ; s ) has a meromorphic continuation to the whole com-plex plane and holomorphic except for simple poles at s = k − and k − − n . Theresidue at s = k − is Res s = k − D ( F, G ; s ) = Res s = k − ( D ( F, G ; s ) π − s Γ( s ) ξ (2 s − k + 2 n + 1))= (1 + δ ,n ) − k − h F, G i . We remark that the case t = n in Corollary 3.4 has been shown in [K-Z 81] (for n = 1)and in [K-K 15] (for n > Proof of Proposition 2.3
In this section we shall prove Proposition 2.3. We use the same notation in §
2. For τ ∈ H n , we decompose τ as τ = (cid:18) τ z ′ t z ′ τ (cid:19) , τ ∈ H n − t , τ ∈ H t , z ′ ∈ C ( n − t,t ) . We write τ = u + iv , τ j = u j + iv j ( j = 1 ,
2) and z ′ = x ′ + iy ′ with matrices u, v, u j , v j , x ′ , y ′ which entries are real numbers. For τ ∈ H n − t , we fix a fundamental domain D t ( τ ) := C ( n − t,t ) / ( τ Z ( n − t,t ) + Z ( n − t,t ) )and put ^ D t ( τ ) := (cid:26) ( z ′ , τ ) ∈ D t ( τ ) × H t (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) τ z ′ t z ′ τ (cid:19) ∈ H n (cid:27) . ANKIN-SELBERG METHOD FOR JACOBI FORMS 19
The group P ,t = (cid:26)(cid:18) A B ( t,t ) t A − (cid:19) ∈ Γ t (cid:27) acts on ^ D t ( τ ) by (cid:18) A B ( t,t ) t A − (cid:19) · ( z ′ , τ ) := ( z ′ t A, τ [ t A ] + B t A )for (cid:18) A B ( t,t ) t A − (cid:19) ∈ P ,t and for ( z ′ , τ ) ∈ ^ D t ( τ ).We put I t ( φ M , ψ M ; s ) := Z Γ n \ H n X R mod Z ( n,r ) (2 M ) f R ( τ ) g R ( τ ) det( v ) k − r E ( n ) t ( s ; τ ) dτ, where f R and g R are denoted in (2.2) through the decompositions of φ M and ψ M withthe theta series, and where we put dτ := det( v ) − n − du dv and du = Q l ≤ m u l,m , dv = Q l ≤ m v l,m . Here u = ( u l,m ) and v = ( v l,m ).We have I t ( φ M , ψ M ; s )= Z P n − t,t \ H n X R mod Z ( n,r ) (2 M ) f R ( τ ) g R ( τ ) det( v ) k − r + s det( v ) − s dτ = 1 + δ t,n Z Γ n − t \ H n − t Z P ,t \ ^ D t ( τ ) X R mod Z ( n,r ) (2 M ) f R ( τ ) g R ( τ ) det( v ) k − r + s − ( n +1) × det( v ) − s dx ′ dy ′ du dv du dv . We write R = (cid:18) R R (cid:19) with R ∈ Z ( n − t,r ) and R ∈ Z ( t,r ) . We take Fourier-Jacobiexpansions of f R and g R : f R ( τ ) = X N ∈ L + t f R,N ( τ , z ′ ) e (cid:18)(cid:18) N − M − [ t R ] (cid:19) τ (cid:19) , (4.1) g R ( τ ) = X N ∈ L + t g R,N ( τ , z ′ ) e (cid:18)(cid:18) N − M − [ t R ] (cid:19) τ (cid:19) , where we have f R,N ( τ , z ′ ) = X N ∈ L + n − t ,N ∈ Z ( n − t,t ) C φ ( (cid:18) N N t N N (cid:19) , R ) × e (( N − M − [ t R ]) τ + ( N − R M − t R ) t z ′ )and g R,N can be written similarly by replacing f R,N (resp. C φ ) by g R,N (resp. C ψ ). Proposition 4.1.
We have the identity I t ( φ M , ψ M ; s )= (1 + δ t,n ) π t ( t − t Y i =1 Γ (cid:18) s + k − r − ( n + 1) + t + 12 − i − (cid:19) × Z Γ n − t \ H n − t Z D t ( τ ) X R mod Z ( n − t,r ) (2 M ) X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) |× det( π (4 N − M − [ t R ])) − k + r − s +( n +1) − t +12 × f R,N ( τ , z ′ ) g R,N ( τ , z ′ ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v − [ y ′ ]) det( v ) k − r − ( n +1) × dx ′ dy ′ du dv . (4.2) Proof.
We obtain I t ( φ M , ψ M ; s )= 1 + δ t,n Z Γ n − t \ H n − t Z P ,t \ ^ D t ( τ ) X R mod Z ( n,r ) (2 M ) X N ,N ′ ∈ L + t f R,N ( τ , z ′ ) g R,N ′ ( τ , z ′ ) × e ( N τ − N ′ τ − √− M − [ t R ] v ) det( v ) k − r + s − ( n +1) det( v ) − s × dx ′ dy ′ du dv du dv . Since Z Sym t ( R ) /Sym t ( Z ) e ( N τ − N ′ τ ) du = δ N ,N ′ and since det( v ) = det( v ) det( v − v − [ y ′ ]), we have I t ( φ M , ψ M ; s )= 1 + δ t,n Z Γ n − t \ H n − t Z GL ( t, Z ) \ D t ( τ ) ′ X R mod Z ( n,r ) (2 M ) X N ∈ L + t f R,N ( τ , z ′ ) g R,N ′ ( τ , z ′ ) × e (2 √− N v − √− M − [ t R ] v )(det( v − v − [ y ′ ])) k − r + s − ( n +1) det( v ) k − r − ( n +1) × dv dx ′ dy ′ du dv , where we put D t ( τ ) ′ := n ( z ′ , √− v ) ∈ ^ D t ( τ ) o and where GL ( t, Z ) acts on D t ( τ ) ′ by A · ( z ′ , √− v ) = ( z ′ t A, √− v [ t A ]) for A ∈ GL ( t, Z ) and for ( z ′ , √− v ) ∈ D t ( τ ) ′ . ANKIN-SELBERG METHOD FOR JACOBI FORMS 21
We substitute v by v + v − [ y ′ ], then I t ( φ M , ψ M ; s )= 1 + δ t,n Z Γ n − t \ H n − t Z GL ( t, Z ) \ D t ( τ ) ′′ X R mod Z ( n,r ) (2 M ) X N ∈ L + t f R,N ( τ , z ′ ) g R,N ′ ( τ , z ′ ) × e ( √− (cid:0) N − M − [ t R ] (cid:1) v ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v − [ y ′ ]) × det( v ) k − r + s − ( n +1) det( v ) k − r − ( n +1) dv dx ′ dy ′ du dv , where we put D t ( τ ) ′′ := { ( z ′ , √− v ) | ( z ′ , v ) ∈ D t ( τ ) × Sym + t ( R ) } and where wedenote by Sym + t ( R ) the positive definite symmetric matrices of size t which entries arereal numbers.Since φ M is a Jacobi form, we obtain C φ ( (cid:18) N N t N N [ A ] (cid:19) , (cid:18) n − t t A (cid:19) R ) = C φ ( (cid:18) N N A − t A − t N N (cid:19) , R )for any A ∈ GL ( t, Z ). Thus, for a fixed R = (cid:18) R R (cid:19) ∈ Z ( n,r ) , we have f (cid:16) n − t t A (cid:17) R,N [ A ] ( τ , z ′ ) = X N ∈ L + n − t ,N ∈ Z ( n − t,t ) C φ ( (cid:18) N N t N N [ A ] (cid:19) , R ) × e (( N − M − [ t R ]) τ + ( N − R M − t R A ) t z ′ )= X N ∈ L + n − t ,N ∈ Z ( n − t,t ) C φ ( (cid:18) N N A − t A − t N N (cid:19) , R ) × e (( N − M − [ t R ]) τ + ( N − R M − t R ) t ( z ′ t A ))= f R,N ( τ , z ′ t A ) . We write N = (cid:18) N R t R M (cid:19) . The summation X R mod Z ( n,r ) (2 M ) X N ∈ L + t equals to the sum-mation X R mod Z ( n − t,r ) (2 M ) X R mod Z ( t,r ) (2 M ) X N ∈ L + t , and the summation X R mod Z ( t,r ) (2 M ) X N ∈ L + t equals to the summation X N ∈ B ∞ t,r ( Z ) \ L + t,r ( M ) , where we put B ∞ t,r ( Z ) := (cid:26)(cid:18) t y r (cid:19) | y ∈ Z ( t,r ) (cid:27) . We remark the isomorphism B ∞ t,r ( Z ) \ B t,r ( Z ) ∼ = GL ( t, Z ).Therefore we have I t ( φ M , ψ M ; s )= 1 + δ t,n Z Γ n − t \ H n − t Z GL ( t, Z ) \ D t ( τ ) ′′ X R mod Z ( n − t,r ) (2 M ) X R mod Z ( t,r ) (2 M ) X N ∈ L + t f R,N ( τ , z ′ ) × g R,N ( τ , z ′ ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v − [ y ′ ]) × det( v ) k − r + s − ( n +1) det( v ) k − r − ( n +1) dv dx ′ dy ′ du dv = 1 + δ t,n Z Γ n − t \ H n − t Z GL ( t, Z ) \ D t ( τ ) ′′ X R mod Z ( n − t,r ) (2 M ) X N ∈ B ∞ t,r ( Z ) \ L + t,r ( M ) f R,N ( τ , z ′ ) × g R,N ( τ , z ′ ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v − [ y ′ ]) × det( v ) k − r + s − ( n +1) det( v ) k − r − ( n +1) dv dx ′ dy ′ du dv , where N = (cid:18) N R t R M (cid:19) . We have I t ( φ M , ψ M ; s )= 1 + δ t,n Z Γ n − t \ H n − t Z GL ( t, Z ) \ D t ( τ ) ′′ X R mod Z ( n − t,r ) (2 M ) X N ∈ B t,r ( Z ) \ L + t,r ( M ) × X diag ( A, r ) ∈ B ∞ t,r ( Z ) \ B t,r ( Z ) /ǫ t,r ( N ) f (cid:16) n − t t A (cid:17) R,N [ A ] ( τ , z ′ ) g (cid:16) n − t t A (cid:17) R,N [ A ] ( τ , z ′ ) × e ( √− (cid:0) N [ A ] − M − [ t R A ] (cid:1) v ) e ( √− (cid:0) N [ A ] − M − [ t R A ] (cid:1) v − [ y ′ ]) × det( v ) k − r + s − ( n +1) det( v ) k − r − ( n +1) dv dx ′ dy ′ du dv = 1 + δ t,n Z Γ n − t \ H n − t Z GL ( t, Z ) \ D t ( τ ) ′′ X R mod Z ( n − t,r ) (2 M ) X N ∈ B t,r ( Z ) \ L + t,r ( M ) X A ∈ GL ( t, Z ) | ǫ t,r ( N ) |× f R,N ( τ , z ′ t A ) g R,N ( τ , z ′ t A ) × e ( √− (cid:0) N − M − [ t R ] (cid:1) v [ t A ]) e ( √− (cid:0) N − M − [ t R ] (cid:1) v − [ y ′ t A ]) × det( v ) k − r + s − ( n +1) det( v ) k − r − ( n +1) dv dx ′ dy ′ du dv = (1 + δ t,n ) Z Γ n − t \ H n − t Z D t ( τ ) Z Sym + t ( R ) X R mod Z ( n − t,r ) (2 M ) X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) |× f R,N ( τ , z ′ ) g R,N ( τ , z ′ ) ANKIN-SELBERG METHOD FOR JACOBI FORMS 23 × e ( √− (cid:0) N − M − [ t R ] (cid:1) v ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v − [ y ′ ]) × det( v ) k − r + s − ( n +1) det( v ) k − r − ( n +1) dv dx ′ dy ′ du dv . As for the last identity, we remark that the set { ( z ′ t A, √− v [ t A ]) | ( z ′ , √− v ) ∈ GL ( t, Z ) \ D t ( τ ) ′′ , A ∈ GL ( t, Z ) } covers D t ( τ ) ′′ twice if t = n and once if t = n .Here we have Z Sym + t ( R ) e ( √− (cid:0) N − M − [ t R ] (cid:1) v ) det( v ) k − r + s − ( n +1) dv = Z Sym + t ( R ) exp ( − π T r (cid:0) N − M − [ t R ] (cid:1) v ) det( v ) k − r + s − ( n +1)+ t +12 det( v ) − t +12 dv = det( π (4 N − M − [ t R ])) − k + r − s +( n +1) − t +12 × Z Sym + t ( R ) exp ( − T r ( v )) det( v ) k − r + s − ( n +1)+ t +12 det( v ) − t +12 dv = det( π (4 N − M − [ t R ])) − k + r − s +( n +1) − t +12 π t ( t − × t Y i =1 Γ (cid:18) s + k − r − ( n + 1) + t + 12 − i − (cid:19) . Here, in the last identity, we used the formula shown by Maass [Ma 71, p.91, l.10-11].Thus we obtain the identity (4.2). ⊓⊔ We denote by D n,r the complex domain H n × C ( n,r ) . Let φ N and ψ N be Fourier-Jacobicoefficients of φ M and ψ M denoted in (1.1).We write N = (cid:18) N R t R M (cid:19) ∈ L + t,r ( M ). For z ′ ∈ C ( n − t,t + r ) we write z ′ = ( z ′ z ′ ) with z ′ ∈ C ( n − t,t ) and z ′ ∈ C ( n − t,r ) . We have theta decompositions of φ N and ψ N as follows: φ N ( τ , z ′ , z ′ ) = X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) ϑ M ,R ,R ( τ , z ′ , z ′ ) ,ψ N ( τ , z ′ , z ′ ) = X R mod Z ( n − t,r ) (2 M ) g R,N ( τ , z ′ ) ϑ M ,R ,R ( τ , z ′ , z ′ ) , where we put ϑ M ,R ,R ( τ , z ′ , z ′ ) := ϑ M ,R ( τ , z ′ R M − + z ′ ) , (4.3)and where R = (cid:18) R R (cid:19) ∈ Z ( n,r ) , and where the function ϑ M ,R is denoted in (2.1). Herewe remark that f R,N (resp. g R,N ) is appeared in (4.1) in the Fourier-Jacobi expansion of f R ( τ ) (resp. g R ( τ )). We omitted the detail of the proof of these decompositions,since the proof is similar to [Ha 18, Lemma 4.1]. In other words, we can say that theFourier-Jacobi expansions and the theta decompositions are compatible.We write τ = u + iv , z ′ j = x ′ j + iy ′ j ( j = 1 ,
2) with matrices u , v ∈ Sym n − t ( R ), x ′ , y ′ ∈ R ( n − t,t ) and x ′ , y ′ ∈ R ( n − t,r ) .We need the following lemma to calculate D t ( φ M , ψ M ; s ). Lemma 4.2.
We have Z D r ( τ ) ϑ M ,R ,R ( τ , z ′ , z ′ ) ϑ M ,R , ˜ R ( τ , z ′ , z ′ ) × e (2 √− M − v − [ 12 y ′ R M − + y ′ ]) dx ′ dy ′ = δ R , ˜ R (det v ) r det(4 M ) − n − t . Proof.
By a straightforward calculation we have Z D r ( τ ) ϑ M ,R ,R ( τ , z ′ , z ′ ) ϑ M ,R , ˜ R ( τ , z ′ , z ′ ) × e (2 √− M − v − [ 12 y ′ R M − + y ′ ]) dx ′ dy ′ = Z D r ( τ ) ϑ M ,R ( τ , z ′ R M − + z ′ ) ϑ M , ˜ R ( τ , z ′ R M − + z ′ ) × e (2 √− M − v − [ 12 y ′ R M − + y ′ ]) dx ′ dy ′ = Z D r ( τ ) ϑ M ,R ( τ , z ′ ) ϑ M , ˜ R ( τ , z ′ ) e (2 √− M − v − [ y ′ ]) dx ′ dy ′ = δ R, ˜ R (det v ) r det(4 M ) − n − t . Here, in the last identity, we used the formula in [Zi 89, p.211, l.19-20]. ⊓⊔ Proposition 4.3.
We have D t ( φ M , ψ M ; s + k − n + ( t − r − / δ t,n M ) − n − t det( M ) − s − k + n − ( t − r − / − r ( n − t )2 t ( s + k − n +( t − r − / × X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) | det(4 N − M − [ t R ]) s + k − n +( t − r − / × Z Γ n − t \ H n − t Z D t ( τ ) X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) g R,N ( τ , z ′ ) × det( v ) k − ( n + r +1) e (2 i (cid:18) N − M − [ t R ] (cid:19) v − [ y ′ ]) dx ′ dy ′ du dv . (4.4) ANKIN-SELBERG METHOD FOR JACOBI FORMS 25
Proof.
We put D r ( τ ) = C ( n − t,r ) / ( τ Z ( n − t,r ) + Z ( n − t,r ) ). We have h φ N , ψ N i = Z Γ Jn − t,t + r \ D n − t,t + r φ N ( τ , z ′ , z ′ ) ψ N ( τ , z ′ , z ′ ) e (2 i N v − [( y ′ y ′ )]) × det( v ) k − ( n + r +1) du dv dx ′ dy ′ dx ′ dy ′ = 1 + δ t,n Z Γ n − t \ H n − t Z D t ( τ ) Z D r ( τ ) φ N ( τ , z ′ , z ′ ) ψ N ( τ , z ′ , z ′ ) e (2 i N v − [( y ′ y ′ )]) × det( v ) k − ( n + r +1) dx ′ dy ′ dx ′ dy ′ du dv = 1 + δ t,n Z Γ n − t \ H n − t Z D t ( τ ) Z D r ( τ ) X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) ϑ M ,R ,R ( τ , z ′ , z ′ ) × X ˜ R mod Z ( n − t,r ) (2 M ) g ˜ R,N ( τ , z ′ ) ϑ M ,R , ˜ R ( τ , z ′ , z ′ ) e (2 i N v − [( y ′ y ′ )]) × det( v ) k − ( n + r +1) dx ′ dy ′ dx ′ dy ′ du dv , where we write R = (cid:18) R R (cid:19) and ˜ R = (cid:18) ˜ R R (cid:19) . Since N = (cid:18) N R t R M (cid:19) = (cid:18) N − M − [ t R ] 00 M (cid:19) (cid:20)(cid:18) t M − t R r (cid:19)(cid:21) , we obtain h φ N , ψ N i = 1 + δ t,n Z Γ n − t \ H n − t Z D t ( τ ) Z D r ( τ ) X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) ϑ M ,R ,R ( τ , z ′ , z ′ ) × X ˜ R mod Z ( n − t,r ) (2 M ) g ˜ R,N ( τ , z ′ ) ϑ M ,R , ˜ R ( τ , z ′ , z ′ ) × e (2 i (cid:18) N − M − [ t R ] (cid:19) v − [ y ′ ]) e (2 i M v − (cid:20) y ′ R M − + y ′ (cid:21) ) × det( v ) k − ( n + r +1) dx ′ dy ′ dx ′ dy ′ du dv . Due to Lemma 4.2 we obtain h φ N , ψ N i = 1 + δ t,n Z Γ n − t \ H n − t Z D t ( τ ) X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) g R,N ( τ , z ′ ) × e (2 i (cid:18) N − M − [ t R ] (cid:19) v − [ y ′ ]) × det( v ) k − ( n + r +1) det(2 M ) − n − t − r ( n − t )2 dx ′ dy ′ du dv . Therefore we have D t ( φ M , ψ M ; s + k − n + ( t − r − / δ t,n M ) − n − t − r ( n − t )2 × X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) | det( N ) s + k − n +( t − r − / × Z Γ n − t \ H n − t Z D t ( τ ) X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) g R,N ( τ , z ′ ) × det( v ) k − ( n + r +1) e (2 i (cid:18) N − M − [ t R ] (cid:19) v − [ y ′ ]) dx ′ dy ′ du dv = 1 + δ t,n M ) − n − t det( M ) − s − k + n − ( t − r − / − r ( n − t )2 t ( s + k − n +( t − r − / × X N ∈ B t,r ( Z ) \ L + t,r ( M ) | ǫ t,r ( N ) | det(4 N − M − [ t R ]) s + k − n +( t − r − / × Z Γ n − t \ H n − t Z D t ( τ ) X R mod Z ( n − t,r ) (2 M ) f R,N ( τ , z ′ ) g R,N ( τ , z ′ ) × det( v ) k − ( n + r +1) e (2 i (cid:18) N − M − [ t R ] (cid:19) v − [ y ′ ]) dx ′ dy ′ du dv . We conclude this proposition. ⊓⊔ By comparing the identity (4.2) with the identity (4.4), we obtain Proposition 2.3.
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