A decomposition of the Fourier-Jacobi coefficients of Klingen Eisenstein series
aa r X i v : . [ m a t h . N T ] J u l A DECOMPOSITION OF THE FOURIER-JACOBICOEFFICIENTS OF KLINGEN EISENSTEIN SERIES
THORSTEN PAUL, RAINER SCHULZE-PILLOT
Abstract.
We investigate the relation between Klingen’s decomposition ofthe space of Siegel modular forms and Dulinski’s analogous decomposition ofthe space of Jacobi forms. Introduction
In analogy to the decomposition of the space of Siegel modular forms of fixed weightand degree into the space of cusp forms and spaces of Eisenstein series of Klingentype associated to cusp forms, Dulinski showed in [3] that the space of Jacobi formsof fixed weight, degree and index admits a natural decomposition into a direct sumof the space of cusp forms and certain spaces of Jacobi Eisenstein series of Klingentype. In [2], B¨ocherer studied how the Fourier-Jacobi coefficients of square freeindex of a Klingen Eisenstein series of degree 2 behave under this decomposition,i. e., how one can identify the components in Dulinski’s decomposition of theseFourier-Jacobi coefficients. In particular, whereas cusp forms have cuspidal Fourier-Jacobi coefficients and the Siegel Eisenstein series has Siegel-Jacobi Eisenstein seriesas Fourier-Jacobi coefficients, he showed that the Fourier-Jacobi coefficients of theKlingen Eisenstein series of degree 2 attached to elliptic cusp forms have both acuspidal and an Eisenstein series part.We continue this investigation here, using a different method, and obtain an explicitdescription of the components for arbitrary degree and index. Again, one sees thatmore than one component appears.This article and the talk at the RIMS workshop “Automorphic Forms and RelatedTopics” in February 2017 on this topic by the second author on which it is basedgive an overview of the work of the first author in his doctoral dissertation [7]written at Universit¨at des Saarlandes under the supervision of the second author.Most of the proofs are only sketched, we refer to the dissertation for full details.All results are due to to the first author, the second author takes responsibility forthe present write-up and all possible mistakes in it. We thank the RIMS and Prof.Nagaoka, who organized the workshop, for the opportunity to present our work.2.
Preliminaries
For the basic notions of the theory of Siegel modular forms we refer to [4, 6], forJacobi forms to [3]. In particular, we consider for k > n + 1 the decomposition M kn = ⊕ nm =0 M kn,m of the space of Siegel modular forms of weight k and degree n for the full modular group Sp n ( Z ) into the spaces M kn,m generated by Eisensteinseries E kn,m ( f ) of Klingen type associated to a cusp form f ∈ M km . For F ∈ M kn wedenote its Fourier coefficient at the symmetric matrix T by A ( F, T ), here T runsover the set d Mat sym n ( Z ) of positive definite half integral symmetric matrices of size n with integral diagonal. For n ′ < n and g = (cid:0) A BC D (cid:1) ∈ Sp n ′ ( R ) ⊆ GL n ′ ( R ) we write g ↑ n = A B
00 1 n − n ′ C D
00 0 0 1 n − n ′ , g ↓ n = n − n ′ A B n − n ′ C D , for U ∈ GL n ( R ) write L ( U ) = (cid:0) t U − U (cid:1) ∈ Sp n ( R ).We let C n,r ⊆ Sp n ( Z ) denote the intersection with Sp n ( Z ) of the maximal parabolic P n,r ( Q ) of Sp n ( Q ) ⊆ GL n ( Q ) characterized as the set of g = ( g ij ) ∈ Sp n ( Q ) with g ij = 0 for i > n + r, j ≤ n + r and J n,r ⊆ C n,r (the Jacobi group of degree( n, r )) asthe set of elements of C n,r with an ( n − r ) × ( n − r ) identity matrix in the lower righthand corner. Notice that, with n = r + r , Dulinski [3] writes J r ,r ⊆ C r + r ,r for this group.For s ≤ r we divide an n × n -matrix into blocks of sizes (cid:18) s × s s × ( r − s ) s × ( n − r )( r − s ) × s ( r − s ) × ( r − s ) ( r − s ) × ( n − r )( n − r ) × s ( n − r ) × ( r − s ) ( n − r ) × ( n − r ) (cid:19) and let Q r,n − rs = { (cid:18) A BC D (cid:19) ∈ Sp n ( Z ) | C = ∗ , D = ∗ ∗ ∗ ∗ ∗ n − r } . With a block division of type (cid:18) s × s s × ( n − r ) s × ( r − s )( n − r ) × s ( n − r ) × ( n − r ) ( n − r ) × ( r − s )( r − s ) × s ( r − s ) × ( n − r ) ( r − s ) × ( r − s ) (cid:19) we let˜ Q r,n − rs = { (cid:18) A BC D (cid:19) ∈ Sp n ( Z ) | C = ∗ , D = ∗ ∗ ∗ r − s ∗ ∗ } . For n = r + r and T ∈ d Mat sym r ( Z ) we denote by J kr ,r ( T ) the space of Jacobi formsof weight k , degree ( r , r ) and index T (which have good transformation behaviorunder the Jacobi group J n,r ). A Siegel modular form then has a Fourier-Jacobiexpansion F ( Z ) = X T ∈ ˜ M sym r ( Z ) φ T ( z , z ) e ( T z ) = X T φ ( T ) ( Z ) , with Fourier-Jacobi coefficients φ T ∈ J kr ,r ( T ) of degree ( r , r ), index T andweight k , where Z = (cid:0) z z t z z (cid:1) is in the Siegel upper half plane H n of degree n with z ∈ H r , z ∈ H r , z ∈ Mat r ,r ( C ).By Theorem 2 of [3] the space J kr ,r ( T ) has a decomposition J kr ,r ( T ) = r M s =0 J k ( r ,r ) ,s ( T ) , where the elements of J k ( r ,r ) ,s ( T ) are Jacobi Eisenstein series of Klingen typeassociated to Jacobi cusp forms of degree ( s, r ) with varying index T ′ for which T ′ [ U ] = T for some integral matrix U . Dulinski defines these Jacobi Eisensteinseries of Klingen type only for index T of maximal rank. For T of rank t < r we notice that by [8] the space J r ,r ( T ) is isomorphic to J r ,r ( (cid:0) T
00 0 (cid:1) ) with a T which is positive definite of size t and that this latter space is isomorphic to J r ,t ( T ). These isomorphisms allow to transfer Dulinski’s definitions to index ofarbitrary rank.Our task is then to identify the components in this decomposition of the Fourier-Jacobi coefficients of an Eisenstein series of Klingen type as explicitly as possible. DECOMPOSITION OF THE FOURIER-JACOBI COEFFICIENTS 3 Partial series of the Klingen Eisenstein series
Lemma 3.1.
For ≤ m < n, ≤ r < n and ≤ t ≤ min( n − m, n − r ) let M tn,m,r denote the set of all g = (cid:0) A BC D (cid:1) ∈ Sp n ( Z ) for which the lower right ( n − m ) × ( n − r ) block C of C has rank t .Then the sets M tn,m,r are left C n,m and right C n,r -invariant, and for fixed j, r their(disjoint) union over ≤ t ≤ min( n − m, n − r ) is Sp n ( Z ) .Proof. This is easily checked, see the proof of Proposition 5.2 of [7]. (cid:3)
Proposition 3.2.
Let f ∈ M km be a cusp form. i) For ≤ m < n, < r < n and ≤ t ≤ min( n − m, n − r ) the partialseries H tn,m,r ( f ; Z ) := X γ ∈ C n,m \ M tn,m,r f ( γ h Z i ∗ ) j ( γ, Z ) − k of the Eisenstein series E kn,m ( f ) of Klingen type is well defined and invari-ant under the action H H | k g of g ∈ J n,r . ii) For ≤ m < n one has for each r with ≤ r < n the decomposition E kn,m ( f ) = min( n − m,n − r ) X t =0 H tn,m,r ( f ) . iii) The partial series H tn,m,r ( f ) has a Fourier-Jacobi decomposition H tn,m,r ( f ; Z ) := X T Ψ ( T ) ,tn,m,r ( f ; Z ) = X T Ψ tn,m,r ; T ( f ; z , z ) e ( T z ) , where the Ψ tn,m,r ; T ( f ; z , z ) are Jacobi forms of degree ( r , n − r ) andindex T .Proof. Obvious. The last assertion follows since both the existence of an expansionas given and the transformation behavior of the coefficients in it hold for functionson H n which are J n,r -invariant but not necessarily Siegel modular forms. (cid:3) Remark 3.3.
Divide a matrix M ∈ Mat n ( R ) for < m, r < n into blocks M , M , M , M of sizes j × r, j × ( n − r ) , ( n − m ) × r, ( n − m ) × ( n − r ) respectively.For γ = (cid:0) A BC D (cid:1) let γ ′ be the ( n + m ) × ( n + r ) matrix obtained from γ by removingthe blocks A , A , B , B , D , D in the second block row and the last blockcolumn. Then it can be shown ( [7, Satz 5.24] ) that the set M tn,m,r is the set of all γ ∈ Sp n ( Z ) for which γ ′ has rank m + r + t . In order to compute the partial series given above one needs explicit coset repre-sentatives for C n,m \ M tn,m,r : Theorem 3.4.
Let R s for s ≤ r denote a set of representatives of the doublecosets in L − ( C m + r + t − s,r − s ) \ GL r − s, ∗ m + r + t − s ( Z ) /L − ( J m + r + t − s,r − s ) and R s a setof representatives of the cosets in (cid:18) ∗ ∗ n − m − t + s − r,m + t + s ∗ (cid:19) ∈ GL n − r ( Z ) }\ GL n − r ( Z ) , where GL r − s, ∗ m + r + t − s ( Z ) denotes the set of matrices in GL m + r + t − s ( Z ) for which the ( r − s ) × ( r − s ) block in the lower left corner has full rank r − s .For u ∈ GL r − s, ∗ m + r + t − s ( Z ) we put ˆ u = s u
00 0 1 n + s − m − t − r ∈ GL n ( Z ) T. PAUL, R. SCHULZE-PILLOT and for u ′ ∈ GL n − r ( Z ) we put ˜ u ′ = (cid:0) r u ′ (cid:1) ∈ GL n ( Z ) .Then a set of representatives of the cosets in C n,m \ M tn,m,r is given by the matrices γ ↑ n L (ˆ u ) γ L (˜ u ′ ) , where for s running from max( r + m + t − n, to min( j, r ) one lets u run through R s and u ′ through R s , γ runs through a set of representatives for C m + t,m \ M tm + t,m,s and γ through a set of representatives of J ↑ n m + t + r − s,r ∩ L (ˆ u − )( ˜ Q r,m + t − ss ) ↑ n L (ˆ u ) \ J ↑ n m + t + r − s,r . Proof.
This is Satz 5.21 of [7]. The rather technical proof occupies most of Section5. (cid:3) The Fourier-Jacobi coefficients of the partial series
Lemma 4.1.
Let f ∈ M km be a cusp form. With the notations of Theorem 3.4 let s, u, u ′ be fixed and let γ , γ run through the sets specified there.Then the partial sum X γ X γ f ( γ ↑ n L (ˆ u ) γ L (˜ u ′ ) h Z i ∗ ) j ( γ ↑ n L (ˆ u ) γ L (˜ u ′ ) , Z ) − k has a Fourier-Jacobi expansion of degree ( r , r ) with coefficients in J ( r ,r ) ,s ( T ′ ) whose index T ′ has rank m + t − s .In particular, for m + t = n and s = r the T ′ occurring have maximal rank r andthe Fourier-Jacobi coefficients are cusp forms.Proof. The first part of the assertion is formulated on p. 57 of [7] before Lemma6.3, its proof uses Lemma 6.3, 6.4, 6.6., where Lemma 6.6 is the second part of ourassertion. (cid:3)
Theorem 4.2. i) The partial series H tn,m,r ( f ) has a Fourier-Jacobi expan-sion whose coefficient Ψ( T ) := Ψ tn,m,r ; T ( f ) at T is in J k ( r ,r ) ,m + t − rk( T ) ( T ) . ii) Let φ ( T ) denote the Fourier-Jacobi coefficient at T ∈ d Mat sym r ( Z ) of de-gree ( r , r ) of the Eisenstein series E n,m ( f ) and let Ψ( T ) be as in i).Then Ψ( T ) is the component φ ( r ,r ) ,m + t − rk( T ) ( T ) of φ ( T ) in the space J k ( r ,r ) ,m + t − rk( T ) ( T ) in Dulinski’s decomposition.Proof. The first assertion is proven in [7] in the calculation following equation (6.2)on page 60 by using the lemma above and carrying out the summation over u, u ′ from the set of representatives given in Theorem 3.4. The second assertion followssince the components in Dulinski’s decomposition are uniquely determined and E n,m ( f ) is the sum of the partial series H tn,m,r ( f ). (cid:3) Remark 4.3.
In particular, we see that only the spaces J k ( r ,r ) ,s ( T ) with m − rk( T ) ≤ s ≤ min( n − rk( T ) , m + r − rk( T )) . For m = n the lower bound and rk( T ) ≤ r give s ≥ r , hence s = r , i.e., the Fourier-Jacobi coefficients of a cuspform are Jacobi cusp forms, which is trivial.For m = 0 we obtain s ≤ r − rk( T ) , so the Fourier-Jacobi coefficients with index ofmaximal rank of the Siegel Eisenstein series are Jacobi Eisenstein series of Siegeltype, which is known from [1] . For rk( T ) < r the Fourier-Jacobi coefficient ofdegree ( r , r ) with index T is essentially the Fourier-Jacobi coefficient of degree ( r , rk( T )) of the Siegel Eisenstein series of degree n − ( r − rk( T )) at a matrix ofmaximal rank, so it is again a Jacobi Eisenstein series of Siegel type. DECOMPOSITION OF THE FOURIER-JACOBI COEFFICIENTS 5 Pullbacks and Fourier expansions
Having identified the components in Dulinski’s decomposition of the Fourier-Jacobiexpansion of the Eisenstein series E n,m ( f ) in terms of the coefficients of the partialseries H tn,m,r we turn now to the task of computing their Fourier expansion ex-plicitly. For this we adapt and refine ideas from [1] to our situation and divide theseries defining the Siegel Eisenstein series of degree n + m into certain subseries inway similar to what we did in Section 3. Lemma 5.1.
Divide for j, r ≤ n a matrix M ∈ Mat n + m ( R ) into blocks of types j × r j × ( n − r ) j × j ( n − m ) × r ( n − m ) × ( n − r ) ( n − m ) × jj × r j × ( n − r ) j × j and denote these blocks by M , . . . , M . For γ = (cid:0) A BC D (cid:1) ∈ Sp n + m ( Z ) let ˆ γ = (cid:18) C C D C C D C C D (cid:19) ∈ Mat n + m,n + r ( Z ) and denote for m + r ≤ v ≤ min( n + m, n + r ) the set of all γ ∈ Sp n + m ( Z ) with rk(ˆ γ ) = v by X vn,m,r .then X vn,m,r is left invariant under C n + m, and right invariant under Sp ↓ n + m m ( Z ) ,and Sp n + m ( Z ) is the disjoint union of the X vn,m,r for m + r ≤ v ≤ min( n + m, n + r ) .Proof. This is Proposition 7.2 of [7]. Since ˆ γ is obtained from γ by deleting n − r columns and n − m rows, its rank v must be between m + r and min( n + m, n + r ).the assertions about left and right invariance are checked easily. (cid:3) We need an explicit set of representatives of the cosets in C n + m, \ X vn,m,r . For thiswe recall that by [5] a set of representatives for C n + m, \ Sp n + m ( Z ) is given by theproducts g j,M ( g ′ j, ) ↑ n + m g ′↑ n + m j (( g ′′ j, ) ↑ m ) ↓ n + m ( g ′′ j ) ↓ n + m , where j runs from 0 to m , and for any such j we let g ′ j, run through Sp j ( Z ), g ′ j through a set of representatives for C n,j \ Sp n ( Z ) and g ′′ through a set of represen-tatives for C m,j \ Sp m ( Z ). Moreover, with M ′ running through the j × j elementarydivisor matrices and M = (cid:0) M ′
00 0 (cid:1) ∈ M m,n ( Z ) we let g j,M = (cid:18) n m t M n M m (cid:19) andΓ j ( M ′ ) := Sp j ( Z ) ∩ (cid:0) M ′− M ′ (cid:1) Sp s ( Z ) (cid:0) M ′− M ′ (cid:1) and let g ′′ j, run through a set ofrepresentatives of Γ j ( M ′ ) \ Sp j ( Z ). Proposition 5.2.
A set of representatives for C n + m, \ X vn,m,r is obtained from therepresentatives above by restricting g ′ j to a set of representatives of C n,j \ M v − r − jn,j,r .Proof. A straightforward computation shows that indeed these are precisely theproducts which are in X vm,n,r , see Satz 7.4 of [7] and the proof given there. (cid:3) Theorem 5.3.
For < s ≤ m let ( f s,ν ) ν be an orthonormal basis of Heckeeigenforms for the space of cusp forms of degree s and weight k . We set A ks := π s ( s − (4 π ) s ( s +1)2 − sk Q si =1 Γ( k − s + i ) and β ( s, k ) = ( − sk s s ( k − s − ) s − Y i =0 π k − i Γ( k − i ) ζ ( k ) − m Y i =1 ζ (2 k − i ) − . For ≤ m, r < n and m + r ≤ v ≤ min( n + m, n + r ) we put G vn,m,r ( Z ) := X γ ∈ C n + m, \ X vn,m,r j ( γ, Z ) − k . T. PAUL, R. SCHULZE-PILLOT
Then for Z ∈ H n , Z ∈ H m the pullback G vn,m,r ( (cid:0) − Z Z (cid:1) ) of G vn,m,r to H n × H m can be written as G vn,m,r (cid:18)(cid:18) − Z Z (cid:19)(cid:19) = m X s =0 c s X ν D f s,ν ( k − s ) E m,s ( f s,ν ; Z ) H v − r − sn,s,r ( f, Z ) , where D f s,ν denotes the standard L -function of the Hecke eigenform f s,ν (and thisfactor doesn’t occur for s = 0 ) and where for s > we put c s = 2 β ( s, k ) A ks and set c = 1 .Proof. This follows from the proof of the theorem in Section 5 of [5] and the explicitevaluation of the constants occurring there in [1]. (cid:3)
Corollary 5.4.
For a Hecke eigenform f ∈ M km of Petersson norm one has H v − r − mn,m,r ( f ; Z ) = λ ( f ) − (cid:28) f ( · ) , G vn,m,r ( (cid:18) − Z · (cid:19) ) (cid:29) with λ ( f ) = 2 β ( m, k ) A km D f ( k − m ) as in the theorem above.Proof. This follows since taking the Petersson product with f singles out the sum-mand containing H v − r − mn,m,r ( f, Z ) from the formula in the theorem. (cid:3) By the corollary we can compute the Fourier expansion of our partial series H v − r − mn,m,r ( f )by computing the Petersson product on the right hand side. We will do this adapt-ing again ideas from [1]. Lemma 5.5. i) Let P n,m = (cid:0) m n (cid:1) . Then for l ≤ n the set M ln + m, ,n L ( P n,m ) ∩ X vn,m,r is nonempty only if l ≤ v and X vn,m,r is contained in the (disjoint)union of the M ln + m, ,n L ( P n,m ) for ≤ l ≤ v . ii) With G v,ln,m,r ( Z ) := X M ln + m, ,n L ( P n,m ) ∩ X vn,m,r j ( γ, Z ) − k one has G vn,m,r ( Z ) = P vl =0 G v,ln,m,r ( Z ) . iii) A set of representatives of C n + m, \ M ln + m, ,n L ( P n,m ) ∩ X vn,m,r is given by the x ↑ n + m L ( U ) y ↓ n + m , where x runs through a set of representatives of C l. \ M ll, , , y through a set of representatives of C m, \ Sp m ( Z ) and U through a set ofrepresentatives of ∗ ∗ ∗ n − l,l ∗ ∗ m,l ∗ ∗ \ u u u u u u u u u ∈ GL n + m ( Z ) | rk (cid:18) u u (cid:19) = v − l, rk (cid:18) u u (cid:19) = m , where U has a block division of type (cid:18) l × r l × ( n − r ) l × m ( n − l ) × r ( n − l ) × ( n − r ) ( n − l ) × mm × r m × ( n − r ) m × m (cid:19) .Proof. This is Satz 8.1 of [7]. For the proof one checks which of the representativesof C n + m, \ M ln + m, ,n L ( P n,m ) obtained from Theorem 3.4 are in X vn,m,r , see [7] fordetails. (cid:3) Lemma 5.6.
Let U run through the set of representatives from the previous lemmaand write a matrix in Mat n + m,l ( Z ) as (cid:18) w w w (cid:19) , where w , w , w have r, n − r, m rowsrespectively. Then the matrix formed by the first l columns of U − runs through aset of representatives of w w w primitive | rk (cid:18) w w (cid:19) = l, rk (cid:18) w w (cid:19) = v − r /GL l ( Z ) . DECOMPOSITION OF THE FOURIER-JACOBI COEFFICIENTS 7
Proof.
This is Lemma 8.4 a) of [7]. The proof uses computations from Lemma 5.7and Remark 5.8 of [7]. (cid:3)
Lemma 5.7.
We denote by a l ( T ) the Fourier coefficient at T of the Siegel Eisen-stein series of degree l and weight k and write A + l for the set of positive definitematrices in d Mat sym l ( Z ) . Then G v,ln,m,r ( Z ) = X T ∈A + l X w ,w X w X y a l ( T ) e ( T ( y ↓ n + m h Z i ) ∗ w w w ) j ( y, z ) − k , where y runs through a set of representatives of C m, \ Sp m ( Z ) , w , w , w are as inthe previous lemma, and z ∈ H m is the lower left m × m corner of Z .Proof. We carry out the summation over the coset representatives given in Lemma5.5, expanding the automorphy factor j using its cocycle relation and j ( L ( U ) , · ) = 1.The summation over x gives then by [1, Lemma 3] X T ∈A + l X U X y a l ( T ) e ( T ( L ( U ) y ↓ n + m h Z i ) ∗ ) j ( Y, z ) − k , Using L ( U ) y ↓ n + m h Z i = y ↓ n + m h Z i [ U − ] and writing the upper left block of U − interms of w , w , w as in the previous lemma, we obtain the assertion. (cid:3) Lemma 5.8.
Write Z m × ls = { w ∈ Mat m,l ( Z ) | rk( w ) = s } , Z m × ls, = { (cid:0) ∗ m − s,l (cid:1) ∈ Z m × ls } .Let GL m ( Z ) s = { (cid:0) ∗ ∗ m − s,s ∗ (cid:1) ∈ GL m ( Z ) } and GL m ( Z ) s = { (cid:0) s ∗ m − s,s ∗ (cid:1) ∈ GL m ( Z ) } .Let w ′ run through a set of representatives of GL m ( Z ) s \ Z m × ls, and w ′′ through aset of representatives of GL m ( Z ) /GL m ( Z ) s . Then every element of Z m × ls has aunique expression as a product w ′′ w ′ , and all these products are in Z m × ls .For w , w fixed, the matrix (cid:18) w w w ′ (cid:19) is primitive if and only if (cid:18) w w w ′′ w ′ (cid:19) is primitive,and one has rk (cid:0) w w ′ (cid:1) = rk (cid:0) w w ′′ w ′ (cid:1) .Proof. This is Lemma 8.4 b) of [7]. It is clear that any u ∈ Z m × ls can be written as ww ′ with w ∈ GL m ( Z ) and w ′ ∈ Z m × ls, , where w ′ is unique up to multiplicationwith an element of GL m ( Z ) s from the left. Moreover, if w ′ is fixed, w is uniqueup to right multiplication by an element of GL m ( Z ) s . The second assertion isobvious. (cid:3) Lemma 5.9. i) With notations as in Lemma 5.7 the sum X w e ( T ( y ↓ n + m h Z i ) ∗ w w w ) j ( y, z ) − k for T, y, w , w fixed is equal to X s X w ′ X w ′′ e ( T ( L ( w ′′− ) ↓ n + m y ↓ n + m h Z i ) ∗ w w w ′ ) j ( y, z ) − k , where s runs from to min( l, m ) , w ′ runs over the set of matrices in Z m × ls for which (cid:0) w w ′ (cid:1) has rank v − r , and w ′′ runs over a set of representativesof GL m ( Z ) /GL m ( Z ) s . T. PAUL, R. SCHULZE-PILLOT ii)
For a block diagonal matrix Z = (cid:0) Z Z (cid:1) with Z ∈ H n , Z ∈ H m one has G v,ln,m,r ( Z ) = X T ∈A + l a ( T ) X w ,w e ( T (cid:20) t (cid:18) w w (cid:19)(cid:21) Z ) min( l,m ) X s =0 ǫ ( s ) X w ′ g km,s ( Z , T [ t w ′ ]) , with ǫ (0) = 1 and ǫ ( s ) = 2 otherwise, where the summations over w , w , w ′ are as before and where the Poincar´e series g km,s ( Z , T [ t w ′ ]) is given by g km,s ( Z , T ↑ ) = X γ ∈ U m,s \ Sp m ( Z ) e ( T ↑ ( γ h Z i )) j ( γ, Z ) − k , where U m,s ⊆ C m, is the group of matrices ( A B D ) ∈ C m, ⊆ Sp m ( Z ) , with A = (cid:0) ± s ∗ ∗ (cid:1) .Proof. For a) we use the decomposition w = w ′′ w ′ from the previous lemma andorder the sum over w ′ by the rank s of w ′ . For b), with U + m,s = { ( A B D ) ∈ U m,s | A = (cid:0) s ∗ ∗ (cid:1) } we see that L ( w ′′− ) runs through a set of representatives of U + m,s \ C m, ,so that L ( w ′′− ) y runs through a set of representatives ˜ y of U + m,s \ Sp m ( Z ) whichsatisfy j (˜ y, Z ) = j ( y, Z ) for ˜ y = L ( w ′′− ) y and Z ∈ H m . For s = 0 one has U m,s = U + m,s , for s > U m,s is the union of two cosets modulo U + m,s , which explains the factor ǫ ( s ). The expression obtained in a) then transforms(with z = Z ) to a l ( T ) e ( T [ t (cid:18) w w (cid:19) ] Z ) X s ǫ ( s ) X w ′ X ˜ y ∈ U m,s \ Sp m ( Z ) e ( T [ t w ′ ]˜ y h Z i ) j (˜ y, Z ) − k , and the sum over ˜ y equals the Poincar´e series g km,s ( Z , T [ t w ′ ]) (notice that T [ t w ′ ]has the block diagonal shape required). (cid:3) Theorem 5.10.
Let f ( Z ) = P S ∈ d Mat sym m ( Z ) b ( S ) e ( SZ ) ∈ M km be a cusp form withFourier coefficients b ( S ) .Then the Fourier coefficient of H tn,m,r ( f ) at R ∈ d Mat sym n ( Z ) with rk( R ) = l is β ( m, k ) − D f ( k − m ) − X T ∈A + l X w ,w X w ′ b ( T [ t w ′ ]) det( T [ t w ′ ]) m +12 − k with β ( m, k ) , D f ( k − m ) as in Theorem 5.3.In the sum, (cid:18) w w w ′ (cid:19) ∈ Mat n + m,l ( Z ) with w ∈ Mat r,l ( Z ) , w ∈ Mat n − r,l ( Z ) , w ′ ∈ Mat m,l ( Z ) runs through those primitive elements of a set of representatives of Mat n + m,l ( Z ) /GL l ( Z ) which satisfy R = T [ t (cid:18) w w (cid:19) ] , rk( w ′ ) = m, rk (cid:18) w w ′ (cid:19) = t + m. Proof.
By our previous results only the Petersson product h f ( · ) , G t + m + r,ln,m,r (cid:0) − Z · (cid:1) i contributes to the Fourier coefficient of H tn,m,r ( f ) at a matrix R of rank l , andwe have reduced the computation of this Petersson product to the product withthe Poincar´e series g km,s ( Z , T [ t w ′ ]). For s < m , these are known to be orthogonalto cusp forms (being Eisenstein series of Klingen type), for s = m the Peters-son product has been computed in [6, p.90,94]. Plugging in that result gives theassertion. (cid:3) Remark 5.11.
It should be noticed that the sum in the formula of the theorem isa finite sum.
DECOMPOSITION OF THE FOURIER-JACOBI COEFFICIENTS 9
Corollary 5.12.
As in Theorem 4.2 denote by φ ( R ) m + t − rk( R ) the component in J k ( r ,r ) ,m + t − rk( T ) ( T ) of the Fourier-Jacobi coefficient at the r × r -symmetric ma-trix R of the Klingen Eisenstein series E kn,m ( f ) .Then the Fourier coefficient at ( R , R ) of φ ( R ) m + t − rk( R ) is given by the formula inthe previous theorem for the Fourier coefficient of H tn,m,r ( f ) at R = (cid:0) R R t R R (cid:1) .Proof. This follows directly from the previous theorem and Theorem 4.2. (cid:3) The case n = 2We consider here r = r = r = m = 1, i.e., we study the Klingen Eisenstein seriesattached to an elliptic cusp form f ( z ) = P ∞ n =1 b ( n ) e ( nz ), which we assume to be aHecke eigenform.One obtains here β ( m, k ) − D f ( k − − = ζ (1 − k ) ζ (2 k − L ( f, k − − , where L ( f, s ) = ζ (2 s − sk + 2) P ∞ n =1 b ( n ) n − s is the symmetric square L -function of f .We have to consider the H t , , for t = 0 , t = 1. For t = 1 our computation inthe previous paragraph shows that H , , has nonzero Fourier coefficients only atmatrices R = (cid:0) r r r r (cid:1) of rank 2. The Fourier coefficient at such an R is thencomputed as12 ζ (1 − k ) ζ (2 k − L ( f, k − − X a,b,d a ( T ) × X u,v b ( u t + uvt + v t )( u t + uvt + v t ) − k , where the summation over a, b, d runs over a, d > ≤ b < a such that T = (cid:18) t t t t (cid:19) = (cid:18) a b d (cid:19) R (cid:18) a b d (cid:19) − ∈ d Mat sym2 ( Z )and the summation over u, v runs over u, v ∈ Z satisfying u = 0 , gcd( u, a ) =gcd( av − ub, d ) = 1. If − det(2 R ) is a fundamental discriminant only a = d = 1occurs, and one checks that this agrees with the result in [2]. One can proceed fromhere to obtain asymptotic formulas as in [2]. For details see [7, Section 9]. References [1] S. B¨ocherer: ¨Uber die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. Math. Z. 183(1983), no. 1, 21–46.[2] S. B¨ocherer: On the Fourier-Jacobi-coefficients of Eisenstein series of Klingen type. Numbertheory, 7–16, Ramanujan Math. Soc. Lect. Notes Ser., 15, Ramanujan Math. Soc., Mysore,2011.[3] J. Dulinski: A decomposition theorem for Jacobi forms. Math. Ann. 303 (1995), no. 3,473–498.[4] E. Freitag: Siegelsche Modulfunktionen. Grundlehren der math. Wiss. 254, Springer-Verlag1983[5] P. Garrett: Pullbacks of Eisenstein series; applications. Automorphic forms of several vari-ables (Katata, 1983), 114–137, Progr. Math., 46, Birkh¨auser Boston, Boston, MA, 1984[6] H. Klingen: Introductory lectures on Siegel modular forms. Cambridge Studies in AdvancedMathematics, 20. Cambridge University Press, Cambridge, 1990.[7] T. Paul:[8] C. Ziegler: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59 (1989),191–224.[1] S. B¨ocherer: ¨Uber die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. Math. Z. 183(1983), no. 1, 21–46.[2] S. B¨ocherer: On the Fourier-Jacobi-coefficients of Eisenstein series of Klingen type. Numbertheory, 7–16, Ramanujan Math. Soc. Lect. Notes Ser., 15, Ramanujan Math. Soc., Mysore,2011.[3] J. Dulinski: A decomposition theorem for Jacobi forms. Math. Ann. 303 (1995), no. 3,473–498.[4] E. Freitag: Siegelsche Modulfunktionen. Grundlehren der math. Wiss. 254, Springer-Verlag1983[5] P. Garrett: Pullbacks of Eisenstein series; applications. Automorphic forms of several vari-ables (Katata, 1983), 114–137, Progr. Math., 46, Birkh¨auser Boston, Boston, MA, 1984[6] H. Klingen: Introductory lectures on Siegel modular forms. Cambridge Studies in AdvancedMathematics, 20. Cambridge University Press, Cambridge, 1990.[7] T. Paul:[8] C. Ziegler: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59 (1989),191–224.