SIEGEL MODULAR FORMS AND THETA SERIES ATTACHED TO QUATERNION ALGEBRAS II

SIEGFRIED BOECHERER AND RAINER SCHULZE-PILLOT

Abstract. We continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

Introduction

We resume in this article our study of Yoshida’s lifting from [5]. This lifting associates a Siegel modular form Y^{(2)}(φ_{1}, φ_{2}) of degree 2 to a pair φ_{1}, φ_{2} of automorphic forms on the adelic multiplicative group of a definite quaternion algebra D over Q. This pair corresponds under Eichler’s correspondence to a pair f, g of elliptic modular forms of weights k_{1} = 2, k_{2} and the same square free level N; it is a theta lifting from the orthogonal group of D equipped with the norm form to the group Sp_{2}. In the case that also k_{2} = 2 holds we proved that this lifting is nonzero if both f, g are cusp forms with the same eigenvalues under the Atkin-Lehner involutions; if φ_{2} is constant (and hence g is an Eisenstein series) the vanishing of Y^{(2)}(φ_{1}, 1) depends on the central critical value of the L-function of f. We extended the construction to arbitrary pairs of weights k_{1}, k_{2} in [6], where we also gave the analogous construction of a lifting Y^{(n)}(φ_{1}, φ_{2}) to Siegel modular forms of higher degree. These higher degree liftings are in general vector valued Siegel modular forms; Y^{(n)}(φ_{1}, φ_{2}) is mapped to Y^{(n−1)}(φ_{1}, φ_{2}) by Siegel’s ϕ-operator. The goal of this article is to extend our nonvanishing results from [5] to this more general situation. The method of proof is essentially the same: To prove the nonvanishing of Y^{(2)}(φ_{1}, φ_{2}) we show that Y^{(3)}(φ_{1}, φ_{2}) is not cuspidal. This is done by studying the analytic behaviour of its standard L-function in s = 1 in two ways: Once by giving an integral representation (assuming Y^{(3)}(φ_{1}, φ_{2}) to be cuspidal) and once by computing the L-function in terms of the Satake parameters of f, g using the construction of Y^{(3)}(φ_{1}, φ_{2}) as a theta lifting. The integral representation which we use is the same as in [5] using the doubling (or pullback) method and an Eisenstein series of weight 2, modified by some differential operator. In the generality needed for our purpose these differential operators have been introduced and studied by Ibukiyama [17]. On the theta lifting side we arrive after a careful computation of the factors occurring at the bad places at the study of the value at s = 1 of the normalized GL_{2} × GL_{2} − L− function (with functional equation under s ↦ 1 − s) associated to f, g. A result of Ogg [23] and Shahidi [24] gives the nonvanishing of this value if f, g are cuspidal and nonproportional, which contradicts the results obtained from the integral representation (and hence our assumption that Y^{(3)}(φ_{1}, φ_{2}) is cuspidal). The case where φ_{2} is constant is more delicate and leads to the L-function condition mentioned. To summarize our results, let f, g be primitive cusp forms of square free levels N_{f}, N_{g} (with N = lcm(N_{f}, N_{g}), gcd(N_{f}, N_{g}) ≠ 1), weights k_{1} ≥ k_{2} and Haupttyp for Γ_{0}(N_{f,g}) (eigenforms of all Hecke operators) and assume that for all p dividing gcd(N_{f}, N_{g}) the eigenvalues of f, g under the Atkin-Lehner-involution w_{p} are the same. Then we can associate to f, g a nonzero vector valued Siegel cusp form of degree 2, whose Satake parameters are computed from those of f, g. If k_{2} = 2 this Siegel modular form is scalar valued of weight 1 + \frac{k_{1}}{2} ; in particular for k_{1} = 4, k_{2} = 2 we obtain cusp forms of weight 3. These are of interest in geometry since they correspond to holomorphic differential forms of top degree on the quotient of the Siegel upper half space of degree 2 by the group Γ_{0}(N). It should be noted that we obtain in fact different constructions for each subset of odd cardinality of the set of prime divisors of gcd(N_{f}, N_{g}) and that the resulting Siegel modular forms are pairwise orthogonal. Most of this article was written while both authors were guests of the MSRI in Berkeley during its special year on automorphic forms. We wish to thank the MSRI for its hospitality and financial support. R. Schulze-Pillot was also supported by Deutsche Forschungsgemeinschaft during a visit of one month at MSRI and was a guest of the Max-Planck-Institut für Mathematik in Bonn in the final stage of the preparation of this manuscript.

Both authors supported by MSRI, Berkeley (NSF-grant DMS-9022140). R. Schulze-Pillot also supported by the Deutsche Forschungsgemeinschaft and by the Max-Planck-Institut für Mathematik, Bonn.

1. Yoshida’s lifting

Yoshida’s lifting [30, 31] associates a Siegel modular form of degree 2 to a pair f, g of elliptic modular forms of even weight, where one of f, g has weight 2. Its generalization to arbitrary even weights considered in [6] yields vector valued Siegel modular forms. As explained in the introduction we want to generalize our results on the nonvanishing of Yoshida’s lifting to this more general situation. For this, we need to recall a few notations and results from [5, 6].

SIEGEL MODULAR FORMS, THETA SERIES, AND QUATERNION ALGEBRAS II 3

Let D be a definite quaternion algebra over Q and R an Eichler order of square free level N in D, i. e. , the completion R_{p} is a maximal order in D_{p} for the primes p ramified in D and is of index p in a maximal order (i. e. R_{p} is conjugate to the set of matrices ( \begin{matrix}a & b \\ c & d \end{matrix}) in M_{2}(Z_{p}) with p | c) for the remaining primes p dividing N. We will usually decompose N as N = N_{1}N_{2} where N_{1} is the product of the primes that are ramified in D. On D we have the involution x ↦ \overline{x}, the (reduced) trace tr(x) = x+\overline{x} and the (reduced) norm n(x) = x\overline{x}. For ν ∈ N let U_{ν}^{(0)} be the space of homogeneous harmonic polynomials of degree ν on R^{3} and view P ∈ U_{ν}^{(0)} as a polynomial on D_{∞}^{(0)} = {x ∈ D_{∞}|tr(x) = 0} by putting P (∑_{i=1}^{3} x_{i}e_{i}) = P (x_{1}, x_{2}, x_{3}) for an orthonormal basis {e_{i}} of D_{∞}^{(0)} with respect to the norm form n. The representations τ_{ν} of D_{∞}^{×}/R^{×} of highest weight (ν) on U_{ν}^{(0)} given by

(τ_{ν}(y))(P)(x) = P (y^{−1}xy)

for ν ∈ N give all the isomorphism classes of irreducible rational representations of D_{∞}^{×}/R^{×}. By ⟨⟨ , ⟩⟩ we denote the suitably normalized invariant scalar product in the representation space U_{ν}^{(0)}. The group of proper similitudes of the quadratic form q(x) = n(x) (with associated symmetric bilinear form B(x, y) = tr(xȳ)) on D is isomorphic to (D^{×} × D^{×})/Z(D^{×}) (as algebraic group) via

(x_{1}, x_{2}) ↦ σ_{x_{1},x_{2}} with σ_{x_{1},x_{2}}(y) = x_{1}yx_{2}^{−1},

with the special orthogonal group being the image of

{(x_{1}, x_{2}) ∈ D^{×} × D^{×} | n(x_{1}) = n(x_{2})}.

We denote by H the orthogonal group of (D, n), by H^{+} the special orthogonal group and by K (resp. K^{+}) the group of units (resp. proper units) of the lattice R in D. Let ν_{1} ≥ ν_{2} be given and let τ_{1} = τ_{ν_{1}}, τ_{2} = τ_{ν_{2}} for i = 1, 2. It is then well known that the H^{+}(R)-space U_{ν_{1}}^{(0)} ⊗ U_{ν_{2}}^{(0)} is isomorphic to the H^{+}(R)-space U_{ν_{1},ν_{2}} of C[X_{1}, X_{2}]-valued harmonic forms on D_{∞}^{2} transforming according to the representation of GL_{2}(R) of highest weight (ν_{1} +ν_{2}, ν_{1} −ν_{2}); an intertwining map Ψ has been given in [6, Section 3] (for the theory of harmonic forms see [10, 19]). It is also well known [19] that the representation λ_{ν_{1},ν_{2}} of H^{+}(R) on U_{ν_{1},ν_{2}} is irreducible of highest weight (ν_{1}+ν_{2}, ν_{1}−ν_{2}). If ν_{1} ≠ ν_{2} it can be extended in a unique way to an irreducible representation of H(R) on the space U_{ν_{1},ν_{2},s} := (U_{ν_{1}}^{(0)} ⊗ U_{ν_{2}}^{(0)}) ⊕ (U_{ν_{2}}^{(0)} ⊗ U_{ν_{1}}^{(0)}), denoted by (τ_{1} ⊗ τ_{2})_{s}, whereas for ν_{1} = ν_{2} there are two possible extensions to representations (τ_{1} ⊗ τ_{2})_{±} on U_{ν_{1},ν_{2}}; we denote this space with the representation (τ_{1} ⊗ τ_{2})_{±} on it by U_{ν_{1},ν_{2},±}. For an irreducible rational representation (V_{τ}, τ) (with τ = τ_{ν} as above) of D_{∞}^{×}/R^{×} we denote by A(D_{A}^{×}, R_{A}^{×}, τ) the space of functions φ : D_{A}^{×} → V_{τ} satisfying φ(γxu) = τ(u_{∞}^{−1})φ(x) for γ ∈ D_{Q}^{×} and u = u_{∞}u_{f} ∈ R_{A}^{×}, where R_{A}^{×} = D_{∞}^{×} × ∏_{p} R_{p}^{×} is the adelic group of units of R. It has been discovered by Eichler that these functions are in correspondence with the elliptic modular forms of weight 2 + 2ν and level N. This correspondence can be described as follows: Let D_{A}^{×} = ∪_{i=1}^{r}D^{×}y_{i}R_{A}^{×} be a double coset decomposition with y_{i,∞} = 1 and n(y_{i}) = 1, put I_{ij} = y_{i}Ry_{j}^{−1}, R_{i} = I_{ii} and let e_{i} be the number of units of the order R_{i}. Recall from §5 of [5] that for each p | N we have an element π_{p} ∈ D_{p}^{×} of norm p normalizing R_{p} and that these π_{p} (together with R_{A,f}^{×}) generate the normalizer of R_{A,f}^{×}Q_{A}^{×}/Q_{A}^{×} in D_{A,f}^{×}Q_{A}^{×}/Q_{A}^{×} and satisfy π_{p}^{2} ∈ Q_{p}R_{p}^{×}. Right translation by π_{p} gives an involution w̃_{p} of A(D_{A}^{×}, R_{A}^{×}, τ), which space then splits into common eigenspaces of all these (pairwise commuting) involutions. On the space A(D_{A}^{×}, R_{A}^{×}, τ) we have for p |̸N Hecke operators T̃(p) defined by

T̃(p)φ(x) = ∫_{D_{p}^{×}} φ(xy^{−1})χ_{p}(y) dy

where χ_{p} is the characteristic function of {y ∈ R_{p}|n(y) ∈ pZ_{p}^{×}}. They commute with

the involutions w̃_{p} and are given explicitly by T̃(p)φ(y_{i}) = ∑_{j=1}^{r} B_{ij}(p)φ(y_{j}), where the

Brandt matrix entry B_{ij}(p) is given as

B_{ij}(p) = B_{ij}^{(ν)}(p) = \frac{1}{e_{j}} ∑_{{x∈y_{j} Ry{−1 \atop i} \atop n(x)=p}} p^{ν}τ(x) ,

hence is itself an endomorphism of the representation space U_{ν}^{(0)} of τ. On the space A(D_{A}^{×}, R_{A}^{×}, τ) we have the natural inner product ⟨ , ⟩ defined by integration, it is explicitly given by

⟨φ, ψ⟩ = ∑_{i=1}^{h} \frac{⟨⟨φ(y_{i}), ψ(y_{i})⟩⟩}{e_{i}}.

By abuse of language we call (in the case ν = 0) forms cuspidal, if they are orthogonal to the constant functions with respect to this inner product. From [8, 14, 25, 18] we know then that the essential part A_{ess}(D_{A}^{×}, R_{A}^{×}, τ) consisting of functions φ that are orthogonal to all ψ ∈ A(D_{A}^{×}, (R′_{A})^{×}, τ) for orders R′ strictly containing R is invariant under the T̃(p) for p |̸N and the w̃_{p} for p |̸N and hence has a basis of common eigenfunctions of all the T̃(p) for p |̸N and all the involutions w̃_{p} for p | N. Moreover in A_{ess}(D_{A}^{×}, R_{A}^{×}, τ) strong multiplicity one holds, i.e., each system of eigenvalues of the T̃(p) for p |̸N occurs at most once, and the eigenfunctions are in one to one correspondence with the newforms in the space S^{2+2ν}(N) of elliptic cusp forms of weight 2 + 2ν for the group Γ_{0}(N) that are eigenfunctions of all Hecke operators (if τ is the trivial representation and R is a maximal order one has to restrict here to functions orthogonal to the constant function 1 on the quaternion side in order to obtain cusp forms on the modular forms side). This correspondence (Eichler’s correspondence) preserves Hecke eigenvalues for p |̸N, and if φ corresponds to f ∈ S^{2+2ν}(N) then the eigenvalue of f under the Atkin-Lehner involution w_{p} is equal to that of φ under w̃_{p} if D splits at p and equal to minus that of φ under w̃_{p} if D_{p} is a skew field. An extension of this correspondence to forms φ as above that are not essential but eigenfunctions of all the involutions w̃_{p} has been given in [13, 7]. Lemma 3.1 of [6] now gives the following setup for the Yoshida lifting: Let A(H_{A}^{+}, K_{A}^{+}, τ_{1} ⊗ τ_{2}) be defined in the same way as A(D_{A}^{×}, R_{A}^{×}, τ_{i}). If φ_{i} ∈ A(D_{A}^{×}, R_{A}^{×}, τ_{i}) for i = 1, 2 (with τ_{i} = τ_{ν_{i}} and ν_{1} ≥ ν_{2}) are eigenfunctions of all the involutions w̃_{p} for p | N and of all the Hecke operators T̃_{p} for p ∤ N, we denote by { \atop φ_{1} ⊗ φ_{2}} the function on the adelic special orthogonal group H_{A}^{+} of (D, n) given by

{ \atop φ_{1} ⊗ φ_{2}}(σ_{y_{1},y_{2}}) = ∫_{K_{f}^{+}} φ_{1}(y_{1}k_{1}) ⊗ φ_{2}(y_{2}k_{2})dk,

where k in the finite part K_{f}^{+} of the adelic group of proper units of the lattice R in D is represented as k = σ_{k_{1},k_{2}}. Then { \atop φ_{1} ⊗ φ_{2}} ≠ 0 if and only if φ_{1} and φ_{2} have the same eigenvalues under all the involutions w̃_{p} for the p | N, and one has { \atop φ_{1} ⊗ φ_{2}} ∈ A(H_{A}^{+}, K_{A}^{+}, τ_{1} ⊗ τ_{2}). Let now φ_{1} and φ_{2} be Hecke eigenforms having the same eigenvalues under all the involutions w̃_{p} for the p | N. If (τ_{1} ⊗τ_{2})_{∗} for ∗ denoting +, − or s denotes the possible extensions of τ_{1} ⊗ τ_{2} to an irreducible representation of H_{R} let (φ_{1} ⊗ φ_{2})_{∗} be the unique function in A(H_{A}, K_{A}, (τ_{1} ⊗ τ_{2})_{∗}) whose value at h ∈ H_{A}^{+} is

\frac{1}{2}({ \atop φ_{1} ⊗ φ_{2}}(h) + (τ_{1} ⊗ τ_{2})_{∗}(ι){ \atop φ_{1} ⊗ φ_{2}}(ιhι)),

where ι ∈ H_{R} is the involution in D. Then for ν_{1} ≠ ν_{2} one has (φ_{1} ⊗ φ_{2})_{s} ≠ 0 if and only if { \atop φ_{1} ⊗ φ_{2}} ≠ 0. For ν_{1} = ν_{2} one has (φ_{1} ⊗ φ_{2})_{+} ≠ 0. Moreover in this case one has (φ_{1} ⊗ φ_{2})_{−} = 0 if and only if φ_{1} = 0 or φ_{2} = aφ_{1} for some a ∈ C. The functions (φ_{1} ⊗ φ_{2})_{∗} on the adelic orthogonal group H_{A} thus constructed can be lifted to Siegel modular forms of degree n, we call this lifting Yoshida’s lifting, denoted by Y^{(n)}(φ_{1}, φ_{2}, ∗). To describe it explicitly we recall from [19, 29] that for any irreducible representation (λ, U_{λ}) of H(R) the space H_{n}(λ) of pluriharmonic polynomials P : M_{m,n}(C) → U_{λ} such that P (h^{−1}x) = λ(h^{t})P (x) for all h ∈ O_{m} is zero or (under the right action of GL_{n}(C) on the variable) isomorphic to an irreducible representation (ρ_{n}(λ), W_{ρ_{n}(λ)}) of GL_{n}(C). In the latter case the space H_{q}(ρ_{n}(λ)) consisting of all q-pluriharmonic polynomials P : M_{m,n}(C) → W_{ρ_{n}(λ)} such that P (xg) = (ρ_{n}(λ)(g^{t}))P (x) for all g ∈ GL_{n}(C) is isomorphic to (U_{λ}, λ) as a representation space of H_{R}.

We denote by P_{n,∗} the (essentially unique) isomorphism from U_{ν_{1},ν_{2},∗} to H_{q}(ρ_{n}((τ_{1} ⊗ τ_{2})_{∗})). Then, again for ∗ denoting +, − or s, we have the n-th Yoshida-lifting (whenever the representation ρ_{n}((τ_{1} ⊗ τ_{2})_{∗}) is defined):

Y^{(n)}(φ_{1}, φ_{2}, ∗) := Θ^{(n)}(φ_{1} ⊗ φ_{2})_{∗}(Z)
= ∫_{H_{Q}^{+}\H_{A}^{+}} ∑_{x∈(hL)^{n}} P_{n,∗}({ \atop (φ_{1} ⊗ φ_{2})}(h))(h_{∞}^{−1}x_{1}, …, h_{∞}^{−1}x_{n}) ×
× exp(2πitr(q(x)Z))dh

It is explicitly given as

Y^{(n)}(φ_{1}, φ_{2}, ∗)(Z) = ∑_{i,j=1}^{r} \frac{1}{e_{i}e_{j}} ∑_{x∈(y_{i}Ry_{j}^{−1})^{n}} P_{n,∗}(Ψ(φ_{1}(y_{i}) ⊗ φ_{2}(y_{j})))(x_{1}, …, x_{n}) ×
× exp(2πitr(q(x)Z))

In particular, if ν_{1} = ν_{2} = ν then Y^{(1)}(φ_{1}, φ_{2}, +) is defined and is an elliptic modular form of weight 2 + 2ν. If φ_{1}, φ_{2} are eigenforms of the T̃_{p} for p ∤ N with different eigenvalues then as in [31] one sees that Y^{(1)}(φ_{1}, φ_{2}, +) = 0. If φ_{1} = φ_{2} = φ ≠ 0 is an eigenform of all the T̃_{p} for p ∤ N and of all the w̃_{p} for the p | N then Y^{(1)}(φ, φ, +) is nonzero, and linear continuation of the map sending an eigenform φ to Y^{(1)}(φ, φ) realizes Eichler’s correspondence described above. For details in the case that φ is not essential see [7]. To investigate the nonvanishing of the lifting in general we need the following generalization of a result of Kitaoka:

Lemma 1.1 Let V be a vector space over Q of dimension m with a positive definite quadratic form q on it, let L_{1}, …, L_{r} be pairwise non-isometric Z-lattices of rank m on V having the same discriminant. Let (W_{ρ}, ρ) be a finite dimensional irreducible rational representation of GL_{n}(C) and let 0 ≠ P_{i} ∈ H_{q}(ρ) for i = 1, …, r be qharmonic forms on V^{m−1} with respect to ρ such that P_{i} is invariant under the action of the group of units O(L_{i}) of L_{i}. Then the theta series

ϑ^{(m−1)}(L_{i}, q, P_{i}, Z) := ∑_{x∈L_{i}^{m−1}} P (x) exp(2πitr(q(x)Z))

(with Z in the Siegel upper half plane of degree m − 1) are linearly independent.

Proof. With his kind permission we give here a proof due to Y. Kitaoka which is simpler and more elegant than our own original proof. We show first that given L on V and 0 ≠ P ∈ H_{q}(ρ) as above there is a global characteristic sublattice M of rank m − 1 of L in the sense of Theorem 6.4.1 of [20] such that P (x_{1}, …, x_{m−1}) ≠ 0 for a basis (and hence for any basis) x_{1}, …, x_{m−1} of M: Write P = ∑_{i=1}^{t} α_{i}Q_{i} where the α_{i} are linearly independent over Q and the nonzero polynomial functions Q_{i} : V^{m−1} → W_{ρ} have rational coefficients (with respect to some fixed bases of V and of W_{ρ}) but are not required to be harmonic forms. We can find an (m − 1)-tuple z_{1}, …, z_{m−1} of linearly independent vectors in L such that Q_{1}(z_{1}, …, z_{m−1}) ≠ 0 and put M′ = Zz_{1} +⋯+Zz_{m−1}. Let S be a finite set of primes containing 2 and the primes dividing the discriminant of L and let ℓ be some odd prime not dividing the discriminant of M. Then by Theorem 6.2.1 of [20] one can, in the same way as in the proof of Proposition 6.4.1 of [20] find linearly independent vectors x_{1}, …, x_{m−1} in L such that

• The x_{i} are as close as we want to the z_{i} in the completion V_{ℓ}

• For the p ∈ S the lattice Z_{p}x_{1}+⋯+Z_{p}x_{m−1} is a p-adic characteristic sublattice of L_{p} in the sense of [20, p. 153]

• There is some prime q ∉ S ∪ {ℓ} such that the discriminant of M := Zx_{1} + ⋯ + Zx_{m−1} is a unit at the primes not in S ∪ {ℓ, q} and exactly divisible by q.

The lattice M is then (again as in the proof of Proposition 6.4.1 of [20]) a global characteristic sublattice of L, and if we chose the x_{i} close enough to the z_{i} at the prime ℓ, we see that Q_{1}(x_{1}, …, x_{m−1}) ≠ 0 holds, hence also P (x_{1}, …, x_{m−1}) ≠ 0, and we have established the existence of the global characteristic sublattice claimed above. As in the proof of Corollary 6.4.1 of [20] we can now apply the above argument to the L_{i} to find an index j, a characteristic sublattice M of L_{j} with basis x_{1}, …, x_{m−1} such that P_{j}(x_{1}, …, x_{m−1}) ≠ 0 and such that M is not represented by any of the L_{i} with i ≠ j. By Theorem 6.4.1 of [20] every embedding of M into L_{j} can be uniquely extended to an isometry in O(L_{j}). Since P_{j} was assumed to be O(L_{j})-invariant, the coefficient at q((x_{1}, …, x_{m−1})) of ϑ^{(m−1)}(L_{j}, P_{j}, q, Z) is nonzero, whereas that of the other ϑ^{(m−1)}(L_{i}, P_{i}, q, Z) is zero. ϑ^{(m−1)}(L_{j}, P_{j}, q, Z) can therefore not appear in any linear relation between the ϑ^{(m−1)}(L_{i}, P_{i}, q, Z), and iterating this argument we get the assertion.

Theorem 1.2 Let ν_{1} ≥ ν_{2} ≥ 0 be integers and let τ_{1} = τ_{ν_{1}}, τ_{2} = τ_{ν_{2}} be the irreducible representations of D_{∞}^{×}/R^{×} of highest weights ν_{1}, ν_{2} respectively. Let φ_{i} ∈ A(D_{A}^{×}, R_{A}^{×}, τ_{i}) for i = 1, 2 be eigenfunctions of the Hecke operators T̃_{p} and of all the involutions w̃_{p}; assume that φ_{1} and φ_{2} have the same eigenvalues for all the w̃_{p}. Then if ν_{1} ≠ ν_{2} and ∗ stands for s the n-th Yoshida-lifting Y^{(n)}(φ_{1}, φ_{2}, ∗) is defined for n ≥ 2; if ν_{1} = ν_{2} and ∗ stands for + as above, it is defined for n ≥ 1. the lifting Y^{(2)}(φ_{1}, φ_{2}, ∗) is in these cases a vector valued Siegel modular form with respect to the representation of highest weight (ν_{1} +ν_{2}, ν_{1} −ν_{2}) of GL_{2}(C) and is cuspidal unless ν_{1} = ν_{2} and φ_{1} and φ_{2} are proportional. Moreover, under the same conditions Y^{(3)}(φ_{1}, φ_{2}, ∗) is a nonzero vector valued Siegel modular form for the group Γ_{0}^{(3)}(N) with respect to the representation of highest weight (ν_{1} + ν_{2} + 2, ν_{1} − ν_{2} + 2, 2) of GL_{3}(C). If ν_{1} = ν_{2} = ν ≠ 0 and φ_{1}, φ_{2} are not proportional, then Y^{(n)}(φ_{1}, φ_{2}, −) is defined for n ≥ 3 and Y^{(3)}(φ_{1}, φ_{2}, −) is a nonzero vector valued Siegel cusp form for the group Γ_{0}^{(3)}(N) with respect to the representation of highest weight (2ν, 3, 3) of GL_{3}(C). If ν_{1} = ν_{2} = 0, then if φ_{1}, φ_{2} are not proportional, Y^{(3)}(φ_{1}, φ_{2}, −) is not defined and Y^{(4)}(φ_{1}, φ_{2}, −) is a nonzero Siegel cusp form for the group Γ_{0}^{(4)}(N) of weight 3.

Proof. The claims about the existence of the Yoshida-lifting follow from the description of the theta correspondence at the infinite place in [19]. Lemma 3.1 of [6] gives then the nonvanishing of the function (φ_{1} ⊗ φ_{2})_{∗} on the adelic orthogonal group H_{A}. By Lemma 1.2 of [6] the Yoshida-lifting of (φ_{1} ⊗ φ_{2})_{∗} is a linear combination of theta series of the lattices I_{ij} with harmonic ploynomials that are invariant under the isometries of the lattices I_{ij} equipped with the norm form. The nonvanishing of the Yoshida-lift follows then from Lemma 1.2 above in the first two cases and is obvious in the last case (notice that the nonvanishing of (φ_{1} ⊗ φ_{2})_{∗} implies in particular that at least one of the I_{ij} has no isometries of determinant −1 in this last case). For the cuspidality properties asserted we notice that from Lemma 1.3 of [6] one sees that Siegel’s Φ-operator sends F := Y^{(n)}(φ_{1}, φ_{2}, ∗) to Y^{(n−1)}(φ_{1}, φ_{2}, ∗) whenever the latter is defined. From [29] one sees that the Φ-operator sends F to zero if the last two entries in the vector (λ_{1}, …, λ_{n}) describing the highest weight of the representation of GL_{n}(C) according to which F transforms are not equal. This covers all cases except that ν_{1} = ν_{2} = ν and ∗ = −. If here ν = 0 then Y^{(4)}(φ_{1}, φ_{2}) is scalar valued of weight 3 for Γ_{0}^{(4)}(N), hence annihilated by Φ. For ν ≠ 0 and n = 3 the form F |Φ transforms under Γ_{0}^{(2)}(N) according to the representation of highest weight (2ν + 2, 3), hence is again zero. The same arguments as in section 8 of [5] apply in the present vector valued situation to show that Y^{(n)}(φ_{1}, φ_{2}, ∗) is cuspidal if and only if its image under the Φ-operator is zero.

Theorem 1.2 settles the question of the nonvanishing of the Yoshida-lift in the cases where ∗ = −. In the remaining cases the argument in the end of the proof reduces the question of the nonvanishing of Y^{(2)}(φ_{1}, φ_{2}, ∗) to the question of the cuspidality of the nonzero Siegel modular form Y^{(3)}(φ_{1}, φ_{2}, ∗) in the same way as in [5] for the case ν_{1} = ν_{2} = 0. We notice this fact as a corollary:

Corollary 1.3 Let the notations be as in Theorem 1.2 and let ∗ denote + if ν_{1} = ν_{2} and s otherwise. Moreover Y^{(2)}(φ_{1}, φ_{2}, ∗) = 0 if and only if Y^{(3)}(φ_{1}, φ_{2}, ∗) is cuspidal.

2. Vector valued Siegel modular forms

In this section we collect some facts about vector-valued Siegel modular forms; unfortunately there is no adequate reference for the things we need (the notes of Godement [12] are still the best reference for basic facts). In most cases we give only sketches of proof, in particular we omit those calculations which are completely analogous to the scalar-valued case.

2.1 Differential operators

Proposition 2.1 Let ρ_{0} : Gl(n, C) → V_{ρ} be an irreducible polynomial representation with highest weight (λ_{1}, ..., λ_{n}). Then there is a nonzero V_{ρ} ⊗ V_{ρ}-valued polynomial with rational coefficients Q(s, X) = Q_{s}(X) in the variables s and X, where X is a symmetric 2n × 2n-matrix of variables X = (x_{ij}), homogeneous of degree ∑ λ_{i} as a polynomial in X with the following property: The associated differential operator Δ_{s}, given by

(Δ_{s}F)(z, w) := (Q_{s}((∂_{ij}))F)(( \begin{matrix}z & 0 \\ 0 & w \end{matrix} ))

with z, w ∈ H_{n} maps holomorphic scalar-valued functions F on H_{2n} to V_{ρ} ⊗V_{ρ}-valued holomorphic functions on H_{n} × H_{n} and satisfies

(2.1)  (Δ_{s}) |_{det^{s} ⊗ρ_{0}}^{z} M_{1} |_{det^{s} ⊗ρ_{0}}^{w} M_{2} = Δ_{s}(F |_{s} M_{1}^{↑}M_{2}^{↓})

for all M_{1}, M_{2} ∈ Sp(n, R), where M^{↑} and M_{↓} denote the standard embeddings

\{ \begin{matrix}Sp(n, R) × Sp(n, R) & ↪ & Sp(2n, R) \\ ( \begin{matrix}A & B \\ C & D \end{matrix} ) × ( \begin{matrix}a & b \\ c & d \end{matrix} ) & ⟼ & ( \begin{matrix}A & 0 & B & 0 \\ 0 & a & 0 & b \\ C & 0 & D & 0 \\ 0 & c & 0 & d \end{matrix} ) \end{matrix}

Proof. It is sufficient to check (2.1) for generators of Sp(n, R) and for “test functions” of type

f_{T} : Z ⟼ e^{tr(T ·Z)}

with T ∈ Sym_{2n}(C) and Z = ( \begin{matrix}z & z_{2} \\ z2^{′} & w \end{matrix} ) ∈ H_{2n} We fix the “weight”s and consider Q_{s}

as a polynomial in the ∂_{ij} with unknown coefficients (depending on s). For generators

of type ( \begin{matrix}1_{n} & R \\ 0_{n} & 1_{n} \end{matrix} ), R ∈ Sym_{n}(R) the equation (2.1) is satisfied, so we only have to

investigateI^{↑}, I^{↓} for I = ( \begin{matrix}0_{n} & −1_{n} \\ 1_{n} & 0_{n} \end{matrix} ). We have to compare

(2.2)  Δ_{s}(f_{T} |_{s} I^{↑})

and

(2.3)  det(z)^{−s} · (ρ_{0}(z) ⊗ id_{V_{ρ_{0}}})^{−1} · (Δ_{s}f_{T})(I^{↑} < Z >)

(and a similar expression with I^{↓} instead of I^{↑}). After multiplication by ρ_{0}(z) ⊗ id_{V_{ρ_{0}}} and a suitable power of det(z) both (2.2) and (2.3) are of the form

e^{tr(−T_{1}·z^{−1}+T_{4}·w)} × (polynomial in z and T )

Here of course T_{1}, ...T_{4} denote the blocks of size n in the matrix T. From this we see that (2.1) for all f_{T} and (M_{1}, M_{2}) = (I, id) or (id, I) is equivalent to a system L_{s} of linear equations for the coefficients of Q_{s}. The coefficients of this system depend polynomially on s. Ibukiyama [17] proved that the space of solutions of L_{s} is onedimensional for s = k ∈ N, k ≥ n. Therefore the corank of L_{s} is generically in s equal to 1. The generic solution of L_{s} (properly normalized, such that the coefficients of ∂_{ij} are polynomials in s with no factor in common) gives us the existence of Δ_{s}; Δ_{s} becomes unique, if we choose some monomial in the ∂_{ij} with non-zero coefficient and require this coefficient to be a monic polynomial in s. Remark: The polynomial Q_{s} automatically satisfies the symmetry relation

(2.4)  Q_{s}(∂_{ij}) = Q_{s} ((∂_{ij}) [( \begin{matrix}0_{n} & 1_{n} \\ 1_{n} & 0_{n} \end{matrix} )])

thanks to the result of Ibukiyama. On V_{ρ_{0}} we have a non-degeneate bilinear form <, > such that

(2.5)  < v_{1}, ρ(g)v_{2} >=< ρ(g^{t})v_{1}, v_{2} >

We denote by ^{•} : V_{ρ} ⊗ V_{ρ} ⟼ V_{ρ} ⊗ V_{ρ}^{⋆} = End(V_{ρ}) the isomorphism given by

(2.6)  (v_{1} ⊗ v_{2})^{•}(v) =< v, v_{1} > v_{2}

Then we have for all g_{1}, g_{2} ∈ Gl(n, C)

(2.7)  (ρ_{o}(g_{1})v_{1} ⊗ ρ_{o}(g_{2})v_{2})^{•} = ρ_{0}(g_{1})(v_{1} ⊗ v_{2})^{•}ρ_{o}(g_{2}^{t})

Denoting by h_{s} the function

h_{s} : \{ \begin{matrix}H_{n} ⟶ C \\ Z ⟼ det(z_{1} + z_{2} + z_{2}^{t} + z_{4})^{−s} \end{matrix}

we claim the following important relation

(2.8)  Δ_{s}^{•}h_{s} = c(s) · ρ_{0}(z_{1} + z_{4})^{−1} · det(z_{1} + z_{4})^{−s}

with a certain scalar c(s) = c_{ρ_{0}}(s).

The proof of this claim is quite similar to the proof of Théoréme 5 in [12, Exp.6]:The function h_{s} is rich of symmetries, which thanks to (2.1) gives rise to enough symmetry properties of Δ_{s}^{•} to prove (2.8). By considering Δ_{s}^{•}h_{s} as a function f_{s}(z_{1} + z_{4}, z_{1} − z_{4}) of the variables z_{1} + z_{4} and z_{1} − z_{4} we see (by applying suitable translations, using (2.1))

f_{s}(z_{1} + z_{4}, z_{1} − z_{4} + 2T) = f_{s}(z_{1} + z_{4}, z_{1} − z_{4})

for all T ∈ Sym_{n}(R), hence

Δ_{s}^{•}h_{s} = g_{s}(z_{1} + z_{4})

with a suitable function g on H_{n}. For g ∈ Gl(n, R)

h_{s} |_{s} ( \begin{matrix}g^{t} & 0 \\ 0 & g^{−1} \end{matrix} )^{↑} |_{s} ( \begin{matrix}g^{t} & 0 \\ 0 & g^{−1} \end{matrix} )^{↓} = h_{s}

hence

Δ_{s}^{•}(h_{s})(z_{1}, z_{4}) = det(g)^{2s}ρ_{0}(g)(Δ_{s}^{•}h_{s})(g^{t}z_{1}g, g^{t}z_{4}g) · ρ_{0}(g^{t})

in particular

g_{s}(iy) = det(y)^{−s} · ρ_{0}(y^{− \frac{1}{2}} ) · g_{s}(i1_{n}) · ρ_{0}(y^{− \frac{1}{2}} )

It remains to show that g_{s}(i1) is of the form scalar×id_{V_{ρo}}. We start from a symmetry relation for h_{s}, namely

(2.9)  h_{s} |_{s} M^{↑} = h_{s} |_{s} M̃^{↓}

for all M ∈ Sp(n, R), where M̃ = M^{−1} [ \begin{matrix}1_{n} & 0_{n} \\ 0_{n} & −1_{n} \end{matrix} ] This relation (2.9) is again

proved by considering generators of Sp(n, R), the of translations being trivial. For M = I we prove

(2.10)  h_{s} |_{s} (I^{↑} ∘ (−I)^{↓}) = h_{s}

To prove (2.10) we use the (elementary) relation

h_{s} = G_{s} |_{s} I^{↑} = G_{s} |_{s} I^{↓}

with

G_{s}(Z) = det (( \begin{matrix}0_{n} & 1_{n} \\ 1_{n} & 0_{n} \end{matrix} ) · Z + ( \begin{matrix}1_{n} & 0_{n} \\ 0_{n} & 1_{n} \end{matrix} ))^{−s}

We obtain

h_{s} |_{s} (I^{↑} ∘ (−I)^{↓}) = G_{s} |_{s} (I^{↑} ∘ I^{↑} ∘ (−I)^{↓})
= G_{s} |_{s} ((−1_{2n})^{↑} ∘ (−I)^{↓})
= G_{s} | ((I^{↓}) ∘ (−1_{4n}))
= G_{s} |_{s} I^{↓}
= h_{s}

Therefore we have for any M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sp(n, R)

Δ_{s}(h_{s} |_{s} M^{↑} ∘ M̃^{−1↓}) = Δ_{s}(h_{s})

which implies

(2.11) (det(cz_{1} + d) det(−cz_{4} + d))^{−s}

× ρ_{0} ((cz_{1} + d)^{−1}) g_{s} (M < z_{1} > +M̃^{−1} < z_{4} >) ρ_{0} ((−cz_{4} + d)^{t})^{−1} = g_{s}(z_{1} + z_{4})

We apply (2.11) to M = ( \begin{matrix}u & −v \\ v & u \end{matrix} ) ∈ Sp(n, R) with u + iv ∈ U(n, C) and z_{1} =

z_{4} = i · 1_{n} :

(2.12)  ρ_{0}(u + iv)^{−1}g_{s}(2i)ρ_{0} ((u − iv))^{t})^{−1} = g_{s}(2i)

This means that g_{s}(2i) must be a scalar matrix, because it commutes with all ρ_{0}(g), g ∈ U(n, C); this follows from (2.12) because of

(u − iv)^{t} = (u + iv)^{−1}

The proof of (2.8) is finished.

2.2 Pullbacks of Eisenstein Series

Using the differential operator Δ_{s} we extend the machinery of “pullbacks of Eisenstein series” to vector valued modular forms. The scalar valued case was previously discussed in [11, 1, 5] and the case of symmetric tensor representations in [3] (and recently in [28]). Our approach is somewhat different fromTakayanagi’s [28], who considers a differential operator independent of s; we avoid some of the combinatorial problems coming up in [28]. We denote by {, }a hermitian scalar product on V_{ρ} such that {ρ(ḡ^{t})v_{1}, v_{2}} = {v_{1}, ρ(g)v_{2}}; after a suitable normalization we may assume

(2.13)  {v_{1}, v_{2}} =< v_{1}, v_{2}^{σ} >

where σ denotes the natural complex conjugation on V_{ρ}. For F ∈ [Γ_{0}^{n}(N), ρ]_{0} with ρ = ρ_{0} ⊗ det^{k} we consider

(2.14) ∫_{Γ_{0}(N)\H_{n}} \{ρ(\sqrt{v})F (w), ρ(\sqrt{v})Δ_{s̄+k}G_{2n}^{k} (( \begin{matrix}w & 0 \\ 0 & −z̄ \end{matrix} ) , s̄)\} det(v)^{s} det(y)^{s}dω_{w}

where

(2.15)

G_{2n}^{k}(Z, s) = ∑_{M∈Γ_{∞}^{2n}\Γ_{0}^{2n}(N)} j((M, Z)^{−k−s}j(M, Z̄)^{−s}
= ∑_{M=( \begin{matrix}⋆ & ⋆ \\ C & D \end{matrix} )} det(CZ + D)^{−k−s} det(CZ̄ + D)^{−s}

To compute this integral, one follows the same line of computation as in [5, section

1] , using the double coset decomposition for

Γ_{∞}^{2n} \ Γ_{0}^{2n}(N)/Γ_{0}^{n}(N)^{↑} × Γ_{0}^{n}(N)^{↓}

as described in [5, Thm.1.1], as well as the description of the corresponding left coset decomposition [5, Thm.1.2]. The standard unfolding procedure yields (using (2.1))

(2.16) ∑_{M} ( ∫_{H_{n}} {ρ(v)F (w), (Δ_{s̄+k}h_{s̄+k}) (w, −z̄) det(w̄ − z)^{−s̄}} det(v)^{s} det(y)^{s}dω)

|T_{N}(M) det(M)^{−k−s}

where the summation is over all integral diagonal matrices

M = ( \begin{matrix}m_{1} \\ & . & . & . \\ & & & & m_{n} \end{matrix} )

with M ≡ 0 modulo N and 0 < m_{1}|m_{2} … |m_{n}, and where T_{N}(M) denotes the Hecke operator associated to the double coset

Γ_{0}(N) ( \begin{matrix}0_{n} & −M^{−1} \\ M & 0_{n} \end{matrix} ) Γ_{0}(N).

In view of (2.13) and (2.8) it remains to investigate (2.17) ∫_{H_{n}} det(−z̄+w)^{−s} ((Δ_{s̄+k}h_{s̄+k}(w, −z̄))^{σ})^{•} (ρ(v)F (w)) det(v)^{s} det(y)^{s}dω_{w}

= \overline{c(s̄+k)} ∫_{H_{n}} \overline{ρ_{0}(−z̄+w)^{−1} det(−z̄+w)^{−k−s̄}} det(−z̄+w)^{−s}ρ(v)F (w) det(v)^{s} det(y)^{s}dω_{w}

= \overline{c(s̄+k)} ∫_{H_{n}} ρ(−z +w̄)^{−1} det(−z +w̄)^{−k−s} det(−z̄+w)^{−s}ρ(v)F (w) det(v)^{s} det(y)^{s}dω_{w}

From [12, Exp.6,Thm.5], applied to the holomorphic(!) function

w ⟼ det(−z̄ + w)^{−s}F (w)

with Re(s) ≫ 0 it is clear that (2.17) is a scalar multiple of F (z); we denote this scalar by H_{ρ_{0}}(s + k). It is actually possible to compute this scalar explicitly by transferring the integral to the generalized unit disc

D_{n} = {ξ ∈ Sym_{n}(C)|1_{n} − ξ̄ξ > 0}

by a Cayley-transform (see e.g.[21]); as a result we get

H_{ρ_{0}}(s + k) = 2^{n^{2}+n−4ns}(\frac{1}{2}i)^{nk+∑ λ_{i}} ∫_{D_{n}} ρ_{o} ⊗ det(1_{n} − ξ̄ξ)^{s+k} \frac{dξ}{det(1_{n} − ξ̄ξ)^{n+1}}

The latter integral (which is easily seen to be of type scalar×id_{V_{ρ}}) is a generalized version of the integrals computed in [16]; using the methods of [16] one can explicitly compute this integral, but we do not do this here. What we shall later on need is a relation between c_{ρ_{0}}(s) · H_{ρ_{0}}(s) and the polynomial

Q_{s} (( \begin{matrix}1_{n} & 1_{n} \\ 1_{n} & 1_{n} \end{matrix} )):

For s ∈ C with Re(s) ≫ 0 we start from the well-known identity

∑_{L∈Sym_{n}(Z)} det(z_{1} + z_{2} + z_{2}^{t} + z_{4})^{−s} = \frac{A}{Γ_{n}(s)} ∑_{T ∈Λ_{n}^{+}} det(T)^{s− \frac{n+1}{2}} e^{2ϕitr(T (z_{1}+z_{2}+z_{2}^{t}+z_{4}))} (2.18)

where Λ_{n}^{+} denotes the set of all half-integral symmetric positive matrices of size n,

Z = ( \begin{matrix}z_{1} & z_{2} \\ z_{2}^{t} & z_{4} \end{matrix} ) is an element of H_{2n} and

A = A(s) = e^{− \frac{1}{2} πins}2^{n(s− \frac{1}{2} (n−1))}π^{sn}

We apply Δ_{s}^{•} to both sides of (2.18):

(2.19)

\frac{A}{Γ_{n}(s)} ∑_{T ∈Λ_{n}^{+}} det(T)^{s− \frac{n+1}{2}} Q_{s} (( \begin{matrix}T & T \\ T & T \end{matrix} ))^{•} e^{2πitr(T (z_{1}+z_{4}))}

= c_{ρ_{o}}(s) ∑_{L∈Sym_{n}(C)} ρ_{o}(z_{1} + z_{4} + L)^{−1} det(z_{1} + z_{4} + L)^{−s}

According to [12, Exp.10,Thm.7,Corollaire] this is equal to

c_{ρ_{0}}(s)H_{ρ_{0}}(s) ∑_{T ∈Λ_{n}^{+}} H_{ρ_{0}⊗det^{s}}(4T)^{−1}e^{2πitr(T (z_{1}+z_{4}))}

with

(2.20)  H_{ρ_{0}⊗det^{s}}(T) = ∫_{Y^{(n)}>0} ρ_{0}(Y) det(Y)^{s}e^{−2πtr(T Y)} det(Y)^{−n−1}dY

Therefore we get for all T ∈ Λ_{n}^{+} and Re(s) ≫ 0 the relation

(2.21) \frac{A}{Γ_{n}(s)} det(T)^{s− \frac{n+1}{2}} Q_{s} (( \begin{matrix}T & T \\ T & T \end{matrix} ))^{•} ∘ H_{ρ_{o}⊗det^{s}}(4T) = c_{ρ_{o}}(s)H_{ρ_{o}}(s) · id_{V_{ρo}}

It is obvious that H_{ρ_{0}⊗det^{s}}(T) has a meromorphic continuation to all of C since it can be obtained from the function

T ⟼ ∫_{Y^{(n)}>0} det(Y)^{s−n−1}e^{−2πitr(T Y)}dY = Γ_{n}(s − \frac{n + 1}{2}) det(2πT)^{−s+ \frac{n+1}{2}}

by applying a suitable differential operator (w.r.t.T). In particular, equation (2.21) provides the meromorphic continuation of c_{ρ_{0}}(s) · H_{ρ_{0}}(s) to all of C (if we do not prefer to compute H_{ρ_{0}} explicitly). Later on we need some information about poles of H_{ρ_{0}⊗det^{s}}(S), which we provide here (S = S^{t} > 0): Let v_{0} ∈ V_{ρ_{0}} be a vector of highest weight, then

< v_{0}, H_{ρ_{0}⊗det^{s}}(S)v_{0} > = ∫_{Y >0} < v_{0}, ρ_{0}(Y)v_{0} > det(Y)^{s−(n+1)}e^{−πtr(SY)}dY

= ∫_{Y >0} (∏_{i=1}^{n} t_{ii}^{2λ_{i}}) det(Y)^{s−(n+1)}e^{−πtr(SY)}dY

with Y = T^{t} · T, T upper triangular. This kind of integrals is studied in [22, p.76], one obtains

< v_{0}, H_{ρ_{0}⊗det^{s}}v_{0} >= π^{\frac{1}{4} n(n−1)} (∏_{ı=1}^{n} Γ(λ_{i} + s − \frac{i}{2} − \frac{n}{2})) · (∏_{i=1}^{n} r_{ii}^{λ_{i}+s− \frac{n+1}{4} − \frac{i}{2}} ) (2.22)

where (πS)^{−1} = R · R^{t} with R upper triangular.

2.3 Hecke Eigenforms We assume now that F ∈ [Γ_{0}^{n}(N), ρ_{0} ⊗ det^{k}]_{0} is an eigenform of all Hecke operators

T_{N}(M), the corresponding eigenvalue will be denoted by λ_{F} (M). If the eigenvalue λ_{F} (N ·1_{n}) is different from zero, we get the following Euler product (see [5, Thm.2.1])

∑_{M} λ_{F} (M) det(M)^{−s} = \frac{λ_{F} (N · 1_{n})}{N^{ns}} · \frac{1}{ζ^{(N)}(s) ∏_{i=1}^{n} ζ^{(N)}(2s − 2i)} · Λ_{N}(s − n) · D_{F}^{(N)}(s − n)

where

D_{F}^{(N)}(s) = ∏_{p|̸N} (\frac{1}{1 − p^{−s}} · ∏_{i=1}^{n} \frac{1}{(1 − α_{ip}p^{−s})(1 − α_{ip}^{−1}p^{−s})})

denotes the (restricted) standard L-function attached to F, the α_{ip} being the Satake parameters attached to F in the usual way; moreover

Λ_{N}(S) = ∏_{q|N} ∏_{i=1}^{n} \frac{1}{1 − β_{iq}q^{−s}}

with β_{iq} being the Satake parameters describing the following one-dimensional representation of the Hecke-algebra attached to the Hecke pair (Gl(n, Z), Gl(n, Z[\frac{1}{q}])):

Gl(n, Z)M_{0}Gl(n, Z) ⟼ \frac{1}{λ_{F} (N · 1_{n})} · λ_{F} (N · M_{0})

for M_{0} ∈ Z^{n,n)}, det(M_{0}) = power of q. For details we refer to [5, section 2]. Summarizing our computation, we obtain for such a Hecke eigenform F ∈ [Γ_{0}^{n}(N), ρ]_{0}, ρ = ρ_{0} ⊗ det^{k}:

(2.23) ∫_{Γ_{0}(N)\H_{n}} \{ρ(\sqrt{v})F (w), ρ(\sqrt{v})Δ_{s̄+k}G_{2n}^{k} (( \begin{matrix}w & 0 \\ 0 & −z̄ \end{matrix} ) , s̄)\} det(v)^{s} det(y)^{s}dω^{w}

= \overline{c_{ρ_{0}}(s̄ + k)} · H_{ρ_{0}}(s + k) · \frac{λ_{F} (N · 1_{n})}{(N^{n})^{k+2s}} · \frac{Λ_{N}(2s + k − n) · D_{F}^{(N)}(2s + k − n)}{ζ^{(N)}(2s + k) ∏_{i=1}^{n} ζ^{(N)}(4s + 2k − 2i)}

2.4 The scalar-valued case revisited

The scalar-valued case (ρ_{0} = det^{ν}) can be made more explicit; the differential operators Δ_{s} = Δ^{(ν)} in this case were constructed in a different way in [2] . We normalize

them by fixing the coefficient of det(∂_{2}) with (∂_{1≤i,j≤2n}) = ( \begin{matrix}∂_{1} & ∂_{2} \\ ∂_{2}^{t} & ∂_{4} \end{matrix} ) to be

C_{n}(s − n + \frac{1}{2} + ν − 1) … C_{n}(s − n + \frac{1}{2})

with

C_{n}(s) = s(s + \frac{1}{2}) … (s + \frac{n − 1}{2}) = \frac{Γ_{n}(s + \frac{n+1}{2})}{Γ_{n}(s + \frac{n−1}{2})}

With this normalization,

(2.24)  c_{ρ}(s) = c_{ν}(s) = \frac{Γ_{n}(s + ν) ⋯ + ν − \frac{n}{2})}{Γ_{n}(s) · Γ_{n}(s − \frac{n}{2})}

Moreover

(2.25) \begin{aligned}H_{ρ_{o}}(s + k) & = H_{ν}(s + k) \\ & = 2^{n^{2}+n−4ns−nk−nν} · i^{n(n+ν} ∫_{D_{n}} det(1 − ξξ̄)^{s+k+ν−n−1}dξ \\ & = i^{n(k+ν)} · 2^{n^{2}+n−4ns−nk−nν− \frac{n(n+1)}{2}} · \frac{Γ_{n}(k + ν + s − \frac{n+1}{2})}{Γ_{n}(k + ν + s)}. \end{aligned}

We need a slightly modified version of (2.23) later on (still in the scalar-valued case):

In [4] it is described, how one gets (essentially from a variant of Shimura’s theory of nearly holomorphic functions) an operator identity (with p_{k}(s), d_{k}(s) being polynomials in s)

(2.26)  p_{k}(s) · det(y_{1})^{s} det(y_{4})^{s} · Δ_{s+k}^{ν} (det(Y)^{−s}×?) = d_{k}(s) · Δ_{k}^{ν} + R

Here R is a finite sum of differential operators of type

(2.27)  (D^{z_{1}} ⊗ D^{z_{4}}) ∘ D

where D is some holomorphic differential operator of the type considered in section 2.1 and D^{z_{1}}(D^{z_{4}} respectively) is a Maaß type differential operator with respect to z_{1} ∈ H_{n} (z_{4} ∈ H_{n} respectively); in (2.27) at least one of the Maaß type operators is non-trivial. From [4] we quote the formula

(2.28)  \frac{d_{k}(s)}{p_{k}(s)} = ∏_{μ=1}^{ν} \frac{C_{n}(k + s − n + ν − \frac{μ}{2})}{C_{n}(k − n + ν − \frac{μ}{2})}

Moreover we use that holomorphic Siegel modular forms are orthogonal (with respect to the Petersson scalar product) to automorphic forms in the image of the Maaß operators. For details on these facts we refer to [4]. For an eigenform F ∈ [Γ_{o}(N), det^{k+ν}]_{0} we obtain now

(2.29) ∫_{Γ_{0}(N)\H_{n}} ∫_{Γ_{0}(N)\H_{n}} F (w)\overline{Δ_{k} (E_{2n}^{k} (( \begin{matrix}w & 0 \\ 0 & −z̄ \end{matrix} ) , s̄)) det(v)^{k+ν}dω_{w}F (z)} det(y)^{k+ν}dω_{z}

=\overline{c_{ν}(s̄ + k)}·H_{ν}(s+k)·\frac{d_{k}(s)}{p_{k}(s)}·\frac{λ_{F} (N · 1_{n})}{(N^{n})^{k+2s}}·\frac{Λ_{N}(2s + k − n) · D_{F}^{(N)}(2s + k − n)}{ζ^{(N)}(2s + k) · ∏_{i=1}^{n} ζ^{(N)}(4s + 2k − 2i)}·⟨F, F ⟩.

Here of course

E_{2n}^{k}(Z, s) = det(Y)^{s} · G_{2n}^{k}(Z, s)

= ∑_{M∈Γ_{∞}^{2n}\Γ_{0}^{2n}(N)} det(Y)^{s} |_{k} M

3. The key proposition

The key to non-vanishing properties of Yoshida-liftings is

Proposition 3.1 Let 0 ≠ F ∈ [Γ_{0}^{3}(N), ρ_{0} ⊗ det^{2}]_{0} be an eigenform of all T_{N}(M),

M ≡ 0 mod N and assume that ρ_{0} is of highest weight (λ_{1}, λ_{2}, 0). Then Λ_{N}(s)·D_{F}^{(N)}(s) is regular in s = 1.

Proof: As in [5, section 9] we start from the fact that G_{6}^{2} has a simple pole in s = 1. In view of (2.23) it is sufficient to show that c_{ρ_{0}}(s) · H_{ρ_{0}}(s) has a pole (of any order)

in s = 3; using (2.21) we investigate Q_{s}^{•} (( \begin{matrix}T & T \\ T & T \end{matrix} )) · H_{ρ_{0}⊗det^{s}}(4T) instead (for any

T ∈ Λ_{n}^{+}): From (2.22) we see that H_{ρ_{0}⊗det^{s}}(4T) must have a pole in s = 3 (i.e. in some matrix realization of H_{ρ_{0}⊗det^{s}}(4T) there is at least one coefficient in this matrix having a pole

in s = 3. We are done, if we can show that Q_{s}^{•} (( \begin{matrix}T & T \\ T & T \end{matrix} )) has determinant different

from zero in s = 3 , when considered as an element of End(V_{ρ_{0}}). It is notationally easier to work with the (real) vectorspace

V_{ρ_{0}}^{+} = {v ∈ V_{ρ_{0}} | v = v^{σ}}

for the moment. The statement in question about det(Q_{s}^{•}) is true more generally:

Propositon 3.2 Let l, n be natural numbers with l ≥ 2n, ρ_{0} a polynomial representation of Gl(n, C) and consider the polynomial function

P_{l} : \{ \begin{matrix}R^{(l,n)} × R^{(l,n)} & ⟶ & V_{ρ_{0}}^{+} ⊗ V_{ρ_{0}}^{+} \\ (X, Y) & ⟼ & Q_{l}(X^{t} · X, X^{t} · Y, Y^{t} · Y) \end{matrix}

Then there is a X_{0} ∈ R^{(l,n)} such that det(P_{l}(X_{0}, X_{0})^{•}) ≠ 0.

It is clear (by continuity) that X_{0} may be chosen to be of maximal rank; by the action

of Gl(n, R) from the left and the right on Q_{l} this implies that Q_{l} (( \begin{matrix}T & T \\ T & T \end{matrix} )) has

non-zero determinant for all positive definite T ∈ Sym_{n}(R).

Proof. As a polynomial in X and Y, P_{l} is an element of Harm(l, n) ⊗_{C} Harm(l, n) and also of Harm^{+}(l, n) ⊗_{R} Harm^{+}(l, n); by its construction (see [17] and Prop.2.1), P_{l} is different from zero and

(3.1)  f ⟼<< f, P_{l}(∗, Y) >>

defines an endomorphism of Harm_{ρ_{0}}^{+}(l, n) commuting with the action of O(n, R); here <<, >> denotes the natural scalar product on Harm_{ρ_{0}}^{+}(l, n) “adapted” to O(l, R).

The endomorphism (3.1) is selfadjoint (because P_{l} is symmetric), hence, by Schur’s lemma, it is scalar, i.e. there is a constant c ≠ 0 such that

(3.2)  P_{l}(X, Y) = c · ∑_{i=1}^{r} f_{i}(X) ⊗ f_{i}(Y)

where f_{i} runs through some orthonormal basis of the r-dimensional space

Harm_{ρ_{0}}^{+}(l, n).

Now we recall from [10] that Harm_{ρ_{0}}^{+}(l, n) is generated by functions of type

(3.3)  X ⟼ ρ_{0}(X^{t} · A)v

with v ∈ V_{ρ_{0}}, A ∈ C^{(l,n)} with A^{t} · A = 0 Therefore there exists X_{0} ∈ C^{(l,n)} such that

(3.4)  {f(X_{0}) | f ∈ Harm_{ρ_{0}}(l, n)} = V_{ρ_{0}}

(take X_{0} in (3.3) such that X_{0}^{t} · A has maximal rank). We can reformulate (3.4) by saying that there isX_{0} such that the matrix

(< f_{i}(X_{0}), v_{j} >_{ij})

has maximal rank, where v_{j} runs through a basis of V_{ρ_{0}} (or V_{ρ_{0}}^{+}). From this it follows immediately that there exists a X_{0} ∈ R^{(l,n)} with

(3.5)  {f(X_{0}) | f ∈ Harm_{ρ_{0}}^{+}(l, n)} = V_{ρ_{o}}^{+}

Now let X_{0} be as in (3.5); then for v ∈ V_{ρ_{0}} the equation P_{l}(X_{0}, X_{0})^{•}(v) = 0 implies ∑_{i} < f_{i}(X_{0}), v > f_{i}(X_{0}) = 0 and hence ∑_{i} |< f_{i}(X_{0}), v) |^{2}= 0, so v must be zero.

4. Hecke action on Yoshida liftings

Throughout this section we let ν_{1} ≥ ν_{2} ≥ 0 as in Section 1 and let φ_{i} ∈ A(D_{A}^{×}, R_{A}^{×}, τ_{ν_{i}}) for i = 1, 2 be given. By A_{ess,p}(D_{A}^{×}, R_{A}^{×}, τ_{ν_{i}}) we denote the p-essential part of A(D_{A}^{×}, R_{A}^{×}, τ_{ν_{i}}), i. e. , the set of φ ∈ A(D_{A}^{×}, R_{A}^{×}, τ_{ν_{i}}) which are orthogonal to all functions invariant under an order R̃ with R̃_{p} strictly containing R_{p}. We assume that for each p | N at least one of the φ_{i} belongs to the p-essential part of A(D_{A}^{×}, R_{A}^{×}, τ_{ν_{i}}). We assume moreover that φ_{1}, φ_{2} are nonproportional eigenfunctions of all the Hecke operators T̃(p) for the primes p ∤ N with eigenvalues λ_{p}^{(i)} and eigenfunctions of the involutions w̃_{p} for the p | N with the same eigenvalues. (If instead of these assumptions we start out with two essential eigenforms φ′ ∈ A(D_{A}^{×}, (R′_{A})^{×}, τ_{ν_{1}}) and φ′′ ∈ A(D_{A}^{×}, (R′′_{A})^{×}, τ_{ν_{2}}) with Eichler orders R′, R′′ of square free levels N′, N′′ in D then we can assume without loss of generality that R′ ∩ R′′ is an Eichler order of level N = lcm(N′, N′′) and go over from φ′, φ′′ to forms φ_{1}, φ_{2} that satisfy the assumptions stated above and have the same Hecke eigenvalues as φ′, φ′′ respectively for the p ∤ N.)

Since the question of nonvanishing of the Yoshida-lift Y^{n}(φ_{1}, φ_{2}, −) in the case ν_{1} = ν_{2} has been settled in Section 1, we let Y^{3}(φ_{1}, φ_{2}) denote Y^{3}(φ_{1}, φ_{2}, +) or Y^{3}(φ_{1}, φ_{2}, s), depending on whether ν_{1} = ν_{2} holds or not. According to the results of the previous section we have to show now that the Yoshidalift Y^{3}(φ_{1}, φ_{2}) is an eigenfunction of all Hecke-operators for the p ∤ N as well as for the operators T_{N}(M) for the bad places and to calculate the Euler factors for p ∤ N of the standard L-function of the Yoshida-lift as well as the factor Λ_{N}(s) for the bad places. The results and proofs are here completely analogous to those of [5] for the case ν_{1} = ν_{2} = 0 (as is clear from an adelic point of view), and we sketch them only very briefly.

Theorem 4.1 Let φ_{i} for i = 1, 2 be as above with different Hecke eigenvalues and put k_{i} = 2 + 2ν_{i}. For primes p ∤ N denote by β_{p}, β_{p}^{−1} respectively β̃_{p}, β̃_{p}^{−1} the Satake parameters of φ_{1} respectively φ_{2} with respect to the Hecke-algebra of GL_{2}(Q_{p}) ≅ D_{p}^{×} (so that λ_{p}^{(1)} = p^{(k_{1}−1)/2}(β_{p}+β_{p}^{−1}), λ_{p}^{(2)} = p^{(k_{2}−1)/2}(β̃_{p}+β̃_{p}^{−1})). Then for ∗ denoting +, − or s and n ≥ 2 the Yoshida-lift F = Y^{(n)} (φ_{1}, φ_{2}) is (if nonzero) an eigenfunction of the p-component of the Hecke-algebra of GSp_{n} whose standard L-function is given for n ≥ 2 by:

D_{F^{(n)}}^{(N)}(s) = ∏_{p∤N}(1 − p^{−s})^{−1}(1 − β_{p}β̃_{p}p^{−s})^{−1}(1 − β_{p}β̃_{p}^{−1}p^{−s})^{−1}(1 − β_{p}^{−1}β̃_{p}p^{−s})^{−1}

·(1 − β_{p}^{−1}β̃_{p}^{−1}p^{−s})^{−1} ∏_{j=1}^{n−2}(1 − p^{−s+j})^{−1}(1 − p^{−s−j})^{−1}

= ζ^{(N)}(s)L_{φ_{1},φ_{2}}^{(N)}(s) · ∏_{j=1}^{n−2} ζ^{(N)}(s − j)ζ^{(N)}(s + j).

Here if φ_{1}, φ_{2} are cuspidal and φ_{i} corresponds to the elliptic cusp form

f_{i}(z) = ∑_{n} a_{i}(n) exp(2πinz)

under Eichler’s correspondence we have

L_{φ_{1},φ_{2}}^{(N)}(s) = L_{f_{1},f_{2}}^{(N)}(s + \frac{k_{1} + k_{2}}{2} − 1),

where L_{f_{1},f_{2}}^{(N)}(s) is the “good” part

ζ^{(N)}(2s + k_{1} + k_{2} − 2) ∑_{(n,N)=1} \frac{a_{1}(n)a_{2}(n)}{n^{s}}

of the tensor product L-function of f_{1}, f_{2}.

Proof. This is proved in the same way as Theorem 6.1 and Corollary 6.1 of [5].

Theorem 4.2 With the notations of Theorem 4.1 assume φ_{i} to be essential for i = 1, 2. Then the function Λ_{N}(s) of Section 2 is given by

Λ_{N}(s) = Λ_{N}^{triv}(s) = ∏_{p|N} ∏_{j=1}^{n}(1 − p^{−s−2+j})^{−1}.

If for some p dividing N only one of the φ_{i} is p-essential (say φ_{1}), then there are precisely two local maximal orders R̃_{p} and R̂_{p} strictly containing R_{p}. We let ψ be the projection of φ_{2} on the space of functions that are right invariant under the order R̃ with completion R̃_{p} at p and R_{l} at the primes l ≠ p, assume without loss of generality ψ ≠ 0, and denote by α_{p}, α_{p}^{−1} the p-Satake-parameters of ψ and by ϵ_{p} the eigenvalue of φ_{2} under w̃_{p}. Then the p-factor in the formula for Λ_{N}(s) above has to be replaced by

(1 + ϵ_{p}α_{p}p^{(−2s−1)/2})^{−1}(1 + ϵ_{p}α_{p}^{−1}p^{(−2s−1)/2})^{−1} ∏_{j=3}^{n}(1 − p^{−s−2+j})^{−1}

Proof. The proof is completely analogous to that of Corollary 7.1 of [5] (see also the note added after this corollary for the case that one of the φ_{i} is not essential). There are two places where the modifications needed to cover the present case are not obvious. Firstly we have to generalize Evdokimov’s [9] calculation of the action of the Hecke-operator

F ↦ ∑_{A∈Z_{sym}^{(n,n)}mod pZ_{sym}^{(n,n)}} F | ( \begin{matrix}1_{n} & A \\ 0 & p1_{n} \end{matrix} ) .

This was used to prove the formula

p^{n(n+1)/2} · s_{p}(S)^{n}γ_{p}^{n} ∑_{i=0}^{r/2}(−1)^{i}p^{i(i−1−n)} ∑_{j=1}^{h(i)} \frac{r(S, S_{m/2−i,j})}{e(S_{m/2−i,j})}ϑ^{(n)}(S_{m/2−i,j})

for the action of the operator K_{p}^{(n)} ( \begin{matrix}0 & −p^{−1}1_{n} \\ p1_{n} & 0 \end{matrix} ) K_{p}^{(n)}

(notation of [5]) on the theta series ϑ^{(n)}(S) (where we denote by S_{ij} representatives of the classes of integral positive definite quadratic forms of rank m and discriminant exactly divisible by p^{2i} which are split over Q_{p}, whose level is not divisible by p^{2}, and that are equivalent over all Z_{ℓ} with ℓ ≠ p to a fixed form S (of square discriminant exactly divisible by p^{m−r}). In the present situation we want to replace the theta series ϑ^{(n)}(S) by a theta series

ϑ^{(n)}(L, q, P, Z) = ∑_{x∈L^{n}} P (x) exp(2πitr(q(x)Z))

with a W_{ρ}-valued harmonic polynomial P as in Section 1, where the lattice L has the Gram matrix S from above with respect to the quadratic form q. A close examination of Evdokimov’s proof shows then that the formula above remains true if one replaces the second sum by

∑_{L_{i}} ϑ^{(n)}(L_{i}, P, q, Z),

where L_{i} runs over the lattices contained in L whose Gram matrix with respect to q is equivalent to one of the S_{ij} from above and where the polynomial P is (as in Section 1) considered as a function on the n-fold sum of the vector space that carries all the lattices involved. We also need a generalization of Lemma 7.5 of [5] describing the commutation relation between Siegel’s Φ-operator and our Hecke operators for the bad places. We notice first that the formula given there should read:

F |T_{n}^{(n)}|Φ = p^{2n−k}F |Φ|T_{n−1}^{(n−1)},
F |T_{i}^{(n)}|Φ = p^{n+i−k}F |Φ|T_{i−1}^{n−1} + p^{2i}F |Φ|T_{i}^{(n−1)} (i < n).

(In [5] the k was replaced by 2). It is easy to see that this remains true if F is a vector valued Siegel modular form transforming according to the representation of highest weight (μ_{1}, …, μ_{n−1}, k). In the case under consideration we have k = 2 (which is also true in the applications of this Lemma in [5]), and the argument proceeds as in loc.cit.. The rest of the proof goes through as in [5], replacing the ordinary theta series used there by the theta series with spherical harmonics used here in the same way as above (i. e. in formulas for the action of some Hecke operator on a theta series ϑ^{(n)}(L, q, P, Z) there appear theta series ϑ^{(n)}(L′, q, P, Z) of lattices L′ on the same vector space V and with the same polynomial P on V^{n}).

5. Nonvanishing

In Section 1 we have reduced the question of the nonvanishing of Yoshida’s lifting Y^{(n)}(φ_{1}, φ_{2}, ∗) to the case where ∗ stands for + or s and where n = 2 and φ_{1}, φ_{2} are non-proportional Hecke eigenforms. We can now put together the results of the previous section and formulate the answer to the nonvanishing problem in the following theorem:

Theorem 5.1 Let φ_{i} ∈ A(D_{A}^{×}, R_{A}^{×}, τ_{i}) for i = 1, 2 be given and assume that for each p | N at least one of φ_{1}, φ_{2} is p-essential. Assume the φ_{i} to be eigenfunctions of the of the involutions w̃_{p} for p | N having the same eigenvalues and of the Hecke operators T̃(p) for the p ∤ N. Assume moreover that the φ_{i} are nonproportional and that they correspond to the normalized newforms f, g of weight k_{i} = 2+2ν_{i} and levels N_{f}, N_{g} with lcm(N_{f}, N_{g}) = N (and gcd(N_{f}, N_{g}) divisible by the ramified primes of D) under Eichler’s correspondence described in Section 1 (see also [7, Section 5] if they are cuspidal. Then for ∗ denoting + or s the Yoshida-lifting Y^{(2)}(φ_{1}, φ_{2}, ∗) is zero if and only if ν_{2} = 0, one of φ_{1}, φ_{2} is constant (without loss of generality φ_{2}), and L(f, k_{1}/2) = 0.

Proof. As shown in Section 1 the lifting Y^{(2)}(φ_{1}, φ_{2}, ∗) is zero if and only if

Y^{(3)}(φ_{1}, φ_{2}, ∗)

is cuspidal. By Proposition 3.1 the latter implies that Λ_{N}(s)D_{F}^{(N)}(s) is regular at s = 1. Theorems 4.1 and 4.2 give that this is equivalent to L_{f,g}((k_{1} + k_{2})/2) = 0 if both of φ_{1}, φ_{2} are cuspidal. But by a result of Shahidi [24] this can not happen; an alternative proof for this in the setting of holomorphic modular forms could be obtained by generalizing Ogg’s result [23] for the value at s = 2 of the tensor product L-function of two cusp forms of weight 2 to arbitrary weights. This proves the nonvanishing statement if φ_{1}, φ_{2} are cuspidal. If φ_{1} is cuspidal and φ_{2} = 1 then L_{φ_{1},φ_{2}}^{(N)}(s) = L^{(N)}(f_{1}, s + ν_{1} + 1)L^{(N)}(f_{1}, s + ν_{1}), and this is zero if and only if the central critical value L(f_{1}, k_{1}/2) is zero. It remains to show that in this case one has indeed that F = Y^{(2)}(φ_{1}, 1, ∗) is zero. For this we can again proceed in the same way as in [5], assuming F ≠ 0. In order to see how to modify the argument let us sketch the argument given there: We used the following version of Siegel’s theorem: The Eisenstein series E_{4}^{2}(Z, s) has a pole of order 1 in s = 1/2, and the residue E_{4}^{2}(Z) is a linear combination (with all coefficients α_{i} nonzero) of the genus theta series ϑ^{(4)}(G_{i}) of degree 4 for the genera G_{i} of positive definite integral quadratic forms of rank 4, square discriminant and level dividing N. We then restricted Z to a block diagonal

(\begin{matrix}Z_{1} & 0 \\ 0 & Z_{2} \end{matrix}) and formed the double Petersson product ⟨⟨F, E_{4}^{2}⟩_{Z_{1}}, F ⟩_{Z_{2}}. By Lemma 9.1

of [5] only the theta series of the genus of the order R gave a contribution to this double product, which then became

c ∑_{i,j} \frac{|⟨F, ϑ^{(2)}(I_{ij})⟩|^{2}}{e_{i}e_{j}}

with c ≠ 0. From this and Theorem 4.1 of [5] (or 2.29 of the present article) we then concluded that Λ_{N}(s)D_{F}^{(N)}(s) has a pole in s = 1 unless F is orthogonal to all the ϑ^{(2)}(I_{ij}), which is excluded since F by construction is a linear combination of these theta series. In our present situation we have to apply the differential operator Δ_{2}^{ν} to all the ϑ^{(4)}(I_{ij}) and obtain as in [1] after evaluating the double Petersson product

c ∑_{i,j,κ} \frac{|⟨F, ϑ^{(2)}(I_{ij}, P_{κ})⟩|^{2}}{e_{i}e_{j}} ,

where P_{κ} runs over a basis of the space of homogenous harmonic forms on D_{∞}^{2} transforming under the action of GL_{2}(R) according to det^{2+ν_{1}} (we notice that as in Lemma 9.1 of [5] F is orthogonal to all the theta series with harmonic polynomials except those belonging to lattices in the genus of R). We conclude again that Λ_{N}(s)D_{F}^{(N)}(s) has a pole in s = 1 unless F is orthogonal to all the ϑ^{(2)}(I_{ij}, P_{κ}), which is excluded since F by construction is a linear combination of these theta series with harmonic polynomials. By our formula for D_{F}^{(N)}(s) in this situation from above we see that this implies L(f_{1}, k_{1}/2) = 0, which proves the rest of our assertion. Remark: A different phrasing of Theorem 5.1 would say: Take any pair of cusp forms f, g of weights 2 + 2ν_{i} (ν_{1} ≥ ν_{2}) with trivial character for groups Γ_{0}(N_{f,g}) with square free N_{f}, N_{g}, gcd(N_{f}, N_{g}) ≠ 1. Assume f, g to be newforms (in particular Hecke eigenforms) and that f, g have the same eigenvalues under the Atkin-Lehner involutions w_{p} for all p | gcd(N_{f}, N_{g}). Then we can construct a nonzero Siegel modular form of degree 2 transforming according to the representation of highest weight (ν_{1} + ν_{2} + 2, ν_{1} − ν_{2} + 2) of GL_{2}(C) under Γ_{0}^{(2)}(N) with N = lcm(N_{f}, N_{g}), whose Satake parameters are related to those of f, g as described in Section 4; this may be viewed as giving an explicit construction of an endoscopic lifting. In order to construct this Siegel modular form we first replace f, g by forms of level N having the same Hecke eigenvalues as f, g for p ∤ N and being eigenfunctions of all the Atkin-Lehner involutions w_{p} for p | N with the same eigenvalues. We then fix a definite quaternion algebra over Q whose ramified primes divide gcd(N_{f}, N_{g}) and use the generalized correspondence of Eichler for an Eichler order of level N in D from Section 1 and [7, Section 5] to find φ_{1}, φ_{2} corresponding to f, g as required in Theorem 5.1 and apply this theorem to the pair φ_{1}, φ_{2}. It should be noted that when gcd(N_{f}, N_{g}) is not prime this allows several possible constructions whose results are pairwise orthogonal by the generalization of Lemma 9.1 of [5] noted in the proof of theorem 5.1. Such a possibility has been discussed under the name of twin forms in [15]. In particular for ν_{1} = 1, ν_{2} = 0 we obtain scalar valued Siegel modular forms of weight 3 and degree 2, which in view of their connection to differential forms on the quotient of H_{2} by the group Γ_{0}^{(2)}(lcm(N_{1}, N_{2})) are of interest in geometry.

6. An example

To illustrate our results we want to discuss an example. The calculations for this example were done using the algorithms from [26] and the PARI number theory package. We put N = 17. The quaternion algebra over Q that is ramified at ∞ and at N only has type number and class number t = h = 2. A basis is given by the f_{i}^{(0)} for i = 0, …, 3 with f_{0}^{(0)} = 1 and (f_{1}^{(0)})^{2} = −17, (f_{2}^{(0)})^{2} = −3, f_{1}^{(0)}f_{2}^{(0)} = −f_{2}^{(0)}f_{1}^{(0)} = f_{3}^{(0)}. The Z-lattice generated by these vectors has index 12 in a maximal order. It is transformed into a maximal order R_{0} containing it by the basis transformation with matrix

(\begin{matrix}1 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{3} \\ 0 & 0 & \frac{1}{2} & \frac{−1}{3} \end{matrix} )

and into another maximal order R_{1} by the basis transformation with matrix

(\begin{matrix}1 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & \frac{1}{4} & \frac{1}{8} & \frac{1}{4} \\ 0 & \frac{−1}{3} & \frac{3}{4} & \frac{−1}{2} \\ 0 & \frac{1}{12} & \frac{−1}{8} & \frac{−1}{4} \end{matrix} ).

The resulting lattice has Gram matrix (with respect to tr(xȳ) and to the new basis {f_{i}})

A_{1} = (\begin{matrix}2 & 1 & 1 & 0 \\ 1 & 4 & −1 & 1 \\ 1 & −1 & 6 & 2 \\ 0 & 1 & 2 & 10 \end{matrix}).

The product in R_{1} of ∑_{i=0}^{3} α_{i}f_{i} and ∑_{i=0}^{3} β_{i}f_{i} is ∑_{i=0}^{3} γ_{i}f_{i} with

γ_{0} = α_{0}β_{0} +((−2α_{1} +(α_{2} +α_{3}))β_{1} +(−2β_{3}α_{1} +((−3α_{2} −3α_{3})β_{2} +(β_{3}α_{2} −5α_{3}β_{3}))))
γ_{1} = α_{1}β_{0} +(β_{1}α_{0} +((α_{1} −α_{3})β_{1} +((β_{2} +β_{3})α_{1} +(3α_{3}β_{2} −3β_{3}α_{2}))))
γ_{2} = α_{2}β_{0} +(β_{2}α_{0} +(−2α_{3}β_{1} +((β_{2} +2β_{3})α_{1} +((α_{2} +α_{3})β_{2} −β_{3}α_{2}))))
γ_{3} = α_{3}β_{0} +(β_{3}α_{0} +((α_{2} +α_{3})β_{1} +(−β_{2}α_{1} +β_{3}α_{2}))).

The other type of maximal order is R_{2}, which is obtained from R_{1} by the basis transformation with matrix

(\begin{matrix}\frac{3}{4} & 1 & \frac{1}{2} & 0 \\ 0 & 0 & −1 & 1 \\ \frac{−1}{2} & 0 & −1 & 0 \\ \frac{1}{4} & 0 & \frac{−1}{2} & −1, \end{matrix})

it has Gram matrix

(\begin{matrix}2 & 1 & 0 & 0 \\ 1 & 2 & −1 & 1 \\ 0 & −1 & 12 & 5 \\ 0 & 1 & 5 & 12. \end{matrix})

An ideal I_{12} with left order R_{2} and right order R_{1} is obtained from R_{1} by the basis transformation with matrix

(\begin{matrix}0 & 1 & −1 & \frac{1}{2} \\ \frac{1}{2} & −1 & \frac{−1}{2} & −1 \\ 0 & 0 & 0 & −1 \\ \frac{1}{2} & 0 & \frac{1}{2} & \frac{1}{2}, \end{matrix})

it has Gram matrix

(\begin{matrix}4 & −2 & 1 & 0 \\ −2 & 4 & 0 & 1 \\ 1 & 0 & 6 & 3 \\ 0 & 1 & 3 & 6 \end{matrix})

with respect to the basis {f′_{i}} obtained. The order R_{2} has 6 units, the order R_{1} has no nontrivial units. We obtain therefore as the unique (up to scalar multiples) nontrivial function in A(D_{A}^{×}, R_{A}^{×}, 1) orthogonal to the constants the function φ_{2} with φ_{2}(y_{1}) = −1, φ_{2}(y_{2}) = 3 (corresponding to the unique normalized cusp form of weight 2 and level 17). The space of cusp forms of weight 4 and level 17 has dimension 4. It is easily checked by looking at the linear forms on D ⊗ R and using Eichler’s correspondence described in Section 1 that only one of these has w_{17}-eigenvalue −1. The corresponding function φ_{1} ∈ A(D_{A}^{×}, R_{A}^{×}, τ_{1}) is given by φ_{2}(y_{2}) = 0 and φ_{2}(y_{1})(∑_{i=0}^{3} α_{i}f_{i}) = α_{3}. Using this we find that the Siegel cusp form Y^{(2)}(φ_{1}, φ_{2}) of weight 3 is given as the sum of the theta series of degree 2 of R_{1} with polynomial

P_{1}(∑_{i=0}^{3} α_{i}f_{i}, ∑_{i=0}^{3} β_{i}f_{i}) = 2α_{3}β_{0} + (−2β_{3}α_{0} + ((2α_{2} + 2α_{3})β_{1} + (−2β_{2} − 2β_{3})α_{1}))

and the theta series of degree 2 of I_{12} with polynomial

P_{12}((∑_{i=0}^{3} α′_{i}f′_{i}, ∑_{i=0}^{3} β′_{i}f′_{i})) = −2α_{2}β_{0} + (2β_{2}α_{0} + (−2α_{3}β_{1} + 2β_{3}α_{1})).

The computation gives the following beginning terms of the Fourier expansion of Y^{(2)}(φ_{1}, φ_{2}) (where the variable in H_{2} is

Z = ( \begin{matrix}τ & z \\ z & τ′ \end{matrix} )

and where we put X_{1} = exp(2πiτ), X_{2} = exp(2πiz), X_{3} = exp(2πiτ′)) so that the

coefficient at X_{1}^{a}X_{2}^{b}X_{3}^{c} is the Fourier coefficient at ( \begin{matrix}a & b/2 \\ b/2 & c \end{matrix} )):

(−32X_{3}^{6}X_{2}^{2} − 64X_{3}^{6}X_{2})X_{1}^{5}
+(−32X_{3}^{5}X_{2}^{3} − 96X_{3}^{6}X_{2}^{2} + (32X_{3}^{6} − 64X_{3}^{5})X_{2})X_{1}^{4}
+((−32X_{3}^{6} + 32X_{3}^{5} + 32X_{3}^{4})X_{2}^{2} − 32X_{3}^{6}X_{2})X_{1}^{3}
+(−32X_{3}^{5} − 32X_{3}^{4} + 32X_{3}^{3})X_{2}X_{1}^{2}.

(We computed the coefficients of X_{1}^{a}X_{2}^{b}X_{3}^{c} for all reduced binary forms [a, b, c] = ax^{2} + bxy + cy^{2} with c ≤ 25) To be sure that this is indeed a Hecke eigenform we computed the action of T (2), T (3) and T (5) following the formulas given in [27]; it turns out that the Hecke eigenvalues are −5, −8 and −4 respectively. Notice that

SIEGEL MODULAR FORMS, THETA SERIES, AND QUATERNION ALGEBRAS II 27

in the formula at the bottom of p. 386 of [27] the term a([m/p, r, np]) should be replaced by a([m/p, −r, np]) (which matters since we are dealing with odd weight).

References

1. S.Böcherer: Siegel modular forms and theta series, Proc. symp. Pure Math. 49, vol. 2, 3-17 (1989)

2. S.Böcherer: Über die Fourier-Jacobientwicklung der Siegelschen Eisensteinreihen II. Mathem.Z.189(1989),81-110

3. S.Böcherer,T.Satoh,T.Yamazaki: On the pullback of a differential operator and its application to vector valued Eisenstein series. Comm.Math.Univ.S.Pauli 41(1992), 1-22

4. S.Böcherer,C.G.Schmidt: p-adic measures attached to Siegel modular forms, Preprint 1995 5. S.Böcherer,R.Schulze-Pillot: Siegel modular forms and theta series attached to quaternion al-

gebras. Nagoya Math.J. 121(1991), 35-96

6. S. Böcherer, R. Schulze-Pillot: Mellin transforms of vector valued theta series attached to quaternion algebras, Math. Nachr. 169 (1994), 31-57

7. S. Böcherer, R. Schulze-Pillot: On the central critical value of the triple product L-function, MSRI Preprint 058-95

8. M. Eichler: The basis problem for modular forms and the traces of the Hecke operators, p. 76-151 in Modular functions of one variable I, Lecture Notes Math. 320, Berlin-Heidelberg-New York 1973

9. S. A. Evdokimov: Action of the irregular Hecke operator of index p on the theta series of a quadratic form, J. Sov. Math. 38 (1987), 2078-2081

10. E.Freitag: Singular Modular Forms and Theta Relations, Lecture Notes Math. 1487. Berlin-Heidelberg 1991

11. P.Garrett: Decomposition of Eisenstein series: Rankin triple products. Annals of Math. 125(1987), 209-235

12. R.Godement: Séminaire Cartan 10 (1957/58), Exp. 4-9

13. K. Hashimoto: On Brandt matrices of Eichler orders, Preprint 1994

14. H. Hijikata, H. Saito: On the representability of modular forms by theta series, p. 13-21 in Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Tokyo 1973

15. R.Howe, I. I. Piatetski-Shapiro: Some examples of automorphic forms on Sp_{4}, Duke Math. J. 50 (1983), 55-106

16. L. K. Hua: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domain, Translations of Mathematical Monographs 6, AMS, Providence 1963

17. T.Ibukiyama: On differential operators on automorphic forms and invariant pluriharmonic polynomials. Preprint 1990

18. H. Jacquet, R. Langlands: Automorphic forms on GL(2), Lect. Notes in Math. 114, Berlin-Heidelberg-New York 1970

19. Kashiwara, M. Vergne: On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1987), 1-44

20. Y. Kitaoka: Arithmetic of Quadratic Forms, Cambridge 1993

21. H.Klingen: Introductory Lectures on Siegel Modular Forms, Cambridge University Press 1990 22. H.Maaß: Siegel’s modular forms and Dirichlet series. Lect. Notes Math. 216 Berlin, Heidelberg,

New York: Springer 1971

23. A. Ogg: On a convolution of L-series, Invent. Math. 7 (1969), 279-312

24. F. Shahidi: On certain L-functions, Amer. J. Math. 103 (1981), 297-355

25. H. Shimizu: Theta series and automorphic forms on GL_{2}. J. of the Math. Soc. of Japan 24 (1972), 638-683

26. R. Schulze-Pillot: an algorithm for computing genera of ternary and quaternary quadratic forms, p. 134-143 in Proceedings of the International Symposium on Symbolic and Algebraic computation (ISSAC) Bonn 1991

27. N.-P. Skoruppa: Computations of Siegel Modular forms of genus two. Math. Comp. 58 (1992), 381-398

28. H. Takayanagi: Vector valued Siegel modular forms and their L-functions; application of a differential operator, Japan. J. Math. 19 (1994), 251- 297

29. R. Weissauer: Vektorwertige Siegelsche Modulformen kleinen Gewichts, J. f. d. reine u. angew. Math. 343 (1983), 184-202

30. H. Yoshida: Siegel’s modular forms and the arithmetic of quadratic forms, Invent. math. 60 (1980), 193-248

31. H. Yoshida: On Siegel modular forms obtained from theta series, J. f. d. reine u. angew. Math. 352 (1984), 184-219

Siegfried Böcherer, Fakultät für Mathematik und Informatik, Universität Mannheim, Seminargebäude A5, D-68131, Mannheim, Germany

Rainer Schulze-Pillot, Fachbereich Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany