Critical values for L-functions on GSp_{2} × Gl_{2}

By Siegfried Böcherer ^{1} and Bernhard E. Heim

Abstract This paper deals with Deligne’s conjecture on the critical values of L-functions. Let Z_{G⊗h}(s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp form h of weight l. Let the first Fourier-Jacobi coefficient of G be not identically zero. Then Deligne’s conjecture is fully proven for Z_{G⊗h}(s), when l ≤ 2k − 2 and partly for the remaining case.

Introduction

The purpose of this paper is to prove parts of Deligne’s conjectures [De79] on the critical values for L-functions attached to Siegel modular forms and the standard representation of the L-group of GSp_{2} × Gl_{2}. Let S_{n}^{k} denote the space of cuspidal Siegel modular forms of genus n and weight k. Let h ∈ S_{1}^{2k̃_{h}}, G ∈ S_{2}^{2k̃_{G}} be eigenforms, whose Fourier coefficients lie in the field generated by the Hecke eigenvalues. Let Z_{G⊗h}(s) be the twisted spinor L-function, the convolution of the spinor L-function of G and Hecke’s L-function of h. We normalize the Petersson scalar product ⟨ , ⟩, such that

⟨F , F ⟩ = ∫_{Sp_{n}(ℤ) \ ℍ_{n}} |f(Z)|^{2} det(Y)^{k−n−1}dX dY,

where F ∈ S_{n}^{k} and Z = X + iY. Deligne’s conjectures (worked out by H. Yoshida [Yo99]) on the critical values of Z_{G⊗h}(s), by assuming the existence of motives M(h) and M(G) attached to the modular forms, can be stated as follows

Conjecture A Let 2k̃_{h} > 4k̃_{G} − 2. Then the critical values of Z_{G⊗h}(s) are given by C_{I} = [4k̃_{G} − 2, 2k̃_{h} − 1] ∩ ℤ. Let m ∈ C_{I}, then

Z_{G⊗h}^{∗}(m) = \frac{Z_{G⊗h}(m)}{π^{4m−8k̃_{G}+6}⟨h , h⟩^{2}} ∈ \overline{ℚ}  and  (1)
(Z_{G⊗h}^{∗}(m))^{σ} = Z_{G^{σ}⊗h^{σ}}^{∗}(m) for all σ ∈ Aut(ℂ).  (2)

1supported by the von Neumann Fund at the Institute for Advanced Study, Princeton

Conjecture B Let 2k̃_{h} < 4k̃_{G} − 2. Then the critical values of Z_{G⊗h}(s) are given by C_{II}^{a} = [2k̃_{G}, 2k̃_{G}+2k̃_{h}−3]∩ℤ if 2k̃_{G} ≥ 2k̃_{h} and C_{II}^{b} = [2k̃_{h}, 4k̃_{G}−3]∩ℤ if 2k̃_{G} < 2k̃_{h}. Let m be a critical value, then

Z_{G⊗h}^{∗}(m) = \frac{Z_{G⊗h}(m)}{π^{4m−4k̃_{G}−2k̃_{h}+4}⟨h , h⟩⟨G , G⟩} ∈ \overline{ℚ}  and  (3)
(Z_{G⊗h}^{∗}(m))^{σ} = Z_{G^{σ}⊗h^{σ}}^{∗}(m) for all σ ∈ Aut(ℂ).  (4)

Supplement: In the case 2k̃_{h} = 4k̃_{G} − 2 there are no critical values.

For a description of this (conjectural) set of critical values in terms of the Γ-factor of the functional equation, see section 5.

Up to now only the following results had been known: Let the weights of G and h be equal. Furusawa [Fu93] proved Z_{G⊗h}^{∗}(4k̃_{h} − 3) is algebraic, provided that there exists a non-trivial Fourier coefficient of G paramatrized by a half-integral positive definit matrix T, such that −4 det T is a fundamental discriminant. Some years later (under similiar assumptions, but using a different integral representation) we [He99] described how the absolute Galois group acts on this critical value. We will always assume that the first Fourier-Jacobi coefficient of G is not identically zero. (Numerical computations of Skoruppa [Sk92] suggest that this assumption may be always satisfied - not just only for Saito-Kurokawa lifts, where it is obvious). In this paper we prove, that then Deligne’s conjecture B is true. Moreover we present some evidence for conjecture A by confirming it for all G, which are Saito-Kurokawa lifts. We mention that conjecture A is similar in nature to the ”unbalanced case” for triple-L-functions, which is still open in general, see [Or87], [G-H93], [H-K91]. The main ingredients in our proof are the integral representation of Z_{G⊗h}(s) found in [He99] (and technical modifications given in [Bo-He99]), Shimura’s concept of nearly holomorphic modular forms [Sh87], and some more or less well known facts on Siegel-type Eisenstein series, special values of Hecke- and Rankin L-series and quotients of scalar products.

Our paper is organized as follows: In the first three sections we collect facts about the L-functions in question, about differential operators and about Eisenstein series. Most of the material presented here is more or less wellknown (at least in principle), but needs to be stated in the form we want to use it. Section 4 contains computations mainly concerned with the role of the auxiliary modular form F from the Maaß space, which occurs in our integral representation of Z_{G⊗h}. In the final section 5 we present and prove our main results (theorems 5.1 and 5.2).

Notation: For matrices M and N we denote by M^{t}, det(M), and tr(M) its transpose, determinant, and trace and we put M[N] = N^{t}MN (whenever it makes sense). For a square matrix Z we put e{Z} = e^{2πitr(Z)}. The symbol Γ_{n} occurs with two meanings, on one hand as the Siegel modular group and on the other hand as a generalized Γ-function.

1 Siegel modular forms and their automorphic L-functions

The symplectic group Sp_{n}(ℝ) acts on Siegel’s upper-half-space IH_{n} of genus n as a group of biholomorphic automorphism by

(M, Z) ⟼ M(Z) = (AZ + B)(CZ + D)^{−1},

where M = \binom{A B}{C D} ∈ Sp_{n}(ℝ) with A, B, C, D ∈ ℝ^{n,n}; we denote by j(M, Z) = det(CZ + D) the standard automorphy factor. For α, β ∈ ℂ we define the slash-operator |_{α,β} for functions f on IH_{n} and M ∈ Sp_{n}(ℝ) by

(f |_{α,β} M) (Z) = j(M, Z)^{−α}\overline{j(M, Z)}^{−β}f(M(Z)).

and we write f |_{k} instead of f |_{k,0}. We denote by M_{n}^{k} (S_{n}^{k}) the space of (cuspidal) Siegel modular forms of degree n and weight k for the Siegel modular group Γ_{n} := Sp_{n}(ℤ); every F ∈ S_{n}^{k} satisfies the transformation law

F (Z) = F (M(Z)) j(M, Z)^{−k} for all M ∈ Γ_{n}, Z ∈ IH_{n}.  (5)

Sometimes we prefer to write 2k̃ instead of k for the weight; the reason for this is that the differential operators ∂_{n,k}^{(μ)} (to be explained below) change the weight by 2μ.

The ℚ-subspaces of those (cuspidal) Siegel modular forms, which have rational Fourier coefficients, will be denoted by M_{n}^{k}(ℚ) and S_{n}^{k}(ℚ). It is well known that M_{n}^{k} = M_{n}^{k} ⊗_{ℚ} ℂ, therefore we have a natural action (σ, f) ↦ f^{σ} of Aut(ℂ) on M_{n}^{k} (by acting on the Fourier coefficients). For arbitrary complex valued functions F and G on IH_{n}, which satisfy (5), we define the Petersson integral, (if convergent) by

⟨F, G⟩ = ∫_{Γ\IH_{n}} F (Z) \overline{G(Z)} δ(Z)^{k}d^{∗}Z.  (6)

Here Z = X +i Y,δ(Z) = det(Y) and d^{∗}Z = det (Y)^{−(n+1)}dXdY denotes the symplectic volume element. Every F ∈ S_{n}^{k} has a Fourier expansion of the form F (Z) = ∑_{T} a_{T}^{F} e{T Z} and a Fourier-Jacobi expansion of the form

F (\begin{matrix}τ & z \\ z^{t} & τ′ \end{matrix}) = ∑_{t=1}^{∞} Φ_{t}^{F} (τ, z) e\{tτ′\},  (τ′ ∈ IH)  (7)

with Fourier-Jacobi coefficients Φ_{t}^{F} (τ, z). The ∑_{T} runs over all half-integral positive definite matrices. These coefficients are Jacobi cusp forms of degree (n − 1, 1), index t and weight k. We denote by ⟨; ⟩_{J} the Petersson scalar product of Jacobi forms (cf. [He99], section 2.3 and 3.4). Further we denote by S_{2}^{k,M} the Maaß space of weight k, which is a well known subspace of S_{2}^{k} (see [He99] Definition 3.14). We will need some embeddings of domains and groups: Let n = n_{1} + … + n_{r} be a partition, then we put

i_{n_{1},... ,n_{r}} : \{ \begin{matrix}IH_{n_{1}} × … × IH_{n_{r}} & ⟶ & IH_{n} \\ (z_{1}, …, n_{r}) & ⟼ & diag(z_{1}, …, z_{r}) \end{matrix}

and we use the same symbol for the corresponding embedding of groups:

i_{n_{1},... ,n_{r}} : \{ \begin{matrix}Sp_{n_{1}}(ℝ) × … × Sp_{n_{r}}(ℝ) & ⟶ & Sp_{n}(ℝ) \\ (g_{1}, …, g_{r}) & ⟼ & ( \begin{matrix}\begin{aligned}a_{1} \\ & . & . & . \\ & & & & a_{r} \end{aligned} & \begin{aligned}b_{1} \\ & . & . & . \\ & & & & b_{r} \end{aligned} \\ \begin{aligned}c_{1} \\ & . & . & . \\ & & & & c_{r} \end{aligned} & \begin{aligned}c_{1} \\ & . & . & . \\ & & & & d_{r} \end{aligned} \end{matrix} ) \end{matrix}

with g_{i} = ( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ).

Let F ∈ S_{n}^{k} be a Hecke eigenform with Satake-parameters (α_{0,p}, α_{1,p}, …, α_{n,p}). Then the spinor zeta function Z_{F} (s) of F is given by

Z_{F}^{n}(s) = ∏_{p} {Q_{p,F}^{n} (p^{−s})}^{−1} , where  (8)

Q_{p,F}^{n} (X) = (1 − α_{0,p}X) ∏_{r=1}^{n} ∏_{1≤i_{1}<···<i_{r}≤n}(1 − α_{0,p}α_{i_{1},p} · . . . · α_{i_{r},p}X).

We call a non-trivial Hecke eigenform F ∈ S_{n}^{k} normalized, if the Fourier coefficients lie in the field generated by the eigenvalues of F; in the special case n = 1 we may (and always will) require in addition that a_{1}^{F} = 1. If f ∈ S_{1}^{k} is a normalized Hecke eigenform, one usually puts α_{p} = α_{0,p} and β_{p} = α_{0,p}α_{1,p} and writes L_{f}(s) for Z_{f}^{1}(s) and L_{p,f}(X) for Q_{p,f}^{1}(X). Thus a_{p}^{f} = α_{p} + β_{p} and L_{p,f}(X) = 1 − a_{p}^{f} X + p^{k−1} X^{2}. Let F ∈ S_{2}^{k,M} be a Hecke eigenform, then the spinor L-function of F splits, i.e.

Z_{F}^{2} (s) = ζ(s − k + 1) ζ(s − k + 2)L_{f}(s)  (9)

where f ∈ S_{1}^{2k−2} corresponds to F by the Saito-Kurokawa lift (cf. [He99], Theorem 3.16). With this notation we introduce the L-function Z_{G⊗h} corresponding to two eigenforms G ∈ S_{2}^{k} and h ∈ S_{1}^{l} with parameters α_{0,p}, α_{1,p}, α_{2,p} and α_{p}, β_{p} for every prime number p:

Z_{G⊗h}(s) = ∏_{p} {Q_{p,G}^{2}(α_{p}p^{−s}) Q_{p,G}^{2}(β_{p}p^{−s})}^{−1}
= ∏_{p} {Z_{p,G⊗h}(p^{−s})}^{−1} .  (10)

In other words, this is the L-function attached to the standard representation of the L-group of GSp_{2} × Gl_{2} and the automorphic representation π_{G} ⊗ π_{h} corresponding to G and h.

Theorem [Bo-He99] Let k_{h} and k_{G} be even natural numbers. Let h be an elliptic cusp form of weight k_{h} and G a Siegel cusp form of degree 2 of weight k_{G}. We assume that h and G are Hecke eigenforms, where the first Fourier-Jacobi coefficient of G is not zero. Let Z_{G⊗h}(s) denote the L-function of G and h attached to the standard representation of the L-group of GSp_{2} × Gl_{2} (cf. (10)). Then Z_{G⊗h}(s) has a meromorphic continuation to the whole complex plane and satisfies the functional equation

{ \atop Z_{G⊗h}}(s) = { \atop Z_{G⊗h}}(k_{h} + 2k_{G} − 3 − s),  (11)

where

{ \atop Z_{G⊗h}}(s) = γ_{∞}(s) Z_{G⊗h}(s)
γ_{∞}(s) = (2π)^{−4s}Γ(s − k_{G} + 2) Γ(s)
×Γ(s − k_{G} + 1) Γ(s − k_{h} + 1).

2 Differential operators and Petersson scalar products

We denote by N_{n}^{(μ)} the space of all polynomials of degree ≤ μ in the entries of Y^{−1} with holomorphic functions on IH_{n} as coefficients. Then N_{n} := ⋃_{μ} N_{n}^{μ} is the space of all nearly holomorphic functions in the sense of Shimura. Furthermore, for integral weight k

N_{n}^{(k,μ)} := {f ∈ N_{n}^{(μ)} | f |_{k} γ = f for all γ ∈ Γ_{n}}

is the (finite dimensional) space of nearly holomorphic modular forms (with the standard additional condition on the Fourier expansion when n = 1). It is an important observation of Shimura, [Sh94, Thm.3.3] that (at least for k > n + μ ) an element of N_{n}^{k,μ} can be written (uniquely) as a

f = h + ∑_{s} L_{k,s}t_{s}  (12)

where h ∈ M_{n}^{k}, the L_{k,s} are certain non-holomorphic differential operators and the t_{s} are nearly holomorphic modular forms (the sum over s degenerates in our case to a single term, because we have only one archimedean place). By studying the adjoint of the operators L_{k,s}, Shimura shows that

⟨f, g⟩ = ⟨h, g⟩ for all g ∈ S_{n}^{k}  (13)

therefore f ⟼ h gives an essentially algebraic construction of a holomorphic projection. In the sequel we write h = f_{hol} for this construction. At this point we mention that for C^{∞} - modular forms satisfying a suitable growth condition there is also an analytic construction of a holomorphic projection, due to J.Sturm [St81] and further elaborated by Panciskin [Pa91]; this construction gives an explicit description of the Fourier coefficients of the holomorphic projection in terms of an integral involving the Fourier coefficient of the C^{∞}- form. We denote this holomorphic projection by f_{ana}. Both constructions coincide for nearly holomorphic cusp forms. We will deal with nearly holomorphic modular forms arising from holomorphic modular forms by applying Maaß-type differential operators. Those operators (a slightly modified version of those introduced by Maaß [Ma53, Ma71]) are defined in the following way: For a variable Z = (z_{ij}) on ℍ_{n}, we put

\frac{d}{dZ} = (d_{ij} \frac{∂}{∂z_{ij}} )_{1≤i,j≤n} ,

where d_{jj} = 1 and d_{ij} = 1 / 2 otherwise. Let α be a non-negative integer. We denote by

∂_{n,α} = (2πi)^{−n}δ(Z)^{\frac{n−1}{2} −α} det ( \frac{d}{dZ} ) δ(Z)^{α− \frac{n−1}{2}} ,

the (modified) Maaß differential operator and for a nonnegative integer μ we put

∂_{n,k}^{(μ)} = ∂_{n,k+2μ−2} ∘ ⋯ ∘ ∂_{n,k}.

This operator satisfies

∂_{n,k}^{(μ)} (f |_{α} g) = (∂_{n,k}^{(μ)}f) |_{α+2μ} g

for all C^{∞}−functions f on IH_{n} and all g ∈ Sp_{n}(ℝ) and it maps N_{n}^{ν} to N_{n}^{ν+nμ}; for the latter statement we refer to the works of Shimura (eg. [Sh94]: there (p.153) such a statement is proved for differential operators of type

D_{ρ} = ρ(Z − Z̄)^{−1} \frac{d}{dZ} ρ(Z − Z̄),

where ρ is any rational representation of Gl_{n}(ℂ). In section 4 of loc.cit. it is then shown how to get our operators ∂_{n,k} from operators of type D_{ρ}, see also [Sh81b]).

For n = 1 we also need the adjoint of ∂_{1,k} with respect to the Petersson scalar product. It is given explicitely by

ˆ∂_{1,k} := −\frac{1}{2πi}y^{2} \frac{∂}{∂z̄}

and we put

ˆ∂_{1,k}^{(μ)} = ˆ∂_{1,k−2μ+2} ∘ ⋯ ∘ ˆ∂_{1,k}.

We will consider the following map, defined on C^{∞}- functions on IH_{5}:

Λ : F ⟼ ∂_{5,k}^{(ν)}F ⟼ (∂_{5,k}^{(ν)}F ) ∘ i_{1,2,2} ⟼ ˆ∂_{1,k+2ν}^{(μ)}((∂_{k}^{(ν)}F ) ∘ i_{1,2,2}).

Only the symbol ˆ∂_{1,k+2ν}^{(μ)} in the last step needs to be explained: It is an operator acting only on the first component z_{1} of (z_{1}, z_{2}, z_{3}) ∈ IH_{1}×IH_{2}×IH_{2}. It is clear that F ∘i_{1,2,2} is nearly holomorphic with respect to all three variables z_{1} ∈ IH_{1}, z_{2}, z_{3} ∈ IH_{2}, if F ∈ N_{5} ; therefore Λ maps M_{n}^{k} to N_{1}^{(k+2ν−2μ,ν−μ)} ⊗ N_{2}^{(k+2ν,2ν)} ⊗ N_{2}^{(k+2ν,2ν)}. The theorem of Shimura mentioned above (see formula (12)) can now be applied to Λ(F), F ∈ M_{5}^{k} (separately for z_{1} ∈ IH_{1}, z_{2} ∈ IH_{2}, z_{3} ∈ IH_{2}), at least as long as

k + 2ν > 2 + 2ν and k + 2ν − 2μ > 1 + ν − μ.

An inspection of the proof shows that the differential operators involved do not depend on F at all ! We can therefore describe the map Λ in the following way:

Λ = D_{hol} + ∑ (L_{i_{1}} ⊗ L_{i_{2}} ⊗ L_{i_{3}}) ∘ D_{i_{1},i_{2},i_{3}}  (14)

All the summands S map functions on IH_{5} to functions on IH_{1} × IH_{2} × IH_{2} and satisfy

S (f |_{k} i_{1,2,2}(g_{1}, g_{2}, g_{3})) = (Sf) |_{k+2ν−2μ} g_{1} |_{k+2μ} g_{2} |_{k+2μ} g_{3}  (15)

for all (g_{1}, g_{2}, g_{3}) ∈ SL_{2}(ℝ) × Sp_{2}(ℝ) × Sp_{2}(ℝ); it should be clear that on the right side of (15) g_{i} is considered as acting on z_{i} (1 ≤ i ≤ 3). The L_{i_{j}} are differential operators of the kind occuring in (12) and acting on functions defined on IH_{1} (for i_{1}) or IH_{2} (for i_{2} and i_{3}) and the D’s are differential operators on IH_{5}, evaluated in i_{1,2,2}(IH_{1} × IH_{2} × IH_{2}), in particular, D_{hol} is a polynomial P in the entries of \frac{d}{dZ} , Z ∈ IH_{5}, evaluated in i_{1,2,2}(IH_{1} × IH_{2} × IH_{2}); the coefficients of this polynomial depend on k, μ and ν. We write Λ = Λ(k, ν, μ), D_{hol} = D_{hol}(k, ν, μ) and P = P(k, ν, μ) if necessary. We summarize some properties of the polynomial P and the map Λ in

Proposition 2.1 With the notations from above, we have

a) P is a polynomial of total degree 5ν − μ

b) The coefficients of (2πi)^{5ν+μ}P are rational numbers

c) (2πi)^{2μ}D_{hol} defines a map

M_{5}^{k}(ℚ) ⟶ M_{1}^{k+2ν−2μ}(ℚ) ⊗ M_{2}^{k+2ν}(ℚ) ⊗ M_{2}^{k+2ν}(ℚ)

d) For all H ∈ M_{5}^{k}, h ∈ S_{1}^{k+2ν−2μ}, F, G ∈ S^{k+2ν} the following identity of threefold Petersson scalar products holds

⟨⟨⟨Λ(H), h⟩, F ⟩, G⟩ = ⟨⟨⟨D_{hol}(H), h, ⟩F ⟩, G⟩

e) Assume in addition that h, F, G from above are normalized eigenforms and H ∈ M_{5}^{k}(ℚ). Then we have for all σ ∈ Aut(ℂ):

(\frac{⟨⟨⟨(2πi)^{2μ}Λ(H), f⟩, F ⟩, G⟩}{⟨f, f⟩⟨F, F ⟩⟨G, G⟩})^{σ} = \frac{⟨⟨⟨(2πi)^{2μ}Λ(H), h^{σ}⟩, F^{σ}⟩, G^{σ}⟩}{⟨f^{σ}, f^{σ}⟩⟨F^{σ}, F^{σ}⟩⟨G^{σ}, G^{σ}⟩}

in particular,

\frac{⟨⟨⟨(2πi)^{2μ}Λ(H), f⟩, F ⟩, G⟩}{⟨f, f⟩⟨F, F ⟩⟨G, G⟩} ∈ K_{f}K_{F} K_{G}

where K_{f}, K_{F}, K_{G} denote the fields generated by the Hecke eigenvalues of f, F and G respectively.

Proof: The statements a), b), c) are obvious, d) follows from (13) and (14); to prove e), one has to substitute (2πi)^{2μ}Λ by (2πi)^{2μ}D_{hol} and to use standard properties of Petersson scalar products, see e.g. [He99, Theorem 1.9]

3 Eisenstein series at negative integers

We begin this section by recalling the definition of the Siegel type Eisenstein series E_{n}^{k}(Z, s) on ℍ_{n} × ℂ: Let P_{n,0} = \{γ = \binom{A B}{C D} ∈ Γ_{n} | C = 0\}, then

E_{n}^{k}(Z, s) = δ(Z)^{s} ∑_{\binom{A B}{C D}∈P_{n,0}(ℤ) \ Sp_{n}(ℤ)}det(CZ + D)^{−k} |det(CZ + D)|^{−2s}(16)

is absolutely convergent for k + 2Re(s) > n + 1 and has a meromorphic continuation on the whole complex plane. It is well known that E_{n}^{k}(Z, −ν) has rational Fourier coefficients for ν = 0 and k > n + 1. But a more subtle statement is true :

Proposition 3.1 Let k > n+1 be even and ν a non-negative integer smaller than (k−n−1)/2. Then E_{n}^{k}(Z, −ν) is a nearly holomorphic function in N_{n}^{k,nν}, more precisely

E_{n}^{k}(Z, −ν) = π^{nν}∂_{n,k−2ν}^{(ν)}F

where F ∈ M_{n}^{k−2ν} is a holomorphic modular form with rational Fourier coefficients. Let k be not congruent to \frac{n+2}{2} or \frac{n+3}{2} modulo 4, then the statement above remains valid for ν ∈ [0, \frac{k}{2} − 1) ∩ ℤ.

Proof: The conditions imposed on k and n guarantee that the Eisenstein series E_{n}^{k−2ν}(Z, s) evaluated at s = 0 is a holomorphic modular form on ℍ_{n}. Moreover, the corresponding Fourier coefficients are rational numbers. This is a classical theorem of Siegel [Si39] if the weight is sufficiently large (k − 2ν > n + 1), in the case of smaller weights it follows from the work of Feit [Fe86], Shimura [Sh83] and Weissauer [We86] on Hecke summation, see also the paper of Haruki [Ha97] for a nice survey of this topic. Now the Proposition can be proven in the following way: We apply the differential operator ∂_{n,k} to the Eisenstein series (16); this gives, after interchanging the summation and the differentiating process (which is allowed), the formula

∂_{n,k−2ν}^{(ν)}E_{n,0}^{k−2ν}(Z, s) = (−4π)^{−nν} ∏_{j=1}^{ν} ε_{n}(s + k − ν − j)E_{n,0}^{k}(Z, s − ν), (17)

for k even and k > 2ν, where ε_{n}(s) = ∏_{j=0}^{n−1}(s− \frac{j}{2}). To prove the proposition, we only have to observe that

(−4π)^{−nν} ∏_{j=1}^{ν} ε_{n}(s + k − ν − j) |_{s=0} ≠ 0.

Combining Propositions 2.1 and 3.1 and slightly changing our notation we obtain

Corollary 3.2 Let h ∈ S_{1}^{2k̃_{h}} and G, F ∈ S^{2k̃_{G}} be normalized Hecke eigenforms and Δ̃ = k̃_{G} − k̃_{h} ≥ 0. Let

Φ(∂_{2k̃_{h}}^{(Δ̃)}h, G, F; s) = ⟨⟨⟨E_{5}^{2k̃_{G}}(∗, s) ∘ i_{1,2,2,} , ∂_{2k̃_{h}}^{(Δ̃)}h, G, F ⟩⟩⟩  (18)

be the iterated Petersson scalar product of an Eisenstein series (restricted to IH_{1} × IH_{2} × IH_{2}) against the automorphic forms ∂_{2k̃_{h}}^{(Δ̃)}h, G and F. Let ν ∈ [0, k̃_{G} − 1) ∩ ℤ. Then we have

Φ^{∗}(h, G, F; −ν) = \frac{Φ(∂_{2k̃_{h}}^{(Δ̃)}h, G, F; s)}{π^{5ν−2Δ̃}⟨h, h⟩⟨G, G⟩⟨F, F ⟩} ∈ K_{h}K_{G}K_{F}  (19)

and

(Φ^{∗}(h, G, F; −ν))^{σ} = Φ^{∗}(h^{σ}, G^{σ}, F^{σ}; ν) for all σ ∈ Aut(ℂ).  (20)

4 Explicit computations

Let h ∈ S_{1}^{2k̃_{h}} and G, F ∈ S_{2}^{2k̃_{G}} be normalized Hecke eigenforms. We assume F to be a Saito-Kurokawa lift of f ∈ S_{1}^{2k̃_{f}}, where 2k̃_{f} = 4k̃_{F} − 2. Let Φ_{1}^{F}, Φ_{1}^{G} be the first Fourier-Jacobi coefficients of F and G and let f̃, g̃ be the corresponding modular forms of half-integral weight. We denote by ⟨ , ⟩_{J}, ⟨ , ⟩_{∼} the Petersson scalar products on the space of Jacobi forms and modular forms of half-integral weight. Let K be a field. We put a ∼_{K^{×}} b for a, b ∈ ℂ, if there exists a k ∈ K^{×}, such that ak = b. It is well known that ⟨Φ_{1}^{F} , Φ_{1}^{G}⟩_{J} ∼_{ℚ^{×}} ⟨f̃, g̃⟩_{∼}. Let F, Φ_{1}^{F} , f, f̃ be given. Then these liftings commute with σ ∈ Aut(ℂ) by fixing F ↦ F^{σ}.

Lemma 4.1 Let f̃, g̃ as before. Then we have ⟨f̃, g̃⟩_{∼} / ⟨f̃, f̃⟩_{∼} ∈ K_{F} K_{G} and

(\frac{⟨f̃, g̃⟩_{∼}}{⟨f̃, f̃⟩_{∼}} )^{σ} = \frac{⟨f̃^{σ}, g̃^{σ}⟩_{∼}}{⟨f̃^{σ}, f̃^{σ}⟩_{∼}}  for all σ ∈ Aut(ℂ).  (21)

This follows directly from [Sh81], Theorem 1.

Lemma 4.2 Let F as before. Then

\frac{L_{f}(2k̃_{G})⟨f̃, f̃⟩_{∼}}{π^{2k̃_{G}}⟨F, F ⟩} = (\frac{L_{f}(2k̃_{G})⟨f̃, f̃⟩_{∼}}{π^{2k̃_{G}}⟨F, F ⟩} )^{σ} = \frac{L_{f^{σ}}(2k̃_{G})⟨f̃^{σ}, f̃^{σ}⟩_{∼}}{π^{2k̃_{G}}⟨F^{σ}, F^{σ}⟩}  (22)

for all σ ∈ Aut(ℂ).

This follows from L_{f}(2k̃_{G}) = a π^{2k̃_{G}}⟨F, F ⟩/ ⟨f̃, f̃⟩, where a is a positive rational number depending only on the weight of F (cf. [Ko84]).

Proposition 4.3 Let F, G as before and ν ∈ [0, k̃_{G} − 1) ∩ ℤ. Then we have

A(F, G, ν) = \frac{⟨Φ_{1}^{F} , Φ_{1}^{G}⟩_{J}L_{f}(4k̃_{G} − 4 − 2ν)}{π^{4k̃_{G}−2ν−4}⟨F, F ⟩} ∈ K_{F} K_{G} and  (23)
A(F, G, ν)^{σ} = A(F^{σ}, G^{σ}, ν)  for all σ ∈ Aut(ℂ).  (24)

Proof:

The claim follows from well-known properties of critical values of Hecke Lfunctions, if we are allowed to replace ⟨F, F ⟩ by ⟨f, f⟩. To justify this replacement we consider B(F, f) := ⟨F, F ⟩/⟨f, f⟩. Up to powers of π we can replace ⟨F, F ⟩ by a critical value of the standard L-function D(F, s) of F and ⟨f, f⟩ by the products of two critical values of the Hecke L function of f with opposite parity. Taking into account that D(F, s) = ζ(s)L(f, s+2k̃_{G}−1)L(s+2k̃_{G}−2) we easily obtain

B(F^{σ}, f^{σ}) = B(F, f)^{σ} σ ∈ Aut(ℂ)

For details we refer to [He99, Prop.3.18].

5 On the proof of Deligne’s conjecture

In this section we present the main results of our paper, proving conjecture B. Before we start we want to make the folllowing comment on the ”critical values”: Let γ_{∞} be the Γ-factor occuring in the functional equation (11) of Z_{G⊗h}. It is easy to see that (in all cases !) the set of m ∈ ℤ for which neither γ_{∞}(s) nor γ_{∞}(k_{h} + 2k_{G} − 3 − s) has a pole, coincides with the set described in conjectures A and B from the introduction (coming from Yoshida’s analysis of Deligne’s conjecture, assuming the existence of motives attached to G and h).

Theorem 5.1 Let k̃_{h}, k̃_{G} be natural numbers and k̃_{h} ≤ k̃_{G}. Let h ∈ S_{1}^{2k̃_{h}}, G ∈ S_{2}^{2k̃_{G}} be normalized Hecke eigenforms and let us assume that the first Fourier-Jacobi coefficient of G is non-zero. Then the critical values of Z_{G⊗h}(s) are given by the integers m ∈ [2k̃_{G}, 2k̃_{G} + 2k̃_{h} − 3]. Moreover we have

Z_{G⊗h}^{∗}(m) = \frac{Z_{G⊗h}(m)}{π^{4m−4k̃_{G}−2k̃_{h}+4}⟨h , h⟩⟨G , G⟩} ∈ \overline{ℚ}  and (25)
(Z_{G⊗h}^{∗}(m))^{σ} = Z_{G^{σ}⊗h^{σ}}^{∗}(m)  for all σ ∈ Aut(ℂ).  (26)

Proof: Let F ∈ S_{2}^{2k̃_{G}} be a normalized Hecke eigenform, assumed to be a Saito-Kurokawa lift of f ∈ S_{1}^{2k̃_{f}}. We choose F in such a way, that ⟨Φ_{1}^{F} , Φ_{1}^{G}⟩_{J} ≠ 0. Then it follows from [Bo-He99]:

Φ(∂_{2k̃_{h}}^{(Δ̃)}h, G, F; s) = a(s, 2k̃_{G}) ⟨Φ_{1}^{F} , Φ_{1}^{G}⟩_{J}Γ_{▷}(s) ζ_{▷}(s)  (27)
×L_{f}(2s + 4k̃_{G} − 4) Z_{G⊗h}(s + 3k̃_{G} + k̃_{h} − 3),

where

a(s, 2k̃_{G}) = 2^{−6s−12k̃_{G}+13}π^{6−s−4k̃_{G}},
ζ_{▷}(s) = {ζ(2s + 2k̃_{G}) ζ(4s + 4k̃_{G} − 2)ζ(4s + 4k̃_{G} − 4)}^{−1} ,
Γ_{▷}(s) = \frac{Γ(2k̃_{G} + s − \frac{3}{2})Γ(2k̃_{G} + s − 2)}{Γ(2k̃_{G} + s)Γ(2k̃_{G} + s − \frac{1}{2})} \frac{Γ(s + k̃_{G} − k̃_{h})}{Γ(s)}

\frac{Γ(s + k̃_{G} + k̃_{h} − 1) Γ(s + 3k̃_{G} + k̃_{h} − 3) Γ(s + 3k̃_{G} − k̃_{h} − 2)}{Γ(2s + 4k̃_{G} − 3)}.

Then a(r, 2k̃_{G}) ∼_{ℚ} π^{6−4k̃_{G}−r} for r ∈ ℤ and

ζ_{▷}(−ν) ∼_{ℚ^{×}} π^{−(−10ν+10k̃_{G}−6)}
Γ_{▷}(−ν) ∼_{ℚ^{×}}  \frac{Γ(−ν + k̃_{G} − k̃_{h})}{Γ(−ν)}  for ν ∈ [0, k̃_{G} − 1) ∩ ℤ.

Hence Γ_{▷}(−ν) ∼_{ℚ} 1 if ν ∈ [k̃_{G} − k̃_{h}, k̃_{G} − 1) ∩ ℤ. Moreover for such ν it is well known that L_{f}(−2ν + 4k̃_{G} − 4) does not vanish. Finally (27) gives a non-trivial relation between an Eisenstein series at negative integers and the twisted spinor L-function Z_{G⊗h}(m) at m ∈ [2k̃_{G}+k̃_{h}−1, 2k̃_{G}+2k̃_{h}−3]. These are only half of the integers we are interested in, nevertheless nothing has been lost, the functional equation of Z_{G⊗h}(s) can be used to get the whole set of the critical values. Let ν ∈ [k̃_{G} − k̃_{h}, k̃_{G} − 1) as before. Then

Φ(∂_{2k̃_{h}}^{(Δ̃)}h, G, F; −ν) ∼_{ℚ^{×}} π^{11ν−14k̃_{G}+12}⟨Φ_{1}^{F} , Φ_{1}^{G}⟩_{J} L_{f}(−2ν + 4k̃_{G} − 4)
Z_{G⊗h}(−ν + 3k̃_{G} + k̃_{h} − 3).

We have already proven in Proposition 4.3 the relation

⟨Φ_{1}^{F} , Φ_{1}^{G}⟩_{J} L_{f}(−2ν + 4k̃_{G} − 4) ∼_{K_{F}^{×}K_{G}^{×}} π^{4k̃_{G}−2ν−4}⟨F , F ⟩.

Putting everything together leads to

Φ(∂_{2k̃_{h}}^{(Δ̃)}h, G, F; −ν) ∼\overline{ℚ}^{×} π^{9ν−10k̃_{G}+8}⟨F , F ⟩ Z_{G⊗h}(−ν + 3k̃_{G} + k̃_{h} − 3).(28)

Moreover we know how Aut(ℂ) acts. Applying corollary 3.2 proves the theorem.

Theorem 5.2 Let k̃_{h}, k̃_{G} be natural numbers with k̃_{G} < k̃_{h} < 2k̃_{G} − 1. Let h ∈ S_{1}^{2k̃_{h}}, G ∈ S_{2}^{2k̃_{G}} be normalized Hecke eigenforms and let us assume that the first Fourier-Jacobi coefficient of G is non-zero. Then the critical values of Z_{G⊗h}(s) are given by the integers m ∈ [2k̃_{h}, 4k̃_{G} − 3]. Moreover we have

Z_{G⊗h}^{∗}(m) = \frac{Z_{G⊗h}(m)}{π^{4m−4k̃_{G}−2k̃_{h}+4}⟨h , h⟩⟨G , G⟩} ∈ \overline{ℚ}  and (29)
(Z_{G⊗h}^{∗}(m))^{σ} = Z_{G^{σ}⊗h^{σ}}^{∗}(m)  for all σ ∈ Aut(ℂ).  (30)

Proof: For convenience we put μ = 2k̃_{h} − 2k̃_{G} > 0. To study analytic properties of the L-function, in the case 2k̃_{h} > 2k̃_{G} we have introduced certain antiholomorphic differential operators A^{∘}D_{n,s}^{μ}. We recall that those operators are defined on IH_{2n}, they are polynomials in the entries of \frac{d}{dZ}, restricted to IH_{n} × IH_{n} ↪ IH_{2n} via i_{n,n}, therefore they map functions on IH_{2n} to functions on IH_{n} × IH_{n}. These operators satisfy

A^{∘}D_{n,β}^{μ} (f |_{α,β} i(g, g′)) = (A^{∘}D_{n,β}^{μ}f) |_{α,β+μ}^{(1)} g |_{α,β+μ}^{(2)} g′

for all g, g′ ∈ Sp(n, ℝ) and all C^{∞}-functions on IH_{2n}; the upper indices

(1) and (2) attached to the slash-operator | indicate to which of the two components of IH_{n} × IH_{n} the |_{α,β+μ} is assigned to. For more details and properties see [Bo-He99, section 4]. We also recall briefly the relevant integral representation [Bo-He99], Theorem 5.3:

⟨⟨⟨δ(z_{1})^{s}δ(z_{2})^{s}δ(z_{3})^{s+μ}A^{∘}D_{2,s}^{μ}(G_{2k̃_{h}+s,s}^{(5)} ∘ i_{1,4}) (z_{1} × z_{2} × z_{3}); h(z_{1})G(z_{2})F (z_{3})⟩⟩⟩
= γ_{∞}^{−}(s) ⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J} ζ_{◁}(s) L_{f}(2s + 2k̃_{G} + 2k̃_{h} − 4)  (31)
Z_{G⊗h}(s + 2k̃_{G} + 2k̃_{h} − 3).

Here

G_{k+s}^{(n)}(Z) = det(Y)^{−s}E_{n}^{k}(Z, s) = ∑_{\binom{A B}{C D}∈P_{n,0}(ℤ) \ Sp_{n}(ℤ)}det(CZ+D)^{−k} |det(CZ+D)|^{−2s},

ζ_{◁}(s) = {ζ(2s + 2k̃_{h}) ζ(4s + 4k̃_{h} − 2)ζ(4s + 4k̃_{h} − 4)}^{−1}

and γ_{∞}^{−}(s) is a certain Γ-factor.

If we assume s = −μ − ν to be an integer, where ν ∈ [0, 2k̃_{G} − k̃_{h} − 2], then formula (31) can be described in a more explicit way:

γ_{∞}^{−}(s) ∼_{ℚ^{×}} π^{−(2k̃_{G}+2k̃_{h}+s−6)}
ζ_{◁}(s) ∼_{ℚ^{×}} π^{−(10s+10k̃_{h}−6)}
⟨Φ_{1}^{F}; Φ_{1}^{G}⟩_{J} L_{f}(2s + 2k̃_{G} + 2k̃_{h} − 4) ∼_{K_{f}^{×}} π^{2s+2k̃_{G}+2k̃_{h}−4}⟨F , F ⟩.

Hence we obtained for the right side of (31):

∼_{K_{f}^{×}} π^{9(μ+ν)−10k̃_{h}+8}Z_{G⊗h}(4k̃_{G} − 3 − ν).

On the other hand, let us assume for the moment that

H_{s}(z_{1}, z_{2}, z_{3}) := δ(z_{1})^{s}δ(z_{2})^{s−μ}δ(z_{3})^{s−μ}A^{o}D_{2,s}^{μ}(G_{2k̃_{h}+s,s}^{(5)})(z_{1} × z_{2} × z_{3})

has at s = −μ − ν a holomorphic projection (w.r.t. all three variables z_{1}, z_{2}, z_{3}); we call this holomorphic projection (in the analytic sense for z_{2} and z_{3}) H_{−μ−ν}(z_{1}, z_{2}, z_{3})_{ana}; it is an element of

π^{3μ+5ν} × M_{1}^{k_{h}}(ℚ) ⊗ S_{2}^{k_{G}}(ℚ) ⊗ S_{2}^{k_{G}}(ℚ).

Then we obtain for the left hand side of equation (31) the expression

a π^{5(μ+ν)−2μ}⟨h , h⟩⟨G , G⟩⟨F , F ⟩,  (32)

where a is an algebraic number. Hence the quotient

\frac{π^{8+2k̃_{h}−12k̃_{G}+4ν}Z_{G⊗h}(4k̃_{G} − 3 − ν)}{⟨h , h⟩⟨G , G⟩}  (33)

is an algebraic number and has the expected behaviour under the action of σ ∈ Aut(ℂ). Again by the functional equation of the L-function, we obtain this for all critical values predicted by Deligne. To finish the proof of theorem5.2, we have to prove the assertion about the holomorphic projection: Using

E_{n}^{k}(Z, s) = δ(Z)^{s}G_{k+s,s}^{(n)}

we get from Prop.3.1 that G_{2k_{h}+s,s}^{(5)} is (at s = −μ − ν) of type

δ(Z)^{μ+ν} × nearly holomorphic function w.r.t. Z

Then

G_{2k̃_{h}+s,s}^{(5)}(z_{1} × Z) , (Z ∈ IH_{4})

is a polynomial of degree ≤ μ + ν in the variable y_{1} = ℑ(z_{1}), and of degree ≤ 2μ + 2ν as a polynomial in the entries of y_{2} = ℑ(z_{2}) and the same w.r.t. the entries of y_{3} = ℑ(z_{3}) and the entries of y = ℑ(z) with

z_{1} × Z = ( \begin{matrix}z_{1} & 0 & 0 \\ 0 & z_{2} & z \\ 0 & z^{t} & z_{3} \end{matrix} ) , Z ∈ IH_{4}

and therefore

A^{o}D_{2,s}^{μ}(G_{2k̃_{h}+s,s}^{(5)})(z_{1} × z_{2} × z_{3})

is a polynomial of degree ≤ μ + ν w.r.t. y_{1}, degree ≤ 2μ + 2ν w.r.t. the entries of y_{2} degree ≤ 2μ + 2ν w.r.t. the entries of y_{3}. with holomorphic functions as coefficients; more precisely, this function has a Fourier expansion of type

∑_{T} P (y_{1}, y_{2}, y_{3}, 2πT)a(T)e^{2πi(t_{1}z_{1}+tr(T_{2}z_{2})+tr(T_{3}z_{3}))}

where

T = ( \begin{matrix}t_{1} & ∗ & ∗ \\ ∗ & T_{2} & ∗ \\ ∗ & ∗ & T_{3} \end{matrix} )

runs over all positive semidefinite half-integral matrices of size 5, a(T) denotes the Fourier coefficient of the degree 5 Siegel Eisenstein series G_{k_{h}−2μ−2ν,0}^{(5)} and P is a polynomial of the type described above (and also a polynomial in the entries of T); as a polynomial in all these variables it has coefficients in ℚ. The polynomial P arises naturally in the following way: We recall first that (by a proof quite similar to the one in Prop.3.1) there is a differential operator M^{(μ+ν)} acting on IH_{5} satisfying

M^{(μ+ν)}G_{2k̃_{h}−2μ−2ν,0}^{(5)} = G_{2k̃_{h}−μ−ν,−μ−ν}^{(5)}.

Up to normalization this is the Maaß-operator introduced by H.Maaß himself [Ma53, Ma71]; the explicit form of this differential operator is not needed here. The polynomial P then comes up, when we apply the differential operators at hand to a function of type f_{T} (Z) = e^{tr(T Z)} with Z ∈ IH_{5}:

A^{o}D_{2,−μ−ν}^{μ}M^{(μ+ν)}f_{T} = P (y_{1}, y_{2}, y_{3}, T)e^{t_{1}z_{1}+tr(T_{2}z_{2})+tr(T_{3}z_{3})}

As a simple consequence of the equivariance properties of the differential operators we obtain

P (T [A^{t}], y_{1}, y_{2}, y_{3}) = det(A)^{μ}P (T, y_{1}, y_{2}[a_{2}], y_{3}[a_{3}])  (34)

for all

A = 1 × a_{2} × a_{3} ∈ GL(5, ℝ) , a_{2}, a_{3} ∈ GL(2, ℝ)

We claim that only the positive definite matrices T_{2} and T_{3} give nontrivial contributions to the series above: This is (for T_{3})a simple consequence of (34) applied to T of type

T = ( \begin{matrix}t_{1} & ∗ & ∗ & 0 \\ ∗ & T_{2} & ∗ & 0 \\ ∗ & ∗ & t & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

and a_{2} = 1_{2}, a_{3} = ( \begin{matrix}1 & 0 \\ 0 & λ \end{matrix} ) with λ ⟶ 0. Exactly the same type of

argument also works for T_{2}. Therefore (by the same argument as in the case of holomorphic cusp forms) the function

H_{−μ−ν}(z_{1}, z_{2}, z_{3}) = δ(z_{1})^{−μ−ν}δ(z_{2})^{−ν}δ(z_{3})^{−ν} ∑_{T} P (y_{1}, y_{2}, y_{3}, 2πT)a(T)e^{2πit_{1}z_{1}+tr(T_{2}z_{2})+tr(T_{3}z_{3})}

is, considered as a (nonholomorphic) modular form w.r.t. z_{2} or z_{3} of rapid decay, in particular the analytic procedure of holomorphic projection as described in [Pa91, St81] can be applied (the condition on the weight is automatically satisfied because we only have to consider weights k_{G} ≥ 10). In this way we obtain a holomorphic cusp form with respect to z_{2} and z_{3}, which as a function of z_{1} is nearly holomorphic; its Fourier coefficients atttached to exponent matrices T_{2} and T_{3} are given by

\frac{π^{4k_{G}−6}}{Γ_{2}(k_{G} − \frac{3}{2})^{2}} det(4T_{2})^{k_{G}− \frac{3}{2}} det(4T_{3})^{k_{G}− \frac{3}{2}} ∑_{T} a(T)

× ∫_{y_{2}} ∫_{y_{3}} y_{1}^{−μ−ν}P (y_{1}, y_{2}, y_{3}, 2πT)e^{−4πtr(T_{2}y_{2})−4πtr(T_{3}y_{3}} det(y_{2})^{k_{G}− \frac{3}{2} −ν} det(y_{3})^{k_{G}− \frac{3}{2} −ν}d^{∗}y_{2}dy_{3}^{∗}

Here T runs over all positive semidefinite half-integral matrices with T_{2} and T_{3} as block matrices on the bottom of its diagonal; furthermore we use the standard notations

Γ_{n}(s) = π^{\frac{n(n−1)}{4}} ∏_{j=0}^{n−1} Γ(s − \frac{j}{2}) = ∫ det(y)^{s}e^{−tr(y)}d^{∗}y

d^{∗}y = det(y)^{− \frac{n+1}{2}} ∏_{i≤j} dy_{ij}

Using

P (y_{1}, y_{2}, y_{3}, 2πT) = π^{2μ}P (πy_{1}, πy_{2}, πy_{3}, 2T)

we obtain that

\frac{1}{Γ_{2}(k_{G} − \frac{3}{2})^{2}} ∫_{y_{2}} ∫_{y_{3}} P (y_{1}, y_{2}, y_{3}, 2πT)e^{−4πtr(T_{2}y_{2})−4πtr(T_{3}y_{3})} det(y_{2})^{k_{G}− \frac{3}{2} −ν} det(y_{3})^{k_{G}− \frac{3}{2} −ν}d^{∗}y_{2}dy_{3}^{∗}

is an element of

π^{2μ+4ν+6−4k_{G}} × ℚ[πy_{1}]

of degree ≤ ν + μ w.r.t. y_{1}. Here we tacitly used that

\frac{1}{Γ_{2}(k_{G} − \frac{3}{2})^{2}} ∫_{y_{2}} ∫_{y_{3}} e^{−4tr(T_{2}y_{2})−4tr(T_{3}y_{3})} det(y_{2})^{k_{G}− \frac{3}{2} −ν} det(y_{3})^{k_{G}− \frac{3}{2} −ν}d^{∗}y_{2}dy_{3}^{∗} =

(\frac{Γ_{2}(k_{G} − \frac{3}{2} − ν)}{Γ_{2}(k_{G} − \frac{3}{2})})^{2} det(4y_{2})^{k_{G}− \frac{3}{2} −ν} det(4y_{3})^{k_{G}− \frac{3}{2} −ν}

and that we may insert the polynomial P in the integrand essentially by differentiation with respect to the entries of T_{2} and T_{3}. Therefore we obtain that the (analytic) holomorphic projection of H_{−μ−ν} with respect to z_{2} and z_{3} is a nearly holomorphic function of z_{1} of the form

π^{3μ+5ν} × ∑_{j=0}^{μ+ν} \frac{1}{(πy_{1})^{j}} h_{j}(z_{1}, z_{2}, z_{3})

with h_{j} being holomorphic periodic functions of all variables with rational Fourier coefficients. The assertion about (H_{−μ−ν})_{ana} follows by considering the holomorphic part of the nearly holomorphic function from above (as described e.g. in [Sh76, lemma 7]) or by the formula (12).

At this point we should remark that we found it more convenient to use the analytic version of holomorphic projection for the ”second case”, i.e. 2k̃_{G} < 2k̃_{h} < 4k̃_{G} − 2. It should also be possible to do it within the framework used for the ”first case”. This however would have required some extra considerations about the differential operators A^{∘}D_{n,s}^{μ} which we prefered to avoid.

Let 2k̃_{h} > 4k̃_{G} − 2. Then γ_{∞}^{−}(s_{0}) ζ_{◁}(s_{0}), for s_{0} chosen to study the critical values m ∈ C_{I}, is zero. This shows that that the method of the present paper cannot be used to study these critical values (we would get the ”wrong” Petersson scalar products anyway !). Nevertheless let G ∈ S_{2}^{k,M} be a normalized eigenform in the Maaß space . Then we are able to prove this conjecture by employing the splitting of Z_{G}(s) given in (9), which relates the critical values of Z_{G⊗h}(s) to critical values of Hecke- and Rankin L-functions attached to elliptic cusp forms. It is then an easy exercise to prove Conjecture A and B directly using only the explicit computations of section 4.

Remark 5.3 We have discovered in [He99] some surprising similiarities between analytic properties of the twisted spinor L-function Z_{G⊗h}(s) and the Rankin triple L-function L_{f_{1}⊗f_{2}⊗f_{3}}(s) attached to three elliptic cuspidal eigenforms. This similarity also appears in the critical values, more precisely, we claim that Conjectures A and B are of exactly the same type as the one stated by Deligne and worked out by Blasius (see [Or87]) for the Rankin triple Lfunction, where a certain triangle condition related to the weights played a fundamental role. By introducing a virtual weight in the context of the twisted spinor L-function, the distinction between conjecture A and B and the relation to the triangle condition can be interpreted in the following way. Let k_{1} = 2k̃_{h} and k_{2} = k_{3} = 2k̃_{G}, where 2k̃_{h} ≤ 2k̃_{G} and k_{1} = k_{2} = 2k̃_{G} and k_{3} = 2k̃_{h} otherwise. Then k_{1} ≤ k_{2} ≤ k_{3} and the hidden triangle condition is given by

k_{3} ≥ k_{1} + k_{2} (Conjecture A)
k_{3} < k_{1} + k_{2} (Conjecture B).

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Siegfried Böcherer^{2} Kunzenhof 4B 79117 Freiburg, Germany boech\begin{matrix}a \\ ◯ \end{matrix}siegel.math.uni-mannheim.de and

2Until may 2000: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton. NJ08540, USA

Bernhard Ernst Heim Max-Planck Institut für Mathematik Vivatsgasse 7 53111 Bonn, Germany heim\begin{matrix}a \\ ◯ \end{matrix}mpim-bonn.mpg.de