Rankin’s lemma of higher genus and explicit formulas for Hecke operators

Alexei Panchishkin^{1} and Kirill Vankov^{2}

1 Institut Fourier, Université de Grenoble I, BP 74, 38402 Saint-Martin d’Hères,

  France Alexei.Pantchichkine@ujf-grenoble.fr

2 Institut Fourier, Université de Grenoble I, BP 74, 38402 Saint-Martin d’Hères,

  France kvankov@fourier.ujf-grenoble.fr

To dear Yuri Ivanovich Manin for his seventieth birthday with admiration

Summary. We develop explicit formulas for Hecke operators of higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebra and in a tensor product of such algebras. In particular, we formulate and prove Rankin’s lemma in genus two. An application to a holomorphic lifting from GSp_{2} × GSp_{2} to GSp_{4} is given using Ikeda-Miyawaki constructions.

1 Introduction: generating series for the Hecke operators

Let p be a prime. The Satake isomorphism [Sa63] relates p-local Hecke algebras of reductive groups over ℚ to certain polynomial rings. Then one can use a computer in order to find interesting identities between Hecke operators, between their eigenvalues, and relations to Fourier coefficients of modular forms of higher degree.

The purpose of the present paper is to extend Rankin’s Lemma to the summation of Hecke series of higher genus using symbolic computations. We refer to [Ma-Pa77], where Rankin’s Lemma was used in the elliptic modular case for multiplicative and additive convolutions of Dirichlet series. That work was further developped in [Pa87], [Pa02], see also [Ma-Pa05].

Recall that a classical method to produce L-functions for an algebraic group G over ℚ uses the generating series

∑_{n=1}^{∞} λ_{f}(n)n^{−s} = ∏_{p primes} ∑_{δ=0}^{∞} λ_{f}(p^{δ})p^{−δs},

of the eigenvalues of Hecke operators on an automorphic form f on G. We study the generating series of Hecke operators T(n) for the symplectic group Sp_{g}, and λ_{f}(n) = λ_{f}(T(n)).

Let Γ = Sp_{g}(ℤ) ⊂ SL_{2g}(ℤ) be the Siegel modular group of genus g, and [p]_{g} = pI_{2g} = T(\underbrace{p, ⋯, p}_{2g}) be the scalar Hecke operator for Sp_{g}. According to

Hecke and Shimura,

D_{p}(X) = ∑_{δ=0}^{∞} T(p^{δ})X^{δ}

= \{\frac{1}{1 − T(p)X + p[p]_{1}X^{2}} , if g = 1 (see [Hecke], and [Shi71], Theorem 3.21), \frac{1 − p^{2}[p]_{2}X^{2}}{1 − T(p)X + \{pT_{1}(p^{2}) + p(p^{2} + 1)[p]_{2}\}X^{2} − p^{3}[p]_{2}T(p)X^{3} + p^{6}[p]_{2}^{2}X^{4}} if g = 2 (see [Shi63], Theorem 2),

where T(p), T_{i}(p^{2}) (i = 1, ⋯, g) are the g+1 generators of the corresponding Hecke ring over ℤ for the symplectic group Sp_{g}, in particular, T_{g}(p^{2}) = [p]_{g}.

It was established by A. N. Andrianov, that there exist polynomials

E(X), F (X) ∈ ℚ[T(p), T_{1}(p^{2}), ⋯, T_{g}(p^{2}), X] such that

D(X) = ∑_{δ=0}^{∞} T(p^{δ})X^{δ} = \frac{E(X)}{F (X)},

with a polynomial F (X) = ∑_{j=0}^{2^{g}} q_{j}X^{j} of degree 2^{g}, and such that E(X) =

∑_{j=0}^{2^{g}−2} u_{j}X^{j} is a polynomial of degree 2^{g} − 2 with the leading term

(−1)^{g−1}p^{g(g+1)2^{g−2}−g^{2}}[p]^{2^{g−1}−1}X^{2^{g}−2}

(as stated in Theorem 6 at p. 451 of [An70] and at p. 61 of § 1.3, [An74]).

In the present paper we study explicit formulas for Hecke operators in higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebras. The question of computing such series explicitly was raised by Prof. S.Friedberg during first author’s talk at the conference "Zeta Functions" (The Independent Moscow University, September 18-22, 2006).

In particular, we formulate and prove Rankin’s lemma in genus two for generating series with coefficients in a tensor product of local Hecke algebras.

We prove in Theorem 4 that for g = 2,

∑_{δ=0}^{∞} T(p^{δ}) ⊗ T(p^{δ}) X^{δ} = \frac{(1 − p^{6}[p] ⊗ [p]X^{2}) · R(X)}{S(X)},

for certain two polynomials

R(X) = 1 + r_{1}X + ⋯ + r_{11}X^{11} + r_{12}X^{12} with r_{1} = r_{11} = 0,

S(X) = 1 + s_{1}X + ⋯ + s_{16}X^{16}

with coefficients explicitly expressed through Hecke operators in Appendix. A motivic interpretation of the polynomial S is given in terms of the tensor product of motives.

For the group GL(n), a simpler result could be obtained using the Tamagawa generating series (see [Tam]) and its expansion into simple fractions:

\frac{1}{(1 − x_{1}X)·. . .·(1 − x_{n}X)} = ∑_{i=1}^{n} \frac{α_{i}}{1 − x_{i}X} where α_{i} = \frac{x_{i}^{n−1}}{∏_{j≠i}^{n}(x_{j} − x_{i})},

see Remark 6.

Also, it is helpful for the reader to remember cases where the Hecke series do have simple numerators: Andrianov (GSp(2), see ([An74]), Tamagawa (GL(n), see [Tam]), Böcherer (for standard-L-function, see [BHam]).

2 Results

2.1 Preparation: a formula for the total Hecke operator T(p^{δ}) of genus 2

We establish first the following useful formula (in spherical variables x_{0}, x_{1}, x_{2}):

Ω_{x}^{(2)}(T(p^{δ})) =  (1)
p^{−1} x_{0}^{δ}(p x_{1}^{(3+δ)} x_{2} − p x_{1}^{(2+δ)} − p x_{1}^{(3+δ)} x_{2}^{(2+δ)} + p x_{1}^{(2+δ)} x_{2}^{(3+δ)}
− p x_{1} x_{2}^{(3+δ)} + p x_{2}^{(2+δ)} + p x_{1} − p x_{2} − x_{1}^{(2+δ)} x_{2}^{2} + x_{1}^{(1+δ)} x_{2}

+ x_{1}^{(2+δ)} x_{2}^{(1+δ)} − x_{1}^{(1+δ)} x_{2}^{(2+δ)} + x_{1}^{2} x_{2}^{(2+δ)} − x_{1} x_{2}^{(1+δ)} − x_{1}^{2} x_{2} + x_{1} x_{2}^{2})/

((1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (x_{1} − x_{2}))
= − p^{−1} x_{0}^{δ}((1 − x_{1} x_{2}) (p x_{1} − x_{2}) x_{1}^{(δ+1)} + (1 − x_{1} x_{2}) (x_{1} − p x_{2}) x_{2}^{(δ+1)}
− (1 − p x_{1} x_{2}) (x_{1} − x_{2}) (x_{1} x_{2})^{(δ+1)} − (p − x_{1} x_{2})(x_{1} − x_{2}))/
((1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (x_{1} − x_{2})).

Andrianov’s generating series

The expression (1) comes from the following Andrianov’s generating series

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{δ})) X^{δ} = \frac{1 − \frac{x_{0}^{2} x_{1} x_{2}}{p} X^{2}}{(1 − x_{0} X) (1 − x_{0} x_{1} X) (1 − x_{0} x_{2} X) (1 − x_{0} x_{1} x_{2} X)}

after developing and a simplification using change of summation.

Note that the formula (1) makes it possible to treat higher generating series of the following type

D_{p,m}(X) = ∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{mδ})) X^{δ} (m = 2, 3, ⋯)

(in spherical variables x_{0}, x_{1}, x_{2}).

2.2 Rankin’s generating series in genus 2

Let us use the spherical variables x_{0}, x_{1}, x_{2} and y_{0}, y_{1}, y_{2} for the Hecke operators.

Note that there are two types of convolutions: the first one is defined through the Fourier coefficients (it was used by [An-Ka] for the analytic continuation of the standard L- function), and the second one is defined through the eigenvalues of Hecke operators, and it is more suitable in order to treat the L-functions attached to tensor products of representations of the Langlands group. However, a link between the two types is known only for g = 1.

In order to state a multiplicative analogue of Rankin’s lemma in genus two we need to write the corrseponding formula for Hecke operator T(p^{δ}) (in spherical variables y_{0}, y_{1}, y_{2}).

Ω_{y}^{(2)}(T(p^{δ})) = p^{−1} y_{0}^{δ}(p y_{1}^{(3+δ)} y_{2} − p y_{1}^{(2+δ)} − p y_{1}^{(3+δ)} y_{2}^{(2+δ)} + p y_{1}^{(2+δ)} y_{2}^{(3+δ)}
− p y_{1} y_{2}^{(3+δ)} + p y_{2}^{(2+δ)} + p y_{1} − p y_{2} − y_{1}^{(2+δ)} y_{2}^{2} + y_{1}^{(1+δ)} y_{2}

+ y_{1}^{(2+δ)} y_{2}^{(1+δ)} − y_{1}^{(1+δ)} y_{2}^{(2+δ)} + y_{1}^{2} y_{2}^{(2+δ)} − y_{1} y_{2}^{(1+δ)} − y_{1}^{2} y_{2} + y_{1} y_{2}^{2})/

((1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (y_{1} − y_{2}))

Then we have that the product of the above polynomials is given by

Rankin’s lemma of higher genus and explicit formulas for Hecke operators  529
Ω_{x}^{(2)}(T(p^{δ})) · Ω_{y}^{(2)}(T(p^{δ})) =

p^{−2} x_{0}^{δ} y_{0}^{δ}(p x_{1}^{(3+δ)} x_{2} − p x_{1}^{(2+δ)} − p x_{1}^{(3+δ)} x_{2}^{(2+δ)} + p x_{1}^{(2+δ)} x_{2}^{(3+δ)} − p x_{1} x_{2}^{(3+δ)} + p x_{2}^{(2+δ)} + p x_{1} − p x_{2} − x_{1}^{(2+δ)} x_{2}^{2} + x_{1}^{(1+δ)} x_{2} + x_{1}^{(2+δ)} x_{2}^{(1+δ)} − x_{1}^{(1+δ)} x_{2}^{(2+δ)} + x_{1}^{2} x_{2}^{(2+δ)} − x_{1} x_{2}^{(1+δ)} − x_{1}^{2} x_{2} + x_{1} x_{2}^{2}) · (p y_{1}^{(3+δ)} y_{2} − p y_{1}^{(2+δ)} − p y_{1}^{(3+δ)} y_{2}^{(2+δ)} + p y_{1}^{(2+δ)} y_{2}^{(3+δ)} − p y_{1} y_{2}^{(3+δ)} + p y_{2}^{(2+δ)} + p y_{1} − p y_{2} − y_{1}^{(2+δ)} y_{2}^{2} + y_{1}^{(1+δ)} y_{2} + y_{1}^{(2+δ)} y_{2}^{(1+δ)} − y_{1}^{(1+δ)} y_{2}^{(2+δ)} + y_{1}^{2} y_{2}^{(2+δ)} − y_{1} y_{2}^{(1+δ)} − y_{1}^{2} y_{2} + y_{1} y_{2}^{2})/ ((1 − x_{1})(1 − x_{2})(1 − x_{1}x_{2})(x_{1} − x_{2})(1 − y_{1})(1 − y_{2})(1 − y_{1}y_{2})(y_{1} − y_{2}))

We wish to compute the generating series

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{δ})) · Ω_{y}^{(2)}(T(p^{δ})) X^{δ} ∈ ℚ[x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}][[X]].

The answer is given by the following multiplicative analogue of Rankin’s lemma in genus two: Theorem 1. The following equality holds

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{δ})) · Ω_{y}^{(2)}(T(p^{δ})) X^{δ} =  (2)

− \frac{(p x_{1} − x_{2}) (1 − p y_{1} y_{2}) x_{1} y_{1} y_{2}}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} x_{1} y_{0} y_{1} y_{2} X)}

+ \frac{x_{2} y_{1} (x_{1} − p x_{2}) (p y_{1} − y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} x_{2} y_{0} y_{1} X)}

+ \frac{x_{2} y_{2} (x_{1} − p x_{2})(y_{1} − p y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{1}) (1 − y_{1}) (1 − y_{2}) (y1 − y_{2}) (1 − x_{0} y_{0} x_{2} y_{2} X)}

− \frac{x_{2} y_{1} y_{2} (x_{1} − p x_{2}) (1 − p y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} x_{2} y_{0} y_{1} y_{2} X)}

− \frac{x_{1} (p x_{1} − x_{2}) (p − y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} x_{1} y_{0} X)}

− \frac{x_{1} x_{2} y_{1}(1 − p x_{1} x_{2}) (p y_{1} − y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} x_{1} x_{2} y_{0} y_{1} X)}

− \frac{x_{1} x_{2} y_{2} (1 − p x_{1} x_{2}) (y_{1} − p y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} x_{1} x_{2} y_{0} y_{2} X)}

+ \frac{y_{1} y_{2} (p − x_{1} x_{2}) (1 − p y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} y_{0} y_{1} y_{2} X)}

+ \frac{x_{1} x_{2} (1 − p x_{1} x_{2}) (p − y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} x_{1} x_{2} y_{0} X)}

− \frac{x_{1} y_{1} (p x_{1} − x_{2}) (p y_{1} − y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} x_{1} y_{0} y_{1} X)}

+ \frac{x_{1} y_{2} (p x_{1} − x_{2}) (y_{1} − p y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} x_{1} y_{0} y_{2} X)}

− \frac{x_{2} (x_{1} − p x_{2}) (p − y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (x_{1} − x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} x_{2} y_{0} X)}

+ \frac{x_{1} x_{2} y_{1} y_{2} (1 − p x_{1} x_{2}) (1 − p y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} x_{1} x_{2} y_{0} y_{1} y_{2} X)}

+ \frac{(p − x_{1} x_{2}) (p − y_{1} y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (1 − y_{1} y_{2}) (1 − x_{0} y_{0} X)}

− \frac{y_{1} (p − x_{1} x_{2}) (p y_{1} − y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} y_{0} y_{1} X)}

− \frac{y_{2} (p − x_{1} x_{2}) (y_{1} − p y_{2})}{p^{2} (1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (1 − y_{1}) (1 − y_{2}) (y_{1} − y_{2}) (1 − x_{0} y_{0} y_{2} X)}

Remark 2 (On the denominator of series (2)). One finds using a computer that the polynomials not depending on X in the denominators of (2) cancel after the simplification in the ring ℚ[x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}][[X]], so that the common denominator becomes

(1 − x_{0} y_{0}X)(1 − x_{0} y_{0} x_{1} X)(1 − x_{0} y_{0} y_{1} X)(1 − x_{0} y_{0} x_{2} X)(1 − x_{0} y_{0} y_{2} X) (1 − x_{0} y_{0} x_{1} y_{1} X)(1 − x_{0} y_{0} x_{1} x_{2} X)(1 − x_{0} y_{0} x_{1} y_{2} X)(1 − x_{0} y_{0} y_{1} x_{2} X) (1 − x_{0} y_{0} y_{1} y_{2} X)(1 − x_{0} y_{0} x_{2} y_{2} X)(1 − x_{0} y_{0} x_{1} y_{1} x_{2} X)(1 − x_{0} y_{0} x_{1} y_{1} y_{2} X) (1 − x_{0} y_{0} x_{1} x_{2} y_{2} X)(1 − x_{0} y_{0} y_{1} x_{2} y_{2} X)(1 − x_{0} y_{0} x_{1} y_{1} x_{2} y_{2} X).

Remark 3 (Comparison with g = 1). It turns out by direct computation that the numerator of the rational fraction (2) is a product of the factor 1 − x_{0}^{2}y_{0}^{2} x_{1}y_{1}x_{2}y_{2}X^{2} by a polynomial of degree 12 in X with coefficients in ℚ[x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}] with the constant term equal to 1 and the leading term

\frac{x_{0}^{12}y_{0}^{12}x_{1}^{6}x_{2}^{6}y_{1}^{6}y_{2}^{6}}{p^{2}} X^{12}.

Moreover, the factor of degree 12 does not contain terms of degree 1 and 11 in X. The factor of degree 2 in X is very similar to one in the case g = 1 (this series was studied and used in [Ma-Pa77] ):

∑_{δ=0}^{∞} Ω_{x}^{(1)}(T(p^{δ})) · Ω_{y}^{(1)}(T(p^{δ})) X^{δ} = ∑_{δ=0}^{∞} \frac{x_{0}^{δ} (1 − x_{1}^{(1+δ)})}{1 − x_{1}} · \frac{y_{0}^{δ} (1 − y_{1}^{(1+δ)})}{1 − y_{1}}X^{δ}

= \frac{1}{(1 − x_{1}) (1 − y_{1}) (1 − x_{0} y_{0} X)} − \frac{y_{1}}{(1 − x_{1}) (1 − y_{1}) (1 − x_{0} y_{0} y_{1} X)}

− \frac{x_{1}}{(1 − x_{1}) (1 − y_{1}) (1 − x_{0} y_{0} x_{1} X)} + \frac{x_{1} y_{1}}{(1 − x_{1}) (1 − y_{1}) (1 − x_{0} y_{0} x_{1} y_{1} X)}

= \frac{1 − x_{0}^{2} y_{0}^{2} x_{1} y_{1} X^{2}}{(1 − x_{0} y_{0} x_{1} y_{1} X) (1 − x_{0} y_{0} x_{1} X) (1 − x_{0} y_{0} y_{1} X) (1 − x_{0} y_{0} X)}

Rankin’s lemma of higher genus and explicit formulas for Hecke operators  531

2.3 Symmetric square generating series in genus 2

Using the same method, one can evaluate the symmetric square generating series and the cubic generating series of higher genus. Note that this series, written here in spherical variables x_{0}, x_{1}, x_{2} is different from one studied by Andrianov-Kalinin, and has the form:

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{2δ}))X^{δ} = (1 − \frac{x_{0}^{2}x_{1}x_{2}}{p} X)×

× \frac{(1 + x_{0}^{2}x_{1}X + x_{0}^{2}x_{2}X + 2x_{0}^{2}x_{1}x_{2}X + x_{0}^{2}x_{1}x_{2}^{2}X + x_{0}^{2}x_{1}^{2}x_{2}X + x_{0}^{4}x_{1}^{2}x_{2}^{2}X^{2})}{(1 − x_{0}^{2} x_{1}^{2} x_{2}^{2} X)(1 − x_{0}^{2} x_{1}^{2} X)(1 − x_{0}^{2} x_{2}^{2} X)(1 − x_{0}^{2} X)}

2.4 Cubic generating series in genus 2

The cubic generating series of higher genus, written here in spherical variables x_{0}, x_{1}, x_{2} has the form:

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{3 δ})) X^{δ} = p^{−1}(−p + x_{0}^{6} x_{1}^{4} x_{2}^{2} X^{2} + x_{0}^{6} x_{1}^{2} x_{2}^{4} X^{2} + 2 x_{0}^{6} x_{1}^{2} x_{2}^{3} X^{2}

− p x_{0}^{6} x_{1}^{4} x_{2}^{4} X^{2} − p x_{0}^{6} x_{1}^{2} x_{2}^{4} X^{2} − 2 p x_{0}^{3} x_{1}^{2} x_{2} X + x_{0}^{6} x_{1} x_{2}^{3} X^{2} + x_{0}^{6} x_{1}^{3} x_{2} X^{2} + x_{0}^{6} x_{1}^{3} x_{2}^{5} X^{2} + x_{0}^{6} x_{1}^{5} x_{2}^{3} X^{2} + 3 x_{0}^{6} x_{1}^{3} x_{2}^{3} X^{2} + x_{0}^{6} x_{1}^{2} x_{2}^{2} X^{2} + 2 x_{0}^{6} x_{1}^{3} x_{2}^{2} X^{2} − p x_{0}^{3} x_{1}^{2} X − p x_{0}^{3} x_{2}^{2} X − p x_{0}^{6} x_{1}^{4} x_{2}^{2} X^{2} − 2 p x_{0}^{3} x_{1} x_{2}^{2} X − p x_{0}^{6} x_{1}^{2} x_{2}^{2} X^{2} + x_{0}^{3} x_{1}^{2} x_{2}^{2} X + x_{0}^{3} x_{1} x_{2} X − p x_{0}^{6} x_{1}^{2} x_{2}^{3} X^{2} − p x_{0}^{6} x_{1}^{3} x_{2}^{2} X^{2} − p x_{0}^{3} x_{1}^{2} x_{2}^{3} X − p x_{0}^{3} x_{1}^{3} x_{2}^{2} X − 2 p x_{0}^{3} x_{1}^{2} x_{2}^{2} X − p x_{0}^{3} x_{1}^{3} x_{2} X + x_{0}^{3} x_{1}^{2} x_{2} X + x_{0}^{9} x_{1}^{4} x_{2}^{4} X^{3} − 2 p x_{0}^{6} x_{1}^{3} x_{2}^{3} X^{2} − 2 p x_{0}^{3} x_{1} x_{2} X + x_{0}^{9} x_{1}^{4} x_{2}^{5} X^{3} + x_{0}^{9} x_{1}^{5} x_{2}^{4} X^{3} − p x_{0}^{6} x_{1}^{3} x_{2}^{4} X^{2} + x_{0}^{3} x_{1} x_{2}^{2} X + x_{0}^{9} x_{1}^{5} x_{2}^{5} X^{3} + x_{0}^{6} x_{1}^{4} x_{2}^{4} X^{2} − p x_{0}^{6} x_{1}^{4} x_{2}^{3} X^{2} − p x_{0}^{3} x_{1} x_{2}^{3} X − p x_{0}^{3} x_{2} X − p x_{0}^{3} x_{1} X + 2 x_{0}^{6} x_{1}^{4} x_{2}^{3} X^{2} + 2 x_{0}^{6} x_{1}^{3} x_{2}^{4} X^{2})/((1 − x_{0}^{3} X)(1 − x_{0}^{3}x_{1}^{3} X)(1 − x_{0}^{3}x_{2}^{3} X)(1 − x_{0}^{3}x_{1}^{3}x_{2}^{3}X)).

3 Proofs: formulas for the Hecke operators of Sp_{g}

3.1 Satake’s spherical map Ω

Our result is based on the use of the Satake spherical map Ω, by applying the spherical map Ω to elements T(p^{δ}) ∈ L_{ℤ} of Hecke ring L_{ℤ} = ℤ[T(p), T_{1}(p^{2}), ⋯, T_{n}(p^{2})] for the symplectic group, see [An87] chapter 3.

• Case T_{1}(p^{2}) In genus 2 (in spherical variables x_{0}, x_{1}, x_{2}), we obtain using Andrianov’s formulas:

Ω(T_{1}(p^{2})) = \frac{x_{0}^{2} ((x_{1}^{2} x_{2} + x_{1} x_{2}^{2}) p^{2} + x_{1} x_{2} p^{2} − x_{1} x_{2} + (x_{1} + x_{2}) p^{2})}{p^{3}}.

• Cases T_{2}(p^{2}) = [p]_{2} and T(p)

Ω(T_{2}(p^{2})) = \frac{x_{0}^{2} x_{1} x_{2}}{p^{3}} , Ω(T(p)) = x_{0}(1 + x_{1})(1 + x_{2}).

3.2 Use of Andrianov’s generating series in genus 2

We refer to [An87], p.164, (3.3.75) for the following celebrated summation formula:

∑_{δ=0}^{∞} Ω^{(2)}(T(p^{δ})) X^{δ} =  (3)

\frac{1 − \frac{x_{0}^{2} x_{1} x_{2}}{p} X^{2}}{(1 − x_{0} X) (1 − x_{0} x_{1} X)(1 − x_{0} x_{2} X) (1 − x_{0} x_{1} x_{2} X)}

gives after development and simplification the following formula

Ω^{(2)}(T(p^{δ})) = p^{−1} x_{0}^{δ}(p x_{1}^{(3+δ)} x_{2} − p x_{1}^{(2+δ)} − p x_{1}^{(3+δ)} x_{2}^{(2+δ)} + p x_{1}^{(2+δ)} x_{2}^{(3+δ)}

− p x_{1} x_{2}^{(3+δ)} + p x_{2}^{(2+δ)} + p x_{1} − p x_{2} − x_{1}^{(2+δ)} x_{2}^{2} + x_{1}^{(1+δ)} x_{2}

+ x_{1}^{(2+δ)} x_{2}^{(1+δ)} − x_{1}^{(1+δ)} x_{2}^{(2+δ)} + x_{1}^{2} x_{2}^{(2+δ)} − x_{1} x_{2}^{(1+δ)} − x_{1}^{2} x_{2} + x_{1} x_{2}^{2})/ ((1 − x_{1}) (1 − x_{2}) (1 − x_{1} x_{2}) (x_{1} − x_{2}))

Then we use two groups of variables: x_{0}, …, x_{n} and y_{0}, …, y_{n} in two copies Ω_{x}, Ω_{y} of the sperical map, in order to treat the tensor product of two local Hecke algebras.

Next, in order to carry out the summation of the series

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{δ})) · Ω_{y}^{(2)}(T(p^{δ})) X^{δ}

on a computer, we used a subdivision of each summand (over δ) into smaller parts. These parts correspond to symbolic monomials in x_{1}^{δ}, y_{1}^{δ}, x_{2}^{δ}, y_{2}^{δ}, (x_{1}x_{2})^{δ}, (y_{1}y_{2})^{δ}.

3.3 Rankin’s Lemma of genus 2 (compare with [Jia96])

Let us compute the series

D_{p}^{(1,1)}(X) = ∑_{δ=0}^{∞} T(p^{δ}) ⊗ T(p^{δ}) X^{δ} ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[[X]]

in terms of the generators of Hecke’s algebra L_{2,ℤ} ⊗L_{2,ℤ} given by the following operators:

T(p) ⊗ 1, T_{1}(p^{2}) ⊗ 1, [p] ⊗ 1, 1 ⊗ T(p), 1 ⊗ T_{1}(p^{2}), 1 ⊗ [p] ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[[X]].

Theorem 4. For g = 2, we have the following explicit representation

D_{p}^{(1,1)}(X) = ∑_{δ=0}^{∞} T(p^{δ}) ⊗ T(p^{δ}) X^{δ} = \frac{(1 − p^{6}[p] ⊗ [p]X^{2}) · R(X)}{S(X)}, where

R(X), S(X) ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[X]

are given by the equalities (4) and (5):

R(X) = 1 + r_{2}X^{2} + ⋯ + r_{10}X^{10} + r_{12}X^{12} ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[X]  (4)
with r_{1} = r_{11} = 0,
S(X) = 1 + s_{1}X + ⋯ + s_{16}X^{16}  (5)

= 1 − (T(p) ⊗ T(p))X + ⋯ + (p^{6}[p] ⊗ [p])^{8}X^{16} ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[X],

with r_{i} and s_{i} given in Appendix. Moreover, there is an easy functional equation (similar to [An87], p.164, (3.3.79)):

s_{16−i} = (p^{6}[p] ⊗ [p])^{8−i}s_{i} (i = 0, ⋯, 8).

Remark 5 (Comparison with the case g = 1). The corresponding result in the case g = 1 written in terms of Hecke operators, looks as follows (see 3):

∑_{δ=0}^{∞} T(p^{δ}) ⊗ T(p^{δ}) X^{δ} = (1 − p^{2}[p] ⊗ [p]X^{2})/

(1 − T(p) ⊗ T(p)X + (p(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}) − 2p^{2}[p] ⊗ [p])X^{2}

− p^{2}T(p)[p] ⊗ T(p)[p]X^{3} + p^{4}[p]^{2} ⊗ [p]^{2}X^{4}).

Indeed this follows directly from Remark 3.

Remark 6. For the group GL(n), a simpler result could be obtained using the Tamagawa generating series (see [Tam]) and its expansion into simple fractions:

\frac{1}{(1 − x_{1}X)·. . .·(1 − x_{n}X)} = ∑_{i=1}^{n} \frac{α_{i}(x)}{1 − x_{i}X} = ∑_{i=1}^{n} ∑_{r=0}^{∞} α_{i}(x)(x_{i}X)^{r},

\frac{1}{(1 − y_{1}X)·. . .·(1 − y_{n}X)} = ∑_{j=1}^{n} \frac{α_{j}(y)}{1 − y_{j}X} = ∑_{j=1}^{n} ∑_{r=0}^{∞} α_{j}(y)(y_{j}X)^{r},

where α_{i}(x) = \frac{x_{i}^{n−1}}{∏_{k≠i}^{n}(x_{k} − x_{i})}, α_{j}(y) = \frac{y_{j}^{n−1}}{∏_{l≠j}^{n}(y_{l} − y_{j})}, hence

∑_{i,j=1}^{n} ∑_{r=0}^{∞} α_{i}(x)α_{j}(y)(x_{i}y_{j}X)^{r} = ∑_{i,j=1}^{n} \frac{α_{i}(x)α_{j}(y)}{1 − x_{i}y_{j}X}.

If n = 3, this gives after simplification the following fraction:

(y_{2}^{2} x_{2}^{2} y_{1}^{2} x_{3}^{2} y_{3}^{2} x_{1}^{2} X^{6} − y_{2} x_{2} y_{1} y_{3}^{2} x_{1} x_{3}^{2} X^{4} − y_{2} x_{2}^{2} y_{1} y_{3}^{2} x_{1} x_{3} X^{4} − y_{2} x_{2} y_{1} x_{1}^{2} x_{3} y_{3}^{2} X^{4} − y_{2}^{2} x_{2} y_{1} y_{3} x_{1} x_{3}^{2} X^{4} − y_{2}^{2} x_{2}^{2} y_{1} y_{3} x_{1} x_{3} X^{4} − y_{2}^{2} x_{2} y_{1} x_{1}^{2} x_{3} y_{3} X^{4} − y_{2} x_{2}^{2} y_{1}^{2} x_{3} y_{3} x_{1} X^{4} − y_{2} x_{2} y_{1}^{2} x_{3}^{2} y_{3} x_{1} X^{4} − y_{2} x_{2} y_{1}^{2} x_{3} x_{1}^{2} y_{3} X^{4} + y_{2} x_{3} y_{1} x_{1}^{2} y_{3} X^{3} + y_{2} x_{2} y_{3}^{2} x_{1} x_{3} X^{3} + y_{2}^{2} x_{2} y_{3} x_{1} x_{3} X^{3} + y_{3} x_{2} y_{1}^{2} x_{3} x_{1} X^{3} + y_{2} x_{2} y_{1} x_{1}^{2} y_{3} X^{3} + y_{2} x_{2} y_{1} y_{3} x_{3}^{2} X^{3} + y_{2} x_{2}^{2} y_{1} y_{3} x_{3} X^{3} + y_{2}^{2} x_{2} y_{1} x_{1} x_{3} X^{3} + 4 y_{2} x_{2} y_{1} y_{3} x_{1} x_{3} X^{3} + y_{2} x_{2}^{2} y_{1} y_{3} x_{1} X^{3} + y_{2} x_{3}^{2} y_{1} y_{3} x_{1} X^{3} + y_{2} x_{2} y_{1}^{2} x_{3} x_{1} X^{3} + x_{2} y_{1} y_{3}^{2} x_{3} x_{1} X^{3} − y_{2} x_{2} y_{1} x_{3} X^{2} − y_{3} x_{2} y_{1} x_{1} X^{2} − y_{2} x_{2} y_{3} x_{1} X^{2} − y_{2} x_{3} y_{3} x_{1} X^{2} − y_{2} x_{3} y_{1} x_{1} X^{2} − y_{3} x_{3} y_{1} x_{1} X^{2} − y_{3} x_{2} y_{1} x_{3} X^{2} − y_{2} x_{2} y_{1} x_{1} X^{2} − y_{2} x_{2} y_{3} x_{3} X^{2} + 1) /((1 − x_{3} y_{3} X) (1 − x_{2} y_{3} X) (1 − x_{1} y_{3} X)(1 − x_{3} y_{2} X) (1 − x_{2} y_{2} X) (1 − x_{1} y_{2} X) (1 − x_{3} y_{1} X) (1 − x_{2} y_{1} X) (1 − x_{1} y_{1} X))

4 Relations with L-functions and motives for Sp_{n}

The modest purpose of this section is to recall the motivic origin of the Lfunction produced by the denominator of our Hecke series.

L-functions, functional equation and motives for Sp_{n} (see [Pa94], [Yosh01]) One defines

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

• R_{f,p}(X) = (1 − X) ∏_{i=1}^{n}(1 − α_{i}^{−1}X)(1 − α_{i}X) ∈ ℚ[α_{0}^{±1}, ⋯, α_{n}^{±1}][X].

Then the spinor L-function L(Sp(f), s) and the standard L-function L(St(f), s, χ) of f (for s ∈ ℂ, and for all Dirichlet characters) χ are defined as the Euler products

• L(Sp(f), s, χ) = ∏_{p} Q_{f,p}(χ(p)p^{−s})^{−1}

• L(St(f), s, χ) = ∏_{p} R_{f,p}(χ(p)p^{−s})^{−1}

Motivic L-functions

Following [Pa94] and [Yosh01], these functions are conjectured to be motivic for all k > n:

L(Sp(f), s, χ) = L(M(Sp(f))(χ), s), L(St(f), s) = L(M(St(f))(χ), s), where

and the motives M(Sp(f) and M(St(f) are pure if f is a genuine cusp form (not coming from a lifting of a smaller genus):

• M(Sp(f)) is a motive over ℚ with coefficients in ℚ(λ_{f}(n))_{n∈ℕ} of rank 2^{n}, of weight w = kn − n(n + 1)/2, and of Hodge type ⊕_{p,q}H^{p,q}, with

p = (k − i_{1}) + (k − i_{2}) + ⋯ + (k − i_{r}),  (6)
q = (k − j_{1}) + (k − j_{2}) + ⋯ + (k − i_{s}), where r + s = n,
1 ≤ i_{1} < i_{2} < ⋯ < i_{r} ≤ n, 1 ≤ j_{1} < j_{2} < ⋯ < j_{s} ≤ n,
{i_{1}, ⋯, i_{r}} ∪ {j_{1}, ⋯, i_{s}} = {1, 2, ⋯, n};

• M(St(f) is a motive over ℚ with coefficients in ℚ(λ_{f}(n))_{n∈ℕ} of rank 2n+1, of weight w = 0, and of Hodge type H^{0,0} ⊕_{i=1}^{n} (H^{−k+i,k−i} ⊕ H^{k−i,−k+i}).

A functional equation

Following general Deligne’s conjecture [De79] on the motivic L-functions, the L-function satisfy a functional equation determined by the Hodge structure of a motive:

Λ(Sp(f), kn − n(n + 1)/2 + 1 − s) = ε(f)Λ(Sp(f), s), where

Λ(Sp(f), s) = Γ_{n,k}(s)L(Sp(f), s), ε(f) = (−1)^{k2^{n−2}},

Γ_{1,k}(s) = Γ_{ℂ}(s) = 2(2π)^{−s}Γ (s), Γ_{2,k}(s) = Γ_{ℂ}(s)Γ_{ℂ}(s − k + 2), and Γ_{n,k}(s) = ∏_{p<q} Γ_{ℂ}(s − p)Γ_{ℝ}^{a_{+}}(s − (w/2))Γ_{ℝ}(s + 1 − (w/2))^{a_{−}} for some nonnegative integers a_{+} and a_{−}, with a_{+} + a_{−} = w/2, and Γ_{ℝ}(s) = π^{−s/2}Γ (s/2)

Motive of the Rankin product of genus g = 2

Let f and g be two Siegel cusp eigenforms of weights k and l, k > l, and let M(Sp(f)) and M(Sp(g)) be the spinor motives of f and g. Then M(Sp(f)) is a motive over ℚ with coefficients in ℚ(λ_{f}(n))_{n∈ℕ} of rank 4, of weight w = 2k − 3, and of Hodge type H^{0,2k−3} ⊕ H^{k−2,k−1} ⊕ H^{k−1,k−2} ⊕ H^{2k−3,0}, and M(Sp(g)) is a motive over ℚ with coefficients in ℚ(λ_{g}(n))_{n∈ℕ} of rank 4, of weight w = 2l−3, and of Hodge type H^{0,2l−3} ⊕H^{l−2,l−1} ⊕H^{l−1,l−2} ⊕H^{2l−3,0}.

The tensor product M(Sp(f)) ⊗ M(Sp(g)) is a motive over ℚ with coefficients in ℚ(λ_{f}(n), λ_{g}(n))_{n∈ℕ} of rank 16, of weight w = 2k + 2l − 6, and of Hodge type

H^{0,2k+2l−6} ⊕ H^{l−2,2k+l−4} ⊕ H^{l−1,2k+l−5} ⊕ H^{2l−3,2k−3}
H^{k−2,k+2l−4} ⊕ H^{k+l−4,k+l−2} ⊕ H_{+}^{k+l−3,k+l−3} ⊕ H^{k+2l−5,k−1}
H^{k−1,k+2l−5} ⊕ H_{−}^{k+l−3,k+l−3} ⊕ H^{k+l−2,k+l−4} ⊕ H^{k+2l−4,k−2}
H^{2k−3,2l−3} ⊕ H^{2k+l−5,l−1} ⊕ H^{2k+l−4,l−2} ⊕ H^{2k+2l−6,0}.

Motivic L-functions: analytic properties

Following Deligne’s conjecture [De79] on motivic L-functions, applied for a Siegel cusp eigenform F for the Siegel modular group Sp_{4}(ℤ) of genus n = 4 and of weight k > 5, one has Λ(Sp(F), s) = Λ(Sp(F), 4k − 9 − s), where

Λ(Sp(F), s) = Γ_{ℂ}(s)Γ_{ℂ}(s − k + 4)Γ_{ℂ}(s − k + 3)Γ_{ℂ}(s − k + 2)Γ_{ℂ}(s − k + 1) × Γ_{ℂ}(s − 2k + 7)Γ_{ℂ}(s − 2k + 6)Γ_{ℂ}(s − 2k + 5)L(Sp(F), s),

(compare this functional equation with that given in [An74], p.115).

On the other hand, for n = 2 and for two cusp eigenforms f and g for Sp_{2}(ℤ) of weights k, l, k > l + 1, Λ(Sp(f) ⊗ Sp(g), s) = ε(f, g)Λ(Sp(f) ⊗ Sp(g), 2k + 2l − 5 − s), |ε(f, g)| = 1, where

Λ(Sp(f) ⊗ Sp(g), s) = Γ_{ℂ}(s)Γ_{ℂ}(s − l + 2)Γ_{ℂ}(s − l + 1)Γ_{ℂ}(s − k + 2)
× Γ_{ℂ}(s − k + 1)Γ_{ℂ}(s − 2l + 3)Γ_{ℂ}(s − k − l + 2)Γ_{ℂ}(s − k − l + 3)
× L(Sp(f) ⊗ Sp(g), s).

We used here the Gauss duplication formula Γ_{ℂ}(s) = Γ_{ℝ}(s)Γ_{ℝ}(s + 1). Notice that a_{+} = a_{−} = 1 in this case, and the conjectural motive M(Sp(f)) ⊗ M(Sp(g)) does not admit critical values.

5 A holomorphic lifting from GSp_{2} × GSp_{2} to GSp_{4}: a conjecture

(compare with constructions in [BFG06], [BFG92], [Jia96] for generic automorphic forms).

Our computation makes it possible to compare the spinor Hecke series of genus 4 computed in [VaSp4] (in variables u_{0}, u_{1}, u_{2}, u_{3}, u_{4}) with the Rankin product of two Hecke series of genus 2 (in variables x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}). It follows from our computation that if we make the substitution u_{0} = x_{0}y_{0}, u_{1} = x_{1}, u_{2} = x_{2}, u_{3} = y_{1}, u_{4} = y_{2} then the denominator of the series

∑_{δ=0}^{∞} Ω_{u}^{(4)}(T(p^{δ}))X^{δ}

coincides with the denominator of the Rankin product

∑_{δ=0}^{∞} Ω_{x}^{(2)}(T(p^{δ})) · Ω_{y}^{(2)}(T(p^{δ})) X^{δ} ∈ ℚ[x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}][[X]].

On the basis of this equality we would like to push forward the following

Conjecture 7 (on a lifting from GSp_{2} × GSp_{2} to GSp_{4}). Let f and g be two Siegel modular forms of genus 2 and of weights k > 4 and l = k − 2. Then there exists a Siegel modular form F of genus 4 and of weight k with the Satake parameters

γ_{0} = α_{0}β_{0}, γ_{1} = α_{1}, γ_{2} = α_{2}, γ_{3} = β_{1}, γ_{4} = β_{2},

for a suitable choice of Satake’s parameters α_{0}, α_{1}, α_{2} and β_{0}, β_{1}, β_{2} of f and g.

Remark 8. An evidence for the conjecture comes from Ikeda-Miyawaki constructions ([Ike01], [Mur02], [Ike06]): let k be an even positive integer, h ∈ S_{2k}(Γ_{1}) a normalized Hecke eigenform of weight 2k, F_{2}(h) ∈ S_{k+1}(Γ_{2}) = Maass(h) the Maass lift of h, and F_{2n} ∈ S_{k+n}(Γ_{2n}) the Ikeda lift of h (we assume k ≡ n mod 2, n ∈ ℕ).

Next let f ∈ S_{k+n+r}(Γ_{r}) be an arbitrary Siegel cusp eigenform of genus r and weight k + n + r, with n, r ≥ 1. Then according to Ikeda-Miyawaki (see [Ike06]) there exists a Siegel eigenform F_{h,f} ∈ S_{k+n+r}(Γ_{2n+r}) such that

L(s, F_{h,f}, St) = L(s, f, St) ∏_{j=1}^{2n} L(s + k + n − j, h)  (7)

(under a non-vanishing condition, see Theorem 2.3 at p.63 in [Mur02]). The form F_{h,f} is given by the integral

F_{h,f}(Z) = ⟨F_{2n+2r}(diag(Z, Z′), f(Z′)⟩_{Z′}

If we take n = 1, r = 2, k := k + 1 then an example of the validity of the conjecture is given by g = F_{2}(h),

(f, g) = (f, F_{2}(h)) ↦ F_{f,h} ∈ S_{k+3}(Γ_{4}),
(f, g) = (f, F_{2}(h)) ∈ S_{k+3}(Γ_{2}) × S_{k+1}(Γ_{2}).

Remark 9. Notice that the Satake parameters of the Ikeda lift F = F_{2m}(h) of h can be taken in the form β_{0}, β_{1}, ⋯, β_{2m}, where

β_{0} = p^{mk−m(m+1)/2}, β_{i} = αp^{i−1/2}, β_{m+i} = α^{−1}p^{i−1/2}, (i = 1, ⋯, m)

and

(1 − αp^{k−1/2}X)(1 − α^{−1}p^{k−1/2}X) = 1 − a(p)X + p^{2k−1}X^{2}, h = ∑_{n=1}^{∞} a(n)q^{n}

see [Mur02].

The L-function of degree 16 in Conjecture 7 is related to the tensor product L-function as in [Jia96]. In the example of Remark 8 it coincides with the product of two shifted L-functions of degree 8 of Boecherer-Heim [BoeH06].

Conjecture 10 (on a lifting from GSp_{2m} × GSp_{2m} to GSp_{4m}).

Here is a version of Conjectire 7 for any even genus r = 2m. Let f and g be two Siegel modular forms of genus 2m and of weights k > 2m and l = k −2m. Then there exists a Siegel modular form F of genus 4m and of weight k with the Satake parameters γ_{0} = α_{0}β_{0}, γ_{1} = α_{1}, γ_{2} = α_{2}, ⋯, γ_{2m} = α_{2m}, γ_{2m+1} = β_{1}, ⋯, γ_{4m} = β_{2m} for suitable choices α_{0}, α_{1}, ⋯, α_{2m} and β_{0}, β_{1}, ⋯, β_{2m} of Satake’s parameters of f and g.

One readily checks that the Hodge types of M(Sp(f)) ⊗ M(Sp(g)) and M(Sp(F)) are again the same (of rank 2^{4m}) (it follows from the above description (6), and from Künneth’s-type formulas).

An evidence for this version of the conjecture comes again from Ikeda-Miyawaki constructions ([Ike01], [Mur02], [Ike06]): let k be an even positive integer, h ∈ S_{2k}(Γ_{1}) a normalized Hecke eigenform of weight 2k, F_{2n} ∈ S_{k+n}(Γ_{2n}) the Ikeda lift of h of genus 2n (we assume k ≡ n mod 2, n ∈ ℕ).

Next let f ∈ S_{k+n+r}(Γ_{r}) be an arbitrary Siegel cusp eigenform of genus r and weight k + n + r, with n, r ≥ 1. If we take in (7) n = m, r = 2m, k := k + m, k + n + r := k + 3m, then an example of the validity of this version of the conjecture is given by

(f, g) = (f, F_{2m}(h)) ↦ F_{h,f} ∈ S_{k+3m}(Γ_{4m}),
(f, g) = (f, F_{2m}) ∈ S_{k+3m}(Γ_{2m}) × S_{k+m}(Γ_{2m}).

Another evidence comes from Siegel-Eiseinstein series

f = E_{k}^{2m} and g = E_{k−2m}^{2m}

of even genus 2m and weights k and k − 2m: we have then

α_{0} = 1, α_{1} = p^{k−2m}, ⋯, α_{2m} = p^{k−1},
β_{0} = 1, β_{1} = p^{k−4m}, ⋯, β_{2m} = p^{k−2m−1},

then we have that

γ_{0} = 1, γ_{1} = p^{k−4m}, ⋯, γ_{2m} = p^{k−1},

are the Satake parameters of the Siegel-Eisenstein series F = E_{k}^{4m}.

Remark 11. If we compare the L-function of the conjecture (given by the Satake parameters γ_{0} = α_{0}β_{0}, γ_{1} = α_{1}, γ_{2} = α_{2}, ⋯, γ_{2m} = α_{2m}, γ_{2m+1} = β_{1}, ⋯, γ_{4m} = β_{2m} for suitable choices α_{0}, α_{1}, ⋯, α_{2m} and β_{0}, β_{1}, ⋯, β_{2m} of Satake’s parameters of f and g), we see that it corresponds to the tensor product of spinor L-functions, and is not of the same type as that of the Yoshida’s lifting [Yosh81], which is a certain product of Hecke’s L-functions.

We would like to mention in this context Langlands’s functoriality: The denominators of our L-series belong to local Langlands L-factors (attached to representations of L-groups). If we consider the homomorphisms

LGSp_{2m} = GSpin(4m + 1) → GL_{2^{2m}}, ^{L}GSp_{4m} = GSpin(8m + 1) → GL_{2^{4m}},

we see that our conjecture is compatible with the homomorphism of L-groups

GL_{2^{2m}} × GL_{2^{2m}} → GL_{2^{4m}}, (g_{1}, g_{2}) ↦ g_{1} ⊗ g_{2}, GL_{n}(ℂ) = ^{L}GL_{n}.

However, it is unclear to us if Langlands’s functoriality predicts a holomorphic Siegel modular form as a lift.

A Appendix: Coefficients of the polynomials R(X) and S(X)

We give here explicit expressions for the coefficients of the polynomials R(X) and S(X) from Theorem 4. From these formulas one can observe some nice divisibility properties (by certain powers of p and the elements [p] ⊗ [p] ∈ L_{2,ℤ} ⊗ L_{2,ℤ}):

R(X) = 1 + r_{2}X^{2} + ⋯ + r_{10}X^{10} + r_{12}X^{12} ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[X]
with r_{1} = r_{11} = 0,
S(X) = 1 + s_{1}X + ⋯ + s_{16}X^{16}

= 1 − (T(p) ⊗ T(p))X + ⋯ + (p^{6}[p] ⊗ [p])^{8}X^{16} ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[X],

with r_{i} and s_{i} given as follows

r_{2} = p^{2}((2p − 1)(p^{2} + 1)[p] ⊗ [p] − (p^{2} − p + 1)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))

− (T_{1}(p^{2}) ⊗ T_{1}(p^{2}) + T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}), r_{3} = p^{3}(p + 1)(2[p] ⊗ [p] + T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T(p) ⊗ T(p) ,

540  Alexei Panchishkin and Kirill Vankov

r_{4} = −p^{5}((p^{7} + 2p^{6} − 2p^{5} + 6p^{4} + p^{3} + 6p^{2} + p + 2)[p]^{2} ⊗ [p]^{2} − (p^{2} + 1)(p^{3} − 3p^{2} − p − 3)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))[p] ⊗ [p] + (p + 4)(p^{2} + 1)T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] − (p^{3} − p^{2} − 1)(T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2}) + (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p(p^{3} + 2p^{2} − p + 2)(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})[p] ⊗ [p] − 2p(T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p] ⊗ [p] + p^{2}(T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2})) + (p + 2)T(p)^{2}[p] ⊗ T(p)^{2}[p]) ,

r_{5} = −p^{7}(2(p + 1)(2p^{4} − p^{3} + p^{2} − 1)[p] ⊗ [p] + (p + 1)(p − 2)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))

− 2T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p(p + 1)(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}))T(p)[p] ⊗ T(p)[p] , r_{6} = −p^{10} (p (p^{2} + 1)(p^{5} − 2p^{3} − 8p^{2} − p − 4)[p]^{3} ⊗ [p]^{3}

− p (p^{5} + 4p^{4} + 2p^{3} + 12p^{2} + p + 6)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))[p]^{2} ⊗ [p]^{2} + p (p − 4)(p^{2} + 1)T_{1}(p^{2})[p]^{2} ⊗ T_{1}(p^{2})[p]^{2} − p (p + 4)(p^{2} + 1)(T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2})[p] ⊗ [p] − p (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] − p (T_{1}(p^{2})^{3} ⊗ [p]^{3} + [p]^{3} ⊗ T_{1}(p^{2})^{3}) − (p^{5} − 4p^{2} − p − 2)(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})[p]^{2} ⊗ [p]^{2} + (p^{2} + 3)(T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p]^{2} ⊗ [p]^{2} + (T(p)^{2}[p] ⊗ T_{1}(p^{2})^{2} + T_{1}(p^{2})^{2} ⊗ T(p)^{2}[p])[p] ⊗ [p] + (p^{3} + 3p^{2} + p + 1)(T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2}))[p] ⊗ [p] + (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] + (p^{2} + 1)T(p)^{2}[p]^{2} ⊗ T(p)^{2}[p]^{2}) ,

r_{7} = −p^{13} (2(p + 1)(p^{3} + p − 1)[p] ⊗ [p] − (p + 1)(p^{2} − 2p + 2)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))

− 2T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − (p + 1)(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}))T(p)[p]^{2} ⊗ T(p)[p]^{2} , r_{8} = −p^{16}(p (2p^{6} + 3p^{5} + 6p^{4} − p^{3} + 6p^{2} − p + 2)[p]^{2} ⊗ [p]^{2}

+ p (p^{2} + 1)(p^{3} + 3p^{2} − p + 3)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))[p] ⊗ [p] + p (p + 4)(p^{2} + 1)T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] + p (p^{2} − p + 1)(T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2}) + p (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p (2p^{3} + p^{2} + 2p − 1)(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})[p] ⊗ [p] − 2p^{2}(T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p] ⊗ [p] + p (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2})) + (2p + 1)T(p)^{2}[p] ⊗ T(p)^{2}[p])[p]^{2} ⊗ [p]^{2} ,

r_{9} = p^{20}(p + 1)(2[p] ⊗ [p] + T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T(p)[p]^{3} ⊗ T(p)[p]^{3} r_{10} = p^{24}((p^{2} + 1)(p^{4} + 2p^{3} − p^{2} − 1)[p] ⊗ [p] + (p^{3} − p^{2} − 1)(T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))

− T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p^{2}(T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}))[p]^{4} ⊗ [p]^{4} r_{11} = 0, r_{12} = p^{34}[p]^{6} ⊗ [p]^{6},

Rankin’s lemma of higher genus and explicit formulas for Hecke operators  541

As for the coefficients of S(X), one has

S(X) = 1 − (T(p) ⊗ T(p))X + ⋯ + (p^{6}[p] ⊗ [p])^{8}X^{16} ∈ L_{2,ℤ} ⊗ L_{2,ℤ}[X], where

s_{1} = −T(p) ⊗ T(p) s_{2} = −p(2 p (p^{2} + 1)^{2} [p] ⊗ [p] + 2 p (p^{2} + 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))

+ 2 p T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − (p^{2} + 1) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}) − (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2}))

s_{3} = p^{2}((2 p^{4} + 4 p^{2} − 1) [p] ⊗ [p] + (2 p^{2} − 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))

− T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}))T(p) ⊗ T(p)) s_{4} = p^{4}((p^{8} + 12 p^{6} + 10 p^{4} + 4 p^{2} + 1) [p]^{2} ⊗ [p]^{2}

+ 2 (3 p^{6} + 5 p^{4} + 3 p^{2} + 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))[p] ⊗ [p] + 4 (p^{2} + 1)^{2} T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] + (3 p^{4} + 2 p^{2} + 1) (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2}) + 2 (p^{2} + 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T_{1}(p^{2}) ⊗ T_{1}(p^{2}) + T_{1}(p^{2})^{2} ⊗ T_{1}(p^{2})^{2} − 2 p (p^{4} + 4 p^{2} + 1) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2}) [p] ⊗ [p] − 4 p (p^{2} + 1) (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p] ⊗ [p] − 2 p (T(p)^{2}[p] ⊗ T_{1}(p^{2})^{2} + T_{1}(p^{2})^{2} ⊗ T(p)^{2}[p]) − 4 p^{3} (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2})) + (p^{2} + 2) T(p)^{2}[p] ⊗ T(p)^{2}[p] + (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))T(p)^{2} ⊗ T(p)^{2} + p^{2} (T(p)^{4} ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{4}))

s_{5} = −p^{6}((6 p^{6} + 2 p^{4} − p^{2} + 2) [p]^{2} ⊗ [p]^{2} + (p^{4} − p^{2} + 3) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2}))[p] ⊗ [p] + (3 p^{2} + 4) T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] − (2 p^{2} − 1) (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2}) + (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p (2 p^{2} + 1) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})[p] ⊗ [p] − 2 p (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p] ⊗ [p] + p (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2})) + T(p)^{2}[p] ⊗ T(p)^{2}[p])T(p) ⊗ T(p))

542  Alexei Panchishkin and Kirill Vankov

s_{6} = −p^{8}(2 p^{2} (p^{8} + 6 p^{6} + 11 p^{4} + 8 p^{2} + 2) [p]^{3} ⊗ [p]^{3} + 2 p^{2} (5 p^{4} + 12 p^{2} + 6) T_{1}(p^{2})[p]^{2} ⊗ T_{1}(p^{2})[p]^{2} + (3 p^{4} + 10 p^{2} − 1) T(p)^{2}[p]^{2} ⊗ T(p)^{2}[p]^{2} − T(p)^{2}T_{1}(p^{2})[p] ⊗ T(p)^{2}T_{1}(p^{2})[p] + 2 p^{2} (3 p^{6} + 11 p^{4} + 12 p^{2} + 4) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) [p]^{2} ⊗ [p]^{2} + 6 p^{2} (p^{2} + 1)^{2} (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2})[p] ⊗ [p] + 6 p^{2} (p^{2} + 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] + 2 p^{2} (p^{2} + 1) (T_{1}(p^{2})^{3} ⊗ [p]^{3} + [p]^{3} ⊗ T_{1}(p^{2})^{3}) + 2 p^{2} (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2})T_{1}(p^{2}) ⊗ T_{1}(p^{2}) − p (5 p^{6} + 13 p^{4} + 10 p^{2} + 2) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})[p]^{2} ⊗ [p]^{2} − p (7 p^{4} + 12 p^{2} + 4) (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p]^{2} ⊗ [p]^{2} − 3p ( p^{2} + 1) (T(p)^{2}[p] ⊗ T_{1}(p^{2})^{2} + T_{1}(p^{2})^{2} ⊗ T(p)^{2}[p])[p] ⊗ [p] − p (T(p)^{2}[p]^{2} ⊗ T_{1}(p^{2})^{3} + T_{1}(p^{2})^{3} ⊗ T(p)^{2}[p]^{2}) − 2 p (3 p^{4} + 4 p^{2} + 1) (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2}))[p] ⊗ [p] − 2 p (3 p^{2} + 1) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})T_{1}(p^{2})[p] ⊗ T_{1}(p^{2})[p] − p (p^{2} + 1) (T(p)^{2}T_{1}(p^{2})^{2} ⊗ [p]^{3} + [p]^{3} ⊗ T(p)^{2}T_{1}(p^{2})^{2}) − p (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2}))T_{1}(p^{2}) ⊗ T_{1}(p^{2}) + (5 p^{2} − 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T(p)^{2}[p] ⊗ T(p)^{2}[p] + 2 p^{2} (p^{2} + 1) (T(p)^{4} ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{4})[p] ⊗ [p] + 2 p^{2} (T(p)^{4} ⊗ T_{1}(p^{2})[p] + T_{1}(p^{2})[p] ⊗ T(p)^{4}) [p] ⊗ [p] − p (T(p)^{4} ⊗ T(p)^{4}[p] + T(p)^{4}[p] ⊗ T(p)^{4}) [p] ⊗ [p])

s_{7} = p^{11}(p (5 p^{6} − 2 p^{4} + 2) T(p)[p]^{3} ⊗ T(p)[p]^{3} + 8 p T(p)T_{1}(p^{2})[p]^{2} ⊗ T(p)T_{1}(p^{2})[p]^{2} + p T(p)^{3}[p]^{2} ⊗ T(p)^{3}[p]^{2} − p (p^{4} − 3) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T(p)[p]^{2} ⊗ T(p)[p]^{2} − p (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2})T(p)[p] ⊗ T(p)[p] + 2 p (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T(p)T_{1}(p^{2})[p] ⊗ T(p)T_{1}(p^{2})[p] − p (T_{1}(p^{2})^{3} ⊗ [p]^{3} + [p]^{3} ⊗ T_{1}(p^{2})^{3})T(p) ⊗ T(p) − (3 p^{4} − 3 p^{2} + 2) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})T(p)[p]^{2} ⊗ T(p)[p]^{2} + (p^{2} − 3) (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})T(p)[p]^{2} ⊗ T(p)[p]^{2} − (T(p)^{2}[p] ⊗ T_{1}(p^{2})^{2} + T_{1}(p^{2})^{2} ⊗ T(p)^{2}[p])T(p)[p] ⊗ T(p)[p] + (2 p^{2} − 1) (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2}))T(p)[p] ⊗ T(p)[p] − (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})T(p)T_{1}(p^{2})[p] ⊗ T(p)T_{1}(p^{2})[p]) s_{8} = p^{14}(2 p^{2} (2 p^{8} + 4 p^{6} + 14 p^{4} + 12 p^{2} + 3) [p]^{4} ⊗ [p]^{4}

+ 4 p^{2} (p^{6} + 7 p^{4} + 9 p^{2} + 3) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) [p]^{3} ⊗ [p]^{3} + 16 p^{2} (p^{2} + 1)^{2} T_{1}(p^{2})[p]^{3} ⊗ T_{1}(p^{2})[p]^{3} + 2 p^{2} (3 p^{4} + 10 p^{2} + 5) (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2}) [p]^{2} ⊗ [p]^{2} + 8 p^{2} (p^{2} + 1) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T_{1}(p^{2})[p]^{2} ⊗ T_{1}(p^{2})[p]^{2} + 4 p^{2} T_{1}(p^{2})^{2}[p]^{2} ⊗ T_{1}(p^{2})^{2}[p]^{2} + 4 p^{2} (p^{2} + 1) (T_{1}(p^{2})^{3} ⊗ [p]^{3} + [p]^{3} ⊗ T_{1}(p^{2})^{3}) [p] ⊗ [p] + p^{2} (T_{1}(p^{2})^{4} ⊗ [p]^{4} + [p]^{4} ⊗ T_{1}(p^{2})^{4}) − 4 p (2 p^{6} + 3 p^{4} + 4 p^{2} + 1) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})[p]^{3} ⊗ [p]^{3} − 8 p (p^{2} + 1)^{2} (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})[p]^{3} ⊗ [p]^{3} − 4 p (p^{2} + 1) (T(p)^{2}[p] ⊗ T_{1}(p^{2})^{2} + T_{1}(p^{2})^{2} ⊗ T(p)^{2}[p])[p]^{2} ⊗ [p]^{2} − 4 p (p^{4} + 4 p^{2} + 1) (T(p)^{2}T_{1}(p^{2}) ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{2}T_{1}(p^{2}))[p]^{2} ⊗ [p]^{2} − 8 p (p^{2} + 1) (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})T_{1}(p^{2})[p]^{2} ⊗ T_{1}(p^{2})[p]^{2} − 4 p (T(p)^{2} ⊗ T_{1}(p^{2}) + T_{1}(p^{2}) ⊗ T(p)^{2})T_{1}(p^{2})[p]^{2} ⊗ T_{1}(p^{2})[p]^{2} − 4 p^{3} (T(p)^{2}T_{1}(p^{2})^{2} ⊗ [p]^{3} + [p]^{3} ⊗ T(p)^{2}T_{1}(p^{2})^{2}) + [p] ⊗ [p] + 2 (5 p^{4} + 2 p^{2} + 2) T(p)^{2}[p]^{3} ⊗ T(p)^{2}[p]^{3} + 2 (p^{2} + 2) (T_{1}(p^{2}) ⊗ [p] + [p] ⊗ T_{1}(p^{2})) T(p)^{2}[p]^{2} ⊗ T(p)^{2}[p]^{2} + 2 T(p)^{2}T_{1}(p^{2})[p]^{2} ⊗ T(p)^{2}T_{1}(p^{2})[p]^{2} + (T_{1}(p^{2})^{2} ⊗ [p]^{2} + [p]^{2} ⊗ T_{1}(p^{2})^{2})T(p)^{2}[p] ⊗ T(p)^{2}[p] + (3 p^{4} + 2 p^{2} + 1) (T(p)^{4} ⊗ [p]^{2} + [p]^{2} ⊗ T(p)^{4})[p]^{2} ⊗ [p]^{2} + 2 (p^{2} + 1) (T(p)^{4} ⊗ T_{1}(p^{2})[p] + T_{1}(p^{2})[p] ⊗ T(p)^{4}) [p]^{2} ⊗ [p]^{2} + (T(p)^{4} ⊗ T_{1}(p^{2})^{2} + T_{1}(p^{2})^{2} ⊗ T(p)^{4}) [p]^{2} ⊗ [p]^{2} − 2 p (T(p)^{2} ⊗ [p] + [p] ⊗ T(p)^{2})T(p)^{2}[p]^{2} ⊗ T(p)^{2}[p]^{2})

Then we find the remaining coefficients s_{9}, ⋯, s_{16}, using an easy functional equation (similar to [An87], p.164, (3.3.79)):

s_{16−i} = (p^{6}[p] ⊗ [p])^{8−i}s_{i} (i = 0, ⋯, 8).

To conclude with, we give the Newton polygons of R(X) and S(X) with respect to powers of p and X (see Figure 1). It follows from our comutation that all slopes are integral. We hope that these polygons could help to find some geometric objects attached to the polynomials R(X) and S(X), in the spirit of a recent work of C.Faber and G.Van Der Geer, [FVdG].

Acknowledgement

We are very grateful to the Institute Fourier (UJF, Grenoble-1) for the permanent excellent working environment.

Fig. 1. Newton polygons of R(X) and S(X) with respect to powers of p and X, of heights 34 and 48, resp.

It is a great pleasure for us to thank Siegfried Boecherer, Stephen Gelbart, Solomon Friedberg and Mikhail Tsfasman for valuable discussions and observations.

Our special thanks go to Yuri Ivanovich Manin, for providing us with advice and encouragement.

References

[An67] Andrianov, A.N., Shimura’s conjecture for Siegel’s modular group of genus 3, Dokl. Akad. Nauk SSSR 177 (1967), 755-758 = Soviet Math. Dokl. 8 (1967), 1474-1478.

[An68] Andrianov, A.N., Rationality of multiple Hecke series of a complete linear group and Shimura’s hypothesis on Hecke’s series of a symplectic group, Dokl. Akad. Nauk SSSR 183 (1968), 9-11 = Soviet Math. Dokl. 9 (1968), 1295 - 1297. MR

[An69] Andrianov, A.N., Rationality theorems for Hecke series and zeta functions of the groups GL_{n} and SP_{n} over local fields, Izv. Akad. Nauk SSSR, Ser. Mat., Tom 33 (1969), No. 3, (Math. USSR – Izvestija, Vol. 3 (1969), No. 3, pp. 439– 476).

[An70] Andrianov, A.N., Spherical functions for GL_{n} over local fields and summation of Hecke series, Mat. Sbornik, Tom 83 (125) (1970), No 3, (Math. USSR Sbornik, Vol. 12 (1970), No. 3, pp. 429–452).

[An74] Andrianov, A.N., Euler products corresponding to Siegel modular forms of genus 2, Russian Math. Surveys, 29:3 (1974), pp. 45–116, (Uspekhi Mat. Nauk 29:3 (1974) pp. 43–110).

[An87] Andrianov, A.N., Quadratic Forms and Hecke Operators, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987.

[An-Ka] Andrianov,A.N., Kalinin, V.L., On Analytic Properties of Standard Zeta Functions of Siegel Modular Forms, Mat. Sbornik 106 (1978), p 323-339 (in Russian).

[AnZh95] Andrianov, A.N., Zhuravlev, V.G., Modular Forms and Hecke Operators, Translations of Mathematical Monographs, Vol. 145, AMS, Providence, Rhode Island, 1995.

[AsSch] Asgari, M., Schmidt, R., Siegel modular forms and representations. Manuscripta Math. 104 (2001), 173-200

[BHam] Böcherer, S., Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe. Abh. Math. Sem. Univ. Hamburg 56 (1986), 35–47

[BoeH06] Böcherer, S. and Heim B.E. Critical values of L-functions on GSp_{2} × GL_{2}. Math.Z, 254, 485-503 (2006).

[BFG06] Bump, D.; Friedberg, S.; Ginzburg, D. Lifting automorphic representations on the double covers of orthogonal groups. Duke Math. J. 131 (2006), no. 2, 363-396.

[BFG92] Bump, D.; Friedberg, S.; Ginzburg, D. Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method. Invent. Math. 109 (1992), no. 1, 55–96.

[CourPa] Courtieu, M., Panchishkin, A.A., Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

[De79] Deligne P., Valeurs de fonctions L et périodes d’intégrales, Proc.Sympos.Pure Math. vol. 55. Amer. Math. Soc., Providence, RI, 1979 , 313-346

[Evd] Evdokimov, S. A., Dirichlet series, multiple Andrianov zeta-functions in the theory of Euler modular forms of genus 3, (Russian) Dokl. Akad. Nauk SSSR 277 (1984), no. 1, 25–29.

[FVdG] Faber, C., van der Geer, G. Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. I, II C. R. Math. Acad. Sci. Paris 338, (2004) No.5, p. 381-384 and No.6, 467-470.

[Hecke] Hecke, E., Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktenwickelung, I, II. Math. Annalen 114 (1937), 1-28, 316-351 (Mathematische Werke. Göttingen: Vandenhoeck und Ruprecht, 1959, 644-707).

[Ike01] Ikeda, T., On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2) 154 (2001), 641-681.

[Ike06] Ikeda, T., Pullback of the lifting of elliptic cusp forms and Miyawaki’s Conjecture Duke Mathematical Journal, 131, 469-497 (2006)

[Jia96] Jiang, D., Degree 16 standard L-function of GSp(2)×GSp(2). Mem. Amer. Math. Soc. 123 (1996), no. 588 (196pp)

[Ku88] Kurokawa, N., Analyticity of Dirichlet series over prime powers. Analytic number theory (Tokyo, 1988), 168–177, Lecture Notes in Math., 1434, Springer, Berlin, 1990.

[La79] R. Langlands, Automorphic forms, Shimura varieties, and L-functions. Ein Märchen. Proc. Symp. Pure Math. 33, part 2 (1979), 205-246

[Maa76] Maass, H. Indefinite Quadratische Formen und Eulerprodukte. Comm. on Pure and Appl. Math, 19, 689-699 (1976)

[Ma-Pa77] Manin, Yu.I. and Panchishkin, A.A., Convolutions of Hecke series and their values at integral points, Mat. Sbornik, 104 (1977) 617–651 (in Russian); Math. USSR, Sb. 33, 539-571 (1977) (English translation). In: Selected Papers of Yu.I.Manin, World Scientific, 1996, pp.325-357 [Ma-Pa05] Manin, Yu.I. and Panchishkin, A.A., Introduction to Modern Number

Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (2nd ed.), Springer-Verlag, 2005, 514 p.

[Mur02] Murokawa, K., Relations between symmetric power L-functions and spinor L-functions attached to Ikeda lifts, Kodai Math. J. 25, 61-71 (2002)

[Pa87] Panchishkin, A., A functional equation of the non-Archimedean Rankin convolution, Duke Math. J. 54 (special volume in Honor of Yu.I.Manin on his 50th birthday)(1987) 77-89

[Pa94] Panchishkin, A., Admissible Non-Archimedean standard zeta functions of Siegel modular forms, Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20–August 2 1991, Seattle, Providence, R.I., 1994, vol.2, 251 – 292

[Pa02] Panchishkin, A., A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal, 2 (2002), Number 2 (special issue in Honor of Yu.I.Manin on his 65th birthday), 1-16

[PaGRFA] Panchishkin, A., Produits d’Euler attachés aux formes modulaires de Siegel. Exposé au Séminaire Groupes Réductifs et Formes Automorphes à l’Institut de Mathématiques de Jussieu, June 22, 2006.

[PaHakuba5] Panchishkin,A.A., Triple products of Coleman’s families and their periods (a joint work with S.Boecherer) Proceedings of the 8th Hakuba conference “Periods and related topics from automorphic forms”, September 25 - October 1, 2005

[PaSerre6] Panchishkin,A.A., p-adic Banach modules of arithmetical modular forms and triple products of Coleman’s families, (for a special volume of Quarterly Journal of Pure and Applied Mathematics dedicated to Jean-Pierre Serre), 2006.

[PaVa] Panchishkin, A., Vankov, K. Explicit Shimura’s conjecture for Sp3 on a computer. Arxiv, math. NT/0607158 (2006).

[SchR03] Schmidt, R., On the spin L-function of Ikeda’s lifts. Comment. Math. Univ. St. Paul 52 (2003), 1–46

[Sa63] Satake, I., Theory of spherical functions on reductive groups over p-adic fields, Publ. mathématiques de l’IHES 18 (1963), 1–59.

[Shi63] Shimura, G., On modular correspondences for Sp(n,Z) and their congruence relations, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 824–828.

[Shi71] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971

[Tam] Tamagawa T., On the ζ-function of a division algebra, Ann. of Math. 77 (1963), 387–405

[VaSp4] Vankov, K., Explicit formula for the symplectic Hecke series of genus four, Arxiv, math.NT/0606492, (2006).

[Yosh81] Yoshida, H., Siegel’s Modular Forms and the Arithmetic of Quadratic Forms, Inventiones math. 60, 193–248 (1980)

[Yosh01] Yoshida, H., Motives and Siegel modular forms, American Journal of Mathematics, 123 (2001), 1171–1197.