##
**Numerical examples of Siegel cusp forms of degree 3 and their zeta- functions.**
*(English)*
Zbl 0780.11022

Let \(\Gamma_ n:= Sp_ n(\mathbb{Z})\) be the Siegel modular group of degree \(n\), and \(M_ k(\Gamma_ n)\) (resp. \(S_ k(\Gamma_ n)\)) be the space of modular (resp. cusp) forms of weight \(k\) with respect to \(\Gamma_ n\). In the present paper the author tries to find something like “Maass space” in the cases of \(S_ k(\Gamma_ 3)\) with \(k=12\) and 14 by examining such spaces numerically. The details will be concisely described as follows.

First, he recalls some basic tools such as Hecke operators, \(L\)-function of Andrianov’s type attached to Siegel cusp form, formal Dirichlet series associated with Hecke operators. Next, he gives explicit constructions of Siegel cusp forms \(F_{12}(Z)\) (resp. \(F_{14}(Z)\)) of degree 3 and weights 12 and 14 respectively by means of theta-series with spherical function associated with two root lattices. He also gives some values of Fourier coefficients to those forms. And he expresses the action of Hecke operators by the relations of Fourier coefficients. This enables him to determine the local factors \(L_ 2(s,F_{12})\) of the \(L\)-function of the above type. With this example the author derives a conjecture that \(L(s,F_{12})= L(s-9,\Delta) L(s-10,\Delta) L(s,\Delta \otimes g_{20})\) would hold. Here \(\Delta\) (resp. \(g_{20}\)) is a cusp form of weight 12 (resp. 20) in the classical modular forms, \(L(s-9,\Delta)\) is the Mellin transform of \(\Delta\) and \(L(s,\Delta\otimes g_{20})\) is the Rankin convolution of two \(L\)-functions. In the same spirit the \(F_{14}\) case is also treated.

In the later part the author proposes a definition of “Maass space” in degree 3 case by using the relation between \(L\)-functions attached to modular forms of various degrees and weights. But he remarks that there is no characterization of such space by means of relations between Fourier coefficients as is done in degree 2 case. Finally some consequences from the conjecture are stated.

First, he recalls some basic tools such as Hecke operators, \(L\)-function of Andrianov’s type attached to Siegel cusp form, formal Dirichlet series associated with Hecke operators. Next, he gives explicit constructions of Siegel cusp forms \(F_{12}(Z)\) (resp. \(F_{14}(Z)\)) of degree 3 and weights 12 and 14 respectively by means of theta-series with spherical function associated with two root lattices. He also gives some values of Fourier coefficients to those forms. And he expresses the action of Hecke operators by the relations of Fourier coefficients. This enables him to determine the local factors \(L_ 2(s,F_{12})\) of the \(L\)-function of the above type. With this example the author derives a conjecture that \(L(s,F_{12})= L(s-9,\Delta) L(s-10,\Delta) L(s,\Delta \otimes g_{20})\) would hold. Here \(\Delta\) (resp. \(g_{20}\)) is a cusp form of weight 12 (resp. 20) in the classical modular forms, \(L(s-9,\Delta)\) is the Mellin transform of \(\Delta\) and \(L(s,\Delta\otimes g_{20})\) is the Rankin convolution of two \(L\)-functions. In the same spirit the \(F_{14}\) case is also treated.

In the later part the author proposes a definition of “Maass space” in degree 3 case by using the relation between \(L\)-functions attached to modular forms of various degrees and weights. But he remarks that there is no characterization of such space by means of relations between Fourier coefficients as is done in degree 2 case. Finally some consequences from the conjecture are stated.

Reviewer: M.Ozeki (Yamagata)

### MSC:

11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |