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Multiplicative properties of Fourier coefficients of Siegel modular forms for principal congruence subgroups of the group Sp(n, \({\mathbb{Z}})\). (Russian) Zbl 0622.10020

Let \(F(Z)=\sum_{T}a(T)e^{2\pi i trace(TZ)}\) be a Siegel modular form of degree n with respect to a congruence subgroup \(\Gamma_ 0(N)\) of Sp(n, \({\mathbb{Z}})\). If F is an eigenform of all Hecke operators, then, for any fixed positive definite half-integral symmetric matrix T of size n, the Dirichlet series \(\sum_{M}a(M' T M)\det (M)^{-s}\) has an Euler product expansion involving the ”standard” L-function attached to F and M runs over all n-rowed integral matrices with (det M,n)\(=1\) up to Gl(n,\({\mathbb{Z}})\)-equivalence. This result is due to A. N. Andrianov [Usp. Mat. Nauk 34, 67-135 (1979; Zbl 0418.10027)].
The purpose of the paper under review is to prove an analogue of that result for Siegel modular forms for principal congruence subgroups of Sp(n, \({\mathbb{Z}})\). The action of Hecke operators on theta series with congruence conditions is considered in the final part of the paper.
Reviewer: S.Böcherer

MSC:

11F27 Theta series; Weil representation; theta correspondences
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)

Citations:

Zbl 0418.10027
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