Combining induction formulas for local densities with a functional equation for the Siegel series, we give an explicit formula for the Siegel series. By this formula, we also give an explicit formula for the Fourier coefficients of the Siegel Eisenstein series.

Abstract. We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.

In this paper, we consider the relationship between the congruence of cuspidal Hecke eigenforms with respect to Sp n (Z) and the special values of their standard zeta functions. In particular, we propose a conjecture concerning the congruence between Saito-Kurokawa lifts and non-Saito-Kurokawa lifts, and prove it under certain condition.

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