### Action of Hecke operators on products of Igusa theta constants with rational characteristics

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Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute the rational fraction in this conjecture. Using formulas for images of double cosets under the Satake spherical map, we first compute the sum of the generating series, which is a rational fraction with polynomial coefficients....

We introduce Hecke operators on de Rham cohomology of compact oriented manifolds. When the manifold is a quotient of a Hermitian symmetric domain, we prove that certain types of such operators are compatible with the usual Hecke operators on automorphic forms.

We prove that the Dirichlet series of Rankin–Selberg type associated with any pair of (not necessarily cuspidal) Siegel modular forms of degree $n$ and weight $k\ge n/2$ has meromorphic continuation to $\u2102$. Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight $k\ge n/2$ may be expressed in terms of the residue at $s=k$ of the associated Dirichlet series.