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Explicit Hecke series for symplectic group of genus 4

Kirill Vankov (2011)

Journal de Théorie des Nombres de Bordeaux

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....

Hecke operators on de Rham cohomology.

Min Ho Lee (2004)

Revista Matemática Complutense

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.

On Dirichlet Series and Petersson Products for Siegel Modular Forms

Siegfried Böcherer, Francesco Ludovico Chiera (2008)

Annales de l’institut Fourier

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 n / 2 has meromorphic continuation to . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight k n / 2 may be expressed in terms of the residue at s = k of the associated Dirichlet series.

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