On Dirichlet Series and Petersson Products for Siegel Modular Forms

Böcherer, Siegfried; Chiera, Francesco Ludovico

Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 801-824.

On démontre que la série de Dirichlet à la Rankin-Selberg associée à toute paire de formes modulaires de Siegel (non nécessairement paraboliques) de degré $n$ et poids $k\ge n/2$ admet un prolongement méromorphe à $ℂ$. En outre, on montre que le produit de Petersson de toute paire de formes modulaires de carré-intégrable et de poids $k\ge n/2$ a une expression en termes du résidu en $s=k$ de la série de Dirichlet associée. Ces résultats sont bien connus pour les formes paraboliques. La méthode que nous adoptons généralise celle qui a été introduite par Maass (dans le cas $n=2$) et se base sur l’utilisation de certains opérateurs différentiels invariants.

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 $ℂ$. 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.

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