On Dirichlet Series and Petersson Products for Siegel Modular Forms

[Sur les séries de Dirichlet et les produits de Petersson pour les formes modulaires de Siegel.]

Böcherer, Siegfried ; Chiera, Francesco Ludovico

Annales de l'Institut Fourier, Tome 58 (2008), p. 801-824 / Harvested from Numdam

Résumé

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 $\mathbb{C}$. 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.

Abstract

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 $\mathbb{C}$. 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.

Détail

Publié le : 2008-01-01

MR 2427511 | Zbl pre05298322

DOI : https://doi.org/10.5802/aif.2370
Classification:  11F46,  11F60,  11F66
Mots clés: méthode de Rankin et Selberg, produit de Petersson, formes modulaires non paraboliques, opérateurs différentielles invariants

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