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.]
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 kn/2 admet un prolongement méromorphe à . En outre, on montre que le produit de Petersson de toute paire de formes modulaires de carré-intégrable et de poids kn/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 kn/2 has meromorphic continuation to . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight kn/2 may be expressed in terms of the residue at s=k of the associated Dirichlet series.

DOI : 10.5802/aif.2370
Classification : 11F46, 11F60, 11F66
Keywords: Rankin-Selberg method, Petersson product, non-cuspidal modular forms, invariant differential operators.
Mot clés : méthode de Rankin et Selberg, produit de Petersson, formes modulaires non paraboliques, opérateurs différentielles invariants
Böcherer, Siegfried 1 ; Chiera, Francesco Ludovico 2

1 Universität Mannheim Fakultät für Mathematik und Informatik A5, 68131 Mannheim(Germany)
2 Università “La Sapienza” di Roma Dipartimento di Matematica P. le A. Moro 2 00185 Rome (Italy)
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Böcherer, Siegfried; Chiera, Francesco Ludovico. On Dirichlet Series and Petersson Products for Siegel Modular Forms. Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 801-824. doi : 10.5802/aif.2370. http://www.numdam.org/articles/10.5802/aif.2370/

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