Böcherer, Siegfried; Chiera, Francesco Ludovico
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 , p. 801-824
MR 2427511 | Zbl pre05298322
doi : 10.5802/aif.2370
URL stable : http://www.numdam.org/item?id=AIF_2008__58_3_801_0

Classification:  11F46,  11F60,  11F66
Mots clés: méthode de Rankin et Selberg, produit de Petersson, formes modulaires non paraboliques, opérateurs différentielles invariants
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.

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