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
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 : 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
@article{AIF_2008__58_3_801_0,
     author = {B\"ocherer, Siegfried and Chiera, Francesco Ludovico},
     title = {On Dirichlet Series and Petersson Products for Siegel Modular Forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {3},
     year = {2008},
     pages = {801-824},
     doi = {10.5802/aif.2370},
     mrnumber = {2427511},
     zbl = {pre05298322},
     language = {en},
     url = {http://http://www.numdam.org/item/AIF_2008__58_3_801_0}
}
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/item/AIF_2008__58_3_801_0/

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