On Dirichlet Series and Petersson Products for Siegel Modular Forms

Böcherer, Siegfried; Chiera, Francesco Ludovico

doi:10.5802/aif.2370

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 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)

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

Résumé
Abstract

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 $k \geq n / 2$ 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 $k \geq n / 2$ 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 $k \geq n / 2$ has meromorphic continuation to $ℂ$ . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight $k \geq n / 2$ 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

Affiliations des auteurs :

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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     title = {On {Dirichlet} {Series} and {Petersson} {Products} for {Siegel} {Modular} {Forms}},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2370/

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