Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms

Reznikov, Andre ¹

¹ Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel

Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 439-477

Résumé

We use the uniqueness of various invariant functionals on irreducible unitary representations of $PGL_2(\mathbb {R})$ in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an application we obtain a subconvexity bound for certain $L$-functions. Our main tool is the notion of a Gelfand pair from representation theory.

Détail

DOI : 10.1090/S0894-0347-07-00581-4

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