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

André Reznikov

Type: Article

Publication Date: 2007-10-04

Citations: 24

DOI: https://doi.org/10.1090/s0894-0347-07-00581-4

Abstract

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.

Locations

Journal of the American Mathematical Society - View - PDF
arXiv (Cornell University) - View - PDF

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