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

Author: Andre Reznikov
Journal: J. Amer. Math. Soc. 21 (2008), 439-477
MSC (2000): Primary 11F67, 22E45; Secondary 11F70, 11M26
Published electronically: October 4, 2007
MathSciNet review: 2373356
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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.

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Additional Information

Andre Reznikov
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel

Keywords: Representation theory, Gelfand pairs, periods, automorphic $L$-functions, subconvexity, Fourier coefficients of cusp forms
Received by editor(s): December 26, 2005
Published electronically: October 4, 2007
Additional Notes: The research for this paper was partially supported by a BSF grant, by the Minerva Foundation, by the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation and the Emmy Noether Institute for Mathematics (the Center of Minerva Foundation of Germany). The paper was mostly written during one of the author’s visits to MPIM at Bonn. It is a pleasure to thank MPIM for its excellent working atmosphere.
Dedicated: To Joseph Bernstein, as a small token of gratitude.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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