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Journal of the American Mathematical Society

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Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms
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by Andre Reznikov PDF
J. Amer. Math. Soc. 21 (2008), 439-477 Request permission

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
  • MR Author ID: 333309
  • Email: reznikov@math.biu.ac.il
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 21 (2008), 439-477
  • MSC (2000): Primary 11F67, 22E45; Secondary 11F70, 11M26
  • DOI: https://doi.org/10.1090/S0894-0347-07-00581-4
  • MathSciNet review: 2373356