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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hypergeometric $\mathbf {{}_3\textit {F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms
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by Ron Evans PDF
Proc. Amer. Math. Soc. 138 (2010), 517-531 Request permission

Abstract:

Let $H$ denote the hypergeometric ${}_3F_2$ function over $\mathbb {F}_p$ whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating $H$ at the argument $1/4$. This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono’s result by evaluating an infinite family of ${}_3F_2(1/4)$ over $\mathbb {F}_q$ in terms of Jacobi sums. Here we present five new ${}_3F_2(1/4)$ over $\mathbb {F}_q$ (involving characters of orders 3, 4, 6, and 8) which are conjecturally evaluable in terms of eigenvalues for Hecke eigenforms of weights 2 and 3. There is ample numerical evidence for these evaluations. We motivate our conjectures by proving a connection between ${}_3F_2(1/4)$ and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.
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Additional Information
  • Ron Evans
  • Affiliation: Department of Mathematics 0112, University of California at San Diego, La Jolla, California 92093-0112
  • MR Author ID: 64500
  • Email: revans@ucsd.edu
  • Received by editor(s): June 22, 2009
  • Published electronically: September 16, 2009
  • Communicated by: Ken Ono
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 517-531
  • MSC (2000): Primary 11T24; Secondary 11F11, 11L05, 33C20
  • DOI: https://doi.org/10.1090/S0002-9939-09-10091-6
  • MathSciNet review: 2557169