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Hypergeometric $ \mathbf{{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms


Author: Ron Evans
Journal: Proc. Amer. Math. Soc. 138 (2010), 517-531
MSC (2000): Primary 11T24; Secondary 11F11, 11L05, 33C20
Published electronically: September 16, 2009
MathSciNet review: 2557169
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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
Email: revans@ucsd.edu

DOI: https://doi.org/10.1090/S0002-9939-09-10091-6
Keywords: Hypergeometric functions over finite fields, Kloosterman sums, Gauss and Jacobi sums, Hecke eigenvalues, Hecke eigenforms, symmetric power, Kloosterman sheaf, Frobenius map
Received by editor(s): June 22, 2009
Published electronically: September 16, 2009
Communicated by: Ken Ono
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.