Hypergeometric 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

DOI:
https://doi.org/10.1090/S0002-9939-09-10091-6

Published electronically:
September 16, 2009

MathSciNet review:
2557169

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the hypergeometric function over whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating at the argument . This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono's result by evaluating an infinite family of over in terms of Jacobi sums. Here we present five new over (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 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.