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

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**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.## References

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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