Hypergeometric βπ β(1/4) evaluations over finite fields and Hecke eigenforms
Ron Evans Β· Proceedings of the American Mathematical Society Β· 2009
Let H H denote the hypergeometric 3 F 2 {}_3F_2 function over F p \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 H at the argument 1 / 4 1/4 . This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Onoβs result by evaluating an infinite family of 3 F 2 ( 1 / 4 ) {}_3F_2(1/4) over F q \mathbb {F}_q in terms of Jacobi sums. Here we present five new 3 F 2 ( 1 / 4 ) {}_3F_2(1/4) over F q \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 3 F 2 ( 1 / 4 ) {}_3F_2(1/4) and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.