A Gaussian hypergeometric series evaluation and Apéry number congruences

Abstract

If p is prime, then let φp denote the Legendre symbol modulo p and let p be the trivial character modulo p. As usual, let n+1Fn(x)p := n+1Fn „ φp, φp, . . . , φp p, . . . , p | x « p be the Gaussian hypergeometric series over Fp. For n > 2 the non-trivial values of n+1Fn(x)p have been difficult to obtain. Here we take the first step by obtaining a simple formula for 4F3(1)p. As a corollary we obtain a result describing the distribution of traces of Frobenius for certain families of elliptic curves. We also find that 4F3(1)p satisfies surprising congruences modulo 32 and 11. We then establish a mod p2 “supercongruence” between Apery numbers and the coefficients of a certain eta-product; this relationship was conjectured by Beukers in 1987. Finally, we obtain many new mod p congruences for generalized Apery numbers.