A Gaussian hypergeometric series evaluation and apéry number congruences

If p is prime, then let φ_p denote the Legendre symbol modulo p and let ε_p be the trivial character modulo p. As usual, let _n+1F_n(x)_p := _n+1F_n(^{φp, φp, ..., φp} _{εp, ..., εp} | x )_p be the Gaussian hypergeometric series over double-struck F_p. For n > 2 the non-trivial values of _n+1F_n(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 Apéry 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 Apéry numbers.