We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function's relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.

Abstract. In examining the relationship between the number of points over Fp on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the p-th Fourier coefficient of a modular form.

Abstract. We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p > 3, that the trace of Frobenius of any elliptic curve over Fp, whose jinvariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over Fp when p ≡ 1 (mod 12).

We define a function which extends Gaussian hypergeometric series to the p-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.