The trace of Frobenius of elliptic curves and thep-adic gamma function

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 j -invariant 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/. 1. Introduction and statement of results Let Fp denote the finite field with p, a prime, elements. Consider E=Q an elliptic curve with an integral model of discriminantA.E/. We denote Ep the reduction of E modulo p. We note that Ep is nonsingular, and hence an elliptic curve over Fp, if and only if p› A.E/, in which case we say p is a prime of good reduction. Regardless, we define