The trace of Frobenius of elliptic curves and the<i>p</i>-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 ‫ކ‬ p , 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 ‫ކ‬ p when p Á 1 .mod12/.