p-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

Abstract

Let [Formula: see text] be an odd prime and [Formula: see text], [Formula: see text]. For positive integers [Formula: see text], let [Formula: see text] denote McCarthy’s [Formula: see text]-adic hypergeometric function. In this paper, we prove an identity expressing a [Formula: see text] hypergeometric function as a sum of two [Formula: see text] hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transformation that relates [Formula: see text] and [Formula: see text] hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of [Formula: see text] and [Formula: see text] hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.