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

Abstract

Let $p$ be an odd prime and $q=p^r$, $r\geq 1$. For positive integers $n$, let ${_n}G_n[\cdots]_q$ denote McCarthy's $p$-adic hypergeometric functions. In this article, we prove an identity expressing a ${_4}G_4[\cdots]_q$ hypergeometric function as a sum of two ${_2}G_2[\cdots]_q$ hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transfomation that relates ${_{n+2}}G_{n+2}[\cdots]_q$ and ${_n}G_n[\cdots]_q$ hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of ${_4}G_4[\cdots]_q$ and ${_6}G_6[\cdots]_q$ hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.