p-adic Gamma function and traces of Frobenius of elliptic curves

TL;DR

This paper explores new expressions for the trace of Frobenius of elliptic curves using the p-adic Gamma function and McCarthy's $_{2}G_{2}[\

Contribution

It introduces two novel formulas connecting the trace of Frobenius with special values of $_{2}G_{2}[\

Findings

Derived two different expressions for Frobenius traces.

Established relations between special values of $_{2}G_{2}[\

Abstract

In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using Teichm\"{u}ller character of finite fields and quotients of $p$-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of $_{2}G_{2}[\cdots]$. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function $_{2}G_{2}[\cdots]$. As a result, we obtain two relations between special values of the function $_{2}G_{2}[\cdots]$ with different parameters.