The trace of Frobenius of elliptic curves and the $p$-adic gamma function

TL;DR

This paper introduces a new $p$-adic gamma function-based approach to express the trace of Frobenius for elliptic curves over finite fields, extending previous hypergeometric function results to a broader setting.

Contribution

It defines a generalized $p$-adic gamma function that relates to the Frobenius trace, broadening the scope of hypergeometric function applications in number theory.

Findings

Expresses Frobenius trace as a special value of the new function

Generalizes previous hypergeometric function evaluations

Applicable to elliptic curves with specific $j$-invariants

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 $\mathbb{F}_p$, 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 $\mathbb{F}_p$ when $p \equiv 1 \pmod {12}$.