Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves

TL;DR

This paper derives a formula linking the Frobenius trace of certain elliptic curves over finite fields to Gaussian hypergeometric series, using intrinsic curve invariants like the j-invariant and discriminant.

Contribution

It introduces a new method to express Frobenius traces of elliptic curves directly in terms of their intrinsic invariants, extending previous approaches.

Findings

Expresses Frobenius trace via Gaussian hypergeometric series.

Relates Frobenius trace to j-invariant and discriminant.

Provides a new intrinsic characterization of elliptic curves.

Abstract

We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve $E$ over $\mathbb{F}_q$ satisfying $j(E)\neq 0,1728$ and $q\equiv 1 \pmod{12}$ in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced in Fuselier's work for one parameter families of curves to express the trace of Frobenius of $E$ as a function of its $j$-invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.