Gaussian Hypergeometric Evaluations of Traces of Frobenius for 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 $j(E)\ne 0,1728$ and $q\equiv 1(\mathrm{mod}12)$ 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.

8 pages

References in corpus (1)

• A Trace Formula for Certain Hecke Operators and Gaussian Hypergeometric Functions