Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves

Catherine Lennon

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arXiv · 1003.4421

Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves

Catherine Lennon

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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.

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Catherine Lennon. 2010-03-23. Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves. https://arxiv.org/abs/1003.4421

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