Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves

Details Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves Gaussian Hypergeometric Evaluations of Traces of Frobenius for Elliptic Curves

2010-03-23

arxiv.org/abs/1003.4421v1

Mathematics

Number Theory

8 pages

Scientific paper

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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