Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves

Lennon, Catherine

doi:10.1090/S0002-9939-2010-10609-3

Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves
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by Catherine Lennon PDF

Proc. Amer. Math. Soc. 139 (2011), 1931-1938 Request permission

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 by Fuselier 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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