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- W4298099452 abstract "A Huff curve over a field $K$ is an elliptic curve defined by the equation $ax(y^2-1)=by(x^2-1)$ where $a,bin K$ are such that $a^2ne b^2$. In a similar fashion, a general Huff curve over $K$ is described by the equation $x(ay^2-1)=y(bx^2-1)$ where $a,bin K$ are such that $ab(a-b)ne 0$. In this note we express the number of rational points on these curves over a finite field $mathbb{F}_q$ of odd characteristic in terms of Gaussian hypergeometric series $displaystyle {_2F_1}(lambda):={_2F_1}left(begin{matrix} phi&phi & epsilon end{matrix}Big| lambda right)$ where $phi$ and $epsilon$ are the quadratic and trivial characters over $mathbb{F}_q$, respectively. Consequently, we exhibit the number of rational points on the elliptic curves $y^2=x(x+a)(x+b)$ over $mathbb{F}_q$ in terms of ${_2F_1}(lambda)$. This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of ${_2F_1}$. Finally, we present the exact value of $_2F_1(lambda)$ for different $lambda$'s over a prime field $mathbb{F}_p$ extending previous results of Greene and Ono." @default.
- W4298099452 created "2022-10-01" @default.
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- W4298099452 date "2018-05-22" @default.
- W4298099452 modified "2023-09-27" @default.
- W4298099452 title "Evaluation of Gaussian hypergeometric series using Huff's models of elliptic curves" @default.
- W4298099452 doi "https://doi.org/10.48550/arxiv.1805.08475" @default.
- W4298099452 hasPublicationYear "2018" @default.
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