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Evaluation Of Gaussian Hypergeometric Series Using Huff’s Models Of Elliptic Curves

Mohammad Sadek, Nermine El-Sissi, Arman Shamsi Zargar, Naser Zamani
Published 2018 · Mathematics
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A Huff curve over a field K is an elliptic curve defined by the equation $$ax(y^2-1)=by(x^2-1)$$ax(y2-1)=by(x2-1) where $$a,b\in K$$a,b∈K are such that $$a^2\ne b^2$$a2≠b2. In a similar fashion, a general Huff curve over K is described by the equation $$x(ay^2-1)=y(bx^2-1)$$x(ay2-1)=y(bx2-1) where $$a,b\in K$$a,b∈K are such that $$ab(a-b)\ne 0$$ab(a-b)≠0. In this note we express the number of rational points on these curves over a finite field $${\mathbb {F}_q}$$Fq of odd characteristic in terms of Gaussian hypergeometric series $$\displaystyle {_2F_1}(\lambda ):={_2F_1}\left( \begin{matrix} \phi &{}\phi \\ &{} \epsilon \end{matrix}\Big | \lambda \right) $$2F1(λ):=2F1ϕϕϵ|λ where $$\phi $$ϕ and $$\epsilon $$ϵ are the quadratic and trivial characters over $${\mathbb {F}_q}$$Fq, respectively. Consequently, we exhibit the number of rational points on the elliptic curves $$y^2=x(x+a)(x+b)$$y2=x(x+a)(x+b) over $${\mathbb {F}_q}$$Fq in terms of $${_2F_1}(\lambda )$$2F1(λ). This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of $${_2F_1}$$2F1. Finally, we present the exact value of $$_2F_1(\lambda )$$2F1(λ) for different $$\lambda $$λ’s over a prime field $${\mathbb {F}_p}$$Fp extending previous results of Greene and Ono.
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