Edwards Curves and Gaussian Hypergeometric Series

Abstract

Let E be an elliptic curve described by either an Edwards model or a twisted Edwards model over F p , namely, E is defined by one of the following equationsWe express the number of rational points of E over F p using the Gaussian hypergeometric where ǫ and φ are the trivial and quadratic characters over F p respectively. This enables us to evaluate |E(F p )| for some elliptic curves E, and prove the existence of isogenies between E and Legendre elliptic curves over F p .

Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves

“…Influenced by the latter articles, research has been targeting the investigation of the various links between elliptic curves described by different equations and Gaussian hypergeometric series. For example, similar links were found for Clausen curves [6] and Edwards curves [11]. Further connections between these series and the number of rational points on higher genus curves were explored, see [1,2,3,12] Herein, we consider Huff curves which are elliptic curves defined by an equation of the form H a,b : ax(y 2 − 1) = by(x 2 − 1), a, b ∈ F × q , a 2 = b 2 .…”

“…x 2 + y 2 = 1 + d 2 x 2 y 2 where d = a − b a + b , see [16, §1]. In [11], the size of the rational points on an Edwards curves is expressed in terms of 2 F 1 over F p . Yet, all the results there are valid over F q .…”

“…  using the counts in [11]. Additionally, these transformations are deployed to evaluate 2 F 1 (λ) at new values of λ extending on the findings of [4] and [9].…”

Moments of Gaussian hypergeometric functions over finite fields

“…Not only they share arithmetic properties with their calssical counterparts, they have also turned out to possess many connections with several arithmeticgeometric objects. Values of Gaussian hypergeometric functions are expressed in terms of traces of families of elliptic curves, [4,5,8,12,14,16]. The number of rational points on certain hyperelliptic curves were linked to values of some Gaussian hypergeometric functions, [6,15].…”

Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves

“…Then we focus on these sums when both m and n are powers of the prime 2. In fact assuming that q satisfies certain congruence relations we make use of elementary identities satisfied by Jacobi sums in order to express ψ Since Greene initiated the study of hypergeometric series over finite fields in [7], many authors have written the number of rational points on algebraic curves over finite fields in terms of different Gaussian hypergeometric series, see for example [1], [2], [3], and [11]. Using identities relating the number of F q -rational points to hypergeometric series one can evaluate these series at specific values, see for example [10].…”