Elliptic curves over the ring F_q[e], e^3 = e^2

Abstract

Let F_q be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted E_a,b(F_q[e]) over the ring F_q[e] where e³ = e² and (a,b) ∈ (F_q[e])². In a first time, we study the arithmetic of the ring F_q[e], e³ = e². In addition, using the Weierstrass equation, we define the elliptic curve E_a,b(F_q[e]). The study of this elliptic curve helped us to define two elliptic curve over the finite field F_q. We finish this paper by given a classification of the elements in E_a,b(F_q[e])