Twisted Hessian curves over the Ring F

Abstract

Let Fq[e] be a finite field of q elements, where q is a power of a prime number p. In this paper, we study the Twisted Hessian curves over the ring Fq[e], where e2 = e, denoted by Ha,d(Fq[e]); (a,d) ∈ (Fq[e])2. Using the Twisted Hessian equation, we define the Twisted Hessian curves Ha,d(Fq[e]) and we will show that Hπ0(a),π0(d)(Fq) and Hπ1(a),π1(d)(Fq) are two Twisted Hessian curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates from Fq[e] to Fq. Precisely, we give a bijection between the sets Ha,d(Fq[e]) and Hπ0(a),π0(d)(Fq)×Hπ1(a),π1(d)(Fq).