Twisted Hessian curves over the ring 𝔽q[∈], ∈2 = 0 Online publication date: Mon, 19-Dec-2022
by Abdelâli Grini; Abdelhakim Chillali; Lhoussain El Fadil; Hakima Mouanis
International Journal of Computer Aided Engineering and Technology (IJCAET), Vol. 18, No. 1/2/3, 2023
Abstract: The goal of this work is to study some arithmetic proprieties of the twisted Hessian curves defined by a equation of type: aX3 + Y3 + Z3 = dXYZ on the local ring R2 = 𝔽q[X]/(X2), where p ≥ 5 is a prime number, q = pd and d ∈ ℕ*, such that -3 is not a square in 𝔽q. This paper consists of an introduction, section, and a conclusion. In the introduction, we review some fundamental arithmetic proprieties of finite local rings R2, which will be used in the remainder of this article. The section is devoted to the study the above mentioned twisted Hessian curves on these finite local rings for restriction to some specific characteristic p ≥ 5. Using these studies, we give essential properties and we define the group H2a,d, these properties, the classification of these elements and a bijection between the sets H2a,d and Ha0,d0 × 𝔽q, where Ha0,d0 is the twisted Hessian curve over the finite field 𝔽q.
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