Courbes de genre 3 avec S3 comme groupe d'automorphismes

TL;DR

This paper investigates genus 3 curves with automorphism group S3 over finite fields, aiming to find optimal curves and establishing connections with elliptic curves and their Jacobians, especially in characteristics 3 and 7.

Contribution

It demonstrates the existence of genus 3 curves with automorphism group S3 whose Jacobians relate to elliptic curves, providing new examples of optimal curves over finite fields.

Findings

Existence of genus 3 curves with Jacobians isogenous to E^3 for elliptic curves E in characteristic 3.

Results on genus 3 curves in characteristic 7 with partial classifications.

Experimental evidence of many optimal curves arising from these constructions.

Abstract

this paper is devoted to the study of curves of genus 3 with group of automorphisms the symmetric group S3, principally over finite fields, in view to obtain optimal curves. For instance, we prove that, over the finite fields of char. 3, for any elliptic curve E with j-invariant not in F_3, there exists a curve of genus 3 whose jacobian is isogeneous to E^3; we have also results in char. 7, and these curves provide many examples of optimal curves in any char. Nous etudions les courbes de genre 3 dont le groupe d'automorphismes est S3, en vue d'obtenir des courbes optimales sur les corps finis. En particulier, en car. 3, si E est unecourbe elliptique dont l'invariant n'est pas dans F_3, il existe une telle courbe de genre 3 dont la jacobienne est isogene a E^3; dans le cas de la caracteristique 7, des resultats plus partiels sont obtenus. Par ailleurs, ces courbes semblent donner experimentalement de nombreux cas de courbes optimales.