Curves of Genus 3 with a Group of Automorphisms Isomorphic to S₃

Abstract

In this talk, we construct curves of genus 3 with automorphism group equal to S₃; we give some applications of this construction to the problem of optimal curves, i.e. of curves over a finite field $\mathbb{F}_q$ having a number of points equal to the Serre-Weil bound M _q ; in particular, we prove that there exists infinitely many fields $\mathbb{F}_{3^n}$ having optimal curves; we prove also that there exists an integer C such that, for any finite field $\mathbb{F}_{7^n}$, there exists a curve of genus 3 defined over having at least M _q  − C points.