Non-hyperelliptic curves of genus three over finite fields of characteristic two

Abstract

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus M 3 nh of the moduli space M 3 of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385–389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198–200] on the hyperelliptic locus we obtain a complete picture of these strata for M 3 .