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

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

2003-12-18

arxiv.org/abs/math/0312366v2

Mathematics

Number Theory

31 pages ; added references ; the analysis of the supersingular locus has been modified

Scientific paper

Let k=F_q 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 normal models and we give 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 knowed results on the hyperelliptic locus we obtain a complete picture of these strata for M_3.

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