Arithmétique des espaces de modules des courbes hyperelliptiques de genre 3 en caractéristique positive

Abstract : The aim of this thesis is to provide an explicit description of the moduli spaces of genus 3 hyperelliptic curves in positive characteristic. Over a field of odd characteristic, a parameterization of these moduli spaces is given via the algebra of invariants of binary forms of degree 8 under the action of the special linear group. Following the work of Lercier and Ritzenthaler, the case of fields of characteristic 3, 5 and 7 are still open. However, in these remaining cases, the classical methods in characteristic zero do not work in providing generators for these algebra of invariants. Hence we provide only separating invariants in characteristic 3 and 7. Furthermore our results in characteristic 5 show that this approach is not suitable. From these results, we describe the stratification of the moduli spaces of genus 3 hyperelliptic curves in characteristic 3 and 7 according to the automorphism groups of the curves and implement algorithms to reconstruct a curve from its invariants. For this reconstruction step, we paid attention to arithmetic issues, like the obstruction to be a field of definition for the field of moduli. Finally, in the case of characteristic 2, we use a different approach where the curves are defined by their Artin-Schreier models. The arithmetic structure of the ramification points of these curves stratifies the moduli space in 5 cases and we define in each case invariants that characterize the isomorphism class of hyperelliptic curves.
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Romain Basson. Arithmétique des espaces de modules des courbes hyperelliptiques de genre 3 en caractéristique positive. Mathématiques [math]. Université de Rennes 1, 2015. Français. ⟨tel-01170922⟩

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