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Prolongement des torseurs via les log schémas

Abstract : Let R be a discrete valuation ring, with field of fractions K and residue field k of characteristic p > 0. Given a finite flat and commutative group scheme G over K and a smooth projective curve C over K with a rational point, we study in this thesis the extension of pointed fppf G-torsors over C to pointed torsors over a regular model of C over R. We already know that an fppf extension of the torsor doesn't always exist. Therefore, we look for a solution in a larger category, namely the category of logarithmic torsors. We prove that extending a G-torsor into a log flat torsor amounts to finding a finite flat model of G over R, for which a certain group scheme morphism to the Jacobian J of the curve extends to the Néron model of J. Assuming that this condition is satisfied, we shall see that we get to an already well-known criterion in the literature for the torsor to admit an fppf extension.In a second step, by generalizing a result by Chiodo, we give a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and this yields interesting examples of commutative group schemes satisfying the criterium of our main theorem. Finally, the last part of this thesis is devoted to the study of some examples of extension of torsors. Let's give an hyperelliptic curve over Q depending on some prime number p and whose Jacobian contains a subgroup isomorphic to (Z/pZ)². This last property implies the data of a mu²p-torsor over the curve. To begin with, we construct a regular model of the curve above Zl, for some prime number l. Then, we will see if the previous torsor extends over this model. For various values of l, we will see two different cases, where in the first one the initial torsor extends into an fppf torsor over the regular model constructed, while in the second one, the torsor admits a logarithmic extension but not an fppf one.
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Submitted on : Monday, April 25, 2022 - 5:36:09 PM
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  • HAL Id : tel-03651499, version 1


Sara Mehidi. Prolongement des torseurs via les log schémas. Analyse fonctionnelle [math.FA]. Université Paul Sabatier - Toulouse III, 2021. Français. ⟨NNT : 2021TOU30181⟩. ⟨tel-03651499⟩



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