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Réduction semi-stable des variétés abéliennes

Abstract : In this thesis, we study the semi-stable property of abelian varieties overnumber fields. More precisely, we study the minimal degree of a field extension that is necessary for an abelian variety of fixed dimension over a number field to obtain semi-stable reduction. We denote by dg this degree,which depends only on the dimension g, and by d(A) the minimal degreefor a given abelian variety A over a number field, which depends only onA. The principal objects of our study are the finite monodromy groups ofA which were introduced by Grothendieck. These groups that we denote byΦA,v for non archimedean places v of K represent the local obstruction tosemi-stability. We start by linking the cardinals of these groups at the placesof bad reduction and the integer d(A). This, with the work of Silverbergand Zarhin, gives that the Minkowski bound M(2g) is divisible by dg. Wecontinue by studying the geometric behaviour of these groups, i.e. their behaviour in the fibers of abelian schemes. By considering a universal abelianscheme provided by Mumford we deduce the existence of abelian varietieswith prescribed finite monodromy groups relative to a number field undersome technical conditions. We finish by building for every odd prime (bytorsion in Galois cohomology) abelian varieties with complex multiplicationthat have maximal finite monodromy at that prime over a same numberfield (in arbitrary dimension). We deduce, for any nonzero integer g, theinequalityM(2g)/2^(g−1)≤ dg ≤ M(2g).
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Submitted on : Wednesday, February 16, 2022 - 2:59:12 PM
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Séverin Philip. Réduction semi-stable des variétés abéliennes. Géométrie algorithmique [cs.CG]. Université Grenoble Alpes [2020-..], 2021. Français. ⟨NNT : 2021GRALM037⟩. ⟨tel-03577086⟩



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