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Theses

Inertia Groups and Jacobian Varieties

Abstract : Let k be an algebraically closed field of characteristic p > 0 and C/k be a projective,smooth, integral curve of genus g > 1 endowed with a p-group of automorphisms G such that |G| > 2p/(p-1)g. The pair (C,G) is called big action. If (C,G) is a big action, then |G|<=4p/(p-1)^2g^2 (*). In this thesis, one studies arithmetical repercussions of geometric properties of big actions. One studies the arithmetic of the maximal wild monodromy extension of curves over a local field K of mixed characteristic p with algebraically closed residue field, with arbitrarily high genus having for potential good reduction a big action achieving equality in (*). One studies the associated Swan conductors. Then, one gives the first examples, to our knowledge, of big actions (C,G) with non abelian derived group D(G). These curves are obtained as coverings of S-ray class fields of P1(Fq) where S is a finite non empty subset of P1(Fq). Finally, one describes a method to compute S-Hilbert class fields of supersingular abelian covers of the projective line having exponent p and one illustrates it for some Deligne-Lusztig curves.
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Pierre Chrétien. Inertia Groups and Jacobian Varieties. General Mathematics [math.GM]. Université Sciences et Technologies - Bordeaux I, 2013. English. ⟨NNT : 2013BOR14785⟩. ⟨tel-00841298⟩

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