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Problèmes arithmétiques relatifs à certaines familles de courbes sur les corps finis

Abstract : This thesis is divided in three parts. The first one deals with the automorphisms group of modular curves X(N), N prime, over F_p, p not equal to N. We prove that for p>3 and X(N) ordinary, this group is PSL_2(Z/NZ). Also the cases N=7,11,13 are completely done. The second part deals with optimal curves. We show that N_3(5)=13 and we study the geometric properties (automorphisms group and coverings) of a curve attaining this bound. In particular, we give explicit coverings (of degree 3 and 4) over elliptic curves. The last part is an extension of the AGM method for points counting in characteristic 2 on an ordinary non hyperelliptic genus 3 curve. We prove a formula linking the quotient of theta constants and the product of the 2-adic unities eigenvalues of the Frobenius. We give an algorithm to compute algebraically the initial quotients, a good model for computations (i.e such that computations stay in a fixed unramified extension of Q_2) and we show how to find the characteristic polynom with LLL.
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Contributor : Christophe Ritzenthaler <>
Submitted on : Tuesday, July 1, 2003 - 2:37:39 PM
Last modification on : Tuesday, July 7, 2020 - 8:53:35 AM
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  • HAL Id : tel-00003070, version 1



Christophe Ritzenthaler. Problèmes arithmétiques relatifs à certaines familles de courbes sur les corps finis. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2003. Français. ⟨tel-00003070⟩



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