Valeur critique de la fonction L adjointe d'une forme modulaire de Hilbert et arithmétique du motif correspondant

Abstract : The aim of this thesis is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention the control of the image of the Galois representation modulo $p$ [Serre, Ribet], Hida's congruence criterion, and the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras [Mazur, Faltings-Jordan]. As an application we relate, in the "minimal" level case, the "algebraic" $p$-part of the adjoint L function of a Hilbert modular newform evaluated at 1 to the cardinality of the corresponding Selmer group. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main tool is the action of the inertia groups at $p$. In order to control this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the $p$-adic (resp. modulo $p$) étale cohomology of the Hilbert modular variety. The cohomological part of our paper builds upon the construction of the arithmetic toroidal compactifications of the universal Hilbert-Blumenthal abelian variety (and of its fiber products) over the arithmetic toroidal compactifications of the Hilbert modular variety of level $\Gamma_1(c,n)$.
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Submitted on : Monday, March 1, 2004 - 11:17:53 PM
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Mladen Dimitrov. Valeur critique de la fonction L adjointe d'une forme modulaire de Hilbert et arithmétique du motif correspondant. Mathématiques [math]. Université Paris-Nord - Paris XIII, 2003. Français. ⟨tel-00005179⟩

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