Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif

Eric Gaudron 1
Abstract : This thesis falls within the theory of linear forms in logarithms. It comprises two parts as well as three annexes. In the first part, we are interested in the general case of an unspecified algebraic commutative group, defined over the algebraic closure of Q. Given such a group G, an hyperplane W of the tangent space at the origin of G and u a complex point of this tangent space, whose image by the exponential map of the Lie group G(C) is an algebraic point, we obtain a lower bound for the distance between u and W, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The demonstration rests on Baker's method as well as a new arithmetic argument (Chudnovsky's process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers, built during the proof. In the second part, we study in details the ``abelian non-homogeneous case'' (in which the group G is the direct product of $\mathbb{G}_{\mathrm{a}}$ by an abelian variety) and we establish a new measure, comparable with the one given in the first part, but totally explicit in function of the invariants of the abelian variety. An important feature of this second part is the implementation, for the first time in this context, of J.-B. Bost's slopes method and some results of Arakelov geometry naturally associated with it.
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Theses
Mathematics. Université Jean Monnet - Saint-Etienne, 2001. French


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Submitted on : Wednesday, February 27, 2002 - 5:15:10 PM
Last modification on : Wednesday, February 27, 2002 - 5:15:10 PM

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Eric Gaudron. Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif. Mathematics. Université Jean Monnet - Saint-Etienne, 2001. French. <tel-00001165>

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