Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif

Abstract

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers $\overline{\mathbb{Q}}$ . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over $\overline{\mathbb{Q}}$ , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction 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.