MESURES D'INDÉPENDANCE LINÉAIRE DE LOGARITHMES DANS UN GROUPE ALGÉBRIQUE COMMUTATIF

Éric Gaudron (1)

1 IF - Institut Fourier

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 $\overline{Q}$. Let $G$ be such a group. Let $W$ be a hyperplane of the tangent space at the origin of $G$, defined over $\overline{Q}$, 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 $W\otimes\mathbf{C}$, 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.

Domaines: Théorie des nombres [math.NT]

Éric Gaudron. MESURES D'INDÉPENDANCE LINÉAIRE DE LOGARITHMES DANS UN GROUPE ALGÉBRIQUE COMMUTATIF. Inventiones Mathematicae, 2005, 162 (1), pp.137-188. ⟨10.1007/s00222-005-0440-5⟩. ⟨hal-00334549⟩