Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif dans le cas rationnel
Volume 543 / 2019
Abstract
We establish new measures of linear independence of logarithms on a commutative algebraic group. Let $G$ be a connected commutative algebraic group over $\overline{{\mathbb{Q}}}$ and let $t_G$ be the tangent space at the origin. We consider a vector $u \in t_G \otimes_{\overline{{\mathbb{Q}}}} {\mathbb{C}}$ such that its image by the exponential map of the Lie group $G({\mathbb{C}})$ is an algebraic point ${\mathbf{p}} \in G(\overline{{\mathbb{Q}}})$. Let $V$ be a hyperplane in $t_G$. We obtain lower bounds for the distance $d(u,V)$ between $u$ and $V \otimes_{\overline{{\mathbb{Q}}}} {\mathbb{C}}$ in the rational case, where $V=t_H$ is the tangent space at the origin of an algebraic connected subgroup of $G$. These lower bounds are the best currently known in terms of the height $h({\mathbf{p}})$ of ${\mathbf{p}}$. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where $V = t_H$ is a linear subspace of any dimension.
