\'Etude du cas rationnel de la th\'eorie des formes lin\'eaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]

Abstract

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u\in Lie(G(C_{v_{0}})) a logarithm of a point p\in G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)\otimes_{k} C_{v_{0}}. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height Log a of p, removing a polynomial term in LogLog a. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of J.-B. Bost) obtained from a lemma by M. Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by D. Roy.