Étude du cas rationnel de la théorie des formes linéaires de logarithmes

Abstract

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called 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 ∈ Lie ( G ( C v 0 ) ) a logarithm of a point p ∈ 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 ) ⊗ 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 log log a . The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy.