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 v0 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 uset membership, variantLie(G(Cv0)) a logarithm of a point pset membership, variantG(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)circle times operatorkCv0. 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 loga of p, removing a polynomial term in logloga. 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.
