Study of the rational case of the linear forms theory of the algorithms

[摘要] 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 nu(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 is an element of Lie(G(C-nu 0)) a logarithm of a point P E 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 a to Lie(H) circle times(k) C-nu 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. (c) 2007 Elsevier Inc. Tous droits reserves.

[发布日期] 2007-12-01