Étude du cas rationnel de la théorie des formes linéaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms] - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Number Theory Année : 2007

Étude du cas rationnel de la théorie des formes linéaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]

Éric Gaudron

Résumé

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.
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Dates et versions

hal-00003000 , version 1 (05-10-2004)
hal-00003000 , version 2 (27-10-2008)

Identifiants

Citer

Éric Gaudron. Étude du cas rationnel de la théorie des formes linéaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]. Journal of Number Theory, 2007, 127 (2), pp.220-261. ⟨10.1016/j.jnt.2007.08.001⟩. ⟨hal-00003000v2⟩

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