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]

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dc.creatorGaudron, Éric
dc.date2004-10-05
dc.date2008-10-27
dc.date.accessioned2026-07-07T12:43:39Z
dc.date.available2026-07-07T12:43:39Z
dc.descriptionWe 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.
dc.descriptionVersion définitive
dc.identifierhttps://arxiv.org/abs/math/0410082
dc.identifierhttp://arxiv.org/abs/math/0410082
dc.identifierJournal of Number Theory 127, 2 (2007) 220-261
dc.identifierdoi:10.1016/j.jnt.2007.08.001
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/220496
dc.subjectNumber Theory
dc.subject11J86, 11J61, 11J13
dc.titleÉtude 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]
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