The rational case in the theory of linear forms in logarithms - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2004

The rational case in 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.
Fichier principal
Vignette du fichier
art3.pdf (597.09 Ko) Télécharger le fichier

Dates et versions

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

Identifiants

Citer

Éric Gaudron. The rational case in the theory of linear forms in logarithms. 2004. ⟨hal-00003000v1⟩
130 Consultations
123 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More