Formes linéaires de logarithmes effectives sur les variétés abéliennes

Éric Gaudron (1)

1 IF - Institut Fourier

We prove new measures of linear independence of logarithms on an abelian variety defined over $\overline{\mathbf{Q}}$, which are \emph{totally explicit} in function of the invariants of the abelian variety (dimension, Faltings height, degree of a polarization). Besides, except an extra-hypothesis on the algebraic point considered and a weaker numerical constant, we improve on earlier results (in particular David's lower bound). We also introduce into the main theorem an algebraic subgroup that leads to a great variety of different lower bounds. An important feature of the proof is the implementation of the \emph{slope method} of Bost and some results of Arakelov geometry naturally associated with it.

Mots clés: Méthode des pentes; formes linéaires de logarithmes; approximations simultanées; méthode de Baker; réduction d'Hirata-Kohno; variété abélienne; fibré vectoriel hermitien; degré d'Arakelov; modèle de Moret-Bailly; hauteur; Slope method; linear forms in logarithms; simultaneous approximations; Baker's method; Hirata-Kohno's reduction; abelian variety; hermitian vector bundle; Arakelov degree; Moret-Bailly model; height

Domaines: Théorie des nombres [math.NT]

Éric Gaudron. Formes linéaires de logarithmes effectives sur les variétés abéliennes. Annales Scientifiques de l'École Normale Supérieure, 2006, 39 (5), pp.699-773. ⟨10.1016/j.ansens.2006.09.001⟩. ⟨hal-00334552⟩