16 research outputs found
\'Etude du cas rationnel de la th\'eorie des formes lin\'eaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]
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.Comment: Version d\'efinitiv
Un critere effectif d'independance algebrique et une mesure d'approximation simultanee de certaines valeurs de la fonction exponentielle
SIGLECNRS T 57192 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc