Minorations de formes linéaires de logarithmes pour les modules de Drinfeld

Abstract

Let k = F q ( T ), k ∞ = F q ((1/ T )), and let us denote by C the completion of an algebraic closure of k ∞ (for the 1/ T -adic valuation), and by K ⊂ C a finite extension of k of degree D . Let ( G a , Φ i ) (1⩽ i ⩽ n ) be n Drinfeld modules of rank ⩾1 defined over K (with exponentials e Φ i ), let u 1 , …, u n ∈ C be such that e Φ i ( u i )∈ K (1⩽ i ⩽ n ), and let β 0 , …, β n be n +1 elements of K . We obtain in this paper a lower bound for the linear form of logarithms β 0 + β 1 u 1 +…+ β n u n (when it is not zero) as a function of the degree D , the heights of the points β i , the absolute values | u i | and the heights of the e Φ i ( u i ), and the heights of the modules ( G a , Φ i ).