Small points on a multiplicative group and class number problem par Francesco

Let V be an algebraic subvariety of a torus Gm ↪→ P and denote by V ∗ the complement in V of the Zariski closure of the set of torsion points of V . By a theorem of Zhang, V ∗ is discrete for the metric induced by the normalized height ĥ. We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields. 1. Lehmer’s problem Let α ∈ Q and let K be a number field containing α. We denote byMK the set of places of K. For v ∈ K, let Kv be the completion of K at v and let | · |v be the (normalized) absolute value of the place v. Hence, if v is an archimedean place associated with the embedding σ : K ↪→ Q |α|v = |σα|, and, if v is a non archimedean place associated with the prime ideal P over the rational prime p, |α|v = p−λ/e, where e is the ramification index of P over p and λ is the exponent of P in the factorization of the ideal (α) in the ring of integers of K. This normalization agrees with the product formula ∏ v∈MK |α|v :Qv ] v = 1 Manuscrit reçu le 31 décembre 2005. 28 Francesco Amoroso which holds for α ∈ K∗. We define the Weil height of α by