Measures of linear independance of logarithms in a commutative algebraic group

• 成果类型: Article
• 署名作者: Gaudron, É
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-005-0440-5
• 发表日期: 2005
• 页码: 137-188
• 关键词: diophantine approximation; forms; points

摘要

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers (Q) over bar. Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over (Q) over bar, and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(C) is an algebraic point. Then we obtain a lower bound for the distance between u and W circle times C, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker's method and Hirata's reduction as well as a new arithmetic argument (Chudnovsky's process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.