Formes linéaires de logarithmes de points algébriques sur les groupes algébriques. (Linear forms in logarithms of algebraic points on algebraic groups)
type: Item
instance of: scholarly article
title: Formes linéaires de logarithmes de points algébriques sur les groupes algébriques. (Linear forms in logarithms of algebraic points on algebraic groups)
zbMATH Open document ID: 0704.11016
DOI: 10.1007/BF01245082
author: Noriko Hirata-Kohno
published in: Inventiones Mathematicae
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143889
review text: We give here a new lower bound for linear forms with algebraic coefficients in logarithms of algebraic points on commutative algebraic groups. Let \(G'\) be a connected commutative algebraic group of dimension d, defined over a number field K and put \(G={\mathbb{G}}_ a\times G'\). We select a base on the tangent space \(T_ G({\mathbb{C}})\) of G at te origin. We consider a point \(u\in T_ G({\mathbb{C}})\) satisfying \(\exp_ G(u)\in G(K)\). We seek for a lower bound for the distance from u to any hyperplane W in \(T_ G({\mathbb{C}})\), defined over K, when u does not belong to W. Such a lower bound corresponds to a quantitative version of a general transcendence theorem due to G. Wüstholz on algebraic groups. Suppose that the height of W is at most B with \(B\geq e^ e\). Our lower bound is: \[ dist(u,W)\quad \geq \quad \exp \{-C(\log B)(\log \log B)^{d+1}\} \] where C is a constant depending only on G,u, a chosen embedding of G into a projective space, the base on \(T_ G({\mathbb{C}})\), and the degree of K. We make in fact explicit the constant C in terms of the degree of K and the point u. Our result is almost best possible as far as the dependence on B is concerned, and improves the previous known estimates in this direction; for example, it improves the following result of P. Philippon and M. Waldschmidt: \[ dist(u,W)\quad \geq \quad \exp \{-C'(\log B)^{d+1}\}. \] In the special case where u is a period of \(\exp_ G\), we obtain a slightly sharper lower bound in that one may replace the factor \((\log \log B)^{d+1}\) by \((\log \log B)^ d\).
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 14L10
zbMATH DE Number: 4154532
zbMATH Keywords: lower bound
zbMATH Keywords: linear forms with algebraic coefficients in logarithms of algebraic points
zbMATH Keywords: commutative algebraic groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4154532