Formes linéaires de logarithmes de points algébriques sur les groupes algébriques. (Linear forms in logarithms of algebraic points on algebraic groups)
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 \((\log \log B)^ d\).
linear forms with algebraic coefficients in logarithms of algebraic points
scientific article; zbMATH DE number 4154532
scientific article