Maculan M. Diophantine applications of geometric invariant theory

Paris: Société Mathématique de France, 2017. — 156 p. — (Memoires de la SMF. Nouvelle serie; No. 152). — ISBN 978-2-85629-865-7.This text consists of two parts. In the frst one we present a proof of Thue-Siegel-Roth’s Theorem (and its more recent variants, such as those of Lang for number felds and that “with moving targets” of Vojta) as an application of Geometric Invariant Theory (GIT). Roth’s Theorem is deduced from a general formula comparing the height of a semi-stable point and the height of its projection on the GIT quotient. In this setting, the role of the zero estimates appearing in the classical proof is played by the geometric semi-stability of the point to which we apply the formula.
In the second part we study heights on GIT quotients. We generalise Burnol’s construction of the height and refne diverse lower bounds of the height of semi-stable points established to Bost, Zhang, Gasbarri and Chen. The proof of Burnol’s formula is based on a non-archimedean version of Kempf-Ness theory (in the framework of Berkovich analytic spaces) which completes the former work of Burnol.