Description

This text consists of two parts. In the first one we present a proof of Thue-Siegel-Roth's Theorem (and its more recent variants, such as those of Lang for
number fields 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-tability 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 refine 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.