Le th\'eor\`eme d'Andr\'e-Chudnovsky-Katz

Abstract

The subject of this Master 2 thesis is the study of the Andr\'e-Chudnovsky-Katz theorem on the structure of the solution of the nonzero differential equation of minimal order with coefficients in $\overline{\mathbb{Q}}(z)$ satisfied by a $G$-function. We begin by presenting the theory of globally nilpotent differential operators, of which the main result is the Katz theorem, which states that they are fuchsian with rational exponents. We then give a full proof of the Chudnovsky theorem implying that the minimal nonzero differential operator with coefficients in $\overline{\mathbb{Q}}(z)$ of a $G$-function satisfies a moderate growth condition on some denominators called the Galochkin condition. We finally outline the proof of the Andr\'e-Bombieri theorem establishing the equivalence between the Galochkin condition and the Bombieri condition, which implies the global nilpotence. This allows us to prove the main point of the Andr\'e-Chudnovsky-Katz theorem.