Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

TL;DR

This paper proves a version of the Ax-Lindemann-Weierstrass theorem with derivatives for genus zero Fuchsian groups, using advanced differential algebra and model theory, with applications to Painlevé equations and the André-Pink conjecture.

Contribution

It establishes the Ax-Lindemann-Weierstrass theorem with derivatives for genus zero Fuchsian groups, extending previous results and applying them to longstanding mathematical questions.

Findings

Proved Ax-Lindemann-Weierstrass with derivatives for genus zero Fuchsian groups.

Answered a question of Painlevé from 1895.

Confirmed cases of the André-Pink conjecture for certain Fuchsian group orbits.

Abstract

We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlev\'e irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences. In particular, we apply our results to answer a question of Painlev\'e (1895). We also answer certain cases of the Andr\'e-Pink conjecture, namely in the case of orbits of commensurators of Fuchsian groups.