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

• 成果类型: Article
• 署名作者: Casale, Guy; Freitag, James; Nagloo, Joel
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.192.3.2
• 发表日期: 2020
• 页码: 721-765
• 关键词: mordell-lang conjecture; shimura curves; points; schanuel; families; geometry; THEOREM

摘要

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 Painleve 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 that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painleve (1895). We also answer certain cases of the Andre-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups.