Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
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 Painlevé irreducibility of certain …