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

We prove the Ax-Lindemann-Weierstrass theorem for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory of Schwarzian equations and machinery from the model theory of differentially closed fields. This result generalizes previous work of Pila-Tsimerman on the j function. (Joint work with James Freitag and Joel Nagloo)