Ax-Schanuel for Shimura varieties

Abstract

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety. A basic version of the theorem concerns the transcendence of the uniformization map from a bounded Hermitian symmetric space to a Shimura variety. We then prove a version of the theorem with derivatives in the setting of jet spaces, and finally a version in the setting of differential fields.

Our method of proof builds on previous work, combined with a new approach that uses higher-order contact conditions to place varieties yielding intersections of excessive dimension in natural algebraic families.