Ax-Schanuel for Shimura varieties


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.


Ngaiming Mok

The University of Hong Kong, Pokfulam, Hong Kong

Jonathan Pila

Mathematical Institute, University of Oxford, Oxford, United Kingdom

Jacob Tsimerman

University of Toronto Toronto, Ontario, Canada