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.
Citation
Ngaiming Mok. Jonathan Pila. Jacob Tsimerman. "Ax-Schanuel for Shimura varieties." Ann. of Math. (2) 189 (3) 945 - 978, May 2019. https://doi.org/10.4007/annals.2019.189.3.7
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