p-adic Hyperbolicity of Shimura Varieties

A theorem of Borel says that any holomorphic map from a complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the (poly) punctured disc to the Baily-Borel compactification of the arithmetic variety has no essential singularity.

I will discuss p-adic analogue of these facts for Shimura varieties of abelian type. Joint with Abhishek Oswal and Ananth Shankar (with an appendix by Anand Patel).