Joint PU/IAS Number Theory

Borel proved that every holomorphic map from a product of punctured unit discs to a complex Shimura variety extends to a map from a product of discs to its Bailey-Borel compactification. In joint work with Oswal, Zhu, and Patel, we proved a p-adic version of this theorem over discretely valued fields for Shimura varieties of abelian type. I will speak about work with Bakker, Oswal, and Yao, where we prove the analogous p-adic extension theorem for compact non-abelian Shimura varieties and geometric period images for large primes p.