p-adic Hyperbolicity of Shimura Varieties

A classical theorem due to Borel states that every holomorphic map from a poly-punctured disk into a Shimura variety (with torsion-free level structure) extends holomorphically across the punctures to the minimal compactification. As a consequence, every holomorphic map from a complex algebraic variety into such a variety is algebraic. In this talk, I shall discuss a p-adic version of the above result. For Shimura varieties of abelian type, this is joint work with Shankar, Zhu and Patel and for exceptional Shimura varieties this is based on ongoing work with Bakker, Shankar and Yao.