A p-adic analogue of an algebraization theorem of Borel

Let S be a Shimura variety such that the connected components of the set of complex points S(C) are of the form D/Γ, where Γ is a torsion-free arithmetic group acting on the Hermitian symmetric domain D. Borel proved that any holomorphic map from any complex algebraic variety into S(C) is an algebraic map. In this talk I shall describe ongoing joint work with Ananth Shankar and Xinwen Zhu, where we prove a p-adic analogue of this result of Borel for compact Shimura varieties of abelian type.