$p$-adic hyperbolicity for Shimura varieties and period images

Abstract

We prove that Shimura varieties and geometric period images satisfy a $p$-adic extension property for large enough primes $p$. More precisely, let ${\mathsf{D}}^{\times }\subset \mathsf{D}$ denote the inclusion of the closed punctured unit disc in the closed unit disc. Let $X$ be either a Shimura variety or a geometric period image with torsion-free level structure. Let $F$ be a discretely valued $p$-adic field containing the number field of definition of $X$, where $p$ is a large enough prime. Then, any rigid-analytic map $f:({\mathsf{D}}^{\times }{)}^a\times {\mathsf{D}}^b\to X_F^{\text{an}}$ defined over $F$ whose image intersects the good reduction locus of $X_F^{\text{an}}$ (with respect to an integral canonical model) extends to a map ${\mathsf{D}}^{a+b}\to X_F^{\text{an}}$. We note that this hypothesis is vacuous if $X$ is proper. We also deduce an application to algebraicity of rigid-analytic maps. Our methods also apply to the more general situation of the rigid generic fiber of formal schemes admitting Fontaine-Laffaile modules which satisfy certain positivity conditions.

We have now removed a good-reduction hypothesis for Shimura varieties