$p$-adic hyperbolicity for moduli spaces of abelian motives

Abstract

We prove that Shimura varieties of abelian type satisfy a $p$-adic Borel-extension property over discretely valued fields. More precisely, let $\mathsf{D}$ denote the rigid-analytic closed unit disc and ${\mathsf{D}}^{\times }=\mathsf{D}\setminus \{0\}$, let $X$ be a smooth rigid-analytic variety, and let $S(G,\mathcal{H}{)}_{\mathsf{K}}$ denote a Shimura variety of abelian type with torsion-free level structure. We prove every rigid-analytic map defined over a discretely valued $p$-adic field ${\mathsf{D}}^{\times }\times X\to S(G,\mathcal{H}{)}_{\mathsf{K}}^{\text{an}}$ extends to an analytic map $\mathsf{D}\times X\to (S(G,\mathcal{H}{)}_{\mathsf{K}}^{\text{BB}}{)}^{\text{an}}$, where $S(G,\mathcal{H}{)}_{\mathsf{K}}^{\text{BB}}$ is the Baily-Borel compactification of $S(G,\mathcal{H}{)}_{\mathsf{K}}$. We also deduce various applications to algebraicity of analytic maps, degenerations of families of abeloids, and to $p$-adic notions of hyperbolicity. Along the way, we also prove an extension result for Rapoport-Zink spaces.