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 $\mathsf{ D}^{ \times} \times X \rightarrow S(G,\mathcal{ H})_{ \mathsf{ K}}^{ \textrm{ an}}$ extends to an analytic map $\mathsf{ D} \times X \rightarrow (S(G,\mathcal{ H})_{ \mathsf{ K}}^{ \textrm{ BB}})^{ \textrm{ an}}$, where $S(G,\mathcal{ H})_{ \mathsf{ K}}^{ \textrm{ BB}}$ 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.
Categories: math.NT math.AG