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

TL;DR

This paper establishes a $p$-adic hyperbolicity property for Shimura varieties of abelian type, showing that certain rigid-analytic maps extend over punctured discs, with implications for algebraicity, degenerations, and hyperbolicity in $p$-adic geometry.

Contribution

It proves a $p$-adic Borel-extension property for Shimura varieties of abelian type and extends this to Rapoport-Zink spaces, advancing understanding of $p$-adic hyperbolicity and degenerations.

Findings

Rigid-analytic maps over punctured discs extend to the Baily-Borel compactification.

Applications to algebraicity of analytic maps and degenerations of abeloids.

Establishment of a $p$-adic hyperbolicity framework for Shimura varieties.

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 $\mathsf{D}^{\times} \times X \rightarrow S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{an}}$ 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.