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

TL;DR

This work develops a $p$-adic extension theory for exceptional Shimura varieties and geometric period images, proving that rigid-analytic maps from $(\mathsf{D}^{\times})^a \times \mathsf{D}^b$ into $X_F^{\mathrm{an}}$ extend to $\mathsf{D}^{a+b}$ provided the image meets the good reduction locus, for large enough primes $p$. The authors bypass the lack of $p$-adic uniformizations by leveraging crystalline $p$-adic local systems, Fontaine–Laffaille structures, and the prismatization framework of analytic prismatic $F$-crystals, establishing a pointwise-constant-to-global-constant phenomenon via constancy results and a $p$-adic Riemann–Hilbert correspondence. A key technical achievement is turning local extension on one-dimensional disks into higher-dimensional polydisk extension using meromorphic extension arguments and Hartogs-type results in the rigid-analytic setting. As a corollary, rigid-analytic maps into these spaces with images in the good reduction locus are algebraic, yielding $p$-adic analogues of Borel extension and algebraicity results without relying on uniformizations, and the framework extends to Fontaine–Laffaille-type setup on formal schemes with positivity assumptions.

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 $f: (\mathsf{D}^{\times})^a \times \mathsf{D}^b \rightarrow X_F^{\textrm{an}}$ defined over $F$ whose image intersects the good reduction locus of $X_F^{\textrm{an}}$ (with respect to an integral canonical model) extends to a map $\mathsf{D}^{a+b}\rightarrow X_F^{\textrm{an}}$. 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.

Theorems & Definitions (64)

• Theorem : Borel extension
• Theorem : Borel algebraicity
• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Definition 3.1
• Theorem 3.2
• Lemma 3.3
• proof : Proof of \ref{['thm: all good or all bad']}