p-adic hypoerbolicity for Shimura varieties and period images

2025/09/29 by Bakker, Benjamin, Oswal, Abhishek, Shankar, Ananth N. +1
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.25461

Abstract

We prove that Shimura varieties and geometric period images satisfy a p-adic extension property for large enough primes p. More precisely, let D^×⊂ 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: (D^×)^a × D^b → X_F^\textrman defined over F whose image intersects the good reduction locus of X_F^\textrman (with respect to an integral canonical model) extends to a map D^a+b→ X_F^\textrman. 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.

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