p-adic Borel extension for local Shimura varieties

TL;DR

This paper proves a $p$-adic Borel extension theorem for moduli spaces of $p$-adic shtukas, including local Shimura varieties, extending previous results and establishing Brody hyperbolicity in the $p$-adic setting.

Contribution

It establishes a $p$-adic Borel extension property for local Shimura varieties, even for exceptional groups, generalizing prior work on Rapoport-Zink spaces.

Findings

Moduli spaces of $p$-adic shtukas satisfy a $p$-adic Borel extension theorem.

All local Shimura varieties satisfy a $p$-adic Brody hyperbolicity.

Extension applies to all local Shimura data, including exceptional groups.

Abstract

We show that the moduli spaces of Scholze's $p$-adic shtukas with framing satisfy a $p$-adic rigid analytic version of Borel's extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G,[b],\{\mu\})$, even for exceptional groups $G$, and extends work of Oswal-Shankar-Zhu-Patel who proved a $p$-adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a $p$-adic rigid analytic version of Brody hyperbolicity.