Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

p-adic Borel extension for local Shimura varieties

Abhishek Oswal, Georgios Pappas

TL;DR

The work proves a $p$-adic Borel extension theorem for moduli spaces of $p$-adic shtukas with framing, covering all local Shimura varieties associated to local shtuka data $(G,[b],\{\mu\})$, including exceptional groups. The main technique reduces to the GL$_n$ case and leverages the $p$-adic Riemann–Hilbert correspondence to extend de Rham local systems across boundaries, together with crystallinity criteria to control Frobenius structures. This yields a $p$-adic Brody hyperbolicity statement: morphisms from rational curves into these spaces are constant, in both diamonds and rigid-analytic settings over discretely valued bases. The results extend prior work of Oswal–Shankar–Zhu–Patel on Rapoport–Zink spaces to the full family of local Shimura varieties and illuminate hyperbolicity phenomena in the $p$-adic analytic landscape, with further questions about non-discretely valued fields and function-field analogues.

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 $(G,[b],\{μ\})$, 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.

p-adic Borel extension for local Shimura varieties

TL;DR

The work proves a -adic Borel extension theorem for moduli spaces of -adic shtukas with framing, covering all local Shimura varieties associated to local shtuka data , including exceptional groups. The main technique reduces to the GL case and leverages the -adic Riemann–Hilbert correspondence to extend de Rham local systems across boundaries, together with crystallinity criteria to control Frobenius structures. This yields a -adic Brody hyperbolicity statement: morphisms from rational curves into these spaces are constant, in both diamonds and rigid-analytic settings over discretely valued bases. The results extend prior work of Oswal–Shankar–Zhu–Patel on Rapoport–Zink spaces to the full family of local Shimura varieties and illuminate hyperbolicity phenomena in the -adic analytic landscape, with further questions about non-discretely valued fields and function-field analogues.

Abstract

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

Paper Structure

This paper contains 16 sections, 17 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. The general set-up
  3. Main results
  4. Discussion of the proofs
  5. Organization
  6. Definitions and preliminaries
  7. General assumptions and notation
  8. Moduli of shtuka and local Shimura varieties
  9. $p$-adic period rings
  10. Shtukas and local systems
  11. Statements and proofs
  12. $p$-adic Borel extension
  13. $p$-adic Brody hyperbolicity
  14. The proof of the extension theorem
  15. The proof of Theorem \ref{['constantIntro']}:
  16. ...and 1 more sections

Key Result

Theorem 1.2.1

Every rigid analytic morphism $a^\times: \mathbb {B}\xspace^{\times s}\times\mathbb {B}\xspace^t \rightarrow \mathcal{M}\xspace_{G,b,\mu, K}$ over ${\mathrm{Sp}}(\breve E)$, extends uniquely to $a: \mathbb {B}\xspace^{s+t} \rightarrow \mathcal{M}\xspace_{G,b,\mu, K}$ over ${\mathrm{Sp}}(

Theorems & Definitions (31)

  • Theorem 1.2.1
  • Theorem 1.2.2
  • Theorem 1.2.3
  • Proposition 2.4.1
  • proof
  • Theorem 3.1.1
  • Corollary 3.1.2
  • Corollary 3.1.3
  • Corollary 3.1.4
  • Corollary 3.1.5
  • ...and 21 more