Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence

Jiahong Yu

TL;DR

This work constructs a sheafified p-adic Riemann–Hilbert correspondence over smooth adic spaces via the de Rham period ring $\mathbb{B}_{\mathrm{dR}}^+$, proving a canonical isomorphism $R^1\nu_*\big(\mathrm{GL}_r(\mathbb{B}_{\alpha,\overline{X}})\big) \cong \mathrm{MIC}_r(X)\{-1\}$ and linking $\mathbb{B}_{\mathrm{dR}}^+$-local systems on $\overline{X}$ to integrable connections on $X$; α=1 recovers Heuer’s p-adic Simpson correspondence. The paper defines moduli prestacks for $v$-vector bundles with $\mathbb{B}_{\mathrm{dR}}^+/({\ker\theta})^\alpha$-coefficients and integrable connections, proving they are small $v$-stacks, thereby enabling moduli-theoretic treatment of p-adic RH data. A central technical advance is the decompletion of toric towers over de Rham period rings via Sen theory, establishing a canonical equivalence between continuous semi-linear $\Gamma$-actions on a perfectoid limit and locally analytic actions on the algebra with toric chart, while identifying obstructions for $\alpha>1$ due to action mismatches. Collectively, these results provide evidence for the geometric p-adic RH conjecture, extend the p-adic Simpson framework, and furnish a robust toolkit for nonabelian p-adic Hodge theory and moduli problems in the adic setting.

Abstract

Let $C$ be a completely algebraic closed non-archimedean field over $\mathbb{Q}_p$ and $α,r$ be two positive integers. Denote by $B_α$ the ring $\mathbb{B}_{\mathrm{dR}}^+(C)/(\kerθ)^α$. This paper first constructs a sheafified $p$-adic Riemann--Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on $X_{\mathrm{\acute{e}t}}$, \[ R^1ν_*\big( \mathrm{GL}_r(\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α) \big) \cong \mathrm{MIC}_{r}(X)\{-1\}, \] where the first term is identified with the sheaf of isomorphism classes of $v$-vector bundles with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α$, and the second term is defined as the sheaf of isomorphism classes of integrable connections of rank $r$. We then define the moduli space of integrable connections on $X$ and the moduli space of $v$-vector bundles on $X$ with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\kerθ)^α$, and prove that they are small $v$-stacks in the sense of Scholze. These constructions generalize Heuer's work on $p$-adic Simpson correspondence.

On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence

TL;DR

This work constructs a sheafified p-adic Riemann–Hilbert correspondence over smooth adic spaces via the de Rham period ring , proving a canonical isomorphism and linking -local systems on to integrable connections on ; α=1 recovers Heuer’s p-adic Simpson correspondence. The paper defines moduli prestacks for -vector bundles with -coefficients and integrable connections, proving they are small -stacks, thereby enabling moduli-theoretic treatment of p-adic RH data. A central technical advance is the decompletion of toric towers over de Rham period rings via Sen theory, establishing a canonical equivalence between continuous semi-linear -actions on a perfectoid limit and locally analytic actions on the algebra with toric chart, while identifying obstructions for due to action mismatches. Collectively, these results provide evidence for the geometric p-adic RH conjecture, extend the p-adic Simpson framework, and furnish a robust toolkit for nonabelian p-adic Hodge theory and moduli problems in the adic setting.

Abstract

Let be a completely algebraic closed non-archimedean field over and be two positive integers. Denote by the ring . This paper first constructs a sheafified -adic Riemann--Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on , where the first term is identified with the sheaf of isomorphism classes of -vector bundles with coefficients in , and the second term is defined as the sheaf of isomorphism classes of integrable connections of rank . We then define the moduli space of integrable connections on and the moduli space of -vector bundles on with coefficients in , and prove that they are small -stacks in the sense of Scholze. These constructions generalize Heuer's work on -adic Simpson correspondence.
Paper Structure (27 sections, 86 theorems, 309 equations)

This paper contains 27 sections, 86 theorems, 309 equations.

Table of Contents

  1. Introduction
  2. Geometric $p$-adic Riemann--Hilbert correspondence
  3. Sheafified $p$-adic Simpson correspondence and Riemann--Hilbert correspondence
  4. Moduli spaces
  5. Decompleting toric towers over a de Rham period ring
  6. Acknowledgement
  7. Notations
  8. Foundations on adic spaces
  9. Foundations on functional analysis over Tate rings
  10. The étale morphisms and the étale topology of adic spaces
  11. Smooth morphisms and complete differential modules
  12. Special cases: strongly noetherian spaces and smoothoid spaces
  13. Foundations of smooth adic spaces over $B_{\alpha}$
  14. The de Rham spectra of perfectoid spaces
  15. Properties of algebras with a toric chart
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $r$ be a positive integer. There is a canonical isomorphism of sheaves where Moreover, when $\alpha=1$, this isomorphism coincides with Heuer's sheafified $p$-adic Simpson correspondence (cf. heuer2022moduli).

Theorems & Definitions (225)

  • Conjecture 1: Geometric $p$-adic Riemann--Hilbert correspondence
  • Remark
  • Remark
  • Theorem 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 1.7: Proposition \ref{['prop:locallyan']}
  • ...and 215 more