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$p$-adic non-abelian Hodge theory for curves via moduli stacks

Ben Heuer, Daxin Xu

Abstract

For a smooth projective curve $X$ over $\mathbb C_p$ and any reductive group $G$, we show that the moduli stack of $G$-Higgs bundles on $X$ is a twist of the moduli stack of v-topological $G$-bundles on $X_v$ in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings' $p$-adic Simpson correspondence for $X$, which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of $p$-adic representations of $π_1(X)$ to an open substack of the stack of semi-stable Higgs bundles of degree $0$.

$p$-adic non-abelian Hodge theory for curves via moduli stacks

Abstract

For a smooth projective curve over and any reductive group , we show that the moduli stack of -Higgs bundles on is a twist of the moduli stack of v-topological -bundles on in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings' -adic Simpson correspondence for , which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of -adic representations of to an open substack of the stack of semi-stable Higgs bundles of degree .
Paper Structure (52 sections, 93 theorems, 231 equations)

This paper contains 52 sections, 93 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. The $p$-adic Simpson correspondence as a twisted isomorphism of moduli spaces
  3. Twisting for the moduli stack of v-topological $G$-torsors
  4. The twisted isomorphism of moduli stacks
  5. The constructible sheaf associated to a lift $\mathbb{X}$, and the exponential
  6. Representation variety
  7. Comparison to the small $p$-adic Simpson correspondence
  8. Comparison to non-abelian Hodge theory over $\mathbb C$ and $\mathbb F_p$
  9. Structure of the article
  10. Notation, conventions and recollections
  11. Setup
  12. Twists by torsors
  13. $G$-bundles
  14. Smoothoid spaces
  15. Higgs bundles on smoothoid spaces
  16. ...and 37 more sections

Key Result

Theorem 1.1.1

There is a natural Zariski-constructible v-sheaf $\mathbf{H}_{\mathbb{X}}\to {\mathbf A}$ that induces a canonical isomorphism Any choice of exponential for $K$ induces a section $\mathop{\mathrm{Exp}}\nolimits: {\mathbf A}(K)\to \mathbf{H}_{\mathbb{X}}(K)$ that induces a homeomorphism

Theorems & Definitions (241)

  • Theorem 1.1.1: \ref{['t:homeom-moduli']}
  • Theorem 1.1.2
  • Theorem 1.1.3: \ref{['t:moduli-spaces-circ']}
  • Theorem 1.3.1: \ref{['p:leray-seq-for-UJ']}
  • Definition 1.3.2
  • Theorem 1.3.3: \ref{['t:fCiso']}
  • Theorem 1.4.1: \ref{['c:exp-splits-L_X']}
  • Corollary 1.4.2
  • Theorem 1.5.2: \ref{['t:profet-locus-open']}
  • Theorem 1.7.1: \ref{['t:comparison-Higgs-stacks']}
  • ...and 231 more