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On the non-Archimedean Hitchin map for $\mathrm{SL}_2(F)$

Jiahuang Chen, Siqi He

TL;DR

The paper develops a non-Archimedean Hitchin theory for SL$_2(F)$ by defining a Hitchin map from the non-Archimedean character variety to holomorphic quadratic differentials via equivariant harmonic maps into the Bruhat–Tits tree $T_F$. It proves the map is continuous and that its image lies in Jenkins–Strebel differentials, leveraging the simplicial nature of $T_F$ to control trajectory structure. A detailed analysis of bounded vs unbounded representations, length functions, and trace data yields a Hausdorffified character variety with a clear division between bounded and unbounded parts, and a dynamical dichotomy on the induced action on $T_F$. The work further links the image of the Hitchin map to JS theory and shows that unbounded irreducible representations lead to non-small actions on the tree, with density notions tied to Zariski/topological closures. Overall, the study provides a foundation for a non-Archimedean analogue of Hitchin fibrations and suggests connections to non-Archimedean Teichmüller–type compactifications.

Abstract

Let $F$ be a non-Archimedean valued field, $Σ$ a closed Riemann surface of genus at least two, and $Γ$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the $\mathrm{SL}_2(F)$-character variety $\mathcal{X}_F(Γ)$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $Σ$. We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $Γ$ on the Bruhat--Tits tree of $\mathrm{SL}_2(F)$ is never small.

On the non-Archimedean Hitchin map for $\mathrm{SL}_2(F)$

TL;DR

The paper develops a non-Archimedean Hitchin theory for SL by defining a Hitchin map from the non-Archimedean character variety to holomorphic quadratic differentials via equivariant harmonic maps into the Bruhat–Tits tree . It proves the map is continuous and that its image lies in Jenkins–Strebel differentials, leveraging the simplicial nature of to control trajectory structure. A detailed analysis of bounded vs unbounded representations, length functions, and trace data yields a Hausdorffified character variety with a clear division between bounded and unbounded parts, and a dynamical dichotomy on the induced action on . The work further links the image of the Hitchin map to JS theory and shows that unbounded irreducible representations lead to non-small actions on the tree, with density notions tied to Zariski/topological closures. Overall, the study provides a foundation for a non-Archimedean analogue of Hitchin fibrations and suggests connections to non-Archimedean Teichmüller–type compactifications.

Abstract

Let be a non-Archimedean valued field, a closed Riemann surface of genus at least two, and its fundamental group. Building on the theory of equivariant harmonic maps into -trees, we study the non-Archimedean Hitchin map from the -character variety , equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on . We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of on the Bruhat--Tits tree of is never small.

Paper Structure

This paper contains 26 sections, 45 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic differentials
  4. $\mathbb{R}$-trees
  5. The length function
  6. Bruhat--Tits trees
  7. Non-Archimedean valued fields
  8. The Bruhat--Tits tree $T_F$ of $\mathrm{SL}_2(F)$
  9. The $\mathrm{SL}_2(F)$-action on $T_F$
  10. Harmonic maps into $\mathbb{R}$-trees
  11. Equivariant harmonic maps into $\mathbb{R}$-trees
  12. Hopf differential and folding map
  13. The non-Archimedean Character variety and the Hitchin map
  14. Types of representations, the length function, and the trace
  15. The trace function
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}_F(\Gamma)$ be equipped with the non-Archimedean topology, and $H^0(\Sigma,\mathcal{K}_\Sigma^{\otimes 2})$ with the Euclidean topology. Then the Hitchin map $\Phi$ is continuous.

Figures (3)

  • Figure 1: Two-level Bruhat--Tits tree for $\mathrm{SL}_2(\mathbb{Q}_3)$ (each vertex has $4$ neighbors).
  • Figure 2: A Jenkins-Strebel differential on a genus-$2$ surface and its leaf space
  • Figure 3: Overlaping geodesics

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • ...and 82 more