Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rank 2 $\ell$-adic local systems and Higgs bundles over a curve

Hongjie Yu

TL;DR

This work proves Deligne's Lefschetz-type predictions for counting rank $2$ $\ell$-adic local systems on a curve over $\mathbb{F}_q$ with prescribed tame ramification, by equating Frobenius-fixed counts to Higgs-bundle moduli counts via the Langlands correspondence and the Arthur–Selberg trace formula. The authors develop a detailed spectral analysis (cuspidal, residual, continuous) for GL$_2$, compute trace contributions with Whittaker models and local intertwining operators, and translate the geometric side into counting parabolic Hitchin bundles, including a residue morphism and independence-of-weights results. A key outcome is that the fixed-point counts are Lefschetz-type functions of $k$, expressed through Higgs-moduli point counts $\mathrm{Higg}_{\mathfrak{R}}(k)$ and various Picard-type terms; in general position these counts reduce to $\mathrm{Higg}_{\mathfrak{R}}(k)$. The genus zero case exhibits an intriguing Simpson-like correspondence between Frobenius-fixed local systems and graded parabolic Higgs bundles, highlighting a Higgs–local-system duality in a function-field setting. The results extend to certain wild ramification scenarios and suggest a broader Lefschetz-type Lefschetz-type framework for arithmetic–geometric counts on moduli spaces of local systems and Higgs bundles.

Abstract

Let $X$ be a smooth, projective, and geometrically connected curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$ different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline{X}$ and $\overline{S}$ be their base changes to an algebraic closure of $\mathbb{F}_q$. We study the number of $\ell$-adic local systems $(\ell\neq p)$ in rank $2$ over $\overline{X}-\overline{S}$ with all possible prescribed tame local monodromies fixed by $k$-fold iterated action of Frobenius endomorphism for every $k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.

Rank 2 $\ell$-adic local systems and Higgs bundles over a curve

TL;DR

This work proves Deligne's Lefschetz-type predictions for counting rank -adic local systems on a curve over with prescribed tame ramification, by equating Frobenius-fixed counts to Higgs-bundle moduli counts via the Langlands correspondence and the Arthur–Selberg trace formula. The authors develop a detailed spectral analysis (cuspidal, residual, continuous) for GL, compute trace contributions with Whittaker models and local intertwining operators, and translate the geometric side into counting parabolic Hitchin bundles, including a residue morphism and independence-of-weights results. A key outcome is that the fixed-point counts are Lefschetz-type functions of , expressed through Higgs-moduli point counts and various Picard-type terms; in general position these counts reduce to . The genus zero case exhibits an intriguing Simpson-like correspondence between Frobenius-fixed local systems and graded parabolic Higgs bundles, highlighting a Higgs–local-system duality in a function-field setting. The results extend to certain wild ramification scenarios and suggest a broader Lefschetz-type Lefschetz-type framework for arithmetic–geometric counts on moduli spaces of local systems and Higgs bundles.

Abstract

Let be a smooth, projective, and geometrically connected curve defined over a finite field of characteristic different from and a subset of closed points. Let and be their base changes to an algebraic closure of . We study the number of -adic local systems in rank over with all possible prescribed tame local monodromies fixed by -fold iterated action of Frobenius endomorphism for every . In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.
Paper Structure (35 sections, 32 theorems, 389 equations)

This paper contains 35 sections, 32 theorems, 389 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. When $g=0$.
  4. Notation
  5. Global and local Langlands correspondence for $GL_2$
  6. Galois representations
  7. Automorphic representations
  8. Spectral side of the trace formula
  9. Explicit spectral decomposition of $J^{1}_{spec}(f)$
  10. Summary of main results of this section
  11. Counting $\ell$-adic local systems in rank 1
  12. Eulerian decomposition and calculations on Whittaker functions
  13. Cuspidal terms
  14. Residuel terms
  15. Continuous terms
  16. ...and 20 more sections

Key Result

Theorem 1.1.1

Suppose that $p\neq 2$. Suppose that Frobe is satisfied, so that $\mathrm{Frob}^\ast$ acts on $E_2(\mathfrak{R})$. Suppose that otherwise $E_2(\mathfrak{R})$ is empty. Then the function is of Lefschetz type. More precisely, we have the following explicit identities that express $\vert E_2(\mathfrak{R})^{\mathrm{Frob}^{*k}} \vert$ following different cases. In the above expressions $\alpha_{\ove

Theorems & Definitions (63)

  • Theorem 1.1.1
  • Remark 1.1.2
  • Corollary 1.1.3
  • Theorem 1.2.1
  • Proposition 3.1.1
  • proof
  • Theorem 3.1.2
  • proof
  • Theorem 3.2.1
  • proof
  • ...and 53 more