Comptage de fibr\'es de Hitchin pour le groupe $\mathrm{SL}(n)$

TL;DR

This paper provides a formula for counting stable Hitchin bundles over finite fields for the group SL(n), linking it to covers of the base curve and deriving explicit formulas for various invariants of the moduli space.

Contribution

It introduces a new counting formula for Hitchin bundles over finite fields, connecting stable bundles over covers to those over the original curve, and computes key invariants of their moduli space.

Findings

Derived a counting formula for stable Hitchin bundles over finite fields.

Obtained explicit formulas for the number of points, Poincaré polynomial, and Euler characteristic of the moduli space.

Connected the counting problem to automorphic induction and the Hitchin fibration support theorem.

Abstract

Let $C$ be a smooth projective curve of genus $g$ over a finite field $\mathbb{F}_q$ and let $D$ be a divisor on $C$ of degree $>2g-2$. We assume that the characteristic of $\mathbb{F}_q$ is sufficiently large. Let $n$ be an integer and let $\beta$ be a line bundle on $C$ of degree $e$, coprime to $n$. We give a formula for the number of stable ($D$-twisted) Hitchin bundles over $C$ of rank $n$ and determinant $\beta$ in terms of the number of stable Hitchin bundles over $C'$ of rank $n/d$ and degree $e$ where $C'$ ranges over cyclic covers $C'$ of $C$ of degree $d$ dividing $n$. Using a work by Mozgovoy-O'Gorman, we derive a closed formula for the following invariants of the moduli space of ($D$-twisted) Hitchin bundles over $C$ of rank $n$, trace $0$ and determinant $\beta$: its number of points over finite extensions of $\mathbb{F}_q$, its $\ell$-adic Poincar\'e polynomial and its Euler-Poincar\'e characteristic. Our main tools are the fundamental lemma of automorphic induction and a support theorem for the relative cohomology of a local system on the Hitchin fibration for the group $\mathrm{GL}(n)$.