Statistics of Moduli Spaces of vector bundles over hyperelliptic curves

TL;DR

This paper derives an asymptotic formula for counting rational points on moduli spaces of stable vector bundles over hyperelliptic curves over finite fields, analyzing error distribution and extending to related moduli spaces.

Contribution

It provides the first asymptotic count of rational points on these moduli spaces and studies error distribution over hyperelliptic curve families, extending to desingularizations and Higgs bundles.

Findings

Asymptotic formula for rational points over fixed determinant moduli spaces

Analysis of error term distribution across hyperelliptic curves

Extension of results to desingularized and Higgs moduli spaces

Abstract

We give an asymptotic formula for the number of $\mathbb{F}_{q}$-rational points over a fixed determinant moduli space of stable vector bundles of rank $r$ and degree $d$ over a smooth, projective curve $X$ of genus $g \geq 2$ defined over $\mathbb{F}_{q}.$ Further, we study the distribution of the error term when $X$ varies over a family of hyperelliptic curves. We then extend the results to the Seshadri desingularisation of the moduli space of semi-stable vector bundles of rank $2$ with trivial determinant, and also to the moduli space of rank $2$ stable Higgs bundles.