Statistics of Moduli Spaces of vector bundles over hyperelliptic curves

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 $g\ge 2$ 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.

This is a corrected and vastly extended version of our previous submission "Statistics of Moduli Spaces of vector bundles II". In particular, the results on the Higgs bundles are new additions. arXiv admin note: substantial text overlap with arXiv:2309.15085