Statistics of Moduli Spaces of vector bundles over hyperelliptic curves

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 ${\mathbb{F}}_q\mathrm{.}$ 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.