Statistics of moduli spaces of vector bundles over hyperelliptic curves

We give an asymptotic formula for the number of $F_q$-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 $F_q$. 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.