Statistics of Moduli Space of vector bundles II

Abstract

Let $X$ be a smooth irreducible projective curve of genus $g \geq 2$ over a finite field $\F_{q}$ of characteristic $p$ with $q$ elements such that the function field $\F_{q}(X)$ is a geometric Galois extension of the rational function field of degree $N.$ Consider $gcd(n,d)=1$, let $M_{L}(n,d)$ be the moduli space of rank $n$ stable vector bundles over $X$ with fixed determinant isomorphic to a $\mathbb F_q$-rational line bundle $L$. Suppose $N_q (M_L(n,d))$ denotes the cardinality of the set of $\F_{q}$-rational points of $M_{L}(n,d)$. We give an asymptotic bound of $\log(N_{q}(M_{L}(n,d)) - (n^2-1)(g-1)\log{q})$ for large genus $g,$ depending on $N$. Further, considering this logarithmic difference as a random variable, we prove a central limit theorem over a large family of hyperelliptic curves with uniform probability measure. Further, over the same family of hyperelliptic curves, we study the distribution of $\F_{q}$-rational points over the moduli space of rank $2$ stable vector bundles with trivial determinant $M^{s}_{\mathcal{O}_{H}}(2,0)$ and it's Seshadri desingularisation ${\widetilde{N}}$ by choosing an appropriate random variable in each case. We also see that the corresponding random variables having standard Gaussian distribution as $g$ and $q$ tends to infinity.