Statistics of Moduli Space of vector bundles II

TL;DR

This paper investigates the asymptotic behavior of the number of rational points on moduli spaces of vector bundles over algebraic curves over finite fields, establishing bounds, distributional results, and a central limit theorem.

Contribution

It provides new asymptotic bounds for rational point counts, proves a central limit theorem for these counts over hyperelliptic curves, and analyzes their distribution approaching Gaussian as genus and field size grow.

Findings

Asymptotic bounds for the logarithm of rational point counts

Central limit theorem for the distribution of point counts

Gaussian distribution of point counts as genus and field size increase

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.