Statistics of moduli space of vector bundles

Abstract

Let X be a smooth projective curve of genus g≥1 over a finite field Fq with q elements, such that the function field Fq(X) is a geometric Galois extension of the rational function field Fq(x) with N=#Gal(Fq(X)/Fq(x)). Let ML(2,1) be the moduli space of rank 2 stable vector bundles over X‾=X×FqFq‾ with fixed determinant L, where L is a line bundle on X‾ of degree 1. Let Nq(ML(2,1)) be the cardinality of the set of Fq-rational points of ML(2,1). We give an estimate of Nq(ML(2,1)) in terms of N,q and g. Further we study the fluctuations of the quantity logNq(ML(2,1))−3(g−1)logq as the curve X as well as L varies over a large family of hyperelliptic curves (N=2) of genus g≥2. We find the limiting distribution of Nq(ML(2,1))−3(g−1)log⁡q as g grows and q is fixed, in terms of its characteristic function. When both g and q grow we see that q3/2(log⁡Nq(ML(2,1))−3(g−1)log⁡q) has a standard Gaussian distribution.