Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2 g ∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/ π 2 )log (2 g ∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2 g ∣ℐ∣ tends to infinity.