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

Dmitry Faifman, Zeev Rudnick

Published 2008-03-25, updated 2008-05-13Version 2

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 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|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 2g|I| tends to infinity.