  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.
