The Distribution of F-q-Points on Cyclic l-Covers of Genus g

We study fluctuations in the number of points of l-cyclic covers of the projective line over the finite field F-q when q = 1 mod l is fixed and the genus tends to infinity. The distribution is given as a sum of q + 1 i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick [7], while statistics were obtained for certain components of the moduli space of l-cyclic covers in [1]. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of l-cyclic covers of genus g. This is achieved by relating l-covers to cyclic function field extensions, and counting such extensions with prescribed ramification and splitting conditions at a finite number of primes.