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We study fluctuations in the number of points of $\ell$-cyclic covers of theprojective line over the finite field $\mathbb{F}_q$ when $q \equiv 1 \mod\ell$ is fixed and the genus tends to infinity. The distribution is given as asum of $q+1$ i.i.d. random variables. This was settled for hyperelliptic curvesby Kurlberg and Rudnick, while statistics were obtained for certain componentsof the moduli space of $\ell$-cyclic covers by Bucur, David, Feigon andLal\'{i}n. In this paper, we obtain statistics for the distribution of thenumber of points as the covers vary over the full moduli space of $\ell$-cycliccovers of genus $g$. This is achieved by relating $\ell$-covers to cyclicfunction field extensions, and counting such extensions with prescribedramification and splitting conditions at a finite number of primes.

The distribution of $\mathbb{F}_q$-points on cyclic $\ell$-covers of genus $g$