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

We study fluctuations in the number of points of $\ell $-cyclic covers of the projective line over the finite field ${\BBF}_q$ when $q \equiv 1\hbox { mod}\,\ell $ 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 ...