ON THE DISTRIBUTION OF THE RATIONAL POINTS ON CYCLIC COVERS IN THE ABSENCE OF ROOTS OF UNITY

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let (Formula presented.) be a prime, (Formula presented.) a prime power and consider the ensemble (Formula presented.) of (Formula presented.) -cyclic covers of (Formula presented.) of genus (Formula presented.). We assume that (Formula presented.). If (Formula presented.), then (Formula presented.) is empty. Otherwise, the number of rational points on a random curve in (Formula presented.) distributes as (Formula presented.) as (Formula presented.), where (Formula presented.) are independent and identically distributed random variables taking the values (Formula presented.) and (Formula presented.) with probabilities (Formula presented.) and (Formula presented.), respectively. The novelty of our result is that it works in the absence of a primitive (Formula presented.) th root of unity, the presence of which was crucial in previous studies.