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 ℓ be a prime, q a prime power and consider the ensemble Hg,ℓ of ℓ ‐cyclic covers of PFq1 of genus g. We assume that q ≢0,1modℓ. If 2g+2ℓ−2 ≢0mod(ℓ−1)ordℓ(q), then Hg,ℓ is empty. Otherwise, the number of rational points on a random curve in Hg,ℓ distributes as ∑i=1q+1Xi as g→∞, where X1,…,Xq+1 are independent and identically distributed random variables taking the values 0 and ℓ with probabilities (ℓ−1)/ℓ and 1/ℓ, respectively. The novelty of our result is that it works in the absence of a primitive ℓ th root of unity, the presence of which was crucial in previous studies. [ABSTRACT FROM AUTHOR]