Weighted distribution of points on cyclic covers of the projective line over finite fields

摘要

For any prescribed finite field F-q and integers m = 2, we study the distribution of the number of F-q-points on (possibly singular) affine curves C-f :y(m) = f(x), where f is randomly chosen from a fixed collection F(F-q) of polynomials in F-q [x]. These equations are affine models of cyclic m-covers of the projective line. Previously, different authors obtained asymptotic results about distributions of F-q-points on curves associated to certain collections of polynomials f. We obtain analogous results for infinitely many new examples of collections F(F-q) and study how the asymptotic distribution depends on the choice of the collection of polynomials F(F-q). (C) 2019 Elsevier Inc. All rights reserved.