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

Given a finite field $\mathbb{F}_{q}$, we study the distribution of the number of $\mathbb{F}_{q}$-points on (possibly singular) affine curves given by the polynomial equations of the form $C_{f} : y^{m} = f(x)$, where $f$ is randomly chosen from a fixed collection $\mathcal{F}(\mathbb{F}_{q})$ of polynomials in $\mathbb{F}_{q}[x]$ with fixed $m \geq 2$. Under some conditions, these equations are affine models of cyclic $m$-covers of the projective line. Previously, different authors obtained asymptotic results about distributions of points on curves associated to certain collections of polynomials $f$ defined by large degree of $f$ or large genus of the smooth, projective, and geometrically irreducible curves $\tilde{C}_{f}$ obtained from the affine equations $C_{f}$, when the degree or genus goes to infinity. We summarize their strategies as a lemma, which gives a sufficient condition on the number of polynomials in a fixed collection $\mathcal{F}(\mathbb{F}_{q})$ with prescribed values, that automatically gives the distribution of points on the affine curves associated to the collection. We give infinitely many new examples of collections $\mathcal{F}(\mathbb{F}_{q})$ which satisfy the sufficient condition and hence produce infinitely many new distributions when a certain invariant goes to infinity. The main object of this paper is to demonstrate how changing the invariant that one takes to infinity changes the resulting distribution of points on curves.