On the distribution of the rational points on cyclic covers in the absence of roots of unity

TL;DR

This paper investigates the distribution of rational points on cyclic covers of the projective line over finite fields, revealing a probabilistic model that applies even without roots of unity, extending previous results.

Contribution

It establishes the distribution of rational points on abelian covers without requiring primitive roots of unity, broadening the scope of prior work.

Findings

Distribution of rational points as sum of i.i.d. variables

Conditions for the ensemble alH_{g,\u03bb} to be non-empty

Probabilistic model valid in absence of roots of unity

Abstract

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: Let $\ell$ be a prime, $q$ a prime power and consider the ensemble $\mathcal{H}_{g,\ell}$ of $\ell$-cyclic covers of $\mathbb{P}^1_{\mathbb{F}_q}$ of genus $g$. We assume that $q\not\equiv 0,1\mod \ell$. If $2g+2\ell-2\not\equiv0\mod (\ell-1){\rm ord}_\ell(q)$, then $\mathcal{H}_{g,\ell}$ is empty. Otherwise, the number of rational points on a random curve in $\mathcal{H}_{g,\ell}$ distributes as $\sum_{i=1}^{q+1} X_i$ as $g\to \infty$, where $X_1,\ldots, X_{q+1}$ are i.i.d.\ random variables taking the values $0$ and $\ell$ with probabilities $\frac{\ell-1}{\ell}$ and $\frac{1}{\ell}$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$-th-root of unity, the presence of which was crucial in previous studies.