On abelian covers of the projective line with fixed gonality and many rational points

TL;DR

This paper demonstrates that for infinitely many genera, abelian covers of the projective line can achieve the maximum number of rational points predicted by gonality constraints, but may not resolve the full conjecture.

Contribution

It constructs infinite sequences of curves attaining the maximum rational points bound via abelian covers and discusses limitations of this approach for the conjecture.

Findings

Achieves the rational points bound for infinitely many genera using abelian covers.

Shows abelian covers may not suffice to prove the full conjecture.

Provides insights into the limitations of abelian covers in this context.

Abstract

A smooth geometrically connected curve over the finite field $\mathbb{F}_q$ with gonality $\gamma$ has at most ${\gamma(q+1)}$ rational points. The first author and Grantham conjectured that there exist curves of every sufficiently large genus with gonality $\gamma$ that achieve this bound. In this paper, we show that this bound can be achieved for an infinite sequence of genera using abelian covers of the projective line. We also argue that abelian covers will not suffice to prove the full conjecture.