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

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.