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

Abstract

A smooth geometrically connected curve over the finite field [Formula: see text] with gonality [Formula: see text] has at most [Formula: see text] rational points. Faber and Grantham conjectured that there exist curves of every sufficiently large genus with gonality [Formula: see text] 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.