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

A smooth geometrically connected curve over the finite field [Formula: see text] with gonality has at most rational points. Faber and Grantham conjectured that there exist curves of every sufficiently large genus achieve this bound. In paper, we show bound can be achieved for an infinite sequence genera using abelian covers projective line. We also argue will not suffice to prove full conjecture.