On the number of rational points on curves over finite fields with many automorphisms

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form y−y = f(x) with f ∈ Fqr [x], on which the additive group Fq acts, and Kummer curves of the form y q−1 e = f(x), which have an action of the multiplicative group Fq . In both cases we can remove a √ q factor from the Weil bound when q is sufficiently large.