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

TL;DR

This paper derives improved bounds for the number of rational points on certain algebraic curves over finite fields with large automorphism groups, using Weil descent to refine classical estimates.

Contribution

It introduces new bounds for rational points on Artin-Schreier and Kummer curves with large automorphism groups, reducing the classical Weil bound by a factor of  when the field size is large.

Findings

Bounds are tighter than Weil bounds for large q.

Applicable to Artin-Schreier and Kummer curves with automorphisms.

Reduces the  factor in the classical estimate.

Abstract

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 $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.