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

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

2010-05-21

arxiv.org/abs/1005.4078v2

Mathematics

Algebraic Geometry

Scientific paper

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.

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