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

2010/05/21 by Antonio Rojas‐León, Antonio Rojas-Leon, Rojas-Leon, Antonio
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1005.4078

openalex publication_date 2010/05/21 · arxiv created 2010/05/27 · arxiv updated 2010/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈\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.

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