Curves with many points over finite fields: the class field theory approach

TL;DR

This paper uses class field theory to construct new algebraic curves over finite fields with many rational points, improving known bounds for certain genera and field sizes.

Contribution

It introduces a novel approach using class field theory to find curves with more points than previously known, focusing on coverings of genus three curves over small finite fields.

Findings

Improved lower bounds on the maximum number of points for certain genera and fields.

Constructed new examples of curves with many points over finite fields.

Enhanced understanding of the distribution of rational points on algebraic curves.

Abstract

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known bounds. More precisely, we improve the lower bounds on the maximal number of points $N_q(g)$ for many values of the genus $g$ and of the cardinality $q$ of the finite field $\mathbb F_q$, by looking at coverings of all genus three smooth projective curves over $\mathbb F_q$, for $q$ is an odd prime less than 19.