Computer search for curves with many points among abelian covers of genus 2 curves

TL;DR

This paper employs class field theory to systematically identify abelian covers of genus 2 curves over finite fields, resulting in new examples of curves with many rational points for various field sizes.

Contribution

It introduces a computational method using class field theory to find curves with many points among abelian covers of genus 2 curves, expanding existing tables.

Findings

New curves with many points identified for fields of size 5,7,9,11,13,16

Enhanced database of curves with many points

Method applicable to other small genus curves

Abstract

Using class field theory one associates to each curve C over a finite field, and each subgroup G of its divisor class group, unramified abelian covers of C whose genus is determined by the index of G. By listing class groups of curves of small genus one may get examples of curves with many points; we do this for all curves of genus 2 over the fields of cardinality 5,7,9,11,13 and 16, giving new entries for the tables of curves with many points (www.manYPoints.org).