Abelian points on algebraic curves

TL;DR

This paper investigates the existence of rational points over the maximal abelian extension on algebraic curves of various genera, providing explicit examples and counterexamples across different fields.

Contribution

It constructs explicit families of curves with or without rational points over the maximal abelian extension, advancing understanding of rational points on algebraic curves.

Findings

Explicit family of diagonal plane cubic curves with Q^{ab}-points

Genus one curve over Q with no K^{ab}-points for any number field K

Genus g curves with no Q^{ab}-points for g ≥ 4

Abstract

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K^{ab} of K. We give: (i) an explicit family of diagonal plane cubic curves with Q^{ab}-points, (ii) for every number field K, a genus one curve C_{/Q} with no K^{ab}-points, and (iii) for every g \geq 4 an algebraic curve C_{/Q} of genus g with no Q^{ab}-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)))^{ab}-points.