Abelian points on algebraic curves
Pete L. Clark

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Advanced Differential Equations and Dynamical Systems
