Abelian points on algebraic curves

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.