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 of K. We give: (i) an explicit family of diagonal plane cubic curves without Q-points, (ii) for every number field K, a genus one curve C/Q with no K -points, and (iii) for every g ≥ 4 an algebraic curve C/Q of genus g with no Q-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)) -points. Convention: All varieties over a field K are assumed to be nonsingular, projective and (as is especially important for what follows) geometrically irreducible.