Abelian Points on Algebraic Varieties

We attempt to determine which classes of algebraic varieties over Q must have points in some abelian extension of Q. We give: (i) for every odd d > 1, an explicit family of degree d, dimension d − 2 diagonal hypersurfaces without Qab-points, (ii) for every number field K, a genus one curve C/Q with no K ab-points, and (iii) for every g ≥ 4 an algebraic curve C/Q of genus g with no Qab-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without Q((t))ab-points. Convention: When we speak of a curve, a surface or a variety over a field K, we shall require it to be nonsingular, projective and (most important of all for what follows) geometrically irreducible. However, by a hypersurface we mean the closed subscheme of projective space defined by any homogeneous polynomial.