Abelian points on algebraic varieties

Abstract

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 Kab-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((f)), obtaining in particular a curve of genus 3 without ℚ((t)) ab-points. © International Press 2007.