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# Mathematical Research Letters

## Volume 14 (2007)

### Number 5

### Abelian points on algebraic varieties

Pages: 731 – 743

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n5.a2

#### Author

#### 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 $\Q^{\ab}$-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.