Mathematical Research Letters

Volume 14, Issue 5, September 2007 pp. 731-743.

Abelian points on algebraic varieties

Authors: Pete L. Clark
Author institution: University of Georgia

Summary: 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.

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