Is there a (non-singular) genus-one curve $E$ over $\mathbb{Q}$ that is known to have no points over any solvable extension?
2026-04-04 12:09:59.1775304599
Is there a genus-one curve over $\mathbb{Q}$ with no points over any solvable extension?
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No, and depending on who you talk to, it is expected that no such curves exist.
The Çiperiani-Wiles theorem says that every genus one curve over $\mathbb{Q}$ with semistable Jacobian and local points everywhere must have a point over some solvable extension. As far as I know, both conditions are expected to be able to be removed, so that every genus one curve has a solvable point.
In fact, Mazur has speculated [Remark 4.6] that every genus one curve over $\mathbb{Q}$ should have a point over a metabelian extension.