Cubic hypersurfaces and a version of the circle method for number fields
Tim Browning, Pankaj Vishe
TL;DR
This paper develops a version of the Hardy-Littlewood circle method tailored for number fields and proves that non-singular cubic hypersurfaces of dimension at least 8 over such fields always possess a rational point.
Contribution
It introduces a novel adaptation of the circle method for number fields and establishes the existence of rational points on high-dimensional cubic hypersurfaces.
Findings
Non-singular cubic hypersurfaces of dimension ≥8 over number fields have rational points.
A new circle method approach applicable to number fields is developed.
The method extends classical results over Q to more general number fields.
Abstract
A version of the Hardy-Littlewood circle method is developed for number fields K/Q and is used to show that non-singular projective cubic hypersurfaces over K always have a K-rational point when they have dimension at least 8.
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