Weak Approximation for Cubic Hypersurfaces of Large Dimension

TL;DR

This paper proves weak approximation for high-dimensional smooth cubic hypersurfaces over number fields, using advanced analytic and geometric methods, extending understanding of rational points on such varieties.

Contribution

It establishes weak approximation for smooth cubic hypersurfaces of dimension at least 17, broadening previous results to include singular cases and higher dimensions.

Findings

Weak approximation holds for smooth cubic hypersurfaces of dimension ≥17.

Uses Hardy-Littlewood circle method and fibration technique.

Extends results to hypersurfaces with arbitrary singular locus.

Abstract

We address the problem of weak approximation for general cubic hypersurfaces defined over number fields, with arbitrary singular locus. In particular, weak approximation is established for the smooth locus of projective, geometrically integral, non-conical cubic hypersurfaces, of dimension at least 17. The proof utilises the Hardy-Littlewood circle method, and the fibration method.