Weak Approximation for Cubic Hypersurfaces of Large Dimension

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.