Weak approximation for cubic hypersurfaces of large dimension. (English) Zbl 1368.11058

Swarbrick Jones, Mike

Algebra Number Theory 7, No. 6, 1353-1363 (2013).

Summary: 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, nonconical cubic hypersurfaces of dimension at least 17. The proof utilises the Hardy-Littlewood circle method and the fibration method.

MSC:

11G35	Varieties over global fields
11D25	Cubic and quartic Diophantine equations
11D72	Diophantine equations in many variables
11P55	Applications of the Hardy-Littlewood method
14G25	Global ground fields in algebraic geometry

Keywords:

cubic hypersurfaces; weak approximation; local-global principles; fibration method; circle method; many variables