Counting rational points on cubic hypersurfaces

TL;DR

This paper establishes an upper bound on the number of rational points of bounded height on certain cubic hypersurfaces over the rationals, extending understanding of rational point distribution on these varieties.

Contribution

It provides a new upper bound of O(B^{D+ ext{epsilon}}) for rational points on cubic hypersurfaces with controlled singularities, generalizing previous results.

Findings

Bound applies to hypersurfaces with singular locus of dimension at most D-4

Number of rational points grows at most like B^{D+epsilon}

Constant depends on epsilon and the defining cubic form

Abstract

Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any \epsilon>0, we show that X contains O(B^{D+\epsilon}) rational points of height at most B. The implied constant in this estimate depends upon the choice of \epsilon and the coefficients of the cubic form defining X.