How often does a cubic hypersurface have a rational point?

TL;DR

This paper investigates the frequency of rational points on cubic hypersurfaces over $Q$, establishing explicit proportions for the existence of rational points in high dimensions and conjecturing results for cubic surfaces.

Contribution

It provides explicit formulas for the proportion of cubic hypersurfaces with rational points for $n geq 4$, extending known results and proposing a conjecture for cubic surfaces.

Findings

Proportion of hypersurfaces with rational points approaches 1 for $n \\geq 9$

Recovers Heath-Brown's result for $n \\geq 10$

Proposes a conjecture for cubic surfaces in $\\mathbb{P}^3$

Abstract

A cubic hypersurface in $\mathbb{P}^n$ defined over $\mathbb{Q}$ is given by the vanishing locus of a cubic form $f$ in $n+1$ variables. It is conjectured that when $n \geq 4$, such cubic hypersurfaces satisfy the Hasse principle. This is now known to hold on average due to recent work of Browning, Le Boudec, and Sawin. Using this result, we determine the proportion of cubic hypersurfaces in $\mathbb{P}^n$, ordered by the height of $f$, with a rational point for $n \geq 4$ explicitly as a product over primes $p$ of rational functions in $p$. In particular, this proportion is equal to 1 for cubic hypersurfaces in $\mathbb{P}^n$ for $n \geq 9$; for $100\%$ of cubic hypersurfaces, this recovers a celebrated result of Heath-Brown that non-singular cubic forms in at least 10 variables have rational zeros. In the $n=3$ case, we give a precise conjecture for the proportion of cubic surfaces in $\mathbb{P}^3$ with a rational point.