On a conjecture of Wooley and lower bounds for cubic hypersurfaces

TL;DR

This paper establishes lower bounds on the number of rational points of bounded height on large-dimensional non-degenerate cubic hypersurfaces, confirming a conjecture of Wooley for sufficiently high dimensions.

Contribution

It proves a lower bound for rational points on cubic hypersurfaces, confirming Wooley's conjecture for dimensions n ≥ 39.

Findings

Lower bound N(X,B) ≫ B^{n-9} for n ≥ 39

Confirms Wooley's conjecture for large enough n

Applicable to non-conical cubic hypersurfaces

Abstract

Let $X \subset \mathbf{P}_{\mathbf{Q}}^{n-1}$ be a cubic hypersurface cut out by the vanishing of a non-degenerate rational cubic form in $n$ variables. Let $N(X,B)$ denote the number of rational points on $X$ of height at most $B$. In this article we obtain lower bounds for $N(X,B)$ for cubic hypersufaces, provided only that $n$ is large enough. In particular, we show that $N(X,B) \gg B^{n-9}$ if $n \geq 39$, thereby proving a conjecture of T. D. Wooley for non-conical cubic hypersurfaces with large enough dimension.