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On a conjecture of Wooley and lower bounds for cubic hypersurfaces
- Publication Year :
- 2024
-
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.<br />Comment: 66 pages. Updated bibliography
- Subjects :
- Mathematics - Number Theory
11D45, 11P55, 14G05, 11D25, 11N36
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2405.04234
- Document Type :
- Working Paper