On a conjecture of Wooley and lower bounds for cubic hypersurfaces

2024/05/07 by V. Vinay Kumaraswamy, Kumaraswamy, V. Vinay, Nick Rome +1
Mathematics · #11D25 #11D45 #11N36 #11P55 #14G05 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2405.04234

openalex publication_date 2024/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X ⊂ P_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) ≫ B^n-9 if n ≥ 39, thereby proving a conjecture of T. D. Wooley for non-conical cubic hypersurfaces with large enough dimension.

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