The distribution of semi-integral points on a class of singular cubic hypersurfaces

TL;DR

This paper derives an asymptotic formula for semi-integral points on a specific class of singular cubic hypersurfaces and confirms its consistency with Manin's conjecture regarding geometric invariants.

Contribution

It provides the first asymptotic count for semi-integral points on these hypersurfaces and verifies the conjectured invariants align with Manin's predictions.

Findings

Asymptotic formula matches Manin's conjecture for the hypersurfaces.

Confirmed the $a$- and $b$-invariants agree with theoretical predictions.

Established distribution patterns of semi-integral points on singular cubic hypersurfaces.

Abstract

Let $k$ be a positive integer and let $X_k$ be the cubic hypersurface defined by the equation $x^3-(y_1^2+\cdots+y_{4k}^2)z=0$. In this paper, we give an asymptotic formula for the counting function of semi-integral points on $X_k$. We also prove that this asymptotic formula agrees with Manin's conjecture for $\mathcal{M}$-points \cite[Conjecture~1.4]{Moe26a} on the $a$-invariant and the $b$-invariant.