Rational Points on a Singular Twisted Cayley Cubic

Definitions and mathematical statement

Let \[ F(x,y,z,w)=x^3+2xyz+xw^2-y^2w+wz^2, \qquad S=\{[x:y:z:w]\in{\mathbb{P}}^3_{{\mathbb{Q}}}:F(x,y,z,w)=0\}. \] The supplied source describes $S$ as a twisted form of the Cayley cubic surface with four singularities. For a real number $B>1$, define \[ N(B)=\#\left\{(x,y,z,w)\in{\mathbb{Z}}^4: \begin{gathered} \max\{|x|,|y|,|z|,|w|\}\le B,\qquad x\ne0,\\ \gcd(|x|,|y|,|z|,|w|)=1,\qquad F(x,y,z,w)=0 \end{gathered} \right\}. \] Thus the boundary of the box is included, and primitive integer vectors are counted individually: a vector and its negative are distinct solutions.

Requested asymptotic. Determine explicit real constants $c_1,c_2>0$ and prove that \[ \lim_{B\to\infty}\frac{N(B)}{B(\log B)^3}=c_1+c_2, \] where $\log$ denotes the natural logarithm. The source requires $c_1$ to be the product of local densities with the value predicted by Peyre, and attributes $c_2$ to a Zariski-dense thin set of rational points on $S$.

Unresolved qualifications. The supplied record gives neither a formula nor normalization for the local densities, and does not identify the thin set or specify a counting decomposition defining its contribution. Consequently, the displayed limit is a precise total-count target, but the intended separate identification of $c_1$ and $c_2$ requires further source clarification, including compatibility with the signed-vector counting convention above. The original excerpt also begins by asking for a proper generalisation to higher dimensions; its antecedent and intended higher-dimensional family are absent, so that question cannot be formulated precisely from this record.

Short English statement

Count primitive integer solutions of the stated cubic equation in the closed box $[-B,B]^4$, with $x\ne0$, counting both signs separately. The requested leading term is $(c_1+c_2)B(\log B)^3$, with explicit positive constants attributed respectively to Peyre's local-density prediction and a Zariski-dense thin set. Their separate normalizations and the thin set are not specified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006468$, title Rational Points on a Singular Twisted Cayley Cubic, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems.
• Supplied source citation: Analytic Number Theory (2026), Oberwolfach Reports, pp. 2701--2764, DOI 10.4171/owr/2025/51. Source URL: https://doi.org/10.4171/owr/2025/51. The supplied literature-sources list labels this same reference "Oberwolfach report 51/2025."
• Attribution: the original excerpt explicitly says "Proposed by Tim Browning," although the structured proposer field is empty. The supplied proposed year is 2025.
• Literature assessment dated 2026-08-22: the supplied curation labels the requested asymptotic open and specifically identifies the thin-set contribution as unresolved. Its listed source is https://doi.org/10.4171/owr/2025/51. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its accompanying explanation limits the repairs to formula delimiters, notation, grammar, and typography, with mathematical content and scope preserved. This label does not establish mathematical correctness or independent source verification.