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

Abstract: Let $k$ be a positive integer and let $X_k$ be the cubic hypersurface defined by the equation $x<sup>3-(y_1<sup>2+\cdots+y_{4k}<sup>2)z=0$. 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.

• The paper presents an explicit asymptotic formula for counting semi-integral points on singular cubic hypersurfaces, linking results to Manin's conjecture.
• It utilizes advanced analytic number theory, including Dirichlet series and residue calculus, to derive error estimates and main term coefficients.
• The research bridges geometric resolution methods with arithmetic conditions, enhancing understanding of rational point distribution on singular varieties.

Asymptotic Distribution of Semi-integral Points on Singular Cubic Hypersurfaces

Overview and Context

The paper "The distribution of semi-integral points on a class of singular cubic hypersurfaces" (2605.22371) investigates the counting problem for semi-integral rational points of bounded height on a specific class of singular cubic hypersurfaces, denoted $X_k$, defined in projective space $\mathbb{P}_{\mathbb{Q}}^{4k+1}$ by the equation: $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$ The main result is a precise asymptotic formula for the counting function $N(B)$, which enumerates semi-integral points up to height $B$ (as defined via an appropriate height function) under a $p$-adic restriction on the coordinates. The paper also verifies the consistency of these asymptotics with Manin's conjecture for rational points, specifically for $\mathcal{M}$-points, referencing the parameterization and invariants in [Moe26a].

Technical Approach

Semi-integral Points and $\mathcal{M}$-Points

Semi-integral points are rational points subject to coprimality and local valuation conditions: $v_p(z) - v_p(x) \neq 1 \quad \forall p \notin S$ where $S$ is a finite set of primes and $\mathbb{P}_{\mathbb{Q}}^{4k+1}$0 denotes $\mathbb{P}_{\mathbb{Q}}^{4k+1}$1-adic valuation. Through a detailed geometric analysis—including resolutions of singularities and intersection theory—the semi-integral condition is aligned with the notion of $\mathbb{P}_{\mathbb{Q}}^{4k+1}$2-points on a smooth model $\mathbb{P}_{\mathbb{Q}}^{4k+1}$3 as developed in the recent $\mathbb{P}_{\mathbb{Q}}^{4k+1}$4-point theory [Moe26a].

Height Function and Counting Problems

For $\mathbb{P}_{\mathbb{Q}}^{4k+1}$5, the height is defined by: $\mathbb{P}_{\mathbb{Q}}^{4k+1}$6 with coordinates chosen to be coprime. The central counting function is: $\mathbb{P}_{\mathbb{Q}}^{4k+1}$7 The asymptotic behavior as $\mathbb{P}_{\mathbb{Q}}^{4k+1}$8 is derived using inclusion-exclusion, analytic number theory (including divisor sums, Dirichlet series, and Perron's formula), and refined error analysis.

Resolution of Singularities and Intersection Multiplicities

A blow-up procedure of $\mathbb{P}_{\mathbb{Q}}^{4k+1}$9 along appropriate loci yields $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$0, a smooth model with well-behaved divisors $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$1 and $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$2. Intersection multiplicities $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$3 for rational points $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$4 and divisors are precisely computed, facilitating the translation from arithmetic conditions to geometric ones. These calculations are instrumental in associating specific semi-integral conditions to $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$5-point definitions.

Analytic Estimates and Main Term Computation

The counting is reduced to evaluating sums of the shape: $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$6 where $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$7 counts representations of $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$8 as a sum of $$x^3 - (y_1^2 + \cdots + y_{4k}^2)z = 0, \quad k \in \mathbb{N}$$9 squares, and $N(B)$0 restricts to allowed $N(B)$1-adic interactions. The analytic machinery, including double Dirichlet series and their holomorphic properties, is leveraged to produce residues leading to the main asymptotic terms.

Main Results and Numerical Asymptotics

Explicit Asymptotic Formula

The principal result is an explicit asymptotic formula: $N(B)$2 where:

• $N(B)$3 is the $N(B)$4-th Bernoulli number,
• $N(B)$5 denotes the Riemann zeta function,
• $N(B)$6 is an explicitly computable Euler product (with local factors detailed in the paper's Proposition~\ref{Gp}).

This formula features a polynomial growth rate with a logarithmic factor, aligning with the expected behavior from Manin's conjecture for Fano varieties.

Agreement with Manin's Conjecture for $N(B)$7-points

The invariants in the asymptotic formula:

• $N(B)$8
• $N(B)$9

correspond precisely to Manin's predicted exponents for the polynomial and logarithmic terms in the counting function for rational points (as refined for $B$0-points in [Moe26a]). The methodology rigorously demonstrates the geometric and arithmetic consistency with the conjectural theory.

Refined Error Terms

The error terms $B$1 are obtained via careful analytic estimates; the paper succeeds in improving these bounds beyond prior methods (cf. [LWZ19]) by constructing suitable auxiliary functions and employing detailed residue calculus for the associated Dirichlet series.

Implications and Theoretical Significance

Advancement in Rational Point Theory

This work contributes substantial evidence toward the general validity of Manin's conjecture for classes of singular cubic hypersurfaces, extending its scope to semi-integral and $B$2-points. The explicit forms and rigorous matching of invariants strengthen the bridge between arithmetic geometry and analytic number theory.

Geometric Interpretation of Semi-integral Data

By converting $B$3-adic semi-integrality into intersection data on the blown-up models, the work illuminates the connection between arithmetic conditions and birational geometry. This is of note for ongoing research into rational points on singular or degenerate varieties, where geometric techniques are crucial for formulating and proving counting conjectures.

Computational Tools and Future Directions

The explicit calculation of local factors and Euler products offers practical algorithms for evaluating constants in analogous problems. The techniques demonstrated here can be adapted to other singular hypersurfaces or more general Campana-type and $B$4-point counting problems.

The results suggest that further development of orbifold and singular geometry, as well as advances in analytic number theory (especially the error analysis for multi-variable zeta functions), will be crucial in extending rational point counting results to broader classes of varieties.

Conclusion

This paper rigorously establishes the asymptotic formula for the counting function of semi-integral points on a family of singular cubic hypersurfaces, matching the predictions from Manin's conjecture for $B$5-points. Through precise analytic calculations, geometric resolution theory, and explicit computation of relevant invariants and local factors, it achieves both a theoretical alignment with conjectural frameworks and a practical improvement in error estimates. The work is an important contribution to the ongoing effort to understand rational and semi-integral point distributions on singular varieties, with implications for future research across arithmetic geometry and analytic number theory.