[Paper Review] Integral points of bounded height on toric varieties

Antoine Chambert-Loir, Yuri Tschinkel|arXiv (Cornell University)|Jun 16, 2010

Algebraic Geometry and Number TheoryMathematics34 references21 citations

TL;DR

This paper establishes an asymptotic formula for the number of integral points of bounded height on quasi-projective toric varieties over number fields, using adelic harmonic analysis and the Poisson summation formula on the associated torus. The leading term is explicitly computed and shown to be positive when integral points exist, with equidistribution of integral points to Peyre’s Tamagawa measure as a key result.

ABSTRACT

We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.

Motivation & Objective

• To derive an asymptotic formula for the number of integral points of bounded height on quasi-projective toric varieties over number fields.
• To extend the framework of height zeta functions and Tamagawa measures to non-projective toric varieties via universal torsors and adelic integration.
• To establish equidistribution of integral points to Peyre’s Tamagawa measure on the adelic points of the variety with Brauer–Manin obstruction removed.
• To compute the leading term of the asymptotic formula explicitly in terms of geometric and arithmetic invariants, including volumes, cohomology, and local strata data.
• To generalize previous results on rational points and integral points on toric varieties to the case of partial equivariant compactifications of tori.

Proposed method

• Apply the Poisson summation formula on the adelic torus associated to the split torus $T$ of the toric variety $X$.
• Use the framework of adelic harmonic analysis and Tamagawa measures to analyze the height zeta function of the variety.
• Employ the parametrization of integral points via universal torsors to reduce the counting problem to lattice point counting on the torsor.
• Compute local Fourier transforms at finite and archimedean places, and analyze their product via the adelic Poisson formula.
• Integrate the Fourier transforms over the adelic torus, identifying the leading pole and its residue to extract the main term.
• Use the analytic Clemens complex $\mathscr{C}^{\text{an}}_v(D)$ at archimedean places and the Galois module $\operatorname{EP}(U)$ to determine the order of the pole and the exponent $b$ in the asymptotic.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of integral points of bounded height on a quasi-projective toric variety over a number field?
• RQ2 How does the leading term in the asymptotic formula depend on geometric and arithmetic invariants of the variety and its boundary?
• RQ3 Can equidistribution of integral points be established with respect to Peyre’s Tamagawa measure on the adelic points of the variety?
• RQ4 What is the precise contribution of the log-anticanonical divisor and the boundary divisor to the asymptotic behavior?
• RQ5 How do the local volumes at archimedean places and the structure of the analytic Clemens complex influence the leading term?

Key findings

• The number of integral points of bounded height on $U$ grows asymptotically as $N(B) \sim \Theta \cdot B (\log B)^{b-1}$ as $B \to \infty$, where $b = r(\operatorname{EP}(U)) + \sum_{v|\infty} (1 + \dim \mathscr{C}^{\text{an}}_v(D))$.
• The leading constant $\Theta$ is explicitly computed and is positive whenever $U(F)$ contains at least one integral point.
• The constant $\Theta$ involves Tamagawa volumes of adelic subsets, local volumes of minimal strata of the boundary divisor $D$, characteristic functions of effective cones, and orders of Galois cohomology groups.
• Integral points on $U$ equidistribute to Peyre’s Tamagawa measure on $X(\mathbb{A}_F)^{\text{Br}(X)}$, the adelic points where the Brauer–Manin obstruction vanishes.
• The result refines classical equidistribution results for rational points in the case $U = X$, and extends them to integral points.
• The framework interpolates between counting rational points (via height zeta functions) and counting integral points (via torsors), unifying both approaches in the toric setting.