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Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Tanush Shaska

TL;DR

This work develops a comprehensive framework for counting rational and algebraic points of bounded weighted height in weighted projective spaces. By introducing a true weighted height $ rak h$, a normalized wgcd, and the Veronese morphism $oldsymbol o$, the authors derive precise asymptotics for points of degree $e$ over number fields, revealing a fundamental arithmetic sparsity captured by the factor $ ext{gcd}(q, ext{phi}(me))^{-1}$. They provide a cohomological interpretation of this sparsity via Kummer torsors, drawing an analogy to Brauer–Manin obstructions and connecting geometric degrees with arithmetic descent. The results refine a weighted version of the Batyrev–Manin conjecture for toric-like varieties and suggest broad implications for moduli theory, arithmetic geometry, and computation on weighted moduli spaces. Overall, the paper advances our understanding of how weighted-graded structures influence point distributions and height asymptotics in arithmetic geometry.

Abstract

This paper investigates the distribution of rational and algebraic points of bounded weighted height in weighted projective spaces over number fields. For a weighted projective space with weights q over a number field k of degree m, we derive an asymptotic formula for the count of such points, featuring a leading term D times X raised to m e Q, plus an error term, where e is the extension degree and Q is the sum of the weights. The constant D combines geometric aspects of the weights with an arithmetic obstruction given by the reciprocal of the gcd of the least common multiple of the weights and Euler's totient of m e. This obstruction stems from the non-surjectivity of the natural morphism from the weighted space to ordinary projective space on rational points, linked to nontrivial torsors under groups of roots of unity. We provide a cohomological interpretation, analogous to the Brauer-Manin obstruction. These findings refine a weighted version of the Batyrev-Manin conjecture and open avenues for applications in moduli theory and arithmetic geometry.

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

TL;DR

This work develops a comprehensive framework for counting rational and algebraic points of bounded weighted height in weighted projective spaces. By introducing a true weighted height , a normalized wgcd, and the Veronese morphism , the authors derive precise asymptotics for points of degree over number fields, revealing a fundamental arithmetic sparsity captured by the factor . They provide a cohomological interpretation of this sparsity via Kummer torsors, drawing an analogy to Brauer–Manin obstructions and connecting geometric degrees with arithmetic descent. The results refine a weighted version of the Batyrev–Manin conjecture for toric-like varieties and suggest broad implications for moduli theory, arithmetic geometry, and computation on weighted moduli spaces. Overall, the paper advances our understanding of how weighted-graded structures influence point distributions and height asymptotics in arithmetic geometry.

Abstract

This paper investigates the distribution of rational and algebraic points of bounded weighted height in weighted projective spaces over number fields. For a weighted projective space with weights q over a number field k of degree m, we derive an asymptotic formula for the count of such points, featuring a leading term D times X raised to m e Q, plus an error term, where e is the extension degree and Q is the sum of the weights. The constant D combines geometric aspects of the weights with an arithmetic obstruction given by the reciprocal of the gcd of the least common multiple of the weights and Euler's totient of m e. This obstruction stems from the non-surjectivity of the natural morphism from the weighted space to ordinary projective space on rational points, linked to nontrivial torsors under groups of roots of unity. We provide a cohomological interpretation, analogous to the Brauer-Manin obstruction. These findings refine a weighted version of the Batyrev-Manin conjecture and open avenues for applications in moduli theory and arithmetic geometry.

Paper Structure

This paper contains 24 sections, 19 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weil Height and Schanuel’s Result
  4. Weighted projective spaces and the Veronese morphism
  5. Weighted Height
  6. Weighted Greatest Common Divisor
  7. Finiteness and Primitive Points
  8. Weighted Primitive Points
  9. Weighted height versus Size
  10. Related Work on Counting Rational Points
  11. Counting Rational Points in Weighted Projective Spaces
  12. The Rational Case
  13. Extension to a General Number Field
  14. Counting Algebraic Points with Fixed Degree Extensions in Weighted Projective Spaces
  15. Preparatory Results
  16. ...and 9 more sections

Key Result

Lemma 1

For $\mathbf{q} = (q_0, \ldots, q_n)$, let $d = \gcd(q_0, \ldots, q_n)$ and $\phi: \mathbb{WP}_{\mathbf{q}}^n(k) \to \mathbb{P}^n(k)$ be the Veronese morphism defined by where $q = \operatorname{lcm}(q_0, \ldots, q_n)$. The map $\phi$ is finite, with degree $d_\phi = \frac{q^n d}{\prod_{i=0}^n q_i}$.

Figures (1)

  • Figure 1: The weighted $\phi$ and its extension to the algebraic closure.

Theorems & Definitions (47)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Example 1
  • ...and 37 more