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Multi-height analysis of rational points of toric stacks

Nicolas Bongiorno

TL;DR

The paper extends the multi-height program for rational points from toric varieties to toric DM stacks by developing a universal stacky-torsor framework. It constructs and lifts the age pairing to universal torsors, defines an orbifold Picard group and its cone of effective divisors, and introduces a refined residue map via twisted sectors to control lifting obstructions. A counting formula is proved for toric stacks with torsion-free Picard, expressing the asymptotic with a Tamagawa-type orbifold measure and linking it to the extended universal torsor parametrization. The work also provides explicit parametrizations and local-height definitions in the extended torsor setting, enabling a coherent arithmetic of rational points on toric stacks and offering tools for future Manin-type conjectures in the orbifold context.

Abstract

We study the multi-height distribution of rational points of toric stacks over $\mathbf{Q}$ using the lift of the number of points to an extended universal torsor.

Multi-height analysis of rational points of toric stacks

TL;DR

The paper extends the multi-height program for rational points from toric varieties to toric DM stacks by developing a universal stacky-torsor framework. It constructs and lifts the age pairing to universal torsors, defines an orbifold Picard group and its cone of effective divisors, and introduces a refined residue map via twisted sectors to control lifting obstructions. A counting formula is proved for toric stacks with torsion-free Picard, expressing the asymptotic with a Tamagawa-type orbifold measure and linking it to the extended universal torsor parametrization. The work also provides explicit parametrizations and local-height definitions in the extended torsor setting, enabling a coherent arithmetic of rational points on toric stacks and offering tools for future Manin-type conjectures in the orbifold context.

Abstract

We study the multi-height distribution of rational points of toric stacks over using the lift of the number of points to an extended universal torsor.
Paper Structure (56 sections, 96 theorems, 352 equations)

This paper contains 56 sections, 96 theorems, 352 equations.

Table of Contents

  1. Introduction
  2. Terminology
  3. Outline of the article
  4. Acknowledgements
  5. Stacky integral points and the residue map
  6. Global extension of a rational point of a nice stack
  7. The stacky Hensel's lemma and its interpretations
  8. Twisted sectors and the residue map
  9. The cyclotomic inertia stack
  10. Relation with the inertia stack
  11. The residue map
  12. The age pairing and the orbifold Picard group
  13. The age pairing
  14. The numerical value associated to the age pairing
  15. The orbifold Picard group
  16. ...and 41 more sections

Key Result

Theorem 1.1

We assume that $X$ is a toric stack over $\mathbf{Q}$. Let $\mathrm{D}_1$ be a compact polyhedron of $\mathop{\mathrm{Pic}}\nolimits_{\mathop{\mathrm{orb}}\nolimits}(X)^{\vee}_{\mathbf{R}}$ and $u$ be an element of the interior of the dual of the effective cone $(\text{C}_{\text{eff},\text{orb}}(X)^ Then the multi-height asymptotic behaviour is of the form: where $\tau_{\mathop{\mathrm{orb}}\noli

Theorems & Definitions (210)

  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • ...and 200 more