Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The multi-height distribution implies the Batyrev-Manin principle

Nicolas Bongiorno

Abstract

We explain how to deduce from the multi-height analysis of rational points on a toric stack (respectively on a toric variety) the asymptotic study of the number of rational points of bounded orbifold anticanonical height (respectively bounded anticanonical height), using a general version of the hyperbola method developed by Marta Pieropan and Damaris Schindler.

The multi-height distribution implies the Batyrev-Manin principle

Abstract

We explain how to deduce from the multi-height analysis of rational points on a toric stack (respectively on a toric variety) the asymptotic study of the number of rational points of bounded orbifold anticanonical height (respectively bounded anticanonical height), using a general version of the hyperbola method developed by Marta Pieropan and Damaris Schindler.
Paper Structure (15 sections, 23 theorems, 117 equations)

This paper contains 15 sections, 23 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Background and previous results
  3. Terminology and the precise statement
  4. Outline of the article
  5. Acknowledgements
  6. Extension of the multi-height analysis to the case bounded above but not below
  7. Estimation of the multi-height behaviour with values in a simplicial subcone
  8. First estimate
  9. An upper bound
  10. Multi-height estimates for the application of the hyperbola method
  11. The hyperbola method
  12. Conclusion
  13. A key property of heights associated with effective line bundles
  14. The decomposition in simplicial subcones and a crucial upper bound
  15. The computation of the precise constant

Key Result

Theorem 1.2

Let $\mathrm{D}_1$ be a finite union of compact polyhedra of $\mathop{\mathrm{Pic}}\nolimits(X)^{\vee}_{\mathbf{R}}$, and let $u$ be an element of the interior of the dual of the effective cone $(\text{C}_{\text{eff}}(X)^{\vee})^{\circ}$. For a real number $B > 1$, we set: Then the multi-height asymptotic behaviour is of the form: where $\tau(X)$ is the Tamagawa number of $X$ (see bongiorno2024m

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • ...and 36 more