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Multi-height analysis of rational points of toric varieties

Nicolas Bongiorno

Abstract

We study the multi-height distribution of rational points of smooth, projective and split toric varieties over $\mathbf{Q}$ using the lift of the number of points to universal torsors.

Multi-height analysis of rational points of toric varieties

Abstract

We study the multi-height distribution of rational points of smooth, projective and split toric varieties over using the lift of the number of points to universal torsors.

Paper Structure

This paper contains 23 sections, 36 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Terminology and the precise statement
  3. Arakelov heights
  4. The multi-height conjecture
  5. The Manin-Peyre conjecture in the case of toric varieties
  6. The notion of universal torsors
  7. Generalities about universal torsors
  8. Local heights defined over the universal torsors
  9. Universal torsors for split, smooth and projective toric varieties
  10. An alternative description of the system of heights over toric varieties
  11. The action of the Neron-Severi torus on each coordinate
  12. A natural covering of the real variety associated to our toric variety
  13. Measure over the universal torsors and Tamagawa number
  14. Adelic measure and Tamagawa number over a variety
  15. Adelic measure over universal torsors
  16. ...and 8 more sections

Key Result

Theorem 2.15

Let $\mathrm{D}_1$ be a finite union of compact polyhedrons in $\mathop{\mathrm{Pic}}\nolimits(X)^{\vee}_{\mathbf{R}}$ and $u$ be an element of the interior of the dual of the effective cone $(C_{\mathop{\mathrm{eff}}\nolimits}(X)^{\vee})^{\circ}$. For a real number $B > 1$, we set: Let $\varepsilon \in ]0,1-\frac{1}{l}[$. Then we have an asymptotic behaviour of the form: where $\nu$ is the meas

Theorems & Definitions (83)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 73 more