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Asymptotic growth of translation-dilation orbits

Victor Y. Wang

TL;DR

The paper resolves Manin's conjecture for sufficiently split smooth equivariant compactifications X of the ax+b group G over \mathbb{Q} by intertwining geometry, height theory, and non-abelian harmonic analysis. Central to the approach is a Clausen-type family of multiple Dirichlet series that, after a careful geometric-parametric analysis of boundary divisors and their special members, yields the main term matching Peyre's constant ${\mathcal A}_{X,\mathsf{H}}$. The archimedean and non-archimedean analyses are coupled through a spectral expansion into Z_0 (main term) and Z_1 (non-abelian oscillations), with the former providing the log-power term and the latter being controlled to an acceptable error. While the leading term is obtained under splitness and SNC-type assumptions, secondary terms remain challenging in general, and certain examples demonstrate the method’s limits beyond the strictly split setting.

Abstract

By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain elusive in general.

Asymptotic growth of translation-dilation orbits

TL;DR

The paper resolves Manin's conjecture for sufficiently split smooth equivariant compactifications X of the ax+b group G over \mathbb{Q} by intertwining geometry, height theory, and non-abelian harmonic analysis. Central to the approach is a Clausen-type family of multiple Dirichlet series that, after a careful geometric-parametric analysis of boundary divisors and their special members, yields the main term matching Peyre's constant . The archimedean and non-archimedean analyses are coupled through a spectral expansion into Z_0 (main term) and Z_1 (non-abelian oscillations), with the former providing the log-power term and the latter being controlled to an acceptable error. While the leading term is obtained under splitness and SNC-type assumptions, secondary terms remain challenging in general, and certain examples demonstrate the method’s limits beyond the strictly split setting.

Abstract

By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain elusive in general.
Paper Structure (13 sections, 45 theorems, 292 equations)

This paper contains 13 sections, 45 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Background
  4. Geometry
  5. Heights
  6. Polar combinatorics
  7. Peyre's constant
  8. Spectral expansion
  9. New geometric and parametric observations
  10. Non-archimedean local calculations
  11. Archimedean endgame
  12. Final reductions
  13. Examples

Key Result

Theorem 1.1

Assume $X$ is strictly split (Definition DEFN:strictly-split-X). If $\mathsf{H}$ is a standard Weil height (Definition DEFN:class-of-standard-Weil-heights) associated to the anticanonical line bundle $K_X^{-1}$, then where $\mathcal{A}_{X,\mathsf{H}}>0$ is Peyre's constant EQN:Peyre-constant. Moreover, if $w\in C^\infty_c(\mathbb{R})$ and $B\ge 2$, then

Theorems & Definitions (107)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2: tanimoto2012height*Proposition 1.2
  • Definition 2.3
  • ...and 97 more