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Galois orbits of torsion points near atoral sets

Vesselin Dimitrov, Philipp Habegger

Abstract

We prove that the Galois equidistribution of torsion points of the algebraic torus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form $\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficients that vanishes on the unit real $d$-torus in a set whose Zariski closure in $\mathbb{G}_m^d$ has codimension at least $2$. Our result includes a power saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih's integrality finiteness conjecture on torsion points for a class of atoral divisors of $\mathbb{G}_m^d$.

Galois orbits of torsion points near atoral sets

Abstract

We prove that the Galois equidistribution of torsion points of the algebraic torus extends to the singular test functions of the form , where is a Laurent polynomial having algebraic coefficients that vanishes on the unit real -torus in a set whose Zariski closure in has codimension at least . Our result includes a power saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih's integrality finiteness conjecture on torsion points for a class of atoral divisors of .

Paper Structure

This paper contains 22 sections, 45 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Ih's conjecture on integral torsion points
  4. Overview of the proof
  5. Final remarks
  6. Notation and preliminaries
  7. Quantitative Galois equidistribution for torsion points
  8. A univariate average
  9. A Galois conjugate near $1$
  10. Theorem of Mahler--Mignotte
  11. A repulsion property of the unit circle
  12. Averages over roots of unity
  13. Geometry of numbers
  14. A preliminary result
  15. Equidistribution
  16. ...and 7 more sections

Key Result

Theorem 1.1

For each essentially atoral $P\in \overline{\mathbb{Q}}[X_1^{\pm 1},\ldots,X_d^{\pm 1}]\backslash \{0\}$ there exists $\kappa >0$ with the following property. Suppose ${\boldsymbol{\zeta}}\in\mathbb{G}_m^d$ has finite order with $\delta({\boldsymbol{\zeta}})$ sufficiently large. Then $P({\boldsymbol as $\delta({\boldsymbol{\zeta}})\rightarrow\infty$, where the implicit constant depends only on $d$

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Theorem 2.1: Dobrowolski--Smyth
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 81 more