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Uniform Bounds on S-Integral Torsion Points for $\mathbb{G}_m$ and Elliptic Curves

Jit Wu Yap

TL;DR

The paper establishes uniform bounds on the number of S-integral torsion points relative to a non-torsion point, first for the multiplicative group and then for elliptic curves over number fields. The authors develop a unified framework based on logarithmic and elliptic logarithmic equidistribution, leveraging Berkovich spaces, energy pairings, and quantitative height-discrepancy estimates to control how close torsion points can be to a fixed non-torsion point as base fields vary. Central to the results are explicit bounds from linear forms in logarithms (Laurent–Mignotte–Nesterenko) for G_m, and David–Hirata-Kohno-type elliptic Baker bounds (with stronger CM-case estimates) for elliptic curves, together with Favre–Rivera-Letelier quantitative equidistribution. The work yields uniform degree bounds for S-integral torsion points, with CM elliptic curves allowing stronger, exception-free statements, and provides a concrete framework potentially applicable to Drinfeld modules and to Ih’s conjecture on preperiodic points.

Abstract

Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or an elliptic curve $E$ defined over $K$, we establish uniformity results on the number of $S$-integral torsion points relative to a non-torsion point $β$, as $β$ varies over number fields of bounded degree. In particular for $\mathbb{G}_m$, if $D$ is a positive integer, we prove a uniform bound on the degree of a torsion point $ζ$ that is $S$-integral relative to a non-torsion point $β$ with degree $\leq D$.

Uniform Bounds on S-Integral Torsion Points for $\mathbb{G}_m$ and Elliptic Curves

TL;DR

The paper establishes uniform bounds on the number of S-integral torsion points relative to a non-torsion point, first for the multiplicative group and then for elliptic curves over number fields. The authors develop a unified framework based on logarithmic and elliptic logarithmic equidistribution, leveraging Berkovich spaces, energy pairings, and quantitative height-discrepancy estimates to control how close torsion points can be to a fixed non-torsion point as base fields vary. Central to the results are explicit bounds from linear forms in logarithms (Laurent–Mignotte–Nesterenko) for G_m, and David–Hirata-Kohno-type elliptic Baker bounds (with stronger CM-case estimates) for elliptic curves, together with Favre–Rivera-Letelier quantitative equidistribution. The work yields uniform degree bounds for S-integral torsion points, with CM elliptic curves allowing stronger, exception-free statements, and provides a concrete framework potentially applicable to Drinfeld modules and to Ih’s conjecture on preperiodic points.

Abstract

Let be a number field, a finite set of places. For or an elliptic curve defined over , we establish uniformity results on the number of -integral torsion points relative to a non-torsion point , as varies over number fields of bounded degree. In particular for , if is a positive integer, we prove a uniform bound on the degree of a torsion point that is -integral relative to a non-torsion point with degree .
Paper Structure (16 sections, 35 theorems, 144 equations)

This paper contains 16 sections, 35 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Statements and Results
  3. Overview of Proofs
  4. Acknowledgements
  5. Quantitative Equidistribution on $\mathbb{P}^1$
  6. The Berkovich Projective Line $\mathbb{P}^1_{\operatorname{Berk},v}$
  7. Quantitative Equidistribution following Favre--Rivera-Letelier
  8. Quantitative Logarithmic Equidistribution for $\mathbb{G}_m$
  9. Bounding distances to torsion points for $\mathbb{G}_m$
  10. Proofs of Theorems \ref{['IntroUniformIh1']} and \ref{['IntroUniformIh3']}
  11. Quantitative Equdistribution for Elliptic Curves
  12. The non-archimedean case.
  13. The archimedean case.
  14. Quantitative Logarithmic Equidistribution for Elliptic Curves
  15. Bounding distances to torsion points for $E$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $K$ be a number field and $S$ a finite set of places including all archimedean places. For each $\beta \in \overline{K}^{\times} \setminus \mu_{\infty}$, the set of $\zeta \in \mu_{\infty}$ such that $\zeta$ is $S$-integral relative to $\beta$ is finite.

Theorems & Definitions (65)

  • Theorem 1.1: Theorem 2.1, BIR08
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 2.1: Theorem 7, FRL06
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 55 more