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

TL;DR

This paper proves uniform bounds on the number and degree of $S$-integral torsion points on $G_m$ and elliptic curves over number fields, relative to a fixed non-torsion point, as the field degree varies.

Contribution

It establishes the first uniform bounds on $S$-integral torsion points for $G_m$ and elliptic curves over number fields, extending previous non-uniform results.

Findings

Uniform bound on the degree of $S$-integral torsion points for $G_m$

Bound on the number of $S$-integral torsion points relative to a non-torsion point

Results hold uniformly over number fields of bounded degree

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 $\beta$, as $\beta$ 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 $\zeta$ that is $S$-integral relative to a non-torsion point $\beta$ with degree $\leq D$.