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

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 $\leq D$.