Uniform bounds on S-integral torsion points for 𝔾m and elliptic curves

Let K be a number field, S a finite set of places. For [Formula: see text] 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 [Formula: see text], as [Formula: see text] varies over number fields of bounded degree. In particular for [Formula: see text], if D is a positive integer, we prove a uniform bound on the degree of a torsion point [Formula: see text] that is S-integral relative to a non-torsion point [Formula: see text] with degree [Formula: see text].