Typically bounding torsion on elliptic curves isogenous to rational $j$-invariant

Abstract

We prove that the family ${\mathcal{I}}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family ${\mathcal{I}}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.

8 pages, to appear in Proceedings of the AMS