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

Abstract

We prove that the family I F 0 \mathcal {I}_{F_0} of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with F 0 F_0 -rational j j -invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family I d 0 \mathcal {I}_{d_0} of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree d 0 d_0 j j -invariant is typically bounded in torsion.