LAUSR

A family F of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups E(F )[tors] of those elliptic curves E/F ∈ F can be made uniformly bounded after removing from F those whose number field degrees lie in a subset of Z with arbitrarily small upper density. For every number field F , we prove unconditionally that the family EF of elliptic curves defined over number fields and with F -rational j-invariant is typically bounded in torsion. For any integer d ∈ Z, we also strengthen a result on typically bounding torsion for the family Ed of elliptic curves defined over number fields and with degree d j-invariant.