Typically bounding torsion on elliptic curves with rational $j$-invariant

Abstract

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 E F of elliptic curves over number fields with F -rational j-invariants is typically bounded in torsion. For any integer d ∈ Z + , we also strengthen a result on typically bounding torsion for the family E d of elliptic curves over number fields with degree d j-invariants.