Typically bounding torsion

We formulate the notion of typical boundedness of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the family all abelian varieties defined over number fields of degree lying in a set of arbitrarily small density. We show that for each fixed g, torsion is typically bounded on the family of all g-dimensional CM abelian varieties. We show that torsion is not typically bounded on the family of all elliptic curves, and we establish results – some unconditional and some conditional – on typical boundedness of torsion of elliptic curves for which the degree of the j-invariant is fixed.