Typically bounding torsion

TL;DR

This paper introduces the concept of typical boundedness of torsion in families of abelian varieties over number fields, showing it holds for CM abelian varieties but not for elliptic curves, with results depending on fixed invariants.

Contribution

It formalizes the notion of typical boundedness of torsion, proving it for CM abelian varieties and analyzing its failure for elliptic curves with fixed invariants.

Findings

Torsion is typically bounded for all CM abelian varieties of fixed dimension.

Torsion is not typically bounded for all elliptic curves.

Results vary depending on fixed invariants like the degree of the j-invariant.

Abstract

We formulate the notion of \emph{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 \emph{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.