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

TL;DR

This paper proves that families of elliptic curves with rational $j$-invariant over number fields are typically bounded in torsion, meaning their torsion subgroups can be uniformly bounded outside a small subset of field degrees.

Contribution

It establishes unconditionally that elliptic curves with rational $j$-invariant over any number field are typically bounded in torsion, extending previous results to a broader family.

Findings

Families with rational $j$-invariant are typically bounded in torsion.

The result applies unconditionally to all number fields.

Strengthens bounds for elliptic curves with fixed degree $j$-invariant.

Abstract

A family $\mathcal{F}$ 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 $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $\mathcal{E}_d$ of elliptic curves defined over number fields and with degree $d$ $j$-invariant.