Typically bounding torsion on elliptic curves isogenous to rational $j$-invariant

TL;DR

This paper proves that elliptic curves over number fields, which are isogenous to curves with rational or fixed-degree $j$-invariants, generally have bounded torsion, under certain conditions.

Contribution

It establishes the boundedness of torsion for families of elliptic curves isogenous to those with rational or fixed-degree $j$-invariants, extending understanding of torsion behavior.

Findings

Torsion is bounded in families isogenous to curves with rational $j$-invariant.

Torsion is bounded in families isogenous to curves with fixed degree $j$-invariant.

Results depend on additional uniformity assumptions.

Abstract

We prove that the family $\mathcal{I}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family $\mathcal{I}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.