Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves

TL;DR

This paper establishes polynomial bounds on the torsion subgroup size of elliptic curves within a fixed geometric isogeny class over number fields, improving understanding of torsion growth relative to field degree.

Contribution

It provides explicit polynomial bounds on torsion points for elliptic curves in a fixed isogeny class over number fields, with bounds depending on the degree of the field.

Findings

Torsion points are bounded by a polynomial in the degree of the field.

The torsion subgroup size grows at most polynomially with field degree.

Results hold uniformly for all elliptic curves in a fixed isogeny class.

Abstract

We show there exist polynomial bounds on torsion of elliptic curves which come from a fixed geometric isogeny class. More precisely, for an elliptic curve $E_0$ defined over a number field $F_0$, for each $\epsilon>0$ there exist constants $c_\epsilon:=c_\epsilon(E_0,F_0),C_\epsilon:=C_\epsilon(E_0,F_0)>0$ such that for any elliptic curve $E_{/F}$ geometrically isogenous to $E_0$, if $E(F)$ has a point of order $N$ then \[ N\leq c_\epsilon\cdot [F:\mathbb{Q}]^{1/2+\epsilon}, \] and one also has \[ \# E(F)[\textrm{tors}] \leq C_\epsilon\cdot [F:\mathbb{Q}]^{1+\epsilon}. \]