On the minimal degree of definition of p-primary torsion subgroups of elliptic curves

TL;DR

This paper investigates the minimal degree of field extensions needed to define p-primary torsion subgroups of elliptic curves over number fields, providing uniform bounds and specific divisibility conditions for the case of Q and p=2.

Contribution

It establishes uniform bounds on the minimal degree of definition for p-primary torsion subgroups, depending on the Galois representation image, and derives optimal divisibility conditions for Q and p=2.

Findings

Uniform bounds for minimal degrees of p-torsion subgroups.

Divisibility conditions for 2-primary torsion subgroups over Q.

Dependence of results on the shape of Galois representations.

Abstract

In this article, we study the minimal degree [K(T):K] of a p-subgroup T <= E(\overline{K})_tors for an elliptic curve E/K defined over a number field K. Our results depend on the shape of the image of the p-adic Galois representation \rho_{E,p^infty}:Gal_K-->GL(2,Z_p). However, we are able to show that there are certain uniform bounds for the minimal degree of definition of T. When the results are applied to K=Q and p=2, we obtain a divisibility condition on the minimal degree of definition of any subgroup of E[2^n] that is best possible.