nLab
torsion points of an elliptic curve

Contents

Idea

Recall that an elliptic curve EE is an abelian variety; in particular, its set of points admits a group structure. A point of EE is a torsion point if it is a torsion element of this group structure.

If EE is defined over a number field FF, the Galois theory of FF interacts very well with the torsion points of EE, as we describe below.

Definition

Definition

Let EE be an elliptic curve over a field FF, and let l2l \geq 2 be an integer. An ll-torsion point of EE is a point xx of EE such that lx=0l x = 0, that is, x+x++x l=0\underbrace{x + x + \cdots + x}_{l} = 0 in the (abelian) group structure on (the set of points of) EE, where 00 denotes the identity element of the group structure (often taken to be the point at infinity).

A torsion point of EE is a point of EE which is an ll-torsion point of EE for some integer l2l \geq 2.

Remark

For a fixed integer l2l \geq 2, the set of ll-torsion points of EE assembles into an abelian group with respect to the group structure of EE. The same is true of the set of all torsion points of EE.

Notation

For a fixed integer l2l \geq 2, the set of ll-torsion points of EE is often denoted E[l]E[l].

Splitting

The following observation is used frequently when working with torsion points of an elliptic curve over a number field.

Proposition

Let EE be an elliptic curve defined over \mathbb{Q}, the rationals. Then for any integer l2l \geq 2, there is an isomorphism of abelian groups E[l]/l/lE[l] \cong \mathbb{Z} / l\mathbb{Z} \oplus \mathbb{Z} / l\mathbb{Z}.

Remark

Thus if ll is a prime, so that /l\mathbb{Z} / l \mathbb{Z} is a field 𝔽 l\mathbb{F}_{l}, we can think of the set of automorphisms of the abelian group E[l]E[l] as the matrix group, specifically the general linear group, GL 2(𝔽 l)GL_2\left( \mathbb{F}_{l} \right).

Galois theory

Given an elliptic curve EE defined over a number field FF, the following observation ties the Galois theory of FF to the torsion points of EE. We shall denote the algebraic closure of FF by F¯\overline{F}.

Proposition

Let EE be an elliptic curve defined over a number field FF. Let FF' be an extension of FF. The Galois group Gal(F/F)Gal\left(F' / F \right) acts on all of (the set of points of) E/FE / F', the (abelian group of) torsion points of E/FE / F', and the (abelian group of) ll-torsion points of E/FE / F' for any fixed integer l2l \geq 2, in the obvious way: given an automorphism σ\sigma of FF' which fixes FF, we send a point xx of E/FE / F' to σ(x)\sigma(x).

Remark

The principal point is that given σ\sigma and xx as in Proposition , one can check that σ(x)\sigma(x) is still a point of EE, and is still an ll-torsion point for some l2l \geq 2 if xx is.

Given Remark , we can, as with any group action, reformulate Proposition in the case of ll-torsion points for a fixed prime l2l \geq 2 as follows.

Corollary

Let l2l \geq 2 be a prime. Let EE be an elliptic curve defined over a number field FF. Let FF' be an extension of FF. Then the application of automorphisms of FF' to points of EE determines a group homomorphism Gal(F/F)GL 2(𝔽 l)Gal\left(F' / F\right) \rightarrow GL_2\left( \mathbb{F}_{l} \right).

Remark

Given the homomorphism of Corollary , we can take its kernel. This subgroup of Gal(F/F)Gal\left( F' / F \right) determines, by Galois theory, a field extension of FF. Let us denote it by KK. In fact, KK is simply the tensor product of E[l]E[l] with FF; from that point of view, we have demonstrated that this tensor product is a finite Galois extension of FF.

Created on March 30, 2021 at 21:23:28. See the history of this page for a list of all contributions to it.