nLab torsion points of an elliptic curve

Idea

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

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

Definition

Definition

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

A torsion point of $E$ is a point of $E$ which is an $l$-torsion point of $E$ for some integer $l \geq 2$.

Remark

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

Notation

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

Splitting

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

Proposition

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

Remark

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

Galois theory

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

Proposition

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

Remark

The principal point is that given $\sigma$ and $x$ as in Proposition , one can check that $\sigma(x)$ is still a point of $E$, and is still an $l$-torsion point for some $l \geq 2$ if $x$ is.

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

Corollary

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

Remark

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