Some Elementary Fun Facts in Elliptic Curves

Main

For a general elliptic curve $y^2 = Ax^3+Bx^2+Cx+D$, where $A,B,C,D\in\mathbb{C}$ are constants, we call it rational, if $A,B,C,D\in\mathbb{Q}$.

As a lemma, we have

[Lemma] Over $\mathbb{CP}^2$ (complex projective plane) a rational line (line whose coefficients rational) cuts rational elliptic curve, if counting multiplicities, in three points.

If two of these points are rational, so is the third one.

As for notation, let $P,Q$ as rational points on $C$. Define $P*Q$ to be the third point in which the line through $P,Q$ meets $C$.

Using the lemma we may construct an abelian group structure in those rational elliptic curves:

[Definition]

The group $\mathcal{G}$ consists of rational points lying on the elliptic curve.

The operation $+$ on $\mathcal{G}$ is given by

$$ P+Q = (P*Q)*\mathbb{O} $$
where $\mathbb{O}$ is a previously arbitarily chosen and fixed, as the identity in $\mathcal{G}$.

[Theorem] The set $\mathcal{G}$ of rational points on a rational elliptic curve forms an abelian group under the operation $+$. The identity elements is previously chosen arbitarily and fixed, namely $\mathbb{O}$.

Here comes the elegant part:

[Theorem] (Mordell’s Theorem) Let $C$ be a nonsingular rational curve in $\mathbb{CP}^2$ having a rational point. Then the group $\mathcal{G}$ of rational points is finitely generated.

Recall that any finitely generated abelian groups is in the form

$$ F\oplus \mathbb{Z}^k $$

where $F$ is a finite abelian group, namely torsion subgroup.

In fact, if we apply this to $\mathcal{G}$, we’ll have

[Theorem] Let $\mathcal{G}$ be the group of rational points on an elliptic curve. Then the torsion subgroup of $\mathcal{G}$ is isomorphic to either

$Z_{l}$, where $l\in[1,10]\cap\mathbb{Z}$.

$Z_{2}\oplus\mathbb{Z}_{2l}$, where $l\in[1,4]\cap\mathbb{Z}$.

Reference:

Algebraic Number Theorey and Fermat’s Last Theorem (Fourth Edition) – Ian Stewart & David Tall, Chapter 13.