§7 Elliptic Curves

Definition 7.1 Let $f(x)$ be a well-behaved cubic polynomial. The elliptic curve defined by $f$ is the set

\mathbb{E}:=\{\mathscr{O}\}\cup\{(x,y)\mid y^{2}=f(x)\}.\tag*{}

Example 7.1 The solutions of the equation are shown below:

Notice that

(x,y)\in\mathbb{E} \quad\Rightarrow\quad (x,-y)\in\mathbb{E}.

Theorem 7.1 Every elliptic curve is an Abelian group.

This is surprising. We define the group operation as follows:

\begin{aligned} \mathbb{E}\times\mathbb{E}&\to\mathbb{E}\\ (P,Q)&\mapsto P+Q \end{aligned}\tag*{}

(1) If $P,Q=\mathscr{O}$ , define

\mathscr{O}+\mathscr{O}=\mathscr{O}.\tag*{}

(2) If $P=\mathscr{O}$ and $Q=(x,y)\in\mathbb{E}$ , define

\mathscr{O}+Q=Q+\mathscr{O}=Q.\tag*{}

(3) If $P,Q$ belong to

\{(x,y)\mid y^{2}=f(x)\},

consider the unique line $L_{PQ}$ containing both of them.

A line $L$ intersects a cubic curve at three points. Let $R=(x,y)$ be the third point of intersection. We define

P+Q:=(x,-y).

Rule: If $L_{PQ}$ is a vertical line, then it has no third point of intersection in $\mathbb{R}^{2}$ . We define the third intersection point $R$ to be the “point at infinity” $\mathscr{O}$ .

(This is really an interpretation coming from projective geometry, where parallel lines—for example, vertical lines—intersect at a point at infinity.)

Notice that

L_{PQ}=L_{QP},

P+Q=Q+P.

Proving that

\mathbb{E}\times\mathbb{E}\to\mathbb{E}

is associative is more difficult.

I suggest choosing three adjacent points.

Proposition 7.2

(P_{1}+P_{2})+P_{3}=P_{1}+(P_{2}+P_{3}).

Here is a wonderful observation. Suppose

f(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}

with $a_i\in\mathbb{Q}$ . If $P,Q\in\mathbb{E}$ are rational points, meaning that both their $x$ - and $y$ -coordinates are rational numbers, then $P+Q$ is also a rational point!

Proof: The equation of $L_{PQ}$ is

y=mx+t.\tag*{}

Since

P,Q\in\mathbb{Q}^{2},

we have

m,t\in\mathbb{Q}.

The intersection

L_{PQ}\cap\mathbb{E}

contains $R$ , whose coordinates satisfy

(mx+t)^{2} = a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\tag*{}

Therefore $P,Q,R$ correspond to roots of a cubic polynomial with rational coefficients.

Hence

(x-x_{1})(x-x_{2})(x-x_{3}) = g(x) = b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}.\tag*{}

Since

x_1,x_2\in\mathbb{Q},

we obtain

x_3\in\mathbb{Q},

because $x_1x_2x_3$ is the constant term of $g(x)$ !

Even better, we can use

x_1+x_2+x_3=b_2

together with

x_1,x_2,b_2\in\mathbb{Q}.

Definition 7.2 If $f$ is a cubic polynomial over $\mathbb{Q}$ , meaning that $a_i\in\mathbb{Q}$ , define

\mathbb{E}(\mathbb{Q})\subset\mathbb{E}

to be the set

(\mathbb{E}\cap(\mathbb{Q}\times\mathbb{Q})) \cup \{\mathscr{O}\}.\tag*{}

That is, it is the set of all points $P$ whose coordinates are rational numbers, together with the point at infinity $\mathscr{O}$ .

Thus we have a subset

\mathbb{E}(\mathbb{Q}) \subset \mathbb{E}\tag*{}

which is closed under addition.

It is also closed under inverses because

P=(x,y)\in\mathbb{Q}\times\mathbb{Q}\tag*{}

implies

-P=(x,-y)\in\mathbb{Q}\times\mathbb{Q}.\tag*{}

Moreover, $\mathbb{E}(\mathbb{Q})$ contains the identity element $\mathscr{O}$ . Therefore:

Proposition 7.3

\mathbb{E}(\mathbb{Q})\subset\mathbb{E}

is a subgroup.

Definition 7.3 A group $G$ is called finitely generated if there exists a finite set $S$ together with a surjective homomorphism

F(S)\to G.\tag*{}

Let us unpack this definition. Suppose

S=\{s_1,\ldots,s_n\}

is a finite set and

\phi:F(S)\to G\tag*{}

is a surjective homomorphism.

The map $\phi$ sends each $s_i$ to some element

g_i=\phi(s_i).\tag*{}

The fact that $\phi$ is surjective means that for every $g\in G$ , there exists a word such that

\phi(w)=g.\tag*{}

That is, $g$ can be expressed as a finite product of the $g_i$ and $g_i^{-1}$ .

In other words, there exists a finite collection

g_1,\ldots,g_n\in G\tag*{}

such that every element of $G$ can be expressed as a product of the $g_i$ and their inverses.

Example 7.2 Every finite group $G$ is finitely generated. Take

S=G\tag*{}

and define

\begin{aligned} F(S)&\to G\\ g&\mapsto g. \end{aligned}\tag*{}

Example 7.3 Every cyclic group is finitely generated. If

G=\langle g\rangle,\tag*{}

take

S=\{g\},

and define

\begin{aligned} F(S)&\to G\\ g&\mapsto g. \end{aligned}\tag*{}

Example 7.4 Any finite product of finitely generated groups is finitely generated:

G=G_1\times\cdots\times G_n.\tag*{}

Take a generating set $S_i$ for each $G_i$ and define

S=S_1\cup\cdots\cup S_n.

\phi_i:F(S_i)\to G_i

is surjective for every $i$ , define

\begin{aligned} \phi:F(S)&\to G\\ a_i&\mapsto (1,\ldots,1,\phi_i(a_i),1,\ldots,1). \end{aligned}\tag*{}

One of the most important theorems about elliptic curves is:

Theorem 7.4 (Mordell's Theorem)

\mathbb{E}(\mathbb{Q})

is finitely generated.

This result is remarkable: there exist finitely many rational points

P_1,\ldots,P_n\in\mathbb{E}(\mathbb{Q})

such that every other rational point can be obtained by adding and subtracting these $P_i$ .

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§7 Elliptic Curves