'Parallelograms' are elliptic curves. And vice versa

The title should actually be: Tori are elliptic curves. And vice versa.

Maybe I’m getting older, maybe I’m more emotionally receptive than I used to be, or maybe this field of mathematics is simply so beautiful and full of surprises.

Because once again, I find this result surprising! A type of parallelogram is isomorphic to an elliptic curve! I don’t recall ever feeling surprised when I was studying mathematics before.

These notes still follow the wonderful Elliptic Curves: Number Theory and Cryptography (opens new window) by Lawrence C. Washington (chapter 9).

Tori are elliptic curves

Complex torus $\mathbb{C}/L$ is isomorphic to the complex points on an elliptic curve.

So, how can we see this isomorphism?

For integers $k \geq 3,$ the Eisenstein series is defined as:

$G_k = \sum_{w \in L, w \neq 0} w^{-k}$

It was discussed last time that such as series is convergent.

The Laurent series for $\wp(z)$ around 0 is:

$\wp(z) = \frac{1}{z^2} + \sum_{j=1}^{\infty} (2j+1) G_{2j+2} z^{2j}$

For $|z| < 1,$ it holds:

$\frac{1}{1-z} = \sum_{i=0}^{\infty}z^i$

$\frac{z}{1-z} = \sum_{i=0}^{\infty}z^{i+1}$

Let’s derivative:

$\frac{1}{(1-z)^2} = \sum_{i=0}^{\infty}(i+1)z^i$

Thus:

$\frac{1}{(z-w)^2} - \frac{1}{w^2} = \frac{1}{w^2}(\frac{1}{(1 - \frac{z}{w})^2} - 1)$

$= \frac{1}{w^2}\sum_{i=1}^{\infty} (i+1) (\frac{z}{w})^i$

So:

$\wp(z) = \frac{1}{z^2} - \sum_{w \in L, w \neq 0}\sum_{i=1}^{\infty} (i+1) \frac{z^i}{w^{i+2}}$

When $i$ is odd, it holds: $\frac{1}{w^{i+2}} = -\frac{1}{(-w)^{i+2}}.$ Hence:

$\wp(z) = \frac{1}{z^2} - \sum_{w \in L, w \neq 0}\sum_{i=1}^{\infty} (2i+1) \frac{z^{2i}}{w^{2i+2}}$

Elliptic function

Theorem: Let $\wp(z)$ be the Weierstrass function for a lattice $L.$ Then:

$\wp'(z)^2 = 4\wp(z)^3 - 60G_4\wp(z) - 140G_6$

To prove this theorem, we observe the Laurent series of the function

$\wp'(z)^2 - 4\wp(z)^3 + 60G_4\wp(z) + 140G_6$

The coefficients are set so that the negative powers of $z$ cancel each other. Additionally, the constant term is 0. Therefore, the function we observe has no poles and no constant term. We know that both $\wp(z)$ and $\wp'(z)$ are doubly periodic. Consequently, the observed function is a constant. Since the constant term is 0, the function itself must be 0.

Let’s now define:

$g_2 = 60G_4$

$g_3 = 140G_6$

The theorem says that:

$\wp'(z)^2 = 4\wp(z)^3 - g_2\wp(z) - g_3$

Discriminant

The discriminant $g_2^3 - 27g_3^2$ is nonzero.

We need a non-singular (smooth) curve, which occurs when when the discriminant is nonzero.

If $4\wp(z)^3 - g_2\wp(z) - g_3$ has three distinct roots, the discriminant is nonzero. Since $\wp'(z)$ is doubly periodic, we have $\wp'(w_i/2) = \wp'(-w_i/2)$ for $i = 1,2,3.$ It’s also odd, so: Moreover, $\wp'(z)$ is an odd function, so: $\wp'(w_i/2) = 0.$

We observe that $\wp(w_i/2)$ are roots of $4\wp(z)^3 - g_2\wp(z) - g_3.$ But are these three roots distinct?

Let’s observe

$h_i = \wp(z) - \wp(w_i/2)$

We know that $\wp(z)$ has only one pole in $F,$ a pole of order two at $0$. The same holds for $h_i(z).$ We also know that $w_i/2$ is the only zero of $h_i,$ as it is of order two. This implies that $w_j/2,$ where $i \neq j,$ cannot be a zero of $h_i.$

Finally, the isomorphism between the tori and elliptic curve

Let $L$ be a lattice and let $E$ be the elliptic curve $y^2 = 4x^3 - g_2x - g_3.$ The map

$\Phi: C/L \rightarrow E(C)$

$z \mapsto (\wp(z), \wp'(z))$

$0 \mapsto \infty$

is an isomorphism of groups.

I will be lazy and skip all the detailed calculations of the proof. However, while quickly going through it, I noticed the significance of divisors again. I will jot down a few notes about it below.

To show that $\Phi$ is a group homomorphism, we consider $z_1, z_2 \in \mathbb{C}.$ Let $\Phi(z_i) = P_i = (x_i, y_i).$

Let $y = ax + b$ be the line passing through $P_1$ and $P_2.$ Let $P_3 = (x_3, y_3)$ be the third point of intersection of this line with $E.$

Consider the following function:

$l(z) = \wp'(z) - a\wp(z) - b$

This function has the pole of order three at $0$. It holds $l(z_i) = 0$ for $i = 1,2,3.$ Since there are no other poles, it follows from the theorem about doubly periodic functions that:

$div(l) = [z_1] + [z_2] + [z_3] - 3[0]$

$z_1 + z_2 + z_3 \in L$

Thus:

$\wp(z_1 + z_2) = \wp(-z_3) = \wp(z_3)$

Now, this by no means proves that:

$\Phi(z_1 + z_2) = \Phi(z_1) + \Phi(z_2)$

It’s just one of the steps to prove it, but I found it interesting.

But also, every elliptic curve over C is a torus

Theorem: Let $y^2 = 4x^3 - Ax - B$ define an elliptic curve $E$ over $\mathbb{C}.$ Then there is a lattice $L$ such that:

$g_2(L) = A \text{ and } g_3(L) = B.$

There is an isomorphism of groups

$\mathbb{C}/L \simeq E(\mathbb{C}).$

I just skimmed through the proof, as my lack of knowledge in complex analysis is a bit disheartening. However, I hope to return to it someday.

Let’s first define the modular j-invariant for the lattice $L.$

Definition:

$j(L) = 1728\frac{g_2(L)^3}{g_2(L)^3 - 27g_3(L)^2}$

The following is a crucial result that aids in proving the theorem above.

Corollary: If $z \in \mathbb{C},$ there is exactly one $\tau \in \mathcal{F}$ such that

$j(\tau Z + Z) = z$

Let’s postpone a discussion of what $\mathcal{F}$ represents until later. For now:

$\mathcal{H} = {x + iy: y > 0; x,y \mathbb{R}}$

$\mathcal{F} = {z \in \mathbb{C}: |Re(z)| \leq \frac{1}{2}, |z| \ge 1}$

$\mathcal{H}$ is referred to as the upper half-plane.

Now, let’s observe the following complex number:

$z_0 = 1728\frac{A^3}{A^3 - 27B^2}$

There exists exactly one $\tau_0$ such that $j(\tau_0Z + Z) = z_0.$

Is lattice $\tau_0Z + Z$ already what we are searching for? That is, does it hold $g_2(\tau_0Z + Z) = A$ and $g_3(\tau_0Z + Z) = B?$

We have:

$\frac{g_2^3}{g_2^3 - 27g_3^2} = \frac{A^3}{A^3 - 27B^2}$

That means:

$g_2^3 = \lambda_0 A^3$

$g_2^3 - 27g_3^2 = \lambda_0 (A^3 - 27B^2)$

So $g_2 = \lambda_0 A$ and $g_3 = \pm\lambda_0 B.$

In general it holds:

$g_2(\lambda L) = \lambda^{-4}g_2(L)$

So

$g_2(\lambda_0^4 (\tau_0 Z+ Z)) = \lambda_0^{-1} \lambda_0 A = A$

Now we have the lattice $\lambda_0^4 (\tau_0 Z+ Z)$, for which $g_2 = A.$ Hence, for this lattice: $g_3 = \pm B.$ If $g_3 = B,$ we are done. Otherwise, the desired lattice is $i\lambda_0^4 (\tau_0 Z+ Z).$