## Moduli

Let $E_1$ and $E_2$ be a pair of elliptic curves. We saw in the previous post that all maps $E_1 \to E_2$ are in fact linear; that is, they arise from linear maps on their universal covers, $\mathbb{C}$. More specifically, they arise from linear maps on $\mathbb{C}$ which preserve the underlying lattices.

The first question which arises is the following.

When are two Elliptic curves isomorphic?

Let’s describe our curves a little more explicitly. We will let $\Lambda_1 = \langle z_1, z_2 \rangle$ and $\Lambda_2 = \langle w_1, w_2\rangle$ denote the underlying lattices of the curves $E_1, E_2$, respectively. There is then the map $z \mapsto z_1^{-1}\cdot z$ which takes the lattice $\Lambda_1$ to the lattice $\Lambda_\tau = \langle 1, \tau\rangle$ where $\tau = z_2/z_1$. In particular, since the map described above is linear, we have an isomorphism

$E_1 \cong E_\tau = \mathbb{C}/\Lambda_\tau$

whose inverse is given by $w \mapsto z_1 \cdot w$. Since isomorphism is an equivalence relation, we see then that $E_1 \cong E_2$ if $z_2 / z_1 = w_2/w_1$. With this being the case, we will often look at the parameter $\tau$ to classify elliptic curves. Due to orientation concerns, we can in fact restrict to the case that $\mathfrak{Im}\tau > 0$. With that in mind, let $\mathbb{H}$ denote the upper half-plane, the collection of those complex numbers with strictly positive imaginary part. We will use this as the starting point to discuss a parameter space, or moduli space for the collection of elliptic curves.

Recall that our question of the day is

When are two Elliptic curves isomorphic?

We have only partially answered this question: We noted that if the ratio of the basis elements are the same, then the curves must be isomorphic. But it seems that the choice of basis elements should also not matter, since an Elliptic curve is defined via a lattice, and not a lattice with a choice of basis. And it turns out that this is indeed true, which has some rather interesting consequences.

Recall that the group $SL_2\mathbb{Z}$ is the group of 2 by 2 invertible integer matrices. This can be thought of as the collection of all possible bases of a rank two free Abelian group. We will use this to identify other isomorphic curves.

Let $E_\tau = \mathbb{C}/\Lambda_\tau$ be an elliptic curve, and let

$\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

be an element of $SL_2\mathbb{Z}$. Note that we have chosen as our basis of $\Lambda_\tau$ the elements $1, \tau$, but we could have just as easily have chosen the elements $c\tau + d, a\tau + b$. That is,

$\langle 1, \tau\rangle = \langle c \tau + d, a\tau + b\rangle$

However, but the same discussion as before, we can multiply this right-hand description by the complex number $(c\tau + d)^{-1}$ to see that the lattices $\Lambda_\tau$ and $\Lambda_{\gamma\tau}$ are isomorphic, where

$\displaystyle\gamma\tau = \frac{a\tau + b}{c\tau + d}$

It follows that if we want to describe the correct moduli space for Elliptic curves, that we must consider instead $\mathbb{H}/SL_2\mathbb{Z}$, where $SL_2\mathbb{Z}$ acts on $\mathbb{H}$ via

$\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot \tau = \frac{a \tau + b}{c \tau + d}$

This quotient space is a coarse moduli space for classifying Elliptic curves.