Moduli Space of Tori

I’m looking at an exercise that reads:

Problem. Let $A = begin{bmatrix}a & b\c & dend{bmatrix} in GL(2, mathbb{C})$ and $Lambda = langle z mapsto z + omega_1, z mapsto z + omega_2rangle$ with $omega_1, omega_2$ linearly independent over $mathbb{R}$. Prove that if $Lambda = langle z mapsto z + omega_1′, z mapsto z + omega_2’rangle$ where
begin{bmatrix}omega_1’\omega_2’end{bmatrix} = Abegin{bmatrix}omega_1\omega_2end{bmatrix}

then $A in GL(2, mathbb{Z})$ with $det(A) = pm1$. Now show that the moduli space of tori, defined to be the space of all conformal equivalence classes of tori, is $mathbb{H}/PSL(2, mathbb{Z})$.

I’ve done the first bit, but not sure how to proceed with the second. In particular, I’m not sure how to visualise $mathbb{H}/PSL(2, mathbb{Z})$ and hence find a bijection between that and the equivalence classes.

Mathematics Asked by bookworm on November 18, 2021

1 Answers

One Answer

Here's an approach that makes clear where the parameterization comes from. By the uniformization theorem, any conformal torus $T$ arises as a quotient $T = mathbb{C}/Lambda$, where $Lambda$ is a discrete subgroup of $text{Isom}(mathbb{C})$. Any such $Lambda$ must be of the form $langle z mapsto z + omega_1, z mapsto z + omega_2rangle cong mathbb{Z}^2$, with $omega_1, omega_2$ linearly independent over $mathbb{R}$, with two conformal tori $mathbb{C}/Lambda, mathbb{C}/Lambda'$ identified if there is a map $A in text{Isom}(mathbb{C})$ so $A(Lambda) = Lambda'$.

Now for any such $Lambda$, fix generators $z mapsto omega_1, z mapsto omega_2$ and identify the generators with $omega_1, omega_2$. By applying an element of $text{Isom}(mathbb{C})$ and swapping the labeling if needed, we can arrange this in a unique way so $omega_1 = 1$ and $omega_2 in mathbb{H}$.

We'd like to parameterize conformal tori by $omega_2$, but this map isn't well-defined because of the choice of generators we made. However, what you've done shows that this choice is well-defined up to the action of elements of $GL(2, mathbb{Z})$ with determinant $pm 1$. The rest of the approach is only sketched, and filling in details is a good exercise.

Track what happens to $omega_2$ when using an element of $GL(2, mathbb{Z})$ with determinant $pm 1$ to act on $1, omega_2$ then re-normalizing. Observe that the action factors to a very familiar action of $PSL(2, mathbb{Z})$ on $mathbb{H}$. Conclude that we can identify conformal structures on tori with the quotient $mathbb{H}/PSL(2, mathbb{Z})$. Thinking of the fundamental domain of this action is a good starting point for thinking about what $mathbb{H}/PSL(2, mathbb{Z})$ looks like.

It's worth mentioning that the transformations mentioned above don't take the generators we originally defined to each other in general. If we consider the moduli space of tori marked with generators in $text{Isom}(mathbb{C})$ of the lattice we get $mathbb{H}$ as our moduli space, the Teichmuller space of the torus.

Answered by Alex Nolte on November 18, 2021

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