### The Modular Flow on the Space of Lattices

#### Posted by Simon Willerton

*Guest post by Bruce Bartlett*

The following is the greatest math talk I’ve ever watched!

- Etienne Ghys (with pictures and videos by Jos Leys), Knots and Dynamics, ICM Madrid 2006. [See below the fold for some links.]

I wasn’t actually *at* the ICM; I watched the online version a few years ago, and the story has haunted me ever since. Simon and I have been playing around with some of this stuff, so let me share some of my enthusiasm for it!

The story I want to tell here is how, via modular flow of lattices in the plane, certain matrices in $\SL(2,\mathbb{Z})$ give rise to knots in the 3-sphere less a trefoil knot. Despite possibly sounding quite scary, this can be easily explained in an elementary yet elegant fashion.

As promised above, here are some links related to Ghys’ ICM talk.

- Video — watch this now!
- Slides — the (ungarbled) slides for the talk
- Web version with animations — a great summary
- Article — the accompanying article with details
- More pictures — from Jos Leys’s webpage
- More animations — from Jos Leys’s webpage
- Article by Dana Lyons — a great summary with historical context and interviews

I’m going to focus on the last third of the talk — the *modular flow on the space of lattices*. That’s what produced the beautiful picture above (credit for this and other similar pics below goes to Jos Leys; the animation is Simon’s.)

## Lattices in the plane

For us, a *lattice* is a discrete subgroup of $\mathbb{C}$. There are three types: the *zero lattice*, the *degenerate lattices*, and the *nondegenerate lattices*:

Given a lattice $L$ and an integer $n \geq 4$ we can calculate a number — the Eisenstein series of the lattice: $G_{n}(L) = \sum _{\omega \in L, \omega \neq 0} \frac{1}{\omega ^{n}}.$ We need $n \geq 3$ for this sum to converge. For, roughly speaking, we can rearrange it as a sum over $r$ of the lattice points on the boundary of a square of radius $r$. The number of lattice points on this boundary scales with $r$, so we end up computing something like $\sum _{r \geq 0} \frac{r}{r^{n}}$ and so we need $n \geq 3$ to make the sum converge.

Note that $G_{n}(L)$ = 0 for $n$ odd since every term $\omega$ is cancelled by the opposite term $-\omega$. So, the first two nontrivial Eisenstein series are $G_{4}$ and $G_{6}$. We can use them to put `Eisenstein coordinates’ on the space of lattices.

Theorem:The map $\begin{aligned} \{ \text{lattices} \} &\rightarrow \mathbb{C}^{2} \\ L & \mapsto (G_{4} (L), \, G_{6}(L)) \end{aligned}$ is a bijection.

The nicest proof is in Serre’s A Course in Arithmetic, p. 89. It is a beautiful application of the Cauchy residue theorem, using the fact that $G_{4}$ and $G_{6}$ define modular forms on the upper half plane $H$. (Usually, number theorists set up their lattices so that they have basis vectors $1$ and $\tau$ where $\tau \in H$. But I want to *avoid* this ‘upper half plane’ picture as far as possible, since it breaks symmetry and mystifies the geometry. The whole point of the Ghys picture is that not breaking the symmetry reveals a beautiful hidden geometry! Of course, sometimes you need the ‘upper half plane’ picture, like in the proof of the above result.)

Lemma:The degenerate lattices are the ones satisfying $20 G_{4}^{3} - 49G_{6}^{2} = 0$.

Let’s prove one direction of this lemma — that the degenerate lattices do indeed satisfy this equation. To see this, we need to perform a computation. Let’s calculate $G_{4}$ and $G_{6}$ of the lattice $\mathbb{Z} \subset \mathbb{C}$. Well, $G_{4}(\mathbb{Z}) = \sum _{n \neq 0} \frac{1}{n^{4}} = 2 \zeta (4) = 2 \frac{\pi ^{4}}{90}$ where we have cheated and looked up the answer on Wikipedia! Similarly, $G_{6}(\mathbb{Z}) = 2 \frac{\pi ^{6}}{945}$.

So we see that $20 G_{4}(\mathbb{Z})^{3} - 49 G_{6}(\mathbb{Z})^{2} = 0$. Now, *every* degenerate lattice is of the form $t \mathbb{Z}$ where $t \in \mathbb{C}$. Also, if we transform the lattice via $L \mapsto t L$, then $G_{4} \mapsto t^{-4} G_{4}$ and $G_{6} \mapsto t^{-6} G_{6}$. So the equation remains true for all the degenerate lattices, and we are done.

Corollary:The space of nondegenerate lattices in the plane of unit area is homeomorphic to the complement of the trefoil in $S^{3}$.

The point is that given a lattice $L$ of unit area, we can scale it $L \mapsto \lambda L$, $\lambda \in \mathbb{R}^{+}$ until $(G_{4}(L), G_{6}(L))$ lies on the 3-sphere $S^{3} = \{ (z,w) : |z|^{2} + |w|^{2} = 1\} \subset \mathbb{C}^{2}$. And the equation $20 z^{3} - 49 w^{2} = 0$ intersected with $S^{3}$ cuts out a trefoil knot… because it is “something cubed plus something squared equals zero”. And the lemma above says that the nondegenerate lattices are precisely the ones which *do not* satisfy this equation, i.e. they represent the complement of this trefoil.

Since we have not divided out by *rotations*, but only by *scaling*, we have arrived at a 3-dimensional picture which is very different to the 2-dimensional moduli space (upper half-plane divided by $\SL(2,\mathbb{Z})$) picture familiar to a number theorist.

## The modular flow

There is an intriguing flow on the space of lattices of unit area, called the *modular flow*. Think of $L$ as sitting in $\mathbb{R}^{2}$, and then act on $\mathbb{R}^{2}$ via the transformation
$\left ( \begin{array}{cc} e^{t} & 0 \\ 0 & e^{-t} \end{array} \right ),$
dragging the lattice $L$ along for the ride. (This isn’t just some formula we pulled out the blue — geometrically this is the ‘geodesic flow on the unit tangent bundle of the modular orbifold’.)

We are looking for *periodic orbits* of this flow.

“Impossible!” you say. “The points of the lattice go off to infinity!” Indeed they do… but disregard the individual points. The *lattice itself* can ‘click’ back into its original position:

How are we to find such periodic orbits? Start with an integer matrix
$A = \left ( \begin{array}{cc} a & b \\ c & d \end{array}\right ) \in \SL(2, \mathbb{Z})$
and assume $A$ is *hyperbolic*, which simply means $|a + d| \geq 2$. Under these conditions, we can diagonalize $A$ over the reals, so we can find a real matrix $P$ such that
$P A P^{-1} = \pm \left ( \begin{array}{cc} e^{t} & 0 \\ 0 & e^{-t} \end{array} \right )$
for some $t \in \mathbb{R}$. Now set $L \coloneqq P(\mathbb{Z}^{2})$. We claim that $L$ is a periodic orbit of period $t$. Indeed:
$\begin{aligned}
L_{t} &= \left ( \begin{array}{cc} e^{t} & 0 \\ 0 & e^{-t} \end{array} \right ) P (\mathbb{Z}^{2}) \\ &= \pm PA (\mathbb{Z}^{2}) \\ &= \pm P (\mathbb{Z}^{2}) \\ &= L.
\end{aligned}$
We have just proved one direction of the following.

Theorem: The periodic orbits of the modular flow are in bijection with the conjugacy classes of hyperbolic elements in $\SL(2, \mathbb{Z})$.

These periodic orbits produce fascinating *knots* in the complement of the trefoil! In fact, they *link* with the trefoil (the locus of degenerate lattices) in fascinating ways. Here are two examples, starting with different matrices $A \in \SL(2, \mathbb{Z})$.

The trefoil is the fixed orange curve, while the periodic orbits are the red and green curves respectively.

Ghys proved the following two remarkable facts about these *modular knots*.

- The linking number of a modular knot with the trefoil of degenerate lattices equals the
*Rademacher function*of the corresponding matrix in $\SL(2, \mathbb{Z})$ (the change in phase of the Dedekind eta function). - The knots occuring in the modular flow are the same as those occuring in the Lorenz equations!

Who would have thought that lattices in the plane could tell the weather!!

I must say I have thought about many aspects of these closed geodesics, but it had never crossed my mind to ask which knots are produced.– Peter Sarnak

## Re: The Modular Flow on the Space of Lattices

That’s a very nice post Bruce.

I first learnt about the space of non-degenerate lattices up-to-homothety being the complement of the trefoil from my friend and old office-mate Jacob Mostovoy.

He has a nice, short paper on how this space is related to subsets of the circle with at most three elements.