Weber modular functions

23 Jun 2016

An quick introduction to modular functions, and the Weber modular functions in particular.

Background

A modular function for a subgroup $\Gamma$ of $\operatorname{SL}_2(\mathbf{Z})$ is a meromorphic function $f$ on the upper half plane $\mathbf{H} =\{x+iy\in \mathbf{C} : y > 0\}$, that is invariant by all matrices in $\Gamma$ when they act via

$$\begin{pmatrix} a&b\\c&d \end{pmatrix} \cdot f(\tau) = f\left(\frac{a\tau+b}{c\tau+d}\right),$$

(we also need $f$ to be “meromorphic at the cusps” too but let’s not worry about this right now).

So for example letting $\Gamma$ be the whole of $\operatorname{SL}_2(\mathbf{Z})$ we see that modular functions for $\operatorname{SL}_2(\mathbf{Z})$ must satisfy $f(\tau) = f(\tau+ 1)$ and $f(\tau) = f(-1/\tau) $ for all $\tau\in\mathbf{H}$, as

$$\begin{pmatrix} 1&1\\0&1 \end{pmatrix} \text{and} \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$$

are both in the modular group. In fact, as the above matrices generate the whole modular group, being invariant under the above two transformations is sufficient for a function on the upper half plane to be modular for $\operatorname{SL}_2(\mathbf{Z})$.

The most well known modular function is undoubtedly the $j$-invariant:

$$j(\tau) = \frac{1}{q} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + \cdots,$$

where $q = e^{2\pi i \tau}$. If you haven’t seen the $j$-invariant before this description as a $q$-expansion is not very helpful, but rest assured this function appears very naturally when considering sublattices of $\mathbf{C}$. In any case the properties of this function are the interesting and important part: first and foremost it is as injective as a modular function possibly can be. What I mean by this is that not only does

$$\begin{pmatrix} a&b\\c&d \end{pmatrix} \cdot j(\tau) = j\left(\frac{a\tau+b}{c\tau+d}\right).$$

for any $\tau$, but if $j(\tau) = j(\sigma)$ then there is some matrix

$$\begin{pmatrix} a&b\\c&d \end{pmatrix} \in \operatorname{SL}_2(\mathbf{Z}) \text{ such that } \sigma = \frac{a\tau+b}{c\tau+d}.$$

This is useful as it allows you to find out if two points in the upper half plane can be transformed into each other by the above matrix action simply by calculating the value of $j$ at these points. This is especially handy when dealing with elliptic curves, which correspond to points in the upper half plane and are isomorphic only when their representative points are equivalent under the above action. This is the reason that $j$ is called the $j$-invariant.

In addition to this another nice fact about $j$ is that it generates the set of all modular functions for the full modular group when we take rational functions of it, that is

$$\{ \text{modular functions for } \operatorname{SL}_2(\mathbf{Z})\} = \mathbf{C}(j).$$

It is no surprise then that $j$ is a sort of poster child for modular functions, and indeed before looking into this topic I don’t think that I knew an interesting example of a modular function other than $j$, well now I do!

Weber modular functions

In order to define some other modular functions we will use an intermediary function, $\eta$, which has $q$-expansion:

$$\eta(\tau) = q^{1/24}\prod_{n=1}^\infty(1-q^n).$$

Now $\eta$ is not a modular function, rather a modular form of weight $\frac{1}{2}$, which means that instead of the condition we had for modular functions we have instead that

$$\eta\left( \frac{az+b}{cz+d} \right) = (cz+d)^{-1/2} \eta(z) \text{ for all }\begin{pmatrix} a&b\\c&d \end{pmatrix}\in\Gamma.$$

One consequence of this is that if we divide $\eta$ by another modular form of weight $\frac12$ we get a modular function. But we have only seen one such function so far, $\eta$, and $\eta/\eta$ is not a very interesting function. We can however apply transformations to $\eta$ similar to those above with elements of the general linear group $\operatorname{GL}_2(\mathbf{Z})$ instead (specifically ones that $\eta$ is not invariant under). This will allow us to create new functions of the same type. For instance we let:

$$\mathfrak{f}(\tau) = e^{\frac{-\pi i}{24}}\frac{\eta\left(\frac{\tau + 1}{2}\right)}{\eta(\tau)} = q^{-\frac{1}{48}}\prod_{n=1}^\infty\left(1+q^{n-\frac{1}{2}}\right),$$ $$\mathfrak{f}_1(\tau) = \frac{\eta\left(\frac{\tau}{2}\right)}{\eta(\tau)} = q^{-\frac{1}{48}}\prod_{n=1}^\infty\left(1-q^{n-\frac{1}{2}}\right),$$ $$\mathfrak{f}_2(\tau) = \sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}= \sqrt{2}q^{-\frac{1}{24}}\prod_{n=1}^\infty\left(1+q^{n}\right).$$

These are Weber modular functions, named after Heinrich Weber who studied them in his Lehrbuch der Algebra. Indeed we could have defined these via their $q$-expansions without mention of $\eta$ at all, but that makes them seem rather arbitrary, which they certainly aren’t!

Notice that we have applied the elements $\begin{pmatrix} 1&1\\0&2 \end{pmatrix}, \begin{pmatrix} 1&0\\0&2 \end{pmatrix} \text{ and } \begin{pmatrix} 2&0\\0&1 \end{pmatrix},$

all of which have determinant $2$, to $\eta$ here.

Until now the only modular function we have looked at is $j$, which was a modular function for the whole of $\operatorname{SL}_2(\mathbf{Z})$ so you will no doubt be pleased to hear that these functions are in fact modular for

$$\Gamma_{48} = \left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbf{Z}) : a\equiv d \equiv 1,\, b\equiv c\equiv 0 \pmod{48}\right\},$$

a proper subgroup of the full modular group.

By virtue of the normalising coefficients we used here, these functions satisfy several nice identities, for example

$$\mathfrak{f}_1(\tau)^8 + \mathfrak{f}_2(\tau)^8 = \mathfrak{f}(\tau)^8.$$

Whilst these are some very lovely looking expressions, such things aren’t quite as exciting in total isolation. So how can we use these functions? One interesting thing we can do with these functions is generate Hilbert class fields, for this sort of work we it will help to look at Shimura reciprocity, which will be the topic of a future post!