## Lecture 12: Monday, September 16, 2013

Last time, we introduced the Fundamental Domain $X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$ in terms of the extended complex plane $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C)$. Felix Klein showed the existence of a function $J: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$, invariant under $SL_2(\mathbb Z)$, giving these isomorphisms. Today, we use Richard Dedekind‘s function $\eta: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$ to create similar functions $\mathbb H^\ast \to \mathbb P^1(\mathbb C)$.

### Covering Spaces

Recall the main result from the previous lecture:

Proposition.
Let $\left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right)$ be a compact, connected Riemann surface. Assume that there exists an $n$-fold covering map $\Pi: X \twoheadrightarrow S^2(\mathbb R)$ such that $\text{Aut}(\Pi)$ is a finite group.

Then, for each $m$-fold covering map $\phi: X \to \mathbb P^1(\mathbb C)$ which is meromorphic on $X$, that is, the composition $\phi \circ {\mu_\alpha}^{-1}$ is the ratio of two analytic functions on the subset $\mu_\alpha(U_\alpha) \subseteq \mathbb P^1(\mathbb C)$ for each index $\alpha \in I$, there exists a rational function $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ which makes the following diagram commute:

$\begin{matrix} X & \overset{\gamma}{\longrightarrow} & X & \overset{\phi}{\longrightarrow} & \mathbb P^1(\mathbb C) & & P & \mapsto & z = \phi(P) \\[10pt] \downarrow^\Pi & & \downarrow^\Pi & & \downarrow^f & & \downarrow & & \downarrow \\[10pt] S^2(\mathbb R) & = & S^2(\mathbb R) & & & & \Pi(P) & & \\[10pt] \downarrow^\simeq & & \downarrow^\simeq & & \downarrow & & \downarrow & & \downarrow \\[10pt] X(1) & = & X(1) & \overset{J}{\longrightarrow} & \mathbb P^1(\mathbb C) & & \tau = \bigl( J^{-1} \circ \sigma^{-1} \circ \Pi \bigr)(P) & \mapsto & J(\tau) = f(z) \end{matrix}$

Eventually, we will call $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ a Belyĭ map. This is a very strange result which seems to have little importance. We explain via a series of examples how this motivated the entire study of modular functions.

### Modular Curve $X_0(2)$

Consider the set $X_0(2) = \bigcup_{\gamma \in I} \gamma \, X(1)$ for the finite set $I = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & -1 \\ 1 & \ \ 0 \end{matrix} \right], \ \left[ \begin{matrix} 0 & -1 \\ 1 & \ \ 1 \end{matrix} \right] \right \}$. We can draw this finite union using Helena Verrill‘s Fundamental Domain drawer, or even Andrzej Kozlowski‘s Wolfram Demonstration. Either way, we see that $X_0(2)$ is a compact, connected, Riemann Surface.

For each subset $I \subseteq SL_2(\mathbb Z)$, we see that Klein’s $J: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$ is invariant under the action $\tau \mapsto \gamma \, \tau$, so that $J: X_0(2) \to \mathbb P^1(\mathbb C)$ is a trivial function: indeed, it factors through the projection $\Pi: X_0(2) \to X(1)$ which sends $\gamma \, \tau \mapsto \tau$. However, a nontrivial function $J_{2,0}: X_0(2) \to \mathbb P^1(\mathbb C)$ is given by

\begin{aligned} J_{2,0}(\tau) & = \left( \dfrac {\eta(\tau)}{\eta(2 \, \tau)} \right)^{24} \\[5pt] & = \left[ \displaystyle q_1 \, \prod_{n=1}^{\infty} \bigl( 1 + q_1^n \bigr)^{24} \right]^{-1} = \dfrac 1{q_1} - 24 + 276 \, q_1 - 2048 \, q_1^2 + \cdots. \end{aligned}
(Recall that $q_h = e^{2 \pi i \tau/h}$.) More information about this function can be found on the Wikipedia page on Classical Modular Curves as well as the On-Line Encyclopedia of Integer Sequences.

Proposition.
Define $J_{2,0}(\tau) = \left( \dfrac {\eta(\tau)}{\eta(2 \, \tau)} \right)^{24}$ in terms of $\eta(\tau) = \displaystyle q_1^{1/24} \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)$.

• $J_{2,0}: X_0(2) \to \mathbb P^1(\mathbb C)$ is a bijection.
• We are able to express the $J$-invariant as a rational function of this function:
$J(\tau) = \dfrac {\bigl( J_{2,0}(\tau) + 256 \bigr)^3}{1728 \, J_{2,0}(\tau)^2} \qquad \text{and} \qquad J(2 \, \tau) = \dfrac {\bigl( J_{2,0}(\tau) + 16 \bigr)^3}{1728 \, J_{2,0}(\tau)}.$

Observe that the projection map $\Pi: X_0(2) \to X(1)$ sending $\gamma \, \tau \mapsto \tau$ is 3-to-1, and the rational function $f(z) = \dfrac {( z + 256)^3}{1728 \, z^2}$ such that $J(\tau) = f(z)$ in terms of $z = J_{2,0}(\tau)$ has $\text{deg}(f) = 3$. This is not a coincidence!

### Modular Curve $X(2)$

Now consider the compact, connected, Riemann Surface $X(2) = \bigcup_{\gamma \in I} \gamma \, X(1)$ for the finite set
$I = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & -1 \\ 1 & \ \ 0 \end{matrix} \right], \ \left[ \begin{matrix} 0 & -1 \\ 1 & \ \ 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} \ \ 0 & 1 \\ -1 & 0 \end{matrix} \right], \ \left[ \begin{matrix} -1 & \ \ 0 \\ \ \ 1 & -1 \end{matrix} \right] \right \}.$
As before, we can draw this finite union using Helena Verrill‘s Fundamental Domain drawer, or even Andrzej Kozlowski‘s Wolfram Demonstration.

We have two projections $X(2) \to X_0(2) \to X(1)$ so that both $J: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$ and $J_{0,2}: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$ are functions from $X(2) \to \mathbb P^1(\mathbb C)$. A nontrivial function which has $X(2)$ as its domain of definition is
\begin{aligned} \lambda(\tau) & = 16 \, \dfrac {\eta(\tau/2)^8 \, \eta(2 \, \tau)^{16}}{\eta(\tau)^{24}} \\[5pt] & = \displaystyle 16 \, q_2 \prod_{n=1}^{\infty} \dfrac { \bigl(1+q_2^{2n} \bigr)^{16}}{\bigl( 1 + q_2^n \bigr)^8} = 16 \, \biggl[ q_2 - 8 \, q_2^2 + 44 \, q_2^3 + \cdots \biggr]. \end{aligned}
(Recall that $q_h = e^{2 \pi i \tau/h}$.) The parameter $q_2 = e^{\pi i \tau}$ is often called the nome, while the function $\lambda(\tau)$ is often called the elliptic modular lambda function. More information about this function can be found on the On-Line Encyclopedia of Integer Sequences.

Proposition.
Define $\lambda(\tau) = 16 \, \dfrac {\eta(\tau/2)^8 \, \eta(2 \, \tau)^{16}}{\eta(\tau)^{24}}$ in terms of $\eta(\tau) = \displaystyle q_1^{1/24} \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)$.

• $\lambda: X(2) \to \mathbb P^1(\mathbb C)$ is a bijection.
• $J_{2,0}(\tau) = 256 \, \dfrac {1 - \lambda(\tau)}{\lambda(\tau)^2}$ and $J(\tau) = \dfrac {4}{27} \, \dfrac {\bigl( \lambda(\tau)^2 - \lambda(\tau) + 1 \bigr)^3}{\lambda(\tau)^2 \, \bigl(\lambda(\tau) - 1 \bigr)^2}$.

The following diagram may be useful to keep track of everything:

$\begin{matrix} X(2) & \longrightarrow & X_0(2) & \longrightarrow & X(1) \\[10pt] \downarrow & & \downarrow & & \downarrow \\[10pt] \mathbb P^1(\mathbb C) & \longrightarrow & \mathbb P^1(\mathbb C) & \longrightarrow & \mathbb P^1(\mathbb C) \end{matrix}$

where we map
$\begin{matrix} \tau & \mapsto & \tau & \mapsto & \tau \\[10pt] \downarrow & & \downarrow & & \downarrow \\[10pt] z= \lambda(\tau) & \mapsto & w = J_{2,0}(\tau) = 256 \, \dfrac {1 - z}{z^2} & \mapsto & J(\tau) = \dfrac {(w + 256)^3}{1728 \, w^2} = \dfrac {4}{27} \, \dfrac {( z^2 - z+ 1 \bigr)^3}{z^2 \, (z- 1)^2} \end{matrix}$

### Relation with Galois Groups

In these examples, we have seen how the projection maps $\Pi: X \twoheadrightarrow X(1)$ yield rational functions $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$, but we have not yet introduced the group $G = \text{Aut}(\Pi)$. Observe that for each $\gamma \in I$ we have
$\lambda(\gamma \, \tau) \in \left \{ \lambda(\tau), \ \dfrac {\lambda(\tau) - 1}{\lambda(\tau)}, \ \dfrac {1}{1 - \lambda(\tau)}, \ 1 - \lambda(\tau), \ \dfrac {\lambda(\tau)}{\lambda(\tau) - 1}, \ \dfrac {1}{\lambda(\tau)} \right \}.$

In fact, we can say more:
\begin{aligned} \lambda(\tau) & = z \\[5pt] \dfrac {\lambda(\tau) - 1}{\lambda(\tau)} & = r(z) \\[5pt] \dfrac {1}{1 - \lambda(\tau)} & = r^2(z) \end{aligned} \qquad \quad \begin{aligned} 1 - \lambda(\tau) & = \bigl( s \circ r \bigr)(z) \\[5pt] \dfrac {\lambda(\tau)}{\lambda(\tau) - 1} & = s(z) \\[5pt] \dfrac {1}{\lambda(\tau)} & = \bigl( r \circ s \bigr)(z) \end{aligned} \qquad \quad \begin{aligned} J_{2,0}(\tau) & = 256 \, \dfrac {1-z}{z^2} \\[5pt] J(\tau) & = \dfrac {4}{27} \, \dfrac {\bigl( z^2 - z + 1 \bigr)^3}{z^2 \, \bigl(z - 1 \bigr)^2} \end{aligned}
in terms of the rational functions $r(z) = \dfrac {z-1}{z}$ and $s(z) = \dfrac {z}{z-1}$. Since $s^2 = r^3 = (s \circ r)^2 = 1$, we see that the following subgroup of $GL_2(\mathbb C)$ acts on $z = \lambda(\tau)$:
$G = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} -1 & 1 \\ -1 & 0 \end{matrix} \right], \ \left[ \begin{matrix} 0 & -1 \\ 1 & -1 \end{matrix} \right], \ \left[ \begin{matrix} -1 & 1 \\ \ \ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & \ \ 0 \\ 1 & -1 \end{matrix} \right], \ \left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right] \right \} \simeq \langle s, \, r \rangle \simeq S_3.$
We call this realization of the symmetric group the Anharmonic Group.

Let’s focus on the subgroups of $S_3$ for the moment. It is easy to check that $J_{2,0}(\tau)$ is invariant under $s$ while $J(\tau)$ is invariant under both $s$ and $r$. In other words, we may think of $S_3 \simeq \langle s, \, r \rangle$ as the Galois group of the extension $\mathbb C(z)$ over $\mathbb C \biggl( \dfrac {4}{27} \, \dfrac {(z^2 - z + 1)^3}{z^2 \, (z-1)^2} \biggr)$; and we may think of the subgroup $Z_2 \simeq \{ 1, \, s \}$ as the Galois group of the extension $\mathbb C(z)$ over $\mathbb C \biggl( \dfrac {4 \, (z-1)}{z^2} \biggr)$. Similarly, we may think of $\{ 1, \, s \, r \}$ and $\{ 1, \, r \, s \}$ as the Galois groups of $\mathbb C(z)$ over $\mathbb C \bigl( 4 \, z \, (1-z) \bigr)$ and $\mathbb C \biggl( \dfrac {4 \, z}{(z+1)^2} \biggr)$, respectively.

### Open Question

What about the subgroup $Z_3 \simeq \{ 1, \, r, \, r^2 \}$? This corresponds to the Galois group $\mathbb C(z)$ over $\mathbb C \biggl( \dfrac {1}{(1 + 2 \, \zeta_3)^3} \, \dfrac {(z + \zeta_3)^3}{z \, (1-z)} \biggr)$. (Recall that $\zeta_n = e^{2 \pi i/n}$ is a root of unity. It appears that the function
\begin{aligned} J_{2,?}(\tau) & = -16 \, \bigl( z + r(z) + r^2(z) \bigr) \\[5pt] & = 16 \, \dfrac {\lambda(\tau)^3 - 3 \, \lambda(\tau) + 1}{\lambda(\tau) \, \bigl( 1-\lambda(\tau) \bigr)} = \dfrac {1}{q_2} - 24 - 492 \, q_2 + \cdots \end{aligned}
is a nontrivial function which has the Riemann surface $X_?(2) = \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] X(1) \cup \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right] X(1)$ as its domain of definition. Is this function a classical one in the literature? The coefficients do not appear in the On-Line Encyclopedia of Integer Sequences.