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.

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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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2 Responses to Lecture 12: Monday, September 16, 2013

  1. Pingback: Lecture 13: Wednesday, September 18, 2013 | Lectures on Dessins d'Enfants

  2. Pingback: MA 59800 Course Syllabus | Lectures on Dessins d'Enfants

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