Lecture 13: Wednesday, September 18, 2013

In Lecture 11 and Lecture 12, we discussed examples of Riemann Surfaces X and their relation with the Riemann Sphere X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R). Today, we put things into perspective by considering the larger picture: we discuss tesselations of the upper half plane, covering spaces, and deck transformations.

The Modular Group

We have seen several functions \mathbb H^\ast \to \mathbb P^1(\mathbb C) which are defined on the extended upper half plane \mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C). For example,
\begin{aligned}   J(\tau) & = \dfrac {\displaystyle \left[ 1 + 240 \sum_{n=1}^{\infty} \sigma_3(n) \, q_1^n \right]^3}{\displaystyle 1728 \, q_1 \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)^{24}} \\[5pt]  J_{2,0}(\tau) & = \left( \dfrac {\eta(\tau)}{\eta(2 \, \tau)} \right)^{24} = \displaystyle \left[ q_1 \, \prod_{n=1}^{\infty} \bigl( 1 + q_1^n \bigr)^{24} \right]^{-1} \\[5pt]  \lambda(\tau) & = 16 \, \dfrac {\eta(\tau/2)^8 \, \eta(2 \, \tau)^{16}}{\eta(\tau)^{24}} = \displaystyle 16 \, q_2 \prod_{n=1}^{\infty} \dfrac { \bigl(1+q_2^{2n} \bigr)^{16}}{\bigl( 1 + q_2^n \bigr)^8} \\[5pt]  \eta(\tau) & = \displaystyle q_1^{1/24} \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)  \end{aligned}
where q_h = e^{2 \pi i \tau/h} and \sigma_h(n) = \sum_{d | n} d^h is the divisor function. These functions were constructed by considering various subsets I in SL_2(\mathbb Z) = \left \{ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in \text{Mat}_{2 \times 2}(\mathbb Z) \, \biggl| \, a \, d - b \, c = 1 \right \}.

In this week’s homework assignment, we see that SL_2(\mathbb Z) is generated by the 2 \times 2 matrices S = \left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right] and T = \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right]. Using the map SL_2(\mathbb Z) \to \text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr), we see that S, S \, T, and T^{-1} map to the Möbius transformations \gamma_0(z) = - \dfrac {1}{z}, \gamma_1(z) = - \dfrac {1}{z+1}, and \gamma_{\infty}(z) = z - 1, respectively. In fact, the image of SL_2(\mathbb Z) is the triangle group
\Gamma(1) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \biggl| \, {\gamma_0}^2 = {\gamma_1}^3 = \gamma_0 \, \gamma_1 \, \gamma_\infty \right \rangle = D(2,3,\infty) \simeq PSL_2(\mathbb Z).
This is called the modular group.

The points P = \zeta_4 = \sqrt{-1}, Q = {\zeta_6}^2 = - \dfrac 12 + i \, \dfrac {\sqrt{3}}{2}, and R = i \, \infty are the unique fixed points of \gamma_0, \gamma_1, and \gamma_\infty, respectively, so that we find a 0^\circ - 60^\circ - 90^\circ hyperbolic triangle V = \{ P, \, Q, \, R \}. The images \gamma \, V = \{ \gamma \, P, \ \gamma \, Q, \ \gamma \, R \} of the triangles for \gamma \in \Gamma(1) yield a tessellation of the extended upper half plane \mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C).

Hecke’s Modular Group

We can completely generalize this. Fix an integer n \geq 3, and consider the root of unity \zeta_{2n} = \cos \dfrac {\pi}{n} + i \, \sin \dfrac {\pi}{n}. Upon defining the positive real number \lambda = \zeta_{2n} + {\zeta_{2n}}^{-1} = 2 \, \cos \dfrac {\pi}{n}, we consider the 2 \times 2 matrices S = \left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right] and T = \left[ \begin{matrix} 1 & \lambda_n \\ 0 & 1 \end{matrix} \right]. The group \left \langle S, \, T \, \biggl| \, S^2 = T^n = -I_2 \right \rangle \subseteq SL_2(\mathbb R), so we consider the same map SL_2(\mathbb R) \to \text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr) as before. We see that S, S \, T, and T^{-1} map to the Möbius transformations \gamma_0(z) = - \dfrac {1}{z}, \gamma_1(z) = - \dfrac {1}{z+ \lambda_n}, and \gamma_{\infty}(z) = z - \lambda_n, respectively. In fact, the image of the group generated by S and T is the triangle group
H_n = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \biggl| \, {\gamma_0}^2 = {\gamma_1}^n = \gamma_0 \, \gamma_1 \, \gamma_\infty \right \rangle = D(2,n,\infty) \hookrightarrow PSL_2(\mathbb R).
This is called the Hecke modular group. Since this is a discrete subgroup of PSL_2(\mathbb R), we call H_n a Fuchsian group.

The points P = \zeta_4, Q = {\zeta_{2n}}^{n-1}, and R = i \, \infty are the unique fixed points of \gamma_0, \gamma_1, and \gamma_\infty, respectively, so that we find a 0^\circ - (180/n)^\circ - 90^\circ hyperbolic triangle V = \{ P, \, Q, \, R \}. The images \gamma \, V = \{ \gamma \, P, \ \gamma \, Q, \ \gamma \, R \} of the triangles for \gamma \in H_n yield a triangulation of the extended upper half plane \mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C).

When n = 3, we see that H_3 = \Gamma(1) is the usual modular group. In fact, the Fundamental Domain for this group is the set
\begin{aligned} X(1) & = \left \{ x + i \, y \in \mathbb C \ \biggl| \ 1 < x^2 + y^2, \ - \dfrac 12 < x < \dfrac 12, \  0 < y \right \} \\[5pt] & \cup \left \{ x + i \, y \in \mathbb C \ \biggl| \ x = - \dfrac 12, \  \dfrac {\sqrt{3}}{2} \leq y \right \} \\[5pt] & \cup \left \{ x + i \, y \in \mathbb C \ \biggl| \ 1 = x^2 + y^2, \ - \dfrac 12 \leq x \leq 0, \  0 < y \right \}  \cup \{ i \, \infty \}. \end{aligned}
When n \geq 4, the description of the Fundamental Domain gets a bit more complicated. A very nice paper by Ronald Evans entitled A fundamental region for Hecke’s modular group sheds some light on this problem.

Covering Spaces

Meromorphic functions in general and elliptic modular functions in particular can be cumbersome and unintuitive, but we introduce a rather technical idea to simplify the discussion.

Let \left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right) and \left( Y, \, \{ V_\beta \}, \, \{ \nu_\beta \} \right) be compact, connected Riemann surfaces for finite indexing sets I and J, respectively. A map \Pi: X \to Y is called an n-fold covering map or an n-fold covering projection if, for each Q \in Y we can find an open set V \subseteq Y containing Q (that is, \nu_\beta(V) \subseteq \mathbb C is open for each \beta \in J) such that the following holds:


  • Surjectivity: The inverse image \Pi^{-1}(V) = U_1 \cup U_2 \cup \cdots \cup U_n is a union of nonempty open sets U_i \subseteq X, that is, \mu_\alpha(U_i) \subseteq \mathbb C is open for each \alpha \in I and 1 \leq i \leq n.

  • Disjoint Sheets: U_i \cap U_j = \emptyset for 1 \leq i, j \leq n with i \neq j.

  • Local Homeomorphisms: \Pi(U_i) \simeq V in Y for 1 \leq i \leq n.

  • Analytic: For every \alpha \in I and \beta \in J, each \bigl( \nu_\beta \circ \Pi \bigr)(U_\alpha) is a connected, open subset of \mathbb C. Moreover, the transition maps \omega_{\beta \alpha} = \nu_\beta \circ \Pi \circ {\mu_\alpha}^{-1} are analytic functions.

    \begin{matrix}  \mathbb C & & X & & Y & & \mathbb C \\[10pt]  \uparrow & & \uparrow & & \uparrow & & \uparrow \\[10pt]  \mu_\alpha \bigl( U_\alpha \cap \Pi^{-1}(V_\beta) \bigr) & \overset{{\mu_\alpha}^{-1}}{\longrightarrow} & U_\alpha \cap \Pi^{-1}(V_\beta) & \overset{\Pi}{\longrightarrow} & \Pi(U_\alpha) \cap V_\beta & \overset{\nu_\beta}{\longrightarrow} & \nu_\beta \bigl( \Pi(U_\alpha) \cap V_\beta \bigr) \end{matrix}

An automorphism \gamma \in \text{Aut}(X) is an invertible map \gamma: X \to X such that the composition \gamma \circ {\mu_\alpha}^{-1} is an analytic function on the subset \mu_\alpha(U_\alpha) \subseteq \mathbb C for each index \alpha \in I. Given a covering map \Pi: X \to Y, we denote the deck transformation group as the set
\text{Aut}(\Pi) = \left \{ \gamma \in \text{Aut}(X) \ \biggl| \ \bigl( \Pi \circ \gamma \bigr)(P) = \Pi(P) \ \text{for all} \ P \in X \right \}.
Elements \gamma \in \text{Aut}(\Pi) are called deck transformations.

Constructing Covering Spaces

The examples discussed in the previous lectures considered covering maps of the Riemann Surface Y = X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R).

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), 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}

In general, we construct the objects above as follows. Fix a finite group G. Let I \subseteq SL_2(\mathbb Z) be a finite set so that the disjoint union X = \bigcup_{\gamma \in I} \gamma \, X(1) is a compact, connected Riemann surface endowed with the canonical surjection \Pi: X \twoheadrightarrow X(1) which sends \gamma \, \tau \mapsto \tau. We seek a surjective, meromorphic function \phi: X \to X(1) such that J = f(z) is a rational function of z = \phi(\tau) having automorphism group G.

For instance, recall that in Lecture 12 we considered the sets
\begin{aligned}  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 \} \\[10pt]  J & = \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 \} \\[5pt]  K & = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \right \}  \end{aligned}

Then we have the covering maps

\begin{matrix} X(2) = \bigcup_{\gamma \in I} \gamma \, X(1) & \longrightarrow & X_0(2)  = \bigcup_{\gamma \in J} \gamma \, X(1) & \longrightarrow & X(1) = \bigcup_{\gamma \in K} \gamma \, 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}

explicitly defined by
\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}

The deck transformations can be associated with those Möbius transformations which fix J = f(z). In the specific case above, the rational function J = f(z) is invariant under 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 J = \dfrac {4}{27} \, \dfrac {( z^2 - z+ 1 \bigr)^3}{z^2 \, (z- 1)^2} is invariant under the symmetric group S_3 = \left \langle r, \, s \, \bigl| \, s^2 = r^3 = (s \circ r)^2 = 1 \right \rangle, while J_{2,0}(\tau) = 256 \, \dfrac {1 - z}{z^2} is invariant under the cyclic group Z_2 = \left \langle 1, \, s \right \rangle. This means we have the following diagram of subfields and corresponding Galois groups:

\begin{matrix}  X(2) & \qquad & \mathbb C(\lambda) & \qquad & \left \langle 1 \right \rangle \\[10pt]  \downarrow & & \uparrow & & \downarrow \\[10pt]  X_0(2) & & \mathbb C(J_{2,0}) & & Z_2 \\[10pt]  \downarrow & & \uparrow & & \downarrow \\[10pt]  X(1) & & \mathbb C(J) & & S_3  \end{matrix}

Motivating Questions

Let Y = S^2(\mathbb R) denote the Riemann Sphere. We have seen that there are n-fold covering maps \Pi: X \to Y for certain values of n. Is there a universal cover \phi: \widetilde{Y} \to Y such that each cover \Pi: X \to Y corresponds to a cover \widetilde{\Pi}: \widetilde{Y} \to X such that \phi = \Pi \circ \widetilde{\Pi}? We will return to this question.

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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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One Response to Lecture 13: Wednesday, September 18, 2013

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

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