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Section 2.5 Riemann Surfaces and Discrete Groups II (Jim)

Subsection 2.5.1 Moduli space of compact Riemann surfaces with genus \(g\)

\(g =0\text{.}\) Uniformization tells us that up to isomorphisms all Riemann surfaces of genus \(0\) are \(\PP^1\) hence the moduli space \(\mathcal M _0 = \{\text{pt}\}\text{.}\)

\(g = 1\text{.}\) Uniformization tells us that each Riemann surface of genus \(1\) is a torus and can be written as \(\CC/ \omega_1\ZZ+ \omega_2\ZZ \to \CC/(\ZZ \oplus \tau \ZZ)\text{,}\) with \(\tau = \pm \omega_1/\omega_2\text{.}\)

Idea: Existence of

\begin{equation*} \CC/ \Lambda_{\tau_1} \xrightarrow{\sim} \CC/\Lambda_{\tau_2} \end{equation*}

with \(\bar T (\lb 0 \rb) = \lb 0 \rb\) is equivalent to the existence of \(T \in \Aut(\CC)\) (choose \(T(z) = wz\)) such that \(w(\ZZ\oplus \tau_1 \ZZ) = \ZZ\oplus \tau_2\ZZ\text{.}\) This in turn is equivalent to the existence of

\begin{equation*} A,A' \in \GL_2(\ZZ) \end{equation*}

s.t. \(\det (A) = \det(A') = \pm 1\) so that

\begin{equation*} \begin{pmatrix} w\\ w\tau_1 \end{pmatrix} A \begin{pmatrix}1\\ \tau_2\end{pmatrix} = A' \begin{pmatrix} w \\ w\tau_1 \end{pmatrix} \end{equation*}
\begin{equation*} \implies \tau_q = A \psi_2 = \frac{a\tau_2 + b}{c\tau_2 + d} \end{equation*}

and \(A \in \PSL_2(\RR)\text{.}\) Implies \(A\in \PSL_2(\ZZ)\) as both \(\tau_1,\tau_2 \in \HH\text{.}\) Conversely if

\begin{equation*} \tau_1 = \frac{a\tau_2 + b}{c\tau_2 + b} \end{equation*}

isomorphism is induced by \(T(z) = (c\tau_2 + d) z\text{.}\)

\(g \gt 1\) \(\mathcal M_g\) is a complex variety of dimension \(3g-3\text{.}\) Uniformization tells us that describing a Riemann surface amounts to specifying \(2g\) real \(2\times 2\) matrices \(\{\gamma_i\}_{i=1}^{2g}\) such that

  1. \(\det(\gamma_i) = 1\) which implies that \(\gamma_i\) depends on 3 real parameters so we have a total of \(6g\text{.}\)
  2. \(\prod_{i=1}^g \lb \gamma_i, \gamma_{g+i}\rb = \begin{pmatrix} 1\amp 0 \\ 0\amp 1\end{pmatrix}\) 3 relations, so \(6g- 3\text{.}\) Since for any \(\gamma \in \PSL_2(\RR)\) \(\Gamma = \langle \gamma_i \rangle\) and \(\gamma\Gamma \gamma\inv\) uniformize isomorphic Riemann surfaces implies \(6g-6\) real parameters, so \(3g-3\) complex parameters.

Subsection 2.5.2 Monodromy

Let \(f\colon S_1 \to S\) be a morphism of degree \(d\) ramified over \(y_1,\ldots, y_n \in S\text{.}\) For \(y\in S\smallsetminus \{y_1,\ldots, y_n\}\) we have a group homomorphism

\begin{equation*} M_f\colon \pi_1(S \smallsetminus \{y_1, \ldots, y_n\} )\to \operatorname{Bij} (f\inv (y)) \end{equation*}
\begin{equation*} \gamma \mapsto M_f(\gamma) = \sigma_\gamma\inv\text{.} \end{equation*}

\(\sigma_\gamma\) is defined as follows:

\begin{equation*} \gamma \in \pi_1(S\smallsetminus \{ y_1,\ldots, y_n\}) \end{equation*}

lifts to a path \(\widetilde \gamma\) from \(x\in f\inv(y)\) to another \(x' \in f\inv(y)\) set \(\sigma_\gamma(x) =x'\text{.}\) If we number the points in \(f\inv (y)\) we may think of \(M_f(\pi_1)\subseteq \Sigma_d\text{,}\) via some \(\phi\colon \{1,\ldots, d\}\to f\inv (y)\text{.}\) \(\operatorname{Mon}(f)\) is the image of \(M_f(\pi_1)\) in \(\Sigma_d\text{.}\)

Monodromy and Fuchsian groups.

Let

\begin{equation*} \pi \colon \HH/\Gamma_1 \to \HH/ \Gamma \end{equation*}

be the Fuchsian group representation of the map

\begin{equation*} f\colon S_1 \to S\ni y\text{.} \end{equation*}

Identifications \(y = \lb z_0 \rb_\Gamma\) for some \(z_0 \in \HH\text{.}\)

\begin{equation*} \pi_1 (S\smallsetminus \{y_1,\ldots, y_n\}) \simeq \Gamma \end{equation*}
\begin{equation*} f\inv(y ) = \{ [\beta z_0]_{\Gamma_1}\} \end{equation*}

where \(\beta\) runs along a set of representatives of \(\Gamma_1\backslash \Gamma\text{.}\)

\begin{equation*} M_f \colon \Gamma \to \operatorname{Bij} (\Gamma_1\backslash \Gamma) \end{equation*}
\begin{equation*} \gamma \mapsto M_f(\gamma) \end{equation*}
\begin{equation*} \implies \gamma \sim \pi_1([z_0,\gamma(z_0)]) \end{equation*}

where \(\lb z_0, \gamma(z_0) \rb\) is a path in \(\HH\text{.}\) Lift this loop to \(\HH/\Gamma_1\) is the path \(\pi_{\Gamma_1}(\beta \lb z_0, \gamma_0(z_0)\rb)\text{.}\) which corresponds to \(\Gamma_1 \beta \gamma\text{,}\) this implies \(\sigma_\gamma(\Gamma_1\beta) = \Gamma_1 \beta\gamma\text{.}\)

characterize morphisms by monodromy. Let \(f_i\) have degree 2, non conjugate.

Special case: Assume \(S = \PP^1\) and \(n=3\text{.}\)

\begin{equation*} \Gamma = \Gamma(2) = \{ A \in \PSL_2(\ZZ) : A = \id \pmod 2\} \end{equation*}
\begin{equation*} = \pi_1 (S' \smallsetminus \{0,1,\infty\}) \end{equation*}

is generated by \(\gamma_1, \gamma_2\) so any map \(M_f \colon \Gamma(2) \to \Sigma_d\) is determined by images of \(\gamma_1, \gamma_2\text{.}\)

Subsection 2.5.3 Galois coverings

Definition 2.5.4.

A covering \(f\colon S_1\to S_2\) is Galois (or regular, or normal) if the covering group

\begin{equation*} \Aut(S,f) = \{ h\in \Aut(S_1) : f\circ h= f\} = G \end{equation*}

acts transitively on each fibre. With this notion we can think of \(S_1 \to S_1/G\text{.}\)

Example 2.5.6.

Hyperelliptic covers of \(\PP^1\) given by

\begin{equation*} S = \{ y^2 = \prod_{i=1}^N (x- a_i)\} \to \PP^1 \end{equation*}
\begin{equation*} (x,y) \to x \end{equation*}

covering group \(G\) is order 2 generated by the involution \(J(x,y) = (x,-y)\text{.}\)

Subsection 2.5.4 Normalization of coverings of \(\PP^1\)

Let \(f\colon S \to \PP^1\) be a cover of \(\deg d \gt 0\) with \(\operatorname{Mon}(f) \le \Sigma_d\text{.}\)

The normalisation

\begin{equation*} \tilde f \colon \tilde S \to \PP^1 \end{equation*}

associated to \(f\) has \(\operatorname{Mon} (f) \cong \Aut( \tilde S, \tilde g\) and \(\tilde f ^* \colon M(\PP^1 ) \to M(\PP^1) \) is the normalisation of the extension

\begin{equation*} f^* \colon M(\PP^1 ) \hookrightarrow M(S) \end{equation*}

Normalization of extensions \(K \hookrightarrow L\) is a Galois extension of \(K \) of lowest possible degree containing \(L\text{.}\)

Definition 2.5.8.

Normalization of \(f\colon S\to \PP^1\) \(\deg d \gt 0 \) is a Galois covering \(\tilde f \colon \tilde S \to \PP^1 \) of lowest possible degree s.t. \(\exists \pi \colon \tilde S \to S\) with the diagram commuting.

Interpretation in terms of Fuchsian groups:

Example 2.5.11.

Let \(F(x,y) = y^2x - (y-1)^3\) consider

\begin{equation*} S_F \to \PP^1 \end{equation*}
\begin{equation*} (x,y) \to x \end{equation*}

\(S_F\) has genus 0. \(S_F \to \PP^1\) is of degree 3 and ramified at most over \(0, \frac{-27}{4}, \infty\text{.}\) \(\operatorname{Mon}(x) \simeq \Sigma_3\) so not a normal covering. Normalization of \((S_F, x)\) is \(S_{\tilde F} , \tilde x)\) where

\begin{equation*} \tilde F (x,y) = y^2 ( 1-y)^2 x + (1-y + y^2) \end{equation*}