Moduli Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 10 September 2012

Elliptic curves

1. $X\left(1\right)$ is the moduli space of elliptic curves $E$.
2. $X_0\left(N\right)$ is the moduli space of $(E,C)$ where $E$ is an elliptic curve and $C\subseteq E\left(N\right)$ is a subgroup which is cyclic of order $N$.
3. $X_1\left(N\right)$ is the moduli space of $(E,p)$ where $E$ is an elliptic curve and $p\in E\left(N\right)$ with $p$ of order $N$.
4. $X\left(N\right)$ is the moduli space of $(E,\phi )$ where $E$ is an elliptic curve and ${\displaystyle \phi :E\left[N\right]⟶\raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$N\mathbb{Z}$}\right.\times \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$N\mathbb{Z}$}\right.}$ is an isomorphism.

Let $G=S{L}_2\left(ℝ\right)$, $K=S{O}_2\left(ℝ\right)$. Let $N\in {\mathbb{Z}}_{>0}$ and define subgroups $\Gamma \left(N\right)$, ${\Gamma }_1\left(N\right)$, ${\Gamma }_0\left(N\right)$ of $S{L}_2\left(\mathbb{Z}\right)$ by

$$\begin{array}{ccc}\Gamma \left(N\right)& =& \{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\equiv \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{mod}N\}\\ \left.\cap \right|\\ {\Gamma }_1\left(N\right)& =& \{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\equiv \left(\begin{array}{cc}1& ✶\\ 0& 1\end{array}\right)\text{mod}N\}\\ \left.\cap \right|\\ {\Gamma }_0\left(N\right)& =& \{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\equiv \left(\begin{array}{cc}✶& ✶\\ 0& ✶\end{array}\right)\text{mod}N\}\end{array}$$

Provide bijections:

$$\begin{array}{ccc}X\left(N\right)& ⟷& \Gamma \left(N\right)\setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\\ X_1\left(N\right)& ⟷& {\Gamma }_1\left(N\right)\setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\\ X_0\left(N\right)& ⟷& {\Gamma }_0\left(N\right)\setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\text{.}\end{array}$$

Abelian varieties [SU, Theorem 2.10 and Proposition 2.12]

Let $d_1,\dots ,d_g\in {\mathbb{Z}}_{>0}$ with $d_1\mid d_2\mid \dots \mid d_g$ and

$$\begin{array}{ccc}\Delta & =& \left(\begin{array}{ccc}{d}_1& & 0\\ & \ddots & \\ 0& & d_g\end{array}\right)\end{array}$$

Let $n\in {\mathbb{Z}}_{>0}$,

1. ${𝒜}_{\Delta }$ be the moduli space of polarized abelian varieties of type $\Delta$, and let
2. ${𝒜}_{\Delta }\left(n\right)$ be the moduli space of level $n$ polarized abelian varieties of type $\Delta$.

The Siegel upper half plane of degree $g$ is

$${𝒢}_g=\{\tau \in M_g\left(ℂ\right)\mid {\tau }^t=\tau \text{and}\text{Im}\tau >0\}$$

Define

$$\begin{array}{ccc}{\Gamma }_{\Delta }& =& \{M\in G{L}_{2g}\left(\mathbb{Z}\right)\mid M\left(\begin{array}{cc}0& \Delta \\ -\Delta & 0\end{array}\right)M^t=\left(\begin{array}{cc}0& \Delta \\ -\Delta & 0\end{array}\right)\}\\ {\Gamma }_{\Delta }\left(n\right)& =& \{M\in {\Gamma }_{\Delta }\mid M={\text{Id}}_{2g}\text{mod}n\}\text{.}\end{array}$$

1. Find $G$ and $K$ such that ${𝒢}_g\cong \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.$.
2. Provide bijections $$\begin{array}{ccc}{𝒜}_{\Delta }& ⟷& {\Gamma }_{\Delta }\setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\\ {𝒜}_{\Delta }\left(n\right)& ⟷& {\Gamma }_{\Delta }\left(n\right)\setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\end{array}$$

From [SU, p125]

$${\displaystyle {𝒢}_g\cong \raisebox{1ex}{$S_p(2g,ℝ)$}\!\left/ \!\raisebox{-1ex}{$B$}\right.}$$

where $B=P\cap G$ and

$$P=\{g\in S_p(2g,ℂ)\mid g{F}_0^p=F_0^p,1\le p\le \text{something}\}\text{.}$$

From [SU, p47 (2.25)]

$${\Gamma }_{\Delta }=\{M\in GL(2g,\mathbb{Z})\mid M\left(\begin{array}{cc}0& \Delta \\ -\Delta & 0\end{array}\right)M^t=\left(\begin{array}{cc}0& \Delta \\ -\Delta & 0\end{array}\right)\}\text{.}$$

Complex tori [SU, (2.5)]

Let $g\in {\mathbb{Z}}_{>0}$ and

$${𝒥}_g\text{the moduli space of complex tori of dimension}g\text{.}$$

Let

$$ℳ=\{\Omega \mid \Omega \text{is a}2g\times g\text{matrix,}\text{det}(\Omega ,\bar{\Omega })\ne 0\}\text{.}$$

1. Find $G$ and $K$ such that $\raisebox{1ex}{$ℳ$}\!\left/ \!\raisebox{-1ex}{$G{L}_g\left(ℂ\right)$}\right.\simeq \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\text{.}$
2. Provide a bijection ${𝒥}_g⟷\Gamma \setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$$}\right.$, $$\text{where}\Gamma =G{L}_{2g}\left(\mathbb{Z}\right)\text{.}$$

Riemann surfaces [SU, p.96] and [SU, p.93]

Let $g\in {\mathbb{Z}}_{>0}$ and

$${ℳ}_g\text{the moduli space of compact Riemann surfaces of genus}g\text{.}$$

1. Find $\Gamma$, $G$ and $K$ and a bijection $${ℳ}_g⟷\Gamma \setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.$$ Let

   ${\pi }_g$ the Trickmuller space of compact Riemann surfaces of genus $g$

   i.e. the space of pairs $(R,H)$ where $R$ is a compact Riemann surface and $H$ is a homotopy class of orientation preserving homeomorphisms $R_0⟶R$ modulo equivalence.

2. Find $\Gamma$, $G$ and $K$ and a bijection $${\pi }_g⟷\Gamma \setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\text{.}$$

Hodge structures [SU, p124], [SU, (3.18)] and [SU, §3.2.2].

Let $H_0$ be a complex vector space. Let

$$D\text{be the moduli space of polarized Hodge structures on}{H}_0$$

1. Find $G$ and $D$ and a bijection $D⟷\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.$.
   Let $S$ be a complex manifold.
   To each polarized variation of $\mathbb{Z}$–Hodge structures of weight $w$, $$\mathscr{H}=(\mathscr{H},ℱ,\Psi ),$$ associate a period mapping $$\varphi :S⟶\Gamma \setminus D\text{.}$$

Principal $G$–bundles on a curve $C$ [SU, p273]

Let $C$ be a curve and let $${\displaystyle M_C^G}$$ be the moduli stack of principal $G$–bundles on $C$.

Let $$G=G\left(ℂ\left(\left(z\right)\right)\right),K=G\left(ℂ\left[\left[z\right]\right]\right),\text{and}\Gamma =G\left(A_C\right),$$ where $$A_C=H^{˚}(C\setminus \left\{Q_0\right\},{𝒪}_C)\hookrightarrow ℂ\left[\left[z\right]\right]\text{.}$$

1. Provide a bijection $$M_C^G⟷\Gamma \setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.\text{.}$$

The definition of a generalized (or non-abelian) theta function is on [SU, p278] (essentially the last sentence of the book)

Notes and References

This is a typed copy of handwritten notes by Arun Ram entitled Moduli Spaces. They were to Norm Do and written on 28.12.2011.

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