## Moduli Spaces

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 $\left(E,C\right)$ 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 $\left(E,p\right)$ 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 $\left(E,\phi \right)$ where $E$ is an elliptic curve and $\phi :\phantom{\rule{0.2em}{0ex}}E\left[N\right]⟶ℤ}{Nℤ}×ℤ}{Nℤ}$ is an isomorphism.

Let $G=S{L}_{2}\left(ℝ\right)$, $K=S{O}_{2}\left(ℝ\right)$. Let $N\in {ℤ}_{>0}$ and define subgroups $\Gamma \left(N\right)$, ${\Gamma }_{1}\left(N\right)$, ${\Gamma }_{0}\left(N\right)$ of $S{L}_{2}\left(ℤ\right)$ by

$Γ(N) = { ( a b c d ) ≡ ( 1 0 0 1 ) modN } ∩ Γ1(N) = { ( a b c d ) ≡ ( 1 ✶ 0 1 ) modN } ∩ Γ0(N) = { ( a b c d ) ≡ ( ✶ ✶ 0 ✶ ) modN }$

Provide bijections:

$X(N) ⟷ Γ(N)∖GK X1(N) ⟷ Γ1(N)∖GK X0(N) ⟷ Γ0(N)∖GK.$

## Abelian varieties [SU, Theorem 2.10 and Proposition 2.12]

Let ${d}_{1},\dots ,{d}_{g}\in {ℤ}_{>0}$ with ${d}_{1}\mid {d}_{2}\mid \dots \mid {d}_{g}$ and

$Δ = ( d1 0 ⋱ 0 dg )$

Let $n\in {ℤ}_{>0}$,

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

$𝒢g= { τ∈Mg(ℂ)∣ τt=τand Imτ>0 }$

Define

$ΓΔ = { M∈GL2g(ℤ) ∣M ( 0 Δ -Δ 0 ) Mt= ( 0 Δ -Δ 0 ) } ΓΔ(n) = { M∈ΓΔ∣ M=Id2gmod n }.$
1. Find $G$ and $K$ such that ${𝒢}_{g}\cong G}{K}$.
2. Provide bijections $𝒜Δ ⟷ ΓΔ∖GK 𝒜Δ(n) ⟷ ΓΔ(n)∖GK$

From [SU, p125]

$𝒢g≅ Sp(2g,ℝ) B$

where $B=P\cap G$ and

$P= { g∈Sp(2g,ℂ) ∣gF0p= F0p,1≤p≤ something } .$

From [SU, p47 (2.25)]

$ΓΔ= { M∈GL(2g,ℤ) ∣M ( 0 Δ -Δ 0 ) Mt= ( 0 Δ -Δ 0 ) } .$

## Complex tori [SU, (2.5)]

Let $g\in {ℤ}_{>0}$ and

$𝒥g the moduli space of complex tori of dimensiong.$

Let

$ℳ= { Ω∣Ω is a2g×gmatrix, det (Ω,Ω‾)≠0 } .$
1. Find $G$ and $K$ such that $ℳ}{G{L}_{g}\left(ℂ\right)}\simeq G}{K}\text{.}$
2. Provide a bijection ${𝒥}_{g}⟷\Gamma \setminus G}{}$, $whereΓ=GL2g (ℤ).$

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

Let $g\in {ℤ}_{>0}$ and

$ℳg the moduli space of compact Riemann surfaces of genusg.$
1. Find $\Gamma$, $G$ and $K$ and a bijection $ℳg⟷Γ∖ GK$ Let
${\pi }_{g}$ the Trickmuller space of compact Riemann surfaces of genus $g$
i.e. the space of pairs $\left(R,H\right)$ 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 $πg⟷Γ∖ GK.$

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

Let ${H}_{0}$ be a complex vector space. Let

$Dbe the moduli space of polarized Hodge structures onH0$
1. Find $G$ and $D$ and a bijection $D⟷G}{K}$.
Let $S$ be a complex manifold.
To each polarized variation of $ℤ$–Hodge structures of weight $w$, $ℋ=(ℋ,ℱ,Ψ),$ associate a period mapping $ϕ:S⟶Γ ∖D.$

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

Let $C$ be a curve and let $MCG$ be the moduli stack of principal $G$–bundles on $C$.

Let $G=G(ℂ((z))), K=G(ℂ[[z]]), and Γ=G(AC),$ where $AC=H˚ ( C∖{Q0}, 𝒪C ) ↪ℂ[[z]].$

1. Provide a bijection $MCG⟷Γ∖ GK.$

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.