# Moduli of abelian varieties

Abelian varieties are a natural generalization of elliptic curves to higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved stack playing the role of a moduli stack for higher-dimensional abelian varieties. One can solve this problem by constructing a moduli stack of abelian varieties equipped with extra structure, such as a principal polarisation. Just as there is a moduli stack of elliptic curves over $\mathbb {C}$ constructed as a stacky quotient of the upper-half plane by the action of $SL_{2}(\mathbb {Z} )$, there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group $\operatorname {Sp} _{2g}(\mathbb {Z} )$. By adding even more extra structure, such as a level n structure, one can go further and obtain a fine moduli space.

## Constructions over characteristic 0
### Principally polarized Abelian varieties

Recall that the Siegel upper half-space $H_{g}$ is the set of symmetric $g\times g$ complex matrices whose imaginary part is positive definite. This an open subset in the space of $g\times g$ symmetric matrices. Notice that if $g=1$, $H_{g}$ consists of complex numbers with positive imaginary part, and is thus the upper half plane, which appears prominently in the study of elliptic curves. In general, any point $\Omega \in H_{g}$ gives a complex torus
   $X_{\Omega }=\mathbb {C} ^{g}/(\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g})$
with a principal polarization $H_{\Omega }$ from the matrix $\Omega ^{-1}$$^{page34}$. It turns out all principally polarized Abelian varieties arise this way, giving $H_{g}$ the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where
   $X_{\Omega }\cong X_{\Omega '}\iff \Omega =M\Omega '$ for $M\in \operatorname {Sp} _{2g}(\mathbb {Z} )$
hence the moduli space of principally polarized abelian varieties is constructed from the stack quotient
   ${\mathcal {A}}_{g}=[\operatorname {Sp} _{2g}(\mathbb {Z} )\backslash H_{g}]$
which gives a Deligne-Mumford stack over $\operatorname {Spec} (\mathbb {C} )$. If this is instead given by a GIT quotient, then it gives the coarse moduli space $A_{g}$.

### Principally polarized Abelian varieties with level n structure

In many cases, it is easier to work with principally polarized Abelian varieties equipped with level n-structure because this breaks the symmetries and gives a moduli space instead of a moduli stack. This means the functor is representable by an algebraic manifold, such as a variety or scheme, instead of a stack. A level n-structure is given by a fixed basis of
    $H_{1}(X_{\Omega },\mathbb {Z} /n)\cong {\frac {1}{n}}\cdot L/L\cong n{\text{-torsion of }}X_{\Omega }$
where $L$ is the lattice $\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g}\subset \mathbb {C} ^{2g}$. Fixing such a basis removes the automorphisms of an abelian variety at a point in the moduli space, hence there exists a bona fide algebraic manifold without a stabilizer structure. Denote
   $\Gamma (n)=\ker[\operatorname {Sp} _{2g}(\mathbb {Z} )\to \operatorname {Sp} _{2g}(\mathbb {Z} )/n]$
and define
   $A_{g,n}=\Gamma (n)\backslash H_{g}$
as a quotient variety.