Moduli of abelian varieties

Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space $$\mathcal{M}_{1,1}$$ over characteristic 0 constructed as a quotient of the upper-half plane by the action of $$SL_2(\mathbb{Z})$$, there is an analogous construction for abelian varieties $$\mathcal{A}_g$$ using the Siegel upper half-space and the symplectic group $$\operatorname{Sp}_{2g}(\mathbb{Z})$$.

Principally polarized Abelian varieties
Recall that the Siegel upper-half plane is given by "$H_g = \{ \Omega \in \operatorname{Mat}_{g,g}(\mathbb{C}) : \Omega^T =\Omega, \operatorname{Im}(\Omega) > 0 \} \subseteq \operatorname{Sym}_g(\mathbb{C})$"which is an open subset in the $$g\times g$$ symmetric matrices (since $$\operatorname{Im}(\Omega) > 0$$ is an open subset of $$\mathbb{R}$$, and $$\operatorname{Im}$$ is continuous). Notice if $$g=1$$ this gives $$1\times 1$$ matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then 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 $$\Omega^{-1}$$ page 34. 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 the moduli space of principally polarized Abelian varieties with level n-structure because it creates a rigidification of the moduli problem which gives a moduli functor 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.