Moduli of abelian varieties

Recall that the Siegel upper-half plane is given by[3]

${\displaystyle H_{g}=\{\Omega \in \operatorname {Mat} _{g,g}(\mathbb {C} ):\Omega ^{T}=\Omega ,\operatorname {Im} (\Omega )>0\}\subseteq \operatorname {Sym} _{g}(\mathbb {C} )}$

which is an open subset in the ${\displaystyle g\times g}$ symmetric matrices (since ${\displaystyle \operatorname {Im} (\Omega )>0}$ is an open subset of ${\displaystyle \mathbb {R} }$, and ${\displaystyle \operatorname {Im} }$ is continuous). Notice if ${\displaystyle g=1}$ this gives ${\displaystyle 1\times 1}$ matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point ${\displaystyle \Omega \in H_{g}}$ gives a complex torus

${\displaystyle X_{\Omega }=\mathbb {C} ^{g}/(\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g})}$

with a principal polarization ${\displaystyle H_{\Omega }}$ from the matrix ${\displaystyle \Omega ^{-1}}$[2]^{page 34}. It turns out all principally polarized Abelian varieties arise this way, giving ${\displaystyle H_{g}}$ the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where

${\displaystyle X_{\Omega }\cong X_{\Omega '}\iff \Omega =M\Omega '}$ for ${\displaystyle M\in \operatorname {Sp} _{2g}(\mathbb {Z} )}$

hence the moduli space of principally polarized abelian varieties is constructed from the stack quotient

${\displaystyle {\mathcal {A}}_{g}=[\operatorname {Sp} _{2g}(\mathbb {Z} )\backslash H_{g}]}$

which gives a Deligne-Mumford stack over ${\displaystyle \operatorname {Spec} (\mathbb {C} )}$. If this is instead given by a GIT quotient, then it gives the coarse moduli space ${\displaystyle A_{g}}$.

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.[4][5] 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

${\displaystyle H_{1}(X_{\Omega },\mathbb {Z} /n)\cong {\frac {1}{n}}\cdot L/L\cong n{\text{-torsion of }}X_{\Omega }}$

where ${\displaystyle L}$ is the lattice ${\displaystyle \Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g}\subset \mathbb {C} ^{2g}}$. 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

${\displaystyle \Gamma (n)=\ker[\operatorname {Sp} _{2g}(\mathbb {Z} )\to \operatorname {Sp} _{2g}(\mathbb {Z} )/n]}$

and define

${\displaystyle A_{g,n}=\Gamma (n)\backslash H_{g}}$

as a quotient variety.