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

Constructions over characteristic 0
Principally polarized 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 $$\mathbb {R}$$ , and Im {\displaystyle \operatorname {Im} } \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 $${\displaystyle \Omega \in H_{g}}$$ 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 $$\Omega ^{{-1}}$$ [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 $$A_{g}.$$
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.[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 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.
References

Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
Arapura, Donu. "Abelian Varieties and Moduli" (PDF).
Birkenhake, Christina; Lange, Herbert (2004). Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften (2 ed.). Berlin Heidelberg: Springer-Verlag. pp. 210–241. ISBN 978-3-540-20488-6.
Mumford, David (1983), Artin, Michael; Tate, John (eds.), "Towards an Enumerative Geometry of the Moduli Space of Curves", Arithmetic and Geometry: Papers Dedicated to I.R. Shafarevich on the Occasion of His Sixtieth Birthday. Volume II: Geometry, Progress in Mathematics, Birkhäuser, pp. 271–328, doi:10.1007/978-1-4757-9286-7_12, ISBN 978-1-4757-9286-7

Level n-structures are used to construct an intersection theory of Deligne–Mumford stacks

Schottky problem
Siegel modular variety
Moduli stack of elliptic curves
Moduli of algebraic curves
Hilbert scheme
Deformation Theory