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 C {\displaystyle \mathbb {C} } constructed as a stacky quotient of the upper-half plane by the action of S L 2 ( Z ) {\displaystyle 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 Sp 2 g ⁡ ( Z ) {\displaystyle \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 the complex numbers

Principally polarized Abelian varieties

Recall that the Siegel upper half-space H g {\displaystyle H_{g}} is the set of symmetric g × g {\displaystyle g\times g} complex matrices whose imaginary part is positive definite. This an open subset in the space of g × g {\displaystyle g\times g} symmetric matrices. Notice that if g = 1 {\displaystyle g=1}, H g {\displaystyle 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 Ω ∈ H g {\displaystyle \Omega \in H_{g}} gives a complex torus

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

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

X Ω ≅ X Ω ′ ⟺ Ω = M Ω ′ {\displaystyle X_{\Omega }\cong X_{\Omega '}\iff \Omega =M\Omega '} for M ∈ Sp 2 g ⁡ ( Z ) {\displaystyle M\in \operatorname {Sp} _{2g}(\mathbb {Z} )}

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

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

which gives a Deligne-Mumford stack over Spec ⁡ ( C ) {\displaystyle \operatorname {Spec} (\mathbb {C} )}. If this is instead given by a GIT quotient, then it gives the coarse moduli space A g {\displaystyle 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 Ω , Z / n ) ≅ 1 n ⋅ L / L ≅ n -torsion of X Ω {\displaystyle H_{1}(X_{\Omega },\mathbb {Z} /n)\cong {\frac {1}{n}}\cdot L/L\cong n{\text{-torsion of }}X_{\Omega }}

where L {\displaystyle L} is the lattice Ω Z g + Z g ⊂ C 2 g {\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

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

and define

A g , n = Γ ( n ) ∖ H g {\displaystyle A_{g,n}=\Gamma (n)\backslash H_{g}}

as a quotient variety.

See also