Moduli of abelian varieties

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.^[4] 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}} ^{[3]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, giving a moduli scheme instead of a moduli stack.^[5][6] In other words, the functor that associates to a test scheme T the set of morphisms X -> T where all geometric fibers are principally polarized abelian varieties with level n-structure (intuitively, a family of objects parameterized by T) is actually representable by a scheme. 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}} . An automorphism of an abelian variety with level structure is an automorphism of the variety that fixes this basis, and there are no such automorphisms other than the identity. Since no points have a stabilizer, the moduli space of principally polarized abelian varieties with level n-structure, which is a priori a stack, is actually a scheme. In this situation, where a family over arbitrary T is the same as a morphism from T into some scheme M (better: arises from a universal family through pullback), we say M is a fine moduli space. 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.