Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space over characteristic 0 constructed as a quotient of the upper-half plane by the action of ,[1] there is an analogous construction for abelian varieties using the Siegel upper half-space and the symplectic group .[2]
Constructions over characteristic 0
Principally polarized Abelian varieties
Recall that the Siegel upper-half plane is given by[3]
which is an open subset in the symmetric matrices (since
is an open subset of
, and
is continuous). Notice if
this gives
matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point
gives a complex torus
with a principal polarization from the matrix
[2]page 34. It turns out all principally polarized Abelian varieties arise this way, giving
the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where
for
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hence the moduli space of principally polarized abelian varieties is constructed from the stack quotient
which gives a Deligne-Mumford stack over . If this is instead given by a GIT quotient, then it gives the coarse moduli space
.
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
where is the lattice
. 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
and define
as a quotient variety.