Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space formula_1 over characteristic 0 constructed as a quotient of the upper-half plane by the action of formula_2, there is an analogous construction for abelian varieties formula_3 using the Siegel upper half-space and the symplectic group formula_4.
Constructions over characteristic 0.
Principally polarized Abelian varieties.
Recall that the Siegel upper-half plane is given byformula_5which is an open subset in the formula_6 symmetric matrices (since formula_7 is an open subset of formula_8, and formula_9 is continuous). Notice if formula_10 this gives formula_11 matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point formula_12 gives a complex torus formula_13with a principal polarization formula_14 from the matrix formula_15page 34. It turns out all principally polarized Abelian varieties arise this way, giving formula_16 the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence whereformula_17 for formula_18hence the moduli space of principally polarized abelian varieties is constructed from the stack quotientformula_19which gives a Deligne-Mumford stack over formula_20. If this is instead given by a GIT quotient, then it gives the coarse moduli space formula_21.
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. 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
 formula_22
where formula_23 is the lattice formula_24. 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. Denoteformula_25and defineformula_26as a quotient variety.