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 constructed as a stacky quotient of the upper-half plane by the action of , there is a moduli space of principally polarised abelian varieties given as a stacky quotient of Siegel upper half-space by the symplectic group . By adding even more extra structure, such as a level n structure, one can go further and obtain a fine moduli space.
Recall that the Siegel upper half-space is the set of symmetric complex matrices whose imaginary part is positive definite. This an open subset in the space of symmetric matrices. Notice that if , 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 gives a complex torus <blockquote></blockquote>with a principal polarization from the matrix <sup>page 34</sup>. 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<blockquote> for </blockquote>hence the moduli space of principally polarized abelian varieties is constructed from the stack quotient<blockquote></blockquote>which gives a Deligne-Mumford stack over . If this is instead given by a GIT quotient, then it gives the coarse moduli space .
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
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<blockquote></blockquote>and define<blockquote></blockquote>as a quotient variety.