Constructing moduli spaces of abelian schemes
I will outline the construction of moduli spaces of abelian schemes with polarization and level n structure, following Mumford's _Geometric Invariant Theory_. The procedure is to find a moduli space for abelian schemes with linear rigidification (i.e. an embedding in projective space), then show indirectly that for sufficiently large n, there exists an appropriate quotient of this space which serves as a fine moduli scheme. Using this result, one can show that for arbitrary n, there is a coarse moduli scheme.