Level Structure

Using the results of the previous sections we now proceed to construct compact moduli stacks for abelian varieties with polarizations and level structure. We present two approaches. The first is based on the theory of logarithmic étale cohomology. The second approach is more in the spirit of Deligne-Rapoport’s construction of compact moduli stacks for elliptic curves with level structure, and is based on our discussion of theta groups in chapter 5. Though we only discuss in this chapter level structures in characteristics prime to the level, we intend in future writings to discuss how to extend this second approach to level structure to compactify moduli spaces also at primes dividing the level. Finally we discuss a modular compactification of certain moduli spaces for polarized abelian varieties with theta level structure as considered in [36].