Reduction of moduli spaces of cyclic Galois covers

In this exposition, I will give an introduction to the problems of compactification and reduction of the moduli spaces of Galois covers. The most classical examples are modular curves with level structures, which over the complex number field $\mathbb{C}$ are classically known as quotients of the upper half plane by various congruence subgroups of $SL_2(\mathbb{Z})$. In the first part of the talk, I will talk about the work of Katz-Mazur of extending the modular curves to $Spec(\mathbb{Z})$ and the geometry of their bad reductions modulo primes dividing the levels. In the later part, I will discuss the higher genus generalization of the compactification problem, with possibly wild ramifications.