Holomorphic differentials of Klein four covers

In this thesis, we combine methods from the representation theory of groups and the geometry of curves. A group is a set with one operation. We focus on a particular group with four elements, called a Klein four group. The representation theory of this group uses linear algebra to understand its structure. Even though this group has only four elements, it turns out that in characteristic two one needs infinitely many matrices to describe the building blocks of every representation, called the indecomposable representations. In this thesis, we focus on representations that are described geometrically. More precisely, we consider the space of holomorphic differentials of a curve in characteristic two on which a Klein four group acts. Under a convenient assumption, we determine the precise building blocks of this representation. Moreover, we analyze our convenient assumption in the context of more general Klein four group actions on curves in characteristic two.