Holomorphic differentials of alternating four covers

This thesis combines methods from two subareas of abstract algebra: group representation theory and arithmetic geometry. A group is a set with one operation. We concentrate on a particular group with twelve elements, called an alternating group on four letters. The representations of this group are given by matrices that represent the group elements in such a way that they preserve the group operation. Even though this group has only twelve elements, in characteristic two one needs infinitely many matrices to describe the so-called indecomposable representations, which are the building blocks of all representations. We focus on particular representations that arise naturally in arithmetic geometry. More specifically, we study the space of holomorphic differentials of a curve in characteristic two on which an alternating group on four letters acts. Under some mild assumptions, we provide precise formulas for the building blocks of this representation and with which frequency they occur.