Abstract:

Classifying the module structure of the space of holomorphic differentials originated from the German mathematician Erich Hecke. His classical problem has been studied in a more general setting. Currently, there is still cases that are open to investigation. We will begin by describing valuations and smooth projective curves. Afterwards, we will discuss the topic of Galois actions on these curves. This will induce a group action on the function field of the curve and its space of holomorphic differentials. Hence, the space of holomorphic differentials has a module structure. Time permitting, we will also try to look at a concrete example.