From hyperelliptic to superelliptic curves
There are two different ways of studying an algebraic curves; function fields $L/k(x)$ or coverings of $\P^1$.
The goal of this book is the study of algebraic curves via coverings in the tradition of Riemann, Clebsch, Hurwitz,
Severi, Grothendieck, et al.
We aim to highlight the theories that can be extended and all the open problems that come with this generalization.
We investigate the correspondence from the group theory data of the cover $f: \X \to \P^1$, via Riemann Existence
Theorem (RET), to the field of moduli of $\X$, relations among the thetanulls on $\Jac (\X)$ and the branch points
of $f: \X \to \P^1$, the Hurwitz spaces of coverings with ramification structure as that of $f$. This enables us to use
the full machinery of group theory to study covers and get more information about the arithmetic aspects of curves.