Isomorphism of meromorphic function fields on Riemann surfaces (Q1309206)

It is well known that two Riemann surfaces are conformally equivalent if and only if their meromorphic function fields are \(\mathbb{C}\)-isomorphic. If Riemann surfaces are compact, however, an abstract field isomorphism does not always imply conformal equivalence. So it is meaningful to consider an equivalence relation in the set of all compact Riemann surfaces defined by field isomorphisms of their meromorphic function fields. In this paper, the author proves that there are countable (infinite) equivalence classes (i.e. equivalent meromorphic function fields on compact Riemann surfaces). In the final section, he studied the above equivalence classes for Riemann surfaces defined by \(y^ n=x(x-1) (x- \alpha)\).