Principal transformations between Riemann surfaces (Q1330250)

Let \(X\) and \(Y\) be compact or open Riemann surfaces, \(M(X)\), \(M(Y)\) be their meromorphic function fields. A bijection \(\Phi\) of \(X\) to \(Y\) is a principal transformation provided that for every divisor \(D\) on \(X\), \(D\) is principal if and only if \(\Phi (D)\) is a principal divisor on \(Y\). We say that a principal transformation is special if it is induced by an abstract field isomorphism of meromorphic function fields of two Riemann surfaces. In this paper the author proves the following two facts: ``For infinitely many pairs of compact Riemann surfaces \(X,Y\) of genus one, there exist nonspecial principal transformations'' and ``Let \(X\) and \(Y\) be compact Riemann surfaces of genus \(g>1\). If there exists a continuous principal transformation \(\Phi\) of \(X\) to \(Y\), then \(X\) and \(Y\) are conformally equivalent and \(\Phi\) is special''.