Isomorphisms between certain function fields over compact Riemann surfaces

Abstract

In a previous paper [1] the author has introduced the concept of a field verifying the Weierstrass property (or, in short, a W-field) in an open connected subset U of a compact Riemann surface :U, in order to study and solve a problem related with the classical theory of functions. We say that a subfield k of the field JI(U) of meromorphic functions on U is a W-field in U, if k contains the field J/e'(~U) of meromorphic functions on ~ (which, by restriction of functions, can be considered as a function field over U) and if, 6 being a given finite divisor on U (i.e. a divisor supported on a finite subset of U), there exists a function belonging to k whose divisor (on U) is 6. Let ~ ' be an open subset of V, whose complementary set is a nonempty finite set. In the above said paper was proved the existence of an isomorphism between every two minimal W-fields in ~U' and, even, the existence of isomorphisms between minimal W-fields corresponding to different sets of the same type of ~ ' . The fundamental purpose of this paper is to prove an analogous theorem for the W-fields, on open sets of ~U' type, generated over ~.( f ) by all meromorphic functions on ~ ' having a finite divisor.