On the immersions between certain function fields on punctured compact Riemann surfaces

Abstract

Let ~r be a compact Riemann surface and ~/r be the complementary of a nonempty finite subset of ~/K. Let K (~/K') be the field generated by all meromorphic functions with finite divisor in "W'. Then, it is known (Cutillas [2]) that there are isomorphisms between any two fields of this type (corresponding to the same Riemann surface ~K) which restricted to the field ~' (~/r of meromorphic functions in ~/U coincide with the identity mapping. In this paper we shall prove that if the genus 9~ of ~ is > 0, then the restriction to Jr of every C-algebra isomorphism between two fields of the said type must be an automorphism of Jr This will be obtained as a consequence of a theorem asserting that the tE-algebra immersions from Jr into a field K (~tr'), corresponding to another compact Riemann surface "U, must take their values in ~ (~) (if 9~r > 0). As other consequences one obtains that K(~K') and K(~F') can only be isomorphic as C-algebras if ~ and ~r are isomorphic Riemann surfaces, and that every ll;-algebra immersion from K (~/K') into K ("U') coincides with the composition of a map naturally induced by a nonconstant holomorphic map from ~ into ~/~, with an isomorphism of Jr In the sequel, ~/r -/Or', K (-/OF,) and 9~r will be as above and G (~K') will be the group of meromorphic functions with finite divisor in ~/K'. Moreover, for any connected open subset U of a Riemann surface, J/Z* (U) will be the multiplicative group of meromorphic functions in U which are not identically zero and E(U) = {eh: h is a holomorphic function in U}.