Valuations on Meromorphic Functions of Bounded Type

Abstract

The primary purpose of this paper is to show that every valuation on the field of meromorphic functions of bounded type on a finitely sheeted unlimited covering Riemann surface is a point valuation if and only if the same is true on its base Riemann surface. The result is then applied to concrete examples and some related results are obtained. Any valuation on the field M(W) of single valued meromorphic functions on a Riemann surface W is a point valuation [9]. What happens to valuations on subfields of M(W)? An especially interesting subfield in this context is the field M°°(W) of meromorphic functions of bounded type on W (cf. [2]). We are thus concerned with the following question in this paper: When is it true that any valuation on nontrivial M°°(W) is a point valuation? The paper consists of five parts. §1 covers preliminaries. The main part, §2, concerns the covering stability. We say that a Riemann surface W is stable if M°°(W) is nontrivial and any valuation on M°°(W) is a point valuation. Then it is shown in this section as the main theorem of this paper that a finitely sheeted unlimited covering surface R oi a Riemann surface S is stable if and only if S is stable. In §3 on stable surfaces, an example due to Forelli [4] of stable surfaces is given among others. In §4 the relation between H°°-maximality and stability of a Riemann surface W and in particular of a plane region W is discussed. We relax the definition of stability in §5 to obtain the notion of weak stability of a Riemann surface W. Here a simple but powerful device of what we call i/°°-barrier is introduced, which is used to exhibit a weakly stable plane region of infinite connectivity. 1. Preliminaries. 1.1. Fields F we consider in this paper are all assumed to be extensions of the complex number field C. We denote by F* the multiplicative group consisting of nonzero elements in F, i.e. F* = F\ {0}. By a valuation v on F we mean a discrete valuation v on F, i.e. a group homomorphism of the multiplicative group F* into the additive group Z of integers such that (1.1) v(f + g)>min(v(f),v(g)) (f,gEF*) where we make the convention that v(0) = +oo. Received by the editors June 16, 1987. 1980 Mathematics Subject Classification. Primary 30H50; Secondary 30F99, 30D50, 46J15. To complete this work the author was supported in part by Grant-in-Aid for Scientific Research, No. 61540094, Japanese Ministry of Education, Science and Culture. ©1988 American Mathematical Society 0002-994T/88 $1.00 + $.25 per page 231 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Cite this paper

@inproceedings{Nakai2010ValuationsOM, title={Valuations on Meromorphic Functions of Bounded Type}, author={Mitsuru Nakai}, year={2010} }