Nevanlinna theory on the p-adic plane

Abstract

Let K be a complete and algebraically closed non-Archimedean valued field. Following ideas of Marc Krasner and Philippe Robba, the author defines K-entire and K-meromorphic functions from K to K, and extends the definitions of Nevanlinna theory to these functions. In this context,he proves versions of the first and second fundamental theorems of Nevanlinna theory, of Picard’s theorem and of Nevanlinna’s five points theorem, as well as other results of Nevanlinna theory. It turns out that in the non-Archimedean case, stronger results often hold. For example, the analogue of Picard’s theorem is: if F is a nonconstant K-entire function, F has no excluded values; and the analogue of the five points theorem is a four points theorem: If F and G are two nonconstant Kmeromorphic functions onKsuch that for distinct a1, a2, a3, a4 we have F(x) = ai () G(x) =ai, i = 1, 2, 3, 4, then F G. This last is an extension of a result ofW.W. Adams and E. G. Straus [Illinois J. Math. 15 (1971), 418–424; who proved similar results in the case of functions defined by power series converging in all of K (of characteristic 0). In the case of “Krasner functions” treated by the author, a kind of “analytic continuation” is used to define the functions, but this is necessarily (because of the properties of non-Archimedean valuations) very different from the usual change of center method in one complex variable. The author states without proof that most of the results on compositional factorization of meromorphic functions of one complex variable carry over to his situation. {Two misprints should be noted: “Oswood” should be “Osgood” throughout, and at the bottom of p. 136 in lines −3 and −6, somewhat mildly startlingly, “f 6= g” appears instead of “f g” in the citation of the Adams and Strauss results.} Reviewed by S. L. Segal