Survey on branched values and exceptional values for p-adic meromorphic functions

摘要

Let K be a complete algebraically closed p-adic field of characteristic zero. We show that a transcendental meromorphic function in K or an"unbounded" meromorphic function inside an open disk cannot admit more than 4 perfectly branched values and a transcendental meromorphic function in K cannot admit more than 3 values aj such that all zeroes of f — aj are multiple. An unbounded analytic function inside an open disk cannot admit more than 2 perfectly branched values. And an entire function cannot admit more than 1 perfectly branched value. Completing a previous result by K. Boussaf and J. Ojeda, we prove that given a transcendental meromorphic function in K, if all zeroes and poles are of order > 2 but finitely many, then the function assumes all non-zero values infinitely often. Similarly, if f is an "unbounded " meromorphic function in an"open" disk, if the residue characteristic p is different from 2 and if all zeroes and poles are of even order, but finitely many, then the function assumes all non-zero values infinitely often.