Branched values and quasi-exceptional values for p-adic meromorphic functions

Abstract. Let K be an algebraically closed field of characteristic 0, com-plete with respect to an ultrametric absolute value. We show that a tran-scendental 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 that 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 f in K, if f admits 0 and ∞ as perfectly branched values, then the function assumes all non-zero values infinitely often. Similarly, if f is an “unbounded ” meromor-phic 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. 1.