Non-Archimedean holomorphic maps and the Ahlfors islands theorem

Abstract. We present a p-adic and non-archimedean version of some classical complex holomorphic function theory. Our main result is an analogue of the Five Islands Theorem from Ahlfors ’ theory of covering surfaces. For non-archimedean holomorphic maps, our theorem requires only two islands, with explicit and nearly sharp constants, as opposed to the three islands without explicit constants in the complex holomorphic theory. We also present non-archimedean analogues of other results from the complex theory, including theorems of Koebe, Bloch, and Landau, with sharp constants. 0. Introduction. In the late nineteenth and early twentieth centuries there was a broad effort devoted to understanding the mapping properties of complex holomorphic and meromorphic functions. In the mid 1930s, much of that study culminated in Ahlfors ’ theory of covering surfaces. The theory is sometimes considered to be a blend of Nevanlinna theory and the Riemann-Hurwitz formula