Finite ramification for preimage fields of post-critically finite morphisms

Given a finite endomorphism phi of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K (phi(-infinity)(alpha)) := boolean OR(n >= 1) K (phi(-n) (alpha)) generated by the preimages of alpha under all iterates of phi. In particular when phi is post-critically finite, i.e., there exists a non-empty, Zariski-open W subset of X such that phi(-1) (W) subset of W and phi : W -> X is etale, we prove that K (phi(-infinity) (alpha)) is rami fied over only finitely many primes of K. This provides a large supply of in finite extensions with restricted rami fication, and generalizes results of Aitken-Hajir-Maire [1] in the case X = A(1) and Cullinan-Hajir, Jones-Manes [7, 13] in the case X = P-1. Moreover, we conjecture that this finite rami fication condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P-1. The proof relies on Faltings' theorem and a local argument.