Arboreal Galois representations and uniformization of polynomial dynamics

Abstract

Given a polynomial f defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z↦ f(z) to the multiplicative action z↦ zdeg(f). The relation between this construction and the Böttcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory. As a consequence, we obtain some corollaries regarding Galois irreducibility of preimages of certain points under certain polynomials, as well as the rationality of preimages in one-parameter families.