Arboreal Galois representations and uniformization of polynomial dynamics
Number Theory
2014-02-26 v1
Abstract
Given a polynomial f of degree d 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->z^d. The relation between this construction and the Bottcher 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.
Keywords
Cite
@article{arxiv.1111.3607,
title = {Arboreal Galois representations and uniformization of polynomial dynamics},
author = {Patrick Ingram},
journal= {arXiv preprint arXiv:1111.3607},
year = {2014}
}