Arboreal Galois representations and uniformization of polynomial dynamics

作者：Ingram Patrick^*

来源：Bulletin of the London Mathematical Society, 2013, 45(2): 301-308.

DOI：10.1112/blms/bds088

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 -%26gt; z(f) to the multiplicative action z -%26gt; z(deg(f)). 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%26apos;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.

• 出版日期2013-4