Galois groups of iterates of some unicritical polynomials

TL;DR

This paper investigates the Galois groups associated with iterates of specific unicritical polynomials, demonstrating their finite index in an infinite wreath product and establishing surjectivity in select cases using advanced local techniques.

Contribution

It proves finite index properties of arboreal Galois representations for certain unicritical polynomials and introduces new surjectivity results for small degree examples.

Findings

Galois groups have finite index in an infinite wreath product.

Surjectivity established for some quadratic polynomial families.

Uses Chabauty-Coleman and Mordell-Weil sieve methods.

Abstract

We prove that the arboreal Galois representations attached to certain unicritical polynomials have finite index in an infinite wreath product of cyclic groups, and we prove surjectivity for some small degree examples, including a new family of quadratic polynomials. To do this, we use a combination of local techniques including the Chabauty-Coleman method and the Mordell-Weil sieve.