Galois uniformity in quadratic dynamics over rational function fields

TL;DR

This paper investigates the Galois groups associated with quadratic polynomials over rational function fields, demonstrating that these groups are of finite index in the automorphism group of the preimage tree, with bounds independent of the polynomial.

Contribution

It establishes finiteness and boundedness of the index of arboreal Galois representations for a broad class of quadratic polynomials over rational function fields.

Findings

Galois representations have finite index in automorphism groups

Index bounds are mostly independent of the polynomial

Results apply to a large class of quadratic polynomials

Abstract

We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that the index is bounded, in most cases, independently of the polynomial.