An arboreal Galois representation is the dynamical analog of the Galois representations attached to elliptic curves. A dynamical analog to Serre's open image theorem, conjectured by Jones, remains largely open. We prove a version of Jones' Conjecture for quadratic and cubic polynomials assuming the abc-Conjecture and Vojta's Conjecture. We also exhibit a family of degree 2 rational maps and give examples of degree 3 polynomial maps whose arboreal representations have finite index in the appropriate group of tree automorphisms.
Arboreal representations for rational maps with few critical points
Michelle Manes, University of Hawaii at ManoaAuthors: Jamie Juul, Holly Krieger, Nicole Looper, Michelle Manes, Bianca Thompson, Laura Walton
2022 AWM Research Symposium
New Directions in Number Theory