A survey on abelian dynamical Galois groups

2022/11/24 by A. Ferraguti, Ferraguti, A.
Arts and Humanities · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.13598

openalex publication_date 2022/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, f∈ K[x] and α∈ K. A recent conjecture of Andrews and Petsche predicts that the dynamical Galois group of the pair (f,α) is abelian if and only if the pair (f,α) is K^ab-conjugated to (g,β), where g is a power or a Chebyshev map and β is ζ or ζ+ζ^-1, respectively, and ζ is a root of unity. We review three completely different approaches that allow to prove several cases of the conjecture.

Related

• Abelian dynamical Galois groups for unicritical polynomials
• Reconstructing global fields from dynamics in the abelianized Galois group
• Galois Extensions of Height-One Commuting Dynamical Systems
• Constraining images of quadratic arboreal representations
• Abelian-by-Central Galois groups of fields I: a formal description