Abelian dynamical Galois groups over global function fields
Abstract
We establish a function field analogue of a recent conjecture of Andrews--Petsche. Our main result characterizes abelian dynamical Galois groups to be precisely the isotrivial ones, whenever the degree of the polynomial is smaller than p, the characteristic of the field. The proof of this characterization is achieved in four independent steps as follows: Abelian ⟹ Finite ramification ⟹ PCF map ⟹ Isotrivial map ⟹ Isotrivial pair, and works more generally for maps with a superattracting fixed point. We observe that this chain of implications is sharp in the degree: as soon as one reaches p there are new non-isotrivial examples coming from Drinfeld modules. We propose a full conjectural classification in all degrees, taking into account all of the new exotic examples coming from Drinfeld modules and their associated Lattès maps.
Related
- Multiplicator freeness for restricted-ramification Galois groups over global function fields
- Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups
- Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions
- Galois groups of Apéry-like series modulo primes
- Infinite free pro-p products as absolute Galois groups
BibTeX
@misc{ferraguti2026abelian,
title = {{Abelian dynamical Galois groups over global function fields}},
author = {Andrea Ferraguti and Patrick Ingram and Carlo Pagano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24920},
eprint = {2609.24920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24920},
}
RIS
TY - UNPB
TI - Abelian dynamical Galois groups over global function fields
AU - Andrea Ferraguti
AU - Patrick Ingram
AU - Carlo Pagano
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24920
UR - https://arxiv.org/abs/2609.24920
AB - We establish a function field analogue of a recent conjecture of Andrews--Petsche. Our main result characterizes abelian dynamical Galois groups to be precisely the isotrivial ones, whenever the degree of the polynomial is smaller than p, the characteristic of the field. The proof of this characterization is achieved in four independent steps as follows: Abelian ⟹ Finite ramification ⟹ PCF map ⟹ Isotrivial map ⟹ Isotrivial pair, and works more generally for maps with a superattracting fixed point. We observe that this chain of implications is sharp in the degree: as soon as one reaches p there are new non-isotrivial examples coming from Drinfeld modules. We propose a full conjectural classification in all degrees, taking into account all of the new exotic examples coming from Drinfeld modules and their associated Lattès maps.
ER -