A Dynamical Analogue of the Criterion of Néron-Ogg-Shafarevich
Ophelia Adams
TL;DR
This work establishes a bridge between anabelian geometry and arithmetic dynamics by formulating a dynamical NOS-type criterion: unramified arboreal representations correspond to maps with a strong form of good reduction centered on the pre-critical locus. Building an anabelian framework via Tamagawa's ideas, it relates the Galois action on a pro-ell étale fundamental group to dynamical ramification and introduces pre-critical incidence graphs as a combinatorial proxy for ramification behavior. The authors develop local dynamical criteria (height, stepwise reduction, and untwisting) to analyze branch ramification, proving equivalences that tie ramification to the presence of directed or undirected cycles in the dynamical portrait. They then derive effective tools for identifying infinitely ramified primes, especially for PCF maps, and explore broader implications for post-critically infinite maps, higher ramification, and abelian dynamical extensions. Overall, the results unify anabelian and dynamical perspectives, offering concrete criteria and examples that illuminate how dynamics controls Galois ramification over number fields and local fields.
Abstract
We introduce an anabelian approach to the study of arboreal Galois representations and apply Tamagawa's anabelian version of the Néron-Ogg-Shafarevich criterion to produce a dynamical analogue of this criterion: unramified representations correspond to rational maps satisfying a strong form of good reduction in terms of their critical locus. Subsequently, we pursue a dynamical anlaogue of the Néron-Ogg-Shafarevich criterion in terms of the more (dynamically) traditional arboreal representations, which relates unramified arboreal representations to a certain separability condition on the dynamical system. Finally, we relate the our criteria: the anabelian criterion corresponds to the dynamical criterion as one varies the base point around the critical locus. Along the way we develop effective criteria to determine which primes are infinitely ramified in arboreal representations over number fields, as well as the asymptotic growth of that ramification; we conclude with examples and applications, especially to dynamical systems over number fields.