A Dynamical Néron–Ogg–Shafarevich Criterion via Orbital Arboreal Representations
Abstract
Let K be a non-archimedean local field and φ: ℙ¹ → ℙ¹ a rational endomorphism of degree d ≥ 2 defined over K. In the tame case (p ∤ d), we give a concise local criterion for strict good reduction on the natural residue étale locus: there exists a nonempty open U₀ ⊆ ℙ¹ \ PC(φ) such that for every x ∈ U₀(O_K) with x̄ ∉ Uₐ, the reduced level-1 fiber polynomial has degree d and is étale; equivalently, all backward preimage extensions K(Xₙ(x))/K are unramified for all n ≥ 1. This work provides an orbital refinement of the pointwise criterion of Benedetto, framing it in terms of a canonical, orbit-invariant Galois object. Complete proofs and explicit examples over ℚ_p are provided.
Type
Publication
Submitted to Research in Number Theory (Springer). arXiv:2510.23097 (12 Nov 2025).
Pérez-Buendía, J. R. (2025). A Dynamical Néron–Ogg–Shafarevich Criterion via Orbital Arboreal Representations. arXiv:2510.23097. Submitted to Research in Number Theory (Springer).
This paper establishes a dynamical Néron–Ogg–Shafarevich type criterion for good reduction of rational maps over non-archimedean local fields, framed in the theory of orbital arboreal representations. The criterion refines the pointwise work of Benedetto and produces a canonical, orbit-invariant Galois object encoding the reduction of the dynamical system.
Arithmetic Dynamics
Arboreal Representations
Good Reduction
Néron-Ogg-Shafarevich
Local Fields
In Review
Authors
Investigador por México · SECIHTI · CIMAT Mérida
Matemático, Investigador por México adscrito al CIMAT Unidad Mérida (Sistema
Nacional de Centros Públicos de Investigación). Afiliado a la Secretaría de
Ciencia, Humanidades, Tecnología e Innovación (SECIHTI). Trabajo en la
intersección entre geometría aritmética, teoría de números y sistemas
dinámicos, con énfasis en métodos p-ádicos y aplicaciones a biología matemática.