A Dynamical Néron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations

2025/10/27 by Pérez-Buendía, J. Rogelio
#11S15 (Primary) #14H25 (Secondary) #37P05 #37P20 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.23097

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 \nmid d) we give a concise local criterion for strict good reduction on the natural residual 'etale locus: there exists a nonempty open U_k⊂ℙ¹_k∖ PC(φ) such that for every x∈ℙ¹(O_K) with x∈ U_k the reduced level-1 fiber has degree d and is 'etale; equivalently, the fiber polynomial has unit leading coefficient and unit discriminant. In particular, all backward preimage extensions K(X_n(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. We provide complete proofs and explicit examples over ℚ_p.

Citations

• A Dynamical Analogue of the Criterion of Néron-Ogg-Shafarevich
• Finite ramification for preimage fields of postcritically finite morphisms
• Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus
• Attaining potentially good reduction in arithmetic dynamics
• A Criterion for Potentially Good Reduction in Non-archimedean Dynamics
• On some notions of good reduction for endomorphisms of the projective line
• Finitely ramified iterated extensions

Related

• A Dynamical Analogue of the Criterion of Néron-Ogg-Shafarevich
• A dynamical Shafarevich theorem for endomorphisms of ℙ^N
• Isotriviality is equivalent to potential good reduction for endomorphisms of \mathbb P^N over function fields
• Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus
• Scarcity of finite orbits for rational functions over a number field