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

TL;DR

The paper addresses a dynamical analogue of the Néron–Ogg–Shafarevich criterion for a rational map $\varphi$ of degree $d\ge2$ over a non-archimedean local field $K$ in the tame case $p\nmid d$ by introducing an orbital colimit $\mathcal{G}_{O^+(x)}$—a canonical Galois object built from the forward orbit. It proves that strict good reduction is equivalent to trivial inertia on the preimage towers along integral points outside the residual postcritical locus, yielding an operational local criterion: there exists a nonempty open $U_k\subset \mathbb{P}^1_k$ such that for $x\in \mathbb{P}^1(\mathcal{O}_K)$ with $\bar{x}\in U_k$, the level-1 fiber is of degree $d$ and étale (equivalently, the discriminant is a unit). The results provide an orbital refinement of Benedetto’s pointwise criterion, give a canonical orbital symmetry target $\text{Aut}^{\text{orb}}(O^+(x))$, and furnish explicit $\mathbb{Q}_p$-examples, including nonpolynomial maps, to illustrate the criterion and its implications for unramified arboreal towers. The framework suggests a global, adelic perspective where inertia is controlled locally away from a finite set by Frobenius statistics in the admissible orbital symmetry group, connecting non-archimedean dynamics to Galois-theoretic ramification phenomena in a basepoint-free, iteration-compatible way.

Abstract

Let $K$ be a non-archimedean local field and $\varphi:\mathbb{P}^1\to\mathbb{P}^1$ a rational endomorphism of degree $d\ge2$ 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 $U_k\subset\mathbb{P}^1_k\setminus \mathrm{PC}(\tilde\varphi)$ such that for every $x\in\mathbb{P}^1(\mathcal{O}_K)$ with $\bar x\in U_k$ 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 $n\ge1$. 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 $\mathbb{Q}_p$.

Theorems & Definitions (42)

• Lemma 2.1: Unit resultant persists under iteration
• proof
• Lemma 2.2: Unit discriminant $\Rightarrow$ unramified
• proof
• Definition 2.3: Postcritical set
• Remark 2.4
• Proposition 2.5: Étale fibers for all iterates $\Leftrightarrow$ non-postcritical
• proof
• Remark 2.6
• Remark 2.7