A Dynamical Néron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations
Authors
J. Rogelio Pérez-Buendía
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 $\mathbb{Q}_p$.
