A Dynamical N\'eron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations

Abstract

Let $K$ be a non-archimedean local field and $\varphi : \mathbb{P}^1 \to \mathbb{P}^1$ a rational endomorphism of degree $d \geq 2$ over $K$. In the tame case ($p \nmid d$), we show that strict good reduction is equivalent to the existence of a nonempty Zariski open subset $U_k \subset \mathbb{P}^1_k \setminus \mathrm{PC}(\widetilde{\varphi})$ over which the canonical residual morphism is finite \'etale of degree $d$. The criterion separates two complementary local invariants of a normalized integral lift: $\mathrm{Res}(F,G)$ controls residual degree drop, while the fiber discriminants $\mathrm{Disc}(F_{n,x})$ control \'etaleness of the residual fibers once full residual degree is ensured. Consequently, for every finite $x \in \mathcal{O}_K$ with $\bar{x} \in U_k$, the extensions $K(X_n(x))/K$ are unramified for all $n \geq 1$. We introduce the orbital preimage tree $T_{O^+(x)} = \varinjlim_n X_\infty(\varphi^n(x))$, the colimit in $G_K$-sets along the forward orbit, and the orbital arboreal Galois image $\mathcal{G}_{O^+(x)} = \mathrm{Im}(G_K \to \mathrm{Aut}(T_{O^+(x)}))$. On the forward-invariant safe locus $U_k^{\mathrm{safe}} = \bigcap_{m \geq 0} \widetilde{\varphi}^{-m}(U_k)$, strict good reduction is captured by the bijectivity of the orbital reduction map $X_n(x_m) \to \widetilde{X}_n(\bar{x}_m)$. This canonical orbit-invariant framework connects with arboreal Galois representations (Boston-Jones, Jones, and others) and yields pointwise and orbit-level reformulations. Explicit examples over $\mathbb{Q}_p$ illustrate the criterion.