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

J. Rogelio P\'erez-Buend\'ia

arxiv: 2510.23097 · v3 · pith:C3GFO3PVnew · submitted 2025-10-27 · 🧮 math.NT · math.AG· math.DS· math.GR

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

J. Rogelio P\'erez-Buend\'ia This is my paper

Pith reviewed 2026-05-21 20:45 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.DSmath.GR

keywords good reductionrational mapsarboreal Galois representationsp-adic fieldsdynamical systemsGalois theoryNéron-Ogg-Shafarevich criterionpreimage trees

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The pith

In the tame case, strict good reduction of a rational endomorphism over a local field is equivalent to its canonical residual morphism being finite étale of degree d on a nonempty open subset.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives a criterion for strict good reduction of a rational map of degree d on the projective line over a non-archimedean local field K. When the residue characteristic does not divide d, this is equivalent to the existence of a nonempty Zariski open set in the reduced projective line minus post-critical points on which the canonical residual morphism is finite and étale of full degree d. The criterion splits the problem into two parts: one invariant controls whether the residual degree drops and another controls whether the residual fibers are étale. If the criterion holds then preimage extensions along orbits remain unramified.

Core claim

Let K be a non-archimedean local field and φ : P¹ → P¹ a rational endomorphism of degree d ≥ 2 over K. In the tame case (p ∤ d), strict good reduction is equivalent to the existence of a nonempty Zariski open subset U_k ⊂ P¹_k ∖ PC(φ̃) over which the canonical residual morphism is finite étale of degree d. The criterion separates Res(F,G) controlling residual degree drop from the fiber discriminants Disc(F_{n,x}) controlling étaleness of the residual fibers once full residual degree is ensured. Consequently, for every finite x ∈ O_K with x̄ ∈ U_k, the extensions K(X_n(x))/K are unramified for all n ≥ 1.

What carries the argument

The canonical residual morphism arising from a normalized integral lift of φ, together with the orbital preimage tree T_{O⁺(x)} and its associated orbital arboreal Galois image G_{O⁺(x)}.

If this is right

On the forward-invariant safe locus U_k^safe, strict good reduction is equivalent to bijectivity of the orbital reduction map X_n(x_m) → X̃_n(x̄_m).
For every finite point x with reduction in U_k, all iterated preimage extensions K(X_n(x))/K remain unramified.
The framework supplies both pointwise and orbit-level reformulations of the good-reduction condition that connect directly to arboreal Galois representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same separation of residual-degree and ramification invariants could be tested on other dynamical systems such as endomorphisms of higher-dimensional varieties.
The orbital arboreal images might be used to compute explicit Galois groups for preimage trees in cases where classical methods are difficult.
Pointwise unramifiedness along safe orbits gives a dynamical analogue of the Néron-Ogg-Shafarevich criterion for unramified extensions.

Load-bearing premise

A normalized integral lift of the rational map exists whose reduction produces a well-defined canonical residual morphism.

What would settle it

An explicit rational map over Q_p in the tame case where the residual morphism is finite étale of degree d on a nonempty open set yet the original map fails to have strict good reduction.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a tame dynamical Néron-Ogg-Shafarevich criterion that ties strict good reduction to residual étaleness on an open set via orbital trees, but the dependence on a normalized integral lift needs checking.

read the letter

The core claim is that for a rational map of degree d on P^1 over a local field, in the tame case, strict good reduction is equivalent to the existence of a Zariski open U_k where the canonical residual morphism is finite étale of degree d. This separates the resultant Res(F,G) for degree drop from the fiber discriminants Disc(F_n,x) for étaleness, and then reformulates the same condition using an orbital preimage tree and its Galois image on the safe locus. The forward orbit invariants are presented as new and tied to existing arboreal representation work. That connection and the explicit separation of the two local invariants are the clearest contributions. The examples over Q_p help ground the statements. The constructions of the orbital tree T_{O^+(x)} and the image G_{O^+(x)} look like genuine additions rather than relabeling of prior objects. The paper does a reasonable job organizing good reduction, residual behavior, and Galois action on preimage trees into one framework. The main soft spot is the reliance on a normalized integral lift whose reduction defines the residual morphism and the quantities Res and Disc. If distinct lifts can produce different post-critical loci or different discriminant behavior, then the open set U_k and the claimed equivalence become choice-dependent instead of intrinsic to the map. The abstract states the separation cleanly, but the actual derivation steps and verification that the lift is canonical are not visible here, so the central equivalence is hard to assess without the proofs. Minor issues like notation or example details would be easy to fix. This is written for people already working in arithmetic dynamics or Galois representations attached to dynamical systems. A reader who knows the classical Néron-Ogg-Shafarevich criterion and the Boston-Jones arboreal literature will see the point quickly and can check the new objects against their own examples. It is coherent enough on its own terms to deserve referee time rather than a desk rejection. I would send it out for review and ask the referees to focus on the lift canonicity and the explicit equivalence proof.

Referee Report

2 major / 2 minor

Summary. The paper proves a dynamical analogue of the Néron-Ogg-Shafarevich criterion for a rational endomorphism φ: ℙ¹ → ℙ¹ of degree d ≥ 2 over a non-archimedean local field K. In the tame case p ∤ d, strict good reduction of φ is equivalent to the existence of a nonempty Zariski open U_k ⊂ ℙ¹_k ∖ PC(φ̃) over which the canonical residual morphism is finite étale of degree d. The argument separates Res(F,G) (controlling residual degree drop) from the fiber discriminants Disc(F_{n,x}) (controlling étaleness of residual fibers). It introduces the orbital preimage tree T_{O⁺(x)} and the orbital arboreal Galois image 𝒢_{O⁺(x)}, and shows that on the forward-invariant safe locus U_k^safe, good reduction is equivalent to bijectivity of the orbital reduction map X_n(x_m) → X̃_n(x̄_m), implying that K(X_n(x))/K is unramified for all n and all x with x̄ in U_k. The framework is illustrated with explicit examples over ℚ_p and connected to existing work on arboreal representations.

Significance. If the central equivalence holds, the result supplies an intrinsic, orbit-level criterion for strict good reduction that cleanly decouples residual degree from ramification via two explicit local invariants of a normalized integral lift. The introduction of orbital preimage trees and safe loci gives a canonical way to pass from pointwise to orbitwise statements, directly linking classical reduction theory to the Galois images studied by Boston-Jones and others. The separation of Res(F,G) and Disc(F_{n,x}) is a structural strength that may extend to other dynamical settings.

major comments (2)

[Definition of normalized integral lift and canonical residual morphism (near the statement of the main theorem)] The equivalence in the tame case is stated to hold for a chosen normalized integral lift whose reduction defines the canonical residual morphism and the post-critical locus PC(φ̃). It is not shown that Res(F,G), the locus U_k, and the finiteness/étaleness property are independent of this choice; if two distinct normalized lifts yield different resultants or different PC(φ̃), the criterion becomes lift-dependent rather than intrinsic to φ. This is load-bearing for the claim that the condition detects strict good reduction.
[Section introducing T_{O⁺(x)} and the safe locus U_k^safe] The construction of the orbital preimage tree T_{O⁺(x)} = lim X_∞(φ^n(x)) and the orbital arboreal Galois image 𝒢_{O⁺(x)} relies on the forward orbit and the safe locus U_k^safe. The bijectivity of the orbital reduction map is asserted to characterize good reduction, but the argument does not explicitly verify that this bijectivity is preserved under different choices of lift or that the colimit commutes with reduction in a lift-independent way.

minor comments (2)

[Abstract and §1] The abstract and introduction use the notation PC(φ̃) without an explicit definition in the provided text; a short paragraph recalling the post-critical locus in the dynamical setting would improve readability.
[Examples section] The examples over ℚ_p are mentioned but their explicit computations of Res(F,G) and Disc(F_{n,x}) are not reproduced in the excerpt; including one fully worked numerical example would make the separation of invariants concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify the intrinsic nature of the criterion. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [Definition of normalized integral lift and canonical residual morphism (near the statement of the main theorem)] The equivalence in the tame case is stated to hold for a chosen normalized integral lift whose reduction defines the canonical residual morphism and the post-critical locus PC(φ̃). It is not shown that Res(F,G), the locus U_k, and the finiteness/étaleness property are independent of this choice; if two distinct normalized lifts yield different resultants or different PC(φ̃), the criterion becomes lift-dependent rather than intrinsic to φ. This is load-bearing for the claim that the condition detects strict good reduction.

Authors: We agree that an explicit verification of independence from the choice of normalized integral lift is necessary to confirm that the criterion is intrinsic to φ. In the manuscript, a normalized integral lift is a pair of homogeneous polynomials (F, G) ∈ O_K[X, Y] of degree d with no common factor over K such that φ = [F : G]. We will add a new lemma (Lemma 3.5) proving that any two such lifts differ by scaling by a unit in O_K^* and a linear change of variables in GL_2(O_K). In the tame case p ∤ d, this preserves both the resultant Res(F, G) up to units and the reduced map φ̃ together with its post-critical locus PC(φ̃). Consequently, the open set U_k ⊂ ℙ¹_k ∖ PC(φ̃) and the finite étale property of the residual morphism are independent of the lift. We will revise the statement of the main theorem (Theorem 1.1) to explicitly note this independence and update the surrounding discussion in Section 3. revision: yes
Referee: [Section introducing T_{O⁺(x)} and the safe locus U_k^safe] The construction of the orbital preimage tree T_{O⁺(x)} = lim X_∞(φ^n(x)) and the orbital arboreal Galois image 𝒢_{O⁺(x)} relies on the forward orbit and the safe locus U_k^safe. The bijectivity of the orbital reduction map is asserted to characterize good reduction, but the argument does not explicitly verify that this bijectivity is preserved under different choices of lift or that the colimit commutes with reduction in a lift-independent way.

Authors: We thank the referee for this observation. The orbital preimage tree T_{O⁺(x)} is the direct limit of the preimage sets along the forward orbit, equipped with the natural G_K-action, and the safe locus U_k^safe is the forward-invariant intersection ∩_{m≥0} φ̃^{-m}(U_k). While the unramifiedness of K(X_n(x))/K is intrinsically defined and independent of any model, we agree that the manuscript should explicitly confirm that bijectivity of the orbital reduction map X_n(x_m) → X̃_n(x̄_m) is likewise independent of the choice of normalized lift. We will add a proposition (Proposition 4.3) showing that, when φ has strict good reduction, the reduction map is bijective on the safe locus for any normalized lift; this follows because the Galois action factors through the unramified quotient of G_K, which does not depend on the integral model. The colimit construction commutes with reduction in a canonical way once the post-critical locus is fixed intrinsically. We will incorporate this verification into Section 4 and add a remark linking it to the existing literature on arboreal representations. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence proved from definitions of residual invariants on normalized lifts

full rationale

The paper states a theorem equating strict good reduction to the existence of a Zariski open U_k on which the canonical residual morphism (arising from any normalized integral lift F,G) is finite étale of degree d, with Res(F,G) and Disc(F_{n,x}) serving as the separating invariants. These quantities are defined directly from the chosen lift and its reduction; the claimed equivalence is therefore a statement about when those invariants vanish or satisfy the étale condition, not a re-labeling of the input data. No equations reduce one quantity to another by construction, no self-citations are invoked as load-bearing uniqueness theorems, and the new objects (orbital preimage tree, safe locus, orbital arboreal image) are introduced as colimits rather than presupposed. The derivation therefore remains self-contained against external benchmarks such as the classical Néron–Ogg–Shafarevich criterion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The abstract invokes standard facts from algebraic geometry (Zariski opens, étale morphisms) and local fields (tame ramification, residue fields) without listing them explicitly; the main new objects are the orbital preimage tree and the orbital arboreal Galois image, introduced to reformulate the criterion.

axioms (2)

domain assumption Existence of a normalized integral lift of the rational map φ over the local field K
Invoked when defining Res(F,G) and the canonical residual morphism; appears in the sentence separating the two local invariants.
standard math Standard properties of finite étale morphisms and Zariski topology on P^1 over the residue field
Used to state that the residual morphism is finite étale of degree d on U_k.

invented entities (2)

orbital preimage tree T_{O^+(x)} no independent evidence
purpose: Colimit of preimage sets X_∞(φ^n(x)) along the forward orbit to capture G_K-action on the entire orbit
Defined as varinjlim_n X_∞(φ^n(x)) and used to define the orbital arboreal Galois image.
orbital arboreal Galois image G_{O^+(x)} no independent evidence
purpose: Image of G_K inside Aut(T_{O^+(x)}) to reformulate good reduction at the orbit level
Introduced as Im(G_K → Aut(T_{O^+(x)})) and connected to the bijectivity of the orbital reduction map on the safe locus.

pith-pipeline@v0.9.0 · 5935 in / 1831 out tokens · 55493 ms · 2026-05-21T20:45:30.083471+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1: strict good reduction ⇔ existence of Uk ⊂ P1k ∖ PC(φ̃) on which canonical residual morphism is finite étale of degree d; Res(F,G) controls residual degree, Disc(Fn,x) controls étaleness
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Orbital colimit GO+(x) := lim→ Im(ρφn(x)) as canonical Galois object; unramifiedness of Iv on GO+(x) equivalent to strict good reduction

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

arXiv:2208.00359

Adams O (2022) A Dynamical Analogue of the Criterion of Néron-Ogg- Shafarevich. arXiv:2208.00359. https://arxiv.org/abs/2208.00359

work page arXiv 2022
[2]

Duke Math

Andreatta F, Iovita A, Kim M (2015) Ap-adic nonabelian criterion for good reduction of curves. Duke Math. J. 164(13):2597–2642. doi:10.1215/00127094- 3146817

work page doi:10.1215/00127094- 2015
[3]

R. L. Benedetto,Dynamics in One Non-Archimedean Variable, GSM 198, AMS,

work page
[4]

Acta Arith

Benedetto RL (2014) A criterion for potentially good reduction in nonar- chimedean dynamics. Acta Arith. 165(3):251–256. doi:10.4064/aa165-3-4

work page doi:10.4064/aa165-3-4 2014
[5]

Benedetto RL (2015) Attaining Potentially Good Reduction in Arith- metic Dynamics. Int. Math. Res. Not. IMRN 2015(22):11828–11846. doi:10.1093/imrn/rnv358

work page doi:10.1093/imrn/rnv358 2015
[6]

Boston N, Jones R (2007) Arboreal Galois representations. Geom. Dedicata 124:27–35. doi:10.1007/s10711-006-9113-9

work page doi:10.1007/s10711-006-9113-9 2007
[7]

Hernandez-Mada G (2021) A monodromy criterion for the good reduction ofK3 surfaces. Rend. Semin. Mat. Univ. Padova 145:73–92. doi:10.4171/rsmup/50

work page doi:10.4171/rsmup/50 2021
[8]

Jones R (2013) Galois representations from preimage trees: an arboreal survey. Publ. Math. Besançon Algèbre Théorie Nr. 2013:107–136. doi:10.5802/pmb.a-154

work page doi:10.5802/pmb.a-154 2013
[9]

Liedtke C, Matsumoto Y (2018) Good reduction of K3 surfaces. Compos. Math. 154(1):1–35. doi:10.1112/S0010437X17007400

work page doi:10.1112/s0010437x17007400 2018
[10]

Pérez-Buendía JR (2019) A Kulikov-type classification theorem for a one param- eter family of K3-surfaces over ap-adic field and a good reduction criterion. Ann. Math. Qué. 43(2):411–434. doi:10.1007/s40316-018-0110-9

work page doi:10.1007/s40316-018-0110-9 2019
[11]

Graduate Texts in Mathematics, vol 67

Serre J-P (1979) Local Fields. Graduate Texts in Mathematics, vol 67. Springer, New York, NY. ISBN: 0-387-90424-7; 978-0-387-90424-5. doi:10.1007/978-1-4757- 5673-9

work page doi:10.1007/978-1-4757- 1979
[12]

Graduate Texts in Mathematics, vol 241

Silverman JH (2007) The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol 241. Springer, New York, NY. ISBN: 978-0-387-69903-5 (hardcover); 978-1-4419-2417-9 (softcover). doi:10.1007/978-0-387-69904-2 15

work page doi:10.1007/978-0-387-69904-2 2007
[13]

Stoll M (1992) Galois groups overQof some iterated polynomials. Arch. Math. (Basel) 59(3):239–244. doi:10.1007/BF01197321 16

work page doi:10.1007/bf01197321 1992

[1] [1]

arXiv:2208.00359

Adams O (2022) A Dynamical Analogue of the Criterion of Néron-Ogg- Shafarevich. arXiv:2208.00359. https://arxiv.org/abs/2208.00359

work page arXiv 2022

[2] [2]

Duke Math

Andreatta F, Iovita A, Kim M (2015) Ap-adic nonabelian criterion for good reduction of curves. Duke Math. J. 164(13):2597–2642. doi:10.1215/00127094- 3146817

work page doi:10.1215/00127094- 2015

[3] [3]

R. L. Benedetto,Dynamics in One Non-Archimedean Variable, GSM 198, AMS,

work page

[4] [4]

Acta Arith

Benedetto RL (2014) A criterion for potentially good reduction in nonar- chimedean dynamics. Acta Arith. 165(3):251–256. doi:10.4064/aa165-3-4

work page doi:10.4064/aa165-3-4 2014

[5] [5]

Benedetto RL (2015) Attaining Potentially Good Reduction in Arith- metic Dynamics. Int. Math. Res. Not. IMRN 2015(22):11828–11846. doi:10.1093/imrn/rnv358

work page doi:10.1093/imrn/rnv358 2015

[6] [6]

Boston N, Jones R (2007) Arboreal Galois representations. Geom. Dedicata 124:27–35. doi:10.1007/s10711-006-9113-9

work page doi:10.1007/s10711-006-9113-9 2007

[7] [7]

Hernandez-Mada G (2021) A monodromy criterion for the good reduction ofK3 surfaces. Rend. Semin. Mat. Univ. Padova 145:73–92. doi:10.4171/rsmup/50

work page doi:10.4171/rsmup/50 2021

[8] [8]

Jones R (2013) Galois representations from preimage trees: an arboreal survey. Publ. Math. Besançon Algèbre Théorie Nr. 2013:107–136. doi:10.5802/pmb.a-154

work page doi:10.5802/pmb.a-154 2013

[9] [9]

Liedtke C, Matsumoto Y (2018) Good reduction of K3 surfaces. Compos. Math. 154(1):1–35. doi:10.1112/S0010437X17007400

work page doi:10.1112/s0010437x17007400 2018

[10] [10]

Pérez-Buendía JR (2019) A Kulikov-type classification theorem for a one param- eter family of K3-surfaces over ap-adic field and a good reduction criterion. Ann. Math. Qué. 43(2):411–434. doi:10.1007/s40316-018-0110-9

work page doi:10.1007/s40316-018-0110-9 2019

[11] [11]

Graduate Texts in Mathematics, vol 67

Serre J-P (1979) Local Fields. Graduate Texts in Mathematics, vol 67. Springer, New York, NY. ISBN: 0-387-90424-7; 978-0-387-90424-5. doi:10.1007/978-1-4757- 5673-9

work page doi:10.1007/978-1-4757- 1979

[12] [12]

Graduate Texts in Mathematics, vol 241

Silverman JH (2007) The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol 241. Springer, New York, NY. ISBN: 978-0-387-69903-5 (hardcover); 978-1-4419-2417-9 (softcover). doi:10.1007/978-0-387-69904-2 15

work page doi:10.1007/978-0-387-69904-2 2007

[13] [13]

Stoll M (1992) Galois groups overQof some iterated polynomials. Arch. Math. (Basel) 59(3):239–244. doi:10.1007/BF01197321 16

work page doi:10.1007/bf01197321 1992