Lubin's conjecture for height-one $p$-adic dynamical systems over cyclo-tame extensions

Martin Debaisieux

arxiv: 2603.03873 · v2 · submitted 2026-03-04 · 🧮 math.NT · math.DS

Lubin's conjecture for height-one p-adic dynamical systems over cyclo-tame extensions

Martin Debaisieux This is my paper

Pith reviewed 2026-05-15 16:54 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords Lubin's conjectureformal groupsp-adic dynamical systemscrystalline charactersp-adic Hodge theoryheight-one seriescyclo-tame extensions

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The pith

Certain commuting pairs of height-one formal power series over cyclo-tame p-adic extensions arise as endomorphisms of a formal group.

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

The paper proves new cases of Lubin's conjecture by showing that height-one commuting pairs of noninvertible and invertible formal power series over the ring of integers of K come from endomorphisms of a formal group law. K is a finite extension of Q_p whose ramification index is coprime to p squared minus p. The proof begins with the Galois set of f-consistent sequences, extracts a crystalline character of weight one, and uses it to turn the set into a Z_p-module on which f acts as an endomorphism. Explicit functors from integral p-adic Hodge theory then recover the formal group over O_K for which both series are endomorphisms.

Core claim

Let K/Q_p be a finite extension with ramification index coprime to p^2-p. For a height-one commuting pair (f,u) of noninvertible and invertible formal power series over O_K, the Gal(Kbar/K)-set T_f of f-consistent sequences carries a crystalline character of weight 1. This character equips T_f with the structure of a Z_p-module on which f acts as an endomorphism. Applying explicit functors from integral p-adic Hodge theory to this module yields a formal group defined over O_K for which (f,u) are endomorphisms. This establishes the conjecture in these new cases.

What carries the argument

The Galois set T_f of f-consistent sequences, equipped with a Z_p-module structure via an extracted crystalline character of weight 1, from which integral p-adic Hodge theory functors recover the formal group law.

If this is right

Such pairs (f,u) generate endomorphisms of a formal group law over O_K.
Lubin's conjecture holds for all height-one commuting pairs satisfying the ramification condition on K.
The Galois representation on T_f determines the formal group via the crystalline character and Hodge functors.
The result applies uniformly to cyclo-tame extensions, including new cases beyond prior work.

Where Pith is reading between the lines

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

The same extraction of a weight-one crystalline character might extend to commuting pairs of higher height.
The construction links p-adic dynamical systems directly to the endomorphism rings of formal groups arising in local class field theory.
Classifying all such height-one pairs reduces to classifying the corresponding formal groups and their endomorphism actions.

Load-bearing premise

The ramification index of K over Q_p must be coprime to p squared minus p.

What would settle it

A height-one commuting pair (f,u) over such a K for which the set T_f admits no Z_p-module structure making f an endomorphism, or for which the functors fail to produce a formal group law over O_K with both series as endomorphisms.

read the original abstract

Let $K/\mathbb{Q}_p$ be a finite extension whose ramification index is coprime to $p^2-p$. We study height-one commuting pairs $(f, u)$ of noninvertible and invertible formal power series defined over the ring of integers $\mathcal{O}_K$ of $K$. We begin by extracting a crystalline character of weight $1$ from the $\mathrm{Gal}(\overline K/K)$-set $T_f$ of $f$-consistent sequences. This character is used in order to equip $T_f$ with a $\mathbb{Z}_p$-module structure for which $f$ is an endomorphism. We then apply explicit functors in integral $p$-adic Hodge theory to $T_f$ to recover a formal group defined over $\mathcal{O}_K$ for which $(f, u)$ is a pair of endomorphisms. This proves new cases of a conjecture of Lubin.

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

This paper proves new cases of Lubin's conjecture for height-one p-adic systems over cyclo-tame extensions by extracting a weight-1 crystalline character and applying integral p-adic Hodge functors.

read the letter

This paper proves new cases of Lubin's conjecture. It handles height-one commuting pairs (f, u) of formal power series over O_K for finite extensions K/Q_p where the ramification index is coprime to p^2 - p. The strategy starts by pulling a crystalline character of weight 1 from the Galois set T_f of f-consistent sequences, equips T_f with a Z_p-module structure so that f acts as an endomorphism, and then applies explicit integral p-adic Hodge functors to recover a formal group over O_K on which both f and u act as endomorphisms. The stress-test confirms the steps follow the abstract without circularity or unverified external results beyond standard tools. The argument is internally consistent and states the ramification hypothesis precisely. The main limitation is that coprimality condition, which rules out some extensions but is necessary for the character extraction and is clearly flagged. No load-bearing gaps appear in the outline, and the constructions are explicit enough to be checkable. This work is for specialists in p-adic dynamics and formal groups. A reader already working on Galois representations or arithmetic geometry over local fields will find the explicit functors and module structures useful. It deserves a serious referee because it advances the conjecture with verifiable new cases and standard but carefully applied tools. I would send it to peer review.

Referee Report

0 major / 1 minor

Summary. The manuscript proves new cases of Lubin's conjecture for height-one p-adic dynamical systems over cyclo-tame extensions. Let K/Q_p be a finite extension with ramification index coprime to p^2-p. For commuting pairs (f,u) of height-one formal power series over O_K (f noninvertible, u invertible), the authors extract a crystalline character of weight 1 from the Gal(Kbar/K)-set T_f of f-consistent sequences. This character is used to equip T_f with a Z_p-module structure making f an endomorphism. Explicit functors from integral p-adic Hodge theory are then applied to T_f to recover a formal group over O_K on which both f and u act as endomorphisms.

Significance. If the result holds, the work is significant because it establishes Lubin's conjecture in new cases under a mild coprimality hypothesis on the ramification index, using only standard tools of integral p-adic Hodge theory. The explicit construction of the Z_p-module structure on T_f and the recovery of the formal group provide a concrete and verifiable advance that strengthens the conjecture's plausibility and may support further generalizations.

minor comments (1)

[Abstract] The abstract sketches the argument clearly but does not state Lubin's conjecture explicitly; adding a one-sentence formulation of the conjecture would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. We are pleased that the new cases of Lubin's conjecture under the coprimality hypothesis on the ramification index, obtained via the Z_p-module structure on T_f and integral p-adic Hodge functors, are viewed as a concrete advance. As no specific major comments appear in the report, we have no point-by-point responses to offer.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external standard tools

full rationale

The paper states a precise ramification hypothesis on K/Q_p and extracts a crystalline character of weight 1 from the Galois set T_f of f-consistent sequences. It then endows T_f with a Z_p-module structure making f an endomorphism and applies explicit integral p-adic Hodge functors to recover a formal group over O_K on which (f,u) act as endomorphisms. These steps invoke standard external machinery from p-adic Hodge theory rather than any self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain. No equation or claim in the provided outline reduces by construction to its own inputs; the central result is a new case of Lubin's conjecture obtained via independent functors. This is the normal non-circular outcome for a proof paper using established external theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the stated ramification condition as a domain assumption and on standard results from integral p-adic Hodge theory; no free parameters or new entities are introduced.

axioms (2)

domain assumption K/Q_p is a finite extension whose ramification index is coprime to p^2-p
This condition is invoked to guarantee the existence of the crystalline character of weight 1 and the subsequent formal group construction.
standard math Standard results of integral p-adic Hodge theory apply to the Z_p-module T_f
The paper applies explicit functors from this theory without re-proving them.

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discussion (0)

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

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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

Tp : Formal Groups of Height 1 → Crystalline Characters of GK of Hodge-Tate Weight 1 is an equivalence

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