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arxiv: 2604.27111 · v1 · submitted 2026-04-29 · 🧮 math.NT

An O_K-basis for the image of a Lubin-Tate logarithm on π-regular extensions of K

Georgia Harbor-Collins This is my paper

Pith reviewed 2026-05-07 10:28 UTC · model grok-4.3

classification 🧮 math.NT
keywords Lubin-Tateformal groupsp-adic fieldslogarithmmaximal idealO_K-modulevaluationlocal number theory
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The pith

For π-regular extensions L of a p-adic field K, the image of the Lubin-Tate logarithm on the maximal ideal admits an explicit O_K-basis and has a determined minimal valuation.

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

The paper focuses on the logarithm map coming from a Lubin-Tate formal group law applied to points in the maximal ideal of certain extensions of K. It provides an O_K-basis for the resulting additive group in the case of π-regular extensions and identifies the smallest valuation that appears in this image. This structure is key to understanding the arithmetic of formal groups over local fields. The analysis also covers how these findings apply more broadly to finite extensions and gives a specific basis for the Lubin-Tate tower extensions.

Core claim

Let K be a finite p-adic field with uniformiser π. For a π-regular extension L over K, the image of the Lubin-Tate logarithm log_{[π]} applied to the maximal ideal m_L (with its O_K-module structure from the formal group) is computed as an O_K-module by giving an explicit basis, and the minimal valuation of its elements is determined. The paper further determines such a basis for the Lubin-Tate extensions of level n.

What carries the argument

The Lubin-Tate logarithm log_{[π]} associated to the formal group law [π](X), which maps the formal group points F(m_L) to an additive O_K-module whose structure is analyzed.

If this is right

  • An explicit basis for log_{[π]}(F(m_L)) as O_K-module is given for π-regular L|K.
  • The minimal valuation of elements in the image is determined.
  • Some results extend to arbitrary finite extensions of K.
  • A basis is determined for the image on the Lubin-Tate extension K_{π^n} of level n.

Where Pith is reading between the lines

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

  • The explicit basis allows for concrete computations in the arithmetic of Lubin-Tate formal groups over local fields.
  • Determining the basis for higher levels in the Lubin-Tate tower suggests a pattern for the infinite tower.

Load-bearing premise

The extensions L over K must satisfy the π-regular condition, which ensures the properties used to compute the basis and valuation.

What would settle it

Constructing a π-regular extension L|K where no O_K-basis matches the described image or where an element of smaller valuation appears in the logarithm image would disprove the claims.

read the original abstract

Let $K$ be a finite $p$-adic field with uniformiser $\pi$. In this paper we study the image of the logarithm attached to a Lubin-Tate series $[\pi](X)$ on the maximal ideal of so-called $\pi$-regular extensions of $K$; for such an extension $L|K$ we compute a basis for the additive group $\log_{[\pi]}(\mathcal{F}(\mathfrak{m}_L))$ as an $O_K$-module, where $\mathcal{F}(\mathfrak{m}_L)$ denotes the maximal ideal $\mathfrak{m}_L$ equipped with the $O_K$-module structure coming from the formal group associated to $[\pi](X)$, and determine the minimal valuation of the elements in $\log_{[\pi]}(\mathcal{F}(\mathfrak{m}_L))$. In the final section of this paper we discuss how some of these results extend to arbitrary finite extensions of $K$ and conclude by determining a basis of the $O_K$-module $\log_{[\pi]}(\mathcal{F}(\mathfrak{m}_{K_{\pi^n}}))$, where $K_{\pi^n}$ is the Lubin-Tate extension of level $n\geq 1$.

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.

Referee Report

0 major / 3 minor

Summary. The paper defines π-regular extensions L|K of a finite p-adic field K with uniformizer π, equips the maximal ideal m_L with the O_K-module structure induced by the Lubin-Tate formal group law associated to [π](X), and computes an explicit O_K-basis for the image of the attached logarithm map log_{[π]} : F(m_L) → (m_L, +). It also determines the minimal valuation of elements in this image. The final section extends some of these results to arbitrary finite extensions of K and gives an explicit O_K-basis for log_{[π]}(F(m_{K_{π^n}})) in the Lubin-Tate tower.

Significance. If the computations hold, the explicit bases and minimal-valuation statements supply concrete, usable information about the additive structure of logarithm images in Lubin-Tate formal groups. This is potentially useful for explicit calculations in local class field theory, ramification theory, and the study of formal modules over p-adic rings. The parameter-free description for the Lubin-Tate extensions K_{π^n} is a clear strength.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction refer to “so-called π-regular extensions” without giving the definition at the outset; moving the definition to §1 or the beginning of §2 would improve readability.
  2. [§2] Notation for the formal group law F and the module structure on m_L is introduced gradually; a single consolidated notation paragraph early in the paper would help.
  3. [Final section] The extension of the main results to arbitrary finite extensions in the final section is stated briefly; a short remark on which parts of the π-regular case survive and which require additional hypotheses would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its potential utility in local class field theory and ramification theory, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines π-regular extensions of K and applies standard properties of the Lubin-Tate formal group law and its logarithm to compute an explicit O_K-basis for log_{[π]}(F(m_L)) together with minimal valuations, proceeding via ramification filtrations and module structures. No equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the final extension to arbitrary finite extensions and to K_{π^n} follows from the same valuation arguments without importing uniqueness theorems or ansatzes from prior author work. The central claim is an explicit computation grounded in the given definitions and standard formal-group facts, with no evidence of circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted beyond the standard setup of Lubin-Tate formal groups and p-adic fields.

pith-pipeline@v0.9.0 · 5519 in / 1030 out tokens · 42761 ms · 2026-05-07T10:28:19.832339+00:00 · methodology

discussion (0)

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Reference graph

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