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

This paper computes an $O_K$ -basis and determines the minimal valuation for the image of a Lubin-Tate logarithm on the maximal ideal of $\pi$ -regular extensions of a finite $p$ -adic field, while also extending these results to arbitrary finite extensions and specifically to Lubin-Tate extensions of level $n$ .

Original authors: Georgia Harbor-Collins

Published 2026-05-01

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CC BY 4.0

Original authors: Georgia Harbor-Collins

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are working in a very specific, high-tech factory called $K$ . This factory produces numbers, but they aren't your usual numbers; they are "p-adic" numbers, which behave like infinite sequences of digits that get smaller and smaller the further out you go.

Inside this factory, there is a special machine called a Lubin-Tate Series. Think of this machine as a unique "shape-shifter." It takes a number, twists it, and turns it into a new number according to a very strict set of rules. Associated with this machine is a Logarithm, which acts like a translator or a decoder ring. Its job is to take these twisted numbers and translate them back into a simpler, straight-line form (an additive group).

The Problem: The Translator Gets Stuck

Usually, if you feed this translator a number from the "maximal ideal" (a specific section of the factory floor), it works perfectly. It translates the number, and you get a result that fits neatly into the factory's inventory system.

However, the author of this paper, Georgia Harbor-Collins, noticed a glitch. When you take numbers from certain extended versions of the factory (called $\pi$ -regular extensions), the translator doesn't always produce a result that fits the standard inventory list. Sometimes the results are "messy" or fall outside the expected range.

The big question was: If we feed the translator a whole bunch of numbers from these extended factories, what exactly does the pile of output look like? Can we describe the "shape" of this pile?

The Solution: Finding the "Lego Bricks"

The paper's main achievement is finding a basis. In simple terms, a basis is like a set of unique Lego bricks. If you have the right set of bricks, you can build any structure in the pile using only those bricks and the factory's standard rules for stacking them.

Harbor-Collins figured out exactly which "bricks" (specific numbers) are needed to build the entire pile of translated numbers for these special extensions.

The Discovery: She found a specific list of numbers (involving roots of the factory's uniformizer $\pi$ and elements from the residue field) that, when run through the translator, create a perfect, non-overlapping set of building blocks.
The Analogy: Imagine you have a messy pile of sand. The author figured out that if you pick up exactly these specific grains of sand, you can reconstruct the entire pile without needing any extra grains or missing any spots.

The "Lowest Point" Discovery

Once she knew the building blocks, she asked: "What is the lowest point in this pile?"

In the world of these numbers, "height" is measured by something called valuation. A higher valuation means the number is "smaller" (closer to zero), and a lower valuation means it's "larger."

The paper calculates the exact "lowest floor" of the translated pile.
The Metaphor: Imagine the translated numbers are a skyscraper. The author determined exactly how many stories down the basement goes. She found a formula that tells you the depth of the basement based on how "twisted" the original factory rules were.

The Special Case: The "Tangled" Factory

The paper also looks at a trickier scenario where the factory extension contains "torsion."

The Analogy: Imagine some of the numbers in the factory are "tangled" or "knotted" (mathematically, they are roots of unity). In the "regular" factories, the numbers were smooth and untangled. In these "tangled" factories, the translator behaves differently because some inputs cancel each other out completely (turning into zero).
The Result: Even in these messy, tangled factories, the author managed to find a new set of "bricks" that build the translated pile. She showed that while the pile is slightly different, it still has a predictable structure, and she provided the exact list of bricks needed for this specific type of factory (known as Lubin-Tate extensions).

Summary of the Journey

The Setup: We have a number translator (Logarithm) in a specialized number factory.
The Issue: When we use it on certain extended factories, we don't know exactly what the output looks like.
The Breakthrough: The author found the exact "Lego set" (basis) needed to build the entire output pile for "clean" (regular) factories.
The Measurement: She calculated the deepest point (minimum valuation) of this pile.
The Expansion: She also solved the puzzle for "tangled" (non-regular) factories, providing a new set of bricks for those cases too.

In essence, this paper turns a messy, unpredictable pile of translated numbers into a neatly organized, fully understood structure, giving mathematicians a precise map of where every number lands.

1. Problem Statement

The paper addresses the problem of determining the structure of the image of the Lubin-Tate logarithm, denoted $\log_{[\pi]}$ , when restricted to the maximal ideal $\mathfrak{m}_L$ of a finite extension $L$ of a $p$ -adic field $K$ .

Context: Let $K$ be a finite extension of $\mathbb{Q}_p$ with uniformizer $\pi$ and residue field size $q$ . Let $[\pi](X)$ be a Lubin-Tate series associated with $\pi$ , and $F(X, Y)$ be the associated formal group law. The logarithm $\log_{[\pi]}$ is a homomorphism from the formal group $F(\mathfrak{m}_{\mathbb{C}_K})$ to the additive group of the field.
Known Results: For sufficiently deep levels of the maximal ideal (specifically where $v_L(x) > \frac{v_L(\pi)}{q-1}$ ), the logarithm is an isomorphism. In the classical multiplicative case ( $K=\mathbb{Q}_p, \pi=p$ ), the image of the $p$ -adic logarithm on principal units is well-understood for extensions not containing primitive $p$ -th roots of unity.
The Gap: When $v_L(\pi) \ge q-1$ , the behavior of the image $\log_{[\pi]}(F(\mathfrak{m}_L))$ becomes complex, particularly regarding its structure as an $\mathcal{O}_K$ -module and the minimal valuation of its elements. While the image is known to be a free $\mathcal{O}_K$ -module of rank $[L:K]$ , an explicit basis and precise bounds on valuations were not fully established for general extensions, especially those containing non-trivial torsion.

2. Methodology

The author employs a combination of formal group theory, valuation theory, and module structure analysis over the ring of integers $\mathcal{O}_K$ .

$\pi$ -Regular Extensions: The core methodology focuses on a specific class of extensions called $\pi$ -regular extensions, defined as finite extensions $L|K$ that contain no non-trivial $[\pi]$ -torsion (i.e., $\mu_{[\pi^\infty]}(L) = \{0\}$ ). This condition ensures the logarithm is injective on the maximal ideal.
Basis Construction via Nakayama's Lemma: The paper constructs a basis for the formal group $F(\mathfrak{m}_L)$ $F (m_{L})$ as an $\mathcal{O}_K$ $O_{K}$ -module. It utilizes the filtration of the maximal ideal and analyzes the induced maps on the quotients $F(\mathfrak{m}_L^i)/F(\mathfrak{m}_L^{i+1})$ $F (m_{L}^{i}) / F (m_{L}^{i + 1})$ .
- The author proves that the $[\pi]$ -series induces isomorphisms between specific quotient layers when $L$ is $\pi$ -regular.
- Using Nakayama's Lemma, the author identifies a minimal generating set for $F(\mathfrak{m}_L)$ by selecting elements from specific valuation layers that are not "killed" or mapped to zero by the $[\pi]$ -action.
Valuation Analysis: The paper derives an explicit formula for the valuation $v_L(\log_{[\pi]}(x))$ based on the number of iterations $\ell_x$ required to map an element $x$ into the domain of convergence of the exponential map ( $D_K(L)$ ).
Extension to Torsion Cases: For extensions containing torsion (where the logarithm has a kernel), the author adapts the spanning set construction, identifying which elements must be added to the generating set to account for the torsion submodule.

3. Key Contributions and Results

A. Definition and Characterization of $\pi$ -Regular Extensions

The paper formalizes the concept of $\pi$ -regularity. An extension $L|K$ is $\pi$ -regular if it contains no non-trivial zeros of $[\pi](X)$ .

Lemma 4.3: Provides an equivalent condition for $\pi$ -regularity based on the valuation $v_L(\pi)$ and the solvability of a specific congruence $u^q + \epsilon u \equiv 0 \pmod{\mathfrak{m}_L}$ .

B. Basis for the Formal Group $F(\mathfrak{m}_L)$

Theorem 4.9: For a $\pi$ -regular extension $L|K$ with uniformizer $\varpi$ and residue degree $f$ , the set
$B_L = \{ \zeta_i \varpi^j \mid 1 \le i \le f, \ 1 \le j \le \frac{q v_L(\pi)}{q-1}, \ q \nmid j \}$
forms an $\mathcal{O}_K$ -basis for $F(\mathfrak{m}_L)$ , where $\zeta_i$ are lifts of a basis for the residue field.

This generalizes the classical result for principal units $1+\mathfrak{m}_L$ in $\mathbb{Q}_p$ (Theorem 4.1).

C. Basis for the Image of the Logarithm

Theorem 5.1: Since the logarithm is an isomorphism on $\pi$ -regular extensions, the image of the basis $B_L$ under $\log_{[\pi]}$ forms a basis for the additive group $\log_{[\pi]}(F(\mathfrak{m}_L))$ :
$\tilde{B}_L = \{ \log_{[\pi]}(\zeta_i \varpi^j) \mid 1 \le i \le f, \ 1 \le j \le \frac{q v_L(\pi)}{q-1}, \ q \nmid j \}$

D. Minimal Valuation Bound

Theorem 5.6: The paper determines the exact minimal valuation of elements in the image. If $v_L(\pi) \ge q-1$ , let $\gamma$ be the unique integer such that $q^{\gamma-1} \le \frac{v_L(\pi)}{q-1} < q^\gamma$ . Then:
$\min_{y \in \log_{[\pi]}(F(\mathfrak{m}_L))} v_L(y) = q^\gamma - \gamma v_L(\pi)$
This result provides a precise lower bound for the "size" of the image, determining the smallest disc containing the logarithm's image.

E. Generalization to Extensions with Torsion

The paper extends these results to arbitrary finite extensions $L|K$ that may contain non-trivial $[\pi]$ -torsion (specifically Lubin-Tate extensions $K_{\pi^n}$ ).

Theorem 6.1: Constructs a spanning set $S_L$ for $F(\mathfrak{m}_L)$ which includes an additional layer of generators corresponding to the torsion level.
Theorem 6.5: For the Lubin-Tate extension $K_{\pi^n}$ , the paper explicitly constructs a basis for the image of the logarithm:
$B_n = \{ \log_{[\pi]}(\lambda^j) \mid 2 \le j \le q^n-1, \ q \nmid j \} \cup \{ \log_{[\pi]}(\lambda^{q^n}) \}$
where $\lambda$ is a generator of the torsion module $\mu_{[\pi^n]}$ .

4. Significance

Structural Clarity: The paper provides the first explicit $\mathcal{O}_K$ -basis for the image of the Lubin-Tate logarithm on a broad class of extensions ( $\pi$ -regular), moving beyond the classical multiplicative case ( $\pi=p$ ).
Precision: It resolves the ambiguity regarding the minimal valuation of the image, offering a formula that depends explicitly on the ramification index and the residue field degree.
Unification: The results unify the treatment of formal groups and classical $p$ -adic logarithms, showing how the presence of torsion (roots of unity) alters the module structure and necessitates specific adjustments to the basis.
Applications: These results are foundational for local class field theory, Iwasawa theory, and the study of $p$ -adic $L$ -functions, where understanding the image of logarithmic maps on specific subgroups of units is crucial.

In summary, Harbor-Collins successfully generalizes classical results on $p$ -adic logarithms to the Lubin-Tate setting, providing rigorous algebraic descriptions (bases) and analytic bounds (valuations) for the image of these logarithms on both regular and torsion-containing extensions.