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On the Image of the $p$ -adic Logarithm on Annuli of Principal Units

This paper provides a self-contained analytic proof, utilizing explicit $p$ -adic logarithmic expansions, to demonstrate that for the cyclotomic extension $K=\mathbb{Q}_p(\zeta_p)$ , the image of the annulus of principal units $(1+\mathfrak{m}_K) \setminus (1+\mathfrak{m}_K^2)$ under the $p$ -adic logarithm is exactly $\mathfrak{m}_K^2$ .

Original authors: Mabud Ali Sarkar

Published 2026-01-27

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CC BY 4.0

Original authors: Mabud Ali Sarkar

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 with a very special kind of number system called $p$ -adic numbers. Think of these not as numbers on a straight line, but as numbers arranged in a complex, branching tree where "closeness" is measured differently than in our usual math.

In this paper, the author, Mabud Ali Sarkar, is investigating a specific tool called the $p$ -adic logarithm.

The Big Picture: A Mathematical Map

Think of the $p$ -adic logarithm as a translator or a map. It takes numbers from one specific neighborhood (called "principal units," which are numbers very close to 1) and translates them into another neighborhood (the "maximal ideal," which are numbers very close to 0).

Mathematicians already knew that this translator works perfectly for numbers that are extremely close to 1. However, there was a "foggy zone" in the middle. When the number system gets a bit more complicated (specifically, when the "ramification index" is high), mathematicians didn't know exactly what the translator would produce for numbers in this middle zone. They knew the destination, but they couldn't prove the map was accurate for every single point in that specific area.

The Specific Puzzle

The author focuses on a very specific, famous type of number system: the cyclotomic extension generated by a special root of unity (think of this as a very symmetric, circular arrangement of numbers).

In this specific system, there is a "ring" of numbers just outside the very center.

The Input: A ring of numbers that are close to 1, but not too close (specifically, they are in the first layer of closeness but not the second).
The Question: When we run these numbers through the $p$ -adic logarithm translator, where do they land?

The Discovery

The paper proves a precise result: The translator sends this entire ring of numbers exactly to the next layer of closeness to zero.

To use an analogy:
Imagine you have a set of keys (the input numbers) that are slightly worn but still fit a specific lock. You put them through a machine (the logarithm) that reshapes them.

Before this paper, mathematicians suspected the machine would turn these keys into a specific shape (the second layer of closeness).
This paper provides a step-by-step manual (an "analytic proof") showing exactly how the machine works. It demonstrates that if you feed in any key from that specific ring, the machine always produces a key that fits perfectly into the next layer of the lock, and it produces every single possible key for that layer.

How They Did It

The author didn't just guess; they built the proof using explicit expansions.
Think of this like taking apart a clock to see how the gears turn. The author wrote out the numbers as long, detailed lists of ingredients (using something called a "Hensel expansion"). They then showed, step-by-step, how the logarithm machine mixes these ingredients.

They proved two things:

No Leaks: Nothing from that input ring escapes to a different destination; everything lands exactly where it should.
Full Coverage: The machine doesn't miss any spots; it can produce every single number in that target layer. In fact, they showed that for every target number, there are exactly $p-1$ different input keys that produce it (like having $p-1$ different colored keys that all open the same door).

Why It Matters (According to the Paper)

The paper mentions that knowing exactly where these numbers land is crucial for arithmetic computations, specifically for calculating something called $p$ -adic regulators.

Think of a "regulator" as a way to measure the "volume" or "size" of a group of numbers. If you don't know exactly where the translator sends your numbers, your volume measurement might be slightly off. This paper removes the guesswork for this specific type of number system, ensuring that the volume measurements are perfectly precise.

Summary

In simple terms, this paper solves a missing piece of a puzzle for a specific, well-known type of number system. It proves that a mathematical translator (the $p$ -adic logarithm) behaves in a perfectly predictable and complete way for a specific ring of numbers, turning a "foggy" area of math into a clear, mapped-out territory. This clarity helps mathematicians calculate precise values for other important number theory problems.

Technical Summary: On the Image of the p-adic Logarithm on Annuli of Principal Units

Problem Statement
The paper addresses the characterization of the image of the group of principal units, $1 + \mathfrak{m}_K$ , under the $p$ -adic logarithm map $\log_p$ , where $K$ is a finite extension of $\mathbb{Q}_p$ with maximal ideal $\mathfrak{m}_K$ . While it is well-established that $\log_p$ induces an isomorphism between $1 + \mathfrak{m}_K^r$ and $\mathfrak{m}_K^r$ for sufficiently large $r$ (specifically $r > e/(p-1)$ , where $e$ is the ramification index), the precise structure of the image $\log_p(1 + \mathfrak{m}_K)$ remains unclear when the ramification index is large. Specifically, the map is not generally an isomorphism on the full group of principal units. The paper focuses on the specific case where $K = \mathbb{Q}_p(\zeta_p)$ , the cyclotomic extension of degree $p-1$ , to resolve the description of the image of the annulus region $(1 + \mathfrak{m}_K) \setminus (1 + \mathfrak{m}_K^2)$ .

Methodology
The author provides a self-contained analytic proof relying on explicit $\pi$ -adic expansions, avoiding reliance on prior results from other works. The methodology proceeds as follows:

Normalization and Setup: The $p$ -adic valuation $v$ is normalized such that $v(p)=1$ . For the cyclotomic extension $K = \mathbb{Q}_p(\zeta_p)$ , a uniformizer $\pi$ is chosen such that $\pi^{p-1} = -p$ . The ring of integers is identified as $\mathcal{O}_K = \mathbb{Z}_p[\pi]$ .
Hensel Expansion: Elements $a \in 1 + \mathfrak{m}_K$ are expanded as $a = 1 + a_1\pi + a_2\pi^2 + \dots$ , where coefficients $a_i$ are in $\{0, \dots, p-1\}$ . The condition $a \in (1 + \mathfrak{m}_K) \setminus (1 + \mathfrak{m}_K^2)$ corresponds to $a_1 \neq 0$ .
Series Expansion and Reduction: The $p$ -adic logarithm $\log_p(1+x)$ is expanded for $x = a_1\pi + a_2\pi^2 + \dots$ . Using the relation $\pi^p = -\pi$ (derived from $\pi^{p-1}=-p$ ), the series is reorganized. The author demonstrates that the linear term $(a_1 - a_1^p)\pi$ vanishes modulo $\pi^2$ because $a_1 - a_1^p$ is divisible by $p$ (and thus by $\pi^{p-1}$ ), effectively shifting the leading term of the logarithm to the $\pi^2$ component.
Surjectivity Construction: To prove the image is exactly $\mathfrak{m}_K^2$ , the author constructs a preimage for an arbitrary element $y \in \mathfrak{m}_K^2$ . This involves an inductive determination of the coefficients $a_i$ of the preimage $a(y)$ . The construction shows that for a given target $y$ , there are exactly $p-1$ choices for the pair $(a_1, a_2)$ satisfying the necessary congruence conditions (related to quadratic residues modulo $p$ ), and subsequent coefficients $a_i$ are uniquely determined to satisfy higher-order congruences.

Key Results

Theorem 2.1: For $p \geq 3$ and $K = \mathbb{Q}_p(\zeta_p)$ , the image of the annulus $(1 + \mathfrak{m}_K) \setminus (1 + \mathfrak{m}_K^2)$ under the $p$ -adic logarithm is exactly $\mathfrak{m}_K^2$ . The mapping is shown to be $(p-1)$ -to-1.
Theorem 3.1: The index of the subgroup $\log_p(1 + \mathfrak{m}_K)$ within the maximal ideal $\mathfrak{m}_K$ is $p$ . This is derived by combining the result of Theorem 2.1 with the known isomorphism $\log_p(1 + \mathfrak{m}_K^2) \cong \mathfrak{m}_K^2$ , leading to the conclusion that $\log_p(1 + \mathfrak{m}_K) = \mathfrak{m}_K^2$ . Since the quotient $\mathfrak{m}_K / \mathfrak{m}_K^2$ is isomorphic to the residue field $\mathbb{F}_p$ , the index is $p$ .

Significance and Applications
The paper claims significance primarily in providing an independent, explicit analytic proof of a result previously established in other literature (cited as reference [1]). The explicit nature of the proof, based on $\pi$ -adic expansions, is valuable for arithmetic computations.

The author highlights a specific application to $p$ -adic regulators in number fields. The $p$ -adic regulator is defined as the covolume of the image of units under the $p$ -adic logarithm embedding. The index $[\mathfrak{m}_{F_v} : \log_p(1 + \mathfrak{m}_{F_v})]$ represents a local distortion of volume. The paper demonstrates that in the case where a local completion $F_v$ is isomorphic to $\mathbb{Q}_p(\zeta_p)$ , this index is exactly $p$ . This explicit determination ensures that the local contribution to the $p$ -adic regulator is "completely controlled" and introduces no ambiguity beyond the size of the residue field, which is crucial for the normalization of these regulators.

Conclusion
The paper successfully characterizes the image of the $p$ -adic logarithm on the specific annulus of principal units in the cyclotomic extension $\mathbb{Q}_p(\zeta_p)$ , proving it coincides with the square of the maximal ideal. This result allows for the precise calculation of the index of the logarithmic image, offering a concrete tool for computations involving $p$ -adic regulators in Iwasawa theory and related arithmetic contexts.