Algebraic number theory and p-adic regulators

Leopoldt's Conjecture

Collaboration beta

Do the p-adic logarithms of a full set of independent units always remain independent?

\ker (E_{K} \otimes_{ℤ} ℚ_{p} \overset{λ_{K, p}}{\to} K \otimes_{ℚ} ℚ_{p}) = 0

Known results and sources

Global units, p-adic logarithms, and the open regulator-rank question.

Research problem

Exact mathematical statement

For a number field $K$ , let $E_K=\mathcal O_K^\times/\mu(K)$ , and for a prime $p$ let

\lambda_{K,p}:E_K\otimes_{\mathbb Z}\mathbb Q_p\longrightarrow K\otimes_{\mathbb Q}\mathbb Q_p

be the diagonal $p$ -adic logarithmic localization map. Leopoldt's conjecture asserts

\ker \lambda_{K,p}=0.

Equivalently, the $p$ -adic regulator has the rank predicted by the ordinary unit theorem.

Problem infographic

Problem at a glance

A four-panel problem-first explainer of Leopoldt's conjecture: units, p-adic logarithm vectors, regulator rank, and known versus open scope. — Leopoldt's conjecture asks whether the p-adic logarithmic regulator always has the rank predicted by the unit theorem.

Current mathematical picture

Where work on Leopoldt's Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: The first self-dual split 2-by-2 analysis covers the full conjecture. Prove a uniform correlation-denominator or mismatch-gcd upper bound and then extend the descent beyond the first self-dual split 2-by-2 atom to all remaining blocks and non-Galois fields.

Route status · Narrowed route

Main reductionExplicit mismatch-gcd target

The source gives an explicit scalar mismatch-gcd upper bound sufficient for its current atom.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Leopoldt's Conjecture in numbers

4.7kretained lines of mathematical investigation4,652 in the current working snapshot

Argument development: 4,032 · 87%
Explored or eliminated routes: 155 · 3%
Computational analysis: 17 · 0%
Open obligations: 114 · 2%
Definitions and setup: 334 · 7%

7selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Prove the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.

Suggested move: Audit dyadic and ramified-place correction ideals, then test the explicit ideal gcd formulation against uniform height and correlation estimates.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.Suggested move: Audit dyadic and ramified-place correction ideals, then test the explicit ideal gcd formulation against uniform height and correlation estimates.

Ready to work on

Treat non-self-dual paired components and larger matrix blocks.Suggested move: Formulate an invariant and denominator control that survives paired and higher-dimensional simple components.

Ready to work on

Complete the non-Galois Hecke/maximal-minor descent.Suggested move: Bind the permutation-module and maximal-minor reductions to explicit local blocks before attempting a global nonvanishing conclusion.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The conjecture is known for fields abelian over Q but remains open for general number fields. Recent work supplies bounds on possible defect rather than a general zero-defect theorem.

[2][1]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
Authoritative summaryAn Oberwolfach report records the conjecture as known for number fields abelian over Q and unknown in general.[2]
2023-11-29
Peer reviewedMaksoud generalized Waldschmidt's bound for Leopoldt's defect over arbitrary extensions; this is a quantitative partial result, not a proof of vanishing.[1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction2 of 72
lemma3 of 73
negative result1 of 71

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included

retained route statementAre the p-adic logarithms of independent global units always independent?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact conjectureintermediate
retained route statementSource-reported projective escapeintermediate
retained route statementExplicit mismatch-gcd targetintermediate
retained route statementAtom scope boundaryintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSource-reported limitationreported failure
Research targetProve the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.open
Research targetTreat non-self-dual paired components and larger matrix blocks.open
Research targetComplete the non-Galois Hecke/maximal-minor descent.open
Research targetNo complete proof in sourcesuperseded
Research targetUniform denominator theoremsuperseded
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: The first self-dual split 2-by-2 analysis covers the full conjecture. Prove a uniform correlation-denominator or mismatch-gcd upper bound and then extend the descent beyond the first self-dual split 2-by-2 atom to all remaining blocks and non-Galois fields.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.

Leopoldt's Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Do the p-adic logarithms of a full set of independent units always remain independent?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
On the rank of Leopoldt’s and Gross’s regulator mapspeer reviewed result · Alexandre Maksoud · EMS Press / Documenta Mathematica · 2023-11-29 · DOI 10.4171/DM/935 · accessed Aug 14, 2026
2
Oberwolfach Report 38/2025authoritative webpage · Mathematisches Forschungsinstitut Oberwolfach / EMS Press · 2025 · accessed Aug 14, 2026

Important qualifications

Bounded collection from two official or publisher-maintained sources; it is not an exhaustive literature review.
The Oberwolfach report is an authoritative current summary, not the original formulation of the conjecture.
No incoming-packet URL was fetched and packet claims were not used as external evidence.
No formalization or computation was independently reproduced.

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Leopoldt's Conjecture

Exact mathematical statement

Problem at a glance

Where work on Leopoldt's Conjecture stands

Leopoldt's Conjecture in numbers

Prove the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier