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Arithmetic geometry · motives · equivariant L-values · algebraic K-theory

Equivariant Tamagawa Number Conjecture

Collaboration beta

A canonical equivariant arithmetic class packages leading L-values, periods, regulators, and integral data. ETNC predicts that this class vanishes in the appropriate relative K-group; the general conjecture remains open despite many special cases.

TΩ(M,Λ)=0
Known results and sources
A motive M with coefficient symmetry Lambda faces an unresolved target panel carrying TΩ of M and Lambda equals zero question mark, with T and Omega shown as adjacent same-height letters.
ETNC predicts vanishing of a canonical equivariant arithmetic class; its exact formulation depends on the motive and coefficient order, and the general conjecture is open.

Research problem

Exact mathematical statement

Let MM be a motive over a number field with an action of a finite-dimensional semisimple ℚ\mathbb Q-algebra AA, and let Λ\Lambda be an order for which the required projective Λ\Lambda-structure is defined. In the Burns–Flach formulation, the Equivariant Tamagawa Number Conjecture predicts that the associated canonical equivariant leading-term class vanishes in the relevant relative algebraic K-group; schematically,

TΩ(M,Λ)=0.T\Omega(M,\Lambda)=0.

The exact class and relative K-group depend on the motive, coefficients, order, and normalization data.

Problem infographic

Problem at a glance

A three-part landscape takes source-bound leading-term data—its order of vanishing, normalized leading term, periods, and regulator—to a canonical relative K-theory class, then marks the TΩ vanishing question open while separating special cases from general ETNC.
The canonical class depends on carefully normalized leading-term data; the visual does not assert an ordinary Taylor-series formula and separates known special cases from general ETNC.

Current mathematical picture

Where work on Equivariant Tamagawa Number Conjecture stands

Open conjecture

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

Useful failureTreating an additive central representative as the canonical multiplicative determinant obstruction

The revision-6 terminal status warns that the additive coordinate is not yet a canonical determinant and that finite descent alone cannot settle the full conjecture. The current work's finite-descent results remain potentially useful once a canonical determinant/K₁ realization and an arithmetic vanishing theorem are supplied without assuming final integrality.

Route status · Narrowed route
Main reductionFour independent global stages

The current work separates rationality, maximal-order ETNC, integral descent, and conductor/K-theory patching.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. 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

Equivariant Tamagawa Number Conjecture in numbers

2.8kretained lines of mathematical investigation2,809 in the current working snapshot
Argument development
2,350 · 84%
Explored or eliminated routes
143 · 5%
Computational analysis
13 · 0%
Open obligations
91 · 3%
Definitions and setup
212 · 8%
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.

Proposed byDavid Burns and Matthias FlachProposed2001Status checkedAug 15, 2026
Source recordedAug 15, 2026

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.

11 selected steps

Scroll horizontally to explore the route

Working route overview for Equivariant Tamagawa Number ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Does the canonical equivariant leading-term class vanish? — Depends on missing premiseDoes the canonicalequivariant leading-termclass…Current reduction — Depends on missing premiseCurrent reductionFour independent global stages — Depends on missing premiseFour independent globalstagesClosing target — Depends on missing premiseClosing targetFinite descent does not prove general ETNC — Depends on missing premiseFinite descent does notprove general ETNCFull ETNC remains unproved — Depends on missing premiseFull ETNC remains unprovedMultiplicative conductor target — Depends on missing premiseMultiplicative conductortargetTreating an additive central representative as the canonical multiplicative determinant obstruction — stoppedTreating an additive centralrepresentative as thecanonical…Construct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient. — OpenConstruct the canonicalrelative-K₁ ordeterminant-torsor…Prove the resulting multiplicative class vanishes for an actual normalized motivic zeta system. — OpenProve the resultingmultiplicative classvanishes…Maintain a global ledger that distinguishes rationality, maximal-order ETNC, descent, conductor compatibility, and patching in every arithmetic family. — OpenMaintain a global ledgerthat distinguishesrationality,…
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 routeTreating an additive central representative as the canonical multiplicative determinant obstruction

The revision-6 terminal status warns that the additive coordinate is not yet a canonical determinant and that finite descent alone cannot settle the full conjecture. The current work's finite-descent results remain potentially useful once a canonical determinant/K₁ realization and an arithmetic vanishing theorem are supplied without assuming final integrality.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Construct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient.Suggested move: State its transformation law under compatible changes of determinant generators and prove the exact coordinate formula.
Ready to work on
02
Prove the resulting multiplicative class vanishes for an actual normalized motivic zeta system.Suggested move: Compute a normalized central coefficient in one critical-Tate family and prove its ratio lies in the required principal-unit subgroup before final integrality.
Ready to work on
03
Maintain a global ledger that distinguishes rationality, maximal-order ETNC, descent, conductor compatibility, and patching in every arithmetic family.Suggested move: For each family, bind a source theorem or explicit assumption to every global stage and do not infer one stage from another.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The general conjecture remains open. Official recent sources prove or conditionally derive specified p-parts or minus parts for restricted Tate-motive settings; those results do not establish ETNC for arbitrary motives and orders.

[1][3][4]
External progress

What the literature has established

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

  1. PreprintDasgupta, Kakde, and Silliman derived a minus-part ETNC result for a Tate motive in a finite abelian CM-extension setting.[4]
  2. Peer reviewedNickel obtained conditional p-part results for specified Tate motives; the hypotheses and restricted setting remain part of the result.[3]
  3. Historical sourceBurns and Flach formulated the leading-term conjecture for motives with noncommutative coefficients and order-dependent integral structures.[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusEquivariant Tamagawa Number Conjecture
Solved special caseSpecified Tate-motive p-parts under additional conjectural hypotheses

These results establish restricted p-parts in specified arithmetic settings, not the general motive-and-order conjecture.

[3]
Solved special caseMinus part for a Tate motive in finite abelian CM extensions

This special case is a substantive result but retains narrower motive, extension, and sign scope.

[4]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal statement fixing the exact motive, semisimple coefficient algebra, order, determinant conventions, and relative K-group.
  • Formalization targetFormal infrastructure for equivariant L-values, periods, regulators, determinant functors, and local complexes.
  • Formalization targetA checked proof of vanishing for the exact canonical class, with rationality, maximal-order, descent, and patching hypotheses explicit.

Research-record corrections

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 updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
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
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
Research-record correctionWe 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
  • lemma2 of 72
  • negative result2 of 72
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 statementDoes the canonical equivariant leading-term class vanish?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementFull ETNC remains unprovedintermediate
  • retained route statementFour independent global stagesintermediate
  • retained route statementMultiplicative conductor targetintermediate
  • retained route statementFinite descent does not prove general ETNCintermediate
  • 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 failureTreating an additive central representative as the canonical multiplicative determinant obstructionreported failure
  • Research targetConstruct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient.open
  • Research targetProve the resulting multiplicative class vanishes for an actual normalized motivic zeta system.open
  • Research targetMaintain a global ledger that distinguishes rationality, maximal-order ETNC, descent, conductor compatibility, and patching in every arithmetic family.open
  • Research targetCanonical relative-K₁ realization opensuperseded
  • Research targetArithmetic vanishing remains opensuperseded
  • Narrowed routeTreating an additive central representative as the canonical multiplicative determinant obstructionThe revision-6 terminal status warns that the additive coordinate is not yet a canonical determinant and that finite descent alone cannot settle the full conjecture. The current work's finite-descent results remain potentially useful once a canonical determinant/K₁ realization and an arithmetic vanishing theorem are supplied without assuming final integrality.
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 bridgeConstruct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient.

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.

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Prepared starting pointConstruct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient.

Equivariant Tamagawa Number Conjecture · ready to start

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Research contextPrepared context for any AI agent

A canonical equivariant arithmetic class packages leading L-values, periods, regulators, and integral data. ETNC predicts that this class vanishes in the appropriate relative K-group; the general conjecture remains open despite many special cases.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Tamagawa numbers for motives with (non-commutative) coefficientsoriginal source · David Burns, Matthias Flach · Documenta Mathematica 6 · 2001 · DOI 10.4171/DM/113 · accessed Aug 15, 2026
  2. 2
    From the Birch & Swinnerton-Dyer Conjecture over the Equivariant Tamagawa Number Conjecture to non-commutative Iwasawa theory — a surveysurvey or monograph · Otmar Venjakob · Cambridge University Press · 2007 · ARXIV math/0507275 · accessed Aug 15, 2026
  3. 3
    On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecturepeer reviewed result · Andreas Nickel · Selecta Mathematica 28 · 2022 · DOI 10.1007/s00029-021-00717-3 · accessed Aug 15, 2026
  4. 4
    On The Equivariant Tamagawa Number Conjecturepreprint · Samit Dasgupta, Mahesh Kakde, Jesse Silliman · arXiv · 2023 · ARXIV 2312.09849 · accessed Aug 15, 2026

Important qualifications

  • The bounded pass checked bibliographic identity, current status, and representative official special-case sources; it was not an exhaustive bibliography or priority review.
  • The private packet and its URLs were not used as external authority. No submitted URL or attachment was fetched, executed, compiled, or rendered.
  • No formal proof of the unrestricted statement was identified in this bounded pass; absence from the pass is not proof of absence.

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