Periods · motives · transcendence theory · arithmetic geometry

Grothendieck Period Conjecture

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Are all algebraic relations among the periods of a motive exactly the relations forced by its motivic structure?

{ev}_{c_{M}} : k [P_{M}] ⟶ C is injective

Known results and sources

The conjecture asks whether a motive's comparison point obeys only the algebraic relations forced by its motivic symmetries.

Research problem

Exact mathematical statement

Let $k=\overline{\mathbf Q}$ . For a motive $M$ , let

\mathcal P_M=\operatorname{Isom}^{\otimes}(\omega_B\otimes k,\omega_{\mathrm{dR}})

be its Betti–de Rham period torsor, and let $c_M\in\mathcal P_M(\mathbf C)$ be the comparison point. The full mixed or motivated Grothendieck Period Conjecture asks whether

\operatorname{ev}_{c_M}:k[\mathcal P_M]\longrightarrow\mathbf C

is injective for every $M$ . This full-torsor statement is stronger than genericity only on the connected component containing $c_M$ . Its first concrete mixed test in the source is the Kummer question: for non-torsion $q\in\overline{\mathbf Q}^{\times}$ , must

\operatorname{trdeg}_{\overline{\mathbf Q}}\overline{\mathbf Q}(2\pi i,\log q)=2?

The Kummer statement and the full conjecture both remain open in this source; even a proof of the Kummer gate would leave further pure, higher-unipotent, connectedness, and cycle-algebraicity gates.

Problem infographic

Problem at a glance

Problem-first diagram of a motive with Betti and de Rham realizations, its tensor-compatible period torsor and comparison point, the injectivity question for evaluation, and the open Kummer test involving 2πi and log q for q algebraic and non-torsion. — For each motive, the comparison isomorphism determines a point of a period torsor; the conjecture says evaluation at that point has no unexpected algebraic kernel. The subordinate Kummer test shown assumes q is algebraic and non-torsion.

Current mathematical picture

Where work on Grothendieck Period Conjecture stands

Open conjecture

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

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A nonzero algebraic one-point pure-character determinant can by itself produce a Liouville-defeating scalar at the Kummer comparison point. Any dense-system survivor would still require rational normalized exponents, reconstruction of integral characters, an exact height audit, and an Archimedean estimate strong enough to beat Liouville; closing Kummer would still not prove the full conjecture.

Route status · Narrowed route

Main reductionFour dense branches

After normalizing n=1 and choosing p_145=1, the current work reports exact singleton-minor nonvanishing that forces c=g=0, bf=0, and dh=0, leaving four dense branches with six reduced fiber equations.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeResolve the four normalized dense Gr(3,6) branches on the authorized rank-two, two-balanced, nonhexagon locus.Task status · Work already reported in progress

Work mapped so far

Grothendieck Period Conjecture in numbers

2.1kretained lines of mathematical investigation2,087 in the current working snapshot

Argument development: 1,751 · 84%
Explored or eliminated routes: 68 · 3%
Computational analysis: 57 · 3%
Open obligations: 72 · 3%
Definitions and setup: 139 · 7%

9selected mapped statements1routes investigated3open questions2contribution-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 byAlexander GrothendieckProposed1966Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Determine whether any surviving normalized component has genuine arithmetic content for the Kummer problem.

Suggested move: Search for rational μ-coordinates, reconstruct integral exponents, restore the n^6 scaling and mixing-gauge height, then compare the best Archimedean size with the complete logarithmic height.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

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

13 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

Determine whether any surviving normalized component has genuine arithmetic content for the Kummer problem.Suggested move: Search for rational μ-coordinates, reconstruct integral exponents, restore the n^6 scaling and mixing-gauge height, then compare the best Archimedean size with the complete logarithmic height.

Ready to work on

Separate a possible Kummer advance from the additional gates in the full period conjecture and independently audit every retained source claim used publicly.Suggested move: Verify the exact internal certificates and imported theorem hypotheses, then state explicitly which pure, connectedness, higher-unipotent, and cycle-algebraicity layers remain after any Kummer result.

Ready to work on

Resolve the four normalized dense Gr(3,6) branches on the authorized rank-two, two-balanced, nonhexagon locus.Suggested move: Generate the exact coefficient ideals branch by branch, apply only the stated saturation or finite-chart conditions, and preserve a reproducible unit certificate or an exact surviving component.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

Open in general over number fields. The classical conjecture predicts that the algebraic relations or transcendence degree of de Rham–Betti periods are exactly those dictated by algebraic cycles and the motivic Galois group. Rigorous theorems cover fixed codimensions for specified varieties, linear relations for 1-motives, and functional analogues over complex function fields, but none proves arbitrary algebraic relations for all motives in the classical arithmetic setting.

[4][7][8]

External progress

What the literature has established

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

2026
Peer reviewedGao, Ullmo, and Yafaev restated the strong algebraic-relations and weak transcendence-degree versions and isolated the universal-period-torsor connectedness condition needed for the weak form to recover the…[8]
2025
Peer reviewedBakker and Tsimerman proved functional transcendence for integrals in algebraic families and a geometric André–Grothendieck period theorem over complex function fields, while describing the classical…[7]
2022
Authoritative summaryHuber and Wüstholz proved that all rational linear relations among 1-periods arise from bilinearity and functoriality for 1-motives and computed dimensions of period spaces for fixed 1-motives.[6]
2017
Authoritative summaryHuber and Müller-Stach developed Nori motives and the formal period torsor, explaining how the Kontsevich–Zagier conjecture is recast in motivic Galois terms.[5]

Mathematical neighborhood

Related results and reusable starting points

Current focusGrothendieck period conjecture

Related problemde Rham–Betti comparison isomorphism

Algebraic de Rham and Betti cohomology become isomorphic over the complex numbers. Matrix entries of this comparison are periods, and cycles on powers create polynomial relations among them.

[1][8]

Stronger or generalized formstrong algebraic-relations period conjecture

The strong form says every polynomial relation among comparison entries is generated by relations from algebraic cycles on all powers. It implies the weak transcendence-degree equality.

[8]

Weaker or relaxed formmotivic Galois transcendence-degree form

The weak form predicts that the transcendence degree of the period field equals the dimension of the motivic Galois group. Recovering the strong form requires connectedness of the universal period torsor.

[7][8]

Equivalent formulationKontsevich–Zagier period conjecture

The Kontsevich–Zagier formal-period conjecture expresses all relations through geometric integral rules. In Nori's framework it is represented by a motivic period torsor; equivalence to genericity statements retains the relevant irreducibility or connectedness hypothesis.

[2][5]

Weaker or relaxed formGPC^k cycle-class formulation

The fixed-codimension form asks a rational de Rham class whose normalized periods are rational to arise from an algebraic cycle. It is a precise algebraicity layer and has significant proved cases, but it does not exhaust the full polynomial-relations conjecture.

[4]

Solved special caseperiod conjecture for 1-motives

For 1-motives, all rational linear relations among 1-periods are generated by the formal motivic rules and the period-space dimensions can be computed. The restriction to 1-motives and linear relations is essential.

[6]

Related problemHodge and Tate conjectures

The fixed-codimension period conjecture is a de Rham–Betti counterpart of the Hodge and Tate conjectures: all three ask whether realization-theoretic classes with special rationality or invariance arise from algebraic cycles.

[4]

Solved special casefunctional and geometric period conjectures

Functional versions over algebraic function fields are theorems through Ayoub and Bakker–Tsimerman. They replace number periods by period functions and therefore do not settle the classical arithmetic specialization.

[3][7]

Formalization opportunities

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

Formalization targetA statement-aligned formalization needs algebraic de Rham cohomology and Betti or singular cohomology for smooth algebraic varieties over number fields, with a verified comparison isomorphism after extension to the complex numbers.
Formalization targetIt needs periods as comparison-matrix entries or pairings, generated fields, algebraic independence and transcendence degree, with careful base-field conventions and Tate-period normalizations.
Formalization targetThe motivic form requires rigid tensor and Tannakian categories, realization fiber functors, torsors of tensor isomorphisms, motivic Galois groups, and dimension computations.
Formalization targetThe strong form additionally needs algebraic cycles on every power, cycle-class and correspondence actions, ideals of polynomial relations, and a proof that the geometric relations are well typed and functorial.
Formalization targetFormal versions must distinguish fixed-codimension GPC^k, linear period relations for 1-motives, strong genericity, weak transcendence degree, and the functional function-field theorem rather than silently identifying them.
Formalization targetNo problem-level formal statement or proof was identified in the scoped current Mathlib documentation and public Formal Conjectures repository search.

Detailed research inventory

Claims, milestones, and routes in the current map

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

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma2 of 92
negative result3 of 93

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementThe comparison point has no algebraic relations beyond those forced by the motivic period torsor.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementVector-layer reductionintermediate
retained route statementPure-character height barrierintermediate
retained route statementFive-character purityintermediate
retained route statementMixed but rank-one hexagonintermediate
retained route statementFour dense branchesintermediate
retained route statementKummer is only the first gateintermediate
Recorded relationshipThe source material 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 work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetResolve the four normalized dense Gr(3,6) branches on the authorized rank-two, two-balanced, nonhexagon locus.in progress reported
Research targetDetermine whether any surviving normalized component has genuine arithmetic content for the Kummer problem.open
Research targetSeparate a possible Kummer advance from the additional gates in the full period conjecture and independently audit every retained source claim used publicly.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: A nonzero algebraic one-point pure-character determinant can by itself produce a Liouville-defeating scalar at the Kummer comparison point. Any dense-system survivor would still require rational normalized exponents, reconstruction of integral characters, an exact height audit, and an Archimedean estimate strong enough to beat Liouville; closing Kummer would still not prove the full conjecture.

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 bridgeResolve the four normalized dense Gr(3,6) branches on the authorized rank-two, two-balanced, nonhexagon locus.

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 pointDetermine whether any surviving normalized component has genuine arithmetic content for the Kummer problem.

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Research contextPrepared context for any AI agent

Are all algebraic relations among the periods of a motive exactly the relations forced by its motivic structure?

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Sources and references10 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
On the de Rham Cohomology of Algebraic Varietiesoriginal source · Alexander Grothendieck · Publications Mathématiques de l'IHÉS · 1966 · DOI 10.1007/BF02684807 · MR MR0199194 · accessed Aug 7, 2026
2
Periodsoriginal source · Maxim Kontsevich, Don Zagier · Springer · 2001 · DOI 10.1007/978-3-642-56478-9_39 · accessed Aug 7, 2026
3
Une version relative de la conjecture des périodes de Kontsevich–Zagierpeer reviewed result · Joseph Ayoub · Annals of Mathematics · 2015 · accessed Aug 7, 2026
4
Some Remarks Concerning the Grothendieck Period Conjecturepeer reviewed result · Jean-Benoît Bost, François Charles · Journal für die reine und angewandte Mathematik · 2016 · ARXIV 1307.1045 · DOI 10.1515/crelle-2014-0025 · accessed Aug 7, 2026
5
Periods and Nori Motivessurvey or monograph · Annette Huber, Stefan Müller-Stach · Springer · 2017 · DOI 10.1007/978-3-319-50926-6 · accessed Aug 7, 2026
6
Transcendence and Linear Relations of 1-Periodssurvey or monograph · Annette Huber, Gisbert Wüstholz · Cambridge University Press · 2022 · ARXIV 1805.10104 · DOI 10.1017/9781009019729 · accessed Aug 7, 2026
7
Functional Transcendence of Periods and the Geometric André–Grothendieck Period Conjecturepeer reviewed result · Benjamin Bakker, Jacob Tsimerman · Forum of Mathematics, Sigma · 2025-06-23 · ARXIV 2208.05182 · DOI 10.1017/fms.2025.10036 · accessed Aug 7, 2026
8
Bi-Qbar Structures on Hermitian Symmetric Spaces and Arithmetic Applicationspeer reviewed result · Ziyang Gao, Emmanuel Ullmo, Andrei Yafaev · Documenta Mathematica · 2026 · accessed Aug 7, 2026
9
Formal Conjecturesformalization · The Formal Conjectures Authors · Google DeepMind public GitHub repository · current main branch accessed 2026-08-07 · accessed Aug 7, 2026
10
Mathematics in mathlibformalization · Mathlib contributors · Lean community · accessed Aug 7, 2026

Important qualifications

The name Grothendieck period conjecture is used for several related strengths: fixed-codimension de Rham–Betti cycle-class algebraicity, genericity of the comparison point, equality of a period-field transcendence degree with a motivic Galois-group dimension, and formal-period injectivity. This record distinguishes them and does not assert unconditional equivalence without each source's hypotheses.
The historical year 1966 marks Grothendieck's published comparison setting and transcendence questions. The modern motivic Galois-group and universal period-torsor formulations were developed later and do not occur verbatim in the 1966 paper.
The classical number-field conjecture remains open in general. Theorems for fixed codimension, 1-motives, linear relations, or function fields are retained at their exact scope and are not presented as solutions of arbitrary algebraic relations for all motives.
Huber and Wüstholz prove the period conjecture for 1-motives as a theorem on all linear relations among 1-periods and derive dimension formulas. This record does not inflate that result into a theorem about arbitrary higher-degree polynomial relations for general motives.
Bakker and Tsimerman prove a functional/geometric André–Grothendieck period theorem over complex function fields. Their paper explicitly distinguishes the classical conjecture over number fields as still very open.
The strong form implies the weak transcendence-degree form. The cited converse requires connectedness or irreducibility of the relevant universal period torsor and is not recorded without that qualification.
The scoped recognition search found no current Clay, Hilbert, Smale, Erdős, Epoch/FrontierMath, named-prize, or comparable selective-list designation. Absence from this bounded record is not a global nonexistence claim.
The scoped formalization search checked current Mathlib documentation and the public Formal Conjectures repository. It found no problem-level statement or proof; this does not establish nonexistence in every proof assistant or private project.
No canonical external finite computation, dataset, or certificate capable of settling the general transcendence conjecture was identified.
No unreviewed source material, submitted mathematical claim, contributor estimate, or packet computation was inspected or used as external authority. This record has no proof, novelty, review, acceptance, or publication authority.

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