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Arithmetic geometry, motives, and equidistribution

Generalized Sato–Tate Conjecture

Collaboration beta

Do normalized Frobenius conjugacy classes spread through the correct compact symmetry group according to Haar measure?

xv⇒Haar measure onSTK†(M)/∼
Known results and sources
Frobenius conjugacy classes distributed around a compact symmetry group, with a visibly open uniformity question.
Generalized Sato–Tate predicts Haar equidistribution of normalized Frobenius classes in the correct compact group.

Research problem

Exact mathematical statement

Let KK be a number field and let M/KM/K be a homogeneous semisimple pure polarizable motive in the source’s stated scope. Form the parity-safe normalized group

GK†=ker(N:HK→Gm),G_K^\dagger=\ker\bigl(N:H_K\to\mathbf G_m\bigr),

and let STK†(M)\operatorname{ST}_K^\dagger(M) be a maximal compact subgroup of GK†(C)G_K^\dagger(\mathbf C). For each good place vv, let xvx_v be the normalized semisimple Frobenius class.

The full parity-safe Generalized Sato–Tate conjecture says that the classes xvx_v, ordered by qvq_v, are equidistributed in the conjugacy-class space of STK†(M)\operatorname{ST}_K^\dagger(M) for the pushforward of Haar probability measure.

Problem infographic

Problem at a glance

A problem-first explainer showing good primes, normalized Frobenius classes, the parity-safe Sato–Tate group, Haar equidistribution, and the unresolved universal scope.
The conjecture compares arithmetic Frobenius data with Haar measure on a compact motivic symmetry group.

Current mathematical picture

Where work on Generalized Sato–Tate Conjecture stands

Partially resolved

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: Meromorphic continuation alone, or pole-freeness of only the original target, is enough to imply full equidistribution. The universal structural and analytic inputs needed to turn the conditional character prime-number theorem into equidistribution are not known for arbitrary motives.

Route status · Narrowed route
Main reductionCurrent reduction

Close three universal inputs: tensor Tate/full group identification, meromorphic continuation near the boundary for all required tests, and ACB3PC pole-freeness at s=1 for noncomponent adaptive constituents.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim.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

Generalized Sato–Tate Conjecture in numbers

2.5kretained lines of mathematical investigation2,542 in the current working snapshot
Argument development
2,186 · 86%
Explored or eliminated routes
36 · 1%
Computational analysis
58 · 2%
Open obligations
66 · 3%
Definitions and setup
196 · 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 byJean-Pierre Serre (generalized formulation)Status checkedAug 14, 2026
Source recordedAug 14, 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 Generalized Sato–Tate ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Do Frobenius classes equidistribute in the correct compact symmetry group? — Depends on missing premiseDo Frobenius classesequidistribute in thecorrect…Adaptive central transfer — Depends on missing premiseAdaptive central transferCurrent reduction — Depends on missing premiseCurrent reductionThree logical layers — Depends on missing premiseThree logical layersClosing target — Depends on missing premiseClosing targetConditional all-base theorem — Depends on missing premiseConditional all-base theoremFull parity-safe statement — Depends on missing premiseFull parity-safe statementSource-reported limitation — stoppedSource-reported limitationProve tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim. — OpenProve tensor Tate for theTannakian category generatedby…Prove meromorphic continuation near the full boundary for every normalized irreducible motivic target and its finite adaptive closure. — OpenProve meromorphiccontinuation near the fullboundary…Prove ACB3PC pole-freeness at s=1 for every nontrivial noncomponent member of the adaptive third-layer family. — OpenProve ACB3PC pole-freenessat s=1 for every nontrivialnoncomponent…
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

The following claim is rejected or insufficient in the recorded route: Meromorphic continuation alone, or pole-freeness of only the original target, is enough to imply full equidistribution. The universal structural and analytic inputs needed to turn the conditional character prime-number theorem into equidistribution are not known for arbitrary motives.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim.Suggested move: Separate a fixed-object fixed-realization theorem from the all-base-change connectedness and component package, and prove the exact invariant comparison needed for the chosen scope.
Ready to work on
02
Prove meromorphic continuation near the full boundary for every normalized irreducible motivic target and its finite adaptive closure.Suggested move: Identify honest automorphic or trace-formula realizations for the exact adaptive constituents, including nonregular and non-self-dual cases.
Ready to work on
03
Prove ACB3PC pole-freeness at s=1 for every nontrivial noncomponent member of the adaptive third-layer family.Suggested move: Audit Section 8 independently, decompose the adaptive projective constituents, and certify constituentwise pole control without relying on virtual cancellation.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusPartially resolved

Many classical and special-family cases are known and substantial group infrastructure exists, but the broad motivic equidistribution statement is not a theorem in full generality.

[1][3][4]
External progress

What the literature has established

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

  1. Authoritative summaryThe maintained database covers rational Sato–Tate groups in stated low weights and degrees; this is finite structural coverage, not universal equidistribution.[2]
  2. PreprintA motivic Serre-group and algebraic Sato–Tate construction is made explicit for a class of motives, with reductions involving identity components.[4]
  3. Authoritative summaryAn expository account presents Serre’s generalization through compact groups, character tests, and L-functions.[3]
5 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusGeneralized Sato–Tate Conjecture
Solved special caseSato–Tate for elliptic curves over totally real or CM fields

The reviewed LMFDB definition records known equidistribution for all elliptic curves over totally real or CM fields.

[1]
Dependency or reductionAlgebraic Sato–Tate group construction

Algebraic and motivic Sato–Tate group identification supplies the compact group in which equidistribution is formulated.

[4]
Related problemWeight-three motive families

Weight-three motives provide classified groups and numerical evidence in selected families.

[5]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • dataset · source linked; not reproduced by ProofAtlasLMFDB Sato–Tate group database

    Maintained low-weight and low-degree group data with explicit stated completeness limits.

    [2]

Formalization opportunities

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

  • Formalization targetA universal formal category of motives and realization comparison matching the conjecture’s intended scope.
  • Formalization targetFormal Haar equidistribution and L-function criteria for all possibly disconnected motivic compact groups.

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 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
  • reduction3 of 73
  • lemma3 of 73
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 statementDo Frobenius classes equidistribute in the correct compact symmetry group?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementFull parity-safe statementintermediate
  • retained route statementThree logical layersintermediate
  • retained route statementAdaptive central transferintermediate
  • retained route statementConditional all-base theoremintermediate
  • 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 tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim.open
  • Research targetProve meromorphic continuation near the full boundary for every normalized irreducible motivic target and its finite adaptive closure.open
  • Research targetProve ACB3PC pole-freeness at s=1 for every nontrivial noncomponent member of the adaptive third-layer family.open
  • Research targetTensor Tate gatesuperseded
  • Research targetContinuation and ACB3PCsuperseded
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Meromorphic continuation alone, or pole-freeness of only the original target, is enough to imply full equidistribution. The universal structural and analytic inputs needed to turn the conditional character prime-number theorem into equidistribution are not known for arbitrary motives.
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 tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim.

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 tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim.

Generalized Sato–Tate Conjecture · ready to start

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Research contextPrepared context for any AI agent

Do normalized Frobenius conjugacy classes spread through the correct compact symmetry group according to Haar measure?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references5 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. 1
    Sato-Tate group (reviewed)authoritative webpage · Kiran S. Kedlaya, Andrew V. Sutherland · LMFDB · reviewed 2021-01-16 · accessed Aug 14, 2026
  2. 2
    Sato-Tate groups databasesoftware or dataset · LMFDB · accessed Aug 14, 2026
  3. 3
    Equidistribution, L-functions, and Sato-Tate groupssurvey or monograph · Francesc Fité · arXiv · 2014 · ARXIV 1405.5162 · accessed Aug 14, 2026
  4. 4
    Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecturepreprint · Grzegorz Banaszak, Kiran S. Kedlaya · arXiv · 2015 · ARXIV 1506.02177 · accessed Aug 14, 2026
  5. 5
    Sato-Tate groups of some weight 3 motivespreprint · Francesc Fité, Kiran S. Kedlaya, Andrew V. Sutherland · arXiv · 2012 · ARXIV 1212.0256 · accessed Aug 14, 2026

Important qualifications

  • The universal motivic formulation varies across conventions; this snapshot records a broad Serre-style equidistribution identity and selected infrastructure.
  • Finite database coverage and special-family results are not evidence for the universal claim.

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