ProofAtlas

Automorphic forms · representation theory · Langlands program

Generalized Ramanujan Conjecture

Collaboration beta

Is every local component of every unitary cuspidal automorphic representation of GL_n over a number field tempered? This is known over function fields and has strong partial bounds and density results over number fields, but the number-field conjecture remains open.

πvis tempered for every placev
Known results and sources
A GL_n emblem and automorphic wave surround a unit-circle spectral motif asking whether every local component is tempered, marked open.
The strong conjecture asks for temperedness at every finite and archimedean place over number fields; it remains open.

Research problem

Exact mathematical statement

Let FF be a number field and let π\pi be a unitary cuspidal automorphic representation of GLn(AF)GL_n(\mathbb A_F). The strong Generalized Ramanujan Conjecture asserts that

πvis tempered for every finite, real, and complex placevofF.\pi_v\text{ is tempered for every finite, real, and complex place }v\text{ of }F.

Problem infographic

Problem at a glance

A four-panel landscape takes a global cuspidal GL_n representation to its local components, asks whether each is tempered, and separates number-field open status from the known function-field analogue.
Function-field results, number-field partial bounds, and density theorems are distinguished from the still-open every-place temperedness statement over number fields.

Current mathematical picture

Where work on Generalized Ramanujan Conjecture stands

Open conjecture

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

Useful failureClassifying every boundary correction by scalar character before testing geometric support

The closed-route ledger records the exact superseded assumptions and requires a concrete failure before reopening them. A paired CPS converse-theorem route remains source-proposed if each actual transform, remainder-support, scheduled nonvanishing, decay, unfolding, and division obligation is proved without circular post-output input.

Route status · Narrowed route
Main reductionConditional exponent-doubling mechanism

The current work reports a formal chain that would double an extremal local exponent until the generic-unitary bound forces it to vanish, conditional on weak automorphic realizations.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct the second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance.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 Ramanujan Conjecture in numbers

2.3kretained lines of mathematical investigation2,258 in the current working snapshot
Argument development
1,809 · 80%
Explored or eliminated routes
55 · 2%
Computational analysis
45 · 2%
Open obligations
181 · 8%
Definitions and setup
168 · 7%
5selected mapped statements1routes investigated4open questions4contribution-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 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

10 selected steps

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

10 selected steps

Scroll horizontally to explore the route

Working route overview for Generalized Ramanujan ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Is every cuspidal GLₙ local component tempered at every place? — Depends on missing premiseIs every cuspidal GLₙ localcomponent tempered at everyplace?Conditional exponent-doubling mechanism — Depends on missing premiseConditionalexponent-doubling mechanismCurrent reduction — Depends on missing premiseCurrent reductionExact strong number-field scope — Depends on missing premiseExact strong number-fieldscopeClosing target — Depends on missing premiseClosing targetClassifying every boundary correction by scalar character before testing geometric support — stoppedClassifying every boundarycorrection by scalarcharacter…Construct the second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance. — OpenConstruct the second partiallocal transform as anisomorphism…Prove the second partial global Poisson formula and a term-by-term distributional support ledger for every correction. — OpenProve the second partialglobal Poisson formula and aterm-by-term…Build a scheduled global identity with nonvanishing, two-sided decay, exact unfolding, and finite unit-ideal division certificates. — OpenBuild a scheduled globalidentity with nonvanishing,two-sided…Arbitrary-rank paired family unresolved — OpenArbitrary-rank paired familyunresolved
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 routeClassifying every boundary correction by scalar character before testing geometric support

The closed-route ledger records the exact superseded assumptions and requires a concrete failure before reopening them. A paired CPS converse-theorem route remains source-proposed if each actual transform, remainder-support, scheduled nonvanishing, decay, unfolding, and division obligation is proved without circular post-output input.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Construct the second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance.Suggested move: Define the transform on a dense subspace, prove continuity and bijectivity, identify its inverse, and record the unramified vector and functional equation.
Ready to work on
02
Prove the second partial global Poisson formula and a term-by-term distributional support ledger for every correction.Suggested move: Test each correction against open-orbit local sections and classify any failure as an explicit residual term rather than calling it boundary by provenance.
Ready to work on
03
Arbitrary-rank paired family unresolved

The principal unresolved theorem is the paired CPS family for a fixed pre-output character and every required test degree and representation.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Build a scheduled global identity with nonvanishing, two-sided decay, exact unfolding, and finite unit-ideal division certificates.Suggested move: Use distinct support places, certify every intermediate zeta value, and separate pole removal from quantitative vertical-strip growth transfer.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The strong generalized Ramanujan conjecture over number fields remains open. Function-field analogues are known, while number-field bounds in low rank and density theorems provide partial progress rather than every-place temperedness.

[1][2][3]
External progress

What the literature has established

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

  1. PreprintThe arXiv record marks v2 as a revision on 2026-08-05, adding a new theorem, numbering changes, and referee comments; this is a revision date, not the initial publication date.[3]
  2. PreprintLichtman and Pascadi initially submitted their density-theorem preprint, which studies GL_n representations over Q that fail generalized Ramanujan at some place.[3]
  3. Authoritative summaryLomelí's survey records the number-field conjecture as open and function-field cases as known.[1]
  4. Peer reviewedBlomer and Brumley extended best-known bounds toward Ramanujan to GL_n for n=2,3,4 over arbitrary number fields.[2]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusGeneralized Ramanujan Conjecture
Solved special caseFunction-field generalized Ramanujan theorem

The survey records known cases over function fields; this is a different global-field setting and does not solve the number-field statement.

[1]
Weaker or relaxed formPointwise bounds toward Ramanujan in ranks 2, 3, and 4

These quantitative bounds are weaker than full temperedness and cover specified ranks.

[2]
Weaker or relaxed formDensity bounds for non-tempered representations

Density theorems quantify exceptional representations but do not prove that no exception exists.

[3][4]

Formalization opportunities

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

  • Formalization targetFormal global and local representation theory for GL_n over number fields, including finite and archimedean places.
  • Formalization targetA formal definition of unitarity, cuspidality, local components, and temperedness compatible with the exact scope.
  • Formalization targetA checked proof covering every n, number field, unitary cuspidal representation, and place.

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
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. 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.

3 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role5 selected mapped statements
  • theorem candidate1 of 51
  • reduction2 of 52
  • lemma1 of 51
  • equivalence1 of 51
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.16 displayed rows · 1 route included
  • retained route statementIs every cuspidal GLₙ local component tempered at every place?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact strong number-field scopeintermediate
  • retained route statementConditional exponent-doubling mechanismintermediate
  • 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
  • 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 failureClassifying every boundary correction by scalar character before testing geometric supportreported failure
  • Research targetConstruct the second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance.open
  • Research targetProve the second partial global Poisson formula and a term-by-term distributional support ledger for every correction.open
  • Research targetBuild a scheduled global identity with nonvanishing, two-sided decay, exact unfolding, and finite unit-ideal division certificates.open
  • Research targetArbitrary-rank paired family unresolvedopen
  • Research targetActual fiber-bundle analysis remains opensuperseded
  • Narrowed routeClassifying every boundary correction by scalar character before testing geometric supportThe closed-route ledger records the exact superseded assumptions and requires a concrete failure before reopening them. A paired CPS converse-theorem route remains source-proposed if each actual transform, remainder-support, scheduled nonvanishing, decay, unfolding, and division obligation is proved without circular post-output input.
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 second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance.

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 second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance.

Generalized Ramanujan Conjecture · ready to start

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Research contextPrepared context for any AI agent

Is every local component of every unitary cuspidal automorphic representation of GL_n over a number field tempered? This is known over function fields and has strong partial bounds and density results over number fields, but the number-field conjecture remains open.

  • 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
    Langlands Program and Ramanujan Conjecture: a surveysurvey or monograph · Luis Alberto Lomelí · arXiv · 2018 · ARXIV 1812.05203 · accessed Aug 15, 2026
  2. 2
    On the Ramanujan conjecture over number fieldspeer reviewed result · Valentin Blomer, Farrell Brumley · Annals of Mathematics · 2011 · ARXIV 1003.0559 · accessed Aug 15, 2026
  3. 3
    Density theorems for GL_n via Rankin–Selberg L-functionspreprint · Jared Duker Lichtman, Alexandru Pascadi · arXiv · 2024-08-24 · ARXIV 2408.13682 · accessed Aug 15, 2026
  4. 4
    Density theorems and applicationsauthoritative webpage · Valentin Blomer · Clay Mathematics Institute · 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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