ProofAtlas

Automorphic forms, Langlands functoriality, and Galois representations

Langlands functoriality: GL4 × GL2 to GL8

Collaboration beta

Do the local tensor products of a cuspidal GL4 representation and a cuspidal GL2 representation come from one global automorphic representation on GL8, with matching parameters at every place?

rec(Π⊠τ)v=rec(Πv)⊗rec(τv)
Known results and sources
Four local strands and two local strands form paired tensor data facing an interrupted global automorphic lattice.
The open problem asks whether local 4×2 tensor parameters globalize to one GL8 automorphic representation at every place.

Research problem

Exact mathematical statement

Let FF be a number field, Π∈A0(GL4(AF))\Pi\in\mathcal A_0(\mathrm{GL}_4(\mathbb A_F)), and τ∈A0(GL2(AF))\tau\in\mathcal A_0(\mathrm{GL}_2(\mathbb A_F)) be cuspidal automorphic representations. Does there exist an isobaric automorphic representation T=Π⊠τ\mathcal T=\Pi\boxtimes\tau on GL8(AF)\mathrm{GL}_8(\mathbb A_F) such that for every place vv,

recFv(Tv)=recFv(Πv)⊗recFv(τv)?\operatorname{rec}_{F_v}(\mathcal T_v)=\operatorname{rec}_{F_v}(\Pi_v)\otimes\operatorname{rec}_{F_v}(\tau_v)?

The strong target requires complete Frobenius-semisimplified Weil–Deligne parameters, including monodromy at finite places and the expected archimedean parameter. The unrestricted number-field problem remains open.

Problem infographic

Problem at a glance

Four input discs and two input discs face a two-by-four tensor tile array, then an octagonal global object across interrupted local-place links.
The eight tensor pairings and octagonal GL8 target remain separated by an open all-places globalization step; a partial dotted inset represents weaker unramified matching.

Current mathematical picture

Where work on Langlands functoriality: GL4 × GL2 to GL8 stands

Open problem

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

Useful failureOrthogonal proper-endoscopy shortcut

The source records that the construction merely restates the triple tensor problem and supplies no independent shortcut. Residual automorphic seeds, safe congruence connectivity, rank-four converse twists, or a new trace-formula or integral construction remain possible at their exact scopes.

Route status · Narrowed route
Main reductionCurrent reduction

For regular algebraic polarized inputs, the source seeks one automorphic tensor seed in each admissible residual box; outside that setting it identifies a rank-four converse-theorem twist as the first analytic self-reference.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting.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

Langlands functoriality: GL4 × GL2 to GL8 in numbers

1.7kretained lines of mathematical investigation1,651 in the current working snapshot
Argument development
1,439 · 87%
Explored or eliminated routes
19 · 1%
Computational analysis
12 · 1%
Open obligations
30 · 2%
Definitions and setup
151 · 9%
7selected 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 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

12 selected steps

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

12 selected steps

Scroll horizontally to explore the route

Working route overview for Langlands functoriality: GL4 × GL2 to GL8A 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 local 4×2 tensor data globalize to GL8 at every place? — Depends on missing premiseDoes the local 4×2 tensordata globalize to GL8 atevery…Current reduction — Depends on missing premiseCurrent reductionResidual-box seed frontier — Depends on missing premiseResidual-box seed frontierClosing target — Depends on missing premiseClosing targetEssentially symplectic weak locus — Depends on missing premiseEssentially symplectic weaklocusLevel-one factor over Q — Depends on missing premiseLevel-one factor over QWeak and strong transfer differ — Depends on missing premiseWeak and strong transferdifferOrthogonal proper-endoscopy shortcut — stoppedOrthogonal proper-endoscopyshortcutConstruct one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting. — OpenConstruct one regularirreducible automorphictensor…Solve the first self-referential rank-four converse-theorem twist without importing the full GL2×GL4 transfer. — OpenSolve the firstself-referential rank-fourconverse-theorem…Extend from arithmetic special loci to arbitrary number-field inputs with complete finite and archimedean parameter matching. — OpenExtend from arithmeticspecial loci to arbitrarynumber-field…Strong GL4×GL2 tensor transfer — OpenStrong GL4×GL2 tensortransfer
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 routeOrthogonal proper-endoscopy shortcut

The source records that the construction merely restates the triple tensor problem and supplies no independent shortcut. Residual automorphic seeds, safe congruence connectivity, rank-four converse twists, or a new trace-formula or integral construction remain possible at their exact scopes.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Construct one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting.Suggested move: Specify one candidate residual pair, lifting theorem, local conditions, and nonvanishing automorphic-cohomology target without assuming the desired transfer.
Ready to work on
02
Solve the first self-referential rank-four converse-theorem twist without importing the full GL2×GL4 transfer.Suggested move: Seek a reduced twist range, direct degree-eight integral representation, or rigorously justified deformation or trace-formula bootstrap.
Ready to work on
03
Strong GL4×GL2 tensor transfer

Construct the GL8 isobaric transfer with complete local-parameter equality at every place.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Extend from arithmetic special loci to arbitrary number-field inputs with complete finite and archimedean parameter matching.Suggested move: Develop a genuinely automorphic construction and separately prove ramified monodromy and infinity-type compatibility.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen problem

Open for arbitrary cuspidal GL4 and GL2 representations over arbitrary number fields with complete all-places parameter matching. Established neighboring and special cases include GL2 × GL3, a self-dual level-one compatible-system setting over Q, and a 2025 weak GL2 × GSp4 to GL8 triality lift; none proves the unrestricted strong target.

[1][2][3]
External progress

What the literature has established

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

  1. PreprintChenevier and Gan used triality to prove weak spin GSp6 to GL8 and weak tensor GL2 × GSp4 to GL8 lifting; this advances the symplectic GL4 locus but not arbitrary GL4.[3]
  2. PreprintArias-de-Reyna, Dieulefait, and Pérez proved automorphy of GL2-compatible-system tensor GLn in a self-dual setting over Q when the GL2 factor is level one, under regularity, irreducibility, and level…[2]
  3. Peer reviewedKim and Shahidi established the lower-rank GL2 × GL3 functorial product and the GL2 symmetric cube, a major precursor rather than the GL2 × GL4 theorem.[1]
3 cited sources4 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusLanglands functoriality for GL4 × GL2 to GL8
Solved special caseGL2 × GL3 functorial product to GL6

The analogous lower-rank GL2 × GL3 tensor product is known and supplies part of the established functoriality landscape, but it does not imply the GL2 × GL4 transfer.

[1]
Solved special caseself-dual level-one GL2 tensor GLn automorphy

A compatible-system tensor is automorphic for a level-one classical modular-form factor and an RACP GLn representation over Q under the paper's regularity, irreducibility, and level assumptions.

[2]
Solved special caseweak GL2 × GSp4 tensor lifting to GL8

Weak GL2 × GSp4 lifting to GL8 covers a symplectic-type source group and almost-all unramified matching; arbitrary GL4 inputs and strong all-places compatibility remain outside the theorem.

[3]
Stronger or generalized formstrong unrestricted GL4 × GL2 tensor transfer

The target record asks for arbitrary cuspidal GL4 and GL2 inputs over a number field and complete local-parameter matching at every place, strictly beyond the cited weak symplectic lift.

[3]

Formalization opportunities

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

  • Formalization targetA formal statement needs number fields, adeles, cuspidal and isobaric automorphic representations of GLn, and the local Langlands correspondence at finite and archimedean places.
  • Formalization targetStrong transfer requires Frobenius-semisimplified Weil–Deligne representations, monodromy operators, tensor products, and exact ramified local compatibility.
  • Formalization targetArithmetic special cases need compatible systems, regular algebraic polarized automorphic representations, local deformation conditions, residual adequacy, and automorphy-lifting theorems.
  • Formalization targetThe weak triality case additionally needs GSp4, its relation to the essentially symplectic GL4 locus, and a precise descent theorem.
  • Formalization targetNo scoped problem-level formal statement or proof was recorded in this bounded pass.

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.

4 standing statements3 proposed statements4 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma4 of 74
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 local 4×2 tensor data globalize to GL8 at every place?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementWeak and strong transfer differintermediate
  • retained route statementEssentially symplectic weak locusintermediate
  • retained route statementLevel-one factor over Qintermediate
  • retained route statementResidual-box seed frontierintermediate
  • 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 failureOrthogonal proper-endoscopy shortcutreported failure
  • Research targetConstruct one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting.open
  • Research targetSolve the first self-referential rank-four converse-theorem twist without importing the full GL2×GL4 transfer.open
  • Research targetExtend from arithmetic special loci to arbitrary number-field inputs with complete finite and archimedean parameter matching.open
  • Research targetStrong GL4×GL2 tensor transferopen
  • Research targetUnrestricted automorphic constructionsuperseded
  • Narrowed routeOrthogonal proper-endoscopy shortcutThe source records that the construction merely restates the triple tensor problem and supplies no independent shortcut. Residual automorphic seeds, safe congruence connectivity, rank-four converse twists, or a new trace-formula or integral construction remain possible at their exact scopes.
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 one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting.

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 one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting.

Langlands functoriality: GL4 × GL2 to GL8 · ready to start

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Research contextPrepared context for any AI agent

Do the local tensor products of a cuspidal GL4 representation and a cuspidal GL2 representation come from one global automorphic representation on GL8, with matching parameters at every place?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 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
    Functorial products for GL2 × GL3 and the symmetric cube for GL2peer reviewed result · Henry H. Kim, Freydoon Shahidi · Annals of Mathematics · 2002 · DOI 10.2307/3062134 · accessed Aug 14, 2026
  2. 2
    Automorphy of GL2 tensor GLn in the self-dual casepreprint · Sara Arias-de-Reyna, Luis Dieulefait, Josu Pérez · arXiv · 2016 · ARXIV 1611.06918 · accessed Aug 14, 2026
  3. 3
    Triality and Functorialitypreprint · Gaëtan Chenevier, Wee Teck Gan · arXiv · 2025 · ARXIV 2510.21169 · accessed Aug 14, 2026

Important qualifications

  • This record concerns the specific tensor L-homomorphism GL4(C) × GL2(C) to GL8(C), not Langlands functoriality in all groups or all representations.
  • Weak almost-all-unramified lifting is distinct from strong equality of complete local parameters, including ramified monodromy and archimedean type. The record does not promote weak results to strong ones.
  • Chenevier and Gan prove a weak GL2 × GSp4 to GL8 lift. Applying it to a GL4 representation requires an essentially symplectic descent and retains that restricted scope.
  • The self-dual compatible-system theorem over Q assumes a level-one GL2 factor and regularity, irreducibility, polarization, and level conditions; it is not the arbitrary number-field transfer.
  • No current problem-level formalization or canonical computation was identified, and no intake-packet claim was used as external status authority.

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