Arithmetic geometry · abelian varieties · Hodge and ℓ-adic monodromy groups

Mumford–Tate Conjecture

Collaboration beta

For an abelian variety over a number field, does the symmetry group seen by every ℓ-adic Galois representation exactly match the symmetry group determined by its rational Hodge tensors?

V = H_{B}^{1} (A_{C}, Q), V_{ℓ} = H_{\overset{´}{e} t}^{1} (A_{\bar{K}}, Q_{ℓ}) .

Known results and sources

The Mumford–Tate conjecture asks whether Hodge tensors and ℓ-adic Galois representations determine exactly the same connected symmetry group.

Research problem

Exact mathematical statement

Let $A/K$ be an abelian variety over a number field, fix an embedding $K\hookrightarrow\mathbf C$ , and put

V=H^1_{\mathrm B}(A_{\mathbf C},\mathbf Q),\qquad V_\ell=H^1_{\acute et}(A_{\overline K},\mathbf Q_\ell).

Let

G=\operatorname{MT}(A)\subseteq\operatorname{GL}(V)

be the Mumford–Tate group, and let

H_\ell=\overline{\rho_\ell(\Gamma_K)}^{\,\mathrm{Zar},\circ}\subseteq\operatorname{GL}(V_\ell)

be the connected $\ell$ -adic monodromy group. Comparison and the Tate realization of Hodge tensors give $H_\ell\subseteq G_{\mathbf Q_\ell}$ . The Mumford–Tate conjecture asks whether

H_\ell=G_{\mathbf Q_\ell}

for every prime $\ell$ . the source studies this general equality and, in detail, the possible cubic $A_1^3$ tensor-cube obstruction for an abelian fourfold with full Mumford–Tate group $\operatorname{GSp}_8$ .

Problem infographic

Problem at a glance

A scientific explainer starts with an abelian variety over a number field, builds its Betti Hodge realization and ℓ-adic étale realization, displays the inclusion Hℓ inside MT(A) over Qℓ and asks whether equality always holds, then separates a cubic A₁³ fourfold obstruction and its open absolute-Hodge alignment bridge from established fact. — Both realizations come from the same abelian variety; the conjecture asks whether their connected symmetry groups agree, while the packet's highlighted fourfold route still lacks a global Hodge alignment theorem.

Current mathematical picture

Where work on Mumford–Tate 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: The current work rejects one-Frobenius recovery of the rank-nine projector because of exact spectral overlap, use of Euler factors or canonical lifts to select parity, automatic passage from rational finite-place data to a Hodge object, transfer of de Rham splitness to the étale quaternion class, and promotion of the Borel-stable positive-root space to a global tensor subobject. Construct one absolute-Hodge hyperdeterminant line, rank-nine projector or Lie algebra, or E-linear adjoint motive realizing the local cubic A₁³ systems, and prove that its finite-place parity torsor has the correct Betti/Hodge position. The general case separately retains the Newton-orbit escape problem.

Route status · Narrowed route

Main reductionCurrent reduction

For each coefficient prime, Faltings commutant equality plus equal-rank rigidity reduces the conjecture to monodromy-rank equality, and Frobenius valuation matrices translate rank into a Newton-cocharacter spanning problem. In the highlighted full-GSp₈ fourfold branch, the remaining proper monodromy candidate is a cubic A₁³ tensor cube whose local embedding is one parity position; closing the branch requires global absolute-Hodge or motivic alignment.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeCertify the algebraic and finite-classification core that isolates the cubic A₁³ residual fourfold branch.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Mumford–Tate Conjecture in numbers

1.8kretained lines of mathematical investigation1,778 in the current working snapshot

Argument development: 1,481 · 83%
Explored or eliminated routes: 46 · 3%
Computational analysis: 38 · 2%
Open obligations: 42 · 2%
Definitions and setup: 171 · 10%

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 byDavid Mumford, John TateProposed1966Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Construct one valid global invariant object realizing the local rank-nine adjoint systems.

Suggested move: Package the local quaternionic Lie algebras as a rank-three E-compatible system with bracket and Killing form, then seek a uniquely characterized algebraic projector or motive in a power of the abelian variety whose restriction of scalars gives the rank-nine adjoint object.

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

Construct one valid global invariant object realizing the local rank-nine adjoint systems.Suggested move: Package the local quaternionic Lie algebras as a rank-three E-compatible system with bracket and Killing form, then seek a uniquely characterized algebraic projector or motive in a power of the abelian variety whose restriction of scalars gives the rank-nine adjoint object.

Ready to work on

Prove the finite-to-archimedean alignment that makes the global invariant object Hodge-theoretic.Suggested move: Describe the local parity positions as an adelic torsor, track its finite marking and étale/de Rham inner forms, and prove that polarization, reciprocity, integral structures, or a separately completed Serre–Tate non-density route selects the required Betti/Hodge point.

Ready to work on

Certify the algebraic and finite-classification core that isolates the cubic A₁³ residual fourfold branch.Suggested move: Turn the tensor-cube and parity results into a convention-fixed lemma package, rerun the exact symbolic checks only in a separately authorized bounded lane, and close the minuscule-table, Cartesian-factor, representation-type, and normalizer-cocycle acceptance gates.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

The equality between the connected l-adic algebraic monodromy group and the Mumford–Tate group is not known for every abelian variety over a number field. Many families, dimensions, Albert types, reduction configurations, and product constructions are known, but none of the cited results removes all hypotheses.

[2][8]

External progress

What the literature has established

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

2023
PreprintGajda and Hindry reported new bad-semistable-reduction cases and a generalization of Hall's toric-dimension-one theorem; the checked source is a preprint.[8]
2019
Peer reviewedCommelin proved that validity for each of two abelian varieties implies validity for their product, a reduction theorem that does not prove the factors.[7]
2016
Peer reviewedLombardo established product-decomposition criteria for nonsimple abelian varieties and applied them in low-dimensional settings.[6]
2008
Peer reviewedVasiu proved the conjecture for specified Shimura types, including orthogonal types; this remains a family of special cases rather than a universal proof.[5]

Mathematical neighborhood

Related results and reusable starting points

Current focusMumford–Tate conjecture

Weaker or relaxed formKnown monodromy-group inclusion

The Deligne–Borovoi–Piatetski-Shapiro inclusion places the connected l-adic monodromy group inside the Mumford–Tate group after scalar extension; equality is the remaining direction.

[2][8]

Solved special caseCM abelian varieties and elliptic curves

The conjecture is known for CM abelian varieties and elliptic curves, among other classical families, but these cases do not cover arbitrary abelian varieties.

[2]

Solved special caseSpecified Albert, Shimura, low-dimensional, product, and reduction-type families

Representation type, Shimura type, dimension, product decomposition, and semistable-reduction hypotheses yield substantial additional cases, with each theorem retaining its own hypotheses.

[4][5]

Dependency or reductionProduct stability

If the conjecture holds for each of two abelian varieties, it holds for their product; this transfers established cases without proving unknown factors.

[7]

Logical consequenceHodge and Tate conjectures for abelian varieties

For abelian varieties, Mumford–Tate combined with the Hodge conjecture yields the corresponding Tate statement, while converse directions in motivic formulations require the stated cycle-conjecture hypotheses; this is not an unconditional equivalence.

[2]

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 of abelian varieties over number fields together with first Betti and l-adic etale cohomology.
Formalization targetContinuous Galois representations, Zariski closures, identity components, and connected reductive algebraic groups over Q and Q_l.
Formalization targetRational Hodge structures, Hodge tensors, and Mumford–Tate groups with checked scalar extension and comparison isomorphisms.
Formalization targetA formal statement connecting the exact classical abelian-variety conjecture to the relevant known special cases without erasing their hypotheses.

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. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

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.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction1 of 91
lemma7 of 97

Selected mathematical clusters3 mathematical clusters

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

retained route statementFor an abelian variety over a number field, does the symmetry group seen by every ℓ-adic Galois representation exactly match the symmetry group determined by its rational Hodge tensors?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementFixed-ℓ rank criterionintermediate
retained route statementFrobenius/Newton rankintermediate
retained route statementResidual cubic tensor cubeintermediate
retained route statementGlobal cubic centroidintermediate
retained route statementEquivalent parity invariantsintermediate
retained route statementAbsolute-Hodge alignment gapintermediate
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 targetCertify the algebraic and finite-classification core that isolates the cubic A₁³ residual fourfold branch.in progress reported
Research targetConstruct one valid global invariant object realizing the local rank-nine adjoint systems.open
Research targetProve the finite-to-archimedean alignment that makes the global invariant object Hodge-theoretic.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: The current work rejects one-Frobenius recovery of the rank-nine projector because of exact spectral overlap, use of Euler factors or canonical lifts to select parity, automatic passage from rational finite-place data to a Hodge object, transfer of de Rham splitness to the étale quaternion class, and promotion of the Borel-stable positive-root space to a global tensor subobject. Construct one absolute-Hodge hyperdeterminant line, rank-nine projector or Lie algebra, or E-linear adjoint motive realizing the local cubic A₁³ systems, and prove that its finite-place parity torsor has the correct Betti/Hodge position. The general case separately retains the Newton-orbit escape problem.

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 bridgeCertify the algebraic and finite-classification core that isolates the cubic A₁³ residual fourfold branch.

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 valid global invariant object realizing the local rank-nine adjoint systems.

Mumford–Tate Conjecture · ready to start

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Research contextPrepared context for any AI agent

For an abelian variety over a number field, does the symmetry group seen by every ℓ-adic Galois representation exactly match the symmetry group determined by its rational Hodge tensors?

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Sources and references8 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
Families of Abelian Varietiesoriginal source · David Mumford · Proceedings of Symposia in Pure Mathematics; author-hosted archive · 1966 · accessed Aug 7, 2026
2
A survey around the Hodge, Tate and Mumford-Tate conjectures for abelian varietiessurvey or monograph · Victoria Cantoral Farfán · arXiv · 2016 · ARXIV 1602.08354 · accessed Aug 7, 2026
3
Endlichkeitssätze für abelsche Varietäten über Zahlkörpernpeer reviewed result · Gerd Faltings · Inventiones Mathematicae · 1983 · DOI 10.1007/BF01388432 · accessed Aug 7, 2026
4
l-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecturepeer reviewed result · Richard Pink · Journal für die reine und angewandte Mathematik · 1998 · DOI 10.1515/crll.1998.018 · accessed Aug 7, 2026
5
Some cases of the Mumford--Tate conjecture and Shimura varietiespeer reviewed result · Adrian Vasiu · Indiana University Mathematics Journal · 2008 · DOI 10.1512/iumj.2008.57.3513 · accessed Aug 7, 2026
6
On the l-adic Galois representations attached to nonsimple abelian varietiespeer reviewed result · Davide Lombardo · Annales de l'Institut Fourier · 2016 · DOI 10.5802/aif.3035 · accessed Aug 7, 2026
7
The Mumford–Tate conjecture for products of abelian varietiespeer reviewed result · Johan Commelin · Algebraic Geometry · 2019 · DOI 10.14231/AG-2019-028 · accessed Aug 7, 2026
8
Remarks on a theorem of Pink in presence of bad reductionpreprint · Wojciech Gajda, Marc Hindry · arXiv · 2023 · ARXIV 2307.10140 · accessed Aug 7, 2026

Important qualifications

This is a bounded representative review, not a complete bibliography of the extensive case-by-case literature.
The target is the classical conjecture for abelian varieties over number fields; motivic, characteristic-p, period, and generalized Shafarevich variants were kept separate.
The 2023 Gajda–Hindry result remains in the current research map at preprint posture, and each special-case result remains qualified by its stated hypotheses.
Negative formalization and computation findings reflect a scoped search and are not proofs of nonexistence.
No packet source or submitted mathematical claim was read or used as authority.

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