Arithmetic geometry · p-adic Galois representations · étale cohomology

Fontaine–Mazur Conjecture

Collaboration beta

Which irreducible p-adic Galois representations satisfying arithmetic finiteness conditions actually come from algebraic geometry?

ρ : G_{K} \to {GL}_{n} (E) continuous, irreducible, finitely ramified, de Rham above p \Rightarrow ρ is geometric

Known results and sources

Arithmetic p-adic representations face an unresolved passage to geometric cohomology; the image identifies the conjecture without presenting the passage as proved.

Research problem

Exact mathematical statement

Let $K$ be a number field, $p$ a prime, and $E/\mathbf Q_p$ a finite extension. the source’s variant asks whether every continuous, irreducible representation

\rho:G_K\to\mathrm{GL}_n(E)

that is unramified outside finitely many places and de Rham at every place $v\mid p$ occurs, after a Tate twist, as a subquotient of

H^i_{\acute et}(X_{\overline K},E)

for some smooth projective variety $X/K$ . The general conjecture remains open. Dimension one is known, and substantial regular two-dimensional cases over $\mathbf Q$ are proved; those special cases do not settle higher dimensions, arbitrary number fields, all irregular cases, or $p=2$ .

Problem infographic

Problem at a glance

Scientific explainer for the Fontaine–Mazur conjecture: a continuous irreducible p-adic representation of a number-field Galois group, unramified outside finitely many places and de Rham above p, faces the open question of occurring after a Tate twist as a subquotient of smooth-projective étale cohomology. A separate status band distinguishes known dimension-one and regular GL2 over Q ranges from the open higher-dimensional, general-number-field, irregular, and p=2 cases. — The conjecture asks whether arithmetic p-adic Galois representations have geometric cohomological origin. Dimension one and major regular GL₂/ℚ ranges are known, while the general case remains open.

Current mathematical picture

Where work on Fontaine–Mazur Conjecture stands

Partially resolved

The v9 packet states the broad geometric Fontaine–Mazur conjecture and reports reductions to residually trivial, cyclotomic-symplectic systems with a Hodge-matched Artin–Tate seed. It also reports a solvable-monodromy special case, exact local-slice and old/new commutative algebra, and conditional numerical support and extraction criteria. None closes the conjecture: seed occurrence, scalar/Eisenstein patching, global support, integral wall control, exact target specialization, and finite-level geometric compatibility remain open. All retained claims are provisional source reports, with no independent review, formal verification, or accepted-result effect.

Strongest supported footholdNumerical and derived support criteria

The source reports exact derived-parity and finite-map freeness criteria, but the scalar/Eisenstein arithmetic objects satisfying them remain unconstructed.

Evidence posture · Reported result

Leading routeInduced-Q seed host

Active host fork: induce the Tate seed to Q, but explicitly realize the permutation-Artin factor in the same Hecke/deformation complex or prove the needed Artin automorphy.

Route status · Active route

Useful failureFixed finite pro-host route H

Tertiary fallback if the exact point cannot be unpatched: one fixed finite smooth-projective complex must contain complete target-lattice torsion subquotients for unbounded exponents.

Route status · Narrowed route

Main reductionResidual and symplectic reductions

The source reports reduction to residually trivial systems and then to a cyclotomic-symplectic semisimple problem with a broad-data-matched Tate seed.

Evidence posture · Reported reduction

Completed special caseSolvable-monodromy special case

The source reports geometricity when the connected algebraic monodromy group is solvable.

Evidence posture · Reported special case

Priority open bridgeProve one global support theorem

After seed occurrence and patching, prove either the numerical defect equality, derived one-parity, or all required arithmetic bridges plus global component connectivity.

Task status · Blocked by the current route

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

Fontaine–Mazur Conjecture in numbers

1.8kretained lines of mathematical investigation1,772 in the current working snapshot

Argument development: 1,412 · 80%
Explored or eliminated routes: 42 · 2%
Computational analysis: 48 · 3%
Open obligations: 129 · 7%
Definitions and setup: 141 · 8%

9selected mapped statements7routes investigated6reported milestones7open 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 byJean-Marc Fontaine, Barry MazurProposed1995Status checkedAug 6, 2026

Source recordedAug 6, 2026

Recommended next task

Realize the Artin–Tate seed in a finite-level scalar/Eisenstein block

Choose the induced-Q or CM-unitary host and construct an actual seed class in the exact finite-level complex that will be patched, retaining local data, induction idempotents, and Hecke/Cayley–Hamilton information.

Suggested move: Prepare a bounded decision packet for Work Order 0 and output a named C_0, a nonzero seed fiber, and explicit Hecke/Cayley–Hamilton data.

What would count as progress

A host route and exact nonregular coefficient system are fixed.
The seed occurs nontrivially in a named finite-level complex at the required scalar residual localization.
Induction idempotents and Frobenius or Cayley–Hamilton data are retained.

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

24 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.

Active routeInduced-Q seed host

Active host fork: induce the Tate seed to Q, but explicitly realize the permutation-Artin factor in the same Hecke/deformation complex or prove the needed Artin automorphy.

Route status · Active route

Active routeCM-unitary seed host

Active alternative: choose CM stabilization and patch the Hodge-matched Tate seed directly through a unitary/PEL host, then descend.

Route status · Active route

Active routeNumerical full-support route N

Preferred after seed occurrence: prove module-finiteness, patched-module Tor vanishing, amplitude zero, and Delta_infinity=0 over the full deformation ring; positive defect or failed hypotheses are stop conditions.

Route status · Active route

Active routeDerived one-parity route P

Strong alternative: prove that the complete derived residual fiber is finite, nonzero, and supported in one parity; ordinary cohomology parity does not qualify.

Route status · Active route

Active routeArithmetic bridges and connectivity route B

Fallback when N/P do not apply: establish every away-p and p-adic arithmetic bridge, kill dual Selmer, and connect seed to target in the global component graph.

Route status · Active route

Active routeExact-point specialization and geometric extraction

Conditional closeout after global support: derived unpatching, target block or positive trace extraction, and finite-level Shimura/Kuga geometric compatibility must all be supplied.

Route status · Active route

Explored alternatives

Other routes

1 recorded

Narrowed routeFixed finite pro-host route H

Tertiary fallback if the exact point cannot be unpatched: one fixed finite smooth-projective complex must contain complete target-lattice torsion subquotients for unbounded exponents.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementCyclotomic-symplectic stabilization and Hodge-matched seed

Hyperbolic doubling and induction reduce the target to a cyclotomic-symplectic semisimple problem over Q; after restriction to M, the current work constructs a geometric Tate seed with matching broad Hodge, inertial, determinant, multiplier, polarization, and residual data, but not known component equality.

Source-reported route statement

Route statementOrdered-root and old/new bridge algebra

For the model node R=B[[x,t]]/(xt), the current work classifies old/new extensions by matrices E in M_r(B), with invertible E equivalent to the multiplicity-one free bridge R^r; applying this to cohomology still requires an arithmetic identification of the extension class.

Source-reported route statement

Route statementDerived residual one-parity freeness criterion

For a bounded finite-cohomology complex over a local ring, finite nonzero derived residual cohomology supported in one parity forces a finite direct sum of shifted free modules in that parity.

Source-reported route statement

Route statementFinite-map numerical freeness criterion

For a module-finite local map S→R and finite R-module M with the stated Tor_1 vanishing, the special-fiber defect Delta_S(M/R) is nonnegative and vanishes exactly when M is free of its minimal number of generators over R.

Source-reported route statement

Route statementConditional exact-point extraction closeout

Once exact target support, derived pointwise unpatching, and finite-level geometric compatibility exist, a correct Cayley–Hamilton block decomposition or positive trace identity can extract one stabilized irreducible constituent; one constituent suffices for descent to the original rho.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks

Realize the Artin–Tate seed in a finite-level scalar/Eisenstein block

Suggested move: Prepare a bounded decision packet for Work Order 0 and output a named C_0, a nonzero seed fiber, and explicit Hecke/Cayley–Hamilton data.

Ready to work on

Compute the integral p-adic rank-two wall

For weights {0,1}, trivial residual representation and inertial type, and cyclotomic determinant, compute the integral potentially semistable ring and identify all branches, nilpotents, and embedded components.

Suggested move: Use an integral Breuil/Kisin or equivalent classification and compare the full ring with the coarse node.

Ready to work on

Prove one global support theorem

After seed occurrence and patching, prove either the numerical defect equality, derived one-parity, or all required arithmetic bridges plus global component connectivity.

Suggested move: Once seed localization exists, test the smallest numerical patched model; preserve any positive defect or higher Tor as a failure witness and keep the bridge route active.

Blocked by the current route

Verify finite-level geometric realization

Place the exact unpatched constituent in a finite-level PEL/Hodge-type Shimura/Kuga complex, or prove a target-block-preserving transfer to one, with action compatibility and descent.

Suggested move: After choosing the global host, verify the exact nonregular local system, Kuga realization, action compatibility, and finite-level descent.

Blocked by the current route

Construct exact target specialization and extraction

At the exact target point, prove a correct semisimple Cayley–Hamilton block decomposition or a positive-multiplicity Frobenius-trace identity, carried through derived unpatching.

Suggested move: In an actual patched setup, retain the faithful Cayley–Hamilton action through unpatching and identify the exact target map.

Blocked by the current route

Prove the first actual away-p arithmetic bridge

Construct the exact old/new filtration in a named arithmetic complex and prove the primitive induction-framed off-diagonal root map is an isomorphism, or prove old-branch freeness plus the opposite generic rank.

Suggested move: After fixing the arithmetic host, verify the Gram normalization and identify the actual extension matrix with the primitive root-Ihara map.

Blocked by the current route

Construct scalar/Eisenstein induction-framed patching

Build a patched complex over the full induction-framed deformation ring that retains boundary, torsion, finite generation, matrix data, and exact derived unpatching.

Suggested move: Begin only after the seed-occurrence obligation produces an actual localized finite-level block.

Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 6, 2026

Current statusPartially resolved

The general geometric Fontaine–Mazur conjecture remains open: an irreducible p-adic representation of the absolute Galois group of a number field that is unramified outside finitely many places and de Rham at places above p is conjectured to occur, up to the customary Tate twist, as a subquotient of étale cohomology of a smooth proper variety. The dimension-one case is known. Major two-dimensional cases over Q are proved; in particular, peer-reviewed work covers the odd distinct-Hodge–Tate-weight case for p at least 5, while a 2024 preprint reports the remaining p=3 regular case and hence the regular case for every odd p. These results do not settle higher dimensions, arbitrary number fields, or all irregular and p=2 cases.

[1][3][5]

External progress

What the literature has established

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

by 2022
Authoritative summaryThe conjecture is known in dimension one over number fields: a geometric character is a Tate twist of a finite-order character and therefore occurs in the cohomology of a zero-dimensional variety.[3]
2026
Peer reviewedPan proved a classicality result: every absolutely irreducible two-dimensional Galois representation that is regular de Rham at p and already appears in completed cohomology of modular curves comes from an…[7]
2024
PreprintZhang's arXiv preprint reports removal of the remaining p=3 restriction and, combined with earlier work, concludes the regular two-dimensional Fontaine–Mazur conjecture over Q for every odd prime p.[6]
2022
Peer reviewedPan treated the residually reducible case; together with earlier work, the peer-reviewed literature covers all odd two-dimensional geometric representations of G_Q with distinct Hodge–Tate weights for p at…[5][3]

Mathematical neighborhood

Related results and reusable starting points

Current focusFontaine–Mazur conjecture

Solved special caseone-dimensional Fontaine–Mazur conjecture

Class field theory and the structure of geometric p-adic characters establish the conjecture in dimension one over number fields.

[3]

Weaker or relaxed formtwo-dimensional modularity formulation over Q

For odd two-dimensional representations of G_Q, the geometric-origin prediction becomes a modularity problem: under the relevant local conditions the representation should arise, up to twist, from a cuspidal Hecke eigenform. Large regular ranges are proved, but this specialization is not the full all-dimension, all-number-field conjecture.

[3][4]

Stronger or generalized formFontaine–Mazur–Langlands conjecture

Combining Fontaine–Mazur geometric origin with Langlands reciprocity predicts that irreducible geometric p-adic Galois representations are automorphic. This adds an automorphy conclusion beyond occurrence in algebraic-geometric cohomology.

[3]

Logical consequenceweak and unramified Fontaine–Mazur conjectures

The weak or unramified Fontaine–Mazur conjecture rules out infinite p-adic analytic quotients of certain everywhere-unramified pro-p Galois groups. A counterexample would yield a finitely ramified p-adic representation that does not come from geometry, so the geometric conjecture implies this group-theoretic consequence.

[8]

Related problemtame and uniform Fontaine–Mazur conjectures

Tame and uniform variants replace the full p-adic representation-theoretic statement with finiteness or nonexistence questions for tame or uniform pro-p Galois groups. They supply special-case evidence but are not equivalent in the full stated generality.

[8]

Related problemstrong Artin and weight-one modularity problems

The equal-Hodge–Tate-weight two-dimensional case is closely connected with weight-one modular forms and the strong Artin conjecture, and has a different literature from the regular distinct-weight case.

[3]

Formalization opportunities

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

Formalization targetNo problem-level formal statement or proof of the Fontaine–Mazur conjecture was identified in the scoped current Mathlib documentation and Google DeepMind Formal Conjectures repository-tree search.
Formalization targetA faithful formal statement requires infrastructure for absolute Galois groups of number fields, continuous finite-dimensional p-adic representations, ramification outside a finite set, de Rham or potentially semistable local representations, étale cohomology of smooth proper varieties, Tate twists, and representation subquotients.
Formalization targetAny formalized finite-group, deformation-ring, or local representation fragment must be linked by statement-aligned definitions and theorems before it can count as a formalization of the global geometric-origin conjecture.
Formalization targetA formal statement would still be statement-only; no checked proof should be inferred from library support or from the extensive two-dimensional literature.

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

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.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

6 mapped milestonesretained argument map

Browse all 6 mapped stages

stage 1Broad geometric Fontaine–Mazur question
stage 2Residual, symplectic, and seed reductions
stage 3Solvable-monodromy special case
stage 4Local node and old/new algebra
stage 5Conditional numerical support criteria
stage 6Open gates and prepared work

Broad geometric Fontaine–Mazur questionThe current work fixes a precise broad geometric formulation and explicitly reports that the full conjecture is not proved.