ProofAtlas

Arithmetic geometry · elliptic curves · mod-p Galois representations · rational isogenies

Frey–Mazur Conjecture

Collaboration beta

For primes greater than 17, does the p-torsion Galois module of an elliptic curve over Q determine its Q-isogeny class?

p>17∧E[p]≃E'[p]asGQ-modules⇒E∼QE'
Known results and sources
Two smooth, nonsingular real cubic loci flank matching mod-p torsion lattices carrying the same Galois-action pattern; an unfinished gold bridge asks whether the corresponding elliptic curves must be rationally isogenous.
Smooth node-free curve loci represent the elliptic curves; for p>17, the conjecture asks whether matching p-torsion Galois modules force a Q-isogeny.

Research problem

Exact mathematical statement

For every prime p>17p>17, if elliptic curves E,E'/QE,E'/\mathbf Q have isomorphic pp-torsion as modules for the absolute Galois group GQG_{\mathbf Q},

E[p]≃E'[p]asGQ-modules,E[p]\simeq E'[p]\quad\text{as }G_{\mathbf Q}\text{-modules},

then EE and E'E' are isogenous over Q\mathbf Q. The isomorphism is a Galois-module isomorphism, and the conclusion is a rational isogeny. The retained source reports reductions and special branches but no complete proof of this exact universal statement.

Problem infographic

Problem at a glance

A problem-first diagram shows elliptic curves E and E' over Q, isomorphic p-torsion Galois modules for p>17, and the open question whether a Q-isogeny must connect the curves.
The hypothesis matches the full G_Q-module structure on p-torsion; the conclusion asks for an isogeny defined over Q.

Current mathematical picture

Where work on Frey–Mazur Conjecture stands

Open conjecture

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

Useful failureRepeating a general Frey–Kani reduction without an arithmetic rigidity theorem

The source says the next task is arithmetic rigidity for the two singularity passports, not another general Frey–Kani reduction, and separately lists the nodal and triple obstructions as open. A formalized passport theorem followed by separate theta-group or polarized-endomorphism rigidity in the triple lane and permutation/quotient arithmetic in the nodal lane remains source-proposed; the Cartan branch and finite audits proceed independently.

Route status · Narrowed route
Main reductionGeometric-isogeny branch split

The source partitions hypothetical counterexamples into geometrically isogenous and geometrically nonisogenous cases with different downstream structures.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeExclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17.Task status · Ready to work on
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

Frey–Mazur Conjecture in numbers

1.4kretained lines of mathematical investigation1,413 in the current working snapshot
Argument development
1,241 · 88%
Explored or eliminated routes
16 · 1%
Computational analysis
10 · 1%
Open obligations
26 · 2%
Definitions and setup
120 · 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 byGerhard Frey and Barry MazurStatus 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 Frey–Mazur ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Above 17, the mod-p Galois representation should determine the rational isogeny class. — Depends on missing premiseAbove 17, the mod-p Galoisrepresentation shoulddetermine…Current reduction — Depends on missing premiseCurrent reductionGeometric-isogeny branch split — Depends on missing premiseGeometric-isogeny branchsplitNodal or triple passport — Depends on missing premiseNodal or triple passportClosing target — Depends on missing premiseClosing targetExact Frobenius window — Depends on missing premiseExact Frobenius windowExact p>17 rigidity target — Depends on missing premiseExact p>17 rigidity targetRepeating a general Frey–Kani reduction without an arithmetic rigidity theorem — stoppedRepeating a generalFrey–Kani reduction withoutan…Exclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17. — OpenExclude all constrainednon-CM rational points onthe…Lift the residual mu_3 packet structure in the large-image triple passport to a genuine geometric endomorphism or theta-group structure. — OpenLift the residual mu_3packet structure in thelarge-image…Rule out the large-image nodal passport arithmetically and complete the five remaining smaller-image prime audits. — OpenRule out the large-imagenodal passportarithmetically…
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 routeRepeating a general Frey–Kani reduction without an arithmetic rigidity theorem

The source says the next task is arithmetic rigidity for the two singularity passports, not another general Frey–Kani reduction, and separately lists the nodal and triple obstructions as open. A formalized passport theorem followed by separate theta-group or polarized-endomorphism rigidity in the triple lane and permutation/quotient arithmetic in the nodal lane remains source-proposed; the Cartan branch and finite audits proceed independently.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Exclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17.Suggested move: Determine the 2,3-supported character from local data and feed the resulting entanglement into an explicit modular-curve argument, while treating p congruent to 3 modulo 4 separately.
Ready to work on
02
Lift the residual mu_3 packet structure in the large-image triple passport to a genuine geometric endomorphism or theta-group structure.Suggested move: Formalize the triple passport with fixed Galois conventions, identify its external scalar partition, and prove that a geometric lift forces the forbidden CM behavior.
Ready to work on
03
Rule out the large-image nodal passport arithmetically and complete the five remaining smaller-image prime audits.Suggested move: Use the SL_2 permutation action on node pairs and the p+1 cyclic quotients to derive an arithmetic contradiction, then audit p=37 and p=19,23,29,31 separately.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The exact Frey–Mazur conjecture for elliptic curves over Q and primes p>17 remains open. A characteristic-zero geometric-function-field analogue is proved, and a bounded LMFDB study reports no p-congruences for p at least 19 in its range; neither result establishes the universal rational-number-field theorem.

[1][2]
External progress

What the literature has established

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

  1. Peer reviewedCremona and Freitas find mod-p congruences for each p at most 17 and none for p at least 19 among LMFDB elliptic curves of conductor below 500,000; this is finite computational evidence, not a universal theorem.[2]
  2. Peer reviewedBakker and Tsimerman prove a characteristic-zero geometric-function-field analogue with a prime bound depending on the gonality of the base curve. Their paper states the rational-number-field p>17 conjecture but does not prove it.[1]
2 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusFrey–Mazur conjecture
Related problemgeometric-function-field Frey–Mazur analogue

The proved monodromy theorem for nonisotrivial elliptic-curve families over complex curves is a geometric-function-field analogue whose bound depends on base gonality.

[1]
Related problemsymplectic type of elliptic-curve congruences

Classifying the symplectic type of mod-p congruences refines the isomorphism data in the conjecture and supplies bounded evidence, while leaving curves outside the database range untouched.

[2]
Weaker or relaxed formuniform Frey–Mazur conjecture over number fields

A uniform-number-field version asks only for some field-dependent threshold; the explicit p>17 rational statement fixes the strongest expected threshold over Q.

[1]

Formal and computational footholds

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

  • dataset · not independently reproducedLMFDB mod-p congruence search below conductor 500,000

    The peer-reviewed study reports no nontrivial p-congruences for p at least 19 in its stated LMFDB range; this staging lane did not rerun the search, and its finite scope cannot settle the conjecture.

    [2]

Formalization opportunities

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

  • Formalization targetA problem-level formal statement needs elliptic curves over Q, finite Galois modules E[p], G_Q-equivariant isomorphisms, and Q-isogenies with the exact p>17 threshold.
  • Formalization targetA universal proof requires arithmetic control beyond finite database enumeration and beyond the geometric-function-field analogue.

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
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

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 statementAbove 17, the mod-p Galois representation should determine the rational isogeny class.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact p>17 rigidity targetintermediate
  • retained route statementGeometric-isogeny branch splitintermediate
  • retained route statementExact Frobenius windowintermediate
  • retained route statementNodal or triple passportintermediate
  • 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 failureRepeating a general Frey–Kani reduction without an arithmetic rigidity theoremreported failure
  • Research targetExclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17.open
  • Research targetLift the residual mu_3 packet structure in the large-image triple passport to a genuine geometric endomorphism or theta-group structure.open
  • Research targetRule out the large-image nodal passport arithmetically and complete the five remaining smaller-image prime audits.open
  • Research targetArithmetic rigidity of both passportssuperseded
  • Research targetFive finite prime auditssuperseded
  • Narrowed routeRepeating a general Frey–Kani reduction without an arithmetic rigidity theoremThe source says the next task is arithmetic rigidity for the two singularity passports, not another general Frey–Kani reduction, and separately lists the nodal and triple obstructions as open. A formalized passport theorem followed by separate theta-group or polarized-endomorphism rigidity in the triple lane and permutation/quotient arithmetic in the nodal lane remains source-proposed; the Cartan branch and finite audits proceed independently.
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 bridgeExclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17.

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

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointExclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17.

Frey–Mazur Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

For primes greater than 17, does the p-torsion Galois module of an elliptic curve over Q determine its Q-isogeny class?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 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
    p-torsion monodromy representations of elliptic curves over geometric function fieldspeer reviewed result · Benjamin Bakker, Jacob Tsimerman · Annals of Mathematics · 2016-09-16 · DOI 10.4007/annals.2016.184.3.2 · MR 3549621 · accessed Aug 14, 2026
  2. 2
    Global methods for the symplectic type of congruences between elliptic curvespeer reviewed result · John E. Cremona, Nuno Freitas · Revista Matemática Iberoamericana / EMS Press · 2021-06-01 · DOI 10.4171/RMI/1269 · accessed Aug 14, 2026

Important qualifications

  • The exact target uses elliptic curves over Q, primes p>17, isomorphic p-torsion G_Q-modules, and Q-isogeny; function-field analogues and finite database searches are not equivalent.
  • The LMFDB search covers the database range described by its authors and is not a universal certificate.
  • No packet claim or submitted URL was used as external status authority.
  • The original unpublished provenance of the conjecture was not assigned an original-source record in this bounded review.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image