Elliptic curves and Galois representations

Serre uniformity conjecture

Collaboration beta

For a non-CM elliptic curve over the rational numbers, the conjecture says that every prime larger than 37 gives the full possible symmetry group on the curve's ell-torsion points.

\forall E / ℚ non-CM, \forall ℓ > 37 prime, {\bar{ρ}}_{E, ℓ} (G_{ℚ}) = {GL}_{2} (F_{ℓ}) .

Known results and sources

Serre uniformity asks whether every non-CM elliptic curve has full mod-ell Galois image for ell greater than 37.

Research problem

Exact mathematical statement

For every elliptic curve $E/\mathbb Q$ without complex multiplication and every prime $\ell>37$ , the residual Galois representation on the $\ell$ -torsion is surjective:

\bar\rho_{E,\ell}(G_{\mathbb Q})=\operatorname{GL}_2(\mathbb F_\ell).

A counterexample would be a pair $(E,\ell)$ with $E/\mathbb Q$ non-CM, $\ell>37$ , and nonsurjective $\bar\rho_{E,\ell}$ . The source explicitly leaves the full conjecture open.

Problem infographic

Problem at a glance

A landscape explainer defines the mod-ell Galois representation, shows the full GL2 image question above 37, and separates the fixed-CM reduction from two unclosed gates. — A strong fixed-CM reduction narrows one branch, but integral rigidity, rational descent, and other branches remain open.

Current mathematical picture

Where work on Serre uniformity conjecture stands

Open conjecture

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

Useful failureGeneric-fiber minimal R=T

A vertical tangent ring can have generic fiber Q_ell while retaining a nonzero mod-ell tangent. A full integral theorem matching the exact local deformation conditions, or a direct Selmer computation, remains viable.

Route status · Narrowed route

Main reductionStrongest fixed-CM branch

The source's strongest branch has K=Q(sqrt(-2)), ell congruent to 13 modulo 24, and residual equality to a fixed level-256 CM curve.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeClose Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.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

Serre uniformity conjecture in numbers

1.5kretained lines of mathematical investigation1,518 in the current working snapshot

Argument development: 1,292 · 85%
Explored or eliminated routes: 37 · 2%
Computational analysis: 55 · 4%
Open obligations: 35 · 2%
Definitions and setup: 99 · 7%

6selected mapped statements2routes 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.

Proposed byJean-Pierre SerreProposed1972Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Close Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.

Suggested move: Match the determinant, finite-flat niveau-two condition at ell, fixed inertial type at 2, irreducibility hypothesis, and full localized level-256 Hecke algebra line by line.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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.

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

2 recorded

Narrowed routeGeneric-fiber minimal R=T

Route status · Narrowed route

Narrowed routeValuation-only rationality test

The valuations are saturated and leave the unit squareclass u undetermined in z^2=uL^2. Computing the descended unit through monodromy, symplectic descent, Tate residues, and the twist involution remains the central target.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Close Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.Suggested move: Match the determinant, finite-flat niveau-two condition at ell, fixed inertial type at 2, irreducibility hypothesis, and full localized level-256 Hecke algebra line by line.

Ready to work on

Compute the unit squareclass controlling rational versus quadratic descent in the one-prime twist-orbit algebra.Suggested move: Construct the anti-invariant algebra explicitly and keep the residual symplectic multiplier independent while combining monodromy and Tate residue data.

Ready to work on

Uniform surjectivity above 37

The conjecture asserts surjectivity onto GL_2(F_ell) for every non-CM elliptic curve over Q and every prime ell greater than 37.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Extend any successful fixed D_K=-8 closure to other class-number-one CM fields, other residue classes of ell, and the remaining global image branches.Suggested move: Rebuild each field-specific local packet and Hecke isolation, then return to order-three inertia and normalization factors in the other congruence classes.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The proposed uniform bound 37 remains unproven for all non-CM elliptic curves over the rationals. Important special families and complementary bounds are published, but they do not settle the full conjecture.

[1][2]

External progress

What the literature has established

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

2026
Peer reviewedGenao, Mayle, and Rouse proved that the smallest surjective prime is at most 7 while explicitly recording that the proposed uniform upper bound 37 for all nonsurjective primes remains unproven.[1]
2019
Peer reviewedLemos proved the bound for non-CM elliptic curves over the rationals admitting a nontrivial rational cyclic isogeny.[3]

Mathematical neighborhood

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 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.

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 statements2 proposed statements4 open questions2 narrowed routes

Statements by mathematical role6 selected mapped statements

theorem candidate1 of 61
reduction2 of 62
lemma2 of 62
negative result1 of 61

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included

retained route statementAre all mod-ell images surjective once ell is greater than 37?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementStrongest fixed-CM branchintermediate
retained route statementUniform CM norm theoremintermediate
retained route statementValuations are saturatedintermediate
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
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 failureGeneric-fiber minimal R=Treported failure
Useful failureValuation-only rationality testreported failure
Research targetClose Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.open
Research targetCompute the unit squareclass controlling rational versus quadratic descent in the one-prime twist-orbit algebra.open
Research targetExtend any successful fixed D_K=-8 closure to other class-number-one CM fields, other residue classes of ell, and the remaining global image branches.open
Research targetUniform surjectivity above 37open
Research targetIntegral minimal-rigidity gatesuperseded
Research targetUnit squareclass obstructionsuperseded
ComputationThe source reports modular-symbol isolation and finite enumerations, but states that the raw computation must be source-reported and saved.Those finite claims are not treated as independently reproduced evidence and do not close the integral rigidity or rational-descent gates. · reported unreproduced
Narrowed routeGeneric-fiber minimal R=TA vertical tangent ring can have generic fiber Q_ell while retaining a nonzero mod-ell tangent. A full integral theorem matching the exact local deformation conditions, or a direct Selmer computation, remains viable.
Narrowed routeValuation-only rationality testThe valuations are saturated and leave the unit squareclass u undetermined in z^2=uL^2. Computing the descended unit through monodromy, symplectic descent, Tate residues, and the twist involution remains the central target.

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 bridgeClose Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.

2 approaches have 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

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Prepared starting pointClose Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.

Serre uniformity conjecture · ready to start

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Research contextPrepared context for any AI agent

For a non-CM elliptic curve over the rational numbers, the conjecture says that every prime larger than 37 gives the full possible symmetry group on the curve's ell-torsion points.

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
A uniform bound on the smallest surjective prime of an elliptic curvepeer reviewed result · Tyler Genao, Jacob Mayle, Jeremy Rouse · Springer Nature · 2026 · DOI 10.1007/s11139-026-01376-8 · accessed Aug 14, 2026
2
On the effective version of Serre's open image theorempeer reviewed result · Jacob Mayle · London Mathematical Society / Wiley · 2024 · DOI 10.1112/blms.13002 · accessed Aug 14, 2026
3
Serre's uniformity conjecture for elliptic curves with rational cyclic isogeniespeer reviewed result · Pedro Lemos · American Mathematical Society · 2019 · ARXIV 1702.01985 · DOI 10.1090/tran/7198 · accessed Aug 14, 2026

Important qualifications

Bounded publisher and official-journal search; not a comprehensive modular-curve census.
No packet-local fixed-CM argument or computational claim was externally validated.

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Serre uniformity conjecture

Exact mathematical statement

Problem at a glance

Where work on Serre uniformity conjecture stands

Serre uniformity conjecture in numbers

Close Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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