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Number theory · elliptic curves · quadratic twists

Goldfeld's Conjecture

Collaboration beta

For a fixed elliptic curve over the rationals, do half of its quadratic twists have analytic rank zero, half rank one, and only density zero have rank at least two?

dens(ran=0)=12,dens(ran=1)=12,dens(ran≥2)=0
Known results and sources
A fixed elliptic curve branches into many quadratic-twist ribbons, half ending at zero-height rank markers and half at one-height markers, with a sparse unresolved tail above them.
Goldfeld predicts a 50–50 split between analytic ranks zero and one across quadratic twists, with higher rank occurring only at density zero.

Research problem

Exact mathematical statement

Fix an elliptic curve E/QE/\mathbf Q. For each nonzero squarefree integer dd, let E(d)E^{(d)} be its quadratic twist, ordered by |d||d|. Goldfeld's conjecture predicts

dens{d:ords=1L(s,E(d))=0}=12,dens{d:ords=1L(s,E(d))=1}=12,\operatorname{dens}\{d: \operatorname{ord}_{s=1}L(s,E^{(d)})=0\}=\frac12,\qquad \operatorname{dens}\{d: \operatorname{ord}_{s=1}L(s,E^{(d)})=1\}=\frac12,

and hence density zero for twists of analytic rank at least two. Equivalently after splitting by root number, almost every sign-+1+1 twist should have analytic rank zero and almost every sign--1-1 twist analytic rank one. The governing source claims no unconditional proof.

Problem infographic

Problem at a glance

A landscape mathematical diagram shows one elliptic curve, a field of squarefree quadratic twists, two equal rank-zero and rank-one populations, and a thin unresolved higher-rank tail.
The conjecture concerns the analytic ranks of every quadratic twist of a fixed elliptic curve when twists are ordered by absolute squarefree discriminant.

Current mathematical picture

Where work on Goldfeld's Conjecture stands

Open conjecture

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

Useful failureInferring rational-prime wedge coverage from centrality, full-module surjectivity, or a nonzero coefficient matrix

Neither a central top kernel, full residual-module evaluation surjectivity, nor a nonzero response coefficient matrix alone proves that rational-prime Frobenius labels supply the visible exterior-square directions needed for contraction. Construct, in one strict self-dual Selmer complex, an arithmetic mixed operator for two rational-prime slots together with noncentral but orbit-visible translation subspaces in coinvariant Frobenius fibers, compute their scalar evaluation…

Route status · Narrowed route
Main reductionFixed-prime closure is conditional

Nine stated hypotheses yield geometric contraction of independent persistent pairs and density zero for persistent dimension at least two among twists prime to the fixed p.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeRealize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex.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

Goldfeld's Conjecture in numbers

999retained lines of mathematical investigation999 in the current working snapshot
Argument development
857 · 86%
Explored or eliminated routes
11 · 1%
Computational analysis
22 · 2%
Open obligations
16 · 2%
Definitions and setup
93 · 9%
6selected mapped statements1routes investigated5open questions5contribution-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 byDorian GoldfeldProposed1979Status 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 Goldfeld's ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.For a fixed elliptic curve over the rationals, do half of its quadratic twists have analytic rank zero, half rank one, and only density zero have rank at least two? — Depends on missing premiseFor a fixed elliptic curveover the rationals, do halfof…Current reduction — Depends on missing premiseCurrent reductionFixed-prime closure is conditional — Depends on missing premiseFixed-prime closure isconditionalTwo twist slots produce a mixed alternating response — Depends on missing premiseTwo twist slots produce amixed alternating responseClosing target — Depends on missing premiseClosing targetVisible wedges give the corrected noncollapse test — Depends on missing premiseVisible wedges give thecorrected noncollapse testInferring rational-prime wedge coverage from centrality, full-module surjectivity, or a nonzero coefficient matrix — stoppedInferring rational-primewedge coverage fromcentrality,…Realize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex. — OpenRealize the strictone-channel mixed operatorand…Prove visible evaluation wedge coverage and uniform balanced completion on every bounded persistent fiber. — OpenProve visible evaluationwedge coverage and uniformbalanced…Convert fixed-prime persistent contraction into the exact analytic-rank density statement. — OpenConvert fixed-primepersistent contraction intothe…Half rank zero and half rank one — OpenHalf rank zero and half rankoneRational-prime labels need orbit-visible geometry — OpenRational-prime labels needorbit-visible geometry
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 routeInferring rational-prime wedge coverage from centrality, full-module surjectivity, or a nonzero coefficient matrix

Neither a central top kernel, full residual-module evaluation surjectivity, nor a nonzero response coefficient matrix alone proves that rational-prime Frobenius labels supply the visible exterior-square directions needed for contraction. Construct, in one strict self-dual Selmer complex, an arithmetic mixed operator for two rational-prime slots together with noncentral but orbit-visible translation subspaces in coinvariant Frobenius fibers, compute their scalar evaluation…

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Realize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex.Suggested move: Choose one odd prime and residual setting, put four twists into a common finite self-dual mapping-cone complex, compute the effective complementary-depth mixed operator including direct curvature, then identify the lower Frobenius element, its coinvariant module, centralizer action, and genuine affine translation subspaces.
Ready to work on
02
Prove visible evaluation wedge coverage and uniform balanced completion on every bounded persistent fiber.Suggested move: For each favorable slot type, prove its scalar evaluation is orbit-invariant, calculate the images A_v and B_w, determine every coefficient κ_v,w, verify that their weighted wedges span the full exterior square, and obtain uniform Chebotarev balance and product-grid completion after truncating the initial Selmer dimension.
Ready to work on
03
Half rank zero and half rank one

The conjecture predicts density one half for analytic rank zero, density one half for analytic rank one, and density zero for analytic rank at least two.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Rational-prime labels need orbit-visible geometry

Chebotarev sees centralizer orbits in a coinvariant quotient, so the translation spaces and their actual evaluation images must be constructed and computed rather than inferred from full-module surjectivity.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Convert fixed-prime persistent contraction into the exact analytic-rank density statement.Suggested move: Identify persistent dimension with divisible p-primary Selmer corank, certify density-one rank-zero and rank-one p-converses prime by prime, prove the infinite usable-prime set and finite-stage cover, and audit sign conventions and root-number equidistribution in the exact positive and negative twist families.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

Goldfeld's conjecture remains open for a general fixed elliptic curve without additional conjectural input. Current primary sources prove important Selmer distributions, special-family rank results, and a general implication from BSD; the 2026 Annals theorem supplies the first even-parity instance for the positive congruent-number twist family, not the full 50–50 analytic-rank statement for arbitrary E.

[1][2][3]
External progress

What the literature has established

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

  1. Peer reviewedBurungale and Tian proved a rank-zero p-converse for CM elliptic curves and, with Smith's Selmer-distribution work, the first even-parity Goldfeld instance for positive twists of the congruent-number curve.[4]
  2. PreprintSmith proved for every elliptic curve over Q that 50 percent of quadratic twists have 2-infinity-Selmer corank zero and 50 percent corank one, and deduced Goldfeld's conjecture conditional on BSD.[3]
  3. PreprintSmith proved the predicted 2-infinity-Selmer distribution for curves with full rational 2-torsion and no rational cyclic subgroup of order four, implying that twists of rank at least two number o(N) in that…[2]
  4. Historical sourceGoldfeld formulated conjectures on elliptic curves over quadratic fields in the Carbondale 1979 proceedings, the historical source for the named quadratic-twist rank conjecture.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusGoldfeld's Conjecture
Solved special caseFull rational 2-torsion quadratic-twist families

The full-rational-2-torsion family has density-zero rank-at-least-two behavior under the source's technical hypotheses. This is a narrower class of elliptic curves than the general fixed-E target.

[2]
Logical consequenceBSD-conditional Goldfeld theorem

The 50–50 2-infinity-Selmer corank theorem holds for arbitrary E/Q, and the preprint derives Goldfeld's analytic-rank statement assuming BSD. The conditional analytic passage remains explicit.

[3]
Solved special caseEven-parity congruent-number twist family

The positive congruent-number twists provide the first even-parity instance: half have analytic rank zero. It does not include the odd-parity half or arbitrary elliptic curves.

[4]

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 elliptic curves over Q, squarefree quadratic twists ordered by absolute discriminant, analytic rank, and natural density.
  • Formalization targetFormal functional equations and root-number parity for the exact twist families, including bad-prime and positive-versus-negative conventions.
  • Formalization targetMachine-checked interfaces between Selmer coranks, p-converse theorems, analytic ranks, and the finite-stage density argument across usable primes.

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 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 statements2 proposed statements5 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction3 of 63
  • lemma2 of 62
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.18 displayed rows · 1 route included
  • retained route statementFor a fixed elliptic curve over the rationals, do half of its quadratic twists have analytic rank zero, half rank one, and only density zero have rank at least two?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementTwo twist slots produce a mixed alternating responseintermediate
  • retained route statementVisible wedges give the corrected noncollapse testintermediate
  • retained route statementFixed-prime closure is conditionalintermediate
  • 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 failureInferring rational-prime wedge coverage from centrality, full-module surjectivity, or a nonzero coefficient matrixreported failure
  • Research targetRealize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex.open
  • Research targetProve visible evaluation wedge coverage and uniform balanced completion on every bounded persistent fiber.open
  • Research targetConvert fixed-prime persistent contraction into the exact analytic-rank density statement.open
  • Research targetHalf rank zero and half rank oneopen
  • Research targetRational-prime labels need orbit-visible geometryopen
  • Narrowed routeInferring rational-prime wedge coverage from centrality, full-module surjectivity, or a nonzero coefficient matrixNeither a central top kernel, full residual-module evaluation surjectivity, nor a nonzero response coefficient matrix alone proves that rational-prime Frobenius labels supply the visible exterior-square directions needed for contraction. Construct, in one strict self-dual Selmer complex, an arithmetic mixed operator for two rational-prime slots together with noncentral but orbit-visible translation subspaces in coinvariant Frobenius fibers, compute their scalar evaluation images, and prove those images cover the full exterior square uniformly on bounded Selmer fibers.
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 bridgeRealize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex.

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 pointRealize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex.

Goldfeld's Conjecture · ready to start

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Research contextPrepared context for any AI agent

For a fixed elliptic curve over the rationals, do half of its quadratic twists have analytic rank zero, half rank one, and only density zero have rank at least two?

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  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 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
    Conjectures on elliptic curves over quadratic fieldsoriginal source · Dorian Goldfeld · Springer, Lecture Notes in Mathematics 751 · 1979 · DOI 10.1007/BFb0062705 · accessed Aug 14, 2026
  2. 2
    2^infinity-Selmer groups, 2^infinity-class groups, and Goldfeld's conjecturepreprint · Alexander Smith · arXiv · 2017-06-07 · ARXIV 1702.02325 · accessed Aug 14, 2026
  3. 3
    The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecturepreprint · Alexander Smith · arXiv · 2025-03-22 · ARXIV 2503.17619 · accessed Aug 14, 2026
  4. 4
    A rank zero p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubinpeer reviewed result · Ashay A. Burungale, Ye Tian · Annals of Mathematics · 2026 · DOI 10.4007/annals.2026.203.1.1 · accessed Aug 14, 2026

Important qualifications

  • The general fixed-elliptic-curve conjecture is kept separate from full-rational-2-torsion families, the congruent-number curve, even-parity subfamilies, and statements conditional on BSD.
  • Smith's 2025 source is a preprint; its 50–50 2-infinity-Selmer result and its conditional implication are not presented as an unconditional analytic-rank proof.
  • The 2026 Annals result proves a rank-zero p-converse for CM curves and an even-parity instance for a specific family, not the full general conjecture.
  • No submitted URL was fetched, and the current work's internal probability estimate was not used as external evidence.
  • No statement-aligned formalization, checked proof certificate, reusable dataset, or independently reproduced computation was identified in this collection.

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