Number theory · elliptic curves · arithmetic geometry · L-functions

Birch and Swinnerton-Dyer Conjecture

Collaboration beta

Does the behavior of an elliptic curve’s L-function at its central point determine the curve’s rational rank and exact arithmetic invariants?

r_{an} = r, \frac{L^{(r)} (E, 1)}{r!} = \frac{Ω_{E} {Reg}_{E} | Sha (E) | \prod_{ℓ} c_{ℓ}}{| E (Q)_{tors} |^{2}}

Clay Millennium Prize Problem

Known results and sources

BSD predicts that the central vanishing of an elliptic curve’s L-function measures its rational points and controls a precise arithmetic leading coefficient.

Research problem

Exact mathematical statement

Let $E/\mathbf Q$ be an elliptic curve, let $r=\operatorname{rank}E(\mathbf Q)$ , and let $r_{\mathrm{an}}=\operatorname{ord}_{s=1}L(E,s)$ . The full Birch and Swinnerton-Dyer conjecture asserts

r_{\mathrm{an}}=r,\qquad \mathrm{Sha}(E)\text{ is finite},

and, with $\Omega_E$ the fixed real period, $\operatorname{Reg}_E$ the Néron–Tate regulator, $c_\ell$ the Tamagawa numbers, and $E(\mathbf Q)_{\mathrm{tors}}$ the rational torsion subgroup,

\frac{L^{(r)}(E,1)}{r!}=\frac{\Omega_E\operatorname{Reg}_E\,|\mathrm{Sha}(E)|\prod_\ell c_\ell}{|E(\mathbf Q)_{\mathrm{tors}}|^2}.

The retained V6 handoff explicitly reports that no proof of the full conjecture has been completed. Its formal and conditional determinant reductions are research architecture, not verification of BSD.

Problem infographic

Problem at a glance

Problem-first scientific diagram showing an elliptic curve over the rationals, its rational-point group and rank, the Hasse–Weil L-function at s equals 1, and the conjectured leading-coefficient balance involving the real period, regulator, Tate–Shafarevich group, Tamagawa numbers, and rational torsion, with the general implication marked open. — For an elliptic curve over the rationals, BSD links the order and first nonzero coefficient of L(E,s) at s=1 to rational rank, heights, local reduction data, torsion, and the Tate–Shafarevich group; the full statement remains open.

Current mathematical picture

Where work on Birch and Swinnerton-Dyer Conjecture stands

Partially resolved

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

Useful failureSource-reported limitation

A determinant-line isomorphism or primitive determinant generator is rank-blind and does not force $W_p=T_p\Sha(E)$ to vanish. If the current work’s global section exists noncircularly, evaluates to the normalized positive rational number Q_E at the real place, lands integrally in every local determinant lattice, and is primitive at every prime, the current work derives analytic rank equality, finiteness of Sha, and the exact leading-coefficient formula.

Route status · Narrowed route

Main reductionQuotient–dual Poincaré descent

The current work reports functorial descent of the Poincaré biextension through a saturated elliptic quotient and invariant dual without the artificial modular multiplier.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeConstruct the canonical Hecke-balanced regularized automorphic determinant family and prove its coupled extension-class identity.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Birch and Swinnerton-Dyer Conjecture in numbers

2.5kretained lines of mathematical investigation2,472 in the current working snapshot

Argument development: 2,064 · 83%
Explored or eliminated routes: 40 · 2%
Computational analysis: 37 · 1%
Open obligations: 71 · 3%
Definitions and setup: 260 · 11%

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 byB. J. Birch, H. P. F. Swinnerton-DyerProposed1965Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Complete the convention-sensitive compact Selmer, inverse-limit, Poitou–Tate, bad-prime, p=2, and real-place audit.

Suggested move: Derive the Milnor sequences and limit terms from finite level, fix the exact duality theorem and local-condition triangles, and prove the adjoint identification used by canonical reduction without any remaining fragile-status qualifier.

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

Complete the convention-sensitive compact Selmer, inverse-limit, Poitou–Tate, bad-prime, p=2, and real-place audit.Suggested move: Derive the Milnor sequences and limit terms from finite level, fix the exact duality theorem and local-condition triangles, and prove the adjoint identification used by canonical reduction without any remaining fragile-status qualifier.

Ready to work on

Construct one noncircular rational fundamental line and section with integral comparison and residual primitivity at every prime.Suggested move: Bind the real scalar and all local evaluation ideals to the same rational section, account explicitly for Manin, congruence, modular-degree, component, and exceptional-zero factors, then prove integral inclusion and residual nonvanishing prime by prime.

Ready to work on

Construct the canonical Hecke-balanced regularized automorphic determinant family and prove its coupled extension-class identity.Suggested move: Specialize by derived saturated quotient–invariant duality, retain Tor and congruence factors, identify the first crossing with the quotient–dual adelic Poincaré class, and prove that the same family has the exactly normalized scalar determinant.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusPartially resolved

Clay continues to classify the Millennium Prize Problem as unsolved. For elliptic curves over Q, the rank equality is known when the analytic order of vanishing is 0 or 1, and the full formula has been rigorously established for many specific curves. The general rank statement and the refined leading-coefficient formula, including the predicted Tate–Shafarevich factor, remain open.

[1][2][8]

External progress

What the literature has established

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

2014
PreprintBhargava, Skinner, and Zhang reported that more than 66% of elliptic curves over Q, ordered by height, satisfy the BSD rank conjecture; this record retains the work as a preprint and distinguishes rank from the refined formula.[9]
2011
Computational resultMiller reported computer-assisted rigorous proofs of the full BSD formula for 16,714 of 16,725 analytic-rank-zero-or-one elliptic curves of conductor below 5,000.[8]
2001
Peer reviewedBreuil, Conrad, Diamond, and Taylor proved that every elliptic curve over Q is modular, extending the analytic-rank-zero-or-one rank theorem to all elliptic curves over Q in that scope.[7][2]
1986–1989
Peer reviewedGross–Zagier related first derivatives to Heegner-point heights, and Kolyvagin's Euler-system work supplied finiteness results; together these establish the BSD rank statement for modular elliptic curves of analytic rank 0 or 1.[5][6]

Mathematical neighborhood

Related results and reusable starting points

Current focusBirch and Swinnerton-Dyer Conjecture

Stronger or generalized formRefined Birch–Swinnerton-Dyer formula

The refined formula identifies the first nonzero Taylor coefficient using the regulator, periods, Tamagawa factors, torsion, and the order of the Tate–Shafarevich group; it is stronger than rank equality alone.

[2][3]

Dependency or reductionTate–Shafarevich group

Finiteness and the predicted order of the Tate–Shafarevich group are central to the refined formula; general finiteness remains unknown.

[2][6]

Stronger or generalized formBSD for abelian varieties and number fields

BSD has formulations for elliptic curves over number fields and for higher-dimensional abelian varieties; results in those broader settings must not be conflated with the exact Clay problem over Q.

[2]

Logical consequenceCongruent number problem

For the congruent-number elliptic curves, even the weak BSD equivalence converts the analytic vanishing question into infinitude of rational points and yields Tunnell's conditional criterion.

[2]

Formal and computational footholds

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

formal library support · partial resource linkedmathlib Weierstrass-curve L-function
mathlib defines local Euler factors, the formal L-function, and the complex L-series of a Weierstrass curve over a number field. The cited module does not state the BSD rank equality or refined leading-coefficient formula.
[10]
computation · not independently reproducedMiller conductor-below-5000 BSD verification
The peer-reviewed study reports rigorous computer-assisted proofs of the full formula for 16,714 specified analytic-rank-zero-or-one curves. ProofAtlas did not rerun the computation.
[8]
software · source linked; not reproduced by ProofAtlasSageMath BSD and descent routines
SageMath documents BSD-oriented rank, descent, Heegner-index, Tate–Shafarevich, regulator, period, and proof routines for elliptic curves. Availability does not mean that ProofAtlas reran them or that they prove the universal conjecture.
[11]
dataset · source linked; not reproduced by ProofAtlasLMFDB reviewed BSD invariants
LMFDB exposes reviewed rank, regulator, period, Tamagawa, torsion, and analytic Tate–Shafarevich invariants for elliptic curves over Q. Individual records are data, not a proof of the conjecture.
[12]

Formalization opportunities

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

Formalization targetA formal Mordell–Weil theorem and a usable formal definition of the rank of E(Q).
Formalization targetAnalytic continuation and central Taylor-order infrastructure for elliptic-curve L-functions at s = 1.
Formalization targetFormal regulators, Néron–Tate heights, Tamagawa factors, periods, and torsion terms in one compatible normalization.
Formalization targetA formal Tate–Shafarevich group and finiteness or order statements sufficient for the refined formula.
Formalization targetFormal Gross–Zagier and Kolyvagin machinery for the analytic-rank-zero-or-one theorem.

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 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 supporting details in the research record. 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.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma5 of 95

Selected mathematical clusters3 mathematical clusters

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

retained route statementCentral analytic vanishing should match rational rank, and the leading coefficient should equal the exact global arithmetic factor.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementReported first-jet formalismintermediate
retained route statementCanonical local arithmetic objectintermediate
retained route statementIntrinsic local Fitting evaluationintermediate
retained route statementQuotient–dual Poincaré descentintermediate
retained route statementKodaira–Spencer bridge formulatedintermediate
retained route statementConditional adelic closureintermediate
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 a formal implication from the stated datum to BSD, while construction of the datum remains nonformal and open; this intake has not independently verified the derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetConstruct the canonical Hecke-balanced regularized automorphic determinant family and prove its coupled extension-class identity.in progress reported
Research targetComplete the convention-sensitive compact Selmer, inverse-limit, Poitou–Tate, bad-prime, p=2, and real-place audit.open
Research targetConstruct one noncircular rational fundamental line and section with integral comparison and residual primitivity at every prime.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
ComputationThe current work proposes exact-algebra scripts for strict complexes, Smith normal forms, Fitting ideals, basis changes, rank jumps, Hecke specialization, and determinant-of-cohomology signs.No attachment or checker was executed by ProofAtlas. Proposed or packet-reported algebraic checks are not formal verification and do not establish BSD. · reported unreproduced
Narrowed routeSource-reported limitationA determinant-line isomorphism or primitive determinant generator is rank-blind and does not force $W_p=T_p\Sha(E)$ to vanish. If the current work’s global section exists noncircularly, evaluates to the normalized positive rational number Q_E at the real place, lands integrally in every local determinant lattice, and is primitive at every prime, the current work derives analytic rank equality, finiteness of Sha, and the exact leading-coefficient formula.

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 bridgeConstruct the canonical Hecke-balanced regularized automorphic determinant family and prove its coupled extension-class identity.

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 pointComplete the convention-sensitive compact Selmer, inverse-limit, Poitou–Tate, bad-prime, p=2, and real-place audit.

Birch and Swinnerton-Dyer Conjecture · ready to start

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Research contextPrepared context for any AI agent

Does the behavior of an elliptic curve’s L-function at its central point determine the curve’s rational rank and exact arithmetic invariants?

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Current routes and known obstacles
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Sources and references12 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
Birch and Swinnerton-Dyer Conjecturemaintained problem list · Clay Mathematics Institute · accessed Aug 7, 2026
2
The Birch and Swinnerton-Dyer Conjecture: Official Problem Descriptionauthoritative webpage · Andrew Wiles · Clay Mathematics Institute · accessed Aug 7, 2026
3
Notes on elliptic curves. IIoriginal source · B. J. Birch, H. P. F. Swinnerton-Dyer · Journal für die reine und angewandte Mathematik · 1965 · DOI 10.1515/crll.1965.218.79 · accessed Aug 7, 2026
4
On the Conjecture of Birch and Swinnerton-Dyerpeer reviewed result · J. Coates, A. Wiles · Inventiones Mathematicae · 1977 · DOI 10.1007/BF01402975 · accessed Aug 7, 2026
5
Heegner points and derivatives of L-seriespeer reviewed result · B. H. Gross, D. B. Zagier · Inventiones Mathematicae · 1986 · DOI 10.1007/BF01388809 · MR MR0833197 · accessed Aug 7, 2026
6
Finiteness of E(Q) and Sha(E,Q) for a subclass of Weil curvespeer reviewed result · V. A. Kolyvagin · Mathematics of the USSR-Izvestiya · 1989 · DOI 10.1070/IM1989v032n03ABEH000779 · accessed Aug 7, 2026
7
On the modularity of elliptic curves over Q: wild 3-adic exercisespeer reviewed result · Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor · Journal of the American Mathematical Society · 2001 · DOI 10.1090/S0894-0347-01-00370-8 · accessed Aug 7, 2026
8
Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and onepeer reviewed result · Robert L. Miller · LMS Journal of Computation and Mathematics · 2011 · ARXIV 1010.2431 · accessed Aug 7, 2026
9
A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecturepreprint · Manjul Bhargava, Christopher Skinner, Wei Zhang · arXiv · 2014-07-07 · ARXIV 1407.1826 · accessed Aug 7, 2026
10
Mathlib.AlgebraicGeometry.EllipticCurve.LFunctionformalization · mathlib community · accessed Aug 7, 2026
11
Birch and Swinnerton-Dyer formulas – Elliptic curvessoftware or dataset · SageMath · accessed Aug 7, 2026
12
Birch and Swinnerton-Dyer invariants of an elliptic curve/Q (reviewed)software or dataset · Duc Khiem Huynh, Paul-Olivier Dehaye, Alina Bucur, David Farmer, John Cremona, Vishal Arul, John Jones · LMFDB · accessed Aug 7, 2026

Important qualifications

The review focuses on elliptic curves over Q, the Clay formulation, the rank-versus-refined distinction, representative milestones, and visible formal or computational resources; it is not an exhaustive survey of number-field, abelian-variety, p-adic, Iwasawa-theoretic, or function-field versions.
Linked computational studies, SageMath routines, and LMFDB data were not independently reproduced by ProofAtlas in this collection run.
A scoped formalization search identified mathlib L-function infrastructure but did not establish the existence or nonexistence of a separate end-to-end BSD formalization.

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