openmath.dev
Esc

Loading…

↑ ↓ to move↵ to open

Appearance
Appearance
← Problems

Number theory · Algebraic geometry · Conjecture

Birch and Swinnerton-Dyer conjecture

For an elliptic curve over the rational numbers, does the rank of its rational points equal the order of vanishing of its L-function at s=1s=1?

Openper Clay · Millennium Problems · checked Oct 10, 2026
Frontier 2 reports →
Cite & export
Open this versionExport record · JSONRecord detailsReport a correction
Use with an agent

Earlier version 13 of this page. View current →

Overview

Known

Rank equality is proved at analytic rank 0 or 1. Further cited results give a conditional converse, selected p-parts and full BSD in specified twist families and finite computations.

Open questions

Rank equality in general, especially analytic rank ≥ 2, remains open. General finiteness of Sha and the complete leading-coefficient formula are stronger unresolved assertions.

Latest

Precise statement

Elliptic curves over Q\mathbb{Q}. The canonical question is rank equality. The stronger BSD formula additionally predicts finite Sha and L(r)(E,1)r! ΩEReg⁡E=#Sha⁡(E/Q)(#E(Q)tors)2⋅∏ℓ finitecℓ(E)\begin{aligned}\frac{L^{(r)}(E,1)}{r!\,\Omega_E\operatorname{Reg}_E}&=\frac{\#\operatorname{Sha}(E/\mathbb{Q})}{\bigl(\#E(\mathbb{Q})_{\mathrm{tors}}\bigr)^2}\\&\quad\cdot\prod_{\ell\ \mathrm{finite}}c_\ell(E)\end{aligned}, with the standard full Hasse–Weil L-function, Néron real period, regulator and finite-prime Tamagawa factors. These aspects are tracked separately.

Definitions

Rank
The number of independent infinite-order rational points on an elliptic curve; BSD compares it with the order of vanishing of the curve’s L-function.

Background

How to read this question

Two ranks count different things

Mordell–Weil rank counts independent rational generators. Analytic rank counts the vanishing order of L(E,s) at 1. BSD predicts these integers agree for every E/Q.

Limit A set of independent points gives a rank lower bound; a rigorous equality also needs an upper bound. Approximate L-values do not certify an exact vanishing order.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Rank equality, p-parts and full BSD have separate scopes

The full formula additionally involves finite Sha, periods, regulator, Tamagawa factors and torsion. A p-part identifies one valuation under specific hypotheses; a family theorem fixes a class of curves.

Limit These results overlap. Matching decimals, one valuation or rank equality alone does not establish the entire normalized formula.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Selmer corank is not just algebraic rank

The new provider proposal assumes the full q-power Selmer corank is 0 or 1. Its exact Kummer sequence also contains the divisible q-primary contribution of Sha, so a Mordell–Weil rank bound alone is insufficient.

Limit The proposal is not independently accepted here. Do not read its low-corank hypothesis as every curve of algebraic rank zero or one.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

Selected coverage

Which parts of BSD are covered

Rank equality with finite Sha, selected prime parts and the full formula have different coverage. Select a result to see the exact hypotheses.

Reading boundary. Rank equality and finite Sha at analytic rank 0 or 1 do not alone prove the full leading-term formula. General analytic rank ≥ 2 remains open. Finite Sha in the converse is assumed. Selected p-parts are not the full formula; scopes overlap and the historical finite enumeration excludes its unresolved curve/prime pairs.

Which parts of BSD are covered
Family / conditionsRank equality / finite ShaSelected p-partsFull leading-term formula
Analytic rank 0 or 1Rank equality + finite Sha · analytic rank 0 or 1● Proved under stated assumptionsNo result indexed in this selectionNo result indexed in this selection
Conditional rank-one converseAlgebraic rank 1 + finite Sha assumed◇ Conditional theoremNo result indexed in this selectionNo result indexed in this selection
Semistable curves · specified primesNo result indexed in this selectionAnalytic rank 1 · good p ≥ 5 · irreducible E[p]● Proved under stated assumptionsAnalytic rank 0 or 1 · supersingular p ≥ 5● Proved under stated assumptionsNo result indexed in this selection
Specified families or finite setsNo result indexed in this selectionNo result indexed in this selectionSelected twists of 46a1● Proved under stated assumptions16,714 certified curves · exceptions excluded● Proved under stated assumptions

● Proved under stated assumptions◇ Conditional theorem— No result indexed in this selection

Text version

For elliptic curves over Q, analytic rank 0 or 1 implies rank equality and finite Sha, but does not alone prove the full leading-term formula. The converse assumes algebraic rank 1, finite Sha and stated reduction hypotheses. Two other results cover selected p-parts under analytic-rank and prime hypotheses. The full formula is covered for specified quadratic twists and a historical certified finite corpus. These scopes overlap and do not partition all curves.

Elliptic curves over Q. Analytic rank, algebraic rank, prime conditions and curve-family restrictions are distinguished in each result.

  1. Rank equality + finite Sha · analytic rank 0 or 1

    Proved

    Rank equality at analytic rank 0 or 1

    Every E/Q of analytic rank 0 or 1.

    Limit. No unrestricted analytic-rank ≥ 2 theorem follows.

    Rank equality at analytic rank 0 or 1 →

  2. Algebraic rank 1 + finite Sha assumed

    Proved

    Rank-one converse assuming finite Sha

    Algebraic rank 1 ⇒ analytic rank 1 in this family if Sha is finite.

    Limit. Finite Sha and the reduction hypotheses must be established for an application.

    Rank-one converse assuming finite Sha →

    Overlap. Runs in the converse direction to the low-analytic-rank theorem. General Sha finiteness is conjectural and is not supplied by assuming algebraic rank 1.
  3. Analytic rank 1 · good p ≥ 5 · irreducible E[p]

    Proved

    Rank-one formula · selected p-parts

    Good-reduction p ≥ 5 with irreducible E[p], for semistable analytic-rank-one curves.

    Limit. The missing primes and unrestricted curves require further results.

    Rank-one formula · selected p-parts →

    Overlap. Adds prime-specific leading-coefficient information beyond low-rank equality. Both good ordinary and supersingular reduction are permitted here; the p=3 extension is not included in this selected Claim.
  4. Analytic rank 0 or 1 · supersingular p ≥ 5

    Proved

    Low-rank formula at supersingular p ≥ 5

    Semistable analytic-rank 0 or 1 at supersingular p ≥ 5.

    Limit. Other primes and higher analytic ranks are not covered by this selected theorem.

    Low-rank formula at supersingular p ≥ 5 →

    Overlap. Overlaps the supersingular rank-one portion of the Jetchev–Skinner–Wan Claim and additionally includes analytic rank zero. Prime parts are not independent percentages of BSD solved.
  5. Selected twists of 46a1

    Proved

    Full BSD in a specified infinite twist family

    An infinite, explicitly conditioned family of twists of 46a1.

    Limit. Does not give full BSD for every twist, every rank-zero curve, or analytic rank ≥ 2.

    Full BSD in a specified infinite twist family →

    Overlap. A full-formula family within the low-analytic-rank case. The 46a1 family is retained after the 2026 Banwait–Huang check; four other base curves in the earlier list have an infinitude caveat.
  6. 16,714 certified curves · exceptions excluded

    Proved

    Computer-assisted full formula · 16,714 curves

    Full formula certified in the paper for 16,714 enumerated curves.

    Limit. The proof does not cover all curves, all conductor ranges or higher analytic ranks.

    Computer-assisted full formula · 16,714 curves →

    Overlap. Strengthens the formula for a finite subset of the classical low-rank regime. This is computer-assisted proof in the source, not AI theorem discovery or a Lean kernel check.
Sources & review

Authored by OpenMath curation on 2026-10-11 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

The experiment that suggested BSD

Birch and Swinnerton-Dyer computed products of Np/p and saw them grow like (log X)r, where r is the rank. Here the slope against log log X grows with the rank of each curve.

02460.511.52log log Xlog ∏ Nₚ/p over p ≤ X11a1 · rank 037a1 · rank 1389a1 · rank 25077a1 · rank 35077a1 · rank 3389a1 · rank 237a1 · rank 111a1 · rank 0

An illustration of the heuristic behind the conjecture for four curves and finitely many primes. It is not evidence that BSD holds, and the products are not known to converge in this form.

Dataset bsd-euler-products v1 · Exact point counts N_p = p + 1 + Σ_x (f(x)/p) for f = 4x³ + b2x² + 2b4x + b6 over odd primes p ≤ 100,000 other than the conductor; running sum of log(N_p/p), sampled about every (log log X) step of 1/160 of the range. Software: numpy 2.3.4, python 3.12.13. Exact integer counts; logarithms rounded to 6 decimal places. SHA-256 ec09bde13238… · Download JSON

Data
SeriesPointsLast value (log log X = 2.44175)
11a1 · rank 0770.837675
37a1 · rank 1773.626527
389a1 · rank 2775.772181
5077a1 · rank 3777.912722

Results

The question 1

  • Open
    BSD rank equality holds for every E/Q

    The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.

Special cases 5

  • Proved
    Rank equality at analytic rank 0 or 1

    Every E/Q of analytic rank 0 or 1.

    Skinner 2020
  • Proved
    Rank-one formula · selected p-parts

    Good-reduction p ≥ 5 with irreducible E[p], for semistable analytic-rank-one curves.

    Jetchev et al. 2015
  • Proved
    Low-rank formula at supersingular p ≥ 5

    Semistable analytic-rank 0 or 1 at supersingular p ≥ 5.

    Burungale et al. 2024
  • Proved
    Full BSD in a specified infinite twist family

    An infinite, explicitly conditioned family of twists of 46a1.

    Burungale et al. 2024
  • Proved
    Computer-assisted full formula · 16,714 curves

    Full formula certified in the paper for 16,714 enumerated curves.

    Miller 2010

Conditional results 1

  • Proved
    Rank-one converse assuming finite Sha

    Algebraic rank 1 ⇒ analytic rank 1 in this family if Sha is finite.

    Skinner 2020

Proposed 2

  • Proposed
    Proposed full BSD at Selmer corank zero or one

    A provider proposal depending on companion low-corank converse and two-primary inputs.

    OpenAI 2026
  • Proposed
    Proposed half-densities in every quadratic-twist family

    A proposed family-frequency theorem, not a theorem for every individual twist.

    OpenAI 2026

Conditional implications 4

  • Proved
    Strong BSD implies its rank and finiteness assertions

    A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.

    per Wiles (official description)
  • Proved
    Rank BSD would decide finite versus infinite rational points analytically

    Under rank equality, the central L-value detects whether E(Q) is finite or infinite.

    per Wiles (official description)
  • Proved
    BSD would complete the odd squarefree congruent-number criterion

    Two finite counts would decide the odd squarefree congruent-number problem, assuming rank BSD for E_n.

    per Wiles (official description)
  • Proved
    Strong BSD would determine Sha’s order from the other invariants

    The leading-term formula would recover Sha’s order from compatible certified invariants.

    per Wiles (official description)

Variants and analogues 2

  • Open
    The full BSD formula holds for every E/Q

    A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.

  • Open
    Sha is finite for every E/Q

    A separately open finiteness assertion, proved in selected low-analytic-rank cases.

Background 1

  • Proved
    Rational points form a finitely generated group

    An unconditional foundation. Knowing finite generation does not compute the rank of an arbitrary curve.

    per Wiles (official description)
Compare statements · 3

Each row is a separate assertion. Its assumptions determine which question it answers.

Connections

How the recorded statements connect
impliesimpliesrelatedrelatedrelatedBSD rank equality holds for every E/Q · OpenBSD rank equality holds forevery E/Q○ OpenBSD would complete the odd squarefree congruent-number criterion · ProvedBSD would complete the oddsquarefree congruent-number…● ProvedSha is finite for every E/Q · OpenSha is finite for every E/Q○ Openthe full BSD formula holds for every E/Q · Openthe full BSD formula holds forevery E/Q○ OpenRank equality at analytic rank 0 or 1 · ProvedRank equality at analytic rank0 or 1● ProvedRank-one converse assuming finite Sha · ProvedRank-one converse assumingfinite Sha● ProvedRank-one formula · selected p-parts · ProvedRank-one formula · selectedp-parts● ProvedLow-rank formula at supersingular p ≥ 5 · ProvedLow-rank formula atsupersingular p ≥ 5● Proved
  • ━ ⇔ equivalent
  • → implies
  • ┅ analogue or related (no logical link)
  • Border colour = status
5 connections · 3 consequences. Select a statement in the graph to show only its entries.

Implications 2

Analogues and related questions 3

Consequences 3

Assuming BSD rank equality holds for every E/QOpen

  • Connect analytic vanishing to rational solutions

    Conditional

    Rank BSD would make the central L-value an exact criterion for finite versus infinite rational points on every elliptic curve over Q.

    Limit This is a mathematical criterion; it does not make floating-point zero testing rigorous or imply a fast implementation.

    Sources & review

    For an elliptic curve E/Q satisfying BSD rank equality, E(Q) is infinite if and only if L(E,1) = 0; if L(E,1) ≠ 0, E(Q) is finite.

    This conclusion assumes rank equality for the curve and uses finite generation.

    Assumptions & sources

    Source-grounded explanation · independent explanation review: pending.

  • Complete a concrete rational-triangle criterion

    Conditional

    For odd positive squarefree n, rank BSD for the congruent-number curve would make Tunnell’s two finite representation counts sufficient as well as necessary for a rational right triangle of area n.

    Limit This selected formula is restricted to odd squarefree n; the even case has a different formula. It does not decide arbitrary Diophantine equations.

    Sources & review

    Supporting statements

    Source-grounded explanation · independent explanation review: pending.

Assuming the full BSD formula holds for every E/QOpen

  • Determine the missing arithmetic factor

    Conditional

    The full formula would tie Sha’s exact order to the central leading term, point-lattice regulator, periods, torsion and local factors.

    Limit Exact compatible inputs are required. This concerns rational elliptic curves and gives no claimed algorithm for finite-field elliptic-curve discrete logarithms.

    Sources & review

    For an E/Q satisfying strong BSD, #Sha(E/Q) = [L^{(r)}(E,1)/r!] · #E(Q)_tors² / [Ω_E Reg_E ∏_{ℓ finite} c_ℓ(E)], with r = rank E(Q) and the stated full-L and Néron conventions.

    This rearranges the full formula under its premise; numerical approximation alone does not certify the order.

    Assumptions & sources

    Source-grounded explanation · independent explanation review: pending.

Approaches

How the problem is attacked, and what each line of attack cannot yet reach.

Heegner points and Euler systems at low analytic rank

Gross–Zagier relates L-derivatives to heights; Kolyvagin’s derived classes constrain ranks and Sha. Together with modularity they establish the selected analytic-rank-0/1 theorem.

Cannot yet reach Higher analytic rank needs further constructions; the low-rank theorem does not identify every leading coefficient.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Iwasawa and p-adic comparisons isolate local arithmetic

The selected rank-one and supersingular formulas compare analytic data with Selmer information and arithmetic factors prime by prime.

Cannot yet reach Reduction type, residual irreducibility, rank and prime bounds are theorem hypotheses. Dropping them requires an additional argument.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Descent and certified finite computations

Descent, rigorous point lattices and theorem-backed computations can establish full BSD for particular curves. Effective family algorithms produce further explicitly sufficient cases.

Cannot yet reach The finite-corpus counts and sufficient-family tests are bounded guarantees. Exploratory numerics and Lean targets containing sorry are separate activities.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Timeline

  1. OpenAI’s proposed full low-Selmer-corank formulaProposal

    The provider proposes the complete leading-term formula and finite Sha when a full q-power Selmer corank is 0 or 1.

    The manuscript and its companion dependencies remain unassessed here. Mordell–Weil rank ≤1 is not the stated hypothesis.

    Also on the Frontier: Goldfeld twist-rank distribution · BSD in Selmer-corank 0/1 cases

  2. Proposed half-densities for analytic ranks in twist familiesProposal

    A separate provider manuscript proposes Goldfeld’s 0/1 half-densities for every E/Q in the signed squarefree ordering.

    An asymptotic family statement allows exceptional twists; mean rank and full leading terms require additional results.

  3. Effective enumeration of sufficient twist subfamiliesAlgorithmic advance

    Banwait–Huang v3 supplies finite algorithms for subfamilies satisfying cited BSD criteria and clarifies the base-curve restrictions.

    The four-base list in the family result is not an unconditional full-BSD assertion for every listed curve; there is no enumeration of all BSD cases.

  4. Supersingular p-parts and a conditioned full-formula familySpecial case proved

    Burungale–Skinner–Tian–Wan v2 gives selected supersingular p-parts and full BSD for the explicitly constrained 46a1 twist family.

    These are two different scopes. The selection omits p=3’s extra conditions and does not cover every quadratic twist.

  5. A conditional algebraic-rank-one converseConditional result

    Skinner’s published theorem obtains analytic rank one from algebraic rank one, finite Sha and the stated semistable reduction pattern.

    Finite Sha is assumed here; this converse cannot be used to prove its own premise.

  6. A rank-one p-part formula under precise hypothesesSpecial case proved

    Jetchev–Skinner–Wan’s author preprint gives the p-part for semistable analytic-rank-one curves at good p ≥5 with irreducible E[p].

    The theorem is prime-specific and hypothesis-specific; its later journal publication is a separate bibliographic event.

  7. Computer-assisted full formulas in a finite corpusSpecial case proved

    Miller’s published proof covers 16,714 of 16,725 enumerated analytic-rank-≤1 curves of conductor <5000.

    The 11 excluded curve/prime pairs describe that paper’s boundary, not today’s list of unsolved cases.

  8. The low-analytic-rank theorem applies to all E/QSpecial case proved

    General modularity, published in 2001, completes the classical scope: analytic rank 0 or 1 gives rank equality and finite Sha for every E/Q.

    This is a synthesis of earlier Gross–Zagier/Kolyvagin inputs with modularity, not a proof of every leading coefficient or higher-rank case.

  9. Heegner points and central derivativesFoundation

    The published Gross–Zagier formula provides the height/L-derivative connection later combined with Euler systems.

    The current analytic-rank-≤1 synthesis also uses Kolyvagin and modularity; the 1986 paper alone is not that full synthesis.

  10. The published conjecture grows from computer experimentsFoundation

    Birch–Swinnerton-Dyer’s Notes II records the rank and L-function prediction motivated by early computations.

    This is a cited publication year. Agreement in finite data did not establish the general formula.

  11. Finite generation makes rank a precise invariantFoundation

    Mordell’s theorem describes rational points as a finite-rank free group plus finite torsion.

    Finite generation does not determine the rank of each elliptic curve.

  12. 2026 · effective families and an earlier-list caveatSource correction

    Banwait–Huang give sufficient-condition algorithms for BSD twist subfamilies and list eligible curves of conductor <500,000. Section 2.7 retains 46a1 but identifies a nonempty-prime-support gap for 62a1, 66b1, 105a1 and 141c1 in an earlier infinitude list.

    The authors do not characterize every BSD-satisfying twist. The flagged four curves are not disproved BSD cases. The code and dataset have not been executed or reproduced by OpenMath.

Full history with sources →

Evidence

Literature

4 published papers · 3 preprints · 2 projects

Status per Clay · Millennium Problems · checked Oct 10, 2026.

Formal

None linked

Reported in sources: No complete Lean proof of the selected BSD statements has been established here. The inspected individual Lean targets contain sorry; they are explicitly classified as placeholders. No external code has an OpenMath Run or reproduction attached.

Computation

None recorded

Review

Not yet reviewed

Collection last checked Oct 10, 2026.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Bibliography 11

Research activity · 3 entries

Formalization

AI-assisted exploration · code and Lean targets

This individual AI-augmented research project reports numerical experiments and proof-search scaffolding. Its phase record states that Lean verification has not started; RankEquality.lean contains sorry targets.

No new BSD theorem, complete formal proof or OpenMath-verified execution is established by this project citation. Its rank-two regulator identity remains conjectural in the project’s own notes.

Related statements · context, not additional proved progress

Kent · AI-augmented BSD research substrate · sources & artifacts →

Computation

2026 · reported rank lower bound 30

The primary record supplies a specific elliptic curve over Q and 30 rational-point witnesses, with a reported rank lower bound of 30.

A lower bound on algebraic rank does not establish equality with analytic rank or BSD. OpenMath has not reproduced the independence certificate. Secondary claims of exact rank under BSD+GRH cannot be used as evidence proving BSD.

ICARM · individual high-rank curve record · sources & artifacts →

Research direction

Company AI proposals: Selmer converses, leading terms and twist densities

The October provider release proposes low-corank converses and exact leading terms, alongside a distinct analytic twist-density theorem. The pinned manuscripts state companion dependencies.

These source-reported proposals are not established progress here. No corresponding external formal-code execution or proof-dependency audit is asserted; asymptotic twist densities do not resolve every curve.

Related statements · context, not additional proved progress

OpenAI · proposed full BSD from low Selmer corank · sources & artifacts →

OpenAI · proposed Goldfeld analytic densities and 2-converse · sources & artifacts →

Selected literature; source observations and coverage limits are retained in the publication details.

Contributors

Company AI manuscripts now indexed as proposals: full BSD from low full-Selmer corank and analytic half-densities in twist families. Their independent assessment and companion proof chains are unassessed. The existing individual AI program and high-rank witness stay separate activities; neither proves general BSD.

Authors & roles

Kent · AI-augmented BSD research substrate →

Owen Kent (repository owner: owenpkent)

README calls the program AI-augmented and contains Claude agent specifications. A precise model version and independently assessed division of labor are not supplied.

ICARM · individual high-rank curve record →

ICARM leaderboard; submitter ranksunbounded

Secondary reports associate this search with Claude and Anthropic researchers; the cited primary curve record itself does not establish that attribution, so it is not treated as verified company authorship.

These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.

Record details
Record version
13
Classification
MSC2020 11G05 / 11G40
Prerequisite level
specialist
Source observations & version history →