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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 = 1?

Openper Clay · Millennium Problems · checked Oct 10, 2026
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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. The canonical question is rank equality. The stronger BSD formula additionally predicts finite Sha and L^(r)(E,1)/(r! Ω_E Reg_E) = #Sha(E/Q) · ∏ c_l / #E(Q)_tors², with the standard full Hasse–Weil L-function, Néron real period, regulator and finite-prime Tamagawa factors. These aspects are tracked separately.

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.

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)

Supporting results 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)

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.

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

Assuming the full BSD formula holds for every E/QOpen

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

Contributions

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.

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.

Record details
Record version
6
Classification
MSC2020 11G05 / 11G40
Prerequisite level
specialist
Source observations & version history →