Clay · Millennium Problems ↗
Unsolved · observed Oct 1, 2026 · superseded
Source status date unavailable · observed Oct 1, 2026
The cited source item and version only; this is not an independent OpenMath assessment.
Publication v7
Record details
Birch and Swinnerton-Dyer conjecture →Coverage & review
Coverage of this curation
Rank equality and finite Sha at analytic rank ≤ 1; one conditional converse; selected p-parts; a corroborated infinite twist family; a historical finite-corpus proof. Recent algorithms and AI/Lean exploratory artifacts are separate context. Includes selected source-dated history, exact Claim connections, interpretations, approaches and grounded conditional impacts.
Formal proofs: No complete Lean proof of the selected BSD Claims 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.
AI-assisted work: 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.
Search scope & review
Clay/Wiles; published Skinner and Jetchev–Skinner–Wan papers; author arXiv revisions through June 2026; public code and datasets; Google DeepMind, Axiom, Harmonic, Anthropic-related and individual AI/formalization searches. Cutoff: 10 October 2026. Not exhaustive. Rechecked source revision dates and the October OpenAI low-Selmer-corank and Goldfeld manuscripts, including their distinct hypotheses and companion dependencies.
Checked Oct 10, 2026 · independent review pending.
- Selected results, not an exhaustive literature survey or a percentage of the conjecture solved.
- Source-attributed theorem statements; independent mathematical and relationship review is pending.
- External papers, programs and Lean placeholders have no OpenMath execution or independent reproduction attached.
- AI involvement is recorded only where the cited source reports it; absence from this selection does not establish absence of work.
- Timeline dates refer to cited research events at their supported precision. Overlapping results and conditional consequences do not define a percentage solved.
General status observations
Clay · Millennium Problems ↗
Unsolved · observed Oct 10, 2026
Source status date unavailable · observed Oct 10, 2026
The cited source item and version only; this is not an independent OpenMath assessment.
Source identities & rights
Clay · Millennium Problems ↗
Clay Mathematics Institute · metadata-link · Not established; links and OpenMath paraphrases only
Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording. Rights / source terms
1 cited entry · exact source identity
- Version
- clay-millennium@observed-2026-10-01
- Revision
- observed-2026-10-01
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-01
- Check cadence
- 30 days
- https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/ ↗
birch-and-swinnerton-dyer-conjecture:entry:0:v1
Clay · Millennium Problems ↗
Clay Mathematics Institute · metadata-link · Not established; links and OpenMath paraphrases only
Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording. Rights / source terms
1 cited entry · exact source identity
- Version
- clay-millennium@observed-2026-10-10
- Revision
- observed-2026-10-10
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/ ↗
birch-and-swinnerton-dyer-conjecture:entry:0:v2
Wiles · official BSD description ↗
Andrew Wiles · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
9 cited entries · exact source identity
- Version
- clay-wiles-bsd@official-description
- Revision
- official-description
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 90 days
- PDF pages 2 and 4, conjecture/refinement and low-rank theorem ↗
birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1 - PDF page 2: rank conjecture; all finite Euler factors restored for the standard L-function ↗
birch-and-swinnerton-dyer-conjecture:bsd-rank-target:v1 - PDF page 2, refinement in Remark 1; standard full-L, total-real-period convention ↗
birch-and-swinnerton-dyer-conjecture:bsd-full-target:v1 - PDF page 1: Mordell theorem and rank definition ↗
birch-and-swinnerton-dyer-conjecture:bsd-mordell-weil:v1 - PDF pages 1–2: Mordell theorem and “In particular” consequence of rank BSD ↗
birch-and-swinnerton-dyer-conjecture:bsd-finite-points-consequence:v1 - §1 Early History, PDF page 3: odd squarefree Tunnell criterion ↗
birch-and-swinnerton-dyer-conjecture:bsd-congruent-number-consequence:v1 - §2 Recent History, PDF page 3; reference [4], Notes on elliptic curves II (1965) ↗
birch-and-swinnerton-dyer-conjecture:bsd-origin-history:v1 - §2, PDF pages 3–4; reference [11], Gross–Zagier (1986) ↗
birch-and-swinnerton-dyer-conjecture:bsd-gross-zagier-history:v1 - PDF page 4, low-rank synthesis; references [1], [24], [25] ↗
birch-and-swinnerton-dyer-conjecture:bsd-modularity-history:v1
Skinner · converse to Gross–Zagier–Kolyvagin ↗
Christopher Skinner · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
2 cited entries · exact source identity
- Version
- skinner-annals-2020@published-online-2020-02-13
- Revision
- published-online-2020-02-13
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2020-02-13
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Introduction, journal page 330 (PDF page 2), classical theorem ↗
birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v2 - Abstract and Theorem A, journal pages 329–331 ↗
birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v2
Jetchev–Skinner–Wan · rank-one BSD p-part ↗
Dimitar Jetchev; Christopher Skinner; Xin Wan · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-1512.06894@arxiv-v1
- Revision
- arxiv-v1
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2015-12-21
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1.2.1, PDF pages 1–2 ↗
birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1
Burungale–Skinner–Tian–Wan · zeta elements ↗
Ashay Burungale; Christopher Skinner; Ye Tian; Xin Wan · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
2 cited entries · exact source identity
- Version
- arxiv-2409.01350@arxiv-v2
- Revision
- arxiv-v2
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2024-09-11
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1.5, PDF page 4 ↗
birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1 - Theorem 1.7, PDF page 4, restricted to base curve 46a1 ↗
birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1
Banwait–Huang · effective BSD twist families ↗
Barinder S. Banwait; Xiaoyu Huang · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
2 cited entries · exact source identity
- Version
- arxiv-2601.16044@arxiv-v3
- Revision
- arxiv-v3
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-06-04
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Lemma 2.18, Algorithms 1–2, Remark 2.20 and Section 2.7, v3 PDF pages 10–13 ↗
birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1 - Section 2.7, v3 PDF page 13 ↗
birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1
Miller · computer-assisted finite BSD cases ↗
Robert L. Miller · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-1010.2431@arxiv-v3
- Revision
- arxiv-v3
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2011-12-21
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1.2 and Table 9, v3 PDF pages 2 and 23 ↗
birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1
Kent · AI-augmented BSD research substrate ↗
Owen P. Kent (repository owner) · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- kent-bsd-project@0e9519aaeb5c8b8c524f95925eecb970d86bc5b1-metadata-2026-10-10
- Revision
- 0e9519aaeb5c8b8c524f95925eecb970d86bc5b1
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-06-14
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Pinned README, PHASE_STATE.md and lean/BSD/RankEquality.lean ↗
birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v2
ICARM · individual high-rank curve record ↗
ICARM leaderboard; submitter ranksunbounded · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- icarm-curve-273@observed-2026-10-10
- Revision
- observed-2026-10-10
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-08-20
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Curve #273: a-invariants, rank (lower bound), witness and update date ↗
birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1
OpenAI · proposed full BSD from low Selmer corank ↗
OpenAI · metadata-link · External full-text redistribution rights not asserted
Only metadata and original OpenMath paraphrases are published; full text stays upstream. Rights / source terms
1 cited entry · exact source identity
- Version
- openai-bsd-low-selmer@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Revision
- fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Content SHA-256
- 64b46603e174462e5bb32d242852eb7f0afe6e519f26534e42fd3f178a16b885
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem 1.1 and normalization, PDF page 3; companion dependencies §2 ↗
birch-and-swinnerton-dyer-conjecture:bsd-openai-selmer:v1
OpenAI · proposed Goldfeld analytic densities and 2-converse ↗
OpenAI · metadata-link · External full-text redistribution rights not asserted
Only metadata and original OpenMath paraphrases are published; full text stays upstream. Rights / source terms
1 cited entry · exact source identity
- Version
- openai-goldfeld@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Revision
- fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Content SHA-256
- c417858c9fe566daacb7738741dc66e211eea148b3da587eb5ee68eaad208d64
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorems 1.1–1.2, PDF page 2; separation of mean-rank assertion, page 3 ↗
birch-and-swinnerton-dyer-conjecture:bsd-openai-goldfeld:v1
Publication history
- Problem ID
- problem-birch-and-swinnerton-dyer-conjecture
- Publication checksum
- 32970d79cf20ed70dfa2252e954a000d392b6b95309919889188a44a5ddeffe1
- Published
- 2026-10-11T03:44:16.623836+00:00
- Publication reason
- Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 1 · Sep 30, 2026
Initial checked pilot publication; source attribution, rights, wording and coverage reviewed by OpenMath staff.
- Version 2 · Oct 10, 2026
Add cited resolution scope and consolidate equivalent prime formulations and supporting Claims; mathematical evidence and execution review remain distinct.
- Version 3 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 4 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 5 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 6 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 7 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 8 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 9 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 10 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 11 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 12 · Oct 12, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 13 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 14 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 15 · Oct 14, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 16 · Oct 15, 2026
Add concise reader labels for the three BSD statements; mathematical statements, assumptions, sources and evidence unchanged.
- Version 17 · Oct 15, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.