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External artifacts and OpenMath execution remain distinct.","withdrawn":false,"draft":{"progress":[{"claim_id":"claim-bsd-analytic-rank-at-most-one","aspect":"Rank equality & finite Sha","summary":"Every E/Q of analytic rank 0 or 1.","remaining":"No unrestricted analytic-rank ≥ 2 theorem follows.","overlaps_with":[],"overlap_note":"The hypothesis is analytic rank ≤ 1, not merely algebraic rank ≤ 1. This does not alone prove the full leading-coefficient formula."},{"claim_id":"claim-bsd-rank-one-converse-finite-sha","aspect":"Conditional rank converse","summary":"Algebraic rank 1 ⇒ analytic rank 1 in this family if Sha is finite.","remaining":"Finite Sha and the reduction hypotheses must be established for an application.","overlaps_with":["claim-bsd-analytic-rank-at-most-one"],"overlap_note":"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."},{"claim_id":"claim-bsd-semistable-rank-one-p-part","aspect":"Leading coefficient · p-part","summary":"Good-reduction p ≥ 5 with irreducible E[p], for semistable analytic-rank-one curves.","remaining":"The missing primes and unrestricted curves require further results.","overlaps_with":["claim-bsd-analytic-rank-at-most-one"],"overlap_note":"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."},{"claim_id":"claim-bsd-supersingular-p-part","aspect":"Leading coefficient · p-part","summary":"Semistable analytic-rank 0 or 1 at supersingular p ≥ 5.","remaining":"Other primes and higher analytic ranks are not covered by this selected theorem.","overlaps_with":["claim-bsd-analytic-rank-at-most-one","claim-bsd-semistable-rank-one-p-part"],"overlap_note":"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."},{"claim_id":"claim-bsd-46a1-quadratic-twists","aspect":"Leading coefficient · full formula","summary":"An infinite, explicitly conditioned family of twists of 46a1.","remaining":"Does not give full BSD for every twist, every rank-zero curve, or analytic rank ≥ 2.","overlaps_with":["claim-bsd-analytic-rank-at-most-one","claim-bsd-supersingular-p-part"],"overlap_note":"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."},{"claim_id":"claim-bsd-miller-finite-computations","aspect":"Leading coefficient · finite corpus","summary":"Full formula certified in the paper for 16,714 enumerated curves.","remaining":"The proof does not cover all curves, all conductor ranges or higher analytic ranks.","overlaps_with":["claim-bsd-analytic-rank-at-most-one","claim-bsd-semistable-rank-one-p-part"],"overlap_note":"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."}],"activities":[{"id":"bsd-effective-families-2026","title":"2026 · effective families and an earlier-list caveat","kind":"source_correction","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1"],"summary":"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.","limitation":"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.","related_claim_ids":["claim-bsd-46a1-quadratic-twists"]},{"id":"bsd-ai-project-kent","title":"AI-assisted exploration · code and Lean targets","kind":"formalization","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v1"],"summary":"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.","limitation":"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_claim_ids":["claim-bsd-analytic-rank-at-most-one"]},{"id":"bsd-high-rank-witness-2026","title":"2026 · reported rank lower bound 30","kind":"computation","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1"],"summary":"The primary record supplies a specific elliptic curve over Q and 30 rational-point witnesses, with a reported rank lower bound of 30.","limitation":"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.","related_claim_ids":[]}],"curation":{"checked_on":"2026-10-10","selection":"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.","search_scope":"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.","formal_status":"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_status":"One self-described AI-augmented individual BSD program is indexed as exploration. High-rank computation is related context, not a BSD proof. No accepted AI-company proof of the general BSD statement was established in this source survey.","independent_review":"pending"},"schema_version":"research-record-v1","claims":[{"statement":"For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/Q) is finite.","id":"claim-bsd-analytic-rank-at-most-one","slug":"bsd-analytic-rank-at-most-one","version":1,"kind":"theorem","title":"Rank equality at analytic rank 0 or 1","assumptions":["E is an elliptic curve over Q.","The full Hasse–Weil L-function has order of vanishing 0 or 1 at s=1."],"scope":"Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.","relation":"special_case","relationship_note":"The hypothesis is analytic rank ≤ 1, not merely algebraic rank ≤ 1. This does not alone prove the full leading-coefficient formula.","result":"proved","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1","birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1"],"report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.","id":"claim-bsd-rank-one-converse-finite-sha","slug":"bsd-rank-one-converse-finite-sha","version":1,"kind":"theorem","title":"Rank-one converse assuming finite Sha","assumptions":["E/Q is semistable.","E has nonsplit multiplicative reduction at at least one odd prime, or split multiplicative reduction at at least two odd primes.","Algebraic rank E(Q) = 1.","Assume Sha(E/Q) is finite."],"scope":"Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.","relation":"conditional_result","relationship_note":"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.","result":"proved","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1"],"report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.","id":"claim-bsd-semistable-rank-one-p-part","slug":"bsd-semistable-rank-one-p-part","version":1,"kind":"theorem","title":"Rank-one formula · selected p-parts","assumptions":["E/Q is semistable (squarefree conductor N).","ord_(s=1)L(E,s) = 1.","p ≥ 5 is prime and p does not divide N.","The mod-p Galois representation on E[p] is irreducible."],"scope":"The p-part of the leading-coefficient formula, not an assertion at every prime. Irreducibility removes p-primary rational torsion.","relation":"special_case","relationship_note":"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.","result":"proved","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1"],"report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"For semistable E/Q of analytic rank 0 or 1 and a good supersingular prime p ≥ 5, the p-part of the BSD leading-coefficient formula holds.","id":"claim-bsd-supersingular-p-part","slug":"bsd-supersingular-p-part","version":1,"kind":"theorem","title":"Low-rank formula at supersingular p ≥ 5","assumptions":["E/Q is semistable.","ord_(s=1)L(E,s) is 0 or 1.","p ≥ 5 is a prime of good supersingular reduction."],"scope":"Theorem 1.5 restricted to p ≥ 5; its additional p=3 conditions and quadratic-twist extension are not asserted here.","relation":"special_case","relationship_note":"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.","result":"proved","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1"],"report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"Let E be Cremona curve 46a1. For squarefree M>1 coprime to 46, if L(E^M,1)≠0 and E has ordinary reduction at every prime dividing M, E^M satisfies full BSD. Infinitely many M meet these conditions.","id":"claim-bsd-46a1-quadratic-twists","slug":"bsd-46a1-quadratic-twists","version":1,"kind":"theorem","title":"Full BSD in a specified infinite twist family","assumptions":["E/Q is the specific Cremona curve 46a1.","M>1 is squarefree and gcd(M,46)=1; E^M is the twist by Q(√M).","L(E^M,1) is nonzero.","E has ordinary reduction at every prime dividing M."],"scope":"Analytic-rank-zero quadratic twists of one named non-CM base curve; rank, finite Sha and the entire leading-coefficient identity.","relation":"special_case","relationship_note":"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.","result":"proved","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1"],"report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1:report"],"formal_declarations":[],"independent_review":"pending"},{"statement":"Miller’s computation-assisted proof establishes full BSD for 16,714 of the 16,725 elliptic curves of analytic rank at most one and conductor <5000 in the paper’s enumeration.","id":"claim-bsd-miller-finite-computations","slug":"bsd-miller-finite-computations","version":1,"kind":"theorem","title":"Computer-assisted full formula · 16,714 curves","assumptions":["E/Q has conductor <5000 and analytic rank 0 or 1.","E belongs to the successfully certified subset in the cited enumeration, excluding the 11 unresolved curve/prime pairs in Table 9.","Computer-assisted calculations are part of the cited proof; OpenMath has not repeated them."],"scope":"Historical finite-corpus proof. Table 9 exceptions are limitations of this paper, not a statement of current unresolved status.","relation":"special_case","relationship_note":"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.","result":"proved","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1"],"report_ids":["birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1:report"],"formal_declarations":[],"independent_review":"pending"}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"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.","report_ids":["birch-and-swinnerton-dyer-conjecture:report:0:v2"],"progress_claim_ids":["claim-bsd-analytic-rank-at-most-one","claim-bsd-rank-one-converse-finite-sha","claim-bsd-semistable-rank-one-p-part","claim-bsd-supersingular-p-part","claim-bsd-46a1-quadratic-twists","claim-bsd-miller-finite-computations"],"remaining":"Rank equality in general, especially analytic rank ≥ 2, remains open. General finiteness of Sha and the complete leading-coefficient formula are stronger unresolved assertions."},"statement_kind":"conjecture","problem_id":"problem-birch-and-swinnerton-dyer-conjecture","slug":"birch-and-swinnerton-dyer-conjecture","title":"Birch and Swinnerton-Dyer conjecture","question":"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?","scope":"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.","aliases":["Birch and Swinnerton-Dyer conjecture"],"fields":["Number theory","Algebraic geometry"],"msc":["11G05","11G40"],"topics":["Scoped progress","Conditional results"],"background":"specialist","scale":"major conjecture","depth":"source-linked","source_entry_ids":["birch-and-swinnerton-dyer-conjecture:entry:0:v1","birch-and-swinnerton-dyer-conjecture:entry:0:v2","birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1","birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1","birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1","birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1","birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1","birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v1","birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1"],"status_report_ids":["birch-and-swinnerton-dyer-conjecture:report:0:v1","birch-and-swinnerton-dyer-conjecture:report:0:v2"],"editorial":null,"literature":[{"title":"Clay problem page and official description","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","note":"Clay reports the general question unsolved, observed 10 October 2026. 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Banwait","Xiaoyu Huang"],"kind":"research_paper","publication":"accepted","publication_note":"arXiv metadata reports acceptance for ANTS XVII. Algorithms enumerate sufficient subfamilies, not all curves or twists satisfying BSD.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Paper PDF · v3","url":"https://arxiv.org/pdf/2601.16044v3","version":"arXiv:2601.16044v3","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"code","title":"Sage algorithms · pinned commit","url":"https://github.com/cocoxhuang/ants_xvii/tree/31fae20c8df3f1f0383f41112b914d4995d5809d/ants_xvii/infinite_bsd","version":"31fae20c8df3f1f0383f41112b914d4995d5809d","assessment":"not_executed","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"dataset","title":"Eligible curve labels · pinned file","url":"https://github.com/cocoxhuang/ants_xvii/blob/31fae20c8df3f1f0383f41112b914d4995d5809d/ants_xvii/infinite_bsd/output/ec_labels_500k.txt","version":"31fae20c8df3f1f0383f41112b914d4995d5809d","assessment":"not_executed","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-2601.16044@arxiv-v3","source_id":"arxiv-2601.16044","title":"Banwait–Huang · effective BSD twist families","owner":"Barinder S. Banwait; Xiaoyu Huang","url":"https://arxiv.org/abs/2601.16044v3","subject_scope":"Banwait–Huang · effective BSD twist families","revision":"arxiv-v3","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-06-04","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2601.16044v3","attribution":"Barinder S. Banwait; Xiaoyu Huang","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Robert L. Miller"],"kind":"research_paper","publication":"published","publication_note":"LMS Journal of Computation and Mathematics 14 (2011), 327–350. This is the scope of that historical paper, not a current unsolved-case list.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Computer-assisted proof PDF · v3","url":"https://arxiv.org/pdf/1010.2431v3","version":"arXiv:1010.2431v3","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-1010.2431@arxiv-v3","source_id":"arxiv-1010.2431","title":"Miller · computer-assisted finite BSD cases","owner":"Robert L. Miller","url":"https://arxiv.org/abs/1010.2431v3","subject_scope":"Miller · computer-assisted finite BSD cases","revision":"arxiv-v3","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2011-12-21","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/1010.2431v3","attribution":"Robert L. Miller","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Owen P. 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A precise model version and independently assessed division of labor are not supplied.","artifacts":[{"kind":"project_notes","title":"Project phase and limitations","url":"https://github.com/owenpkent/birch-swinnerton-dyer/blob/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1/PHASE_STATE.md","version":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","assessment":"external_report","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"code","title":"Computational experiments","url":"https://github.com/owenpkent/birch-swinnerton-dyer/tree/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1/experiments","version":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","assessment":"not_executed","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."},{"kind":"formal_code","title":"Lean rank-equality targets","url":"https://github.com/owenpkent/birch-swinnerton-dyer/blob/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1/lean/BSD/RankEquality.lean","version":"0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","assessment":"placeholder","note":"The inspected weakBSD and low-rank theorem bodies contain sorry. This is a target skeleton, not a completed formal proof."}]},"id":"kent-bsd-project@0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","source_id":"kent-bsd-project","title":"Kent · AI-augmented BSD research substrate","owner":"Owen P. 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recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-08-20","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://elliptic-rank.icarm.cloud/curve/273","attribution":"ICARM leaderboard; submitter ranksunbounded","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90}],"entries":[{"id":"birch-and-swinnerton-dyer-conjecture:entry:0:v1","source_version_id":"clay-millennium@observed-2026-10-01","item_id":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","wording":"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?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:entry:0:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","wording":"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?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":"birch-and-swinnerton-dyer-conjecture:entry:0:v1"},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1","source_version_id":"skinner-annals-2020@annals-191-2-2020","item_id":"Introduction, journal page 330 (PDF page 2), classical theorem","url":"https://doi.org/10.4007/annals.2020.191.2.1","wording":"For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/Q) is finite.","wording_kind":"paraphrase","locator":"Introduction, journal page 330 (PDF page 2), classical theorem","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-official-rank-description:v1","source_version_id":"clay-wiles-bsd@official-description","item_id":"PDF pages 2 and 4, conjecture/refinement and low-rank theorem","url":"https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf","wording":"Wiles distinguishes the rank statement from the refined leading-coefficient formula and records the analytic-rank-zero-or-one theorem.","wording_kind":"paraphrase","locator":"PDF pages 2 and 4, conjecture/refinement and low-rank theorem","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1","source_version_id":"skinner-annals-2020@annals-191-2-2020","item_id":"Abstract and Theorem A, journal pages 329–331","url":"https://doi.org/10.4007/annals.2020.191.2.1","wording":"For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.","wording_kind":"paraphrase","locator":"Abstract and Theorem A, journal pages 329–331","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","source_version_id":"arxiv-1512.06894@arxiv-v1","item_id":"Theorem 1.2.1, PDF pages 1–2","url":"https://arxiv.org/abs/1512.06894v1","wording":"For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.","wording_kind":"paraphrase","locator":"Theorem 1.2.1, PDF pages 1–2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","source_version_id":"arxiv-2409.01350@arxiv-v2","item_id":"Theorem 1.5, PDF page 4","url":"https://arxiv.org/abs/2409.01350v2","wording":"For semistable E/Q of analytic rank 0 or 1 and a good supersingular prime p ≥ 5, the p-part of the BSD leading-coefficient formula holds.","wording_kind":"paraphrase","locator":"Theorem 1.5, PDF page 4","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","source_version_id":"arxiv-2409.01350@arxiv-v2","item_id":"Theorem 1.7, PDF page 4, restricted to base curve 46a1","url":"https://arxiv.org/abs/2409.01350v2","wording":"Let E be Cremona curve 46a1. For squarefree M>1 coprime to 46, if L(E^M,1)≠0 and E has ordinary reduction at every prime dividing M, E^M satisfies full BSD. Infinitely many M meet these conditions.","wording_kind":"paraphrase","locator":"Theorem 1.7, PDF page 4, restricted to base curve 46a1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-effective-families-2026:v1","source_version_id":"arxiv-2601.16044@arxiv-v3","item_id":"Lemma 2.18, Algorithms 1–2, Remark 2.20 and Section 2.7, v3 PDF pages 10–13","url":"https://arxiv.org/abs/2601.16044v3","wording":"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.","wording_kind":"paraphrase","locator":"Lemma 2.18, Algorithms 1–2, Remark 2.20 and Section 2.7, v3 PDF pages 10–13","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-corroboration:v1","source_version_id":"arxiv-2601.16044@arxiv-v3","item_id":"Section 2.7, v3 PDF page 13","url":"https://arxiv.org/abs/2601.16044v3","wording":"The 2026 eligible-family list retains Cremona curve 46a1; the discrepancy concerns four other base curves.","wording_kind":"paraphrase","locator":"Section 2.7, v3 PDF page 13","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","source_version_id":"arxiv-1010.2431@arxiv-v3","item_id":"Theorem 1.2 and Table 9, v3 PDF pages 2 and 23","url":"https://arxiv.org/abs/1010.2431v3","wording":"Miller’s computation-assisted proof establishes full BSD for 16,714 of the 16,725 elliptic curves of analytic rank at most one and conductor <5000 in the paper’s enumeration.","wording_kind":"paraphrase","locator":"Theorem 1.2 and Table 9, v3 PDF pages 2 and 23","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-ai-project-kent:v1","source_version_id":"kent-bsd-project@0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","item_id":"Pinned README, PHASE_STATE.md and lean/BSD/RankEquality.lean","url":"https://github.com/owenpkent/birch-swinnerton-dyer/tree/0e9519aaeb5c8b8c524f95925eecb970d86bc5b1","wording":"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.","wording_kind":"paraphrase","locator":"Pinned README, PHASE_STATE.md and lean/BSD/RankEquality.lean","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-high-rank-witness-2026:v1","source_version_id":"icarm-curve-273@observed-2026-10-10","item_id":"Curve #273: a-invariants, rank (lower bound), witness and update date","url":"https://elliptic-rank.icarm.cloud/curve/273","wording":"The primary record supplies a specific elliptic curve over Q and 30 rational-point witnesses, with a reported rank lower bound of 30.","wording_kind":"paraphrase","locator":"Curve #273: a-invariants, rank (lower bound), witness and update date","supersedes":null}],"reports":[{"id":"birch-and-swinnerton-dyer-conjecture:report:0:v1","entry_id":"birch-and-swinnerton-dyer-conjecture:entry:0:v1","state":"reported_open","source_label":"Unsolved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-01","evidence_url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:report:0:v2","entry_id":"birch-and-swinnerton-dyer-conjecture:entry:0:v2","state":"reported_open","source_label":"Unsolved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-10","evidence_url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","supersedes":"birch-and-swinnerton-dyer-conjecture:report:0:v1"},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-analytic-rank-at-most-one:v1","state":"resolved","source_label":"Rank equality at analytic rank 0 or 1","scope":"Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.","source_date":"2020-03-01","observed_on":"2026-10-10","evidence_url":"https://doi.org/10.4007/annals.2020.191.2.1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-rank-one-converse-finite-sha:v1","state":"resolved","source_label":"Rank-one converse assuming finite Sha","scope":"Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.","source_date":"2020-03-01","observed_on":"2026-10-10","evidence_url":"https://doi.org/10.4007/annals.2020.191.2.1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-semistable-rank-one-p-part:v1","state":"resolved","source_label":"Rank-one formula · selected p-parts","scope":"The p-part of the leading-coefficient formula, not an assertion at every prime. Irreducibility removes p-primary rational torsion.","source_date":"2015-12-21","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/1512.06894v1","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-supersingular-p-part:v1","state":"resolved","source_label":"Low-rank formula at supersingular p ≥ 5","scope":"Theorem 1.5 restricted to p ≥ 5; its additional p=3 conditions and quadratic-twist extension are not asserted here.","source_date":"2024-09-11","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2409.01350v2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-46a1-quadratic-twists:v1","state":"resolved","source_label":"Full BSD in a specified infinite twist family","scope":"Analytic-rank-zero quadratic twists of one named non-CM base curve; rank, finite Sha and the entire leading-coefficient identity.","source_date":"2024-09-11","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/2409.01350v2","supersedes":null},{"id":"birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1:report","entry_id":"birch-and-swinnerton-dyer-conjecture:bsd-miller-finite-computations:v1","state":"resolved","source_label":"Computer-assisted full formula · 16,714 curves","scope":"Historical finite-corpus proof. Table 9 exceptions are limitations of this paper, not a statement of current unresolved status.","source_date":"2011-12-21","observed_on":"2026-10-10","evidence_url":"https://arxiv.org/abs/1010.2431v3","supersedes":null}]},"status":{"state":"open","label":"Open","extent":"general_case","summary":"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.","basis":"source","attributed_to":"Clay · Millennium Problems","as_of":"2026-10-10","progress_label":"Special cases & conditional results","progress_claim_ids":["claim-bsd-analytic-rank-at-most-one","claim-bsd-rank-one-converse-finite-sha","claim-bsd-semistable-rank-one-p-part","claim-bsd-supersingular-p-part","claim-bsd-46a1-quadratic-twists","claim-bsd-miller-finite-computations"],"remaining":"Rank equality in general, especially analytic rank ≥ 2, remains open. General finiteness of Sha and the complete leading-coefficient formula are stronger unresolved assertions.","views":["open","partial"]}},"selected_context":null}}