Proved
Computer-assisted full formula · 16,714 curves ↗
Miller · computer-assisted finite BSD cases · Theorem 1.2 and Table 9, v3 PDF pages 2 and 23 →
Source dated Dec 21, 2011 · Observed Oct 10, 2026
Theorem · version 1
Computer-assisted full formula · 16,714 curves
Earlier version 1. View current version 2 →
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.
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.
Evidence
Source reports
Connections
In this problem · publication v5 · Selected context
Birch and Swinnerton-Dyer conjecture →
Special case · Proved
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.
Scope: 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.
Related results · 2
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v10
Birch and Swinnerton-Dyer conjecture →
Special case · Proved
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.
Scope: 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.
Related results · 2
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Miller · computer-assisted finite BSD cases →
- Theorem 1.2 and Table 9, v3 PDF pages 2 and 23 → · Original passage ↗
Recorded 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.
paraphrase · No upstream full text or code is republished by this record.
Robert L. Miller · Published
LMS Journal of Computation and Mathematics 14 (2011), 327–350. This is the scope of that historical paper, not a current unsolved-case list.
Revision & attribution
Source revision: arxiv-v3. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Dec 21, 2011.
Wiles · official BSD description →
- PDF pages 2 and 4, conjecture/refinement and low-rank theorem → · Original passage ↗
Recorded wording
Wiles distinguishes the rank statement from the refined leading-coefficient formula and records the analytic-rank-zero-or-one theorem.
paraphrase · No upstream full text or code is republished by this record.
Andrew Wiles · Published
Official Clay problem description; separates rank equality from its refined formula.
Revision & attribution
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Skinner · converse to Gross–Zagier–Kolyvagin →
- Introduction, journal page 330 (PDF page 2), classical theorem → · Original passage ↗
Recorded 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.
paraphrase · No upstream full text or code is republished by this record.
Christopher Skinner · Published
Annals of Mathematics 191 (2020), 329–354; published online 13 February 2020. Introduction also states the classical low-analytic-rank theorem.
Revision & attribution
Source revision: published-online-2020-02-13. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Feb 13, 2020.
Jetchev–Skinner–Wan · rank-one BSD p-part →
- Theorem 1.2.1, PDF pages 1–2 → · Original passage ↗
Recorded 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.
paraphrase · No upstream full text or code is republished by this record.
Dimitar Jetchev · Christopher Skinner · Xin Wan · Published
Published in Cambridge Journal of Mathematics 5 (2017), 369–434; the cited author preprint supplies Theorem 1.2.1.
Revision & attribution
Source revision: arxiv-v1. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Dec 21, 2015.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Computer-assisted full formula · 16,714 curves ↗
Source dated Dec 21, 2011 · Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-bsd-miller-finite-computations
- Statement version
- 1
- SHA-256
- 83f94df37be3e28db51e6f9c8954ef9e5efb32375145994e2b64b95fb2df5c9e