Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Theorem · version 1
Rank BSD would decide finite versus infinite rational points analytically
Earlier version 1. View current version 2 →
Statement
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.
Assumptions
- BSD rank equality for the particular E/Q.
- Known analytic continuation and Mordell–Weil finite generation.
Scope
A proved implication under the rank-equality premise, not an unconditional arbitrary-curve decision algorithm.
Evidence
Source reports
Connections
In this problem · publication v6 · Selected context
Birch and Swinnerton-Dyer conjecture →
Conditional implication · Proved
Scope: Under rank equality, the central L-value detects whether E(Q) is finite or infinite.
Limit: Rank equality for arbitrary E/Q remains open; numerical near-zero values are not exact certificates.
A consequence under an open premise, not an additional resolved part of that premise.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v10
Birch and Swinnerton-Dyer conjecture →
Conditional implication · Proved
Scope: Under rank equality, the central L-value detects whether E(Q) is finite or infinite.
Limit: Rank equality for arbitrary E/Q remains open; numerical near-zero values are not exact certificates.
A consequence under an open premise, not an additional resolved part of that premise.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Wiles · official BSD description →
- PDF pages 1–2: Mordell theorem and “In particular” consequence of rank BSD → · Original passage ↗
Recorded wording
Under BSD rank equality, L(E,1) = 0 exactly when E(Q) is infinite.
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.
Official description PDF ↗Manuscript / exposition · external
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.
Official description PDF: official-description. External evidence linked; no OpenMath execution or independent reproduction is asserted.
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
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-bsd-finite-points-consequence
- Statement version
- 1
- SHA-256
- d82661d6c78a5bfa9db99e1dc9aae1924ba94fd5e3952c6dcab7a4faf0038c3e