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 2
Strong BSD would determine Sha’s order from the other invariants
Statement
For an E/Q satisfying strong BSD, #Sha(E/Q) = [L^{(r)}(E,1)/r!] · #E(Q)_tors² / [Ω_E Reg_E ∏_{ℓ finite} c_ℓ(E)], with r = rank E(Q) and the stated full-L and Néron conventions.
Assumptions
- Strong BSD for the particular E/Q, including Sha finiteness.
- Compatible normalizations and certified values of the other inputs are required to recover an exact order.
Scope
An algebraic rearrangement of the conjectural leading-term formula under its premise; approximating an L-value does not certify the integer.
Evidence
Source reports
Connections
In this problem · publication v12 · Selected context
Birch and Swinnerton-Dyer conjecture →
Conditional implication · Proved
Source-grounded deduction from the normalized strong formula, not an independently proved general formula for Sha.
This rearranges the full formula under its premise; numerical approximation alone does not certify the order.
Scope: The leading-term formula would recover Sha’s order from compatible certified invariants.
Limit: The full formula and finiteness remain unproved in general. A decimal comparison alone is insufficient.
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 v17
Birch and Swinnerton-Dyer conjecture →
Conditional implication · Proved
Source-grounded deduction from the normalized strong formula, not an independently proved general formula for Sha.
This rearranges the full formula under its premise; numerical approximation alone does not certify the order.
Scope: The leading-term formula would recover Sha’s order from compatible certified invariants.
Limit: The full formula and finiteness remain unproved in general. A decimal comparison alone is insufficient.
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 page 2, refinement in Remark 1; standard full-L, total-real-period convention → · Original passage ↗
Recorded wording
Strong BSD predicts rank equality, finite Sha, and the leading coefficient formula with real period, regulator, Tamagawa factors and torsion.
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-sha-order-consequence
- Statement version
- 2
- SHA-256
- 0f8797aa89f487ba4bd6ce9b07f4f7dd67a64ae67fd50842540a1607f8475586