Open
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
Conjecture · version 1
the full BSD formula holds for every E/Q
Earlier version 1. View current version 3 →
Statement
For every E/Q, let r = rank_Z E(Q). Then ord_{s=1}L(E,s) = r, Sha(E/Q) is finite, and L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors².
Assumptions
- Full Hasse–Weil L-function; Ω_E = ∫_{E(R)} |ω_E| for a global minimal Néron differential, including both real components when present.
- Reg_E is the Néron–Tate regulator of a basis of E(Q)/torsion, with Reg_E = 1 at rank zero.
- c_ℓ are finite-prime Tamagawa numbers; E(Q)_tors is rational torsion.
Scope
A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.
Evidence
Source reports
Connections
In this problem · publication v7 · Selected context
Birch and Swinnerton-Dyer conjecture →
Different statement · related variant · Open
One-way implication
The strong assertion includes the rank statement and additional arithmetic information.
Proving rank equality alone does not prove finite Sha or the leading coefficient.
Details & sources
the full BSD formula holds for every E/Q →
Open · version 1
For every E/Q, let r = rank_Z E(Q). Then ord_{s=1}L(E,s) = r, Sha(E/Q) is finite, and L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors².
- Full Hasse–Weil L-function; Ω_E = ∫_{E(R)} |ω_E| for a global minimal Néron differential, including both real components when present.
- Reg_E is the Néron–Tate regulator of a basis of E(Q)/torsion, with Reg_E = 1 at rank zero.
- c_ℓ are finite-prime Tamagawa numbers; E(Q)_tors is rational torsion.
Scope: A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.
BSD rank equality holds for every E/Q →
Open · version 1
For every elliptic curve E over Q, rank_Z E(Q) = ord_{s=1} L(E,s), where L is the full Hasse–Weil L-function with all finite Euler factors.
- Elliptic curve over Q, with its rational identity point.
- The full Hasse–Weil L-function, whose analytic continuation is known by modularity.
Scope: The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.
Strong BSD implies its rank and finiteness assertions →
Relationship statement · Proved · version 1
If the full BSD assertion holds for every E/Q, then BSD rank equality holds for every E/Q and Sha(E/Q) is finite for every E/Q.
- Strong BSD includes rank equality, Sha finiteness and the normalized leading-term formula.
Scope: A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →One-way implication
The full BSD scenario explicitly includes finiteness of the whole Tate–Shafarevich group.
Finiteness is stronger than a finite p-primary check for one prime and does not follow from rank equality alone.
Details & sources
the full BSD formula holds for every E/Q →
Open · version 1
For every E/Q, let r = rank_Z E(Q). Then ord_{s=1}L(E,s) = r, Sha(E/Q) is finite, and L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors².
- Full Hasse–Weil L-function; Ω_E = ∫_{E(R)} |ω_E| for a global minimal Néron differential, including both real components when present.
- Reg_E is the Néron–Tate regulator of a basis of E(Q)/torsion, with Reg_E = 1 at rank zero.
- c_ℓ are finite-prime Tamagawa numbers; E(Q)_tors is rational torsion.
Scope: A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.
Sha is finite for every E/Q →
Open · version 1
The Tate–Shafarevich group Sha(E/Q) is finite for every elliptic curve E over Q.
- Sha parametrizes everywhere locally soluble principal homogeneous spaces up to equivalence.
Scope: A separately open finiteness assertion, proved in selected low-analytic-rank cases. It is included in the strong BSD scenario.
Strong BSD implies its rank and finiteness assertions →
Relationship statement · Proved · version 1
If the full BSD assertion holds for every E/Q, then BSD rank equality holds for every E/Q and Sha(E/Q) is finite for every E/Q.
- Strong BSD includes rank equality, Sha finiteness and the normalized leading-term formula.
Scope: A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v9
Birch and Swinnerton-Dyer conjecture →
Different statement · related variant · Open
One-way implication
The strong assertion includes the rank statement and additional arithmetic information.
Proving rank equality alone does not prove finite Sha or the leading coefficient.
Details & sources
the full BSD formula holds for every E/Q →
Open · version 1
For every E/Q, let r = rank_Z E(Q). Then ord_{s=1}L(E,s) = r, Sha(E/Q) is finite, and L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors².
- Full Hasse–Weil L-function; Ω_E = ∫_{E(R)} |ω_E| for a global minimal Néron differential, including both real components when present.
- Reg_E is the Néron–Tate regulator of a basis of E(Q)/torsion, with Reg_E = 1 at rank zero.
- c_ℓ are finite-prime Tamagawa numbers; E(Q)_tors is rational torsion.
Scope: A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.
BSD rank equality holds for every E/Q →
Open · version 1
For every elliptic curve E over Q, rank_Z E(Q) = ord_{s=1} L(E,s), where L is the full Hasse–Weil L-function with all finite Euler factors.
- Elliptic curve over Q, with its rational identity point.
- The full Hasse–Weil L-function, whose analytic continuation is known by modularity.
Scope: The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.
Strong BSD implies its rank and finiteness assertions →
Relationship statement · Proved · version 1
If the full BSD assertion holds for every E/Q, then BSD rank equality holds for every E/Q and Sha(E/Q) is finite for every E/Q.
- Strong BSD includes rank equality, Sha finiteness and the normalized leading-term formula.
Scope: A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →One-way implication
The full BSD scenario explicitly includes finiteness of the whole Tate–Shafarevich group.
Finiteness is stronger than a finite p-primary check for one prime and does not follow from rank equality alone.
Details & sources
the full BSD formula holds for every E/Q →
Open · version 1
For every E/Q, let r = rank_Z E(Q). Then ord_{s=1}L(E,s) = r, Sha(E/Q) is finite, and L^{(r)}(E,1)/r! = Ω_E Reg_E · #Sha(E/Q) · ∏_{ℓ finite} c_ℓ(E) / #E(Q)_tors².
- Full Hasse–Weil L-function; Ω_E = ∫_{E(R)} |ω_E| for a global minimal Néron differential, including both real components when present.
- Reg_E is the Néron–Tate regulator of a basis of E(Q)/torsion, with Reg_E = 1 at rank zero.
- c_ℓ are finite-prime Tamagawa numbers; E(Q)_tors is rational torsion.
Scope: A stronger open assertion. Normalizations must be consistent; rank equality and one p-part do not imply the full formula.
Sha is finite for every E/Q →
Open · version 1
The Tate–Shafarevich group Sha(E/Q) is finite for every elliptic curve E over Q.
- Sha parametrizes everywhere locally soluble principal homogeneous spaces up to equivalence.
Scope: A separately open finiteness assertion, proved in selected low-analytic-rank cases. It is included in the strong BSD scenario.
Strong BSD implies its rank and finiteness assertions →
Relationship statement · Proved · version 1
If the full BSD assertion holds for every E/Q, then BSD rank equality holds for every E/Q and Sha(E/Q) is finite for every E/Q.
- Strong BSD includes rank equality, Sha finiteness and the normalized leading-term formula.
Scope: A one-way implication by the definition of the stronger conjecture; no converse from rank equality is asserted.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →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.
- PDF page 2: rank conjecture; all finite Euler factors restored for the standard L-function → · Original passage ↗
Recorded wording
For every elliptic curve E/Q, rank E(Q) equals ord_{s=1} L(E,s).
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.
Open
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-full-target
- Statement version
- 1
- SHA-256
- d58b619aa3d7a33f2577a022716b77a493bc868ca8c479c7b65f481223be52f8