Strong BSD implies its rank and finiteness assertions

Statement

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.

Assumptions

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.

Evidence

Source reports

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.

Wiles · official BSD description · PDF page 2, refinement in Remark 1; standard full-L, total-real-period convention →

Observed Oct 10, 2026

Connections

In this problem · publication v10 · Selected context

Birch and Swinnerton-Dyer conjecture →

Conditional implication · Proved

One-way implication

the full BSD formula holds for every E/Q

implies ↓BSD rank equality holds for every E/Q

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 2

For every $E/\mathbb{Q}$ , let $r=\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q})$ . Then $\operatorname{ord}_{s=1}L(E,s)=r$ , $\operatorname{Sha}(E/\mathbb{Q})$ is finite, and $\begin{aligned}\frac{L^{(r)}(E,1)}{r!}&=\Omega_E\operatorname{Reg}_E\\&\quad\cdot\frac{\#\operatorname{Sha}(E/\mathbb{Q})\prod_{\ell\ \mathrm{finite}}c_\ell(E)}{\bigl(\#E(\mathbb{Q})_{\mathrm{tors}}\bigr)^2}\end{aligned}$ .

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.

Wiles · official BSD description →

BSD rank equality holds for every E/Q →

Open · version 2

For every elliptic curve $E$ over $\mathbb{Q}$ , $\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)$ , 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.

Wiles · official BSD description →

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.

Wiles · official BSD description →

Wiles · official BSD description →

Relationship not independently reviewed.

Connection in Birch and Swinnerton-Dyer conjecture →

One-way implication

the full BSD formula holds for every E/Q

implies ↓Sha is finite for every E/Q

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 2

For every $E/\mathbb{Q}$ , let $r=\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q})$ . Then $\operatorname{ord}_{s=1}L(E,s)=r$ , $\operatorname{Sha}(E/\mathbb{Q})$ is finite, and $\begin{aligned}\frac{L^{(r)}(E,1)}{r!}&=\Omega_E\operatorname{Reg}_E\\&\quad\cdot\frac{\#\operatorname{Sha}(E/\mathbb{Q})\prod_{\ell\ \mathrm{finite}}c_\ell(E)}{\bigl(\#E(\mathbb{Q})_{\mathrm{tors}}\bigr)^2}\end{aligned}$ .

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.

Wiles · official BSD description →

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.

Wiles · official BSD description →

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.

Wiles · official BSD description →

Wiles · official BSD description →

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.

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-strong-implies-rank
Statement version: 1
SHA-256: 4986d6f72504fd566a4a3b96f1b8e07ad82538784c68495354b09564ec95b160

Publication records

Birch and Swinnerton-Dyer conjecture · publication v10 · JSON →