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Conjecture · version 2

Sha is finite for every E/Q

Statement

The Tate–Shafarevich group Sha(E/Q) is finite for every elliptic curve E over Q.

Assumptions

  • 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.

Evidence

Source reports

Connections

In this problem · publication v15 · Selected context

Birch and Swinnerton-Dyer conjecture →

Different statement · related variant · Open

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 3

For every E/QE/\mathbb{Q}, let r=rank⁡ZE(Q)r=\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q}). Then ord⁡s=1L(E,s)=r\operatorname{ord}_{s=1}L(E,s)=r, Sha⁡(E/Q)\operatorname{Sha}(E/\mathbb{Q}) is finite, and L(r)(E,1)r!=ΩEReg⁡E⋅#Sha⁡(E/Q)∏ℓ finitecℓ(E)(#E(Q)tors)2\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.

Sha is finite for every E/Q →

Open · version 2

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 2

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 v17

Birch and Swinnerton-Dyer conjecture →

Different statement · related variant · Open

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 3

For every E/QE/\mathbb{Q}, let r=rank⁡ZE(Q)r=\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q}). Then ord⁡s=1L(E,s)=r\operatorname{ord}_{s=1}L(E,s)=r, Sha⁡(E/Q)\operatorname{Sha}(E/\mathbb{Q}) is finite, and L(r)(E,1)r!=ΩEReg⁡E⋅#Sha⁡(E/Q)∏ℓ finitecℓ(E)(#E(Q)tors)2\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.

Sha is finite for every E/Q →

Open · version 2

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 2

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 →

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.

Record details
Claim ID
claim-bsd-sha-finiteness-target
Statement version
2
SHA-256
eae1e525c2b9cb4e9c9580eff30464635a95ea0a928cd49bf2532580ab431da2

Publication records