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Conjecture · version 3

BSD rank equality holds for every E/Q

Statement

For every elliptic curve EE over Q\mathbb{Q}, rank⁡ZE(Q)=ord⁡s=1L(E,s)\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.

Assumptions

  • 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.

Evidence

Source reports

Connections

In this problem · publication v13 · Selected context

Birch and Swinnerton-Dyer conjecture →

Primary formulation · 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 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.

BSD rank equality holds for every E/Q →

Open · version 3

For every elliptic curve EE over Q\mathbb{Q}, rank⁡ZE(Q)=ord⁡s=1L(E,s)\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.

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 →

Related statements

Rank BSD specialized to E_n would supply the missing sufficient direction of Tunnell’s exact counting criterion.

This implication is restricted to the stated odd squarefree congruent-number curves.

Details & sources
BSD rank equality holds for every E/Q →

Open · version 3

For every elliptic curve EE over Q\mathbb{Q}, rank⁡ZE(Q)=ord⁡s=1L(E,s)\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.

BSD would complete the odd squarefree congruent-number criterion →

Proved · version 2

For every odd positive squarefree integer n, assuming BSD rank equality for E_n: y² = x³−n²x, n is the area of a right triangle with positive rational side lengths if and only if #{(x,y,z)∈Z³: 2x²+y²+8z²=n} = 2·#{(x,y,z)∈Z³: 2x²+y²+32z²=n}.

  • n is positive, odd and squarefree.
  • BSD rank equality for the congruent-number elliptic curve E_n.

Scope: A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.

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 →

Primary formulation · 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 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.

BSD rank equality holds for every E/Q →

Open · version 3

For every elliptic curve EE over Q\mathbb{Q}, rank⁡ZE(Q)=ord⁡s=1L(E,s)\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.

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 →

Related statements

Rank BSD specialized to E_n would supply the missing sufficient direction of Tunnell’s exact counting criterion.

This implication is restricted to the stated odd squarefree congruent-number curves.

Details & sources
BSD rank equality holds for every E/Q →

Open · version 3

For every elliptic curve EE over Q\mathbb{Q}, rank⁡ZE(Q)=ord⁡s=1L(E,s)\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.

BSD would complete the odd squarefree congruent-number criterion →

Proved · version 2

For every odd positive squarefree integer n, assuming BSD rank equality for E_n: y² = x³−n²x, n is the area of a right triangle with positive rational side lengths if and only if #{(x,y,z)∈Z³: 2x²+y²+8z²=n} = 2·#{(x,y,z)∈Z³: 2x²+y²+32z²=n}.

  • n is positive, odd and squarefree.
  • BSD rank equality for the congruent-number elliptic curve E_n.

Scope: A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.

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-rank-target
Statement version
3
SHA-256
d67e9eff19fa042fe4524d50a6f54626deb7ae8bf354857d141401a2a6083dd9

Publication records