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Theorem · version 4

Rank equality at analytic rank 0 or 1

Statement

For E/QE/\mathbb{Q} with ord⁡s=1L(E,s)\operatorname{ord}_{s=1}L(E,s) equal to 0 or 1, rank⁡ZE(Q)\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q}) equals that analytic rank and Sha⁡(E/Q)\operatorname{Sha}(E/\mathbb{Q}) is finite.

Assumptions

  • E is an elliptic curve over Q.
  • The full Hasse–Weil L-function has order of vanishing 0 or 1 at s=1.

Scope

Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.

Evidence

Connections

In this problem · publication v14 · Selected context

Birch and Swinnerton-Dyer conjecture →

Special case · Proved

The hypothesis is analytic rank ≤ 1, not merely algebraic rank ≤ 1. This does not alone prove the full leading-coefficient formula.

Scope: Every E/Q of analytic rank 0 or 1.

Limit: No unrestricted analytic-rank ≥ 2 theorem follows.

Related statements

The first starts from analytic rank ≤1; Skinner’s converse starts from algebraic rank one, finite Sha and specified semistable reduction.

Do not reverse the first implication without the additional converse hypotheses.

Details & sources
Rank equality at analytic rank 0 or 1 →

Proved · version 4

For E/QE/\mathbb{Q} with ord⁡s=1L(E,s)\operatorname{ord}_{s=1}L(E,s) equal to 0 or 1, rank⁡ZE(Q)\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q}) equals that analytic rank and Sha⁡(E/Q)\operatorname{Sha}(E/\mathbb{Q}) is finite.

  • E is an elliptic curve over Q.
  • The full Hasse–Weil L-function has order of vanishing 0 or 1 at s=1.

Scope: Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.

Rank-one converse assuming finite Sha →

Proved · version 3

For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.

  • E/Q is semistable.
  • E has nonsplit multiplicative reduction at at least one odd prime, or split multiplicative reduction at at least two odd primes.
  • Algebraic rank E(Q) = 1.
  • Assume Sha(E/Q) is finite.

Scope: Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.

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 →

Special case · Proved

The hypothesis is analytic rank ≤ 1, not merely algebraic rank ≤ 1. This does not alone prove the full leading-coefficient formula.

Scope: Every E/Q of analytic rank 0 or 1.

Limit: No unrestricted analytic-rank ≥ 2 theorem follows.

Related statements

The first starts from analytic rank ≤1; Skinner’s converse starts from algebraic rank one, finite Sha and specified semistable reduction.

Do not reverse the first implication without the additional converse hypotheses.

Details & sources
Rank equality at analytic rank 0 or 1 →

Proved · version 4

For E/QE/\mathbb{Q} with ord⁡s=1L(E,s)\operatorname{ord}_{s=1}L(E,s) equal to 0 or 1, rank⁡ZE(Q)\operatorname{rank}_{\mathbb{Z}}E(\mathbb{Q}) equals that analytic rank and Sha⁡(E/Q)\operatorname{Sha}(E/\mathbb{Q}) is finite.

  • E is an elliptic curve over Q.
  • The full Hasse–Weil L-function has order of vanishing 0 or 1 at s=1.

Scope: Gross–Zagier and Kolyvagin, with modularity for all E/Q: rank equality and finite Sha in the low-analytic-rank regime.

Rank-one converse assuming finite Sha →

Proved · version 3

For semistable E/Q with nonsplit multiplicative reduction at an odd prime or split multiplicative reduction at two odd primes, algebraic rank 1 and finite Sha imply analytic rank 1.

  • E/Q is semistable.
  • E has nonsplit multiplicative reduction at at least one odd prime, or split multiplicative reduction at at least two odd primes.
  • Algebraic rank E(Q) = 1.
  • Assume Sha(E/Q) is finite.

Scope: Skinner’s converse in the stated semistable family; a proved implication with a finiteness assumption.

Relationship not independently reviewed.

Connection in Birch and Swinnerton-Dyer conjecture →

Relationship not independently reviewed. This concerns its use in this problem.

Sources

Skinner · converse to Gross–Zagier–Kolyvagin →

Christopher Skinner · Published

Annals of Mathematics 191 (2020), 329–354; published online 13 February 2020. Introduction also states the classical low-analytic-rank theorem.

Revision & attribution

Source revision: published-online-2020-02-13. Observed Oct 10, 2026.

The cited work does not report AI involvement; this is not an assertion that no AI was used.

Source updated Feb 13, 2020.

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-analytic-rank-at-most-one
Statement version
4
SHA-256
3f5169fceea017e89dff2beaaab7350fd02159a9f882b5969ecc7ea671d45c38

Publication records