Proved
Rank equality at analytic rank 0 or 1 ↗
Source dated Mar 1, 2020 · Observed Oct 10, 2026
Theorem · version 1
Rank equality at analytic rank 0 or 1
Earlier version 1. View current version 4 →
Statement
For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/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
Source reports
Connections
In this problem · publication v3 · 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.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Skinner · converse to Gross–Zagier–Kolyvagin →
- Introduction, journal page 330 (PDF page 2), classical theorem → · Original passage ↗
Recorded wording
For E/Q with ord_(s=1)L(E,s) equal to 0 or 1, rank E(Q) equals that analytic rank and Sha(E/Q) is finite.
paraphrase · No upstream full text or code is republished by this record.
Christopher Skinner · Published
Annals of Mathematics 191 (2020), 329–354. Introduction also states the classical low-analytic-rank theorem.
Revision & attribution
Source revision: annals-191-2-2020. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Mar 1, 2020.
Wiles · official BSD description →
- PDF pages 2 and 4, conjecture/refinement and low-rank theorem → · Original passage ↗
Recorded wording
Wiles distinguishes the rank statement from the refined leading-coefficient formula and records the analytic-rank-zero-or-one theorem.
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
Rank equality at analytic rank 0 or 1 ↗
Source dated Mar 1, 2020 · Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-bsd-analytic-rank-at-most-one
- Statement version
- 1
- SHA-256
- 5567644fba9bf14c558747089211fec669a5764dcfdfdf59546db987595a99ea