Proved
Rank-one converse assuming finite Sha ↗
Skinner · converse to Gross–Zagier–Kolyvagin · Abstract and Theorem A, journal pages 329–331 →
Source dated Mar 1, 2020 · Observed Oct 10, 2026
Theorem · version 1
Rank-one converse assuming finite Sha
Earlier version 1. View current version 3 →
Statement
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.
Assumptions
- 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.
Evidence
Source reports
Connections
In this problem · publication v3
Birch and Swinnerton-Dyer conjecture →
Conditional result · Proved
Runs in the converse direction to the low-analytic-rank theorem. General Sha finiteness is conjectural and is not supplied by assuming algebraic rank 1.
Scope: Algebraic rank 1 ⇒ analytic rank 1 in this family if Sha is finite.
Limit: Finite Sha and the reduction hypotheses must be established for an application.
Related results · 1
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Skinner · converse to Gross–Zagier–Kolyvagin →
- Abstract and Theorem A, journal pages 329–331 → · Original passage ↗
Recorded wording
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.
paraphrase · No upstream full text or code is republished by this record.
- 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.
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.
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-one converse assuming finite Sha ↗
Source dated Mar 1, 2020 · Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-bsd-rank-one-converse-finite-sha
- Statement version
- 1
- SHA-256
- bf4ec9c94cb6c8b9e9b6db46065f7501b9a2ac6916d9e1fd62e5eae92550e639