Proved
Rank-one formula · selected p-parts ↗
Jetchev–Skinner–Wan · rank-one BSD p-part · Theorem 1.2.1, PDF pages 1–2 →
Source dated Dec 21, 2015 · Observed Oct 10, 2026
Theorem · version 1
Rank-one formula · selected p-parts
Earlier version 1. View current version 2 →
Statement
For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.
Assumptions
- E/Q is semistable (squarefree conductor N).
- ord_(s=1)L(E,s) = 1.
- p ≥ 5 is prime and p does not divide N.
- The mod-p Galois representation on E[p] is irreducible.
Scope
The p-part of the leading-coefficient formula, not an assertion at every prime. Irreducibility removes p-primary rational torsion.
Evidence
Source reports
Connections
In this problem · publication v3 · Selected context
Birch and Swinnerton-Dyer conjecture →
Special case · Proved
Adds prime-specific leading-coefficient information beyond low-rank equality. Both good ordinary and supersingular reduction are permitted here; the p=3 extension is not included in this selected Claim.
Scope: Good-reduction p ≥ 5 with irreducible E[p], for semistable analytic-rank-one curves.
Limit: The missing primes and unrestricted curves require further results.
Related results · 1
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v10
Birch and Swinnerton-Dyer conjecture →
Special case · Proved
Adds prime-specific leading-coefficient information beyond low-rank equality. Both good ordinary and supersingular reduction are permitted here; the p=3 extension is not included in this selected Claim.
Scope: Good-reduction p ≥ 5 with irreducible E[p], for semistable analytic-rank-one curves.
Limit: The missing primes and unrestricted curves require further results.
Related results · 1
Related statements
related toLow-rank formula at supersingular p ≥ 5
These formulas overlap at selected supersingular rank-one curves; the supersingular theorem also covers analytic rank zero in its scope.
An equality of p-valuations leaves other primes and hypotheses to be checked; neither statement is the general full formula.
Details & sources
Rank-one formula · selected p-parts →
Proved · version 1
For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.
- E/Q is semistable (squarefree conductor N).
- ord_(s=1)L(E,s) = 1.
- p ≥ 5 is prime and p does not divide N.
- The mod-p Galois representation on E[p] is irreducible.
Scope: The p-part of the leading-coefficient formula, not an assertion at every prime. Irreducibility removes p-primary rational torsion.
Low-rank formula at supersingular p ≥ 5 →
Proved · version 1
For semistable E/Q of analytic rank 0 or 1 and a good supersingular prime p ≥ 5, the p-part of the BSD leading-coefficient formula holds.
- E/Q is semistable.
- ord_(s=1)L(E,s) is 0 or 1.
- p ≥ 5 is a prime of good supersingular reduction.
Scope: Theorem 1.5 restricted to p ≥ 5; its additional p=3 conditions and quadratic-twist extension are not asserted here.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Jetchev–Skinner–Wan · rank-one BSD p-part →
- Theorem 1.2.1, PDF pages 1–2 → · Original passage ↗
Recorded wording
For semistable E/Q of analytic rank 1 and a good-reduction prime p ≥ 5 with E[p] irreducible, the p-adic valuation of L′(E,1)/(Ω_E Reg_E) equals that of #Sha(E/Q) ∏ c_l.
paraphrase · No upstream full text or code is republished by this record.
Dimitar Jetchev · Christopher Skinner · Xin Wan · Published
Published in Cambridge Journal of Mathematics 5 (2017), 369–434; the cited author preprint supplies Theorem 1.2.1.
Revision & attribution
Source revision: arxiv-v1. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Dec 21, 2015.
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.
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.
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; 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.
Burungale–Skinner–Tian–Wan · zeta elements →
- Theorem 1.5, PDF page 4 → · Original passage ↗
Recorded wording
For semistable E/Q of analytic rank 0 or 1 and a good supersingular prime p ≥ 5, the p-part of the BSD leading-coefficient formula holds.
paraphrase · No upstream full text or code is republished by this record.
Ashay Burungale · Christopher Skinner · Ye Tian · Xin Wan · Preprint
Author preprint at the cited revision; journal acceptance has not been established by this curation.
Revision & attribution
Source revision: arxiv-v2. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Sep 11, 2024.
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 formula · selected p-parts ↗
Source dated Dec 21, 2015 · Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-bsd-semistable-rank-one-p-part
- Statement version
- 1
- SHA-256
- d9f528560617685bb2212659d7ef3a4d2163c362a6b4abf0ce141c3786c465f0