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Theorem · version 1

Full BSD in a specified infinite twist family

Earlier version 1. View current version 2 →

Statement

Let E be Cremona curve 46a1. For squarefree M>1 coprime to 46, if L(E^M,1)≠0 and E has ordinary reduction at every prime dividing M, E^M satisfies full BSD. Infinitely many M meet these conditions.

Assumptions

  • E/Q is the specific Cremona curve 46a1.
  • M>1 is squarefree and gcd(M,46)=1; E^M is the twist by Q(√M).
  • L(E^M,1) is nonzero.
  • E has ordinary reduction at every prime dividing M.

Scope

Analytic-rank-zero quadratic twists of one named non-CM base curve; rank, finite Sha and the entire leading-coefficient identity.

Evidence

Connections

In this problem · publication v6 · Selected context

Birch and Swinnerton-Dyer conjecture →

Special case · Proved

A full-formula family within the low-analytic-rank case. The 46a1 family is retained after the 2026 Banwait–Huang check; four other base curves in the earlier list have an infinitude caveat.

Scope: An infinite, explicitly conditioned family of twists of 46a1.

Limit: Does not give full BSD for every twist, every rank-zero curve, or analytic rank ≥ 2.

Related results · 2

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

In this problem · publication v10

Birch and Swinnerton-Dyer conjecture →

Special case · Proved

A full-formula family within the low-analytic-rank case. The 46a1 family is retained after the 2026 Banwait–Huang check; four other base curves in the earlier list have an infinitude caveat.

Scope: An infinite, explicitly conditioned family of twists of 46a1.

Limit: Does not give full BSD for every twist, every rank-zero curve, or analytic rank ≥ 2.

Related results · 2

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

Sources

Banwait–Huang · effective BSD twist families →

Barinder S. Banwait · Xiaoyu Huang · Acceptance reported

arXiv metadata reports acceptance for ANTS XVII. Algorithms enumerate sufficient subfamilies, not all curves or twists satisfying BSD.

Revision & attribution

Source revision: arxiv-v3. Observed Oct 10, 2026.

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

Source updated Jun 4, 2026.

Burungale–Skinner–Tian–Wan · zeta elements →

  • Theorem 1.7, PDF page 4, restricted to base curve 46a1 → · Original passage ↗
    Recorded wording

    Let E be Cremona curve 46a1. For squarefree M>1 coprime to 46, if L(E^M,1)≠0 and E has ordinary reduction at every prime dividing M, E^M satisfies full BSD. Infinitely many M meet these conditions.

    paraphrase · No upstream full text or code is republished by this record.

  • 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.

Wiles · official BSD description →

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 →

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.

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.

Source updates

2026 · effective families and an earlier-list caveat →

Banwait–Huang give sufficient-condition algorithms for BSD twist subfamilies and list eligible curves of conductor <500,000. Section 2.7 retains 46a1 but identifies a nonempty-prime-support gap for 62a1, 66b1, 105a1 and 141c1 in an earlier infinitude list.

The authors do not characterize every BSD-satisfying twist. The flagged four curves are not disproved BSD cases. The code and dataset have not been executed or reproduced by OpenMath.

Birch and Swinnerton-Dyer conjecture · publication v6

2026 · effective families and an earlier-list caveat →

Banwait–Huang give sufficient-condition algorithms for BSD twist subfamilies and list eligible curves of conductor <500,000. Section 2.7 retains 46a1 but identifies a nonempty-prime-support gap for 62a1, 66b1, 105a1 and 141c1 in an earlier infinitude list.

The authors do not characterize every BSD-satisfying twist. The flagged four curves are not disproved BSD cases. The code and dataset have not been executed or reproduced by OpenMath.

Birch and Swinnerton-Dyer conjecture · publication v10

Record details
Claim ID
claim-bsd-46a1-quadratic-twists
Statement version
1
SHA-256
ddc2f7e4b086da3a29d76c3eb4f066383f6572bd804df4ea48471fa72cef78b4

Publication records