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The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one

OpenAI · October 6, 2026

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Abstract

We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Comments and discussion

  • Ashay Burungale, BSD and the p-converse (guest post on Persiflage) (2026-10-15). Survey of the announced family results that mentions this paper only as the source of the 2-primary component of the leading-term formula; it says correctness is still under analysis. [family-level]
  • Persiflage, A Brief Digest of Some New OpenAI Results, item 6 "Results on BSD" (galoisrepresentations.org, 2026-10-08). First impression that the BSD results are "technical improvements on results already in the literature" rather than progress on higher rank; the item links the companion leading-term paper. [family-level]
  • David Loeffler, comment on the Persiflage digest (2026-10-09). Agrees with the blog's assessment and says the BSD results do not seem to go much beyond recent work of Skinner, Castella, Burungale et al. [family-level]
  • Alexander Smith (@Alexand11428651), post on X (2026-10-07). Says BSD in Selmer corank 0 and 1 was an active research area "likely to reach this point 'soon'", counts it as no progress on BSD, calls it "unwisely not formalized", and compares the 2-converse argument to work of Kriz and Nordentoft. [family-level]
  • Pierre Colmez (@ColmezPierre), post on X (2026-10-09). Says that at this point the result was "basically a matter of chasing powers of 2: a non trivial task, but far from a progress towards the millenium problem." [family-level]
  • Hara qui rit (@humanismetoussa), post on X (2026-10-07). Explains that full BSD in Selmer corank 0 and 1 combined with the Goldfeld conjecture gives full BSD for 100% of quadratic twists of any elliptic curve over Q, "still far from BSD." [family-level]

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Cite

@misc{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026,
  author = {{OpenAI}},
  title = {{The two-primary Birch--Swinnerton-Dyer formula in Selmer corank at most one}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf}{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026}},
  year = {2026}
}

Versions

This version updates citations to revised manuscripts in this collection; see the previous version.

Family 002: The full BSD formula from low Selmer corank

Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.

Other papers in this family: