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

OpenAI · September 24, 2026

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This is an earlier edition (September 24, 2026). Current edition: The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, October 6, 2026.

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Cite

@misc{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026,
  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-September-24-2026/paper.pdf}{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026}},
  year = {2026}
}

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.

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