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Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank

OpenAI · October 3, 2026

Earlier edition

This is an earlier edition (October 3, 2026). Current edition: Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank, October 7, 2026.

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Cite

@misc{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026,
  author = {{OpenAI}},
  title = {{Exact Birch--Swinnerton-Dyer Formula from Low Selmer Corank}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf}{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026}},
  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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