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The Selmer converse for elliptic curves at every prime

OpenAI · October 7, 2026

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Abstract

We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤp-corank $r\in\{0,1\}$, then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $\ell\equiv4,7,8\pmod9$, the cubic $X^3+Y^3=\ell Z^3$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.

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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). Discusses this paper (cited as [OA1]) in detail against prior Iwasawa-theoretic p-converse results, covering its consequences for Goldfeld's and Sylvester's conjectures and its rank-one strategy via tame congruences and Heegner points; it says correctness is still under analysis. [family-level]
  • L-function apprentice, comment on Burungale's guest post (2026-10-15). Reads this paper's argument as "more like a Skinner style argument" that avoids Iwasawa-theoretic difficulties; a related comment (#comment-28605) discusses the strict case and Heegner-point vanishing.
  • AB, reply on Burungale's guest post (2026-10-16). Describes the strategy as "Skinner-like with vanishing understood in the tame parameter t."
  • 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). Says the BSD results do not seem to go much beyond recent work of Skinner, Castella, Burungale et al., and notes that the Sylvester cube-sum application (proved in this paper's last section) "has already been announced multiple times by different groups in the last 6 months." [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]

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Cite

@misc{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026,
  author = {{OpenAI}},
  title = {{The Selmer converse for elliptic curves at every prime}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf}{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026}},
  year = {2026}
}

Versions

This version updates a citation to a revised manuscript 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.

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