Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank

Follow-up works

• Ruihan Chen, On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$ (arXiv:2610.09370, 2026-10-07). Computes the $2^\infty$-Selmer corank of this family and, taking the specialization at 2 of this paper's main theorem as an explicit hypothesis (Hypothesis 1.1), deduces rank equality, finiteness of Sha and the exact leading-term formula for $p\equiv3,5,7\pmod8$. [cites earlier version]
• First Use Labs, Faster BSD certification (local benchmark, 2026-10-09). Reports a computational adapter that applies Theorem 1.1 of the October 7 revision after a finite 2-Selmer bound, stating that it accepts the theorem as an unaudited premise.

Comments and discussion

• Ashay Burungale, BSD and the p-converse (guest post on Persiflage) (2026-10-15). Survey of BSD and the p-converse that describes the announced family results and the division of leading-term results between the companions; it cites this paper as treating "the odd-primary components of the BSD formula" and says correctness is still being analysed. [family-level]
• Persiflage, A Brief Digest of Some New OpenAI Results, item 6 "Results on BSD" (galoisrepresentations.org, 2026-10-08). Links this paper and gives a first impression that the BSD results are "technical improvements on results already in the literature" rather than progress on higher rank; the author says they have not studied the paper and invites a guest post. [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 cube-sum application has already been announced by several groups. [family-level]
• Álvaro Lozano-Robledo, quoted in the Proofs and Prompts reactions roundup (2026-10-08). Notes the release targeted BSD but obtained partial results ("BSD, not Goldfeld's 50-50").
• 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 (2026-10-07). Contrasts the release's BSD and Riemann items, calling the claimed progress on BSD "pure PR, no content"; a later reply describes the result as "basically a matter of chasing powers of 2". [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]

Related prior literature mentioned by others

• Samit Dasgupta and Mahesh Kakde, On the Brumer–Stark Conjecture (author manuscript, 2022-02-15). Burungale's exposition and its discussion identify this work as a precursor to the family's tame congruence methods. [family-level]

Possibly relevant literature (found by the agent, not checked)

• Wei Zhang, Selmer groups and the indivisibility of Heegner points (Cambridge J. Math. 2 (2014)). Proves the p-part of the refined BSD formula in analytic rank one for certain good ordinary $p\ge5$, relevant to this paper's odd-primary rank-one valuation; it is not cited here (the two companion papers cite it).
• Francesc Castella, On the p-part of the Birch-Swinnerton-Dyer formula for multiplicative primes (arXiv:1704.06608, 2017). Extends the Jetchev–Skinner–Wan rank-one p-part result to primes of multiplicative reduction, relevant to the paper's odd-prime leading-term claim; it is not in the bibliography.

Digests and expositions

• Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics, §9.1.1 (arXiv:2610.17090, 2026-10-12). Survey entry on family 002 that summarizes the three-paper chain, extracts tame variables, integral character division and the U(3,1) theta implication, relates them to Taylor–Wiles–Kisin, Scholze–Manning and Skinner–Urban/Castella–Liu–Wan, and lists caveats including a Kato-divisibility step at every odd prime. [AI-generated per source] [family-level]