Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 002-03. The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
Manuscript map · lines 2475–2484
OpenAI@2026-10-07#rem:oai-p-002-03
Authors: OpenAI. Family 002.
Title: The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one.
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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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- section: Manuscript map
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- remark: Remark: 002-02. The Selmer converse for elliptic curves at every prime
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- remark: Remark: 003-02. The Quasi-Riemann Hypothesis (alternate 11/12 proof)
- remark: Remark: 003-03. Uniform exclusion of Landau–Siegel zeros
- remark: Remark: 004-01. Hilbert’s tenth problem over the rational numbers