The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
Source: github.com/openai/math/tree/main/preprints/The-two-primary-…
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.
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BSD and the p-converse (Guest Post: Ashay Burungale) | Persiflagegaloisrepresentations.orgAshay Burungale
Here is the first of a series of guest blog posts by number theorists on the recent OpenAI results, this one by Ashay Burungale .) Let \(E/\mathbf{Q}\) be an elliptic curve. Its rational points form a finitely generated abelian group. The Birch–Swinnerton-Dyer (BSD) conjecture, formulated in the…