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Number theory · Oct 6, 2026

BSD in Selmer-corank 0/1 cases

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.

Scope and what remains

Not every elliptic curve or the full unrestricted BSD conjecture.

Contributors

OpenAI source publisher and an unnamed unreleased internal model. Exact paper-level human roles are not disclosed by the family summary.

AI systems: Unreleased OpenAI model

Sources

More on Birch and Swinnerton-Dyer conjecture

October 2026 1

New proofNumber theoryopenai/math #006

Goldfeld twist-rank distribution

Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.

Unreleased OpenAI modelProblem page →
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