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Number theory · Oct 6, 2026

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.

Scope and what remains

Twist order, exceptional sets, analytic/algebraic rank and dependence on companion BSD results must be explicit.

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 #002

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.

Unreleased OpenAI modelProblem page →
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