Goldfeld’s conjecture: densities and mean analytic rank
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
Goldfeld's analytic density conjecture and the 2-converse for elliptic curves
We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the 2^∞-Selmer corank of E is zero or one, then it…
Full source abstract
We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the 2^∞-Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.
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The mean analytic rank of quadratic twists of elliptic curves
For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
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