Goldfeld’s conjecture: densities and mean analytic rank

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.

Manuscripts (2)

Goldfeld's analytic density conjecture and the 2-converse for elliptic curves

October 7, 2026 · source · cite as \cite{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026}

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^\infty-Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.

The mean analytic rank of quadratic twists of elliptic curves

October 6, 2026 · source · cite as \cite{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026}

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.

Result 006 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.