{"id":"openai-math/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves","pdf":"https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026","tags":["number-theory","elliptic-curves","quadratic-twists","bsd-conjecture"],"score":0,"title":"Goldfeld's analytic density conjecture and the 2-converse for elliptic curves","domain":"Number theory","family":"006","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-07","reactions":{},"formalized":false,"family_title":"Goldfeld’s conjecture: densities and mean analytic rank"}