Goldfeld's density conjecture for quadratic twists
Let be an elliptic curve over . Let be the set of Dirichlet characters of order , and interpret density in using conductor ordering.
Goldfeld's conjecture. For every elliptic curve , those for which
have density in .
This is a conjecture about the distribution of central vanishing orders in quadratic twists. The paper uses it as a standard analytic-rank expectation in discussing obstructions to expressing Mordell–Weil rank modulo an integer as a sum of local invariants.
References
Primary source
Tim Dokchitser and Vladimir Dokchitser, “A note on the Mordell-Weil rank modulo n”, arXiv:0910.4588 (2009).
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI.See full solution
Claimed by OpenAI.
For every elliptic curve over Q, the manuscript claims analytic ranks zero and one each have density 1/2 among signed squarefree quadratic-twist parameters ordered by absolute value; hence analytic rank at least two has density zero in that ordering. It also claims that full 2-power Selmer corank zero or one equals the analytic and Mordell–Weil ranks and forces finiteness of the whole Tate–Shafarevich group. This MathDB question instead orders quadratic Dirichlet characters by conductor. The attachment is related progress; equivalence of these two orderings is not asserted here.
Repository: https://github.com/openai/math
- OpenAI-006-01-Goldfeld-s-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves.pdfOpen