Folklore conjecture on ranks with prescribed torsion

Let GG be a torsion group, and let EG\mathcal{E}_G denote the family of elliptic curves over Q\mathbb{Q} with torsion subgroup prescribed by GG. Folklore conjecture on ranks with prescribed torsion. The proportion of elliptic curves with rank 00 in EG\mathcal{E}_G and the proportion of elliptic curves with rank 11 in EG\mathcal{E}_G are both 12\frac 12. The conjecture predicts the minimalist rank distribution within every prescribed-torsion family; the source says that evenly distributed root numbers and the authors' one-level density results provide small evidence toward it, but gives no resolution.

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Primary source

Peter J. Cho and Keunyoung Jeong, “Average analytic rank of elliptic curves with prescribed torsion”, arXiv:2005.06862 (2021).

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