Folklore conjecture on ranks with prescribed torsion

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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.

References

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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