Stevenhagen’s conjecture on the proportion of solvable negative Pell equations

Determine the limiting proportion, among square‑free integers d with all odd prime divisors p≡1 (mod 4), of parameters d for which the negative Pell equation x^2−dy^2=−1 is solvable in integers. Specifically, prove that the limit exists and equals c_Pell=1−∏_{j=1}^{∞}(1−2^{−(2j−1)}).

Background

The appendix recalls Stevenhagen’s conjecture on the solvability of the negative Pell equation in the natural subfamily where the local norm obstruction is removed (all odd prime divisors of d are ≡1 mod 4). The conjecture predicts a precise irrational constant c_Pell≈0.58057… as the limiting proportion.

The authors reinterpret Stevenhagen’s heuristic in Selmer‑group language: the condition x2−dy2=−1 reduces to asking whether −1 arises from the global norm inside the Selmer group Sel_2(O×(χ)/Q), and the conjectural proportion is tied to the distribution of the dimension of Sel_2+(O×(χ)/Q).

References

In light of this observation, it is natural to ask not for what proportion of all d′ (negative Pell) is soluble, but for what proportion of d′∈F. The expected answer to this question is given by a beautiful conjecture of Stevenhagen. Conjecture (Stevenhagen) Let F_{Pell} denote the set of square-free integers d′ for which (negative Pell) is soluble. Then the limit lim_{X→∞} #{d′∈F_{Pell}: d′<X}/#{d′∈F: d′<X} exists and is equal to the irrational number c_{Pell}=1−∏_{j=1}{∞}(1−2{−(2j−1)})=0.58057….

— Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups (2508.14026 - Bartel et al., 19 Aug 2025) in Appendix, Conjecture (Stevenhagen)