Extended ray class fields of orders conjecture for SICs

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Fix d≥4d\geq4, write Δ=(d+1)(d−3)=f2Δ0\Delta=(d+1)(d-3)=f^2\Delta_0 with Δ0\Delta_0 a fundamental discriminant, and let f′∣ff'\mid f. Let M\mathcal M satisfy the extended order-to-multiplet conjecture, and let Ed,f′E_{d,f'} and E~d,f′\widetilde E_{d,f'} be the ray class fields associated with the order O(f′)2Δ0\mathcal O_{(f')^2\Delta_0}, with levels dO′d\mathcal O' and d′O′d'\mathcal O' respectively, where d′=dd'=d for odd dd and d′=2dd'=2d for even dd. Extended Ray Class Fields of Orders Conjecture. If [S]∈M(f′)[S]\in\mathcal M(f'), then \fieldtripS=Ed,f′\fieldtrip{S}=E_{d,f'} and \fieldvecS=\fieldprojS=E~d,f′\fieldvec{S}=\fieldproj{S}=\widetilde E_{d,f'}. This extends the original ray class prediction from the maximal order to every order indexing a SIC multiplet; it remains open.

References

Primary source

Gene S. Kopp and Jeffrey C. Lagarias, “SIC-POVMs and orders of real quadratic fields”, arXiv:2407.08048 (2026).

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