Equality of SIC-generated fields with ray class fields

Prove that for any admissible tuple \(t=(d,r,Q)\), letting \(E = H^{\mathcal{O}_f}_{\bar{d}\,\infty_1\,\infty_2}\) be the ray class field of the order \(\mathcal{O}_f\) with modulus \(\bar{d}\) and both real places ramified, the SIC-generated field \(E_t\) satisfies \(\hat{E}_t = E_t = E\).

Background

After establishing conditional containments of SIC-generated fields in certain ray class fields, the authors conjecture exact equality. This would identify the fields produced by the SIC construction precisely with ray class fields of orders, completing the link to explicit class field theory.

References

Empirically, it seems that E_t is actually equal to the ray class field E in \Cref{thm:RayClassField2}, and indeed a similar statement may be made when Q is not fundamental. As we do not know how to prove this from any form of the Stark conjectures in the literature, we state it as a separate conjecture. Let t=(d,r,Q)\sim(K,j,m,Q) be an admissible tuple, let d=d_{j,m}, and let f be the conductor of Q. Let E= H{\OO_f}_{\db\infty_1\infty_2} be the ray class field with level datum (\mcl{O}_f; \db \mcl{O}_f,(\infty_1,\infty_2)), as defined by \Cref{thm:rayclassfield}. Then \hat{E}_t = E_t = E.

— A Constructive Approach to Zauner's Conjecture via the Stark Conjectures  (2501.03970 - Appleby et al., 7 Jan 2025) in Conjecture 4.9, Section 4.2 (The main theorems: class fields attained)

The statements regarding the algebraic nature of these complex numbers remain conjectural.

— Geometric families of multiple elliptic Gamma functions and arithmetic applications, III  (2610.12169 - Morain, 8 Oct 2026) in Conjecture [BCG, Conjecture], Section 2.2; concluding discussion of the cubic case

The study of SIC fields has led to several conjectures, such as the so-called Ray Class Field Conjecture: Let d > 3, and let K_d = Q(√((d + 1)(d − 3))). Then E_{Π(v)} is an Abelian extension of K_d. Furthermore, there is at least one SIC-POVM whose corresponding SIC field is equal to R, the ray class field of K_d. Moreover, every other SIC field in dimension d is a finite Galois extension of Q containing R.

— Real Fiducial Vectors Only Exist in Dimensions $d\equiv 3 \pmod{4}$  (2610.13168 - Loo et al., 9 Oct 2026) in Section 1, Introduction, immediately before Conjecture 1 (the Ray Class Field Conjecture)