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Precise identification of SIC-generated subfields with ray class fields

Prove that, for any fiducial datum \(s=(d,r,Q,G,g)\) and associated admissible tuple \(t=(d,r,Q)\), the following equalities hold: (1) \(E_s^{(1)} = H^{\mathcal{O}_f}_{\bar{d}\,\infty_1}\); (2) \(E_t^{(2)} = H^{\mathcal{O}_f}_{\bar{d}\,\infty_2}\); (3) \(E_t = H^{\mathcal{O}_f}_{\bar{d}\,\infty_1\,\infty_2}\).

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Background

This conjecture refines the identification of SIC-generated fields, asserting each of the fields built from overlaps, ghost overlaps, and their union equals the corresponding ray class field with specific infinity ramification. It complements the broader equality conjecture E^t=Et=E\hat{E}_t = E_t = E.

References

Let s=(d,r,Q,G,g)\sim(K,j,m,Q,G,g) be a fiducial datum, let t=(d,r,Q)\sim(K,j,m,Q), and let d=d_{j,m}. Suppose that \disc(Q) is fundamental. Choose \p_1 = \smcoltwo{p_{11}{p_{12}} such that (p_{12}\rho_t-p_{11})\OO_1 is coprime to d\OO_1 as \OO_1-ideals. (1) The ray class field H{\OO_1}_{d\infty_1} is equal to the field extension of K generated by the numbers {\overlap_s(\mbf{p}) : 0\le p_1,p_2 < d, \mbf{p}\neq 0} and is also equal to \Q(\overlap_s(\p_1)2). (2) The ray class field H{\OO_1}_{d\infty_2} is equal to the field extension of K generated by the numbers {\tilde\mu_t(\mbf{p})2 : 0\le p_1,p_2 < d, \mbf{p}\neq 0} and is also equal to \Q(\tilde\mu_t(\p_1)2). (3) The ray class field H{\OO_1}_{\bar{d}\infty_1\infty_2} is equal to the field extension of K generated by the numbers {\tilde\mu_t(\mbf{p})2 : 0\le p_1,p_2 < d, \mbf{p}\neq 0} together with \xi_d.

— A Constructive Approach to Zauner's Conjecture via the Stark Conjectures (2501.03970 - Appleby et al., 7 Jan 2025) in Conjecture 4.12, Section 4.2 (The main theorems: class fields attained)