CM fields and Hilberts 12th problem

Question

According to the main theorem of CM, for every abelian variety $A$ associated to a CM field $K$, one obtains a certain unramified abelian extension of the reflex field $K^\times$ given by the field of moduli of $A$. Similarly, the fields of moduli of ideal section points generate certain ramified extensions. At many places (for instance, in Ogg's review of Shimura's book) it is mentioned that in contrast to the case of imaginary quadratic $K$, in general not all abelian extensions are obtained in this way.

My question: Is there an easy description of the abelian extensions which are obtained by the above construction, just in terms of the idele class group of $K^\times$? And furthermore, is there some kind of survey article available which reports on the recent progress towards a complete solution of Hilbert's 12th problem over CM fields? On the arxiv there is an article from 2006 by Sixin Zeng, "Notes on Hilberts 12th problem" which claims to have a solution based on mirror symmetry. However, it doesn't seem to be published somewhere, and doesn't seem to have received much attention.

Answer 1

Beginning with the work of Taniyama, Shimura, and Weil in the late fifties, the theory of elliptic curves and elliptic modular curves has been successfully generalized to higher dimensions. In this theory, an elliptic curve with complex multiplication by an imaginary quadratic field is replaced by an abelian variety with complex multiplication by a CM field, that is, a quadratic totally imaginary extension $K$ of a totally real field $F$, and an elliptic modular function by an automorphic function.

Philosophically, one expects that, with the exception of $\mathbb{Q}$, one can not obtain abelian extensions of totally real fields by adjoining special values of automorphic functions. However, it is known that, roughly speaking, one does obtain the largest possible abelian extension of a CM-field $K$ consistent with this restriction.

More precisely, let $K$ be a CM-field and let $F$ be the largest totally real subfield of $K$. Then $G=Gal(\mathbb{Q}^{\mathrm{al}} /K)$ is a subgroup of index $2$ in $G^{\prime}=Gal(\mathbb{Q}^{\mathrm{al}}/F)$, and the corresponding Verlagerung is a homomorphism $V:G^{\prime\mathrm{ab}}\rightarrow G^{\mathrm{ab}}$. In this case, $V$ has a very simple description.

Theorem: Let $K$ be a CM-field, and let $F$ be the totally real subfield of $K$ with $[K:F]=2$. Let $H$ be the image of the Verlagerung map $Gal(F^{\mathrm{al}}/F)^{\mathrm{ab}}\rightarrow Gal(K^{\mathrm{al}% }/K)^{\mathrm{ab}}. $ Then the extension of $K$ obtained by adjoining the special values of all automorphic functions defined on canonical models of Shimura varieties with rational weight is $(K^{\mathrm{ab}})^{H}\cdot\mathbb{Q}^{\mathrm{ab}}$.

See the 1993 thesis of Wafa Wei (University of Michigan).

Wei, Wafa, Weil numbers and generating large field extensions, 1993, Available at the library of the University of Michigan, Ann Arbor,

Wei, Wafa, Moduli fields of CM-motives applied to Hilbert's 12-th problem, 1994, 18pp; http://www.mathematik.uni-bielefeld.de/sfb343/preprints/pr94070.ps.gz

(Copied from Milne's Class Field Theory notes, where everything is described in terms of ideles.)

Answer 2

You can refer to https://arxiv.org/abs/1912.08128
(Form class groups and class fields of CM fields).