CM fields and Hilberts 12th problem

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.)