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 seem to have drawn much attention.

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.

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 $K$? 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 seem to have drawn much attention.

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 seem to have drawn much attention.