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In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary qudratic extenstion field over $F$, and $D$ a quaternion algebra over $F$ satisfying certain conditions (see also Henri Carayol's paper "Sur la mauvaise reduction des courbes de Shimura"). My question is : What is the condition (any classification?) for the possible CM-fields(algebras) of the CM points (maximal commutative subalgebras of endomorpisms of the corresponding abelian varieties of CM-type.) on this shimura varieties (any reference?)? Thank you!

I guess it may have the following condition: (i)Any such CM_field must contain $L\otimes_FE$, with $L$ a imaginary qudratic extensiton of $F$ which can be embded in $D$.

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  • $\begingroup$ @TOM: OK, Deligne's paper is on my table, opened on Section 6 "Modèles étranges". Still I do not understand your question. Deligne constructs a Shimura variety ${\rm Sh}_{\mathbb{C}}(G,h_0)$, which is not of PEL-type. In order to construct a canonical model ${\rm Sh}(G,h_0)$ of this Shimura variety, he choses a totally imaginary quadratic extension $Z/F$ and embeds ${\rm Sh}_{\mathbb{C}}(G,h_0)$ into a Shimura variety of PEL-type constructed using $Z$. $\endgroup$
    – Mikhail Borovoi
    Commented Feb 2, 2013 at 14:22
  • $\begingroup$ (Continued) Is your question about CM-points of ${\rm Sh}(G,h_0)$ or of this auxiliary Shimura variety of PEL-type, or of ${\rm Sh}(G,h_0)$, but with respect to the weakly canonical model constructed using this embedding? Please elaborate! $\endgroup$
    – Mikhail Borovoi
    Commented Feb 2, 2013 at 14:25
  • $\begingroup$ @Mikhail: In my notation E is just your Z, and The PEL type shimura variety I mentioned is exactly the "auxiliary Shimura variety of PEL-type". And My question is about the CM-point of this "auxiliary Shimura variety of PEL-type", In fact I even wonder is there any classification of CM-fields for given PEL-type shimura datum. $\endgroup$
    – TOM
    Commented Feb 2, 2013 at 15:45
  • $\begingroup$ @TOM: OK, now what do you mean by the possible CM-fields (algebras) of the CM-points? Do you mean the algebras of endomorphisms of the corresponding abelian varieties of CM-type, or maximal commutative subalgebras of these algebras, or maybe the "dual fields"? $\endgroup$
    – Mikhail Borovoi
    Commented Feb 4, 2013 at 2:45
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    $\begingroup$ @TOM: I agree with your condition (i). Why don't you answer your own question? Try to write explicitly, what semisimple group $G$ you consider, what is a maximal torus $T\subset G$ (compact over $\mathbb{R}$), such a torus gives $L$. You get $E$ from the construction of the family of abelian varieties. Note that the dimension of $L\otimes_F E$ over $\mathbb{Q}$ is $4\cdot[F:\mathbb{Q}]$ - two times the dimension of an abelian variety. $\endgroup$
    – Mikhail Borovoi
    Commented Feb 7, 2013 at 10:21

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