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 $D$.