Serre's tensor construction and moduli of abelian schemes

Shimura varieties of PEL-type attached to unitary groups of signature (p,q) often have integral models that represent moduli problems of PEL abelian schemes. I will define these moduli problems, and then restrict to the case of signature (n,0), where the corresponding Shimura variety is zero-dimensional. I will describe how this family of abelian schemes can be systematically constructed out of CM abelian varieties and certain positive-definite hermitian modules of rank n via Serre's tensor construction. In the quadratic imaginary case this can be succinctly formulated as an isomorphism of stacks involving a tensor product of categories. Along the way one obtains that the description, given by the main theorem of complex multiplication, of the action of the absolute Galois group on CM abelian varieties also applies to the family of abelian varieties in consideration. This implies the zero-dimensional Shimura varieties in question admit integral canonical models.