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Abelian variety of CM-type manipulate analytic functions of several complex variables. The formal definition is that the tensor product of End("A") with the rational number field Q, should contain a commutative subring of dimension 2"d" over Q. When "d" = 1 this can only be a quadratic field, and one recovers the cases where End("A") is an order in an imaginary quadratic field. For "d" > 1 there are comparable cases for CM-fields, the complex quadratic extensions of totally real fields. There are other cases that reflect that "A" may not be a simple abelian variety (it might be a cartesian product of elliptic