3 added 171 characters in body

A CM type for K is a choice of one out of every pair of complex conjugate embeddings of K. For each CM type on K, there is an abelian variety of that type. This is a complex abelian variety B of dimension g with an action of K (rather, its integers) such that K acts on the tangent space of B through the g chosen complex embeddings. Now you can take A to be a product of B's corresponding to whichever CM types you choose. That certainly gives you a lot of possibilities for the multiplicities.

Added: In fact, that gives all multiplicities subject to the obvious condition that the multiplicity of an embedding and its conjugate add to n/g. However, the questioner points out that he wants an abelian variety A such that $K=End(A)\otimes Q$. Take an A as constructed in the first paragraph. Then $V=H_1(A,Q)$ is a K-vector space with a Hodge structure and a compatible Riemann form, which we can use to define a Shimura variety (= moduli variety). The problem now is to prove that (as expected), the family of abelian varieties on this variety contains one satisfying the condition...condition. For this check that each abelian variety with endomorphism algebra strictly larger than K lies in a (one of countably many) Shimura subvarieties of strictly lower dimension.

2 added 598 characters in body

A CM type for K is a choice of one out of every pair of complex conjugate embeddings of K. For each CM type on K, there is an abelian variety of that type. This is a complex abelian variety B of dimension g with an action of K (rather, its integers) such that K acts on the tangent space of B through the g chosen complex embeddings. Now you can take A to be a product of B's corresponding to whichever CM types you choose. That certainly gives you a lot of possibilities for the multiplicities.

Added: In fact, that gives all multiplicities subject to the obvious condition that the multiplicity of an embedding and its conjugate add to n/g. However, the questioner points out that he wants an abelian variety A such that $K=End(A)\otimes Q$. Take an A as constructed in the first paragraph. Then $V=H_1(A,Q)$ is a K-vector space with a Hodge structure and a compatible Riemann form, which we can use to define a Shimura variety (= moduli variety). The problem now is to prove that (as expected), the family of abelian varieties on this variety contains one satisfying the condition....

1

A CM type for K is a choice of one out of every pair of complex conjugate embeddings of K. For each CM type on K, there is an abelian variety of that type. This is a complex abelian variety B of dimension g with an action of K (rather, its integers) such that K acts on the tangent space of B through the g chosen complex embeddings. Now you can take A to be a product of B's corresponding to whichever CM types you choose. That certainly gives you a lot of possibilities for the multiplicities.