Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End\_{\mathbb{C}}(A) \otimes\_{\mathbb{Z}} \mathbb{Q}$. (So $n$ is a multiple of $g$.)


Then, the tangent space to the identity of $A$ defines an $n$-dimensional complex representation $\Phi$ of $K$ (which Shimura calls the type of $(A, \theta)$). Write


$\tau\_1, \tau\_2, \ldots, \tau\_g, \rho \tau\_1, \rho \tau\_2, \ldots, \rho \tau\_g$


for the different embeddings of $K$ into $\mathbb{C}$, where $\rho$ denotes complex complex conjugation. Then (as shown by Shimura), the representation $\Phi$ decomposes as a direct sum


$\Phi = \bigoplus\_{\nu = 1}^g (r\_\nu \cdot \tau\_\nu \oplus s\_\nu \cdot \rho \tau\_\nu)$ where $r\_1 + s\_1 = r\_2 + s\_2 = \cdots = r\_g + s\_g = \frac{n}{g}$.


What are the possible values of $(r\_1, s\_1, r\_2, s\_2, \ldots, r\_g, s\_g)$ as $A$ ranges over all complex abelian varieties, while $K$ remains fixed? Can they be arbitrary nonnegative integers satisfying the above constraint, or does fixing $K$ impose further conditions?



possible CM-types of abelian varieties
There are certain additional conditions on ``multiplicities" that are spelled out in the original Shimura's paper. Let me discuss a couple of interesting cases.


The case of CM type, i.e. when $n=g$. Let $\Phi$ be the corresponding CM type, which is a $n$-element set of embeddings of $K$ into the field $C$ of complex numbers. For each pair of complex-conjugate embeddings $K \to C$ exactly one of them lies in $\Phi$.
If $K$ does not contain a proper CM subfield then everything is fine, i.e., the endomorphism algebra of $A$ is $K$. However, if $K$ contains a proper CM subfield $L$ then a ``bad" choice of $\Phi$ would imply that the endomorphism algebra of $A$ is a matrix algebra over $L$ rather than $K$. In order to guarantee that the endomorphism algebra is $K$, one should require that if $\tau\_1, \tau\_2 \in \Phi$ and their restrictions to the maximal totally real subfield $L\_0$ of $L$ coincide then their restrictions to $L$ also coincide. (Of course, one should require it for all CM subfields.)
For example, if $n=g=2$ and a quartic CM field $K$ contains an imaginary quadratic subfield then $K$ must contain another imaginary quadratic subfield as well! In this case one may check that the endomorphism algebra of $A$ is always bigger than $K$ (and $A$ is isogenous to a square of a CM elliptic curve).


The case of an abelian surface $A$ and an imaginary quadratic field $K$, i.e., $g=1, n=2$. It is known that the endomorphism algebra of an abelian surface is never an imaginary quadratic field. (Actually, as was pointed out by Frans Oort, this assertion remains true in characteristic p.) More precisely, suppose we are given an embedding of $K$ in the endomorphism algebra of $A$. Then either $A$ is isogenous to a square of an elliptic curve with multiplication by $K$ or the endomorphism algebra of $A$ is an indefinite quaternion algebra over the rationals.


