abelian varieties with the same CM type are isogenous Does anybody have a reference for the following fact? 
All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$? 
Here abelian variety with complex multiplication means an abelian variety A over $\overline{\mathbb{Q}}$ such that there exists a CM number field $K$ of degree twice the dimension of A and an embedding of $K$ into $End(A) \otimes \mathbb{Q}$. The CM type is obtained by looking at the action of $K$ into $H^0(A, \Omega^1_A)$.  
 A: This is true over $\mathbb{C}$, and for any algebraically closed subfield $k$ of $\mathbb{C}$ the functor from CM abelian varieties over $k$ to CM abelian varieties over $\mathbb{C}$ is an equivalence of categories. All of this can be found in the notes on Complex Multiplication on Milne's webpage.
A: I do not know a reference, but it is easy to prove. Let $K/F$ be a totally imaginary quadratic extension $K$ of a totally real number field $F$ of degree $n$ over $\mathbb Q$. Let ${\mathfrak a}$ be a nonzero fractional ideal in $K$. Now, $K\otimes _{\mathbb Q} {\mathbb R}$ is isomorphic as an $\mathbb R$-algebra to the $n$ fold product of $\mathbb C$ with itself. Then $A={\mathbb C}^d/{\mathfrak a}$ is an abelian variety and its ring of complex multiplications tensored with ${\mathbb Q}$ is precisely $K$. \vskip 5mm 
Moreover all the abelian varieties whose ring of endomorphisms tensored with $\mathbb Q$ is $K$ is obtained in this way, up to isogeny. This is proved by showing that a lattice in ${\mathbb C}^d$ which is stable under an order $R$ in $K$ is isogenous to  ${\mathbb C}^d/R$.  
