Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, that is, a subset $\Phi$ of $\mathrm{Hom}(K, \overline{\mathbb{Q}})$ such that $\Phi \amalg \bar{\Phi}=\mathrm{Hom}(K, \overline{\mathbb{Q}}).$

If now $B$ is another CM abelian variety with the same CM type, one can show that $A$ and $B$ are isogenous over $\overline{\mathbb{Q}}$. In the reference I have found this seems to be an "if and only if". However, I'm confused by the example of an elliptic curve, where you have an imaginary quadratic field, hence two CM types, but only one isogeny class of elliptic curves (if I understood correctly, taking the other CM type you get the dual elliptic curve).

So what is the precise relation between CM types and isogeny classes?

  • $\begingroup$ The isogeny from $A$ to $B$ is required to be $K$-linear (and then is unique up to $K^{\times}$)! It is not true that CM elliptic curves $E$ and $E'$ over $\overline{\mathbf{Q}}$ with specified CM by $K$ are $K$-linearly isogenous. But after precomposing the "action" of $K$ on $E'$ with complex conjugation on $K$ if necessary, we can ensure $E$ and $E'$ have the same CM type and so are $K$-linarly isogenous (so $E$ and $E'$ are isogenous always!). In higher dimension the CM types on $K$ are generally not a single ${\rm{Aut}}(K)$-orbit, so the gimmick in dimension 1 does not carry over. $\endgroup$
    – grghxy
    Jul 1 '15 at 1:11

You can find what you are looking for in Milne's notes on complex multiplication:


The overall idea is that for a CM-field $K$ you can consider a pair $(A,i)$, where $A$ is an abelian variety together with an isomorphism $i : K \to \mathrm{End}^0(A) := \mathrm{End}(A) \otimes \mathbb{Q}$. For another pair $(A',i')$ (for perhaps a different CM field $K'$), there is a natural notion of isogeny of pairs $(A,i) \to (A',i')$, which is an isogeny of abelian varieties that respects the embeddings $i$, $i'$.

We can also consider two CM-types $(K,\Phi)$ and $(K',\Phi')$, and Milne defines an "isomorphism of CM-pairs" to be an automorphism $\alpha : K \to K'$ such that $\Phi = \Phi' \circ \alpha$. What you want then is the following (see Prop. 3.12 in Milne):

There is a bijection between isogeny classes of pairs $(A,i)$, where $A$ is an abelian variety with CM, and isomorphism classes of CM-pairs $(K,\Phi)$.

In the case of elliptic curves, the two CM-types you are describing are isomorphic in this context.


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