# how to see CM types as functions on the Galois group?

Let $K$ be a CM field, that is, an imaginary quadratic extension of a totally real number field. Its degree $[K: \mathbb{Q}]$ is a en even number $2n$.

(1) For me a CM type is a subset $\Phi \subset \mathrm{Hom}(K, \mathbb{C})$ consisting of $n$ elements such that, if $\sigma$ belongs to $\Phi$, then its complex conjugate $\bar{\sigma}$ doesn't.

(2) But I've seen that people consider a CM type as a locally constant function $\Phi: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathbb{Q}$.

So: how to pass from (1) to (2)?

And there is some condition under which a function like in (2) comes from a CM type like in (1)?

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I think (1) is the universally used definition. Could you give a reference for (2)? – abx Jan 25 '14 at 15:38
This is e.g. the point of view of Colmez in "Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe", link.springer.com/article/10.1023/A:1000390105495#page-1 – totto Jan 25 '14 at 17:23
In §2 of that paper, Colmez explains quite clearly the equivalence between his definition and the classical one. – abx Jan 25 '14 at 17:33
Not clearly enough for me. But this is my problem I guess. Maybe the best solution would be to quit mathematics – totto Jan 25 '14 at 17:59
That... seems like an overreaction. – arsmath Jan 26 '14 at 7:30

The characteristic function of a CM-type in the sense (1) on a subfield of $\bar{\mathbb{Q}}$ is a locally constant function $Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to \{0,1\}\subset\mathbb{Z}$. CM-types in the sense (1) classify CM abelian varieties (up to isogeny) and elements in the $\mathbb{Z}$-module spanned by their characteristic functions classify CM motives (these are locally constant function $Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to \mathbb{Z}$; I don't know where the $\mathbb{Q}$ comes from).