# Reference for: CM Hilbert Modular forms arise from Hecke characters

For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a standard reference.)

I am looking for a reference for the analogous result for Hilbert modular forms over a totally real field F. In particular, if the form has CM then it arises from a Hecke character on a quadratic imaginary extension $K$ (over $F$.) I believe, for the converse, Yoshida/Hida is the reference. Thanks

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Just be clear, is your definition of CM that there is some totally imaginary quadratic extension $K$ of $F$ such that for a set of prime ideals of $F$ of density 1 the coefficient $a_\mathfrak{p}$ is zero if and only if $\mathfrak{p}$ is inert in $K/F$? –  Rob Harron May 15 '12 at 15:14
@Rob: Yes, this is the definition. –  unramified May 15 '12 at 15:20
I don't know a reference for this result. Personally, I'd take the fact that a modular form is induced from a Hecke character of $K$ to be the definition of CM, but that doesn't help answer your question. Have you tried generalizing Ribet's argument? Otherwise, this seems like the type of thing that would show up somewhere in a paper/book of Hida's. –  Rob Harron May 15 '12 at 16:12
FWIW non-algebraic Hecke characters can give rise to HMFs with weights which aren't congruent mod 2 and hence don't have associated Galois representations. This presents an obstruction to proving the result using arguments on the Galois side in this generality, which presumably you can try to get around by using some symmetric square argument. –  Kevin Buzzard May 15 '12 at 21:14