In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a totally real field, which is not of CM type. I could not find any definition or reference for "CM type" in that paper. But I expect it should correspond to CM elliptic curve in the classical modular case.

My question is :

  1. What is the precise definition for "an automorphic representation of CM type", both in the $GL_2$ case here and for general reductive group over number fields.

    I prefer a definition "purely" in terms of representation-theory, not of arithmetic-geometry.

  2. Why is the CM case excluded in that paper ?

Any comments or references will be very welcome. Thanks

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    $\begingroup$ Dear unknown, The distribution law for Hecke eigenvalues is different in the CM and non-CM cases (e.g. because the Mumford--Tate groups are quite different in the two cases). This is probably discussed in various expository articles about Sato--Tate, such as the one by Mazur. Regards, Matthew $\endgroup$ – Emerton Jun 26 '12 at 3:09
  • $\begingroup$ @Emerton: Thank you ! Has the distribution law in the CM case been already known in general ? $\endgroup$ – user4245 Jun 26 '12 at 3:41
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    $\begingroup$ A "pure" representation-theory definition: a $GL_2$ automorphic representation $\pi$ is of CM type if there is a quadratic idele class character $\eta$ such that $\pi \simeq \pi \otimes \eta$. $\endgroup$ – David Hansen Jun 26 '12 at 3:56
  • $\begingroup$ David Hansen: Thanks ! It looks like a characterization rather than a definition. $\endgroup$ – user4245 Jun 26 '12 at 6:27
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    $\begingroup$ Dear unknown, The distribution in the CM case has been known since Hecke, and is much simpler. (The relevant $L$-functions reduce to abelian, or Hecke, $L$-functions for the CM extension, whose analytic properties were established by Hecke.) Regards, $\endgroup$ – Emerton Jun 26 '12 at 12:32

1.-- in the $Gl_2$-case, $\pi$ is of CM type if it is the automorphic induction of a Grossencharacter of a CM extension K of $F$. In terms of the Galois representation of $Gal(\bar F/F)$ attached to $\pi$, that means that $\pi$ is not the induced representation from a character of a subgroup $Gal(\bar F/K)$ of index $2$, where $K$ is a CM extension of $F$.

In the general case, the notion of CM stratifies into many different notions. Read things about the Mumford-Tate groups for more about this.

2.-- because already in the $F=\mathbb{Q}$-case, the Sato-Tate conjecture excludes the CM case.

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