Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
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 –  Emerton Jun 26 '12 at 3:09
@Emerton: Thank you ! Has the distribution law in the CM case been already known in general ? –  user4245 Jun 26 '12 at 3:41
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 $\pi \simeq \pi \otimes \eta$. –  David Hansen Jun 26 '12 at 3:56
David Hansen: Thanks ! It looks like a characterization rather than a definition. –  user4245 Jun 26 '12 at 6:27
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, –  Emerton Jun 26 '12 at 12:32

1 Answer 1

up vote 5 down vote accepted

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.