Let $f$ be a classical modular form of weight $k \geq 2$ and $M_f$ be the motive attached to it. If $M_f$ is a CM motive then is it true that $f$ has CM? For weight $2$, I believe this is a result of Shimura. He shows that if the variety $A_f$ has CM then it can be written as a product of several copies of an elliptic curve E with CM by $K$ and $f$ arises from the Hecke character associated to $K$.
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$\begingroup$ Hello unramified. I am sure that the answer is yes. T he proof seems straightforward to me, so probably I am missing something. Could you precise how you define "$M_f$ is a CM motive"? $\endgroup$– JoëlCommented Nov 30, 2011 at 2:00
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$\begingroup$ If $M_f$ is the motive attached to $f$ then $M_f$ has CM if $2dim(M_f) = End(M_f) \otimes \mathbb{Q}$. $\endgroup$– unramifiedCommented Nov 30, 2011 at 9:54
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$\begingroup$ I'm not sure how to edit comments but there should be a dim on the right side as well. $\endgroup$– unramifiedCommented Nov 30, 2011 at 12:58
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$\begingroup$ Not sure I understand your definition yet. $dim(M_f)$ is 2, no? So you want $End(M_f)$ to be of dimension $4$ ?? $\endgroup$– JoëlCommented Nov 30, 2011 at 16:20
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$\begingroup$ And also, how do you define $End(M_f)$? In which category of motives are you working on? I am not asking which construction (Chow motives, homological motives, etc.), but over which base field? $\mathbb{Q}$? Thanks. I am not asking just for the sake of precision: I do believe that when the definitions are written down, the proof will be obvious to anyone. $\endgroup$– JoëlCommented Nov 30, 2011 at 16:26
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1 Answer
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The answer is yes. Motives $M$ with CM are determined by algebraic Hecke characters $\chi$. Hence their associated modular forms $f$ have CM. References that contain detailed discussions about this relation include articles by Anderson (1986), and Blasius (1986), as well as Schappacher's contribution to the Motives volume edited by Jannsen, Kleiman and Serre, AMS 1994.
