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Dear friends,

I have some trouble finding a precise definition of what a modular form with complex multiplication. Could anyone provide such a definition and references for the study of CM modular forms and its main properties? I would be grateful

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2 Answers 2

Let $f$ be a newform of level $N$ and weight $k\geq 2$. We say $f$ has CM by the quadratic field $K$ is there exists a quadratic extension $K/\mathbb Q$ such that if $\eta_{K/\mathbb Q}$ is the quadratic character whose kernel is $G_{K}$ then the automorphic representation $\pi(f)$ of $\operatorname{GL}(2,\mathbb A_{\mathbb Q})$ is isomorphic to $\pi(f)\otimes\eta_{K/\mathbb Q}$. If this is true, then $K$ has to be an imaginary quadratic extension. More generally, if $F$ is a totally real field and $\pi$ is an automorphic representation (EDIT: as wccanard points out, here again the condition that the weight should be greater than $2$ has to be included) of $\operatorname{GL}(2,\mathbb A_{F})$ isomorphic to $\pi\otimes\eta_{K/F}$ for $K/F$ quadratic then $K$ is a CM extension (a totally imaginary quadratic extension of $F$).

As Marc Palm writes, when $f$ has CM by $K$ there exists a character $\chi$ of $\mathbb A_{K}^{\times}/K^{\times}$ such that for all finite place $v$, the $L$-factor $L_{v}(f,s)$ of $f$ is equal to the product $\underset{w|v}{\prod}L_{w}(\chi,s)$ of $L$-factors of $\chi$ over places of $K$ above $v$. A highbrow version of this last statement is that $\pi(f)$ is isomorphic to the automorphic induction of $\chi$ from $K$ to $F$.

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Thanks, this is a great answer –  mod78 Feb 13 '13 at 9:34
    
Let me add a reference. You can read Motives and automorphic forms: the potentially abelian case, available on L.Fargues webpage. This is a modern exploration of the topic (which contains much much more than the answer to your question). –  Olivier Feb 13 '13 at 10:31
    
Olivier -- your assertion about $K$ being CM is false in the generality that you write it, in the totally real case (at least at the time I am writing this comment); there are for example Hilbert modular forms of parallel weight 1, and also automorphic representations of Artin type attached to non-holomorphic $\pi$s, which are isomorphic to a quadratic twist of themselves where the associated quadratic extension is not $CM$. In fact in the totally real case $K$ may be neither totally real nor totally imaginary. –  user30035 Feb 13 '13 at 20:49
    
Dear wccanard (!), I did write that $k$ should be greater than 2, but now I realize I did not repeat the condition when passing to $F$. Thanks for pointing this out. –  Olivier Feb 14 '13 at 8:48
    
OK I'm happy :-) If $K$ is any quadratic extension of $F$, totally real or totally imaginary or otherwise, and if $\chi$ is a grossencharacter of $K$ then of course you can automorphically induce $\chi$ up to $GL(2)/F$ by standard converse theorems and Hecke/Tate. The point is that if $K$ isn't CM then you have far less choice about what you can do at infinity because of units. Kevin –  user30035 Feb 14 '13 at 20:46

One indirect way to define it would be that it is the Inverse Mellin tranform of the Hasse-Weil L-function of an elliptic curve with complex multiplication.

For more direct things look at the references suggested on pg. 118 and pg.166ff. in Shimura's "Abelian Varities with CM"

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Is that also true for modular forms of weight $>2$? –  mod78 Feb 13 '13 at 9:21
    
Ah okay, my answer applies only to weight 2 things. I leave it though, since Olivier's perfect anwer addresses mine. –  plusepsilon.de Feb 13 '13 at 9:56

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