Definition of CM modular form

2013-02-13T08:43:26Z

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

Answer by Marc Palm for Definition of CM modular form

2013-02-13T09:13:35Z

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"

Answer by Olivier for Definition of CM modular form

2013-02-13T09:29:56Z

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 $F$.

Definition of CM modular form - MathOverflow

Definition of CM modular form

Answer by Marc Palm for Definition of CM modular form

Answer by Olivier for Definition of CM modular form