Definition of CM modular form - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:14:59Z http://mathoverflow.net/feeds/question/121688 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121688/definition-of-cm-modular-form Definition of CM modular form mod78 2013-02-13T08:43:26Z 2013-02-14T08:49:41Z <p>Dear friends,</p> <p>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 </p> http://mathoverflow.net/questions/121688/definition-of-cm-modular-form/121691#121691 Answer by Marc Palm for Definition of CM modular form Marc Palm 2013-02-13T09:13:35Z 2013-02-13T09:13:35Z <p>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.</p> <p>For more direct things look at the references suggested on pg. 118 and pg.166ff. in Shimura's "Abelian Varities with CM"</p> http://mathoverflow.net/questions/121688/definition-of-cm-modular-form/121693#121693 Answer by Olivier for Definition of CM modular form Olivier 2013-02-13T09:29:56Z 2013-02-14T08:49:41Z <p>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 $F$).</p> <p>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 $F$.</p>