Timeline for Motive of CM elliptic curve and modular forms
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2022 at 14:57 | comment | added | David Loeffler | You are right, sorry, I slipped up calculating the field of definition of the components. | |
| May 19, 2022 at 14:54 | history | edited | David Loeffler | CC BY-SA 4.0 |
added 55 characters in body
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| May 19, 2022 at 14:44 | comment | added | François Brunault | I think it depends on the level $\Gamma \subset \mathrm{GL}_2(\mathbb{A}_f)$: some building blocks in $J_1(N)$ are defined over fields which are not totally real. | |
| May 19, 2022 at 14:38 | comment | added | David Loeffler | No, this isn't the reason. If $E$ is a CM elliptic curve defined over $\mathbf{Q}$, coming from a CM-field $K$ with class number one, then the CM-endomorphisms are all defied over $K$, which is quadratic and thus certainly abelian over $\mathbf{Q}$. But I think the field of definition of the Hecke correspondences is the real subfield $\mathbf{Q}(\zeta_N)^+$ of $\mathbf{Q}(\zeta_N)$. | |
| May 19, 2022 at 14:36 | comment | added | François Brunault | Maybe the reason is that the complex multiplication endomorphisms of $E$ are not defined over an abelian field in general. | |
| May 19, 2022 at 14:31 | comment | added | François Brunault | Allowing the more general Hecke correspondences attached to double cosets $KgK$ in $\mathrm{GL}_2(\mathbb{A}_f)$, we get correspondences defined over the cyclotomic field $\mathbb{Q}(\zeta_N)$ where $N$ is the level of $K$ (e.g. the building blocks of a modular abelian variety $A_f$ are not defined over $\mathbb{Q}$). | |
| May 19, 2022 at 6:37 | history | answered | David Loeffler | CC BY-SA 4.0 |