Timeline for Reference for "Gal represenations attached to CM eigenforms"
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2011 at 7:43 | vote | accept | unramified | ||
| Oct 4, 2011 at 2:35 | answer | added | Emerton | timeline score: 7 | |
| Oct 3, 2011 at 20:54 | comment | added | Rob Harron | it leads. Maybe it leads to Shimura. Or maybe he did it independently. Try also checking the references in Ribet's paper. Or I could be completely wrong. | |
| Oct 3, 2011 at 20:50 | comment | added | Rob Harron | Attaching $p$-adic Galois representations to algebraic Hecke characters was done by Weil in the paper in which he introduced algebraic Hecke characters On a certain type of characters of the idèle-class group of an algebraic number-field (1955). The Galois representation of a CM modular form is just going to be the induction of the Galois representation of the associated Hecke character. Now maybe people had already done such things in the special case of im. quad. fields (Weil doesn't mention any such work I don't think), but if not, you could try starting with that paper and seeing ... | |
| Oct 3, 2011 at 20:45 | comment | added | Rob Harron | @David: a CM form is one that is isomorphic to its twist by a quadratic character attached to an imaginary quadratic field. These are the modular forms that arise as automorphic induction from Hecke characters of type $(k-1,0)$ on an imaginary quadratic field ($k\geq1$). Tommaso's suggestion of Ribet's article (dx.doi.org/10.1007/BFb0063943) is a terrific one. | |
| Oct 3, 2011 at 19:33 | comment | added | Tommaso Centeleghe | You can take a look to a paper of Ribet (especially sections 3 and 4), appearing in one of the Antwerp volumes (Modular Functions of one variable VI, LNM 601). There he explains how to construct CM forms from Groessencharacters and the other way around (Thm 4.5). | |
| Oct 3, 2011 at 19:07 | comment | added | unramified | My understanding is that CM eigenforms were originally defined as arising from Hecke characters and then "someone" later showed that this is equivalent to the form being attached to an abelian variety with CM.The Gal repn then just falls out like you said... | |
| Oct 3, 2011 at 18:57 | comment | added | David E Speyer | This may be a dumb question but: How do you define an eigenform having CM, if not by saying that it is attached to an abelian variety with CM? And once you have the abelian variety, the galois representation is obvious -- you just act on the Tate module of the abelian variety. The significance of Eichler-Shimura (and I think also the other papers, though I'm less familiar with them) is showing how to attach an abelian variety to an eigenform. If you say the eigenform has CM, it seems to me that the question is already solved. Apologies if I am missing something here. | |
| Oct 3, 2011 at 18:37 | history | asked | unramified | CC BY-SA 3.0 |