Timeline for Relationship among class field theory, modularity theorem, and the langlands program
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Nov 21, 2016 at 8:55 | comment | added | David Benjamin Lim | @libofmath For dimension 1, try to understand the following statement. There is a continuous bijection between finite index Hecke characters $I_K \to \Bbb{C}^\times$ and continuous representations $G_K \to \Bbb{C}^\times$ where $I_K$ is the idele class group. The bijection essentially follows from existence and properties of the Global Artin map. | |
| Nov 21, 2016 at 8:12 | comment | added | znt | In particular one might note that if $E$ is an elliptic curve over $Q$ and $\pi$ is the corresponding automorphic representation, then Langlands' most general conjecture would predict the existence of a 2-dimensional complex representation of a group that we can't define attached to $\pi$; the image (appropriately normalised) would I believe land in $U(2)$, and $Frob_\ell$ would be sent to a complex 2x2 matrix with char poly $X^2-a_\ell/\ell X+1$; Sato-Tate is that these Frobenii are equidistributed in the image. Nothing to do with the Tate module at all and still open (because no defn of gp). | |
| Nov 21, 2016 at 8:07 | comment | added | libofmath | @znt Thank you for the clarification. I knew that there should be more to Frankel's comment. | |
| Nov 21, 2016 at 7:58 | comment | added | znt | As for class field theory, the answer is yes: Dirichlet characters give auto reps of GL_1, this is a good exercise (but you'll need to know the proper adelic definition of an auto rep so p-adics, adeles, relationship to class groups etc). | |
| Nov 21, 2016 at 7:58 | comment | added | libofmath | @NoahSchweber I will erase the mse question for now. I understand that I should have waited, but I'm in a situation where I might not be able to use the internet in a few days. Thank you for the suggestion. | |
| Nov 21, 2016 at 7:55 | comment | added | znt | There is a subtlety, often glossed over in introductory expositions. There are several correspondences, all conjectural. There's a correspondence between $\pi$'s "of Artin type" and $\sigma$ as in your question (complex Galois reps). There's another one between "algebraic" $\pi$'s and motives (e.g. elliptic curves), via $p$-adic Galois representations, and a third between all $\pi$'s and complex representations of the global Langlands group (which doesn't yet exist). Strictly speaking the answer to your second question is no, because $\sigma$ coming from etale cohomology isn't a complex rep. | |
| Nov 21, 2016 at 7:49 | comment | added | Noah Schweber | The mse question is here. To the OP, I would in general wait more than a day before pushing up to MO (disclaimer - this isn't my field, I can't tell whether this question is appropriate for MO or not, so I'm just saying this as a matter of general practice). | |
| Nov 21, 2016 at 7:41 | review | First posts | |||
| Nov 21, 2016 at 8:00 | |||||
| Nov 21, 2016 at 7:40 | history | asked | libofmath | CC BY-SA 3.0 |