Timeline for Non-abelian class field theory and fundamental groups
Current License: CC BY-SA 3.0
10 events
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| Nov 30, 2011 at 5:58 | history | edited | Emerton | CC BY-SA 3.0 |
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| Jul 15, 2010 at 0:51 | history | edited | Emerton | CC BY-SA 2.5 |
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| Jul 14, 2010 at 10:38 | comment | added | Minhyong Kim | To answer my own question, I guess the answer is 'yes', by what we know about unramified extensions of $\mathbb{Q}$. So the only reasonable question would be 'is it impossible for purely automorphic reasons' in a sense I hope is clear enough. | |
| Jul 14, 2010 at 6:11 | comment | added | Minhyong Kim | Regarding the point that's been made about discrete series, let me just reveal my ignorance by asking the stupid question that gave me reason to worry. If you look at the unramified discrete series cusp forms of the sort constructed by Shin, is it impossible for it to transfer to a non-discrete one on some $GL_n$ corresponding to an Artin representation? | |
| Jul 14, 2010 at 5:48 | comment | added | Victor Protsak | The Langlands Program grew out of Langlands' realization that Class Field Theory may be reformulated as an equivalence between 1-dim representations of the Galois group and automorphic forms on $GL_1.$ Thus "linear structure", in the sense described by Matt, is intrinsic to the LP. If you want something that does not involve it, a better choice would be a theory outputting some anabelian version of the Galois group (e.g. nilpotent completions have been studied). | |
| Jul 14, 2010 at 5:12 | comment | added | Emerton | I agree, but unfortunately it seems to really investigate it as a completion (!), passing through each possible dimension in turn. | |
| Jul 14, 2010 at 5:09 | comment | added | Minhyong Kim | But still, a portion of the Langlands programme is about the $\mathbb{C}$-algebraic completion of $\pi_1(O_K)$, and it would be nice if it could tell us whether or not the algebraic completion is trivial. | |
| Jul 14, 2010 at 5:04 | vote | accept | Minhyong Kim | ||
| Jul 14, 2010 at 4:45 | history | edited | Emerton | CC BY-SA 2.5 |
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| Jul 14, 2010 at 4:38 | history | answered | Emerton | CC BY-SA 2.5 |