Timeline for Non-abelian class field theory and fundamental groups
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22 events
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| Jul 26, 2010 at 17:55 | answer | added | user631 | timeline score: 31 | |
| Jul 14, 2010 at 5:04 | vote | accept | Minhyong Kim | ||
| Jul 14, 2010 at 4:38 | answer | added | Emerton | timeline score: 29 | |
| Jul 14, 2010 at 4:26 | comment | added | Minhyong Kim | Matthew: I guess I'd heard this in a lecture of Lafforgue before, but never understood how it goes. How does one prove finiteness of automorphic representations of bounded weight (which I presume is defined in terms of the 'Hodge numbers') and fixed level? Perhaps you could briefly discuss this in your answer? | |
| Jul 14, 2010 at 4:12 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Jul 14, 2010 at 3:47 | comment | added | Emerton | One more thing re. discrete series: even in the $GL_2(\mathbb R)$ case, discrete series are never relevant to Galois-type cuspforms (classical weight 1 is limit of discrete series, not discrete series, as James Newton noted). This is what lies behind David Hansen's comment. The trace formula methods, such as the one of Shin that Minhyong refers to, don't work very well in non-discrete series situations. Nevertheless, as Selberg's computation (mentioned above) shows, they can yield some information! So I don't think that (for fixed $n$) this approach is intrinsically ridiculous. | |
| Jul 14, 2010 at 3:44 | comment | added | Emerton | ... there is any particular way to study all $n$ simultaneously from this point of view. In other words, for a given $K$, if we compute all the way up to $n = 10^6$, we can't rule out that there isn't some fascinating unramified extension waiting for us whose lowest degree irrep. just happens to be of enormous dimension. | |
| Jul 14, 2010 at 3:42 | comment | added | Emerton | ... irred. $n$-dim'l rep. I don't think that it is "in principle" impossible to calculate this list, for $K$ fixed; for example, we can compute that there are no holomorphic weight one forms fairly easily, and Selberg's original form of the trace formula was used (by him) to show that the smallest eigenvalue of a Maass form of level one was well above 1/4 (closer to 90, maybe?). So when $K = \mathbb Q$ and $n = 2$, this approach is practical. What seems much harder is to deal with the fact is that this point of view forces one to treat each $n$ separately, and it doesn't seem that ... | |
| Jul 14, 2010 at 3:39 | comment | added | Emerton | ... for each prime $v$ lying over $\infty$. (These are the cuspidal rep's called "Galois type", especially in the older literature.) Since the $\pi_{v}$ for $v|\infty$ are prescribed to lie in a finite list, and since we are interested in the ramification at finite primes being trivial, we see that there will only be finitely many such $\pi$ (in classical terms, the weight is restricted to a bounded range and the level is fixed to equal 1), and so we conclude that for $K$ fixed, there are only finitely many everywhere unramified extensions $L$ such that $Gal(L/K)$ admits an ... | |
| Jul 14, 2010 at 3:35 | comment | added | Emerton | Regarding the Artin rep. subthread: there are no discrete series for $GL_n(\mathbb R)$ when $n > 2$, or $GL_n(\mathbb C)$ when $n > 1$, so we can ignore that particular issue. More helpfully: for each $n$ there is a certain, precise, and finite list of candidates $\pi_{\infty}$ for automorphic representations corresponding to Artin rep's of dim'n $n$ (so that in the Langlands paradise, with all conjectures proved, the n-dim'l irred. Artin reps. of $Gal(\bar{K}/K)$ are in bijection with the cuspidal automorphic reps. of $GL_n(\mathbb A_K)$ having $\pi_{v}$ in this particular list, ... | |
| Jul 13, 2010 at 16:11 | comment | added | jnewton | So I guess what I wrote above is just saying that the automorphic reps corresponding to Artin reps are those with infinitesimal character zero at $\infty$. By the way, I don't think these will be discrete series at $\infty$ - for GL_2(R) they're either limit of discrete series (the holomorphic weight one case) or principal series (the Maass form case). | |
| Jul 13, 2010 at 15:35 | comment | added | jnewton | @Minhyong: My impression was that the automorphic representations corresponding to Artin reps are characterised by their local factors at archimedean places e.g. for GL_2/Q, if the Artin rep is odd it comes from a holomorphic modular form of weight 1, if it is even it comes from a non-holomorphic Maass form with Laplacian eigenvalue 1/4. I guess the local factors at arch. places should match under local Langlands with arch. Weil group reps which factor through the local Galois groups, given by the actions of complex conjugations on the Artin rep. | |
| Jul 13, 2010 at 14:31 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Jul 12, 2010 at 20:47 | comment | added | Dror Speiser | Does anyone know of an answer when we change "Langlands programme"'s role in the above to any other set of mostly-believed conjectures? | |
| Jul 12, 2010 at 20:38 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Jul 12, 2010 at 17:20 | comment | added | David Hansen | @jnewton: It is indeed very difficult; the relevant automorphic representations cannot be counted by any naive application of the trace formula. This dichotomy is already apparent in classical modular forms - forms of weight $\geq 2$ and a given level can be counted precisely, but counting forms of weight $1$ is very difficult. | |
| Jul 12, 2010 at 17:08 | comment | added | jnewton | I guess the (strong) Artin conjecture tells you that given $L/K$ unramified, then non-trivial irreducible n-dimensional (complex) representations of $Gal(L/K)$ arise from cuspidal automorphic representations of $GL_n(\mathbb{A}_K)$ of `level 1' (i.e. unramified at all finite places) and certain type at the archimedean places. So if we can rule out the existence of any of these, we can conclude that there's no non-trivial unramified $L$ - this may well be difficult to do though. | |
| Jul 12, 2010 at 15:54 | comment | added | David Hansen | Minhyong, KConrad gives some further interesting examples of fields with no unramified extensions here: mathoverflow.net/questions/26491 | |
| Jul 12, 2010 at 15:24 | comment | added | Andy Putman | This is a fantastic question, but maybe it could have a more descriptive title? | |
| Jul 12, 2010 at 14:35 | history | edited | Minhyong Kim | CC BY-SA 2.5 |
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| Jul 12, 2010 at 11:53 | history | edited | Robin Chapman |
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| Jul 12, 2010 at 11:49 | history | asked | Minhyong Kim | CC BY-SA 2.5 |