Timeline for Insolvable number fields ramified only at one (small) prime
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2011 at 13:02 | answer | added | JSE | timeline score: 9 | |
| Oct 18, 2011 at 6:57 | vote | accept | Chandan Singh Dalawat | ||
| Oct 18, 2011 at 6:55 | comment | added | Kevin Buzzard | @Francois/Minhyong: exactly! One needs bigger groups, so one has to wait until we can compute Hilbert modular forms better, and that's what happened, thanks mostly to Dembele. | |
| Oct 18, 2011 at 6:48 | comment | added | François Brunault | One could try to adapt the argument by looking at Artin reps with image $\mathrm{GL}_2(F)$ where $F$ is a finite field of char. $p \in \{2,3\}$. But Serre and Tate showed that every such representation has to be ramified outside $p$ (this is one of the first steps of the proof of Serre's conjecture). So I guess one has to look at other kinds of automorphic forms. | |
| Oct 18, 2011 at 6:45 | answer | added | Kevin Buzzard | timeline score: 12 | |
| Oct 18, 2011 at 6:39 | comment | added | Minhyong Kim | OK, forget all my silly comments. $GL2(F_3)$ is still solvable, of course. But anyways, I guess we can figure out what needs to be done and that we will need automorphic forms on bigger groups. That's where recent work comes in, I suppose. As usual, I will leave my comments up, so others can benefit from my stupidity. | |
| Oct 18, 2011 at 6:30 | comment | added | Minhyong Kim | I may be getting confused, but I guess the 2-adic rep of $\Delta$ would be solvable, so forget my comment. But maybe the 3-adic one still works? If it does, one can take a suitable finite quotient. | |
| Oct 18, 2011 at 6:28 | history | edited | Chandan Singh Dalawat | CC BY-SA 3.0 |
added 7 characters in body
|
| Oct 18, 2011 at 6:27 | comment | added | Minhyong Kim | By the way, one really striking recent application of a similar flavor is this paper of Clozel and Chenevier: ams.org/journals/jams/2009-22-02/S0894-0347-08-00617-6/… | |
| Oct 18, 2011 at 6:22 | comment | added | Minhyong Kim | These must be things roughly like 2-adic or 3-adic reps associated to the $\Delta$ function. I guess these are not too recent. But it's curious that there aren't suppose to be elementary constructions of finite extensions of the right sort. Is this really true? | |
| Oct 18, 2011 at 5:17 | history | asked | Chandan Singh Dalawat | CC BY-SA 3.0 |