Timeline for $A_5$-extension of number fields unramified everywhere
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2015 at 23:42 | history | edited | GH from MO |
edited tags
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| Mar 14, 2011 at 11:33 | answer | added | Chandan Singh Dalawat | timeline score: 9 | |
| Nov 4, 2010 at 13:14 | vote | accept | Olivier | ||
| Nov 4, 2010 at 12:22 | answer | added | David E Speyer | timeline score: 24 | |
| Nov 4, 2010 at 11:23 | answer | added | Robin Chapman | timeline score: 20 | |
| Nov 4, 2010 at 11:08 | answer | added | Kevin Buzzard | timeline score: 13 | |
| Nov 4, 2010 at 11:04 | comment | added | Kevin Buzzard | You'll need the base to contain $Q(\zeta_5)$ or something, to stop it being ramified at 5. | |
| Nov 4, 2010 at 11:00 | comment | added | François Brunault | I don't know whether this is relevant here, but since $A_5$ is isomorphic to $\mathrm{PSL}_2(\mathbf{F}_5)$, one could try to start with an elliptic curve over some number field with good reduction everywhere, and look at the Galois representation on the 5-torsion points. | |
| Nov 4, 2010 at 10:50 | comment | added | Kevin Buzzard | The problem with using Hilbert modular forms is that it is very common that the associated mod $p$ representation is ramified at $p$, making the problem computationally very hard. For example if you want to compute in parallel weight 2 then even the determinant of the mod $p$ representation will be ramified at $p$ unless $p=2$. This rules out using, say, standard tables of elliptic curves of conductor 1. | |
| Nov 4, 2010 at 10:47 | comment | added | Kevin Buzzard | I've forgotten the standard example, but I think there's a standard example. You write down some explicit polynomial of degree 5 with rational coefficients (it's something like $x^5+x+1$ but it's not exactly that) and its Galois group over $\mathbf{Q}$ is $S_5$, and then you let the base be the quadratic subextension. | |
| Nov 4, 2010 at 10:33 | history | asked | Olivier | CC BY-SA 2.5 |