Timeline for Computing (on a computer) higher ramification groups and/or conductors of representations.
Current License: CC BY-SA 2.5
7 events
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| Mar 6, 2010 at 22:28 | comment | added | Kevin Buzzard | Pete: I think what's going is the following. If you work globally then at times you might have to solve global problems (e.g. factoring discriminants), but on the other hand all your data is exact. If you work locally then you only know all your data mod p^{50}, which is a huge saving if p=2 and your discriminant is O(10^{1000}), but a big loss if your discriminant is O(10^{10}). My global field is sufficiently simple that apparently a global approach is more efficient. OTOH if it were impossible to compute the integers in my global field in finite time, it might be a different story. | |
| Mar 6, 2010 at 22:00 | comment | added | Pete L. Clark | So it turns out to be a mistake to try to compute ramification groups using local arithmetic, rather than global arithmetic? That's weird. | |
| Mar 6, 2010 at 20:24 | vote | accept | Kevin Buzzard | ||
| Mar 6, 2010 at 10:47 | comment | added | Kevin Buzzard | So in fact what you're saying is that my mistake was to try and do the calculation locally; I should never have used p-adic fields at all! | |
| Mar 6, 2010 at 10:23 | comment | added | David Loeffler |
You can get this in Sage using G.ramification_group(p, n), and in Magma by using RamificationGroup(p, n), where in each case p is the prime of you number field above 2. If you like the answer, please vote it up, I've been stuck on zero reputation for months.
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| Mar 6, 2010 at 10:13 | comment | added | Kevin Buzzard | David this is great---thanks. It's even more efficient than dke's answer. I wonder if I was just using magma incorrectly, as it can obviously do number fields as well. Can it actually tell me the groups? Both answers have only told me the breaks. I guess I can just look this up myself. | |
| Mar 6, 2010 at 10:04 | history | answered | David Loeffler | CC BY-SA 2.5 |