Timeline for Upper bound on order of finite subgroups of GL_n(Z_p)?
Current License: CC BY-SA 3.0
9 events
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| Jun 22, 2022 at 8:13 | history | edited | CommunityBot |
replaced http://math.uga.edu/~pete with http://alpha.math.uga.edu/~pete
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| Apr 21, 2013 at 20:49 | vote | accept | Jim Humphreys | ||
| Apr 21, 2013 at 17:46 | comment | added | Pete L. Clark | @Geoff: I'm sorry, evidently I haven't been clear enough on this point: the bound does not only depend on $n$ and $p$. It fully depends on $n$, $p$, the ramification index $e$ and the inertial degree $f$: if you fix any three of them and let the fourth go to infinity, the supremum will be infinite. | |
| Apr 21, 2013 at 17:26 | comment | added | Geoff Robinson | @Pete: This works for a fixed $\mathbb{K}$ or $R.$ But I am not sure that the bound only depends on n and p, as Jim sought. I do not think there is a bound which only depends on n and p, as you can't a priori bound the number of p′-roots of unity in R just in terms of n and p. | |
| Apr 21, 2013 at 13:54 | comment | added | Jim Humphreys | @Pete: This looks very helpful, though I'll need to sort references out more carefully. I've tracked down my ancient copy of Serre's 1964 Harvard notes, where I found to my surprise that I had once underlined with red pencil some passages in the LG part including Theorem 1 in Appendix 3, page LG 4.35, based on earlier theorems. For some reason I looked at Bourbaki's Chapter 3 without getting any help there. I've always found Serre's notes hard to navigate, so didn't think to look there. | |
| Apr 21, 2013 at 2:31 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Apr 21, 2013 at 1:56 | history | edited | Pete L. Clark | CC BY-SA 3.0 |
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| Apr 21, 2013 at 1:11 | comment | added | Pete L. Clark | Also, although I certainly don't claim that any of these bounds are sharp, I believe, in line with the above comments, that easy examples will show that if you fix any two of $e$ (the ramification index of $K/\mathbb{Q}_p$), $f$ (the residual degree of $K/\mathbb{Q}_p$) and $n$ and let the third one grow, then there will be no uniform bound. | |
| Apr 21, 2013 at 0:54 | history | answered | Pete L. Clark | CC BY-SA 3.0 |