Timeline for maximal subgroups of $GL_2(Z/p^kZ)$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2012 at 19:19 | comment | added | Marusia Rebolledo | sorry : I never tanked for your answers so I just do it : thank you very much! | |
| Mar 5, 2012 at 15:39 | comment | added | Tim Dokchitser | @Marusia: If I remember correctly, this particular type of "bizarre" behaviour does not occur for $p>3$: In the paper that you mention, Serre has a group-theoretic argument to prove that when a subgroup of $GL_2(Z_p)$ surjects onto $SL_2(F_p)$, it must contain the whole of $SL_2(Z_p)$. So it easy to classify maximal subgroups that are "not proper at level $p$", just by looking at the determinant. For $p=2$ and 3 there are indeed proper maximal subgroups that surject onto $GL_2(F_p)$ - e.g. this is in arxiv.org/abs/1104.5031 for $p=2$ and Elkies' paper referenced in there for $p=3$. | |
| Mar 5, 2012 at 15:02 | comment | added | Leandro Vendramin | I am not sure, but maybe this paper is useful. For example, it contains a big list of references: staff.ncl.ac.uk/o.h.king/KingBCC05.pdf | |
| Mar 5, 2012 at 9:36 | comment | added | Marusia Rebolledo | @colin : yes but I am interested in a description at the level $p^k$ for a fixed $k$ (in the same flavour that the description for $k=1$). Maybe it is easy, please tell me. I am in trouble by some "bizarreries", for instance of course a proper maximal subgroup at the level $p^k$ can be not yet proper for $p^j, j<k$. | |
| Mar 5, 2012 at 9:24 | comment | added | Marusia Rebolledo | @leandro : for $k=1$ for instance Serre J. P. Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. | |
| Mar 4, 2012 at 23:32 | comment | added | Colin Reid | You can ask for all $k$ at once by asking for the maximal open subgroups of $GL_2(\mathbb{Z}_p)$. There are only finitely many of them as this group is finitely generated and virtually pro-$p$. | |
| Mar 4, 2012 at 21:32 | comment | added | Leandro Vendramin | Do you have a reference for the classification you mention related to the case $k=1$? | |
| Mar 4, 2012 at 20:39 | history | asked | Marusia Rebolledo | CC BY-SA 3.0 |