Hello, is there any classification of proper maximal subroups of $GL_2(\mathbb{Z}/p^k\mathbb{Z})$ for $k>1$ (analogous to the one which exist for $GL_2(\mathbb{Z}/p\mathbb{Z})$)? Could you give me some references? thank you very much.
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1$\begingroup$ Do you have a reference for the classification you mention related to the case $k=1$? $\endgroup$– Leandro VendraminCommented Mar 4, 2012 at 21:32
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2$\begingroup$ 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-$p$. $\endgroup$– Colin ReidCommented Mar 4, 2012 at 23:32
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$\begingroup$ @leandro : for $k=1$ for instance Serre J. P. Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. $\endgroup$– Marusia RebolledoCommented Mar 5, 2012 at 9:24
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$\begingroup$ @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 $p^j, j<k$. $\endgroup$– Marusia RebolledoCommented Mar 5, 2012 at 9:36
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1$\begingroup$ @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 $p=3$. $\endgroup$– Tim DokchitserCommented Mar 5, 2012 at 15:39
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