Let $F$ be a non-archimedean local field, $\mathfrak o$ its ring of units and $\mathfrak p$ its unique maximal ideal. I would like to show that $K=GL_2(\mathfrak o)$ is the unique maximal compact open subgroup of $GL_2(F)$ up to conjugacy.
The hints I am given suggest to relate this to lattices, and I don't see much the relation. The hints are as follow:
- show that there exists a $K$-stable $\mathfrak o$-lattice (here we can take whatever $\mathfrak o$-lattice $\Lambda$ and then $K\Lambda$ is $\mathfrak o$-stable)
- show that the only $GL_2(\mathfrak o)$-stable lattices are $\mathfrak p^j \oplus \mathfrak p^j$ (I guess this is merely a matter of operating on line/columns and looking at valuations)
However, what next? I don't see how to relate the properties of these lattices to a relation between $K$ and $GL_2(\mathfrak o)$. Thanks in advance for your help.
If you have two lattices $L, L'$, try to find $g \in GL_2(F)$ such that $g(L)=L'$.
Now if $K, K'$ are the stabiliser subgroups corresponding to the respective lattices, how are they related, using the above $g$?