My setting is the $p$-adic field. $K$ (finite extension of $\Bbb Q_p$.) Proposition 2.1.18 seems to claim that
Any open subgroup of $GL_n(K)$. contains $U_n(K)$ the unipotent upper triangular matrices.
Is this true? If so, is there a reference?
$\begingroup$
$\endgroup$
3
-
3$\begingroup$ This is true if the subgroup is normal. This not true in general: e.g. take a congruence subgroup. $\endgroup$– Paul BroussousCommented Aug 23, 2021 at 19:33
-
$\begingroup$ Thanks, is there a reference for the normal case? $\endgroup$– Bryan ShihCommented Aug 23, 2021 at 20:31
-
3$\begingroup$ I don't know a reference, but the proof is short: You can conjugate any unipotent upper triangular matrix by a diagonal matrix (with $i$th diagonal entry $p^{ni}$, say) to obtain a matrix which is congruent to $1$ modulo an arbitrarily high power of $p$, hence in an arbitrary open neighborhood of the identity. $\endgroup$– Will SawinCommented Aug 23, 2021 at 21:02
Add a comment
|