Take the 2-minute tour ×  Question: Say $p$ is an odd prime, and
take two matrices $A,B\in GL_n({\mathbb Z}_p)$ of finite order
$m$. Is it true that if their reductions mod $p$ are conjugate
in $GL_n({\mathbb F}_p)$ then they are conjugate in
$GL_n({\mathbb Q}_p)$?  This is a follow-up to the question
whether $GL_n({\mathbb F}_p)$-conjugacy implies $GL_n({\mathbb
Z}_p)$-conjugacy, for which the answer is "No".  For $p=2$ this
is not true, although I would be very much interested to know
whether conjugacy mod 4 implies conjugacy over ${\mathbb Q}_2$.
(Note that if the answer is "Yes", then by character theory any
two representations $\rho, \rho': G\to GL_n({\mathbb Z}_p)$ that
are equivalent mod $p$ are equivalent in $GL_n({\mathbb Q}_p)$.)
share|improve this question This seems relevant:
mathoverflow.net/questions/69578/… –  Igor Rivin Aug 9 '11 at
18:53 I think my answer, for the other question (I misread) is
now correct for the direction of $F_p$ conjugacy implying $Q_p$
conjugacy. Namely, given a general