Niko Naumann
Quasi-isogenies and Morava stabilizer groups
niko.naumann@mathematik.uni-regensburg.de

For every prime $p$ and integer $n\ge 3$ we explicitly construct an
abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable
prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power
degree is canonically a dense subgroup of the $n$-th Morava stabilizer
group at $p$.  We also give a variant of this result taking into account
a polarization.  This is motivated by a perceivable generalization of
topological modular forms to more general topological automorphic forms.

For this, we prove some results about approximation of local units in
maximal orders which is of independent interest. For example, it gives a
precise solution to the problem of extending automorphisms of the
$p$-divisible group of a simple abelian variety over a finite field to
quasi-isogenies of the abelian variety of degree divisible by as few
primes as possible.