For every prime p and integer n ≥ 3 we explicitly construct an abelian variety A/Fpn of dimension n such that for a suitable prime l the group of quasi-isogenies of A/Fpn 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 the recent construction of topological automorphic forms which generalizes topological modular forms [BL1].
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.