Divisibility properties of integer quaternions and adelic
equidistribution<br> <br>

Manfred Einsiedler, ETH Z"urich <br> <br>

Abstract: <br>
In joint work with Shahar Mozes we derive results of the following
type: "Given a primitive integer quaternion a
very few other integer quaternions are not divisible by a." In the
proof these results depend
on measure rigidity of higher rank Cartan actions respectively adelic
equidistribution of adelic
torus actions. Of course, care must be taken in formulation of the
result as there are actually
many integer quaternions that are not divisible by a. Here are the
two main theorems: <br> <br>

Given a as above there exists some rational integer A such that
every integer quaternion whose norm is divisible
by A is divisible by a.<br> <br>

Given two split primes p and q, define the sub semi group of
quaternions whose norm is a product of powers of p and
q. Then for any primitive element a in this semi group, there
exists some rational integer A such that
the set of integer quaternions in the semi group that are not
divisible by a but whose norm is divisible by A is
of sub-exponential growth.