Class invariants in a non-archimedean setting

One of the goals of explicit class field theory is to compute 
generating polynomials of abelian extensions of a given number field.
For imaginary quadratic fields, the theory of complex multiplication
provides us with an elegant solution. The classical approach has two
improvements. Firstly, one can use `smaller' functions than the classical
j-function, leading to smaller polynomials. This leads to the theory of 
class invariants, which was initiated by Weber. Secondly, one can work in a
non-archimedean setting, avoiding the problem of rounding errors in the
classical approach. In this talk we will combine both improvements, i.e.,
we will show how to use class invariants in a non-archimedean setting.
Examples will be given.