The Inverse Galois Problem for Convergent Arithmetic Power Series

In the late eighties, D. Harbater proved that every finite group is a Galois group over the fraction field of a ring of convergent arithmetic power series, namely the ring of power series in Z[[T]] that converge on the complex open unit disc. We give a new geometric proof of this result by constructing a cover of the open unit disk in the affine Berkovich line over Z.