  We generalize the Ahlfors-Bers theory to the adelic Riemann sphere. In
particular, after defining the appropriate notion of a Beltrami differential in
the solenoidal context, we give a sufficient condition on it such that the
corresponding Beltrami equation has a quasiconformal homeomorphism solution;
i.e. The Ahlfors-Bers Theorem in the solenoidal case. This additional condition
on the solenoidal Beltrami differentials can be written as a Banach norm in a
subspace of solenoidal differentials. Moreover, this subspace is the completion
under this norm of those solenoidal differentials locally constant at the
fiber. As a toy example, we show how this technique works on a linear problem:
We generalize the diophantine equation complex analytic extension problem to
the respective solenoidal space.
