Adelic solenoid II: Ahlfors-Bers theory

TL;DR

This paper extends the Ahlfors-Bers theory to the adelic Riemann sphere, establishing conditions for quasiconformal solutions to Beltrami equations in this new setting, and demonstrating the approach with a linear example.

Contribution

It introduces a generalized Beltrami differential framework on the adelic Riemann sphere and proves an Ahlfors-Bers type theorem in this context.

Findings

Defined Beltrami differentials in the adelic solenoidal setting.

Proved existence of quasiconformal solutions under a Banach norm condition.

Applied the theory to a solenoidal version of a diophantine equation problem.

Abstract

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.