Uniformization as Tannakian Reconstruction

Abstract

We formulate hyperbolic uniformization as a Tannakian reconstruction theorem. For a hyperbolic log-orbi curve C, we construct an intrinsic canonical maximal parahoric PSL2-Higgs object. A tensor-functorial parahoric non-abelian Hodge-Riemann-Hilbert correspondence identifies its Betti realization with a representation of the orbifold fundamental group whose image is the uniformizing cofinite Fuchsian lattice. This construction quasi-inverts the compactified quotient functor from cofinite Fuchsian lattices in PSL2(R) to hyperbolic log-orbi curves. We also prove a Galois enhancement, identifying finite etale covers of C with finite continuous sets for the profinite completion of the reconstructed lattice. As an application, for a one-variable complex function field F, we recover its absolute Galois group as the inverse limit of etale fundamental groups of orbifold models and identify the hyperbolic sector with a compatible Fuchsian-lattice-valued pro-system.