Uniformization as Tannakian Reconstruction

Abstract

Classical hyperbolic uniformization identifies every hyperbolic log-orbi curve with a compactified quotient of the upper half-plane by a cofinite Fuchsian lattice. The lattice is unique up to conjugacy. We reconstruct it intrinsically. For each hyperbolic log-orbi curve C we construct a canonical maximal principal PSL2-Higgs object. Etale-locally it comes from the standard square-root SL2-model. The central mu2 ambiguity disappears after passage to PSL2. Using vector tame non-abelian Hodge theory and regular-singular Riemann--Hilbert as input we assemble the required principal realizations Tannakianly. Parahoric structures encode the orbifold and cusp data on the coarse curve. After choosing a base point and conjugating the Betti realization is represented by a discrete faithful finite-covolume representation whose image is the uniformizing lattice. Compatibility with finite etale pullback makes the lattice construction a quasi-inverse to the compactified quotient functor. Thus classical uniformization is recast as an intrinsic Tannakian reconstruction theorem. We also identify finite etale covers with finite continuous sets for the profinite completion of the reconstructed lattice. After fixing a separable closure and the resulting geometric generic point we recover the absolute Galois group of the function field of C as the inverse limit of the based etale fundamental groups of orbifold models over C.