Uniformization of p-adic curves via Higgs-de Rham flows

Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve X-1 defined over k, there exists a lifting X of the curve to the ring W(k) of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over X / W(k). These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group pi(1) (X-K) of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmuller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin-Simpson's uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.