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

Abstract

Abstract 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 {X_{1}} defined over k, there exists a lifting X of the curve to the ring W ⁢ ( k ) {W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over X / W ⁢ ( k ) {X/W(k)} . These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group π 1 ⁢ ( X K ) {\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 Teichmüller 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.