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

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 $X_1$ 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 $X/W$. As a consequence, it gives 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 with the canonical curve and its associated canonical crystalline representation in the $p$-adic Teichm\"{u}ller 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.