  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.
