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 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.

To appear in Crelle's Journal

References in corpus (4)

• Logarithmic nonabelian Hodge theory in characteristic p
• Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
• Nonabelian Hodge theory in positive characterstic via exponential twisting
• On the deformation of a Barsotti-Tate group over a curve

Cited by in corpus (2)

• Nearby cycles and semipositivity in positive characteristic
• Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups