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

2014/04/02 by Lan, Guitang, Sheng, Mao, Yang, Yanhong +1
#14D07 #14F30 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1404.0538

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 X₁ 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 π₁(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.

Related

• Parallel transport for Higgs bundles over p-adic curves
• Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
• Higgs Bundles in Geometry and Arithmetic
• ℓ-adic local systems and Higgs bundles: the generic case
• Strongly semistable reduction of syzygy bundles on plane curves