Higgs-de Rham flow in the non-abelian p-adic Hodge theory and a p-adic analogue of the uniformization theory of a hyperbolic Riemann surface

Abstract:

The notion ``Higgs-de Rham flow'' over schemes  X/W(k) over the ring of Witt vectors of finite field k of characteristic p  introduced in a recent paper by Lan-Sheng-Zuo has found some interesting applications in arithmetic geometry.

Higgs-de Rham flow induces a correspondence between semistable graded Higgs bundles with c_i=0 and crystalline representations of the algebraic fundamental group of the generic fibre of X, an p-adic analogue of the well known correspondence  between polystable graded Higgs bundles of c_i=0 and polarized complex variation of Hodge structures. In my lecture I shall talk about  application of  Higgs-de Rham flow on uniformization of hyperbolic p-adic curves, which is  closely related to S. Mochizuki's p-adic Teichmueller thoery. This is a joint work with Lan-Sheng-Yang.