Hodge-Tate theory for p-adic character varieties

p-adic character varieties are rigid analytic moduli spaces for p-adic
representations of fundamental groups. For curves, I will explain how
these varieties lead to a non-abelian generalisation of the p-adic
Hodge-Tate decomposition, by way of an arithmetic Hitchin fibration. I
will then report on joint work in progress with Magnus Carlson and Ruth
Wild which aims to use this perspective for a new approach to anabelian
geometry for curves.