Notes on the local p-adic Simpson correspondence

Abstract

We complete the theory of the local p-adic Simpson correspondence developed by Faltings for rational coefficients. It asserts that there is an equivalence between the category of small \(\mathbb {Q}_p\)-generalized representations of the geometric fundamental group and that of small \(\mathbb {Q}_p\)-Higgs bundles for a certain kind of log smooth affine scheme over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. The difficulty lies in the construction of the latter from the former, and we give it via a generalized Sen’s theory for the log smooth affine scheme, which depends on Faltings’ almost purity theorem. Inspired by a recent work by R. Liu and X. Zhu, we also give a formulation of local p-adic Simpson correspondence for \(\mathbb {Q}_p\)-generalized representations of the arithmetic fundamental group, and a characterization of Hodge-Tate generalized representations in terms of the correspondence.