Twisting Higgs Modules and Functorial Aspects of the p-adic Simpson Correspondence
The classical Simpson correspondence describes complex linear representations of the fundamental group of a smooth complex projective variety in terms of linear algebra objects, namely Higgs bundles. Its p-adic analogue, introduced by G. Faltings, aims to understand continuous p-adic representations of the geometric fundamental group of a smooth projective variety over a p-adic local field. The main goal of this work is to establish a robust framework for studying the functoriality of the p-adic Simpson correspondence. We introduce a new method for twisting Higgs modules via Higgs-Tate algebras. This construction builds on one of our earlier approaches to the p-adic Simpson correspondence, which it recovers as a special case. The resulting framework yields twisted pullbacks and higher direct images of Higgs modules, thereby enabling a systematic study of the functoriality of the p-adic Simpson correspondence under arbitrary pullbacks and proper (log)smooth direct images, including for morphisms that do not admit liftings to the infinitesimal deformations used in the construction of the correspondence. We also clarify how this new twisting relates to the constructions of Heuer and Heuer-Xu, involving line bundles on the spectral variety.