| Abstract: |
The classical Simpson correspondence describes complex linear representations of the fundamental group of a smooth complex projective variety in terms of linear algebraic objects, namely Higgs bundles. Inspired by this, Gerd Faltings initiated in 2005 a p-adic analogue, with the aim of understanding continuous p-adic representations of the geometric fundamental group of a smooth projective variety over a p-adic local field.
I will present joint work with M. Gros and T. Tsuji, whose goal is to establish a robust framework for a broader functoriality of the p-adic correspondence. We introduce a new method for twisting Higgs modules via Higgs–Tate algebras. This construction builds on our earlier joint work with M. Gros on the p-adic Simpson correspondence, which it recovers as a special case.
The resulting framework provides twisted pullbacks and higher direct images of Higgs modules, making it possible to study the functoriality of the p-adic Simpson correspondence under arbitrary pullbacks and proper (log-)smooth direct images, even for morphisms that do not lift to the infinitesimal deformations used in the construction of the correspondence.
In this lecture, I will present our twisting approach to the p-adic Simpson correspondence, restricted to the local picture, and illustrate it with a new construction of Sen endomorphism.
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