Reconstructing D-cap from p-adic Hodge theory

Thesis

We construct a solution functor within the context of a still hypothetical p-adic analytic Riemann-Hilbert correspondence. Our approach relies on the overconvergent de Rham period sheaf, obtained from an ind-Banach completion of the infinitesimal period sheaf along the kernel of Fontaine’s map. A key result in this thesis is establishing a bimodule structure on the overconvergent de Rham period structure sheaf over D-cap and the overconvergent de Rham period sheaf. Here, D-cap denotes the sheaf of infinite order differential operators introduced by Ardakov-Wadsley; notably, the analogous statement does not hold for Scholze’s de Rham period sheaf. We explain how this leads to a solution functor for D-cap modules and propose conjectures about its compatibility with Scholze’s horizontal sections functor and the reconstruction of D-cap-modules from their solutions.

Authors: Wiersig, F

• DOI: 10.5287/ora-erm9wdje6
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: D-cap modules; p-adic geometric representation theory; p-adic Hodge theory
• Subjects: Mathematics