A p-adic Riemann-Hilbert functor

par Vadim Vologodsky (University of Chicago & IHES)

mardi 12 mai 2026, 14:30 → 16:00 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

The parallel transport construction can be used to produce an equivalence of categories between the category of representations of the fundamental group of a smooth connected manifold and the category of flat bundles over this manifold. I will discuss an analogue of this construction when the field of real numbers is replaced by the field of p-adic numbers. Given a smooth rigid space X over ℚ_p, consider the ring D of differential operators on the base change of X to Fontaine's period ring B_dR⁺. Let D_t be the subalgebra spanned by functions and vector fields multiplied by a uniformizer t in B_dR⁺. Thus, D_t is an algebra over B_dR⁺, whose mod t reduction is the commutative algebra of functions on the cotangent space, and which is isomorphic to D after inverting t. The category of modules over D_t can be twisted by any μ^p∞ gerbe over the cotangent space. I will construct a functor from the category of étale B_dR⁺-local systems on X_Cp to the category of modules over D_t twisted by the Simpson gerbe. The composition of this functor with the mod t reduction recovers the p-adic Simpson functor of Bhatt and Zhang.

 

This is a joint work in progress with Bhargav Bhatt, Ben Heuer and Alexander Petrov.

 

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