A Motivic Approach to p-adic Hodge Theory

by Tess Bouis (Universität Regensburg)

Thursday May 22, 2025, 11:00 AM → 12:30 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

A category of motives is an axiomatic framework in which several cohomology theories, which typically appear in algebraic geometry, are represented. While Voevodsky's classical framework of motivic homotopy theory focused on $\mathbb{A}^1$-invariant cohomology theories, such as $\ell$-adic étale cohomology, the more recent developments in integral $p$-adic Hodge theory have motivated lots of progress towards a more general theory of non-$\mathbb{A}^1$-motives in which $p$-adic cohomology theories, such as crystalline or prismatic cohomology, are also represented. In this talk, I want to explain how the Beilinson--Lichtenbaum phenomenon in non-$\mathbb{A}^1$-invariant motivic cohomology can be used to shed some light on the proof of Fontaine's crystalline conjecture in $p$-adic Hodge theory. This is based on a joint work with Arnab Kundu, where we develop a version in families of Gabber's presentation lemma to prove such a Beilinson--Lichtenbaum phenomenon over general valuation rings. 

 

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