Abstract

Modifications of vector bundles over the Fargues-Fontaine curve play a central role in recent developments in p-adic Hodge theory. Let K be a p-adic field and let C_p be the completed algebraic closure of K. In this talk, we will first review how Fargues use the Fargues-Fontaine curve to define the category of p-adic Hodge structures over C_p, then I will talk about how does the category of classical p-adic Hodge structures over K is embedded into this category. Moreover, I will talk about how to characterize this embedding using Breuil-Kisin-Fargues modules. We will also present how to relate this with the result of Emerton-Gee on moduli of (phi, Gamma)-modules.