Bloch-Kato groups over perfectoid fields and Galois theory of $p$-adic periods

TL;DR

This paper explores the relationship between Bloch-Kato groups and the Galois theory of p-adic periods over perfectoid fields, providing new computations and insights using vector bundle classification on the Fargues-Fontaine curve.

Contribution

It establishes a connection between Bloch-Kato groups and the Galois theory of p-adic periods over perfectoid fields, extending previous results and answering open questions.

Findings

Computed Bloch-Kato groups over new perfectoid field cases.

Linked Bloch-Kato groups to the Galois theory of p-adic periods.

Utilized classification of vector bundles on the Fargues-Fontaine curve.

Abstract

We relate the structure of the Bloch-Kato groups associated with a de Rham Galois representation over a perfectoid field to the Galois theory of the ring $\mathbf{B}_\mathrm{dR}^+$ of $p$-adic periods. As an application, we answer the question raised by Coates and Greenberg and motivated by Iwasawa theory to compute the Bloch-Kato groups over perfectoid fields in new cases, generalising results of Coates and Greenberg and the author. Our method relies on the classification of vector bundles over the Fargues-Fontaine curve.