Topics on Geometric and Representation Theoretic Aspects of Period Rings I

TL;DR

This paper explores advanced topics in the geometric and representation theoretic aspects of period rings within a broader framework involving analytic adic and perfectoid spaces, aiming to generalize existing theories.

Contribution

It extends the framework of Hodge-Iwasawa theory to more general base spaces, including analytic adic and perfectoid spaces, and discusses potential further generalizations to topological rings.

Findings

Generalization of Hodge-Iwasawa theory to analytic adic spaces

Discussion of perfectoid spaces and their relation to topological rings

Insights into potential extensions to broader classes of spaces

Abstract

We consider more general framework than the corresponding one considered in our previous work on the Hodge-Iwasawa theory. In our current consideration we consider the corresponding more general base spaces, namely the analytic adic spaces and analytic perfectoid spaces in Kedlaya's AWS Lecture notes. We hope our discussion will also shed some light on further generalization to even more general spaces such as those considered by Gabber-Ramero namely one just considers certain topological rings which satisfy the Fontaine-Wintenberger idempotent correspondence and calls them perfectoid generalizing the notions from Scholze, Fontaine, Kedlaya-Liu and Kedlaya (AWS Lecture notes). Actually some of the discussion we presented here is already in some more general form for this purpose (although we have not made enough efforts to write all the things).