The $p$-adic monodromy theorem over algebraic-affinoid algebras

TL;DR

This paper extends the $p$-adic monodromy theorem to families of $G_K$-equivariant vector bundles over algebraic-affinoid algebras, generalizing previous work and removing certain restrictions.

Contribution

It introduces a generalized $p$-adic monodromy theorem for families of vector bundles, removing the need for a freeness condition used in prior classifications.

Findings

Proves the $p$-adic monodromy theorem in this new setting.

Classifies families of $G_K$-equivariant line bundles without freeness assumptions.

Generalizes previous results by bypassing Kedlaya--Pottharst--Xiao's methods.

Abstract

In the previous paper of the author, motivated by the categorical $p$-adic local Langlands correspondence, the author studied families of $G_K$-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid $\mathbb{Q}_{p,\square}$-algebras (e.g., $\mathbb{Q}_p[T]$). In this paper, we study the $p$-adic Hodge theoretic properties of such families. More precisely, we define the notions of Hodge-Tate, de Rham, and semistable representations for such families, and then prove the $p$-adic monodromy theorem ("de Rham" implies "potentially semistable") in this setting. This is a generalization of the work of Berger-Colmez. As an application, we prove the classification of families of $G_K$-equivariant line bundles. While a similar classification was previously obtained in the previous paper under a certain freeness condition by relying on the results of Kedlaya--Pottharst--Xiao, the approach in the present paper removes this freeness condition and entirely bypasses their results.