Paraboline variation of $p$-adic families of $(\varphi,\Gamma)$-modules

TL;DR

This paper investigates how triangulations of $p$-adic families of $(, abla)$-modules vary, extending known results and analyzing ramification of weight parameters in these families.

Contribution

It generalizes previous results on extending triangulations to affinoid neighborhoods and studies ramification in $p$-adic families of $(, abla)$-modules.

Findings

Canonical sub-filtrations extend to affinoid neighborhoods of crystalline points

Generalization of previous extension results for triangulations

Analysis of ramification of weight parameters in $p$-adic families

Abstract

We study the $p$-adic variation of triangulations over $p$-adic families of $(\varphi,\Gamma)$-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. As an application, we study the ramification of weight parameters over natural $p$-adic families.