Slope filtration on Banach-Colmez spaces

TL;DR

This paper provides a new proof of a fundamental theorem relating weakly admissible and admissible p-adic representations, using the framework of Banach-Colmez spaces with applications to crystalline representations.

Contribution

It introduces a novel proof of the weakly admissible implies admissible theorem and studies the structure of Banach-Colmez spaces, including their filtration by positive rationals.

Findings

New proof of the weakly admissible implies admissible theorem

Dimension and height functions on Banach-Colmez spaces

Filtration of crystalline representations by positive rationals

Abstract

We give a new proof of the "weakly admissible implies admissible" theorem of Colmez and Fontaine describing the semi-stable p-adic representations. We study Banach-Colmez spaces, i.e. p-adic Banach spaces with the extra data of a C_p-algebra of analytic functions. The "weakly admissible" theorem is then a result of the existence of dimension and height functions on these objects. Furthermore, we show that the subcategory of Banach-Colmez spaces corresponding to crystalline representations is naturally filtered by the positive rationals.