In this chapter, we study the metric properties of differential modules over nonarchimedean differential rings. The principal invariant that we will identify is a familiar quantity from functional analysis known as the spectral radius of a bounded endomorphism. When applied to the derivation acting on a differential module, we obtain a quantity which can be related to the least slope of the Newton polygon of the corresponding twisted polynomial.

We can give meaning to the other slopes as well, by proving that over a complete nonarchimedean differential field any differential module decomposes into components whose spectral radii are computed by the various slopes of the Newton polygon. However, this theorem will provide somewhat incomplete results when we apply it to p-adic differential modules in Part III; we will have to remedy the situation using Frobenius descendants and antecedents.

This chapter provides important foundational material for much of what follows, but on its own it may prove indigestably abstract at first. The reader arriving at this opinion is advised to read Chapter 7 in conjunction with this one, to see how the constructions of this chapter become explicit in a simple but important class of examples.

Spectral radii of bounded endomorphisms

Before considering differential operators, let us recall the difference between the operator norm and the spectral radius of a bounded endomorphism of an abelian group.

Hypothesis 6.1.1. Throughout this section, let G be a nonzero abelian group equipped with a norm | · |, and let T : G → G be a bounded endomorphism of G.