In this chapter, we recall some basic facts about norms (absolute values), primarily of the nonarchimedean sort, on groups, rings, fields, and modules. We also briefly discuss the phenomenon of spherical completeness, which is peculiar to the nonarchimedean setting. Our discussion is not particularly comprehensive; the reader new to nonarchimedean analysis is directed to [191] for a fuller treatment.

Several proofs in this chapter make forward references to Chapter 2. There should be no difficulty in verifying the absence of circular references.

Convention 1.0.1. In this book, a ring means a commutative ring unless commutativity is suppressed explicitly by describing the ring as “not necessarily commutative” or implicitly by its usage in certain phrases, e.g., a ring of twisted polynomials (Definition 5.5.1).

Notation 1.0.2. For R a ring, we denote by R× the multiplicative group of units of R.

Norms on abelian groups

Let us start by recalling some basic definitions from analysis, before specializing to the nonarchimedean case.

Definition 1.1.1. Let G be an abelian group. A seminorm (or semiabsolute value) on G is a function | · | : G → [0,+∞) satisfying the following conditions.

(a) We have |0| = 0.

(b) For f, g ∈ G, |f − g∈ ≤ |f| + |g|. (Equivalently, |g| = |−g| and | f + g| ≤ | f| + |g|. This condition is usually called the triangle inequality.)

We say that the seminorm | · | is a norm (or absolute value) if the following additional condition holds.

It has been suggested several times in this book that the study of p-adic differential equations is deeply connected with the theory of p-adic cohomology for varieties over finite fields. In particular, the Frobenius structures arising on Picard–Fuchs modules, discussed in the previous chapter, appear within this theory.

In this chapter, we introduce a little of the theory of rigid p-adic cohomology, as developed by Berthelot and others. In particular, we illustrate the role played by the p-adic local monodromy theorem in a fundamental finiteness problem in the theory.

Isocrystals on the affine line

We start with a concrete description of p-adic cohomology in a very special case, namely the cohomology of “locally constant” coefficient systems on the affine line over a finite field. This is due to Crew [62].

Definition 23.1.1. Let k be a perfect (for simplicity) field of characteristic p > 0. Let K be a complete discrete (again for simplicity) nonarchimedean field of characteristic 0 with kK = k. An overconvergent F-isocrystal on the affine line over k (with coefficients in K) is a finite differential module with Frobenius structure on the ring A = A, ∪β>1K〈t/β〉, for some absolute Frobenius lift ϕ; as in Proposition 17.3.1 the resulting category is independent of the choice of Frobenius lift.

Definition 23.1.2. Let M be an overconvergent F-isocrystal on the affine line over k. Let ℛ be a copy of the Robba ring with series parameter t−1, so that we can identify A as a subring of ℛ.

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.

This book is an outgrowth of a course, taught by the author at MIT during fall 2007, on p-adic ordinary differential equations. The target audience was graduate students with some prior background in algebraic number theory, including exposure to p-adic numbers, but not necessarily with any background in p-adic analytic geometry (of either the Tate or Berkovich flavors).

Custom would dictate that ordinarily this preface would continue with an explanation of what p-adic differential equations are, and why they matter. Since we have included a whole chapter on this topic (Chapter 0), we will devote this preface instead to a discussion of the origin of the book, its general structure, and what makes it different from previous books on the subject.

The subject of p-adic differential equations has been treated in several previous books. Two that we used in preparing the MIT course, and to which we make frequent reference in the text, are those of Dwork, Gerotto, and Sullivan [80] and of Christol [42]. Another existing book is that of Dwork [78], but it is not a general treatise; rather, it focuses in detail on hypergeometric functions.

However, this book develops the theory of p-adic differential equations in a manner that differs significantly from most prior literature. Key differences include the following.

• We limit our use of cyclic vectors. This requires an initial investment in the study of matrix inequalities (Chapter 4) and lattice approximation arguments (especially Lemma 8.6.1), but it pays off in significantly stronger results.

• We introduce the notion of a Frobenius descendant (Chapter 10). This complements the older construction of Frobenius antecedents, particularly in dealing with certain boundary cases where the antecedent method does not apply.

We come now to the subject of metric properties of matrices over a field complete for a given norm. While this topic is central to our study of differential modules over nonarchimedean fields, it is based on ideas which have their origins largely outside number theory. We have thus opted to present the main points first in the archimedean setting and then to repeat the presentation for nonarchimedean fields.

The main theme is the relationship between the norms of the eigenvalues of a matrix, which are core invariants but depend on the entries of the matrix in a somewhat complicated fashion, and some less structured but more readily visible invariants. The latter are the singular values of a matrix, which play a key role in numerical linear algebra in controlling the numerical stability of certain matrix operations (including the extraction of eigenvalues). Their role in our work is similar.

Before proceeding, we set some basic notation and terminology for matrices.

Notation 4.0.1. Let Diag(σ1,…, σn) denote the n × n diagonal matrix D with Dii = σi for i = 1,…,n.

Notation 4.0.2. For A a matrix, let AT denote the transpose of A. For A an invertible square matrix, let A−T denote the inverse transpose of A.

The theory of ordinary differential equations is a fundamental instrument of continuous mathematics, in which the central objects of study are functions involving real numbers. It is not immediately apparent that this theory has anything useful to say about discrete mathematics in general or number theory in particular.

In this book we consider ordinary differential equations in which the role of the real numbers is instead played by the field of p-adic numbers, for some prime number p. The p-adics form a number system with enough formal similarities to the real numbers to permit meaningful analogues of notions from calculus, such as continuity and differentiability. However, the p-adics incorporate data from arithmetic in a fundamental way; two numbers are p-adically close together if their difference is divisible by a large power of p.

In this chapter, we first indicate briefly some ways in which p-adic differential equations appear in number theory. We then focus on an example of Dwork, in which the p-adic behavior of Gauss's hypergeometric differential equation relates to the manifestly number-theoretic topic of the number of points on an elliptic curve over a finite field.

Since this chapter is meant only as an introduction, it is full of statements for which we give references instead of proofs. This practice is not typical of the rest of this book, except for the discussions in Part VI.

Whyp-adic differential equations?

Although the very existence of a highly developed theory of p-adic ordinary differential equations is not entirely well known even within number theory, the subject is actually almost 50 years old.

In this chapter, we begin to approach a fundamental question peculiar to the study of nonarchimedean differential modules. It was pointed out in Chapter 0 that a differential module over a nonarchimedean disc can fail to have horizontal sections even in the absence of singularities. The radius of convergence of local horizontal sections is thus an important numerical invariant, whose control is a key factor in being able to produce solutions of p-adic differential equations.

Unfortunately, the radius of convergence is often difficult to compute directly. One of Dwork's fundamental insights is that one can get much better control over the radius of convergence around a so-called generic point. The properties of the generic radius of convergence can then be used to infer information about the actual convergence of horizontal sections. For instance, Dwork's transfer theorem asserts that the radius of convergence of a differential module over a nonarchimedean disc is no less than the generic radius of convergence at the boundary of the disc.

However, both the radius of convergence and the generic radius of convergence are rather coarse invariants. Just as the notion of the spectral radius is refined by the notion of the full spectrum, we can introduce subsidiary radii of convergence and subsidiary generic radii of convergence, which detect whether some local horizontal sections at a point converge further than others. We will devote much effort in the remainder of this part of the book to analyzing the behavior of these refined invariants.

In the next part of the book, which begins with Chapter 8, we will use the results from the previous chapters to make a detailed analysis of ordinary differential equations over nonarchimedean fields of characteristic 0, the motivating case being that of positive residual characteristic. However, before doing so it may be helpful to demonstrate how the results apply in a somewhat simpler setting.

In this chapter, we reconstruct some of the traditional Fuchsian theory of regular singular points of meromorphic differential equations. (The treatment is modeled on [80, §3].) We first introduce a quantitative measure of the irregularity of a singular point. We then recall how, in the case of a regular singularity (i.e., a singularity with irregularity equal to zero), one has an algebraic interpretation, using the notion of exponents, of the eigenvalues of the monodromy operator around the singular point. We then describe how to compute formal solutions of meromorphic differential equations and go on to sketch the proof of Fuchs's theorem, that the formal solutions of a regular meromorphic differential equation actually converge in some disc. We finally establish the Turrittin–Levelt–Hukuhara decomposition theorem, which gives a decomposition of an arbitrary formal differential module that is analogous to the eigenspace decomposition of a complex linear transformation. The search for an appropriate p-adic analogue of this result will lead us in Part V to the p-adic local monodromy theorem.

In this chapter, we discuss some effective bounds on the solutions of p-adic differential equations with nilpotent singularities. These come in two forms. We start by discussing bounds that make no reference to a Frobenius structure; these are due to Christol, Dwork, and Robba. They could have been presented earlier, and indeed one was invoked in Chapter 13; we chose to postpone them until this point so that we could better contrast them with the bounds available in the presence of a Frobenius structure. The latter are original, though strongly inspired by some recent results of Chiarellotto and Tsuzuki.

These results carry both theoretical and practical interest. Besides their application in the study of p-adic exponents mentioned above (and in the proof of the unit-root p-adic local monodromy theorem to follow; see Theorem 19.3.1), another theoretical point of interest is their use in the study of the logarithmic growth of horizontal sections at a boundary. We will discuss some recent advances in this subject due to André, Chiarellotto, and Tsuzuki. (An area of application that we will not discuss is the theory of G-functions, as found in [80].)

A point of practical interest is that effective convergence bounds are useful for carrying out rigorous numerical calculations, e.g., in the machine computation of zeta functions of varieties over finite fields. See the notes for Chapter 23 for further discussion.

Hypothesis 18.0.1. In this chapter, we will drop the running restriction that K is discretely valued, imposing it only when we discuss Frobenius structures.

In this chapter we give a proof of the p-adic local monodromy theorem, at the full level of generality at which we stated it (Theorem 20.1.4). After some initial reductions, we start with the case of a module of differential slope 0, i.e., one satisfying the Robba condition. We describe how this case can be treated using either the p-adic Fuchs theorem for Christol–Mebkhout annuli (Theorem 13.6.1) or the slope filtration theorem of Kedlaya (Theorem 16.4.1). We then treat the rank 1 case using the classification of rank 1 solvable modules from Chapter 12. We then show that any module of rank greater than 1 and prime to p can be made reducible, by comparing the module with its top exterior power and using properties of refined differential modules. We finally handle the case of a module M of rank divisible by p by considering M∨ ⊗? M instead.

The reader may notice some similarities to the proof of the Turrittin–Levelt–Hukuhara decomposition theorem (Theorem 7.5.1). In fact, this theorem is also known as the p-adic Turrittin theorem for this reason.

Besides the running hypothesis for this part of the book (Hypothesis 14.0.1) and the hypothesis from the previous chapter (Hypothesis 20.0.1), it will be convenient to set several more hypotheses.

Running hypotheses

We are going to make a number of calculations under the same hypotheses. Rather than repeat the hypotheses each time, we enunciate them once and for all here.

In this chapter, we introduce Dwork's technique of Frobenius descent to analyze the generic radius of convergence and subsidiary radii of a differential module, in the range where Newton polygons do not apply. In one direction we introduce a somewhat refined form of the Frobenius antecedents introduced by Christol and Dwork. These fail to apply in an important boundary case; we remedy this by introducing the new notion of Frobenius descendants, which covers the boundary case.

Using these results, we are able to improve a number of results from Chapter 6 in the special case of differential modules over Fρ. For instance we get a full decomposition by spectral radius, extending the visible decomposition theorem (Theorem 6.6.1) and the refined visible decomposition theorem (Theorem 6.8.2). We will use these results again to study the variation of subsidiary radii, and decomposition by subsidiary radii, in the remainder of this part.

Notation 10.0.1. Throughout this chapter we retain Hypothesis 9.0.1. We also continue to use Fρ to denote the completion of K(t) for the ρ-Gauss norm viewed as a differential field with respect to d = d/dt, unless otherwise specified.

Notation 10.0.2. Throughout this chapter we also assume p > 0 unless otherwise specified.

Why Frobenius descent?

Remark 10.1.1. It may be helpful to review the current state of affairs, in order to clarify why we need to descend along a Frobenius morphism.