In this chapter, we study the relationship between a complete nonarchimedean field and its finite extensions; this relationship involves the residue fields, value groups, and Galois groups of the fields in question. We distinguish some important types of extensions, the unramified and tamely ramified extensions. We also briefly discuss the special case of discretely valued fields with perfect residue field, in which one can say much more. We introduce the standard ramification filtrations on the Galois groups of extensions of local fields; these will not reappear again until Part IV, at which point they will relate to the study of convergence of solutions of p-adic differential equations made in Part III.

In this chapter, we introduce Dwork’s technique of descent along Frobenius in order to analyze the generic radius of convergence and subsidiary radii of a differential module, primarily 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. Using these results, we are able to improve a number of results from Chapter 6 in the special case of differential modules over ??_??. For instance, we get a full decomposition by spectral radius, extending the visible decomposition theorem and the refined visible decomposition theorem. We will use these results again to study variation of subsidiary radii, and decomposition by subsidiary radii, in the remainder of Part III.

In this chapter, we apply the tools developed in the preceding chapters to study the variation of the generic radius of convergence and the subsidiary radii associated to a differential module on a disc or annulus. We have already seen some instances where this study is needed to deduce consequences about convergence of solutions of p-adic differential equations. The statements we formulate are modeled on statements governing the variation of the Newton polygon of a polynomial over a ring of power series, as we vary the choice of a Gauss norm on the power series ring. The guiding principle is that in the visible spectrum, one should be able to relate variation of subsidiary radii to variation of Newton polygons via matrices of action of the derivation on suitable bases. This includes the relationship between subsidiary radii and Newton polygons for cyclic vectors, but trying to use that approach directly creates no end of difficulties because cyclic vectors only exist in general for differential modules over fields. We implement the guiding principle in a somewhat more robust manner, using the work of Chapter 6 based on matrix inequalities.

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.

In the previous chapter, we established a number of important variational properties of the subsidiary radii of a differential module over a disc or annulus. In this chapter, we continue the analysis by showing that under suitable conditions, one can separate a differential module into components of different subsidiary radii. That is, we can globalize the decompositions by spectral radius provided by the strong decomposition theorem, in case a certain numerical criterion is satisfied. As in the previous chapter, our discussion begins with some observations about power series, in this case identifying criteria for invertibility. We use these in order to set up a Hensel lifting argument to give the desired decompositions; again we must start with the visible case and then extend using Frobenius descendants. We end up with a number of distinct statements, covering open and closed discs and annuli, as well as analytic elements. As a corollary of these results, we recover an important theorem of Christol and Mebkhout. That result gives a decomposition by subsidiary radii on an annulus in a neighborhood of a boundary radius at which all of the intrinsic subsidiary radii tend to 1.

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

We come now to the subject of metric properties of matrices over a field complete for a specified 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 of number theory. We have thus opted to first present the main points in the archimedean setting, then 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 numerical stability of certain matrix operations (including the extraction of eigenvalues). Their role in our work is similar.

In this appendix, we revisit the territory of Chapter 0, briefly discussing how Picard–Fuchs modules give rise to differential equations with Frobenius structures, and what this has to do with zeta functions.

In this chapter, we study p-adic differential modules in a situation left untreated by our preceding analysis, namely when the intrinsic generic radius of convergence is equal to 1 everywhere (the Robba condition). This setting is loosely analogous to the study of regular singularities of formal meromorphic differential modules considered in Chapter 7; in particular, there is a meaningful theory of p-adic exponents in this setting. However, some basic considerations indicate that p-adic exponents must necessarily be more complicated than the exponents considered in Chapter 7. For instance, the p-adic analogue of the Fuchs theorem can fail unless we impose a further condition: the difference between exponents must not be p-adic Liouville numbers. With this in mind, we may proceed to construct p-adic exponents for differential modules satisfying the Robba condition. Such modules carry an action of the group of p-power roots of unity via Taylor series; under favorable circumstances, the module splits into isotypical components for the characters of this group. We may identify these characters with elements of Z_??, and these give the exponents.

Having introduced the general formalism of difference algebra, and made a more careful study over a complete nonarchimedean field, we specialize to the sort of power series rings over which we studied differential algebra. Most of the rings are ones we have seen before, but we encounter a couple of new variations, notably the Robba ring. This ring consists of power series convergent on some annulus of outer radius 1, but with unspecified inner radius which may vary with the choice of the series. This may seem to be a strange construction at first, but it is rather natural from the point of view of difference algebra: the endomorphisms we will consider (Frobenius lifts) do not preserve the region of convergence of an individual series, but do act on the Robba ring as a whole. This chapter serves mostly to set definitions and notation for what follows. That said, one nontrivial result here is the behavior of the Newton polygon under specialization.

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, due to Christol, Dwork, and Robba. These could have been presented earlier; we chose to postpone them until this point so that we can better contrast them against the bounds available in the presence of a Frobenius structure. The latter are original, though they are 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, another theoretical point of interest is their use in the study of logarithmic growth of horizontal sections at a boundary. We discuss some recent advances in this study due to André, Chiarellotto–Tsuzuki, and Ohkubo. 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 Appendix B for further discussion.

In this chapter, we introduce some basic formalism of differential algebra. This may viewed as a mild perturbation of commutative algebra, in which we consider commutative rings equipped with the additional noncommutative structure of a derivation. This allows us to manipulate differential equations and differential systems in a manner that keeps a bit more useful structure visible, though we will need to convert back and forth from this point of view. A particularly important result we introduce is the cyclic vector theorem, which gives a compact but highly noncanonical way to represent a finite differential module over a field. While the cyclic vector theorem will prove indispensable at a few key points in our treatment of p-adic differential equations, we will ultimately make more progress by limiting its use.

In this chapter, we recall the traditional theory of Newton polygons for polynomials over a nonarchimedean field. In the process, we introduce a general framework which will allow us to consider Newton polygons in a wider range of circumstances; this framework is based on a version of Hensel’s lemma that applies in not necessarily commutative rings. As a first application, we fill in a few missing proofs from Chapter 1.

In this chapter, we first survey some ways in which ??-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.