p-adic Differential Equations

In this chapter we construct a class of examples of differential modules on open annuli which carry Frobenius structures and hence are solvable at a boundary. These modules are derived from continuous linear representations of the absolute Galois group of a positive-characteristic local field.

We first construct a correspondence between Galois representations and differential modules over ℇ carrying a unit-root Frobenius structure. The basic mechanism for producing these modules is to tensor with a large ring carrying a Galois action and then take Galois invariants. This mechanism will reappear when we turn to p-adic Hodge theory, at which point we will attempt to simulate this situation using the Galois group of a mixed-characteristic local field. See Chapter 24.

Then we refine the construction to compare Galois representations having finite image of inertia with differential modules over ℇ± carrying a unit-root Frobenius structure; the main result here is an equivalence of categories due to Tsuzuki. It is generalized by the absolute case of the p-adic local monodromy theorem (Theorem 20.1.4 below) and indeed can be used together with the slope filtration theorem (Theorem 17.4.3) to prove the monodromy theorem in the absolute case. This result also has an analogue in p-adic Hodge theory; see Theorem 24.2.5.

We finally describe (without proof) a numerical relationship between the wild ramification of a Galois representation and the convergence of solutions of p-adic differential equations. Besides making explicit the analogy between the wild ramification of Galois representations and the irregularity of meromorphic differential systems, it also suggests an approach to higherdimensional ramification theory.