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.