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.