18 - Effective convergence bounds

Summary

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.