9 - Radius and generic radius of convergence

Summary

In this chapter, we begin to approach a fundamental question peculiar to the study of nonarchimedean differential modules. It was already pointed out in Chapter 0 that a differential module over a nonarchimedean disc can fail to have horizontal sections even in the absence of singularities. The radius of convergence of local horizontal sections is thus an important numerical invariant, the control of which is a key factor in the production of solutions of p-adic differential equations. Unfortunately, the radius of convergence is often difficult to compute directly. One of Dwork’s fundamental insights is that one can get much better control over the radius of convergence around a so-called generic point. The properties of the generic radius of convergence can then be used to infer information about the actual convergence of horizontal sections. For instance, Dwork’s transfer theorem asserts that the radius of convergence of a differential module over a nonarchimedean disc is no less than the generic radius of convergence at the boundary of the disc.