In this chapter, we begin to approach a fundamental question peculiar to the study of nonarchimedean differential modules. It was 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, whose control is a key factor in being able to produce 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.

However, both the radius of convergence and the generic radius of convergence are rather coarse invariants. Just as the notion of the spectral radius is refined by the notion of the full spectrum, we can introduce subsidiary radii of convergence and subsidiary generic radii of convergence, which detect whether some local horizontal sections at a point converge further than others. We will devote much effort in the remainder of this part of the book to analyzing the behavior of these refined invariants.