In this chapter, we introduce Dwork's technique of Frobenius descent to analyze the generic radius of convergence and subsidiary radii of a differential module, in the range where Newton polygons do not apply. In one direction we introduce a somewhat refined form of the Frobenius antecedents introduced by Christol and Dwork. These fail to apply in an important boundary case; we remedy this by introducing the new notion of Frobenius descendants, which covers the boundary case.

Using these results, we are able to improve a number of results from Chapter 6 in the special case of differential modules over Fρ. For instance we get a full decomposition by spectral radius, extending the visible decomposition theorem (Theorem 6.6.1) and the refined visible decomposition theorem (Theorem 6.8.2). We will use these results again to study the variation of subsidiary radii, and decomposition by subsidiary radii, in the remainder of this part.

Notation 10.0.1. Throughout this chapter we retain Hypothesis 9.0.1. We also continue to use Fρ to denote the completion of K(t) for the ρ-Gauss norm viewed as a differential field with respect to d = d/dt, unless otherwise specified.

Notation 10.0.2. Throughout this chapter we also assume p > 0 unless otherwise specified.

Why Frobenius descent?

Remark 10.1.1. It may be helpful to review the current state of affairs, in order to clarify why we need to descend along a Frobenius morphism.