8 - Rings of functions on discs and annuli

Summary

In Part III, we focus our attention specifically on p-adic ordinary differential equations (although most of our results apply also to compete nonarchimedean fields of residual characteristic 0). To do this at the ideal level of generality, one would first need to introduce a category of geometric spaces over which to work. This would require a fair bit of discussion of either rigid analytic geometry in the manner of Tate, or nonarchimedean analytic geometry in the manner of Berkovich, neither of which we want to assume or introduce. Fortunately, since we only need to consider one-dimensional spaces, we can manage by working completely algebraically, considering differential modules over appropriate rings. In this chapter, we introduce those rings and collect their basic algebraic properties. This includes the fact that they carry Newton polygons analogous to those for polynomials. Another key fact is that one has a form of the approximation lemma valid over some of these rings.