  This paper works out the structure of singular points of p-adic differential
equations (i.e. differential modules over the ring of functions analytic in
some annulus with external radius 1). Surprisingly results look like in the
formal case (differential modules over a one variable power series field) but
proofs are much more involved. However, unlike in the Turritin theorem, even
after ramification, in the p-adic theory there are irreducible objects of rank
>1. The first part is devoted to the definition of p-adic slopes and to a
decomposition along p-adic slopes theorem. The case of slope 0 (p-adic analogue
of the regular singular case) was already studied in the paper with the same
title but number II [Ann. of Math. (2) 146 (1997), 345-410]. The second part
states several index existence theorems and index formulas. As a consequence,
vertices of the Newton polygon built from p-adic slopes are proved to have
integral components (analogue of the Hasse-Arf theorem). After the work of the
second author, existence of index implies finitness of p-adic
(Monsky-Washnitzer) cohomology for affine varieties over finite fields. The end
of the paper outlines the construction of a p-adic-coefficient category over
curves (over a finite field) with all needed finitness properties.
  In the paper with the same title but number IV [Invent. Math. 143 (2001),
629-672], further insights are given.
