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**On the index theorem for \(p\)-adic differential equations. III.
(Sur le théorème de l’indice des équations différentielles \(p\)-adiques. III.)**
*(French)*
Zbl 1078.12500

Summary: 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 Part II of this series of papers [Ann. Math. (2) 146, 345–410 (1997; Zbl 0929.12003)].

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 finiteness 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 finiteness properties.

In Part IV [Invent. Math. 143, 629–672 (2001; Zbl 1078.12502)], further insights are given.

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 Part II of this series of papers [Ann. Math. (2) 146, 345–410 (1997; Zbl 0929.12003)].

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 finiteness 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 finiteness properties.

In Part IV [Invent. Math. 143, 629–672 (2001; Zbl 1078.12502)], further insights are given.