## 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.

### MSC:

 12H25 $$p$$-adic differential equations 14F30 $$p$$-adic cohomology, crystalline cohomology 14G20 Local ground fields in algebraic geometry

### Citations:

Zbl 0834.12005; Zbl 0929.12003; Zbl 1078.12502
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