## Modules différentiels sur les couronnes (Differential modules over annuli).(French)Zbl 0859.12004

The goal of the paper is a proof of a first step in the direction of solving the following conjecture: Every differential polynomial in $$\overline \mathbb{Q} [x,d/dx]$$ is an operator with index on the space of analytic functions in $$C_p[[x]]$$ convergent in the open unit disk. This first step consists in studying free modules of finite rank over $${\mathcal H}[d/dx]$$ where $${\mathcal H}$$ is the ring of analytic elements in the annulus $$r_1< |x|<r_2$$ of $$C_p$$. For each $$r\in [r_1,r_2]$$ a generic radius of convergence $$R(r)$$ is defined and it is shown that this is a continuous function. Moreover, existence and unicity of a so-called Frobenius antecedent are studied.

### MSC:

 12H25 $$p$$-adic differential equations 13N05 Modules of differentials 14F30 $$p$$-adic cohomology, crystalline cohomology
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