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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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References:

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