Abstract

We introduce the notion of a twisted differential operator of given radius relative to an endomorphism $\sigma$ of an affinoid algebra $A$. We show that this notion is essentially independent of the choice of the endomorphism $\sigma$. As a particular case, we obtain an explicit equivalence between modules endowed with a usual integrable connection (i.e., differential systems) and modules endowed with a $\sigma$-connection of the same radius (this concept generalizes both finite difference and $q$-difference systems). Moreover, this equivalence preserves cohomology and in particular solutions.

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Mathematical Subject Classification 2010

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