#### Vol. 304, No. 2, 2020

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Twisted calculus on affinoid algebras

### Bernard Le Stum and Adolfo Quirós

Vol. 304 (2020), No. 2, 523–560
##### 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.

##### Keywords
arithmetic difference equations, twisted differential operators, twisted affinoid algebras, deformation, confluence, radius of convergence
##### Mathematical Subject Classification 2010
Primary: 12H10, 12H25, 14G22