Twisted calculus on affinoid algebras

Bernard Le Stum and Adolfo Quirós

Vol. 304 (2020), No. 2, 523–560

DOI: 10.2140/pjm.2020.304.523

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

Milestones

Received: 15 December 2017

Revised: 24 July 2019

Accepted: 2 October 2019

Published: 12 February 2020