Unipotent monodromy and arithmetic D-modules

TL;DR

This paper develops a theory of arithmetic D-modules with potentially-unipotent and quasi-unipotent monodromy, extending stability properties and including overconvergent isocrystals with Frobenius structure within Berthelot's framework.

Contribution

It introduces the concepts of potentially-unipotent and quasi-unipotent monodromy for arithmetic D-modules, expanding the class of objects stable under Grothendieck's six operations.

Findings

Constructed stable coefficient categories containing overconvergent isocrystals.

Proved stability of these categories under Grothendieck's six operations.

Included objects with potentially unipotent monodromy in the overholonomic class.

Abstract

In the framework of Berthelot's theory of arithmetic $\mathcal{D}$-modules, we introduce the notion of arithmetic $\mathcal{D}$-modules having potentially-unipotent monodromy. For example, from Kedlaya's semistable reduction theorem, overconvergent isocrystals with Frobenius structure have potentially unipotent monodromy. We construct some coefficients stable under Grothendieck's six operation, containing overconvergent isocrystals with Frobenius structure and whose object have potentially unipotent monodromy. On the other hand, we introduce the notion of arithmetic $\mathcal{D}$-modules having quasi-unipotent monodromy. These objects are overholonomic, contain the isocrystals having potentially unipotent monodromy and are stable under Grothendieck's six operations and under base change.