Unipotent monodromy and arithmetic D-modules

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.

10 pages

References in corpus (5)

• Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations
• Theory of weights in p-adic cohomology
• Langlands correspondence for isocrystals and the existence of crystalline companions for curves
• Overconvergent F-isocrystals and differential overcoherence
• On Beilinson's equivalence for $p$-adic cohomology

Cited by in corpus (1)

• Arithmetic $\mathcal{D}$-modules over Laurent series fields: absolute case