Around the nearby cycle functor for arithmetic $\mathscr{D}$-modules

TL;DR

This paper develops a formalism for nearby and vanishing cycles in arithmetic $ ext{D}$-modules, linking smooth objects to overconvergent isocrystals, advancing the understanding of arithmetic $ ext{D}$-module theory.

Contribution

It introduces a nearby and vanishing cycle formalism for arithmetic $ ext{D}$-modules, establishing an equivalence with overconvergent isocrystals.

Findings

Formalism for nearby and vanishing cycles established.

Category of smooth objects shown to be equivalent to overconvergent isocrystals.

Framework advances the theory of arithmetic $ ext{D}$-modules.

Abstract

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$-module theory following Beilinson's philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$-modules whose category is equivalent to the category of overconvergent isocrystals.