Rings of microdifferential operators for arithmetic $\mathscr{D}$-modules

TL;DR

This paper develops a theory of microdifferential operators for arithmetic $$-modules, introducing sheaves of operators at various levels and addressing their interrelations, with a conjecture on characteristic varieties supported by proof in the curve case.

Contribution

It introduces a new framework for microdifferential operators in arithmetic $$-modules, including intermediate operators and sheaves, advancing microlocal analysis in arithmetic geometry.

Findings

Defined sheaves of microdifferential operators of arbitrary levels.

Introduced intermediate differential operators to connect different levels.

Proved the characteristic variety conjecture in the curve case.

Abstract

The aim of this paper is to develop a theory of microdifferential operators for arithmetic $\mathscr{D}$-modules. We first define the sheaves of microdifferential operators of arbitrary levels on arbitrary smooth formal schemes. A difficulty lies in the fact that there are no homomorphisms between sheaves of microdifferential operators of different levels. To remedy this, we define the intermediate differential operators, and using these, we define the sheaf of microdifferential operators for $\mathscr{D}^\dag$. We conjecture that the characteristic variety of a $\mathscr{D}^\dag$-module is computed as the support of the microlocalization of a $\mathscr{D}^\dag$-module, and prove it in the curve case.