Bernstein--Sato Theory for D-modules in Positive Characteristic

TL;DR

This paper develops a positive characteristic analogue of Bernstein--Sato theory for D-modules, defining roots as p-adic integers and proving finiteness and rationality in certain cases.

Contribution

It introduces a new positive characteristic Bernstein--Sato theory for D-modules, extending Bitoun's work and developing a related Cartier module theory.

Findings

Bernstein--Sato roots are p-adic integers for these D-modules.

Roots are finite and rational when arising from certain F^e-modules.

Develops a related Cartier module theory in positive characteristic.

Abstract

In this article, we develop a positive characteristic analogue of the Bernstein--Sato theory for holonomic D-modules in the complex setting. We work with D-modules on a Noetherian regular $F$-finite $\mathbb{F}_p$-scheme $X$, and define their Bernstein--Sato roots as $p$-adic integers. When the D-module is the structure sheaf $O_X$, this recovers Bitoun's definition. When the D-module arises from a locally finitely generated unit $F^e$-module and $X$ is of finite type over an $F$-finite field, we show that the roots are finite and rational, generalizing Bitoun's result. In the course of the proof, we also develop a related theory for Cartier modules.