Holonomic $\mathcal{D}$-modules of arithmetic type and middle convolution

TL;DR

This paper develops a framework for studying arithmetic properties of holonomic $ abla$-modules over algebraic varieties, establishing stability of middle convolution and proving conjectures related to rigid Fuchsian systems.

Contribution

It extends classes of $G$-connections to holonomic $ abla$-modules, proves stability of middle convolution, and confirms conjectures on arithmetic properties of rigid Fuchsian systems.

Findings

Grothendieck six-functor formalism for extended classes of $ abla$-modules

Stability of middle convolution with respect to global inverse radii

Equivalence of arithmetic properties in rigid Fuchsian systems

Abstract

The aim of the present paper is to study arithmetic properties of $\mathcal{D}$-modules on an algebraic variety over the field of algebraic numbers. We first provide a framework for extending a class of $G$-connections (resp., globally nilpotent connections; resp., almost everywhere nilpotent connections) to holonomic $\mathcal{D}$-modules. It is shown that the derived category of $\mathcal{D}$-modules in each of such extended classes carries a Grothendieck six-functor formalism. This fact leads us to obtain the stability of the middle convolution for $G$-connections with respect to the global inverse radii. As a consequence of our study of middle convolution, we prove equivalences between various arithmetic properties on rigid Fuchsian systems. This result gives, for such systems of differential equations, an affirmative answer to a conjecture described in a paper written by Y. Andr\'{e} and F. Baldassarri.