Fourier-Muka\"i transform for $\mathcal{D}^{(0)}$-modules over formal abelian schemes

TL;DR

This paper extends the Fourier-Mukai transform for D-modules to abelian formal schemes over a discrete valuation ring, broadening its applicability to arithmetic geometry.

Contribution

It introduces a Fourier-Mukai transform for D-modules on abelian formal schemes over a discrete valuation ring, generalizing classical results to an arithmetic setting.

Findings

Defined a Fourier-Mukai transform for D-modules on abelian formal schemes.

Extended classical Fourier-Mukai results to the arithmetic case.

Discussed implications for D-modules over formal schemes.

Abstract

In 1996, Rothstein and Laumon simultaneously constructed a Fourier-Mukai transform for D-modules over a locally noetherian base of characteristic 0. This functor induces an equivalence of categories between quasi-coherent sheaves of D-modules over an abelian variety A and quasicoherent sheaves of O-modules over its universal vectorial extension A. In this article, we define a Fourier-Mukai transform for D-modules on an abelian formal scheme A/S = Spf (V), where V is a discrete valuation ring, and we discuss the extension of the classical results of Fourier-Mukai transform to this arithmetic case.