Integral \(p\)-adic differential modules. (English) Zbl 1158.13009

Matzat, B. H.

Bertrand, Daniel (ed.) et al., Groupes de Galois arithmétique et différentiels. Paris: Société Mathématique de France (ISBN 978-2-85629-222-8/pbk). Séminaires et Congrès 13, 263-292 (2006).

Summary: An integral (or bounded) local D-module is a differential module over a local D-ring \(R\) having congruence solution bases over \(R\). In case \(R\) is equipped with an iterative derivation, such a D-module is an iterative differential module (ID-module) over \(R\). In this paper we solve the connected inverse Galois problem for integral D-modules over fields of analytic elements \(K\{t\}\). In case the residue field of \(K\) is algebraically closed, we are able to additionally solve the non-connected inverse Galois problem. Further, we study the behaviour of ID-modules by reduction of constants.
For the entire collection see [Zbl 1108.12001].

MSC:

13N05	Modules of differentials
12H25	\(p\)-adic differential equations

Keywords:

\(p\)-adic differential equations; locally bounded D-modules; iterative differential modules; inverse Galois theory; reduction