A local-global problem for linear differential equations

Marius van der Put and Marc Reversat

Vol. 238 (2008), No. 1, 171–199

DOI: 10.2140/pjm.2008.238.171

Abstract

An inhomogeneous linear differential equation Ly = f over a global differential field can have a formal solution for each place without having a global solution. The vector space lgl(L) measures this phenomenon. This space is interpreted in terms of cohomology of linear algebraic groups and is computed for abelian differential equations and for regular singular equations. An analogue of Artin reciprocity for abelian differential equations is given. Malgrange’s work on irregularity is reproved in terms cohomology of linear algebraic groups.

Keywords

Galois differential groups, abelian differential extensions, local and global solutions of differential equations

Mathematical Subject Classification 2000

Primary: 34A30

Secondary: 34M15

Milestones

Received: 23 November 2007

Revised: 20 April 2008

Accepted: 5 June 2008

Published: 1 November 2008