A local-global problem for linear differential equations

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 of cohomology of linear algebraic groups.