A local-global problem for linear differential equations

TL;DR

This paper investigates the discrepancy between local and global solutions of linear differential equations over global fields, using cohomology of algebraic groups, and provides new computations and analogies to class field theory.

Contribution

It introduces the space lgl(L) to measure local-global obstructions and computes it for specific classes of differential equations, extending the understanding of irregularity and reciprocity.

Findings

lgl(L) is computed for abelian and regular singular equations

An analogue of Artin reciprocity is established for abelian differential equations

Malgrange's irregularity is reinterpreted via cohomology of algebraic groups

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 of cohomology of linear algebraic groups.