Formal Local Theory

Abstract

In this chapter we will classify linear differential equations over the field of formal Laurent series \( \hat K = k((z)) \) and describe their differential Galois groups. Here k is an algebraically closed field of characteristic 0. For most of what follows the choice of the field k is immaterial. In the first two sections one assumes that k = C. This has the advantage that the roots of unity have the convenient description e2πiλ with λ ∈ Q. Moreover, for k = C one can compare differential modules over \( \hat K \) with differential modules over the field of convergent Laurent series C((z)). In the third section k is an arbitrary algebraically closed field of characteristic 0. Unless otherwise stated the term differential module will refer in this chapter to a differential module over \( \hat K \)