Differential Equations and Finite Groups

This note is devoted to linear differential equations with finite Galois groups. It is a famous conjecture due to A. Grothendieck that the finiteness of the differential Galois group should be equivalent to the triviality of the p-curvature for almost all p. The p-curvature is just the first integrability obstruction for the reduced differential equation in characteristic p. In the case all such integrability obstructions vanish in characteristic p we obtain a so-called iterative differential equation or iterative differential module, respectively. For these a nice Picard-Vessiot theory has been developed by M. van der Put and the author. In particular, the differential Galois groups are linear algebraic groups and there is a Galois correspondence. Thus a natural question arises, wether there exists a reasonable reduction theory preserving Galois groups etc. The corresponding objects in characteristic zero are iterative differential modules over iterative differential rings. The latter are suitable Dedekind subrings of algebraic function fields over number fields, here called global differential rings. These and the corresponding global differential modules are studied in Chapter 1. Chapter 2 presents the construction of global Picard-Vessiot rings (PV-rings) over global differential rings and proves that such PV-rings are generated by globally bounded power series as introduced by G. Christol. In Chapter 3 the reduction of global differential modules and their PV-rings is studied. The main result is that a global PV-ring in characteristic zero is algebraic if and only if for almost all primes p the reduced PV-ring is algebraic. Moreover, for almost all p the reduced PV-ring and the PV-ring of the modulo p reduced global differential module coincide. According to Grothendieck's p-curvature conjecture all global PV-rings are algebraic. Using the result above, this fact might be proven directly. This would already imply a nice algebraicity criterion for formal power series over number fields used by G. Eisenstein and could become a significant step towards the proof of Grothendieck's conjecture