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Abstract
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 pcurvature for almost all p. The pcurvature 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 socalled iterative differential equation or iterative differential module, respectively. For these a nice PicardVessiot 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 PicardVessiot rings (PVrings) over global differential rings and proves that such PVrings are generated by globally bounded power series as introduced by G. Christol. In Chapter 3 the reduction of global differential modules and their PVrings is studied. The main result is that a global PVring in characteristic zero is algebraic if and only if for almost all primes p the reduced PVring is algebraic. Moreover, for almost all p the reduced PVring and the PVring of the modulo p reduced global differential module coincide. According to Grothendieck's pcurvature conjecture all global PVrings 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.
Item Type:  Preprint 

Series Name:  IWRPreprints 
Date Deposited:  17. Jan 2005 13:55 
Date:  2005 
Faculties / Institutes:  Service facilities > Interdisciplinary Center for Scientific Computing 
Subjects:  510 Mathematics 
Controlled Keywords:  differential algebra, Galois groups, PicardVessiot theory, iterative differential equations, pcurvature 