21 - The p-adic local monodromy theorem: proof

Summary

In this chapter, we give a proof of the p-adic local monodromy theorem, at the full level of generality at which we have stated it in the previous chapter. After some initial reductions, we start with the case of a differential module satisfying the Robba condition. We describe how this case can be treated using either the p-adic Fuchs theorem for annuli or the slope filtration theorem. We then treat the rank 1 case using the classification of rank 1 solvable modules from Chapter 12. We then show that any module of rank greater than 1 and prime to ?? can be made reducible, by comparing the module with its top exterior power and using properties of refined differential modules. We finally handle the case of a module of rank divisible by ?? by considering its adjoint instead.