Local-to-global extensions of representations of fundamental groups

Abstract

Let K be a field of characteristic p>0, C a proper, smooth, geometrically connected curve over K, and 0 and ∞ two K-rational points on C. We show that any representation of the local Galois group at ∞ extends to a representation of the fundamental group of C-{0,∞} which is tamely ramified at 0, provided either that K is separately closed or that C is P 1 . In the latter case, we show there exists a unique such extension, called “canonical”, with the property that the image of the geometric fundamental group has a unique p-Sylow subgroup. As an application, we give a global cohomological construcion of the Swan representation in equal characteristic.