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Local-to-global extensions of representations of fundamental groups

Nicholas M. Katz — 1986

Annales de l'institut Fourier

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...

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