Local-to-global extensions of representations of fundamental groups

Katz, Nicholas M.. "Local-to-global extensions of representations of fundamental groups." Annales de l'institut Fourier 36.4 (1986): 69-106. <http://eudml.org/doc/74739>.

@article{Katz1986,
abstract = {Let $K$ be a field of characteristic $p&gt;0$, $C$ a proper, smooth, geometrically connected curve over $K$, and 0 and $\infty $ two $K$-rational points on $C$. We show that any representation of the local Galois group at $\infty $ extends to a representation of the fundamental group of $C-\lbrace 0,\infty \rbrace $ which is tamely ramified at 0, provided either that $K$ is separately closed or that $C$ is $\{\bf P\}^ 1$. 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.},
author = {Katz, Nicholas M.},
journal = {Annales de l'institut Fourier},
keywords = {characteristic p; rational points; representation of local Galois group; fundamental group of curve; monodromy; Swan representation},
language = {eng},
number = {4},
pages = {69-106},
publisher = {Association des Annales de l'Institut Fourier},
title = {Local-to-global extensions of representations of fundamental groups},
url = {http://eudml.org/doc/74739},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Katz, Nicholas M.
TI - Local-to-global extensions of representations of fundamental groups
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 4
SP - 69
EP - 106
AB - Let $K$ be a field of characteristic $p&gt;0$, $C$ a proper, smooth, geometrically connected curve over $K$, and 0 and $\infty $ two $K$-rational points on $C$. We show that any representation of the local Galois group at $\infty $ extends to a representation of the fundamental group of $C-\lbrace 0,\infty \rbrace $ which is tamely ramified at 0, provided either that $K$ is separately closed or that $C$ is ${\bf P}^ 1$. 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.
LA - eng
KW - characteristic p; rational points; representation of local Galois group; fundamental group of curve; monodromy; Swan representation
UR - http://eudml.org/doc/74739
ER -