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Local-to-global extensions of representations of fundamental groups. (English) Zbl 0564.14013
Let \(K\) be a field of characteristic \(p>0\), \(C\) 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 \(C\setminus\{0,\infty\}\) which is tamely ramified at \(0\), provided either that \(K\) is separately closed or that \(C\) is \(\mathbb P^1\). 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 construction of the Swan representation in equal characteristic.

14H30 Coverings of curves, fundamental group
14H25 Arithmetic ground fields for curves
11S20 Galois theory
14G20 Local ground fields in algebraic geometry
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