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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.

MSC:
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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