Local-to-global extensions of representations of fundamental groups. (English) Zbl 0564.14013

Katz, Nicholas M.

Ann. Inst. Fourier 36, No. 4, 69-106 (1986).

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.

Reviewer: Nicholas M. Katz (Princeton)

MSC:

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

Keywords:

characteristic p; rational points; representation of local Galois group; fundamental group of curve; monodromy; Swan representation

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