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I'm trying to understand whether there is a sophisticated reason that forces the transfer map to play its role in class field theory or not. Because, at least in Neukirch's proof (at his book ANT) on the "compatibility" of reciprocity map with the transfer map, everything seems to me just like a coincidence. Is there a higher reason that forces Ver to sit where it is? Or may be answer of a such question could make me understand what is going on at all: Is there a role of verlagerung in the higher dimensional class field theories as the role of itself played in the number fields case?

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this might be also related: mathoverflow.net/questions/18719/… –  safak Dec 2 '10 at 0:37
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A lot of Neukirch's (fantastic) book is a "proof of concept" for his point of view on class field theory, so the fact that trasnfer appears auxiliary there is likely just an artifice of it being hidden behind alternative definitions. The transfer map is of certain importance with respect to Artin and Furtwangler's proof of the principal ideal theorem. –  Cam McLeman Dec 2 '10 at 1:00
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While I'm here, depending on your point of view, a "source" for the transfer is that it agrees with an edge map in the Hochschild-Serre spectral sequence, so if you're of the mindset that class field theory comes from (is?) Galois cohomology, it's forced on you almost from the start. –  Cam McLeman Dec 2 '10 at 3:08

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The transfer map even makes sense in the abelian case, where it encodes pieces of the decomposition law (in other words: of the reciprocity law). Gauss's Lemma in his fifth proof of the quadratic reciprocity law is basically a computation of the transfer from the rationals to the maximal real subfield in the p-th cyclotomic field. This is more of a lower than a higher reason, but I'm afraid I can't say anything enlightening; after all, this is a map coming up (naturally) in the cohomology of group extensions, and many of these maps "are just there".

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