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?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
4
1
|
||||||||||||||
|
|
5
|
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". |
||
|
|
