# Lubin-Tate vs cohomological approach to local CFT

Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes http://www.jmilne.org/math/CourseNotes/cft.html).

Going either way, one can ultimately see that both approaches yield the same reciprocity map

K* --> Gal(K^al,K)^ab

This can for example be done since both approaches show that the reciprocity map has certain properties (e.g. norm compatibility, uniformizing elements go to Frobenius lifts,...).

It seems a bit unfortunate that one seems to be able to see the equality of the approaches only at such a late stage of the development.

Is there a way to see already at an earlier stage how these approaches are connected? For example I've seen papers computing Galois cohomology of formal group laws, is this a bridge to the cohomological approach?

Moreover, Is there a formalism of "space" that would allow me to treat a formal group law like a geometric object, as the analogy to elliptic curves would suggest?

I've seen papers of Vostokov et al lifting the Hilbert symbol to formal group laws. Does this have a geometric interpretation? I mean, I always imagine formal group laws as elliptic curves, so maybe this is some sort of avatar of a pairing of cohomology groups of the formal group law (interpreted as a "space" in some way)?

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Your title reminded me of an amusing and ongoing linguistic class. I'm sure you know that physicists use the word "field" for something entirely different than the dominant algebraic definition. In that language, the "C" in "CFT" is always short for "conformal". And there is much research into "local CFT" and "cohomological CFT". So I followed the link to your question expecting a very different topic. I don't think you should make any changes — both languages are highly entrenched — but I was amused nonetheless. – Theo Johnson-Freyd May 8 '12 at 21:41
class = clash :) – Theo Johnson-Freyd May 13 '12 at 2:47
@Theo Johnson-Freyd : The title is perfectly clear. Since "Lubin-tate" has nothing to do with conformal field theory. – user4245 Jun 30 '12 at 9:43
I am interested in reading the paper you mentioned that computes the Galois cohomology of formal group laws. Could provide the paper? Thanks very much in adcvance. – awllower Mar 8 '13 at 6:08