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)?