Mathematical dreams

[posic]

Perhaps there should be a theory which is an analogue of the Milnor-Bloch-Kato business for one-dimensional Galois cohomology with trivial (or "almost trivial") finite noncommutative coefficients. It would stand in the same position with respect to the Langlands program as Kummer's theory of abelian extensions does with respect to the class field theory of number fields. It would feature representations of Galois groups in groups of points of algebraic groups over, approximately, the same fields (as finite groups of "coefficients" would be embedded into groups of points as a way of computing their cohomology, etc.) Ideally, this theory might shed new light on all questions of the improper embedding problem, the behaviour of (commutative high-dimensional) Galois cohomology in finite field extensions, Bogomolov's conjecture, anabelian conjectures, etc.

Comments

[roma.livejournal.com]

ty by togo, pojasnil.
What is the "position of Kummer theory with respect to CFT"?
Doen't CFT describe abelian extensions completely, and Kummer theory -- only some of them?

[posic.livejournal.com]

Kummer's theory describes abelian extensions of arbitrary fields containing roots of unity, in a uniform way over all fields (i.e., the answer doesn't depend on the dimension of the field and all such.) Class field theory describes abelian extensions of number fields, and apparently it can be extended to fields of higher dimension, but the answer then will depend on the dimension of the field.

The answers given by Kummer and c.f.t. are related via Tate's duality in cohomology, which depends heavily on the dimension and other special properties of the field. E.g., the local and the global case are usually treated separately and the answers obtained for them in the c.f.t. are quite different. (Perhaps there exists a unified treatment of the local and global cases, but I am not aware of it right now.)