October 2nd, 2003

Mathematical dreams

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.