For an abelian variety A of dimension d with good ordinary reduction over a complete discrete valuation field K, the norm map along a totally ramified Zp-extension L/K has a cokernel A(K)/N(A(L)) that sits in an exact sequence whose left-hand term is built from the twist matrices of the formal group of A. Classically this was known only when the residue field is finite; the general construction indexes topological generators of the Galois group of the maximal unramified p-extension by a set J and carries a whole family of twist matrices, so the left-hand term becomes a direct sum over J. Studying it teaches how cohomological triviality of unit groups, the Snake Lemma, and a change of basis over Fp combine to compute norm cokernels, and how the resulting isomorphism plays the role that the reciprocity map plays in local class field theory.
The setting is a local field K, meaning a complete discrete valuation field of characteristic 0 whose residue field k is perfect of characteristic p greater than 0, and an abelian variety A of dimens…