Here K̂ is the `-completed periodic complex K-theory spectrum, Λ is the ring of operations [K̂, K̂], and Λ′F is the Iwasawa algebra associated to the `-adic cyclotomic extension F∞ obtained by adjoining all `-power roots of unity. The action of Λ′F on these roots of unity gives an embedding Λ′F ⊂ Λ. The Λ′F -module M∞ is the “basic Iwasawa module”. It can be defined as the étale homology group H1(...

We study the Iwasawa theory of fine Selmer group associated to certain Galois representations. The vanishing $\mu$-invariant is shown follow in some cases from a natural property satisfied by deformation rings. outline conditions under which $\mu=0$ conjecture hold for various representations interest.

For a local system on a compact hyperbolic threefold, under a cohomological assumption, we will show that the order of its twisted Alexander polynomial and of the Ruelle L function at s = 0 coincide. Moreover we will show that their leading constant are also identical. These results may be considered as a solution of a geomeric analogue of the Iwasawa conjecture in the algebraic number theory. 1 2

For a unitary local system of rank one on a complete hyperbolic threefold of a finite volume with only one cusp, we will compare the order of its Alexander invariant at t = 1 and one of the Ruelle L function at s = 0. Our results may be considered as a solution of a geomeric analogue of the Iwasawa main conjecture in the algebraic number theory. 1

We introduce the apollonian metric in Carnot groups using capacity. Extending Beardon’s result for euclidean space, we give an equivalent definition using the cross ratio in Iwasawa groups. We also show that the apollonian metric is bounded above by twice the quasihyperbolic metric in domains in Iwasawa groups.

We consider the computation of the Iwasawa decomposition of a symplectic matrix via the QR factorization. The algorithms presented improve on the method recently described by T.-Y. Tam in [Computing Iwasawa decomposition of a symplectic matrix by Cholesky factorization, Appl. Math. Lett. (in press) doi:10.1016/j.aml.2006.03.001]. c © 2006 Elsevier Ltd. All rights reserved.