Yasushi Mizusawa

ON THE ℤp-RANKS OF TAMELY RAMIFIED IWASAWA MODULES

For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic ℤp-extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, we give the explicit formulae of the ℤp-ranks of the S-ramified Iwasawa modules by using Brumers p-adic version of Bakers theorem on the linear independence of logarithms of algebraic numbers.

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

Based on the analogy between links and primes, we present an analogue of the Iwasawas class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

On unramified Galois

We prove that the Galois groups of the maximal unramified pro-2-extensions over the cyclotomic ℤ 2 -extensions of certain real quadratic fields are metacyclic pro-2 groups, and we give some criteria for the finiteness and examples relating to Greenbergs conjecture.

2

For an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p , we consider the Galois group of the maximal pro- p -extension unramified outside S over the

-groups over

{\mathbb Z}_p

\mathbb Z_2

-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro- p group.

-extensions of real quadratic fields

By regarding a finite field as a vector space over the prime field with a basis consisting of powers of an element, a Hamming distance is defined on the finite field with respect to the power basis. We consider the existence of isometric homomorphisms between such finite fields, and characterize the isometric embeddings for even characteristic by arithmetical conditions. Moreover, a canonical Hamming metric is defined in a certain infinite dimensional algebraic extension of a finite field.