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 Z p -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 Z p -ranks of the S-ramified Iwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.
Based on the analogy between links and primes, we present an analogue of the Iwasawa's 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.
For a number field, we consider the Galois group of the maximal tamely ramified pro-2-extension with restricted ramification. Providing a general criterion for the metacyclicity of such Galois groups in terms of 2-ranks and 4-ranks of ray class groups, we classify all finite sets of odd prime numbers such that the maximal pro-2-extension unramified outside the set has prometacyclic Galois group over the
Z
2
\mathbb Z_2
-extension of the rationals. The list of such sets yields new affirmative examples of Greenberg’s conjecture.
We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field Q( √ −3), following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over Q( √ −3) with restricted ramification, which we construct concretely in the form similar to Rédei's dihedral extension over Q. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.
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