Tame pro-2 Galois groups and the basic ℤ₂-extension

Abstract

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.

Some remarks on pseudo-null submodules of tamely ramified Iwasawa modules

“…However, Y. Mizusawa remarked that the same assertion does not hold for the cyclotomic Z 2extension of Q. That is, for a certain set S of finite primes of Q (which does not contain 2), the S-ramified Iwasawa module of the cyclotomic Z 2extension of Q is infinite and does not contain a non-trivial finite submodule (see [18,Theorem 7.3]). It was also found (after Mizusawa's remark) that there are an odd prime p, a totally real number field H, and a finite set S of finite primes of H (which does not contain any prime lying above p) such that the S-ramified Iwasawa module of the cyclotomic Z p -extension of H is infinite and does not contain a non-trivial finite submodule.…”

Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules

“…In [4], it was shown that if p is odd and X S (Q c ) = 0, then X S (Q c ) always contains a non-trivial finite submodule. On the other hand, when p = 2, Mizusawa's result [16,Theorem 7.3] implies the existence of the case that X S (Q c ) ∼ = Z 2 as a Z 2 -module. Hence, the case when p = 2 is more complicated.…”

On the $λ$-stability of p-class groups along cyclic p-towers of a number field

“…(ii) Let's give examples for cyclic cubic fields k. For f = 5023, because of the strict increasing of the #C n , n ∈ [0, 4] (structures [12,3], [12,12,12,3], [36,12,12,3,3,3]), the stabilization is unknown (if any). (iv) If K/Q is abelian with ∆ := Gal(k/Q) of prime to p order, similar statements hold for the isotopic Z p [∆]-components of the X n 's, using fixed points formulas [8,9].…”

“…For known results (all relative to λ = 0), one may cite Fukuda [2] using Iwasawa's theory, Li-Ouyang-Xu-Zhang [10, § 3] working in a non-abelian Galois context, in Kummer towers, via the use of the fixed points formulas [3,4], then Mizusawa [12] above Z 2 -extensions, and Mizusawa-Yamamoto [13] for generalizations, including ramification and splitting conditions, via the Galois theory of pro-p-groups.…”