On the Cohomological Dimension of Some Pro-p-Extensions above the Cyclotomic ℤp-Extension of a Number Field

On pro-𝑝 link groups of number fields

“…(For example, P 0 = ∅ if p = 2, and P 0 = P if p = 2 and k = Q. cf. [2,Lemma 2.2] and [44,Proposition 4.8 (4)]. ) As in [44, §4], the five term exact sequence induces the exact rows of commutative diagrams…”

“…by Corollary 3.9, and hence (2,113,593) is also a triple of (proper) Borromean primes modulo 2 in the sense of [36,52].…”

“…Since the pro-p group G S (k) is also finitely presented (cf. [2,44]), we can know the structure of any subquotient in principle if we obtain the explicit presentation of G S (k) by generators and relations. In particular when S = ∅, (k cyc ) ∅ is the union of the p-class field towers of intermediate fields of k cyc /k, and the closed subgroup G ∅ (k cyc ) of G ∅ (k) is a central object in nonabelian Iwasawa theory of Z p -extensions ( [40]).…”

“…In Section 2, we recall Salle's Shafarevich type formula on the generator rank and the relation rank of G S (k), by refering the results and arguments of [2,44] with a little refinement in the case where p = 2.…”

“…In particular, the commutator subgroup of G {∞} (K) is G ∅ (K cyc ( √ −ℓ 1 )), and G {∞} (K cyc ) ab is an abelian group of type [2,2]. Since [4,21]), the maximal subgroup G ∅ (K cyc ( √ −ℓ 1 )) of G {∞} (K cyc ) has infinite abelian quotient.…”