1-smooth pro-p groups and Bloch-Kato pro-p groups

Abstract

Let p be a prime. We study pro-p groups G endowed with a continuous homomorphism G → 1+pZp satisfying a formal version of Hilbert 90. These pro-p groups are particularly important in Galois theory because by Kummer theory maximal pro-p Galois groups of fields containing a root of 1 of order p, together with the cyclotomic character, satisfy such property. In particular, we prove that De Clerq-Florence's "Smoothness Conjecture", which states that the Bloch-Kato conjecture follows from this formal version of Hilbert 90, holds true in the class of p-adic analytic pro-p groups.

Frattini-injectivity and Maximal pro-$p$ Galois groups

“…More recently, 1-smooth cyclotomic pro-p pairs, a formal version of Hilbert 90 for pro-p groups (for a precise definition, see Section 7), have been investigated in an attempt to abstract essential features of maximal pro-p Galois groups (see [4], [13], [30], [31] and [32]).…”

“…The following two corollaries, in particular, subsume Theorem 1.8 and Theorem 1.9. They were recently proved by Quadrelli [31], [32].…”

“…Given a cyclotomic pro-p pair G = (G, θ), let Z p (1) be the G-module with underlying abelian group Z p and G action defined by g • v = θ(g)v for all g ∈ G and v ∈ Z p . The pair G is said to be 1-smooth (or 1-cyclotomic; see [4], [13], [30], [31] and [32]), if for every open subgroup U of G and every n ∈ N, the quotient map Z p (1)/p n Z p (1) → Z p (1)/pZ p (1) (considered as a U-module homomorphism) induces an epimorphism…”

Chasing maximal pro-p Galois groups via 1-cyclotomicity

“…4]). More recently, cohomologically Kummerian and 1-cyclotomic oriented pro-p groups were formally defined and investigated (see [3,13,30,31,38]) to study maximal pro-p Galois groups: these works suggest that 1-cyclotomicity is a very restrictive property, and therefore pro-p groups which may be completed into 1-cyclotomic oriented pro-p groups "approximate" quite well maximal pro-p Galois groups. This provides a very strong motivation for studying cohomologically Kummerian and 1-cyclotomic oriented pro-p groups.…”

“…Remark 1.5. Part of the research carried out in this manuscript was originally made public in the preprint [30], published on arXiv in April 2019, and submitted to a refereed journal (in particular, Theorems 1.1-1.2 were [30, Thm. 1.1-1.2]).…”