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

Abstract

Let $p$ be a prime. A pro‑$p$ group $G$ is said to be $1$-smooth if it can be endowed with a homomorphism of pro‑$p$ groups of the form $G \to 1 + p \mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro‑$p$ Galois groups of fields containing a root of $1$ of order $p$, together with the cyclotomic character, are $1$-smooth. We prove that a finitely generated padic analytic pro‑$p$ group is $1$-smooth if, and only if, it occurs as the maximal pro‑$p$ Galois group of a field containing a root of $1$ of order $p$. This gives a positive answer to De Clercq–Florence’s “Smoothness Conjecture” — which states that the surjectivity of the norm residue homomorphism (i.e., the “surjective half” of the Bloch–Kato Conjecture) follows from $1$-smoothness — for the class of finitely generated $p$-adic analytic pro‑$p$ groups.