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

## TL;DR

This paper characterizes 1-smooth pro-$p$ groups, showing they are precisely the maximal pro-$p$ Galois groups of fields with roots of unity, and confirms the Smoothness Conjecture for finitely generated $p$-adic analytic groups.

## Contribution

It proves that finitely generated $p$-adic analytic pro-$p$ groups are 1-smooth if and only if they are maximal pro-$p$ Galois groups, confirming the Smoothness Conjecture in this case.

## Key findings

- Characterization of 1-smooth pro-$p$ groups as Galois groups.
- Proof that 1-smoothness is equivalent to being a Galois group for finitely generated $p$-adic analytic groups.
- Confirmation that the Smoothness Conjecture holds for this class of 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 $G\to1+p\mathbb{Z}_p$ 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 $p$-adic 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 Clerq-Florence's "Smoothness Conjecture" - which states that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$ groups.