Profinite Groups with a Cyclotomic $p$-Orientation

Quadrelli, Claudio ; Weigel, Thomas S.

Documenta mathematica, Tome 25 (2020), pp. 1881-1916.

Résumé

Let $p$ be a prime. A continuous representation $\theta\colon G\to\text{GL}_1(\mathbb{Z}_p)$ of a profinite group $G$ is called a cyclotomic $p$-orientation if for all open subgroups $U\subseteq G$ and for all $k,n\geq1$ the natural maps $H^k(U,\mathbb{Z}_p(k)/p^n)\to H^k(U,\mathbb{Z}_p(k)/p)$ are surjective. Here $\mathbb{Z}_p(k)$ denotes the $\mathbb{Z}_p$-module of rank 1 with $U$-action induced by $\theta\vert_U^k$. By the Rost-Voevodsky theorem, the cyclotomic character of the absolute Galois group $G_{\mathbb{K}}$ of a field $\mathbb{K}$ is, indeed, a cyclotomic $p$-orientation of $G_{\mathbb{K}} $. We study profinite groups with a cyclotomic $p$-orientation. In particular, we show that cyclotomicity is preserved by several operations on profinite groups, and that Bloch-Kato pro-$p$ groups with a cyclotomic $p$-orientation satisfy a strong form of Tits' alternative and decompose as semi-direct product over a canonical abelian closed normal subgroup.

Détail

Classification : 12G05, 20E18, 12F10
Keywords: absolute Galois groups, Rost-Voevodsky theorem, elementary type conjecture