On tamely ramified infinite Galois extensions

TL;DR

This work analyzes the finitely generated pro-$p$ quotients of the Galois group $G^{\rm ta}_K$ of the maximal tamely ramified extension of a number field $K$, under the standing assumption $\mu_p\nsubseteq K$. The authors introduce the notion of stably inertially generated pro-$p$ groups, show that such groups occur as quotients of $G^{\rm ta}_K$ via a careful local-global lifting strategy, and develop a unified cohomological framework to control obstructions through Shafarevich and Selmer groups. They provide concrete realizations, including quotients like $\mathrm{SL}_m^k(\mathbb{Z}_p)$ and congruence subgroups over $\mathbb{Z}_p[[T_1,\dots,T_n]]$, and establish criteria (via toral/pluperfect Lie algebras) for when no nontrivial toral uniform quotients can arise, touching on tame Fontaine–Mazur-type phenomena. The paper also develops a Lie-algebraic dictionary for $p$-adic analytic groups, extends lifting results to complete local Noetherian rings, and shows how these ideas yield infinite splitting primes in tame extensions, thereby enriching the landscape of possible tame Galois groups and informing conjectures about ramification and Galois representations.

Abstract

For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $μ_p \not \subset K$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of~$G^{\rm ta}$. We give a {unified point of view} by introducing the notion of {\it stably inertially generated} pro-$p$ groups~$G$, for which linear groups are archetypes. This key notion {is compatible} with local {\it tame liftings} as used in the Scholz-Reichardt Theorem. We realize every finitely generated pro-$p$ group~$G$ which is stably inertially generated as a quotient of $G^{\rm ta}$. Further examples of groups that we realize as quotients of $G^{\rm ta}$ include congruence subgroups of special linear groups over ${\mathbb Z}_p[[ T_1,\cdots, T_n ]]$. Finally, we give classes of groups which cannot be realized as quotients of $G^{\rm ta}_{\mathbb Q}$.

Theorems & Definitions (32)

• Theorem A
• Corollary B
• Corollary C
• Corollary D
• Theorem E
• Theorem F
• proof
• proof
• proof
• proof