Bockstein Spectral Sequences and Applications to the Tame Fontaine Mazur Conjecture

Abstract

For a number field $K$, a prime $p$ and a finite set of tame places $S$ we consider the groups $G_{K,S}$ - the Galois groups of the maximal pro-$p$ extension of $K$ unramified outside $S$. The tame Fontaine--Mazur Conjecture predicts that these groups have no nontrivial uniformly powerful pro-$p$ quotients. In this paper we develop a new approach to this problem using Bockstein spectral sequences and Lie-theoretic tools. This allows us to extend and refine an earlier method due to J.~Labute, who was only able to consider the case where $p^2\nmid N(\mathfrak{q})-1$ for each $\mathfrak{q}\in S$. Based on this we develop a method to verify the uniform Fontaine--Mazur property for many $G_{K,S}$ with $|S|=3$ arbitrary. Under mild conditions on $K$ we show that for infinitely many triples $S$, the groups $G_{K,S}$ have no nontrivial uniform quotients. We also exhibit a large class of these groups, which are infinite. Finally, we present numerical evidence indicating that the criteria developed here detect the uniform Fontaine--Mazur property with very high probability for $|S|=3$.