On the analyticity of the maximal extension of a number field with prescribed ramification and splitting

TL;DR

This paper classifies all $p$-adic analytic groups that can serve as Galois groups of maximal pro-$p$ extensions of number fields with specific ramification and splitting conditions, expanding beyond the Tame Fontaine-Mazur conjecture.

Contribution

It provides a complete characterization of realizable $p$-adic analytic Galois groups under certain assumptions, advancing understanding of Galois representations with prescribed ramification.

Findings

Classified all realizable $p$-adic analytic Galois groups

Extended beyond the Tame Fontaine-Mazur conjecture assumptions

Identified conditions for prescribed ramification and splitting

Abstract

We determine all the $p$-adic analytic groups that are realizable as Galois groups of the maximal pro-$p$ extensions of number fields with prescribed ramification and splitting under an assumption which allows us to move away from the Tame Fontaine-Mazur conjecture.