Remarks on the Boston Unramified Fontaine-Mazur Conjecture, II

TL;DR

This paper explores a group-theoretic approach to Boston's unramified Fontaine-Mazur conjecture, reducing it to specific p-adic analytic groups and establishing principles linking local and global cases.

Contribution

It offers a new reduction of the conjecture to distinguished p-adic analytic groups and interprets it via the virtually Golod-Shafarevich property, along with local-global principles.

Findings

Reduction to p-adic analytic groups and algebraic groups over local fields

Group-theoretic interpretation via Golod-Shafarevich property

Established local-global and prime-to-adjoint principles

Abstract

In this paper, we investigate Boston's generalization of the unramified Fontaine-Mazur conjecture for Galois representations. From a group-theoretic perspective, we first show that the conjecture can be reduced to the case of certain distinguished classes of $p$-adic analytic groups and $\mathbb{F}_{p}[[T]]$-adic analytic groups. Specifically, these are open subgroups of the groups of integral points of absolutely simple algebraic groups defined over non-Archimedean local fields. Furthermore, we provide a group-theoretic interpretation of the conjecture in terms of the virtually Golod-Shafarevich property. Finally, we establish a local-global principle and a prime-to-adjoint principle for the conjecture.