Remarks on the Boston Unramified Fontaine-Mazur Conjecture, II
Yufan Luo
TL;DR
This work advances the understanding of the unramified Fontaine–Mazur framework by formulating a precise reduction of the Boston BUFM conjecture to two distinguished classes of $A$-linear pro-$p$ groups of simple type, one $p$-adic and one mod-$p$-adic. It interprets BUFM through the Virtually Golod–Shafarevich lens, showing that a virtual GS property suffices to derive finiteness results and revealing deep connections to GS-pro-$p$ group theory and branch group phenomena. The paper also proves a local-global principle and a prime-to-adjoint principle, demonstrating that finiteness of images can be controlled by local Frobenius data and by residual-adjoint interactions, respectively. These results collectively reduce the conjecture to a finite set of structural cases and provide robust strategies for deducing finiteness of Galois representations in unramified or tamely ramified settings. The findings have potential implications for understanding which infinite tamely ramified pro-$p$ extensions can occur and for guiding future verification of BUFM in concrete arithmetic contexts.
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.