On the Boston's Unramified Fontaine-Mazur Conjecture

Abstract

This paper studies the Unramified Fontaine-Mazur Conjecture for $ p $-adic Galois representations and its generalizations. We prove some basic cases of the conjecture and provide some useful criterions for verifying it. In addition, we propose several different strategies to attack the conjecture and reduce it to some special cases. We also prove many new results of the conjecture in the two-dimensional case. Furthermore, we also study the unramified Galois deformation rings. Assuming the Unramified Fontaine-Mazur conjecture, we prove that the generic fiber of the unramified deformation ring is a finite direct product of fields. In particular, the unramified deformation ring has only finitely many $\overline{\mathbb{Q}}_{p}$-valued points. We also give some counterexamples to the so-called dimension conjecture for Galois deformation rings assuming the conjecture.