From Fontaine-Mazur conjecture to analytic pro-p groups -- A survey

TL;DR

This survey reviews progress on the Fontaine-Mazur conjecture, focusing on group-theoretic approaches to understanding its special cases and related questions in arithmetic geometry.

Contribution

It compiles and discusses developments in group-theoretic methods applied to the Fontaine-Mazur conjecture and highlights open problems in the area.

Findings

Group-theoretic methods have advanced understanding of special cases.

Several formulations of the conjecture have been explored since 1993.

Open questions remain in the interplay between group theory and arithmetic geometry.

Abstract

Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic geometry. Several formulations were given since its original statement in 1993, and various angles have been adopted by numerous authors to try to tackle it. Boston's seminal paper in 1992 gave a range of purely group-theoretic methods rather than representation-theoretic ones to prove some special cases of this conjecture. Such methods have been later successfully carried on by Maire and his co-authors, and brings different informations on the objects involved in the conjecture. This survey article aims to review what is known in this direction and to present some interesting related questions the authors work on.