From Fontaine–Mazur Conjecture to Analytic Pro-p-groups: A Survey

The Fontaine–Mazur Conjecture is a core statement in modern arithmetic geometry. Several formulations of this conjecture were given since its original statement in 1993, and various angles have been adopted by numerous authors to try to tackle it. Among those, a range of tools that is not so well known among young arithmetic geometers goes back to Boston’s seminal 1992 paper and relies on 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 to bring different information on the objects involved in the conjecture. This chapter aims to review what is known in this direction and to present some interesting, related questions the authors (aim to) work on.