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

Ramla Abdellatif, Supriya Pisolkar, Marine Rougnant, Lara Thomas

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.

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

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.
Paper Structure (15 sections, 19 theorems, 31 equations)

This paper contains 15 sections, 19 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. General notations
  3. Preliminaries on uniform pro-$p$ groups
  4. Some basic definitions related to profinite groups
  5. Pro-$p$-groups: definition and basic properties
  6. The (uniform) power of pro-$p$-groups
  7. A key example: uniform pro-$p$ groups and principal congruence subgroups
  8. Boston's proof of a special case of Fontaine-Mazur Conjecture
  9. Some results on Tame Fontaine-Mazur conjecture and its uniform version
  10. Motivation and background
  11. Class field theory implies Tame Fontaine-Mazur Conjecture for $n=1$
  12. Equivalence of TFMC and TFMC-U
  13. The small dimensional cases of TFMC ($d = 1$ and $d = 2$)
  14. Tame ramification and co-primality matters
  15. Some future directions

Key Result

Theorem 1.5

Given a prime number $p$ and a number field $F$, let $K$ be a normal extension of $F$ of prime degree $\ell \neq p$ and such that $p$ does not divide $h(F)$, the class number of $F$. Then there is no infinite, everywhere unramified, Galois, pro-$p$ extension $L$ of $K$ such that $L/F$ is Galois and

Theorems & Definitions (47)

  • Conjecture 1.1: Fontaine, Mazur
  • Conjecture 1.2: Fontaine-Mazur Conjecture for $n = 2$
  • Conjecture 1.3: Weak Fontaine-Mazur Conjecture
  • Conjecture 1.4: Uniform Fontaine-Mazur Conjecture (UFMC)
  • Theorem 1.5: Boston
  • Conjecture 1.6: Tame Fontaine-Mazur Conjecture
  • Conjecture 1.7: Tame Fontaine-Mazur Conjecture - Uniform version
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 37 more