# From Fontaine-Mazur conjecture to analytic pro-$p$-groups : a survey

Ramla Abdellatif  Laboratoire Amiénois de Mathématique Fondamentale et
Appliquée (LAMFA) – Université de Picardie Jules Verne, 33 rue Saint-Leu , 80
039 Amiens Cedex 1, France<EMAIL_ADDRESS>,  Supriya Pisolkar
Indian Institute of Science, Education and Research (IISER), Homi Bhabha Road,
Pashan, Pune - 411008, India<EMAIL_ADDRESS>,  Marine Rougnant
Laboratoire de Mathématiques de Besançon, UFR Sciences et techniques, 16,
route de Gray, 25030 Besançon cedex, France<EMAIL_ADDRESS>and  Lara Thomas  Institut Camille Jordan - Lyon/Saint-Etienne – Université
Lyon 1, 43, boulevard du 11 Novembre 1918, 69622 Villeurbanne, France
<EMAIL_ADDRESS>
(Date: November 3rd, 2021 – V3)

###### 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. Among those, a range of tools that is not so well-known among young
arithmetic geometers, goes back to Boston’s seminal paper in 1992, 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, 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 (aim to) work on.

###### Contents

  1. 1 Introduction
  2. 2 General notations
  3. 3 Preliminaries on uniform pro-$p$ groups
    1. 3.1 Some basic definitions related to profinite groups
    2. 3.2 Pro-$p$-groups: definition and basic properties
    3. 3.3 The (uniform) power of pro-$p$-groups
    4. 3.4 A key example: uniform pro-$p$ groups and principal congruence subgroups
  4. 4 Boston’s proof of a special case of Fontaine-Mazur Conjecture
  5. 5 Some results on Tame Fontaine-Mazur conjecture and its uniform version
    1. 5.1 Motivation and background
    2. 5.2 Class field theory implies Tame Fontaine-Mazur Conjecture for $n=1$
    3. 5.3 Equivalence of TFMC and TFMC-U
    4. 5.4 The small dimensional cases of TFMC ($d=1$ and $d=2$)
    5. 5.5 Tame ramification and co-primality matters
  6. 6 Some future directions

##  1\. Introduction

Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic
geometry. Several formulations were given since its original statement (as
appeared in [8]), and various angles has been adopted by numerous authors to
try to tackle it. To state the original Fontaine-Mazur Conjecture (FMC), we
need to introduce some definitions and notations. Let $K$ be a number field,
let $\overline{K}$ be a fixed separable closure of $K$ and let
$G_{K}:=\operatorname{Gal}(\overline{K}/K)$ be the corresponding absolute
Galois group. Given a prime number $p$, a finite extension $F$ of the field
$\mathbb{Q}_{p}$ of $p$-adic numbers and a profinite group $G$, an
$F$-representation of $G$ is a finite-dimensional vector space over $F$
equipped with a continuous and linear action of $G$. When $F=\mathbb{Q}_{p}$,
we call it a $p$-adic representation of $G$. A $p$-adic representation $\rho$
of $G_{K}$ is called geometric if it ramifies only at a finite number of
places of $K$, and if for each place $v$ of $K$ above $p$, the restriction of
$\rho$ to $G_{v}$ is potentially semi-stable in the sense of Fontaine [9,
Section 1.8]. Finally, for any integer $r$, we let $\mathbb{Q}_{p}(r)$ denote
the $r^{th}$ Tate twist of $\mathbb{Q}_{p}$, as defined in [27, §3].

######  Conjecture 1.1 (Fontaine, Mazur).

An irreducible $p$-adic representation of $G_{K}$ is geometric if, and only
if, it is isomorphic to a subquotient of an étale cohomology group with
coefficients in $\mathbb{Q}_{p}(r)$, for some $r\in\mathbb{Z}$, of a
(projective, smooth) algebraic variety over $K$.

In short, Fontaine-Mazur conjecture predicts that $p$-adic representations of
global Galois groups that are potentially semi-stable at primes dividing $p$
and unramified outside finitely many places ought to come from algebraic
geometry.

In the recent years, substantial progress has been made using $p$-adic
representations and deformation theory, allowing for instance Kisin [15] to
prove the original conjecture for $2$-dimensional representations. Applied to
$K=\mathbb{Q}$, this special case of the conjecture asserts that potentially
semi-stable representations with odd determinant come from modular forms.

######  Conjecture 1.2 (Fontaine-Mazur Conjecture for $n=2$).

Let $\rho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to
GL_{2}(\mathbb{Q}_{p})$ be an odd, irreducible representation that is
unramified outside finitely many primes and whose restriction to the
decomposition group at $p$ is potentially semistable with distinct Hodge-Tate
weights. Then $\rho$ is the twist of a Galois representation associated to a
modular form of weight $k\geq 2$.

Kisin’s proof relies on an intimate connection between modularity lifting
theorems, the Breuil-Mézard conjecture [2], and Breuil’s $p$-adic local
Langlands correspondence for $\operatorname{GL}_{2}$ [3, 4]. We are here
interested in a different approach, which can be motivated as follows. First
note that, together with a well-known conjecture of Tate, Conjecture 1.1
implies the following conjecture, which is elementary to state but completely
out of reach for now.

######  Conjecture 1.3 (Weak Fontaine-Mazur Conjecture).

Every unramified pro-$p$-extension of $K$ whose Galois group is $p$-adic
analytic is finite.

Now recall that in [10], Golod and Shafarevich proved the existence of a
number field $K$ and a prime $p$ such that $K$ admits an everywhere unramified
infinite pro-$p$ extension $L$. Conjecture 1.3 then claims that the Galois
group of this extension $L/K$ cannot be an infinite analytic pro-$p$ group
(i.e. something that is isomorphic to a closed subgroup of
$GL_{n}(\mathbb{Z}_{p})$ for some positive integer $n$). Hence the idea here
is that a counter-example to Conjecture 1.3 would produce an everywhere
unramified Galois representation with infinite image, which cannot “come from
algebraic geometry” in the sense of Conjecture 1.1.

From Lazard’s seminal work on $p$-adic analytic groups [17, III, 3.4.3], we
know that any finite-dimen- sional $p$-adic analytic group contains a finite
index open uniform subgroup. Thus we can reformulate Conjecture 1.3 as
follows.

######  Conjecture 1.4 (Uniform Fontaine-Mazur Conjecture (UFMC)).

There is no number field $K$ for which there exists an infinite everywhere
unramified Galois pro-$p$ extension $L$ such that $\operatorname{Gal}(L/K)$ is
uniform.

A major advantage in considering uniform groups is that they have a simple
characterisation (in terms of filtration by their subgroups) that analytic
pro-$p$-groups do not satisfy in general. (Recall that a finitely generated
pro-$p$-group is analytic if, and only if, it has a powerful subgroup of
finite index.)

The first evidence for the truth of Conjecture 1.4 were given by N. Boston in
[5, 6], using purely group-theoretic methods based on the connections between
powerful pro-$p$-groups and uniform pro-$p$-groups instead of representation-
theoretic tools. The goal of this paper is to introduce these purely group-
theoretic tools, which are not so well-known among (young) arithmetic
geometers, and to review some of the main results they bring about various
conjectures related to Conjectures 1.1 and 1.4. We also present some related
questions on which is based current and future works of the authors. We think
that this survey paper may be of interest to anyone willing to have a
different viewpoint on Fontaine-Mazur Conjecture, which does not require
advanced knowledge in $p$-adic representation theory and underlines the
number-theoretic nature of the problem.

This paper is organised as follows. After gathering in Section 3 the
definitions and basic results we need about (uniform) pro-$p$-groups, we
devote Section 4 of this paper to study the proofs of the main result of [5],
which is the following special case of Conjecture 1.4, and its generalisation
to cyclic extensions of degree prime to $p$, which is the main result of [6].

######  Theorem 1.5 (Boston).

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
$\operatorname{Gal}(L/K)$ is uniform.

The main ingredient of the group-theoretic methods used to prove these results
is the cyclic action on $\operatorname{Gal}(L/K)$ of a generator $\sigma$ of
the Galois group $\operatorname{Gal}(K/F)$. This works quite well when the
action of $\sigma$ on $\operatorname{Gal}(L/K)$ has no non-trivial fixed
point. The study of such actions is in the spirit of what did later Hajir and
Maire in [12], where they attempted to extend Boston’s strategy to the case of
(tamely) ramified extensions $L/K$. The challenge here is to handle the fixed
points introduced by ramification and, as a consequence, the constraints posed
on the arithmetic of $L/K$.

This work of Hajir and Maire motivates our interest in the two following
Fontaine-Mazur-style conjectures, where we consider $p$-adic representations
that are finitely and tamely ramified: they are respectively known as Tame
Fontaine-Mazur Conjecture (TFMC) [8, 5a] and Tame Fontaine-Mazur Conjecture -
Uniform version (or Uniform Tame Fontaine-Mazur Conjecture, UTFMC).

######  Conjecture 1.6 (Tame Fontaine-Mazur Conjecture).

Let $K$ be a number field and $S$ be a finite set of places of $K$ that are
prime to $p$. Let $G_{S}:=\operatorname{Gal}(K_{S}/K)$, where $K_{S}$ denotes
the maximal pro-$p$-extension of $K$ that is unramified outside $S$. Then any
$p$-adic representation of $G_{S}$ has finite image.

The idea behind this statement is that the eigenvalues of a Frobenius element
must become roots of unity under the action of a finitely tamely ramified
$p$-adic representation. In this case, the image of such a representation is
solvable, hence finite by class field theory. Class field theory also helps to
prove that the conjecture holds for one-dimensional representations: we do
this in Section 5.2. For higher-dimensional representations, Conjecture 1.6
seems for now out of reach in general.

######  Conjecture 1.7 (Tame Fontaine-Mazur Conjecture - Uniform version).

Let $K$ be a number field, and $\Gamma$ be a uniform pro-$p$ group of
dimension $d>2$ (hence infinite). Then there does not exist a finitely and
tamely ramified Galois extension $L/K$ whose Galois group
$\operatorname{Gal}(L/K)$ is isomorphic to $\Gamma$.

We elaborate on how these statements connect in Section 5, and on how the
assumptions made matter. In particular, we discuss a counterexample to
Fontaine-Mazur Conjecture when ramification at $p$ is allowed. In Section 6,
we expose some connected questions that are of interest to us.

##  2\. General notations

From now on, we fix a prime integer $p$. We let $\mathbb{Q}_{p}$ be the field
of $p$-adic numbers and $\mathbb{Z}_{p}$ be its ring of integers. Since the
ring $\mathbb{Z}_{p}$ is isomorphic to the projective limit
$\lim\limits_{\begin{subarray}{c}\longleftarrow\\\ k\geq
1\end{subarray}}\mathbb{Z}/p^{k}\mathbb{Z}$, there exists, for any integer
$k\geq 1$, a natural ring isomorphism
$\psi_{k}:\mathbb{Z}_{p}/p^{k}\mathbb{Z}_{p}\simeq\mathbb{Z}/p^{k}\mathbb{Z}$.
These ring isomorphisms allows us to define principal congruence subgroups in
this $p$-adic setting as follows. Given any integer $n\geq 2$, we consider the
special linear group $\operatorname{SL}_{n}(\mathbb{Z}_{p})$, which is a
maximal open compact subgroup of $\operatorname{SL}_{n}(\mathbb{Q}_{p})$. For
any integer $k\geq 1$, we can then define the $k^{th}$-principal congruence
subgroup of $\operatorname{SL}_{n}(\mathbb{Z}_{p})$ as
$\Gamma_{n,k}:=\ker\left(\operatorname{SL}_{n}(\mathbb{Z}_{p})\twoheadrightarrow\operatorname{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})\right)$.
Note that we use this $\Gamma_{n,k}$ notation instead of the classical
$K_{n}(k)$ notation to make the connection with the Galois context more
apparent in the sequel.

Assume that we are given a group $G$. For any subsets $X,Y$ of $G$ and any
positive integer $n$, we write $X^{n}$ (resp. $[X,Y]$) for the subgroup of $G$
generated by $\\{x^{n}|x\in X\\}$ (resp. by $\\{[x,y]:=x^{-1}y^{-1}xy,\
(x,y)\in X\times Y\\}$). As usual, for any subgroup $H$ of $G$, we let $[G:H]$
denote the index of $H$ in $G$.

Also, if $\Gamma$ is a topological space and if $\Omega$ is a subset of
$\Gamma$, then $\overline{\Omega}$ denotes the closure of $\Omega$ in
$\Gamma$. If $\Gamma$ is moreover a topological group, a closed subgroup of
$\Gamma$ is called topologically characteristic when it is stable under all
continuous group automorphisms of $\Gamma$.

##  3\. Preliminaries on uniform pro-$p$ groups

Our main reference for this section is [7]. The goal here is to recollect all
the information we need on uniform pro-$p$-groups to understand how they
connect to Fontaine-Mazur conjecture in the view of Boston’s methods. The
reader who is already familiar with uniform pro-$p$-groups can hence skip this
section and directly go to Section 4.

###  3.1. Some basic definitions related to profinite groups

We start by recalling some basic definitions, coming from [7, Chapter 1],
which are fundamental for our purpose. The first definition we make is exactly
[7, Definition 1.1].

######  Definition 3.1.

A profinite group is a compact Hausdorff topological group whose open
subgroups form a basis of neighbourhoods of the identity.

According to [7, Proposition 1.3], this is equivalent to the usual definition
of profinite groups as inverse limits of finite groups.

######  Definition 3.2.

A profinite group $G$ is finitely generated if it contains a finite subset $X$
such that $X$ topologically generates $G$, i.e. such that $G$ is the smallest
closed subgroup of $G$ containing $X$.

Given a profinite group, we can also define its Frattini subgroup, which is
one of the key tools in the forthcoming study of uniform pro-$p$-groups, as
follows (see [7, Definition 1.8]).

######  Definition 3.3.

Let $G$ be a profinite group. The Frattini subgroup is defined as
$\Phi(G):=\bigcap M$, where the intersection runs over all maximal proper open
subgroups of $G$.

###  3.2. Pro-$p$-groups: definition and basic properties

Among all profinite groups, we will have a special interest in those coming as
projective limits of $p$-groups: the so-called pro-$p$-groups, which we define
now, following [7, Definition 1.10].

######  Definition 3.4.

A pro-$p$ group is a profinite group in which every open normal subgroup has
index equal to a power of $p$.

Note that [7, Proposition 1.12] ensures that pro-$p$-groups are exactly groups
that are isomorphic to inverse limits of $p$-groups. (Recall that a $p$-group
is just a finite group of $p$-power order.)

A nice characterisation of finitely generated pro-$p$-groups comes from the
topology of their Frattini subgroup, as proven in [7, Proposition 1.14].

######  Proposition 3.5.

A pro-$p$-group $G$ is finitely generated if, and only if, its Frattini
subgroup $\Phi(G)$ is open in $G$.

In the case of pro-$p$-groups, we can interestingly go beyond Frattini
subgroups via the notion of lower $p$-series, defined as follows in [7,
Definition 1.15].

######  Definition 3.6.

The lower $p$-series of a pro-$p$-group $G$ is the series $(P_{i}(G))_{i\geq
1}$ of topologically characteristic subgroups of $G$ defined by $P_{1}(G)=G$
and: $\forall\ i\geq 1,\ P_{i+1}(G):=\overline{P_{i}(G)^{p}[P_{i}(G),G]}$.

The next key proposition shows how the Frattini subgroup of a pro-$p$-group is
actually encoded in the lower $p$-series [7, Proposition 1.13].

######  Proposition 3.7.

For any pro-$p$-group $G$, we have $\Phi(G)=P_{2}(G)=\overline{G^{p}[G,G]}$.

If $G$ is a finitely generated pro-$p$-group, then the quotient $G/\Phi(G)$ is
a finite-dimensional $\mathbb{F}_{p}$-vector space (see [7, Proposition 3.7]
for further details on this statement). We can hence define the following
integer.

######  Definition 3.8.

For any finitely generated pro-$p$-group $G$, we let $d(G)$ be the dimension
of $G/\Phi(G)$ as $\mathbb{F}_{p}$-vector space:

$\displaystyle d(G):=\dim_{\mathbb{F}_{p}}G/\Phi(G)\ .$

According to [7, Proposition 3.7], this integer is also equal to the minimal
cardinality of a topological generating set for $G$. We will see later that,
in good cases, $d(G)$ is also equal to the dimension of $G$ as a $p$-adic
analytic group, provided that such a structure exists on $G$.

###  3.3. The (uniform) power of pro-$p$-groups

We still consider the case of (pro-)$p$-groups, and we introduce now the
following notion of powerfulness (as in [7, Definition 2.1]).

######  Definition 3.9.

We say that a $p$-group $G$ is powerful when one of the following holds:

  * •

either $p$ is odd, and $G/G^{p}$ is an abelian group;

  * •

or $p=2$, and $G/G^{4}$ is an abelian group.

We say that a pro-$p$-group $G$ is powerful when the analoguous alternative,
with $G^{p}$ and $G^{4}$ respectively replaced by their closure
$\overline{G^{p}}$ or $\overline{G^{4}}$ in $G$, holds.

According to [7, Corollary 3.3], powerful pro-$p$-groups can equivalently be
defined as inverse limits of powerful $p$-groups in which all transition maps
are surjective.

We are now ready to define the first of our key notions, namely uniform
pro-$p$-groups, as follows.[7, Definition 4.1].

######  Definition 3.10.

A uniformly powerful pro-$p$ group $G$ (or: uniform pro-$p$-group) is a
powerful, finitely generated pro-$p$-group $G$ such that: $\forall\ i\geq 1,\
[P_{i}(G)\colon P_{i+1}(G)]=[G\colon P_{2}(G)]$.

Note that the assumptions in Definition 3.10 ensure that $P_{2}(G)=\Phi(G)$ is
open in $G$. Another exciting feature of uniform pro-$p$-groups is that they
are automatically endowed with a structure of $p$-adic analytic group. The
reader interested in more information about the general setting of $p$-adic
analytic groups should refer to Lazard’s seminal paper [17], which remains to
our knowledge the best reference so far on this topic. For now, we only need
the following existence result [7, Theorem 8.32].

######  Theorem 3.11.

A topological group has a structure of $p$-adic analytic group if, and only
if, it contains an open subgroup which is a uniform pro-$p$-group.

In particular, this theorem ensures that uniform pro-$p$-groups themselves
always have a structure of $p$-adic analytic group. According to [7, Theorem
8.36], their dimension as analytic group is then equal to the integer $d(G)$
given by Definition 3.8.

###  3.4. A key example: uniform pro-$p$ groups and principal congruence
subgroups

The goal of this subsection is to prove the following result [7, Theorem 5.2],
where the principal congruence subgroups $\Gamma_{n,k}$ are those we defined
in Section 2.

######  Theorem 3.12.

Let $n\geq 2$ be an integer.

  1. (1)

If $p$ is odd, then the first principal congruence subgroup
$\Gamma_{n,1}\subset\operatorname{SL}_{n}(\mathbb{Z}_{p})$ is a uniform
pro-$p$-group of dimension $n^{2}-1$.

  2. (2)

If $p=2$, then the second principal congruence subgroup
$\Gamma_{n,2}\subset\operatorname{SL}_{n}(\mathbb{Z}_{2})$ is a uniform
pro-$p$-group of dimension $n^{2}-1$.

For instance, we obtain for $n=2$ that for $p\geq 3$,
$\Gamma_{1}:=\ker\left(\operatorname{SL}_{2}(\mathbb{Z}_{p})\twoheadrightarrow\operatorname{SL}_{2}(\mathbb{F}_{p})\right)$
is a uniform pro-$p$-group of dimension $3$, while
$\Gamma_{2}:=\ker\left(\operatorname{SL}_{2}(\mathbb{Z}_{2})\twoheadrightarrow\operatorname{SL}_{2}(\mathbb{Z}/4\mathbb{Z})\right)$
is a pro-$2$-group of dimension $3$.

Also note that, for $n=1$, $\Gamma_{1,m}$ is the trivial group for any integer
$m\geq 1$, and any prime $p$, which explains why we assume $n\geq 2$ in the
previous theorem.

Though the proof of this fact is essentially given in [7, Theorem 5.2], we
collect here the relevant facts and definitions on which it is based to ease
later reference. For convenience, once $n$ is fixed, we write $\Gamma_{m}$
instead of ${\Gamma}_{n,m}$ to avoid this cumbersome notation as often as
possible. Otherly said, we now fix an integer $n\geq 2$ and a prime $p$ and we
set:

$\forall\ m\geq 1,\
\Gamma_{m}:=\ker(\operatorname{SL}_{n}(\mathbb{Z}_{p})\to\operatorname{SL}_{n}(\mathbb{Z}/p^{m}\mathbb{Z}))\
.$

######  Lemma 3.13.

For every $m\geq 1$, the group $\Gamma_{m}$ is a pro-$p$ group. In fact, we
have

$\displaystyle\Gamma_{m}\cong\varprojlim_{r\geq 1}\Gamma_{m}/\Gamma_{m+r},$

with each $\Gamma_{m}/\Gamma_{m+r}$ being a finite $p$-group. Moreover,
$\Gamma_{m}/\Gamma_{m+1}$ is isomorphic to the additive group
$\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$ of matrices in
$M_{n}(\mathbb{F}_{p})$ with trace 0.

###### Proof.

Let $m$ be a positive integer. The obvious projection map
$\displaystyle\Gamma_{m}\to\varprojlim_{r\geq 1}\Gamma_{m}/\Gamma_{m+1}$ is
injective since we have $\displaystyle\bigcap_{r\geq
1}\Gamma_{m+r}=\\{I_{n}\\}$. Conversely, any compatible sequence in
$\displaystyle\varprojlim_{r\geq 1}\Gamma_{m}/\Gamma_{m+r}$ provides an
element of $M_{n}(\mathbb{Z}_{p})$ that is congruent to $I_{n}$ modulo
$p^{m}$. By continuity of the determinant function, this element must have
determinant $1$, so we have the desired isomorphism.

We now show that each $\Gamma_{m}/\Gamma_{m+r}$ is a (finite) $p$-group. By
induction on $r$, it is enough to prove that $\Gamma_{m}/\Gamma_{m+1}$ is a
finite $p$-group. To do this, we consider the map $\varphi_{m}:\Gamma_{m}\to
M_{n}(\mathbb{Z}_{p})$ defined by
$\varphi_{m}(x):=p^{-m}\left(x-I_{n}\right)$. One directly checks that
$\varphi_{m}(\Gamma_{m+1})$ is contained in $M_{n}(p\mathbb{Z}_{p})$, so we
have a well-defined function
$\overline{\varphi}_{m}:\Gamma_{m}/\Gamma_{m+1}\to
M_{n}(\mathbb{Z}_{p})/M_{n}(p\mathbb{Z}_{p})\simeq M_{n}(\mathbb{F}_{p})$.

We claim that $\overline{\varphi_{m}}$ is a group homomorphism, for the usual
additive group structure on $M_{n}(\mathbb{F}_{p})$. Indeed, let
$I_{n}+p^{m}a$ and $I_{n}+p^{m}b$ be elements of $\Gamma_{m}$: as
$(I_{n}+p^{m}a)(I_{n}+p^{m}b)=I_{n}+p^{m}(a+b)+p^{2m}ab$, we have that

$\displaystyle\varphi_{m}(I_{n}+p^{m}a)(I_{n}+p^{m}b))=\varphi_{m}(I_{n}+p^{m}(a+b)+p^{2m}ab)=a+b+p^{m}ab\
.$

Since $a+b+p^{m}ab\equiv a+b\bmod p$, we obtain as expected that

$\displaystyle\overline{\varphi_{m}}\left(I_{n}+p^{m}a)(I_{n}+p^{m}b)\right)=a+b\bmod
p=\overline{\varphi_{m}}\left(I_{n}+p^{m}a)\right)+\overline{\varphi_{m}}\left(I_{n}+p^{m}b)\right)\
.$

Now assume that $[I_{n}+p^{m}a]\in\Gamma_{m}/\Gamma_{m+1}$ is in
$\ker\overline{\varphi_{m}}$. This means that $a$ belongs to
$M_{n}(p\mathbb{Z}_{p})$, which implies that $I_{n}+.p^{m}a=I_{n}$ belongs to
$\Gamma_{m+1}$, hence that $[I_{n}+p^{m}a]\in\Gamma_{m}/\Gamma_{m+1}$ is
trivial and that $\overline{\varphi}_{m}$ is injective. Since
$M_{n}(\mathbb{F}_{p})$ is a finite-dimensional $\mathbb{F}_{p}$-vector space,
we obtain that $\Gamma_{m}/\Gamma_{m+1}\simeq\mathrm{Im}\
\overline{\varphi_{m}}$ is a finite $p$-group.

To conclude, we are left to check that the image of $\overline{\varphi_{m}}$
is isomorphic to $\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$. First,
consider an element $I_{n}+p^{m}a$ of $\Gamma_{m}$ and let
$f_{a}(X)=\displaystyle\sum_{k=0}^{n}\alpha_{k}X^{k}\in\mathbb{Z}_{p}[X]$ be
the characteristic polynomial of $a$. Then the characteristic polynomial of
$-p^{m}a$ is equal to $\displaystyle
f_{-p^{m}a}(X)=\sum_{k=0}^{n}\alpha_{k}(-p^{m})^{n-k}X^{k}$. So we have in
particular that $1=\det(I_{n}+p^{m}a)=f_{-p^{m}a}(1)\
=\displaystyle\sum_{k=0}^{n}\alpha_{n-k}(-p^{m})^{k}$. Reducing this equality
mod $p^{m}$ shows that $\alpha_{0}-1\in p^{m}\mathbb{Z}_{p}$, hence
subtracting $1$ to it and dividing the result by $p^{m}$ gives that

$\displaystyle-\alpha_{n-1}+\sum_{k=2}^{n}\alpha_{n-k}(-p^{m})^{k-1}=0\ .$

This proves in particular that $-\alpha_{n-1}=\operatorname{tr}(a)$ is in
$p^{m}\mathbb{Z}_{p}$, hence that $\operatorname{tr}(a)\equiv 0\bmod p$, which
proves that $\overline{\varphi_{m}}\left(I_{n}+p^{m}a\right)$ lies in
$\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$ as required.
Conversely, let us check that the image of $\overline{\varphi_{m}}$ is equal
to $\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$. First recall that,
since $\overline{\varphi_{m}}$ is injective, it is also a homomorphism of
$\mathbb{F}_{p}$-vector spaces. We are hence left to prove that a basis of
$\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$ is contained in
$\mathrm{Im}\ \overline{\varphi_{m}}$ to conclude. For $1\leq i,j\leq n$, let
$e_{ij}$ be the matrix whose $(i,j)$-th entry is $1$ and all other entries are
$0$. For $1\leq i\leq n-1$, set $d_{i}:=e_{ii}-e_{nn}$. Then an
$\mathbb{F}_{p}$-basis for $\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$
is given by

$\displaystyle\left\\{e_{ij}\colon 1\leq i\neq j\leq
n\right\\}\sqcup\left\\{d_{i}\colon 1\leq i\leq n-1\right\\}\ .$

As $\varphi_{m}(I_{n}+p^{m}e_{ij})=e_{ij}$ for all $1\leq i\neq j\leq n$ while
$\varphi_{m}(I_{n}+p^{m}d_{i})=d_{i}$ for all $1\leq i\leq n-1$, we obtain
that
$\varphi_{m}(\Gamma_{m}/\Gamma_{m+1})=\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$,
as claimed. ∎

Until the end of this section, we essentially follow [7, Section 5.1], making
the necessary changes to pass from $\operatorname{GL}_{n}$ to
$\operatorname{SL}_{n}$ and filling in some missing details. First note that,
if $x\in\Gamma_{m}$ with $m\geq 1$, then $x^{p}\in\Gamma_{m+1}$. Indeed, we
can write $x=I_{n}+p^{m}a$ for some $a\in M_{n}(\mathbb{Z}_{p})$, and as $a$
and $I_{n}$ commute, we have

$\displaystyle x^{p}=(I_{n}+p^{m}a)^{p}=I_{n}+\sum_{k=1}^{p}{p\choose
k}p^{mk}a^{m}\equiv I_{n}\bmod p^{m+1}\ ,$

since $p$ divides ${p\choose k}$ for all $1\leq k\leq p-1$. Following [7,
Lemma 5.1], we will now see that, unless $p=2$ and $m=1$, the $p$-th power map
$\Gamma_{m}\to\Gamma_{m+1}$ is actually surjective.

######  Lemma 3.14.

Let $p$ be a prime and $m$ be a positive integer. If $p$ is odd, or if $p=2$
and $m\geq 2$, then every element of $\Gamma_{m+1}$ is the $p$-th power of an
element of $\Gamma_{m}$.

###### Proof.

Let $a\in M_{n}(\mathbb{Z}_{p})$ be such that $\det(I_{n}+p^{m+1}a)=1$. We are
looking for some $x\in M_{n}(\mathbb{Z}_{p})$ such that $\det(I_{n}+p^{m}x)=1$
and $(I_{n}+p^{m}x)^{p}=I_{n}+p^{m+1}a$. We will find it by successive
approximations, which means that we will produce a sequence $(x_{k})_{k\geq
0}$ of elements of $M_{n}(\mathbb{Z}_{p})$ such that:

$\displaystyle\forall\ k\geq 0,\ \det(I_{n}+p^{m}x_{k})\equiv 1\bmod
p^{m+1+k}\ \text{ and }\ (I_{n}+p^{m}x_{k})^{p}\equiv I_{n}+p^{m}a\bmod
p^{m+1+k}\ .$

Then $\displaystyle x:=\lim_{r\to\infty}x_{r}$ will satisfy the expected
conditions. First note that the congruence condition
$I_{n}+p^{m}a\equiv(I_{n}+p^{m-1}x_{r})^{p}\bmod p^{m+r}$ together with the
equality $\det(I_{n}+p^{m}a)=1$ already ensures that
$\det(I_{n}+p^{m-1}x_{r})^{p}\equiv 1\bmod p^{m+r}$. Since we know that
$\det(I_{n}+p^{m-1}a)\in 1+p\mathbb{Z}_{p}$, it follows that
$\det(I_{n}+p^{m-1}x_{r})\equiv 1\bmod p^{m+r}$. This shows that we do not
need to check whether the determinant congruence holds when constructing the
sequence $(x_{r})_{r\geq 1}$ as it will automatically be true.

We start by constructing $x_{1}$. Note that, so long as $m$ is in the supposed
range, we have

$(I_{n}+p^{m-1}a)^{p}=I_{n}+p^{m}a+\sum_{k=2}^{p}{p\choose
k}(p^{m-1}a)^{k}\equiv I_{n}+p^{m}a\bmod p^{m+1}\ .$

Thus we may set $x_{1}:=a$.

Now suppose that there exists some integer $r\geq 1$ for which we built some
$x_{r}\in M_{n}(\mathbb{Z}_{p})$ satisfying
$I_{n}+p^{m}a\equiv(I_{n}+p^{m-1}x_{r})^{p}\bmod p^{m+r}$. Then there exists
some $c\in M_{n}(\mathbb{Z}_{p})$ such that

$(I_{n}+p^{m-1}x_{r})^{p}=I_{n}+p^{m}a+p^{m+r}c\ .$

Now note that we could expand the left-hand side of this equality as a
$\mathbb{Z}$-linear combination of powers of $x_{r}$: this shows that both $a$
and $c$ can be expressed as a $\mathbb{Q}_{p}$-linear combination of powers of
$x_{r}$, which implies that all of $a,c$ and $x_{r}$ commute. Let us set

$z:=(I_{n}+p^{m-1}x_{r})^{-(p-1)}c\text{ and }x_{r+1}:=x_{r}-p^{r}z\ .$

Then $x_{r+1}$ satisfies the desired congruence, as we have:

$\displaystyle(I_{n}+p^{m-1}x_{r+1})^{p}$
$\displaystyle=((I_{n}+p^{m-1}x_{r})-p^{m+r-1}z)^{p}$
$\displaystyle=\sum_{k=0}^{p}{p\choose
k}(I_{n}+p^{m-1}x_{r})^{p-k}(-p^{m+r-1}z)^{k}$
$\displaystyle=(I_{n}+p^{m-1}x_{r})^{p}+\sum_{k=1}^{p}(-1)^{k}{p\choose
k}(I_{n}+p^{m-1}x_{r})^{p-k}p^{(m+r-1)k}z^{k}$
$\displaystyle=I_{n}+p^{m}a+p^{m+r}c+\sum_{k=1}^{p}(-1)^{k}{p\choose
k}(I_{n}+p^{m-1}x_{r})^{p-k}p^{(m+r-1)k}z^{k}\ ,$

which implies that

$\displaystyle(I_{n}+p^{m-1}x_{r+1})^{p}$ $\displaystyle\equiv
I_{n}+p^{m}a+p^{m+r}c-p(I_{n}+p^{m-1}x_{r})^{p-1}p^{m+r-1}z\bmod p^{m+r+1}$
$\displaystyle\text{ i.e. }(I_{n}+p^{m-1}x_{r+1})^{p}$ $\displaystyle\equiv
I_{n}+p^{m}a\bmod p^{m+r+1}.$

As we have $x_{r+1}-x_{r}=-p^{r}z$ by construction, $\displaystyle
x:=\lim_{r\to\infty}x_{r}$ is well-defined, which ends the proof. ∎

The two next results come from [7, Theorem 5.2].

######  Corollary 3.15.

Let $p$ be a prime integer.

  * •

If $p$ is odd, then $P_{i}(\Gamma_{1})=\Gamma_{i}$ for all $i\geq 1$.

  * •

If $p=2$, then $P_{i}(\Gamma_{2})=\Gamma_{i+1}$ for all $i\geq 1$.

###### Proof.

Set $\varepsilon:=0$ if $p$ is odd (resp. $1$ if $p=2$). We show by induction
on $i\geq 1$ that $P_{i}(\Gamma_{1+\varepsilon})=\Gamma_{i+\varepsilon}$. When
$i=1$, this follows from the definition of $P_{1}$, so we can suppose that
$P_{i}(\Gamma_{1+\varepsilon})=\Gamma_{i+\varepsilon}$ for some $i\geq 1$.
Note that Lemma 3.14 then ensures that
$P_{i}(\Gamma_{1+\varepsilon})^{p}=\Gamma_{i+\varepsilon}^{p}=\Gamma_{i+1+\varepsilon}$.

We claim that
$[P_{i}(\Gamma_{1+\varepsilon}),\Gamma_{1+\varepsilon}]=[\Gamma_{i+\varepsilon},\Gamma_{1+\varepsilon}]\subseteq\Gamma_{i+1+\varepsilon}$.
Indeed, let $I_{n}+p^{i+\varepsilon}\in\Gamma_{i+\varepsilon}$ and
$I_{n}+p^{1+\varepsilon}b$ be two elements of $\Gamma_{1+\varepsilon}$. Note
that we have

$(I_{n}+p^{i+\varepsilon}a)^{-1}=I_{n}+\sum_{k=1}^{\infty}(-1)^{k}p^{(i+\varepsilon)k}a^{k}\equiv
I_{n}-p^{i+\varepsilon}a\bmod p^{i+1+\varepsilon}.$

This hence implies that we have

$\displaystyle[I_{n}+p^{i+\varepsilon}a,I_{n}+p^{i+\varepsilon}b]$
$\displaystyle\equiv(I_{n}+p^{i+\varepsilon}a+pb)(1-p^{i+\varepsilon}a)(I_{n}+pb)^{-1}\bmod
p^{i+1+\varepsilon}$ $\displaystyle\equiv(I_{n}+pb)(I_{n}+pb)^{-1}\bmod
p^{i+1+\varepsilon}$ $\displaystyle\equiv I_{n}\bmod p^{i+1+\varepsilon}\ ,$

hence shows that
$[P_{i}(\Gamma_{1+\epsilon}),\Gamma_{1+\epsilon}]\subseteq\Gamma_{i+1+\epsilon}$,
and thus we have
$P_{i}(\Gamma_{1+\varepsilon})[P_{i}(\Gamma_{1+\varepsilon}),\Gamma_{1+\varepsilon}]\subseteq\Gamma_{i+1+\varepsilon}$.
Since $P_{i}(\Gamma_{1+\varepsilon})=\Gamma_{i+1+\varepsilon}$, we must have
equality. As $\Gamma_{i+1+\varepsilon}$ is open (hence closed) in
$\Gamma_{1+\varepsilon}$, it follows that
$P_{i+1}(\Gamma_{1+\varepsilon})=\Gamma_{i+1+\varepsilon}$, as expected. ∎

######  Theorem 3.16.

Let $p$ be a prime integer.

  * •

If $p$ is odd, then $\Gamma_{1}$ is a uniform pro-$p$-group of dimension
$n^{2}-1$.

  * •

If $p=2$, then $\Gamma_{2}$ is a uniform pro-$2$-group of dimension $n^{2}-1$.

###### Proof.

As in the proof of Lemma 3.14, we will give a uniform proof for the two cases
by setting $\varepsilon:=0$ if $p$ is odd (resp. $1$ if $p=2$). According to
Lemma 3.13, $\Gamma_{1+\varepsilon}$ is a pro-$p$ group. Propositions 3.5 and
3.7 hence ensure that $\Gamma_{1+\varepsilon}$ is finitely generated if, and
only if, $P_{2}(\Gamma_{1+\varepsilon})$ is open in $\Gamma_{1+\varepsilon}$.
But we know from Corollary 3.15 that
$P_{2}(\Gamma_{1+\varepsilon})=\Gamma_{2+\varepsilon}$, which is open in
$\Gamma_{1+\varepsilon}$, so the finite type condition is satisfied.

Now recall that Lemma 3.14 ensures that
$\Gamma_{1+\varepsilon}^{p}=\Gamma_{2+\varepsilon}$. From Lemma 3.13, we get
that
$\Gamma_{1+\varepsilon}/\Gamma_{2+\varepsilon}\cong\operatorname{\mathfrak{sl}}_{2}(\mathbb{F}_{p})$
is abelian, hence $\Gamma_{1+\varepsilon}$ is powerful. Using similarly that
Corollary 3.15 gives that
$P_{i}(\Gamma_{1+\varepsilon})=\Gamma_{i+\varepsilon}$ while Lemma 3.13 shows
that
$\Gamma_{j}/\Gamma_{j+1}\cong\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$,
we obtain that $P_{i}(\Gamma_{1+\varepsilon})$ also satisfies the necessary
condition from Definition 3.10 for all $i\geq 1$. All this finally shows that
$\Gamma_{1+\epsilon}$ is uniform.

Finally, we calculate the dimension using Lemma 3.13, Corollary 3.15 and
Definition 3.10 as follows: from these statements follows that we have
$\Gamma_{1+\varepsilon}/\Phi(\Gamma_{1_{\varepsilon}})=\Gamma_{1+\varepsilon}/P_{2}(\Gamma_{1+\varepsilon})=\Gamma_{1+\varepsilon}/\Gamma_{2+\varepsilon}\cong\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})$.
Since
$\displaystyle\dim_{\mathbb{F}_{p}}\operatorname{\mathfrak{sl}}_{n}(\mathbb{F}_{p})=n^{2}-1$,
we are done. ∎

We end this section by a well-known result that will be used in Section 5. Its
proof is as outlined in [19, Page 1271]. Let $G$ be a finitely generated
pro-$p$-group. For any $i\geq 1$, set $H^{i}(G):=H^{i}(G,\mathbb{F}_{p})$ and
let $d_{p}(G):=\dim_{\mathbb{F}_{p}}H^{1}(G)$ be the $p$-rank of $G$.

######  Theorem 3.17.

Any uniform subgroup of dimension 1 or 2 has a quotient isomorphic to
$\mathbb{Z}_{p}$.

Proving this result requires the following result of Lazard [17, V,
Proposition (2.5.7.1)].

######  Theorem 3.18.

Let $G$ be a uniform group of positive dimension $d$. Then, for all $i\geq 1$,
one has

$\displaystyle H^{i}(G)\cong\bigwedge^{i}(H^{1}(G))\ ,$

where the exterior product is induced by the cup product.

###### Proof of Theorem 3.17.

Suppose that $G$ is a uniform pro-$p$-group of dimension $d\geq 1$. If $d=1$,
then $G$ is clearly isomorphic to $\mathbb{Z}_{p}$ so we are done. If $\dim
G=\dim_{\mathbb{F}_{p}}H^{1}(G,\mathbb{F}_{p})=2$, then Theorem 3.18 implies
that
$\dim_{\mathbb{F}_{p}}H^{2}(G,\mathbb{F}_{p})=\dim_{\mathbb{F}_{p}}\bigwedge^{2}H^{1}(G,\mathbb{F}_{p}))=1$,
i.e. that $H^{2}(G,\mathbb{F}_{p})$ is actually isomorphic to
$\mathbb{F}_{p}$. Now note that
$H^{1}(G,\mathbb{F}_{p})=H^{1}(G^{ab},\mathbb{F}_{p})$. Since $G^{ab}$ is a
finitely generated $\mathbb{Z}_{p}$-module, we can write it as

(3.1) $\displaystyle
G^{ab}\simeq\mathbb{Z}_{p}^{r}\times\prod_{s=1}^{n}\mathbb{Z}/p^{i_{s}}\mathbb{Z}$

for some nonnegative integers $r,n,i_{1},\ldots,i_{n}$. Recall that the short
exact sequence

$\displaystyle
1\longrightarrow\mathbb{F}_{p}\longrightarrow\mathbb{Q}_{p}/\mathbb{Z}_{p}\stackrel{{\scriptstyle\times
p}}{{\longrightarrow}}\mathbb{Q}_{p}/\mathbb{Z}_{p}\longrightarrow 1$

leads to the following long exact sequence of cohomology:

$0\longrightarrow H^{1}(G^{ab},\mathbb{F}_{p})\longrightarrow
H^{1}(G^{ab},\mathbb{Q}_{p}/\mathbb{Z}_{p})\stackrel{{\scriptstyle\times
p}}{{\longrightarrow}}H^{1}(G^{ab},\mathbb{Q}_{p}/\mathbb{Z}_{p})\longrightarrow
H^{2}(G,\mathbb{F}_{p})\longrightarrow\ldots$

Let $K$ and $C$ respectively denote the kernel and the cokernel of $\times p$,
and let $d_{p}(K)$ and $d_{p}(C)$ denote their respective dimensions as
$\mathbb{F}_{p}$-vector spaces. From what we said above directly follows that

$\displaystyle d_{p}(C)\leq d_{p}(H^{2}(G,\mathbb{F}_{p}))\ \text{ and }\
d_{p}(K)\geq d_{p}(H^{1}(G,\mathbb{F}_{p}))\ ,$

hence we have $d_{p}(K)-d_{p}(C)\geq
d_{p}(H^{1}(G,\mathbb{F}_{p}))-d_{p}(H^{2}(G,\mathbb{F}_{p}))=1$. Since (3.1)
also ensures that $d_{p}(K)-d_{p}(C)=r=\mathrm{rk}_{\mathbb{Z}_{p}}G^{ab}$, we
obtain that $\mathrm{rk}_{\mathbb{Z}_{p}}(G^{ab})\geq 1$, hence $G^{ab}$
surjects onto $\mathbb{Z}_{p}$. ∎

##  4\. Boston’s proof of a special case of Fontaine-Mazur Conjecture

In this section, we study the proof of Theorem 1.5 as given by Boston in [5,
6]. It heavily relies on Lazard’s extensive study of $p$-adic analytic Lie
groups, as given in [17]. Recall in particular that Lazard defined the notion
of $p$-saturated groups and used it to give an algebraic characterisation of
$p$-adic analytic groups, as topological groups containing a topologically
finitely generated, open, $p$-saturated pro-$p$-group with an integer-valued
filtration. In [7], the notion of uniform pro-$p$-groups is used to transpose
Lazard’s work in a group-theoretic maneer and to obtain a analogue
characterization of $p$-adic analytic groups for odd $p$,where uniform
pro-$p$-groups play the same role as $p$-saturated pro-$p$-groups with
integer-valued filtration do in Lazard’s statement [7, Page 81]. Note that in
[5], Boston used Lazard’s terminology ($p$-saturated with integer values)
instead of the uniform group one. Let us recall here the main statement of [5]
(namely Theorem 1), as originally stated.

######  Theorem 4.1.

Let $K$ be a normal extension of prime degree $\ell\neq p$ of a number field
$F$ such that $p\nmid h(F)$, the class number of $F$. Then there is no
infinite everywhere unramified Galois pro-$p$ extension $L$ of $K$ such that
$L$ is Galois over $F$ and $\operatorname{Gal}(L/K)$ is $p$-saturated with
integer values.

This result gives some evidence for Conjecture 1.3 in a special case. Using
the strength of fixed point free automorphisms, Boston generalised this result
to the case of cyclic extensions $L/K$ where $[L:K]$ is not necessarily a
prime, see [6, Theorem 1]. To do this, he introduced a class of ‘self-similar’
groups, which contains the class of ‘uniform’ groups, and showed that, under
some condition $H(G,n)$ related to fixed-point free automorphisms (that is
conjectured to always hold), Theorem 4.1 carries over for cyclic finite
extensions of degree co-prime to $p$, with ‘uniform’ replaced by ‘self-
similar’. As it appeared above, stating this more general result requires to
define some new notions.

######  Definition 4.2.

A pro-$p$-group $G$ is said self-similar when it has a filtration by open,
characteristic subgroups $G=G_{1}\subseteq G_{2}\subseteq\cdots$ such that
$G_{i}/G_{i+1}$ is abelian for all $i\geq 1$ and $\displaystyle\bigcap
G_{i}=\\{1\\}$, together with a family of group isomorphisms

$\displaystyle\phi_{i}:G_{i}/G_{i+1}\to G_{i+1}/G_{i+2}$

that commute with every continuous automorphism of $G$.

Note that the commutativity of all quotients $G_{i}/G_{i+1}$ ensures that we
only have to check the second condition for outer automorphisms of $G$. Also
note that Definition 4.2 implies that self-similar groups are always infinite,
since
$\displaystyle\lim_{i\to\infty}|G/G_{i}|=\lim_{i\to\infty}|G/G_{2}|^{i-1}=\infty$.

The next proposition justifies our interest in this notion.

######  Proposition 4.3.

Any uniform group is a self-similar group.

###### Proof.

Let $G$ be a uniform pro-$p$-group. For any $i\geq 1$, set $G_{i}:=P_{i}$,
where $P_{i}$ is as in Definition 3.6. Then $(G_{i})_{i\geq 1}$ is a
filtration as in Definition 4.2 and the map $[x\mapsto x^{p}]$ induces group
isomorphisms $\varphi_{i}:G_{i-1}/G_{i}\to G_{i}/G_{i+1}$ that commute with
any continuous automorphism of $G$, hence $G$ is indeed a self-similar group.
∎

We now make explicit the aforementioned condition $H(G,n)$ defined by Boston
in [6, Definition 2].

######  Definition 4.4.

Let $G$ be a pro-$p$ group and $n$ be a positive integer. We say that $H(G,n)$
holds when there is a function of $n$ that is an upper bound for the derived
length of every finite quotient of $G$ that admits a fixed point free
automorphism of order $n$.

We already mentioned above that $H(G,n)$ is conjectured to hold for any
pro-$p$-group $G$ and any integer $n\geq 1$. So far, it is known to hold when:

  * •

$n$ is either a prime integer or $n=4$, and $G$ arbitrary;

  * •

$G$ is such that the rank of all its finite quotients is bounded, and $n$
arbitrary.

The latter case covers the uniform groups case, see [7, p. 52], hence for $G$
being a uniform group, there is a (uniform) bound on the derived length of the
quotients $G/P_{i}$ for $i\geq 1$.

We can now state the generalisation of Theorem 4.1 as proven in [6, Theorem
1].

######  Theorem 4.5 (Boston).

Let $F$ be a number field such that $p$ does not divide the class number of
$F$. Let $K$ be a cyclic extension of $F$ of degree $n\geq 2$ co-prime to $p$.
Then there is no infinite, everywhere unramified, Galois pro-$p$-extension $L$
of $K$ such that $L$ is Galois over $F$ and ${\rm Gal}(L/K)$ is self-similar
and satisfies the property $H({\rm Gal}(L/K),n)$ as stated in Definition 4.4.

Before proving this theorem, we recall a classical result of Schur and
Zassenhaus [24, Chapter 4].

######  Theorem 4.6 (Schur-Zassenhaus).

Let $1\to\Gamma\to\mathcal{G}\to\mathcal{G}/\Gamma\to 1$ be a short exact
sequence of profinite groups, with $\Gamma$ a finitely generated pro-$p$-group
and $\mathcal{G}/\Gamma$ of finite order co-prime to $p$. Then $\mathcal{G}$
contains a subgroup $\Delta_{0}$ isomorphic to the quotient
$\Delta:=\mathcal{G}/\Gamma$, and $\Delta_{0}$ is unique up to conjugation in
$\mathcal{G}$. In particular, $\mathcal{G}$ is isomorphic to a semi-direct
product of $\Delta$ and $\Gamma$:
$\mathcal{G}=\Gamma\rtimes\Delta_{0}\cong\Gamma\rtimes\Delta$.

###### Proof of Theorem 4.5.

To make things more confortable to the reader, we only deal here with uniform
groups, but the general case of self-similar groups can be proven along the
same lines, as done in [6]. Suppose by contradiction that there exists an
infinite, everywhere unramified, Galois pro-$p$-extension $L$ of $K$ such that
$L/F$ is Galois and $G:={\rm Gal}(L/K)$ is uniform and satisfies condition
$H(G,n)$, where $n:=[L:K]$. By Schur-Zassenhaus theorem, the following
extension splits:

$\displaystyle 1\longrightarrow G\to{\rm Gal}(L/F)\longrightarrow{\rm
Gal}(K/F)\longrightarrow 1\ .$

Let us pick an element $\sigma$ of ${\rm Gal}(L/F)$ that lifts a generator of
the cyclic group ${\rm Gal}(K/F)$ under the splitting above. As $G$ is a
normal subgroup of ${\rm Gal}(L/F)$, $\sigma$ induces an action by conjugation
on $G$. Now, since $G$ is a uniform group, we can consider its filtration by
its characteristic subgroups $P_{i}$ (as given in Definition 3.6). Each of the
$P_{i}$ is preserved under the action by conjugation of $\sigma$ (as it is a
characteristic subgroup of $G$), hence we get a conjugation action of $\sigma$
on each of the quotients $G/P_{i}$.

Suppose that, for all $i\geq 1$, $\sigma$ has no non-trivial fixed point in
$G/P_{i}$. As $G$ satisfies $H(G,n)$, there is a uniform bound on the derived
length of each of the quotients $G/P_{i}$ as $i$ goes to $\infty$. Such a
bound cannot exists since a repeated use of the finiteness of class number
shows that the maximal unramified pro-$p$-extension of any fixed derived
length must be a finite extension.

This shows that there exists a positive integer $i$ such that $\sigma$ does
not act fixed-point free on $G/P_{i}$. Assume that $i$ is minimal for this
property and let $\tau$ be a non-trivial fixed point of $\sigma$ in $G/P_{i}$:
$\sigma\cdot\tau=\tau$. By minimality of $i$, $\tau$ must maps to the identity
element in $G/P_{i-1}$, i.e. lands in $P_{i-2}/P_{i-1}$, since we have
compatibility of the action of $\sigma$ with $P_{i-1}\supseteq P_{i}$ and
$G/P_{i}\twoheadrightarrow G/P_{i-1}$. Now recall that Proposition 4.3 ensures
that the isomorphism $\phi_{i-1}:P_{i-2}/P_{i-1}\to P_{i-1}/P_{i+2}$ is
$\sigma$-equivariant, hence $\varphi_{i-1}^{-1}(\tau)$ should define a non-
trivial fixed point for $\sigma$ in $P_{i-2}$, which contradicts the
minimality of $i$ if $i\geq 3$.

If $i=2$, then our assumption is that the action by conjugation of $\sigma$ on
$G/P_{2}$ has a non-trivial fixed point. By definition, we have
$P_{2}=\overline{G^{p}[G,G]}$, hence $G/P_{2}$ is naturally an
$\mathbb{F}_{p}$-vector space. Otherly said, it defines a representation over
a field of characteristic $p$ of the group $\langle\sigma\rangle$ generated by
$\sigma$. As this group is, by definition of $\sigma$, of cardinality
$\mid\langle\sigma\rangle\mid=\mid\mathrm{Gal}(K/F)\mid=n$, which is prime to
$p$, we can apply Maschke’s theorem to decompose $G/P_{2}$ as a direct sum of
irreducible representations of $\langle\sigma\rangle$ over $\mathbb{F}_{p}$.
Since $\tau$ is fixed under the action of $\sigma$, this decomposition can be
written as $G/P_{2}=\mathbb{F}_{p}\tau\oplus W$ for some representation $W$ of
$\langle\sigma\rangle$ over $\mathbb{F}_{p}$.

The direct sum decomposition ensures that $W$ also defines a subgroup and a
quotient of $G/P_{2}$, so we can define an extension $M$ of $K$ such that
$\mathrm{Gal}(M/K)=\langle\tau\rangle$ by letting $M$ be the fixed field (in
$L$) of $W$ over $K$. The field extension $M/K$ is then an abelian extension
of prime degree $p$, and we have a tower of fields of the form $F\subset
K\subset M$.

We now claim that there exists a cyclic extension of degree $p$ over $F$ that
is contained in $M$. Indeed, recall that
$\mathrm{Gal}(K/F)\simeq\langle\sigma\rangle$ acts on $\mathrm{Gal}(M/K)$ by
conjugation. Since $K/F$ is assumed to be a normal extension,
$\mathrm{Gal}(M/K)$ must be a normal subgroup of $\mathrm{Gal}(M/F)$. We can
hence apply Schur-Zassenhaus theorem to write $\mathrm{Gal}(M/F)$ as a semi-
direct product of the following form:

(4.1) $\displaystyle\mathrm{Gal}(M/F)\ \simeq\
\mathrm{Gal}(M/K)\rtimes\mathrm{Gal}(K/F)\ .$

As $\tau$ is fixed under $\sigma$, the action of $\sigma$ on
$\mathrm{Gal}(M/K)=\langle\tau\rangle$ is trivial, hence $\mathrm{Gal}(K/F)$
is also normal in $\mathrm{Gal}(M/F)$, which turns (4.1) into a direct product
of the two groups appearing on the right hand side. As $\mathrm{Gal}(M/K)$ is
of order $p$ while $\mathrm{Gal}(K/F)$ is of order $n$, with $n$ and $p$ co-
prime, we can conclude that there exists a Galois extension $N$ of $F$ of
degree $p$ (hence cyclic) in $M$.

$F$$K$$N$$M$$n$$p$

Since $M/F$ is unramified, so is $N/F$, which is in contradiction with the
fact that $p$ is supposed to be co-prime to $h(F)$. In all cases ($i\geq 3$ or
$i=2$), we have a contradiction, which proves that there exists no
pro-$p$-extension $L/K$ that is everywhere unramified and has for Galois group
a uniform group. ∎

##  5\. Some results on Tame Fontaine-Mazur conjecture and its uniform version

For this section, we will mostly refer to [11, 12, 22]. The goal of this
section is to present some results attached to the Tame Fontaine-Mazur
Conjecture (TFMC for short) and the Tame Fontaine-Mazur Conjecture-Uniform
version (TFMC-U for short), as well as related questions that some of the
authors are currently working on.

###  5.1. Motivation and background

We start by recalling some results that are at the core of this section,
namely the two versions of Fontaine-Mazur Conjectures mentioned above that
will be proven below to be equivalent to each other.

As explained in Section 1, TFMC is an analogue of Conjecture 1.3 when
considering $p$-adic representations that are finitely and tamely ramified.
Since a theorem of Grothendieck [26, Appendix] ensures that tame
representations are automatically potentially semi-stable, Fontaine-Mazur
conjecture must imply (under some standard conjectures in algebraic geometry,
see [16] for more details) that the following statement holds [8, Conjecture
5a].

Fix a prime $p$. For a number field $K$ and a (finite) set of places $S$ of
$K$ all of which are co-prime to $p$, let $K_{S}$ be the maximal
pro-$p$-extension of $K$ that is unramified outside of $S$. Write
$G_{S}:=\operatorname{Gal}(K_{S}/K)$.

######  Conjecture 5.1 (Tame Fontaine-Mazur Conjecture).

Let $p$ be a prime integer and $K$ be a number field. Let $S$ be a finite set
of places of $K$ that are all co-prime to $p$ and let $K_{S}$ be the maximal
pro-$p$-extension of $K$ that is unramified outside $S$. Set
$G_{S}:=\mathrm{Gal}(K_{S}/K)$. Then, for any positive integer $n$, any
continuous Galois representation $\rho:G_{S}\to GL_{n}(\mathbb{Q}_{p})$ has
finite image.

In the next section (Section 5.2), we will see that when $n=1$, TFMC follows
by class field theory. For $n>1$, this conjecture appears in general
completely out of reach, though some preliminary evidence exists, as those
given by [6, 11, 28]. When $K=\mathbb{Q}$ and $n=2$, we pointed out in Section
1 that the main contribution so far is due to Kisin by different methods [15].

Recall that Theorem 3.11 claims that any finitely generated $p$-adic analytic
group contains a uniform open subgroup. Moreover, Theorem 3.17 asserts that
any uniform group of dimension $1$ or $2$ admits a quotient isomorphic to
$\mathbb{Z}_{p}$. We can hence rephrase TFMC as follows.

######  Conjecture 5.2 (Tame Fontaine-Mazur Conjecture - Uniform version for
$(K,d)$).

Let $K$ be a number field and $\Gamma$ be a uniform pro-$p$-group of dimension
$d>2$ (hence infinite). Then there is no finitely and tamely ramified Galois
extension of $K$ whose Galois group is isomorphic to $\Gamma$.

In the light of Conjectures 1.6 and 1.7 for $n=2$, we would like to underline
the major contribution due to Hajir and Maire in [12] before going further.
Among several results of importance, they proved an analogue of Theorem 1.5
for odd $p$ by the mean of purely group-theoretic and arithmetic methods such
as the effect of a semi-simple cyclic action with fixed points on the group
structure, the rigidity of uniform groups, the existence of Minkowski units,
some arithmetic properties of Galois groups, etc.

To make this more precise, we need to introduce some notation and definitions,
following [12, 1.2]. With the notation above, let $T$ be an auxiliary finite
set of places of $K$ such that $T\cap S=\emptyset$. Define $K^{T}_{S}$ as the
maximal pro-$p$-extension of $K$ that is unramified outside $S$ and in which
any place in $T$ splits completely, and let
$G_{S}^{T}:=\mathrm{Gal}(K^{T}_{S}/K)$ be the corresponding Galois group. Note
that $K_{S}^{T}$ is a subfield of $K_{S}$ while $G^{T}_{S}$ is a quotient of
$G_{S}$, and that $K_{S}^{\emptyset}=K_{S}$.

We can now state one of the main results of Hajir-Maire [12, Section 1.2,
Theorem], which can be seen as a special case of more general statements
proven in [12, Section 2].

######  Theorem 5.3 (Hajir-Maire).

Let $K/k$ be a quadratic extension with Galois group $\langle\sigma\rangle$.
Assume that the odd prime $p$ does not divide the class number of $k$. Suppose
that for any finite set $\Sigma$ of places of $k$ all co-prime to $p$, there
is no continuous Galois representation
$G_{\Sigma}(k)\twoheadrightarrow\Gamma_{2,1}$. Then there exist infinitely
many disjoint finite sets $S$ and $T$ of primes of $K$ such that any place of
$S$ is prime to $p$, $|S|$ is arbitrarily large and:

  1. (1)

$G^{T}_{S}$ is finite;

  2. (2)

$G^{T}_{S}/\Phi(G^{T}_{S})$ has $|S|$ independent fixed points under the
action of $\sigma$;

  3. (3)

there is no continuous representation
$\rho:G^{T}_{S}\twoheadrightarrow\Gamma_{2,1}$ whose image is a uniform group.

Note that in the last assertion, the uniformness assumption is actually
strengthened into a notion of $\sigma$-uniformness that gives further
condition on the action of $\sigma$ on $G^{T}_{S}$, see [12, Definition 1.1].
Let us also mention that, though it is not so obvious at first sight, this
result is actually tightly connected to Fontaine-Mazur Conjecture for uniform
groups of constant type for order $2$ automorphisms. The interested reader can
check [12, Corollary 2.6] for more details on this topic.

###  5.2. Class field theory implies Tame Fontaine-Mazur Conjecture for $n=1$

The goal of this section is to prove the tame Fontaine-Mazur Conjecture
(Conjecture 1.6) for $n=1$. This may be known to experts in the field, but we
think this is important to have a reference where this claim is actually
proven, hence we now give a full proof of the following statement.

######  Theorem 5.4 (TFMC for $n=1$).

Let $K$ be a number field and $p$ be a rational prime. Let $S$ be a finite set
of place of $K$ in which no place has residue characteristic $p$. Let $K_{S}$
denote the maximal pro-$p$-extension of $K$ that is unramified outside $S$ and
$G_{S}:=\mathrm{Gal}(K_{S}/K)$ be its Galois group. Then every continuous
group homomorphism

$\displaystyle\chi:G_{S}\to\operatorname{GL}_{1}(\mathbb{Q}_{p})=\mathbb{Q}_{p}^{\times}$

has finite image.

We keep the notation above throughout the rest of this section and prove
(using class field theory) that $\chi$ must have finite image. First note
that, since $G_{S}$ is a compact group (as it is a pro-$p$-group), is image
under the continuous map $\chi$ must be a compact subgroup of
$\mathbb{Q}_{p}^{\times}$, hence lands into $\mathbb{Z}_{p}^{\times}$. Also
note that, since $\mathbb{Q}_{p}^{\times}$ is an abelian group, $\chi$ must
factor through the abelianisation $G_{S}^{\operatorname{ab}}$ of $G_{S}$,
which motivates the connection with class field theory. The latter provides
indeed an injective group homomorphism with dense image (called Artin
reciprocity law)

$\rho:\mathbb{A}_{K}^{\times}/K^{\times}\hookrightarrow\operatorname{Gal}(\overline{K}/K)^{\operatorname{ab}}\
,$

that becomes an isomorphism when $\mathbb{A}_{K}^{\times}/K^{\times}$ is
replaced by its profinite completion. We are hence reduced to show that the
composite group homomorphism
$\tilde{\chi}:=\chi\circ\rho:\mathbb{A}_{K}^{\times}/K^{\times}\to\mathbb{Z}_{p}^{\times}$
has finite image.

Saying that $\chi$ is unramified outside $S$ means that $\chi$ is trivial on
inertia groups for places outside $S$, hence so is $\tilde{\chi}$. Now let $v$
be a place in $S$. We need to distinguish between the finite and archimedean
cases as follows.

If $v$ is a finite place, then the inertia group at $v$ corresponds to the
subgroup of $\mathbb{A}_{K}^{\times}$ that is trivial at all places outside of
$v$ and equal at $v$ to the subgroup $\mathcal{O}_{K_{v}}^{\times}$ of
$K_{v}^{\times}$. As all places in $S$ are co-prime to $p$, the group of
$1$-units in $\mathcal{O}_{K_{v}}^{\times}$ is a pro-$\ell$-group for some
rational prime $\ell\not=p$. Since any continuous group homomorphism from a
pro-$\ell$-group into a pro-$p$-group is necessarily trivial, knowing that the
$1$-units of $\mathbb{Z}_{p}^{\times}$ is a pro-$p$-group ensures that the
image by $\tilde{\chi}$ of $\mathcal{O}_{K_{v}}^{\times}$ is finite. Letting
$S^{\infty}$ denote the subset of finite places in $S$, we hence have proven
that $\tilde{\chi}\left(\prod_{v\in
S^{\infty}}\mathcal{O}_{K_{v}}^{\times}\right)$ is finite.

We now study what happens on $\displaystyle\prod_{v{\text{ finite
place}}}\varpi_{v}^{\mathbb{Z}}$, where $\varpi_{v}$ denotes a uniformising
element of $\mathcal{O}_{v}$. First note that the product is actually a
restricted product, which means only finitely many places are such that
$\varpi_{v}$ has non-trivial image in $\mathbb{Z}_{p}^{\times}$. Also recall
that any element $x=(\varpi_{v}^{e_{v}})_{v\text{ finite }}$ of
$\displaystyle\prod_{v{\text{ finite place}}}\varpi_{v}^{\mathbb{Z}}$ defines
a fractional ideal $\mathfrak{a}_{x}$ of $K$, namely
$\displaystyle\prod_{v}\mathfrak{p}_{v}^{e_{v}}$, where $\mathfrak{p}_{v}$
denotes the prime ideal of $\mathcal{O}_{K}$ corresponding to $v$. (This is
well-defined as only finitely many $e_{v}$ can be non-zero.) If $h$ denotes
the class number of $K$, then $\mathfrak{a}_{x}^{h}$ is a non-zero principal
ideal of $\mathcal{O}_{K}$ by definition of $h$. This implies that $x\mod
K^{\times}$ must have finite order dividing $h$. Thus we obtain that the
quotient of $\displaystyle\prod_{v{\text{ finite
place}}}\varpi_{v}^{\mathbb{Z}}$ by the diagonal copy of $K^{\times}$ (in
$\mathbb{A}_{K}^{\times}/K^{\times}$) has finite exponent dividing $h$, hence
its image by $\tilde{\chi}$ must satisfy the same condition in
$\mathbb{Q}_{p}^{\times}$. As $\mathbb{Q}_{p}^{\times}$ only contains finitely
many elements of order dividing $h$, we can conclude that this image by
$\tilde{\chi}$ is finite, as expected.

Finally assume that $v$ is an archimedean place in $S$. As
$\mathbb{Z}_{p}^{\times}$ is a totally disconnected space, the identity
component of $K_{v}^{\times}$ (that is either $\mathbb{R}_{>0}$ if $v$ is
real, or $\mathbb{C}^{\times}$ is $v$ is complex) has trivial image under the
continuous group homomorphism $\tilde{\chi}$. Letting $S_{\infty}$ denote the
set of archimedean places in $S$, we are hence left to see that
$\displaystyle\prod_{v\in S_{\infty}\text{ real
}}\mathbb{R}^{\times}/\mathbb{R}_{>0}$ has finite image under $\tilde{\chi}$,
which is straightforward as $S$ and
$\mathbb{R}^{\times}/\mathbb{R}_{>0}\simeq\mathbb{Z}/2\mathbb{Z}$ are finite
sets, so the proof is complete.

###  5.3. Equivalence of TFMC and TFMC-U

This subsection aims to prove the equivalence of Conjectures 1.6 (TFMC) and
1.7 (TFMC-U). First note that we have the following nice result coming from
[7, Theorem 4.5], which makes unnecessary the assumption that $G$ is infinite
in TFMC-U.

######  Proposition 5.5.

Any uniform pro-$p$-group is torsion-free.

###### TFMC $\implies$ TFMC-U.

Let $G$ be a uniform pro-$p$ group, and suppose that there exists a finitely
and tamely ramified Galois extension $L/K$ with $\operatorname{Gal}(L/K)\cong
G$. To prove that $G$ is necessarily finite, we will use for $G$ the following
key fact, proven in [7, Section 7.3].

######  Proposition 5.6.

Every pro-$p$-group of finite rank admits a faithful representation into
$\operatorname{GL}_{n}(\mathbb{Q}_{p})$ for some integer $n\geq 1$.

Hence, there is an integer $n\geq 1$ for which there exists a faithful
representation $\iota\colon
G\hookrightarrow\operatorname{GL}_{n}(\mathbb{Q}_{p})$. Fixing an isomorphism
$\psi$ between $\mathrm{Gal}(L/K)$ and $G$ as given in the statement, the
composition of $\iota$ by $\psi$ and by the natural projection map
$\operatorname{Gal}(\overline{K}/K)\to G$ provides a Galois representation

$\displaystyle\rho\colon\operatorname{Gal}(\overline{K}/K)\to\operatorname{GL}_{n}(\mathbb{Q}_{p})\
.$

Denote by $S$ the set of places in $K$ that ramify in $L$: by assumption, $S$
is a finite set and $\rho$ factors through $G_{S}$. We are now left to prove
that any place in $S$ is prime to $p$, as TFMC would hence be applicable to
$\rho$ and give that $\mathrm{Im}(\rho)\simeq G$ is finite, as expected.

Recall that $L/K$ is assumed to be tamely ramified. This ensures that for any
prime $\mathfrak{p}$ of $K$ and any prime $\mathfrak{q}$ of $L$ above
$\mathfrak{p}$, the extension $L_{\mathfrak{q}}/K_{\mathfrak{p}}$ is tamely
ramified, and there exists a unique intermediate Galois extension
$K_{\mathfrak{p}}\hookrightarrow F\hookrightarrow L_{q}$ such that $L_{q}/F$
is totally ramified and $F/K_{p}$ is unramified. The ‘tame quotient’ of the
decomposition group of $\mathfrak{q}$ above $\mathfrak{p}$ is then defined by
$\operatorname{Gal}(L_{q}/F)$, and it is in particular of (pro-)order prime-
to-$p$. Since $G$ is a pro-$p$-group, this implies that $L/K$ must be
unramified at places above $p$, hence that any place in $S$ is prime to $p$,
and we are done. ∎

###### TFMC-U $\implies$ TFMC.

Conversely, assume that $K$ is a number field and that $S$ is a finite set of
places of $K$ that are all prime to $p$, and let
$\rho:G_{S}\to\operatorname{GL}_{n}(\mathbb{Q}_{p})$ be a continuous Galois
representation. To use TFMC-U, we need to get a uniform pro-$p$-group our of
this setting. First note that since $G_{S}$ is compact while every maximal
compact subgroup of $GL_{n}(\mathbb{Q}_{p})$ is conjugate to
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$, we can assume without loss of
generality that $\rho$ takes values in
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$.

Set $G:=\mathrm{Im}(\rho)\subset\operatorname{GL}_{n}(\mathbb{Z}_{p})$. We
claim that $G$ is a $p$-adic analytic group. Indeed, [7, Corollary 8.34]
ensures that it is enough to prove that $G$ contains an open subgroup $H$ that
is a pro-$p$-group of finite rank. Let us set

$\mathcal{G}:=\ker(\operatorname{GL}_{n}(\mathbb{Z}_{p})\to\operatorname{GL}_{n}(\mathbb{Z}/p\mathbb{Z}))\
\text{ and }\ H:=G\cap\mathcal{G}\ .$

Then $H$ is certainly a pro-$p$-group (as $G$ is), and is open in $G$ (as
$\mathcal{G}$ is open in $\operatorname{GL}_{n}(\mathbb{Z}_{p})$), so we are
left to check that $H$ is of finite rank. Recall that the rank of $H$ is
defined as

$\displaystyle\mathrm{rk}(H):=\sup_{J\leq_{c}H}d(J)\ ,$

where $d(J)$ denotes the minimal number of elements needed to topologically
generate $J$, for $J$ a closed subgroup of $H$ (compare with [7, Proposition
3.11 and Definition 3.12]).

Note that $H$ is also closed in $\operatorname{GL}_{n}(\mathbb{Z}_{p})$, as
intersection of two closed subgroups of
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$, namely $G$ (that is the image of a
compact set under a continuous map) and $\mathcal{G}$ (that is open in
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$, hence closed). As a consequence, any
closed subgroup of $H$ is also closed in $\mathcal{G}$. Now recall that
$\mathcal{G}$ is a uniform pro-$p$-group of dimension
$d(\mathcal{G})=\operatorname{rk}(\mathcal{G})=n^{2}$ (see [7, Theorem 5.2],
which follows the same path as the proof of Theorem 3.12 given above). This
implies, thanks to [7, Theorem 3.8], that any closed subgroup $J$ of $H$
(hence of $\mathcal{G}$ too) satisfies

$d(J)\leq d(\mathcal{G})=n^{2}.$

In particular, this shows that $H$ has finite rank, hence that $G$ is a
$p$-adic analytic group. Applying [7, Corollary 8.34] then shows that $G$
contains an open normal uniform pro-$p$-subgroup $\Gamma$ of finite index.
Denote by $F$ the subfield extension of $K_{S}^{\ker\rho}/K$ on which $\Gamma$
acts trivially. As $\Gamma$ is normal of finite index in
$G=\mathrm{Im}(\rho)$, $F/K$ is a finite Galois extension. We can now apply
TFMC-U to $F/K$ and $\Gamma$ to obtain that $\Gamma$ has to be finite (and
actually trivial because of Proposition 5.5), hence so does
$G=\mathrm{Im}(\rho)$. Note that the ramification conditions of TFMC-U are
satisfied since $\rho$ is unramified outside $S$ while the places in $S$ are
all prime to $p$, so we are done. ∎

###  5.4. The small dimensional cases of TFMC ($d=1$ and $d=2$)

In this section, we will briefly see why we do not have to consider the cases
$d\leq 2$ in Conjecture 1.7. First recall that any uniform pro-$p$-group of
dimension $1$ is isomorphic to $\mathbb{Z}_{p}$, hence can be represented into
$\operatorname{GL}_{1}(\mathbb{Z}_{p})=\mathbb{Z}_{p}^{\times}$ using the map
$p\mapsto 1+p$. This reduces us to the case $n=1$ of TFMC, which was proven
from class field theory in Section 5.2 (see Theorem 5.4 in particular).

Now assume that $\Gamma$ is a uniform pro-$p$-group of dimension $2$. Using
[7, Exercise 3.11], we know that $\Gamma$ is meta-procyclic, which means that
there exists an element $\gamma$ in $\Gamma$ such that
$\overline{\langle\gamma\rangle}$ is a normal subgroup of $\Gamma$ and the
quotient $\Gamma/\overline{\langle\gamma\rangle}$ is pro-cyclic. Suppose that
there exists a finitely and tamely ramified Galois extension $L/K$ with
$\mathrm{Gal}(L/K)$ isomorphic to $\Gamma$. Quotienting $\Gamma$ by
$\overline{\langle\gamma\rangle}$ then provides a
$\Gamma/\overline{\langle\gamma\rangle}\simeq\mathbb{Z}_{p}$-extension of $K$
that is finitely and tamely ramified. This cannot happen as we just proved
that TFMC-U holds for $d=1$, so $L/K$ cannot exist and TFMC-U is proven for
$d=2$.

###  5.5. Tame ramification and co-primality matters

The goal of this section is to discuss why the two main assumptions in TFMC
(tame ramification and co-primality with $p$ of the places in $S$) cannot be
dropped. We start by discussing the ramification assumption and give a simple
example that shows why tame ramification is necessary in the statement of
Conjecture 1.6, i.e. we explain how to produce an interesting representation
$\rho\colon G_{\mathbb{Q},S}\to\operatorname{GL}_{2}(\mathbb{Q}_{p})$ that is
wildly ramified at $p$ and whose image is infinite.

Let $\mathcal{E}$ be a non-CM elliptic curve defined over $\mathbb{Q}$ and $p$
be any prime integer. By Serre’s open image theorem [25], we know that the
$p$-adic representation of $G_{\mathbb{Q}}$ attached to $\mathcal{E}$ has open
image in $\operatorname{GL}_{2}(\mathbb{Z}_{p})$ (and is actually surjective
for almost all $p$). Also note that this representation is unramified away
from $p$ and from the primes dividing the conductor of $\mathcal{E}$, which
certainly provides a finite set $S$. We are hence left to see that this
representation is wildly ramified at $p$ to conclude. To do so, we notice that
the determinant of this representation is the $p$-adic cyclotomic character
$\chi_{p}\colon G_{\mathbb{Q}}\to\mathbb{Z}_{p}^{\times}$. As the tame
quotient of $G_{\mathbb{Q}_{p}}$ is prime to $p$ while
$\mathbb{Z}_{p}^{\times}$ is (up to a finite part) a pro-$p$-group, we get
that $\chi_{p}$ is wildly ramified at $p$, hence so is the $p$-adic
representation of $G_{\mathbb{Q}}$ attached to $\mathcal{E}$.

One can now wonder whether one can get rid of this wild ramification by simply
twisting away the determinant. To deal with this, we need to set up a bit more
notation first. We fix a prime integer $p>2$ and let $\rho\colon
G_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathbb{Z}_{p})$ be the $p$-adic
representation attached to $\mathcal{E}$: we hence have $\det\rho=\chi_{p}$.
Let $\langle\ \cdot\ \rangle\colon\mathbb{Z}_{p}^{\times}\to
1+p\mathbb{Z}_{p}$ denote the projection of $\mathbb{Z}_{p}^{\times}$ onto its
pro-$p$ part. As $p$ is odd, every element of $1+p\mathbb{Z}_{p}$ has a unique
squareroot that is congruent to $1$ mod $p$, so we get a character
$\varepsilon:G_{\mathbb{Q}}\to 1+p\mathbb{Z}_{p}$ whose square equals
$\langle\chi_{p}\rangle$. If we set
$\rho^{\prime}:=\rho\otimes\varepsilon^{-1}$, then $\det\rho^{\prime}$ is of
finite order, so we cannot use the previous argument to show that
$\rho^{\prime}$ is wildly ramified at $p$. Nevertheless, note that the
surjectivity of $\rho$ would ensure that $\mathrm{Im}(\rho^{\prime})$ contains
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$, and otherwise
$\mathrm{Im}(\rho^{\prime})$ will at least contain an open subgroup of
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$, so $\mathrm{Im}(\rho^{\prime})$
cannot be finite, and TFMC predicts that $\rho^{\prime}$ must be wildly
ramified at $p$. We can prove this last assertion without using TFMC: indeed,
first observe that if the image of the inertia contains an open subgroup of a
unipotent subgroup, i.e. the conjugate of an open subgroup of

$U=\left\\{\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}\colon
u\in\mathbb{Z}_{p}\right\\}\ ,$

then so does $\mathrm{Im}(\rho^{\prime})$ as $\det(U)=\\{1\\}$. As $U$ is a
pro-$p$-group while the tame inertia quotient has pro-order prime-to-$p$,
$\rho^{\prime}$ cannot be tamely ramified at $p$, hence must be wildly
ramified at $p$, as expected.

######  Remark 5.7.

There are many one-dimensional counterexamples: for instance, any integer $k$
defines a one-dimensional representation $\chi_{p}^{(p-1)k}$ that is
unramified outside $p$ and wildly ramified at $p$. Nevertheless, such
representations do not arise from geometry (since they are not De Rham
representations), which is not very satisfying in the context of Fontaine-
Mazur conjectures, the latter being motivated by understanding representations
coming from geometry.

Let us now focus on the other main assumption, namely that the places in $S$
must be co-prime to $p$. First recall that any $\mathbb{Z}_{p}$-extension must
be unramified at primes with residue characteristic different from $p$. As
$\mathbb{Z}_{p}$ is abelian, the finiteness of the class number requires that
$\mathbb{Z}_{p}$-extensions cannot be everywhere unramified, hence there must
be some ramification at places above $p$. In particular, this implies that the
requirement claiming that $S$ contains no place above $p$ is necessary in
Conjecture 1.6.

Following Fontaine and Mazur [8, page 44], one can wonder whether the
finiteness of $S$ is automatic, or if it is an actual assumption, at least for
semisimple finite-dimensional $p$-adic representations. Ramakrishna answered
this question when $n=2$, proving in [21] that under GRH, there exists an
explicit irreducible $2$-dimensional $p$-adic representation that ramifies at
infinitely many primes but is potentially semistable at $p$. Let us moreover
mention that in [14], Khare and Ramakrishna proved that the conditions $S$ is
finite and $S$ does not contain places above $p$ required in the statement of
Conjecture 1.6 are actually independent.

Since the finiteness of $S$ is not automatic even in small dimension, one can
wonder whether it is a necessary condition for Conjecture 1.6 to hold. Again,
this question has been addressed by Ramakrishna and his co-authors in [13].
More precisely, recall that a $p$-adic representation is said deeply ramified
at a given prime when it does not vanish on any of the corresponding higher
ramification groups of finite index. With this definition in mind, one can
rephrase our question as follows: can we find a $p$-adic representation that
ramifies at infinitely many primes of a number field $K$ but is not deeply
ramified at $p$? The answer is yes, at least when $n=2$ and $p>5$, as proven
by Khare-Larsen-Ramakrishna in [13, Main Theorem].

##  6\. Some future directions

We end this paper by giving some future directions of research, some of them
being work in progress, others being longer-term projects. First recall that
the results of Hajir-Maire on which are based Section 5, but also the most
recent results connected to these questions obtained by Maire [20], require
the prime $p$ to be odd. A first natural question we hence want to explore in
the future is the following one.

######  Question 6.1.

To which extend do the previous results remain valid for $p=2$?

Another interesting question, standing at the core of an ongoing project of
two of the authors [1], is to study an explicit example of biquadratic field
pointed out by Boston and Buell [5, page 290] as being close to providing an
actual counterexample to Conjecture 1.3. More precisely, we are interested in
the following question.

######  Question 6.2.

Set $K=\mathbb{Q}(\sqrt{-104},\sqrt{229})$, $p=3$ and let $L$ be the maximal
everywhere unramified pro-$p$-extension of $K$. Does $\operatorname{Gal}(L/K)$
have an infinite $p$-adic analytic pro-$p$-quotient?

Note that Boston’s method cannot be used to test whether Conjecture 1.3 holds
for this biquadratic field and this $p$, since we are not dealing with a
cyclic extension. This hence leads to wonder whether we could extend Boston’s
method beyond the cyclic case, i.e. when $\operatorname{Gal}(K/\mathbb{Q})$ is
a non-cyclic extension.

#### Acknowledgments

We want to thank the organizers of the workshop Women in Numbers Europe 3 for
giving us this opportunity to work on this project. We also warmly thank
Jaclyn Lang for many stimulating discussions and for earlier work on this
project.

## References

  * [1] R. Abdellatif, S. Pisolkar, On a question of Nigel Boston about Fontaine-Mazur Conjecture, preprint available upon request (2021).
  * [2] C. Breuil, A. Mézard, Multiplicités modulaires et représentations de $GL_{2}(\mathbb{Z}_{p})$ et de $\operatorname{Gal}(\mathbb{Q}_{p}/\mathbb{Q}_{p})$ en $l=p$, Duke Math. J. 115 (2002), 205–310, with an appendix by G. Henniart.
  * [3] C. Breuil, Sur quelques représentations modulaires et $p$-adiques de $GL_{2}(Q_{p})$ I, Compositio Math. 138 (2003), 165–188.
  * [4] C. Breuil, Sur quelques représentations modulaires et $p$-adiques de $GL_{2}(Q_{p})$ II, J. Inst. Math. Jussieu 2 (2003), 1–36.
  * [5] N. Boston, Some cases of the Fontaine-Mazur conjecture, J. Number Theory 42, (3) (1992), 285–291.
  * [6] N. Boston, Some cases of the Fontaine-Mazur conjecture. II, J. Number Theory 75 (2) (1999), 161–169.
  * [7] J.D. Dixon, J. D, M.P.F. du Sautoy, A. Mann, D. Segal, Analytic pro-p groups, Second edition, Cambridge Studies in Advanced Mathematics, 61 (1999), Cambridge University Press, Cambridge.
  * [8] J.-M. Fontaine, B. Mazur, Geometric Galois representations, in “Elliptic Curves and Modular Forms” (J. H. Coates and S. T. Yau, Eds.), Proceedings of a Conference in Hong-Kong, December 18–21, 1993; Ser. Number Theory, I (1995), Int. Press, Cambridge, MA.
  * [9] J.-M. Fontaine, Représentations $p$-adiques semi-stables (Exposé III), Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque No. 223 (1994), 59–111.
  * [10] E. Golod, I. Shafarevich, On class field towers, Izv. Aka. Nauk SSSR 28 (1964), 261–272.
  * [11] F. Hajir, On the growth of $p$-class groups in $p$-class field towers, J. Algebra 188 (1) (1997), 256–271.
  * [12] F. Hajir, C. Maire, Analytic Lie extensions of number fields with cyclic fixed points and tame ramification, preprint (2017).
  * [13] C. Khare, M. Larsen, R. Ramakrishna, Constructing semisimple $p$-adic Galois representations with prescribed properties, Amer. J. Math. 127 (4) (2005), 709–734.
  * [14] C. Khare, R. Ramakrishna, Finiteness of Selmer groups and deformation rings, Invent. Math. 154 (1) (2003), 179–198.
  * [15] M. Kisin, The Fontaine-Mazur conjecture for $GL_{2}$, J. Amer. Math. Soc. 22 (3) (2009), 641–690.
  * [16] M. Kisin, S. Wortmann, A note on Artin motives, Math. Res. Lett. 10 (2-3) (2003), 375–389.
  * [17] M. Lazard, Groupes analytiques $p$-adiques, Publ. Math. Inst. Hautes Études Sci. No. 26 (1965), 389–603.
  * [18] Maire, Christian; Cohomology of number fields and analytic pro-p-groups, Mosc. Math. J. 10 (2010), no. 2, 399–414, 479.
  * [19] C. Maire, On the quotients of the maximal unramified $2$-extension of a number field, Doc. Math. 23 (2018), 1263–1290.
  * [20] C. Maire, On Galois representations with large image, preprint (2021).
  * [21] D.E. Rohrlich, Root numbers, in “Arithmetic of L-functions”, IAS/Park City Math. Ser. 18 (2011), Amer. Math. Soc., Providence, RI, 353–448.
  * [22] R. Ramakrishna, Deforming an even representation, Invent. Math. 132 (3) (1998), 563–580.
  * [23] B. Poonen, A brief summary of the main statements of class field theory, Expository paper (2012), see www-math.mit.edu/~poonen/papers/cft.pdf
  * [24] L. Ribes, P. Zalesskii, Profinite groups, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 40 (2010), Springer-Verlag, Berlin.
  * [25] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.
  * [26] J.-P. Serre, J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (2) (1968), 492–517.
  * [27] J. Tate, Relations between $K_{2}$ and Galois cohomology, Invent. Math. 36 (1976), 257–274.
  * [28] K. Wingberg, On the Fontaine-Mazur conjecture for CM-fields, Compositio Math.131 (3) (2002), 341–354.


#

⋆ Instituto de Física, Universidade Federal do Rio de Janeiro, Av. Athos da
Silveira Ramos, 149, Cidade Universitária, 21941-972, Rio de Janeiro, RJ,
Brazil  ‡

Comment on “Universal and accessible entropy estimation using a compression
algorithm”

In a recent Letter Avinery  a framework for estimating entropy was introduced
and applied to one-dimensional and two-dimensional systems. In this Comment we
show that the method is not well suited for estimating entropy in
bidimensional systems presenting long-range correlations.

In Ref. Avinery , entropy is evaluated following this scheme: 1) discretize
the considered configurations 2) store them in a 1D file 3) measure the
compressed file size with a lossless compression algorithm 4) estimate the
incompressibility and map it to the asymptotic entropy $S_{A}$. Here we test
this scheme for binary variables, and we do not consider the complication
produced by the coarse-graining of continuous variables.

In general, lossless compression algorithms are known to present slow entropy
convergence and alternative more efficient methods are used Grassberger96 .
There is a rich literature studying 1D systems but the analyses of
multidimensional patterns are very few, and they are expected to present new
features compared to the 1D case. Some methods are known to apply to patterns
of arbitrary dimension (e.g. block entropies). In contrast, mapping
multidimensional patterns to a one-dimensional sequence can be path-dependent,
loses bidimensional correlations, and can even produce spurious long-range
correlations Grassberger86 . The use of a locality-preserving curve, like
Hilbert’s one, does not guarantee to solve these difficulties.

Systems which present long-range correlations can be used to test these
effects. In the original Letter, long-range correlations are matched only in
the 2D NN Ising model near the critical point. In that region, the estimated
$S_{A}$ displays a relative error between 5% and 10%. Unfortunately, this
Ising model is a particular exception where the reduction to a 1D string is
known to not affect the statistics that determine entropy estimation Ising1D .
For this reason, this system is not a good benchmark for testing entropy
estimation in 2D patterns with long-range correlations. For testing the
Avinery’s framework in this situation, here we consider a set of 68 different
patterns obtained from built form maps of cities around the world nos2 . These
binary matrices of size $1000\times 1000$ are particularly well suited for the
test. The subextensive part of their block entropies diverges nos2 ; nos1 ,
implying the presence of very long-range correlations that make the entropy
estimation particularly difficult. For these symbolic sequences we can not
obtain the exact entropy values with analytical methods. Instead, we can use a
reliable block-entropies method Feldman  which estimates the entropy by using
differential entropies ($S_{B}$). Results are robust, as alternative block-
entropies methods give equivalent outcomes nos2 .

Figure 1: Scatter plot of the $S_{A}$ and $S_{B}$ values. $S_{A}$ are
estimated by using the method in Avinery  with the Hilbert’s curve. $S_{B}$
are obtained following the scheme of equation 3 in nos2 , for blocks of size
up to $15$. The continuous line represents the linear fitting: $y=1.25x+0.15$.
In the inset, the distribution of ($S_{A}$-$S_{B}$)/$S_{B}$.

Fig.1 compares the results obtained by the classical block-entropies method
($S_{B}$) with the ones generated by the compression algorithm ($S_{A}$). This
second approach dramatically overestimates the entropy. The median of the
percentage error is 91% and the scattering plot shows that the difference
between $S_{A}$ and $S_{B}$ grows for larger $S_{B}$ values. The reduction of
2D patterns to a 1D string significantly destroys the involved bidimensional
structures, which are particularly significant in systems with long-range
correlations, and generates a substantial overestimation of the entropy.

Even if Avinery’s method is indeed elegant and computationally effective, it
can not be considered accurate for general 2D systems.

E. Brigatti

> Instituto de Física, Universidade Federal do Rio de Janeiro, Av. Athos da
> Silveira Ramos, 149, 21941-972, Rio de Janeiro, RJ, Brasil.

F.N.M. de Sousa Filho

> Instituto de Computação, Universidade Federal do Rio de Janeiro, Av. Athos
> da Silveira Ramos, 274, 21941-916, Rio de Janeiro, RJ, Brasil.

## References

  * (1) R. Avinery, M. Kornreich, and R. Beck, Universal and accessible entropy estimation using a compression algorithm, Phys. Rev. Lett. 123, 178102 (2019).
  * (2) Schürmann T, Grassberger P. Entropy estimation of symbol sequences. Chaos, 6, 414 (1996).
  * (3) Grassberger P. Toward a quantitative theory of self-generated complexity. Intl. J. Theo. Phys., 25, 907 (1986).
  * (4) Z Alexandrowicz. Stochastic models for the statistical description of lattice systems. J. Chem. Phys., 55 2765, (1971); S. Goldstein, R. Kuik, and A.G. Schlijper, Entropy and global Markov properties, Commun. Math. Phys., 126, 469 (1990).
  * (5) E Brigatti, VM Netto, FNM de Sousa Filho , and C Cacholas, Entropy and hierarchical clustering: characterizing the morphology of the urban fabric in different spatial cultures. Chaos, 31, 113138 (2021).
  * (6) Netto, VM, Brigatti E, Meirelles J, Ribeiro FL, Pace B, Cacholas C, Sanches P. Cities, from Information to Interaction. Entropy, 20, 834 (2018).
  * (7) Feldman DP, Crutchfield JP. Structural information in two-dimensional patterns: Entropy convergence and excess entropy. Phys. Rev. E, 67, 051104 (2003).