# Can we dream of a
1-adic Langlands correspondence?

Xavier Caruso  CNRS; IMB, Université de Bordeaux, 351 cours de la Libération,
33405 Talence, France<EMAIL_ADDRESS>Agnès David  LMB,
Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France;
IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France
<EMAIL_ADDRESS>Ariane Mézard  DMA, École Normale Supérieure PSL, 45
rue d’Ulm, 75005 Paris, France<EMAIL_ADDRESS>
###### Abstract

After observing that some constructions and results in the $p$-adic Langlands
programme are somehow independent from $p$, we formulate the hypothesis that
this astonishing uniformity could be explained by a $1$-adic Langlands
correspondence.

To Catriona Byrne.

###### Contents

  1. 1 Combinatorics around the Breuil-Mézard conjecture
    1. 1.1 Review on the Breuil-Mézard conjecture
    2. 1.2 Encoding representations with combinatorial data
    3. 1.3 Explicit computations of $R^{\lambda,{\rm t}}_{\overline{\rho}}$
  2. 2 The field with one element
    1. 2.1 Clues in favor of a $1$-adic Breuil-Mézard conjecture
    2. 2.2 Major challenges
    3. 2.3 Conclusion
  3. A Remarks on Galois theory in characteristic $1$
    1. A.1 Brief review of geometry over $\mathbb{F}_{1}$
    2. A.2 Galois theory over $\mathbb{F}_{1}$
    3. A.3 Galois theory over ${\mathbb{Q}}_{1}$

The Langlands programme is a far-reaching and influential web of theorems and
conjectures which has motivated a lot of research in Number Theory and
Arithmetic Geometry for more than fifty years. Very roughly, it stipulates a
profound and meaningful correspondence between representations of Galois
groups on the one hand and representations of reductive groups on the other
hand. Many variations on this theme are actually possible, depending on which
base field (number field, function field, $p$-adic field, _etc_.) and which
category of coefficients we are working with.

The pioneer works in the Langlands programme were mostly concerned with
$\mathbb{C}$-valued representations. However, since the beginning of the 21st
century, a purely $p$-adic version of Langlands correspondence has emerged
under the impulsion of Breuil. Nowadays, this $p$-adic correspondence is fully
established for $2$-dimensional representations of
$\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/{\mathbb{Q}}_{p}$) but, beyond
this, little is known. Many examples have however been worked out and several
conjectures have been proposed—and sometimes proved—throughout the years. One
of them is the Breuil-Mézard conjecture, which predicts that the geometrical
properties of some Galois deformations spaces are directly related to the
decomposition properties of some representations of the corresponding
reductive group.

Looking more carefully into the aforementioned works, we notice that, in many
cases, the underlying prime number $p$ often plays a figurative role in the
calculations. Typically, the relevant reductive groups are usually defined
over ${\mathbb{Z}}$ and a significant part of the constructions and arguments
can be carried out at this level. On the Galois side, this constancy is not so
obvious but it is nevertheless visible; indeed, even though a prime number
needs to be fixed from the very beginning, we often observe, at the end of the
day, that the results of the computations are mostly independent from it.

We make the hypothesis that these strong properties of uniformity with respect
to $p$ could have a deep meaning and all be the consequences of a new type of
Langlands correspondence, which should be considered as the common denominator
of the $p$-adic Langlands correspondences when $p$ varies. We call this new
hypothetic correspondence the _$1$ -adic Langlands correspondence_ because we
believe that the natural language to formulate it is the mysterious theory of
characteristic one whose main protagonist is the famous field with one
element.

The aim of this note is to bring the reader to the agreement that our
hypothesis is not crazy but has conceivable foundations and deserves
consideration. We start our argumentation by reviewing in §1 some recent
developments towards the Breuil-Mézard conjecture, with the objective to
highlight the places where the arguments and/or the notions take a
combinatorial flavour in which we have the feeling that the underlying prime
number $p$ plays a secondary role. Then, in §2, we briefly recall the
philosophy of the field with one element and show that it is incredibly
appropriate for interpreting many objects and carrying out many constructions
encountered in §1. Finally, in order to give more substance to our dream, we
conclude this article by an appendix in which we share some thoughts towards
the development of a Galois theory in characteristic one, which is certainly a
prerequisite for a $1$-adic Langlands correspondence.

##  1 Combinatorics around the Breuil-Mézard conjecture

Let $p>2$ be a prime number. Throughout this section, we fix a finite
extension $K$ of ${\mathbb{Q}}_{p}$ and write
$G_{K}=\operatorname{Gal}(\overline{{\mathbb{Q}}}_{p}/K)$ for its absolute
Galois group. The Breuil-Mézard conjecture is a concrete statement relating
deformations spaces of representations of $G_{K}$, on the one hand, and
representations of $p$-adic reductive groups, on the other hand. The aim of
this section is, firstly, to recall the formulation of this conjecture and,
secondly, to emphasize that, in many cases, it can be approched using
combinatorial arguments and constructions. These observations will be the key
to build bridges with the field with one element in §2.

###  1.1 Review on the Breuil-Mézard conjecture

We denote by ${\mathcal{O}}_{K}$ (resp $k_{K}$) the ring of integers (resp.
the residue field) of $K$. Let
$\overline{\rho}:G_{K}\to\operatorname{GL}_{n}(\overline{\mathbb{F}}_{p})$ be
a continuous $\overline{\mathbb{F}}_{p}$-representation of $G_{K}$ of
dimension $n$ and let $R_{\overline{\rho}}$ denote the
$\overline{\mathbb{Z}}_{p}$-algebra parametrizing the deformations of
$\overline{\rho}$. In [Ki3], Kisin proved that $R_{\overline{\rho}}$ admits
quotients with strong arithmetical interest. More precisely, given in addition
the two following data:

  * •

a _Hodge type_ $\lambda$, that is, by definition, the datum of a tuple
$(\lambda_{1},\ldots,\lambda_{n})\in{\mathbb{Z}}^{n}$ with
$\lambda_{1}\geq\cdots\geq\lambda_{n}$ for all embedding
$\iota:k_{K}\hookrightarrow\overline{\mathbb{F}}_{p}$,

  * •

an _inertial type_ ${\rm t}$, that is, by definition, a finite dimensional
$\overline{\mathbb{Q}}_{p}$-representation of the inertia subgroup
$I_{K}\subset G_{K}$ having open kernel and admitting an extension to $G_{K}$,

Kisin constructed a surjective morphism of
$\overline{\mathbb{Z}}_{p}$-algebras $R_{\overline{\rho}}\to R^{\lambda,{\rm
t}}_{\overline{\rho}}$ that parametrizes the lifts of $\overline{\rho}$ which
are potentially crystalline with Hodge Tate weights
$(\lambda_{i}+n-i)_{i,\,\iota}$ and inertial type ${\rm t}$.

The Breuil-Mézard conjecture is a numerical relation between the Hilbert-
Samuel multiplicity of the special fibre of $R^{\lambda,{\rm
t}}_{\overline{\rho}}$, denoted by $e(R^{\lambda,{\rm
t}}_{\overline{\rho}}\otimes_{\overline{\mathbb{Z}}_{p}}\overline{\mathbb{F}}_{p})$,
and invariants coming from the representation theory of
$\operatorname{GL}_{n}$. Precisely, let $L_{\lambda}$ be the
$\overline{\mathbb{Z}}_{p}$-representation of
$\operatorname{GL}_{n}({\mathcal{O}}_{K})$ of highest weight $\lambda$. After
[He, ScZi, CEG+], we know that there is a finite dimensional smooth
irreducible $\overline{\mathbb{Q}}_{p}$-representation $\sigma({\rm t})$ of
$\operatorname{GL}_{n}({\mathcal{O}}_{K})$ associated to ${\rm t}$. We choose
a $\operatorname{GL}_{n}({\mathcal{O}}_{K})$-stable
$\overline{\mathbb{Z}}_{p}$-lattice $L_{{\rm t}}$ in $\sigma({\rm t})$, form
the tensor product $L_{\lambda,{\rm t}}=L_{{\rm
t}}\otimes_{\overline{\mathbb{Z}}_{p}}L_{\lambda}$ and write its semi-
simplification modulo $p$ as follows:

$\big{(}L_{\lambda,{\rm
t}}\otimes_{\overline{\mathbb{Z}}_{p}}\overline{\mathbb{F}}_{p}\big{)}^{\text{ss}}\simeq\bigoplus_{\sigma\in{\mathcal{D}}}\sigma^{n_{\lambda,{\rm
t}}(\sigma)}$

where the sums runs over the set ${\mathcal{D}}$ of Serre weights $\sigma$,
that is the set of (isomorphism classes of) irreducible
$\overline{\mathbb{F}}_{p}$-representations of $\operatorname{GL}_{n}(k_{K})$.

######  Conjecture 1 (Breuil-Mézard).

There exists a family of integers
$(\mu_{\overline{\rho}}(\sigma))_{\overline{\rho},\sigma}$, called _intrinsic
multiplicities_ , such that the following numerical equality holds:

$e\big{(}R_{\overline{\rho}}^{\lambda,{\rm
t}}\otimes_{\overline{\mathbb{Z}}_{p}}\overline{\mathbb{F}}_{p}\big{)}=\sum_{\sigma\in{\mathcal{D}}}n_{\lambda,{\rm
t}}(\sigma)\>\mu_{\overline{\rho}}(\sigma)$ (1)

for all triple $(\overline{\rho},\lambda,{\rm t})$ as above.

The Breuil-Mézard conjecture was first formulated for $2$-dimensional
representations in [BrMé1]. Since then, it has attracted a lot of attention.
Kisin [Ki1] proved it when $K={\mathbb{Q}}_{p}$ (and $n=2$) by making use of
the $p$-adic local Langlands correspondence for
$\operatorname{GL}_{2}({\mathbb{Q}}_{p})$ and the (global) Taylor-Wiles-Kisin
patching argument. Sander [Sa] and Paškūnas [Pa] gave a purely local
alternative proof which has been extended later on by Hu and Tan to nonscalar
split residual representations [HuTa]. For a general $K$, but still assuming
$n=2$, the conjecture was proved by Gee and Kisin [GeKi] (see also [CEGS,
Appendix C]) when $\lambda=(0,0)$ for each embedding (which corresponds to
potentially Barsotti–Tate deformations).

The extension of the Breuil-Mézard conjecture to higher $n$ came later. The
case of $3$-dimensional representations was considered and partially solved by
Herzig, Le and Morra [HLM] and Le, Le Hung, Levin and Morra [LLHLM1, LLHLM2].
The formulation stated in dimension $n$ in Conjecture 1 is due to Emerton and
Gee [EmGe] (see also [GHS]).

###  1.2 Encoding representations with combinatorial data

It turns out that, in some cases, all the objects that intervene in the
statement of the Breuil-Mézard conjecture can be entirely described by
combinatorial data. Besides, these explicit descriptions provide us with a new
viewpoint on the conjecture, quite useful for attacking it.

#### The case of $2$-dimensional representations

We start with the case of $\operatorname{GL}_{2}$ for which encodings are
simpler and more is known. For simplicity, we assume further that
$K/{\mathbb{Q}}_{p}$ is unramified, _i.e._ $K={\mathbb{Q}}_{p^{f}}$ for some
positive integer $f$. In dimension $2$, the _irreducible_ continuous
representations
$\overline{\rho}:G_{K}\to\operatorname{GL}_{2}(\overline{\mathbb{F}}_{p})$ all
take the form:

$\overline{\rho}=\operatorname{Ind}_{G_{K^{\prime}}}^{G_{K}}\Big{(}\omega_{2f}^{h}\cdot\text{\rm
nr}^{\prime}(\theta)\Big{)}$ (2)

where $K^{\prime}$ is the unique unramified extension of degree $2$ of $K$,
$\omega_{2f}$ is the fundamental character of $G_{K^{\prime}}$ of level $2f$
and $\text{\rm nr}^{\prime}(\theta)$ denotes the unique unramified character
of $G_{K^{\prime}}$ sending the arithmetic Frobenius to $\theta$. The
parameters $h$ and $\theta$ are an integer defined modulo $p^{2f}{-}1$ and an
element of $\overline{\mathbb{F}}_{p}$ respectively.

Similarly, we have a complete description of _tame_ inertial types, that are,
by definition, those inertial types ${\rm
t}:I_{K}\to\operatorname{GL}_{2}(\overline{\mathbb{F}}_{p})$ that factor
through the tame inertia. Depending on whether they are reducible or not, they
are of the form:

${\rm
t}=\omega_{f}^{\gamma}\oplus\omega_{f}^{\gamma^{\prime}}\quad\text{or}\quad{\rm
t}=\operatorname{Ind}_{I_{K^{\prime}}}^{I_{K}}\omega_{2f}^{\gamma}$ (3)

with obvious notations. In the first case, we say that ${\rm t}$ has level
$f$; otherwise, that it has level $2f$.

We now assume further that $\lambda=(0,0)$ for each embedding. In this case,
the integers $n_{\lambda,{\rm t}}(\sigma)$ are always $0$ or $1$ and we define
${\mathcal{D}}({\rm t})$ as the set of Serre weights $\sigma\in{\mathcal{D}}$
for which $n_{\lambda,{\rm t}}(\sigma)=1$. Similarly, it is conjectured that
$\mu_{\overline{\rho}}(\sigma)$ is always $0$ or $1$ as well and we let
${\mathcal{D}}(\overline{\rho})\subset{\mathcal{D}}$ be the locus over which
$\mu_{\overline{\rho}}(\sigma)$ is strictly positive. Understanding the
summation in the Breuil-Mézard conjecture (see Eq. (1)) then amounts to
understanding the set ${\mathcal{D}}({\rm
t},\overline{\rho})={\mathcal{D}}({\rm t})\cap{\mathcal{D}}(\overline{\rho})$.

It turns out that ${\mathcal{D}}({\rm t})$ and
${\mathcal{D}}(\overline{\rho})$ admit very explicit combinatorial
descriptions in terms of the parameters $h$, $\gamma$ and $\gamma^{\prime}$ we
introduced earlier. These descriptions first appeared in [BDJ, BrPa] and were
then simplified in [Da]. Very roughly, once ${\rm t}$ (resp.
$\overline{\rho}$) is fixed, the weights in ${\mathcal{D}}({\rm t})$ (resp. in
${\mathcal{D}}(\overline{\rho})$) are parametrized by tuples
$\underline{\varepsilon}=\\{\varepsilon_{0},\ldots,\varepsilon_{f-1}\\}\in\\{0,1\\}^{f}$.
Each such tuple produces a Serre weight by a simple recipe and the set
${\mathcal{D}}({\rm t})$ (resp. ${\mathcal{D}}(\overline{\rho})$) is finally
obtained by putting together all such weights. We underline that
${\mathcal{D}}({\rm t})$ and ${\mathcal{D}}(\overline{\rho})$ have usually
cardinality strictly less than $2^{f}$ because some $\underline{\varepsilon}$
may actually fail to produce a weight and it also happens that two different
$\underline{\varepsilon}$ lead to the same weight.

#### The gene

Building on the previous results, we gave in [CDM3] a purely combinatorial
description of the intersection ${\mathcal{D}}({\rm
t},\overline{\rho})={\mathcal{D}}({\rm t})\cap{\mathcal{D}}(\overline{\rho})$
in terms of the parameters $h$, $\gamma$ and $\gamma^{\prime}$, assuming that
the tame inertial type is reducible. More precisely, setting $q=p^{f}$ for
simplicity, we considered the quantity $h-(q{+}1)\gamma^{\prime}\text{ mod
}q^{2}-1$ and wrote its decomposition in radix $p$:

$h-(q{+}1)\gamma^{\prime}\equiv
p^{2f-1}v_{0}+p^{2f-2}v_{1}+\cdots+pv_{2f-2}+v_{2f-1}\pmod{q^{2}-1}.$ (4)

From the $v_{i}$’s, we then formed a periodic sequence
$\mathbb{X}=(X_{i})_{i\in{\mathbb{Z}}}$ of period $2f$ assuming values in the
finite set $\\{\text{\tt A},\text{\tt B},\text{\tt AB},\text{\tt O}\\}$. The
sequence $\mathbb{X}$ is called the _gene_ of $({\rm t},\overline{\rho})$ and
it satisfies the following rules (see [CDM3, Lemma B.1.3]):

  * •

if $v_{i}=0$ and $X_{i+1}=\text{\tt O}$, then $X_{i}=\text{\tt AB}$;

  * •

if $v_{i}=0$ and $X_{i+1}\neq\text{\tt O}$, then $X_{i}=\text{\tt A}$;

  * •

if $v_{i}=1$ and $X_{i+1}=\text{\tt O}$, then $X_{i}=\text{\tt O}$;

  * •

if $v_{i}=1$ and $X_{i+1}\neq\text{\tt O}$, then $X_{i}=\text{\tt B}$;

  * •

if $v_{i}\geq 2$, then $X_{i}=\text{\tt O}$.

To each gene $\mathbb{X}$, we attached a set ${\mathcal{W}}(\mathbb{X})$ of
_combinatorial weights_ , which are sequences of length $f$ with values in
$\\{0,1\\}$. We then proved (see [CDM3, Theorem 3.1.2]) that, if $\mathbb{X}$
denotes the gene of $({\rm t},\overline{\rho})$, there is a canonical
bijection:

${\mathcal{W}}(\mathbb{X})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{D}}({\rm
t},\overline{\rho}).$ (5)

Beyond yielding an explicit description of ${\mathcal{D}}({\rm
t},\overline{\rho})$ and opening concrete and algorithmical perspectives on
the Breuil-Mézard conjecture, the above result raises new questions because it
somehow shows that the dependence of ${\mathcal{D}}({\rm t},\overline{\rho})$
in ${\rm t}$, $\overline{\rho}$ and even in the underlying prime number $p$
itself, is very weak, given that the gene only retains few information about
these data. In some sense, one can interpret the gene as the “skeleton” of the
pair $({\rm t},\overline{\rho})$ that captures its most fundamental
combinatorial properties in view of the Breuil-Mézard conjecture. In this
perspective, the construction $\mathbb{X}\mapsto{\mathcal{W}}(\mathbb{X})$
should be thought of as the core factory of Serre weights, while the
bijections (5), for varying ${\rm t}$, $\overline{\rho}$ and $p$, appear as
many tangible incarnations of this manufacture.

#### Higher dimension and group-theoretic formulation

When we are moving to higher dimensions, the numerical descriptions we used
previously cannot continue to be that simple but, interestingly, they have
analogues which can be formulated in the language of group theory. In what
follows, we continue to assume that $K={\mathbb{Q}}_{p^{f}}$ for some positive
integer $f$. In this setting, the relevant algebraic group is:

$G=\big{(}\text{\rm
Res}_{{\mathcal{O}}_{K}/{\mathbb{Z}}_{p}}\operatorname{GL}_{n/{\mathcal{O}}_{K}}\big{)}\times_{{\mathbb{Z}}_{p}}\overline{\mathbb{Z}}_{p}\simeq\prod_{\mathfrak{J}}\operatorname{GL}_{n/\overline{\mathbb{Z}}_{p}}$

where $\mathfrak{J}$ is the set of embeddings
${\mathcal{O}}_{K}\hookrightarrow\overline{\mathbb{Z}}_{p}$ (or, equivalenty,
$k_{K}\hookrightarrow\overline{\mathbb{F}}_{p}$) and will be identified with
${\mathbb{Z}}/f{\mathbb{Z}}$ in what follows. We denote by $T$ the diagonal
maximal torus of $G$. We let $R$ be the set of roots of $(G,T)$ and $W$ be the
corresponding Weyl group. The Borel $B\subset G$ of upper triangular matrices
determines a subset $R^{+}\subset R$ of positive roots. The group of
characters $X^{\star}(T)$ can be canonically identified with
$({\mathbb{Z}}^{n})^{\mathfrak{J}}=({\mathbb{Z}}^{n})^{f}$ and the Weyl group
$W$ is isomorphic to $\mathfrak{S}_{n}^{f}$. We shall also need the extended
Weyl group $\widetilde{W}$ of $G$, defined by $\widetilde{W}=W\ltimes
X^{\star}(T)\simeq(\mathfrak{S}_{n}\ltimes{\mathbb{Z}}^{n})^{f}$.

In this setting, a Serre weight is an (isomorphism class of) irreducible
$\overline{\mathbb{F}}_{p}$-representation of
$\operatorname{GL}_{n}(\mathbb{F}_{p^{f}})$. It follows from a somehow
classical argument of the theory of representation of reductive groups that
Serre weights can be parametrized by certain characters of $G$. Precisely,
after [GHS, Lemma 9.2.4], we know that if we set:

$\displaystyle X_{0}(T)$ $\displaystyle=\big{\\{}\,\lambda\in
X^{\star}(T)\text{ s.t. }\langle\lambda,\alpha^{\vee}\rangle=0\text{ for all
}\alpha\in R^{+}\,\big{\\}}$ $\displaystyle X_{1}(T)$
$\displaystyle=\big{\\{}\,\lambda\in X^{\star}(T)\text{ s.t.
}0\leq\langle\lambda,\alpha^{\vee}\rangle\leq p-1\text{ for all }\alpha\in
R^{+}\,\big{\\}}$

and let $\pi$ denote the shift
$(x_{i})_{i\in{\mathbb{Z}}/f{\mathbb{Z}}}\mapsto(x_{i+1})_{i\in{\mathbb{Z}}/f{\mathbb{Z}}}$
on $X^{\star}(T)\simeq({\mathbb{Z}}^{n})^{f}$, there is a bijection:

$F:X_{1}(T)/(p{-}\pi)X_{0}(T)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{D}}$
(6)

taking a character $\lambda$ to the restriction to $\text{\rm
Res}_{{\mathcal{O}}_{K}/{\mathbb{Z}}_{p}}\operatorname{GL}_{n/{\mathcal{O}}_{K}}(\mathbb{F}_{p})=\operatorname{GL}_{n}(\mathbb{F}_{p^{f}})$
of the representation of $G(\mathbb{F}_{p})$ induced by the algebraic
representation of $G$ of highest weight $\lambda$.

We also have a description of tame inertial types in terms of elements of the
group $\widetilde{W}$ [LMS, LLHLM1, BHHMS]. Let
$(s,\mu)\in\widetilde{W}=W\ltimes X^{\star}(T)$ and write
$s=(s_{0},\ldots,s_{f-1})$ with $s_{i}\in\mathfrak{S}_{n}$ for all $i$. Let
$r$ be the order of $s_{0}s_{f-1}\cdots s_{1}\in\mathfrak{S}_{n}$. We consider
the unramified extension $K^{\prime}$ of $K$ of degree $r$ and let
$k_{K^{\prime}}$ denote its residue field. We let
$\tilde{\omega}_{rf}:I_{K}=I_{K^{\prime}}\to\overline{\mathbb{Z}}_{p}^{\times}$
be the Teichmüller lift of the Serre fundamental character of level $rf$. For
$j\in\mathfrak{J}$, we put:

$\eta_{j}=\left\\{\begin{array}[]{ll}(n-1,\ldots,1,0)&j\mbox{-th
coordinate}\\\ (0,\ldots,0)&\mbox{elsewhere}\end{array}\right.$

and $\eta=\sum_{j\in\mathfrak{J}}\eta_{j}$. We define
$\alpha^{\prime}_{(s,\mu)}\in
X^{\star}(T)^{\operatorname{Hom}_{k_{K}\text{-alg}}(k_{K^{\prime}},\overline{\mathbb{F}}_{p})}\simeq
X^{\star}(T)^{r}\simeq({\mathbb{Z}}^{n})^{rf}$ by:

$\alpha^{\prime}_{(s,\mu),j}=s_{1}^{-1}s_{2}^{-1}\cdots
s_{j}^{-1}(\mu_{j}+\eta_{j})$

and finally set:

$\tau(s,\mu+\eta)=\bigoplus_{1\leq i\leq
n}\tilde{\omega}_{rf}^{\sum_{j^{\prime}=0}^{rf-1}\alpha^{\prime}_{(s,\mu),j^{\prime},i}p^{j^{\prime}}}.$
(7)

It is our tame inertial type.

Irreducible representations
$\overline{\rho}:G_{K}\to\operatorname{GL}_{n}(\overline{\mathbb{F}}_{p})$ can
be encoded in a similar fashion. Moreover, when $n=2$, it turns out that the
explicit descriptions of ${\mathcal{D}}({\rm t})$ and
${\mathcal{D}}(\overline{\rho})$ we have mentioned earlier can be rephrased in
the language of group theory which was briefly sketched above (at least under
sufficiently generic assumptions). We do not reproduce here the corresponding
recipes (which involve the so-called _$p$ -dot product_) but refer to
Propositions 2.4.2 and 2.4.3 of [BHHMS] for more details. So far, we do not
have any candidate for being a plausible replacement of the gene when $n>2$.
However, all the above constructions tend to show that, even though they now
need to be formulated in the language of group theory, combinatorics is still
here (and is maybe even more ubiquitous) in higher dimensions.

###  1.3 Explicit computations of $R^{\lambda,{\rm t}}_{\overline{\rho}}$

The Breuil-Mézard conjecture is concerned with the special fibre of
$R^{\lambda,{\rm t}}_{\overline{\rho}}$ but, of course, obtaining a complete
description of the ring $R^{\lambda,{\rm t}}_{\overline{\rho}}$ has also
interest for its own. For example, explicit presentations of some
$R^{\lambda,{\rm t}}_{\overline{\rho}}$ have been used by Emerton, Gee and
Savitt [EGS] to prove important conjectures stated by Breuil in [Br] about
lattices in the cohomology of Shimura curves. In this subsection, we outline
the standard strategy that is used to approach $R^{\lambda,{\rm
t}}_{\overline{\rho}}$ and report on the results of some explicit
computations.

#### Review on Kisin’s construction of $R^{\lambda,{\rm t}}_{\overline{\rho}}$

The main theoretical ingredient for studying deformations of $\overline{\rho}$
with prescribed Hodge type and inertial type is the theory of Breuil-Kisin,
which provides a description of those deformations by means of semi-linear
algebra. In our setting and assuming in addition that ${\rm t}$ is tame of
level $f$ and $K={\mathbb{Q}}_{p^{f}}$ as we already did previously, a Breuil-
Kisin module is a projective module $\mathfrak{M}$ over
$\overline{\mathbb{Z}}_{p}\otimes_{{\mathbb{Z}}_{p}}{\mathcal{O}}_{K}[[u]]\simeq\overline{\mathbb{Z}}_{p}[[u]]^{\mathfrak{J}}$
equipped with two additional structures:

  * •

a _Frobenius map_ $\varphi_{\mathfrak{M}}:\mathfrak{M}\to\mathfrak{M}$ which
is semi-linear (with respect to the endomorphism of
$\overline{\mathbb{Z}}_{p}\otimes_{{\mathbb{Z}}_{p}}{\mathcal{O}}_{K}[[u]]$
acting by the identity on $\overline{\mathbb{Z}}_{p}$, by the Frobenius of
${\mathcal{O}}_{K}$ and taking $u$ to $u^{p}$) and satisfy additional
properties,

  * •

a _descent data_ , that is a linear action of the group
$\operatorname{Gal}(K[\sqrt[e]{p}]/K)\simeq{\mathbb{Z}}/e{\mathbb{Z}}$ (with
$e=p^{f}-1)$ that commutes with the Frobenius.

A famous theorem of Kisin [Ki2] indicates that those modules are in
correspondence with $\overline{\mathbb{Z}}_{p}$-representations of $G_{K}$
that become crystalline over the extension $K[\sqrt[e]{p}]$. Besides, the
Hodge type (resp. the inertial type) of the latter can be easily read off on
the form of the Frobenius map (resp. of the descent data) on the former. Even
better, the reduction modulo $p$ of the representation associated to a Breuil-
Kisin module $\mathfrak{M}$ is uniquely and entirely described by the module
$\mathfrak{M}\otimes_{{\mathcal{O}}_{K}[[u]]}k_{K}((u))$ equipped with its
additional structures.

The Breuil-Kisin theory then looks particularly well fitted for the study of
the deformations rings $R^{\lambda,{\rm t}}_{\overline{\rho}}$ and it turns
out that it indeed is. In [Ki3], Kisin constructed a scheme
$\mathcal{G}\mathcal{R}^{\lambda,{\rm t}}_{\overline{\rho}}$ parametrizing the
Breuil-Kisin modules $\mathfrak{M}$ of Hodge type $\lambda$, inertial type
${\rm t}$ and having the additional property that
$\mathfrak{M}\otimes_{{\mathcal{O}}_{K}[[u]]}k_{K}((u))$ corresponds to the
given representation $\overline{\rho}$. This scheme is moreover equipped with
a morphism $\mathcal{G}\mathcal{R}^{\lambda,{\rm
t}}_{\overline{\rho}}\to\operatorname{Spec}R_{\overline{\rho}}$ whose
schematic image has closure $\operatorname{Spec}R^{\lambda,{\rm
t}}_{\overline{\rho}}$. One should be careful however that the morphism:

$\kappa:\mathcal{G}\mathcal{R}^{\lambda,{\rm
t}}_{\overline{\rho}}\to\operatorname{Spec}R^{\lambda,{\rm
t}}_{\overline{\rho}}$ (8)

is _not_ an isomorphism in general because two $p$-torsion Breuil-Kisin
modules may correspond to the same Galois representation. It is however always
an isomorphism in generic fibre. The special fibre of
$\mathcal{G}\mathcal{R}^{\lambda,{\rm t}}_{\overline{\rho}}$ is denoted by
$\overline{\mathcal{G}\mathcal{R}}^{\lambda,{\rm t}}_{\overline{\rho}}$ and is
called the _Kisin variety_ ; in some sense, it measures the default for
$\kappa$ to be an isomorphism.

#### Examples in dimension $2$: the generic case

The first examples of explicit calculations of certain rings $R^{\lambda,{\rm
t}}_{\overline{\rho}}$ have been carried out by Breuil and Mézard in [BrMé1]
and [BrMé2]. They considered the case where $\overline{\rho}$ is
$2$-dimensional and absolutely irreducible, $\lambda=(0,0)$ for each embedding
and ${\rm t}$ is tame of level $f$. Under some additional assumptions of
genericity on $\overline{\rho}$, they obtained, when ${\mathcal{D}}({\rm
t},\overline{\rho})$ is not empty:

$R^{\lambda,{\rm
t}}_{\overline{\rho}}\simeq\frac{\overline{\mathbb{Z}}_{p}[[X_{i},Y_{i},i\in\mathfrak{J}_{\text{\rm
II}},Z_{j},j\in\mathfrak{J}\setminus\mathfrak{J}_{\text{\rm
II}}]]}{\left(X_{i}Y_{i}-p,i\in\mathfrak{J}_{\text{\rm II}}\right)}.$ (9)

for a certain subset $\mathfrak{J}_{\text{\rm II}}$ of $\mathfrak{J}$ (which
depends on $\lambda$, ${\rm t}$ and $\overline{\rho}$). The aforementioned
genericity assumptions play a quite important role in Breuil and Mézard’s
argument. In fact, they imply that the underlying Kisin variety is reduced to
one point, which itself ensures that the morphism $\kappa$ of Eq. (8) is an
isomorphism. The computation of $R^{\lambda,{\rm t}}_{\overline{\rho}}$ then
directly reduces to that of $\mathcal{G}\mathcal{R}^{\lambda,{\rm
t}}_{\overline{\rho}}$.

Before moving to nongeneric cases, it is important to comment on the subset
$\mathfrak{J}_{\text{\rm II}}$ which appeared in Eq. (9). The triviality of
the Kisin variety indicates that the module over
$\overline{\mathbb{F}}_{p}\otimes_{{\mathbb{F}}_{p}}k_{K}((u))\simeq\overline{\mathbb{F}}_{p}((u))^{\mathfrak{J}}$
corresponding to $\overline{\rho}$ contains a unique lattice
$\mathfrak{M}(\overline{\rho})$ that is a Breuil-Kisin module of type
$(\lambda,{\rm t})$. Breuil and Mézard then defined the _shape_ of
$\mathfrak{M}(\overline{\rho})$: it is a finite sequence
$(g_{0},\ldots,g_{f-1})$ assuming values in the finite set $\\{\text{\rm
I},\text{\rm II}\\}$ which, roughly speaking, is obtained by looking at the
form of the matrix of $\varphi_{\mathfrak{M}(\overline{\rho})}$ in bases
diagonalizing the action of the descent data. The set $\mathfrak{J}_{\text{\rm
II}}$ is then formed by the indices $i$ for which $g_{i}$ is II.

The shape also plays a key role on the $\operatorname{GL}_{2}$-side of the
Breuil-Mézard conjecture: in our setting, the cardinality of $\mathcal{D}({\rm
t},\overline{\rho})$ is $2^{\text{Card}(\mathfrak{J}_{\text{\rm II}})}$ (which
is the Hilbert-Samuel multiplicity of the special fibre of the ring
$R^{\lambda,{\rm t}}_{\overline{\rho}}$ given by Eq. (9)) and, more precisely,
we can explicitely parametrize the weights in $\mathcal{D}({\rm
t},\overline{\rho})$ by subsets of $\mathfrak{J}_{\text{\rm II}}$.

#### Examples in dimension $2$: the nongeneric case

Nongeneric cases are more complicated because they usually correspond to
nontrivial Kisin varieties. In [CDM2], we computed those Kisin varieties when,
as above, $\overline{\rho}$ is absolutely irreducible, $\lambda=(0,0)$ for
each embedding and ${\rm t}$ is tame of level $f$. We recall that, in this
setting, we have attached to the pair $({\rm t},\overline{\rho})$ its gene
$\mathbb{X}$ (see §1.2). Our results show that the Kisin variety is entirely
determined by the gene. Being a little bit more precise, we showed that
$\overline{\mathcal{G}\mathcal{R}}^{\lambda,{\rm t}}_{\overline{\rho}}$ is a
closed subscheme of
$(\mathbb{P}^{1}_{\overline{\mathbb{F}}_{p}})^{{\mathbb{Z}}/f{\mathbb{Z}}}$
defined by equations of the form:

$\lambda_{i}\>x_{i}\>y_{i+1}=\mu_{i}\>x_{i+1}\>y_{i}$ (10)

where $[x_{i}:y_{i}]$ denotes the projective coordinates on the $i$-th copy of
$\mathbb{P}^{1}_{\overline{\mathbb{F}}_{p}}$ and $\lambda_{i}$ and $\mu_{i}$
are elements of $\\{0,1\\}$ that can be read off on the gene $\mathbb{X}$.

It is important to observe that the notion of shape can be extended to the
nongeneric case as well. Indeed, each $\overline{\mathbb{F}}_{p}$-point $x$ of
$\overline{\mathcal{G}\mathcal{R}}^{\lambda,{\rm t}}_{\overline{\rho}}$
corresponds, by definition, to a Breuil-Kisin module and so has a well defined
shape $g(x)=(g_{0}(x),\ldots,g_{f-1}(x))\in\\{\text{\rm I},\text{\rm
II}\\}^{f}$ in the sense of Breuil and Mézard. In full generality, the shape
is thus no longer a unique element in $\\{\text{\rm I},\text{\rm II}\\}^{f}$
but a function on the $\overline{\mathbb{F}}_{p}$-points of the Kisin variety
taking values in $\\{\text{\rm I},\text{\rm II}\\}^{f}$. We proved moreover
that this function is lower-continuous (for the partial ordering on the
codomain defined by $\text{\rm I}<\text{\rm II}$) and thus defined a
stratification on $\overline{\mathcal{G}\mathcal{R}}^{\lambda,{\rm
t}}_{\overline{\rho}}$ by locally closed subschemes. As well as the Kisin
variety, the shape stratification is entirely determined by the gene.

We then proposed the following conjecture.

######  Conjecture 2.

  1. (i)

The generic fibre of $R^{\lambda,{\rm t}}_{\overline{\rho}}$ is determined by
the Kisin variety equipped with its shape stratification.

  2. (ii)

The ring $R^{\lambda,{\rm t}}_{\overline{\rho}}$ is determined by the gene.

Regarding the first item of the conjecture, we were actually much more precise
and exhibited a candidate for being the generic fibre of $R^{\lambda,{\rm
t}}_{\overline{\rho}}$. Besides, our candidate is rather explicit: it is
defined as the formal neighborhood of the Kisin variety in a certain blow-up
of
$(\mathbb{P}^{1}_{\overline{\mathbb{Z}}_{p}})^{{\mathbb{Z}}/f{\mathbb{Z}}}$.
We refer to [CDM2, §5.4] for a complete exposition of this construction.

Conjecture 2 is known is most cases where the Kisin variety is trivial. It has
also been checked in [CDM1] in one example where the Kisin variety is
isomorphic to $\mathbb{P}^{1}_{\overline{\mathbb{F}}_{p}}$; in this case, the
deformations ring we obtained is
$\overline{\mathbb{Z}}_{p}[[X,Y,Z]]/(XY-p^{2})$.

#### Higher dimension and group-theoretic formulation

In dimension $3$, tamely potentially crystalline deformation rings for small
Hodge-Tate weights and generic Galois representations have been studied in
[LLHLM2]. The authors also obtained explicit presentations of the deformations
rings $R^{\lambda,{\rm t}}_{\overline{\rho}}$ in several cases. The equations
they found are often quite similar to the one given in Eq. (9). The case of
rank $2$ unitary group has also been considered in [KoMo]. In this setting, it
turns out that the explicit computations of the corresponding $R^{\lambda,{\rm
t}}_{\overline{\rho}}$’s boil down to the case of $\operatorname{GL}_{2}$
(with an additional polarization structure on the Breuil-Kisin modules). The
final equations they get are then again similar to Eq. (9).

In all these situations, although computations are certainly much more
difficult, it is important to underline that the basic ingredients are the
same: we continue to have a Kisin variety, together with a shape
stratification and we hope that those two objects strongly govern the final
form of the deformations space. In full generality (_i.e._ for any reductive
group, even not necessarily $\operatorname{GL}_{n}$), the Kisin variety and
the shape stratification are defined using techniques coming from group
theory: the Kisin variety is a subspace of the affine Grassmannian defined by
an explicit condition, which can be formulated in terms of the Cartan
decomposition, while the shape function $x\mapsto g(x)$ is defined by means of
the Iwahori-Cartan decomposition (and it now takes values in the Iwahori-Weyl
group). After [CaLe], we know moreover that Kisin varieties are related with
Pappas-Zhu local models [PaZh]; in this connection, the shape stratification
corresponds to the canonical stratification by affine Schubert varieties.

##  2 The field with one element

The theory of the field with one element ($\mathbb{F}_{1}$) starts with an
observation of Tits [Ti] who notices inquiring numerical coincidences in the
theory of algebraic linear groups. The most fundamental example is given by
the group $\operatorname{GL}_{n}$ itself. Indeed, its number of points over
the finite field $\mathbb{F}_{q}$ is given by:

$\displaystyle\operatorname{Card}\operatorname{GL}_{n}(\mathbb{F}_{q})$
$\displaystyle=(q^{n}-1)\cdot(q^{n}-q)\cdots(q^{n}-q^{n-1})$
$\displaystyle=(q-1)^{n}\cdot
q^{n(n-1)/2}\cdot[n]_{q}\cdot[n{-}1]_{q}\cdots[1]_{q}$

where $[i]_{q}=1+q+\cdots+q^{i-1}$ is the $q$-analogue of $i$ and letting $q$
tends to $1$, we find:

$\operatorname{Card}\operatorname{GL}_{n}(\mathbb{F}_{q})\sim_{q\to
1}(q-1)^{n}\cdot n!.$ (11)

What is surprising is that $n!$ can be interpreted as the cardinality of the
symmetric group $\mathfrak{S}_{n}$, which is nothing but the Weyl group of
$\operatorname{GL}_{n}$. Similar results hold more generally for a large
family of groups, including the orthogonal groups, the sympletic groups and
their scalar restrictions. After these observations, Tits asked if these
numerical matchings could have deeper roots and proposed to build a geometry
of the so-called _field with one element_ with the objective to give a
systematical and geometrical understanding of all the combinatorial structures
and constructions which appear in the theory of Lie groups or algebraic
groups.

Tits’ vision was then popularized by Soulé who came up in [So] with a first
tentative definition of affine varieties over $\mathbb{F}_{1}$. Later on,
other constructions were proposed and the subject has attracted more and more
attention for the last two decades [De1, ToVa, CCM, CoCo, Co, LPL, Lo1, BBK];
see [Lo2] for a recent review on this topic. Nowadays, the theory of
$\mathbb{F}_{1}$ is not well estalished yet. However, several definitions for
the category of schemes over $\mathbb{F}_{1}$ have been proposed over the
years and significant progress towards Tits’ initial dream have been realized.
Besides, geometry over $\mathbb{F}_{1}$ have been extended to new contexts and
have nowadays close interactions with Arakelov geometry and $p$-adic geometry.
In particular, Bambozzi, Ben-Bassat and Kremnizer introduced in [BBK] analytic
geometry over $\mathbb{F}_{1}$; they managed notably to construct a model of
the Fargues-Fontaine curve [FaFo] in their theory. Up to our knowledge, this
was the first connexion between the theory of Galois representations
(incarnated here by $p$-adic Hodge theory) and the field with one element.

###  2.1 Clues in favor of a $1$-adic Breuil-Mézard conjecture

The main reason why we believe that a $1$-adic version of the Breuil-Mézard
conjecture is possible is that many objects involved in the formulation and/or
the resolution of this conjecture do have natural seeds in
$\mathbb{F}_{1}$-geometry. In this subsection, we list the most important of
them and comment on their $\mathbb{F}_{1}$-aspects.

First of all, we notice that the Weyl group $W$ and its extended version
$\widetilde{W}$, which both play a quite important role in the construction of
Serre’s weights and inertial types, do have natural interpretations in
characteristic one: the Weyl group is the set of $\mathbb{F}_{1}$-points (this
property is Tits’ dream, which is the main guide of the theory) while the
extended Weyl group may be interpreted as the set of points over
$\mathbb{F}_{1}(X)$ (see Eq. (12) in Appendix A.1) Moreover, the recipes used
for constructing Serre’s weights and inertial types from an element of
$\widetilde{W}$ (see Eqs. (6) and (7)) are purely combinatorial and it is
quite likely that they can be reformulated by means of
$\mathbb{F}_{1}$-geometry.

The equations of the Kisin varieties we obtained in [CDM2] (see Eq. (10)) show
that all of them are defined over $\mathbb{F}_{1}$. Similarly, the
deformations spaces computed in [BrMé1, BrMé2] and [LLHLM2] all appear as
product of discs and annuli (see particularly Eq. (9)); as a consequence, they
all come throught scalar extensions from analytic spaces over $\mathbb{F}_{1}$
in the sense of [BBK]. Moreover, although the construction of blow-ups and
formal neighborhoods was not addressed in [BBK], it looks quite plausible that
the candidates for deformation spaces we introduced in [CDM2] are defined over
$\mathbb{F}_{1}$ as well.

All the above examples show furthermore a strong uniform behaviour with
respect to $p$. In the language of $\mathbb{F}_{1}$-geometry, this uniformity
means that a whole family of Kisin varieties (resp. of deformation spaces)
parametrized by $p$ comes by scalar extension to $\mathbb{F}_{p}$ (resp. to
${\mathbb{Q}}_{p}$) from a _unique_ variety (resp. analytic variety) over
$\mathbb{F}_{1}$. This result might suggest the existence of a common
denominator of the theory of Kisin varieties (resp. deformations spaces) which
is defined in characteristic one and underpins some of their features we
observe over the $p$-adics. This expectation is strengthened by the fact that
Kisin varieties have a deep group-theoretic interpretation (see last paragraph
of §1.3). The same remark is valid for the shape stratification as well: they
have good chance to be visible in characteristic one, given that thay are
closely connected to affine Schubert varieties, which are themselves known to
be defined over $\mathbb{F}_{1}$ [LPL].

The recipe of [CDM3], giving a combinatorial description of the set of common
Serre’s weights in terms of the corresponding gene, has also a strong
$\mathbb{F}_{1}$-flavour. Concretely, what we expect is that:

  1. (1)

the gene is a sort of $\mathbb{F}_{1}$-encoding of the pair $({\rm
t},\overline{\rho})$,

  2. (2)

the combinatorial weights of [CDM3] are the mirror of a notion of Serre’s
weights in characteristic $1$,

  3. (3)

the association

$\text{gene}\,\mapsto\,\text{set of combinatorial weights}$

is the $\mathbb{F}_{1}$-incarnation of the construction $({\rm
t},\overline{\rho})\mapsto{\mathcal{D}}({\rm t},\overline{\rho})$.

Beyond the justifications coming from the constructions of [CDM3], we
underline that there are other evidences supporting that Serre’s weights in
characteristic $1$ should have something to do with combinatorial weights
(which are, we recall, sequences of length $f$ assuming values in
$\\{0,1\\}$). Indeed, mimicing the usual definition in characteristic $p$, we
expect Serre’s weights in characteristic $1$ to be interpreted as
$\bar{\mathbb{F}}_{1}$-representations (whatever it means) of the group
$\operatorname{GL}_{2}(\mathbb{F}_{1^{f}})$. But, following Tits’ vision, we
can write:

$\operatorname{GL}_{2}(\mathbb{F}_{1^{f}})=\big{(}\text{\rm
Res}_{\mathbb{F}_{1^{f}}/\mathbb{F}_{1}}\operatorname{GL}_{2}\big{)}(\mathbb{F}_{1})=\text{Weyl}\big{(}\text{\rm
Res}_{\mathbb{F}_{p^{f}}/\mathbb{F}_{p}}\operatorname{GL}_{2}\big{)}=({\mathbb{Z}}/2{\mathbb{Z}})^{f}$

and we already see the set $\\{0,1\\}$ entering into the scene. More
precisely, we can define $\operatorname{Sym}^{k}\mathbb{F}_{1}^{2}$ as the set
$\\{X^{k},XY^{k-1},\ldots,Y^{k}\\}$ and let
$\operatorname{GL}_{2}(\mathbb{F}_{1})={\mathbb{Z}}/2{\mathbb{Z}}$ act on it
by letting its unique nontrivial element operate by swapping $X$ and $Y$ (see
Appendix A.1 for a justification of this definition). Similarly, if
$\underline{k}=(k_{0},\ldots,k_{f-1})$ is a tuple of integers, we let
$\operatorname{Sym}^{\underline{k}}\mathbb{F}_{1}^{2}$ be the cartesian
product of the $\operatorname{Sym}^{k_{i}}\mathbb{F}_{1}^{2}$ equipped with
the induced action of
$\operatorname{GL}_{2}(\mathbb{F}_{1^{f}})=({\mathbb{Z}}/2{\mathbb{Z}})^{f}$.
It is then an easy exercise to check that
$\operatorname{Sym}^{\underline{k}}\mathbb{F}_{1}^{2}$ is irreducible (_i.e._
the action is transitive) if and only if $k_{i}\in\\{0,1\\}$ for all $i$.

We underline that Conjecture 2 is also in line with the above vision: roughly
speaking, it stipulates that the mapping $({\rm t},\overline{\rho})\mapsto
R^{{\rm t}}_{\overline{\rho}}$ descends over $\mathbb{F}_{1}$.

Combining the previous observations and being quite optimistic, one might hope
that the Pappas-Rapoport spaces [PaRa] and/or Emerton-Gee stacks [EmGe]
themselves have a model over $\mathbb{F}_{1}$ and that the irreducible
components of the special fibre of the latter will be related to the set of
Serre’s weights in characteristic $1$.

###  2.2 Major challenges

We do not hide that, if possible, devising a $1$-adic Langlands correspondence
(or a $1$-adic Breuil-Mézard conjecture) will definitely not be a simple task.
Actually, although the geometry over $\mathbb{F}_{1}$ has already been
developed quite a lot, a lot of fundamental ingredients and objects of the
Langlands programme are missing in characteristic $1$.

To start with, we notice that extensions of $\mathbb{F}_{1}$, usually referred
to as $\mathbb{F}_{1^{n}}$, have been already considered by several authors
[So, KaSm, Co] but they have never been systematically studied. Moreover, in
the above references, $\mathbb{F}_{1^{n}}$ is defined as the cyclotomic
extension of $\mathbb{F}_{1}$ whose Galois group is isomorphic to
$({\mathbb{Z}}/n{\mathbb{Z}})^{\times}$, and not ${\mathbb{Z}}/n{\mathbb{Z}}$.
This means in particular that we do not have apparently a nice analogue of the
Frobenius endomorphism, which sounds annoying. An option for fixing this issue
is to work with another version of $\mathbb{F}_{1^{n}}$ on which we impose by
design the existence of a Frobenius of order $n$ (see Appendix A.2 for a first
rough proposal).

Similarly, the field of $1$-adic numbers ${\mathbb{Q}}_{1}$ and its extensions
have not attracted much attention so far. We mention however that Connes
introduced the ring of Witt vectors over $\mathbb{F}_{1}$ in [Co] but we are
afraid that Connes’ treatement does not perfectly fit with our perspectives
since it is eventually related to Banach algebras over the reals, and not over
the $p$-adics. In Appendix A.3, using a (certainly too) naive definition of
${\mathbb{Q}}_{1}$, we start exploring its theory of finite extensions.

On a different note, it seems that the theory of representations of reductive
groups have not been systematically studied yet. This could sound surprising
given that representations of reductive groups over finite fields have been
attracted a lot of attention for more than fifty years and the underlying
theory includes a lot of combinatorial parts that have good chance to be
“defined” over $\mathbb{F}_{1}$.

###  2.3 Conclusion

Although that, clearly, many locks still need to be unlocked, we continue to
believe that the $1$-adic Langlands correspondence is possible. Besides,
according to us, trying to develop it could be, at the same time, a wonderful
motivation and guide for exploring the $1$-adic world and for inspiring the
$p$-adic Langlands correspondence by separating the universal combinatorial
structures on the one hand and the more usual arithmetical properties on the
other hand. We thus warmly encourage all contributions to this topic.

##  Appendix A Remarks on Galois theory in characteristic $1$

In this appendix, we start exploring the Galois properties of the field with
one element $\mathbb{F}_{1}$ and the field of $1$-adic numbers
${\mathbb{Q}}_{1}$. Our aim is not at all to elaborate a complete and coherent
theory but to share our intuition and point out some difficulties. In §A.1, we
review briefly the usual constructions of geometries over $\mathbb{F}_{1}$. We
then successively address the Galois theory of $\mathbb{F}_{1}$ and
${\mathbb{Q}}_{1}$ in §A.2 and §A.3 respectively.

###  A.1 Brief review of geometry over $\mathbb{F}_{1}$

Most of modern theories of geometry over $\mathbb{F}_{1}$ start with defining
the category $\mathbb{F}_{1}\textbf{-Vect}$ of $\mathbb{F}_{1}$-vector spaces.
After Tits’ observation that $\operatorname{GL}_{d}(\mathbb{F}_{1})$ should be
isomorphic to $\mathfrak{S}_{d}$, it is tempting to define a vector space over
$\mathbb{F}_{1}$ simply as a set, this cardinality corresponding to its
dimension over $\mathbb{F}_{1}$. By definition, a $\mathbb{F}_{1}$-linear
morphism $V\to W$ is a set-theoretical _partially defined_ function $f:V\to
W$. The direct sum (resp. the tensor product) of two vector spaces is their
disjoint union (resp. their cartesian product). Besides, with this point of
view, the standard $\mathbb{F}_{1}$-vector space of dimension $d$, namely
$\mathbb{F}_{1}^{d}$, is represented by the set $\\{1,\ldots,d\\}$; its group
of automorphisms then coincides with $\mathfrak{S}_{d}$, _i.e._
$\operatorname{GL}_{d}(\mathbb{F}_{1})=\mathfrak{S}^{d}$ as expected.

After vector spaces over $\mathbb{F}_{1}$, one introduces
$\mathbb{F}_{1}$-algebras: they are, by definition, objects in commutative
monoids in the category $\mathbb{F}_{1}\textbf{-Vect}$, _i.e._ sets equipped
with a partially defined law of commutative monoids which is usually denoted
with the multiplicative convention. Examples of $\mathbb{F}_{1}$-algebras
include $X^{\mathbb{N}}=\\{1,X,X^{2},\ldots\\}$ and $X^{\mathbb{Z}}$ which
should be thought of as $\mathbb{F}_{1}[X]$ and $\mathbb{F}_{1}(X)$
respectively. Similarly, the $\mathbb{F}_{1}$-algebra
$\mathbb{F}_{1}[X_{1},\ldots,X_{d}]$ is realized by the monoid
$X_{1}^{\mathbb{N}}X_{2}^{\mathbb{N}}\cdots X_{d}^{\mathbb{N}}$ whose elements
are monomials in $X_{1},\ldots,X_{d}$.

If $M$ is a $\mathbb{F}_{1}$-algebra, it makes sense to define $M$-modules:
they are sets endowed with an action of $M$. The standard free module of rank
$d$ over $M$ is $M^{\oplus d}=\\{1,\ldots,d\\}\times M$ where $M$ acts by
multiplication on the second coordinate. It is an easy exercise to check that
the group of $M$-linear automorphisms of $M^{\oplus d}$ is the semi-direct
product $\mathfrak{S}_{d}\ltimes(M^{\text{gp}})^{d}$ where $M^{\text{gp}}$
denotes the subgroup of invertible elements of $M$. In particular, we get:

$\operatorname{GL}_{d}\big{(}\mathbb{F}_{1}(X)\big{)}=\mathfrak{S}_{d}\ltimes{\mathbb{Z}}^{d}$
(12)

and thus obtain, at least in the case of $\operatorname{GL}_{d}$, a
$\mathbb{F}_{1}$-style interpretation of the extended Weyl group.

Until this point, (almost) all theories agree on definitions but, when we are
coming to $\mathbb{F}_{1}$-schemes, points of view start to diverge.
Chronologically, the first approach is due to Deitmar [De1] and it closely
follows the classical theory of schemes: Deitmar introduced spectra of
monoids, equipped them with a topology and a notion of sheaves and finally
glued them to get $\mathbb{F}_{1}$-schemes. Soon after, Toen and Vaquié [ToVa]
developed the functorial point of view. Starting from the category
$\mathbb{F}_{1}\textbf{-Vect}$ (or, more generally, with an abstract monoidal
symmetric category $\mathcal{C}$), they defined the category
$F_{1}\textbf{-Alg}$ as we did before, introduced a notion of Zariski covering
on it and finally defined $\mathbb{F}_{1}$-schemes as sheaves on
$F_{1}\textbf{-Alg}$ for this Grothendieck topology. In [Ve], Vezzani proved
(in a slightly different context) that Deitmar’s construction on the one hand
and Toen-Vaquié’s approach on the other hand are equivalent, in the sense that
they give rise to the same category of $\mathbb{F}_{1}$-schemes. Besides, both
viewpoints include a functor of scalar extension:

$\mathbb{F}_{1}\textbf{-Sch}\to{\mathbb{Z}}\textbf{-Sch},\quad X\mapsto
X_{\mathbb{Z}}$

deriving from the construction $M\to{\mathbb{Z}}[M]$ at the level of monoids.
Deitmar observed that toric varieties are defined over $\mathbb{F}_{1}$ but he
also proved that there are essentialy the only ones [De2].

Another point of view on $\mathbb{F}_{1}$-schemes, which in some sense goes
back to Soulé’s original definition, was proposed in [CoCo] by Connes and
Consani. They suggested to define a $\mathbb{F}_{1}$-scheme as a triple
$(\tilde{A},X_{\mathbb{Z}},e_{X})$ where $\tilde{A}$ is a
$\mathbb{F}_{1}$-algebra, $X_{\mathbb{Z}}$ is a classical scheme and
$e_{X}:(\operatorname{Spec}\tilde{A})_{\mathbb{Z}}\to X_{\mathbb{Z}}$ is a
morphism of schemes inducing a bijection on $k$-points for any field $k$. In
some sense, this approach separates the purely combitorial part, which is
encoded by $\tilde{A}$, and the geometrical part, which is delegated to the
classical theory throught the scheme $X_{\mathbb{Z}}$. López Peña and
Lorscheid [LPL] proved that this framework is more flexible in the sense that
it allows for defining a much larger panel of varieties over $\mathbb{F}_{1}$;
those include grassmannians, split reductive groups, Schubert varieties,
_etc._ Besides, in a subsequent paper, Lorscheid [Lo1] concretized Tits’
premonition by realizing the Weyl group of a split reductive group as its set
of points over $\mathbb{F}_{1}$.

Beyond the field with one element, what we need for the purpose of this paper
is the field of $1$-adic numbers. Fortunately, this question has already been
touched in the literature by several authors. In [Co], Connes came with a
definition of Witt vectors over $\mathbb{F}_{1}$. However, Connes’
construction looks a bit disconnected to our needs as it comes equipped with a
scalar extension functor assumption assuming values in Banach algebras over
the reals, and not over the $p$-adics. In a different direction, Bambozzi,
Ben-Bassat and Kremnizer [BBK] laid the foundations of the theory of analytic
varieties over $\mathbb{F}_{1}$. Roughly speaking, their construction is
similar to the ones we briefly sketched above except that, instead of starting
with the category $\mathbb{F}_{1}\textbf{-Vect}$, they consider various
categories of sets $X$ equipped with a function $|\cdot|:X\to\mathbb{R}^{+}$
whose purpose is to model the norm map. They showed that balls and annuli of
rigid geometry come from analytic varieties over $\mathbb{F}_{1}$; this is
thus also the case for all the deformation spaces we listed in §1.

###  A.2 Galois theory over $\mathbb{F}_{1}$

It is a natural expectation that $\mathbb{F}_{1}$ should have a finite
extension of degree $n$ for all $n$ with Galois group isomorphic to
${\mathbb{Z}}/n{\mathbb{Z}}$. In the literature [So, KaSm, Co], this extension
$\mathbb{F}_{1^{n}}$ is usually defined as the cyclotomic extension of
$\mathbb{F}_{1}$, that is the $\mathbb{F}_{1}$-algebra represented by the
monoid $X^{{\mathbb{Z}}/n{\mathbb{Z}}}$. However, it appears that this point
of view is not perfectly suited to our purpose for at least two reasons:

  1. (1)

_The Galois theory is not the expected one._ Indeed, the group of
automorphisms of the monoid $X^{{\mathbb{Z}}/n{\mathbb{Z}}}$ is
$({\mathbb{Z}}/n{\mathbb{Z}})^{\times}$ and not ${\mathbb{Z}}/n{\mathbb{Z}}$;
in particular, we do not have a distinguished Frobenius endomorphism. This
issue is maybe even more visible when we extend scalars to ${\mathbb{Z}}$ or
${\mathbb{F}}_{p}$. Indeed, with the above definition, one would get:

$\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{F}_{p}=\mathbb{F}_{p}[X]/(X^{n}-1)$

which is certainly _not_ $\mathbb{F}_{p^{n}}$ and which besides has a
different Galois group.

  2. (2)

_The formation of $\mathbb{F}_{1^{n}}$-points does not behave as desired._ If
$X$ is a scheme defined over $\mathbb{F}_{1^{n}}$, it seems reasonable to
expect that the set of $\mathbb{F}_{1^{n}}$-points of $X$ agrees with the set
of $\mathbb{F}_{1}$-points of $\text{\rm
Res}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}X$. For $X=\operatorname{GL}_{d}$,
this results in:

$\operatorname{GL}_{d}(\mathbb{F}_{1^{n}})=\text{Weyl}\big{(}\text{\rm
Res}_{\mathbb{F}_{p^{n}}/\mathbb{F}_{p}}\operatorname{GL}_{d}\big{)}=(\mathfrak{S}_{d})^{n}$

where $p$, here, denotes any auxiliary prime number. This expectation is
reinforced by the fact that the group $(\mathfrak{S}_{d})^{n}$ plays a quite
important role in the Breuil-Mézard conjecture as recalled in §1. However, if
one lets $\mathbb{F}_{1^{n}}$ be the $\mathbb{F}_{1}$-algebra corresponding to
the monoid $X^{{\mathbb{Z}}/n{\mathbb{Z}}}$, one would obtain
$\operatorname{GL}_{d}(\mathbb{F}_{1^{n}})=\mathfrak{S}_{d}\ltimes({\mathbb{Z}}/n{\mathbb{Z}})^{d}$
(see discussion before Eq. (12)) which is certainly _not_ isomorphic to
$(\mathfrak{S}_{d})^{n}$ since even cardinalities differ!

The conclusion of these observations is that, although the cyclotomic
extension of $\mathbb{F}_{1}$ is undoubtedly an interesting object, it is
probably not the $\mathbb{F}_{1^{n}}$ we need for the applications we have in
mind. Moreover, one checks that there is unfortunately no
$\mathbb{F}_{1}$-algebra meeting all our requirements. Instead, we propose to
define from scratch a theory of $\mathbb{F}_{1^{n}}$-vector spaces as we did
previously for $\mathbb{F}_{1}$, trying as much as possible to incorporate the
Frobenius action and keep its desired properties.

######  Definition A.1.

A _$\mathbb{F}_{1^{n}}$ -vector space_ is a set. Given two
$\mathbb{F}_{1^{n}}$-vector spaces $V$ and $W$, a _$\mathbb{F}_{1^{n}}$
-linear morphism_ $f:V\to W$ is the datum of $n$ partially defined set-
theoretical functions $f_{1},\ldots,f_{n}:V\to W$.

Again, the standard $\mathbb{F}_{1^{n}}$-vector space of dimension $d$ is
represented by the set $\\{1,\ldots,d\\}$. We denote it by
$(\mathbb{F}_{1^{n}})^{d}$, or simply $\mathbb{F}_{1^{n}}$ when $d=1$, in what
follows. It is obvious from the definition that the automorphism group of
$(\mathbb{F}_{1^{n}})^{d}$ is $(\mathfrak{S}_{d})^{n}$, _i.e._ we have the
expected equality
$\operatorname{GL}_{d}(\mathbb{F}_{1^{n}})=(\mathfrak{S}_{d})^{n}$. Similarly,
extending the definition of $\mathbb{F}_{1}(X)$ to our new setting, one checks
that
$\operatorname{GL}_{d}(\mathbb{F}_{1^{n}}(X))=(\mathfrak{S}_{d}\ltimes{\mathbb{Z}}^{d})^{n}$.

Moreover, we have an obvious scalar extension functor
$\mathbb{F}_{1}\textbf{-Vect}\to\mathbb{F}_{1^{n}}\textbf{-Vect}$ acting on
objects by $V\mapsto V$ and on morphisms by $f\mapsto(f,f,\ldots,f)$. In what
follows we shall use the notation
$\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}V$ to denote the scalar extension
of $V$ from $\mathbb{F}_{1}$ to $\mathbb{F}_{1^{n}}$. Regarding scalar
restriction, there are two different options to define it, namely:

$\begin{array}[]{rcl}\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}:\,\,\mathbb{F}_{1^{n}}\textbf{-Vect}&\longrightarrow&\mathbb{F}_{1}\textbf{-Vect}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ V&\mapsto&V^{\oplus n}\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ \text{\rm
mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}:\,\,\mathbb{F}_{1^{n}}\textbf{-Vect}&\longrightarrow&\mathbb{F}_{1}\textbf{-Vect}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ V&\mapsto&V^{\otimes n}.\end{array}$

(We recall that the direct sum and the tensor product over $\mathbb{F}_{1}$
are defined as the disjoint union and the cartesian product respectively.) The
functor $\text{\rm aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}$ (resp.
$\text{\rm mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}$) will be referred to as
the _additive_ (resp. the _multiplicative_) scalar restriction from
$\mathbb{F}_{1^{n}}$ to $\mathbb{F}_{1}$; hence the notation. Both versions
look interesting given than they both appear as adjoints of the scalar
extensions. Precisely, for $V\in\mathbb{F}_{1^{n}}\textbf{-Vect}$ and
$W\in\mathbb{F}_{1}\textbf{-Vect}$, we have:

$\displaystyle\operatorname{Hom}_{\mathbb{F}_{1^{n}}\textbf{-Vect}}(V,\,\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}W)$
$\displaystyle=\operatorname{Hom}_{F_{1}\textbf{-Vect}}(\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(V),\,W),$
$\displaystyle\operatorname{Hom}_{\mathbb{F}_{1^{n}}\textbf{-Vect}}(\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}W,\,V)$
$\displaystyle=\operatorname{Hom}_{\mathbb{F}_{1}\textbf{-Vect}}(W,\,\text{\rm
mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(V)).$

Besides, $\text{\rm aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(V)$ and
$\text{\rm mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(V)$ are both equipped
with a Frobenius which acts by permuting cyclically the summands/factors. More
concretely, the Frobenius action on $\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(V)\simeq{\mathbb{Z}}/n{\mathbb{Z}}\times
V$ is given by $(i,x)\mapsto(i{+}1,x)$ and it is given on $\text{\rm
mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(V)\simeq V^{n}$ by
$(x_{1},\ldots,x_{n})\mapsto(x_{2},\ldots,x_{n},x_{1})$. We observe in
particular that $\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(\mathbb{F}_{1^{n}})\simeq{\mathbb{Z}}/n{\mathbb{Z}}$.
In this sense, our definition meets the most standard presentation of
$\mathbb{F}_{1^{n}}$ [So, KaSm, Co]; the main difference is that we do not
retain the group structure on ${\mathbb{Z}}/n{\mathbb{Z}}$ but replace it by a
Frobenius structure given by the shift. This slight modification in the point
of view is actually enough to retrieve a cyclic Galois group of order $n$.

######  Proposition A.2.

The group of automorphisms of $\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(\mathbb{F}_{1^{n}})$ commuting with
the Frobenius action is the cyclic group of order $n$ generated by the
Frobenius.

###### Proof.

An automorphism of $\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(\mathbb{F}_{1^{n}})$ is, by
definition, a bijection
$f:{\mathbb{Z}}/n{\mathbb{Z}}\to{\mathbb{Z}}/n{\mathbb{Z}}$. Requiring that it
commutes with the Frobenius amounts to saying that $f(x+1)=f(x)+1$ for all
$x\in{\mathbb{Z}}/n{\mathbb{Z}}$. Clearly, a function satisfying this
condition must be of the form $x\mapsto x+a$, _i.e._ $f$ is a power of the
Frobenius. ∎

Proposition A.2 does not extend _verbatim_ if we replace $\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}$ by $\text{\rm
mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}$; indeed, given that $\text{\rm
mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(\mathbb{F}_{1^{n}})$ is reduced to
one point, its group of automorphisms is trivial as well. However, we can
recover the expected group if we consider all objects
$V\in\mathbb{F}_{1^{n}}\textbf{-Vect}$ at the same time: the group of
Frobenius-preserving automorphisms of the _functor_ $\text{\rm
mRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}$ is cyclic of order $n$ and
generated by the Frobenius.

Passing to the limit, we can similarly set up a theory of vector spaces over
$\bar{\mathbb{F}}_{1}=\varinjlim_{n}\mathbb{F}_{1^{n}}$.

######  Definition A.3.

A _$\bar{\mathbb{F}}_{1}$ -vector space_ is a set. Given two
$\bar{\mathbb{F}}_{1}$-vector spaces $V$ and $W$, a _$\bar{\mathbb{F}}_{1}$
-linear morphism_ $f:V\to W$ is the datum of a sequence $(f_{i})_{i\geq 0}$ of
partially defined functions from $V$ to $W$ such that for all $x\in V$, the
sequence $(f_{i}(x))_{i\geq 0}$ is periodic.

As before, we have a scalar extension functor
$\mathbb{F}_{1}\textbf{-Vect}\to\bar{\mathbb{F}}_{1}\textbf{-Vect}$ which acts
trivially on objects and takes a morphism $f$ in
$\mathbb{F}_{1}\textbf{-Vect}$ to the constant sequence $(f,f,\ldots)$. More
generally, there is a functor
$\mathbb{F}_{1^{n}}\textbf{-Vect}\to\bar{\mathbb{F}}_{1}\textbf{-Vect}$
mapping a $\mathbb{F}_{1^{n}}$-linear morphism $(f_{1},\ldots,f_{n})$ to the
sequence $(f_{i\text{ mod }n})_{i\geq 0}$. If $V$ is finite dimensional over
$\bar{\mathbb{F}}_{1}$ (_i.e._ if $V$ is a finite set), any
$\bar{\mathbb{F}}_{1}$-linear morphism with domain $V$ comes from a
$\mathbb{F}_{1^{n}}$-linear morphism for some $n$. Restriction of scalars also
exist in this context. Writing
$\hat{\mathbb{Z}}=\varprojlim_{n}{\mathbb{Z}}/n{\mathbb{Z}}$, they are given
by:

$\begin{array}[]{rcl}\text{\rm
aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}:\quad\bar{\mathbb{F}}_{1}\textbf{-Vect}&\longrightarrow&\mathbb{F}_{1}\textbf{-Vect}\\\
V&\mapsto&\hat{\mathbb{Z}}\times V\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\text{\rm
mRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}:\quad\bar{\mathbb{F}}_{1}\textbf{-Vect}&\longrightarrow&\mathbb{F}_{1}\textbf{-Vect}\\\
V&\mapsto&\\{\,\text{periodic sequences with values in }V\,\\}.\end{array}$

Furthermore, $\text{\rm aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}(V)$ and
$\text{\rm mRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}(V)$ are both equipped
with a Frobenius endomorphism: on $\text{\rm
aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}(V)$, it is $(i,x)\mapsto(i{+}1,x)$
while it acts by shifting the sequence by $1$ on $\text{\rm
mRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}(V)$. One checks that
$\operatorname{Aut}_{\varphi}(\text{\rm
aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}(\bar{\mathbb{F}}_{1}))\simeq\operatorname{Aut}_{\varphi}(\text{\rm
aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}})\simeq\operatorname{Aut}_{\varphi}(\text{\rm
mRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}})\simeq\hat{\mathbb{Z}}$ where
$\operatorname{Aut}_{\varphi}$ means the Frobenius-preserving automorphisms.

###  A.3 Galois theory over ${\mathbb{Q}}_{1}$

In what follows, we view ${\mathbb{Z}}_{1}$ (resp. ${\mathbb{Q}}_{1})$ as the
$\mathbb{F}_{1}$-analytic algebra (in the sense of [BBK]) corresponding to the
monoid $\varpi^{\mathbb{N}}$ (resp. $\varpi^{\mathbb{Z}}$) endowed with the
norm $\|\varpi^{v}\|=r^{v}$ where $r$ is a fixed real number in $(0,1)$. Here
$\varpi$ is a formal notation for the uniformizer of ${\mathbb{Z}}_{1}$ and
does not have further meaning. Our definition of ${\mathbb{Z}}_{1}$ might
sound too naive as it seems to identify ${\mathbb{Z}}_{1}$ with
$\mathbb{F}_{1}[[\varpi]]$; however, at least for the properties we want to
illustrate in this appendix, making this confusion will not have undesirable
consequences.

We now aim at defining several families of extensions of ${\mathbb{Q}}_{1}$
and studying their Galois properties. We start with unramified extensions: we
set
${\mathbb{Q}}_{1^{n}}=\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}{\mathbb{Q}}_{1}$
for any positive integer $n$ and ${\mathbb{Q}}_{1}^{\text{\rm
ur}}=\bar{\mathbb{F}}_{1}\otimes_{\mathbb{F}_{1}}{\mathbb{Q}}_{1}$. They are
equipped with a Frobenius structure coming from the Frobenius on
$\mathbb{F}_{1^{n}}$ (resp. $\bar{\mathbb{F}}_{1}$) and with a structure of
${\mathbb{Q}}_{1}$-algebra materialized by the multiplication morphism by
$\varpi$ at the level of monoids. One checks that the group of automorphisms
of $\text{\rm aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}({\mathbb{Q}}_{1^{n}})$
(resp. of $\text{\rm
aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}({\mathbb{Q}}_{1}^{\text{\rm
ur}})$) commuting with both structures is isomorphic to
${\mathbb{Z}}/n{\mathbb{Z}}$ (resp. to $\hat{\mathbb{Z}}$). We then get the
expected Galois group.

We now come to the analogue of the tower of tamely ramified extensions. When
$p$ is an actual prime number, this tower is obtained by extracting $e$-th
roots of the uniformizer for $e$ coprime with $p$. There exists an obvious
analogue of this construction over ${\mathbb{Q}}_{1}$: for any positive
integer $e$ (without any condition of coprimality), we consider the
$\mathbb{F}_{1}$-analytic algebra ${\mathbb{Q}}_{1}[\sqrt[e]{\varpi}]$ defined
by the underlying monoid $\varpi^{(1\\!/\\!e)\cdot{\mathbb{Z}}}$ equipped with
the norm $\|\varpi^{v}\|=r^{v}$ ($v\in\frac{1}{e}{\mathbb{Z}}$). Clearly
${\mathbb{Q}}_{1}[\sqrt[e]{\varpi}]$ is an extension of ${\mathbb{Q}}_{1}$ and
we can define generally
${\mathbb{Q}}_{1^{n}}[\sqrt[e]{\varpi}]=\mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}{\mathbb{Q}}_{1}[\sqrt[e]{\varpi}]$
and ${\mathbb{Q}}_{1}^{\text{\rm
ur}}[\sqrt[e]{\varpi}]=\bar{\mathbb{F}}_{1}\otimes_{\mathbb{F}_{1}}{\mathbb{Q}}_{1}[\sqrt[e]{\varpi}]$.
It is also possible to take the limit on $e$ and define
${\mathbb{Q}}_{1}[\sqrt[\infty]{\varpi}]$ as the $\mathbb{F}_{1}$-analytic
algebra associated to the monoid $\varpi^{\mathbb{Q}}$ with norm
$\|\varpi^{v}\|=r^{v}$. We write ${\mathbb{Q}}_{1}^{\text{\rm
tr}}=\bar{\mathbb{F}}_{1}\otimes_{\mathbb{F}_{1}}{\mathbb{Q}}_{1}[\sqrt[\infty]{\varpi}]$;
it is our candidate for being the maximal tamely ramified extension (or even
an algebraic closure?) of ${\mathbb{Q}}_{1}$.

Devising a decent Galois theory for the extension ${\mathbb{Q}}_{1}^{\text{\rm
tr}}/{\mathbb{Q}}_{1}^{\text{\rm ur}}$ looks more difficult. Indeed, given
that a morphism of monoids ${\mathbb{Q}}\to{\mathbb{Q}}$ which acts by the
identity on ${\mathbb{Z}}$ needs to be trivial, we conclude that there is no
nontrivial morphism of ${\mathbb{Q}}_{1}^{\text{\rm ur}}$-algebras of
${\mathbb{Q}}_{1}^{\text{\rm tr}}$ in the sense of Definition A.1. In order to
explain how this issue can be tackled, it will be more convenient to work with
finite extensions. For any positive integer $n$, we define
$K_{n}=\mathbb{F}_{1^{n}}[\sqrt[n]{\varpi}]$; it is the
$\mathbb{F}_{1^{n}}$-algebra represented by the monoid $\eta^{\mathbb{Z}}$
where we have set $\eta=\sqrt[n]{\varpi}$ for simplicity. As we said earlier,
there are no nontrivial automorphisms of ${\mathbb{Q}}_{1^{n}}$-algebras of
$K_{n}$. The subtlety is that such automorphisms do exist after restricting
scalars to $\mathbb{F}_{1}$. An explicit example is given by the morphism

$\sigma_{n}:\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(K_{n})\to\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(K_{n})$

corresponding to the map:

$\begin{array}[]{rcl}\sigma_{n}^{\sharp}:\,\,{\mathbb{Z}}/n{\mathbb{Z}}\times\eta^{\mathbb{Z}}&\longrightarrow&{\mathbb{Z}}/n{\mathbb{Z}}\times\eta^{\mathbb{Z}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
(i,\,\eta^{j})&\mapsto&(i{+}j,\,\eta^{j}).\end{array}$

One checks that $\sigma_{n}^{\sharp}$ is a morphism of monoids (where the
factor ${\mathbb{Z}}/n{\mathbb{Z}}$ is endowed with its additive structure)
acting by the identity on $\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}({\mathbb{Q}}_{1^{n}})$. Therefore,
$\sigma_{n}$ has all the virtues to be considered as an element of the Galois
group $\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})$, although it is not
clear to us, here, why we need to retain the monoid structure on
${\mathbb{Z}}/n{\mathbb{Z}}$ whereas we argued earlier that it should be
discarded in favor of the Frobenius structure. In any case, we have the
following proposition.

######  Proposition A.4.

The group of automorphisms of monoids of
${\mathbb{Z}}/n{\mathbb{Z}}\times\eta^{\mathbb{Z}}$ acting trivially on the
submonoid ${\mathbb{Z}}/n{\mathbb{Z}}\times\varpi^{\mathbb{Z}}$ is cyclic of
order $n$, generated by $\sigma_{n}^{\sharp}$.

###### Proof.

Let $f$ be an automorphism of
${\mathbb{Z}}/n{\mathbb{Z}}\times\eta^{\mathbb{Z}}$ satisfying the conditions
of the proposition. By assumption $f$ fixes $(1,1)$ and $(0,\varpi)$. Write
$f(0,\eta)=(a,\eta^{b})$ with $a\in{\mathbb{Z}}/n{\mathbb{Z}}$ and
$b\in{\mathbb{Z}}$. Since $f$ is a morphism of monoids, we must have
$(na,\eta^{nb})=(0,\varpi)$, showing that $b=1$. Hence $f$ takes the form
$(i,\eta^{j})\mapsto(i{+}aj,\,\eta^{j})$ and the proposition follows. ∎

After what precedes, we are tempted to write:

$\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})=\langle\sigma_{n}\rangle\simeq{\mathbb{Z}}/n{\mathbb{Z}}.$

Moreover, noticing that $\sigma_{n}$ commutes with the Frobenius, we conclude
that:

$\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1})=\langle\varphi,\sigma_{n}\rangle\simeq({\mathbb{Z}}/n{\mathbb{Z}})^{2}.$

Passing to the limit, we would end up with
$\operatorname{Gal}({\mathbb{Q}}_{1}^{\text{\rm
tr}}/{\mathbb{Q}}_{1}^{\text{\rm ur}})\simeq\hat{\mathbb{Z}}$ and
$\operatorname{Gal}({\mathbb{Q}}_{1}^{\text{\rm
tr}}/{\mathbb{Q}}_{1})\simeq\hat{\mathbb{Z}}^{2}$.

In order to give more credit to this conclusion, we would like to make the
comparison with the classical case of ${\mathbb{Q}}_{p}$ (where $p$ is an
actual prime number). For each positive integer $n$, we set
$K_{p,n}={\mathbb{Q}}_{p^{n}}[p^{1/(p^{n}-1)}]$; it is the maximal totally and
tamely ramified Galois extension of ${\mathbb{Q}}_{p^{n}}$. As a consequence,
the extensions $K_{p,n}$ are cofinal in the maximal tamely ramified extension
of ${\mathbb{Q}}_{p}$. Besides, the Galois group of $K_{p,n}/{\mathbb{Q}}_{p}$
sits in the following exact sequence:

$1$ $\operatorname{Gal}(K_{p,n}/{\mathbb{Q}}_{p^{n}})$
$\operatorname{Gal}(K_{p,n}/{\mathbb{Q}}_{p})$
$\operatorname{Gal}({\mathbb{Q}}_{p^{n}}/{\mathbb{Q}}_{p})$ $1$ cyclic of
order $p^{n}{-}1$ cyclic of order $n$

which admits a section and provides a presentation of
$\operatorname{Gal}(K_{p,n}/{\mathbb{Q}}_{p})$ as a semi-direct product
${\mathbb{Z}}/n{\mathbb{Z}}\ltimes{\mathbb{Z}}/(p^{n}{-}1){\mathbb{Z}}$ where
$a\in{\mathbb{Z}}/n{\mathbb{Z}}$ acts on
${\mathbb{Z}}/(p^{n}{-}1){\mathbb{Z}}$ by multiplication by $p^{a}$.

Setting naively $p=1$ in what precedes, we find $p^{n}{-}1=0$ which does not
really make sense since $\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})$
cannot be decently of cardinality zero. Remembering what we did in Eq. (11),
we instead write the factorization:

$p^{n}-1=(p-1)\cdot(1+p+\cdots+p^{n-1})=(p-1)\cdot[n]_{p}.$

At the level of groups, the above factorization reflects the fact that
$\operatorname{Gal}(K_{p,n}/{\mathbb{Q}}_{p^{n}})$ admits a subgroup of order
$[n]_{p}$, namely $\operatorname{Gal}(K_{p,n}/K_{p,1})$. When $p$ goes to $1$,
the factor $[n]_{p}$ converges to $n$ and then, passing to the limit, we
expect the group
$\operatorname{Gal}(K_{n}/K_{1})=\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})$
to be cyclic of order $n$, which is exactly what we have found earlier.
Furthermore, when $p$ tends to $1$, the action of ${\mathbb{Z}}/n{\mathbb{Z}}$
on ${\mathbb{Z}}/(p^{n}{-}1){\mathbb{Z}}$ (and consequently on all its
subgroups) becomes trivial, confirming our prediction that
$\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1})$ should be a direct product of
$\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})$ by
$\operatorname{Gal}({\mathbb{Q}}_{1^{n}}/{\mathbb{Q}}_{1})$.

######  Remark A.5.

There is however one small annoying point in what we have said: why is it
legitimate to get rid of the factor $(p{-}1)$? If instead of discarding it
without further discussion, we try to keep it, we come to the conclusion that
there should be between ${\mathbb{Q}}_{1^{n}}$ and $K_{1}$ an extension of
degree $0$ or, say, of infinitesimal degree. This suggests that the extensions
${\mathbb{Q}}_{1^{n}}$ and $K_{1}$ need to be considered as different objects,
which could be a way to explain that the prefactor
${\mathbb{Z}}/n{\mathbb{Z}}$ should be endowed with its Frobenius structure in
the former case and with its monoid structure in the latter one (see
discussion before Proposition A.4).

In the similar fashion that the factor $(q{-}1)^{n}$ in Eq. (11) corresponds
to the $n$-dimensional torus of $\operatorname{GL}_{n}$, it is tempting to
interpret the Galois group of the ghost infinitesimal extension
$K_{1}/{\mathbb{Q}}_{1^{n}}$ as the algebraic group $\mathbb{G}_{m}$ over
$\mathbb{F}_{1}$. Similarly, it sounds plausible to interpret the cyclic group
${\mathbb{Z}}/n{\mathbb{Z}}$ (which is supposed to be the Galois group of
$K_{n}/K_{1}$) as the group of $\mathbb{F}_{1}$-points of an algebraic group,
maybe $\text{\rm aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(\mathbb{G}_{a})$.
All of this suggests that $\operatorname{Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})$
could just be the pale reflection of an algebraic group $\text{\bf
Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})$ defined over $\mathbb{F}_{1}$ and sitting in
an exact sequence of the form:

$1\to\text{\rm
aRes}_{\mathbb{F}_{1^{n}}/\mathbb{F}_{1}}(\mathbb{G}_{a})\to\text{\bf
Gal}(K_{n}/{\mathbb{Q}}_{1^{n}})\to\mathbb{G}_{m}\to 1.$

And similarly, passing to the limit:

$1\to\text{\rm
aRes}_{\bar{\mathbb{F}}_{1}/\mathbb{F}_{1}}(\mathbb{G}_{a})\to\text{\bf
Gal}({\mathbb{Q}}_{1}^{\text{\rm tr}}/{\mathbb{Q}}_{1}^{\text{\rm
ur}})\to\mathbb{G}_{m}\to 1.$

Beyond its own interest, this interpretation would provide us with a natural
Frobenius structure (given by $i\mapsto i+1$) on
$\operatorname{Gal}({\mathbb{Q}}_{1}^{\text{\rm
tr}}/{\mathbb{Q}}_{1}^{\text{\rm ur}})\simeq\hat{\mathbb{Z}}$ after taking
$\mathbb{F}_{1}$-points.

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