# The infinite fern in higher dimensions

Valentin Hernandez  Université Paris-Saclay, CNRS, Laboratoire de
mathématiques d’Orsay, 91405, Orsay, France<EMAIL_ADDRESS>Benjamin Schraen  Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay, France  Institut Universitaire de France
<EMAIL_ADDRESS>
##  1 Introduction

Let $p$ be a prime number, $E$ a number field and

$
\overline{\rho}:\operatorname{Gal}(\overline{E}/E)\longrightarrow\operatorname{GL}_{n}(\overline{\mathbb{F}}_{p}),$

a continuous (i.e. which factors through a finite extension) representation.
It was shown by Mazur that the set of deformations of $\overline{\rho}$, with
minor technical conditions, with values in finite extensions of
$\mathbb{Q}_{p}$, can be arranged in a natural rigid space
$\mathcal{X}_{\overline{\rho}}$. Conjecturally, and now in many known cases
([CHLN11, CH13, Shi14, HLTT16, Sch15]), we can use automorphic representations
to construct Galois representations, and in particular points in
$\mathcal{X}_{\overline{\rho}}$ that we call _of automorphic nature_ , or just
_automorphic_. It is then natural to wonder which structure has this set of
automorphic points in $\mathcal{X}_{\overline{\rho}}$: is it an algebraic
subspace ? a closed one ? is it Zariski dense ?

The first example beyond the case of characters was studied by Gouvêa and
Mazur. In this situation $E=\mathbb{Q}$ and $n=2$, and $\overline{\rho}$ is
irreducible, modular and unobstructed, so that $\mathcal{X}_{\overline{\rho}}$
is a 3-dimensional open ball. In that case, automorphic points are related to
modular forms. In [GM98], Gouvêa and Mazur show that automorphic points are
Zariski dense in $\mathcal{X}_{\overline{\rho}}$ using the so called _infinite
fern_. Let us explain this name: up to twisting by powers of the cyclotomic
character, we can replace $\mathcal{X}_{\overline{\rho}}$ by a two dimensional
open ball. Modular forms (of finite slope) can be interpolated by a geometric
object, the Coleman–Mazur _Eigencurve_ $\mathcal{E}$ ([CM98]), which is a
rigid-analytic curve, whose points are refined $p$-adic modular forms of
finite slope. Generically, a classical modular forms has two refinements, thus
gives rise to two distinct points in the Eigencurve. Moreover the points
corresponding to refined classical modular form are Zariski-dense in the
Eigencurve. By $p$-adic interpolation, it is possible to associate a
$2$-dimensional $p$-adic representation of
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to a point of
$\mathcal{E}$. The points giving rise to deformations of $\overline{\rho}$
form a union $\mathcal{E}(\overline{\rho})$ of connected components of
$\mathcal{E}$ and the universal property of $\mathcal{X}_{\overline{\rho}}$
implies the existence of a map

$\mathcal{E}(\overline{\rho})\longrightarrow\mathcal{X}_{\overline{\rho}}.$

Generically, each modular point $f$ in $\mathcal{X}_{\overline{\rho}}$ has two
preimage in $\mathcal{E}$, giving rise to two distincts small curves around
those preimages, whose image in $\mathcal{X}_{\overline{\rho}}$ meet only at
$f$. By density, each of these two small curves has a Zariski-dense set of
modular points, and for each of these points there is another small curve
passing through, and so on, giving a fractal-like object which we picture as
follows :

giving a justification for the name of the _infinite fern_.

This article deals with a generalization of this result to more general number
fields and greater values of $n$. First, we need to assume that the number
field $E$ is a CM field, with totally real field $F$, in order to be able to
associate Galois representations to automorphic representations. Second, it is
expected that for general $n$ the automorphic points are not Zariski dense in
$\mathcal{X}_{\overline{\rho}}$, thus we reduce to the case of _$\chi$
-polarized_ Galois representations, for a character
$\chi:G_{E}\longrightarrow\overline{\mathbb{Q}_{p}}^{\times}$, i.e. continuous
group homomorphisms
$\rho:\operatorname{Gal}(\overline{E}/E)\longrightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{p})$
such that

$ \rho^{\vee}\simeq\rho^{c}\otimes\chi\varepsilon^{n-1},\quad\text{where
}\rho^{c}:=\rho(c\cdot c^{-1}),$

where $\varepsilon$ is the cyclotomic character, and
$c\in\operatorname{Gal}(\overline{E}/F)$ is a lift of the unique non trivial
element of $\operatorname{Gal}(E/F)$. Fix $S$ a finite set of primes of $E$
containing all primes above $p$. In this situation, assume that
$\overline{\rho}$ is $\chi$-polarized, absolutely irreducible (for simplicity)
and unramified away from $S$. Let $R_{\overline{\rho}}^{\chi-\mathrm{pol}}$ be
the complete noetherian local algebra parametrizing deformations of
$\overline{\rho}$ which are $\chi$-polarized and unramified away from $S$. Its
rigid fiber $\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$ is a rigid
space of dimension at least $[F:\mathbb{Q}]\frac{n(n+1)}{2}$. A natural source
of automorphic points in $\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$
is given by the regular, algebraic, essentially polarized, cuspidal
automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{E})$, by
work of many authors ([HT01, CHT08, CHLN11, CH13, Shi14] for example). In this
paper, we make the following hypothesis,

######  Hypothesis 1.1.

  1. 1.

$\overline{\rho}$ is conveniently modular (see Definition 6.1),

  2. 2.

All primes above $p$ in $F$ are unramified, and split in $E$,

  3. 3.

the character $\chi$ satisfies $\chi=\chi^{c}$ and satisfies a sign condition
(see Hypothesis 5.2 and section 4)

Under the previous hypothesis, we prove the following result:

######  Theorem 1.2.

The Zariski-closure of automorphic points contains a (non empty) union of
irreducible components of $\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$,
each of which are of dimension $[F:\mathbb{Q}]\frac{n(n+1)}{2}$.

We say that our deformation problem is unobstructed if
$H^{2}(G_{F,S},\operatorname{ad}(\overline{r}))=\\{0\\}$ where $\overline{r}$
is some extension of $\overline{\rho}$ to $G_{F,S}$ the Galois group of the
maximal unramified extension of $F$ (see section 5). In this situation, we
know that $\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$ is a rigid open
unit ball in $[F:\mathbb{Q}]\frac{n(n+1)}{2}$-variables.

######  Corollary 1.3.

Under the previous hypothesis, if moreover $\overline{\rho}$ is unobstructed,
then automorphic points are Zariski dense in
$\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$.

######  Remark 1.4.

In [Gui20] Giraud proved that if $\pi$ is an extremely regular automorphic
representation of $\operatorname{GL}_{n}(\mathbb{A}_{E})$ (see [BLGGT14a]
section 2.1), then there exists a density 1 set of primes $\lambda$ of $E$
such that $\rho_{\pi,\lambda}$ is unobstructed. As we have assumed
$\overline{\rho}$ to be conveniently modular, we can actually find some
extremely regular $\pi$, so that
$\overline{\rho}=\overline{\rho}_{\pi,\lambda}$ for $\lambda|p$, and thus
using [Gui20], up to change $\lambda$ in a density 1 set, we can assume
unobstructedness. In particular, under our assumption, $\overline{\rho}$ is
part of a compatible system for which in a density 1 set of primes, we have
Zariski density of automorphic points in the associated deformation spaces.

Before explaining the strategy of proof, let us say what was known. The first
case was the non-polarized case, $n=2$ and $E=F=\mathbb{Q}$, when
unobstructed, which was proven by Gouvêa-Mazur [GM98], and generalised by
Böckle [B0̈1]. The non-polarised case for $n=2$ and totally real fields $E=F$,
and the polarised case of $n=3$ (and general CM fields $E/F$) was proved by
Chenevier ([Che11]). A generalisation for greater $n$; but under restrictive
hypothesis (of Taylor–Wiles type) on $E$ and $\overline{\rho}$, was proven
recently by Hellmann–Margerin–Schraen ([HMS22]). All of these proofs use the
analogue (in higher dimensions) of the infinite fern. Now we explain our
strategy together with the relation to the previous works.

The Galois representations that we study can be viewed as $p$-adic ($L-$ or
$C-$) parameters of a reductive group: $\operatorname{GL}_{2}/\mathbb{Q}$ in
the situation of Gouvêa-Mazur and $U_{E/F}(n)$, or $GU_{E/F}(n)$, or one of
its inner forms, a (similitude) unitary group in $n$-variables, for polarized
deformation problems. A natural source of automorphic points is given by
automorphic representations of these groups. It turns out that these groups
give rise to Shimura data, that we can use to construct $p$-adic refined
families of automorphic forms, that is $p$-adic automophic eigenforms together
with the extra data of a refinement. These families generalize the Eigencurve
of Coleman–Mazur and are called Eigenvarieties (see [CM98, Che04, Urb11,
Eme06, AIP15, Her22] for example). Let us assume for simplicity $\chi=1$ for
the rest of the introduction.

For general $n$, a given automorphic form $f$ has at most $n![F:\mathbb{Q}]$
refinements $f_{i}$, and generically exactly $n![F:\mathbb{Q}]$ refinements.
Moreover the Eigenvariety $\mathcal{E}$ has equidimension $n[F:\mathbb{Q}]$,
$\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$ has dimension at least
(but conjecturally exactly) $\frac{n(n+1)}{2}[F:\mathbb{Q}]$ and there is a
map
$\mathcal{E}(\overline{\rho})\longrightarrow\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$111an
open-closed subvariety of $\mathcal{E}$ which forgets the refinement.

######  Definition 1.5.

The image of the map
$\mathcal{E}(\overline{\rho})\longrightarrow\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$
is called the _infinite fern_ and is denoted by
$\mathcal{F}(\overline{\rho})$.

Actually we can make our main theorem more precise:

######  Theorem 1.6.

Under the previous hypothesis, the Zariski closure of the infinite fern
$\mathcal{F}(\overline{\rho})$ in
$\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$ is a non-empty union of
irreducible components, each of which are of dimension
$[F:\mathbb{Q}]\frac{n(n+1)}{2}$.

Let us comment the various hypothesis we made. The main advantage compared to
[HMS22] is that we don’t have any hypothesis on $p$, $E/F$ or $n$, and
moreover we don’t need to assume $\overline{\rho}$ absolutely irreducible (or
has big image) if we use Chenevier’s determinants, which we do (see section
5). The hypothesis of being conveniently modular is necessary to expect the
infinite fern to be non-empty, and is in practice very close to the usual
modularity hypothesis which is anyway necessary. The hypothesis on the
splitting of primes above $p$ is technical, and we hope to come back on this
question soon.

Following the strategy of Chenevier, the main goal is to prove that for a
Zariski dense set of points $\rho$ in the infinite fern, the part of the
tangent space at $\rho$ in $\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$
coming from $\mathcal{E}$ has dimension at least
$\frac{n(n+1)}{2}[F:\mathbb{Q}]$. This will imply that the closure of the
infinite fern $\mathcal{F}(\overline{\rho})$, has dimension at least
$\frac{n(n+1)}{2}[F:\mathbb{Q}]$. As, by construction, automorphic points are
Zariski-dense in $\mathcal{E}$, thus in $\mathcal{F}(\overline{\rho})$, this
will prove that the Zariski closure of automorphic points has dimension at
least $\frac{n(n+1)}{2}[F:\mathbb{Q}]$. Thus to prove the assertion on the
tangent space, we need to show that the tangent spaces $T_{f_{i}}\mathcal{E}$
are “transverse”, more precisely that the sum of the images of the tangent
spaces $T_{f_{i}}\mathcal{E}$ in
$T_{\rho_{f}}\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$, for well
chosen automorphic forms $f$, has large enough dimension. But, clearly, as
soon as $n\geq 3$ it is not sufficient that these tangent spaces are pairwise
distinct, and this is the main difficulty to extend the proof of Gouvêa and
Mazur. To overcome this problem, Chenevier suggested a strategy which he
applied successfully when $n=3$ and which can be sketched as follows:

  1. 1.

find a good Zariski-dense subset $D$ of the infinite fern
$\mathcal{F}(\overline{\rho})$ ;

  2. 2.

show that the analogue of the transversality on the tangent spaces of points
in $D$ but for local deformation rings is valid ;

  3. 3.

show that the Global situation “embeds well” in the local situation, and thus
gives the result.

For the first part, Chenevier suggests to look at automorphic points $\rho$
which he calls generic: they are crystalline at $p$ and all their refinements
are non-critical. More precisely, if $\rho$ is crystalline at a place $v\mid
p$, its restriction $\rho_{v}$ to a decomposition group at $v$ is
characterized by a $n$-dimensional vector space
$V=D_{\mathrm{cris}}(\rho|_{\operatorname{Gal}(\overline{E}_{v}/E_{v})})$ with
its Hodge-Tate filtration $\mathcal{F}_{HT}$ (a complete flag) and Frobenius
operator $\varphi$. The refinements of $\rho$ correspond to the complete flags
of $V$ stable by $\varphi$. We say that a refinement of $\rho$ is non critical
it is opposite to $\mathcal{F}_{HT}$. Actually, Chenevier proved that the
second step works for crystalline points which have $n$ _well-positioned_ non-
critical flags and call those points _weakly-generic_. He moreover proved that
weakly generic points (with some extra but harmless conditions) are Zariski
dense in the infinite fern when $n=3$ and uses those as the subset $D$.

Concerning the second point, the weakly generic condition is used to carry an
induction in the local situation and prove that tangent spaces of local,
refined, deformations problem spawn the tangent space of the full local
deformation ring. This is where the definition of _well positioned_
refinements comes from.

For the last point, actually it is enough to embed the situation at the level
of tangent spaces. Chenevier proves that for all preimages of points in $D$,
the map $\mathcal{E}\longrightarrow\mathcal{W}$ to the weight space is étale,
and deduces that some Selmer group vanishes at those points, allowing to
_embed_ infinitesimally the global situation into the local one, transversally
to the cristalline locus, and thus deduce the result. This last argument is
classical in the Taylor-Wiles method.

The main issue to generalise Chenevier’s strategy in higher dimensions is that
it is completely unclear that weakly generic points satisfies some density
assumptions when $n\geq 4$ (see remark 7.2).

The strategy of [HMS22] is different but shares some similarities: for the
first point they choose points which are crystalline with some genericity
assumption222precisely on the Frobenius eigenvalues and Hodge-Tate weights
which are less restrictive than being _generic_ or _weakly-generic_. Their set
$D$ is then automatically Zariski dense. For the second point, they use a
local model for the local deformations spaces, which is of purely geometric
nature, and a rather evolved but completely elementary argument allows to
conclude in the second point, using not only all refinements but also
_companion points_ , which are extra-points appearing when the refinement is
critical (whose existence is proved in [BHS19]) and relies on the Taylor-Wiles
hypothesis.

Then the third point is the most delicate one and is proved by
Taylor–Wiles–Kisin method via “patched eigenvarieties” (see [BHS19]). The
second and third point relies deeply on the Taylor-Wiles hypothesis, in
particular it crucially needs $\overline{\rho}$ to have adequate image.

In this article we use a strategy closer to Chenevier’s, but using the local
model of [BHS19] as in [HMS22], but we never need any Taylor-Wiles assumption.
Namely, using the local model and a careful study of its geometry, we first
prove the second point without using companion points but rather generalizing
Chenevier’s transversality result at critical refinements (see section 3). For
the first point, we show that setting for $D$ the set of crystalline points
satisfying genericity conditions as in [HMS22] and which have moreover
_enormous_ image are actually Zariski dense in the infinite fern ; we call
those points _almost-generic_ (see Definition 7.4) because they will replace
Chenevier’s generic points in our argument. The density of these points is far
from being automatic and the argument is originally due to Bellaïche–Chenevier
and Taïbi (see section 7). Then, for the third point, we show that using the
enormous image, and a result of Newton–Thorne, we have the vanishing of the
expected Selmer group at points of $D$. We then show that this can be used to
relate the global situation to the local one. As a byproduct, we obtain that
our Eigenvariety is smooth at those points, as it was the case in other
situations (see [BHS19, Ber20]) (see section 8). Then, a local calculation
which was previously carried out in [All16], we show that our almost generic
points are smooth points of
$\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$ of the expected dimension.

The results of Chenevier were combined with those by Allen ([All19]) (who
proves that under some hypothesis every component contains an automorphic
point) to prove full density of the infinite fern when $n\leq 3$, without
assuming unobstructedness. We can adapt this generalisation also here,

######  Corollary 1.7 (Allen).

Assume hypothesis 8.12, then the infinite fern is Zariski dense in
$\mathcal{X}^{\chi-\mathrm{pol}}_{\overline{\rho}}$.

The only thing we need to check for this corollary is that we use classical
points which are automorphic representations for a similitude unitary group,
which moreover contributes to the coherent $H^{0}$, whereas Allen’s proof a
priori only provides an (essentially) polarised autormorphic representation of
$\operatorname{GL}_{n}$.

Acknowledgements: We would like to warmly thank Gaëtan Chenevier for very
helpful suggestions concerning this work. We would also like to thank George
Boxer, Laurent Clozel and Olivier Schiffmann for many interresting
discussions. Finally, we would like to thank Anne vaugon and Marc Mezzarobba
for their help for using SAGE to compute the local tangent spaces.

##  2 A lemma on Borel enveloppes

In this section we prove a technical, but essential, generalization of the
linear algebra result [HMS22, Thm. 2.3]. The main result of this section is
Lemma 2.1.

Fix $n$ an integer, $k$ a field, $G=\operatorname{GL}_{n/k}$, $B$ the upper
triangular Borel, $T$ its diagonal torus,
$\mathfrak{g},\mathfrak{b},\mathfrak{t}$ their respective Lie algebras. We
also set $\mathfrak{u}$ the nilpotent radical of $\mathfrak{b}$. Let $W$ be
the Weyl group of the pair $(G,T)$ that we identify to $\mathfrak{S}_{n}$. Let
$w_{0}\in W\simeq\mathfrak{S}_{n}$ be the longuest element for the order given
by $\mathfrak{b}$. For $g,h\in G(k)$, we set
$\mathfrak{b}_{g}=g^{-1}\mathfrak{b}g$ and
$(\mathfrak{b}_{g}\cap\mathfrak{b}_{h})^{gr=w_{0}}$ the $k$-linear subspace of
$M\in\mathfrak{b}_{g}\cap\mathfrak{b}_{h}$ such that, in $\mathfrak{t}$,

$hMh^{-1}\pmod{\mathfrak{u}}=\operatorname{Ad}(w_{0})(gMg^{-1}\pmod{\mathfrak{u}})\in\mathfrak{t}.$

Let $\mathfrak{C}_{n}$ be the set of “full” cycles:

$\mathfrak{C}_{n}\coloneqq\\{c_{i,j}:=(i,i-1,\dots,j+1,j)\in\mathfrak{S}_{n}|i\geq
j\\}.$

######  Lemma 2.1.

For every Borel subalgebra $\mathfrak{b}^{\prime}\subset\mathfrak{g}$, we have

$\mathfrak{b}^{\prime}=\sum_{w\in\mathfrak{C}_{n}w^{\prime}}(\mathfrak{b}^{\prime}\cap\mathfrak{b}_{w})^{gr=w_{0}},$

for some element $w^{\prime}\in\mathfrak{S}_{n}$ depending on
$\mathfrak{b}^{\prime}$. Moreover if
$\mathfrak{b^{\prime}}=\mathfrak{b}_{w_{0}b}$ for some $b\in B(k)$, we can
choose $w^{\prime}=1$.

###### Proof.

We can write $\mathfrak{b}^{\prime}=\mathfrak{b}_{g}$ for $g\in G(k)$. Let
$g=uls$, $u\in\mathfrak{b},l\in\mathfrak{b}_{w_{0}}$ lower triangular i.e.
$l=w_{0}bw_{0}$ with $b\in B(k)$, and $s\in W$. Thus
$\mathfrak{b}_{g}=(\mathfrak{b}_{w_{0}})_{bq}$, for $q=w_{0}s\in W$. Up to
conjugate by $bq$, we check at once that it is enough to show

$\mathfrak{b}_{w_{0}}=\sum_{w\in\mathfrak{C}_{n}}(\mathfrak{b}_{w_{0}}\cap\mathfrak{b}_{wb^{-1}})^{gr=w_{0}}.$

Thus we reduce to show the following lemma :

######  Lemma 2.2.

For all $i\geq j$, there exists $x_{\ell}\in k$ such that

$a^{i,j}:=\delta_{i,j}+\sum_{\ell=j+1}^{i}x_{\ell}\delta_{\ell,j}\in(\mathfrak{b}_{w_{0}}\cap\mathfrak{b}_{c_{i,j}b^{-1}})^{gr=w_{0}}.$

###### Proof.

We follow the proof of [HMS22]. For $i\geq j$, let $a^{i,j}$ be the element
constructed at the beginning of the proof of Lemma 2.1 in _loc. cit._ For the
convenience of the reader we recall its construction. Let $e_{1},\dots,e_{n}$
be the standard basis of $k^{n}$ and let $V_{\bullet}$ be the standard flag of
$k^{n}$. Let $\mathcal{B}$ be the basis

$b(e_{1}),b(e_{2}),\dots,b(e_{j-1}),e_{j},b(e_{j+1}),\dots,b(e_{i}),e_{i+1},\dots,e_{n}$

of $k^{n}$. Then $a^{i,j}$ is the matrix, in the standard basis, of the
endomorphism $\pi$ of $k^{n}$ defined by $\pi(x)=0$ if
$x\in\mathcal{B}\setminus\\{e_{j}\\}$ and $\pi(e_{j})=e_{i}$. As in _loc.
cit._ we check that

  1. (i)

$e_{\ell}\in\ker(\pi)$ if $\ell<j$ or $\ell>i$;

  2. (ii)

$\mathrm{Im}(\pi)\subset ke_{i}$ and $\pi(e_{j})=e_{i}$;

  3. (iii)

the endomorphism $b^{-1}\pi b$ stabilizes the flag $c_{i,j}^{-1}V_{\bullet}$.

The first two points are checked in _loc. cit._ The third point follows from
the fact that

$b^{-1}\pi b(c_{i,j}^{-1}V_{\ell})=\begin{cases}0&\text{if
}\ell=1,\dots,i-1\\\ kb^{-1}(e_{i})&\text{if }\ell\geq i\end{cases}$ (1)

and $kb^{-1}(e_{i})\in V_{i}=c_{i,j}^{-1}V_{i}$. This shows that the matrix
$a^{i,j}$ is an element of
$\mathfrak{b}_{w_{0}}\cap\mathfrak{b}_{c_{i,j}b^{-1}}$ and has the form

$\delta_{i,j}+\sum_{\ell=j+1}^{i}x_{\ell}\delta_{i,k}$ (2)

for some $x_{\ell}\in k$.

It remains to check that
$a^{i,j}\in(\mathfrak{b}_{w_{0}}\cap\mathfrak{b}_{c_{j,i}b^{-1}})^{gr=w_{0}}$
which is equivalent to check that the diagonal elements of $a^{i,j}$ and
$\operatorname{Ad}(c_{i,j}b^{-1})a^{i,j}$ are the same. It follows from (1)
that

$(\operatorname{Ad}(c_{i,j}b^{-1})\pi)(e_{\ell})=\begin{cases}0&\text{if
}k<i\\\ c_{i,j}b^{-1}(e_{i})&\text{if }k=i\end{cases}$

and $(\operatorname{Ad}(c_{i,j}b^{-1}\pi))(e_{\ell})\in V_{i}$ if $\ell>i$.
Therefore the diagonal of $\operatorname{Ad}(c_{i,j}b^{-1})a^{i,j}$ is zero
except for the coefficient $(i,i)$. On the other hand, we see by (2) that the
diagonal entries of $a^{i,j}$ are all zero except $(i,i)$. As the matrices
$a^{i,j}$ and $\operatorname{Ad}(c_{i,j}b^{-1})a^{i,j}$ are conjugated they
have the same trace and thus the same diagonal. ∎

∎

##  3 Local deformation rings

The aim of this section is to prove Proposition 3.7 which says that the
tangent space of the deformation ring of a crystalline $\varphi$-regular and
Hodge-Tate regular $(\varphi,\Gamma)$-module is generated by the tangent space
of some quasi-trianguline deformation subspaces.

Let $k$ be a field of characteristic $0$. Let $G$ be a split reductive group
over $k$, $B\subset G$ a Borel subgroup of $G$, $T\subset B$ a maximal split
torus of $G$, $U$ the unipotent radical of $B$ and $U^{-}$ the unipotent
radical of the opposite Borel subgroup to $B$ with respect to $T$ (in
particular $U^{-}\cap B=\\{1\\}$). Let $W\coloneqq N_{G}(T)/T$ be the Weyl
group of $(G,T)$. We denote $w_{0}$ the longest element of $W$ with respect to
the Bruhat order induced by the choice of $B$. Let $\mathfrak{g}$,
$\mathfrak{b}$, $\mathfrak{t}$, $\mathfrak{u}$, $\mathfrak{u}^{-}$ be the
respective Lie algebras of $G$, $B$, $T$, $U$, $U^{-}$. Let
$\tilde{\mathfrak{g}}\subset\mathfrak{g}\times G/B$ be the Grothendieck
simultaneous resolution of $\mathfrak{g}$ and
$X\coloneqq\tilde{\mathfrak{g}}\times_{\mathfrak{g}}\tilde{\mathfrak{g}}$. We
recall that $X$ has irreducible components $X_{w}$ which are indexed by the
elements of the Weyl group $W$ (see [BHS19, Def. 2.2.3]). The map
$\tilde{\mathfrak{g}}\rightarrow\mathfrak{t}$ sending $(\psi,gB)$ to the
projection of $\operatorname{Ad}(g)^{-1}\psi$ on $\mathfrak{t}$ via
$\mathfrak{b}/\mathfrak{u}\simeq\mathfrak{t}$ gives rise to two different maps
$\kappa_{1},\kappa_{2}:X\rightarrow\mathfrak{t}$ corresponding to the two
projections $X\rightarrow\tilde{\mathfrak{g}}$ and to a map
$\kappa\coloneqq(\kappa_{1},\kappa_{2}):X\rightarrow\mathfrak{t}\times_{\mathfrak{t}/W}\mathfrak{t}$.
If $w\in W$, we let $\mathfrak{t}^{w}\subset\mathfrak{t}$ be the subspace of
elements fixed by $w$ and
$T_{w}\subset\mathfrak{t}\times_{\mathfrak{t}/W}\mathfrak{t}$ be the
irreducible component

$T_{w}\coloneqq\\{(x_{1},x_{2})\in\mathfrak{t}\times\mathfrak{t}\mid
x_{1}=\operatorname{Ad}(w)x_{2}\\}.$

The space $X$ has a partition by locally closed subschemes $V_{w}$ defined as
inverse images of the Bruhat strata $U_{w}\subset G/B\times G/B$ by the map
$\pi:X\rightarrow G/B\times G/B$ and $X_{w}=\overline{V_{w}}$. We have an
inclusion $\kappa(X_{w})\subset T_{w}$ ([BHS19, Lem. 2.5.1]).

######  Proposition 3.1.

Let $x=(g_{1}B,0,g_{2}B)\in X_{w_{0}}(k)\subset
G(k)/B(k)\times\mathfrak{g}\times G(k)/B(k)$ be a $k$-point. Let $w\in W$ be
such that $x\in V_{w}$ and assume that $w_{0}w^{-1}$ is a product of distinct
simple reflexions. Then we have an equality of $k$-vector spaces

$T_{x}X_{w_{0}}=T_{x}\kappa^{-1}(T_{w_{0}}).$

###### Proof.

The inclusion $X_{w_{0}}\subset\kappa^{-1}(T_{w_{0}})$ induces an inclusion
$T_{x}X_{w_{0}}\subset T_{x}\kappa^{-1}(T_{w_{0}})$. We will prove that these
two $k$-vector spaces have the same dimension.

Let $k[\varepsilon]\coloneqq k[X]/(X^{2})$. The tangent space
$T_{x}(\kappa^{-1}(t_{w_{0}}))$ is the set of $k[\varepsilon]$-points
$(\tilde{g}_{1}B,\varepsilon A,\tilde{g}_{2}B)$ of $X$ specialising to $x$
such that moreover

$\operatorname{Ad}(\tilde{g}_{1})^{-1}(\varepsilon
A)=\operatorname{Ad}(w_{0})\operatorname{Ad}(\tilde{g}_{2})^{-1}(\varepsilon
A)$ (3)

in $\mathfrak{t}\otimes_{k}k[\varepsilon]$. Let
$\tilde{x}=(\tilde{g}_{1}B,\varepsilon A,\tilde{g}_{2}B)$ be such a point. We
can write $\tilde{g}_{i}=g_{i}(1+\varepsilon h_{i})$, where
$h_{i}\in\mathfrak{u}^{-}$. Using $\varepsilon^{2}=0$, the condition
$\tilde{x}\in X(k[\varepsilon])$ is equivalent to
$\operatorname{Ad}(g_{i}^{-1})A\in\mathfrak{b}$ for $i\in\\{1,2\\}$. The
condition (3) is then equivalent to
$\operatorname{Ad}(g_{1}^{-1})A=\operatorname{Ad}(w_{0})\operatorname{Ad}(g_{2}^{-1})A$
in $\mathfrak{t}$. Note that, up to changing $x$ by a point of its
$G(k)$-orbit, we can assume, without changing the dimensions of the tangent
spaces, that $g_{1}=1$ and $g_{2}=w$. The conditions above are then equivalent
to

$A\in\mathfrak{t}^{w_{0}w^{-1}}+(\mathfrak{u}\cap\operatorname{Ad}(w)\mathfrak{u})$

which is a $k$-vector space of dimension
$\dim_{k}\mathfrak{t}^{w_{0}w^{-1}}+\lg(w_{0}w^{-1})$. As $w_{0}w^{-1}$ is a
product of distinct simple reflexions, we have ([Car72, Lem. 2 & 3])

$\dim_{k}\mathfrak{t}^{w_{0}w^{-1}}=\dim_{k}\mathfrak{t}-\lg(w_{0}w^{-1}).$

Namely, we have a $W$-equivariant isomorphism
$\mathfrak{t}\simeq\operatorname{Hom}(X^{*}(T),k)$, so that it is sufficient
to prove that

$\dim_{\mathbb{R}}(X^{*}(T)\otimes\mathbb{R})^{w^{\prime}}=\dim_{\mathbb{R}}(X^{*}(T)\otimes\mathbb{R})-\lg(w^{\prime})$

when $w^{\prime}$ is a product of simple reflexions. Let $V$ be the subspace
of $X^{*}(T)\otimes\mathbb{R}$ generated by the roots of $(G,T)$. It is stable
under $W$ and has a direct summand on which $W$ is acting trivially. It is
therefore sufficient to prove that

$\dim_{\mathbb{R}}V^{w^{\prime}}=\dim_{\mathbb{R}}V-\lg(w^{\prime}).$

As $W$ is a finite group, $\dim_{\mathbb{R}}V-\dim_{\mathbb{R}}V^{w^{\prime}}$
is equal to the number of eigenvalues of $w^{\prime}$ acting on $V$ which are
different from $1$. By [Car72, Lem. 2], this number is equal to the number
$l(w^{\prime})$ of _loc. cit._ (which is a priori _not_ $\lg(w^{\prime})$). By
[Car72, Lem. 3], we have $l(w^{\prime})=\lg(w^{\prime})$ since the set of
simple roots is a set of linearly independent vectors of $V$.

Finally we deduce that

$\dim_{k}T_{x}(\kappa^{-1}(T_{w_{0}}))\leq\dim_{k}(G/B\times
G/B)+\dim_{k}\mathfrak{t}=\dim G.$

On the other hand, we know that $X_{w_{0}}$ is irreducible of dimension $G$.
Consequently we have

$\dim
G\leq\dim_{k}T_{x}X_{w_{0}}\leq\dim_{k}T_{x}\kappa^{-1}(T_{w_{0}})\leq\dim G$

so that $T_{x}X_{w_{0}}=\dim_{k}T_{x}\kappa^{-1}(T_{w_{0}})$. ∎

######  Lemma 3.2.

Let $w\in W$ and $b\in B$. Then the point $wB$ is in the closure of the
$T$-orbit of $bwB$ in $G/B$.

###### Proof.

Let $\nu$ be a cocharacter of $T$ such that $\langle\nu,\alpha\rangle>0$ for
all positive root $\alpha$ of $(G,B,T)$. Then the map
$\mathbb{G}_{m}\rightarrow G$ defined by $\nu u\nu^{-1}$ for $u$ in the
unipotent radical of $B$ extends to a map $\mathbb{A}^{1}\rightarrow G$
sending $0$ to $1$, thus as does the map $\nu b\nu^{-1}$. Consequently, as $w$
normalises $T$, $wB$ is in the closure of the image of the map

$t\mapsto\nu(t)bwB=\nu(t)b\nu(t)^{-1}wB.\qed$

######  Lemma 3.3.

Let $(w_{1},w_{2})\in W^{2}$ and let $b\in B$. If $(w_{1}B,bw_{2}B)\in
U_{w}\subset G/B\times G/B$, we have $w_{1}^{-1}w_{2}\leq w$ in the Bruhat
order.

###### Proof.

If $t\in T$, we have $tw_{1}B=w_{1}B$ in $G/B$ so that $(w_{1}B,tbw_{2}B)\in
U_{w}$. It follows from Lemma 3.2 that $w_{2}B$ is in the closure of the set
$\\{tbw_{2}B\mid t\in T\\}$ so that $(w_{1}B,w_{2}B)$ is in the closure if
$U_{w}$. As the closure of $U_{w}$ is the union of the $U_{w}^{\prime}$ with
$w^{\prime}\leq w$ and $(w_{1}B,w_{2}B)\in U_{w_{1}^{-1}w_{2}}$ we obtain the
result. ∎

From now on we consider $K/\mathbb{Q}_{p}$ a finite extension, and denote
$\Gamma=\mathrm{Gal}(K(\zeta_{p^{\infty}})/K)$. We fix $L$ a finite extension
of $\mathbb{Q}_{p}$ that splits $K$, i.e.

$L\otimes_{\mathbb{Q}_{p}}K\simeq L^{[K:\mathbb{Q}_{p}]}$

and we denote by $k_{L}$ its residue field. We follow to the notations of
[KPX14] concerning $(\varphi,\Gamma)$-modules over Robba rings. Let
$\mathcal{R}(\pi_{K})$ be the Robba ring for $K$ (see [KPX14] definition
2.2.2). We define $t\in\mathcal{R}(\pi_{K})$ by $t=\log(1+\pi_{K})$. Let
$\mathcal{C}_{L}$ be the category local artinian $\mathcal{O}_{L}$-algebra $A$
with maximal ideal $\mathfrak{m}_{A}$ such that the natural map
$k_{L}\rightarrow A/\mathfrak{m}_{A}$ is an isomorphism. If $A$ is an object
of $\mathcal{C}_{L}$, we denote $\mathcal{R}_{A}(\pi_{K})\coloneqq
A\otimes_{\mathbb{Q}_{p}}\mathcal{R}(\pi_{K})$. We refer to [KPX14, Def.
2.2.12] for the notion of $(\varphi,\Gamma)$-module over
$\mathcal{R}_{A}(\pi_{K})$. Let $D$ be a $(\varphi,\Gamma)$-module over
$\mathcal{R}_{L}(\pi_{K})$. We denote

$\mathfrak{X}_{D}:\mathcal{C}_{L}\longrightarrow\mathrm{Sets}$

the deformation functor of $D$, i.e. for an object $A$ of $\mathcal{C}_{L}$,
$\mathfrak{X}_{D}(A)$ is the set of isomorphism classes of pairs
$(D_{A},i_{A})$ where $D_{A}$ is a $(\varphi,\Gamma)$-modules over
$\mathcal{R}_{A}(\pi_{K})$ and $i_{A}:L\otimes_{A}D_{A}\simeq D$ is an
isomorphism of $(\varphi,\Gamma)$-modules. If $(\rho,V)$ is a continuous
representation of $G_{K}$ on a finite dimensional $L$-vector space, the
functor $\mathrm{D}_{\mathrm{rig}}$ of [Ber02] induces an isomorphism of
deformation functors (see [HMS22, §3.6] for details)

$\mathrm{D}_{\mathrm{rig}}:\mathfrak{X}_{V}\overset{\sim}{\longrightarrow}\mathfrak{X}_{\mathrm{D}_{\mathrm{rig}}(V)}.$

Let
$\mathcal{F}=(0=\mathcal{F}_{0}\subsetneq\cdots\subsetneq\mathcal{F}_{n}=D[t^{-1}])$
be a complete filtration of $D[t^{-1}]$ by sub-$(\varphi,\Gamma)$-modules over
$\mathcal{R}_{L}(\pi_{K})[t^{-1}]$ which are direct factors as
$\mathcal{R}(\pi_{K})[t^{-1}]$-modules (we will call such a filtration a
_quasi-triangulation_). We define similarly

$\mathfrak{X}_{D,\mathcal{F}}:\mathcal{C}_{L}\longrightarrow\text{Sets}$

the deformation functor of the pair $(D,\mathcal{F})$, i.e. for $A$ in
$\mathcal{C}_{L}$, the set $\mathfrak{X}_{D,\mathcal{F}}(A)$ is the set of
isomorphism classes of triples $(D_{A},\mathcal{F}_{A},i_{A})$ where
$(D_{A},i_{A})\in\mathfrak{X}_{D}(A)$ and $\mathcal{F}_{A}$ is a filtration of
$D_{A}[t^{-1}]$ by $(\varphi,\Gamma)$-stable
$\mathcal{R}_{A}(\pi_{K})[t^{-1}]$-submodules which are direct factors of
$D_{A}[t^{-1}]$ in the category of $\mathcal{R}_{A}(\pi_{K})[t^{-1}]$-modules
and such that $i_{A}(L\otimes_{A}\mathcal{F}_{A,i})=\mathcal{F}_{i}$ for all
$i\in\mathbb{Z}$.

We recall some notations of [BHS19] section 3 and we refer the reader to _loc.
cit._ for more precisions. Let $W$ be an
$L\otimes_{\mathbb{Q}_{p}}B_{\mathrm{dR}}$-representation of $G_{K}$ which is
almost de Rham. Let $W^{+}$ be a $G_{K}$-stable
$L\otimes_{\mathbb{Q}_{p}}B_{\mathrm{dR}}^{+}$-lattice of $W$. Let
$\mathfrak{X}_{W}:\mathcal{C}_{L}\longrightarrow\text{Sets}$ be the
deformation functor of $W$, which means that $\mathfrak{X}_{W}(A)$ is the set
of isomorphism classes of pairs $(W_{A}^{+},i_{A})$ such that $W_{A}^{+}$ is a
finite free $A\otimes_{\mathbb{Q}_{p}}B_{\mathrm{dR}}^{+}$-module endowed with
a continuous semilinear action of $G_{K}$ and $i_{A}$ is a $G_{K}$-equivariant
isomorphism $L\otimes_{A}W_{A}^{+}\simeq W^{+}$ of
$L\otimes_{\mathbb{Q}_{p}}B_{\mathrm{dR}}^{+}$-modules. If we fix an
$L\otimes_{\mathbb{Q}_{p}}K$-linear isomorphism
$\alpha:(L\otimes_{\mathbb{Q}_{p}}K)^{n}\overset{\sim}{\longrightarrow}D_{\mathrm{pdR}}(W)$
we can define
$\mathfrak{X}_{W^{+}}^{\Box}:\mathcal{C}_{L}\longrightarrow\text{Sets}$ the
deformation functor of the pair $(W^{+},\alpha)$. Let $F_{\bullet}$ be a
$G_{K}$-stable flag of $L\otimes_{\mathbb{Q}_{p}}B_{\mathrm{dR}}$-submodules
of $W$, we define $\mathfrak{X}_{W^{+},{F}_{\bullet}}$ the deformation functor
of the pair $(W^{+},{F}_{\bullet})$ and
$\mathfrak{X}_{W^{+},{F}_{\bullet}}^{\Box}$ the deformation functor of the
triple $(W^{+},{F}_{\bullet},\alpha)$.

######  Remark 3.4.

Assume that $D$ is a crystalline, $\varphi$-generic, $(\varphi,\Gamma)$-module
(see [HMS22, §3.3]). In this case, it can be convenient to choose $\alpha$
compatible with the Frobenius structure. The isocrystal
$(D_{\mathrm{cris}}(D),\varphi)$ has exactly $n!$-refinements, i.e. complete
flags stable by $\varphi$, which are in natural bijection with the orderings
of the eigenvalues of the linearized Frobenius $\varphi^{f}$. By
$\varphi$-genericity, there exists an isomorphism
$\alpha_{0}:(L\otimes_{\mathbb{Q}_{p}}K_{0})^{n}\simeq
L\otimes_{\mathbb{Q}_{p}}D_{\mathrm{cris}}(D)$ sending the canonical basis on
an eigenbasis of $\varphi$. Then $\alpha_{0}$ induces a bijection between the
complete flags of $(L\otimes_{\mathbb{Q}_{p}}K_{0})^{n}$ stable under the
group of diagonal matrices and the set of refinements of
$D_{\mathrm{cris}}(D)$. Denote
$\alpha=\alpha_{0}\otimes_{K_{0}}\operatorname{Id}_{K}:(L\otimes_{\mathbb{Q}_{p}}K)^{n}\simeq
D_{\mathrm{dR}}(D)=D_{\mathrm{pdR}}(D)\simeq
D_{\mathrm{cris}}(D)\otimes_{K_{0}}K$. By [HMS22] Lemma 3.7, the map
$\mathcal{F}\mapsto D_{\mathrm{cris}}(\mathcal{F}[1/t])$ induces a bijection
between the set of triangulations of $D$ and the set of $n!$ refinements of
$D_{\mathrm{cris}}(D)$. If $\mathcal{F}$ is a triangulation of $D$ and
$w\in\mathfrak{S}_{n}\subset\operatorname{GL}_{n}(L\otimes_{\mathbb{Q}_{p}}K)$
we denote $w\cdot\mathcal{F}$ the unique triangulation of $D$ such that
$w\alpha^{-1}(D_{\mathrm{dR}}(\mathcal{F}[1/t]))=\alpha^{-1}(D_{\mathrm{dR}}(w\cdot\mathcal{F}[1/t]))$.
This defines a simply transitive action of the group $\mathfrak{S}_{n}$ on the
set of triangulations of $D$ (which depends on the choice of $\alpha$).

Now we fix $G=\operatorname{GL}_{n,K}$, $B\subset G$ the Borel subgroup of
upper triangular matrices and $T\subset B$ the maximal torus of diagonal
matrices. We recall that $\mathfrak{g}$ is the $K$-Lie algebra of $G$ and
$X=\widetilde{\mathfrak{g}}\times_{\mathfrak{g}}\widetilde{\mathfrak{g}}$. We
also note $X_{K/\mathbb{Q}_{p}}$ and
$\widetilde{\mathfrak{g}}_{K/\mathbb{Q}_{p}}$ their Weil restrictions from $K$
to $\mathbb{Q}_{p}$. If $A$ is an object of $\mathcal{C}_{L}$ and
$(W_{A}^{+},\mathcal{F}_{\bullet,A},\alpha_{A})$ is an element of
$\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{\Box}(A)$, we can produce an
element of $X_{K/\mathbb{Q}_{p}}(A)$ by sending
$(W_{A}^{+},\mathcal{F}_{\bullet,A},\alpha_{A})$ to
$x_{A}\coloneqq(\alpha^{-1}(\mathrm{D}_{\mathrm{pdR}}(\mathcal{F}_{\bullet})),N_{W_{A}},\alpha^{-1}(\operatorname{Fil}_{W_{A}^{+}}^{\bullet}))$.
By [BHS19] Corollary 3.5.8, this map is a bijection. This implies that the
functor $\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{\Box}$ is pro-represented
by the complete local ring of $X$ at $x_{L}$.

Let $w\in\mathfrak{S}_{n}^{[K:\mathbb{Q}_{p}]}$. Recall that
$X_{K/\mathbb{Q}_{p},w}$ is the irreducible component of
$(X_{K/\mathbb{Q}_{p}})_{L}\simeq(X\times_{K}L)^{[L:K]}$ associated to $w$.
Let $D$ be a crystalline $(\varphi,\Gamma)$-module over
$\mathcal{R}_{L}(\pi_{K})$, together with a filtration $F_{\bullet}$ of
$D[1/t]$. Let $W^{+}=W_{\mathrm{dR}}^{+}(D)$, $W=W^{+}[t^{-1}]$ and
$\mathcal{F}_{\bullet}=W_{\mathrm{dR}}(F_{\bullet})$. As $D$ is crystalline,
the $B_{\mathrm{dR}}$-representation $W$ is de Rham and thus almost de Rham.
The functors $W_{\mathrm{dR}}$ and $W_{\mathrm{dR}}^{+}$ induce a morphism of
functors
$\mathfrak{X}_{D,F_{\bullet}}\rightarrow\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}$.
If $D$ is moreover assumed to be $\varphi$-regular, this morphism if formally
smooth by [BHS19, Cor. 3.5.4].

We define $\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{w,\Box}$ as the
subfunctor of $\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{\Box}$ pro-
represented by the quotient of $\widehat{\mathcal{O}}_{X_{L},x_{L}}$
corresponding to the complete local ring of $X_{K/\mathbb{Q}_{p},w}$ at
$x_{L}$ (with the convention that it is empty if $x_{L}\notin X_{w}$). We also
define
$\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{w}\subset\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}$
as the image of $\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{\Box,w}$ via
$\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{\Box}\longrightarrow\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}$
and we define
$\mathfrak{X}_{D,F_{\bullet}}^{w}\subset\mathfrak{X}_{D,F_{\bullet}}$ as the
inverse image of $\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}^{w}$ by
$\mathfrak{X}_{D,F_{\bullet}}\rightarrow\mathfrak{X}_{W^{+},\mathcal{F}_{\bullet}}$.

We assume from now on that $D$ is crystalline and $\varphi$-generic (see
[HMS22, §3.3]). Let $\mathfrak{X}_{D}^{\mathrm{cris}}\subset\mathfrak{X}_{D}$
be the subfunctor of crystalline deformations of $D$. Let $\mathcal{F}$ be a
triangulation of $D$, we use the same symbol for the filtration induced on
$D[1/t]$.

######  Lemma 3.5.

We have a inclusion
$\mathfrak{X}_{D}^{\mathrm{cris}}\subset\mathfrak{X}_{D,\mathcal{F}}^{w_{0}}$.

###### Proof.

It follows from [HMS22, §3.3] that
$\mathfrak{X}_{D}^{\mathrm{cris}}\subset\mathfrak{X}_{D,\mathcal{F}}$. We fix
an isomorphism of $L\otimes_{\mathbb{Q}_{p}}K_{0}$-modules
$\beta:(L\otimes_{\mathbb{Q}_{p}}K_{0})^{n}\simeq D_{\mathrm{cris}}(D)$ such
that $\beta\otimes\operatorname{Id}_{K}=\alpha$. Let
$\mathfrak{X}_{D}^{\mathrm{cris},\Box}$ be the functor of crystalline
deformations of the pair $(D,\alpha)$. Let’s consider the composite

$\mathfrak{X}_{D}^{\mathrm{cris},\Box}\longrightarrow\mathfrak{X}_{D,F_{\bullet}}^{\Box}\longrightarrow\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D),W_{\mathrm{dR}}(F_{\bullet})}^{\Box}.$

Let $A\in\mathcal{C}_{L}$ and let
$(D_{A},\alpha_{A})\in\mathfrak{X}_{D}^{\mathrm{cris},\Box}(A)$ and let
$(D_{A},\alpha_{A},F_{\bullet,A})$ be its image in
$\mathfrak{X}_{D,F_{\bullet}}^{\Box}(A)$. As $D_{A}$ is crystalline, the
operator $\nu_{A}$ on $W_{\mathrm{dR}}(D_{A})$ is zero.

Now we remark that the schematic inverse image of $\\{0\\}$ by the natural map
$(X_{K/\mathbb{Q}_{p}})_{L}\rightarrow(\mathfrak{g}_{K/\mathbb{Q}_{p}})_{L}$
of $L$-schemes is contained in the irreducible component
$X_{K/\mathbb{Q}_{p},w_{0}}$. Namely it is sufficient to prove the inverse
image $Z$ of $\\{0\\}$ by the natural map of $K$-schemes
$X\rightarrow\mathfrak{g}$ is contained in $X_{w_{0}}$. But $Z$ is
$G/B\times\\{0\\}\times G/B$ which is the Zariski closure of
$V_{w_{0}}\cap(G/B\times\\{0\\}\times G/B)$, so that $Z\subset X_{w_{0}}$.

This implies that the image of $(D_{A},\alpha_{A},F_{\bullet,A})$ in
$\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D),W_{\mathrm{dR}}(F_{\bullet})}(A)$ is
contained in
$\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D),W_{\mathrm{dR}}(F_{\bullet})}^{w_{0}}(A)$
and finally that
$\mathfrak{X}_{D}^{\mathrm{cris},\Box}\subset\mathfrak{X}_{D,F_{\bullet}}^{w_{0},\Box}$
and
$\mathfrak{X}_{D}^{\mathrm{cris}}\subset\mathfrak{X}_{D,F_{\bullet}}^{w_{0}}$.
∎

######  Definition 3.6.

Let $D$ be a crystalline $\varphi$-generic and HT regular
$(\varphi,\Gamma)$-module over $\mathcal{R}_{L}(\pi_{K})$. Let $F_{\bullet}$
be a triangulation of $D$. Let
$w_{(D,F_{\bullet}})\in\mathfrak{S}_{n}^{[K:\mathbb{Q}_{p}]}$ be the element
measuring the relative position of $F_{\bullet}$ and the Hodge filtration of
$D$ (see [BHS19] before Proposition 3.6.4). We say that $(D,F_{\bullet})$ is
_associated to a product of distinct transpositions_ if
$w_{0}w_{D,F_{\bullet}}$ is a product of distinct simple transpositions.
Moreover we say that the triangulation $F_{\bullet}$ is _non-critical_ if
$w_{D,F_{\bullet}}=w_{0}$.

We can now prove the main result of this section.

######  Proposition 3.7.

Let $D$ be a $\varphi$-generic, regular, crystalline,
$(\varphi,\Gamma)$-module over $\mathcal{R}_{L}(\pi_{K})$. Denote
$\operatorname{\mathrm{Tri}}(D)$ the set of triangulations of $D$.

  1. (i)

The following $L$-linear map is surjective

$\bigoplus_{\mathcal{F}\in\operatorname{\mathrm{Tri}}(D)}T\mathfrak{X}^{w_{0}}_{D,\mathcal{F}[1/t]}\longrightarrow
T\mathfrak{X}_{D}.$

Assume moreover that there exists $\mathcal{F}^{\mathrm{nc}}$ a non-critical
triangulation of $D$.

  1. (ii)

For any $c\in\mathfrak{C}_{n}$, the pair $(D,c\cdot\mathcal{F}^{\mathrm{nc}})$
is associated to a product of simple transpositions.

  2. (iiii)

The following $L$-linear map is surjective:

$\bigoplus_{c\in\mathfrak{C}_{n}}T\mathfrak{X}^{w_{0}}_{D,(c\cdot\mathcal{F}^{\mathrm{nc}})[1/t]}\longrightarrow
T\mathfrak{X}_{D}.$

###### Proof.

Let $\Sigma$ be the set of embeddings $\tau:K\rightarrow L$. Let $U$ be the
kernel of the map $T\mathfrak{X}_{D}\rightarrow
T\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D)}$. It follows from Lemma 3.5 as in
[HMS22, Cor. 3.13] that the following sequence is exact for any triangulation
$\mathcal{F}$ of $D$:

$0\longrightarrow U\longrightarrow
T\mathfrak{X}_{D,\mathcal{F}[\frac{1}{t}]}^{w_{0}}\longrightarrow
T\mathfrak{X}_{W_{\mathrm{dR}}^{+}(D),W_{\mathrm{dR}}(\mathcal{F}[\frac{1}{t}])}^{w_{0}}\longrightarrow
0.$

Therefore, if $\mathcal{S}\subset\operatorname{\mathrm{Tri}}(D)$ is any
subset, we have the following commutative diagram

${0}$${0}$${\bigoplus_{\mathcal{F}\in\mathcal{S}}U}$${U}$${0}$${\bigoplus_{\mathcal{F}\in\mathcal{S}}T\mathfrak{X}^{w_{0}}_{D,\mathcal{F}[\frac{1}{t}]}}$${T\mathfrak{X}_{D}}$${\bigoplus_{\mathcal{F}\in\mathcal{S}}T\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D),W_{\mathrm{dR}}(\mathcal{F}[\frac{1}{t}])}^{w_{0}}}$${T\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D)}}$${0}$$\scriptstyle{\sum}$$\scriptstyle{W^{+}_{\mathrm{dR}}}$

Thus to prove that the middle horizontal arrow is surjective, it is sufficient
to prove that the bottom horizontal arrow is surjective. As
$\mathfrak{X}_{D}^{\Box}\rightarrow\mathfrak{X}_{D}$ is formally smooth, it is
sufficient to prove that the map

$\bigoplus_{\mathcal{F}\in\mathcal{S}}T\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D),W_{\mathrm{dR}}(\mathcal{F}[\frac{1}{t}])}^{w_{0},\Box}\longrightarrow
T\mathfrak{X}_{W^{+}_{\mathrm{dR}}(D)}^{\Box}$

is surjective. Fix $\alpha:(L\otimes_{\mathbb{Q}_{p}}K)^{n}\simeq
D_{\mathrm{dR}}(D)$ an isomorphism compatible with Frobenius as in Remark 3.4.
Let $\mathcal{F}$ the triangulation of $D$ such that
$\alpha^{-1}(D_{\mathrm{dR}}(\mathcal{F}[1/t]))$ is the standard flag of
$(L\otimes_{\mathbb{Q}_{p}}K)^{n}$ and let $F_{\bullet}\coloneqq
D_{\mathrm{dR}}(\mathcal{F}[1/t])$. We denote
$x_{\mathcal{F}}\coloneqq(\alpha^{-1}(F_{\bullet}),0,\alpha^{-1}(\operatorname{Fil}_{\mathrm{dR}}^{\bullet}))\in
X_{K/\mathbb{Q}_{p}}(L)$. It follows from [BHS19, Thm. 3.2.5 & Cor. 3.5.9]
that the vertical arrows in the following commutative diagram are isomorphisms

${\mathfrak{X}_{W^{+},F}^{\Box}}$${\mathfrak{X}_{W^{+}}^{\Box}}$${\widehat{(X_{K/\mathbb{Q}_{p},L})}_{x_{\mathcal{F}}}}$${(\widehat{\widetilde{\mathfrak{g}}}_{K/\mathbb{Q}_{p},L})_{\pi_{2}(x_{\mathcal{F}})}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\pi_{2}}$

($\pi_{2}:\widetilde{\mathfrak{g}}\times_{\mathfrak{g}}\widetilde{\mathfrak{g}}\rightarrow\widetilde{\mathfrak{g}}$
is the second projection ).

Recall that we have a decomposition
$X_{K/\mathbb{Q}_{p},L}\simeq\prod_{\tau\in\Sigma}X_{\tau}$ where
$X_{\tau}\simeq L\times_{K,\tau}X$ and
$\widetilde{\mathfrak{g}}_{K/\mathbb{Q}_{p},L}\simeq\prod_{\tau\in\Sigma}\widetilde{\mathfrak{g}}_{\tau}$
and the map $\pi_{2}$ is of the form $(\pi_{2,\tau})_{\tau\in\Sigma}$ with
$\pi_{2,\tau}$ the base change of the second projection
$X\rightarrow\widetilde{\mathfrak{g}}$. Moreover the irreducible component of
$X_{K/\mathbb{Q}_{p},L}$ corresponding to the longest element is isomorphic to
$\prod_{\tau\in\Sigma}X_{w_{0},\tau}$ with $w_{0}$ the longest element of
$\mathfrak{S}_{n}$.

Therefore we have to prove that the map

$\bigoplus_{\tau\in\Sigma}\bigoplus_{\mathcal{F}\in\mathcal{S}}\widehat{X}_{w_{0},\tau,x_{\mathcal{F},\tau}}\longrightarrow\bigoplus_{\tau\in\Sigma}\widehat{\widetilde{\mathfrak{g}}}_{\tau,\pi_{2,L}(x_{\mathcal{F},\tau})}$
(4)

is surjective at the level of tangent spaces. As the formation of tangent
spaces commutes with finite products, it is sufficient to prove that for a
fixed embedding $\tau\in\Sigma$, the following map is surjective

$\bigoplus_{\mathcal{F}\in\mathcal{S}}\widehat{X}_{w_{0},\tau,x_{\mathcal{F},\tau}}\longrightarrow\widehat{\widetilde{\mathfrak{g}}}_{\tau,x_{\mathcal{F},\tau}}.$
(5)

Let $g_{\tau},h_{\tau}\in\operatorname{GL}_{n}(L)$ such that
$x_{\mathcal{F},\tau}=(g_{\tau}B(L),0,h_{\tau}B(L))$. By the choice of
$\alpha$, we have $g_{\tau}=1$, for all $\tau$.

The desired surjectivity is equivalent to the following equality

$\operatorname{Im}(\sum_{w\in
S}T_{(wB(L),0,h_{\tau}B(L))}X_{w_{0},\tau}\longrightarrow
T_{(0,h_{\tau}B(L))}\widetilde{\mathfrak{g}}_{\tau})=T_{(0,h_{\tau}B(L))}\widetilde{\mathfrak{g}}_{\tau}$

for a well chosen $S=\\{w\in\mathfrak{S}_{n}\mid
w\mathcal{F}\in\mathcal{S}\\}$. By Proposition 3.1, we deduce that, for any
$w\in\mathfrak{S}_{n}$,

$T_{(wB(L),0,h_{\tau}B(L))}X_{w_{0},\tau}=T_{(wB(L),0,h_{\tau}B(L))}\kappa^{-1}(T_{w_{0}})\\\
=T_{wB(L)}G/B\oplus(w\mathfrak{b}w^{-1}\cap
h_{\tau}\mathfrak{b}h_{\tau}^{-1})^{gr=w_{0}}\oplus T_{h_{\tau}B(L)}G/B.$

By Lemma 2.1 there exists an element $w_{\tau}\in\mathfrak{S}_{n}$ such that

$\operatorname{Im}(\sum_{c\in\mathfrak{C}_{n}}T_{(cw_{\tau}B(L),0,h_{\tau}B(L))}X_{\tau,w_{0}}\longrightarrow
T_{(0,h_{\tau}B(L))}\widetilde{\mathfrak{g}}_{\tau})\\\
=\left(\sum_{c\in\mathfrak{C}_{n}}(cw_{\tau}\mathfrak{b}w_{\tau}^{-1}c^{-1}\cap
h_{\tau}\mathfrak{b}h_{\tau}^{-1})^{gr=w_{0}}\right)\oplus
T_{h_{\tau}B}G/B=h_{\tau}\mathfrak{b}h_{\tau}^{-1}\oplus
T_{h_{\tau}B}G/B=T_{(0,h_{\tau}B)}\widetilde{\mathfrak{g}}_{\tau}.$

This implies the surjectivity of the map (5) when
$\mathcal{S}=\\{cw_{\tau}\mathcal{F}\mid c\in\mathfrak{C}_{n}\\}$, and in
particular when $\mathcal{S}=\operatorname{\mathrm{Tri}}(D)$ (which is
independant of $\tau$). This implies the surjectivity of the map (4) and our
first statement.

Now we assume that $\mathcal{F}$ is non critical. By non criticality of
$\mathcal{F}$, $h_{\tau}\in B(L)w_{0}$. Let $c\in\mathfrak{C}_{n}$ and let
$\sigma\in\mathfrak{S}_{n}$ be such that $(cB(L),0,h_{\tau}B(L))\in
U_{\sigma}$. We claim that $w_{0}\sigma^{-1}$ is a product of distinct simple
reflections. Namely it follows from Lemma 3.3 that $c^{-1}w_{0}\leq\sigma$ so
that $w_{0}\sigma^{-1}\leq c$ and, as $c$ is a product of simple reflections,
so is $w_{0}\sigma^{-1}$. This proves part (i).

Moreover as $h_{\tau}\in B(L)w_{0}$, we can choose $w_{\tau}=1$ by Lemma 2.1.
This implies that the surjectivity of (5) is true with
$\mathcal{S}=\\{c\mathcal{F}\mid c\in\mathfrak{C}_{n}\\}$ for all
$\tau\in\Sigma$. This proves the surjectivity of (4) for this $\mathcal{S}$
and part (ii). ∎

##  4 A remark on signs

By a theorem of Artin, all elements of order $2$ are conjugate in
$G_{\mathbb{Q}}$. Let $C_{\infty}\subset G_{\mathbb{Q}}$ be their conjugacy
class and let $H\subset G_{\mathbb{Q}}$ be the closed subgroup generated by
$C_{\infty}$. There is a unique continuous morphism
$\varepsilon:H\rightarrow\\{\pm 1\\}$ such that $\varepsilon(c)=-1$ for all
$c\in C_{\infty}$. Let $K$ be a number field. Then $K$ is totally real if and
only if $H\subset G_{K}$. Let $E$ be a CM field with totally real subfield
$F$, we have $H_{1}:=\ker\varepsilon\subset G_{E}$ and $G_{F}=G_{E}H$. Let
$c\in C_{\infty}$. We can consider the action of $c$ on $G_{E}$ by conjugacy,
and we have $G_{F}=G_{E}\rtimes\\{1,c\\}$. If $\rho$ is a morphism of $G_{E}$
in some group and $c\in C_{\infty}$, we set $\rho^{c}=\rho(c^{-1}\cdot
c)=\rho(c\cdot c)$. The $c$-conjugacy induces an automorphism $c$ of
$G_{E}^{\mathrm{ab}}$. As $H_{1}\subset G_{E}$, this automorphism does not
depend on the choice of $c\in C_{\infty}$.

######  Lemma 4.1.

Let $\chi:G_{F}\rightarrow\overline{\mathbb{Q}_{p}}^{\times}$ be a continuous
character. Then $\chi|_{G_{E}}=(\chi|_{G_{E}})^{c}$. If there exists a
continuous character $\psi:G_{E}\rightarrow\overline{\mathbb{Q}_{p}}^{\times}$
such that $\psi\psi^{c}=\chi|_{G_{E}}$, then the element $\chi(c)$ for $c\in
C_{\infty}$ does not depend on the choice of $c\in C_{\infty}$. Conversely if
we assume that $\chi$ is algebraic and that $\chi(c)$ does not depend on the
choice of $c\in C_{\infty}$, then there exists an algebraic continuous
character $\psi:G_{E}\rightarrow\overline{\mathbb{Q}_{p}}^{\times}$ such that
$\psi\psi^{c}=\chi|_{G_{E}}$. If moreover $\chi$ is unramified outside a
finite set $S$ of places of $F$, we can assume that $\psi$ is unramified
outside $S$.

###### Proof.

The first statement is clear as $\chi(G_{F})$ is abelian. Assume that
$\chi|_{G_{E}}=\psi\psi^{c}$ for some $\psi$ and let $c_{1}$ and $c_{2}$ be
two elements of $C_{\infty}$. Then

$\displaystyle\chi(c_{1}c_{2}^{-1})$
$\displaystyle=\chi(c_{1}c_{2})=\psi(c_{1}c^{2}c_{2})\psi(cc_{1}c_{2}c)$
$\displaystyle=\psi(c_{1}c)\psi(cc_{2})\psi(cc_{1})\psi(c_{2}c)=\psi^{-1}(cc_{1})\psi(cc_{1})\psi(cc_{2})\psi^{-1}(cc_{2})=1.$

The last statements are then direct consequences of [CHT08, Lem. 4.1.1 &
4.1.4]. ∎

Note that a continuous morphism $\chi:G_{E}\rightarrow G$ extends to a
morphism $G_{F}\rightarrow G$ if and only if $\chi=\chi^{c}$. The fact that
the extension of $\chi$ to $G_{F}$ satisfies the assumptions of Lemma 4.1
depends only on the restriction of $\chi$ to $G_{E}$ and is equivalent to the
fact that $\chi|_{H_{1}}$ is trivial.

Let $\chi:G_{F}\rightarrow\overline{\mathbb{Q}_{p}}^{\times}$ be a continuous
character. We say that a continuous representation
$\rho:G_{E}\longrightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}_{p}})$ is
polarized by $\chi$ if

$\rho^{\vee}\simeq\rho^{c}\otimes(\chi|_{G_{E}}\varepsilon^{n-1}).$ (6)

If $\rho$ is irreducible then we can define its sign, with respect to
$\chi|_{G_{E}}\varepsilon^{n-1}$, $\lambda\in\\{\pm 1\\}$ as in [BC11, 1.1].
This is the sign of the pairing appearing in (6).

Fix an isomorphism $\iota:\mathbb{C}\simeq\overline{\mathbb{Q}_{p}}$. Let
$\rho:G_{E}\longrightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}_{p}})$ be
a semi-simple continuous representation. Assume that there exists a cuspidal
regular algebraic automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{E})$ such that $\rho$ is strongly
associated with $\Pi$ (see Definition A.1) with respect to $\iota$. We will
say that $\Pi$ is _polarized by $\chi$_ if the pair
$(\Pi,\chi\circ\mathrm{Art}_{F})$ is polarized in the sense of [BLGGT14b,
2.1], which means
$\Pi^{\vee}\simeq\Pi^{c}\otimes(\iota^{-1}\circ\chi|_{G_{E}}\circ\mathrm{Art}_{E})$.
Then $\rho$ is polarized by $\chi$ and $\rho$ satisfies the properties of
[BGGT14, Thm. 2.1].

The following result is essentially [BC11, Thm. 1.2].

######  Theorem 4.2 (Bellaïche-Chenevier).

Assume that $\Pi$ is conjugate self-dual and regular algebraic and that
$\chi|_{H_{1}}$ is trivial. Let
$\psi:G_{E}\longrightarrow\overline{\mathbb{Q}_{p}}$ be a continuous character
such that $\psi\psi^{c}=\chi|_{G_{E}}$. Then every irreducible constituent $r$
of $\rho\otimes\psi^{-1}$ satisfying (6) has sign $\lambda=+1$ with respect to
$\chi|_{G_{E}}\varepsilon^{n-1}$.

###### Proof.

If $\psi=1$ this is [BC11] Theorem 1.2. We deduce the general case from [BC11]
Lemma 2.1. ∎

##  5 Deformation spaces

Denote by $k$ a topological field and $\mathcal{O}$ a complete noetherian
local $\mathbb{Z}_{p}$-algebra with residue field $k$.

Fix $E$ a totally imaginary CM number field with maximal totally real subfield
$F$ and fix $S$ a finite set of finite places of $E$ containing the places
above $p$, and the ramified places of $E$. Denote

$G_{E,S}=\operatorname{Gal}(E_{S}/E),$

the Galois group of the maximal unramified outside $S$ extension $E_{S}/E$.

Suppose given

$\overline{\rho}:G_{E,S}\longrightarrow\operatorname{GL}_{n}(k),$

a continuous semi-simple Galois representation. From now on we choose $c\in
G_{F}\backslash G_{E}$ such that $c^{2}=1$ a complex conjugation, denoting
similarly its image in $\operatorname{Gal}(E/F)$. As $\mathcal{O}$ is a
$\mathbb{Z}_{p}$-algebra, we have a continuous ring homomorphism
$\mathbb{Z}_{p}\longrightarrow k$ and we denote $\overline{\varepsilon}$ its
composition with the cyclotomic character. We also assume that
$\overline{\rho}$ is polarized by $\overline{\chi}$, i.e.

$\overline{\rho}^{\vee}\simeq\overline{\rho}^{c}\otimes({\overline{\varepsilon}}^{n-1}\overline{\chi}),$

for some continuous character $\overline{\chi}:G_{E,S}\longrightarrow
k^{\times}$ satisfying $\overline{\chi}^{c}=\overline{\chi}$. We denote
$\mathcal{C}_{\mathcal{O}}$, or $\mathcal{C}$ if the context is clear, the
category of artinian local $\mathcal{O}$-algebras with residue field $k$.

######  Hypothesis 5.1.

From now on in this section, we suppose that we are in either one of the two
situations:

  * •

$k\subset\overline{\mathbb{F}}_{p}$ with the discrete topology, $\mathcal{O}$
a finite totally ramified extension of $W(k)$

  * •

$k\subset\overline{\mathbb{Q}_{p}}$ a finite extension of $\mathbb{Q}_{p}$
with its $p$-adic topology and in this case we set $\mathcal{O}=k$.

In the second case $\overline{\varepsilon}=\varepsilon$ is just the
$\mathbb{Z}_{p}^{\times}$-valued cyclotomic character.

Denote by $\operatorname{tr}\overline{\rho}$ the Determinant (in the sense of
Chenevier [Che14, Définition 1.5]) of $\overline{\rho}$. As $\overline{\rho}$
is semi-simple it is completely determined by
$\operatorname{tr}\overline{\rho}$ (by [Che14, Cor. 2.13]). We fix once and
for all a continuous character
$\chi:G_{E,S}\longrightarrow\mathcal{O}^{\times}$ lifting $\overline{\chi}$
and such that $\chi^{c}=\chi$. We will need sometimes to assume the following
hypothesis on $\chi$.

######  Hypothesis 5.2.

The character $\chi$ is algebraic at $p$ and $\chi|_{H_{1}}$ is trivial.

######  Definition 5.3.

We denote by $\mathcal{F}^{\chi-\mathrm{pol}}_{\overline{\rho}}$ the functor
that associates to any object $A$ of $\mathcal{C}$ the set of continuous
determinants $D$ lifting $\operatorname{tr}\overline{\rho}$ such that
$D^{\vee}=D^{c}\otimes\chi\varepsilon^{n-1}$. It is pro-representable by a
ring $R^{\chi-\mathrm{pol}}_{\overline{\rho}}$ ([Che14, Prop. 3.3]333for
$R_{\overline{\rho}}$, and then
$R^{\chi-\mathrm{pol}}_{\overline{\rho}}=R_{\overline{\rho}}/I$ with
$I=(D^{univ,\vee}(g)-D^{univ,c}(g)\chi\varepsilon^{n-1}(g),g\in G)$). We
denote the associated formal scheme
$\mathfrak{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}=\operatorname{Spf}(R_{\overline{\rho}}^{\chi-\mathrm{pol}})$.
When $k$ is a finite field of characteristic $p$, we denote the generic fibre
of $R^{\chi-\mathrm{pol}}_{\overline{\rho}}$ by

$\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}:=\operatorname{Spf}(R^{\chi-\mathrm{pol}}_{\overline{\rho}})^{rig}.$

If $\overline{\rho}$ is absolutely irreducible, this coincides with the rigid
fiber of the polarized-by-$\chi$ deformation space of $\overline{\rho}$.

Recall that we fixed an isomorphism
$\iota:\mathbb{C}\xrightarrow{\sim}\overline{\mathbb{Q}_{p}}$.

######  Definition 5.4.

A point of
$x\in\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}(\overline{\mathbb{Q}_{p}})$
is $GL_{n}$-modular444Compare with Definition 6.1. if there exists a polarized
by $\chi$ automorphic representation $\Pi$ of
$\operatorname{GL}_{n}(\mathbb{A}_{E})$ such that
$\rho_{\Pi,\iota}\simeq\rho_{x}$.

It follows from Lemma 4.1 that there exists a finite extension
$\mathcal{O}^{\prime}/\mathcal{O}$ of discrete valuation rings and a
continuous algebraic character
$\psi_{0}:G_{E,S}\longrightarrow(\mathcal{O}^{\prime})^{\times}$. We fix such
a character $\psi_{0}$.

We start with the following reduction to lighten slightly the notations in the
rest of the text.

######  Lemma 5.5.

Assume $\chi$ is as before, and $\overline{\rho}$ satisfies
$\overline{\rho}^{\vee}\simeq\overline{\rho}^{c}\otimes\varepsilon^{n-1}$.
Then there is an isomorphism, which identifies modular points,

$\mathcal{X}_{\overline{\rho}}^{1-\mathrm{pol}}\otimes_{\mathcal{O}[1/p]}\mathcal{O}^{\prime}[1/p]\overset{\sim}{\longrightarrow}\mathcal{X}_{\overline{\rho}\psi_{0}^{-1}}^{\chi-\mathrm{pol}}\otimes_{\mathcal{O}[1/p]}\mathcal{O}^{\prime}[1/p].$

In particular it is enough to prove theorem 1.6 for $\chi=1$.

###### Proof.

The character $\psi_{0}\circ\mathrm{Art}_{E}$ is automorphic as
$\psi_{0}:G_{E,S}\longrightarrow(\mathcal{O}^{\prime})^{\times}$ is algebraic.
Moreover, the isomorphism is given by

$\rho\longmapsto\rho\psi_{0}^{-1}.$

This is obviously an isomorphism, and because $\psi_{0}\circ\mathrm{Art}_{E}$
is automorphic it identifies ($\operatorname{GL}_{n}$-)modular points on both
sides. ∎

Our goal is to understand the geometry of the set of modular points in
$\mathcal{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}$ when
$k\subset\overline{\mathbb{F}}_{p}$, $\mathcal{O}=\mathcal{O}_{K}$,
$K/\mathbb{Q}_{p}$ finite. However we will need some result concerning the
situation where $k=\mathcal{O}$ is a finite extension of $\mathbb{Q}_{p}$ but
for completeness we prove them in the two situations of Hypothesis 5.1.
Following [CHT08] we introduce

$\mathcal{G}_{n}=(\operatorname{GL}_{n}\times\operatorname{GL}_{1})\rtimes\operatorname{Gal}(E/F),$

where $c\in\operatorname{Gal}(E/F)$ acts on
$(g,x)\in\operatorname{GL}_{n}\times\operatorname{GL}_{1}$ via
$(x{{}^{t}}g^{-1},x)$. We denote $\nu$ the homomorphism
$\mathcal{G}_{n}\longrightarrow\operatorname{GL}_{1}$ sending $(g,x)$ to $x$
and $c$ to $-1$.

Finally we assume until the end of this section that $\overline{\rho}$ is
absolutely irreducible and denote $\lambda$ its sign (as defined in section
4). Then we can extend $\chi$, which satisfies $\chi^{c}=\chi$, to
$G_{F}\simeq G_{E}\rtimes\operatorname{Gal}(E/F)$ by setting
$\chi(c):=(-1)^{n}\lambda$ and we extend compatibly $\overline{\chi}$ to
$G_{F}$, so that $\mu:=\chi\varepsilon^{n-1}$ satisfies $\mu(c)=-\lambda$. By
[CHT08] Lemma 2.1.1 we can thus extend $\overline{\rho}$ to a continuous
homomorphism

$\overline{r}:G_{F,S}\longrightarrow\mathcal{G}_{n}(k),$

such that $c\in G_{F}$ is sent to $c\in\operatorname{Gal}(E/F)$ via
$\overline{r}$ and projection and
$\nu\circ\overline{r}=\overline{\chi}^{-1}\overline{\varepsilon}^{1-n}$ (as
extended before to $G_{F}$).

######  Definition 5.6.

Let $\operatorname{Def}_{\overline{r}}^{\chi}$ be the functor that associates
to any object $R$ of $\mathcal{C}$ the set
$\operatorname{Def}_{\overline{r}}^{\chi}(R)$ of lifts
$r:G_{F,S}\rightarrow\mathcal{G}_{n}(R)$ of $\overline{r}$ such that $\nu\circ
r=\overline{\chi}^{-1}\overline{\varepsilon}^{1-n}$ considered up to
$1+\mathfrak{m}_{R}M_{n}(R)$-conjugation. As in [CHT08, Prop. 2.2.9]555There
the field $k$ is finite, but we can check that everything carries over in our
setting, as already remarked in [Kis09b], this functor is pro-represented by a
local complete noetherian $\mathcal{O}$-algebra $R_{\overline{r}}^{\chi}$.
When $k$ is a finite field of characteristic $p$, we denote by
$\mathcal{X}^{\chi}_{\overline{r}}$ the generic fiber of the formal scheme
$\mathfrak{X}^{\chi}_{\overline{r}}=\operatorname{Spf}(R_{\overline{r}}^{\chi})$.

In the following, all cohomology groups are continuous cohomology groups.

######  Proposition 5.7.

Assume that $\overline{\rho}$ is absolutely irreducible,
$\operatorname{char}k\neq 2$, $\chi(c^{\prime})=(-1)^{n}$ for all complex
conjugacy $c^{\prime}$ and, if $k$ is of characteristic $p$,
$R^{\chi}_{\overline{r}}[1/p]\neq 0$. Then

$\dim(R^{\chi}_{\overline{r}}[1/p])\geq\dim_{k}H^{1}(G_{F,S},\operatorname{ad}(\overline{r}))-\dim_{k}H^{2}(G_{F,S},\operatorname{ad}(\overline{r}))=\frac{n(n+1)}{2}[F:\mathbb{Q}].$

Moreover the topological $\mathcal{O}[1/p]$-algebra
$R^{\chi}_{\overline{r}}[1/p]$ is formally smooth of relative dimension
$\frac{n(n+1)}{2}[F:\mathbb{Q}]$ if
$H^{2}(G_{F,S},\operatorname{ad}(\overline{r}))=0$.

###### Proof.

This appeared already in [CHT08, All16], let us give the argument. As
$\overline{\rho}$ is absolutely irreducible,
$\operatorname{ad}(\overline{r})^{G_{F}}=H^{0}(G_{F},\operatorname{ad}(\overline{r}))=0$
by [CHT08, Lem. 2.1.7(3)]. For each place $v|\infty$, we have ([CHT08, Lem.
2.1.3])

$\dim_{k}H^{0}(G_{F_{v}},\operatorname{ad}(\overline{r}))=\frac{n(n+\overline{\chi}\overline{\varepsilon}^{n-1}(c_{v}))}{2}=\frac{n(n-1)}{2}$

(where $c_{v}$ is the complex conjugation in $F_{v}$).

Now the equality

$\dim_{k}H^{1}(G_{F,S},\operatorname{ad}(\overline{r}))-\dim_{k}H^{2}(G_{F,S},\operatorname{ad}(\overline{r}))=\frac{n(n+1)}{2}[F:\mathbb{Q}]$

follows from [CHT08, Lem. 2.3.3]666Note that in [CHT08, 2.3], it is supposed
that the places of $S$ are split in $E$ but this is not used in their Lemma
2.3.3. when $k$ is a finite field and from [All16, Lem. 1.3.4] when $k$ is a
finite extension of $\mathbb{Q}_{p}$.

When $k$ is a finite extension of $\mathbb{Q}_{p}$, the result follows from
the analogue of [CHT08, Cor. 2.2.12] (but without the $+1$ since here
$\mathcal{O}=k$). When $k$ is a finite field, it follows from [CHT08, Cor.
2.2.12] that

$\dim(R_{\overline{r}}^{\chi})\geq 1+\frac{n(n+1)}{2}[F:\mathbb{Q}].$

Let $x\in\operatorname{Spec}(R_{\overline{r}}^{\chi}[1/p])$ be a closed point,
$\mathfrak{p}_{x}$ the corresponding prime ideal and
$r_{x}:G_{F}\rightarrow\mathcal{G}_{n}(k(x))$ the corresponding
representation. It follows from [All16, Prop. 1.3.11(1)] that the
localization-completion of $R_{\overline{r}}^{\chi}$ at $\mathfrak{p}_{x}$ is
isomorphic to $R_{r_{x}}^{\chi}$. It follows that

$\dim(R_{\overline{r}}^{\chi}[1/p])\geq\dim(R_{r_{x}}^{\chi})\geq\frac{n(n+1)}{2}[F:\mathbb{Q}]$

using the case where $k$ has characteristic $0$.

The assertion concerning the formal smoothness follows from [CHT08, Cor.
2.2.12] and [All16, Prop. 1.3.11] ∎

######  Remark 5.8.

The hypothesis on the sign of $\chi$ in Proposition 5.7 will be satisfied in
the rest of the text where we choose $\chi=\psi_{0}\psi_{0}^{c}$ for some
$\psi_{0}:G_{E}\rightarrow\overline{\mathbb{Q}_{p}}^{\times}$, actually
because of the sign theorem 4.2. Indeed, by [CHT08] Lemma 2.1.1 we need to
extend $\mu=\chi\varepsilon^{n-1}$ to $G_{F,S}$, for an absolutely irreducible
$\rho$, by sending $c$ to $-\lambda$. Thus the previous hypothesis is
equivalent to $\lambda=1$.

######  Proposition 5.9.

Denote $\overline{\rho}$ as before. Suppose it is absolutely irreducible, and
denote $\overline{r}$ the chosen $\mathcal{G}_{n}$-extension as before.
Suppose $\operatorname{Char}(k)\neq 2$. Then the natural map

$R_{\overline{r}}^{\chi}\longrightarrow
R_{\overline{\rho}}^{\chi-\mathrm{pol}}$

is an isomorphism.

###### Proof.

This is also [All19, Prop. 2.2.3]. Denote $\rho$, $\rho^{\prime}$ resp.
$R,R^{\prime}$ valued points of
$\mathcal{F}^{\chi-\mathrm{pol}}_{\overline{\rho}}$, with
$R^{\prime}\twoheadrightarrow R$, and $\rho^{\prime}\otimes R=\rho$. Suppose
we have a fixed pairing
$<,>:\rho\otimes\rho^{c}\longrightarrow\chi^{-1}\varepsilon^{1-n}$ inducing
$r:G_{E,S}\longrightarrow\mathcal{G}_{n}(R)$ by [CHT08] Lemma 2.1.1. Choose
any pairing fixing $<,>^{\prime}_{0}$ for $\rho^{\prime}$. Then reducing to
$R$ this gives a pairing for $\rho$, but as $\overline{\rho}$ is absolutely
irreducible, $\rho$ is also and thus there is only one pairing up to scalar
for $\rho$, i.e. $<,>^{\prime}_{0}\otimes R=\alpha<,>$ for some $\alpha$ in
$R^{\times}$. Choose a lift $\beta$ of $\alpha^{-1}$, then set
$<,>^{\prime}:=\beta<,>^{\prime}_{0}$, then $<,>^{\prime}$ reduces to $<,>$
and to $(\rho^{\prime},\chi,<,>^{\prime})$ is associated by [CHT08] Lemma
2.1.1 an $r^{\prime}:G_{E,S}\longrightarrow\mathcal{G}_{n}(R^{\prime})$,
reducing to $r$. Let $r^{\prime\prime}$ another point over $R^{\prime}$,
inducing $\rho^{\prime}$ and reducing to $r$ , then it corresponds to
$\gamma<.,.>^{\prime}$ with $\gamma\equiv 1\pmod{m_{R^{\prime}}}$, thus
writing $\gamma=1+m$ with $m\in m_{R^{\prime}}$ we have
$\gamma=(1+\frac{1}{2}m)^{2}\pmod{m_{R^{\prime}}^{2}}$ and as $R^{\prime}$ is
artinian, a direct induction shows that $\gamma$ is a square in $R^{\prime}$,
thus $r^{\prime}=r^{\prime\prime}$.777[CHT08] is written over a field, but
their proof applies here

The same argument with $\overline{r}$ for $r$ and $r$ for $r^{\prime}$ shows
that we can actually choose $r$ inside $\mathfrak{X}_{\overline{r}}^{\chi}$
(and thus automatically for any $r^{\prime}$ above) and thus proves etaleness,
and surjectivity. As the map is an isomorphism in special fiber, this is an
isomorphism. ∎

By [CHT08, 2.1], the adjoint action $ad(r)$ for some
$r:G_{F}\longrightarrow\mathcal{G}_{n}(k)$ coincides with the extension of
$\operatorname{ad}(\rho)$ to $G_{F}$ as in [NT19] (when $\chi=1$), when $\rho$
corresponds to $r$ by [CHT08, Lemma 2.1.1]. Thus we will often denote
abusively $H^{1}(G_{F},\operatorname{ad}(\rho))$ instead of
$H^{1}(G_{F},\operatorname{ad}(r))$.

##  6 Eigenvarieties and the infinite fern

There are at least two ways to define Eigenvarieties as explained in [BC09]
which, at least, in our case of interest end up to be the same. By the Lemma
5.5, we can assume $\chi=1$.

In this section $(\mathcal{O},k)$ are as in the first case of Hypothesis 5.1.
We fix $\overline{\rho}:G_{E}\rightarrow\operatorname{GL}_{n}(k)$ a semi-
simple polarised-by-$1$ continuous representation, i.e.
$\overline{\rho}^{\vee}\simeq\overline{\rho}^{c}\otimes\overline{\varepsilon}^{n-1}$.
Let
$\mathfrak{X}_{\overline{\rho}}^{\mathrm{pol}}:=\mathfrak{X}_{\overline{\rho}}^{1-\mathrm{pol}}$
be its polarised pseudodeformation space. Let $G$ be the quasi-split
similitude unitary group of dimension $n$ over $\mathbb{Q}$ whose $R$-points,
for $R$ a $\mathbb{Q}$-algebra, are:

$G(R)=\\{(g,\nu)\in\operatorname{GL}_{n}(R\otimes_{\mathbb{Q}}E)\times
R^{\times}\mid{}^{t}c(g)Jg=\nu J\\}$

where $J$ is the $n\times n$ matrix $\left(\begin{smallmatrix}0&\cdots&1\\\
\vdots&\iddots&\vdots\\\ 1&\cdots&0\end{smallmatrix}\right)$. Moreover let
$G_{1}$ be the kernel of the morphism $\nu:G\rightarrow\mathbb{G}_{m}$. As $p$
is unramified in $E$, we also fix a reductive model $G_{\mathbb{Z}_{p}}$ over
$\mathbb{Z}_{p}$ of $G$ defined by the similar formula (replacing
$R\otimes_{\mathbb{Q}}E$ by $R\otimes_{\mathbb{Z}}\mathcal{O}_{E}$).

We fix embeddings $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_{p}}$ that we use to
identify the embeddings of $E$ (resp. $F$) in $\overline{\mathbb{Q}}_{p}$ with
the set $\Sigma_{E}$ (resp. $\Sigma_{F}$) of embeddings of $E$ (resp. $F$) in
$\mathbb{C}$. We fix a CM type $\Phi$ for $E$. For $\sigma\in\Sigma_{E}$, we
use the notation $\overline{\sigma}=\sigma\circ c$. If $\tau\in\Sigma_{F}$,
let $\sigma_{\tau}\in\Sigma_{E}$ be the unique element such that
$\tau=\sigma_{\tau}|_{F}$ and $\sigma_{\tau}\in\Phi$.

We fix a PEL datum $(E,c,V,\langle\cdot,\cdot\rangle,h)$ for the previous
group $G$ and denote its signature
$(p_{\sigma_{\tau}},q_{\sigma_{\tau}})_{\tau\in\Sigma_{F}}$ at
infinity888Because $G$ is quasi-split these integers are explicit, but we keep
the slightly general notation as we think it is a bit clearer.. In particular
we have that $p_{\sigma_{\tau}}+q_{\sigma_{\tau}}=n$ doesn’t depend on $\tau$.
We define more generally $(p_{\sigma})_{\sigma\in\Sigma_{E}}$ by
$p_{\sigma}=p_{\sigma_{\tau}}$ if $\sigma=\sigma_{\tau}$ and $p_{\sigma}
=q_{\sigma_{\tau}}=n-p_{\sigma_{\tau}}$ if $\sigma=\overline{\sigma_{\tau}}$.
We also sometimes abuse notation and write $p_{\tau},q_{\tau}$ for
$p_{\sigma_{\tau}},q_{\sigma_{\tau}}$. Let $(G,h)$ be a Shimura datum
associated to $G$. We let $\mathcal{S}=(S_{K})_{K}$ be the tower of Shimura
varieties for $(G,h)$ ([Lan13] or [Her22] which we will use later). Let
$\mu:\mathbb{G}_{m}\rightarrow G_{\mathbb{C}}$ be the cocharacter associated
to $h$ and let $P$ be the parabolic subgroup fixing the Hodge filtration
associated to $\mu$. Let $M$ be the Levi subgroup of $P$ fixing the Hodge
decomposition of $V_{\mathbb{C}}$ (defined over some extension $L$ of the
reflex field). Let $\mathfrak{p}$ be the Lie algebra of $P$ and let
$K_{\infty}$ be the centralizer of $h(i)$ in $G_{1}(\mathbb{R})$.

######  Definition 6.1.

We say that a polarised-by-$1$ representation

$\rho:G_{E}\longrightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}_{p}}),$

is _modular_ if $\rho$ is (strongly essentially) associated to a cuspidal
algebraic automorphic representation $\pi$ for $G$ as in Definition A.3. We
say that $\rho$ is _holomorphically modular_ if its Hecke eigensystem appears
in the space of cuspidal sections of some coherent automorphic sheaf on some
Shimura variety of $\mathcal{S}$. This is equivalent to the fact that $\pi$ is
cuspidal and holomorphic at infinity; i.e.
$H^{0}(\mathfrak{p},K_{\infty},\pi_{\infty}\otimes\sigma)\neq 0$ for some
algebraic representation $\sigma$ of $K_{\infty}$ 999These hypothesis are here
to assure a concrete (= computable) way to verify if our $\pi$ ”appears” in an
Eigenvariety. We could introduce the notion of $p$-adically modular for which
we ask for a Hecke eigensystem appearing in the considered Eigenvariety
$\mathcal{E}$ whose associated trace is $\operatorname{tr}\rho$. It is enough
to assume $p$-adic modularity for $\overline{\rho}$ to get Theorem 1.6 by
[Har90b] Proposition 5.4.2.

We say that $\overline{\rho}$ is modular if it admits a lift $\rho$ which is
modular. We say that $\overline{\rho}$ is _conveniently modular_ if it has a
lift $\rho$ associated to a cuspidal automorphic representation $\pi$ which
can be chosen unramified at $p$ and outside $S$ and its Hecke eigensystem
appears in $i$-th interior coherent cohomology group on some Shimura variety
of $\mathcal{S}$, with values in some coherent automorphic sheaf, for some
$i\geq 0$.

If $K^{p}$ is a compact open open subgroup of $G(\mathbb{A}^{p,\infty})$, we
say that $\overline{\rho}$ is _conveniently modular of tame level $K^{p}$_ if
$\pi$ can be moreover chosen such that $\pi^{K^{p}}\neq 0$.

######  Hypothesis 6.2.

For the rest of the article, we assume that every $v|p$ in $F$ is unramified,
and splits in $E$. Moreover we assume that $\overline{\rho}$ is conveniently
modular.

In particular, if $v$ is a place of $F$ dividing $p$, among the two places
$w,\overline{w}$ of $E$ above $v$ only one, say $w$, corresponds to an element
of $\Phi$. We fix this choice, which allows us to identify $E_{w}$ with
$F_{v}$. Choose a sufficiently large $p$-adic field $L$ such that $M$ and $P$
are defined over $L$ and $L$ splits $E$, i.e.
$E\otimes_{\mathbb{Q}_{p}}L=\prod_{w|p\in E}L$. Let $\mathcal{T}$ be the rigid
space over $L$ given by
$\prod_{v|p}\operatorname{Hom}((F_{v}^{\times})^{n},\mathbb{G}_{m})$, and
$\mathcal{W}=\prod_{v|p}\operatorname{Hom}((\mathcal{O}_{F_{v}}^{\times})^{n},\mathbb{G}_{m})$
the weight space. There is thus a restriction map

$\mathcal{T}\longrightarrow\mathcal{W}.$

From now on we fix a finite field $k$ of characteristic $p$ and a continuous
semisimple representation
$\overline{\rho}:G_{E}\rightarrow\operatorname{GL}_{n}(k)$ which polarized by
$\overline{\varepsilon}^{n-1}$ and is conveniently modular. We fix a tame
level outside of $p$, $K^{p}$, which is hyperspecial outside $S$ and deep
enough so that $\overline{\rho}$ is conveniently modular of tame level
$K^{p}$.

Let
$\mathcal{Z}^{\prime}_{K^{p}}\subset\mathfrak{X}_{\overline{\rho}}^{\chi-\mathrm{pol}}(\overline{\mathbb{Q}_{p}})\times\mathcal{T}(\overline{\mathbb{Q}_{p}})$
the set of pairs $(D,\delta)$ where $D$ is the determinant associated to (the
Galois representation of) a cuspidal, regular, algebraic, unramified at $p$
automorphic form $\pi$ for $G$ appearing in degree 0 coherent cohomology
101010this means that $H^{0}(\mathfrak{p},K_{\infty},\pi_{\infty}\otimes
V)\neq 0$ for a finite dimensionnal representation of $K_{\infty}$. In
particular $D$ is holomorphically modular. by Corollary A.9, of level $K^{p}$
outside $p$, of Hodge-Tate weights
$k_{v,\tau,1}>k_{v,\tau,2}>\dots>k_{v,\tau,n}$111111We choose the convention
for which the cyclotomic character has Hodge-Tate weight +1 for each $v|p$ in
$F$,$\tau$, and $\delta$ such that for all $v,i$, $\delta_{v,\tau,i}$
coincides on $\mathcal{O}_{F_{v}}^{\times}$ with
$\prod_{\tau}\tau^{k_{v,\tau,i}}$ and sends $p$ to $\phi_{v,i}$, where
$\phi_{v,1},\dots,\phi_{v,n}$ is an admissible refinement for $\pi_{v}$ (and
obviously such that $D$ lifts $\overline{\rho}$).121212for these local data at
$p$ we have used the implicit choice of $w|v$

######  Remark 6.3.

By [Box15] Theorem D, [PS16] or [GK15] Theorem I.3.1, we have that, under the
hypothesis 6.2, $\mathcal{Z}_{K^{p}}^{\prime}$ is non empty, i.e. we can
choose a lift of $\overline{\rho}$ that is holomorphically modular.

######  Definition 6.4.

The Eigenvariety for $G$, $\overline{\rho}$, $\chi=1$ and $K^{p}$ is the
Zariski closure

$\mathcal{E}_{K^{p}}(\overline{\rho})\subset\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}\times\mathcal{T},$

of $\mathcal{Z}^{\prime}_{K^{p}}$. The infinite fern
$\mathcal{F}_{K^{p}}(\overline{\rho})$ is the image of
$\mathcal{E}_{K^{p}}(\overline{\rho})$ by the first projection.

As $G$ is a unitary similitude group with similitude in $\mathbb{Q}$, thus
giving rise to a PEL Shimura datum, and $p$ is unramified in $E$, we also
constructed in [Her22] an Eigenvariety for $G$, for any type $K^{p}$ outside
$p$. Actually these two constructions compare, and allow us to deduce the
following proposition.

######  Remark 6.5.

Actually we could take $G$ any similitude unitary group with similitude factor
in $\mathbb{Q}$ instead of the quasi-split one. Indeed, as long as $p$ is
unramified for $G$ the construction of [Her22] applies and we get the
following proposition. In particular, if we have a result analogous to [Her22]
for ramified primes (i.e. for primes $v|p$ in $F$ which are ramified, but
still assuming $v=w\overline{w}$ in $E$) then all the methods of this article
applies (see [BP20]). For the moment, we still need our $p$-adic group to be
(a product of) $GL_{n}$ to use results on the trianguline variety, but we hope
to come back on this question in the future.

######  Proposition 6.6.

The rigid space $\mathcal{E}_{K^{p}}(\overline{\rho})$ is equidimensional of
dimension $n[F:\mathbb{Q}]$. The map

$h:\mathcal{E}_{K^{p}}(\overline{\rho})\longrightarrow\mathcal{W},$

is locally, on the goal and the source, finite. In particular the image of any
irreducible component of $\mathcal{E}_{K^{p}}(\overline{\rho})$ is open in
$\mathcal{W}$. Moreover, for all $C>0$, if $\mathcal{Z}_{C}
\subset\mathcal{Z}_{K^{p}}^{\prime}$ consists of classical points, crystalline
at $p$, which are moreover $C$-very regular (i.e. its Hodge-Tate weights
satisfies $k_{v,\tau,i}>k_{v,\tau,i+1}+C$ for all $v,\tau,i$), then
$\mathcal{Z}_{C}$ is Zariski dense in $\mathcal{E}_{K^{p}}(\overline{\rho})$
and accumulation at every point of $\mathcal{Z}_{K^{p}}^{\prime}$.

We need to introduce a few notations. Let $T$ be the diagonal torus of
$G_{\mathbb{Z}_{p}}$. Its group of $\mathbb{Q}_{p}$-points has the following
description,

$T(\mathbb{Q}_{p})=\left\\{\left(\begin{array}[]{ccc}a_{1,w}&&\\\ &\ddots&\\\
&&a_{n,w}\end{array}\right)\in\prod_{w|p\text{
in}E}\operatorname{GL}_{n}(E_{w}),\exists
r\in\mathbb{Q}_{p}^{\times},a_{i,w}a_{n-i+1,\overline{w}}=r,\forall
i,w\right\\}.$

Denote by $T^{1}$ the subtorus with trivial similitude character (i.e. $r=1$).
We identify $\mathcal{T}$ with the space of characters of
$T^{1}(\mathbb{Q}_{p})$ using the isomorphism $(F_{v}^{\times})^{n}\simeq
T^{1}(\mathbb{Q}_{p})$ sending $(a_{1,v},\dots,a_{n,v})$ to the diagonal
matrix of $\operatorname{GL}_{n}(E_{w})$ with diagonal
$(a_{1,v},\dots,a_{n,v})$, via the identification $E_{w}\simeq F_{v}$ where
$w\mid v$ and $w\in\Phi$. $T(\mathbb{Q}_{p})$ (resp. $T^{1}(\mathbb{Q}_{p})$)
can be identified also with a subgroup of the $L$-points of the torus (resp.
subtorus of $r=1$ elements) of $M\simeq\mathbb{G}_{m}\times\prod_{v|p\text{ in
}F}\prod_{\sigma_{\tau}\in\operatorname{Hom}(F_{v},\overline{\mathbb{Q}_{p}})}\operatorname{GL}_{p_{\sigma_{\tau}}}\times\operatorname{GL}_{q_{\sigma_{\tau}}}$,
using,

$\left(\begin{array}[]{ccc}a_{w,1}&&\\\ &\ddots&\\\
&&a_{w,n}\end{array}\right)_{w}\longmapsto\left(r,\left(\left(\begin{array}[]{ccc}\tau(a_{w,p_{\sigma_{\tau}}})^{-1}&&\\\
&\ddots&\\\
&&\tau(a_{w,1})^{-1}\end{array}\right),\left(\begin{array}[]{ccc}\overline{\tau}(a_{\overline{w},q_{\sigma_{\tau}}})^{-1}&&\\\
&\ddots&\\\
&&\overline{\tau}(a_{\overline{w},1})^{-1}\end{array}\right)\right)_{\tau}\right)$

######  Definition 6.7.

Let $\kappa=(k_{\sigma,i})_{\begin{subarray}{c}\sigma\in\Sigma_{E}\\\ 1\leq
i\leq p_{\sigma}\end{subarray}}\in\mathbb{Z}^{n[F:\mathbb{Q}]}$. We say that a
character $\chi\in\mathcal{W}(\mathbb{C}_{p})$ is _algebraic of coherent
weight $\kappa$_ if

$\forall z=(z_{v,i})\in\prod_{v\mid
p}(\mathcal{O}_{F_{v}}^{\times})^{n},\quad\chi(z)=\prod_{v\mid
p}\prod_{i=1}^{p_{\tau}=p_{\sigma_{\tau}}}\sigma_{\tau}(z_{v,i})^{k_{\sigma_{\tau},i}}\prod_{i=1}^{q_{\tau}=p_{\overline{\sigma_{\tau}}}}\sigma_{\tau}(z_{n+1-i})^{-k_{\overline{\sigma_{\tau}},i}}.$
(7)

We say that an algebraic character of coherent weight $\kappa$ is _$M$
-dominant_ if $k_{\sigma,i}\geq k_{\sigma,i+1}$ for $\sigma\in\Sigma_{E}$ and
$1\leq i\leq p_{\sigma}-1$.

This corresponds to the choice of the upper triangular Borel for $M$, in the
sense that if we have character $\kappa$ of $M$ which is dominant for the
upper Borel of $M$, then its restriction to $T^{1}(\mathbb{Q}_{p})$ via the
previous embedding gives a M-dominant $\chi$ in the previous sense. Suppose
$\chi$ is algebraic for some coherent weight $\kappa$. For
$h=(h_{\tau,i})_{\tau\in\Sigma_{F},1\leq i\leq
n}\in\mathbb{Z}^{n[F:\mathbb{Q}]}$, if131313This just means that
$k_{\sigma_{\tau},i}=h_{\tau,i},i\leq p_{\tau}$ and
$-k_{\overline{\sigma_{\tau}},i}=h_{\tau,n+1-i}$ for $i>p_{\tau}$.

$\chi(z)=\prod_{\tau\in\Sigma_{F}}\sigma_{\tau}(z_{i})^{h_{\tau,i}},$

then we say that $\chi$ is of _infinitesimal weight_ $h$. We say that such a
$\chi$ is _dominant_ (or $G$-_dominant_) if $h_{\tau,1}\geq
h_{\tau,2}\geq\dots\geq h_{\tau,n}$ for all $\tau$.

###### Proof.

Let $\mathcal{E}^{\prime}$ together with a locally finite map
$w:\mathcal{E}^{\prime}\longrightarrow\mathcal{W}$ the Eigenvariety for $G$ of
tame level $K^{p}$ constructed in [Her22]. It is an equidimensionnal rigid
space of dimension $n[F:\mathbb{Q}]$.

Let $S$ be a finite set of places of $F$ containing the places dividing $p$
and the places where $K^{p}$ is not hyperspecial. For $v\notin S$, let
$\mathcal{H}_{v}$ be the spherical Hecke algebra
$\mathbb{Z}[G(F_{v})/\\!/K^{p}_{v}]$ of $G$ and let
$\mathcal{H}^{S}=\bigotimes_{v\notin S}\mathcal{H}^{v}$. For $v|p$, let
$\mathcal{A}_{v}$ be the (commutative) $\mathbb{Z}$-algebra generated by
$T_{n}^{-}(F_{v})$ and their inverses with $T_{n}$ the diagonal torus of
$\operatorname{GL}_{n}$, and $T_{n}^{-}(F_{v})$ the subgroup of matrices
$\operatorname{Diag}(p^{a_{1}},\dots,p^{a_{n}})$ with $a_{i}\geq a_{i+1}$. Let
$\mathcal{A}(p)=\bigotimes_{v|p}\mathcal{A}(v)$ be the Atkin-Lehner algebra.
It follows from [Her22, §7]141414There $\mathcal{H}^{S}$ is denoted
$\mathcal{H}_{K^{p}}$. See also remark 7.12 of [Her22] that there exists
homomorphism
$\lambda:\mathcal{H}^{S}\rightarrow\Gamma(\mathcal{E}^{\prime},\mathcal{O}_{\mathcal{E}^{\prime}}^{+})$
and
$\mathcal{A}(p)\rightarrow\Gamma(\mathcal{E}^{\prime},\mathcal{O}_{\mathcal{E}^{\prime}})$,
sending $\operatorname{Diag}(\underbrace{1,\dots,1}_{i\text{
times}},p^{-1},\dots,p^{-1})_{v}$ to a Hecke operator $U_{v,i}$ at $v$, such
that, if $z\in\mathcal{E}^{\prime}(\mathbb{C}_{p})$, the evaluation of these
morphisms at $z$ induces a non-zero eigenspace in
$H^{0}(S_{K}^{tor}(v),\omega^{w(z),\dagger}(-D))$, with $K=K^{p}I$, and $I$ a
Iwahori subgroup at $p$, (which is a space of overconvergent cuspidal forms,
defined in [Her22] Definition 6.12). Moreover if
$\kappa\in\mathcal{W}(\mathbb{C}_{p})$ is an algebraic character of
$M$-dominant coherent weight, then the action of $\mathcal{H}^{S}$ preserves
the subspace of classical forms $H^{0}(S_{K}^{tor},\omega^{\kappa}(-D))$ and
coincides with the “usual” action of $\mathcal{H}^{S}$ on
$H^{0}(S_{K}^{tor},\omega^{\kappa}(-D))$.

We remind now that $\mathcal{E}^{\prime}$ contains an accumulation and Zariski
dense subspace of automorphic points that we will call very regular small
slope classical points.

Let $\mathcal{Z}\subset\mathcal{E}^{\prime}(\mathbb{C}_{p})$ be the set of
points $z$ satisfying [Her22] Proposition 8.2, a slightly stronger form of
Theorem 8.3, namely

$\max(n_{\tau}+v_{p}(\alpha_{\tau}),0)<\inf_{\tau}(k_{\sigma_{\tau},p_{\tau}}+k_{\overline{\sigma_{\tau}},q_{\tau}}),\quad\forall\tau\in\Sigma_{F}$
(8)

where $w(z)$ is (thus) a $G$-dominant algebraic character of coherent weight
$(k_{\sigma,i})_{\begin{subarray}{c}\sigma\in\Sigma_{E}\\\ 1\leq i\leq
p_{\sigma}\end{subarray}}$ and $\alpha_{\tau}$ is the eigenvalue for the
operator $U_{v,\min(p_{\tau},q_{\tau})}$ and $n_{\tau}$ is a normalisation
constant depending only on $(p_{\tau},q_{\tau})_{\tau}$, asking that $w(z)$ is
moreover far from the walls as in [Har90a] Lemma 3.6.1. For $C>0$, we define
$\mathcal{Z}_{C}\subset\mathcal{Z}$ adding the condition
$k_{\sigma,i}-k_{\sigma,i+1}>C$ for all $i$. For $C>\\!>0$, these points give
rise to crystalline representations at $p$ as we will see. Each of the sets
$\mathcal{Z}_{C}$ is accumulation in $\mathcal{Z}$, we can thus prove the
claim with $\mathcal{Z}_{C}$ replaced by $\mathcal{Z}$. By [Bij16] (see
[Her22, Thm 9.4]), if $z\in\mathcal{Z}$, the system of eigenvalues
corresponding to $z$ has an eigenvector in
$H^{0}(S_{K}^{tor},\omega^{\kappa}(-D))$. This implies that actually
$z\in\mathcal{E}^{\prime}(\overline{\mathbb{Q}}_{p})$. It follows from [SU02],
or [Har90a], that there exists a cuspidal automorphic form $\pi$ of
$G(\mathbb{A}_{F})$ such that $\pi_{f}^{K^{p}}\neq 0$, the Satake parameter of
$\pi_{v}$, $v\nmid p$, corresponds to $\lambda|_{\mathcal{H}_{v}}\otimes k(z)$
and $\pi_{\infty}$ is tempered of weight
$((k_{\sigma_{\tau},p_{\tau}},\cdots,k_{\sigma_{\tau},1},-k_{\overline{\sigma_{\tau}},1},\cdots,-k_{\overline{\sigma_{\tau}},q_{\tau}}))_{\tau\in\Sigma_{F}}-\rho_{G}-w_{0,M}\rho_{G}$,
with $\rho_{G}$ the half-sum of positive roots, as we will explain.

To be able to clearly label the weights, let $P_{h}$151515also called
$P_{\mu}^{std}$ in other references be the parabolic corresponding to
$\mathfrak{p}$, and choose a Borel, equivalently a set $\Phi^{+}$ of positive
roots such that if $\Phi^{+}_{c}$ is the set of positive roots contained in
the Levi of $\mathfrak{p}$, then
$\Phi^{+}_{nc}:=\Phi^{+}\backslash\Phi^{+}_{c}$ is chosen to be included in
$\mathfrak{g}_{\mathbb{C}}/\mathfrak{p}$. Equivalently $B\subset
P_{h}^{opp}=P_{\mu}$, the parabolic opposite to $P_{h}$. This allows us to
label similarly (classical, dominant) weights of representations of
$K_{\infty}$ (with respect to $\Phi^{+}_{c}$) and of $G$ (with respect to
$\Phi^{+}$). Let us be more precise for these choices. Let $G(\mathbb{Q}_{p})$
our unitary group, thus given by the hermitian form $J$, and let $T$ be its
diagonal torus, and $T^{1}$ the subtorus of elements of similitude 1. We have
an embedding

$ \begin{array}[]{ccc}T^{1}(\mathbb{Z}_{p})&\longrightarrow&\prod_{\tau
\in\Phi}\operatorname{GL}_{p_{\tau},K}\times\operatorname{GL}_{q_{\tau},K}\\\
\left(\begin{array}[]{ccc}a_{1}&&\\\ &\ddots&\\\
&&a_{n}\end{array}\right)&\longmapsto&\left(\left(\begin{array}[]{ccc}\sigma_{\tau}(a_{p_{\tau}})^{-1}&&\\\
&\ddots&\\\
&&\sigma_{\tau}(a_{1})^{-1}\end{array}\right),\left(\begin{array}[]{ccc}\overline{\sigma_{\tau}}(a_{q_{\tau}})^{-1}&&\\\
&\ddots&\\\
&&\overline{\sigma_{\tau}}(a_{1})^{-1}\end{array}\right)\right)_{\tau}\end{array}$

where $a_{i}\in\mathcal{O}_{E}\otimes\mathbb{Z}_{p}$ and
$a_{i}\overline{a_{n+1-i}}=1$. Writing $a_{i}=(z_{i},t_{i})$ (using the choice
of $w|v$ for all $v$ and $z_{i}$ corresponding to $w$ in $\Phi$), we can
rewrite the previous embedding

$\iota:\begin{array}[]{ccc}((\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times})^{n}&\longrightarrow&\prod_{\tau
\in\Phi}\operatorname{GL}_{p_{\tau},K}\times\operatorname{GL}_{q_{\tau},K}\\\
\left(\begin{array}[]{ccc}z_{1}&&\\\ &\ddots&\\\
&&z_{n}\end{array}\right)&\longmapsto&\left(\left(\begin{array}[]{ccc}\tau(z_{p_{\tau}})^{-1}&&\\\
&\ddots&\\\
&&\tau(z_{1})^{-1}\end{array}\right),\left(\begin{array}[]{ccc}\tau(z_{p_{\tau}+1})&&\\\
&\ddots&\\\ &&\tau(z_{n})\end{array}\right)\right)_{\tau}\end{array}$

We choose the diagonal torus and upper Borel for
$M\simeq\prod_{\tau}\operatorname{GL}_{p_{\tau}}\times\operatorname{GL}_{q_{\tau}}$161616This
is actually the subgroup of $M$ of element with similitude factor 1, but in
all this discussion we ignore the similitude factor to simplify the
notations., which we see as the Levi for
$P_{\mu}\subset\prod_{\tau\in\Phi}\operatorname{GL}_{p_{\tau}+q_{\tau}}$, the
standard lower parabolic with blocs $p_{\tau},q_{\tau}$. Thus the choice of
$\Phi^{+}$ for $G$ corresponds to the standard upper Borel. Denote $\rho_{G}$
the half-sum of positive roots in $\Phi^{+}$.

Thus if we choose $\kappa:=(k_{\sigma,i})_{\sigma\in\Sigma_{E},1\leq i\leq
p_{\sigma}}$ a dominant (integral) weight for $M$ for the previous upper
triangular Borel, then the algebraic representation of highest weight $\kappa$
is constructed using $-w_{0,M}\kappa$, which induces the weight $\chi$ of
$((\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times})^{n}$ which is algebraic of
coherent weight $\kappa$ in the sense of equation (7). This weight $\chi$ is
$\Phi^{+}$-dominant if and only if $k_{\sigma_{\tau},p_{\tau}}\geq-
k_{\overline{\sigma_{\tau}},q_{\tau}}$.

Let $z\in\mathcal{Z}$, which corresponds to a classical automorphic form
(itself giving an automorphic representation $\pi$) appearing in
$H^{0}(S_{K}^{tor},\omega^{\kappa})$ for
$\kappa=(k_{\sigma,i})_{\sigma\in\Sigma_{E},1\leq i\leq p_{\sigma}}$ as
before, which is a classical (and $M$-dominant) weight. Then
$\chi=w(z)\in\mathcal{W}$ is the algebraic character of weight $\kappa$, i.e.
is $-w_{0,M}\kappa\circ\iota$. This sheaf $\omega^{\kappa}$ coincides with the
coherent sheaf $V_{s}$ over $\mathbb{C}$ defined by Harris ([Har90b]),
associated to the highest weight $s$ representation of
$M=\prod_{\tau\in\Phi}\operatorname{GL}_{p_{\tau}}\times\operatorname{GL}_{q_{\tau}}$,
with
$s=(-k_{\tau,p_{\tau}},\dots,-k_{\tau,1},k_{\overline{\tau},1},\dots,k_{\overline{\tau},q_{\tau}})$
with the previous identifications. This calculation is the one done in [FP19]
section 7.4, based on [Gol14]. Remark that if $\chi$ is algebraic of weight
$\kappa$ and dominant, then the dominant representative of $-s$ is given by
$\operatorname{wt}(\chi)=(k_{\sigma_{\tau},1},\dots,k_{\sigma_{\tau},p_{\tau}},-k_{\overline{\sigma_{\tau}},q_{\tau}},\dots,-k_{\overline{\sigma_{\tau}},1})$.
In particular as the Hecke eigensystem corresponding to $z$ appears in
$H^{0}(S^{tor}_{K},\omega^{\kappa})$ thus in
$H^{0}(S^{tor}_{K}(\mathbb{C}),V_{s})$ this means that

$H^{0}(\mathfrak{p},K_{\infty},\pi_{\infty}\otimes V_{s})\neq\\{0\\},$

i.e. that the infinitesimal character of $\pi_{\infty}$ is $-s-\rho_{G}$ (up
to reordering) by e.g. [Har90b] Proposition 4.3.2 (see also [BP20] Proposition
5.37). But if $z\in\mathcal{Z}$ and $\pi$ an automorphic representation
corresponding to its system of eigenvalues $\lambda(z)$ of $\mathcal{H}^{S}$,
then using that $w(z)$ is far from the walls, by Corollary A.9 there is a
semisimple representation
$\rho^{u}=\rho^{u}_{z}:G_{E}\rightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{p})$
such that $\rho^{u}(\operatorname{Frob}_{v})$ is associated to the semi-simple
conjugacy class at $v$ determined by $\lambda$, for all $v\notin S$ and
satisfying moreover

$(\rho^{u})^{\vee}\simeq(\rho^{u})^{c}\otimes\varepsilon^{n-1}.$

By for example [BGGT14], the previous calculation of the infinitesimal weight
means that $\rho^{u}$, associated to $\pi$, has Hodge-Tate weights given by
$-s-\rho_{G}-\underline{\frac{n-1}{2}}$ i.e. the $v,\tau$ Hodge-Tate weights
of $\rho^{u}_{z}$ are (up to order)

$\displaystyle\left(k_{v,\tau,p_{\tau}}+1-n,k_{v,\tau,p_{\tau}-1}+2-n,\dots,k_{v,\tau,1}+p_{\tau}-n,-k_{v,\overline{\tau},1}+p_{\tau}-n+1,\dots,-k_{v,\overline{\tau},q_{\tau}}\right)$

which we can reorder to be dominant for very regular $z\in\mathcal{Z}$,

$(k_{v,\tau,1}+(p_{\tau}-n),k_{v,\tau,2}+(p_{\tau}-n-1),\dots,k_{v,\tau,p_{\tau}}+1-n,-k_{v,\overline{\tau},q_{\tau}},\dots,-k_{v,\overline{\tau},1}+(p_{\tau}-n+1)).$
(9)

We first construct a union of connected components of $\mathcal{E}^{\prime}$
and a map from this subspace to
$\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$. As in [Che04], we construct a
determinant

$D:G_{E}\longrightarrow\mathcal{O}_{\mathcal{E^{\prime}}}.$

Let $z\in\mathcal{Z}$ and $\pi$ an automorphic representation corresponding to
its system of eigenvalues $\lambda(z)$ of $\mathcal{H}^{S}$, as we have seen,
by Corollary A.9 there is a semisimple representation
$\rho^{u}:G_{E}\rightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{p})$
associated to $\lambda(z)$ such that

$(\rho^{u})^{\vee}\simeq(\rho^{u})^{c}\otimes\varepsilon^{n-1}.$

Let $D_{z}$ the pseudo-representation of $\rho^{u}_{z}$. The continuous map
$(D_{z})_{z\in\mathcal{Z}}$ from $G_{E}$ to $\prod_{x\in\mathcal{Z}}k(x)$
factors actually through
$\Gamma(\mathcal{E}^{\prime},\mathcal{O_{\mathcal{E^{\prime}}}^{+}})$ and
gives rise to a pseudo deformation $D$ on
$\Gamma(\mathcal{E}^{\prime},\mathcal{O}_{\mathcal{E^{\prime}}}^{+})$. By
continuity, we have $D^{\vee}\simeq D^{c}\varepsilon^{n-1}$.

As there is only a finite number of possible reductions modulo $p$ of $D$,
there is $\mathcal{E^{\prime}}(\overline{\rho})$ an open and closed subset of
$\mathcal{E^{\prime}}$ of points whose reduction of $D$ is
$(\operatorname{tr})\overline{\rho}$. This is non empty by Hypothesis 6.2. In
particular the restriction of the previous $D$ to
$\mathcal{E^{\prime}}(\overline{\rho})$ induces a morphism of rigid analytic
spaces

$\mathcal{E^{\prime}}(\overline{\rho})\longrightarrow\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}.$

Now we construct a rigid analytic map
$\mathcal{E}^{\prime}\rightarrow\mathcal{T}$.

Denote $w=(w_{v,1},\dots,w_{v,n})_{v}$ the universal character of
$((\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times})^{n}$. In [Her22, Section
7.2.2] we constructed Hecke operators which are in
$\mathcal{O}(\mathcal{E}^{\prime})^{\times}$, denoted by $U_{v,i}$ for $v|p$
in $F$ and $i=0,\dots,n$. The operator $U_{v,i}$ coincides up to normalisation
(this normalisation is made in order to vary in family) with the double class,

$ \left(\left(\begin{array}[]{cc}I_{i}&\\\
&p^{-\delta_{v=v^{\prime}}}I_{n-i}\end{array}\right)_{w},\left(\begin{array}[]{cc}I_{n-i}&\\\
&p^{-\delta_{v=v^{\prime}}}I_{i}\end{array}\right)_{\overline{w}},\right)_{w\overline{w}=v^{\prime}|
p}\in G(\mathbb{Q}_{p})\subset\prod_{w\overline{w}=v^{\prime}|p\text{ in
F}}\operatorname{GL}_{n}(E_{w})\times\operatorname{GL}_{n}(E_{\overline{w}}).$

If $U_{v,i}^{class}$ denotes the action of the (classical, i.e. non
normalised) Hecke operator corresponding to the previous Iwahori double class
acting on global sections of the classical automorphic sheaf, for (fixed)
algebraic weight $w\in\mathcal{W}$, then the normalisation is

$ U_{v,i}=\widetilde{w}_{v}\left(\begin{array}[]{cccccc}1&&\\\ &\ddots&\\\
&&1\\\ &&&p^{-1}\\\ &&&&\ddots\\\
&&&&&p^{-1}\end{array}\right)U_{v,i}^{class},$

where $\widetilde{w}_{v}$ is the (unique) algebraic extension of $w_{v}$ as a
character of $(F_{v}^{\times})^{n}$171717This normalisation factor, which is a
power of $p$, comes when trying to express a character of $T$, the maximal
torus of $G$, as one of $F\otimes\mathbb{Q}_{p}^{\times}$. The key point is
that the double class corresponding to $U_{v,i}$ has similitude factor
$p^{-1}$, thus is not in $T^{1}$. But remark that then the Hecke operator
$F_{v,i}$ has similitude factor 1., and there is $i$ times 1 (and $n-i$ times
$p^{-1}$) appearing in the matrix (see before and remark 7.5, together with
remark 8.3 of [Her22]). For all $i\in\\{1,\dots,n\\}$, we set

$F_{v,i}:=U_{v,i}U_{v,i-1}^{-1}.$

It corresponds, up to normalisation, to the Hecke operator in
$\mathcal{A}(p)$181818But not the the double class associated to the following
matrix, as the Hecke algebra at Iwahori level is not commutative! See e.g.
[BC09] Proposition 6.4.1 and the remark that follows.

$ F_{v,i}^{cl}:=\left(\left(\begin{array}[]{ccc}I_{i-1}&&\\\
&p^{\delta_{v=v^{\prime}}}&\\\
&&I_{n-i}\end{array}\right)_{w},\left(\begin{array}[]{ccc}I_{n-i}&\\\
&p^{-\delta_{v=v^{\prime}}}&\\\
&&I_{i-1}\end{array}\right)_{\overline{w}},\right)_{w\overline{w}=v^{\prime}|
p}\in G(\mathbb{Q}_{p}),$

and the normalisation is the following, for $w$ algebraic of infinitesimal
weight $h=(h_{\tau,i})_{\tau,i}$

$ F_{v,i}=\widetilde{w}_{v}\left(\begin{array}[]{ccccc}1&&\\\ &\ddots&\\\
&&p\\\ &&&\ddots\\\
&&&&1\end{array}\right)F_{v,i}^{cl}=p^{\sum_{\tau}h_{i,\tau}}F_{v,i}^{cl},$

where $p$ is in position $i$. The all point is that
$F_{v,i}^{cl}\notin\mathcal{O}(\mathcal{E}^{\prime})$, i.e. they don’t
interpolate, whereas $F_{v,i}\in\mathcal{O}(\mathcal{E}^{\prime})^{\times}$.
We construct characters
$\delta^{0}_{v,i}:F_{v}^{\times}\longrightarrow\mathcal{O}(\mathcal{E}^{\prime})^{\times}$
by setting $\delta^{0}_{v,i}(p):=F_{v,i}$ and

$(\delta^{0}_{v,i})_{|\mathcal{O}_{F_{v}}^{\times}}=w_{v,i},$

and we finally set

$\delta_{v,i}:=\delta_{v,i}^{0}\times\prod_{\tau}x_{\tau}^{s_{\tau}(i)}\times|.|^{\frac{1-n}{2}},$
(10)

where for $\tau:F_{v}\hookrightarrow\mathbb{C}_{p}$,
${x_{\tau}}_{|\mathcal{O}_{F_{v}}}=\tau_{\mathcal{O}_{F_{v}}}$ and
$x_{\tau}(p)=1$, and $s_{\tau}(i)=\frac{1-n}{2}+p_{\tau}-i$ if $1\leq i\leq
p_{\tau}$, and $s_{\tau}(i)=\frac{n-1}{2}-(i-p_{\tau}-1)$ if
$i>p_{\tau}$191919Remark that actually in our quasi split situation we have
$p_{\tau},q_{\tau}$ which doesn’t depend on $\tau\in\Phi$. In any case
$(s_{\tau}(1),\dots,s_{\tau}(n))_{\tau}=w_{0,M}(0,\dots,n-1)_{\tau}+\underline{\frac{1-n}{2}}=-w_{0,M}\rho_{G}$,
with $\rho_{G}$ defined in the next paragraph.. Thus the characters
$(\delta_{v,i})_{v,i}$ gives a map

$ \mathcal{E}^{\prime}\longrightarrow\mathcal{T}.$

Still denote $\mathcal{Z}$ for
$\mathcal{Z}\cap\mathcal{E^{\prime}}(\overline{\rho})$, which is Zariski dense
and accumulation. Now we are reduced to prove that the two constructed maps
$\mathcal{E^{\prime}}(\overline{\rho})\longrightarrow\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$
and $\mathcal{E^{\prime}}(\overline{\rho})\longrightarrow\mathcal{T}$ are
compatible, in the sense that for $z\in\mathcal{Z}$ the second map is the
parameter of a triangulation for the image of $z$ via the first map. By local
global compatibility at $v$ for $\pi$ and $\rho^{u}$, we have that, using
$\pi_{v}^{I}\neq\\{0\\}$ by construction of $\mathcal{E}^{\prime}$, that
$\pi_{v}$ is a subquotient of the Borel induction of an unramified character
$\chi$ of $(F_{v}^{\times})^{n}$ (e.g. [BC09] Proposition 6.4.3 and 6.4.4)
with $\chi$ related to the eigenvalues of $F_{v,i}^{class}$ at $z$
($\chi=(\varphi_{1},\dots,\varphi_{n})=(F_{v,1}^{cl},\dots,F_{v,i}^{cl})\delta_{B}^{1/2}$).
But $F_{v,i}$ has a locally constant valuation (thus not $F_{v,i}^{cl}$ !), so
up to choose another point of $\mathcal{Z}$ close to $z$, we can assume that
this induced representation is irreducible, and thus unramified. By local
global compatibility this proves that there is an accumulation subset of
$\mathcal{Z}$, which accumulates at any point of $\mathcal{E}$ with algebraic
weight, consisting of points $z$ with representation $\rho_{z}$ semi simple
corresponding to $D_{z}$, crystalline at every $v|p$ and such that
$D_{\mathrm{cris}}(\rho_{v})$ has all its refinement, one of which is given by
$(D_{\mathrm{cris}}(\delta_{v,i}))_{1\leq i\leq n}$, i.e.
$(F_{v,i}^{cl})_{i}$. Moreover, the calculation for $z\in\mathcal{Z}$ we did
in equation 9, together with the definition of the weight of $\delta$ in
equation 10 implies that the Hodge-Tate weights of $\rho_{z}$ are given by
$\delta_{|(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}}$, in the right
order ! This means that the map

$D\times\delta:\mathcal{E}(\overline{\rho})\longrightarrow\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}\times\mathcal{T},$

sends a dense subset of $\mathcal{Z}$ (namely the previous one where points
are crystalline) into $\mathcal{Z}^{\prime}_{K^{p}}$, but conversely by
construction of $\mathcal{E}^{\prime}$, all points of
$\mathcal{Z}^{\prime}_{K^{p}}$ are in the image of the previous map.

Moreover $D\times\delta$ is a closed immersion. Indeed, by construction
$\mathcal{E^{\prime}}$ is (locally) constructed as the image of
$\mathcal{H}\otimes\mathcal{O}_{\mathcal{W}}=\mathcal{H}^{S}\otimes\mathcal{O}_{\mathcal{T}}$
where $\mathcal{H}=\mathcal{H}^{S}\otimes\mathcal{A}(p)$ acting on some space
of overconvergent, locally analytic modular forms (of finite slope). Let
$U\subset\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}\times\mathcal{T}$ be an
affinoid, in particular it is quasi-compact thus the slopes of
$\mathcal{A}(p)$ on $U$ are bounded (say by $\alpha$). Thus
$(D\times\delta^{-1})(U)$ is included in $\mathcal{E}_{v,w}^{\leq\alpha}$ for
some $v,w$ (see [Her22]) and then, by local-global compatibility, it is clear
that $(D\times\delta)^{-1}(U)\longrightarrow U$ is a closed immersion. As
explained, $\mathcal{Z}$ accumulates to any point with classical weight of
$\mathcal{E^{\prime}}$, thus to $\mathcal{Z}^{\prime}$. If we denote by $h$
the composite of the map

$\mathcal{E}\longrightarrow\mathcal{T}\longrightarrow\mathcal{W},$

it coincides with a map

$\mathcal{E}\overset{w}{\longrightarrow}\mathcal{W}\overset{\phi}{\longrightarrow}\mathcal{W},$

where $\phi$ is the isomorphism of $\mathcal{W}$ given by the definition (10).
The properties of the map $h$ thus comes from the analogous one for $w$,
proven in [Her22] Theorem 9.5 (see also [Che04] which was the first to prove
those properties). ∎

######  Corollary 6.8.

If $\pi$ is a cuspidal, algebraic, regular automorphic form (of tame level
$K^{p}$), which appears in degree 0 coherent cohomology and which is finite
slope at $p$, then $\rho_{\pi}$ appear in
$\mathcal{F}_{K^{p}}(\overline{\rho}_{\pi})$. If moreover $\delta$ is an
accessible refinement for $\rho_{\pi}$, then
$(\rho_{\pi},\delta)\in\mathcal{E}_{K^{p}}(\overline{\rho}_{\pi})$. In
particular, $\mathcal{E}_{K^{p}}(\overline{\rho})$ is the closure in
$\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}\times\mathcal{T}$ of
homomorphically modular (for a cuspidal, algebraic, regular automorphic
representation of tame level $K^{p}$), finite slope at $p$, representations
$\rho$, together with an accessible refinement.

###### Proof.

As any finite slope automorphic form has an accessible refinement, it is
enough to prove the second part of the statement. This is a direct consequence
of the fact that $\mathcal{E}(\overline{\rho})$ coincides with the
Eigenvariety $\mathcal{E}^{\prime}(\overline{\rho})$ constructed in [Her22].
We take the notation of [Her22] section 6. Let $\pi$ be an automorphic
representation as in the statement and assume given $n$ and $f\in
H^{0}(\mathcal{X}_{1}(p^{n})^{tor},\omega^{\kappa})$ be a cuspidal holomorphic
modular form corresponding to $\pi$ which is a finite slope eigenvector at
$p$, where $\mathcal{X}_{1}(p^{n})^{tor}$ is a toroidal compactification of
the Shimura variety of level at $p$,

$ \Gamma_{1}(p^{n})=\\{ M\in G(\mathbb{Z}_{p})| M\in
U(\mathbb{Z}_{p})\pmod{p^{n}}\\},$

with $U\subset B\subset G_{\mathbb{Z}_{p}}$ are unipotent and Borel subgroup.
Then there is an open
$\mathcal{X}_{1}(p^{n})^{tor}(v)\subset\mathcal{X}_{1}(p^{n})^{tor}$, for $v$
small enough (a strict neighborhood of the ordinary locus), and for the
Shimura variety $\mathcal{X}_{0}(p^{n})^{tor}$ of level

$\Gamma_{0}(p^{n})=\\{ M\in G(\mathcal{\mathbb{Z}}_{p})| M\in
B(\mathbb{Z}_{p})\pmod{p^{n}}\\}.$

we have a corresponding open $\mathcal{X}_{0}(p^{n})^{tor}(v)$, such that over
$\mathcal{X}_{0}(p^{n})^{tor}(v)$ there exists the sheaf
$\omega^{\chi{\dagger}}_{w}(-D)$ of $w$-analytic cuspidal overconvergent
modular forms of (any $n$-analytic) $p$-adic weight $\chi$. By definition,
finite slope (at $p$) sections of
$H^{0}(\mathcal{X}_{0}(p^{n})^{tor}(v),\omega_{w}^{\chi{\dagger}}(-D))$ which
are eigenvectors for the Hecke operators do appear in the Eigenvariety
$\mathcal{E}_{K^{p}}$. By construction, there exists a rigid open
$\mathcal{IW}_{w}^{+}\subset\mathcal{T}^{\times}/U$ in the torsor of
trivialisations of $\omega$ above $\mathcal{X}_{1}(p^{n})^{tor}(v)$, so that
$H^{0}(\mathcal{X}_{0}(p^{n})^{tor}(v),\omega_{w}^{\chi{\dagger}}(-D))$ is the
set of sections
$H^{0}(\mathcal{IW}_{w}^{+},\mathcal{O}_{\mathcal{IW}_{w}^{+}}[\chi](-D))$
which are $\chi$-equivariant for the action of $T(\mathbb{Z}_{p})$202020A
warning : this action is twisted compared to the one on
$\mathcal{T}^{\times}/U$, see [Her22] see Proposition 6.16 and its proof. on
$\mathcal{IW}_{w}^{+}$ above $\mathcal{X}_{0}(p^{n})^{tor}(v)$ (see [Her22]
Definition 6.10). Let $\varepsilon$ be a finite order character of
$T(\mathbb{Z}_{p})$ trivial on $T(1+p^{n}\mathbb{Z})$, i.e. a character of
$\Gamma_{0}(p^{n})/\Gamma_{1}(p^{n})$. We claim that there is an injection,
for any algebraic $\kappa$

$
H^{0}(\mathcal{X}_{1}(p^{n})^{tor},\omega^{\kappa}(-D))(\varepsilon)\hookrightarrow
H^{0}(\mathcal{IW}_{w}^{+},\mathcal{O}_{\mathcal{IW}_{w}^{+}}[\varepsilon\kappa](-D)).$

But this comes from the identification (see [Her22, Definition 3.1])

$
H^{0}(\mathcal{X}_{1}(p^{n})^{tor},\omega^{\kappa})=H^{0}(\mathcal{T}^{\times}/U,\mathcal{O}[\kappa^{\vee}]),$

the fact that the restriction to the open $\mathcal{X}_{1}(p^{n})^{tor}(v)$ is
injective by analytic continuation, and that the restriction from
$\mathcal{T}^{\times}/U\times_{\mathcal{X}_{1}(p^{n})^{tor}}\mathcal{X}_{1}(p^{n})^{tor}(v)$
to $\mathcal{IW}_{w}^{+}$ is also injective again by analytic continuation.
Moreover all those maps are equivariant for the action of the Atkin-Lehner
algebra $\mathcal{A}(p)$. More precisely, the action of $\mathcal{A}(p)$, i.e.
of matrices of the form $\operatorname{Diag}(p^{a_{1}},\dots,p^{a_{n}})$, on
$\mathcal{E}$, together with the map
$T(\mathbb{Z}_{p})\longrightarrow\Gamma(\mathcal{W},\mathcal{O}_{\mathcal{W}})$,
thus giving a map
$T(\mathbb{Z}_{p})\longrightarrow\Gamma(\mathcal{E},\mathcal{O}_{\mathcal{E}})$,
can be packaged together as an map
$T(\mathbb{Q}_{p})\longrightarrow\Gamma(\mathcal{E},\mathcal{O}_{\mathcal{E}})$.
Then the previous injection is equivariant for this action of
$T(\mathbb{Q}_{p})$. But we can assume that $f$ is an eigenvector for
$\Gamma_{0}(p^{n})/\Gamma_{1}(p^{n})$, of some character $\varepsilon$ as
before. Thus the associated refinement at $p$ of $f$ gives a character
$\delta$ of $T(\mathbb{Q}_{p})$ depending on the refinement, such that
$\delta_{T(\mathbb{Z}_{p})}=\kappa\varepsilon$. Then the previous injection
assure that $f$ with this $\delta$ appears in $\mathcal{E}^{\prime}_{K^{p}}$.

For the last part of the statement, since any crystalline point at $p$ is
indeed finite slope, and finite slope holomorphic points are already in
$\mathcal{E}_{K^{p}}(\overline{\rho})$ as we just proved, we get the result. ∎

From now on, to lighten notations denote
$\mathcal{E}(\overline{\rho}):=\mathcal{E}_{K^{p}}(\overline{\rho})$,
$\mathcal{F}(\overline{\rho}):=\mathcal{F}_{K^{p}}(\overline{\rho})$
$\mathcal{Z}^{\prime}:=\mathcal{Z}^{\prime}_{K^{p}}$
accordingly212121Everything we will say is still dependant of this level
$K^{p}$. At the end of the article, in corollary 8.10, we show that there
exists an optimal level $K^{p}$, but we can’t choose it right away. Compare
[Che11] Lemma 2.4.

##  7 Automorphic forms, infinite fern and big image

Let $K$ be finite extension of $\mathbb{Q}_{p}$ and $\overline{K}$ an
algebraic closure of $K$. Denote $v_{p}$ the $p$-adic absolute value of
$\overline{K}$ such that $v_{p}(p)=1$. Let $K_{0}\subset K$ be the maximal
unramified extension of $\mathbb{Q}_{p}$ with Frobenius operator $\sigma$ and
set $f\coloneqq[K_{0}:\mathbb{Q}_{p}]$. Let $L$ be a finite extension of
$\mathbb{Q}_{p}$ such that $K\otimes_{\mathbb{Q}_{p}}L\simeq
L^{[K:\mathbb{Q}_{p}]}$. If $(\rho,V)$ is a crystalline representation of
$\operatorname{Gal}(\overline{K}/K)$, we denote
$(D_{\mathrm{cris}}(V),\varphi)$ its associated $\varphi$-module. It is a
finite dimensional free $K_{0}\otimes_{\mathbb{Q}_{p}}L$-module of rank
$n=\dim_{L}V$ with a $\sigma\otimes\operatorname{Id}_{L}$-linear automorphism
$\varphi$. Its de Rham module $D_{\mathrm{dR}}(V)$ is a filtered finite free
$K\otimes_{\mathbb{Q}_{p}}L$-module. The _Hodge-Tate type_ of $V$ is the
$[K:\mathbb{Q}_{p}]$-uple $(k_{1,\tau}\geq\cdots\geq
k_{n,\tau})_{\tau:K\hookrightarrow L}$ where the $k_{i,\tau}$ are the integers
$m$ such that $\operatorname{gr}^{-m}(D_{\mathrm{dR}}(V)\otimes_{K,\tau}L)\neq
0$ counted with multiplicity.

######  Definition 7.1.

We say that a crystalline Galois representation $(\rho,V)$ over $L$ is _Hodge-
Tate regular_ (or simply _HT-regular_) if, for all $\tau:K\hookrightarrow L$,
the integers $k_{i,\tau}$ are pairwise distinct. It is said to be _$\varphi$
-generic_ if the _linear_ endomorphism $\varphi^{f}$ of the finite free
$K_{0}\otimes_{\mathbb{Q}_{p}}L$-module $D_{\mathrm{cris}}(V)$ is split
semisimple regular ie has $\dim_{L}V$ pairwise distinct eigenvalues
$\varphi_{i}$ in $L$ such that $\varphi_{i}/\varphi_{j}\notin\\{1,p^{f}\\}$
(note that these eigenvalues are in $L$ since $\varphi^{f}$ commutes to the
semilinear action of $\operatorname{Gal}(K_{0}/\mathbb{Q}_{p})$ on
$D_{\mathrm{cris}}(V)$ given by $\varphi$).

######  Remark 7.2.

In [Che11, §3] Chenevier introduced the notion of weakly generic crystalline
$(\varphi,\Gamma)$-module, i.e. crystalline $(\varphi,\Gamma)$-modules for
which all refinements are non-critical, and weakly-generic crystalline
$(\varphi,\Gamma)$-module for which $n$ (the rank of the
$(\varphi,\Gamma)$-module) well-positioned refinements are non critical. It is
possible to deduce from the results of [Che11] that if the classical points of
$\mathcal{E}$ or $\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$ which are
weakly-generic at $p$ are Zariski-dense, then modular points are dense in
$\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$ (or the closure has at least
the expected dimension). He moreover proves that if $n=3$, every
$\varphi$-generic, HT regular absolutely irreducible $(\varphi,\Gamma)$ is
automatically weakly generic. Unfortunately it does not seem to be true
anymore even for $n=4$ that absolutely irreducible points are weakly generic,
as shown in the following example, and thus it does not seem any easier to
prove that weakly generic points are dense in
$\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$ when $n\geq 4$ than proving the
analogous result for generic points.

######  Example 7.3.

Let $(V,\varphi,\operatorname{Fil}^{\bullet})$ be the filtered
$\varphi$-module of an irreducible, $\varphi$-generic, HT regular, crystalline
4 dimensional representation of $G_{\mathbb{Q}_{p}}$ with $L$-coefficients.
Choosing $L$ big enough, we can assume that there exists a basis
$(f_{1},f_{2},f_{3},f_{4})$ of $V$ such that
$\varphi(f_{i})=\varphi_{i}f_{i}$. Refinements of $V$ are thus given by a
permutation $\sigma$ of this basis. By irreducibility and weak-admissibility,
we check that it is impossible for $\operatorname{Fil}^{k}V\neq\\{0\\},V$ to
be $\varphi$-stable. For example, suppose that the HT weights are
${-k_{4}<-k_{3}<-k_{2}<-k_{1}}$ (i.e. the jumps of the filtration on $V$ are
$k_{1}<k_{2}<k_{3}<k_{4}$) with

  * •

$\operatorname{Fil}^{k_{1}}V=<f_{2},f_{3},f_{1}+f_{4}>=<f_{2},f_{2}+f_{3},f_{1}+f_{4}>$

  * •

$\operatorname{Fil}^{k_{2}}V=<f_{1}+f_{4},f_{2}+f_{3}>$

  * •

$\operatorname{Fil}^{k_{3}}V=<f_{1}+f_{4}>$

  * •

$\operatorname{Fil}^{k_{4}}V=\\{0\\}.$

We can check that the non critical refinements are given by
$\sigma=\operatorname{id},(23),(14),(14)(23)$, and they don’t form a weakly-
nested sequence. Moreover, for generic choices of $k_{i}$ and
$v(\varphi_{i})$, the associated $p$-adic representation $V(D)$ will be
irreducible. Indeed, if
$v(\varphi_{1})=v(\varphi_{2})=v(\varphi_{3})=16,v(\varphi_{4})=12,k_{1}=0,k_{2}=10,k_{3}=20,k_{4}=30$,
then we can check that no non-trivial $\varphi$-stable submodule of $D$ is
weakly-admissible. In particular if we do not already know that generic points
are Zariski dense, it is not likely to prove that weakly generic ones are.

Moreover we can check that locally the image of the tangeant space of those
refined points doesn’t cover all the tangent space for the corresponding point
in the local deformation ring (i.e. at this point the analogous of proposition
3.7 only for non-critical refinements isn’t true). In the following we will
use a replacement of those generic points by the so-called _almost generic_
ones, which are Zariski dense in the fern and for which we can apply
proposition 3.7 and thus Chenevier’s stategy. Remark that some irreducible
weakly-admissible filtered $\varphi$-modules of dimension 4 which admits a
critical refinement are weakly-generic, but we don’t know how to discriminate
these from the previous example on the deformation rings (or on the infinite
fern).

Recall that $k$ is a finite field of characteristic $p$ and that
$\overline{\rho}:G_{E}\rightarrow\operatorname{GL}_{n}(k)$ is a continuous
semisimple representation which is conveniently modular of tame level $K^{p}$.

Denote by $\overline{\mathcal{F}(\overline{\rho})}$ the Zariski-closure of the
image
$\mathcal{E}(\overline{\rho})\rightarrow\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$,
i.e. the Zariski closure of the infinite fern $\mathcal{F}(\overline{\rho})$.

######  Definition 7.4.

We say that a Galois representation
$\rho:G_{E}\longrightarrow\operatorname{GL}_{n}(\overline{\mathbb{Q}_{p}})$
has enormous image, if $\rho(G_{E(\zeta_{p^{\infty}})})$ is enormous, in the
sense of [NT19] Definition 2.27. We say that a point $x$ of
$\mathcal{E}(\overline{\rho})$ (resp. of
$\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$) is almost generic if it is in
$\mathcal{Z}^{\prime}=\mathcal{Z}^{\prime}_{K^{p}}$ (resp. in the image of
$\mathcal{Z}^{\prime}$ in $\mathcal{X}_{\overline{\rho}}^{\mathrm{pol}}$), the
associated Galois representation $\rho$ has enormous image, and if
$\rho_{|G_{E_{v}}}$ is crystalline, $\varphi$-generic, HT-regular, posses an
irreducible refinement and is absolutely irreducible for all $v|p$.

Let $v$ be a place of $E$ dividing $p$. Let $(\rho,V)$ be a continuous finite
dimensional representation of $G_{E_{v}}$ over $L$. It follows from the
compactness of $G_{E_{v}}$ that $\rho(G_{E_{v}})$ is a closed subgroup of
$\operatorname{GL}_{L}(V)$. The $p$-adic analogue of Cartan’s Theorem shows
that $\rho(G_{E_{v}})$ is a $p$-adic Lie subgroup of
$\operatorname{GL}_{L}(V)$ so that we can define
$\mathfrak{g}_{\rho}\coloneqq\operatorname{Lie}(\rho(G_{E_{v}}))$ (see [Ser89,
Rem. I.1.1]) and $\mathfrak{g}_{\rho,L}$ the $L$-span of $\mathfrak{g}_{\rho}$
in $\operatorname{End}_{L}(V)$. Our goal is to prove that almost generic
points are Zariski-dense in the infinite fern $\mathcal{F}(\overline{\rho})$.
Such a result was proven by Taïbi ([Tai16]) in a slightly different (and more
difficult) context (improving results of [BC09]). As the case we consider is
easier, and for convenience of the reader, we repeat the argument in our
context.

Denote by $K_{n}/E_{v}$ the compositum of extensions of degree dividing $n$,
this is a finite Galois extension.

######  Proposition 7.5.

Let $(\rho,V)$ be some continuous $n$-dimensional representation of
$G_{E_{v}}$ over $L$ and assume that $(\rho|_{G_{K_{n}}},V)$ is absolutely
irreducible. Then $V$ is a simple $\mathfrak{g}_{\rho,L}$-module,
$\mathfrak{g}_{\rho,L}$ is a reductive Lie algebra and
$\mathfrak{h}_{\rho,L}$, the semisimple part of $\mathfrak{g}_{\rho,L}$, is
isomorphic to a sub Lie algebra of $\mathfrak{sl}(V)$ of semisimple rank at
most $\dim_{L}V-1$.

###### Proof.

Let $K$ be a finite extension of $E_{v}$, we claim that $\rho|_{G_{K}}$ is
absolutely irreducible. We can assume that $K/E_{v}$ is Galois. Suppose that
$\rho|_{G_{K}}$ is not absolutely irreducible, then as

$\rho:G_{E_{v}}\longrightarrow\operatorname{GL}_{L}(V)\simeq\operatorname{GL}_{n}(L)$

is absolutely irreducible and $G_{K}$ is normal in $G_{E_{v}}$, we have, up to
enlarge $L$, a decomposition into absolutely irreducible
$G_{K}$-representations

$\rho|_{G_{K}}=\bigoplus_{k=1}^{r}W_{k},$

and $G_{E_{v}}$ permutes these representations, in particular they have the
same dimension. Let $H\subset G_{E_{v}}$ be the stabiliser of $W_{1}$, then
$G_{E_{v}}/H$ acts transitively on the $W_{k}$ and thus
$[G_{E_{v}}:H]\dim(W_{1})=n$. In particular $H=G_{K^{\prime}}$ with
$K^{\prime}$ a finite extension of $E_{v}$ of degree dividing $n$, thus
$K^{\prime}\subset K_{n}$, but $\rho|_{G_{K_{n}}}$ is irreducible, thus as is
$\rho|_{G_{K^{\prime}}}$ and thus $r=1$ i.e. $\rho|_{G_{K}}$ is irreducible.

For each open subgroup $H\subset G_{E_{v}}$, the representation
$(\rho|_{H},V)$ is irreducible, so that $V$ is a simple
$\mathfrak{g}_{\rho,L}$-module. Thus $\mathfrak{g}_{\rho,L}$ has a simple
faithful module, which implies that $\mathfrak{g}_{\rho,L}$ is a reductive Lie
algebra. Let
$\mathfrak{g}_{\rho,L}=\mathfrak{a}_{\rho,L}\oplus\mathfrak{h}_{\rho,L}$ be
the decomposition of $\mathfrak{g}_{\rho,L}$ as direct sum of an abelian and
of a semisimple Lie algebras. As $V$ is an absolutely simple
$\mathfrak{g}_{\rho,L}$-module, the Lie algebra $\mathfrak{a}_{\rho,L}$ acts
on $V$ by scalars and $V$ is an absolutely simple
$\mathfrak{h}_{\rho,L}$-module. Then $\mathfrak{a}_{\rho,L}\subset
L\operatorname{Id}_{V}$ and, as $\mathfrak{h}_{\rho,L}$ is semisimple,
$\mathfrak{h}_{\rho,L}\subset\mathfrak{g}_{\rho,L}\cap\mathfrak{sl}(V)$. As a
consequence $\mathfrak{h}_{\rho,L}=\mathfrak{g}_{\rho,L}\cap\mathfrak{sl}(V)$
is a semisimple Lie algebra and $V$ is an absolutely simple
$\mathfrak{h}_{\rho,L}$-module. As $\mathfrak{sl}(V)$ has rank $\dim_{L}V-1$,
the rank of $\mathfrak{h}_{\rho,L}$ is at most $\dim_{L}V-1$. ∎

######  Proposition 7.6.

There exist finitely many nonzero $\mathbb{Q}$-linear forms
$\Lambda_{1},\dots,\Lambda_{r}$ on $\mathbb{Q}^{n}$ such that the following is
true: let $(\rho,V)$ be a crystalline $n$-dimensional representation of
$G_{F_{v}}$ over $L$, with Hodge-Tate weights $(k_{1,\sigma}\leq\dots\leq
k_{n,\sigma})_{\sigma}$ such that $(\rho|_{G_{K_{n}}},V)$ is absolutely
irreducible and such that there exists at least one
$\sigma:F_{v}\hookrightarrow L$ such that for all $1\leq i\leq r$,
$\Lambda_{i}(k_{1,\sigma},k_{2,\sigma},\dots,k_{n,\sigma})\neq 0$, then
$\rho(G_{F_{v}})$ contains an open subgroup of $\operatorname{SL}(V)$.

###### Proof.

Let $C$ be some algebraically closed field of characteristic $0$. The
classification of semisimple Lie algebras and their representations shows that
all semisimple Lie algebras and their finite dimensional simple modules are
defined over $\mathbb{Q}$, that there are finitely many isomorphism classes of
semisimple Lie algebras of bounded rank and that each of them has finitely
many semisimple modules of bounded rank. Consequently, for a fixed $n\geq 2$,
there exist a finite number of pairs $(\mathfrak{h}_{i},\theta_{i})$ where
$\mathfrak{h}_{i}$ is a semisimple Lie algebra and $\theta_{i}$ an embedding
of $\mathfrak{h}_{i}$ in $\mathfrak{sl}_{n,\mathbb{Q}}$ such that for each
semisimple Lie subalgebra $\mathfrak{h}\subset\mathfrak{sl}_{n,C}$, there
exists $i$ such that $\mathfrak{h}\simeq\mathfrak{h}_{i}\otimes_{\mathbb{Q}}C$
and the inclusion is $\operatorname{GL}_{n}(C)$-conjugated to
$\theta_{i}\otimes\operatorname{Id}_{C}$. As a consequence a Cartan subalgebra
of $\mathfrak{h}$ is conjugated to one of finitely many $\mathbb{Q}$-linear
subspaces of the space of diagonal matrices in $\mathfrak{sl}_{n,C}$. Moreover
it follows from [Bou, VIII.§3 Prop.2.(ii)] and from Borel-de Siebenthal
Theorem ([Kan01, Thm. 12.1]) that a semisimple Lie subalgebra of
$\mathfrak{sl}_{n,C}$ containing $\mathfrak{h}$ is equal to
$\mathfrak{sl}_{n,C}$ or of rank strictly less than $n-1$. Thus there exist
finitely many nonzero $\mathbb{Q}$-linear forms
$\Lambda_{1}^{\prime},\dots,\Lambda_{s}^{\prime}$ on $\mathbb{Q}^{n}$ such
that if $\mathfrak{h}$ a semisimple subalgebra of $\mathfrak{sl}_{n,C}$ of
rank strictly less than $n-1$ and $x\in\mathfrak{h}$ is a semisimple element
of eigenvalues $\lambda_{1},\dots,\lambda_{n}$ (counted with multiplicities),
then there exists $1\leq i\leq s$ and $w\in\mathfrak{S}_{n}$ such that
$w(\Lambda_{i}^{\prime})(\lambda_{1},\dots,\lambda_{n})\coloneqq\Lambda_{i}^{\prime}(\lambda_{w(1)},\dots,\lambda_{w(n)})=0$.
We set

$\\{\Lambda_{1},\dots,\Lambda_{r}\\}=\\{w(\Lambda_{i}^{\prime})\mid 1\leq
i\leq s,\,w\in\mathfrak{S}_{n}\\}.$

Let
$\Theta\in\operatorname{End}_{\mathbb{C}_{p}\otimes_{\mathbb{Q}_{p}}L}(\mathbb{C}_{p}\otimes_{\mathbb{Q}_{p}}V)$
the Sen operator of $V$. As $(\rho,V)$ is Hodge-Tate, it follows from [Sen73,
Thm. 1] that $\Theta$ belongs to
$\mathbb{C}_{p}\otimes_{\mathbb{Q}_{p}}\mathfrak{g}_{\rho,L}\subset\mathbb{C}_{p}\otimes_{\mathbb{Q}_{p}}\operatorname{End}_{\mathbb{Q}_{p}}V$.

Suppose $L$ is big enough so that $F_{v}\otimes_{\mathbb{Q}_{p}}L\simeq
L^{[F_{v}:\mathbb{Q}_{p}]}$. Then

$\mathbb{C}_{p}\otimes_{\mathbb{Q}_{p}}L=\prod_{\sigma:F_{v}\hookrightarrow
L}\mathbb{C}_{p}\otimes_{F_{v},\sigma}L,$

decomposes over all embeddings of $F_{v}$ and let $\Theta_{\sigma}$ be the
$\mathbb{C}_{p}\otimes_{F_{v},\sigma}L$-linear endomorphism of
$\mathbb{C}_{p}\otimes_{F_{v},\sigma}V$ induced by $\Theta$. The eigenvalues
of $\Theta_{\sigma}$ are the $\sigma$-Hodge-Tate weights
$(k_{1,\sigma}\leq\cdots\leq k_{n,\sigma})$ of $(\rho,V)$ (counted by
multiplicities).

Assume that $\Lambda_{i}(k_{1,\sigma},\dots,k_{n,\sigma})\neq 0$ for all
$1\leq i\leq r$. Then by what preceeds, the element $\Theta_{\sigma}$ can’t be
contained in a strict semisimple Lie subalgebra of
$\mathbb{C}_{p}\otimes_{F_{v},\sigma}\mathfrak{sl}(V)$ so that
$\mathbb{C}_{p}\otimes_{F_{v},\sigma}\mathfrak{h}_{\rho,L}=\mathbb{C}_{p}\otimes_{F_{v},\sigma}\mathfrak{sl}(V)$.
For dimension reasons, we have $\mathfrak{h}_{\rho,L}=\mathfrak{sl}(V)$. We
conclude that $\rho(G_{F_{v}})$ contains an open subgroup of
$\operatorname{SL}(V)$. ∎

######  Proposition 7.7.

The set of points $x\in\overline{\mathcal{F}(\overline{\rho})}$ such that
$\operatorname{Tr}\rho_{x}|_{G_{K_{n}}}$ is absolutely irreducible is a
Zariski-dense and Zariski-open subset of
$\overline{\mathcal{F}(\overline{\rho})}$.

###### Proof.

The fact that the absolutely irreducible locus is Zariski-open is a
consequence of [Che14, §4.2]. In order to prove that it is Zariski-dense, it
is then sufficient to prove that each Zariski-open subset $U$ of
$\mathcal{F}(\overline{\rho})$ contains a point $x$ such that
$\operatorname{Tr}\rho_{x}|_{G_{K_{n}}}$ is absolutely irreducible.

Now we follow the strategy of [BC11] and [Tai16]. Let us fix some notation. If
$x=(\rho_{x},\delta_{x})\in\mathcal{Z}^{\prime}\subset\mathcal{E}(\overline{\rho})$
and $\sigma:K_{n}\hookrightarrow\overline{\mathbb{Q}_{p}}$, we let
$(k_{\sigma,1}(x)\leq\cdots\leq k_{\sigma,n}(x))$ the Hodge-Tate weights at
$\sigma$ of $\rho_{x}|_{G_{K_{n}}}$ and $(\phi_{1}(x),\cdots,\phi_{n}(x))\in
k(x)^{n}$ the ordered eigenvalues of the linearized Frobenius of
$D_{\mathrm{cris}}(\rho_{x}|_{G_{K_{n}}})$ corresponding to the refinement of
$\rho_{x}|_{G_{F_{v}}}$ defined by $\delta_{x,v}$. We also set
$k_{i}(x)=\sum_{\sigma}k_{\sigma,i}(x)$. Let $e$ be the ramification index of
$K_{n}/\mathbb{Q}_{p}$. The functions $x\mapsto
v_{p}(\phi_{i}(x))+e^{-1}k_{i}(x)$ are therefore locally constant on
$\mathcal{Z}^{\prime}$. Now fix $U$ a Zariski open non empty subset of
$\overline{\mathcal{F}(\overline{\rho})}$. We have
$U\cap\mathcal{F}(\overline{\rho})\neq\emptyset$ so that the inverse image $V$
of $U$ in $\mathcal{E}(\overline{\rho})$ is a non empty Zariski-open subset.
Let $x\in V\cap\mathcal{Z}^{\prime}$. Let $c\geq\max_{i}\lvert
v_{p}(\phi_{i}(x))+e^{-1}k_{i}(x)\rvert$ and larger than $n^{2}+1$ and let
$\mathcal{Z}_{c}^{\prime\prime}$ be the subset of point
$z\in\mathcal{Z}^{\prime}$ such that
$k_{i+1}(z)-k_{i}(z)>cen(k_{i}(z)-k_{i-1}(z))$ $2\leq i\leq n-1$, $2\leq i\leq
n-1$, and $k_{2}(z)-k_{1}(z)>3cen$. If $z\in\mathcal{Z}_{c}^{\prime\prime}$,
then $\lvert\sum_{i\in I}k_{i}(z)-\sum_{i\in J}k_{i}(z)\rvert>cen$ for all
distinct non empty proper subsets $I$ and $J$ of the same cardinal in
$\\{1,\dots,n\\}$ (by the same proof than [BC09, Lem. 4.5.5]). Then
$\mathcal{Z}^{\prime\prime}_{c}$ is Zariski dense and accumulates at
$\mathcal{Z}^{\prime}$ in $\mathcal{E}(\overline{\rho})$. Therefore there
exists a point $y\in\mathcal{Z}^{\prime\prime}_{c}\cap V$ such that moreover
$\max_{i}\lvert v_{p}(\phi_{i}(y))+e^{-1}k_{i}(y)\rvert\leq c$. Then, for
$i\neq j$,

$\displaystyle\lvert
v_{p}(\phi_{i}(y))-v_{p}(\phi_{j}(y))\rvert\geq\frac{1}{e}\lvert
k_{i}(y)-k_{j}(y)\rvert-\lvert
v_{p}(\phi_{i}(y)+\frac{k_{i}(y)}{e}\rvert-\lvert
v_{p}(\phi_{j}(y))+\frac{k_{j}(y)}{e}\rvert$ $\displaystyle>3cn-c-c>1.$

In particular, $\phi_{i}(y)/\phi_{j}(y)\neq p$, so if $\Pi$ is an automorphic
representation corresponding to $y$, we have that $\Pi_{v}$ is an irreducible
principal series. In particular, all its refinements are accessible. Now we
choose a transitive permutation $\sigma$ of $\\{1,\dots,n\\}$ and, since all
the refinements of $\Pi_{v}$ are accessible, there exists
$z_{0}\in\mathcal{Z}^{\prime}$ such that $\rho_{y}\simeq\rho_{z_{0}}$ and
$(\phi_{1}(z_{0}),\dots,\phi_{n}(z_{0}))=(\phi_{\sigma(1)}(y),\dots,\phi_{\sigma(n)}(y))$.
As in the proof of [BC11, Lem. 3.3], we deduce that $\sum_{i\in
I}v_{p}(\phi_{i}(z_{0}))+e^{-1}k_{i}(z_{0}))\neq 0$ for any non empty proper
subset $I$ of $\\{1,\dots,n\\}$. Let $C=\max_{i}\left\lvert
v_{p}(\phi_{i}(z_{0}))+e^{-1}k_{i}(z_{0})\right\rvert$. Choose $V^{\prime}$ a
neighborhood of $z_{0}$ in $\mathcal{E}(\overline{\rho})$. Let
$\mathcal{Z}^{\prime}_{C}$ be the subset of point $z\in\mathcal{Z}^{\prime}$
such that $k_{\sigma,j+1}(z)-k_{\sigma,j}(z)>C$ for any
$\sigma\in\operatorname{Hom}(K_{n},\overline{\mathbb{Q}_{p}})$ and $1\leq
i\leq n-1$. The set $\mathcal{Z}^{\prime}_{C}$ is Zariski-dense and
accumulates at $\mathcal{Z}^{\prime}$ in $\mathcal{E}(\overline{\rho})$ so
that there exists a point $z\in V^{\prime}\cap\mathcal{Z}^{\prime}_{C}$ such
that $\sum_{i\in I}(v_{p}(\phi_{i}(z))+e^{-1}k_{i}(z))\neq 0$ for any non
empty proper subset $I$ of $\\{1,\dots,n\\}$. For any of those $z$, by [Tai16,
Lem. 3.2.3], if $D^{\prime}\subset D_{\mathrm{cris}}(\rho_{z}|_{G_{K_{n}}})$
is a nonzero proper weakly admissible sub-$\varphi$-module, then there exists
a non empty proper subset $I\subset\\{1,\dots,n\\}$ such that

$\sum_{i\in I}v_{p}(\phi_{i}(z))+e^{-1}\sum_{i\in I}k_{i}(z)=0$

which is not possible. Therefore $\rho_{z}|_{G_{K_{n}}}$ is absolutely
irreducible. ∎

######  Proposition 7.8.

Let $x\in\mathcal{E}$ be a point which is crystalline at $v|p$. Then there
exists $C>0$ such that for any $z\in\mathcal{Z}^{\prime}_{C}$ sufficiently
close to $x$ the local representation $\rho_{z,v}$ admits a non-critical
refinement (see Definition 3.6).

###### Proof.

Let $x$ be a point as in the statement. As the maps $y\mapsto
v_{p}(\phi_{i}(y))+e^{-1}k_{i}(y)$ are locally constant on
$\mathcal{Z}^{\prime}$, there exists an open subset $U\subset\mathcal{E}$
containing $x$ such that these are constant on $U\cap\mathcal{Z}^{\prime}$.
Let

$C=e\max(1,\max\\{v_{p}(\phi_{1}(x))+e^{-1}k_{1}(x)+\dots+v_{p}(\phi_{i}(x))+e^{-1}k_{i}(x)|{i=2,\dots,n}\\})$

and $\mathcal{Z}^{\prime}_{C}$ the set of points $z\in\mathcal{Z}^{\prime}$
such that $\lvert k_{i,\tau}-k_{i+1,\tau}\rvert>C$ for all $1\leq i\leq n-1$
and $\tau\in\Phi$. We claim that any $z\in\mathcal{Z}^{\prime}_{C}\cap U$, $z$
has a non-critical refinement at $v$ (in fact even the refinement given by
$\mathcal{E}$ at $z$ is non-critical)222222Those $z$ are _numerically non-
critical_ as in [BC09, Remark 2.4.6, (ii)]. Let
$z\in\mathcal{Z}_{C}^{\prime}\cap U$ and let $F_{\bullet}\subset
D(\rho_{z,v})$ be the refinement defined by $z$. We note $F_{i}$ its
$i$-dimensional part. The (linearized) Frobenius eigenvalues on $F_{i}$ are
$\phi_{1}(z),\dots,\phi_{i}(z)$. Assume that the refinement $F_{\bullet}$ is
critial. This means that the there exists some $1\leq i\leq n-1$ and some
$\sigma:F_{v}\hookrightarrow\overline{\mathbb{Q}_{p}}$ and such that the gaps
of the Hodge-Tate filtration on
$\overline{\mathbb{Q}_{p}}\otimes_{F_{v,0},\sigma}D_{\mathrm{cris}}(\rho_{z,v})$
are _not_ $-k_{1,\sigma}(z),\dots,-k_{i,\sigma}(z)$. Choose $i$ minimal for
this property. This implies that the sum of these gaps is greater or equal
than $-(k_{1,\sigma}(z)+\dots+k_{i-1,\sigma}(z)+k_{i+1,\sigma}(z))$. Moreover
if $\tau\neq\sigma$, the sum of the gaps of the Hodge-Tate filtration on
$\overline{\mathbb{Q}_{p}}\otimes_{F_{v,0},\sigma}D_{\mathrm{cris}}(\rho_{z,v})$
is greater or equal than
$-(k_{1,\tau}(z)+\dots+k_{i-1,\tau}(z)+k_{i,\tau}(z))$. As $F_{i}$ is
$\varphi$-stable, the weak admissibility of $D_{\mathrm{cris}}(\rho_{z,v})$
implies that

$v_{p}(\phi_{1}(z))+\dots+v_{p}(\phi_{i}(z))\geq-\frac{1}{e}\left(\sum_{\tau\neq\sigma}\left(\sum_{j\leq
i}k_{j,\tau}(z)\right)+\sum_{j=1}^{i-1}k_{j,\sigma}(z)+k_{\sigma,i+1}(z)\right),$

thus

$\sum_{j=1}^{i}\left(v_{p}(\phi_{j}(z))+\frac{1}{e}k_{j}(z)\right)\geq\frac{1}{e}(k_{i,\sigma}(z)-k_{i+1,\sigma}(z))>C,$

contradicting $z\in\mathcal{Z}_{C}^{\prime}$. Therefore $F_{\bullet}$ is non-
critical. ∎

Let $\mathcal{X}^{\mathrm{mod},\mathrm{ag}}$ be the set of almost generic
points in $\mathcal{F}(\overline{\rho})$.

######  Theorem 7.9.

The set of points $x$ in $\overline{\mathcal{F}(\overline{\rho})}$ which are
in the image of the set $\mathcal{Z}^{\prime}$ such that $\rho_{x,v}$ is
crystalline $\varphi$-generic, HT-regular, admits an non-critical refinement,
and such that $\rho_{x}(G_{F_{v}})$ contains an open subgroup of
$\operatorname{SL}(V_{x})$ is a Zariski dense accumulation subset. Moreover