# The geometric Breuil–Mézard conjecture for two-dimensional potentially
Barsotti–Tate Galois representations

Ana Caraiani<EMAIL_ADDRESS>Department of Mathematics, Imperial
College London, London SW7 2AZ, UK  ,  Matthew Emerton
<EMAIL_ADDRESS>Department of Mathematics, University of Chicago,
5734 S. University Ave., Chicago, IL 60637, USA  ,  Toby Gee
<EMAIL_ADDRESS>Department of Mathematics, Imperial College London,
London SW7 2AZ, UK  and  David Savitt<EMAIL_ADDRESS>Department of
Mathematics, Johns Hopkins University

###### Abstract.

We establish a geometrisation of the Breuil–Mézard conjecture for potentially
Barsotti–Tate representations, as well as of the weight part of Serre’s
conjecture, for moduli stacks of two-dimensional mod $p$ representations of
the absolute Galois group of a $p$-adic local field. These results are first
proved for the stacks of [CEGS20b, CEGS20c], and then transferred to the
stacks of [EG22] by means of a comparison of versal rings.

The first author was supported in part by NSF grant DMS-1501064, by a Royal
Society University Research Fellowship, and by ERC Starting Grant 804176. The
second author was supported in part by the NSF grants DMS-1303450,
DMS-1601871, DMS-1902307, DMS-1952705, and DMS-2201242. The third author was
supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie
Career Integration Grant 303605, ERC Starting Grant 306326, and a Royal
Society Wolfson Research Merit Award. The fourth author was supported in part
by NSF CAREER grant DMS-1564367 and NSF grant DMS-1702161.

###### Contents

  1. 1 Introduction
  2. 2 Notation and conventions
  3. 3 Moduli stacks of Breuil–Kisin modules and étale $\varphi$-modules
  4. 4 Generic reducedness of $\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$
  5. 5 A case of the classical geometric Breuil–Mézard Conjecture
  6. 6 The geometric Breuil–Mézard conjecture for the stacks ${\mathcal{Z}}^{\mathrm{dd},1}$
  7. 7 The geometric Breuil–Mézard conjecture for the stacks ${\mathcal{X}}_{2,\operatorname{red}}$
  8. A A lemma on formal algebraic stacks

##  1\. Introduction

Let $K/{\mathbf{Q}}_{p}$ be a finite extension with residue field $k$, let
$\overline{K}$ be an algebraic closure of $K$, and let $d\geq 1$ be a positive
integer. Two of us (M.E. and T.G. [EG22]) have constructed moduli stacks of
representations of the absolute Galois group
$G_{K}:=\operatorname{Gal}(\overline{K}/K)$, globalizing Mazur’s classical
deformation theory of Galois representations. These stacks are expected to be
the backbone of a categorical $p$-adic Langlands correspondence, playing the
role anticipated by the stacks of [DHMK20, Zhu20] in the $\ell\neq p$ setting.

To be precise, the book [EG22] defines the category $\mathcal{X}_{d}$ fibred
in groupoids over $\operatorname{Spf}{\mathbf{Z}}_{p}$ whose $A$-valued
points, for any $p$-adically complete ${\mathbf{Z}}_{p}$-algebra $A$, are the
groupoid of rank $d$ projective étale $(\varphi,\Gamma)$-modules with
$A$-coefficients. Then the finite type points of $\mathcal{X}_{d}$ correspond
to representations
$\overline{r}:G_{K}\to\operatorname{GL}_{d}(\overline{{\mathbf{F}}}_{p})$, the
versal rings of $\mathcal{X}_{d}$ at finite type points recover classical
Galois deformation rings, and one has the following, which is one of the main
results of [EG22].

######  Theorem 1.1.

[EG22, Thm. 1.2.1] Each $\mathcal{X}_{d}$ is a Noetherian formal algebraic
stack. Its underlying reduced substack $\mathcal{X}_{d,\operatorname{red}}$ is
an algebraic stack of finite type over ${\mathbf{F}}_{p}$, and is
equidimensional of dimension $[K:{\mathbf{Q}}_{p}]\binom{d}{2}$. The
irreducible components of $\mathcal{X}_{d,\operatorname{red}}$ have a natural
bijective labeling by Serre weights.

Recall that a _Serre weight_ in this context is an irreducible
$\overline{{\mathbf{F}}}_{p}$-representation of $\operatorname{GL}_{d}(k)$ (or
rather an isomorphism class thereof). The description in [EG22] of the
labeling of components of $\mathcal{X}_{d,\operatorname{red}}$ by Serre
weights is to some extent combinatorial. Namely, it is shown that each
irreducible component of $\mathcal{X}_{d,\operatorname{red}}$ has a dense set
of $\overline{{\mathbf{F}}}_{p}$-points which are successive extensions of
characters, with extensions as non-split as possible. The restrictions of
these characters to the inertia group yield discrete data (their tame inertia
weights) which, together with some further information about peu and très
ramifiee extensions, amounts precisely to the data of (a highest weight of) a
Serre weight.

It is expected, however, that there is another description of the irreducible
components of $\mathcal{X}_{d,\operatorname{red}}$ that is more precise and
more informative from the perspective of the $p$-adic Langlands program. The
weight part of Serre’s conjecture, as described for instance in [GHS18, Sec.
3], associates to each
$\overline{r}:G_{K}\to\operatorname{GL}_{d}(\overline{{\mathbf{F}}}_{p})$ a
set of Serre weights $W(\overline{r})$. One expects for each Serre weight
$\sigma$ that there is a set of components of
$\mathcal{X}_{d,\operatorname{red}}$, including the irreducible component
labelled by $\sigma$ in [EG22], the union of whose
$\overline{{\mathbf{F}}}_{p}$-points are _precisely_ the representations
$\overline{r}$ with $\sigma\in W(\overline{r})$. Equivalently, after adding
additional labels to some of the components (so that components will be
labelled by a set of weights, rather than a single weight), the set
$W(\overline{r})$ is precisely the collection of labels of the various
components of $\mathcal{X}_{d,\operatorname{red}}$ on which $\overline{r}$
lies.

One of the aims of this paper is to establish this expectation in the case
$d=2$, taking as input the weight part of Serre’s conjecture for
$\operatorname{GL}_{2}$ [GLS15] and the Breuil–Mézard conjecture for two-
dimensional potentially Barsotti–Tate representations [GK14], and thus obtain
a description of _all_ the finite type points of each irreducible component of
$\mathcal{X}_{2,\operatorname{red}}$, as opposed to just a dense set of
points. Indeed we have the following theorem, which can be regarded as a
geometrisation of the weight part of Serre’s conjecture for
$\operatorname{GL}_{2}$. If $\sigma$ is a Serre weight, let
$\mathcal{X}^{\sigma}_{d,\operatorname{red}}$ denote the irreducible component
of $\mathcal{X}_{d,\operatorname{red}}$ labelled by $\sigma$ in [EG22]. (We
refer the reader to Section 2 for any unfamiliar notation or terminology in
what follows.)

######  Theorem 1.2.

Suppose $p>2$. We define:

  * •

${\mathcal{Z}}^{\sigma}=\mathcal{X}_{2,\operatorname{red}}^{\sigma}$, if the
weight $\sigma$ is not Steinberg, while

  * •

${\mathcal{Z}}^{\chi\otimes\mathrm{St}}=\mathcal{X}_{2,\operatorname{red}}^{\chi}\cup\mathcal{X}_{2,\operatorname{red}}^{\chi\otimes\mathrm{St}}$
if the weight $\sigma\cong\chi\otimes\mathrm{St}$ is Steinberg.

Then $\sigma\in W(\overline{r})$ if and only if $\overline{r}$ lies on
${\mathcal{Z}}^{\sigma}$.

Indeed a stronger statement is true: the cycles
$Z^{\sigma}=\mathcal{X}_{2,\operatorname{red}}^{\sigma}$ (for $\sigma$ non-
Steinberg) and
$Z^{\chi\otimes\mathrm{St}}=\mathcal{X}_{2,\operatorname{red}}^{\chi}+\mathcal{X}_{2,\operatorname{red}}^{\chi\otimes\mathrm{St}}$
constitute the cycles in a geometric version of the Breuil–Mézard conjecture
(to be explained below).

We emphasise that the existence of such a geometric interpretation of the sets
$W(\overline{r})$ is far from obvious, and indeed we know of no direct proof
using any of the explicit descriptions of $W(\overline{r})$ in the literature;
it seems hard to understand in any explicit way which Galois representations
arise as the limits of a family of extensions of given characters, and the
description of the sets $W(\overline{r})$ is very complicated (for example,
the description in [BDJ10] relies on certain $\operatorname{Ext}$ groups of
crystalline characters). Our proof is indirect, and ultimately makes use of a
description of $W(\overline{r})$ given in [GK14], which is in terms of
potentially Barsotti–Tate deformation rings of $\overline{r}$ and is motivated
by the Taylor–Wiles method. We interpret this description in the geometric
language of [EG14], which we in turn interpret as the formal completion of a
“geometric Breuil–Mézard conjecture” for our stacks.

The proof of Theorem 1.2 entwines the main results of the book [EG22] with the
results of our papers [CEGS20b, CEGS20c]. Indeed Theorem 1.2 is (more or less)
stated at [EG22, Thm. 8.6.2], but the argument given there makes reference to
(an earlier version of) this paper111The reference [CEGS19, Thm. 5.2.2] in
[EG22] is Theorem 6.2 of this paper, while the reference [CEGS19, Lem. B.5] in
[EG22] is [CEGS20b, Lem. A.4].. We should therefore explain more precisely
what are the contributions of this paper.

For each Hodge type $\lambda$ and inertial type $\tau$, the book [EG22]
constructs a closed substack
$\mathcal{X}_{d}^{\lambda,\tau}\subset\mathcal{X}_{d}$ parameterizing
$d$-dimensional potentially crystalline representations of $G_{K}$ of Hodge
type $\lambda$ and inertial type $\tau$. When $d=2$, $\lambda$ is trivial, and
$\tau$ is tame, these are stacks of potentially Barsotti–Tate representations
of type $\tau$, and we write $\mathcal{X}_{2}^{\tau,\operatorname{BT}}$
instead.

The papers [CEGS20b, CEGS20c] construct and study another stack
${\mathcal{Z}}^{\mathrm{dd}}$ which can be regarded as a stack of _tamely_
potentially Barsotti–Tate representations; as well as a closed substack
${\mathcal{Z}}^{\tau}\subset{\mathcal{Z}}^{\mathrm{dd}}$, for each tame type
$\tau$, of potentially Barsotti–Tate representations of type $\tau$. Our
stacks ${\mathcal{Z}}^{\tau}$ are presumably isomorphic to the stacks
$\mathcal{X}_{2}^{\tau,\operatorname{BT}}$, but literally they are different
stacks, constructed differently: the stacks ${\mathcal{Z}}^{\tau}$ are stacks
of étale $\varphi$-modules with tame descent data, constructed by taking the
scheme-theoretic image of a stack ${\mathcal{C}}^{\tau,\operatorname{BT}}$ of
Breuil–Kisin modules with tame descent data; whereas the
$\mathcal{X}_{2}^{\tau,\operatorname{BT}}$ are stacks of étale
$(\varphi,\Gamma)$-modules, constructed by taking the scheme-theoretic image
of a stack of Breuil–Kisin–Fargues modules satisfying a descent condition.

The properties of ${\mathcal{Z}}^{\mathrm{dd}}$ and ${\mathcal{Z}}^{\tau}$
that we will use in this paper are recalled in detail in Section 3, but we
mention two crucial properties now.

  * •

It is proved in [CEGS20b] by a local model argument that the special fibre of
${\mathcal{C}}^{\tau,\operatorname{BT}}$ is reduced. As a consequence so is
its scheme-theoretic image ${\mathcal{Z}}^{\tau,1}$ in ${\mathcal{Z}}^{\tau}$.
The stack ${\mathcal{Z}}^{\mathrm{dd},1}$, the scheme-theoretic image in
${\mathcal{Z}}^{\mathrm{dd}}$ of the special fibre of
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}$, is similarly reduced.

  * •

It is shown in [CEGS20c] that the irreducible components of
${\mathcal{Z}}^{\tau,1}$ are in bijection with the Jordan–Hölder factors of
$\overline{\sigma}(\tau)$; the component corresponding to $\sigma$ has a dense
set of $\overline{{\mathbf{F}}}_{p}$-points $\overline{r}$ such that
$W(\overline{r})=\\{\sigma\\}$.

Here $\sigma(\tau)$ is the representation of
$\operatorname{GL}_{2}({\mathcal{O}}_{K})$ corresponding to $\tau$ under the
inertial local Langlands correspondence, and $\overline{\sigma}(\tau)$ is its
reduction modulo $p$. Note the similarity between the second of these two
properties, and the labeling by Serre weights in Theorem 1.1.

These properties are combined in Section 4 to prove that the special fibre of
${\mathcal{Z}}^{\tau}$ is generically reduced. (Note that in general the
special fibre of ${\mathcal{Z}}^{\tau}$ need not be the same as
${\mathcal{Z}}^{\tau,1}$; and similarly for the special fibre of
${\mathcal{Z}}^{\mathrm{dd}}$ vis-à-vis ${\mathcal{Z}}^{\mathrm{dd},1}$.) From
this we deduce the following theorem about the special fibres of potentially
Barsotti–Tate deformation rings, which seems hard to prove purely in the
setting of formal deformations. Let ${\mathcal{O}}$ be the ring of integers in
a finite extension of ${\mathbf{Q}}_{p}$, with residue field ${\mathbf{F}}$.

######  Theorem 1.3.

Let $\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}})$ be a continuous
representation, and $\tau$ a tame type. Let
$R^{\tau,\operatorname{BT}}_{\overline{r}}$ be the universal framed
deformation ${\mathcal{O}}$-algebra parameterising potentially Barsotti–Tate
lifts of $\overline{r}$ of type $\tau$. Then
$R^{\tau,\operatorname{BT}}_{\overline{r}}\otimes_{{\mathcal{O}}}{\mathbf{F}}$
is generically reduced.

We anticipate that this result will be of independent interest. For example,
one of us (A.C.), in joint work in progress with James Newton, has used this
result in the proof a modularity lifting theorem in the Barsotti–Tate case for
$\operatorname{GL}_{2}$ over a CM field.

We remark that très ramifiée representations do not have tamely potentially
Barsotti–Tate lifts, hence do not correspond to finite type points on
${\mathcal{Z}}^{\mathrm{dd},1}$. Equivalently, the Jordan–Hölder factors of
$\overline{\sigma}(\tau)$ for tame types $\tau$ are never Steinberg, and
therefore the stacks ${\mathcal{Z}}^{\tau,1}$ and
${\mathcal{Z}}^{\mathrm{dd},1}$ do not have irreducible components
corresponding to Steinberg weights; so, although
${\mathcal{Z}}^{\mathrm{dd},1}$ and ${\mathcal{X}}_{2,\operatorname{red}}$
cannot be isomorphic, we anticipate (but do not prove) that there is an
isomorphism between ${\mathcal{Z}}^{\mathrm{dd},1}$ and the union of the non-
Steinberg components of ${\mathcal{X}}_{2,\operatorname{red}}$.

If $\sigma$ is a non-Steinberg weight, then $\sigma$ can be written as a
virtual linear combination of representations $\overline{\sigma}(\tau)$ in the
Grothendick group of $\operatorname{GL}_{2}(k)$. In Section 5 this observation
is translated into a special case of the classical geometric Breuil–Mézard
conjecture [EG14]; we globalize this in Section 6 to prove the following
theorem, which is the main result of this paper.

######  Theorem 1.4.

The irreducible components of ${\mathcal{Z}}^{\mathrm{dd},1}$ are in bijection
with non-Steinberg Serre weights; write $\overline{{\mathcal{Z}}}(\sigma)$ for
the component corresponding to $\sigma$. Then:

  1. (1)

The finite type points of $\overline{{\mathcal{Z}}}(\sigma)$ are precisely the
representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$ having
$\sigma$ as a Serre weight.

  2. (2)

The stack ${\mathcal{Z}}^{\tau,1}$ is equal to
$\cup_{\sigma\in\operatorname{JH}(\overline{\sigma}(\tau))}\overline{{\mathcal{Z}}}(\sigma)$.

Part (1) of the theorem is the analogue of Theorem 1.2 for the stacks
${\mathcal{Z}}^{\mathrm{dd},1}$, while part (2) is a geometrisation of the
Breuil–Mézard conjecture for our tamely potentially Barsotti–Tate stacks.
Theorem 1.3 is used crucially in the proof of the theorem, to confirm that
each component $\overline{{\mathcal{Z}}}(\sigma)$ contributes with
multiplicity at most one to the cycle of
$R^{\tau,\operatorname{BT}}\otimes_{{\mathcal{O}}}{\mathbf{F}}$. We emphasize
that, to this point, our results are independent from those of [EG22].

As explained above, our construction excludes the très ramifiée
representations, which are twists of certain extensions of the trivial
character by the mod $p$ cyclotomic character. From the point of view of the
weight part of Serre’s conjecture, they are precisely the representations
which admit a twist of the Steinberg representation as their only Serre
weight. In accordance with the picture described above, this means that the
full moduli stack of 2-dimensional representations of $G_{K}$ can be obtained
from our stack by adding in the irreducible components consisting of the très
ramifiée representations. This is carried out by extending our results to the
stacks of [EG22], using the full strength of _loc. cit._

In particular, it is proved in [EG22] that the classical (numerical)
Breuil–Mèzard conjecture is equivalent to a geometrised Breuil–Mézard
conjecture for the stacks ${\mathcal{X}}_{d}^{\lambda,\tau}$ of [EG22]. Taking
the Breuil–Mézard conjecture for potentially Barsotti–Tate representations
[GK14] as input, they obtain the following theorem.

######  Theorem 1.5 ([EG22]).

There exist effective cycles $Z^{\sigma}$ (elements of the free group on the
irreducible components of ${\mathcal{X}}_{2,\operatorname{red}}$, with
nonnegative coefficients) such that for all inertial types $\tau$, the cycle
of the special fibre of ${\mathcal{X}}_{2}^{\tau,\operatorname{BT}}$ is equal
to $\sum_{\sigma}m_{\sigma}(\tau)\cdot Z^{\sigma}$, where
$\overline{\sigma}(\tau)=\sum_{\sigma}m_{\sigma}(\tau)\cdot\sigma$ in the
Grothendieck group of $\operatorname{GL}_{2}(k)$.

We stress that this theorem of [EG22] is for all inertial types, in contrast
to the Breuil–Mézard result of Theorem 1.4(2) which is only for tame types; in
particular the cycles for Steinberg weights $\sigma$ do occur. In fact the
theorem can be (and is) extended to cover potentially semistable
representations of Hodge type $0$ as well.

It remains to prove that the cycles $Z^{\sigma}$ are as in Theorem 1.2, i.e.,
to check that $Z^{\sigma}={\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ when
$\sigma$ is non-Steinberg, whereas
$Z^{\chi\otimes\mathrm{St}}={\mathcal{X}}^{\chi}_{2,\operatorname{red}}+{\mathcal{X}}^{\chi\otimes\mathrm{St}}_{2,\operatorname{red}}$.
This is where the the results of the present paper enter. The point is that
one can transfer results from ${\mathcal{Z}}^{\tau}$ to
${\mathcal{X}}_{2}^{\tau,\operatorname{BT}}$ by consideration of versal rings,
without comparing the two stacks directly. In particular the ring
$R^{\tau,\operatorname{BT}}_{\overline{r}}$ is a versal ring to
${\mathcal{X}}^{\tau,\operatorname{BT}}$ at the point corresponding to
$\overline{r}$; and so the formula
$Z^{\sigma}={\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ in the non-
Steinberg case will follow by an application of Theorem 1.3 for a suitably
chosen $\overline{r}$. The Steinberg case is handled directly using a
semistable deformation ring. This completes the proof.

##  2\. Notation and conventions

### Galois theory

Let $p>2$ be a prime number, and fix a finite extension $K/{\mathbf{Q}}_{p}$,
with residue field $k$ of cardinality $p^{f}$. In this paper we will study
various stacks that are closely related to the representation theory of
$G_{K}$, the absolute Galois group of $K$.

Our representations of $G_{K}$ will have coefficients in
$\overline{{\mathbf{Q}}}_{p}$, a fixed algebraic closure of ${\mathbf{Q}}_{p}$
whose residue field we denote by $\overline{{\mathbf{F}}}_{p}$. Let $E$ be a
finite extension of ${\mathbf{Q}}_{p}$ contained in
$\overline{{\mathbf{Q}}}_{p}$. Write ${\mathcal{O}}$ for the ring of integers
in $E$, with uniformiser $\varpi$ and residue field
${\mathbf{F}}\subset\overline{{\mathbf{F}}}_{p}$.

As is often the case, we assume that our coefficients are “sufficiently
large”. Specifically, if $L$ is the quadratic unramified extension of $K$, we
assume that $E$ admits an embedding of $K^{\prime}=L(\pi^{1/(p^{2f}-1)})$ for
some uniformizer $\pi$ of $K$. Write $l$ for the residue field of $L$.

Fix an embedding $\sigma_{0}:k\hookrightarrow{\mathbf{F}}$, and recursively
define $\sigma_{i}:k\hookrightarrow{\mathbf{F}}$ for all $i\in{\mathbf{Z}}$ so
that $\sigma_{i+1}^{p}=\sigma_{i}$. For each $i$ we define the fundamental
character $\omega_{\sigma_{i}}$ to be the composite

$\textstyle{I_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{K}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{i}}$$\textstyle{\overline{{\mathbf{F}}}_{p}^{\times},}$

where the map $I_{K}\to{\mathcal{O}}_{K}^{\times}$ is induced by the
restriction of the inverse of the Artin map, which we normalise so that
uniformisers correspond to geometric Frobenius elements.

### Inertial local Langlands

A two-dimensional _tame inertial type_ is (the isomorphism class of) a tamely
ramified representation
$\tau:I_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Z}}}_{p})$ that extends
to a representation of $G_{K}$ and whose kernel is open. Such a representation
is of the form $\tau\simeq\eta\oplus\eta^{\prime}$, and we say that $\tau$ is
a _tame principal series type_ if $\eta,\eta^{\prime}$ both extend to
characters of $G_{K}$. Otherwise, $\eta^{\prime}=\eta^{q}$, and $\eta$ extends
to a character of $G_{L}$. In this case we say that $\tau$ is a _tame cuspidal
type_. In either case $\tau|_{I_{K^{\prime}}}$ is trivial, since $\tau$ is
tame.

Henniart’s appendix to [BM02] associates a finite dimensional irreducible
$E$-representation $\sigma(\tau)$ of
$\operatorname{GL}_{2}({\mathcal{O}}_{K})$ to each inertial type $\tau$; we
refer to this association as the inertial local Langlands correspondence.
Since we are only working with tame inertial types, this correspondence can be
made very explicit, as in [CEGS20c, Sec. 1.2]. (Since we will not directly use
the explicit description in this paper, we will not repeat it here.)

### Serre weights and tame types

By definition, a _Serre weight_ is an irreducible
${\mathbf{F}}$-representation of $\operatorname{GL}_{2}(k)$. Then, concretely,
a Serre weight is of the form

$\overline{\sigma}_{\vec{t},\vec{s}}:=\otimes^{f-1}_{j=0}(\det{\\!}^{t_{j}}\operatorname{Sym}^{s_{j}}k^{2})\otimes_{k,\sigma_{j}}{\mathbf{F}},$

where $0\leq s_{j},t_{j}\leq p-1$ and not all $t_{j}$ are equal to $p-1$. We
say that a Serre weight is _Steinberg_ if $s_{j}=p-1$ for all $j$, and _non-
Steinberg_ otherwise.

Let $\tau$ be a tame inertial type. Write $\overline{\sigma}(\tau)$ for the
semisimplification of the reduction modulo $p$ of a
$\operatorname{GL}_{2}({\mathcal{O}}_{K})$-stable ${\mathcal{O}}$-lattice in
$\sigma(\tau)$. The action of $\operatorname{GL}_{2}({\mathcal{O}}_{K})$ on
$\overline{\sigma}(\tau)$ factors through $\operatorname{GL}_{2}(k)$, so the
Jordan–Hölder factors $\operatorname{JH}(\overline{\sigma}(\tau))$ of
$\overline{\sigma}(\tau)$ are Serre weights. By the results of [Dia07], these
Jordan–Hölder factors of $\overline{\sigma}(\tau)$ are pairwise non-
isomorphic, and are parametrised by a certain set ${\mathcal{P}}_{\tau}$ that
we now recall.

Suppose first that $\tau=\eta\oplus\eta^{\prime}$ is a tame principal series
type. Set $f^{\prime}=f$ in this case. We define $0\leq\gamma_{i}\leq p-1$
(for $i\in{\mathbf{Z}}/f{\mathbf{Z}}$) to be the unique integers not all equal
to $p-1$ such that
$\eta(\eta^{\prime})^{-1}=\prod_{i=0}^{f-1}\omega_{\sigma_{i}}^{\gamma_{i}}$.

If instead $\tau=\eta\oplus\eta^{\prime}$ is a cuspidal type, set
$f^{\prime}=2f$. We define $0\leq\gamma_{i}\leq p-1$ (for
$i\in{\mathbf{Z}}/f^{\prime}{\mathbf{Z}}$) to be the unique integers such that
$\eta(\eta^{\prime})^{-1}=\prod_{i=0}^{f^{\prime}-1}\omega_{\sigma^{\prime}_{i}}^{\gamma_{i}}$.
Here $\sigma^{\prime}_{0}:l\to\overline{{\mathbf{F}}}_{p}^{\times}$ is a fixed
choice of embedding extending $\sigma_{0}$,
$(\sigma^{\prime}_{i+1})^{p}=\sigma^{\prime}_{i}$ for all $i$, and the
fundamental characters
$\omega_{\sigma^{\prime}_{i}}:I_{L}\to\overline{{\mathbf{F}}}_{p}^{\times}$
for each $\sigma^{\prime}_{i}:l\to\overline{{\mathbf{F}}}_{p}^{\times}$ are
defined in the same way as the $\omega_{\sigma_{i}}$.

If $\tau$ is scalar then we set ${\mathcal{P}}_{\tau}=\\{\varnothing\\}$.
Otherwise we have $\eta\neq\eta^{\prime}$, and we let ${\mathcal{P}}_{\tau}$
be the collection of subsets $J\subset{\mathbf{Z}}/f^{\prime}{\mathbf{Z}}$
satisfying the conditions:

  * •

if $i-1\in J$ and $i\notin J$ then $\gamma_{i}\neq p-1$, and

  * •

if $i-1\notin J$ and $i\in J$ then $\gamma_{i}\neq 0$

and, in the cuspidal case, satisfying the further condition that $i\in J$ if
and only if $i+f\not\in J$.

The Jordan–Hölder factors of $\overline{\sigma}(\tau)$ are by definition Serre
weights, and are parametrised by ${\mathcal{P}}_{\tau}$ as follows (see
[EGS15, §3.2, 3.3]). For any $J\subseteq{\mathbf{Z}}/f^{\prime}{\mathbf{Z}}$,
we let $\delta_{J}$ denote the characteristic function of $J$, and if
$J\in{\mathcal{P}}_{\tau}$ we define $s_{J,i}$ by

$s_{J,i}=\begin{cases}p-1-\gamma_{i}-\delta_{J^{c}}(i)&\text{if }i-1\in J\\\
\gamma_{i}-\delta_{J}(i)&\text{if }i-1\notin J,\end{cases}$

and we set $t_{J,i}=\gamma_{i}+\delta_{J^{c}}(i)$ if $i-1\in J$ and $0$
otherwise.

In the principal series case we let
$\overline{\sigma}(\tau)_{J}:=\overline{\sigma}_{\vec{t},\vec{s}}\otimes\eta^{\prime}\circ\det$;
the $\overline{\sigma}(\tau)_{J}$ are precisely the Jordan–Hölder factors of
$\overline{\sigma}(\tau)$.

In the cuspidal case, one checks that $s_{J,i}=s_{J,i+f}$ for all $i$, and
also that the character
$\eta^{\prime}\cdot\prod_{i=0}^{f^{\prime}-1}(\sigma^{\prime}_{i})^{t_{J,i}}:l^{\times}\to{\mathbf{F}}^{\times}$
factors as $\theta\circ N_{l/k}$ where $N_{l/k}$ is the norm map. We let
$\overline{\sigma}(\tau)_{J}:=\overline{\sigma}_{0,\vec{s}}\otimes\theta\circ\det$;
the $\overline{\sigma}(\tau)_{J}$ are precisely the Jordan–Hölder factors of
$\overline{\sigma}(\tau)$.

###  $p$-adic Hodge theory

We normalise Hodge–Tate weights so that all Hodge–Tate weights of the
cyclotomic character are equal to $-1$. We say that a potentially crystalline
representation $r:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Q}}}_{p})$
has _Hodge type_ $0$, or is _potentially Barsotti–Tate_ , if for each
$\varsigma:K\hookrightarrow\overline{{\mathbf{Q}}}_{p}$, the Hodge–Tate
weights of $r$ with respect to $\varsigma$ are $0$ and $1$. (Note that this is
a more restrictive definition of potentially Barsotti–Tate than is sometimes
used; however, we will have no reason to deal with representations with non-
regular Hodge-Tate weights, and so we exclude them from consideration. Note
also that it is more usual in the literature to say that $r$ is potentially
Barsotti–Tate if it is potentially crystalline, and $r^{\vee}$ has Hodge type
$0$.)

We say that a potentially crystalline representation
$r:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Q}}}_{p})$ has _inertial
type_ $\tau$ if the traces of elements of $I_{K}$ acting on $\tau$ and on

$\operatorname{D_{pcris}}(r)=\varinjlim_{K^{\prime}/K}(\mathrm{B}_{\mathrm{cris}}\otimes_{{\mathbf{Q}}_{p}}V_{r})^{G_{K^{\prime}}}$

are equal (here $V_{r}$ is the underlying vector space of $V_{r}$). A
representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$ _has
a potentially Barsotti–Tate lift of type $\tau$_ if and only if $\overline{r}$
admits a lift to a representation
$r:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{Z}}}_{p})$ of Hodge type
$0$ and inertial type $\tau$.

### Serre weights of mod $p$ Galois representations

Given a continuous representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$,
there is an associated (nonempty) set of Serre weights $W(\overline{r})$,
defined to be the set of Serre weights $\overline{\sigma}_{\vec{t},\vec{s}}$
such that $\overline{r}$ has a crystalline lift whose Hodge–Tate weights are
as follows: for each embedding $\sigma_{j}:k\hookrightarrow{\mathbf{F}}$ there
is an embedding
$\widetilde{\sigma}_{j}:K\hookrightarrow\overline{{\mathbf{Q}}}_{p}$ lifting
$\sigma_{j}$ such that the $\widetilde{\sigma}_{j}$-labeled Hodge–Tate weights
of $r$ are $\\{-s_{j}-t_{j},1-t_{j}\\}$, and the remaining $(e-1)f$ pairs of
Hodge–Tate weights of $r$ are all $\\{0,1\\}$.

There are in fact several different definitions of $W(\overline{r})$ in the
literature; as a result of the papers [BLGG13, GK14, GLS15], these definitions
are known to be equivalent up to normalisation. The normalisations in this
paper are the same as those of [CEGS20b, CEGS20c]; see [CEGS20c, Sec. 1.2] for
a detailed discussion of these normalisations. In particular we have
normalised the set of Serre weights so that $\overline{r}$ has a potentially
Barsotti–Tate lift of type $\tau$ if and only if
$W(\overline{r})\cap\operatorname{JH}(\overline{\sigma}(\tau))\neq\varnothing$
([CEGS20b, Lem. A.4]).

### Stacks

We follow the terminology of [Sta13]; in particular, we write “algebraic
stack” rather than “Artin stack”. More precisely, an algebraic stack is a
stack in groupoids in the _fppf_ topology, whose diagonal is representable by
algebraic spaces, which admits a smooth surjection from a scheme. See [Sta13,
Tag 026N] for a discussion of how this definition relates to others in the
literature, and [Sta13, Tag 04XB] for key properties of morphisms
representable by algebraic spaces.

For a commutative ring $A$, an _fppf stack over $A$_ (or _fppf_ $A$-stack) is
a stack fibred in groupoids over the big _fppf_ site of
$\operatorname{Spec}A$. Following [Eme, Defs. 5.3, 7.6], an fppf stack in
groupoids ${\mathcal{X}}$ over a scheme $S$ is called a formal algebraic stack
if there is a morphism $U\to{\mathcal{X}}$, whose domain $U$ is a formal
algebraic space over $S$ (in the sense of [Sta13, Tag 0AIL]), and which is
representable by algebraic spaces, smooth, and surjective.

Let $\operatorname{Spf}{\mathcal{O}}$ denote the affine formal scheme (or
affine formal algebraic space, in the terminology of [Sta13]) obtained by
$\varpi$-adically completing $\operatorname{Spec}{\mathcal{O}}$. A formal
algebraic stack ${\mathcal{X}}$ over $\operatorname{Spec}{\mathcal{O}}$ is
called $\varpi$-adic if the canonical map
${\mathcal{X}}\to\operatorname{Spec}{\mathcal{O}}$ factors through
$\operatorname{Spf}{\mathcal{O}}$, and if the induced map
${\mathcal{X}}\to\operatorname{Spf}{\mathcal{O}}$ is algebraic, i.e.
representable by algebraic stacks (in the sense of [Sta13, Tag 06CF] and [Eme,
Def. 3.1]).

##  3\. Moduli stacks of Breuil–Kisin modules and étale $\varphi$-modules

The main object of study in this paper is the $\varpi$-adic formal algebraic
stack ${\mathcal{Z}}^{\mathrm{dd}}$ that was introduced and studied in
[CEGS20b, CEGS20c], and whose $\overline{{\mathbf{F}}}_{p}$-points are
naturally in bijection with the continuous representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$
admitting a potentially Barsotti–Tate lift of some tame type. In this section
we review the construction and known properties of
${\mathcal{Z}}^{\mathrm{dd}}$, as well as those of several other closely
related stacks.

### Stacks of Breuil–Kisin modules

For each tame type $\tau$, there is a $\varpi$-adic formal algebraic stack
${\mathcal{C}}^{\tau,\operatorname{BT}}$ whose
$\operatorname{Spf}({\mathcal{O}}_{E^{\prime}})$-points, for any finite
extension $E^{\prime}/E$, are the Breuil–Kisin modules correpsonding to
$2$-dimensional potentially Barsotti–Tate representations of type $\tau$. This
stack is constructed in several steps, which we review in brief. (We refer the
reader to [CEGS20b] as well as to the summary in [CEGS20c, Sec. 2.3] for
complete definitions, recalling here only what will be used in this paper.)

For each integer $a\geq 1$, we write ${\mathcal{C}}^{\mathrm{dd},a}$ for the
fppf stack over ${\mathcal{O}}/\varpi^{a}$ which associates to any
${\mathcal{O}}/\varpi^{a}$-algebra $A$ the groupoid
${\mathcal{C}}^{\mathrm{dd},a}(A)$ of rank $2$ Breuil–Kisin modules of height
at most $1$ with $A$-coefficients and descent data from $K^{\prime}$ to $K$.
Set ${\mathcal{C}}^{\mathrm{dd}}=\varinjlim_{a}{\mathcal{C}}^{\mathrm{dd},a}$.
The closed substack ${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}$ of
${\mathcal{C}}^{\mathrm{dd}}$ is cut out by a Kottwitz-type determinant
condition, which can be thought of (on the Galois side) as cutting out the
tamely potentially Barsotti–Tate representations from among all tamely
potentially crystalline representations with Hodge–Tate weights in
$\\{0,1\\}$. By [CEGS20b, Cor. 4.2.13] the stack
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}$ then decomposes as a disjoint
union of closed substacks ${\mathcal{C}}^{\tau,\operatorname{BT}}$, one for
each tame type $\tau$, consisting of Breuil–Kisin modules with descent data of
type $\tau$. Finally, for each $a\geq 1$ we write
${\mathcal{C}}^{\tau,\operatorname{BT},a}={\mathcal{C}}^{\tau,\operatorname{BT}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$,
and similarly for ${\mathcal{C}}^{\mathrm{dd},\operatorname{BT},a}$. The
following properties are established in [CEGS20b, Cor. 3.1.8, Cor. 4.5.3,
Prop. 5.2.21].

######  Theorem 3.1.

The stacks ${\mathcal{C}}^{\mathrm{dd},a}$,
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT},a}$, and
${\mathcal{C}}^{\tau,\operatorname{BT},a}$ are algebraic stacks of finite type
over ${\mathcal{O}}$, while the stacks ${\mathcal{C}}^{\mathrm{dd}}$,
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT}}$, and
${\mathcal{C}}^{\tau,\operatorname{BT}}$ are $\varpi$-adic formal algebraic
stacks. Moreover:

  1. (1)

${\mathcal{C}}^{\tau,\operatorname{BT}}$ is analytically normal,
Cohen–Macaulay, and flat over ${\mathcal{O}}$.

  2. (2)

The stacks ${\mathcal{C}}^{\mathrm{dd},\operatorname{BT},a}$ and
${\mathcal{C}}^{\tau,\operatorname{BT},a}$ are equidimensional of dimension
$[K:{\mathbf{Q}}_{p}]$.

  3. (3)

The special fibres ${\mathcal{C}}^{\mathrm{dd},\operatorname{BT},1}$ and
${\mathcal{C}}^{\tau,\operatorname{BT},1}$ are reduced.

### Galois moduli stacks

Let ${\mathcal{R}}^{\mathrm{dd},a}$ be the _fppf_ ${\mathbf{F}}$-stack which
associates to any ${\mathcal{O}}/\varpi^{a}$-algebra $A$ the groupoid
${\mathcal{R}}^{\mathrm{dd},a}(A)$ of rank $2$ étale $\varphi$-modules with
$A$-coefficients and descent data from $K^{\prime}$ to $K$. Inverting $u$ on
Breuil–Kisin modules gives a proper morphism
${\mathcal{C}}^{\mathrm{dd},a}\to{\mathcal{R}}^{\mathrm{dd},a}$, which then
restricts to proper morphisms
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT},a}\to{\mathcal{R}}^{\mathrm{dd},a}$
as well as
${\mathcal{C}}^{\tau,\operatorname{BT},a}\to{\mathcal{R}}^{\mathrm{dd},a}$ for
each $\tau$.

The paper [EG21] develops a theory of scheme-theoretic images of proper
morphisms ${\mathcal{X}}\to{\mathcal{F}}$ of stacks over a locally Noetherian
base-scheme $S$, where ${\mathcal{X}}$ is an algebraic stack which is locally
of finite presentation over $S$, and the diagonal of ${\mathcal{F}}$ is
representable by algebraic spaces and locally of finite presentation. This
theory applies in particular to each of the morphisms of the previous
paragraph (even though ${\mathcal{R}}^{\mathrm{dd},a}$ is _not_ algebraic). We
define ${\mathcal{Z}}^{\mathrm{dd},a}$ and ${\mathcal{Z}}^{\tau,a}$ to be the
scheme-theoretic images of the morphisms
${\mathcal{C}}^{\mathrm{dd},\operatorname{BT},a}\to{\mathcal{R}}^{\mathrm{dd},a}$
and
${\mathcal{C}}^{\tau,\operatorname{BT},a}\to{\mathcal{R}}^{\mathrm{dd},a}$,
respectively. Set
${\mathcal{Z}}^{\mathrm{dd}}=\varinjlim_{a}{\mathcal{Z}}^{\mathrm{dd},a}$ and
${\mathcal{Z}}^{\tau}=\varinjlim_{a}{\mathcal{Z}}^{\tau,a}$. The following
theorem combines [CEGS20b, Thm. 5.1.2, Prop. 5.1.4, Lem. 5.1.8, Prop. 5.2.20].

######  Theorem 3.2.

The stacks ${\mathcal{Z}}^{\mathrm{dd},a}$ and ${\mathcal{Z}}^{\tau,a}$ are
algebraic stacks of finite type over ${\mathcal{O}}$, while the stacks
${\mathcal{Z}}^{\mathrm{dd}}$ and ${\mathcal{Z}}^{\tau}$ are $\varpi$-adic
formal algebraic stacks. Moreover:

  1. (1)

The stacks ${\mathcal{Z}}^{\mathrm{dd},a}$ and ${\mathcal{Z}}^{\tau,a}$ are
equidimensional of dimension $[K:{\mathbf{Q}}_{p}]$.

  2. (2)

The stacks ${\mathcal{Z}}^{\mathrm{dd},1}$ and ${\mathcal{Z}}^{\tau,1}$ are
reduced.

  3. (3)

The $\overline{{\mathbf{F}}}_{p}$-points of ${\mathcal{Z}}^{\mathrm{dd},1}$
are naturally in bijection with the continuous representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$
which are not a twist of a très ramifiée extension of the trivial character by
the mod $p$ cyclotomic character. Similarly, the
$\overline{{\mathbf{F}}}_{p}$-points of ${\mathcal{Z}}^{\tau,1}$ are naturally
in bijection with the continuous representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}(\overline{{\mathbf{F}}}_{p})$
which have a potentially Barsotti–Tate lift of type $\tau$.

In particular the stack ${\mathcal{Z}}^{\mathrm{dd},1}$ is the underlying
reduced substack of ${\mathcal{Z}}^{\mathrm{dd},a}$ for each $a\geq 1$, as
well as of ${\mathcal{Z}}^{\mathrm{dd}}$; and similarly for the stacks
${\mathcal{Z}}^{\tau,1}$.

######  Remark 3.3.

We stress that the morphism
${\mathcal{Z}}^{\mathrm{dd},a}\hookrightarrow{\mathcal{Z}}^{\mathrm{dd}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$
need not be an isomorphism _a priori_ , and we have no reason to expect that
it is. However, our results in the next section will prove that it is
_generically_ an isomorphism for all $a\geq 1$.

### Versal rings and deformation rings

Let $x$ be an ${\mathbf{F}}^{\prime}$-point of ${\mathcal{Z}}^{\tau,a}$,
corresponding to the representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$. We will
usually write $R_{\overline{r}}^{\tau,\operatorname{BT}}$ for the reduced and
$p$-torsion free quotient of the universal framed deformation ring of
$\overline{r}$ whose $\overline{{\mathbf{Q}}}_{p}$-points correspond to the
potentially Barsotti–Tate lifts of $\overline{r}$ of type $\tau$. (In Section
5 we will denote this ring instead by $R_{\overline{r},0,\tau}$, for ease of
comparison with the paper [GK14].)

It is explained in [CEGS20b, Sec. 5.2] that there are versal rings
$R^{\tau,a}_{x}$ to ${\mathcal{Z}}^{\tau,a}$ at the point $x$, such that the
following holds. (These rings are denoted $R^{\tau,a}$ in [CEGS20b]; we
include the subscript $x$ here to emphasize the dependence on the point $x$.)

######  Proposition 3.4 ([CEGS20b, Prop. 5.2.19]).

We have $\varprojlim
R^{\tau,a}_{x}=R^{\tau,\operatorname{BT}}_{\overline{r}}$; thus
$R^{\tau,\operatorname{BT}}_{\overline{r}}$ is a versal ring to
${{\mathcal{Z}}^{\tau}}$ at $x$.

Similarly there is a versal ring $R^{\mathrm{dd},a}_{x}$ to
${\mathcal{Z}}^{\mathrm{dd},a}$ at $x$, and each $R^{\tau,a}_{x}$ is a
quotient of $R^{\mathrm{dd},a}_{x}$.

### Irreducible components of ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and
${\mathcal{Z}}^{\tau,1}$

Fix a tame type $\tau$, and recall that we set $f^{\prime}=f$ if the type
$\tau$ is principal series, while $f^{\prime}=2f$ if the type $\tau$ is
cuspidal. We say that a subset $J\subset{\mathbf{Z}}/f^{\prime}{\mathbf{Z}}$
is a _shape_ if:

  * •

$\tau$ is scalar and $J=\varnothing$,

  * •

$\tau$ is a non-scalar principal series type and $J$ is arbitrary, or

  * •

$\tau$ is cupsidal and $J$ has the property that $i\in J$ if and only if
$i+f\not\in J$.

Thus there are exactly $2^{f}$ shapes if $\tau$ is non-scalar. The set
${\mathcal{P}}_{\tau}$ introduced in Section 2 is a subset of the set of
shapes.

To each shape $J$, the discussion in [CEGS20c, Sec. 4.2.7] associates a closed
substack $\overline{\mathcal{C}}(J)$ of
${\mathcal{C}}^{\tau,\operatorname{BT},1}$. The stack
$\overline{{\mathcal{Z}}}(J)$ is then defined to be the scheme-theoretic image
of $\overline{{\mathcal{C}}}(J)$ under the map
${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{Z}}^{\tau,1}$.

The following description of the irreducible components of
${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and ${\mathcal{Z}}^{\tau,1}$ is
proved in [CEGS20c, Prop. 5.1.13, Thm. 5.1.17, Cor. 5.3.3, Thm. 5.4.3]; the
description of the components of ${\mathcal{Z}}^{\tau,1}$ in part (3) of the
theorem is analogous to the description of the components of
${\mathcal{X}}_{2,\operatorname{red}}$ of Theorem 1.1.

######  Theorem 3.5.

The irreducible components of ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and
${\mathcal{Z}}^{\tau,1}$ are as follows.

  1. (1)

The irreducible components of ${\mathcal{C}}^{\tau,1}$ are precisely the
$\overline{{\mathcal{C}}}(J)$ for shapes $J$, and if $J\neq J^{\prime}$ then
$\overline{{\mathcal{C}}}(J)\neq\overline{{\mathcal{C}}}(J^{\prime})$.

  2. (2)

The irreducible components of ${\mathcal{Z}}^{\tau,1}$ are precisely the
$\overline{{\mathcal{Z}}}(J)$ for shapes $J\in{\mathcal{P}}_{\tau}$, and if
$J\neq J^{\prime}$ then
$\overline{{\mathcal{Z}}}(J)\neq\overline{{\mathcal{Z}}}(J^{\prime})$.

  3. (3)

For each $J\in{\mathcal{P}}_{\tau}$ there is a dense open substack
${\mathcal{U}}$ of $\overline{{\mathcal{C}}}(J)$ such that the map
$\overline{{\mathcal{C}}}(J)\to\overline{{\mathcal{Z}}}(J)$ restricts to an
open immersion on ${\mathcal{U}}$.

  4. (4)

For each $J\in{\mathcal{P}}_{\tau}$, there is a dense set of finite type
points of $\overline{{\mathcal{Z}}}(J)$ with the property that the
corresponding Galois representations have $\overline{\sigma}(\tau)_{J}$ as a
Serre weight, and which furthermore admit a unique Breuil–Kisin model of type
$\tau$.

######  Remark 3.6.

If $\overline{\sigma}(\tau)_{J}=\overline{\sigma}_{\vec{t},\vec{s}}$, then the
dense set of finite type points of $\overline{{\mathcal{Z}}}(J)$ produced in
the proof of [CEGS20c, Thm. 5.1.17], as claimed in Theorem 3.5(4), consists of
points corresponding to reducible representations $\overline{r}$ such that
$\overline{r}|_{I_{K}}$ is an extension of
$\overline{\varepsilon}^{-1}\prod_{i=0}^{f-1}\omega_{\sigma_{i}}^{t_{i}}$ by
$\prod_{i=0}^{f-1}\omega_{\sigma_{i}}^{s_{i}+t_{i}}$ (necessarily peu ramifié
in case the ratio of the two characters is cyclotomic).

######  Remark 3.7.

We emphasize in Theorem 3.5 that the components of ${\mathcal{Z}}^{\tau,1}$
are indexed by shapes $J\in{\mathcal{P}}_{\tau}$, _not_ by all shapes. If
$J\not\in{\mathcal{P}}_{\tau}$, then by [CEGS20c, Thm. 5.1.12] the stack
$\overline{{\mathcal{Z}}}(J)$ has dimension strictly smaller than
$[K:{\mathbf{Q}}_{p}]$, and so is properly contained in some component of
${\mathcal{Z}}^{\tau,1}$.

######  Remark 3.8.

Strictly speaking, in the principal series case Theorem 3.5 is proved in
[CEGS20c] for stacks of Breuil–Kisin modules and of étale $\varphi$-modules
with descent data from $K(\pi^{1/(p^{f}-1)})$ to $K$, rather than from our
$K^{\prime}$ to $K$. But by [CEGS20b, Prop. 4.3.1(2)], together with the
analogous statement for stacks of étale $\varphi$-modules (which is easier,
since multiplication by $u$ on an étale $\varphi$-module is bijective) we can
replace $K(\pi^{1/(p^{f}-1)})$ with any extension of prime-to-$p$ degree that
remains Galois over $K$, such as $K^{\prime}$, without changing the resulting
stacks.

##  4\. Generic reducedness of
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$

Fix a Galois representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$, where
${\mathbf{F}}^{\prime}/{\mathbf{F}}$ is a finite extension. Our goal in this
section is to prove that the scheme
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$ is
generically reduced; this will be a key ingredient in the proof of our
geometric Breuil–Mézard result in Section 6. Recall that a scheme is
generically reduced if it contains an open reduced subscheme whose underlying
topological space is dense; in the case of a Noetherian affine scheme
$\operatorname{Spec}A$, this is equivalent to requiring that the localisation
of $A$ at each of its minimal primes is reduced.

We may of course suppose that $R^{\tau,\operatorname{BT}}_{\overline{r}}\neq
0$, so that $\overline{r}$ has a potentially Barsotti–Tate lift of type
$\tau$, and so corresponds to a finite type point
$x:\operatorname{Spec}{\mathbf{F}}^{\prime}\to{\mathcal{Z}}^{\tau,a}$. It
follows from Proposition 3.4 that $\operatorname{Spec}R^{\tau,a}_{x}$ is a
closed subscheme of
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi^{a}$, but
we have no reason to believe that equality holds. It follows from Theorem
3.2(2), together with Lemma 4.5 below, that
$\operatorname{Spec}R^{\tau,1}_{x}$ is the underlying reduced subscheme of
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$, so that
equality holds in the case $a=1$ if and only if
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$ is
reduced; however, again, we have no reason to believe that this holds in
general.

Neverthelss it _is_ the case that
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$ is
generically reduced;  _cf._ Theorem 4.6 below. We will deduce Proposition 4.6
from the following global statement.

######  Proposition 4.1.

Let $\tau$ be a tame type. There is a dense open substack ${\mathcal{U}}$ of
${\mathcal{Z}}^{\tau}$ such that ${\mathcal{U}}_{/{\mathbf{F}}}$ is reduced.

###### Proof.

The proposition will follow from an application of Lemma A.6, and the key to
this application will be to find a candidate open substack ${\mathcal{U}}^{1}$
of ${\mathcal{Z}}^{\tau,1}$, which we will do using our study of the
irreducible components of ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ and
${\mathcal{Z}}^{\tau,1}$.

Recall that, for each shape $J$, we let $\overline{{\mathcal{Z}}}(J)$ denote
the scheme-theoretic image of $\overline{{\mathcal{C}}}(J)$ under the proper
morphism ${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{Z}}^{\tau,1}$.
Each $\overline{{\mathcal{Z}}}(J)$ is a closed substack of
${\mathcal{Z}}^{\tau,1}$, and so, if we let ${\mathcal{V}}(J)$ be the
complement in ${\mathcal{Z}}^{\tau,1}$ of the union of the
$\overline{{\mathcal{Z}}}(J^{\prime})$ for all shapes $J^{\prime}\neq J$ then
${\mathcal{V}}(J)$ is a dense open open substack of ${\mathcal{Z}}^{\tau,1}$,
by Theorem 3.5(2) and Remark 3.7 (the former in consideration of
$J^{\prime}\in{\mathcal{P}}_{\tau}$, the latter for
$J^{\prime}\not\in{\mathcal{P}}_{\tau}$). The preimage ${\mathcal{W}}(J)$ of
${\mathcal{V}}(J)$ in ${\mathcal{C}}^{\tau,\operatorname{BT},1}$ is therefore
a dense open substack of $\overline{{\mathcal{C}}}(J)$. Possibly shrinking
${\mathcal{W}}(J)$ further, we may suppose by Theorem 3.5(3) that the morphism
${\mathcal{W}}(J)\to{\mathcal{Z}}^{\tau,1}$ is a monomorphism.

Write $|\cdot|$ for the underlying topological space of a stack. The
complement $|\overline{{\mathcal{C}}}(J)|\setminus|{\mathcal{W}}(J)|$ is a
closed subset of $|\overline{{\mathcal{C}}}(J)|$, and thus of
$|{\mathcal{C}}^{\tau,\operatorname{BT},1}|$, and its image under the proper
morphism ${\mathcal{C}}^{\tau,\operatorname{BT},1}\to{\mathcal{Z}}^{\tau,1}$
is a closed subset of $|{\mathcal{Z}}^{\tau,\operatorname{BT},1}|$, which is
(e.g. for dimension reasons) a proper closed subset of
$|\overline{{\mathcal{Z}}}(J)|$; so if we let ${\mathcal{U}}(J)$ be the
complement in ${\mathcal{V}}(J)$ of this image, then ${\mathcal{U}}(J)$ is
open and dense in $\overline{{\mathcal{Z}}}(J)$, and the morphism
${\mathcal{C}}^{\tau,\operatorname{BT},1}\times_{{\mathcal{Z}}^{\tau,1}}{\mathcal{U}}(J)\to{\mathcal{U}}(J)$
is a monomorphism. Set ${\mathcal{U}}^{1}=\cup_{J}{\mathcal{U}}(J)$. Since the
${\mathcal{U}}(J)$ are pairwise disjoint by construction,
${\mathcal{C}}^{\tau,\operatorname{BT},1}\times_{{\mathcal{Z}}^{\tau,1}}{\mathcal{U}}^{1}\to{\mathcal{U}}^{1}$
is again a monomorphism. By construction (taking into account Theorem 3.5(2)),
${\mathcal{U}}^{1}$ is dense in ${\mathcal{Z}}^{\tau,1}$.

Now let ${\mathcal{U}}$ denote the open substack of ${\mathcal{Z}}^{\tau}$
corresponding to ${\mathcal{U}}^{1}$. Since
$|{\mathcal{Z}}^{\tau}|=|{\mathcal{Z}}^{\tau,1}|$, we see that ${\mathcal{U}}$
is dense in ${\mathcal{Z}}^{\tau}$. We have seen in the previous paragraph
that the statement of Lemma A.6 (5) holds (taking $a=1$,
${\mathcal{X}}={\mathcal{C}}^{\tau,\operatorname{BT}}$, and
${\mathcal{Y}}={\mathcal{Z}}^{\tau}$); so Lemma A.6 implies that, for each
$a\geq 1$, the closed immersion
${\mathcal{U}}\times_{{\mathcal{Z}}^{\tau}}{\mathcal{Z}}^{\tau,a}\hookrightarrow{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$
is an isomorphism.

In particular, since the closed immersion
${\mathcal{U}}^{1}={\mathcal{U}}\times_{{\mathcal{Z}}^{\tau}}{\mathcal{Z}}^{\tau,1}\to{\mathcal{U}}_{/{\mathbf{F}}}$
is an isomorphism, we may regard ${\mathcal{U}}_{/{\mathbf{F}}}$ as an open
substack of ${\mathcal{Z}}^{\tau,1}$. Since ${\mathcal{Z}}^{\tau,1}$ is
reduced, by Theorem 3.2(2), so is its open substack
${\mathcal{U}}_{/{\mathbf{F}}}$. This completes the proof of the proposition.
∎

######  Corollary 4.2.

Let $\tau$ be a tame type. There is a dense open substack ${\mathcal{U}}$ of
${\mathcal{Z}}^{\tau}$ such that we have an isomorphism
${\mathcal{C}}^{\tau,\operatorname{BT}}\times_{{\mathcal{Z}}^{\tau}}{\mathcal{U}}\buildrel\sim\over{\longrightarrow}{\mathcal{U}},$
as well as isomorphisms

${\mathcal{U}}\times_{{\mathcal{Z}}^{\tau}}{\mathcal{C}}^{\tau,\operatorname{BT},a}\buildrel\sim\over{\longrightarrow}{\mathcal{U}}\times_{{\mathcal{Z}}^{\tau}}{\mathcal{Z}}^{\tau,a}\buildrel\sim\over{\longrightarrow}{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a},$

for each $a\geq 1$.

###### Proof.

This follows from Proposition A.6 and the proof of Proposition 4.1. ∎

######  Remark 4.3.

More colloquially, Corollary 4.2 shows that for each tame type $\tau$, there
is an open dense substack ${\mathcal{U}}$ of ${\mathcal{Z}}^{\tau}$ consisting
of Galois representations which have a unique Breuil–Kisin model of type
$\tau$.

######  Lemma 4.4.

If ${\mathcal{U}}$ is an open substack of ${\mathcal{Z}}^{\tau}$ satisfying
the condition of Proposition 4.1, and if
$T\to{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$ is a smooth morphism whose source
is a scheme, then
$T\times_{{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}}{\mathcal{U}}_{/{\mathbf{F}}}$
is reduced, and is a dense open subscheme of $T$.

###### Proof.

Since ${\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$ is a Noetherian algebraic stack
(being of finite presentation over $\operatorname{Spec}{\mathbf{F}}$), the
open immersion

${\mathcal{U}}_{/{\mathbf{F}}}\to{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$

is quasi-compact ([Sta13, Tag 0CPM]). Since
$T\to{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$ is flat (being smooth, by
assumption), the pullback
$T\times_{{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}}{\mathcal{U}}_{/{\mathbf{F}}}\to
T$ is an open immersion with dense image; here we use the fact that for a
quasi-compact morphism, the property of being scheme-theoretically dominant is
preserved by flat base-change, together with the fact that an open immersion
with dense image induces a scheme-theoretically dominant morphism after
passing to underlying reduced substacks. Since the source of this morphism is
smooth over the reduced algebraic stack ${\mathcal{U}}_{/{\mathbf{F}}}$, it is
itself reduced. ∎

The following result is standard, but we recall the proof for the sake of
completeness.

######  Lemma 4.5.

Let $T$ be a Noetherian scheme, all of whose local rings at finite type points
are $G$-rings. If $T$ is reduced (resp. generically reduced), then so are all
of its complete local rings at finite type points.

###### Proof.

Let $t$ be a finite type point of $T$, and write $A:={\mathcal{O}}_{T,t}$.
Then $A$ is a (generically) reduced local $G$-ring, and we need to show that
its completion $\widehat{A}$ is also (generically) reduced. Let
$\widehat{\mathfrak{p}}$ be a (minimal) prime of $\widehat{A}$; since
$A\to\widehat{A}$ is (faithfully) flat, $\widehat{\mathfrak{p}}$ lies over a
(minimal) prime $\mathfrak{p}$ of $A$ by the going-down theorem.

Then $A_{\mathfrak{p}}$ is reduced by assumption, and we need to show that
$\widehat{A}_{\widehat{\mathfrak{p}}}$ is reduced. By [Sta13, Tag 07QK], it is
enough to show that the morphism $A\to\widehat{A}_{\widehat{\mathfrak{p}}}$ is
regular. Both $A$ and $\widehat{A}$ are $G$-rings (the latter by [Sta13, Tag
07PS]), so the composite

$A\to\widehat{A}\to(\widehat{A}_{\widehat{\mathfrak{p}}})^{\widehat{}}$

is a composite of regular morphisms, and is thus a regular morphism by [Sta13,
Tag 07QI].

This composite factors through the natural morphism
$A_{\mathfrak{p}}\to(\widehat{A}_{\widehat{\mathfrak{p}}})^{\widehat{}}$, so
this morphism is also regular. Factoring it as the composite

$A_{\mathfrak{p}}\to\widehat{A}_{\widehat{\mathfrak{p}}}\to(\widehat{A}_{\widehat{\mathfrak{p}}})^{\widehat{}},$

it follows from [Sta13, Tag 07NT] that
$A_{\mathfrak{p}}\to\widehat{A}_{\widehat{\mathfrak{p}}}$ is regular, as
required. ∎

Finally, we are ready to prove the main result of this section.

######  Theorem 4.6.

For any tame type $\tau$, the scheme
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$ is
generically reduced, with underlying reduced subscheme
$\operatorname{Spec}R^{\tau,1}_{x}$.

###### Proof.

By Proposition 3.4, we have a versal morphism

$\operatorname{Spf}R_{\overline{r}}^{\tau,\operatorname{BT}}/\varpi\to{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$

at the point of ${\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$ corresponding to
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$. Since
${\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$ is an algebraic stack of finite
presentation over ${\mathbf{F}}$ (as ${\mathcal{Z}}^{\tau}$ is a $\varpi$-adic
formal algebraic stack of finite presentation over
$\operatorname{Spf}{\mathcal{O}}$), we may apply [Sta13, Tag 0DR0] to this
morphism so as to find a smooth morphism
$V\to{\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$ with source a finite type
${\mathcal{O}}/\varpi$-scheme, and a point $v\in V$ with residue field
${\mathbf{F}}^{\prime}$, such that there is an isomorphism
$\widehat{{\mathcal{O}}}_{V,v}\cong
R^{\tau,\operatorname{BT}}_{\overline{r}}/\varpi$, compatible with the given
morphism to ${\mathcal{Z}}^{\tau}_{/{\mathbf{F}}}$. Proposition 4.1 and Lemma
4.4 taken together show that $V$ is generically reduced, and so the result
follows from Lemma 4.5. ∎

##  5\. A case of the classical geometric Breuil–Mézard Conjecture

In this section, by combining the methods of [EG14] and [GK14] we prove a
special case of the classical geometric Breuil–Mézard conjecture [EG14, Conj.
4.2.1]. This result is “globalised” in Section 6.

Let $\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}})$ be a continuous
representation, and let $R^{\square}_{\overline{r}}$ be the universal framed
deformation ${\mathcal{O}}$-algebra for $\overline{r}$. In this section we
write $R^{\square}_{\overline{r},0,\tau}$ for the quotient of
$R^{\square}_{\overline{r}}$ that elswhere we have denoted
$R^{\tau,\operatorname{BT}}_{\overline{r}}$. We use the more cumbersome
notation $R^{\square}_{\overline{r},0,\tau}$ here to make it easier for the
reader to refer to [EG14] and [GK14].

By [EG14, Prop. 4.1.2], $R^{\square}_{\overline{r},0,\tau}/\varpi$ is zero if
$\overline{r}$ has no potentially Barsotti–Tate lifts of type $\tau$, and
otherwise it is equidimensional of dimension $4+[K:{\mathbf{Q}}_{p}]$. Each
$\operatorname{Spec}R^{\square}_{\overline{r},0,\tau}/\varpi$ is a closed
subscheme of $\operatorname{Spec}R^{\square}_{\overline{r}}/\varpi$, and we
write $Z(R^{\square}_{\overline{r},0,\tau}/\varpi)$ for the corresponding
cycle, as in [EG14, Defn. 2.2.5]. This is a formal sum of the irreducible
components of $\operatorname{Spec}R^{\square}_{\overline{r},0,\tau}/\varpi$,
weighted by the multiplicities with which they occur.

######  Lemma 5.1.

If $\sigma$ is a non-Steinberg Serre weight of $\operatorname{GL}_{2}(k)$,
then there are integers $n_{\tau}(\sigma)$ such that
$\sigma=\sum_{\tau}n_{\tau}(\sigma)\overline{\sigma}(\tau)$ in the
Grothendieck group of mod $p$ representations of $\operatorname{GL}_{2}(k)$,
where the $\tau$ run over the tame types.

###### Proof.

This is an immediate consequence of the surjectivity of the natural map from
the Grothendieck group of $\overline{{\mathbf{Q}}}_{p}$-representations of
$\operatorname{GL}_{2}(k)$ to the Grothendieck group of
$\overline{{\mathbf{F}}}_{p}$-representations of $\operatorname{GL}_{2}(k)$
[Ser77, §III, Thm. 33], together with the observation that the reduction of
the Steinberg representation of $\operatorname{GL}_{2}(k)$ is precisely
$\overline{\sigma}_{\vec{0},\vec{p-1}}$. ∎

Let $\sigma$ be a non-Steinberg Serre weight of $\operatorname{GL}_{2}(k)$, so
that by Lemma 5.1 we can write

(5.1.1) $\sigma=\sum_{\tau}n_{\tau}(\sigma)\overline{\sigma}(\tau)$

in the Grothendieck group of mod $p$ representations of
$\operatorname{GL}_{2}(k)$. Note that the integers $n_{\tau}(\sigma)$ are not
uniquely determined; however, all our constructions elsewhere in this paper
will be (non-obviously!) independent of the choice of the $n_{\tau}(\sigma)$.
We also write

$\overline{\sigma}(\tau)=\sum_{\sigma}m_{\sigma}(\tau)\sigma;$

since $\overline{\sigma}(\tau)$ is multiplicity-free, each $m_{\sigma}(\tau)$
is equal to $0$ or $1$. Then

$\sigma=\sum_{\sigma^{\prime}}\left(\sum_{\tau}n_{\tau}(\sigma)m_{\sigma^{\prime}}(\tau)\right)\sigma^{\prime},$

and therefore

(5.1.2)
$\sum_{\tau}n_{\tau}(\sigma)m_{\sigma^{\prime}}(\tau)=\delta_{\sigma,\sigma^{\prime}}.$

For each non-Steinberg Serre weight $\sigma$, we set

${\mathcal{C}}_{\sigma}:=\sum_{\tau}n_{\tau}(\sigma)Z(R^{\square}_{\overline{r},0,\tau}/\varpi),$

where the sum ranges over the tame types $\tau$, and the integers
$n_{\tau}(\sigma)$ are as in (5.1.1). By definition this is a formal sum with
(possibly negative) multiplicities of irreducible subschemes of
$\operatorname{Spec}R^{\square}_{\overline{r}}/\varpi$; recall that we say
that it is _effective_ if all of the multiplicities are non-negative.

######  Theorem 5.2.

Let $\sigma$ be a non-Steinberg Serre weight. Then the cycle
${\mathcal{C}}_{\sigma}$ is effective, and is nonzero precisely when
$\sigma\in W(\overline{r})$. It is independent of the choice of integers
$n_{\tau}(\sigma)$ satisfying (5.1.1). For each tame type $\tau$, we have

$Z(R^{\square}_{\overline{r},0,\tau}/\varpi)=\sum_{\sigma\in\operatorname{JH}(\overline{\sigma}(\tau))}{\mathcal{C}}_{\sigma}.$

###### Proof.

We will argue exactly as in the proof of [EG14, Thm. 5.5.2] (taking $n=2$),
and we freely use the notation and definitions of [EG14]. Since $p>2$, we have
$p\nmid n$ and thus a suitable globalisation $\overline{\rho}$ exists provided
that [EG14, Conj. A.3] holds for $\overline{r}$. Exactly as in the proof of
[EG14, Thm. 5.5.4], this follows from the proof of Theorem A.1.2 of [GK14]
(which shows that $\overline{r}$ has a potentially Barsotti–Tate lift) and
Lemma 4.4.1 of _op.cit_. (which shows that any potentially Barsotti–Tate
representation is potentially diagonalizable). These same results also show
that the equivalent conditions of [EG14, Lem. 5.5.1] hold in the case that
$\lambda_{v}=0$ for all $v$, and in particular in the case that
$\lambda_{v}=0$ and $\tau_{v}$ is tame for all $v$, which is all that we will
require.

By [EG14, Lem. 5.5.1(5)], we see that for each choice of tame types
$\tau_{v}$, we have

(5.2.1)
$Z(\overline{R}_{\infty}/\varpi)=\sum_{\otimes_{v|p}\sigma_{v}}\prod_{v|p}m_{\sigma_{v}}(\tau_{v})Z^{\prime}_{\otimes_{v|p}\sigma_{v}}(\overline{\rho}).$

Now, by definition we have

(5.2.2)
$Z({\bar{R}_{\infty}}/\varpi)=\prod_{v|p}Z(R^{\square}_{\overline{r},0,\tau_{v}}/\varpi)\times
Z({\mathbf{F}}[[[x_{1},\dots,x_{q-[F^{+}:{\mathbf{Q}}]n(n-1)/2},t_{1},\dots,t_{n^{2}}]]).$

Fix a non-Steinberg Serre weight $\sigma=\otimes_{v}\sigma_{v}$, and sum over
all choices of types $\tau_{v}$, weighted by
$\prod_{v|p}n_{\tau_{v}}(\sigma_{v})$. We obtain

$\sum_{\tau}\prod_{v|p}n_{\tau_{v}}(\sigma_{v})\prod_{v|p}Z(R^{\square}_{\overline{r},0,\tau_{v}}/\varpi)\times
Z({\mathbf{F}}[[[x_{1},\dots,x_{q-[F^{+}:{\mathbf{Q}}]n(n-1)/2},t_{1},\dots,t_{n^{2}}]])$
$=\sum_{\tau}\prod_{v|p}n_{\tau_{v}}(\sigma_{v})\sum_{\otimes_{v|p}\sigma^{\prime}_{v}}\prod_{v|p}m_{\sigma^{\prime}_{v}}(\tau_{v})Z^{\prime}_{\otimes_{v|p}\sigma^{\prime}_{v}}(\overline{\rho})$

which by (5.1.2) simplifies to

(5.2.3) $\prod_{v|p}{\mathcal{C}}_{\sigma_{v}}\times
Z({\mathbf{F}}[[[x_{1},\dots,x_{q-[F^{+}:{\mathbf{Q}}]n(n-1)/2},t_{1},\dots,t_{n^{2}}]])=Z^{\prime}_{\otimes_{v|p}\sigma_{v}}(\overline{\rho}).$

Since $Z^{\prime}_{\otimes_{v|p}\sigma_{v}}(\overline{\rho})$ is effective by
definition (as it is defined as a positive multiple of the support cycle of a
patched module), this shows that every $\prod_{v|p}{\mathcal{C}}_{\sigma_{v}}$
is effective. We conclude that either every ${\mathcal{C}}_{\sigma}$ is
effective, or that every $-{\mathcal{C}}_{\sigma}$ is effective.

Substituting (5.2.3) and (5.2.2) into (5.2.1), we see that

(5.2.4) $\prod_{v|p}Z(R^{\square}_{\overline{r},0,\tau_{v}}/\varpi)\times
Z({\mathbf{F}}[[[x_{1},\dots,x_{q-[F^{+}:{\mathbf{Q}}]n(n-1)/2},t_{1},\dots,t_{n^{2}}]])$
$=\prod_{v|p}\left(\sum_{\sigma\in\operatorname{JH}(\overline{\sigma}(\tau))}{\mathcal{C}}_{\sigma_{v}}\right)\times
Z({\mathbf{F}}[[[x_{1},\dots,x_{q-[F^{+}:{\mathbf{Q}}]n(n-1)/2},t_{1},\dots,t_{n^{2}}]],$

and we deduce that either
$Z(R^{\square}_{\overline{r},0,\tau}/\varpi)=\sum_{\sigma}m_{\sigma}(\tau){\mathcal{C}}_{\sigma}$
for all $\tau$, or
$Z(R^{\square}_{\overline{r},0,\tau}/\varpi)=-\sum_{\sigma}m_{\sigma}(\tau){\mathcal{C}}_{\sigma}$
for all $\tau$.

Since each $Z(R^{\square}_{\overline{r},0,\tau}/\varpi)$ is effective, the
second possibility holds if and only if every $-{\mathcal{C}}_{\sigma}$ is
effective (since either all the $-{\mathcal{C}}_{\sigma}$ are effective, or
all the ${\mathcal{C}}_{\sigma}$ are effective). It remains to show that this
possibility leads to a contradiction. Now, if
$Z(R^{\square}_{\overline{r},0,\tau}/\varpi)=-\sum_{\sigma}m_{\sigma}(\tau){\mathcal{C}}_{\sigma}$
for all $\tau$, then substituting into the definition
${\mathcal{C}}_{\sigma}=\sum_{\tau}n_{\tau}(\sigma)Z(R^{\square}_{\overline{r},0,\tau}/\varpi)$,
we obtain

${\mathcal{C}}_{\sigma}=\sum_{\sigma^{\prime}}\left(\sum_{\tau}n_{\tau}(\sigma)m_{\sigma^{\prime}}(\tau)\right)\left(-{\mathcal{C}}_{\sigma^{\prime}}\right),$

and applying (5.1.2), we obtain
${\mathcal{C}}_{\sigma}=-{\mathcal{C}}_{\sigma}$, so that
${\mathcal{C}}_{\sigma}=0$ for all $\sigma$. Thus all the
${\mathcal{C}}_{\sigma}$ are effective, as claimed.

Since $Z^{\prime}_{\otimes_{v|p}\sigma_{v}}(\overline{\rho})$ by definition
depends only on (the global choices in the Taylor–Wiles method, and)
$\otimes_{v|p}\sigma_{v}$, and _not_ on the particular choice of the
$n_{\tau}(\sigma)$, it follows from (5.2.3) that ${\mathcal{C}}_{\sigma}$ is
also independent of this choice.

Finally, note that by definition
$Z^{\prime}_{\otimes_{v|p}\sigma_{v}}(\overline{\rho})$ is nonzero precisely
when $\sigma_{v}$ is in the set $W^{\operatorname{BT}}(\overline{r})$ defined
in [GK14, §3]; but by the main result of [GLS15], this is precisely the set
$W(\overline{r})$. ∎

######  Remark 5.3.

As we do not use wildly ramified types elsewhere in the paper, we have
restricted the statement of Theorem 5.2 to the case of tame types; but the
statement admits a natural extension to the case of wildly ramified inertial
types (with some components now occurring with multiplicity greater than one),
and the proof goes through unchanged in this more general setting.

##  6\. The geometric Breuil–Mézard conjecture for the stacks
${\mathcal{Z}}^{\mathrm{dd},1}$

We now prove our main results on the irreducible components of
${\mathcal{Z}}^{\mathrm{dd},1}$. We do this by a slightly indirect method,
defining certain formal sums of these irreducible components which we then
compute via the geometric Breuil–Mézard conjecture, and in particular the
results of Section 5.

By Theorem 3.2, ${\mathcal{Z}}^{\mathrm{dd},1}$ is reduced and
equidimensional, and each ${\mathcal{Z}}^{\tau,1}$ is a union of some of its
irreducible components. Let $K({\mathcal{Z}}^{\mathrm{dd},1})$ be the free
abelian group generated by the irreducible components of
${\mathcal{Z}}^{\mathrm{dd},1}$. We say that an element of
$K({\mathcal{Z}}^{\mathrm{dd},1})$ is _effective_ if the multiplicity of each
irreducible component is nonnegative. We say that an element of
$K({\mathcal{Z}}^{\mathrm{dd},1})$ is _reduced and effective_ if the
multiplicity of each irreducible component is $0$ or $1$.

Let $x$ be a finite type point of ${\mathcal{Z}}^{\mathrm{dd},1}$,
corresponding to a representation
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$, and
recall that $R^{\mathrm{dd},1}_{x}$ is a versal ring to
${\mathcal{Z}}^{\mathrm{dd},1}$ having each $R^{\tau,1}_{x}$ as a quotient.
Since ${\mathcal{Z}}^{\tau,1}$ is a union of irreducible components of
${\mathcal{Z}}^{\mathrm{dd},1}$, $\operatorname{Spec}R^{\tau,1}_{x}$ is a
union of irreducible components of $\operatorname{Spec}R^{\mathrm{dd},1}_{x}$.

Let $K(R^{\mathrm{dd},1}_{x})$ be the free abelian group generated by the
irreducible components of $\operatorname{Spec}R^{\mathrm{dd},1}_{x}$. By
[Sta13, Tag 0DRB,Tag 0DRD], there is a natural multiplicity-preserving
surjection from the set of irreducible components of
$\operatorname{Spec}R^{\mathrm{dd},1}_{x}$ to the set of irreducible
components of ${\mathcal{Z}}^{\mathrm{dd},1}$ which contain $x$. Using this
surjection, we can define a group homomorphism

$K({\mathcal{Z}}^{\mathrm{dd},1})\to K(R^{\mathrm{dd},1}_{x})$

in the following way: we send any irreducible component ${\mathcal{Z}}$ of
${\mathcal{Z}}^{\mathrm{dd},1}$ which contains $x$ to the formal sum of the
irreducible components of $\operatorname{Spec}R^{\mathrm{dd},1}_{x}$ in the
preimage of ${\mathcal{Z}}$ under this surjection, and we send every other
irreducible component to $0$.

######  Lemma 6.1.

An element $\overline{{\mathcal{T}}}$ of $K({\mathcal{Z}}^{\mathrm{dd},1})$ is
effective if and only if for every finite type point $x$ of
${\mathcal{Z}}^{\mathrm{dd},1}$, the image of $\overline{{\mathcal{T}}}$ in
$K(R^{\mathrm{dd},1}_{x})$ is effective. We have $\overline{{\mathcal{T}}}=0$
if and only if its image is $0$ in every $K(R^{\mathrm{dd},1}_{x})$.

###### Proof.

The “only if” direction is trivial, so we need only consider the “if”
implication. Write
$\overline{{\mathcal{T}}}=\sum_{\overline{{\mathcal{Z}}}}a_{\overline{{\mathcal{Z}}}}\overline{{\mathcal{Z}}}$,
where the sum runs over the irreducible components $\overline{{\mathcal{Z}}}$
of ${\mathcal{Z}}^{\mathrm{dd},1}$, and the $a_{\overline{{\mathcal{Z}}}}$ are
integers.

Suppose first that the image of $\overline{{\mathcal{T}}}$ in
$K(R^{\mathrm{dd},1}_{x})$ is effective for every $x$; we then have to show
that each $a_{\overline{{\mathcal{Z}}}}$ is nonnegative. To see this, fix an
irreducible component $\overline{{\mathcal{Z}}}$, and choose $x$ to be a
finite type point of ${\mathcal{Z}}^{\mathrm{dd},1}$ which is contained in
$\overline{{\mathcal{Z}}}$ and in no other irreducible component of
${\mathcal{Z}}^{\mathrm{dd},1}$. Then the image of $\overline{{\mathcal{T}}}$
in $K(R^{\mathrm{dd},1}_{x})$ is equal to $a_{\overline{{\mathcal{Z}}}}$ times
a nonempty sum of irreducible components of
$\operatorname{Spec}R^{\mathrm{dd},1}_{x}$. By hypothesis, this must be
effective, which implies that $a_{\overline{{\mathcal{Z}}}}$ is nonnegative,
as required.

Finally, if the image of $\overline{{\mathcal{T}}}$ in
$K(R^{\mathrm{dd},1}_{x})$ is $0$, then $a_{\overline{{\mathcal{Z}}}}=0$; so
if this holds for all $x$, then $\overline{{\mathcal{T}}}=0$. ∎

For each tame type $\tau$, we let ${\mathcal{Z}}(\tau)$ denote the formal sum
of the irreducible components of ${\mathcal{Z}}^{\tau,1}$, considered as an
element of $K({\mathcal{Z}}^{\mathrm{dd},1})$. By Lemma 5.1, for each non-
Steinberg Serre weight $\sigma$ of $\operatorname{GL}_{2}(k)$ there are
integers $n_{\tau}(\sigma)$ such that
$\sigma=\sum_{\tau}n_{\tau}(\sigma)\overline{\sigma}(\tau)$ in the
Grothendieck group of mod $p$ representations of $\operatorname{GL}_{2}(k)$,
where the $\tau$ run over the tame types. We set

${\mathcal{Z}}(\sigma):=\sum_{\tau}n_{\tau}(\sigma){\mathcal{Z}}(\tau)\in
K({\mathcal{Z}}^{\mathrm{dd},1}).$

The integers $n_{\tau}(\sigma)$ are not necessarily unique, but it follows
from the following result that ${\mathcal{Z}}(\sigma)$ is independent of the
choice of $n_{\tau}(\sigma)$, and is reduced and effective.

######  Theorem 6.2.

  1. (1)

Each ${\mathcal{Z}}(\sigma)$ is an irreducible component of
${\mathcal{Z}}^{\mathrm{dd},1}$.

  2. (2)

The finite type points of ${\mathcal{Z}}(\sigma)$ are precisely the
representations
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$ having
$\sigma$ as a Serre weight.

  3. (3)

For each tame type $\tau$, we have
${\mathcal{Z}}(\tau)=\sum_{\sigma\in\operatorname{JH}(\overline{\sigma}(\tau))}{\mathcal{Z}}(\sigma)$.

  4. (4)

Every irreducible component of ${\mathcal{Z}}^{\mathrm{dd},1}$ is of the form
${\mathcal{Z}}(\sigma)$ for some unique Serre weight $\sigma$.

  5. (5)

For each tame type $\tau$, and each $J\in{\mathcal{P}}_{\tau}$, we have
${\mathcal{Z}}(\overline{\sigma}(\tau)_{J})=\overline{{\mathcal{Z}}}(J)$.

###### Proof.

Let $x$ be a finite type point of ${\mathcal{Z}}^{\mathrm{dd},1}$
corresponding to
$\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$, and write
${\mathcal{Z}}(\sigma)_{x}$, ${\mathcal{Z}}(\tau)_{x}$ for the images in
$K(R^{\mathrm{dd},1}_{x})$ of ${\mathcal{Z}}(\sigma)$ and
${\mathcal{Z}}(\tau)$ respectively. Each $\operatorname{Spec}R^{\tau,1}_{x}$
is a closed subscheme of $\operatorname{Spec}R^{\square}$, the universal
framed deformation ${\mathcal{O}}_{E^{\prime}}$-algebra for $\overline{r}$, so
we may regard the ${\mathcal{Z}}(\tau)_{x}$ as formal sums (with
multiplicities) of irreducible subschemes of
$\operatorname{Spec}R^{\square}/\pi$.

By definition, ${\mathcal{Z}}(\tau)_{x}$ is just the underlying cycle of
$\operatorname{Spec}R^{\tau,1}_{x}$. By Theorem 4.6, this is equal to the
underlying cycle of
$\operatorname{Spec}R_{\overline{r}}^{\tau,\operatorname{BT}}/\varpi$.
Consequently, ${\mathcal{Z}}(\sigma)_{x}$ is the cycle denoted by
${\mathcal{C}}_{\sigma}$ in Section 5. It follows from Theorem 5.2 that:

  * •

${\mathcal{Z}}(\sigma)_{x}$ is effective, and is nonzero precisely when
$\sigma$ is a Serre weight for $\overline{r}$.

  * •

For each tame type $\tau$, we have
${\mathcal{Z}}(\tau)_{x}=\sum_{\sigma\in\operatorname{JH}(\overline{\sigma}(\tau))}{\mathcal{Z}}(\sigma)_{x}$.

Applying Lemma 6.1, we see that each ${\mathcal{Z}}(\sigma)$ is effective, and
that (3) holds. Since ${\mathcal{Z}}^{\tau,1}$ is reduced,
${\mathcal{Z}}(\tau)$ is reduced and effective, so it follows from (3) that
each ${\mathcal{Z}}(\sigma)$ is reduced and effective. Since $x$ is a finite
type point of ${\mathcal{Z}}(\sigma)$ if and only if
${\mathcal{Z}}(\sigma)_{x}\neq 0$, we have also proved (2).

Since every irreducible component of ${\mathcal{Z}}^{\mathrm{dd},1}$ is an
irreducible component of some ${\mathcal{Z}}^{\tau,1}$, in order to prove (1)
and (4) it suffices to show that for each $\tau$, every irreducible component
of ${\mathcal{Z}}^{\tau,1}$ is of the form
${\mathcal{Z}}(\overline{\sigma}(\tau)_{J})$ for some $J$, and that each
${\mathcal{Z}}(\overline{\sigma}(\tau)_{J})$ is irreducible. Since $\tau$ is
fixed for the rest of the argument, let us simplify notation by writing
$\overline{\sigma}_{J}$ for $\overline{\sigma}(\tau)_{J}$. Now, by Theorem
3.5(2), we know that ${\mathcal{Z}}^{\tau,1}$ has exactly
$\\#{\mathcal{P}}_{\tau}$ irreducible components, namely the
$\overline{{\mathcal{Z}}}(J^{\prime})$ for
$J^{\prime}\in{\mathcal{P}}_{\tau}$. On the other hand, the
${\mathcal{Z}}(\overline{\sigma}_{J})$ are reduced and effective, and since
there certainly exist representations admitting $\overline{\sigma}_{J}$ as
their unique Serre weight, it follows from (2) that for each $J$, there must
be a $J^{\prime}\in{\mathcal{P}}_{\tau}$ such that
$\overline{{\mathcal{Z}}}(J^{\prime})$ contributes to
${\mathcal{Z}}(\overline{\sigma}_{J})$, but not to any
${\mathcal{Z}}(\overline{\sigma}_{J^{\prime\prime}})$ for
$J^{\prime\prime}\neq J$.

Since ${\mathcal{Z}}(\tau)$ is reduced and effective, and the sum in (3) is
over $\\#{\mathcal{P}}_{\tau}$ weights $\sigma$, it follows that we in fact
have
${\mathcal{Z}}(\overline{\sigma}_{J})=\overline{{\mathcal{Z}}}(J^{\prime})$.
This proves (1) and (4), and to prove (5), it only remains to show that
$J^{\prime}=J$. To see this, note that by (2),
${\mathcal{Z}}(\overline{\sigma}_{J})=\overline{{\mathcal{Z}}}(J^{\prime})$
has a dense open substack whose finite type points have
$\overline{\sigma}_{J}$ as their unique non-Steinberg Serre weight (namely the
complement of the union of the ${\mathcal{Z}}(\sigma^{\prime})$ for all
$\sigma^{\prime}\neq\overline{\sigma}_{J}$). By Theorem 3.5(4), it also has a
dense open substack whose finite type points have
$\overline{\sigma}_{J^{\prime}}$ as a Serre weight. Considering any finite
type point in the intersection of these dense open substacks, we see that
$\overline{\sigma}_{J}=\overline{\sigma}_{J^{\prime}}$, so that
$J=J^{\prime}$, as required. ∎

##  7\. The geometric Breuil–Mézard conjecture for the stacks
${\mathcal{X}}_{2,\operatorname{red}}$

We now explain how to transfer our results from the stacks
${\mathcal{Z}}^{\mathrm{dd},1}$ to the stacks
${\mathcal{X}}_{2,\operatorname{red}}$ of [EG22]. The book [EG22] establishes
an equivalence between the classical “numerical” Breuil–Mézard conjecture and
the geometric Breuil–Mézard conjecture for the stacks
${\mathcal{X}}_{2,\operatorname{red}}$. (Indeed the implication from the
former to the latter only requires the classical Breuil–Mézard conjecture at a
single sufficiently generic point of each component of
${\mathcal{X}}_{2,\operatorname{red}}$.)

The Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate
respresentations is established in [GK14], and this is extended to two-
dimensional potentially semistable representations of Hodge type $0$ in [EG22,
Thm. 8.6.1]. The arguments of [EG22] translate this into the following
theorem. Here ${\mathcal{X}}_{2}^{\tau,\operatorname{BT},\mathrm{ss}}$ denotes
the stack of two-dimensional potentially semistable representations of Hodge
type $0$ constructed in [EG22], while $\sigma^{\mathrm{ss}}(\tau)$ is as in
[EG22, Thm. 8.2.1].

######  Theorem 7.1 ([EG22]).

There exist effective cycles $Z^{\sigma}$ (elements of the free group on the
irreducible components of ${\mathcal{X}}_{2,\operatorname{red}}$, with
nonnegative coefficients) such that for all inertial types $\tau$,

  * •

the cycle of the special fibre of ${\mathcal{X}}_{2}^{\tau,\operatorname{BT}}$
is equal to $\sum_{\sigma}m_{\sigma}(\tau)\cdot Z^{\sigma}$, while

  * •

the cycle of the special fibre of
${\mathcal{X}}_{2}^{\tau,\operatorname{BT},\mathrm{ss}}$ is equal to
$\sum_{\sigma}m^{\mathrm{ss}}_{\sigma}(\tau)\cdot Z^{\sigma}$.

Here $\overline{\sigma}(\tau)=\sum_{\sigma}m_{\sigma}(\tau)\cdot\sigma$ and
$\overline{\sigma}^{\mathrm{ss}}(\tau)=\sum_{\sigma}m^{\mathrm{ss}}_{\sigma}(\tau)\cdot\sigma$
in the Grothendieck group of $\operatorname{GL}_{2}(k)$.

######  Corollary 7.2 ([EG22]).

Let $\overline{r}:G_{K}\to\operatorname{GL}_{2}({\mathbf{F}}^{\prime})$ be a
continuous Galois representation, corresponding to a finite type point of
${\mathcal{X}}_{2,\operatorname{red}}$. For each Serre weight $\sigma$ we have
$\sigma\in W(\overline{r})$ if and only if $\overline{r}$ lies in the support
of $Z^{\sigma}$.

###### Proof.

This follows by the argument in [EG22, §8.4]: the Breuil–Mézard multiplicity
$\mu_{\sigma}(\overline{r})$ is nonzero if and only if $Z^{\sigma}$ is
supported at $\overline{r}$. More precisely, this shows that $\overline{r}$
lies in the support of $Z^{\sigma}$ if and only if $\sigma$ is an element of
the weight set
$W_{\operatorname{BT}}(\overline{r})=\\{\sigma:\mu_{\sigma}(\overline{r})>0\\}$.
But $W_{\operatorname{BT}}(\overline{r})=W(\overline{r})$ by the main results
of [GLS15]. ∎

######  Remark 7.3.

The cycles $Z^{\sigma}$ are constructed as follows in the proof of Theorem
7.1. For each Serre weight $\sigma^{\prime}$, the smooth points of
${\mathcal{X}}^{\sigma^{\prime}}_{2,\operatorname{red}}$ that furthermore do
not lie on any other component of ${\mathcal{X}}_{2,\operatorname{red}}$ are
dense. Choose such a point $\overline{r}_{\sigma^{\prime}}$ and let
$\\{\mu_{\sigma}(\overline{r}_{\sigma^{\prime}})\\}$ be the multiplicities in
the Breuil–Mézard conjecture for $\overline{r}_{\sigma^{\prime}}$. Then
$Z^{\sigma}:=\sum_{\sigma^{\prime}}\mu_{\sigma}(\overline{r}_{\sigma^{\prime}})\cdot{\mathcal{X}}^{\sigma^{\prime}}_{2,\operatorname{red}}$.

It remains to compute the cycles $Z^{\sigma}$. We begin with the following
observation, which could be proved with modest effort by calculating
dimensions of families of extensions and using the results of [GLS15], but is
also easily deduced from the results of Section 6.

######  Lemma 7.4.

Let $\sigma,\sigma^{\prime}$ be Serre weights and suppose that
$\sigma^{\prime}$ is non-Steinberg. Then
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ contains at least one finite
type point corresponding to a representation $\overline{r}$ with
$\sigma^{\prime}\not\in W(\overline{r})$.

###### Proof.

Suppose first that $\sigma$ is non-Steinberg. By Theorem 6.2 the component
${\mathcal{Z}}(\sigma)$ of ${\mathcal{Z}}^{\mathrm{dd},1}$ has a dense open
set $U(\sigma)$ whose finite type points $\overline{r}$ have no non-Steinberg
Serre weights other than $\sigma$: take
${\mathcal{Z}}(\sigma)\setminus\cup_{\sigma^{\prime}\neq\sigma}{\mathcal{Z}}(\sigma^{\prime})$.
The finite type points of ${\mathcal{Z}}(\sigma)$ described in Remark 3.6 are
also dense; therefore at least one of them (indeed a dense set of them) lies
in $U(\sigma)$. Let $\overline{r}$ be such a representation. The finite type
points of ${\mathcal{Z}}(\sigma)$ described in Remark 3.6 are precisely the
family of niveau $1$ representations in the description of
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ of Theorem 1.1 (see also the
construction in [EG22, §5.5]). So $\overline{r}$ lies on
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ as well, and by construction
the only non-Steinberg weight in $W(\overline{r})$ is $\sigma$. This completes
the non-Steinberg case.

If instead $\sigma$ is Steinberg, then by construction
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ contains representations
$\overline{r}$ that are très ramifié. But très ramifié representations have no
non-Steinberg Serre weights by [CEGS20b, Lem. A.4]. ∎

######  Remark 7.5.

Once we have proved Theorem 7.6 below, “at least one finite type point” in the
statement of Lemma 7.4 can be promoted to “a dense set of finite type points”
by taking
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}\setminus\cup_{\sigma^{\prime}\neq\sigma}{\mathcal{X}}^{\sigma^{\prime}}_{2,\operatorname{red}}$.

We now reach our main theorem.

######  Theorem 7.6.

Suppose $p>2$. We have:

  * •

$Z^{\sigma}=\mathcal{X}_{2,\operatorname{red}}^{\sigma}$, if the weight
$\sigma$ is not Steinberg, while

  * •

$Z^{\chi\otimes\mathrm{St}}=\mathcal{X}_{2,\operatorname{red}}^{\chi}+\mathcal{X}_{2,\operatorname{red}}^{\chi\otimes\mathrm{St}}$
if the weight $\sigma\cong\chi\otimes\mathrm{St}$ is Steinberg.

In particular $\sigma\in W(\overline{r})$ if and only if $\overline{r}$ lies
on ${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ if $\sigma$ is not
Steinberg, or on
$\mathcal{X}_{2,\operatorname{red}}^{\chi}\cup\mathcal{X}_{2,\operatorname{red}}^{\chi\otimes\mathrm{St}}$
if $\sigma\cong\chi\otimes\mathrm{St}$ is Steinberg.

Essentially the same statement appears at [EG22, Thm 8.6.2], and the argument
given there invokes an earlier version of this paper222As mentioned in the
introduction, the reference [CEGS19, Thm. 5.2.2] in [EG22] is Theorem 6.2 of
this paper, while the reference [CEGS19, Lem. B.5] in [EG22] is [CEGS20b, Lem.
A.4].. The proof we give below has the same major beats as the argument in
[EG22], but is rearranged to delay until as late as possible making any
references to the earlier parts of this paper. Indeed the proof now only
invokes the generic reducedness of Theorem 4.6 (but certainly invokes it in a
crucial way); we hope that this clarifies the various dependencies involved.

However, we also take the opportunity to repair a small gap in the argument
given at [EG22, Thm 8.6.2]. It is claimed there that [CEGS19, Thm. 5.2.2(2)]
(i.e., our Theorem 6.2(2)) implies that
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ has a dense set of finite type
points corresponding to representations $\overline{r}$ whose only non-
Steinberg Serre weight is $\sigma$. Although the conclusion is certainly true,
the deduction is incorrect, or at least seems to presume that
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ can be identified with our
${\mathcal{Z}}(\sigma)$. We replace this claim with an argument using Lemma
7.4, which does follow from Theorem 6.2 (though as previously noted can also
be proved without it). The claim about points of
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$ will follow once Theorem 7.6
is proved, as explained in Remark 7.5.

###### Proof of Theorem 7.6.

Consider first the non-Steinberg case. The finite type points in the support
of the effective cycle $Z^{\sigma}$ are precisely the representations having
$\sigma$ as a Serre weight. By Lemma 7.4 each component
${\mathcal{X}}^{\sigma^{\prime}}_{2,\operatorname{red}}$ with
$\sigma^{\prime}\neq\sigma$ has a finite type point for which $\sigma$ is not
a Serre weight; therefore
${\mathcal{X}}^{\sigma^{\prime}}_{2,\operatorname{red}}$ cannot occur in the
cycle $Z^{\sigma}$. It follows that
$Z^{\sigma}=\mu_{\sigma}(\overline{r}_{\sigma})\cdot{\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$
with $\overline{r}_{\sigma}$ as in Remark 7.3, and that
$\mu_{\sigma}(\overline{r}_{\sigma^{\prime}})=0$ for all
$\sigma^{\prime}\neq\sigma$. Note that we can already deduce that $\sigma\in
W(\overline{r})$ if and only if $\overline{r}$ lies on
${\mathcal{X}}^{\sigma}_{2,\operatorname{red}}$.

Now consider the Steinberg case; by twisting it will suffice to consider the
weight $\sigma=\mathrm{St}$. The type $\sigma^{\mathrm{ss}}(\mathrm{triv})$ is
the Steinberg type, and so by Theorem 7.1 the cycle of the special fibre of
${\mathcal{X}}^{\mathrm{triv},\operatorname{BT},\mathrm{ss}}_{2}$ is equal to
$Z^{\mathrm{St}}$. In particular the finite type points of $Z^{\mathrm{St}}$
are precisely the representations $\overline{r}$ having a semistable lift of
Hodge type $0$. Such a lift is either crystalline, in which case the trivial
Serre weight is a Serre weight for $\overline{r}$, and by the previous
paragraph $\overline{r}$ is a finite type point of
${\mathcal{X}}^{\mathrm{triv}}_{2,\operatorname{red}}$ ; or else the lift is
semistable non-crystalline, in which case $\overline{r}$ is an unramified
twist of an extension of the inverse of the cyclotomic character by the
trivial character. Such an extension is either peu ramifiée, in which case
again $\mathrm{triv}\in W(\overline{r})$ and $\overline{r}$ lies on
${\mathcal{X}}^{\mathrm{triv}}_{2,\operatorname{red}}$; or else it is tres̀
ramifiée, and is a finite type point of
${\mathcal{X}}^{\mathrm{St}}_{2,\operatorname{red}}$ (as a member of the
family of niveau $1$ representations defining
${\mathcal{X}}^{\mathrm{St}}_{2,\operatorname{red}}$). Since all the finite
type points of the support of $Z^{\mathrm{St}}$ are contained in
${\mathcal{X}}^{\mathrm{triv}}_{2,\operatorname{red}}\cup{\mathcal{X}}^{\mathrm{St}}_{2,\operatorname{red}}$
it follows that
$Z^{\mathrm{St}}=\mu_{\mathrm{St}}(\overline{r}_{\mathrm{triv}}){\mathcal{X}}^{\mathrm{triv}}_{2,\operatorname{red}}+\mu_{\mathrm{St}}(\overline{r}_{\mathrm{St}}){\mathcal{X}}^{\mathrm{St}}_{2,\operatorname{red}}$,
and that $\mu_{\mathrm{St}}(\overline{r}_{\sigma^{\prime}})=0$ for all
$\sigma^{\prime}\neq\mathrm{triv},\mathrm{St}$.

It remains to determine the multiplicities
$\mu_{\sigma}(\overline{r}_{\sigma})$,
$\mu_{\mathrm{St}}(\overline{r}_{\mathrm{triv}})$, and
$\mu_{\mathrm{St}}(\overline{r}_{\mathrm{St}})$. Each of these is positive
because $\overline{r}_{\sigma}$ does have $\sigma$ as a Serre weight, and
similarly $\overline{r}_{\mathrm{triv}}$ has $\mathrm{St}$ as a Serre weight.
Suppose that $\sigma$ is non-Steinberg. Choose any tame type $\tau$ such that
$\overline{\sigma}(\tau)$ has $\sigma$ as a Jordan–Hölder factor. The ring
$R^{\tau,\operatorname{BT}}_{\overline{r}_{\sigma}}$ is versal to
${\mathcal{X}}^{\tau,\operatorname{BT}}_{2}$ at $\overline{r}_{\sigma}$, and
since $\overline{r}_{\sigma}$ is a smooth point of
${\mathcal{X}}_{2,\operatorname{red}}$ the underlying reduced of
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}_{\sigma}}/\varpi$
is smooth. But
$\operatorname{Spec}R^{\tau,\operatorname{BT}}_{\overline{r}_{\sigma}}/\varpi$
is generically reduced by Theorem 4.6. We deduce that the Hilbert–Samuel
multiplicity of $R^{\tau,\operatorname{BT}}_{\overline{r}_{\sigma}}/\varpi$ is
$1$, and therefore $\mu_{\sigma}(\overline{r}_{\sigma})\leq 1$. Since
$\mu_{\sigma}(\overline{r}_{\sigma})$ is positive it must be equal to $1$.

The point $\overline{r}_{\mathrm{triv}}$ may be assumed to be an extension of
an unramified twist of the inverse cyclotomic character by a _different_
unramified character (certainly these are dense in
${\mathcal{X}}^{\mathrm{triv}}_{2,\operatorname{red}}$). Then
$\overline{r}_{\mathrm{triv}}$ does not have any semistable non-crystalline
lifts, and the semistable Hodge type $0$ deformation ring of
$\overline{r}_{\mathrm{triv}}$ is simply a crystalline deformation ring,
indeed one of the flat deformation rings studied by Kisin in [Kis09]. The
argument in the previous paragraph showed that the Hilbert–Samuel multiplicity
of $R^{\mathrm{triv},\operatorname{BT}}_{\overline{r}_{\mathrm{cris}}}/\varpi$
is $1$. It follows that $\mu_{\mathrm{St}}(\overline{r}_{\mathrm{triv}})\leq
1$, and since it is positive it must be precisely $1$. Similarly the
semistable Hodge type $0$ deformation ring of the très ramifiée representation
$\overline{r}_{\mathrm{St}}$ is an ordinary deformation ring, hence formally
smooth, and we obtain $\mu_{\mathrm{St}}(\overline{r}_{\mathrm{St}})=1$.∎

######  Remark 7.7.

The finite type points of $X^{\mathrm{St}}_{2,\operatorname{red}}$ are
precisely those $\overline{r}$ having a semistable non-crystalline lift, i.e.,
the unramified twists of an extension of the inverse of the cyclotomic
character by the trivial character; for the details see [EG22, Lem. 8.6.4].

##  Appendix A A lemma on formal algebraic stacks

We suppose given a commutative diagram of morphisms of formal algebraic stacks

$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Spf}{\mathcal{O}}}$

We suppose that each of ${\mathcal{X}}$ and ${\mathcal{Y}}$ is quasi-compact
and quasi-separated, and that the horizontal arrow is scheme-theoretically
dominant, in the sense of [Eme, Def. 6.13]. We furthermore suppose that the
morphism ${\mathcal{X}}\to\operatorname{Spf}{\mathcal{O}}$ realises
${\mathcal{X}}$ as a finite type $\varpi$-adic formal algebraic stack.

Concretely, if we write
${\mathcal{X}}^{a}:={\mathcal{X}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a},$
then each ${\mathcal{X}}^{a}$ is an algebraic stack, locally of finite type
over $\operatorname{Spec}{\mathcal{O}}/\varpi^{a}$, and there is an
isomorphism
$\varinjlim_{a}{\mathcal{X}}^{a}\buildrel\sim\over{\longrightarrow}{\mathcal{X}}.$
Furthermore, the assumption that the horizontal arrow is scheme-theoretically
dominant means that we may find an isomorphism
${\mathcal{Y}}\cong\varinjlim_{a}{\mathcal{Y}}^{a}$, with each
${\mathcal{Y}}^{a}$ being a quasi-compact and quasi-separated algebraic stack,
and with the transition morphisms being thickenings, such that the morphism
${\mathcal{X}}\to{\mathcal{Y}}$ is induced by a compatible family of morphisms
${\mathcal{X}}^{a}\to{\mathcal{Y}}^{a}$, each of which is scheme-theoretically
dominant. (The ${\mathcal{Y}}^{a}$ are uniquely determined by the requirement
that for all $b\geq a$ large enough so that the morphism
${\mathcal{X}}^{a}\to{\mathcal{Y}}$ factors through
${\mathcal{Y}}\otimes_{\mathcal{O}}{\mathcal{O}}/\varpi^{b}$,
${\mathcal{Y}}^{a}$ is the scheme-theoretic image of the morphism
${\mathcal{X}}^{a}\to{\mathcal{Y}}\otimes_{\mathcal{O}}{\mathcal{O}}/\varpi^{b}$.
In particular, ${\mathcal{Y}}^{a}$ is a closed substack of
${\mathcal{Y}}\times_{\mathcal{O}}{\mathcal{O}}/\varpi^{a}$.)

It is often the case, in the preceding situation, that ${\mathcal{Y}}$ is also
a $\varpi$-adic formal algebraic stack. For example, we have the following
result. (Note that the usual graph argument shows that the morphism
${\mathcal{X}}\to{\mathcal{Y}}$ is necessarily algebraic, i.e. representable
by algebraic stacks, in the sense of [Sta13, Tag 06CF] and [Eme, Def. 3.1].
Thus it makes sense to speak of it being proper, following [Eme, Def. 3.11].)

######  Proposition A.1.

Suppose that the morphism ${\mathcal{X}}\to{\mathcal{Y}}$ is proper, and that
${\mathcal{Y}}$ is locally Ind-finite type over
$\operatorname{Spec}{\mathcal{O}}$ (in the sense of [Eme, Rem. 8.30]). Then
${\mathcal{Y}}$ is a $\varpi$-adic formal algebraic stack.

###### Proof.

This is an application of [Eme, Prop. 10.5]. ∎

A key point is that, because the formation of scheme-theoretic images is not
generally compatible with non-flat base-change, the closed immersion

(A.2)
${\mathcal{Y}}^{a}\hookrightarrow{\mathcal{Y}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$

is typically not an isomorphism, even if ${\mathcal{Y}}$ is a $\varpi$-adic
formal algebraic stack. Our goal in the remainder of this discussion is to
give a criterion (involving the morphism ${\mathcal{X}}\to{\mathcal{Y}}$) on
an open substack ${\mathcal{U}}\hookrightarrow{\mathcal{Y}}$ which guarantees
that the closed immersion
${\mathcal{U}}\times_{{\mathcal{Y}}}{\mathcal{Y}}^{a}\hookrightarrow{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$
induced by (A.2) is an isomorphism.

We begin by establishing a simple lemma. For any $a\geq 1$, we have the
$2$-commutative diagram

(A.3)
$\textstyle{{\mathcal{X}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Y}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}}$

Similarly, if $b\geq a\geq 1$, then we have the $2$-commutative diagram

(A.4)
$\textstyle{{\mathcal{X}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Y}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Y}}^{b}}$

######  Lemma A.5.

Each of the diagrams (A.3) and (A.4) is $2$-Cartesian.

###### Proof.

We may embed the diagram (A.3) in the larger $2$-commutative diagram

$\textstyle{{\mathcal{X}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Y}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$(A.2)$\textstyle{{\mathcal{Y}}\otimes_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}}$

Since the outer rectangle is manifestly $2$-Cartesian, and since (A.2) is a
closed immersion (and thus a monomorphism), we conclude that (A.3) is indeed
$2$-Cartesian.

A similar argument shows that (A.4) is $2$-Cartesian. ∎

We next note that, since each of the closed immersions
${\mathcal{Y}}^{a}\hookrightarrow{\mathcal{Y}}$ is a thickening, giving an
open substack ${\mathcal{U}}\hookrightarrow{\mathcal{Y}}$ is equivalent to
giving an open substack ${\mathcal{U}}^{a}\hookrightarrow{\mathcal{Y}}^{a}$
for some, or equivalently, every, choice of $a\geq 1$; the two pieces of data
are related by the formulas
${\mathcal{U}}^{a}:={\mathcal{U}}\times_{{\mathcal{Y}}}{\mathcal{Y}}^{a}$ and
$\varinjlim_{a}{\mathcal{U}}^{a}\buildrel\sim\over{\longrightarrow}{\mathcal{U}}.$

######  Lemma A.6.

Suppose that ${\mathcal{X}}\to{\mathcal{Y}}$ is proper. If ${\mathcal{U}}$ is
an open substack of ${\mathcal{Y}}$, then the following conditions are
equivalent:

  1. (1)

The morphism
${\mathcal{X}}\times_{{\mathcal{Y}}}{\mathcal{U}}\to{\mathcal{U}}$ is a
monomorphism.

  2. (2)

The morphism
${\mathcal{X}}\times_{{\mathcal{Y}}}{\mathcal{U}}\to{\mathcal{U}}$ is an
isomorphism.

  3. (3)

For every $a\geq 1$, the morphism
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a}$
is a monomorphism.

  4. (4)

For every $a\geq 1$, the morphism
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a}$
is an isomorphism.

  5. (5)

For some $a\geq 1$, the morphism
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a}$
is a monomorphism.

  6. (6)

For some $a\geq 1$, the morphism
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a}$
is an isomorphism.

Furthermore, if these equivalent conditions hold, then the closed immersion
${\mathcal{U}}^{a}\hookrightarrow{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$
is an isomorphism, for each $a\geq 1$.

###### Proof.

The key point is that Lemma A.5 implies that the diagram

$\textstyle{{\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{U}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\times_{{\mathcal{Y}}}{\mathcal{U}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{U}}$

is $2$-Cartesian, for any $a\geq 1$, and similarly, that if $b\geq a\geq 1,$
then the diagram

$\textstyle{{\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{U}}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{b}\times_{{\mathcal{Y}}^{b}}{\mathcal{U}}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{U}}^{b}}$

is $2$-Cartesian. Since the vertical arrows of this latter diagram are finite
order thickenings, we find (by applying the analogue of [Sta13, Tag 09ZZ] for
algebraic stacks, whose straightforward deduction from that result we leave to
the reader) that the top horizontal arrow is a monomorphism if and only if the
bottom horizontal arrow is. This shows the equivalence of (3) and (5). Since
the morphism
${\mathcal{X}}\times_{{\mathcal{Y}}}{\mathcal{U}}\to{\mathcal{U}}$ is obtained
as the inductive limit of the various morphisms
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a},$
we find that (3) implies (1) (by applying e.g. [Eme, Lem. 4.11 (1)], which
shows that the inductive limit of monomorphisms is a monomorphism), and also
that (4) implies (2) (the inductive limit of isomorphisms being again an
isomorphism).

Conversely, if (1) holds, then the base-changed morphism

${\mathcal{X}}\times_{{\mathcal{Y}}}({\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a})\to{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$

is a monomorphism. The source of this morphism admits an alternative
description as ${\mathcal{X}}^{a}\times_{{\mathcal{Y}}}{\mathcal{U}},$ which
the $2$-Cartesian diagram at the beginning of the proof allows us to identify
with ${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}$. Thus we
obtain a monomorphism

${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\hookrightarrow{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}_{/}\varpi^{a}.$

Since this monomorphism factors through the closed immersion
${\mathcal{U}}^{a}\hookrightarrow{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a},$
we find that each of the morphisms of (3) is a monomorphism; thus (1) implies
(3). Similarly, (2) implies (4), and also implies that the closed immersion
${\mathcal{U}}^{a}\hookrightarrow{\mathcal{U}}\times_{{\mathcal{O}}}{\mathcal{O}}/\varpi^{a}$
is an isomorphism, for each $a\geq 1$.

Since clearly (4) implies (6), while (6) implies (5), to complete the proof of
the proposition, it suffices to show that (5) implies (6). Suppose then that
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a}$
is a monomorphism. Since ${\mathcal{U}}^{a}\hookrightarrow{\mathcal{Y}}^{a}$
is an open immersion, it is in particular flat. Since
${\mathcal{X}}^{a}\to{\mathcal{Y}}^{a}$ is scheme-theoretically dominant and
quasi-compact (being proper), any flat base-change of this morphism is again
scheme-theoretically dominant, as well as being proper. Thus we see that
${\mathcal{X}}^{a}\times_{{\mathcal{Y}}^{a}}{\mathcal{U}}^{a}\to{\mathcal{U}}^{a}$
is a scheme-theoretically dominant proper monomorphism, i.e. a scheme-
theoretically dominant closed immersion, i.e. an isomorphism, as required. ∎

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