On some mod p representations of quaternion algebra over ℚ_p

Abstract

Let $F$ be a totally real field in which $p$ is unramified and let $B$ be a quaternion algebra over $F$ which splits at at most one infinite place. Let $\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses. Assume that for some fixed place $v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $\mathbb {Q}_p$ and $\overline {r}$ is generic at $v$. We prove that the admissible smooth representations of the quaternion algebra over $\mathbb {Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand–Kirillov dimension $1$. As an application we prove that the degree-two Scholze's functor (which is defined by Scholze [On the $p$-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 811–863]) vanishes on generic supersingular representations of $\mathrm {GL}_2(\mathbb {Q}_p)$. We also prove some finer structure theorems about the image of Scholze's functor in the reducible case.

1. Introduction

Let $p$ be a prime number. The mod $p$ (and $p$-adic) Langlands program has been emerged starting from the fundamental work of Breuil [Bre03]. Up to present, the correspondence in the case of $\mathrm {GL}_2(\mathbb {Q}_p)$ has been well-understood in various aspects, by the work of [Bre03, Col10, Eme11, Paš13]. Recently, there have been significant progress towards a mod $p$ Langlands correspondence for $\mathrm {GL}_2(L)$, when $L$ is a finite unramified extension of $\mathbb {Q}_p$ (see [BHHMS23, HW22, BHHMS21]). However, a mod $p$ Jacquet–Langlands correspondence is still largely unknown, even in the case of $\mathrm {GL}_2(\mathbb {Q}_p)$.

Inspired by the local–global compatibility results [Eme11, BDJ10], it is natural to search for the correspondence in the cohomology of Shimura curves. To explain this, let $F$ be a totally real extension of $\mathbb {Q}$ in which $p$ is unramified. Let $B$ be a quaternion algebra over $F$, which we assume to be split at only one infinite place in this introduction (in the text, we will also treat the case where $B$ is definite). If $U$ is a compact open subgroup of $(B\otimes _F\mathbb {A}_{F,f})^{\times }$, let $X_U$ be the associated smooth projective Shimura curve over $F$. Let $\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a continuous absolutely irreducible representation. Fix a place $v$ above $p$ and a compact open subgroup $U^v\subset (B\otimes _{F}\mathbb {A}_{F,f}^{\{v\}})^{\times }$, where $\mathbb {A}_{F,f}^{\{v\}}$ denotes the ring of finite adèles of $F$ outside $v$. We define

\[ \pi_v^B(\overline{r}):=\varinjlim_{U_v} \operatorname{{\mathrm{Hom}}}_{\operatorname{{\mathrm{Gal}}}(\overline{F}/F)} (\overline{r},H^1_{\mathrm{\acute{e}t}}(X_{U^vU_v}\times_F\overline{F},\overline{\mathbb{F}}_p), \]

where $U_v$ runs over compact open subgroups of $B_v^{\times } :=(B\otimes _{F}F_v)^{\times }$. In this way, we obtain an admissible smooth representation of $B_v^{\times }$. We assume that $B$ ramifies at $v$ from now on.

Assume that $\pi _v^B(\overline {r})$ is nonzero, i.e. $\overline {r}$ is modular for $B$ and $U^v$; we also need to impose some extra assumptions on $\overline {r}$, see § 5 for details. Then it is known that $\pi ^B_v(\overline {r})$ is infinite-dimensional (cf. [BD14, Corollary 3.5.4] and [Scho18, Theorem 1.4]). On the other hand, since $B_v^{\times }$ is compact modulo its centre, irreducible smooth mod $p$ representations of $B_v^{\times }$ (with a fixed central character) are easy to classify. Actually, such a representation always has dimension $\leq 2$ and there are only finitely many isomorphism classes. This implies that $\pi _v^B(\overline {r})$ is necessarily of infinite length, and is built out by infinitely many pieces of a finite number of isomorphism classes of irreducible representations of $B_v^{\times }$ in a highly non-semisimple way. A natural way to study such a representation is to look at its socle filtration. More conceptually, there is a standard invariant which measures the growth of the dimension of this socle filtration, called Gelfand–Kirillov dimension (cf. § 1.1).

In this paper, we study the Gelfand–Kirillov dimension of $\pi _v^B(\overline {r})$ in the case $F_v\cong \mathbb {Q}_p$. We make this assumption and assume $p\geq 5$ from now on; the reason for this restriction will be explained below after more notation is introduced.

Let $\overline {\rho }:=\overline {r}_v(1)$. We make the following assumption on $\overline {\rho }$.

1. (H1) Assume that $\overline {\rho }$ has one of the following forms:

   • • $\overline {\rho }$ is absolutely irreducible and up to twist $\overline {\rho }|_{I(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)} \sim \big (\begin{smallmatrix} \omega _{2}^{r+1} & 0 \\ 0 & \omega _2^{p(r+1)} \end{smallmatrix}\big )$, with $2\leq r\leq p-3$, where $\omega _2$ is Serre's fundamental character of niveau 2;

   • • $\overline {\rho }$ is reducible nonsplit and up to twist $\overline {\rho }|_{I(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)} \sim \big (\begin{smallmatrix} \omega ^{r+1} & * \\ 0 & 1 \end{smallmatrix}\big )$, with $0\leq r \leq p-3$, where $\omega$ is the mod $p$ cyclotomic character of $\operatorname {{\mathrm {Gal}}}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$.

The following is our main result.

Theorem 1.1 Keep the above assumptions on $F$, $B$ and $\overline {r}$. Then $\pi _v^B(\overline {r})$ has Gelfand–Kirillov dimension $1$.

An analogue of Theorem 1.1 was previously proved by Paškūnas [Paš22] when $\overline {\rho }$ is reducible, using Scholze's functor (introduced in [Scho18]) and a result of Ludwig [Lud17]. Combining with some argument of [Paš22], Theorem 1.1 implies some vanishing result on Scholze's functor, see Theorem 1.2 below.

The proof of Theorem 1.1 follows the innovative method of [BHHMS23] (which treats the case of $\mathrm {GL}_2$ over an unramified extension of $\mathbb {Q}_{p}$), but has several differences in technique. To explain this, recall that one key step in [BHHMS23] is to compare some potentially crystalline deformation rings of $\overline {\rho }$ of different (tame) types, and use it to gain information about the first three steps of the socle filtration of certain $\overline {\mathbb {F}}_p$-representations of $\mathrm {GL}_2$ with respect to the Iwahori subgroup. In [BHHMS23], the relevant deformation rings are explicitly worked out by complicated computations, but unfortunately in doing this a stronger genericity condition on $\overline {\rho }$ is imposed, for example $12\leq r\leq p-15$ when $\overline {\rho }$ is reducible. One may wonder, assuming this stronger genericity condition, if (the analogue of) Theorem 1.1 remains true when $F_v$ is an unramified extension of $\mathbb {Q}_p$, namely if $\pi _v^B(\overline {r})$ has Gelfand–Kirillov dimension equal to $[F_v:\mathbb {Q}_p]$. We believe this should be true and provable using the method of [BHHMS23]. In fact, we do give a criterion for controlling the Gelfand–Kirillov dimension in this generality, see Corollary 2.12 (which is an analogue of [BHHMS23, Corollary 5.3.5]). However, we caution that using only the deformation rings computed in [BHHMS23] may not be enough to prove this statement, because by the classical Jacquet–Langlands correspondence only those involving discrete series inertial types are useful to obtain information about $\pi _v^B(\overline {r})$. Namely, to check the condition of Corollary 2.12, one possibly needs to compute extra deformation rings (of discrete series inertial type), even when $F_v=\mathbb {Q}_p$.

For the above reason and also with the wish to weaken as much as possible the genericity condition in Theorem 1.1, we have chosen to restrict to the case $F_v \cong \mathbb {Q}_p$. The point is that in this case there is an alternative construction of Kisin's potentially semistable deformation rings, due to Paškūnas [Paš15]. This construction works only for two-dimensional representations of $\operatorname {{\mathrm {Gal}}}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$ and, in general, does not allow us to determine the explicit form of these rings, but it fits perfectly with our aim for the following two reasons.

• – First, to carry out the strategy in [BHHMS23], we do not really need the explicit form of these deformation rings, but only certain congruence relations between them (cf. [BHHMS23, Proposition 4.3.3]). In Paškūnas’ construction, these congruence relations can be proved by congruence relations between suitably chosen integral lattices inside the corresponding types.

• – Second, this construction closely relates the structure of the deformation rings to the structure of $\pi (\overline {\rho })$, the admissible smooth representation of $\mathrm {GL}_2(\mathbb {Q}_p)$ associated to $\overline {\rho }$ by the mod $p$ local Langlands correspondence (see § 4.2 for the precise definition). Thus, we may make use of the results of [BL94, Bre03, Mor11, Mor17] on $\pi (\overline {\rho })$ to study these deformation rings; see Theorem 4.15 for such an example.

In addition, in [BHHMS23] they use potentially crystalline deformation rings of Hodge–Tate weights $(-1,2)$ (and of $(0,1)$), while we use deformation rings of Hodge–Tate weights $(0,2)$. This also allows a further (minor) improvement on the genericity condition.

Theorem 1.1 can be applied to study Scholze's functors. Let $L$ be a finite extension of $\mathbb {Q}_p$ (not necessarily unramified). Let $D$ be the central division algebra over $L$ of dimension $n^2$ and invariant $1/n$, Scholze [Scho18] has constructed a cohomological covariant $\delta$-functor $\{\mathcal {S}^i, i\geq 0\}$ from the category of admissible smooth representations of $\mathrm {GL}_n(L)$ over $\overline {\mathbb {F}}_p$ to admissible smooth representations of $D^{\times }$ which carry a continuous and commuting action of $\operatorname {{\mathrm {Gal}}}(\overline {L}/L)$. If $\pi$ is an admissible smooth representation of $\mathrm {GL}_n(L)$ over $\overline {\mathbb {F}}_p$, then $\mathcal {S}^i(\pi )$ is defined as the cohomology group $H^i_{\mathrm {\acute {e}t}}(\mathbb {P}_{\mathbb {C}_p}^{n-1},\mathcal {F}_{\pi })$, where $\mathcal {F}_{\pi }$ is a certain Weil-equivariant sheaf on the adic space $\mathbb {P}_{\mathbb {C}_p}^{n-1}$. His construction is expected to realize both $p$-adic local Langlands and Jacquet–Langlands correspondences. In general, these cohomology groups seem very difficult to compute, but Scholze has computed $\mathcal {S}^0(\pi )$ and showed that $\mathcal {S}^i(\pi )$ vanishes whenever $i>2(n-1)$. Specializing to $n=2$, the case we are interested in, we have $\mathcal {S}^i(-)=0$ for $i>2$. Later on, Ludwig proved that $\mathcal {S}^2(\pi )=0$ if either $\pi$ is principal series or special series of $\mathrm {GL}_2(\mathbb {Q}_p)$, using the geometry of perfectoid modular curves [Lud17]. Since it is easy to compute $\mathcal {S}^2(\pi )$ if $\pi$ is one-dimensional, this leaves only the case of supersingular representations for $\mathcal {S}^2$.

By Breuil's classification [Bre03], any supersingular representation of $\mathrm {GL}_2(\mathbb {Q}_p)$ with a central character is up to twist isomorphic to

\[ \big (\mathrm{c}\textrm{-}\mathrm{Ind}_{\mathrm{GL}_2(\mathbb{Z}_p)\mathbb{Q}_p^{\times}}^{\mathrm{GL}_2(\mathbb{Q}_p)}\mathrm{Sym}^{r}\overline{\mathbb{F}}_p^2\big )/T, \]

where $0\leq r\leq p-1$ and $T$ is a certain Hecke operator [BL94]. As an application of Theorem 1.1, we have the following result.

Theorem 1.2 Let $\pi$ be a supersingular representation of $\mathrm {GL}_2(\mathbb {Q}_p)$ as above and assume $2\leq r\leq p-3$. Then $\mathcal {S}^2(\pi )=0$.

Our proof of Theorem 1.2 is inspired by Paškūnas’ work [Paš22], where he has used Ludwig's vanishing result of $\mathcal {S}^2$ to prove Theorem 1.1 in the case $\overline {\rho }$ is reducible. We observe that his argument can actually go in reverse direction, namely the vanishing of $\mathcal {S}^2$ on supersingular $\pi$ can be deduced from the Gelfand–Kirillov dimension of $\mathcal {S}^1(\pi )$ (see Proposition 7.4). Thus, Theorem 1.2 follows from Theorem 1.1 and a local–global compatibility result à la Emerton [Eme11, DLB17].

Another reason for focusing on the case of $\mathrm {GL}_2(\mathbb {Q}_p)$ is that we can prove some finer results on the structure of $\mathcal {S}^1(\pi (\overline {\rho }))$. We put

\[ \mathrm{JL}(\overline{\rho})=\left\{\begin{array}{@{}lll} \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi\omega^{-1},\mathcal{S}^1(\pi(\overline{\rho}))) & \mathrm{if}\ \overline{\rho}\sim \bigg(\begin{matrix} {\chi} & {*}\\ {0} & {\chi\omega}\end{matrix}\bigg), \\ \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\overline{\rho}\otimes\omega^{-1},\mathcal{S}^1(\pi(\overline{\rho}))) & \mathrm{otherwise}. \end{array}\right. \]

Theorem 1.3 Let $\overline {\rho }$ be as in (H1).

1. (i) Assume $\overline {\rho }\nsim \big (\begin{smallmatrix} {\chi } & {*}\\ {0} & {\chi \omega }\end{smallmatrix}\big )$ for any character $\chi$. Then $\mathcal {S}^1(\pi (\overline {\rho }))\cong (\overline {\rho }\otimes \omega ^{-1})\otimes \mathrm {JL}(\overline {\rho })$ as representations of $\operatorname {{\mathrm {Gal}}}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)\times B_v^{\times }$.

2. (ii) Assume $\overline {\rho }$ is reducible. Denote by $\overline {\rho }^{\rm ss}$ the semisimplification of $\overline {\rho }$.

   1. (a) Assume $\overline {\rho }^{\rm ss}\nsim \chi \oplus \chi \omega$ for any $\chi$. Then $\mathrm {JL}(\overline {\rho })$ depends only on $\overline {\rho }^{\rm ss}$.

   2. (b) Let $\overline {\rho }_1\sim \big (\begin{smallmatrix} {\omega } & {*}\\ {0} & {1}\end{smallmatrix}\big )$ and $\overline {\rho }_2\sim \big (\begin{smallmatrix} {1} & {*}\\ {0} & {\omega }\end{smallmatrix}\big )$ be nonsplit extensions. Then there exists an admissible $\overline {\mathbb {F}}_p$-representation $V$ of $B_v^{\times }$ such that

      \begin{gather*} 0\rightarrow \mathbf{1}_{D^{\times}}\rightarrow \mathrm{JL}(\overline{\rho}_1)\rightarrow V\rightarrow0,\\ 0\rightarrow V\rightarrow \mathrm{JL}(\overline{\rho}_2)\rightarrow (\mathbf{1}_{D^{\times}})^{\oplus 2}\rightarrow 0. \end{gather*}

It may look surprising that the representation $\mathrm {JL}(\overline {\rho })$ does not determine $\overline {\rho }$, but only $\overline {\rho }^{\rm ss}$, in case (a) of Theorem 1.3(ii); see Remark 8.13 for an explanation. It would be interesting to describe the precise structure of $\mathrm {JL}(\overline {\rho })$. We plan to come back to this question in future work.

We now give a brief overview of the contents of each section. In § 2, we study the structure of the $p$-adic group $B_v^{\times }$ and prove a criterion for controlling the Gelfand–Kirillov dimension of its representations (analogous to [BHHMS23, § 5]). In § 3 we study the structure of integral lattices in various locally algebraic types of $\mathrm {GL}_2(\mathbb {Z}_p)$. In § 4, we use Paškūnas’ technique to study potentially crystalline deformation rings of tame type and Hodge–Tate weights $(0,2)$. In §§ 5 and 6, we carry out the gluing process for $B_v^{\times }$-representations and prove our main result, Theorem 1.1. Finally, we study Scholze's functors, and prove Theorem 1.2 in § 7 and Theorem 1.3 in § 8.

1.1 Notation

We fix a prime number $p\geq 5$. Let $E\subset \overline {\mathbb {Q}}_p$ be a finite unramified extension of $\mathbb {Q}_p$, with ring of integers $\mathcal {O}$ and residue field $\mathbb {F}$. We will assume without further comment that $\mathbb {F}$ is sufficiently large.

If $F$ is a field, let $G_F := \operatorname {{\mathrm {Gal}}}({\overline {F}}/F)$ denote its absolute Galois group. Let $\varepsilon$ denote the $p$-adic cyclotomic character of $G_F$, and $\omega$ the mod $p$ cyclotomic character.

If $F$ is a $p$-adic field, $V$ is a de Rham $p$-adic representation of $G_F$ over $E$, and $\kappa : F\hookrightarrow E$, then we will write ${\rm HT}_{\kappa } (V)$ for the multiset of Hodge–Tate weights of $V$ with respect to $\kappa$. By definition, ${\rm HT}_{\kappa } (V)$ consists of $- i$ with multiplicity $\dim _E(V \otimes _{\kappa, F} \widehat {{\overline {F}}}(i))^{G_F}$, e.g. ${\rm HT}_{\kappa } (\varepsilon ) = \{1\}$ at all embedding $\kappa$.

If $G$ is a $p$-adic analytic group, we denote by $\operatorname {\mathrm {Mod}}_G^{\rm sm}(\mathcal {O})$ the category of smooth representations of $G$ on $\mathcal {O}$-torsion modules. Let $\operatorname {\mathrm {Mod}}_G^{\rm l.adm}(\mathcal {O})$ (respectively, $\operatorname {\mathrm {Mod}}_G^{\rm adm}(\mathcal {O})$) denote the full subcategory of locally admissible (respectively, admissible) representations. If $\zeta : Z_G \to \mathcal {O}^{\times }$ is a continuous character of the centre of $G$, then we denote by $\operatorname {\mathrm {Mod}}_{G,\zeta }^{\rm sm}(\mathcal {O})$ (respectively, $\operatorname {\mathrm {Mod}}_{G,\zeta }^{\rm l.adm}(\mathcal {O})$, respectively, $\operatorname {\mathrm {Mod}}_{G,\zeta }^{\rm adm}(\mathcal {O})$) the full subcategory of $\operatorname {\mathrm {Mod}}_G^{\rm sm}(\mathcal {O})$ consisting of smooth (respectively, locally admissible, respectively, admissible) representations on which $Z_G$ acts by the character $\zeta$.

The Pontryagin duality $M \mapsto M^{\vee }: = \operatorname {{\mathrm {Hom}}}_{\mathcal {O}}^{\rm cont}(M , E/\mathcal {O})$ induces an anti-equivalence between the category of discrete $\mathcal {O}$-modules and the category of pseudo-compact $\mathcal {O}$-modules. Under this duality the category $\operatorname {\mathrm {Mod}}_G^{\rm sm}(\mathcal {O})$ is anti-equivalent to the category of profinite augmented $G$-representations over $\mathcal {O}$ which is denoted by $\operatorname {\mathrm {Mod}}_G^{\rm pro}(\mathcal {O})$. Let $\frak {C}_{G}(\mathcal {O})$ (respectively, $\frak {C}_{G,\zeta }(\mathcal {O})$) denote the full subcategory of $\operatorname {\mathrm {Mod}}_G^{\rm pro}(\mathcal {O})$ which is anti-equivalent to $\operatorname {\mathrm {Mod}}_{G}^{\rm l.adm}(\mathcal {O})$ (respectively, $\operatorname {\mathrm {Mod}}_{G,\zeta }^{\rm l.adm}(\mathcal {O})$) under the Pontryagin duality. Note that on an object in $\frak {C}_{G,\zeta }(\mathcal {O})$ the centre is acting by $\zeta ^{-1}$.

Let $(R,\mathfrak {m})$ be a complete noetherian local commutative $\mathcal {O}$-algebra with residue field $\mathbb {F}$. We define the category $\operatorname {\mathrm {Mod}}_{G}^{\rm sm}(R)$ of smooth $R[G]$-modules, and the category $\operatorname {\mathrm {Mod}}_{G}^{\rm l.adm}(R)$ of locally admissible smooth $R[G]$-modules as in [Paš13, § 2]. Let $\frak {C}_{G}(R)$ be the dual category of $\operatorname {\mathrm {Mod}}_{G}^{\rm l.adm}(R)$ under the Pontryagin duality. If $\zeta : Z_G \to \mathcal {O}^{\times }$ is a continuous character of the centre of $G$, we can similarly define $\operatorname {\mathrm {Mod}}_{G,\zeta }^{\rm l.adm}(R)$ and its dual category $\frak {C}_{G,\zeta }(R)$.

If $M$ is a torsion-free linear-topological $\mathcal {O}$-module, $M^{d}$ denotes its Schikhof dual $\operatorname {{\mathrm {Hom}}}^{\rm cont}_{\mathcal {O}} (M, \mathcal {O})$. The functor $M \mapsto M^d$ induces an anti-equivalence of categories between the category of pseudo-compact torsion-free linear-topological $\mathcal {O}$-modules and the category of $\varpi$-adically complete and separated torsion-free $\mathcal {O}$-modules.

If $R$ is a ring and $M$ is a left $R$-module, we denote by $\operatorname {{\mathrm {soc}}}_R(M)$ (respectively, $\mathrm {cosoc}_R(M)$) the socle (respectively, cosocle) of $M$. Inductively, we define the socle (respectively, cosocle) filtration of $M$. If $M$ has finite length, we denote by $\operatorname {{\mathrm {JH}}}(M)$ the set of Jordan–Hölder factors of $M$.

The grade $j_{R}(M)$ of $M$ over $R$ is defined by

\[ j_{R}(M)=\mathrm{inf}\{i \in \mathbb{N} \,|\, \operatorname{{\mathrm{Ext}}}^i_{R}(M,R)\neq0\}. \]

Assume $R$ is noetherian. The ring $R$ is called Auslander–Gorenstein if it has finite left and right injective dimension and the following Auslander condition holds: for any $R$-module $M$, any integer $m\geq 0$ and any $R$-submodule $N$ of $\operatorname {{\mathrm {Ext}}}^m_R(M,R)$, we have $j_{R}(N)\geq m$. An Auslander–Gorenstein ring is called Auslander regular if it has finite global dimension. If $R$ is an Auslander regular ring and $M$ is a finitely generated $R$-module, define the dimension

\[ \delta_R(M) : = {\rm gld}(R) - j_R(M), \]

where ${\rm gld}(R)$ is the global dimension of $R$.

Let $G_0$ be a compact $p$-adic analytic group. The ring-theoretic properties of $\mathcal {O} [\![G_0]\!]$ are established by the fundamental works of Lazard [Laz65] and Venjakob [Ven02]. In particular, if $G_0$ has no element of order $p$, then $\mathcal {O} [\![G_0]\!]$ is an Auslander regular ring of dimension $1+\dim _{\mathbb {Q}_p} G_0$, where $\dim _{\mathbb {Q}_p} G_0$ is the dimension of $G_0$ as a $p$-adic analytic group. If $M$ is nonzero, we have

\[ 0 \leq j_{ \mathcal{O} [\![G_0]\!]} (M) \leq 1+\dim_{\mathbb{Q}_p} G_0, \]

and $\delta _{ \mathcal {O} [\![G_0]\!]}( M) = 1+\dim _{\mathbb {Q}_p} G_0 - j_{ \mathcal {O} [\![G_0]\!]}(M)$. If $G$ is a $p$-adic analytic group with a fixed open compact subgroup $G_0 \subseteq G$ and $M$ is a finitely generated $\mathcal {O} [\![G_0]\!]$-module equipped with a compatible $G$-action, we define $j_{G}(M)$ (respectively, $\delta _G(M)$) as $j_{\mathcal {O}[\![G_0]\!]}(M)$ (respectively, $\delta _{\mathcal {O}[\![G_0]\!]}(M)$); this does not depend on the choice of $G_0$.

If $\pi$ is an admissible smooth representation of $G$ over $\mathbb {F}$, then $\pi ^{\vee }$ is finitely generated over $\mathcal {O} [\![G_0]\!]$. The Gelfand–Kirillov dimension of $\pi$ is defined by (see [BHHMS23, Remark 5.1.1])

\[ \dim_G(\pi) := \delta_G(\pi^{\vee}) . \]

2. The $p$-adic Lie group $D^{\times }$

2.1 Results of Kohlhaase

We recall and extend some results of [Koh13].

Let $L = \mathbb {Q}_{p^f}$ be the unramified extension of degree $f$ over $\mathbb {Q}_p$. Let $D$ be the unique central division algebra of dimension $4$ over $L$. For $a\in D$, define $v_D(a) := v_p ({\rm Nrd}_D(a))$, where $v_p$ is the $p$-adic valuation on $L$ normalized so that $v_p(p) = 1$, and ${\rm Nrd}_D : D \to L$ is the reduced norm map; this gives a non-archimedean valuation on $D$. Let ${\mathcal {O}}_D : =\{a\in D\,|\, v_D(a) \geq 0\}$ be the ring of integers and $\frak {p}_D: = \{a\in D\,|\, v_D(a) \geq 1\}$ the maximal ideal, which can be generated by a uniformizer $\varpi _D$. The residue field $k_D : = {\mathcal {O}}_D/\frak {p}_D$ is isomorphic to $\mathbb {F}_{q^2}$, where $q:= p^f$. Let $L'$ be the unramified quadratic extension of $L$ in $\overline {\mathbb {Q}}_p$. We denote by $\sigma : L' \to L'$ a lift of the Frobenius map $x \mapsto x^q$ on $\mathbb {F}_{q^2}$. Let $L'\langle X \rangle$ denote the non-commutative polynomial ring in one variable over $L'$ satisfying the relation $X a = \sigma (a) X,~ \forall a \in L'$. Then the homomorphism $L'\langle X \rangle \to D$, $X\mapsto \varpi _D$ induces an isomorphism of $L$-algebras

(2.1)\begin{equation} L'\langle X \rangle / ( X^2 - p) \cong D. \end{equation}

Let $D^{\times }$ (respectively, $\mathcal {O}_D^{\times }$) denote the group of invertible elements of $D$ (respectively, $\mathcal {O}_D$) and

(2.2)\begin{equation} U^n_D: = 1+\varpi_D^n {\mathcal{O}}_D,\quad n\geq 1, \end{equation}

which are compact open normal (pro-$p$) subgroups of $D^{\times }$. We have

\[ D^{\times} = {\mathcal{O}}^{\times}_D \rtimes \varpi_D^{\mathbb{Z}},\quad {\mathcal{O}}_D^{\times} / U^1_D \cong \mathbb{F}_{q^2}^{\times}. \]

Let $Z_D$ denote the centre of $D^{\times }$ which is isomorphic to $L^{\times }$. Then $Z_D{\mathcal {O}}_D^{\times }$ is of index $2$ in $D^{\times }$. Let $Z^1_D = Z_D \cap U^1_D$.

Assume $p \geq 5$. Let $\omega : U^1_D \backslash \{1\} \to (0, \infty )$ be the map defined by $\omega (g) : = \frac {1}{2} v_D (g - 1)$, and set $\omega (1):=\infty$. As in [Schn11, Example 23.2], one shows that $\omega$ is a $p$-valuation on $U^1_D$ in the sense of Lazard [Laz65, III.2.1.2]. For any real number $\nu > 0$, let

\[ (U^1_D)_{\nu} : = \{ g\in U^1_D\,|\, \omega (g)\geq \nu\},\quad (U^1_D)_{\nu+} : = \{ g\in U^1_D\,|\, \omega (g)> \nu\}. \]

We set

\[ \operatorname{{\mathrm{gr}}} U^1_D: = \bigoplus_{\nu>0} (U^1_D)_{\nu} / (U^1_D)_{\nu+} . \]

It is easy to see that $U_D^i=(U^1_D)_{{i}/{2}}$ and $U^{i+1}_D = (U^1_D)_{ ({i}/{2}) +},$ so we have

\[ \operatorname{{\mathrm{gr}}} U^1_D = \bigoplus_{i\geq 1} U^i_D / U^{i+1}_D. \]

We say a nonzero homogeneous element $t\in \operatorname {{\mathrm {gr}}} U^1_D$ is of degree $i$ if $t \in U^i_D/ U^{i+1}_D$.

As explained in [Schn11, § 25], $\operatorname {{\mathrm {gr}}} U^1_D$ is a graded Lie algebra over the polynomial ring $\mathbb {F}_p[\varepsilon ]$ by setting

\[ [g U^{i+1}_D , g' U^{j+1}_D] : = g g' g^{-1} g'^{-1} U^{i+j+1}_D,\quad g \in U^{i}_D,\enspace g' \in U^{j}_D, \]

and

\[ \varepsilon (g U^{i+1}_D) : = g^p U^{i+3}_D,\quad g \in U^i_D. \]

Note that $U^i_D/ U^{i+1}_D \cong (\mathbb {F}_{q^2}, + )$ is an $\mathbb {F}_q$-vector space by setting

\[ \lambda\cdot (1 + \varpi_D^i a) U^{i+1}_D := (1 + \varpi_D^i [\lambda]a) U^{i+1}_D, \]

where $[\lambda ] \in \mathcal {O}_L$ is the Teichmüller lift of $\lambda \in \mathbb {F}_q$. One checks that the Lie bracket on $\operatorname {{\mathrm {gr}}} U^1_D$ is $\mathbb {F}_q$-bilinear, hence $\operatorname {{\mathrm {gr}}} U^1_D$ becomes a graded Lie algebra over the polynomial ring $\mathbb {F}_q[\varepsilon ]$.

Proposition 2.1 The natural map $\mathbb {F}_q[\varepsilon ] \otimes _{\mathbb {F}_q} (U^1_D/U^2_D \oplus U^2_D / U^3_D) \to \operatorname {{\mathrm {gr}}} U^1_D$ is an isomorphism of $\mathbb {F}_q[\varepsilon ]$-modules.

Proof. The proof of [Koh13, Lemma 3.12] (when $L=\mathbb {Q}_p$) extends to the general case.

Let $\overline {\operatorname {{\mathrm {gr}}} U^1_D} : = \operatorname {{\mathrm {gr}}} U^1_D \otimes _{\mathbb {F}_q[\varepsilon ]} \mathbb {F}_q$ where the map $\mathbb {F}_q[\varepsilon ] \to \mathbb {F}_q$ sends $\varepsilon$ to $0$. We first determine the Lie algebra structure of $\overline {\operatorname {{\mathrm {gr}}} U^1_D}$. Fix $\xi \in \mathbb {F}_{q^2} \setminus \mathbb {F}_q$ and set

\[ \gamma_{1} : = 1+\varpi_D,\quad \gamma_{2} : = 1+ \varpi_D [\xi],\quad \gamma_{3} := \gamma_{1} \gamma_{2} \gamma_{1}^{-1} \gamma_{2}^{-1},\quad \gamma_{4} : = 1+p, \]

where $[\xi ] \in \mathcal {O}_{L'}$ is the Teichmüller lift of $\xi$. We have $\omega (\gamma _{1}) = \omega (\gamma _{2}) = 1/2$ and $\omega (\gamma _{3} ) = \omega (\gamma _{4}) = 1$.¹ Let $\overline {\gamma }_{1},\overline {\gamma }_{2}\in U^1_D/ U^2_D$ be the images of $\gamma _1$ and $\gamma _2$ and let $\overline {\gamma }_{3},\overline {\gamma }_{4}\in U^2_D/ U^3_D$ be the images of $\gamma _3$ and $\gamma _4$. Then $\overline {\gamma }_{1}$, $\overline {\gamma }_{2}$, $\overline {\gamma }_{3}$, $\overline {\gamma }_{4}$ form an $\mathbb {F}_q$-basis of $U_D^1/U_D^2\oplus U_D^2/U_D^3$, hence also an $\mathbb {F}_q$-basis of $\overline {\operatorname {{\mathrm {gr}}} U^1_D}$. They satisfy (in $\overline {\operatorname {{\mathrm {gr}}} U^1_D}$, i.e. after modulo $\varepsilon$)

(2.3)\begin{equation} [\overline{\gamma}_{1},\overline{\gamma}_{2}] = \overline{\gamma}_{3},\quad [\overline{\gamma}_{1},\overline{\gamma}_{3}]=[\overline{\gamma}_{2}, \overline{\gamma}_{3}] = [\overline{\gamma}_{4},\overline{\gamma}_{1}] = [\overline{\gamma}_{4},\overline{\gamma}_{2}] = [\overline{\gamma}_{4},\overline{\gamma}_{3}] = 0, \end{equation}

see the discussion after [Koh13, Remark 3.15].

Passing to the quotient group $U^1_D /Z^1_D$, we can consider $\overline {\operatorname {{\mathrm {gr}}} U^1_D/Z^1_D} : = \operatorname {{\mathrm {gr}}} U^1_D/Z^1_D \otimes _{\mathbb {F}_q[\varepsilon ]} \mathbb {F}_q$, with the induced filtration on $U^1_D/Z^1_D$. Then $\overline {\operatorname {{\mathrm {gr}}} U^1_D/Z^1_D}$ is isomorphic to $\overline {\operatorname {{\mathrm {gr}}} U^1_D}/ (\overline {\gamma }_{4} )$ as graded Lie algebras over $\mathbb {F}_q$, where $(\overline {\gamma }_{4}) : = \mathbb {F}_q \overline {\gamma }_4$ is the sub-Lie algebra of $\overline {\operatorname {{\mathrm {gr}}} U^1_D}$ generated by $\overline {\gamma }_{4}$.

Let $\frak {g}_{\mathbb {F}_p} = \mathbb {F}_p e \oplus \mathbb {F}_p f \oplus \mathbb {F}_p h$ be the graded Lie algebra of dimension $3$ over $\mathbb {F}_p$, with $e$ and $f$ in degree $1$, $h$ in degree $2$ and satisfying the relations

\[ [e,f] = h,\quad [h,e] = [h,f] =0. \]

From (2.3) we easily deduce the following result.

Corollary 2.2 The graded Lie algebra $\overline {\operatorname {{\mathrm {gr}}} U^1_D/Z_D^1}$ is isomorphic to $\frak {g}_{\mathbb {F}_q} : =\mathbb {F}_q \otimes _{\mathbb {F}_p} \frak {g}_{\mathbb {F}_p}$.

Remark 2.3 One can also deduce the structure of the Lie algebra $\overline {\operatorname {{\mathrm {gr}}} U^1_D/Z^1_D} \cong \overline {\operatorname {{\mathrm {gr}}} U^1_D}/ (\overline {\gamma }_{4} )$ from the results of [BHHMS23, § 5.3] by comparing with the pro-$p$-Iwahori subgroup of $\mathrm {GL}_2$ over $\mathcal {O}_{L'}$.

2.2 The graded group algebra

Let $\mathbb {Z}_p[\![ U^1_D ]\!]=\varprojlim _{i\geq 1}\mathbb {Z}_p[U_D^1/U_D^i]$ be the Iwasawa algebra of $U^1_D$ over $\mathbb {Z}_p$. It is a pseudo-compact local $\mathbb {Z}_p$-algebra. For $\nu \geq 0$, let $J_{\nu }$ denote the smallest closed $\mathbb {Z}_p$-submodule of $\mathbb {Z}_p[\![ U^1_D]\!]$ which contains all elements of the form $p^{\ell } (h_1 - 1)\cdots (h_s - 1)$ with $\ell, s \geq 0$, $h_1,\ldots, h_s \in U^1_D$ and

\begin{align*} \ell + \omega(h_1) + \cdots +\omega(h_s) \geq \nu. \end{align*}

Let $J_{\nu +} : = \bigcup _{\nu ' > \nu } J_{\nu '}$. Let

\[ \operatorname{{\mathrm{gr}}}_J \mathbb{Z}_p[\![ U^1_D ]\! ] := \bigoplus_{\nu\geq 0} J_{\nu}/J_{\nu+}, \]

which is an associative graded algebra over $\operatorname {{\mathrm {gr}}} \mathbb {Z}_p := \bigoplus _{i\geq 0} p^i \mathbb {Z}_p/p^{i+1} \mathbb {Z}_p$. It naturally has a graded Lie algebra structure.

The homomorphism of abelian groups $\mathcal {L}_{\nu } : \operatorname {{\mathrm {gr}}}_{\nu } U^1_D \to J_{\nu } / J_{\nu +}$, $g (U^1_D)_{\nu + } \mapsto (g-1) + J_{\nu +}$ extends to a homomorphism of graded $\mathbb {F}_p[\varepsilon ]$-Lie algebras $\mathcal {L} : \operatorname {{\mathrm {gr}}} U^1_D \to \operatorname {{\mathrm {gr}}} \mathbb {Z}_p[\![ U^1_D ]\! ]$, where the $\mathbb {F}_p[\varepsilon ]$-algebra structure on $\operatorname {{\mathrm {gr}}}\mathbb {Z}_p[\![U_D^1]\!]$ is given through the isomorphism $\mathbb {F}_p[\varepsilon ] \xrightarrow {\sim } \operatorname {{\mathrm {gr}}} \mathbb {Z}_p$, $\varepsilon \mapsto p+ p^2 \mathbb {Z}_p \in \operatorname {{\mathrm {gr}}}^1 \mathbb {Z}_p$. Let $U_{\mathbb {F}_p[\varepsilon ]}(\operatorname {{\mathrm {gr}}} U^1_D)$ be the universal enveloping algebra of $\operatorname {{\mathrm {gr}}} U^1_D$ over $\mathbb {F}_p[\varepsilon ]$. By the universal property of $U_{\mathbb {F}_p[\varepsilon ]}(\operatorname {{\mathrm {gr}}} U_D^1)$, we have a homomorphism of associative $\operatorname {{\mathrm {gr}}} \mathbb {Z}_p$-algebras

(2.4)\begin{equation} \widetilde{\mathcal{L}} : U_{\mathbb{F}_p[\varepsilon]}(\operatorname{{\mathrm{gr}}} U^1_D) \to \operatorname{{\mathrm{gr}}}_J \mathbb{Z}_p[\![ U^1_D ]\!]. \end{equation}

By [Schn11, Theorem 28.3], $\widetilde {\mathcal {L}}$ is an isomorphism.

In practice, we will consider the Iwasawa algebra associated to the quotient group $U_D^1/Z_D^1$. Let $\mathbb {Z}_p[\![U_D^1/Z_D^1]\!]$ (respectively, $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$) be the Iwasawa algebra of $U_D^1/Z_D^1$ over $\mathbb {Z}_p$ (over $\mathbb {F}_p$). We have $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]=\mathbb {Z}_p[\![U_D^1/Z_D^1]\!]\otimes _{\mathbb {Z}_p}\mathbb {F}_p$. The filtration $\{J_{\nu }, \nu \geq 0\}$ induces a filtration on $\mathbb {Z}_p[\![U_D^1/Z_D^1]\!]$ and on $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$. On the other hand, letting $\frak {m}_{D}$ denote the maximal ideal of $\mathbb {F}_p[\![ U^1_D/Z^1_D ]\!]$, we may consider the $\mathfrak {m}_{D}$-adic filtration on $\mathbb {F}_p[\![ U^1_D/Z^1_D ]\!]$. The following result shows that these two filtrations coincide up to rescaling indices.

Lemma 2.4 Denote by $\overline {J}_{\nu }$ the image of $J_{\nu }$ in $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$. Then $\overline {J}_{i/2}=\mathfrak {m}_{D}^i$ for any $i\geq 0$.

Proof. The proof of [Koh13, Lemma 3.13] (when $L=\mathbb {Q}_p$) extends to the general case.

One checks that $J_{\nu }\neq J_{\nu +}$ exactly when $\nu = {i}/{2}$ for some $i\geq 0$. Thus, by Lemma 2.4 the graded algebra

(2.5)\begin{equation} \operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}_p[\![ U^1_D /Z^1_D ]\! ] := \bigoplus_{i\geq 0} \frak{m}_{D}^i / \frak{m}_{D}^{i+1} \end{equation}

is identical to $\bigoplus _{\nu \geq 0}\overline {J}_{\nu }/\overline {J}_{\nu +}$.

Proposition 2.5 There is an isomorphism of graded $\mathbb {F}_p$-algebras

\[ \operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}_p[\![ U^1_D /Z^1_D ]\!] \cong U_{\mathbb{F}_p}(\frak{g}_{\mathbb{F}_{q}}). \]

Proof. By the above discussion, the result is a direct consequence of Corollary 2.2 via (2.4).

Let $\mathbb {F}$ be a finite extension of $\mathbb {F}_p$ such that $\mathbb {F}_q$ embeds into $\mathbb {F}$. Let $\mathcal {J}$ denote the set of embeddings $\mathbb {F}_{q} \hookrightarrow \mathbb {F}$ and fix $\sigma _0\in \mathcal {J}$. We label the embeddings $\sigma _j = \sigma _0 \circ \varphi ^j$, so that $\mathcal {J}$ is identified with $\{0,\ldots, f-1\}$. Let $\frak {g}_j: = \mathbb {F} \otimes _{\mathbb {F}_{q}, \sigma _j} \frak {g}_{\mathbb {F}_{q}}$. We then have $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_{q}} = \bigoplus _{j=0}^{f-1} \frak {g}_j$. Let $e_j,f_j,h_j \in \frak {g}_j$ denote $1\otimes e$, $1\otimes f$, $1\otimes h \in \mathbb {F} \otimes _{\mathbb {F}_{q}, \sigma _j} \frak {g}_{\mathbb {F}_{q}}$.

We again denote by $\frak {m}_{D}$ the maximal ideal of $\mathbb {F}[\![ U^1_D/Z^1_D ]\!]=\mathbb {F}\otimes _{\mathbb {F}_p}\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$. Then Proposition 2.5 implies that

(2.6)\begin{equation} \operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}[\![ U^1_D /Z^1_D ]\!] =\mathbb{F}\otimes_{\mathbb{F}_p} (\operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}_p[\![ U^1_D /Z^1_D ]\! ] ) \cong U_{\mathbb{F}}(\mathbb{F}\otimes_{\mathbb{F}_p} \frak{g}_{\mathbb{F}_q})\cong \bigotimes_{j=0}^{f-1} U_{\mathbb{F}} (\frak{g}_j). \end{equation}

In particular, we have $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D}^1 \mathbb {F}[\![ U^1_D /Z^1_D ]\!]=\bigoplus _{j=0}^{f-1}(\mathbb {F} e_j\oplus \mathbb {F} f_j)$.

Theorem 2.6

1. (i) The graded ring $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]$ is Auslander regular.

2. (ii) The sequence $(h_0,\ldots, h_{f-1})$ is a regular sequence of central elements of $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]$. The quotient $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]/(h_0,\ldots, h_{f-1})$ is commutative and is isomorphic to the polynomial ring $\mathbb {F} [e_j,f_j;\ 0\leq j \leq f-1]$.

Proof. The proof is the same as that of [BHHMS23, Theorem 5.3.4].

Theorem 2.6 is not enough for the application to Gelfand–Kirillov dimension, namely Corollary 2.12 below. We shall find eigenbases of $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_q}$ for the $\mathbb {F}_{q^2}^{\times }$-action in the next subsection.

2.3 Gelfand–Kirillov dimension

We regard $\mathbb {F}_{q^2}^{\times }$ as a subgroup of $\mathcal {O}_{L'}^{\times }$ via the Teichmüller lifting map, and then as a subgroup of $\mathcal {O}_D^{\times }$ via the fixed embedding $L'\hookrightarrow D$. It normalizes $U_D^1$, thus acts on $\overline {\operatorname {{\mathrm {gr}}} U_D^1}$ and on $\frak {g}_{\mathbb {F}_{q}}$. In practice, we need a basis of $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_q}$ consisting of eigenvectors for the action of $\mathbb {F}_{q^2}^{\times }$. Note that $e_j$ and $f_j$ are only eigenvectors for the action of $\mathbb {F}_q^{\times }$, but not for $\mathbb {F}_{q^2}^{\times }$.

Choose an embedding $\mathbb {F}_{q^2}\hookrightarrow \mathbb {F}$ which extends the fixed embedding $\sigma _0:\mathbb {F}_q\hookrightarrow \mathbb {F}$; we again denote it by $\sigma _0$ and let $\sigma _j=\sigma _0\circ \varphi ^j$ for $0\leq j\leq 2f-1$.

For $0\leq j\leq 2f-1$, define the following elements in $\mathbb {F}[\![U_D^1/Z_D^1]\!]$:

\[ Y_j:=\sum_{\lambda\in\mathbb{F}_{q^2}^{\times}}\sigma_j(\lambda)^{-1}(1+\varpi_D[\lambda]), \]

where the term $1+\varpi _D[\lambda ]$ is considered as an element in the group $U_D^1/Z_D^1$. Since $\sum _{\lambda \in \mathbb {F}_{q^2}^{\times }}\sigma _{j}(\lambda )^{-1}=0$, we have $Y_j\in \mathfrak {m}_{D}$. If $\mu \in \mathbb {F}_{q^2}^{\times }$, then one checks that

(2.7)\begin{equation} \mu\cdot Y_j:=[\mu]Y_j[\mu]^{-1}=\alpha_j(\mu)Y_j, \end{equation}

where $\alpha _j:\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ denotes the character defined by

(2.8)\begin{equation} \alpha_j(x):=\sigma_j(\overline{x})^{q-1}. \end{equation}

Note that $\alpha _{j+f}=\alpha _j^q=\alpha _j^{-1}$.

For $0\leq j\leq 2f-1$, let $y_j:= Y_j+ \frak {m}_D^2 \in \operatorname {{\mathrm {gr}}}^1_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$.

Lemma 2.7

1. (i) The elements $\{Y_j, 0\leq j\leq 2f-1\}$ generate the ideal $\mathfrak {m}_{D}$.

2. (ii) The elements $\{y_j, 0\leq j\leq 2f-1\}$ form a basis of $\operatorname {{\mathrm {gr}}}^1_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$.

Proof. (i) This is equivalent to checking that the images of $Y_j$ in $\mathfrak {m}_{D}/\mathfrak {m}_{D}^2$ are linearly independent (over $\mathbb {F}$). This is proved by a standard technique; see the proof of [Schr15, Proposition 2.13] for a similar argument.

(ii) This is clear, because $\operatorname {{\mathrm {gr}}}^1_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$ has dimension $2f$ (with a basis $\{e_j,f_j, 0\leq j\leq f-1\}$).

Lemma 2.8 For $g\in U_D^i/(U_D^i\cap Z_D^1)$ and $h\in U_D^j/(U_D^j\cap Z_D^1)$, we have

\[ gh-1\equiv (g-1)+(h-1) \mod \mathfrak{m}_D^{i+j}. \]

Proof. Using Lemma 2.4, this is a consequence of the equality $(g-1)(h-1)=(gh-1)- (g-1)-(h-1)$.

For $t\in \mathbb {F}_{q^2}^{\times }$, write

(2.9)\begin{equation} g_{t}:=1+p [t]\in U_D^1/Z_D^1. \end{equation}

Note that $\omega (g_t)=1$, so $g_t-1\in \mathfrak {m}_D^2$ by Lemma 2.4. Let $u_t$ denote the image of $g_t-1$ in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$.

Proposition 2.9

1. (i) We have $[y_i,y_j]=0$ for any pair $(i,j)$ with $i-j\neq f$ (in $\mathbb {Z}/2f\mathbb {Z}$).

2. (ii) Set $h_j':=[y_j,y_{f+j}]$ for $0\leq j\le f-1$. Then $\{h_j',\ 0\leq j\leq f-1\}$ are linearly independent in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$ and they span the same subspace as $\{h_j, 0\leq j\leq f-1\}$.

Proof. A direct computation shows

\[ Y_iY_{j}=\sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}}\sigma_i(\lambda)^{-1} \sigma_{j}(\mu)^{-1} (1+\varpi_D[\lambda]+\varpi_D[\mu]+p[\lambda^{q}\mu]). \]

We may write (in $U_D^1$)

\[ 1+\varpi_D[\lambda]+\varpi_D[\mu]+p[\lambda^q\mu]=(1+\varpi_D[\lambda]+ \varpi_D[\mu])(1+p[\lambda^q\mu]+x) \]

with $x\in \varpi _D^3\mathcal {O}_D$ and note that $(1+p[\lambda ^q\mu ]+x)-1$ has the same image as $(1+p[\lambda ^q\mu ])-1$ in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$ by Lemma 2.8. Using Lemma 2.8 again, we have

\[ (1+\varpi_D[\lambda]+\varpi_D[\mu]+p[\lambda^q\mu])-1\equiv (h_{\lambda,\mu}-1)+(g_{\lambda^q\mu}-1) \mod \mathfrak{m}_D^3, \]

where $h_{\lambda,\mu }:=1+\varpi _D[\lambda ]+\varpi _D[\mu ]$ and $g_{\lambda ^q\mu }$ is defined by (2.9). Similarly, we have

\begin{align*} Y_jY_i\equiv\sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}}\sigma_i(\lambda)^{-1}\sigma_{j}(\mu)^{-1} ((h_{\lambda,\mu}-1)+(g_{\lambda\mu^q}-1))\mod \mathfrak{m}_D^3 \end{align*}

and so

\[ [Y_i,Y_j]\equiv \sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}} \sigma_i(\lambda)^{-1}\sigma_j(\mu)^{-1}((g_{\lambda^q\mu}-1)-(g_{\lambda\mu^q}-1))\mod \mathfrak{m}_D^3. \]

Taking the image in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$ and noting that $\sigma _i(\lambda )=\sigma _{i-f}(\lambda ^q)$, we obtain

\[ [y_i,y_{j}]=\sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}} \frac{\sigma_j(\lambda^q)}{\sigma_{i-f}(\lambda^q)} \sigma_{j}(\lambda^q\mu)^{-1}(u_{\lambda^q\mu}-u_{\lambda\mu^q}). \]

The map

\[ \mathbb{F}_{q^2}^{\times}\times \mathbb{F}_{q^2}^{\times}\rightarrow \mathbb{F}_{q^2}^{\times},\quad (\lambda,\mu)\mapsto \lambda^q\mu \]

is surjective and each fibre is bijective to $\mathbb {F}_{q^2}^{\times }$ (by projecting to the second component), thus

\[ [y_i,y_{j}]=\sum_{t\in\mathbb{F}_{q^2}^{\times}} \bigg(\sum_{\lambda\in\mathbb{F}_{q^2}^{\times}} \frac{\sigma_j(\lambda^q)}{\sigma_{i-f}(\lambda^q)}\bigg)\cdot \sigma_j(t)^{-1}(u_{t}-u_{t^q}). \]

If $i-j\neq f$, then $\sum _{\lambda \in \mathbb {F}_{q^2}^{\times }} ({\sigma _j(\lambda ^q)}/{\sigma _{i-f}(\lambda ^q)})=0$ and so $[y_i,y_{j}]=0$, proving part (i). If $i-j=f$, then the last sum equals to $-1$, and so

\[ [y_{j+f},y_j]=-\sum_{t\in\mathbb{F}_{q^2}^{\times}}\sigma_j(t)^{-1}(u_t-u_{t^q}). \]

To prove part (ii), one could argue as in Lemma 2.7, but this needs to make explicit the $h_j$. Nonetheless, we can conclude by the following observation: since $y_i$ lies in $\bigoplus _{0\leq j\leq f-1}(\mathbb {F} e_j\oplus \mathbb {F} f_j)$, $[y_i,y_j]$ lies in the subspace spanned by $h_j=[e_j,f_j]$ (recall $[e_i,e_j]=[e_i,f_j] = [f_i,f_j]=0$ whenever $i\neq j$), and vice versa by part (i) and Lemma 2.7(ii).

To make the notation more transparent, we write $z_i:=y_{i+f}$ for $0\leq i\leq f-1$. Lemma 2.7 and Proposition 2.9 imply that the Lie algebra $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_q}$ has another basis over $\mathbb {F}$ given by $\{y_j,z_j,h_j'; 0\leq j\leq f-1\}$, with $y_j$ and $z_j$ in degree $1$, $h_j'$ in degree $2$ and satisfying the relations

\[ h_j'=[y_j,z_j],\quad [y_i,z_j]=0 \quad \mathrm{if}\ i\neq j,\quad [y_i,y_j]=[z_i,z_j]=[y_i,h_j']=[z_i,h_j']=0. \]

Let $I_D$ be the left ideal of $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]$ generated by the degree-two elements $y_jz_j$ and $h'_j$ for all $0\leq j\leq f-1$. The ideal $I_D$ is, in fact, a two-sided ideal of $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]$; it is also the left ideal generated by $(y_j z_j, h_j;\ 0\leq j \leq f-1)$ by Proposition 2.9(ii).

Corollary 2.10

1. (i) The sequence $(h_0',\ldots,h_{f-1}')$ is a regular sequence of central elements of $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$. The quotient $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]/(h_0',\ldots,h_{f-1}')$ is commutative and is isomorphic to the polynomial ring $\mathbb {F}[y_j,z_j;0\leq j\leq f-1]$.

2. (ii) The quotient $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!] / I_D$ is isomorphic to $\mathbb {F} [y_j,z_j; 0\leq j\leq f-1]/(y_jz_j; 0\leq j \leq f-1)$.

Proof. The proof of [BHHMS23, Theorem 5.3.4] goes through by the above discussion.

Let $\chi : \mathcal {O}_D^{\times } \to \mathbb {F}^{\times }$ be a smooth character. Let $\operatorname {{\mathrm {Proj}}}_{\mathbb {F}[\![\mathcal {O}_{D}^{\times } / Z^1_D ]\!]} \chi$ denote the projective envelope of $\chi$ in the category of $\mathbb {F}[\![\mathcal {O}_{D}^{\times } / Z^1_D]\!]$-modules. For $n\geq 1$, let

(2.10)\begin{equation} W_{\chi_,n}: = (\operatorname{{\mathrm{Proj}}}_{\mathbb{F}[\![\mathcal{O}_{D}^{\times} / Z^1_D]\!]} \chi) / \frak{m}_{D}^n. \end{equation}

It is clear that $W_{\chi,n}\cong \chi \otimes W_{\mathbf {1},n}$, where $\mathbf {1}$ denotes the trivial character. The module $W_{\chi,3}$ is of particular importance to us.

Corollary 2.11 The module $W_{\mathbf {1},3}$ has the following graded structure:

\begin{gather*} \operatorname{{\mathrm{gr}}}^0 W_{\mathbf{1},3} = \mathbb{F},\quad \operatorname{{\mathrm{gr}}}^1W_{\mathbf{1},3} = \bigoplus_{i=0}^{f-1}\mathbb{F}\alpha_i \oplus \mathbb{F}\alpha_i^{-1},\\ \operatorname{{\mathrm{gr}}}^2 W_{\mathbf{1},3} = \mathbb{F}^{2f}\oplus \bigoplus_{0\leq i\leq j\leq f-1}\mathbb{F}\alpha_i\alpha_j\oplus \bigoplus_{0\leq i\leq j\leq f-1}\mathbb{F}\alpha_{i}^{-1}\alpha_{j}^{-1}\oplus \bigoplus_{0\leq i\neq j\leq f-1} \mathbb{F}\alpha_i\alpha_j^{-1}, \end{gather*}

where $\alpha _j:\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ is the character defined in (2.8).

Proof. It follows from Corollary 2.10 using (2.7); cf. [BHHMS23, (44)].

We have the following criterion which allows us to control the Gelfand–Kirillov dimension of an admissible smooth $\mathbb {F}$-representation of $\mathcal {O}_D^{\times }/Z_D^1$. It is an analogue of [BHHMS23, Corollary 5.3.5]. Let $\overline {W}_{\chi,3}$ denote the quotient of $W_{\chi,3}$ by the sum of characters which occur in $\operatorname {{\mathrm {gr}}}^2W_{\chi,3}$ and non-isomorphic to $\chi$. For example, if $L=\mathbb {Q}_p$, then $\dim _{\mathbb {F}}\overline {W}_{\chi,3}=5$ and has a socle filtration as follows (with $\alpha =\alpha _0$):

\[ (\chi\oplus\chi)\ \textbf{---}\ (\chi\alpha\oplus\chi\alpha^{-1})\ \textbf{---}\ \chi. \]

Corollary 2.12 Let $\pi$ be an admissible smooth representation of $\mathcal {O}_D^{\times } / Z_D^1$ over $\mathbb {F}$. Assume for each character $\chi$ such that $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\chi, \pi ) \neq 0$, the natural injection

(2.11)\begin{equation} \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}(\chi , \pi) \hookrightarrow \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}( \overline{W}_{\chi,3} , \pi) \end{equation}

is an isomorphism. Then $\dim _{\mathcal {O}_{D}^{\times }} (\pi ) \leq f$, where $\dim _{\mathcal {O}_{D}^{\times }}(\pi )$ is the Gelfand–Kirillov dimension of $\pi$ over $\mathcal {O}_{D}^{\times }$.

Proof. The Pontryagin dual $\pi ^{\vee }$ is naturally a finitely generated module over $\mathbb {F}[\![U_D^1/Z_D^1]\!]$ as $\pi$ is admissible, so the graded module $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D}(\pi ^{\vee })$ is finitely generated over $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$. The condition (2.11) implies that $\operatorname {{\mathrm {gr}}}^0_{\frak {m}_D} (\pi ^{\vee })$ is killed by $y_jz_j$ and $h_j'$ (for $1\leq j\leq f-1$), hence also by $I_D$. The result then follows from Corollary 2.10; see [BHHMS23, Corollary 5.3.5] for details.

2.4 $\operatorname {{\mathrm {Ext}}}^i$ groups when $L=\mathbb {Q}_p$

We assume $L=\mathbb {Q}_p$ with $p\geq 5$. We write $\alpha =\alpha _0$.

Proposition 2.13 Let $\psi,\chi :\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ be two smooth characters. Then $\operatorname {{\mathrm {Ext}}}^1_{\mathcal {O}^{\times }_D/Z_D^1}(\psi,\chi )$ is nonzero if and only if $\psi =\chi \alpha$ or $\psi =\chi \alpha ^{-1}$. Moreover,

\[ \dim_{\mathbb{F}}\operatorname{{\mathrm{Ext}}}^1_{\mathcal{O}_D^{\times}/Z_D^1}(\chi\alpha,\chi)=\dim_{\mathbb{F}} \operatorname{{\mathrm{Ext}}}^1_{\mathcal{O}_D^{\times}/Z_D^1}(\chi\alpha^{-1},\chi)=1. \]

Proof. This is a consequence of Corollary 2.11.

Proposition 2.14 Let $\tau _1,\tau _2$ be finite-dimensional smooth representations of $\mathcal {O}_D^{\times }/Z_D^1$. Then there is an isomorphism $\operatorname {{\mathrm {Ext}}}^i_{\mathcal {O}_D^{\times }/Z_D^1}(\tau _1,\tau _2)\cong \operatorname {{\mathrm {Ext}}}^{3-i}_{\mathcal {O}_D^{\times }/Z_D^1}(\tau _2,\tau _1)^{\vee }$ for $0\leq i\leq 3$.

Proof. First, we have isomorphisms

\[ \operatorname{{\mathrm{Ext}}}^i_{\mathcal{O}^{\times}_D/Z_D^1}(\tau_1,\tau_2)\cong\operatorname{{\mathrm{Ext}}}^i_{\mathcal{O}_D^{\times}/Z_D^1} (\mathbf{1},\tau_1^{\vee}\otimes\tau_2)\cong H^i(\mathcal{O}_D^{\times}/Z_D^1,\tau_1^{\vee}\otimes\tau_2) \cong H^i(U_D^1/Z_D^1,\tau_1^{\vee}\otimes\tau_2)^{\mathbb{F}_{p^2}^{\times}}. \]

Second, since $U_D^1/Z_D^1$ is a Poincaré group of dimension three (cf. [Ser02, § 4.5]), Poincaré duality induces an isomorphism

\[ H^i(U_D^1/Z_D^1,\tau)\cong H^{3-i}(U_D^1/Z_D^1,\tau^{\vee})^{\vee} \]

for $0\leq i\leq 3$ and any finite-dimensional representation $\tau$. The result easily follows.

3. Lattices in some locally algebraic representations of $\mathrm {GL}_2(\mathbb {Z}_p)$

Let $K: = \mathrm {GL}_2(\mathbb {Z}_p)$, $\Gamma : = \mathrm {GL}_2(\mathbb {F}_p)$, and $K_1: = \operatorname {{\mathrm {Ker}}}(K \twoheadrightarrow \Gamma )$. Let $I$ (respectively, $I_1$) denote the upper Iwahori (respectively, pro-$p$ Iwahori) subgroup of $K$. Let $Z$ denote the centre of $G$, $Z_1 : = Z\cap K_1$. Let

\[ H : = \bigg\{\!\!\begin{pmatrix}{[a]} & {0}\\ {0} & {[d]}\end{pmatrix},\ a,d\in \mathbb{F}_p^{\times}\!\bigg\}. \]

Let $\alpha : H \to \mathbb {F}^{\times }$ be the character of $H$ sending $\big (\begin{smallmatrix} [a] & 0 \\ 0 & [d] \end{smallmatrix}\big )$ to $a d^{-1}$. By abuse of notation we also denote the image of $H$ in $\Gamma$ by the same letter. If $\chi$ is a character of $H$, we denote by $\chi ^s$ the character sending $h$ to $\chi (shs)$, where $s:=\big (\begin{smallmatrix} {0} & {1}\\ {1} & {0}\end{smallmatrix}\big )$. We regard a character of $H$ as a character of $I$ via the quotient map $I\twoheadrightarrow H$; note that any smooth $\mathbb {F}$-valued character of $I$ arises in this way.

For $m,n\in \mathbb {N}$, we denote

\[ \sigma_{m,n} := \mathrm{Sym}^m \mathbb{F}^2 \otimes {\det}^n \]

which are naturally representations of $\Gamma$ over $\mathbb {F}$. We also regard them as representations of $K$ via the natural projection $K\twoheadrightarrow \Gamma$. Up to isomorphism the set $\{\sigma _{m,n},\ 0\leq m \leq p-1,\ 0\leq n \leq p-2\}$ forms a complete list of irreducible representations of $\Gamma$ (and of $K$) over $\mathbb {F}$.

We choose the standard basis of $\sigma _{m,n}$ to be $\{X^iY^{m-i};\ 0\leq i\leq m\}$, with the action of $\Gamma$ given by

\[ \begin{pmatrix}{a} & {b}\\ {c} & {d}\end{pmatrix} X^iY^{m-i}=(aX+cY)^i(bX+dY)^{m-i}. \]

It is well-known that $\sigma _{m,n}^{I_1}$ is one-dimensional (spanned by $X^m$), on which $H$ acts via the character sending $\big (\begin{smallmatrix} {[a]} & 0 \\ 0 & {[d]} \end{smallmatrix}\big )$ to $a^{m+n} d^n$, which we denote by $\chi _{m,n}$. Similarly, the space of coinvariants $(\sigma _{m,n})_{I_1}$ is one-dimensional on which $H$ acts via $\chi _{m,n}^s$.

Recall $E := W(\mathbb {F})[1/p]$, where $\mathcal {O}:=W(\mathbb {F})$ is the ring of Witt vectors in $\mathbb {F}$. If $V$ is a finite-dimensional representation of $K$ over $E$, then $V^{\circ }$ will denote a $K$-stable $\mathcal {O}$-lattice in $V$ and $\overline {V^{\circ }}$ its reduction modulo $p$. We will write $\overline {V}^{\rm ss}$ for the semisimplification of $\overline {V^{\circ }}$. Following [EGS15], we say $V$ is residually multiplicity free if any of the Jordan–Hölder factors of $\overline {V}^{\rm ss}$ occurs with multiplicity one. In this section, a lattice always means a $K$-stable $\mathcal {O}$-lattice.

3.1 Preliminaries

Denote by $U(\mathbb {Z}_p)$ (respectively, $B(\mathbb {Z}_p)$) the (upper) unipotent (respectively, Borel) subgroup of $K$. Note that $H$ normalizes $U(\mathbb {Z}_p)$.

Proposition 3.1 Let $W$ be a finite-dimensional $\mathbb {F}$-representation of $B(\mathbb {Z}_p)$, of dimension $\geq 2$.

1. (i) Assume that $W^{U(\mathbb {Z}_p)}$ is one-dimensional and isomorphic to $\chi$ as an $H$-representation. Then

   \[ (\mathrm{Sym}^{1}\mathbb{F}^2\otimes W)^{U(\mathbb{Z}_p)}\cong \chi\chi_{1,0}\oplus \chi\chi_{1,0}^s. \]

2. (ii) Assume that $W_{U(\mathbb {Z}_p)}$ is one-dimensional and isomorphic to $\chi$ as an $H$-representation. Then

   \[ (\mathrm{Sym}^1\mathbb{F}^2\otimes W)_{U(\mathbb{Z}_p)}\cong\chi\chi_{1,0}\oplus \chi\chi_{1,0}^s. \]

Proof. (i) Let $W_0:=W^{U(\mathbb {Z}_p)}\cong \chi$. We first prove that $(W/W_0)^{U(\mathbb {Z}_p)}$ is one-dimensional and isomorphic to $\chi \alpha ^{-1}$ as an $H$-representation. Since $\dim _{\mathbb {F}}W\geq 2$ by assumption, $W/W_0$ is nonzero, hence $(W/W_0)^{U(\mathbb {Z}_p)}$ is also nonzero because $U(\mathbb {Z}_p)$ is a pro-$p$-group. On the other hand, we have an $H$-equivariant injection

\[ 0\rightarrow (W/W_0)^{U(\mathbb{Z}_p)}\rightarrow H^1(U(\mathbb{Z}_p),W_0), \]

which is actually an isomorphism because $H^1(U(\mathbb {Z}_p),\chi )\cong \chi \alpha ^{-1}$ is one-dimensional (see, e.g., [Paš10, Lemma 5.5]). This proves the claim.

Any element $w\in \mathrm {Sym}^1\mathbb {F}^2\otimes W$ can be written as $Y\otimes w_0+X\otimes w_1$ for (unique) $w_0,w_1\in W$. Let $g=\big (\begin{smallmatrix} {1} & {t}\\ {0} & {1}\end{smallmatrix}\big ) \in U(\mathbb {Z}_p)$. Then

\[ gw=(\bar{t}X+Y)\otimes gw_0+X\otimes gw_1 =Y\otimes gw_0+X\otimes(\bar{t}\cdot gw_0+gw_1). \]

Hence, $w$ is fixed by $U(\mathbb {Z}_p)$ if and only if

\[ \left\{\begin{array}{@{}ll} gw_0=w_0,\\ gw_1=w_1-\bar{t}gw_0. \end{array}\right. \]

We have two cases.

1. (a) If $w_0=0$, then the above condition becomes $gw_1=w_1$, i.e. $w_1\in W_0$.

2. (b) If $w_0\neq 0$, then $w_0\in W_0$ and $w_1\in (W/W_0)^{U(\mathbb {Z}_p)}$. Moreover, $(W/W_0)^{U(\mathbb {Z}_p)}$ is one-dimensional as seen above and the condition $gw_1=w_1-\bar {t}w_0$ determines uniquely $w_1$ (whenever $w_0\neq 0$ is fixed).

The result easily follows.

(ii) This follows from part (i) via the fact that $(W_{U(\mathbb {Z}_p)})^{\vee }\cong (W^{\vee })^{U(\mathbb {Z}_p)}$, and similarly for

\[ \mathrm{Sym}^1\mathbb{F}^2\otimes W. \]

Corollary 3.2 Let $V=\operatorname {{\mathrm {Ind}}}_{I}^{K}\chi$ for some smooth character $\chi :I\rightarrow \mathbb {F}^{\times }$. Then

\begin{gather*} (\mathrm{Sym}^1\mathbb{F}^2\otimes V)^{U(\mathbb{Z}_p)}\cong \chi^s\chi_{1,0}\oplus\chi^s\chi_{1,0}^s\oplus \chi\chi_{1,0},\\ (\mathrm{Sym}^1\mathbb{F}^2\otimes V)_{U(\mathbb{Z}_p)}\cong \chi^s\chi_{1,0}\oplus\chi^s\chi_{1,0}^s\oplus \chi\chi_{1,0}^s. \end{gather*}

Proof. Mackey's decomposition theorem gives an isomorphism $V|_{I}\cong \chi \oplus V'$, where $V':= \operatorname {{\mathrm {Ind}}}_{HK_1}^{I}\chi ^s$. It is easy to see that $V'$ has dimension $p$, and $V'^{U(\mathbb {Z}_p)}\cong V'_{U(\mathbb {Z}_p)}\cong \chi ^s$. Thus, Proposition 3.1 applies to $\mathrm {Sym}^{1}\mathbb {F}^2\otimes V'$. The results then follow by noting that $(\mathrm {Sym}^1\mathbb {F}^2\otimes \chi )^{U(\mathbb {Z}_p)}\cong \mathbb {F} X\otimes \chi$ and $(\mathrm {Sym}^1\mathbb {F}^2\otimes \chi )_{U(\mathbb {Z}_p)}\cong \mathbb {F} Y\otimes \chi$.

Consider the following situation: $V_1, V_2$ are two irreducible locally algebraic representations of $K$, and $L_i\subset V_i$ is a lattice for $i=1,2$. Assume that we are given an $\mathbb {F}[K]$-module $W$, together with $K$-equivariant morphisms $r_i:L_i\rightarrow W$. Let $L$ be the fibered product of $r_1$ and $r_2$, namely

(3.1)\begin{equation} 0\rightarrow L\rightarrow L_1\oplus L_2 {\buildrel {r_1 - r_2} \over \longrightarrow} W. \end{equation}

Then $L$ is a lattice in $V_1\oplus V_2$. We also call $L$ the gluing lattice of $L_1$ and $L_2$ along $W$. We remark that, if either $r_1$ or $r_2$ is surjective, then so is $r_1-r_2$.

Lemma 3.3 Assume that $r_1$ is surjective.

1. (i) There exists a short exact sequence

   \[ 0\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)/p\operatorname{{\mathrm{Ker}}}(r_1)\rightarrow L/pL\rightarrow L_2/pL_2\rightarrow0. \]

2. (ii) Let $r_L$ denote the composite morphism $L\rightarrow L/pL\rightarrow L_2/pL_2$, where the second map is as in part (i). Then $\operatorname {{\mathrm {Ker}}}(r_L)=\operatorname {{\mathrm {Ker}}}(r_1)+pL$ and

   \[ \operatorname{{\mathrm{Ker}}}(r_L)/p\operatorname{{\mathrm{Ker}}}(r_L)\cong \operatorname{{\mathrm{Ker}}}(r_1)/p\operatorname{{\mathrm{Ker}}}(r_1)\oplus pL_2/p^2L_2. \]

Proof. (i) We have the following commutative diagram.

By the snake lemma, it induces a short exact sequence $0\rightarrow \operatorname {{\mathrm {Ker}}}(r_1)\rightarrow L\rightarrow L_2\rightarrow 0$. We obtain the result by taking mod $p$ reduction (as $L_2$ is $\mathcal {O}$-flat).

(ii) It is clear from part (i) that $\operatorname {{\mathrm {Ker}}}(r_L)=\operatorname {{\mathrm {Ker}}}(r_1)+pL$, so we have a short exact sequence

\[ 0\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)\cap pL\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)\oplus pL\rightarrow \operatorname{{\mathrm{Ker}}}(r_L)\rightarrow0. \]

Taking mod $p$ reduction and noting that $\operatorname {{\mathrm {Ker}}}(r_1)\cap pL=p\operatorname {{\mathrm {Ker}}}(r_1)$ by part (i), we obtain an exact sequence

\[ 0\rightarrow p\operatorname{{\mathrm{Ker}}}(r_1)/p^2\operatorname{{\mathrm{Ker}}}(r_1)\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)/p\operatorname{{\mathrm{Ker}}}(r_1)\oplus pL/p^2L\rightarrow \operatorname{{\mathrm{Ker}}}(r_L)/p\operatorname{{\mathrm{Ker}}}(r_L)\rightarrow0. \]

But the map $p\operatorname {{\mathrm {Ker}}}(r_1)/p^2\operatorname {{\mathrm {Ker}}}(r_1)\rightarrow \operatorname {{\mathrm {Ker}}}(r_1)/p\operatorname {{\mathrm {Ker}}}(r_1)$ is identically zero, so the result follows from part (i).

Lemma 3.4 Assume that both $r_1$ and $r_2$ are surjective. Assume moreover that:

1. (a) $\mathrm {cosoc}(L_1)=\mathrm {cosoc}(W)$;

2. (b) $\mathrm {cosoc}(\operatorname {{\mathrm {Ker}}}(r_1))$ and $\mathrm {cosoc}(\operatorname {{\mathrm {Ker}}}(r_2))$ do not admit common Jordan–Hölder factors.

Then $\mathrm {cosoc}(L)\cong \mathrm {cosoc}(L_2)$.

Proof. We need to show that the natural map

\[ \operatorname{{\mathrm{Hom}}}_{\mathcal{O}[K]}(L_2,\sigma)\rightarrow\operatorname{{\mathrm{Hom}}}_{\mathcal{O}[K]}(L,\sigma) \]

is an isomorphism for any Serre weight $\sigma$. By applying $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}[K]}(-,\sigma )$ to (3.1) we obtain a long exact sequence

\begin{align*} 0\rightarrow \operatorname{{\mathrm{Hom}}}(W,\sigma)\rightarrow\operatorname{{\mathrm{Hom}}}(L_1,\sigma)\oplus \operatorname{{\mathrm{Hom}}}(L_2,\sigma) &\rightarrow \operatorname{{\mathrm{Hom}}}(L,\sigma)\\ &\rightarrow \operatorname{{\mathrm{Ext}}}^1(W,\sigma)\rightarrow \operatorname{{\mathrm{Ext}}}^1(L_1,\sigma)\oplus \operatorname{{\mathrm{Ext}}}^1(L_2,\sigma). \end{align*}

By assumption (a), the surjection $r_1:L_1\twoheadrightarrow W$ induces an isomorphism $\operatorname {{\mathrm {Hom}}}(W,\sigma )\xrightarrow {\sim } \operatorname {{\mathrm {Hom}}}(L_1,\sigma )$. To conclude we need to show that the morphism

\[ \operatorname{{\mathrm{Ext}}}^1(W,\sigma)\rightarrow \operatorname{{\mathrm{Ext}}}^1(L_1,\sigma)\oplus \operatorname{{\mathrm{Ext}}}^1(L_2,\sigma) \]

is injective. For this, it is enough to prove that either $\operatorname {{\mathrm {Hom}}}(\operatorname {{\mathrm {Ker}}}(r_1),\sigma )$ or $\operatorname {{\mathrm {Hom}}}(\operatorname {{\mathrm {Ker}}}(r_2),\sigma )$ vanishes, which is a consequence of assumption (b).

Finally, we record a result which will be used later on.

Proposition 3.5 Let $V$ be an irreducible smooth representation of $K$ over $E$. Then the $K$-representation $\mathrm {Sym}^1 E^2 \otimes V$ is again irreducible.

Proof. This is [STP01, Proposition 3.4].

3.2 Lattices in tame types

We consider the following representations of $\Gamma$ over $E$, and view them as smooth representations of $K$ via the projection $K \twoheadrightarrow \Gamma$.

1. • Let $\chi _1,\chi _2: \mathbb {F}_{p}^{\times } \to E^{\times }$ be two characters. Let $I(\chi _1,\chi _2)$ denote the principal series representation $\operatorname {{\mathrm {Ind}}}_{B(\mathbb {F}_p)}^{\Gamma }\chi _1\otimes \chi _2$, where $B(\mathbb {F}_p)$ is the (upper) Borel subgroup of $\Gamma$. It is well-known that $I(\chi _1,\chi _2)$ is irreducible if $\chi _1 \neq \chi _2$. If $\chi _1 =\chi _2 =\chi$, then

   \[ I(\chi, \chi) \cong (\chi \circ\det) \oplus\ ({\rm sp} \otimes\chi\circ\det), \]
   where ${\rm sp}$ denotes the Steinberg representation.

2. • Let $\psi :\mathbb {F}_{p^2}^{\times } \to E^{\times }$ be a character which does not factor through the norm map $\mathbb {F}_{p^2}^{\times } \to \mathbb {F}_{p}^{\times }$. This is equivalent to requiring $\psi \neq \psi ^p$. There is an irreducible $(p-1)$-dimensional representation $\Theta (\psi )$ characterized by the isomorphism $\Theta (\psi )\otimes {\rm sp} \cong \operatorname {{\mathrm {Ind}}}_{\mathbb {F}_{p^2}^{\times }}^{\mathrm {GL}_2(\mathbb {F}_p)} \psi$, where $\mathbb {F}_{p^2}^{\times } \hookrightarrow \mathrm {GL}_2(\mathbb {F}_p)$ is a fixed group embedding. For two such characters $\psi,\psi '$, $\Theta (\psi ) \cong \Theta (\psi ')$ if and only if $\psi '\in \{\psi, \psi ^p\}$.

The Jordan–Hölder factors of the reduction mod $p$ of any lattice in the above representations are determined in [Dia07]. We recall the results in the next proposition.

Let $x: \mathbb {F}_p \hookrightarrow \mathbb {F}$ denote the natural embedding and $[x]: \mathbb {F}_p \to \mathcal {O}$ be the Teichmüller lift of $x$ which will be viewed as a multiplicative character of $\mathbb {F}_p^{\times }$. Let $\xi : \mathbb {F}_{p^2} \hookrightarrow \mathbb {F}$ be an embedding extending $x$. Let $\xi ':= \xi ^p$ and $\zeta : = \xi \xi '$. Let $[\xi ] : \mathbb {F}_{p^2} \to \mathcal {O}$ be the Teichmüller lift of $\xi$ which will be viewed as a multiplicative character of $\mathbb {F}_{p^2}^{\times }$. We have $[x]^{p-1} = \mathbf {1}$ and $[\xi ]^{p+1} = [x]$.

Proposition 3.6

1. (i) Let $0\leq a \leq p-1$ and $0\leq b\leq p-2$. Then

   \[ \overline{I([x]^b , [x]^{b+a})}^{\rm ss} \cong \sigma_{a,b} \oplus \sigma_{p-1-a, a+b}. \]

2. (ii) Let $\psi :\mathbb {F}_{p^2}^{\times } \to E^{\times }$ with $\psi \neq \psi ^p$. Write $\psi = [\xi ]^{a+1 + (p+1)b}$ with $0\leq a \leq p-1$ and $0\leq b\leq p-2$. Then

   \[ \overline{\Theta(\psi)}^{\rm ss} \cong \sigma_{a-1, b+1} \oplus \sigma_{p-2-a, a+b+1}, \]
   with the convention that $\sigma _{-1, b} = 0$.

3. (iii) The representations $I([x]^b , [x]^{b+a})$ and $\Theta (\psi )$ are residually multiplicity free.

Proof. Part (i) follows from [Dia07, Proposition 1.1]; part (ii) follows from [Dia07, Proposition 1.3]. Part (iii) follows directly from parts (i) and (ii).

We recall Lemma 4.1.1 of [EGS15] on the lattices of finite-dimensional irreducible residually multiplicity-free $E$-representations of $K$.

Proposition 3.7 [EGS15]

Let $V$ be a finite-dimensional irreducible representation of $K$ over $E$ which is residually multiplicity free. Let $\sigma$ be a Jordan–Hölder factor of $\overline {V}^{\rm ss}$. Then there is up to homothety a unique lattice $V^{\circ }_{\sigma }$ in $V$ such that the socle of $\overline {V^{\circ }_{\sigma }}$ is $\sigma$. Similarly, there is up to homothety a unique lattice $V^{\circ,\sigma }$ in $V$ such that the cosocle of  $\overline {V^{\circ,\sigma }}$ is $\sigma$.

3.3 Lattices in $ \underline {\mathrm {Sym}}^1E^2 \otimes \Theta (\psi )$

Let $\mathrm {pr}:\mathbb {Q}_p^{\times }\rightarrow 1+p\mathbb {Z}_p$ denote the projection sending $p$ to $1$. As $p>2$, we can define the square root on $1+p\mathbb {Z}_p$ by the usual binomial formula. Define

(3.2)\begin{equation} \underline{\mathrm{Sym}}^1E^2:=\mathrm{Sym}^{1}E^2\otimes (\mathrm{pr}\circ\det)^{-1/2}. \end{equation}

The reason to introduce the twist is to make the central character of $\underline {\mathrm {Sym}}^1E^2$ to be trivial on $Z_1$. Note that the mod $p$ reduction of $\underline {\mathrm {Sym}}^1\mathcal {O}^2:=\mathrm {Sym}^{1}\mathcal {O}^2\otimes (\mathrm {pr}\circ \det )^{-1/2}$ still gives $\mathrm {Sym}^1\mathbb {F}^2$.

Let $\psi : \mathbb {F}_{p^2}^{\times } \to E^{\times }$ be a character with $\psi \neq \psi ^p$. Write $\psi = [\xi ]^{a+1 + (p+1)b}$ with $0\leq a \leq p-1$ and $0\leq b\leq p-2$. By Proposition 3.6, $\overline {\Theta (\psi )}^{\rm ss}$ is multiplicity free and has two (respectively, one) Jordan–Hölder factors if $1 \leq a \leq p-2$ (respectively, if $a\in \{0,p-1\}$).

Assume first $1\leq a\leq p-2$. By Propositions 3.6 and 3.7, there are two lattices $T,T'$ in $\Theta (\psi )$ such that

(3.3)$$\begin{gather} 0\rightarrow \sigma_{p-2-a, a+b+1} \rightarrow T/pT\rightarrow \sigma_{a-1, b+1}\rightarrow 0 , \end{gather}$$

(3.4)$$\begin{gather}0\rightarrow \sigma_{a-1, b+1} \rightarrow T'/pT'\rightarrow \sigma_{p-2-a, a+b+1}\rightarrow 0, \end{gather}$$

where both extensions are nonsplit. Note that $T/pT$ and $T'/pT'$ are $\Gamma$-representations as $\Theta (\psi )$ itself is. Moreover, if we fix $T$ and normalize $T'$ (by a scalar) so that $T'\subset T$ and $T'\nsubseteq pT$, then by [EGS15, Proposition 5.2.3(1)] we have

(3.5)\begin{equation} pT\subset T'\subset T. \end{equation}

Lemma 3.8

1. (i) We have $(T/pT)^{I_1}\cong \chi _{p-2-a,a+b+1}$ and $(T'/pT')^{I_1}\cong \chi _{a-1,b+1}$.

2. (ii) We have $(T / pT)_{I_1} = \chi ^s_{ a -1, b +1}$ and $(T'/pT')_{I_1}\cong \chi _{p-2-a,a+b+1}^{s}$.

Proof. (i) We only give the proof in the case of $T/pT$. Using (3.3) we obtain an exact sequence

\[ 0\rightarrow (\sigma_{p-2-a,a+b+1})^{I_1} \to (T/pT)^{I_1} \to (\sigma_{a-1 ,b+1})^{I_1}. \]

Assume for a contradiction that $(T / pT)^{I_1}$ is two-dimensional. Then we would obtain

\[ (T/pT)^{I_1}\cong \chi_{p-2-a,a+b+1}\oplus \chi_{a-1,b+1}, \]

and consequently an $I$-equivariant injection $\chi _{a-1,b+1}\hookrightarrow T/pT$. By Frobenius reciprocity, we would get a nonzero $K$-equivariant map

\[ \operatorname{{\mathrm{Ind}}}_{I}^{K} \chi_{a-1,b+1}\rightarrow T/pT. \]

By comparing the Jordan–Hölder factors, this map cannot be injective and must have image isomorphic to $\sigma _{a-1,b+1}$ (see [BP12, Lemma 2.3]). This gives a contradiction because the sequence (3.3) is nonsplit.

(ii) This is proved in a similar way as part (i). Alternatively, it can be deduced from part (i) by taking dual.

Recall that $E$ is unramified over $\mathbb {Q}_p$. Consider $\underline {\mathrm {Sym}}^1\mathcal {O}^2:=\mathcal {O} Y\oplus \mathcal {O} X$, the standard lattice in $\underline {\mathrm {Sym}}^1E^2$ and set

(3.6)\begin{gather} L:= \underline{\mathrm{Sym}}^1\mathcal{O}^2\otimes_{\mathcal{O}} T , \end{gather}

(3.7)\begin{gather}L' := \underline{\mathrm{Sym}}^1\mathcal{O}^2\otimes_{\mathcal{O}} T'. \end{gather}

Then we have²

\begin{align*} L / p L &\cong \mathrm{Sym}^1\mathbb{F}^2\otimes T/pT,\\ L' / p L' &\cong \mathrm{Sym}^1\mathbb{F}^2\otimes T'/p T', \end{align*}

and (3.5) implies $p L \subset L' \subset L$.

Lemma 3.9 The group $K_1$ acts trivially on $L/pL$ and $L'/pL'$.

Proof. This is because $K_1$ acts trivially on both $\mathrm {Sym}^1\mathbb {F}^2$ and $\Theta (\psi )$.

Lemma 3.10

1. (i) We have $(L/pL)^{I_1}\cong \chi _{p-1-a,a+b+1}\oplus \chi _{p-3-a,a+b+2}$ and $(L'/pL')^{I_1}\cong \chi _{a,b+1}\oplus \chi _{a-2,b+2}$.

2. (ii) We have $(L / p L)_{I_1} \cong \chi ^s_{ a, b +1} \oplus \chi ^s_{ a - 2, b +2}$ and $(L'/pL')_{I_1}\cong \chi _{p-1-a,a+b+1}^s\oplus \chi _{p-3-a.a+b+2}^s$.

Proof. By Lemma 3.9, we have $(L/pL)^{I_1}=(L/pL)^{U(\mathbb {Z}_p)}$ and $(L/pL)_{I_1}=(L/pL)_{U(\mathbb {Z}_p)}$, so the results follow from Proposition 3.1 and Lemma 3.8.

Proposition 3.11 Assume $1\leq a \leq p-2$.

1. (i) We have that $L/pL$ is multiplicity free and has a two-step socle (and cosocle) filtration

   (3.8)\begin{equation} (\sigma_{p-3-a, a+b+2} \oplus\sigma_{p-1-a, a+b+1}) \ \textbf{---}\ (\sigma_{a, b+1} \oplus \sigma_{a-2, b+2}) \end{equation}
   (with the convention $\sigma _{-1,b+1} = \sigma _{-1,b+2} =0$). Moreover, the following nonsplit extensions
   \begin{align*} E_1 &= (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}),\\ E_2 &= (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}),\\ E_3 &= (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a, b+1}) \end{align*}
   occur in $L/pL$ as subquotients, with the exception that $E_1$ (respectively, $E_2$) does not exist if $a=p-2$ (respectively, $a=1$).

2. (ii) We have that $L'/pL'$ is multiplicity free and has a two-step socle (and cosocle) filtration

   \[ (\sigma_{a, b+1} \oplus \sigma_{a-2, b+2}) \ \textbf{---}\ (\sigma_{p-3-a, a+b+2} \oplus\sigma_{p-1-a, a+b+1}). \]
   (with the convention $\sigma _{-1,b+1} = \sigma _{-1,b+2} =0$). Moreover, the following nonsplit extensions
   \begin{align*} E'_1 &= (\sigma_{a, b+1} \ \textbf{---}\ \sigma_{p-3-a, a+b +2}),\\ E'_2 & = ( \sigma_{a-2, b+2} \ \textbf{---}\ \sigma_{p-1-a, a+b+1}),\\ E'_3 & = (\sigma_{a, b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1}) \end{align*}
   occur in $L'/pL'$ as subquotients, with the exception that $E'_1$ (respectively, $E'_2$) does not exist if $a=p-2$ (respectively, $a=1$).

Proof. It suffices to prove part (i). Recall the following facts (see [BP12, Lemma 3.8])

\begin{align*} \mathrm{Sym}^1\mathbb{F}^2\otimes \sigma_{a-1, b+1} &\cong \sigma_{a, b+1} \oplus \sigma_{a-2, b+2}, \\ \mathrm{Sym}^1\mathbb{F}^2\otimes\sigma_{p-2-a,a+b+1} &\cong \sigma_{p-1-a,a+b+1}\oplus\sigma_{p-3-a,a+b+2}, \end{align*}

with the convention $\sigma _{-1,b+1} = \sigma _{-1,b+2}= 0$. Using (3.3) this implies that $L/pL\cong \mathrm {Sym}^1\mathbb {F}^2\otimes T/pT$ has a two-step filtration as claimed in (3.8), and is multiplicity free. By Lemma 3.10, the filtration gives exactly the socle (and cosocle) filtration. This also completes the proof if $a\in \{1,p-2\}$.

Assume $2\leq a \leq p-3$ in the rest of the proof. For a Serre weight $\sigma$, denote by ${\rm Inj}_{\Gamma }(\sigma )$ the injective envelope of $\sigma$ in the category of $\mathbb {F}[\Gamma ]$-modules; we remark that $\mathrm {Inj}_{\Gamma }(\sigma )$ is also projective. Let $W_1$ denote the image of the unique (up to scalar) nonzero map ${\rm Inj}_{\Gamma }(\sigma _{a-2,b+2}) \to L/ p L$. Since $\sigma _{p-3-a, a+b+2}$ is not a Jordan–Hölder factor of ${\rm Inj}_{\Gamma }(\sigma _{a-2,b+2})$ (see [BP12, Lemma 3.2]), $W_1$ does not admit $\sigma _{p-3-a,a+b+2}$ as a subquotient. Since $\mathrm {cosoc}(W_1)\cong \sigma _{a-2,b+2}$ by construction, we deduce from (3.8) that

\[ W_1\cong (\sigma_{p - 1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}), \]

i.e. the nonsplit extension $E_2$ occurs in $L/pL$. Consequently, the cokernel of the inclusion $W_1\hookrightarrow L/ p L$, denoted by $W_2$, has $\{\sigma _{a,b+1},\sigma _{p-3-a,a+b+2}\}$ as the set of Jordan–Hölder factors, hence is isomorphic to the nonsplit extension $E_1=(\sigma _{p- 3-a, a+b+2} \ \textbf {---}\ \sigma _{a , b+1})$ because $\sigma _{p-3-a,a+b+2}$ does not occur in the cosocle of $W_2$ by (3.8).

We are left to show that $L / pL$ is a nonsplit extension of $W_2$ by $W_1$ (this implies that $E_3$ occurs in $L/pL$). Assume for a contradiction that $L / pL \cong W_1 \oplus W_2$. Let $V$ denote the principal series $\operatorname {{\mathrm {Ind}}}_{I}^{K}\chi _{a+1,b}^s$ which is isomorphic to the (unique) nonsplit extension $(\sigma _{a+1,b}\ \textbf {---}\ \sigma _{p-2-a,a+b+1})$. By [BP12, § 3], there exists a short exact sequence

\[ 0 \to T / pT \to {\rm Inj}_{\Gamma} (\sigma_{p - 2 - a, a+b +1}) \to V \to 0, \]

which induces a short exact sequence

\[ 0 \to L/pL \to \mathrm{Sym}^1 \mathbb{F}^2 \otimes {\rm Inj}_{\Gamma} (\sigma_{p - 2 - a, a+b +1}) \to \mathrm{Sym}^1 \mathbb{F}^2 \otimes V \to 0. \]

By Lemma 3.12 below, if $2 \leq a \leq p-4$, then

\[ \mathrm{Sym}^1 \mathbb{F}^2 \otimes {\rm Inj}_{\Gamma} (\sigma_{p - 2 - a, a+b +1}) = {\rm Inj}_{\Gamma} (\sigma_{p -1-a, a+b+1}) \oplus {\rm Inj}_{\Gamma} (\sigma_{p-3-a,a+b+2}). \]

Comparing the socles, it is clear that $W_2\cap {\rm Inj}_{\Gamma } (\sigma _{p -1-a, a+b+1})\,{=}\,0$, thus $W_2\!\hookrightarrow\! {\rm Inj}_{\Gamma } (\sigma _{p-3-a,a+b+2})$. Moreover, we have

\[ L/pL \cap \mathrm{Inj}_{\Gamma}\sigma_{p-3-a,a+b+2}=W_2, \]

which induces a (nonzero) morphism

\[ {\rm Inj}_{\Gamma} (\sigma_{p-3-a,a+b+2}) / W_2 \rightarrow\mathrm{Sym}^1 \mathbb{F}^2 \otimes V. \]

However, by [BP12, § 3] we have

\[ {\rm Inj}_{\Gamma} (\sigma_{p-3-a,a+b+2}) / W_2 \cong \operatorname{{\mathrm{Ind}}}_{B(\mathbb{F}_p)}^{\Gamma}\chi_{p-3-a,a+b+2}, \]

so by Frobenius reciprocity we obtain a nonzero $I$-equivariant morphism

\[ \chi_{p-3-a,a+b+2}\rightarrow \mathrm{Sym}^1\mathbb{F}^2\otimes V. \]

But this contradicts Corollary 3.2, by which $(\mathrm {Sym}^{1}\mathbb {F}^2\otimes V)^{I_1}\cong \chi _{a+2,b}\oplus \chi _{a,b+1}\oplus \chi _{p-1-a,a+b+1}$.

The case $a = p-3$ is a little subtle. By Lemma 3.12 below we have

\[ \mathrm{Sym}^1 \mathbb{F}^2 \otimes {\rm Inj}_{\Gamma} (\sigma_{1, b -1}) = {\rm Inj}_{\Gamma} (\sigma_{2, b- 1}) \oplus {\rm Inj}_{\Gamma} (\sigma_{0,b}) \oplus \sigma_{p-1,b}. \]

Comparing the socles, one checks that $W_2$ embeds into ${\rm Inj}_{\Gamma } (\sigma _{0,b})$ and actually

\[ L/pL\cap \mathrm{Inj}_{\Gamma}(\sigma_{0,b})=W_2. \]

Hence, we obtain a nonzero morphism from ${\rm Inj}_{\Gamma } (\sigma _{0,b})/ W_2 \cong \sigma _{0,b}$ to $\mathrm {Sym}^1\mathbb {F}^2 \otimes V$. On the other hand, $\sigma _{p-1,b}$ also occurs in $\mathrm {Sym}^1\mathbb {F}^2\otimes V$ and, in fact, is a direct summand because $\sigma _{p-1,b}$ is an injective $\mathbb {F}[\Gamma ]$-module. Thus, there exists an embedding

\[ \sigma_{0,b}\oplus\sigma_{p-1,b}\hookrightarrow \mathrm{Sym}^1\mathbb{F}^2\otimes V. \]

However, by Corollary 3.2 we have $(\mathrm {Sym}^{1}\mathbb {F}^2\otimes V)^{I_1}\cong \chi _{p-1,b}\oplus \chi _{p-3,b+1}\oplus \chi _{2,b-1}$, in which the character $\chi _{p-1,b}$ ($=\chi _{0,b}$) occurs only once. This gives a contradiction and finishes the proof.

Recall the following facts (see [BP12, § 3]): ${\rm Inj}_{\Gamma } (\sigma _{a , b})$ is of dimension $2p$ if $1\leq a \leq p-2$; ${\rm Inj}_{\Gamma }(\sigma _{p-1,b}) \cong \sigma _{p-1 , b}$ is of dimension $p$; ${\rm Inj}_{\Gamma } (\sigma _{0,b}) \cong (\sigma _{0,b} \ \textbf {---}\ \sigma _{p-3, b+1} \ \textbf {---}\ \sigma _{0,b})$ is of dimension $p$.

Lemma 3.12

1. (i) If $a = 1$, then $\mathrm {Sym}^1 \mathbb {F}^2 \otimes {\rm Inj}_{\Gamma } (\sigma _{1, b}) \cong {\rm Inj}_{\Gamma } (\sigma _{2, b})\oplus {\rm Inj}_{\Gamma }(\sigma _{0, b+1}) \oplus \sigma _{p-1, b+1}$.

2. (ii) If $a = p-2$, then $\mathrm {Sym}^1 \mathbb {F}^2 \otimes {\rm Inj}_{\Gamma } (\sigma _{p-2, b}) \cong \sigma _{p-1, b}\oplus \sigma _{p-1, b} \oplus {\rm Inj}_{\Gamma }(\sigma _{p-3, b+1})$.

3. (iii) If $0 \leq a \leq p - 1$ and $a\notin \{1,p-2\}$, then $\mathrm {Sym}^1 \mathbb {F}^2 \otimes {\rm Inj}_{\Gamma } (\sigma _{a, b}) \!=\! {\rm Inj}_{\Gamma } (\sigma _{a+1, b})\oplus {\rm Inj}_{\Gamma }(\sigma _{a - 1, b+1})$ with the convention ${\rm Inj}_{\Gamma }(\sigma _{- 1, b+1}) = {\rm Inj}_{\Gamma }(\sigma _{p, b}) =0$.

Proof. Using the fact that $\mathrm {Sym}^{1}\mathbb {F}^2\otimes {\rm Inj}_{\Gamma }\sigma$ is an injective object in the category of $\mathbb {F}[\Gamma ]$-modules, the results can be easily deduced from [BP12, Lemma 3.8].

Finally, we treat the case $a\in \{0,p-1\}$.

Proposition 3.13 Assume $a\in \{0,p-1\}$. There are two lattices (unique up to homothety) $L, L'$ of $\underline {\mathrm {Sym}}^1 E^2\otimes \Theta (\psi )$ such that $pL \subset L' \subset L$ and

\begin{align*} L/pL &\cong \sigma_{p-1, b+1} \oplus \sigma_{p-3, b+2},\\ L'/pL' &\cong (\sigma_{p-1, b+1} \ \textbf{---}\ \sigma_{p-3, b+2}). \end{align*}

The lattice $pL$ is then identified with the kernel of the natural projection $L' \twoheadrightarrow \sigma _{p-3,b+2}$. Moreover, $(L'/pL')_{K_1} \cong \sigma _{p-3, b+2}$.

Proof. Let $T$ be any lattice in $\Theta (\psi )$ and $L: = \underline {\mathrm {Sym}}^1 \mathcal {O}^2 \otimes T$. Then $T/pT \cong \sigma _{p-2, b+1}$ by Proposition 3.6, and consequently $L/pL \cong \sigma _{p-1, b+1} \oplus \sigma _{p-3,b+2}$ by [BP12, Lemma 3.8]. Let $L'$ be the kernel of the composition $L \to L/pL \overset {p_1}{\twoheadrightarrow } \sigma _{p-1,b+1}$. Then $pL \subsetneq L'\subsetneq L$. Moreover, we have a short exact sequence

(3.9)\begin{equation} 0\to pL/pL' \to L'/pL' \to \sigma_{p-3,b+2} \to 0. \end{equation}

We claim that (3.9) induces an isomorphism $(L'/pL')_{K_1} \xrightarrow {\sim } \sigma _{p-3,b+2};$ this will imply that $L'/pL'$ is a nonsplit extension of $\sigma _{p-3,b+2}$ by $\sigma _{p-1,b+1}$.

The proof of Proposition 3.1 shows that there exist $\overline {w}_0,\overline {w}_1\in T/pT$ such that $X\otimes \overline {w}_0$ and $Y\otimes \overline {w}_0+X\otimes \overline {w}_1$ span $(L/pL)^{I_1}$. Comparing the $H$-action, we must have

\[ (\sigma_{p-1,b+1})^{I_1}=\mathbb{F} (X\otimes \overline{w}_0),\quad (\sigma_{p-3,b+2})^{I_1}=\mathbb{F} (Y\otimes \overline{w}_0+X\otimes \overline{w}_1). \]

Let $w_0,w_1 \in T$ be a lift of $\overline {w}_0,\overline {w}_1$, respectively. From the definition of $L'$ we see that $Y\otimes w_0+X\otimes w_1\in L'$. As $K_1$ acts trivially on $T$, we have

\[ \bigg(\bigg(\begin{matrix} {1} & {p}\\ {0} & {1}\end{matrix}\bigg) - 1\bigg) (Y\otimes w_0 + X \otimes w_1) = (pX )\otimes w_0 \in pL. \]

Since $(pX )\otimes \overline {w}_0$ generates $pL/pL'$, the claim follows.

The uniqueness of $L'$ (up to homothety) follows from Proposition 3.7. Since $pL$ is identified with the kernel of the natural projection $L' \twoheadrightarrow \sigma _{p-3,b+2}$, the uniqueness of $L$ follows.

3.3.1 Sublattices in $L$

In this subsection we specify some sublattices in $L$ in the case $1 \leq a \leq p-2$. Recall that $\sigma _{-1, b+1} = \sigma _{-1, b+2} = 0$ by our convention.

Let $L_1 : = \operatorname {{\mathrm {Ker}}}( L \twoheadrightarrow L/p L \twoheadrightarrow \sigma _{a-2, b+2} )$. It is clear that $pL\subset L_1\subset L$.

Proposition 3.14 The following nonsplit extensions

\begin{align*} & (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}),\\ & (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a, b+1}),\\ & (\sigma_{a-2, b+2} \ \textbf{---}\ \sigma_{p-1-a, a+b+1}) \end{align*}

occur in $L_1/ p L_1$ as subquotients. Consequently, $L_1/pL_1$ has a cosocle filtration

\[ \sigma_{a-2, b+2} \ \textbf{---}\ ( \sigma_{p-1-a, a+b+1} \oplus \sigma_{p-3-a, a+b+2} ) \ \textbf{---}\ \sigma_{a, b+1} \]

and $L_1$ is the unique (up to homothety) lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi )$ whose reduction has cosocle $\sigma _{a, b+1}$. Moreover, we have

\[ (L_1/ p L_1)_{K_1} = ( \sigma_{p-1-a, a+b+1} \oplus \sigma_{p-3-a, a+b+2} ) \ \textbf{---}\ \sigma_{a, b+1}. \]

Proof. By construction, we have $pL\subset L_1$ and $L_1 / pL = \operatorname {{\mathrm {Ker}}} (L/pL \to \sigma _{a-2, b+2})$. By Proposition 3.11(i), the cosocle filtration of $L_1/pL$ is

\[ (\sigma_{p-1-a,a+b+1}\oplus\sigma_{p-3-a,a+b+2}) \ \textbf{---}\ \sigma_{a,b+1}, \]

thus the nonsplit extensions $(\sigma _{p-3-a, a+b +2} \ \textbf {---}\ \sigma _{a, b+1})$ and $(\sigma _{p-1-a, a+b+1} \ \textbf {---}\ \sigma _{a, b+1})$ occur in $L_1/pL$, hence also in $L_1/pL_1$.

We need to show that the nonsplit extension $(\sigma _{a-2, b+2} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ also occurs in $L_1/pL_1$. For this we note that $pL_1 \subset L' \subset L_1$, where $L'$ is defined in (3.7). Consequently, $L' / pL_1$ is a subrepresentation of $L_1 / pL_1$, and it is easy to see that

\[ \operatorname{{\mathrm{JH}}}(L'/pL_1)=\{\sigma_{a-2,b+2},\sigma_{p-1-a,a+b+1},\sigma_{p-3-a,a+b+2}\}. \]

As a quotient of $L' / pL'$, $L' / pL_1$ admits the nonsplit extension $(\sigma _{a-2, b+2} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ as a subquotient, see Proposition 3.11(ii). The structure of $(L_1 / pL_1)_{K_1}$ and other statements easily follow.

Let $L_1':=\operatorname {{\mathrm {Ker}}}(L'\twoheadrightarrow L'/pL'\twoheadrightarrow \sigma _{p-3-a,a+b+2})$, where $L'$ is defined in (3.7). Then $pL'\subset L_1'\subset L'$. Alternatively, $L_1'$ is characterized by the following exact sequence:

(3.10)\begin{equation} 0\rightarrow pL\rightarrow L_1'\rightarrow \sigma_{p-1-a,a+b+1}\rightarrow0. \end{equation}

In a similar way to Proposition 3.14, we have the following result.

Proposition 3.15 The nonsplit extensions

\begin{align*} & (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}), \\ & (\sigma_{a, b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1} ), \\ & (\sigma_{a-2, b+2} \ \textbf{---}\ \sigma_{p-1-a, a+b+1}) \end{align*}

occur in $L'_1/ p L'_1$. Consequently, $L'_1/pL'_1$ has a cosocle filtration

\[ \sigma_{p-3-a, a+b+2} \ \textbf{---}\ (\sigma_{a-2, b+2}\oplus \sigma_{a, b+1}) \ \textbf{---}\ \sigma_{p-1-a, a+b+1}, \]

and $L_1'$ is the unique (up to homothety) lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi )$ whose reduction has cosocle $\sigma _{p-1-a, a+b+1}$. Moreover, we have

\[ (L'_1/ p L'_1)_{K_1} = (\sigma_{a-2, b+2}\oplus \sigma_{a, b+1} ) \ \textbf{---}\ \sigma_{p-1-a, a+b+1}. \]

Let $L_2 : = \operatorname {{\mathrm {Ker}}}( L \twoheadrightarrow L/p L \twoheadrightarrow (\sigma _{p-3-a, a+b+2} \ \textbf {---}\ \sigma _{a, b+1}) )$. Then $pL\subset L_2\subset L$ and there is a short exact sequence

(3.11)\begin{equation} 0\rightarrow pL\rightarrow L_2\rightarrow (\sigma_{p-1-a,a+b+1}\ \textbf{---}\ \sigma_{a-2,b+2})\rightarrow0. \end{equation}

Proposition 3.16 Assume $2\leq a \leq p-2$. Then the nonsplit extensions

\begin{align*} & (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}),\\ & (\sigma_{a,b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1} ),\\ & (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}) \end{align*}

occur in $L_2/ p L_2$. Consequently, $L_2/pL_2$ has a cosocle filtration

\[ \sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a,b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}, \]

and $L_2$ is the unique (up to homothety) lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi )$ whose reduction has cosocle $\sigma _{a-2, b+2}$. Moreover, we have

\[ (L_2 / pL_2)_{K_1} = (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}). \]

Proof. Since $L_2 / pL_2$ is an extension of $L_2 / pL$ by $pL / pL_2$, it suffices to show that the nonsplit extension $(\sigma _{a, b+1} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ occurs in $L_2 / pL_2$.

It follows from (3.10) and (3.11) that $pL\subset L'_1\subset L_2$ and there is a short exact sequence

\[ 0\rightarrow L_1'\rightarrow L_2\rightarrow \sigma_{a-2,b+2}\rightarrow0. \]

This implies that

\[ L_1'/pL_2\cong (L_1'/pL_1')/(\sigma_{a-2,b+2}), \]

and the nonsplit extension $(\sigma _{a, b+1} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ occurs in $L'_1 / pL_2$ by Proposition 3.15. Since $L_1'/pL_2$ embeds in $L_2/pL_2$, this nonsplit extension also occurs in $L_2/pL_2$.

The sublattices $L_1$ and $L_2$ of $L$ satisfy the following property.

Proposition 3.17 Assume $3\leq a\leq p-2$. Fix $i\in \{1,2\}$. Then for every $x\in p L$, there exist $r\in \mathbb {N}$, $k_1,\ldots, k_r \in K_1$, $y_1,\ldots, y_r \in L_i$, such that

\[ x= (k_1 -1) y_1 + \cdots + (k_r - 1) y_r. \]

The proof of Proposition 3.17 requires a technique introduced in [BHHMS23, § 7], so we first recall some notation. Let $\frak {sl}_{2,\mathbb {F}_p}$ be the Lie algebra consisting of trace-zero $2\times 2$ matrices with coefficients in $\mathbb {F}_p$. It is a three-dimensional vector space over $\mathbb {F}_p$ with a basis

\[ e = \bigg(\begin{array}{cc} 0 & 1 \\ 0 & 0\\ \end{array}\bigg),\quad f = \bigg(\begin{array}{cc} 0 & 0 \\ 1 & 0\\ \end{array}\bigg),\quad h = \bigg(\begin{array}{cc} 1 & 0 \\ 0 & -1\\ \end{array}\bigg) \]

subject to the Lie bracket relations

\[ [e,f] = h,\quad [h,e] = 2e,\quad [h,f] = -2f. \]

Let $(V,\rho )$ be a continuous finite-dimensional representation of $K/Z_1$ over $E$. Assume that $V^{\circ }$ is a $K$-stable $\mathcal {O}$-lattice in $V$ such that $K_1$ acts trivially on $V^{\circ } / p V^{\circ }$. Breuil et al. [BHHMS23, § 7.1] defined a Lie algebra action of $\frak {sl}_{2,\mathbb {F}_p}$ on $V^{\circ }/pV^{\circ }$, which induces an $\mathbb {F}$-linear map

\[ \beta_{V^{\circ}} : \frak{sl}_{2,\mathbb{F}_p} \otimes_{\mathbb{F}_p} (V^{\circ}/pV^{\circ} )\to V^{\circ}/pV^{\circ} \]

sending $x \otimes v$ to $p^{-1} (\rho (\exp (p \widetilde {x})) \widetilde {v} - \widetilde {v}) \pmod {p V^{\circ }}$, where $\widetilde {x} \in \frak {sl}_{2,\mathbb {Z}_p}$ is a trace-zero $2\times 2$ matrix with coefficients in $\mathbb {Z}_p$ lifting $x\in \frak {sl}_{2,\mathbb {F}_p}$, and $\widetilde {v} \in V^{\circ }$ is a lift of $v \in V^{\circ }/pV^{\circ }$. The definition does not depend on the choice of the lifts. Moreover, letting $K$ act on $\frak {sl}_{2,\mathbb {F}_p}$ by conjugation $k\cdot x:=\overline {k}x\overline {k}^{-1}$ (for $k\in K$ and $k\mapsto \overline {k}\in \mathrm {GL}_2(\mathbb {F}_p)$), $\beta _{V^{\circ }}$ is $K$-equivariant. Indeed,

\begin{align*} \beta_{V^{\circ}}(k(x\otimes v))=\beta_{V^{\circ}} (\overline{k}x\overline{k}^{-1}\otimes kv) &= p^{-1} (\rho(\exp(pk\widetilde{x}k^{-1}))k\widetilde{v}-k\widetilde{v}) \pmod{p V^{\circ}}\\ &=p^{-1}(\rho(k\exp(p\widetilde{x}))\widetilde{v}-k\widetilde{v})\pmod{p V^{\circ}}\\ &=k\beta_{V^{\circ}}(x\otimes v). \end{align*}

In the special case $V=\underline {\mathrm {Sym}}^1E^2$ and $V^{\circ }=\underline {\mathrm {Sym}}^1\mathcal {O}^2$, we easily check that the map $\beta _{\underline {\mathrm {Sym}}^1\mathcal {O}^2}:\frak {sl}_{2,\mathbb {F}_p}\otimes \mathrm {Sym}^1\mathbb {F}^2\rightarrow \mathrm {Sym}^2\mathbb {F}^2$ is given by

(3.12)\begin{equation} \beta_{\underline{\mathrm{Sym}}^1\mathcal{O}^2}(\alpha\otimes \ell(X,Y))=\ell(a_{11}X+a_{21}Y,a_{12}X-a_{11}Y) \end{equation}

for $\alpha = \big (\begin{smallmatrix} a_{11} & a_{12} \\ a_{21} & -a_{11} \end{smallmatrix}\big ) \in \frak {sl}_{2,\mathbb {F}_p}$ and $\ell (X,Y) \in \mathrm {Sym}^1 \mathbb {F}^2$.

Proof of Proposition 3.17 We only give the proof for $L_2$ (which will be used in the proof of Proposition 3.19). Take $V=\underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi )$ and $V^{\circ }=L=\underline {\mathrm {Sym}}^1\mathcal {O}^2\otimes T$ in the above discussion. Since $K_1$ acts trivially on $T$, the map $\beta _{L}:\frak {sl}_{2,\mathbb {F}_p}\otimes \mathrm {Sym}^1\mathbb {F}^2\otimes T/pT\rightarrow \mathrm {Sym}^1\mathbb {F}^2\otimes T/pT$ is given by (cf. [BHHMS23, Remark 7.1.3])

(3.13)\begin{equation} \beta_{L}=\beta_{\underline{\mathrm{Sym}}^1\mathcal{O}^2}\otimes \mathrm{Id}_{T/pT}. \end{equation}

Let $W_1=(\sigma _{p - 1-a, a+b+1} \ \textbf {---}\ \sigma _{a-2, b+2})$ be the subrepresentation of $L/ pL$ defined in the proof of Proposition 3.11. Then $L_2$ is exactly the preimage of $W_1$ in $L$ under the surjection $L \twoheadrightarrow L/pL$. Taking $W= W_1$ (and $V^\circ = L$) in [BHHMS23, Lemma 7.1.4], we obtain the following commutative diagram in which both rows are exact.

To prove the proposition, it suffices to check that the dotted map $\varphi$, which is the composite

(3.14)\begin{equation} \varphi: \frak{sl}_{2, \mathbb{F}_p} \otimes W_1 \hookrightarrow \frak{sl}_{2,\mathbb{F}_p} \otimes L / pL {\buildrel {\beta_{L}} \over \longrightarrow} L/ pL, \end{equation}

is surjective, since this implies that the images of $pL$ and $p^2L$ in $(L_2)_{K_1}$ coincide. Recall that $T/pT$ fits into a short exact sequence

\[ 0\rightarrow \sigma_{p-2-a,a+b+1}\rightarrow T/pT\rightarrow \sigma_{a-1,b+1}\rightarrow0 \]

and $W_1\cap (\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{p-2-a,a+b+1})=\sigma _{p-1-a,a+b+1}$ (see the proof of Proposition 3.11). Using (3.13), we see that $\varphi$ sends $\frak {sl}_{2,\mathbb {F}_p}\otimes \sigma _{p-1-a,a+b+1}$ to $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{p-2-a,a+b+1}$, so that $\varphi$ induces a $K$-equivariant map

\[ \overline{\varphi}: \frak{sl}_{2,\mathbb{F}_p}\otimes \sigma_{a-2,b+2}\hookrightarrow \frak{sl}_{2,\mathbb{F}_p}\otimes \mathrm{Sym}^1\mathbb{F}^2\otimes\sigma_{a-1,b+1}\rightarrow \mathrm{Sym}^1\mathbb{F}^2\otimes\sigma_{a-1,b+1}, \]

where the second map is given by $\beta _{\underline {\mathrm {Sym}}^1\mathcal {O}^2}\otimes \mathrm {Id}_{\sigma _{a-1,b+1}}$. By Proposition 3.11(i), $\mathrm {cosoc}(L/pL)=\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}$. Hence, to prove $\varphi$ is surjective it suffices to prove $\overline {\varphi }$ is surjective.

We prove that $\overline {\varphi }$ is surjective by a direct computation (analogous to [BHHMS23, Lemma 7.2.1]). Fix a nonzero element $v\in (\sigma _{a-1,b+1})^{U(\mathbb {Z}_p)}$. Then $(\sigma _{a-1,b+1})^{U(\mathbb {Z}_p)} = \mathbb {F} v$ and the group $H = \{\big (\begin{smallmatrix} {[a]} & {0}\\ {0} & {[d]}\end{smallmatrix}\big ),\ a,d\in \mathbb {F}_p^{\times }\}$ acts on $v$ by $\chi _{a-1,b+1}$. Recall that there is a natural action of $\frak {sl}_{2,\mathbb {F}_p}$ on $\sigma _{a-1,b+1}$ and the set $\{v,f(v),\ldots, f^{a-1}(v)\}$ forms an $\mathbb {F}$-basis of $\sigma _{a-1,b+1}$, see [Paš04, § 4.2.1]. By construction, we have $e(v)=0$ and $h(v)=(a-1)v$. Consider the following nonzero elements of $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}$:

\[ l_1 : = X \otimes v,\quad l_2: = Y\otimes v - \frac{1}{a-1} X \otimes f(v). \]

It is easy to see that $H$ acts on $l_1$ (respectively, $l_2$) by $\chi _{ a, b +1}$ (respectively, $\chi _{ a - 2, b +2}$). Moreover, using the fact $ef(v)=fe(v)+h(v)=(a-1)v$, one checks that $e(l_1)=e(l_2)=0$, so that $l_1$ and $l_2$ are fixed by $U(\mathbb {Z}_p)$. Since $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}\cong \sigma _{a,b+1}\oplus \sigma _{a-2,b+2}$, we deduce that $l_1\in (\sigma _{a,b+1})^{U(\mathbb {Z}_p)}$ and $l_2\in (\sigma _{a-2,b+2})^{U(\mathbb {Z}_p)}$ under this decomposition. In particular, $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}$ is generated by $l_1$ and $l_2$ as a $K$-representation.

Since $\overline {\varphi }$ is $K$-equivariant, to finish the proof it suffices to prove that both $l_1$ and $l_2$ lie in the image of $\overline {\varphi }$. We let (recall $a\geq 3$)

\[ w_1 := e \otimes l_2,\quad w_2 : = -h\otimes l_2 - \frac{2}{a-2} e\otimes f(l_2) \]

be elements of $\frak {sl}_{2,\mathbb {F}_p} \otimes \sigma _{a-2,b+2}$ and claim that $\overline {\varphi }(w_i) = l_i$, $i=1,2$. Indeed, as $\beta _{\underline {\mathrm {Sym}}^1 \mathcal {O}^2}(e\otimes X)=0$ and $\beta _{\underline {\mathrm {Sym}}^1 \mathcal {O}^2}(e\otimes Y)=X$ by (3.12),

\begin{align*} \overline{\varphi} (w_1) &= \beta_L(e \otimes l_2) = \beta_L\bigg(e\otimes \bigg(Y\otimes v - \frac{1}{a-1} X \otimes f(v)\bigg)\bigg) \\ &= \beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2} (e\otimes Y) \otimes v - \frac{1}{a-1} \beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2} (e\otimes X) \otimes f(v) = X\otimes v = l_1. \end{align*}

Similarly, $\overline {\varphi }(w_2) = \beta _L(-h\otimes l_2 - ({2}/({a-2})) e\otimes f(l_2))$. We have

\begin{align*} f (l_2) &= f \bigg(Y\otimes v - \frac{1}{a-1} X \otimes f(v)\bigg)\\ &= Y \otimes f(v) - \frac{1}{a-1} (f(X) \otimes f(v) + X \otimes f^2(v))\\ &= \frac{a-2}{a-1}Y\otimes f(v) - \frac{1}{a-1}X \otimes f^2 (v), \end{align*}

hence

\begin{align*} \beta_L( e\otimes f (l_2)) &= \frac{a-2}{a-1}\beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2} (e\otimes Y) \otimes f(v) - \frac{1}{a-1} \beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2}(e\otimes X) \otimes f^2 (v) \\ &= \frac{a-2}{a-1} X\otimes f(v). \end{align*}

As $h(X)=X$ and $h(Y)=-Y$ by (3.12), we obtain

\begin{align*} \overline{\varphi} (w_2) &=- \beta_{L} \bigg(h \otimes \bigg(Y\otimes v - \frac{1}{a-1}X \otimes f(v)\bigg)\bigg) - \frac{2}{a-2}\cdot \frac{a-2}{a-1} X\otimes f(v) \\ &= Y \otimes v + \frac{1}{a-1}X \otimes f(v) - \frac{2}{a-1} X\otimes f(v) = l_2. \end{align*}

This proves the claim and finishes the proof of the proposition.

3.4 Gluing lattices

Assume $1 \leq a \leq p-3$. Consider the following three characters of $\mathbb {F}_{p^2}^{\times }$:

(3.15)\begin{equation} \psi_1 = [\xi]^{a+2 + (p+1)b},\quad \psi_{2} = [\xi]^{a+3 + (p+1)(b-1)},\quad \psi_{3} = [\xi]^{a+1 + (p+1)b}. \end{equation}

In this subsection, we construct a lattice $\widetilde {R}$ in

\[ \Theta(\psi_1) \oplus (\underline{\mathrm{Sym}}^1 E^2 \otimes\Theta(\psi_2)) \oplus (\underline{\mathrm{Sym}}^1 E^2 \otimes\Theta(\psi_3)) \]

such that $\widetilde {R} / p\widetilde {R}$ is killed by $\frak {m}_{K_1}^2$ and $\mathrm {cosoc} (\widetilde {R} / p\widetilde {R}) = \sigma _{a,b+1}$. We divide the construction into two cases: $1\leq a\leq p-4$ and $a=p-3$.

3.4.1 The case $1\,{\leq}\, a \,{\leq}\, p-4$

Denote by $W$ the nonsplit $\Gamma$-extension $( \sigma _{p- 3 - a, a + b + 2} \ \textbf {---}\ \sigma _{a, b+1})$.

1. (1) Let $R_1$ be the unique (up to homothety) lattice in $\Theta (\psi _1)$ such that $\mathrm {cosoc}(R_1/ p R_1) = \sigma _{a,b+1}$. Then $R_1/ p R_1\cong W$. Let $r_1$ denote the composite

   \[ r_1: R_1 \twoheadrightarrow R_1 / pR_1 \cong W. \]

2. (2) By Proposition 3.6 and [BP12, Lem. 3.8], we have

   \[ \operatorname{{\mathrm{JH}}}(\overline{\underline{\mathrm{Sym}}^1E^2\otimes\Theta(\psi_2)}^{\rm ss})= \{\sigma_{a,b+1},\sigma_{p-3-a,a+b+2},\sigma_{a+2,b},\sigma_{p-5-a,a+b+3}\} \]
   with the convention $\sigma _{-1,b}=0$. Let $R_2\subset \underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi _2)$ be the unique (up to homothety) lattice such that $\mathrm {cosoc}(R_2/ p R_2) = \sigma _{a,b+1}$. The structure of $R_2 / p R_2$ is given by Proposition 3.16, i.e.
   (3.16)\begin{equation} R_2/pR_2\cong(\sigma_{p-5-a,a+b+3}\ \textbf{---}\ \sigma_{a+2,b}\ \textbf{---}\ \sigma_{p-3-a,a+b+2}\ \textbf{---}\ \sigma_{a,b+1}). \end{equation}

3. (3) By Proposition 3.6 and [BP12, Lemma 3.8], we have

   \[ \operatorname{{\mathrm{JH}}}(\overline{\underline{\mathrm{Sym}}^1E^2\otimes\Theta(\psi_3)}^{\rm ss})= \{\sigma_{a,b+1},\sigma_{p-3-a,a+b+1},\sigma_{a-2,b+2},\sigma_{p-1-a,a+b+1}\} \]
   with the convention $\sigma _{-1,b+2}=0$. Let $R_3\subset \underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi _3)$ be the unique (up to homothety) lattice such that $\mathrm {cosoc}(R_3)\cong \sigma _{a,b+1}$. By Proposition 3.14, $R_3/pR_3$ has a cosocle filtration
   (3.17)\begin{equation} \sigma_{a-2, b+2} \ \textbf{---}\ ( \sigma_{p-1-a, a+b+1} \oplus \sigma_{p-3-a, a+b+2} ) \ \textbf{---}\ \sigma_{a, b+1}. \end{equation}

Note that there exists a surjection $R_2\twoheadrightarrow W$ which we denote by $r_2$; let $R_2':=\operatorname {{\mathrm {Ker}}}(r_2)$. The structure of $R_2'/pR_2'$ is determined in Proposition 3.11(i). Precisely, it has a two-step socle and cosocle filtration

(3.18)\begin{equation} (\sigma_{p-3-a,a+b+2}\oplus \sigma_{p-5-a,a+b+3})\ \textbf{---}\ (\sigma_{a+2,b}\oplus\sigma_{a,b+1}) \end{equation}

and all possible extensions do occur.

Similarly, there exists a surjection $R_3\twoheadrightarrow W$ which we denote by $r_3$; let $R_3':=\operatorname {{\mathrm {Ker}}}(r_3)$. The structure of $R_3'/pR_3'$ is also determined in Proposition 3.11(i). Precisely, it has a cosocle filtration

(3.19)\begin{equation} \sigma_{p-3-a, a+b+2} \ \textbf{---}\ ( \sigma_{a, b+1} \oplus \sigma_{a-2, b+2} ) \ \textbf{---}\ \sigma_{p-1-a, a+b+1} \end{equation}

and all possible extensions do occur.

3.4.2 Glue $R_1$ and $R_2$ $(1\leq a\leq p-4)$

Let $R$ be the lattice in $\Theta (\psi _1) \oplus (\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi _2))$ obtained by gluing $R_1$ and $R_2$ along $W$, i.e. $R$ is given by the short exact sequence

(3.20)\begin{equation} 0 \to R \to R_1 \oplus R_2 {\buildrel {r_1 - r_2} \over \longrightarrow} W \to 0. \end{equation}

Let $r_{R}$ denote the composition $R \twoheadrightarrow R/pR \twoheadrightarrow R_1/pR_1 \cong W$.

Lemma 3.18 We have that:

1. (i) $r_R$ induces a short exact sequence

   (3.21)\begin{equation} 0 \to R_2' / p R_2' \rightarrow R /p R \to W \to 0 ; \end{equation}
   in particular, $R/ pR$ is killed by $\frak {m}_{K_1}^2$;

2. (ii) $\operatorname {{\mathrm {Ker}}}(r_R)=R_2'+pR$ and

   \[ \operatorname{{\mathrm{Ker}}}(r_R)/p\operatorname{{\mathrm{Ker}}}(r_R)\cong R_2'/pR_2'\oplus W. \]

Proof. This is a special case of Lemma 3.3 applied to $L_1=R_2$ and $L_2=R_1$.

Proposition 3.19 The short exact sequence (3.21) induces an isomorphism $(R/pR)_{K_1} \cong W$. In particular, $\mathrm {cosoc} (R/ p R) = \sigma _{a,b+1}$.

Proof. By Lemma 3.18(i), it suffices to show that for any $x \in R_2'$, there exist $r\in \mathbb {N}$, $k_1,\ldots, k_r \in K_1$, $v_1,\ldots, v_r \in R$ such that $x= (k_1 -1) v_1 + \cdots + (k_r - 1) v_r$. By Proposition 3.17 (applied to $L=p^{-1}R_2'$ and $L_2=R_2$), there exist $r\in \mathbb {N}$, $k_1,\ldots, k_r \in K_1$, $y_1,\ldots, y_r \in R_2$ such that

\[ x = (k_1 -1) y_1 + \cdots + (k_r - 1) y_r. \]

For $1\leq i \leq r$, choose $z_i \in R_1$ such that $r_1(z_i) = r_2 (y_i)$ and let $v_i = (z_i,y_i)\in R$. Since $K_1$ acts trivially on $R_1$, we have

\[ (0, x) = \bigg(\sum_{i=1}^r (k_i -1) z_i, \sum_{i=1}^r (k_i -1) y_i\bigg) = \sum_{i=1}^r (k_i - 1) v_i, \]

giving the result.

3.4.3 Glue $R$ and $R'_3$ $(1\leq a\leq p-4)$

We define $\widetilde {R}$ to be the lattice in $\Theta (\psi _1)\oplus (\underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi _2))\oplus (\underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi _3))$ obtained by gluing $R$ and $R_3$ along $W$, i.e.

(3.22)\begin{equation} 0\rightarrow \widetilde{R}\rightarrow R\oplus R_3 {\buildrel {r_R-r_3} \over \longrightarrow} W\rightarrow0. \end{equation}

Proposition 3.20

1. (i) There exists a short exact sequence

   \[ 0 \to \operatorname{{\mathrm{Ker}}}(r_R) / p \operatorname{{\mathrm{Ker}}}(r_R) \rightarrow \widetilde{R} /p \widetilde{R} \to R_3/pR_3 \to 0. \]

2. (ii) We have $\mathrm {cosoc} (\widetilde {R}/ p \widetilde {R}) = \sigma _{a,b+1}$.

Proof. (i) This is a special case of Lemma 3.3.

(ii) This is a special case of Lemma 3.4, with $L_1=R$ and $L_2=R_3$. First, condition (a) in Lemma 3.4 holds by Proposition 3.19. Second, we have

\[ \mathrm{cosoc}(\operatorname{{\mathrm{Ker}}}(r_R))=\mathrm{cosoc}(W)\oplus \mathrm{cosoc}(R'_2)=\sigma_{a,b+1}\oplus \sigma_{a,b+1}\oplus \sigma_{a+2,b} \]

by (3.18) and Lemma 3.18(ii), and

\[ \mathrm{cosoc}(\operatorname{{\mathrm{Ker}}}(r_3))\cong\sigma_{p-1-a,a+b+1} \]

by (3.19), hence condition (b) in Lemma 3.4 also holds.

Proposition 3.21 Let $V$ denote the quotient of $R_3/pR_3$ by $\sigma _{a-2,b+2}$ via (3.17). Then there exists a short exact sequence

(3.23)\begin{equation} 0 \to R_2'/ p R_2'\oplus W \oplus \sigma_{a-2, b+2} \to \widetilde{R} / p\widetilde{R} \to V \to 0. \end{equation}

In particular, $\widetilde {R} / p\widetilde {R}$ is killed by $\frak {m}_{K_1}^2$.

Proof. By definition, we have

\[ 0\rightarrow \sigma_{a-2,b+2}\rightarrow R_3/pR_3\rightarrow V\rightarrow0. \]

Note that $\sigma _{a-2, b+2}$ has no nontrivial extensions with any Jordan–Hölder factor of $W$ and of $R_2 / pR_2$, using [BP12, Corollary 5.6] and Lemma 3.22 below. The result easily follows by Proposition 3.20.

Lemma 3.22 Assume $2\leq a\leq p-2$. Then $\operatorname {{\mathrm {Ext}}}^1_{K}(\sigma _{a-2,b+2}, \sigma _{a,b+1})=0$.

Proof. We have a short exact sequence $0\rightarrow \sigma _{p+1-a,a+b}\rightarrow \operatorname {{\mathrm {Ind}}}_I^{K}\chi _{a-2,b+2}\rightarrow \sigma _{a-2,b+2}\rightarrow 0$. Since $\operatorname {{\mathrm {Ext}}}^1_{K}(\sigma _{p+1-a,a+b},\sigma _{a,b+1})=0$ by [BP12, Corollary 5.6], we are reduced to proving

\[ \operatorname{{\mathrm{Ext}}}^1_K(\operatorname{{\mathrm{Ind}}}_I^K\chi_{a-2,b+2},\sigma_{a,b+1})=0, \]

equivalently $\operatorname {{\mathrm {Ext}}}^1_{I}(\chi _{a-2,b+2},\sigma _{a,b+1})=0$ by Frobenius reciprocity.

Consider an $I$-extension $0\rightarrow \sigma _{a,b+1}|_I\rightarrow \mathcal {E}\rightarrow \chi _{a-2,b+2}\rightarrow 0$. We first prove that it splits as $U(\mathbb {Z}_p)$-representation. Since $\sigma _{a,b+1}$ is a cyclic $\mathbb {F}[\![U(\mathbb {Z}_p)]\!]$-module, we have $H^1(U(\mathbb {Z}_p),\sigma _{a,b+1})\cong H^1(U(\mathbb {Z}_p),\chi _{a,b+1}^s)$, where $\chi _{a,b+1}^s$ is identified with the $U(\mathbb {Z}_p)$-cosocle of $\sigma _{a,b+1}$. As seen in the proof of Proposition 3.1, we get

\[ H^1(U(\mathbb{Z}_p),\sigma_{a,b+1})\cong\chi_{a,b+1}^s\alpha^{-1}. \]

As $2\leq a\leq p-2$, this implies $\chi _{a-2,b+2}\neq \chi _{a,b+1}^s\alpha ^{-1}$, and so $\operatorname {{\mathrm {Ext}}}^1_{U(\mathbb {Z}_p)}(\chi _{a-2,b+2},\sigma _{a,b+1})=0$.

As a consequence, we may choose $v\in \mathcal {E}$ which is fixed by $U(\mathbb {Z}_p)$ and on which $H$ acts via $\chi _{a-2,b+2}$. Next, as in the proof of [Paš10, Proposition 7.2], we show that $v$ is actually fixed by $I_1$, showing that $\mathcal {E}$ splits as $I$-representation. This finishes the proof.

We obtain the following corollary.

Corollary 3.23 We have that $\widetilde {R} / p \widetilde {R}$ has cosocle $\sigma _{a,b+1}$ and is a quotient of $(\operatorname {{\mathrm {Proj}}}_{\mathcal {O}[\![K/Z_1]\!]}\sigma _{a,b+1}) / \frak {m}_{K_1}^2$.

3.4.4 The case $a = p-3$

In this case, we need to slightly modify the above construction. We only sketch the construction and leave the detail to the reader.

1. (1) Let $R_1$ be the unique (up to homothety) lattice in $\Theta (\psi _1)$ such that $\mathrm {cosoc}(R_1/ p R_1) = \sigma _{p-3,b+1}$. Then

   \[ R_1/ p R_1\cong (\sigma_{0,b} \ \textbf{---}\ \sigma_{p-3,b+1}) = : W. \]
   Let $r_1$ denote the projection $R_1 \twoheadrightarrow \sigma _{p-3,b+1}$.

2. (2) By Proposition 3.6 and [BP12, Lemma 3.8], we have

   \[ \operatorname{{\mathrm{JH}}}(\overline{\underline{\mathrm{Sym}}^1 \mathcal{O}^2 \otimes \Theta(\psi_2)}^{\rm ss})=\{\sigma_{p-1,b},\sigma_{p-3,b+1}\}. \]
   Let $R_2$ be the unique lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi _2)$ such that $\mathrm {cosoc} (R_2/pR_2) = \sigma _{p-3, b+1}$. Then
   \[ R_2/pR_2\cong(\sigma_{p-1,b}\ \textbf{---}\ \sigma_{p-3,b+1}). \]
   Let $r_2$ denote the projection $R_2\twoheadrightarrow \sigma _{p-3,b+1}$ and $R_2':=\operatorname {{\mathrm {Ker}}}(r_2)$. Proposition 3.13 implies that
   \[ R_2'/pR_2'\cong \sigma_{p-3,b+1}\oplus\sigma_{p-1,b}. \]

3. (3) Let $R_3$ and $R'_3$ be the lattices in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi _3)$ constructed as in the case $1\leq a\leq p-4$. Namely, $R_3$ has cosocle $\sigma _{p-3,b+1}$, and $R_3':=\operatorname {{\mathrm {Ker}}}(r_3)$ where $r_3$ denotes the projection $R_3\twoheadrightarrow W$.

We first glue $R_1$ and $R_2$ along $\sigma _{p-3,b+1}$, namely

\[ 0\rightarrow R\rightarrow R_1 \oplus R_2 {\buildrel {r_1-r_2} \over \longrightarrow} \sigma_{p-3,b+1}\rightarrow0. \]

Then by Lemma 3.3(i) there is a short exact sequence

\[ 0\rightarrow R_2'/pR_2'\rightarrow R/pR {\buildrel {r_R} \over \longrightarrow} W\rightarrow0. \]

Moreover, as in the proof of Proposition 3.19 one can show that $r_R$ induces an isomorphism $(R/pR)_{K_1} \cong W$. In particular, $\mathrm {cosoc}(R/pR)\cong \sigma _{p-3,b+1}$.

The gluing of $R$ and $R_3$ is exactly as in the case $1\leq a \leq p-4$. Let $\widetilde {R}$ be defined by

\[ 0\rightarrow \widetilde{R}\rightarrow R\oplus R_3 {\buildrel {r_R-r_3} \over \longrightarrow} W\rightarrow0. \]

One can follow the proof of Proposition 3.21 and Corollary 3.23, to show the following result.

Proposition 3.24

1. (i) We have that $\widetilde {R} / p \widetilde {R}$ has cosocle $\sigma _{p-3,b+1}$.

2. (ii) Let $V$ denote the quotient of $R_3/pR_3$ by $\sigma _{p-5,b+2}$. Then there is a short exact sequence

   (3.24)\begin{equation} 0 \to W \oplus R'_2/ p R'_2 \oplus \sigma_{p-5, b+2} \to \widetilde{R} / p\widetilde{R} \to V \to 0. \end{equation}
   As a consequence, $\widetilde {R} / p \widetilde {R}$ is a quotient of $(\operatorname {{\mathrm {Proj}}}_{ \mathcal {O}[\![K/Z_1]\!]}\sigma _{p-3,b+1}) / \frak {m}_{K_1}^2$.

4. Galois deformation rings

Assume $p\geq 5$. Let $\overline {\rho }: G_{\mathbb {Q}_p}\to \mathrm {GL}_2(\mathbb {F})$ be a two-dimensional continuous representation of $G_{\mathbb {Q}_p} = \operatorname {{\mathrm {Gal}}}(\overline {\mathbb {Q}}_p / \mathbb {Q}_p)$. In this section, we study the congruence relation of Galois deformation rings of different (tame) types. Our method does not allow us to determine the precise structure of the Galois deformation rings, but is enough for application in § 6.3.

4.1 Preliminaries

We collect some results on the set of Serre weights associated to $\overline {\rho }$ and some results of Paškūnas and of Morra. We prove in § 4.2.2 a criterion for some Galois deformation rings to be regular.

4.1.1 Serre weights

Let $\omega$ (respectively, $\omega _2$) be the mod $p$ cyclotomic character (respectively, Serre's fundamental character of niveau 2) of $G_{\mathbb {Q}_p}$. Up to isomorphism, $\overline {\rho }$ has one of the following forms:

1. Case 1. $\overline {\rho }$ is absolutely irreducible and $\overline {\rho }|_{I_p} \sim \big (\begin{smallmatrix} \omega _{2}^{r+1} & 0 \\ 0 & \omega _2^{p(r+1)} \end{smallmatrix} \big ) \otimes \omega ^{s+1}$, $0\leq r\leq p-1$, $0\leq s \leq p-2$;

2. Case 2. $\overline {\rho } \sim \big (\begin{smallmatrix} {\rm unr}_1 \omega ^{r+1} & * \\ 0 & {\rm unr}_2 \end{smallmatrix} \big ) \otimes \omega ^{s+1}$ is reducible nonsplit, where ${\rm unr}_1$, ${\rm unr}_2$ are unramified characters, and $0\leq r \leq p-2$, $0\leq s \leq p-2$;

3. Case 3. $\overline {\rho } \sim \big (\begin{smallmatrix} {\rm unr}_1 \omega ^{r+1} & 0 \\ 0 & {\rm unr}_2 \end{smallmatrix} \big ) \otimes \omega ^{s+1}$ is reducible split, where ${\rm unr}_1$, ${\rm unr}_2$ are unramified characters, and $0 \leq r\leq p-2$, $0\leq s \leq p-2$.

Let $W(\overline {\rho })$ be the set of Serre weights associated to $\overline {\rho }$ in [BDJ10]. We have the following explicit description of $W(\overline {\rho })$.

Theorem 4.1 [BDJ10, Theorem 3.17]

1. (i) Assume $\overline {\rho }$ is in case 1. Then $W(\overline {\rho })= \{\sigma _{r,s+1},\ \sigma _{p-1-r, r+s+1}\}$.

2. (ii) Assume $\overline {\rho }$ is in case 2.

   1. (a) If $r \neq 0$, then $W(\overline {\rho })= \{\sigma _{r, s+1}\}$.

   2. (b) If $r=0$, ${\rm unr}_1 = {\rm unr}_2$ and $\overline {\rho }$ is très ramifié, then $W(\overline {\rho }) =\{ \sigma _{p-1,s+1}\}$.

   3. (c) For other $\overline {\rho }$, $W(\overline {\rho })= \{ \sigma _{0,s+1},\ \sigma _{p-1,s+1}\}$.

3. (iii) Assume $\overline {\rho }$ is in case 3.

   1. (a) If $1\leq r\leq p-4$, then $W(\overline {\rho })=\{ \sigma _{r,s+1},\ \sigma _{p-3-r, r+s+2}\}$.

   2. (b) If $r = 0$, then $W(\overline {\rho }) =\{ \sigma _{0,s+1},\ \sigma _{p-3, s+2},\ \sigma _{p-1, s+1} \}$.

   3. (c) If $r = p-3$, then $W(\overline {\rho })=\{ \sigma _{0, s},\ \sigma _{p-3,s+1},\ \sigma _{p-1, s} \}$.

   4. (d) If $r = p-2$, then $W(\overline {\rho })=\{ \sigma _{p-2, s+1}\}$.

4.2 Mod $p$ representations of $\mathrm {GL}_2(\mathbb {Q}_p)$

Assume that $\overline {\rho }$ satisfies $\operatorname {{\mathrm {End}}}_{G_{\mathbb {Q}_p}}(\overline {\rho })\cong \mathbb {F}$. We associate to $\overline {\rho }$ an admissible smooth $\mathbb {F}$-representation $\pi (\overline {\rho })$ of $G:=\mathrm {GL}_2(\mathbb {Q}_p)$ as follows.

1. Case 1. If $\overline {\rho }$ is absolutely irreducible, then $\pi (\overline {\rho })$ is the irreducible supersingular representation of $G$ associated to $\overline {\rho }$ by the mod $p$ local Langlands correspondence defined in [Bre03].

2. Case 2. If $\overline {\rho } \sim \big (\begin{smallmatrix} \chi _1 & * \\ 0 & \chi _2 \end{smallmatrix}\big )$ with $\chi _1 \chi _2^{-1} \neq \omega ^{\pm 1}, \mathbf {1}$, then there is an exact nonsplit sequence

   \[ 0 \to \operatorname{{\mathrm{Ind}}}_{B(\mathbb{Q}_p)}^{G} \chi_2\otimes \chi_1 \omega^{-1} \to \pi(\overline{\rho}) \to \operatorname{{\mathrm{Ind}}}_{B(\mathbb{Q}_p)}^{G} \chi_1\otimes \chi_2 \omega^{-1} \to 0. \]
   If $\overline {\rho } \sim \big (\begin{smallmatrix} \chi & * \\ 0 & \chi \omega \end{smallmatrix}\big )$, then there is an exact nonsplit sequence
   \[ 0\to \operatorname{{\mathrm{Ind}}}_{B(\mathbb{Q}_p)}^{G} \chi\omega \otimes \chi\omega^{-1} \to \pi(\overline{\rho}) \to \tau_1 \otimes \chi\circ \det \to 0, \]
   where ${\rm Sp}$ is the Steinberg representation of $G$ and $\tau _1$ is a nonsplit extension $0 \to {\rm Sp} \to \tau _1 \to \mathbf {1}_G^{\oplus 2} \to 0$ with $\operatorname {{\mathrm {soc}}}_G(\tau _1)=\operatorname {{\mathrm {Sp}}}$.

   If $\overline {\rho }\sim \big (\begin{smallmatrix} {\chi \omega } & {*}\\ {0} & {\chi }\end{smallmatrix}\big )$, then $\pi (\overline {\rho })$ is the representation defined in [Paš15, Lemma 6.7] (denoted by $\beta$ there). Its precise structure will be recalled in § 8.3.

We remark that the representation $\pi (\overline {\rho })$ is just the representation corresponding to $\overline {\rho }$ in the mod $p$ local Langlands correspondence for $\mathrm {GL}_2(\mathbb {Q}_p)$, except in the case $\overline {\rho }\sim \big (\begin{smallmatrix} {\chi } & {*}\\ {0} & {\chi \omega }\end{smallmatrix}\big )$, $\pi (\overline {\rho })$ has one extra copy of $\chi \circ \det$ than the usual form.

The following theorem is a consequence of results of Morra [Mor11, Mor17].

Theorem 4.2 Assume $\overline {\rho }$ is either in case 1 of § 4.1.1 with $r\notin \{1,p-2\}$ or $\overline {\rho }$ is in case 2 of § 4.1.1 with $1 \leq r\leq p-3$.³ Then for any $\sigma \in W(\overline {\rho })$, $\sigma$ occurs in $\pi (\overline {\rho }) [\frak {m}_{K_1}^2]$ with multiplicity one.

Proof. If $\overline {\rho }$ is absolutely irreducible, then $\pi (\overline {\rho })$ is the representation $\pi (\overline {\rho })$ in [Mor11] whose $K$-socle filtration is given by [Mor11, Theorem 1.1]. If $\overline {\rho }$ is reducible nonsplit and $\overline {\rho } \nsim \big (\begin{smallmatrix} \mathbf {1} & * \\ 0 & \omega \end{smallmatrix}\big ) \otimes \chi$, then $\pi (\overline {\rho })$ is equal to the representation $A_{r,\lambda }$ (in [Mor17, Theorem 1.1]) for some $\lambda \in \mathbb {F}^{\times }$. If $\overline {\rho } \sim \big (\begin{smallmatrix} \mathbf {1} & * \\ 0 & \omega \end{smallmatrix}\big ) \otimes \chi$, then $\pi (\overline {\rho })$ has an extra copy of $\chi \circ \det$ than the representation $A_{r,\lambda }$. However, in this case $(\chi \circ \det )|_K$ is not a Serre weight of $\overline {\rho }$. Thus, for any $\sigma \in W(\overline {\rho })$ the multiplicity of $\sigma$ in $\pi (\overline {\rho }) [\frak {m}_{K_1}^2]$ is equal to the multiplicity of $\sigma$ in $A_{r,\lambda } [\frak {m}_{K_1}^2]$. The $K$-socle filtration of $A_{r,\lambda }$ is given by [Mor17, Theorem 1.1] and [Mor11, Theorem 1.2], from which the result follows.

4.2.1 Results of Paškūnas

Recall that $\overline {\rho }$ is called generic in the sense of [Paš15] if either $\overline {\rho }$ is absolutely irreducible or $\overline {\rho } \sim \big (\begin{smallmatrix} \chi _1 & * \\ 0 & \chi _2 \end{smallmatrix}\big )$ is reducible nonsplit with $\chi _1 \chi _2^{-1} \neq \omega, \mathbf {1}$. We assume $\overline {\rho }$ is generic, so in particular $\operatorname {{\mathrm {End}}}_{G_{\mathbb {Q}_p}} (\overline {\rho }) = \mathbb {F}$. Let $\eta : G_{\mathbb {Q}_p} \to \mathcal {O}^{\times }$ be a character such that $\eta \pmod {\varpi } = \det \overline {\rho }$. Let $R_{\overline {\rho }}^{\eta }$ denote the universal deformation ring of $\overline {\rho }$ with determinant $\eta$ and let $\rho ^{\rm univ}$ denote the universal object over $R_{\overline {\rho }}^{\eta }$.

Let $\psi = \eta \varepsilon ^{-1}$. According to [Paš15, § 6.1], there exists $N \in \frak {C}_{G,\psi }(\mathcal {O})$ with a faithful continuous action of $R^{\eta }_{\overline {\rho }}$ which commutes with the action of $G$ such that:

1. (N0) $\mathbb {F}\widehat {\otimes }_{R_{\overline {\rho }}^{\eta }} N$ is of finite length in $\frak {C}_{G,\psi }(\mathcal {O})$ and is finitely generated over $\mathcal {O}[\![ K ]\!];$

2. (N1) $\operatorname {{\mathrm {Hom}}}_{\mathrm {SL}_2(\mathbb {Q}_p)}(\mathbf {1}_G,N^{\vee })=0$;

3. (N2) $\operatorname {{\mathrm {End}}}_{\frak {C}_{G,\psi }(\mathcal {O})} (N) \cong R_{\overline {\rho }}^{\eta }$ and $\check {\mathbf {V}}(N)$ is isomorphic to $\rho ^{\rm univ}$ as $R_{\overline {\rho }}^{\eta }[\![ G_{\mathbb {Q}_p} ]\!]$-modules, where $\check {\mathbf {V}}$ is the modified Colmez functor in [Paš15, § 3];

4. (N3) $N$ is projective in $\frak {C}_{G,\psi }(\mathcal {O})$, and there exists $x\in R^{\eta }_{\overline {\rho }}$ such that $N/xN$ is isomorphic to a projective envelope of $\oplus _{\sigma \in W(\overline {\rho })}\sigma ^{\vee }$ in $\operatorname {\mathrm {Mod}}^{\rm pro}_{K ,\psi }(\mathcal {O})$.

Remark 4.3 Under our assumption on $\overline {\rho }$, $N$ is just a projective envelope of $\mathbb {F}\widehat {\otimes }_{R_{\overline {\rho }}^{\eta }} N$ in $\frak {C}_{G,\psi }(\mathcal {O})$. Hence, (N3) follows from [Paš15, Theorem 5.2].

Proposition 4.4 Assume $\overline {\rho }$ is generic. Then there is an isomorphism $\mathbb {F}\widehat {\otimes }_{R_{\overline {\rho }}^{\eta }}N\cong \pi (\overline {\rho })^{\vee }$.

Proof. See the proof of [Paš15, Proposition 6.1].⁴

If $\Theta$ (respectively, $\sigma$) is a finite free $\mathcal {O}$-module (respectively, $\mathbb {F}$-module) equipped with a continuous action of $K$, we define

\[ M(\Theta) := \operatorname{{\mathrm{Hom}}}^{\rm cont}_{\mathcal{O} [\![ K ]\!]} (N , \Theta^{d})^{d}\quad (\mathrm{resp}.\ M(\sigma) := \operatorname{{\mathrm{Hom}}}^{\rm cont}_{\mathcal{O} [\![ K ]\!]} (N , \sigma^{\vee})^{\vee}). \]

Then $M(\Theta )$ (respectively, $M(\sigma )$) is a finitely generated $R_{\overline {\rho }}^{\eta }$-module by (N0).

Let $\mathbf {w} = (a, b)$ be a pair of integers with $a< b$ and $\tau : I_{\mathbb {Q}_p} \to \mathrm {GL}_2(E)$ be an inertial type, where $I_{\mathbb {Q}_p}$ is the inertia subgroup of $G_{\mathbb {Q}_p}$. Let

\begin{gather*} \sigma(\mathbf{w},\tau):= \mathrm{Sym}^{b-a-1} E^2 \otimes {\det}^a\otimes \sigma(\tau),\\ \sigma^{\rm cr}(\mathbf{w},\tau):=\mathrm{Sym}^{b-a-1} E^2 \otimes {\det}^a\otimes \sigma^{\rm cr}(\tau), \end{gather*}

where $\sigma (\tau )$ and $\sigma ^{\rm cr}(\tau )$ are finite-dimensional representations of $K$ over $E$ associated to $\tau$ by the inertial local Langlands correspondence [Hen02] (see § 5.1 for details). Let $R^{\eta }_{\overline {\rho }}( \mathbf {w},\tau )$ (respectively, $R^{\eta,{\rm cr}}_{\overline {\rho }}(\mathbf {w}, \tau )$) denote the reduced $p$-torsion-free quotient of $R_{\overline {\rho }}^{\eta }$ which parametrizes potentially semistable (respectively, potentially crystalline) deformations of $\overline {\rho }$ of Hodge–Tate weights $\mathbf {w}$ and type $\tau$. It is well-known that these rings are nonzero only if $\eta \varepsilon ^{-(a+b)}|_{I_{\mathbb {Q}_p}}\sim \det \tau$, in which case they have Krull dimension $2$. This requires, in particular, that $\eta$ is locally algebraic.

Recall the following theorem of Paškūnas.

Theorem 4.5 Let $\mathbf {w}, \tau$ be as above. Let $\Theta$ be any $K$-stable $\mathcal {O}$-lattice in $\sigma (\mathbf {w},\tau )$ (respectively, $\sigma ^{\rm cr}(\mathbf {w},\tau )$). Then $R_{\overline {\rho }}^{\eta } /\operatorname {{\mathrm {Ann}}}_{R^{\eta }_{\overline {\rho }}}( M (\Theta ))$ is equal to $R_{\overline {\rho }}^{\eta }(\mathbf {w},\tau )$ (respectively, $R_{\overline {\rho }}^{\eta,{\rm cr}}(\mathbf {w}, \tau )$).

Proof. See [Paš15, Corollary 6.5].

Let $\delta :G_{\mathbb {Q}_p}\rightarrow \mathcal {O}^{\times }$ be a continuous character that is trivial modulo $p$, not necessarily locally algebraic. Twisting by $\delta$ induces a natural isomorphism of $\mathcal {O}$-algebras

(4.1)\begin{equation} \mathrm{tw}_{\delta}:R_{\overline{\rho}}^{\eta\delta^2}\xrightarrow{\sim} R_{\overline{\rho}}^{\eta}. \end{equation}

By a similar discussion as in [CEG⁺18, § 6.1], we have the following variant of Theorem 4.5.

Corollary 4.6 Assume $\eta \delta ^2\varepsilon ^{-(a+b)}|_{I_{\mathbb {Q}_p}}\sim \det \tau$. Let $\Theta$ be any $K$-stable $\mathcal {O}$-lattice in $\sigma (\mathbf {w},\tau )\otimes \delta ^{-1}\circ \det$ (respectively, $\sigma ^{\rm cr}(\mathbf {w},\tau )\otimes \delta ^{-1}\circ \det$). Then $R_{\overline {\rho }}^{\eta }/\mathrm {Ann}_{R_{\overline {\rho }}^{\eta }}(M(\Theta ))$ is equal to $\mathrm {tw}_{\delta }(R_{\overline {\rho }}^{\eta \delta ^2}(\mathbf {w},\tau ))$ (respectively, $\mathrm {tw}_{\delta }(R_{\overline {\rho }}^{\eta \delta ^2, {\rm cr}}(\mathbf {w},\tau ))$). As a consequence, we have isomorphisms of $\mathcal {O}$-algebras

(4.2)\begin{equation} \mathrm{tw}_{\delta}: R_{\overline{\rho}}^{\eta\delta^2}(\mathbf{w},\tau) \xrightarrow{\sim} R_{\overline{\rho}}^{\eta}/\mathrm{Ann}_{R_{\overline{\rho}}^{\eta}}(M(\Theta)) \end{equation}

(respectively, for $R_{\overline {\rho }}^{\eta \delta ^2,{\rm cr}}(\mathbf {w},\tau )$).

If $\sigma$ is a finite-dimensional $\mathbb {F}[K]$-module, by Proposition 4.4 we have

(4.3)\begin{equation} \mathbb{F}\widehat{\otimes}_{R_{\overline{\rho}}^{\eta}} M(\sigma) = \operatorname{{\mathrm{Hom}}}_{K} (\sigma , \pi(\overline{\rho})). \end{equation}

It follows from (N3) and Nakayama's lemma that $M(\sigma )\neq 0$ if and only if $\sigma$ admits at least one Jordan–Hölder factor lying in $W(\overline {\rho })$.

4.2.2 A criterion for regularity

Lemma 4.7 Let $\sigma \in \operatorname {\mathrm {Mod}}_{K}^{\rm sm}(\mathbb {F})$ be of finite length. Assume that, taking into account multiplicities, $\operatorname {{\mathrm {JH}}}(\sigma )$ contains exactly one element in $W(\overline {\rho })$. Then $M(\sigma )$ is a cyclic $R_{\overline {\rho }}^{\eta }$-module and isomorphic to $\mathbb {F}[\![x]\!]$, where $x\in R_{\overline {\rho }}^{\eta }$ is as in (N3).

Proof. See (the end of) the proof of [Paš15, Theorem 6.6].

Recall that $\mathcal {O}$ is unramified over $\mathbb {Z}_p$.

Proposition 4.8 Let $\mathbf {w}, \tau$ be as above. Assume that there exist two $K$-stable $\mathcal {O}$-lattices $\Theta _1, \Theta _2$ in $\sigma (\mathbf {w},\tau )$ (respectively, $\sigma ^{\rm cr}(\mathbf {w},\tau )$) such that the following conditions hold:

1. (a) $p\Theta _1\subset \Theta _2\subset \Theta _1$ and $\dim _{\mathbb {F}}\operatorname {{\mathrm {Hom}}}_{K}(\Theta _i/p\Theta _i,\pi (\overline {\rho }))=1$ for $i=1,2$;

2. (b) taking into account multiplicities, $\operatorname {{\mathrm {JH}}}(\Theta _1/\Theta _2)$ contains exactly one element in $W(\overline {\rho })$.

Then $R^{\eta }_{\overline {\rho }}(\mathbf {w},\tau )$ (respectively, $R_{\overline {\rho }}^{\eta,{\rm cr}}(\mathbf {w},\tau )$) is a regular local ring.

Proof. We only treat the case for $R^{\eta }_{\overline {\rho }}(\mathbf {w},\tau )$. By Nakayama's lemma and (4.3), condition (a) implies that $M(\Theta _1)$ and $M(\Theta _2)$ are both cyclic modules over $R_{\overline {\rho }}^{\eta }$. Hence, $M(\Theta _1)$ and $M(\Theta _2)$ are isomorphic to $R^{\eta }_{\overline {\rho }}(\mathbf {w},\tau )$ by Theorem 4.5.

The exact sequence $0\rightarrow \Theta _2\rightarrow \Theta _1\rightarrow \Theta _1/\Theta _2\rightarrow 0$ induces a sequence of $R_{\overline {\rho }}^{\eta }$-modules

\[ 0\rightarrow M(\Theta_2) {\buildrel {f} \over \longrightarrow} M(\Theta_1)\rightarrow M(\Theta_1/\Theta_2)\rightarrow0, \]

which is again exact by (N3). Since both $M(\Theta _1)$ and $M(\Theta _2)$ are isomorphic to $R_{\overline {\rho }}^{\eta }(\mathbf {w},\tau )$, the morphism $f$ is equal to the multiplication by some element $y\in R_{\overline {\rho }}^{\eta }(\mathbf {w},\tau )$. On the other hand, by Lemma 4.7, condition (b) implies that $M(\Theta _1/\Theta _2)$ is isomorphic to $\mathbb {F}[\![x]\!]$. This means that $R_{\overline {\rho }}^{\eta }(\mathbf {w},\tau )/(y)$ is a regular local ring of Krull dimension $1$. Since $R_{\overline {\rho }}^{\eta }(\mathbf {w},\tau )$ has Krull dimension $2$, it is also regular.

4.3 Potentially crystalline deformation rings of tame supercuspidal inertial types

In this subsection, we assume $\overline {\rho }$ is of one of the following forms:

1. (C1) $\overline {\rho }$ is in case 1 of § 4.1.1 with $2 \leq r\leq p-3;$

2. (C2) $\overline {\rho }$ is in case 2 of § 4.1.1 with $1 \leq r\leq p-3$.

In particular, $\overline {\rho }$ is generic (see § 4.2.1). We study the properties of deformation rings of tame supercuspidal inertial types and Hodge–Tate weights $(0,1)$ and $(0,2)$ in the cases (C1) and (C2) separately. The main result is Theorem 4.15.

Recall that given a pair of integers $(a,b)$ with $1\leq a \leq p-3$, we can associate:

1. • characters $\psi _i$, $1 \leq i \leq 3$, introduced in (3.15);

2. • tame supercuspidal inertial types $\tau _i=\psi _i\oplus \psi _i^p$ satisfying $\sigma (\tau _i) = \Theta (\psi _i)$ (cf. Lemma 5.1);

3. • lattices $R_1, R_2, R_3, R , \widetilde {R}$ introduced in § 3.4 satisfying ${\rm cosoc}(\mathcal {R} / p \mathcal {R}) = \sigma _{a,b+1}$ for any $\mathcal {R} \in \{R_1, R_2, R_3, R, \widetilde {R}\}$.

We choose $(a,b)$ as follows:

1. • in the case (C1), let $(a, b) \in \{(r,s), (p-1-r, r+s)\};$

2. • in the case (C2), let $(a,b) = (r,s)$.

Then $\sigma _{a,b+1}$ lies in $W(\overline {\rho })$ by Theorem 4.1. For $\mathcal {R} \in \{R_1, R_2, R_3, R , \widetilde {R}\}$, we denote by

(4.4)\begin{equation} I_{\mathcal{R}}: = \operatorname{{\mathrm{Ann}}}_{R_{\overline{\rho}}^{\eta}} (M(\mathcal{R})) \end{equation}

the annihilator of $M(\mathcal {R})$ in $R_{\overline {\rho }}^{\eta }$.

Proposition 4.9 We have that $M(\mathcal {R})$ is a (nonzero) cyclic $R_{\overline {\rho }}^{\eta }$-module for $\mathcal {R} \in \{R_1,R_2,R_3,$ $R,\widetilde {R}\}$. As a consequence $M(\mathcal {R}) \cong R_{\overline {\rho }}^{\eta } / I_{\mathcal {R}}$.

Proof. By Nakayama's lemma, it suffices to show $M(\mathcal {R})/\frak {m}$ is of dimension $1$ over $\mathbb {F}$, where $\frak {m}$ denotes the maximal ideal of $R_{\overline {\rho }}^{\eta }$. Since $\sigma _{a,b+1}\in W(\overline {\rho })$ is a quotient of $\mathcal {R}/p\mathcal {R}$, we always have $\dim _{\mathbb {F}} M(\mathcal {R}/p\mathcal {R})/ \mathfrak {m} \geq 1$ by (N3) of § 4.2.1.

To show $\dim _{\mathbb {F}} M(\mathcal {R}/p\mathcal {R})/ \mathfrak {m} \leq 1$, we note that $\mathcal {R}/p\mathcal {R}$ is a quotient of $(\operatorname {{\mathrm {Proj}}}_{\mathbb {F}[\![K / Z_1]\!]} \sigma _{a, b+1}) /\frak {m}_{K_1}^2$ by Lemma 3.18, Corollary 3.23 and Proposition 3.24. Hence, by (N3) of § 4.2.1

\[ \dim_{\mathbb{F}} M (\mathcal{R}/ p\mathcal{R} )/ \mathfrak{m} \leq \dim_{\mathbb{F}} M ((\operatorname{{\mathrm{Proj}}}_{\mathbb{F}[\![K / Z_1]\!]} \sigma_{a, b+1})/\frak{m}_{K_1}^2)/ \mathfrak{m} . \]

If $\overline {\rho }$ satisfies either case (C1) or case (C2), then by (4.3) and Theorem 4.2, we have

\[ \dim_{\mathbb{F}} M ((\operatorname{{\mathrm{Proj}}}_{\mathbb{F}[\![K / Z_1]\!]} \sigma_{a, b+1})/\frak{m}_{K_1}^2) / \frak{m} = \dim_{\mathbb{F}} \operatorname{{\mathrm{Hom}}}_{K} ((\operatorname{{\mathrm{Proj}}}_{\mathbb{F}[\![K / Z_1]\!]} \sigma_{a, b+1}) / \frak{m}_{K_1}^2, \pi(\overline{\rho})) = 1. \]

Hence, $\dim _{\mathbb {F}} M (\mathcal {R}/ p\mathcal {R} )/\frak {m} = 1$.

Remark 4.10 For $i\in \{2,3\}$, we have constructed $K$-stable $\mathcal {O}$-lattices $L, L'$ in $\underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi _i)$ in Proposition 3.11. The cosocle of $L/pL$ (respectively, $L'/pL'$) need not be irreducible, but $M(L)$ (respectively, $M(L')$) is still cyclic over $R_{\overline {\rho }}^{\eta }$.

Indeed, if $\operatorname {{\mathrm {JH}}}(\overline {\underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi _i)}^{\rm ss})\cap W(\overline {\rho })$ consists of one element, then the claim is obvious. Otherwise, $\overline {\rho }$ satisfies case (C1) and $W(\overline {\rho })$ consists of two elements, say $W(\overline {\rho })=\{\sigma _1,\sigma _2\} \subset \operatorname {{\mathrm {JH}}}(\overline {\underline {\mathrm {Sym}}^{1}E^2\otimes \Theta (\psi _i)}^{\rm ss})$. By Proposition 3.11, one of the nonsplit extensions, $E=(\sigma _1\ \textbf {---}\ \sigma _2)$ or $E'=(\sigma _2\ \textbf {---}\ \sigma _1)$, occurs in $L/pL$ (respectively, $L'/pL'$). As in the proof of Proposition 4.9, $M(E)$ and $M(E')$ are cyclic over $R_{\overline {\rho }}^{\eta }$, from which the claim follows as $M(\sigma )=0$ for $\sigma \notin W(\overline {\rho })$.

Corollary 4.11 We have:

1. (i) $I_{R_1} + I_{R_2} = (p, I_{R_1})$ and $I_R = I_{R_1} \cap I_{R_2}$;

2. (ii) $I_{R} + I_{R_3} = (p, I_{R_1})$ and $I_{\widetilde {R}} = I_{R_1}\cap I_{R_2} \cap I_{R_3}$.

Proof. Recall the following lemma from [HW22, Lemma 8.11].

Lemma 4.12 Let $(A,\mathfrak {m}_A)$ be a commutative noetherian local ring with $k=A/\mathfrak {m}_A$. Let $\mathcal {I}_0, \mathcal {I}_1,\mathcal {I}_2$ be ideals of $A$ such that $\mathcal {I}_1,\mathcal {I}_2\subset \mathcal {I}_0\subset \mathfrak {m}_A$. Consider the natural surjective homomorphism $A/\mathcal {I}_1\oplus A/\mathcal {I}_2\twoheadrightarrow A/\mathcal {I}_0$. Then $\operatorname {{\mathrm {Ker}}}(A/\mathcal {I}_1\oplus A/\mathcal {I}_2\twoheadrightarrow A/\mathcal {I}_0)$ is a cyclic $A$-module if and only if $\mathcal {I}_1+\mathcal {I}_2=\mathcal {I}_0$.

By (N3), the sequence (3.20) induces a short exact sequence

\[ 0 \to M(R) \to M(R_1) \oplus M(R_2) \to M(R_1 / pR_1) \to 0. \]

Since $M(R)$ is cyclic over $R_{\overline {\rho }}^{\eta }$ by Proposition 4.9, we deduce part (i) using Lemma 4.12 and the fact that⁵

\[ \mathrm{Ann}_{R_{\overline{\rho}}^{\eta}}(M(R_1/pR_1))=\mathrm{Ann}_{R_{\overline{\rho}}^{\eta}}(M(R_1)/p)=(p,I_{R_1}). \]

Similarly, we obtain part (ii) by using the short exact sequence (3.22).

Let $\delta :G_{\mathbb {Q}_p}\rightarrow \mathcal {O}^{\times }$ denote the character, via the local class field theory, sending $x\in \mathbb {Q}_p^{\times }\mapsto \mathrm {pr}(x)^{1/2}\in 1+p\mathbb {Z}_p$. By Theorem 4.5 and Corollary 4.6, we have

(4.5)\begin{align} R_{\overline{\rho}}^{\eta} / I_{R_1} = R_{\overline{\rho}}^{\eta }((0,1), \tau_1),\quad R_{\overline{\rho}}^{\eta} / I_{R_2} \overset{\mathrm{tw}_{\delta}^{-1}}{\cong} R_{\overline{\rho}}^{\eta\delta^2}((0,2),\tau_2),\quad R_{\overline{\rho}}^{\eta} / I_{R_3} \overset{\mathrm{tw}_{\delta}^{-1}}{\cong} R_{\overline{\rho}}^{\eta\delta^2 }((0,2),\tau_3). \end{align}

Proposition 4.13 The ring $R_{\overline {\rho }}^{\eta }((0,1), \tau _1)$ is a regular local ring.

Proof. Recall from § 3.3 that there exist two $K$-stable $\mathcal {O}$-lattices $T,T'\subset \sigma (\tau _1)$ such that $pT\subset T'\subset T$ and $T/T'\cong \sigma _{a,b+1}$ and $\mathrm {cosoc}(T/pT)\cong \sigma _{a,b+1}$. Here, if $\operatorname {{\mathrm {JH}}} (\overline {\sigma (\tau _1)}^{\rm ss})\cap W(\overline {\rho })$ consists of only one element, then we take $T'=pT$. In any case, the cosocle of $T'/pT'$ is irreducible. Using Theorem 4.2, it is easy to check that

\[ \dim_{\mathbb{F}}\operatorname{{\mathrm{Hom}}}_K(T/pT,\pi(\overline{\rho}))=\dim_{\mathbb{F}}\operatorname{{\mathrm{Hom}}}_K(T'/pT',\pi(\overline{\rho}))=1. \]

The result then follows from Proposition 4.8.

Remark 4.14 If $\overline {\rho }$ is generic in the sense of [BP12, Definition 11.7], Proposition 4.13 is a direct consequence of [EGS15, Theorem 7.2.1].

Theorem 4.15 The rings $R_{\overline {\rho }}^{\eta \delta ^2}((0,2),\tau _2)$ and $R_{\overline {\rho }}^{\eta \delta ^2}((0,2),\tau _3)$ are regular local rings.

Proof. Assume $\overline {\rho }$ is in case (C1). For $R_{\overline {\rho }}^{\eta \delta ^2 }((0,2),\tau _2)$, it is equivalent to proving that $R^{\eta }_{\overline {\rho }}/I_{R_2}$ is a regular local ring by (4.5). Note that $\sigma _{a,b+1}$ is the unique Serre weight in the intersection $\operatorname {{\mathrm {JH}}} (\overline {\underline {\mathrm {Sym}}^1E^2 \otimes \sigma (\tau _2)}^{\rm ss}) \cap W(\overline {\rho })$. The assertion follows from Proposition 4.8, by choosing any $K$-stable $\mathcal {O}$-lattice $\Theta _1$ in $\underline {\mathrm {Sym}}^1E^2\otimes \sigma (\tau _2)$, and taking $\Theta _2=p\Theta _1$ in Proposition 4.8.

We now consider $R_{\overline {\rho }}^{\eta \delta ^2}((0,2),\tau _3)$, equivalently $R_{\overline {\rho }}^{\eta }/I_{R_3}$ via (4.5). By Proposition 3.11, there are $K$-stable $\mathcal {O}$-lattices $L,L'$ of $\underline {\mathrm {Sym}}^1 E^2\otimes \Theta (\psi _3)$ such that $pL \subset L' \subset L$ and

\[ L / L' = \sigma_{a, b+1} \oplus \sigma_{a-2, b+2}. \]

Note that $\sigma _{a-2,b+2}\notin W(\overline {\rho })$. Using Remark 4.10, the result follows from Proposition 4.8.

Assume $\overline {\rho }$ is in case (C2). Then $\overline {\rho }$ has only one Serre weight $\sigma _{a,b+1}$, and we conclude as in the first paragraph.

4.4 Endomorphism rings and faithfulness

In this subsection, we assume $\overline {\rho }$ is reducible nonsplit and isomorphic to $\big (\begin{smallmatrix} {1} & {*}\\ {0} & {\omega }\end{smallmatrix}\big )$. Let $N\in \mathfrak {C}_{G/Z_G}(\mathcal {O})$ be as in § 4.2.1. In this case $N$ is isomorphic to a projective envelope of $(\operatorname {{\mathrm {Ind}}}_{B(\mathbb {Q}_p)}^{G}\omega \otimes \omega ^{-1})^{\vee }$ in $\mathfrak {C}_{G/Z_G}(\mathcal {O})$.

Let $(A,\mathfrak {m}_A)$ be a pseudo-compact flat local $\mathcal {O}$-algebra with residue field $\mathbb {F}$. Set $R:=A\widehat {\otimes }_{\mathcal {O}}R_{\overline {\rho }}^{\eta }$ and

\[ M:=R\widehat{\otimes}_{R_{\overline{\rho}}^{\eta}}N \cong A\widehat{\otimes}_{\mathcal{O}}N. \]

Then $M\in \frak {C}_{G/Z_G}(R)$. In fact, as in [CEG⁺18, Lemma 4.9] we show that $M^{\vee }$ is admissible in $\operatorname {\mathrm {Mod}}_G^{\rm sm}(R)$, and so $M\in \mathfrak {C}_{G/Z_G}(\mathcal {O})$.

Lemma 4.16 The object $M$ is a projective object in $\mathfrak {C}_{G/Z_G}(\mathcal {O})$.

Proof. By assumption $A$ is $\mathcal {O}$-flat. Since pseudo-compact flat $\mathcal {O}$-modules are projective (see, e.g., [Bru66, Proposition 3.1]), $A$ is a projective $\mathcal {O}$-module. By the definition of $M$ we have

(4.6)\begin{equation} \operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(\mathcal{O})}(M,-)\cong \operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(\mathcal{O})} (A\widehat{\otimes}_{\mathcal{O}}N,-)\cong\operatorname{{\mathrm{Hom}}}_{\mathcal{O}}^{\rm cont}(A,\operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(\mathcal{O})}(N,-)) \end{equation}

from which the result follows.

Lemma 4.17 We have $\operatorname {{\mathrm {Hom}}}_{\frak {C}_{G/Z_G}(R)}(M,\mathbf {1}_G^{\vee })=0$ and $\operatorname {{\mathrm {Ext}}}^1_{\frak {C}_{G/Z_G}(R)}(M,\mathbf {1}_G^{\vee })=0$.

Proof. The first assertion follows from (4.6) because $\operatorname {{\mathrm {Hom}}}_{\frak {C}_{G/Z_G}(\mathcal {O})}(N,\mathbf {1}_G^{\vee })=0$, see (N1) in § 4.2.1. For the second, we work on the dual side and show $\operatorname {{\mathrm {Ext}}}^1_{R[G]}(\mathbf {1}_G,M^{\vee })=0$. By Lemma 4.16, $M$ is a projective object in $\frak {C}_{G/Z_G}(\mathcal {O})$, so dually $M^{\vee }$ is an injective object in $\operatorname {\mathrm {Mod}}_{G/Z_G}^{\rm l.adm}(\mathcal {O})$. Consider an extension

\[ 0\rightarrow M^{\vee}\rightarrow \mathcal{E}\rightarrow \mathbf{1}_G\rightarrow0 \]

in $\operatorname {\mathrm {Mod}}_{G/Z_G}^{\rm l.adm}(R)$. It must split in $\operatorname {\mathrm {Mod}}_{G/Z_G}^{\rm l.adm}(\mathcal {O})$, so we may find $v\in \mathcal {E}$ such that $\langle \mathcal {O}[G].v\rangle \cong \mathbf {1}_G$. It suffices to show that $R$ acts on $v$ via the quotient $R\twoheadrightarrow R/\mathfrak {m}_R\cong \mathbb {F}$. This is clear, since if $x\in \mathfrak {m}_{R}$, then $x\cdot v\in M^{\vee }$ and, if it were nonzero, then it would generate a subrepresentation of $M^{\vee }$ isomorphic to $\mathbf {1}_G$, which is not possible by the first assertion.

Proposition 4.18 For any compact $R$-module $\mathrm {m}$, the natural map $v\mapsto (m\mapsto (v\widehat {\otimes } m))$ (where $v\in \mathrm {m}$ and $m\in M$) induces an isomorphism

\[ \mathrm{m}\xrightarrow{\sim}\operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(R)}(M,\mathrm{m}\widehat{\otimes}_{R}M). \]

Remark 4.19 Note that $M$ is not projective in $\frak {C}_{G/Z_G}(R)$ so that we cannot apply [Paš13, Lemma 2.9].

Proof. The proof is similar to [HP19, Proposition 3.12]. As in [HP19, Proposition 3.12], we may assume that $\mathrm {m}$ is of finite length. In particular, the completed tensor product $\mathrm {m}\widehat {\otimes }_{R}M$ coincides with the usual one.

We proceed by induction on the length of $\mathrm {m}$. Note that since $R$ is a local ring, any $R$-module of length $1$ is isomorphic to $R/\mathfrak {m}_R\cong \mathbb {F}$. If $\mathrm {m}\cong \mathbb {F}$, we need to show that $\operatorname {{\mathrm {Hom}}}_{\frak {C}_{G/Z_G}(R)}(M,\mathbb {F}\otimes _{R}M)\cong \mathbb {F}$. But any morphism $M\rightarrow \mathbb {F}\otimes _{R}M$ in $\frak {C}_{G/Z_G}(R)$ factors through

\begin{align*} M\twoheadrightarrow \mathbb{F}\otimes_{R} M\rightarrow \mathbb{F}\otimes_{R}M, \end{align*}

so the assertion is reduced to

\[ \operatorname{{\mathrm{End}}}_{\frak{C}_{G/Z_G}(\mathcal{O})}(\mathbb{F}\otimes_{R}M)=\operatorname{{\mathrm{End}}}_{\frak{C}_{G/Z_G}(\mathcal{O})} (\mathbb{F}\otimes_{R_{\overline{\rho}}^{\eta}}N)\cong\mathbb{F}, \]

which is a direct consequence of Proposition 4.4. If the length of $\mathrm {m}$ is $\geq 2$, let $\mathrm {m}_1\subset \mathrm {m}$ be a proper $R$-submodule such that $\mathrm {m}_2:=\mathrm {m}/\mathrm {m}_1$ has length $1$, i.e. $\mathrm {m}_2\cong \mathbb {F}$. We then obtain a long exact sequence

(4.7)\begin{equation} \operatorname{{\mathrm{Tor}}}_1^{R}(\mathrm{m}_2,M)\rightarrow \mathrm{m}_1\otimes_{R}M\rightarrow\mathrm{m} \otimes_{R}M\rightarrow \mathrm{m}_2\otimes_{R}M\rightarrow0. \end{equation}

Since $\mathrm {m}_2\cong \mathbb {F}$ and $R$ is flat over $R_{\overline {\rho }}^{\eta }$ by construction, we have

\[ \operatorname{{\mathrm{Tor}}}_1^{R}(\mathrm{m}_2,M)= \operatorname{{\mathrm{Tor}}}_1^{R}(\mathbb{F},R\widehat{\otimes}_{R_{\overline{\rho}}^{\eta}}N)\cong \operatorname{{\mathrm{Tor}}}_1^{R_{\overline{\rho}}^{\eta}}(\mathbb{F},N)\cong (\mathbf{1}_{G}^{\vee})^{\oplus 2}, \]

where the last isomorphism follows from [Hu21, Proposition 3.30]. By applying $\operatorname {{\mathrm {Hom}}}_{\frak {C}_{G/Z_G}(R)}(M,-)$ to (4.7) and using Lemma 4.17, we obtain the following short exact sequence

\[ 0\rightarrow \operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(R)}(M,\mathrm{m}_1\otimes_{R}M)\rightarrow \operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(R)}(M,\mathrm{m}\otimes_{R}M)\rightarrow \operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(R)}(M,\mathrm{m}_2\otimes_{R}M). \]

By inductive hypothesis, we have $\mathrm {m}_i\xrightarrow {\sim } \operatorname {{\mathrm {Hom}}}_{\frak {C}_{G/Z_G}(R)}(M,\mathrm {m}_i\otimes _{R}M)$ for $i\in \{1,2\}$, hence the result using the snake lemma.

Corollary 4.20 We have $\operatorname {{\mathrm {End}}}_{\frak {C}_{G/Z_G}(R)}(M)\cong R$. In particular, $R$ acts faithfully on $M$.

Proposition 4.21 Let $x_1,\ldots,x_g\in R$ be an $M$-regular sequence. Then $(x_1,\ldots,x_g)$ is also $R$-regular and

\[ \operatorname{{\mathrm{End}}}_{\frak{C}_{G/Z_G}(R)}(M/(x_1,\ldots,x_i) M)\cong R/(x_1,\ldots,x_i)R \]

for any $1\leq i\leq g$.

Proof. The proof is analogous to [Hu21, Proposition 5.11].

Since $R$ acts faithfully on $M$ by Corollary 4.20 and $x_1$ is $M$-regular, $x_1$ is also $R$-regular. By Proposition 4.18, we have

\[ R/x_1R\xrightarrow{\sim} \operatorname{{\mathrm{Hom}}}_{\frak{C}_{G/Z_G}(R)}(M,M/x_1M)=\operatorname{{\mathrm{End}}}_{\frak{C}_{G/Z_G}(R)}(M/x_1M). \]

This, in turn, shows that $R/x_1R$ acts faithfully on $M/x_1M$, hence $x_2$ is $R/x_1R$-regular because it is $M/x_1M$-regular by assumption. We may thus continue the above argument to conclude.

Remark 4.22 Proposition 4.21 could be used to prove a big ‘$R=\mathbb {T}$’ theorem, see the proof of Proposition 8.20 below. Such a result is proved in [GN22, Theorem B(3)] when $R_{\overline {\rho }}^{\eta }$ is formally smooth, by first proving that a suitable patched module $M_{\infty }$ is faithfully flat over the patched ring $R_{\infty }$ and then passing to the quotient. However, when $\overline {\rho } \sim \big (\begin{smallmatrix} {1} & {*}\\ {0} & {\omega }\end{smallmatrix}\big ) \otimes \chi$, $R_{\overline {\rho }}^{\eta }$ is not formally smooth and the patched module $M_{\infty }$ is not flat over $R_{\infty }$, so the argument in [GN22] does not apply. In addition, this case is also excluded in [Eme11, Theorem 1.3], so Proposition 4.21 may be of independent interest.

5. Automorphic forms and big patched modules

Let $F$ be a totally real extension of $\mathbb {Q}$ in which $p$ is unramified, and let ${\mathcal {O}}_F$ be its ring of integers. Let $\Sigma _p$ denote the set of places of $F$ dividing $p$ and let $\Sigma _{\infty }$ denote the set of infinite places of $F$. For any place $v$ of $F$, let $F_v$ denote the completion of $F$ at $v$ with ring of integers ${\mathcal {O}}_{F_v}$, uniformizer $\varpi _v$ and residue field $k_{F_v}$. Let $q_v$ denote the cardinality of $k_{F_v}$. Let ${\mathbb {A}}_{F,f}$ denote the ring of finite adèles of $F$. If $S$ is a finite set of finite places of $F$, let ${\mathbb {A}}^S_{F,f}$ denote the ring of finite adèles outside $S$. Recall $G_F=\operatorname {{\mathrm {Gal}}}({\overline {F}}/F)$ and $G_{F_v}=\operatorname {{\mathrm {Gal}}}({\overline {F}}_v/F_v)$. By fixing an embedding ${\overline {F}}\hookrightarrow {\overline {F}}_v$, $G_{F_v}$ is identified with the decomposition group at $v$. We let $\operatorname {{\mathrm {Frob}}}_v\in G_{F_v}$ denote a (lift of the) geometric Frobenius element, and let $\operatorname {{\mathrm {Art}}}_{F_v}$ denote the local Artin map, normalized so that it sends $\varpi _v$ to $\operatorname {{\mathrm {Frob}}}_v$. The global Artin map is denoted by $\operatorname {{\mathrm {Art}}}_F$ which is compatible with the local Artin map. We denote by $\mathrm {rec}_{v}$ the local Langlands correspondence normalized as in the introduction of [HT01], so that if $\pi$ is a smooth irreducible $\overline {\mathbb {Q}}_p$-representation of $\mathrm {GL}_2(F_{v})$, then $\mathrm {rec}_{v}(\pi )$ is a Weil–Deligne representation of the Weil group $W_{F_v}$ defined over $\overline {\mathbb {Q}}_p$. Recall that $\mathbb {F}$ is a sufficiently large finite extension of $\mathbb {F}_p$, $\mathcal {O} = W(\mathbb {F})$ and $E = \mathcal {O}[1/p]$. We prepare the global setup we need in this section.

5.1 Tame types and the inertial local Langlands correspondence

Let $I_{F_{v}}$ be the inertia subgroup of $G_{F_{v}}$. An inertial type at $v$ is a two-dimensional representation $\tau : I_{F_{v}} \to \mathrm {GL}_2(\overline {\mathbb {Q}}_p)$ with open kernel which extends to a representation of $G_{F_{v}}$. We say $\tau$ is a discrete series inertial type if it is either scalar, or extends to an irreducible representation of $G_{F_{v}}$. In the latter case, we call $\tau$ supercuspidal. We say $\tau$ is tame if it is trivial on the wild inertia subgroup. Under Henniart's inertial local Langlands correspondence [Hen02] (cf. also [Kis09]), there is a unique finite-dimensional irreducible representation $\sigma (\tau )$ (respectively, $\sigma ^{\rm cr}(\tau )$) of $\mathrm {GL}_2(\mathcal {O}_{F_{v}})$ over $\overline {\mathbb {Q}}_p$-vector spaces, called types, satisfying if $\pi$ is an infinite-dimensional smooth irreducible representation of $\mathrm {GL}_2(F_{v})$, then $\operatorname {{\mathrm {Hom}}}_{\mathrm {GL}_2(\mathcal {O}_{F_{v}})}(\sigma (\tau ),\pi ) \neq 0$ (respectively, $\operatorname {{\mathrm {Hom}}}_{\mathrm {GL}_2(\mathcal {O}_{F_{v}})}(\sigma ^{\rm cr}(\tau ),\pi ) \neq 0$) if and only if $\mathrm {rec}_{v}(\pi )|_{I_{F_{v}}} \cong \tau$ (respectively, $\mathrm {rec}_{v}(\pi )|_{I_{F_{v}}} \cong \tau$ and the monodromy operator $N$ on $\mathrm {rec}_{v}(\pi )$ is zero), in which case the space $\operatorname {{\mathrm {Hom}}}_{\mathrm {GL}_2(\mathcal {O}_{F_{v}})}(\sigma (\tau ),\pi )$ (respectively, $\operatorname {{\mathrm {Hom}}}_{\mathrm {GL}_2(\mathcal {O}_{F_{v}})}(\sigma ^{\rm cr}(\tau ),\pi )$) is one-dimensional. We always have $\sigma (\tau ) = \sigma ^{\rm cr}(\tau )$ except when $\tau = \chi \oplus \chi$, in which case $\sigma (\chi \oplus \chi ) = {\rm sp}\otimes \chi \circ \det$ (here ${\rm sp}$ denotes the Steinberg representation of $\mathrm {GL}_2(k_{F_v})$ over $\overline {\mathbb {Q}}_p$) and $\sigma ^{\rm cr}(\chi \oplus \chi ) = \chi \circ \det$. Let $\psi : \mathbb {F}_{q_v^2}^{\times } \to \overline {\mathbb {Q}}_p^{\times }$ be a character such that $\psi \neq \psi ^{q_v}$. Let $\Theta (\psi )$ be the irreducible cuspidal $\overline {\mathbb {Q}}_p$-representation of $\mathrm {GL}_2(k_{F_v})$ associated to $\psi$ as in [Dia07]. A tame supercuspidal type is an irreducible $\overline {\mathbb {Q}}_p$-representation of $\mathrm {GL}_2(\mathcal {O}_{F_{v}})$ that arises by inflation from $\Theta (\psi )$ for some $\psi$ as above.

In [GG15], Gee and Geraghty developed an analogous theory for $D^{\times }$, where $D$ is the nonsplit central quaternion algebra over $F_{v}$. Let $\mathrm {JL}$ denote the Jacquet–Langlands correspondence giving a bijection from irreducible smooth representations of $D^{\times }$ over $\overline {\mathbb {Q}}_p$ to discrete series representations of $\mathrm {GL}_2(F_v)$ over $\overline {\mathbb {Q}}_p$. Let $\tau$ be a discrete series inertial type. By the Jacquet–Langlands correspondence, there is an irreducible smooth representation $\pi _{D,\tau }$ of $D^{\times }$ such that $\mathrm {rec}_{v}(\mathrm {JL}(\pi _{D,\tau }))|_{I_{F_{v}}} \cong \tau$. Since $F_{v}^{\times }\mathcal {O}_D^{\times }$ has index two in $D^{\times }$, $\pi _{D,\tau }|_{\mathcal {O}_D^{\times }}$ is either irreducible or a sum of two irreducible representations which are conjugate under the uniformizer $\varpi _D$ of $D$. Let $\sigma _D(\tau )$ be one of the irreducible components of $\pi _{D,\tau }|_{\mathcal {O}_D^{\times }}$. If $\pi _D$ is a smooth irreducible $\overline {\mathbb {Q}}_p$-representation of $D^{\times }$, then $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\sigma _D(\tau ),\pi _D) \neq 0$ if and only if $\mathrm {rec}_{v}(\mathrm {JL}(\pi _D))|_{I_{F_{v}}} \cong \tau$, in which case, $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\sigma _D(\tau ),\pi _D)$ is one-dimensional. If $\tau$ is a tame inertial type, then $\sigma (\tau )$ and $\sigma _D(\tau )$ can be defined over $E$ once $E$ is taken sufficiently large (and unramified), see the proof of [EGS15, Lemma 3.1.1]. Recall the following lemma.

Lemma 5.1 Let $\psi : \mathbb {F}_{q_v^2}^{\times } \to E^{\times }$ with $\psi \neq \psi ^{q_v}$. Let $\tau : = \psi \oplus \psi ^{q_v}$ be the supercuspidal inertial type associated to $\psi$, where we denote by $\psi$ the composition $I_{F_v} \twoheadrightarrow \mathbb {F}_{q_v^2}^{\times } {\buildrel {\psi } \over \longrightarrow} E^{\times }$. Then $\sigma (\tau ) = \Theta (\psi )$ and $\pi _{D,\tau }|_{\mathcal {O}_{D}^{\times }} = \psi \oplus \psi ^{q_v}$.

Proof. The assertion on $\sigma (\tau )$ follows from Henniart's construction in [Hen02]. The assertion on $\sigma _D(\tau )$ follows from the classical Jacquet–Langlands correspondence; see, for example, [BH06, Chapter 13].

5.2 Automorphic forms, Galois representations and the big patched modules

We define the space of automorphic forms. Let $B$ be a quaternion algebra over $F$. Fix a maximal order $\mathcal {O}_B$ of $B$. Let $\Sigma _B$ be the set of primes $v$ in $F$ at which $B$ is ramified. Let $\infty _{F}$ be a fixed infinite place of $F$. We say $B$ is definite if it is ramified at all infinite places; $B$ is indefinite if it splits at $\infty _F$ and ramifies at all other infinite places. If $v$ is a finite place of $F$, let $\mathcal {O}_{B_{v}}^{\times }$ denote the maximal compact subgroup of $B_{v}^{\times } : = (B\otimes _F F_v)^{\times }$. For $v \notin \Sigma _B$, we fix an isomorphism $B_{v}^{\times } \cong \mathrm {GL}_2(F_v)$ so that $\mathcal {O}_{B_{v}}^{\times }$ is identified with $\mathrm {GL}_2(\mathcal {O}_{F_v})$. Let $\psi :F^\times \setminus {\mathbb {A}}_{F,f}^\times \to \mathcal {O}^\times$ be a continuous character. Via the global Artin map, $\psi$ induces a continuous character $G_{F}\to \mathcal {O}^{\times }$ which, by abuse of notation, is again denoted by $\psi$. Assume, moreover, that $(F, B) \neq (\mathbb {Q}, \mathrm {GL}_{2})$.

Let $U$ be a compact open subgroup of $(B\otimes _{F} {\mathbb {A}}_{F,f})^\times$. We denote by $Y^B_U$ the finite set $B^\times \setminus (B\otimes _{F} {\mathbb {A}}_{F,f})^\times /U$ if $B$ is definite. If $B$ is indefinite, let $Y^B_U$ denote the quotient of $X^B_U$ by the action of the finite group $\mathbb {A}_{F,f}^{\times } / (F^{\times } (\mathbb {A}_{F,f}^{\times } \cap U))$, where $X_U^B$ is the associated Shimura curve as in [BD14], which is the same convention used in [Eme11, Scho18] but is different from the convention used in [BDJ10].

From now on until the end of the paper, we assume that $\Sigma _B$ and $\Sigma _p$ intersect at a unique place $v$ above $p$. Fix $U^p = \prod _{w \nmid p} U_w$ a compact open subgroup of $(B\otimes _{F} \mathbb {A}^{\Sigma _p}_{F,f})^{\times }$. For each place $w \in \Sigma _p \setminus \{v\}$, let $\sigma _w$ be a finite free $\mathcal {O}$-module equipped with a continuous action of $U_w$ such that $F_w^{\times }\cap U_w$ acts by $\psi ^{-1}|_{F_w^{\times }}$. Denote

\[ \sigma_p^v = \otimes_{ w \in \Sigma_p \setminus \{v\} } \sigma_w. \]

Let $U^{v}= U^p U_p^{v} \subset (B \otimes _F \mathbb {A}^{\{v\}}_{F,f})^{\times }$. Then $\sigma _p^v$ is equipped with an action of $U^v$ via the projection $U^v \twoheadrightarrow U^{v}_p$. We extend this action to $U^v \mathbb {A}_{F,f}^{\times }$ by letting $\mathbb {A}_{F,f}^{\times }$ act by $\psi ^{-1}$. Assume that $U_v$ is a compact open subgroup of $\mathrm {GL}_2(\mathcal {O}_{F_v})$ such that $\psi |_{U_{v}\cap \mathcal {O}_{F_v}^\times }=1$. Then $\sigma _p^v$ admits an action of $U^v U_v\mathbb {A}_{F,f}^{\times }$ by letting $U_v$ act trivially.

If $B$ is definite, set

\begin{align*} \widetilde{H}^{0,B}_{\sigma_p^v , \psi} (U^{v}, \mathcal{O}) &:= \{f: B^\times\setminus (B\otimes_F {\mathbb{A}}_{F,f})^\times \to \sigma_p^v \,|\, f\ \text{is}\ \textit{continuous}\ \text{and}\ f(g u) = u^{-1} f(g), \\ &\quad\ \ \forall g\in (B\otimes_F {\mathbb{A}}_{F,f})^\times, \forall u \in U^v U_v \mathbb{A}_{F,f}^{\times}\}. \end{align*}

If $B$ is indefinite, let $\mathcal {F}_{\sigma _p^v/\varpi ^s}$ be the local system over $Y_{U^{v}U_{v}}^{B}$ associated to $\sigma _p^v/\varpi ^s$ (see [Eme06]), and set

\begin{align*} \widetilde{H}^{1,B}_{\sigma_p^v , \psi} (U^{v}, \mathcal{O}) := \varprojlim_{s} \varinjlim_{U_{v}} H^1_{{\unicode{x00E9}}\textrm{t}} (Y_{U^{v}U_{v}}^{B} , \mathcal{F}_{\sigma_p^v/\varpi^s}). \end{align*}

Both $\widetilde {H}^{0,B}_{\sigma _p^v , \psi } (U^{v}, \mathcal {O})$ and $\widetilde {H}^{1,B}_{\sigma _p^v , \psi } (U^{v}, \mathcal {O})$ carry an action of $(B\otimes _{F} F_{v})^\times$.

Let $S$ be a set of places of $F$ containing all places in $\Sigma _{\infty } \cup \Sigma _B\cup \Sigma _p$, all places where $\psi$ is ramified, and all places $w$ such that $U_w$ is not $\mathcal {O}_{B_w}^{\times }$. Let $\overline {r}: G_{F} \to \mathrm {GL}_2(\mathbb {F})$ be an absolutely irreducible totally odd representation. Assume $\overline {r}$ is unramified outside $S$. Assume $\overline {\psi } := \psi \pmod {\varpi }$ is equal to $\omega \det \overline {r}$. Denote $\overline {r}_w {\overset {\rm {def}}{=}}\, \overline {r}|_{G_{F_w}}$. We make the following assumptions on $\overline {r}$:

1. (a) $\overline {r}$ is modular in the sense of [BD14, § 3.1], $\overline {r}|_{G_{F(\sqrt [p]{1})}}$ is absolutely irreducible and, if $p = 5$, the image of $\overline {r}(G_{F(\sqrt [p]{1})})$ in ${\rm PGL}_2(\mathbb {F})$ is not isomorphic to ${\rm PSL}_2(\mathbb {F}_5)$;

2. (b) for $w \in S \setminus \Sigma _p$ the framed deformation ring of $\overline {r}_w$ is formally smooth over $\mathcal {O}$ (cf. [BHHMS23, Remark 8.1.1]);

3. (c) if $w\nmid p$ and $w \in \Sigma _B$, then $\overline {r}_w$ is either irreducible or a twist of an extension of the trivial representation by $\overline {\varepsilon }$;

4. (d) if $w|p$, $w\neq v$, then $\overline {r}|_{I_{F_w}}$ is generic in the sense of [BP12, Definition 11.7] (which is different from the genericity used in § 4.2.1).

Assumption (c) is often called the compatibility condition between $B$ and $\overline {r}$. By [BD14, Corollaire 3.2.3], the above assumptions guarantee the non-vanishing of $\pi ^B(\overline {r})$, where $\pi ^B(\overline {r})$ is defined in (5.4). For each $w \in \Sigma _p\setminus \{v\}$, we fix a tame inertial type $\tau _w$ over $E$ such that $\det (\tau _w)|_{I_{F_w}}=\psi |_{I_{F_w}}$ and $\operatorname {{\mathrm {JH}}}( \overline {\sigma (\tau _w)}^{\rm ss})$ contains exactly one Serre weight in $W(\overline {r}_w(1))$ [EGS15, Proposition 3.5.1]. This is possible by our assumption (d) and (the proof of) [EGS15, Proposition 3.5.1]. Let $\sigma ^{\circ }(\tau _w)$ be an $\mathcal {O}_{B_w}^{\times }$-stable $\mathcal {O}$-lattice in $\sigma (\tau _w)$ and

(5.1)\begin{equation} \sigma^v_p := \otimes_{w\in \Sigma_p\setminus \{v \}} \sigma^{\circ}(\tau_w)^d. \end{equation}

For $w$ a finite place of $F$, let $R_w$ denote the universal framed deformation ring of $\overline {r}_w$ over ${\mathcal {O}}$. Let $R_{w}^{\psi \varepsilon ^{-1}}$ denote the quotient of $R_{w}$ corresponding to liftings with determinant $(\psi |_{F^\times _w})\varepsilon ^{-1}$. If $w \in S\setminus \Sigma _p$, $R^{\psi \varepsilon ^{-1}}_w$ is a formal power series ring in $3$-variables over $\mathcal {O}$ by our assumption (b). If $w\,|\,p,\ w\neq v$, let $R_{w}^{\psi \varepsilon ^{-1}}((-1,0)_{\kappa },\tau _w)$ denote the reduced $p$-torsion-free quotient of $R_{w}^{\psi \varepsilon ^{-1}}$ corresponding to potentially crystalline (framed) deformations of inertial type $\tau _w$ and Hodge–Tate weights $(-1,0)$ for all embeddings $\kappa : F_w\hookrightarrow E$. By the choice of $\tau _w$, $R_{w}^{\psi \varepsilon ^{-1}}((-1,0)_{\kappa },\tau _w)$ is a formal power series ring in $(3+[F_w:\mathbb {Q}_p])$-variables over $\mathcal {O}$ (see [EGS15, Theorem 7.2.1]). Let

\[ R_S := \widehat{\otimes}_{w\in S}R_w^{\psi\varepsilon^{-1}} \]

and

\[ R^{ \operatorname{{\mathrm{loc}}}}:= R_v^{\psi\varepsilon^{-1}} \widehat{\otimes } (\widehat{\otimes}_{w\,|\,p, w\neq v } R_w^{\psi\varepsilon^{-1} }((-1,0)_{\kappa},\tau_w))\widehat{\otimes } (\widehat{\otimes}_{w\in S\setminus \Sigma_p }R_w^{\psi\varepsilon^{-1}}). \]

Let $R^{\Box, \psi \varepsilon ^{-1}}_{\overline {r},S}$ (respectively, $R^{\psi \varepsilon ^{-1}}_{\overline {r},S}$) be the framed (respectively, universal) deformation ring of $\overline {r}$ parametrizing liftings (respectively, deformations) of $\overline {r}$ which are unramified outside $S$ with determinant $\psi \varepsilon ^{-1}$ as in [GK14, § 5.4.1]. Let $r^{\rm univ}$ denote the universal deformation of $\overline {r}$ over $R^{\psi \varepsilon ^{-1}}_{\overline {r} , S}$. Define $R^{\Box,\psi \varepsilon ^{-1}, \operatorname {{\mathrm {loc}}}}_{\overline {r},S}:=R^{\Box, \psi \varepsilon ^{-1}}_{\overline {r},S}\widehat {\otimes }_{R_S}R^{ \operatorname {{\mathrm {loc}}}}$. Let $R^{\psi \varepsilon ^{-1},\operatorname {{\mathrm {loc}}}}_{\overline {r} , S }$ denote the image of $R^{\psi \varepsilon ^{-1}}_{\overline {r},S}$ in $R^{\Box,\psi \varepsilon ^{-1},\operatorname {{\mathrm {loc}}}}_{\overline {r},S}$.

By [DDT97, Lemma 4.11], there is a finite place $w_1\notin S$ with the following properties:

• – $q_{w_1}\not \equiv 1 \pmod {p}$;

• – the ratio of the eigenvalues of $\overline {r}(\operatorname {{\mathrm {Frob}}}_{w_1})$ is not equal to $q_{w_1}^{\pm 1}$;

• – the residue characteristic of $w_1$ is sufficiently large such that for any nontrivial root of unity $\zeta$ in a quadratic extension of $F$, $w_1$ does not divide $\zeta +\zeta ^{-1}-2$.

Let $U = \prod _w U_w \subset (B \otimes _F \mathbb {A}_{F,f})^{\times }$ be a compact open subgroup satisfying:

• – $U_w = \mathcal {O}_{B_w}^{\times }$ for $w \notin S \cup \{w_1\};$

• – $U_{w_1}$ is contained in the subgroup of $(\mathcal {O}_B)_{w_1}^{\times } = \mathrm {GL}_2(\mathcal {O}_{F_{w_1}})$ consisting of matrices that are upper-triangular and unipotent modulo $\varpi _{w_1};$

• – for places over $p$, $U_w = 1 +\varpi _w M_2(\mathcal {O}_{F_w})$ if $w\,|\,p$, $w\neq v;$ $U_v$ is the subgroup $U^1_{B_v}$ defined in (2.2).

By the choice of $U_{w_1}$, $U$ is sufficiently small in the sense of [CHT08, § 3.3].

[CEG⁺16] and [Scho18] extend the Taylor–Wiles–Kisin method to construct the big patched modules. The detailed construction for Shimura curves in the minimal case is given in [DL21, § 6]. By the arguments of [DL21], replacing $K^v$ in [DL21] by $U^v$, the representation $V = \bigotimes _{w \in S, w\neq v} V_w$ of $K^v$ in [DL21] by the representation $\sigma ^v_p$ of $U^v$, forgetting the Hecke operators $T_w$ at places $w\in S'$, and allowing $B$ possibly ramifies at some places above $p$, the same patching arguments produce a ‘big’ patched module $M_{\infty }^B$ with the following data. (Let $j:=4 \# S - 1$ and let $g,q$ be positive integers such that $q = g+ [F:\mathbb {Q}] - \# S +1$.)

• • A formal power series ring in $q$-variables $\mathcal {O}[\![z_1,\ldots,z_q]\!]$ with a homomorphism

  \[ \mathcal{O}[\![z_1,\ldots,z_q]\!] \to R^{\psi\varepsilon^{-1},\operatorname{{\mathrm{loc}}}}_{\overline{ r}, S} \]
  which extends to a homomorphism from $S_{\infty }:= \mathcal {O}[\![z_1,\ldots,z_q,y_1,\ldots,y_j]\!]$ to $R^{\Box, \psi \varepsilon ^{-1},\operatorname {{\mathrm {loc}}}}_{\overline { r}, S}$.

• • There is a surjective homomorphism

  \[ R_{\infty}^{ \psi\varepsilon^{-1}} \twoheadrightarrow R^{\Box, \psi\varepsilon^{-1},\operatorname{{\mathrm{loc}}}}_{\overline{r}, S}, \]
  where $R_{\infty }^{ \psi \varepsilon ^{-1}}:= R^{ \operatorname {{\mathrm {loc}}}}[\![x_1,\ldots,x_g]\!]$. Let $\frak {m}_{\infty }$ be the maximal ideal of $R_{\infty }^{ \psi \varepsilon ^{-1}}$.

• • An ${\mathcal {O}}$-algebra homomorphism $S_{\infty }\to R_{\infty }^{ \psi \varepsilon ^{-1}}$ such that

  \[ R_{\infty}^{ \psi\varepsilon^{-1}}/\frak{a}_{\infty}\cong R^{\psi\varepsilon^{-1},\operatorname{{\mathrm{loc}}}}_{\overline{r}, S }, \]
  where $\frak {a}_{\infty }$ denotes the ideal $(z_1,\ldots,z_{q},y_1,\ldots,y_j)$ of $S_{\infty }$.

• • A finitely generated Cohen–Macaulay $S_{\infty }[\![ \mathcal {O}_{B_v}^{\times } ]\!]$-module $M^{B}_{\infty }$ equipped with an action of $R_{\infty }^{ \psi \varepsilon ^{-1}}$, so that the action of $S_{\infty }$ factors through $R_{\infty }^{ \psi \varepsilon ^{-1}}$. The module $M^B_{\infty }$ is also Cohen–Macaulay over $R_{\infty }^{ \psi \varepsilon ^{-1}}[\![ \mathcal {O}_{B_v}^{\times } ]\!]$ by [GN22, Corollary A29]. Moreover, $M^{B}_{\infty }$ is projective in the category $\frak {C}_{\mathcal {O}_{B_v}^{\times },\psi }(S_{\infty })$. Note that projectivity in the case where $B$ ramifies at $v$ follows from the proof of [CEG⁺16, Proposition 2.10] using [New13, Proposition 5.6]. Let $\frak {m}_{\overline {r}}$ be the maximal ideal of the abstract Hecke algebra associated to $\overline {r}$ as in [Scho18, § 5]. We have

  (5.2) \begin{equation} M^{B}_{\infty} / \frak{a}_{\infty} = \left\{{\begin{array}{@{}ll} \widetilde{H}^{0,B}_{\sigma_p^v, \psi} (U^{v}, \mathcal{O})^d_{\frak{m}_{\overline{r}}} & \text{if}\ \text{B is definite}, \\ \operatorname{{\mathrm{Hom}}}_{{\mathbb{T}}(U^v)_{\frak{m}_{\overline{r}}}[ G_{F}]} (r_{\frak{m}}, \widetilde{H}^{1,B}_{\sigma_p^v , \psi} (U^{v}, \mathcal{O})_{\frak{m}_{\overline{r}}})^d & \text{if}\ B\ \text{is indefinite,} \end{array}}\right. \end{equation}
  where ${\mathbb {T}}(U^v)_{\frak {m}_{\overline {r}}}$ denotes the Hecke algebra defined in the paragraph before [Scho18, Proposition 5.7] (by taking $\frak {p} = v$ and $\frak {m} = \frak {m}_{\overline {r}}$ in [Scho18]), and $r_{\frak {m}}$ denotes the composite
  \[ G_F {\buildrel {r^{\rm univ}} \over \longrightarrow} \mathrm{GL}_2(R^{\psi\varepsilon^{-1}}_{\overline{r},S}) \to \mathrm{GL}_2({\mathbb{T}}(U^v)_{\frak{m}_{\overline{r}}}). \]

Remark 5.2 In the indefinite case, there is a variant of $M_{\infty }^B$ denoted by $N_{\infty }^B$, which is obtained by patching $\widetilde {H}^{1,B}_{\sigma _p^v , \psi } (U^{v}, \mathcal {O}/\varpi ^s)_{\frak {m}_{\overline {r}}}$ but without factorizing out $r^{\rm univ}$. Namely, we have

\[ N_{\infty}^B/\mathfrak{a}_{\infty}\cong (\widetilde{H}^{1,B}_{\sigma_p^v , \psi} (U^{v}, \mathcal{O})_{\frak{m}_{\overline{r}}})^d. \]

Let ${\rm Mod}_{\mathcal {O}^{\times }_{B_v},\psi }^{\rm fin}$ denote the category of finite $\mathcal {O}$-modules with a continuous action of $\mathcal {O}_{B_v}^{\times }$ such that the $\mathcal {O}_{B_v}^{\times }$-action has central character $\psi |_{F_v^{\times }}$. Define a functor $M_{\infty }^B(-)$ from ${\rm Mod}_{\mathcal {O}^{\times }_{B_v},\psi }^{\rm fin}$ to the category of finitely generated $R_{\infty }^{\psi \varepsilon ^{-1}}$-modules by letting

(5.3)\begin{equation} M^{B}_{\infty}(\sigma):= \operatorname{{\mathrm{Hom}}}_{ \mathcal{O}_{B_v}^{\times} }^{\rm cont} ( M_{\infty }^{B}, \sigma^{\vee} )^{\vee}. \end{equation}

By the projectivity of $M^{B}_{\infty }$ in $\frak {C}_{\mathcal {O}_{B_v}^{\times },\psi }(\mathcal {O})$, $M^{B}_{\infty }(-)$ is an exact functor. Define

(5.4)\begin{align} \pi^B (\overline{ r}) := (M^{B}_{\infty}/ \frak{m}_{\infty})^{\vee}. \end{align}

By definition, we have

(5.5)\begin{equation} (M^{B}_{\infty}(\sigma) / \frak{m}_{\infty})^{\vee} \cong \operatorname{{\mathrm{Hom}}}_{ \mathcal{O}_{B_v}^{\times} }(\sigma,\pi^B (\overline{ r})). \end{equation}

At the place $v$, let $\tau _v : I_{F_v}\to \mathrm {GL}_2(E)$ be an inertial type and $\mathbf {w} = (a_{\kappa } , b_{\kappa })_{\kappa : F_v \hookrightarrow E}$ be a Hodge type with $a_{\kappa } < b_{\kappa }$ for all $\kappa$. Assume $\tau _v$ is a discrete series inertial type if $B$ ramifies at $v$. Let $R^{\psi \varepsilon ^{-1}}_{v}(\mathbf {w}, \tau _v)$ (respectively, $R^{\psi \varepsilon ^{-1}, {\rm cr}}_{v}(\mathbf {w}, \tau _v)$) denote the reduced $p$-torsion-free quotient of $R_v^{\psi \varepsilon ^{-1}}$ which parametrizes potentially semistable (respectively, potentially crystalline) liftings of $\overline {r}_v$ of Galois type $\tau _v$ and Hodge–Tate weights $\mathbf {w}$. Following [GG15] let $R^{\psi \varepsilon ^{-1}, {\rm ds}}_{v}(\mathbf {w}, \tau _v)$ denote the maximal reduced $p$-torsion-free quotient of $R^{\psi \varepsilon ^{-1}}_{v}(\mathbf {w} ,\tau _v)$ which is supported on the irreducible components where the associated Weil–Deligne representation is generically of discrete series type. Let $R_{\infty }^{ \psi \varepsilon ^{-1}}(\mathbf {w},\tau _{v}):= R_{v}^{\psi \varepsilon ^{-1} }(\mathbf {w},\tau _{v}) \widehat {\otimes }_{R_{v}^{\psi \varepsilon ^{-1}}} R_{\infty }^{ \psi \varepsilon ^{-1}}$, $R_{\infty }^{\psi \varepsilon ^{-1} ,{\rm cr}}(\mathbf {w}, \tau _{v}):= R_{v}^{\psi \varepsilon ^{-1}, {\rm cr}}(\mathbf {w} ,\tau _{v}) \widehat {\otimes }_{R_{v}^{\psi \varepsilon ^{-1}}} R_{\infty }^{ \psi \varepsilon ^{-1}}$ and $R_{\infty }^{ \psi \varepsilon ^{-1}, {\rm ds}}(\mathbf {w},\tau _{v}):= R_{v}^{\psi \varepsilon ^{-1}, {\rm ds}}(\mathbf {w}, \tau _{v}) \widehat {\otimes }_{R_{v}^{\psi \varepsilon ^{-1}}} R_{\infty }^{ \psi \varepsilon ^{-1}}$.

Lemma 5.3

1. (i) If $\tau _v$ is a supercuspidal inertial type, then $R^{\psi \varepsilon ^{-1},{\rm ds}}_{v}(\mathbf {w}, \tau _v) = R^{\psi \varepsilon ^{-1} }_{v} (\mathbf {w},\tau _v)= R^{\psi \varepsilon ^{-1} ,{\rm cr}}_{v}(\mathbf {w}, \tau _v )$.

2. (ii) If $\tau _v$ is a scalar type, then $R^{\psi \varepsilon ^{-1},{\rm ds}}_{v} (\mathbf {w}, \tau _v)$ corresponds to the closure of potentially semistable but not potentially crystalline points in $\operatorname {{\mathrm {Spec}}} R^{\psi \varepsilon ^{-1} }_{v}(\mathbf {w},\tau _v )$.

Proof. See [GG15, § 5].

We assume $B$ ramifies at $v$ and $F_v= \mathbb {Q}_p$ for the rest of this section. Let $\tau _v$ be supercuspidal and $\mathbf {w} = (a,b)$ be as above satisfying

(5.6)\begin{equation} \varepsilon^{b+a-1}|_{I_{F_v}}\det(\tau_v) |_{I_{F_v}} \sim \psi|_{I_{F_v}}. \end{equation}

We have a natural action of $B_{v}^{\times }$ on $\mathrm {Sym}^{b - a -1} E^2 \otimes {\det }^{a}$ as follows: we fix an embedding $B_{v}^{\times } \hookrightarrow \mathrm {GL}_2(\mathbb {Q}_{p^2})$. Then $B_v^{\times }$ acts by the composite $B_{v}^{\times } \hookrightarrow \mathrm {GL}_2(\mathbb {Q}_{p^2}) \hookrightarrow \mathrm {GL}_2(E)$. Let $\Theta$ be any $\mathcal {O}_{B_{v}}^{\times }$-stable $\mathcal {O}$-lattice in

\[ \sigma_{B_v}(\mathbf{w},\tau):=\sigma_{B_v}(\tau_{v}) \otimes \mathrm{Sym}^{b - a -1} E^2 \otimes {\det}^{a}. \]

The homomorphism $R_{\infty }^{\psi \varepsilon ^{-1}} \to \operatorname {{\mathrm {End}}}(M^B_{\infty }(\Theta ))$ factors through $R_{\infty }^{ \psi \varepsilon ^{-1}, {\rm ds}}(\mathbf {w},\tau _{v})$, which is $R_{\infty }^{\psi \varepsilon ^{-1}}(\mathbf {w}, \tau _{v} )$ by Lemma 5.3(i), by the global Jacquet–Langlands correspondence and local–global compatibility. Since $S_{\infty }$ and $R_{\infty }^{ \psi \varepsilon ^{-1}}(\mathbf {w}, \tau _{v})$ have the same Krull dimension, $M^B_{\infty }(\Theta )$ is maximal Cohen–Macaulay over $R_{\infty }^{\psi \varepsilon ^{-1}}(\mathbf {w}, \tau _{v} )$ by the same argument of the proof of [CEG⁺16, Lemma 4.18].

Let $\delta :F^\times \setminus {\mathbb {A}}_{F,f}^\times \to \mathcal {O}^\times$ be a continuous character trivial on $U^{v}\cap {\mathbb {A}}_{F,f}^\times$ and trivial mod $\varpi$. Sending a lifting $r_w$ of $\overline {r}_w$ with determinant $\varepsilon ^{-1}\psi |_{F^\times _w}$ to $r_w\otimes \delta |_{F^{\times }_w}$ gives an isomorphism $\mathrm {tw}_{\delta |_{F_w^{\times }}}: R^{\psi \delta ^2\varepsilon ^{-1}}_w \xrightarrow {\sim } R_w^{\psi \varepsilon ^{-1}}$. We hence have an isomorphism $\mathrm {tw}_{\delta }:= \otimes _w \mathrm {tw}_{\delta |_{F_w^{\times }}}: R_{\infty }^{\psi \delta ^2 \varepsilon ^{-1}} \xrightarrow {\sim } R_{\infty }^{\psi \varepsilon ^{-1}}$. We have the following analogue of Corollary 4.6.

Lemma 5.4 Let $(\mathbf {w},\tau _v)$ be as above satisfying $\varepsilon ^{b+a-1}|_{I_{F_v}}\det (\tau _v) |_{I_{F_v}} \sim (\psi \delta ^2)|_{I_{F_v}}$. Let $\Theta$ be any $\mathcal {O}_{B_{v}}^{\times }$-stable $\mathcal {O}$-lattice in $\sigma _{B_v}(\mathbf {w},\tau _v) \otimes (\delta |_{F_v^{\times }}\circ {\rm Nrd})^{-1}$. Then $R_{\infty }^{\psi \varepsilon ^{-1}}/\operatorname {{\mathrm {Ann}}}_{R_{\infty }^{\psi \varepsilon ^{-1}}}(M^B_{\infty }(\Theta ))$ is equal to $\mathrm {tw}_{\delta } (R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} ))$ if $R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} )$ is an integral domain.

Proof. Note that $M_{\infty }^{B,\delta ^{-1}}: = M_{\infty }^B \otimes \delta ^{-1} \circ {\rm Nrd}$ is a ‘big’ patched module, which is finitely generated and Cohen–Macaulay over $S_{\infty }[\![ \mathcal {O}_{B_v}^{\times } ]\!]$ equipped with a compatible action of $R_{\infty }^{ \psi \delta ^2\varepsilon ^{-1}}$. This is because $\delta$ is trivial mod $\varpi$, and when $B$ is definite, the map $f\mapsto [g\mapsto f(g)(\delta \circ {\rm Nrd})^{-1}(g)]$ induces an isomorphism

(5.7)\begin{equation} \widetilde{H}^{0,B}_{\sigma_p^v , \psi\delta^2} (U(N)^{v}, \mathcal{O}) \cong \widetilde{H}^{0,B}_{\sigma_p^v , \psi} (U(N)^{v}, \mathcal{O})\otimes \delta\circ{\rm Nrd}, \end{equation}

which is compatible with the action of the Hecke algebra. Here $U(N)^v$ denotes the group $U(N)^{\frak {p}}$ in the proof of [DPS23, Theorem 8.10]. The isomorphism similar to (5.7) in the indefinite case follows from the proof of [BDJ10, Lemma 2.3]. Since $\Theta \otimes \delta |_{F_v^{\times }}\circ {\rm Nrd}$ is an $\mathcal {O}_{B_v}^{\times }$-stable lattice in the locally algebraic representation $\sigma _{B_v}(\mathbf {w},\tau _v)$, the action of $R_{\infty }^{ \psi \delta ^2\varepsilon ^{-1}}$ on $M_{\infty }^{B,\delta ^{-1}}(\Theta \otimes \delta |_{F_v^{\times }}\circ {\rm Nrd})$ factors through $R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} )$. Then the $R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} )$-module

\[ M_{\infty}^{B,\delta^{-1}}(\Theta\otimes \delta|_{F_v^{\times}} \circ {\rm Nrd}) = M_{\infty}^B (\Theta ) \otimes \delta^{-1}|_{F_v^{\times}}\circ {\rm Nrd} \]

is supported on a union of irreducible components of $\operatorname {{\mathrm {Spec}}}(R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} ))$, which is all of $\operatorname {{\mathrm {Spec}}}(R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} ))$ if $R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} )$ is an integral domain. The construction of the ‘big’ patched module and the relation with the associated Galois representation imply that the quotient ring $R_{\infty }^{\psi \varepsilon ^{-1}}/\operatorname {{\mathrm {Ann}}}_{R_{\infty }^{\psi \varepsilon ^{-1}}}(M^B_{\infty }(\Theta ))$ is equal to $\mathrm {tw}_{\delta } (R_{\infty }^{\psi \delta ^2\varepsilon ^{-1}}(\mathbf {w}, \tau _{v} ))$.

6. The Gelfand–Kirillov dimension of $\pi ^B (\overline {r})$

In this section, we maintain the assumptions made in § 5. In particular, $B$ and $\overline {r}$ satisfy the compatibility condition (H0) of [BD14], which implies that $\pi ^B(\overline {r})\neq 0$ by [BD14, Corollaire 3.2.3]. Since our main applications are for the quaternion algebra over $\mathbb {Q}_p$, we assume further that $F_v \cong \mathbb {Q}_p$, where $v$ is the unique place over $p$ at which $B$ is ramified. We denote by $D:= B_v$ the quaternion algebra over $\mathbb {Q}_p$. We prove our main results on the Gelfand–Kirillov dimension of $\pi ^B (\overline {r})$ which is defined by (5.4). Assume $p\geq 5$.

6.1 Serre weights for quaternion algebras

Let $W_B(\overline {r})$ denote the set of modular quaternionic Serre weights at $v$ defined in [BD14, § 3.1]. Recall that an irreducible smooth representation of $\mathcal {O}_{D}^{\times }$ over $\mathbb {F}$, equivalently a smooth character $\chi :\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$, is in $W_B(\overline {r})$ if

\[ \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_{D}^{\times}} (\chi, \pi^B(\overline{r})) \neq 0, \]

equivalently $M_{\infty }^B(\chi ) \neq 0$ by (5.5). Moreover, $\dim _{\mathbb {F}} \operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }} (\chi, \pi ^B(\overline {r})) = \dim _{\mathbb {F}} M_{\infty }^B (\chi ) / \frak {m}_{\infty }$.

Let $\overline {\rho }: = \overline {r}_v(1)$. Note that by our assumption $\overline {\rho }$ is a two-dimensional continuous representation of $G_{\mathbb {Q}_p}$. We recall the definition of $W_{D}(\overline {\rho })$, the set of predicted quaternionic Serre weights for $\overline {\rho }$, which is denoted by $W^?(\overline {\rho })$ in [GS11, Definition 3.4]. A character $\psi : \mathcal {O}_D^{\times }\twoheadrightarrow \mathbb {F}_{p^2}^{\times }\to \mathbb {F}^{\times }$ is in $W_D(\overline {\rho })$ if and only if $\overline {\rho }$ has a potentially Barsotti–Tate lift of type $[\psi ] \oplus [\psi ]^{p}$ if $\psi \neq \psi ^p$, and $\overline {\rho }$ has a potentially semistable lift of Hodge–Tate weights $(0,1)$ and type $[\psi ] \oplus [\psi ]$ which is not potentially crystalline if $\psi = \psi ^p$.

We have the following description of the set $W_D(\overline {\rho })$.

Proposition 6.1 Recall $\xi :\mathbb {F}_{p^2}^{\times }\hookrightarrow \mathbb {F}^{\times }$ the character introduced in § 3.2. Let $\zeta$ denote the character $\xi ^{p+1}$, and let $\alpha$ denote the character $\xi ^{p-1}$.

1. (i) Assume $\overline {\rho }$ is in case 1 of § 4.1.1. We have the following.

   1. (a) If $r \!\neq\! 0,p-1$ then $\chi \!\in\! W_D(\overline {\rho })$ if and only if $\chi \in \{ \xi ^r \zeta ^{s+1},\ \xi ^{pr} \zeta ^{s+1},\ \xi ^{r}\alpha ^{-1} \zeta ^{s+1},\ \xi ^{pr} \alpha \zeta ^{s+1}\}$.

   2. (b) If $r = 0$ or $p-1$, then $\chi \in W_D(\overline {\rho })$ if and only if $\chi \in \{\alpha ^{-1} \zeta ^{s+1},\ \alpha \zeta ^{s+1}\}$.

2. (ii) Assume $\overline {\rho }$ is in case 2 of § 4.1.1. We have the following.

   1. (a) If $r=0$, ${\rm unr}_1 = {\rm unr}_2$ and $\overline {\rho }$ is très ramifié, then $\chi \in W_D(\overline {\rho })$ if and only if $\chi = \zeta ^{s+1}$.

   2. (b) If $r=0$, ${\rm unr}_1 = {\rm unr}_2$ and $\overline {\rho }$ is peu ramifié, then $\chi \in W_D(\overline {\rho })$ if and only if $\chi \in \{ \zeta ^{s+1},\ \alpha ^{-1} \zeta ^{s+1},\ \alpha \zeta ^{s+1} \}$.

   3. (c) For other $\overline {\rho }$, $\chi \in W_D(\overline {\rho })$ if and only if $\chi \in \{ \xi ^{r} \alpha ^{-1} \zeta ^{s+1},\ \xi ^{pr}\alpha \zeta ^{s+1} \}$.

3. (iii) Assume $\overline {\rho }$ is in case 3 of § 4.1.1. We have the following.

   1. (a) If $r= 0$ and ${\rm unr}_1 = {\rm unr}_2$, then $\chi \in W_D(\overline {\rho })$ if and only if $\chi \in \{\zeta ^{s+1}, \alpha ^{-1} \zeta ^{s+1}, \alpha \zeta ^{s+1} \}$.

   2. (b) For other $\overline {\rho }$, $\chi \in W_D(\overline {\rho })$ if and only if $\chi \in \{ \xi ^{r} \alpha ^{-1} \zeta ^{s+1},\ \xi ^{pr}\alpha \zeta ^{s+1}\}$.

Proof. This follows from the definition of $W_D(\overline {\rho })$. More precisely, the Breuil–Mézard conjecture [BM02], proved in [Kis09, Paš15, HT15, San16], states exactly when the involved deformation rings are nonzero in terms of $W(\overline {\rho })$ (cf. § 4.1.1). We take the case (ii)(b) as an example, so that $W(\overline {\rho })=\{\sigma _{0,s+1},\sigma _{p-1,s+1}\}$. Up to twist we may assume $s=0$. It follows from the Breuil–Mézard conjecture and Proposition 3.6(ii) that $\overline {\rho }$ has a potentially Barsotti–Tate lift of type $[\xi ^2]\oplus [\xi ^{2p}]$, so we obtain $\alpha ^{-1}\zeta,\alpha \zeta \in W_D(\overline {\rho })$. On the other hand, $\overline {\rho }$ has a potentially semistable lift of Hodge–Tate weights $(0,1)$ and type $[\zeta ]\oplus [\zeta ]$ which is not potentially crystalline (see [BM02, Théorème 1.2]), which gives $\zeta \in W_D(\overline {\rho })$. Conversely, using the Breuil–Mézard conjecture again one checks that these exhaust all the Serre weights in $W_{D}(\overline {\rho })$.

Proposition 6.2 We have $W_B(\overline {r}) = W_D(\overline {\rho })$.

Proof. The inclusion $W_B(\overline {r}) \subseteq W_D(\overline {\rho })$ follows from [GS11, Lemma 3.3]. Note that [GS11] works only with definite $B$ which ramifies at all places above $p$, but the argument also works in our case. By [GS11, Theorem 8.3], the two sets $W_B(\overline {r})$ and $W_D(\overline {\rho })$ are identical in most cases with exception possibly when $\overline {\rho }$ is an unramified twist of $\big (\begin{smallmatrix} \omega & * \\ 0 & 1 \end{smallmatrix}\big ) \otimes \omega ^{s+1}$ and $\chi = \zeta ^{s+1} \in W_D(\overline {\rho })$. In this exceptional case, $\overline {\rho }$ has a potentially semistable lift of type $[\chi ] \oplus [\chi ]$ which is not potentially crystalline. Applying [BD14, Théorème 3.2.2] (by taking $[r_v,N_v] = [[\chi ] \oplus [\chi ], N_v\neq 0]$ there), there is a Hilbert modular form over $F$ of parallel weight $(2,\ldots,2)$ special at $v$ which gives $\overline {r}$. By global Jacquet–Langlands correspondence, as in the proof of [GS11, Lemma 3.3], we have $\chi \in W_B(\overline {r})$.

Remark 6.3 The question of determining the quaternionic Serre weights is first studied by Khare [Kha01]. More precisely, [Kha01, Theorem 7] proves that if $B_0$ denotes the definite quaternion algebra over $\mathbb {Q}$ which is ramified exactly at $p$ and $\infty$, then $W_{B_0}(\overline {r}) = W_D(\overline {\rho })$.

6.2 Lattices in some locally algebraic representations of $\mathcal {O}_D^{\times }$

Let $\chi$ be any character of $\mathcal {O}_D^{\times }$ over $\mathbb {F}$. Recall that $W_{\chi,n}$ denotes $(\operatorname {{\mathrm {Proj}}}_{ \mathbb {F}[\![\mathcal {O}_D^{\times }/Z^1_D]\!]}\chi ) / \frak {m}_{D}^3$ for $n\geq 1$, where $\frak {m}_{D}$ denotes the maximal ideal of the Iwasawa algebra $\mathbb {F}[\![U^1_D / Z^1_D]\!]$. We construct suitable lattices $\mathcal {L}$ in locally algebraic representations of $\mathcal {O}_{D}^{\times }$ over $E$ so that $\mathcal {L} / p\mathcal {L}$ is a quotient of $W_{\chi, 3} = (\operatorname {{\mathrm {Proj}}}_{\mathbb {F}[\![\mathcal {O}_D^{\times }/Z^1_D]\!]}\chi ) / \frak {m}_{D}^3$. The construction of these lattices is much easier than the case considered in § 3.

Recall that $\mathcal {O}_D^{\times }$ embeds into $\mathrm {GL}_2(\mathbb {Z}_{p^2})$ and then embeds into $\mathrm {GL}_2(\mathcal {O})$ via the embedding $\mathrm {GL}_2(\mathbb {Z}_{p^2}) \subset \mathrm {GL}_2(\mathcal {O})$. An explicit embedding is given by (cf. (2.1))

\[ \varpi_D\mapsto \begin{pmatrix}{0} & {1}\\ {p} & {0}\end{pmatrix},\quad a\mapsto \begin{pmatrix}{a} & {0}\\ {0} & {\sigma(a)}\end{pmatrix},\quad a\in\mathbb{Q}_{p^2}. \]

Let $\mathcal {O}_D^{\times }$ act on $\mathrm {Sym}^1\mathcal {O}^2$ and $\mathrm {Sym}^1 E^2$ via the above embedding. Precisely, for $a,b\in \mathbb {Z}_{p^2}$,

(6.1)\begin{equation} (a+\varpi_Db)\cdot X=aX+p\sigma(b)Y,\quad (a+\varpi_Db)\cdot Y=bX+\sigma(a)Y. \end{equation}

Equipped with this action, $\mathrm {Sym}^1 \mathcal {O}^2$ and $\mathrm {Sym}^1 E^2$ are continuous representations of $\mathcal {O}_D^{\times }$. Let $\mathrm {pr}:\mathbb {Q}_p^{\times }\rightarrow 1+p\mathbb {Z}_p$ denote the projection sending $p$ to $1$. Let $\underline {\mathrm {Sym}}^1 \mathcal {O}^2$ (respectively, $\underline {\mathrm {Sym}}^1 E^2$) denote the continuous $\mathcal {O}_D^{\times }$-module $\mathrm {Sym}^1\mathcal {O}^2\otimes (\mathrm {pr}\circ {\rm Nrd}_D)^{-1/2}$ (respectively, $\mathrm {Sym}^1 E^2\otimes (\mathrm {pr}\circ {\rm Nrd}_D)^{-1/2}$), where ${\rm Nrd}_D:D^{\times } \to \mathbb {Q}_p^{\times }$ is the reduced norm. One checks that $Z_D^1$ acts trivially on $\underline {\mathrm {Sym}}^1 E^2$. Note that $(\underline {\mathrm {Sym}}^1 \mathcal {O}^2) /p \cong \mathrm {Sym}^1 \mathbb {F}^2$ with semisimplification $(\mathrm {Sym}^1 \mathbb {F}^2)^{\rm ss} = \chi _1 \oplus \chi _2$, where $\chi _1,\chi _2$ are characters of $\mathcal {O}_D^{\times }$ determined by $\chi _1 (t) = t$, $\chi _2 (t) = t^p$ for all $t \in \mathbb {F}_{p^2}^{\times }$. In particular, $\chi _1=\chi _2\alpha ^{-1}$. By Proposition 2.13 we have

\[ \dim_{\mathbb{F}}\operatorname{{\mathrm{Ext}}}^1_{\mathcal{O}_D^{\times}/Z_D^1}(\chi_1,\chi_2)= \dim_{\mathbb{F}}\operatorname{{\mathrm{Ext}}}^1_{\mathcal{O}_D^{\times}/Z_D^1}(\chi_2,\chi_1)=1, \]

so there exist (up to isomorphism) unique nonsplit extensions $(\chi _1\ \textbf {---}\ \chi _2)$ and $(\chi _2\ \textbf {---}\ \chi _1)$.

Lemma 6.4 There exist $\mathcal {O}_D^{\times }$-stable $\mathcal {O}$-lattices $L,L'$ in $\underline {\mathrm {Sym}}^1 E^2$ such that:

1. (a) $pL\subset L'\subset L$;

2. (b) $L/pL\cong (\chi _1\ \textbf {---}\ \chi _2)$ and $L'/pL'\cong (\chi _2\ \textbf {---}\ \chi _1)$.

Proof. We take $L=\underline {\mathrm {Sym}}^1\mathcal {O}^2=\mathcal {O}( Y \otimes 1)\oplus \mathcal {O} (X\otimes 1)$ and $L'=\mathcal {O} (X\otimes 1)\oplus p\mathcal {O} (Y\otimes 1)$. The properties are easily checked using (6.1).

Let $\chi : \mathcal {O}_D^{\times } \to \mathbb {F}^{\times }$ be a character. Then there exist integers $-2 \leq a \leq p-2$, $b\in \mathbb {Z}$ such that

\[ [\chi] = [\xi]^{a+2 + (p+1)b}, \]

where $[-]$ denotes the Teichmüller lift. We write

(6.2)\begin{equation} \psi_1 := [\chi] = [\xi]^{a+2 + (p+1)b},\quad \psi_{2} := [\xi]^{a+3 + (p+1)(b-1)},\quad \psi_{3} := [\xi]^{a+1 + (p+1)b}. \end{equation}

Let $\Theta _1: = \psi _1$, viewed as an $\mathcal {O}_D^{\times }$-stable lattice in $V_1 := \psi _1 \otimes _{\mathcal {O}} E$. For $i =2,3$, let

(6.3)\begin{equation} V_i: = \underline{\mathrm{Sym}}^1 E^2 \otimes \psi_i. \end{equation}

Note that $Z_D\cap \mathcal {O}_D^{\times }$ acts on $V_i$ by the same character $[\chi ]$. The $\mathcal {O}_D^{\times }$-representations $V_i,~1\leq i\leq 3$ are irreducible and

\[ \overline{V_1} = \chi, \quad \overline{V_2}^{\rm ss} = \chi \oplus \chi \alpha^{-1},\quad \overline{V_3}^{\rm ss} = \chi \oplus \chi \alpha. \]

An analogue of Proposition 3.7 implies that there exists a unique (up to homothety) $\mathcal {O}_D^{\times }$-stable $\mathcal {O}$-lattice in $V_i$, say $\Theta _i$, such that $\mathrm {cosoc}_{\mathcal {O}_D^{\times }}(\Theta _i / p \Theta _i) = \chi$ for $i =2,3$. We have surjective maps

\begin{align*} & r_1 : \Theta_1 \twoheadrightarrow \Theta_1 / p\Theta_1 \cong \chi, \\ & r_i : \Theta_i \twoheadrightarrow \Theta_i / p\Theta_i \twoheadrightarrow \mathrm{cosoc}(\Theta_i / p\Theta_i) \cong \chi,\quad i=2,3. \end{align*}

Let $\Theta _i':=\operatorname {{\mathrm {Ker}}}(r_i)$ for $i=2,3$. Then by Lemma 6.4 we have

\[ \Theta_2'/p\Theta_2'\cong (\chi\ \textbf{---}\ \chi\alpha^{-1}),\quad \Theta_3'/p\Theta_3'\cong (\chi\ \textbf{---}\ \chi\alpha). \]

Since every irreducible representation of $\mathcal {O}_D^{\times }$ over $\mathbb {F}$ is one-dimensional, $\Theta _1 / p\Theta _1$ is killed by $\frak {m}_{D}$, while $\Theta _i / p\Theta _i$ and $\Theta '_i / p\Theta '_i$ are killed by $\frak {m}^2_{D}$ for $i=2,3$. By construction, $\Theta _1$, $\Theta _2$ and $\Theta _3$ are quotients of $\operatorname {{\mathrm {Proj}}}_{ \mathcal {O}[\![\mathcal {O}_D^{\times }/Z^1_D]\!]}\chi$.

We now glue the three lattices $\Theta _1$, $\Theta _2$ and $\Theta _3$. We first glue $\Theta _1$ and $\Theta _2$ along $\chi$, namely define $\Theta$ by the short exact sequence

(6.4)\begin{equation} 0\rightarrow \Theta\rightarrow \Theta_1\oplus \Theta_2 {\buildrel {r_1-r_2} \over \longrightarrow} \chi\rightarrow0. \end{equation}

Proposition 6.5

1. (i) There is a short exact sequence $0\rightarrow \Theta _2'/p\Theta _2'\rightarrow \Theta /p\Theta \rightarrow \chi \rightarrow 0$.

2. (ii) The cosocle of $\Theta /p\Theta$ is isomorphic to $\chi$. Moreover, the cosocle filtration of $\Theta /p\Theta$ is

   \[ \chi\ \textbf{---}\ \chi\alpha^{-1}\ \textbf{---}\ \chi. \]

Proof. Clearly, Lemmas 3.3 and 3.4 remain true if we are considering $\mathcal {O}_D^{\times }$-representations instead of $\mathrm {GL}_2(\mathbb {Z}_p)$-representations. The results follow from them.

Let $r$ denote the map $\Theta \twoheadrightarrow \Theta /p\Theta \twoheadrightarrow \chi$ where the second map is as in Proposition 6.5(i). Denote by $\widetilde {\Theta }$ the lattice in $V_1 \oplus V_2 \oplus V_3$ obtained by gluing $\Theta$ and $\Theta _3$ along $\chi$. Namely, $\widetilde {\Theta }$ is defined by the following short exact sequence

(6.5)\begin{align} 0\rightarrow \widetilde{\Theta}\rightarrow \Theta\oplus \Theta_3 {\buildrel {r-r_3} \over \longrightarrow} \chi\rightarrow0. \end{align}

Proposition 6.6

1. (i) The cosocle of $\widetilde {\Theta }/p\widetilde {\Theta }$ is isomorphic to $\chi$.

2. (ii) We have that $\widetilde {\Theta }/p \widetilde {\Theta }$ is a quotient of $W_{\chi,3}$. More precisely, $\widetilde {\Theta }/p\widetilde {\Theta }$ is isomorphic to $\overline {W}_{\chi,3}:=W_{\chi,3}/(\chi \alpha ^2\oplus \chi \alpha ^{-2})$.

Proof. (i) Note that the cosocle of $\operatorname {{\mathrm {Ker}}}(r)$ is $\chi \oplus \chi \alpha ^{-1}$, while that of $\operatorname {{\mathrm {Ker}}}(r_3)$ is $\chi \alpha$, so the result follows from Lemma 3.4.

(ii) It follows from Lemma 3.3 that there are short exact sequences

\begin{gather*} 0\rightarrow \operatorname{{\mathrm{Ker}}}(r)/p\operatorname{{\mathrm{Ker}}}(r)\rightarrow \widetilde{\Theta}/p\widetilde{\Theta}\rightarrow \Theta_3/p\Theta_3\rightarrow0,\\ 0\rightarrow \Theta_3'/p\Theta_3'\rightarrow \widetilde{\Theta}/p\widetilde{\Theta}\rightarrow \Theta/p\Theta\rightarrow0. \end{gather*}

Using Proposition 6.5(ii), we deduce that $\widetilde {\Theta }/p\widetilde {\Theta }$ admits both the nonsplit extensions $(\chi \alpha ^{-1}\ \textbf {---}\ \chi )$ and $(\chi \alpha \ \textbf {---}\ \chi )$ as quotients. Combining with part (i), this implies that $\widetilde {\Theta }/p\widetilde {\Theta }$ admits a quotient isomorphic to $W_{\chi,2}$; let $\operatorname {{\mathrm {Ker}}}$ be the corresponding kernel. Comparing the Jordan–Hölder factors, we have $(\operatorname {{\mathrm {Ker}}})^{\rm ss}\cong \chi \oplus \chi$. However, we know $\operatorname {{\mathrm {Ext}}}^1_{\mathcal {O}_D^{\times }/Z_D^1}(\chi,\chi )=0$ by Proposition 2.13, hence $\operatorname {{\mathrm {Ker}}} \cong \chi \oplus \chi$. In particular, $\widetilde {\Theta }/p\widetilde {\Theta }$ is killed by $\mathfrak {m}_{D}^3$. The last statement is a consequence of Corollary 2.11.

6.3 The Gelfand–Kirillov dimension

Assume $\overline {\rho } := \overline {r}_v(1)$ is of the form (C1) or (C2) in § 4.3. Recall that for any character $\chi : \mathcal {O}_D^{\times } \to \mathbb {F}^{\times }$, we have constructed $\mathcal {L} \in \{\Theta _1, \Theta _2, \Theta _3, \Theta, \widetilde {\Theta }\}$ such that $\mathrm {cosoc} (\mathcal {L} / p\mathcal {L}) = \chi$. The construction depends on the choice of $(a,b)$ in (6.2). From now on, we assume $\chi \in W_D(\overline {\rho })$, and make our choice of $(a,b)$ as follows:

\[ \left\{{\begin{array}{@{}ll} (a,b) = (r,s) & \text{if}\ \chi = \xi^{r}\alpha^{-1} \zeta^{s+1}; \\ (a,b) = (p-3-r,r+s+1) & \text{if}\ \chi = \xi^{pr} \alpha \zeta^{s+1}; \\ (a,b) = (r - 2,s+1) & \text{if}\ \chi = \xi^{r} \zeta^{s+1}; \\ (a,b) = (p-1-r,r+s) & \text{if}\ \chi = \xi^{pr} \zeta^{s+1}. \end{array}}\right. \]

Let $\psi _i$ be given by (6.2) for $i=0,1,2$. Then one may check directly that $\psi _i \neq \psi _i^p$. Let $\tau _i$ be a tame supercuspidal inertial type so that $\sigma (\tau _i) = \Theta (\psi _i)$. Let $\psi :F^\times \setminus {\mathbb {A}}_{F,f}^\times \to \mathcal {O}^\times$ be a continuous character as in § 5 satisfying $\psi |_{Z_D\cap \mathcal {O}_D^{\times }} = \psi _1|_{Z_D\cap \mathcal {O}_D^{\times }}$. For $\mathcal {L} \in \{\Theta _1, \Theta _2, \Theta _3, \Theta, \widetilde {\Theta }\}$, let $I_{\mathcal {L} }: = \operatorname {{\mathrm {Ann}}}_{R_{\infty }^{\psi \varepsilon ^{-1}}}(M_{\infty }^B(\mathcal {L}))$ denote the annihilator of $M_{\infty }^B(\mathcal {L})$ in $R_{\infty }^{\psi \varepsilon ^{-1}}$.

Let $R$ be any commutative ring and $M$ be an $R$-module. Following [BHHMS23, § 8.2] we say $M$ is free of rank $m$ over its scheme-theoretic support if it is isomorphic to $(R/ \operatorname {{\mathrm {Ann}}}_R(M))^m$.

Proposition 6.7 Assume $\chi \in W_B(\overline {r})$. Let $m := \dim _{\mathbb {F}} \operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }} (\chi, \pi ^B(\overline {r}))$. Then the $R_{\infty }^{ \psi \varepsilon ^{-1}}$-module $M_{\infty }^B(\Theta _1)$ (respectively, $M_{\infty }^B(\Theta _2)$, respectively, $M_{\infty }^B(\Theta _3)$) is free of rank $m$ over its scheme-theoretic support. In particular, $I_{\Theta _1} = I_{R_1} R_{\infty }^{ \psi \varepsilon ^{-1}}$, $I_{\Theta _2} = I_{R_2} R_{\infty }^{ \psi \varepsilon ^{-1}}$ and $I_{\Theta _3} = I_{R_3} R_{\infty }^{ \psi \varepsilon ^{-1}}$, where $I_{R_1}$, $I_{R_2}$ and $I_{R_3}$ are given in (4.4).

Proof. Twisting by the cyclotomic character and taking into account the framed variables, we have an isomorphism

\[ R_v^{\psi\varepsilon^{-1}} ( (-1,0),\tau_1(-1)) \cong R_{\overline{\rho}}^{\psi\varepsilon} ((0,1),\tau_1)[\![ X_1,X_2, X_3]\!], \]

where $R_{\overline {\rho }}^{\psi \varepsilon }( (0,1),\tau _1)$ is a regular local ring by Proposition 4.13. Hence, $R_{\infty }^{\psi \varepsilon ^{-1}}( (-1,0),\tau _1(-1))$, as a formal power series ring over $R_v^{\psi \varepsilon ^{-1}}( (-1,0),\tau _1(-1))$, is also a regular local ring. Since $M_{\infty }^B(\Theta _1)$ is finite maximal Cohen–Macaulay over $R_{\infty }^{\psi \varepsilon ^{-1} }( (-1,0),\tau _1(-1))$, the Auslander–Buchsbaum formula implies that $M_{\infty }^B (\Theta _1)$ is finite free over $R_{\infty }^{\psi \varepsilon ^{-1}}( (-1,0),\tau _1(-1))$ of rank $m$ and, hence, $I_{\Theta _1} = I_{R_1} R^{\psi \varepsilon ^{-1}}_{\infty }$.

Assume $\mathcal {L} \in \{\Theta _2, \Theta _3\}$. Let $\delta : \mathbb {Q}_p^{\times } \to \mathcal {O}^{\times }$ be the character sending $x\in \mathbb {Q}_p^{\times }\mapsto \mathrm {pr}(x)^{1/2}\in 1+p\mathbb {Z}_p,$ viewed also as a character of $F^\times \setminus {\mathbb {A}}_{F,f}^\times$. We also view $\delta$ as a character of $G_{\mathbb {Q}_p}$, via the local class field theory. Then $(\mathcal {L}\otimes \delta \circ {\rm Nrd})[1/p]$ is a locally algebraic representation of $\mathcal {O}_{B_v}^{\times }$. By Theorem 4.15 and Lemma 5.4, $R_{\infty }^{ \psi \varepsilon ^{-1} }/I_{\mathcal {L}}$ is a regular local ring. We show as above that $M_{\infty }^B (\mathcal {L})$ is finite free over $R_{\infty }^{\psi \varepsilon ^{-1} }/I_{\mathcal {L}}$ of some rank $n$. It follows from Lemma 5.4 and Corollary 4.6 that $I_{\mathcal {L}}$ has the description as in the statement of the proposition. We are left to prove $n = m$.

If $\operatorname {{\mathrm {JH}}}(\mathcal {L}/p \mathcal {L}) \cap W_D(\overline {\rho }) = \{\chi \}$, then

\[ M_{\infty}^B(\mathcal{L} / p\mathcal{L}) \xrightarrow{\sim} M_{\infty}^B(\chi), \]

which is free of rank $m$ over its scheme-theoretic support. Hence, $n = m$.

Now assume both Jordan–Hölder factors of $\mathcal {L} / p \mathcal {L}$ are in $W_D(\overline {\rho })$. By Proposition 6.1 this can only happen when $\overline {\rho }$ is absolutely irreducible. We assume $\chi = \xi ^{r}\alpha ^{-1} \zeta ^{s+1}$ and $\mathcal {L} = \Theta _3$, the other cases can be handled in the same way. Since $\delta \equiv 1\pmod {\varpi }$ and the following discussion only involves $\mathbb {F}$-representations, the twisting by $\delta$ will not change anything.

Since

\[ \Theta_3 / p \Theta_3 = (\xi^{r} \zeta^{s+1} \ \textbf{---}\ \xi^{r}\alpha^{-1} \zeta^{s+1}). \]

Applying the patching functor $M_{\infty }^B(-)$, we obtain a short exact sequence

(6.6)\begin{equation} 0 \to M_{\infty}^B(\xi^{r} \zeta^{s+1}) \to M_{\infty}^B(\Theta_3 / p \Theta_3) \to M_{\infty}^B(\xi^{r}\alpha^{-1} \zeta^{s+1}) \to 0, \end{equation}

where all the modules in the sequence are finite free over their scheme-theoretic support. We must show the modules have the same rank. For this, we use the knowledge on $\mathrm {GL}_2$-side to study their support.

According to Proposition 3.11, there exists a $K$-stable $\mathcal {O}$-lattice $L$ in $\underline {\mathrm {Sym}}^1 E^2\otimes \Theta (\psi _3)$ such that $L/pL$ is a nonsplit extension of $(\sigma _{p-3-r, r+s+2} \ \textbf {---}\ \sigma _{r, s+1})$ by $( \sigma _{p-1-r, r+s+1} \ \textbf {---}\ \sigma _{r-2, s+2} )$. Let $\sigma ^{\circ }(\tau _1)$ denote the unique (up to homothety) $K$-stable $\mathcal {O}$-lattice in $\Theta (\psi _1)$ so that $\sigma ^{\circ }(\tau _1)/ p \sigma ^{\circ }(\tau _1) = (\sigma _{p-3-r, r+s+2} \ \textbf {---}\ \sigma _{r, s+1})$. Let $\tau$ be a tame supercuspidal inertial type so that there is a $K$-stable $\mathcal {O}$-lattice $\sigma ^{\circ }(\tau )$ of $\sigma (\tau )$ satisfying $\sigma ^{\circ }(\tau ) / p\sigma ^{\circ }(\tau ) = ( \sigma _{p-1-r, r+s+1} \ \textbf {---}\ \sigma _{r-2, s+2} )$. Applying Paškūnas’ functor $M(-)$ in § 4.2.1, we obtain a short exact sequence

(6.7)\begin{equation} 0\to M(\sigma^{\circ}(\tau)/ p \sigma^{\circ}(\tau)) \to M(L/pL) \to M(\sigma^{\circ}(\tau_1)/ p \sigma^{\circ}(\tau_1)) \to 0. \end{equation}

Note that the three $R_{\overline {\rho }}^{\psi \varepsilon }$-modules in the above short exact sequence are all cyclic by Lemma 4.7 and Remark 4.10. Then by Theorem 4.5 we obtain the following short exact sequence:

(6.8)\begin{equation} 0\to R_{\overline{\rho}}^{\psi\varepsilon}( (0,1),\tau) \otimes_{\mathcal{O}} \mathbb{F} \to R_{\overline{\rho}}^{\psi\varepsilon}( (0,2),\tau_3) \otimes_{\mathcal{O}} \mathbb{F} \to R_{\overline{\rho}}^{\psi\varepsilon}( (0,1),\tau_1) \otimes_{\mathcal{O}} \mathbb{F} \to 0. \end{equation}

On the other hand, $\sigma _{r-2,s+2},\sigma _{p-3-r,r+s+2}\notin W(\overline {\rho })$ and the extension $(\sigma _{p-1-r, r+s+1}\ \textbf {---}\ \sigma _{r, s+1})$ occurs in $L/pL$ by Proposition 3.11. Let $\tau '$ denote a tame inertial type so that $\sigma (\tau ')$ is isomorphic to the principal series tame type $I([x]^{s+1}, [x]^{r+s+1})$ defined in Proposition 3.6. Let $\sigma ^{\circ }(\tau ')$ be the unique (up to homothety) $K$-stable $\mathcal {O}$-lattice in $I([x]^{s+1}, [x]^{r+s+1})$ such that $\sigma ^{\circ }(\tau ')/ p \sigma ^{\circ }(\tau ') = (\sigma _{p-1-r, r+s+1}\ \textbf {---}\ \sigma _{r, s+1})$. Then the short exact sequence (6.7) can be identified with the following short exact sequence:

(6.9)\begin{equation} 0\to M(\sigma_{p-1-r, r+s+1}) \to M(\sigma^{\circ}(\tau')/p\sigma^{\circ}(\tau')) \to M(\sigma_{r, s+1}) \to 0. \end{equation}

The short exact sequence (6.8) becomes

(6.10)\begin{align} 0\to R_{\overline{\rho}}^{\psi\varepsilon ,{\rm cr} }( (r+s+1,p+s+1), \mathbf{1}) &\otimes_{\mathcal{O}} \mathbb{F} \to R_{\overline{\rho}}^{\psi\varepsilon}((0,1),\tau') \nonumber\\ &\otimes_{\mathcal{O}} \mathbb{F} \to R_{\overline{\rho}}^{\psi\varepsilon ,{\rm cr}}( (s+1,r+s+2),\mathbf{1}) \otimes_{\mathcal{O}} \mathbb{F} \to 0. \end{align}

By [EGS15, Theorem 7.2.1] $R_{\overline {\rho }}^{\psi \varepsilon }( (0,1),\tau ') \otimes _{\mathcal {O}} \mathbb {F}$ is isomorphic to a formal power series ring over $\mathbb {F}[\![ X,Y]\!]/(XY)$, and $R_{\overline {\rho }}^{\psi \varepsilon,{\rm cr}} ((r+s+1,p+s+1),\mathbf {1})\otimes \mathbb {F}$ (respectively, $R_{\overline {\rho }}^{\psi \varepsilon, {\rm cr}} ( (s+1,r+s+2), \mathbf {1}) \otimes _{\mathcal {O}} \mathbb {F}$) is the quotient of $R_{\overline {\rho }}^{\psi \varepsilon }( (0,1),\tau ') \otimes _{\mathcal {O}} \mathbb {F}$ by $X$ (respectively, $Y$). Therefore, $\operatorname {{\mathrm {Spec}}} (R^{\psi \varepsilon }_{\overline {\rho }} ((0,2) ,\tau _3) \otimes _{\mathcal {O}} \mathbb {F})$ has two irreducible components. By Lemma 5.4 and Corollary 4.6, $\operatorname {{\mathrm {Spec}}}(( R_{\infty }^{ \psi \varepsilon ^{-1} }/I_{\Theta _3}) \otimes _{\mathcal {O}}\mathbb {F})$ also has two irreducible components.

Back to the short exact sequence (6.6). By the discussion of the first paragraph of the proof, $M_{\infty }^B(\xi ^{r} \zeta ^{s+1})$ and $M_{\infty }^B(\xi ^{r}\alpha ^{-1} \zeta ^{s+1})$ are supported on $\operatorname {{\mathrm {Spec}}} (R_{\infty }^{\psi \varepsilon ^{-1}}((-1,0),\tau (-1)) \otimes _{\mathcal {O}} \mathbb {F})$ and $\operatorname {{\mathrm {Spec}}}(R_{\infty }^{\psi \varepsilon ^{-1}}( (-1,0),\tau _1(-1)) \otimes _{\mathcal {O}} \mathbb {F})$, respectively. Hence, $M_{\infty }^B(\xi ^{r} \zeta ^{s+1})$ and $M_{\infty }^B(\xi ^{r}\alpha ^{-1} \zeta ^{s+1})$ are supported on different irreducible components of $\operatorname {{\mathrm {Spec}}}((R_{\infty }^{ \psi \varepsilon ^{-1} }/I_{\Theta _3}) \otimes _{\mathcal {O}}\mathbb {F})$. We deduce that

(6.11)\begin{align} {\rm rank}_{R_{\infty}^{ \psi\varepsilon^{-1} }((-1,0),\tau_1(-1)) \otimes_{\mathcal{O}} \mathbb{F}} (M_{\infty}^B(\xi^{r}\alpha^{-1} \zeta^{s+1})) &= {\rm rank}_{R_{\infty}^{ \psi\varepsilon^{-1}}((-1,0),\tau(-1)) \otimes_{\mathcal{O}} \mathbb{F}} (M_{\infty}^B(\xi^{r} \zeta^{s+1})) \nonumber\\ &= {\rm rank}_{(R_{\infty}^{ \psi\varepsilon^{-1} }/I_{\Theta_3}) \otimes_{\mathcal{O}}\mathbb{F}} (M_{\infty}^B(\Theta_3 / p \Theta_3)). \end{align}

Consequently $m=n$.

Corollary 6.8 For any $\chi _1, \chi _2 \in W_D(\overline {\rho })$, we have

\[ \dim_{\mathbb{F}}\operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}(\chi_1,\pi^B(\overline{r}))= \dim_{\mathbb{F}}\operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}(\chi_2,\pi^B(\overline{r})). \]

Proof. In view of (6.11), it remains to treat the case $\chi _2 = \chi _1^p$, equivalently $\chi _2$ is equal to the conjugation of $\chi _1$ by $\varpi _D$. But this is clear since $\pi ^B(\overline {r})$ is a representation of $D^{\times }$, hence is stable under taking the conjugation by $\varpi _D$. See also [GS11, Lemma 2.3] which is based on an observation of Serre.

Theorem 6.9 The $R_{\infty }^{ \psi \varepsilon ^{-1}}$-module $M_{\infty }^B(\widetilde {\Theta })$ is free of rank $m$ over its scheme-theoretic support with $I_{\widetilde {\Theta }} = I_{\Theta _1}\cap I_{\Theta _2} \cap I_{\Theta _3}$.

Proof. We first prove $M_{\infty }^B(\Theta )$ is free of rank $m$ over its scheme-theoretic support with $I_{\Theta } = I_{\Theta _1} \cap I_{\Theta _2}$. By the short exact sequence (6.4) and the exactness of the functor $M_{\infty }^B(-)$, we have

\[ M_{\infty}^B(\Theta) \xrightarrow{\sim} M_{\infty}^B(\Theta_1 )\times_{M_{\infty}^B(\Theta_1/p \Theta_1 )} M_{\infty}^B( \Theta_2). \]

As for a commutative ring $A$ and two ideals $I_1,I_2 \subset A$,

\[ A/ I_1\cap I_2 \cong A/I_1 \times_{A/ (I_1+I_2)} A/I_2, \]

by Proposition 6.7 we are reduced to checking

\[ I_{\Theta_1} + I_{\Theta_2} = (p,I_{\Theta_1}) = \operatorname{{\mathrm{Ann}}}_{R_{\infty}^{ \psi\varepsilon^{-1}}} (M_{\infty}^B (\Theta_1/ p \Theta_1)). \]

Using Corollary 4.11(i) and Proposition 6.7 again, we have

\[ I_{\Theta_1} + I_{\Theta_2} = (I_{R_1 } + I_{R_2}) R_{\infty}^{ \psi\varepsilon^{-1}} = (p, I_{R_1}) R_{\infty}^{ \psi\varepsilon^{-1}} = (p, I_{\Theta_1}) = \operatorname{{\mathrm{Ann}}}_{R_{\infty}^{ \psi\varepsilon^{-1}}}(M_{\infty}^B (\Theta_1/ p \Theta_1)). \]

In particular, we obtain

(6.12)\begin{equation} I_{\Theta}=I_{\Theta_1} \cap I_{\Theta_2} = I_{R_1} R_{\infty}^{ \psi\varepsilon^{-1}} \cap I_{R_2} R_{\infty}^{ \psi\varepsilon^{-1}} =(I_{R_1} \cap I_{R_2}) R_{\infty}^{\psi\varepsilon^{-1}} = I_R R_{\infty}^{ \psi\varepsilon^{-1}}, \end{equation}

where the third equality holds by [Mat89, Theorem 7.4(ii)] because $R_{\infty }^{\psi \varepsilon ^{-1}}$ is flat over $R_{\overline {\rho }}^{\psi \varepsilon }$.

Now we prove $M_{\infty }^B(\widetilde {\Theta })$ is free of rank $m$ over its scheme-theoretic support. Using (6.5) we have similarly

\[ M_{\infty}^B(\widetilde{\Theta}) \xrightarrow{\sim} M_{\infty}^B(\Theta ) \times_{M_{\infty}^B(\Theta_1/p \Theta_1 )} M_{\infty}^B( \Theta_3) \]

and it suffices to check

\[ I_{\Theta} + I_{\Theta_3} = \operatorname{{\mathrm{Ann}}}_{R_{\infty}^{ \psi\varepsilon^{-1}}} (M_{\infty}^B (\Theta_1/ p \Theta_1)) = (p, I_{\Theta_1}) . \]

This easily follows from (6.12) and Corollary 4.11(ii).

Corollary 6.10 For any $\chi \in W_D(\overline {\rho })$, the natural inclusion

(6.13)\begin{equation} \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}} (\chi,\pi^B (\overline{r}) ) \hookrightarrow \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}} (\overline{W}_{\chi, 3 },\pi^B (\overline{r}) ) \end{equation}

is an isomorphism, where the structure of $\overline {W}_{\chi, 3 }$ is given in Proposition 6.6.

Proof. The mod $p$ reduction of the lattice $\widetilde {\Theta }$ is isomorphic to $\overline {W}_{\chi,3}$ by Proposition 6.6. The result then follows from Theorem 6.9.

The main result of this section is the following.

Theorem 6.11 Maintain all the assumptions we have made on $F$, $B$, and $\overline {r}$. Assume $\overline {\rho } = \overline {r}_v(1)$ satisfies (C1) or (C2) in § 4.3. Then $\dim _{\mathcal {O}_D^{\times }} (\pi ^B (\overline {r})) = 1$.

Proof. Since $\pi ^B (\overline {r})$ is of infinite dimension over $\mathbb {F}$ by [BD14, Corollary 3.2.4] (or [Scho18, Theorem 7.8]), $\dim _{\mathcal {O}_D^{\times }} (\pi ^B (\overline {r}))$ is at least one. The other inequality follows from Corollaries 6.10 and 2.12.

Remark 6.12 Although we have excluded the case $r=0$ in (C2), this case (at least when $B$ is indefinite) can be deduced from the case $r=p-3$. The proof uses Scholze's functor introduced in [Scho18] and the mod $p$ local–global compatibility (à la Emerton), see Corollary 7.9.

6.4 The graded module $\operatorname {{\mathrm {gr}}}(\pi ^B(\overline {r})^{\vee })$

Following [BHHMS21, § 3.1], we consider the category $\mathcal {C}$ of admissible smooth representations $\pi$ of $D^{\times }$ over $\mathbb {F}$ with a central character and such that there exists a good filtration on the $\pi ^{\vee }$ such that the $\operatorname {{\mathrm {gr}}}\mathbb {F}[\![U_D^1/Z_D^1]\!]$-module $\operatorname {{\mathrm {gr}}} (\pi ^{\vee })$ is annihilated by some power of the ideal $(yz,zy)$, where $y=y_0$, $z=z_0$ are as in § 2.3.⁶ It is clear that $\mathcal {C}$ is an abelian category and is stable under subquotients and extensions.

Definition 6.13 For each $\chi \in W_D(\overline {\rho })$, we define an ideal $\mathfrak {a}(\chi )$ of $\mathbb {F}[y,z]$ as follows.

• • If $\chi \alpha ^{-1}\in W_D(\overline {\rho })$, then $\mathfrak {a}(\chi ):=(y)$; if $\chi \alpha \in W_D(\overline {\rho })$, then $\mathfrak {a}(\chi ):=(z)$.

• • If neither of $\chi \alpha$, $\chi \alpha ^{-1}$ lies in $W_D(\overline {\rho })$, then $\mathfrak {a}(\chi ):=(yz)$.

Theorem 6.14 Maintain all the assumptions we have made on $F$, $B$ and $\overline {r}$. Assume $\overline {r}_v$ satisfies (C1) or (C2) in § 4.3. Then there exists a surjective graded morphism

\[ \bigg(\bigoplus_{\chi\in W_D(\overline{\rho})}\chi^{\vee}\otimes \mathbb{F}[y,z]/\mathfrak{a}(\chi)\bigg)^{\oplus m} \twoheadrightarrow \operatorname{{\mathrm{gr}}}(\pi^B(\overline{r})^{\vee}), \]

where the integer $m$ is as in Proposition 6.7.

Proof. This is an easy consequence of Corollary 6.10.

7. Application to Scholze's functor

7.1 Results of Scholze and Paškūnas

Let $L$ be a finite extension of $\mathbb {Q}_p$, $G: = \mathrm {GL}_n(L)$ and $G_L= \operatorname {{\mathrm {Gal}}}(\overline {L}/L)$. Let $D$ be the central division algebra over $L$ of dimension $n^2$ and invariant $1/n$. To any $\pi \in \operatorname {\mathrm {Mod}}_G^{\rm adm}(\mathcal {O})$, Scholze [Scho18] associated a Weil-equivariant sheaf $\mathcal {F}_{\pi }$ on the étale site of the adic space $\mathbb {P}^{n-1}_{\mathbb {C}_p}$. We collect some results of Scholze [Scho18] and Paškūnas [Paš22].

Theorem 7.1 Let $\pi \in \operatorname {\mathrm {Mod}}_{G}^{\rm adm}(\mathcal {O})$.

1. (i) For any $i\geq 0$ the étale cohomology group $H^i_{\mathrm {\acute {e}t}}(\mathbb {P}^{n-1}_{\mathbb {C}_p}, \mathcal {F}_{\pi })$ carries a continuous $G_L \times D^{\times }$-action. Moreover, the restriction of $H^i_{\mathrm {\acute {e}t}}(\mathbb {P}^{n-1}_{\mathbb {C}_p}, \mathcal {F}_{\pi })$ to $D^{\times }$ is an admissible smooth representation of $D^{\times }$.

2. (ii) We have $H^i_{\mathrm {\acute {e}t}}(\mathbb {P}^{n-1}_{\mathbb {C}_p}, \mathcal {F}_{\pi }) = 0$ for $i > 2(n-1)$.

3. (iii) Assume $\pi$ admits central character $\psi : Z_G \to \mathbb {F}^{\times }$. If $\pi$ is injective in $\operatorname {\mathrm {Mod}}_{\mathrm {GL}_n(\mathcal {O}_L),\psi }^{\rm adm}(\mathcal {O})$, then $H^i_{\mathrm {\acute {e}t}}(\mathbb {P}^{n-1}_{\mathbb {C}_p}, \mathcal {F}_{\pi }) = 0$ for $i > n-1$.

4. (iv) The natural map

   \[ H^0_{\mathrm{\acute{e}t}}(\mathbb{P}^{n-1}_{\mathbb{C}_p}, \mathcal{F}_{\pi^{\mathrm{SL}_n(L)}}) \hookrightarrow H^0_{\mathrm{\acute{e}t}}(\mathbb{P}^{n-1}_{\mathbb{C}_p}, \mathcal{F}_{\pi}) \]
   is an isomorphism. In particular, if $\pi ^{\mathrm {SL}_n(L)} = 0$, then $H^0_{\mathrm {\acute {e}t}}(\mathbb {P}^{n-1}_{\mathbb {C}_p}, \mathcal {F}_{\pi }) = 0$.

5. (v) If $\pi = \mathbf {1}_{G}$ is the trivial representation of $G$ over $\mathbb {F}$, then

   \[ H^i_{\mathrm{\acute{e}t}}(\mathbb{P}^{n-1}_{\mathbb{C}_p}, \mathcal{F}_{\mathbf{1}_G}) = \left\{{\begin{array}{@{}ll} \omega^{-i/2}\otimes \mathbf{1}_{D^{\times}} & \text{if}\ i\ \text{is even and}\ 0\leq i \leq 2(n-1); \\ 0 & \text{if}\ i\ \text{is odd.} \end{array}}\right. \]

Proof. Parts (i) and (ii) are proved in [Scho18, Theorem 3.2].

For part (iii),⁷ it is proved in [Scho18, Theorem 3.2] that if $\pi \in \operatorname {\mathrm {Mod}}_{G}^{\rm adm}(\mathcal {O})$ such that $\pi |_{\mathrm {GL}_n(\mathcal {O}_L)}$ is injective, then $H^i_{\mathrm {\acute {e}t}}(\mathbb {P}^{n-1}_{\mathbb {C}_p}, \mathcal {F}_{\pi }) = 0$ for $i > n-1$. We need to prove a similar result for $\pi$ which admits a central character. Twisting by a character, we may assume the central character of $\pi$ is trivial. Examining the proof of [Scho18, Theorem 3.2], it suffices to show that $\mathcal {M}_{\infty }/L^{\times }$ is a perfectoid space, where $\mathcal {M}_{\infty }$ is the infinite level Lubin–Tate space. Passing to the connected components, it suffices to show that $\mathcal {M}^{(0)}_{\infty }/\mathcal {O}_L^{\times }$ is a perfectoid space, where $\mathcal {M}^{(0)}_{\infty }$ is the perfectoid space denoted by $\mathcal {M}^{(0)}_{\bf 1}$ in [JLH21, § 4.1]. We will deduce this from [JLH21, Proposition 4.1.1] which proves that $\mathcal {M}^{(0)}_{\infty }/P(\mathcal {O}_L)$ is a perfectoid space, where $P\subset \mathrm {GL}_n$ is the parabolic subgroup of block form $(n-1,1)$.

We use freely the notation of [JLH21, § 4.1]. Note that $\mathcal {M}_{\infty }$ is denoted by $\mathcal {M}_{\bf 1}$ in [JLH21, § 4.1]. Let $H\subseteq \mathrm {GL}_n(\mathcal {O}_L)$ be a closed subgroup. As in [JLH21, § 4.1], we set

\[ \mathcal{M}^{(0)}_{H} : =\varprojlim_{U\supseteq H} (\mathcal{M}_U^{(0)})^{\diamondsuit}, \]

where $U$ ranges over open subgroups of $\mathrm {GL}_n(\mathcal {O}_L)$ containing $H$. By [JLH21, Proposition 4.1.1], $\mathcal {M}^{(0)}_{P(\mathcal {O}_L)}$ is the quotient $\mathcal {M}^{(0)}_{\infty }/ P(\mathcal {O}_L)$ in Huber's category $\mathcal {V}$, and is a perfectoid space. Now assume $H\subset \mathrm {GL}_n(\mathcal {O}_L)$ is a closed subgroup contained in $P(\mathcal {O}_L)$. The same argument as in the proof of [JLH21, Proposition 3.2.1] shows that $\mathcal {M}^{(0)}_{H}$ is a perfectoid space. More precisely, if $H$ is of finite index in $P(\mathcal {O}_L)$, then $\mathcal {M}^{(0)}_{H}$ is finite étale over $\mathcal {M}^{(0)}_{P(\mathcal {O}_L)}$, and the result then follows. In general, $\mathcal {M}^{(0)}_{H} = \varprojlim _{H'} \mathcal {M}^{(0)}_{H'}$ where $H'$ ranges over closed subgroups with $H\subseteq H' \subseteq P(\mathcal {O}_L)$ and $H'\subseteq P(\mathcal {O}_L)$ has finite index, and the result follows. We can then argue as in the proof of [JLH21, Lemma 3.3.4] to prove that $\mathcal {M}^{(0)}_{\infty }$ is an $H$-torsor over $\mathcal {M}^{(0)}_{H}$. Finally, it follows from [JLH21, Lemma 3.3.5] that $\mathcal {M}^{(0)}_{H}$ is the quotient $\mathcal {M}^{(0)}_{\infty }/ H$ in Huber's category $\mathcal {V}$. We finish the proof by taking $H = \mathcal {O}_L^{\times }$.

Part (iv) is proved in [Scho18, Proposition 4.7]. For part (v), we note that $\mathcal {F}_{\mathbf {1}_G}$ is the trivial local system on $\mathbb {P}^{n-1}_{\mathbb {C}_p}$. It follows from [Hub96, Theorem 3.8.1] that the cohomology of $\mathbb {P}^{n-1}_{\mathbb {C}_p}$ (with the Galois action) is as in the classical case. As $D^{\times }$ acts on $\mathbb {P}^{n-1}_{\mathbb {C}_p}$ via an embedding $D^{\times }\hookrightarrow \mathrm {GL}_n(L^{\rm un})$, $D^{\times }$ acts trivially on the cohomology.

Let $\pi$ be a locally admissible $\mathcal {O}$-torsion representation of $G$. The construction of the sheaf $\mathcal {F}_{\pi }$ in [Scho18, Proposition 3.1] extends to such $\pi$. Write $\pi = \varinjlim _{\pi '} \pi '$, where the limit is taken over all admissible subrepresentations of $\pi$. By [Paš22, (9)], we have

\[ H^i_{\mathrm{\acute{e}t}}(\mathbb{P}^{n-1}_{\mathbb{C}_p}, \mathcal{F}_{\pi}) = \varinjlim_{\pi'}H^i_{\mathrm{\acute{e}t}}(\mathbb{P}^{n-1}_{\mathbb{C}_p}, \mathcal{F}_{\pi'}) . \]

We denote by $\mathcal {S}^i$ the cohomological covariant $\delta$-functor

\[ \mathcal{S}^i:\operatorname{\mathrm{Mod}}_G^{\rm l.adm}(\mathcal{O}) \to \operatorname{\mathrm{Mod}}_{G_{L}\times D^{\times}}^{\rm l.adm}(\mathcal{O}),\quad \pi \mapsto H^i_{\mathrm{\acute{e}t}} (\mathbb{P}^1_{\mathbb{C}_p}, \mathcal{F}_{\pi}), \]

where $\operatorname {\mathrm {Mod}}_{G_{L}\times D^{\times }}^{\rm l.adm}(\mathcal {O})$ is the category of locally admissible representations of $D^{\times }$ on $\mathcal {O}$-torsion modules equipped with a continuous commuting $G_{L}$-action. As in [Paš22], it is more convenient to work on pseudo-compact modules rather than smooth representations via the Pontryagin duality. Namely, we consider the covariant homological $\delta$-functor $\{\check {\mathcal {S}}^i \}_{i \geq 0}$ defined by

\[ \check{\mathcal{S}}^i : \frak{C}_G(\mathcal{O}) \to \frak{C}_{G_{L}\times D^{\times}}(\mathcal{O}),\quad M \mapsto H^i_{\mathrm{\acute{e}t}} (\mathbb{P}^1_{\mathbb{C}_p}, \mathcal{F}_{M^{\vee}})^{\vee}. \]

If $R$ is a complete local noetherian $\mathcal {O}$-algebra with residue field $\mathbb {F}$, we extend the $\delta$-functor $\check {\mathcal {S}}^i$ to $\frak {C}_G(R)$ (defined in § 4.4) in a similar way.

7.2 Local–global compatibility (à la Scholze)

From now on, we follow the notation of § 5. Let $B$ be an indefinite quaternion algebra over the totally field $F$ such that $B$ is ramified at the fixed place $v$ above $p$. Let $B'$ be the definite quaternion algebra over $F$ which splits at $v$ and has the same ramification behavior as $B$ at all the other finite places. Fix an isomorphism $B^{\times }(\mathbb {A}_{F,f}^{v}) \cong B'^{\times }(\mathbb {A}_{F,f}^{v})$. Fix an open compact subgroup $U^{v}\subset B^{\times }(\mathbb {A}_{F,f}^{v}) \cong B'^{\times }(\mathbb {A}_{F,f}^{v})$. Let $\overline {r} : G_{F} \to \mathrm {GL}_2(\mathbb {F})$ be a modular Galois representation and let $\frak {m}_{\overline {r}}$ be the non-Eisenstein maximal ideal associated to $\overline {r}$ as in § 5. Let $\sigma _p^{v}$ be the finite $\mathcal {O}[\![U^{v}]\!]$-module as in (5.1). If $A$ is a topological $\mathcal {O}$-algebra, let

\begin{align*} \widetilde{S}_{\sigma_p^v, \psi} (U^{v} , A) &:= \widetilde{H}^{0,B'}_{\sigma_p^v , \psi} (U^{v}, \mathcal{O}) \otimes_{\mathcal{O}} A,\\ \widetilde{H}^i_{\sigma_p^{v}, \psi} (U^{v} , A) &:= \widetilde{H}^{i,B}_{\sigma_p^v , \psi} (U^{v}, \mathcal{O}) \otimes_{\mathcal{O}} A,\quad i\geq 0. \end{align*}

The Hecke algebra ${\mathbb {T}}(U^v)_{\frak {m}_{\overline {r}}}$ acts faithfully and continuously on $\widetilde {S}_{\sigma _p^v, \psi } (U^{v}, \mathcal {O})_{\frak {m}_{\overline {r}}}$ and $\widetilde {H}^i_{\sigma _p^{v}, \psi } (U^{v}, \mathcal {O})_{\frak {m}_{\overline {r}}}$ (see [Scho18, Corollary 7.3]).

Let $M_{\infty }$ denote the big patched module $M_{\infty }^{B'}$ in § 5, so that

\[ M_{\infty}/\mathfrak{a}_{\infty}\cong \widetilde{S}_{\sigma_p^v,\psi}(U^v,\mathcal{O})_{\mathfrak{m}_{\overline{r}}}^d \quad \text{and}\quad M_{\infty}/\mathfrak{m}_{\infty}\cong \widetilde{S}_{\sigma_p^v,\psi}(U^v,\mathbb{F})[\mathfrak{m}_{\overline{r}}]^{\vee}, \]

where $\mathfrak {a}_{\infty }$ denotes the ideal $(z_1,\ldots,z_q,y_1,\ldots,y_j)$ of $S_{\infty }$ and $\mathfrak {m}_{\infty }$ denotes the maximal ideal of $R_{\infty }^{\psi \varepsilon ^{-1}}$.

On the other hand, let $N_{\infty }$ be the variant of $M_{\infty }^B$ which is obtained by the same patching process as $M_{\infty }^{B}$, but without ‘factorizing out’ the Galois representation, see Remark 5.2. Similarly to (5.2) we have

\[ N_{\infty} / \frak{a}_{\infty} = \widetilde{H}^{1}_{\sigma_p^v, \psi} (U^{v}, \mathcal{O})^d_{\frak{m}_{\overline{r}}},\quad N_{\infty}/\mathfrak{m}_{\infty}\cong \widetilde{H}^{1}_{\sigma_p^v, \psi} (U^{v}, \mathbb{F})[\frak{m}_{\overline{r}}]^{\vee}. \]

Theorem 7.2 Denote the restriction of $\psi$ to $F_v^{\times }$ again by $\psi$. Let $G := \mathrm{GL}_2 (F_v)$.

1. (i) We have that $\widetilde {S}_{\sigma _p^v,\psi }(U^{v}, E/\mathcal {O})_{\frak {m}_{\overline {r}}}$ lies in $\operatorname {\mathrm {Mod}}_{G,\psi }^{\rm adm}(\mathcal {O})$, and its restriction to $K:= \mathrm {GL}_2(\mathcal {O}_{F_v})$ is injective in $\operatorname {\mathrm {Mod}}_{K,\psi }^{\rm sm}(\mathcal {O})$. Equivalently by taking dual, $\widetilde {S}_{\sigma _p^v ,\psi }(U^{v} , \mathcal {O})^d_{\frak {m}_{\overline {r}}}$ is finitely generated over $\mathcal {O}[\![K]\!]$ and is projective in $\operatorname {\mathrm {Mod}}_{K,\psi }^{\rm pro}(\mathcal {O})$.

2. (ii) We have that $\widetilde {H}^1_{\sigma _p^v, \psi } (U^{v} , E / \mathcal {O})_{\frak {m}_{\overline {r}}}$ lies in $\operatorname {\mathrm {Mod}}_{D^{\times },\psi }^{\rm adm}(\mathcal {O})$, and its restriction to $\mathcal {O}_{D}^{\times }$ is injective in $\operatorname {\mathrm {Mod}}_{ \mathcal {O}_{D}^{\times },\psi }^{\rm sm}(\mathcal {O})$. Equivalently, $\widetilde {H}^1_{\sigma _p^v, \psi } (U^{v}, \mathcal {O})^d_{\frak {m}_{\overline {r}}}$ is a finitely generated $\mathcal {O}[\![\mathcal {O}_{D}^{\times }]\!]$-module and is projective in $\operatorname {\mathrm {Mod}}_{\mathcal {O}_{D}^{\times } ,\psi }^{\rm pro}(\mathcal {O})$.

3. (iii) For $0\leq i\leq 2$, there is a canonical isomorphism of ${\mathbb {T}}(U^{v})_{\frak {m}_{\overline {r}}}[G_{F_v} \times D^{\times }]$-modules

   \[ \check{\mathcal{S}}^i (\widetilde{S}_{\sigma_p^v, \psi}(U^{v} , \mathcal{O})^d_{\frak{m}_{\overline{r}}}) \cong \widetilde{H}^i_{\sigma_p^v, \psi} (U^{v} , \mathcal{O})^d_{\frak{m}_{\overline{r}}}. \]

4. (iv) There is a canonical $R_{\infty }^{\psi \varepsilon ^{-1}}[G_{F_{v}} \times D^{\times }]$-equivariant isomorphism

   \[ \check{\mathcal{S}}^1 ( M_{\infty} ) = N_{\infty}. \]

Proof. Part (i) is [Paš22, Lemma 5.3, Proposition 5.4]. Part (ii) is proved in [New13, Proposition 5.6] and [Paš22, Proposition 6.4]. Part (iii) is [Paš22, Proposition 6.3]. Part (iv) follows from (the proof of) [Scho18, Corollary 9.3], see [DPS23, Theorem 8.10 (4)] for details.

Lemma 7.3 We have $\check {\mathcal {S}}^0(\widetilde {S}_{\sigma _p^v , \psi } (U^{v}, \mathcal {O})^d_{\frak {m}_{\overline {r}}})=0$ and $\check {\mathcal {S}}^0(M_{\infty })=0$.

Proof. The first statement is a direct consequence of Theorem 7.2(iii) because $\widetilde {H}^0_{\sigma _p^v, \psi } (U^{v} , \mathcal {O})^d_{\frak {m}_{\overline {r}}}=0$ (as $\frak {m}_{\overline {r}}$ is non-Eisenstein). The second statement follows from this and the patching construction (cf. [Scho18, Corollary 9.3]).

Define

\[ \pi^{B'}(\overline{r}):= (M_{\infty}/\mathfrak{m}_{\infty})^{\vee},\quad \pi^{B} (\overline{r}): = \operatorname{{\mathrm{Hom}}}_{G_F}(\overline{r}, (N_{\infty}/\mathfrak{m}_{\infty})^{\vee}). \]

Note that $(N_{\infty }/\mathfrak {m}_{\infty })^{\vee }$ is $\overline {r}$-typic, so we have a $G_{F}\times D^{\times }$-equivariant isomorphism $(N_{\infty }/\mathfrak {m}_{\infty })^{\vee }\cong \overline {r}\otimes \pi ^B(\overline {r})$. The following result is motivated by [Paš22, Propositions 3.7, 4.1].

Proposition 7.4 Assume that $R_{v}^{\psi \varepsilon ^{-1}}$ is formally smooth and that $\dim _{K}(\pi ^{B'}(\overline {r}))=[F_v:\mathbb {Q}_p]$. Then $M_{\infty }$ is a flat $R_{\infty }^{\psi \varepsilon ^{-1}}$-module. Moreover, the following statements are equivalent:

1. (i) $\dim _{\mathcal {O}_D^{\times }} (\pi ^B (\overline {r})) = [F_v:\mathbb {Q}_p]$;

2. (ii) $N_{\infty }$ is flat over $R_{\infty }^{\psi \varepsilon ^{-1}}$;

3. (iii) $\mathcal {S}^2 (\pi ^{B'} (\overline {r}) ) = 0$.

Proof. Since $R_{v}^{\psi \varepsilon ^{-1}}$ is formally smooth by assumption, it is isomorphic to a power series ring in $(3 + 3[F_v:\mathbb {Q}_p])$-variables over $\mathcal {O}$. Consequently, $R_{\infty }^{\psi \varepsilon ^{-1}}$ is a regular local ring of Krull dimension equal to $\dim S_{\infty } + 2 [F_v : \mathbb {Q}_p]$.

Since $M_{\infty }$ is finite projective over $S_{\infty }[\![K/Z_1]\!]$, where $Z_1$ is the centre of $K_1$, $\delta _{S_{\infty }[\![K]\!]}(M_{\infty }) = \dim S_{\infty } + \dim _{\mathbb {Q}_p} (K/Z_1)$ by [GN22, Lemma A.15], see § 1.1 for the notation. Since $\dim _{\mathbb {Q}_p}(K/Z_1)=3[F_v:\mathbb {Q}_p]$ and $\dim _K(\pi ^{B'}(\overline {r}))=[F_v:\mathbb {Q}_p]$ by assumption, we deduce

\[ \dim_{K} (\pi^{B'}(\overline{r})) + \dim R_{\infty}^{\psi \varepsilon^{-1}} = \delta_{S_{\infty}[\![K]\!]}(M_{\infty}). \]

It follows from the miracle flatness criterion [GN22, Proposition A.30] that $M_{\infty }$ is flat over $R_{\infty }^{\psi \varepsilon ^{-1}}$.

Now we prove the equivalence between the three statements. The equivalence (i) $\Leftrightarrow$ (ii) is proved as above by replacing $K/Z_1$ by $\mathcal {O}_D^{\times }/Z_D^1$ and noting that $\dim _{\mathbb {Q}_p}\mathcal {O}_D^{\times }/Z_D^1 =3[F_v:\mathbb {Q}_p]$.

We prove part (ii) implies part (iii). Since $R_{\infty }^{\psi \varepsilon ^{-1}}$ is regular, we may choose a regular system of parameters of $\mathfrak {m}_{\infty }$, say $\underline {s}$. Since $M_{\infty }$ is flat over $R_{\infty }^{\psi \varepsilon ^{-1}}$, the Koszul complex $K_{\bullet }(\underline {s}, M_{\infty })$ gives a resolution of $\pi ^{B'}(\overline {r})^{\vee }=M_{\infty }/\mathfrak {m}_{\infty }$:

\[ \cdots\to K_2(\underline{s}, M_{\infty}) {\buildrel {d_2} \over \longrightarrow} K_1(\underline{s}, M_{\infty}) {\buildrel {d_1} \over \longrightarrow} K_0(\underline{s}, M_{\infty}) {\buildrel {d_0} \over \longrightarrow} M_{\infty}/\mathfrak{m}_{\infty} \to 0. \]

It follows from Lemma 7.3(ii) that $\check {\mathcal {S}}^0(K_i(\underline {s},M_{\infty }))=0$ for any $i$, hence $\check {\mathcal {S}}^0(\mathrm {Im}(d_i))=0$ as well. It is then easy to deduce that the sequence

(7.1)\begin{equation} \check{\mathcal{S}}^1(K_2(\underline{s}, M_{\infty})) \to\check{\mathcal{S}}^1(K_1(\underline{s}, M_{\infty})) \to \check{\mathcal{S}}^1(Q) \to 0 \end{equation}

is exact, where $Q:=\mathrm {Im}(d_1)=\operatorname {{\mathrm {Ker}}}(d_0)$. On the other hand, the functor $\check {\mathcal {S}}^1$ is $R_{\infty }^{\psi \varepsilon ^{-1}}$-equivariant, so the complex $\check {\mathcal {S}}^1 (K_{\bullet }(\underline {s}, M_{\infty }))$ is isomorphic to $K_{\bullet }(\underline {s},\check {\mathcal {S}}^1 ( M_{\infty }))$, the Koszul complex with respect to $\underline {s}$ and $\check {\mathcal {S}}^1 (M_{\infty })$. Since $\check {\mathcal {S}}^1 (M_{\infty })\cong N_{\infty }$ is flat over $R_{\infty }^{\psi \varepsilon ^{-1}}$ by part (ii), the complex $\check {\mathcal {S}}^1 (K_{\bullet }(\underline {s}, M_{\infty }))$ is again exact. Together with (7.1) this implies that the map

(7.2)\begin{equation} \check{\mathcal{S}}^1 (Q) \to \check{\mathcal{S}}^1( K_0(\underline{s} , M_{\infty})) \end{equation}

is injective.

The short exact sequence $0\to Q \to K_0(\underline {s}, M_{\infty }) {\buildrel {d_0} \over \longrightarrow} M_{\infty }/\mathfrak {m}_{\infty }\to 0$ induces an exact sequence

\[ \check{\mathcal{S}}^2( K_0(\underline{s}, M_{\infty})) \to \check{\mathcal{S}}^2( M_{\infty}/\mathfrak{m}_{\infty}) \to\check{\mathcal{S}}^1 (Q) \to \check{\mathcal{S}}^1( K_0(\underline{s}, M_{\infty})) \]

in which the first morphism is surjective by the injectivity of (7.2). Since $M_{\infty }^{\vee }|_{K}$ is injective in $\operatorname {\mathrm {Mod}}^{\rm sm}_{K,\psi }(\mathcal {O})$, Theorem 7.1(iii) implies $\check {\mathcal {S}}^2( K_0(\underline {s} , M_{\infty }) )=0$, thus $\check {\mathcal {S}}^2 (M_{\infty }/\mathfrak {m}_{\infty }) = 0$ as required.

We prove part (iii) implies part (ii). This essentially follows from the above argument. Indeed, we deduce from part (iii) the injectivity of (7.2), which together with (7.1) implies the exactness of

\[ \check{\mathcal{S}}^1(K_2(\underline{s} , M_{\infty})) \to\check{\mathcal{S}}^1( K_1(\underline{s} , M_{\infty})) \to \check{\mathcal{S}}^1(K_0(\underline{s} , M_{\infty})).\]

In other words, the Koszul complex $\check {\mathcal {S}}^1 (K_{\bullet }(\underline {s}, M_{\infty }))$ is exact at degree $1$, thus $\underline {s}$ is $N_{\infty }$-regular by a standard argument.

Remark 7.5 Under some (stronger) genericity condition on $\overline {r}|_{G_{F_v}}$, the assumption on $\dim _K(\pi ^{B'}(\overline {r}))$ of Proposition 7.4 is verified in [BHHMS23, HW22].

We recall the following important result of Scholze.

Proposition 7.6 There is a $G_{\mathbb {Q}_p}\times D^{\times }$-equivariant inclusion

\[ \mathcal{S}^1(\pi^{B'}(\overline{r}))\subset (\overline{r}|_{G_{F_v}})\otimes \pi^B(\overline{r}), \]

whose cokernel is annihilated by $(\mathcal {O}_{D}^{\times })_1$, where $(\mathcal {O}_{D}^{\times })_1$ denotes the reduced norm $1$ elements of $\mathcal {O}_D^{\times }$. As a consequence, the cokernel is finite-dimensional over $\mathbb {F}$ and $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1(\pi ^{B'}(\overline {r}))=\dim _{\mathcal {O}_D^{\times }}\pi ^B(\overline {r})$.

Proof. The first assertion is a restatement of [Scho18, Proposition 7.7] and [Paš22, Lemma 6.1]. The second assertion follows from the first (note that we have fixed the central character).

7.3 Local–global compatibility (à la Emerton)

In this subsection, we assume $F_v\cong \mathbb {Q}_p$. Assume $\operatorname {{\mathrm {End}}}_{G_{\mathbb {Q}_p}}(\overline {\rho })= \mathbb {F}$ where $\overline {\rho }=\overline {r}_v(1)$, and let $\pi (\overline {\rho })$ be the admissible smooth representation of $G=\mathrm {GL}_2(\mathbb {Q}_p)$ attached to $\overline {\rho }$ (cf. § 4.2).

Theorem 7.7 We have $\pi ^{B'}(\overline {r})\cong \pi (\overline {\rho })^{\oplus d}$ for some $d\geq 1$.

Proof. If $\overline {\rho }\nsim \big (\begin{smallmatrix} {\chi } & {*}\\ {0} & {\chi \omega }\end{smallmatrix}\big )$ for any character $\chi$, the result is essentially a consequence of [Eme11] (which treats the case of $\mathrm {GL}_{2/\mathbb {Q}}$). In the definite quaternion algebra setting, the proof is carried out in [DLB17, Appendix]. Note that in [DLB17] the quaternion algebra is assumed to be over $\mathbb {Q}$, but the argument goes through in our setting, under the assumption that $F_v$ is isomorphic to $\mathbb {Q}_p$. Another assumption made in [DLB17] is that $\overline {\rho }$ is irreducible, but the only places where this assumption is needed are as follows.

• – Page 403, the proof of Lemma 13.6. In our case, the proof goes through if we replace the vector $v$ (in [DLB17]) by a finite-dimensional subspace which generates $\pi (\overline {\rho })$ over $G$ (compare the proof of [Eme11, Theorem 6.3.12]).

• – Page 404, the proof of Lemma 13.9. To ensure that $r(\mathfrak {p})|_{G_{\mathbb {Q}_p}}$ is absolutely irreducible for $\mathfrak {p}$ in a suitable set $\mathcal {C}$ defined before Lemma 13.8. But this can be avoided by replacing $\mathcal {C}$ by the subset of ‘allowable’ points as in [Eme11, Definition 5.4.7].

• – Page 405, the proof of the injectivity of

  \[ \pi(\overline{\rho})\otimes \operatorname{{\mathrm{Hom}}}_{G}(\pi(\overline{\rho}),\widetilde{S}_{\sigma_p^v, \psi} (U^{v}, \mathbb{F})_{\mathfrak{m}_{\overline{r}}})\rightarrow\widetilde{S}_{\sigma_p^v, \psi} (U^{v} , \mathbb{F})_{\mathfrak{m}_{\overline{r}}}. \]
  This can be proved as in the proof of [Eme11, Theorem 6.4.16] for $\overline {\rho }\nsim \big (\begin{smallmatrix} {\chi } & {*}\\ {0} & {\chi \omega }\end{smallmatrix}\big )$.

If $\overline {\rho }\sim \big (\begin{smallmatrix} {\chi } & {*}\\ {0} & {\chi \omega }\end{smallmatrix}\big )$, the result follows from [CEG⁺18, § 4]. We remark that a multiplicity one assumption is made in [CEG⁺18, § 4], but the necessary modification is given in [GN22, § 5].

Remark 7.8 In fact, [CEG⁺18, Theorem 4.32] and its generalization [GN22, Corollary 5.3.2] prove a much stronger statement than Theorem 7.7. Namely, assuming moreover that $\overline {\rho }\nsim \big (\begin{smallmatrix} {\chi \omega } & {*}\\ {0} & {\chi }\end{smallmatrix}\big )$ for any character $\chi$, there is an isomorphism in $\frak {C}_{G,\psi }(R_{\infty }^{\psi \varepsilon ^{-1}})$

\[ M_{\infty}\cong R_{\infty}^{\psi \varepsilon^{-1}}\widehat{\otimes}_{R_{\overline{\rho}}^{\psi\varepsilon}}N^{\oplus d}, \]

where $N:= N_{\psi }\in \frak {C}_{G,\psi }(\mathcal {O})$ is the object attached to $\overline {\rho }$ in § 4.2.1, and $d$ is the integer in Theorem 7.7.

Corollary 7.9 Maintain the global assumptions we have made in Theorem 6.11, and assume up to twist $\overline {\rho } \sim \big (\begin{smallmatrix} {\mathrm {unr}_1 \omega } & {*}\\ {0} & {\mathrm {unr}_2}\end{smallmatrix}\big )$. Then $\dim _{\mathcal {O}_D^{\times }}(\pi ^B(\overline {r}))=1$.

Proof. As in the proof of Theorem 6.11, it suffices to prove $\dim _{\mathcal {O}_D^{\times }}(\pi ^B(\overline {r}))\leq 1$. We reduce the result to a situation covered by Theorem 6.11.

Let $\overline {\rho }'\sim \big (\begin{smallmatrix} {\mathrm {unr}_2} & {*}\\ {0} & {\mathrm {unr}_1\omega }\end{smallmatrix}\big )$ with $*\neq 0$. Choose a global setup, namely a totally real field $\widetilde {F}$, an indefinite quaternion algebra $\widetilde {B}$ over $\widetilde {F}$ which is ramified at $v$, and a modular absolutely irreducible Galois representation $\overline {r}'$ as in Theorem 6.11, such that $\overline {\rho }'\cong \overline {r}_v'(1)$. Then $\overline {\rho }'$ satisfies (C2) in § 4.3, and so $\dim _{\mathcal {O}_D^{\times }}(\pi ^{\widetilde {B}}(\overline {r}'))=1$ by Theorem 6.11(ii). Combining Theorem 7.7 and Proposition 7.6, we deduce that $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1(\pi (\overline {\rho }'))\leq 1$. The structure of $\pi (\overline {\rho }')$ is recalled in § 4.2. In particular, the set $\operatorname {{\mathrm {JH}}}(\pi (\overline {\rho }'))$ consists of non-supersingular representations. Using Theorem 7.1(iv) and Ludwig's result [Lud17], we have

(7.3)\begin{equation} \dim_{\mathcal{O}_D^{\times}}\mathcal{S}^0(\pi)=\dim_{\mathcal{O}_D^{\times}}\mathcal{S}^2(\pi)=0 \end{equation}

for any non-supersingular irreducible representation $\pi$ of $G$. We deduce that $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1(\pi )\leq 1$ for any $\pi \in \operatorname {{\mathrm {JH}}}(\pi (\overline {\rho }'))$. It is clear from the definition of $\pi (\overline {\rho })$ (see § 4.2 or Proposition 8.14) that $\operatorname {{\mathrm {JH}}}(\pi (\overline {\rho }))$ differs from $\operatorname {{\mathrm {JH}}}(\pi (\overline {\rho }'))$ by at most one-dimensional representations. Hence, using (7.3) we obtain

\[ \dim_{\mathcal{O}_D^{\times}}\mathcal{S}^1(\pi(\overline{\rho}))\leq 1. \]

By Theorem 7.7 and Proposition 7.6 again, this implies $\dim _{\mathcal {O}_D^{\times }}(\pi ^B(\overline {r}))\leq 1$.

7.4 Vanishing for supersingular representations

Ludwig [Lud17] has proved that $\mathcal {S}^2(\pi )=0$ if $\pi$ is a principal series of $\mathrm {GL}_2(\mathbb {Q}_p)$. Together with Theorem 6.11, we deduce the following vanishing result when $\pi$ is supersingular.

Corollary 7.10 Assume that $\pi = \pi (\overline {\rho })$ is supersingular with $2\leq r\leq p-3$ in the notation of (C1) in § 4.3. Then $\mathcal {S}^2(\pi )=0$. Moreover, we have $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1(\pi )=1$.

Proof. As $\overline {\rho }$ is irreducible, the ring $R_{v}^{\psi \varepsilon ^{-1}}$ is formally smooth. It is proved in [Paš22, Lemma 5.16] that $\dim _{K}(\pi )=1$, so the assumptions of Proposition 7.4 hold via Theorem 7.7. The existence of a suitable $B'$ and $\overline {r}$ is well-known; see, for example, [DT94]. Thus, the vanishing of $\mathcal {S}^2(\pi )$ follows from this and Proposition 7.4. Finally, Theorem 6.11 and Proposition 7.6 imply that $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1(\pi ^{B'}(\overline {r}))=1$, hence also $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1(\pi )=1$ via Theorem 7.7 again.

8. Further studies on Scholze's functor

In this section, we study the behavior of $\mathcal {S}^i$ on some non-supersingular representations of $\mathrm {GL}_2(\mathbb {Q}_p)$. Recall that $p\geq 5$.

8.1 Preparations

8.1.1 Some definitions

Recall that $D$ is the nonsplit quaternion algebra over $\mathbb {Q}_p$ and $U_D^1=1+\mathfrak {p}_D$.

Definition 8.1 Given an admissible smooth $\mathbb {F}$-representation $V$ of $D^{\times }$, let $V_{\rm fd}$ be the largest finite-dimensional quotient of $V$.

Remark 8.2 That $V_{\rm fd}$ is well-defined can be seen as follows. Let $V^{\vee }$ be the Pontryagin dual of $V$. Then by the general theory of finitely generated modules over $\mathbb {F}[\![U_D^1]\!]$ (see, e.g., [Ven02, § 3.1]), $V^{\vee }$ has a largest submodule of $\delta$-dimension $0$ (i.e. finite dimensional over $\mathbb {F}$). Clearly this submodule is $D^{\times }$-stable because $V^{\vee }$ carries a compatible action of $D^{\times }$. Taking the dual back gives $V_{\rm fd}$ in Definition 8.1.

We give some basic properties of $(\cdot )_{\rm fd}$. For $i\geq 0$ and $M$ a finitely generated $\mathbb {F}[\![U_D^1]\!]$-module, set

\[ \mathrm{E}^i(-):=\operatorname{{\mathrm{Ext}}}^i_{\mathbb{F}[\![U_D^1]\!]}(-,\mathbb{F}[\![U_D^1]\!]). \]

Note that $\mathrm {E}^i(-)=0$ for $i\geq 5$, as $\mathbb {F}[\![U_D^1]\!]$ is an Auslander regular ring of global dimension $4$. Also recall that $M$ (when it is nonzero) is called Cohen–Macaulay if there exists exactly one $i$ such that $\mathrm {E}^i(M)\neq 0$; in this case $i$ equals to the grade of $M$.

Lemma 8.3 Let $V$ be an admissible smooth $\mathbb {F}$-representation of $D^{\times }$. If $V$ is infinite-dimensional (as an $\mathbb {F}$-vector space) and $V^{\vee }$ is Cohen–Macaulay as an $\mathbb {F}[\![U_D^1]\!]$-module, then $V_{\rm fd}=0$.

Proof. By assumption, $V^{\vee }$ is Cohen–Macaulay with $\dim _{\mathcal {O}_D^{\times }}(V)\geq 1$, thus $\mathrm {E}^4(V^{\vee }) = 0$. If $V_{\rm fd}\neq 0$, then the inclusion $(V_{\rm fd})^{\vee }\hookrightarrow V^{\vee }$ induces a surjection

\[ \mathrm{E}^4(V^{\vee})\rightarrow \mathrm{E}^4((V_{\rm fd})^{\vee})\rightarrow0. \]

This gives a contradiction as $\mathrm {E}^4((V_{\rm fd})^{\vee })\neq 0$.

Lemma 8.4 If $0\rightarrow V'\rightarrow V\rightarrow V''\rightarrow 0$ is a short exact sequence of admissible smooth $\mathbb {F}$-representations of $D^{\times }$, then $(V')_{\rm fd}\rightarrow V_{\rm fd}\rightarrow (V'')_{\rm fd}\rightarrow 0$ is exact. If, moreover, $V''$ is finite-dimensional over $\mathbb {F}$, then

\[ 0\rightarrow (V')_{\rm fd}\rightarrow V_{\rm fd}\rightarrow (V'')_{\rm fd}\rightarrow0 \]

is exact.

Proof. It is obvious from Definition 8.1.

Lemma 8.5 Let $V$ be an admissible smooth $\mathbb {F}$-representation of $D^{\times }$. Assume that $V$ carries an $\mathbb {F}$-linear continuous action of $G_{\mathbb {Q}_p}$ which commutes with the action of $D^{\times }$. Then $V_{\rm fd}$ is also stable under $G_{\mathbb {Q}_p}$.

Proof. Consider the Pontryagin dual $V^{\vee }$, so that $(V_{\rm fd})^{\vee }$ is identified with the largest finite-dimensional submodule of $V^{\vee }$, see Remark 8.2. It suffices to prove the following statement: if $x\in V^{\vee }$ such that $\langle D^{\times }. x\rangle$ is finite-dimensional, then so is $\langle D^{\times }. (gx)\rangle$ for any $g\in G_{\mathbb {Q}_p}$. This is clear because the actions of $G_{\mathbb {Q}_p}$ and of $D^{\times }$ commute.

We now recall the notion of being $\sigma$-typic from [Scho18, Definition 5.2], adapted to our situation. Let $G$ be a group, $\sigma :G\rightarrow \mathrm {GL}_n(\mathbb {F})$ be an $n$-dimensional representation and $M$ an $\mathbb {F}[G]$-module. Then $M$ is said to be $\sigma$-typic if one can write $M$ as a tensor product

\[ M=\sigma\otimes_{\mathbb{F}}M_0, \]

such that $G$ acts on $\sigma \otimes _{\mathbb {F}}M_0$ through its action on $\sigma$.

Lemma 8.6 Assume that $\operatorname {{\mathrm {End}}}_{\mathbb {F}[G]}(\sigma )=\mathbb {F}$.

1. (i) If $M$ is $\sigma$-typic, then $M_0\cong \operatorname {{\mathrm {Hom}}}_{\mathbb {F}[G]}(\sigma,M)$.

2. (ii) Let $M'\subset M$ be $\mathbb {F}[G]$-modules and assume that $M'$ is a direct summand of $M$. If $M$ is $\sigma$-typic, then so is $M'$.

Proof. (i) This follows from the same proof of [Scho18, Proposition 5.3]. In [Scho18, Proposition 5.3] $\sigma$ is assumed to be absolutely irreducible, but in the proof only the assumption $\operatorname {{\mathrm {End}}}_{\mathbb {F}[G]}(\sigma )=\mathbb {F}$ is needed.

(ii) Since $M$ is $\sigma$-typic by assumption, the natural map $\sigma \otimes \operatorname {{\mathrm {Hom}}}_{\mathbb {F}[G]}(\sigma,M)\rightarrow M$ is an isomorphism. Since $M'$ is a direct summand of $M$, the map

\[ \sigma\otimes\operatorname{{\mathrm{Hom}}}_{\mathbb{F}[G]}(\sigma,M')\rightarrow M' \]

is also an isomorphism by functoriality.

8.1.2 Complements on Scholze's functor

Keep the notation from § 7 and assume $F_v\cong \mathbb {Q}_p$. To simplify notation we write

\[ \widetilde{S}(U^v,\mathbb{F})=\widetilde{S}_{\sigma_p^v, \psi} (U^{v}, \mathbb{F}),\quad \widetilde{H}^1(U^{v}, \mathbb{F})= \widetilde{H}^1_{\sigma_p^v, \psi} (U^{v}, \mathbb{F}). \]

It is a consequence of [Scho18, Proposition 5.8] that $\widetilde {H}^1(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}]$ is $\overline {r}$-typic, so

(8.1)\begin{equation} \widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]\cong\overline{r}\otimes \operatorname{{\mathrm{Hom}}}_{G_{F}}(\overline{r},\widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]). \end{equation}

It is proved in [Scho18, Proposition 7.7] and [Paš22, Lemma 6.1] that there is a $G_{\mathbb {Q}_p}\times D^{\times }$-equivariant inclusion

(8.2)\begin{equation} \mathcal{S}^1(\widetilde{S}(U^v,\mathbb{F})[\mathfrak{m}_{\overline{r}}])\subset \widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}], \end{equation}

whose cokernel is finite-dimensional over $\mathbb {F}$, cf. Proposition 7.6.

Corollary 8.7 If $(\widetilde {H}^1(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}])^{\vee }$ is a Cohen–Macaulay $\mathbb {F}[\![U_D^{1}]\!]$-module, then (8.2) becomes an equality.

Proof. Since $\widetilde {H}^1 (U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}]$ is always infinite-dimensional, see [BD14, Corollary 3.2.4] or [Scho18, Theorem 7.8], the assumption implies that $(\widetilde {H}^1(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}])_{\rm fd}=0$ by Lemma 8.3. The result follows.

Remark 8.8 Paškūnas [Paš22, Lemma 6.1] also proves a criterion for (8.2) to be an equality. Corollary 8.7 can be viewed as a complement to it.

Proposition 8.9 Assume that $R_{v}^{\psi \varepsilon ^{-1}}$ is formally smooth and

\[ \dim_{\mathcal{O}_D^{\times}}(\widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}])=1. \]

Then $(\widetilde {H}^1(U^{v}, \mathcal {O})_{\mathfrak {m}_{\overline {r}}})^{d}$ is a faithfully flat $\mathbb {T}(U^v)_{\mathfrak {m}_{\overline {r}}}$-module, and $(\widetilde {H}^1(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}])^{\vee }$ is a Cohen–Macaulay $\mathbb {F}[\![U_D^{1}]\!]$-module. In particular, (8.2) becomes an equality. Moreover,

\[ \mathcal{S}^2(\widetilde{S}(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}])=0. \]

Proof. The faithful flatness is proved by the same argument of [GN22, Theorem B(3)].

Since $N_{\infty }$ is a projective object in $\frak {C}_{\mathcal {O}_{D}^{\times },\psi }(S_{\infty })$, it is a Cohen–Macaulay $S_{\infty }[\![U_D^1]\!]$-module, thus is also Cohen–Macaulay over $R_{\infty }[\![U_D^1]\!]$ by [GN22, Lemma A.29]. The formal smoothness of $R_{v}^{\psi \varepsilon ^{-1}}$ ensures that $R_{\infty }$ is formally smooth, namely its maximal ideal $\mathfrak {m}_{\infty }$ is generated by a regular sequence. By the proof of [GN22, Proposition A.30], $(\widetilde {H}^1(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}])^{\vee }\cong N_{\infty }/\mathfrak {m}_{\infty }$ is a Cohen–Macaulay $\mathbb {F}[\![U_D^1]\!]$-module.

The last assertion follows from Proposition 7.4.

For simplicity and clarity we make the following assumption in §§ 8.2 and 8.3 below. The general case will be treated in § 8.4.

(H) Assume $d=1$ in Theorem 7.7, i.e. $\widetilde {S}(U^v,\mathbb {F})[\mathfrak {m}_{\overline {r}}]\cong \pi (\overline {\rho })$.

For notational convenience, we make the following definition. Let $\overline {\rho } : = \overline {r}_v(1)$.

Definition 8.10 We define

(8.3)\begin{equation} \mathrm{JL}(\overline{\rho}):=\operatorname{{\mathrm{Hom}}}_{G_{F}}(\overline{r},\widetilde{H}^1(U^{v} , \mathbb{F})[\mathfrak{m}_{\overline{r}}]), \end{equation}

which is an admissible smooth $\mathbb {F}$-representation of $D^{\times }$. Then (8.1) restricts to a $G_{\mathbb {Q}_p}\times D^{\times }$-isomorphism

(8.4)\begin{equation} \widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]\cong \overline{\rho}(-1)\otimes \mathrm{JL}(\overline{\rho}). \end{equation}

Finally we recall the following important results which will be repeatedly used later on.

Theorem 8.11 Let $\pi$ be an admissible smooth $\mathbb {F}$-representation of $G$.

1. (i) The natural morphism $\mathcal {S}^0(\pi ^{\mathrm {SL}_2(\mathbb {Q}_p)})\rightarrow \mathcal {S}^0(\pi )$ is an isomorphism.

2. (ii) If $\pi \cong \operatorname {{\mathrm {Ind}}}_{B(\mathbb {Q}_p)}^G\chi$ is a principal series (for some smooth character $\chi :B(\mathbb {Q}_p)\rightarrow \mathbb {F}^{\times }$), then $\mathcal {S}^2(\pi )=0$.

Proof. Part (i) is a special case of Theorem 7.1(iv) and part (ii) is [Lud17, Theorem 4.6].

8.2 The generic case in the minimal case

In this subsection, we assume $\overline {\rho } \sim \big (\begin{smallmatrix} {\chi _1} & {*}\\ {0} & {\chi _2}\end{smallmatrix}\big )$ is reducible nonsplit such that $\chi _1 \chi _2^{-1} \neq \mathbf {1},~\omega ^{\pm 1}$.

Theorem 8.12 Let $\overline {\rho }$ be as above. Then $\mathrm {JL}(\overline {\rho })$ depends only on $\overline {\rho }^{\rm ss}$.

Proof. Write $\overline {\rho }_1$ (respectively, $\overline {\rho }_2$) for the nonsplit extension $\big (\begin{smallmatrix} {\chi _1} & {*}\\ {0} & {\chi _2}\end{smallmatrix}\big )$ (respectively, $\big (\begin{smallmatrix} {\chi _2} & {*}\\ {0} & {\chi _1}\end{smallmatrix}\big )$). Combining Theorem 7.7 and [Paš22, Proposition 6.7],⁸ we see that $\dim _{\mathcal {O}_D^{\times }}\mathcal {S}^1 (\widetilde {S}(U^v,\mathbb {F})[\mathfrak {m}_{\overline {r}}])=1$, hence

\[ \dim_{\mathcal{O}_D^{\times}}(\widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}])=1 \]

by (8.2). By Proposition 8.9, (8.4) and assumption (H), we obtain for $i\in \{0,1,2\}$

(8.5)\begin{equation} \mathcal{S}^1(\pi(\overline{\rho}_i))=\overline{\rho}_i(-1)\otimes\mathrm{JL}(\overline{\rho}_i). \end{equation}

Recall from § 4.2 that there exist exact sequences

\begin{gather*} 0\rightarrow \pi_1\rightarrow \pi(\overline{\rho}_1)\rightarrow \pi_2\rightarrow0,\\ 0\rightarrow \pi_2\rightarrow \pi(\overline{\rho}_2)\rightarrow \pi_1\rightarrow 0, \end{gather*}

where $\pi _1:=\operatorname {{\mathrm {Ind}}}_{B(\mathbb {Q}_p)}^G\chi _2\otimes \chi _1\omega ^{-1}$ and $\pi _2:=\operatorname {{\mathrm {Ind}}}_{B(\mathbb {Q}_p)}^G\chi _1\otimes \chi _2\omega ^{-1}$. Note that $\mathcal {S}^0(\pi _i)=\mathcal {S}^2(\pi _i)=0$ for $i\in \{1,2\}$, by Theorem 8.11. Hence, by applying the functor $\mathcal {S}^i$ and using (8.5), we obtain

(8.6)\begin{gather} 0\rightarrow \mathcal{S}^1(\pi_1)\overset{\iota_1}{\rightarrow} \overline{\rho}_1(-1)\otimes\mathrm{JL}(\overline{\rho}_1)\rightarrow \mathcal{S}^1(\pi_2)\rightarrow 0 , \end{gather}

(8.7)\begin{gather}0\rightarrow \mathcal{S}^1(\pi_2)\overset{\iota_2}{\rightarrow} \overline{\rho}_2(-1)\otimes\mathrm{JL}(\overline{\rho}_2)\rightarrow \mathcal{S}^1(\pi_1)\rightarrow0. \end{gather}

Since $\overline {\rho }_1$ is nonsplit, we have

\[ \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_2,\overline{\rho}_1\otimes \mathrm{JL}(\overline{\rho}_1))=0. \]

As a consequence, $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _2\omega ^{-1},\mathcal {S}^1(\pi _1))=0$ by (8.6) and applying $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _2\omega ^{-1},-)$ to (8.7) gives isomorphisms

\[ \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_2\omega^{-1},\mathcal{S}^1(\pi_2))\cong \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_2\omega^{-1},\overline{\rho}_2(-1)\otimes\mathrm{JL}(\overline{\rho}_2))\cong \mathrm{JL}(\overline{\rho}_2), \]

where the last isomorphism follows from the definition of $\overline {\rho }_2$. This gives a $G_{\mathbb {Q}_p}\otimes D^{\times }$-equivariant embedding

(8.8)\begin{equation} \chi_2\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_2)\hookrightarrow \mathcal{S}^1(\pi_2). \end{equation}

One checks that its composition with $\iota _2$ (in (8.7)) coincides with the morphism obtained by tensoring the inclusion $\chi _2\omega ^{-1}\hookrightarrow \overline {\rho }_2(-1)$ with $\mathrm {JL}(\overline {\rho }_2)$. Combining with the short exact sequence

\[ 0\rightarrow \chi_2\omega^{-1}\otimes\mathrm{JL}(\overline{\rho}_2)\rightarrow \overline{\rho}_2(-1)\otimes\mathrm{JL}(\overline{\rho}_2)\rightarrow \chi_1\omega^{-1}\otimes\mathrm{JL}(\overline{\rho}_2)\rightarrow0, \]

a diagram chasing gives a surjection

(8.9)\begin{equation} \chi_1\omega^{-1}\otimes\mathrm{JL}(\overline{\rho}_2)\twoheadrightarrow \mathcal{S}^1(\pi_1). \end{equation}

In particular, when restricted to $G_{\mathbb {Q}_p}$, $\mathcal {S}^1(\pi _1)$ is semisimple and any irreducible subquotient of $\mathcal {S}^1(\pi _1)$ is isomorphic to $\chi _1\omega ^{-1}$.

On the other hand, the same argument as above implies an embedding (analogous to (8.8))

(8.10)\begin{align} \chi_1\omega^{-1}\otimes\mathrm{JL}(\overline{\rho}_1)\hookrightarrow \mathcal{S}^1(\pi_1). \end{align}

We claim that (8.10) is an isomorphism. Indeed, $\iota _1$ in (8.6) induces a $G_{\mathbb {Q}_p}\times D^{\times }$-equivariant embedding

\[ \mathcal{S}^1(\pi_1)/(\chi_1\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1)) \hookrightarrow (\overline{\rho}_1(-1)\otimes\mathrm{JL}(\overline{\rho}_1))/ (\chi_1\omega^{-1}\otimes\mathrm{JL}(\overline{\rho}_1))\cong \chi_2\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1). \]

However, as shown in the last paragraph, $\mathcal {S}^1(\pi _1)$ admits only $\chi _1\omega ^{-1}$ as irreducible subquotient (when restricted to $G_{\mathbb {Q}_p}$), while $\chi _2\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho }_1)$ admits only $\chi _2\omega ^{-1}$ as irreducible subquotients. Since $\chi _1\neq \chi _2$, this forces $\mathcal {S}^1(\pi _1)/(\chi _1\omega ^{-1}\otimes \mathrm {JL}( \overline {\rho }_1))=0$, proving the claim. In a similar way, the embedding (8.8) is also an isomorphism and consequently (8.9) is an isomorphism.

In summary, we have proven that

\[ \chi_1\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_2)\overset{(8.9)}{\cong} \mathcal{S}^1(\pi_1)\overset{(8.10)}{\cong} \chi_1\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1). \]

Hence, by applying $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _1\omega ^{-1},-)$ we obtain a $D^{\times }$-equivariant isomorphism $\mathrm {JL}(\overline {\rho }_1)\cong \mathrm {JL}(\overline {\rho }_2)$.

Remark 8.13 It might be strange that $\mathrm {JL}(\overline {\rho }_i)$ only carries the information of $\overline {\rho }^{\rm ss}$. This can be explained as follows. On the one hand, since $\mathcal {S}^1(\pi (\overline {\rho }_i))=\overline {\rho }_i(-1)\otimes \mathrm {JL}(\overline {\rho }_i)$, the information of $\overline {\rho }_i$ is indeed caught by the functor $\mathcal {S}^1$. On the other hand, comparing the quaternionic Serre weights (cf. Propositions 6.1 and 6.2), $\mathrm {JL}(\overline {\rho }_1)$ and $\mathrm {JL}(\overline {\rho }_2)$ have the same set of quaternionic Serre weights. However, we do not expect this phenomenon happens once $L\neq \mathbb {Q}_p$.

8.3 The non-generic case in the minimal case

In this subsection, we extend the result in § 8.2 to the case $\overline {\rho }^{\rm ss}\sim \omega \oplus \mathbf {1}$ (up to twist). In the following, we will denote by $\mathbf {1}_{G_{\mathbb {Q}_p}}$, $\mathbf {1}_{G}$ and $\mathbf {1}_{D^{\times }}$ the trivial representation of $G_{\mathbb {Q}_p}$, $G$ and $D^{\times }$, respectively; sometimes we will omit the subscript if no confusion is caused.

Let $\overline {\rho }_1\sim \big (\begin{smallmatrix} {\omega } & {*}\\ {0} & {\mathbf {1}}\end{smallmatrix}\big )$ be a nonsplit extension of $\mathbf {1}$ by $\omega$; we do not make assumptions on the extension type of $\overline {\rho }_1$ (i.e. peu ramifié or très ramifié). On the other hand, $\operatorname {{\mathrm {Ext}}}^1_{G_{\mathbb {Q}_p}}(\omega,\mathbf {1})$ is one-dimensional; let $\overline {\rho }_2\sim \big (\begin{smallmatrix} {\mathbf {1}} & {*}\\ {0} & {\omega }\end{smallmatrix}\big )$ be the unique nonsplit extension of $\omega$ by $\mathbf {1}$.

Let $\tau _1$ be the universal extension of $\mathbf{1}_{G}^{\oplus2}$ by $\operatorname {{\mathrm {Sp}}}$, i.e.

(8.11)\begin{equation} 0\rightarrow \operatorname{{\mathrm{Sp}}}\rightarrow \tau_1\rightarrow \mathbf{1}_{G}^{\oplus2}\rightarrow0 \end{equation}

with $\operatorname {{\mathrm {soc}}}_{G}\tau _1=\operatorname {{\mathrm {Sp}}}$. Recall from § 4.2 that there is a short exact sequence

\[ 0\rightarrow \pi_{\alpha}\rightarrow\pi(\overline{\rho}_2)\rightarrow\tau_1\rightarrow0, \]

where $\pi _{\alpha }:=\operatorname {{\mathrm {Ind}}}_{B(\mathbb {Q}_p)}^{G}(\omega \otimes \omega ^{-1})$.

It is shown in [Paš13, § 10.1] that $\dim _{\mathbb {F}}\operatorname {{\mathrm {Ext}}}^1_{G/Z_G}(\pi _{\alpha },\mathbf {1}_G)=1$. Thus, there exists a unique (up to isomorphism) nonsplit extension

(8.12)\begin{equation} 0\rightarrow \mathbf{1}_{G}\rightarrow \kappa\rightarrow \pi_{\alpha}\rightarrow0. \end{equation}

On the other hand, there is a natural isomorphism $\operatorname {{\mathrm {Ext}}}^1_{G/Z_G}(\mathbf {1}_G,\operatorname {{\mathrm {Sp}}})\cong \operatorname {{\mathrm {Hom}}}(\mathbb {Q}_p^{\times },\mathbb {F})$ by [Col10, Theorem VII.4.18]; we denote by $E_{\phi }$ the extension corresponding to $\phi \in \operatorname {{\mathrm {Hom}}}(\mathbb {Q}_p^{\times },\mathbb {F})$. The next result gives the structure of $\pi (\overline {\rho }_1)$.

Proposition 8.14 We have $\operatorname {{\mathrm {soc}}}_G\pi (\overline {\rho }_1)\cong \operatorname {{\mathrm {Sp}}}$ and there exist nonsplit extensions

\begin{gather*} 0\rightarrow E_{\phi}\rightarrow \pi(\overline{\rho}_1)\rightarrow \pi_{\alpha}\rightarrow0,\\ 0\rightarrow \operatorname{{\mathrm{Sp}}}\rightarrow \pi(\overline{\rho}_1)\rightarrow \kappa\rightarrow0. \end{gather*}

Proof. See [Paš15, Lemma 6.7].

Proposition 8.15 The following statements hold:

1. (i) $\mathcal {S}^0(\mathbf {1}_G)\cong \mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }}$, $\mathcal {S}^1(\mathbf {1}_G)=0$, $\mathcal {S}^2(\mathbf {1}_G)\cong \omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$;

2. (ii) $\mathcal {S}^0(\operatorname {{\mathrm {Sp}}})=\mathcal {S}^2(\operatorname {{\mathrm {Sp}}})=0$;

3. (iii) $\mathcal {S}^0(\pi _{\alpha })=\mathcal {S}^2(\pi _{\alpha })=0$.

Proof. Statement (i) follows from Theorem 7.1(v). Statements (ii) and (iii) are special cases of Theorem 8.11, except for $\mathcal {S}^2(\operatorname {{\mathrm {Sp}}})$ which is [Lud17, Corollary 4.7].

Corollary 8.16 Let $\pi \in \operatorname {\mathrm {Mod}}_{G/Z}^{\rm l.adm}(\mathcal {O})$. Assume that each of the irreducible subquotients of $\pi$ lies in $\{\operatorname {{\mathrm {Sp}}},\mathbf {1}_G,\pi _{\alpha }\}$. Then $\mathcal {S}^0(\pi )$ (respectively, $\mathcal {S}^2(\pi )$) admits only $\mathbf {1}_{G_{\mathbb {Q}_p}}$ (respectively, $\omega ^{-1}$) as subquotients when restricted to $G_{\mathbb {Q}_p}$.

Proof. This is a direct consequence of Proposition 8.15.

Proposition 8.17

1. (i) We have that $\mathrm {JL}(\overline {\rho }_1)^{\vee }$ is a Cohen–Macaulay $\mathbb {F}[\![U_D^1]\!]$-module.

2. (ii) We have $\mathcal {S}^1(\pi (\overline {\rho }_1))=\overline {\rho }_1(-1)\otimes \mathrm {JL}(\overline {\rho }_1)$ and $\mathcal {S}^2(\pi (\overline {\rho }_1))=0$.

Proof. Since $R_{\overline {\rho }_1}^{\psi \varepsilon }$ is formally smooth, the assertions follow from Corollary 7.9 and Proposition 8.9.

Corollary 8.18 We have $\mathcal {S}^0(\kappa )\cong \mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }}$ and $\mathcal {S}^2(\kappa )=0$.

Proof. Since $\mathcal {S}^0(\pi _{\alpha })=0$, the first assertion is a direct consequence of Proposition 8.15(i) via (8.12). Since $\kappa$ is a quotient of $\pi (\overline {\rho }_1)$, the second assertion is a consequence of Proposition 8.17(ii).

By Propositions 8.15 and 8.17 and Corollary 8.18, the sequence $0\rightarrow \operatorname {{\mathrm {Sp}}}\rightarrow \pi (\overline {\rho }_1)\rightarrow \kappa \rightarrow 0$ (see Proposition 8.14) induces an exact sequence

(8.13)\begin{equation} 0\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes \mathbf{1}_{D^{\times}}\rightarrow \mathcal{S}^1(\operatorname{{\mathrm{Sp}}})\rightarrow \overline{\rho}_1(-1)\otimes \mathrm{JL}(\overline{\rho}_1)\rightarrow \mathcal{S}^1(\kappa)\rightarrow0. \end{equation}

Similarly, the sequence $0\rightarrow E_{\phi }\rightarrow \pi (\overline {\rho }_1)\rightarrow \pi _{\alpha }\rightarrow 0$ induces an exact sequence

(8.14)\begin{equation} 0\rightarrow \mathcal{S}^1(E_{\phi})\rightarrow \overline{\rho}_1(-1)\otimes \mathrm{JL}(\overline{\rho}_1)\rightarrow \mathcal{S}^1(\pi_{\alpha})\rightarrow \omega^{-1}\otimes\mathbf{1}_{D^{\times}}\rightarrow0. \end{equation}

Lemma 8.19 We have $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},\mathcal {S}^1(\operatorname {{\mathrm {Sp}}}))=0$, and $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},\mathcal {S}^1(\tau _1))$ is finite-dimensional.

Proof. As $\overline {\rho }_1\sim \big (\begin{smallmatrix} {\omega } & {*}\\ {0} & {\mathbf {1}}\end{smallmatrix}\big )$ is assumed to be nonsplit, we have $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},\overline {\rho }_1(-1)\otimes \mathrm {JL}(\overline {\rho }_1))=0$, which implies the first assertion via (8.13). For the second assertion, we note that the short exact sequence $0\rightarrow \operatorname {{\mathrm {Sp}}}\rightarrow \tau _1\rightarrow (\mathbf {1}_{G})^{\oplus 2}\rightarrow 0$ induces an exact sequence

(8.15)\begin{equation} 0\rightarrow (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathbf{1}_{D^{\times}})^{\oplus 2}\rightarrow \mathcal{S}^1(\operatorname{{\mathrm{Sp}}})\rightarrow\mathcal{S}^1(\tau_1)\rightarrow0 \end{equation}

by Proposition 8.15(i). By applying $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},-)$ to (8.15), we obtain

\[ 0=\operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\omega^{-1},\mathcal{S}^1(\operatorname{{\mathrm{Sp}}}))\rightarrow\operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\omega^{-1},\mathcal{S}^1(\tau_1)) \rightarrow \operatorname{{\mathrm{Ext}}}^1_{G_{\mathbb{Q}_p}}(\omega^{-1},\mathbf{1}_{G_{\mathbb{Q}_p}}^{\oplus2}) \]

from which the result easily follows.

Proposition 8.20 There exists a short exact sequence

(8.16)\begin{equation} 0\rightarrow \mathcal{S}^1(\pi(\overline{\rho}_2))\rightarrow \overline{\rho}_2(-1)\otimes \mathrm{JL}(\overline{\rho}_2)\rightarrow (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathbf{1}_{D^{\times}})^{\oplus 2}\rightarrow0. \end{equation}

As a consequence, $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1}, \mathcal {S}^1(\pi (\overline {\rho }_2)))\cong \mathrm {JL}(\overline {\rho }_2)$.

Proof. We need to show that the cokernel of (8.2) is isomorphic to $(\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }})^{\oplus 2}$. For this we need a refined version of [Scho18, Proposition 7.7], which we put separately in Lemma 8.21 below. In our situation with $A=\mathbb {T}(U^v)_{\mathfrak {m}_{\overline {r}}}$, $I=\mathfrak {m}_{\overline {r}}$ and $P:=(\widetilde {S}(U^{v}, \mathcal {O})_{\mathfrak {m}_{\overline {r}}})^{d}$, we are left to show

(8.17)\begin{equation} \operatorname{{\mathrm{Tor}}}_1^{\mathbb{T}(U^v)_{\mathfrak{m}_{\overline{r}}}}(\mathbb{T}(U^v)_{\mathfrak{m}_{\overline{r}}}/ \mathfrak{m}_{\overline{r}},P)\cong(\mathbf{1}_G^{\vee})^{\oplus 2} \end{equation}

by Proposition 8.15(i) (here we use [Paš22, Proposition 5.4] to ensure that $P$ satisfies assumption (c) of Lemma 8.21). This is a consequence of [Hu21, Proposition 3.30], as we explain below. After enlarging $\mathbb {F}$, we may assume $\mathbb {T}(U^v)_{\mathfrak {m}_{\overline {r}}}/\mathfrak {m}_{\overline {r}}\cong \mathbb {F}$.

To be able to apply [Hu21, Proposition 3.30], we need to relate $P$ with $N$, where $N$ is the object introduced in § 4.2.1 for $\overline {\rho }_2$. We do this by passing to $M_{\infty }$. On the one hand, by Remark 7.8 and assumption (H) we have $M_{\infty }\cong R_{\infty }^{\psi \varepsilon ^{-1}}\widehat {\otimes }_{R_{\overline {\rho }_2}^{\psi \varepsilon }}N$. Since $R_{\infty }^{\psi \varepsilon ^{-1}}$ is flat over $R_{\overline {\rho }_2}^{\psi \varepsilon }$, we deduce

(8.18)\begin{equation} \operatorname{{\mathrm{Tor}}}_1^{R_{\overline{\rho}_2}^{\psi\varepsilon}}(\mathbb{F},N)\cong \operatorname{{\mathrm{Tor}}}_1^{R_{\infty}^{\psi \varepsilon^{-1}}}(\mathbb{F},M_{\infty}). \end{equation}

On the other hand, $R_{\infty }^{\psi \varepsilon ^{-1}}$ acts on $P$ via the isomorphism (5.2) $M_{\infty }/\frak {a}_{\infty }\cong P$, and the action factors through

\[ R_{\infty}^{\psi \varepsilon^{-1}}\twoheadrightarrow R_{\infty}^{\psi \varepsilon^{-1}}/\mathfrak{a}_{\infty} \cong R_{\overline{r},\mathcal{S}}^{\psi\varepsilon^{-1}}\twoheadrightarrow\mathbb{T}(U^v)_{\mathfrak{m}_{\overline{r}}}. \]

Recall that $\frak {a}_{\infty }$ is generated by an $M_{\infty }$-regular sequence $z_1,\ldots,z_q, y_1,\ldots,y_j$. By Proposition 4.21, this sequence is also $R_{\infty }^{\psi \varepsilon ^{-1}}$-regular and $R_{\infty }^{\psi \varepsilon ^{-1}}/\mathfrak {a}_{\infty }$ acts faithfully on $P$. But $\mathbb {T}(U^v)_{\mathfrak {m}_{\overline {r}}}$ also acts faithfully on $P$, so the surjection $R_{\infty }^{\psi \varepsilon ^{-1}}/\mathfrak {a}_{\infty }\twoheadrightarrow \mathbb {T}(U^v)_{\mathfrak {m}_{\overline {r}}}$ is actually an isomorphism.⁹ Consequently,

\[ \operatorname{{\mathrm{Tor}}}_1^{R_{\infty}^{\psi \varepsilon^{-1}}}(\mathbb{F},M_{\infty})\cong \operatorname{{\mathrm{Tor}}}_1^{\mathbb{T}(U^v)_{\mathfrak{m}_{\overline{r}}}}(\mathbb{F},P). \]

Combining this with (8.18), we deduce (8.17) from [Hu21, Proposition 3.30].

Lemma 8.21 Let $(A,\mathfrak {m})$ be a complete noetherian local $\mathcal {O}$-algebra with $A/\mathfrak {m}\cong \mathbb {F}$ and $P\in \frak {C}_{G/Z_G}(A)$. Assume that:

1. (a) $P$ is projective in the category of pseudo-compact $\mathcal {O}[\![K/Z_1]\!]$-modules;

2. (b) $P_{\mathrm {SL}_2(\mathbb {Q}_p)}=0$;

3. (c) each of the irreducible subquotients of $P^{\vee }$ lies in $\{\operatorname {{\mathrm {Sp}}},\mathbf {1}_G,\pi _{\alpha }\}$.

Let $I$ be an ideal of $A$. Then there exists an exact sequence

\[ 0\rightarrow \check{\mathcal{S}}^0(\operatorname{{\mathrm{Tor}}}_1^A(A/I,P))\rightarrow A/I \otimes_A \check{\mathcal{S}}^1(P)\rightarrow \check{\mathcal{S}}^1(A/I\otimes_AP)\rightarrow 0. \]

Proof. Choose a finite free resolution of $A/I$: $\cdots \rightarrow F_1\rightarrow F_0\rightarrow A/I\rightarrow 0$. By applying $-\otimes _AP$ to it, we obtain a chain complex

(8.19)\begin{equation} \cdots \overset{d_2}{\rightarrow} F_1\otimes_A P\overset{d_1}{\rightarrow} F_0\otimes_AP\overset{d_0}{\rightarrow}A/I\otimes_A P\rightarrow0 \end{equation}

whose homology computes $\operatorname {{\mathrm {Tor}}}_i^A(A/I,P)$. Since each $F_i$ is a finite free $A$-module (for $i\geq 0$), assumption (a) implies that each $F_i\otimes _A P$ is projective when restricted to $K$, hence $\check {\mathcal {S}}^2(F_i\otimes _A P)=0$ by Theorem 7.1(iii). Assumption (b) implies that $\check {\mathcal {S}}^0(F_i\otimes _A P)=0$ by Theorem 7.1(iv). As a consequence, $\check {\mathcal {S}}^0(\mathrm {Im}(d_i))=0$ for any $i\geq 0$. On the other hand, since $\check {\mathcal {S}}^3(-)=0$, we have $\check {\mathcal {S}}^2(\mathrm {Im}(d_i))=0$ for $i\geq 1$.

We may split (part of) the complex (8.19) as

\[ 0\rightarrow \mathrm{Im}(d_1)\rightarrow F_0\otimes_AP\rightarrow A/I\otimes_A P\rightarrow0,\quad 0\rightarrow \operatorname{{\mathrm{Ker}}}(d_1)\rightarrow F_1\otimes_A P\rightarrow \mathrm{Im}(d_1)\rightarrow0 \]

from which we deduce long exact sequences

\begin{gather*} 0\rightarrow \check{\mathcal{S}}^2(A/I\otimes_A P)\rightarrow\check{\mathcal{S}}^1(\mathrm{Im}(d_1)) \overset{f}{\rightarrow}\check{\mathcal{S}}^1(F_0\otimes_AP)\rightarrow \check{\mathcal{S}}^1(A/I\otimes_A P)\rightarrow 0,\\ 0\rightarrow\check{\mathcal{S}}^1(\operatorname{{\mathrm{Ker}}}(d_1))\rightarrow\check{\mathcal{S}}^1(F_1\otimes_A P)\overset{g}{\rightarrow} \check{\mathcal{S}}^1(\mathrm{Im}(d_1))\rightarrow \check{\mathcal{S}}^0(\operatorname{{\mathrm{Ker}}}(d_1))\rightarrow0. \end{gather*}

Note that $\check {\mathcal {S}}^1(F_i\otimes _A P)\cong F_i\otimes _A \check {\mathcal {S}}^1(P)$ (as $F_i$ is a finite free $A$-module), and that there is an exact sequence

\[ F_1\otimes_A\check{\mathcal{S}}^1( P)\overset{f\circ g}{\longrightarrow} F_0\otimes\check{\mathcal{S}}^1(P)\rightarrow A/I\otimes_A\check{\mathcal{S}}^1(P)\rightarrow0 \]

by tensoring the sequence $F_1\rightarrow F_0\rightarrow A/I\rightarrow 0$ with $\check {\mathcal {S}}^1(P)$. Recall that a variant of the snake lemma shows that there is a long exact sequence

\[ 0\rightarrow \operatorname{{\mathrm{Ker}}}(g)\rightarrow \operatorname{{\mathrm{Ker}}}(f\circ g)\rightarrow \operatorname{{\mathrm{Ker}}}(f)\overset{\partial}{\rightarrow} \operatorname{{\mathrm{Coker}}}(g)\rightarrow \operatorname{{\mathrm{Coker}}}(f\circ g)\rightarrow \operatorname{{\mathrm{Coker}}}(f)\rightarrow0. \]

In our situation, this gives (by considering the last four nonzero terms)

\[ \check{\mathcal{S}}^2(A/I\otimes_A P)\overset{\partial}{\rightarrow}\check{\mathcal{S}}^0(\operatorname{{\mathrm{Ker}}}(d_1))\rightarrow A/I \otimes_A\check{\mathcal{S}}^1(P)\rightarrow \check{\mathcal{S}}^1(A/I\otimes_A P)\rightarrow0. \]

By Corollary 8.16, assumption (c) implies that $\partial$ is identically zero. Hence, we are left to show

\[ \check{\mathcal{S}}^0(\operatorname{{\mathrm{Ker}}}(d_1))=\check{\mathcal{S}}^0(\operatorname{{\mathrm{Tor}}}_1^A(A/I,P)), \]

which follows from the exact sequence $0\rightarrow \mathrm {Im}(d_2)\rightarrow \operatorname {{\mathrm {Ker}}}(d_1)\rightarrow \operatorname {{\mathrm {Tor}}}_1^A(A/I,P)\rightarrow 0$ (recall $\check {\mathcal {S}}^0(\mathrm {Im}(d_2))=0$ from the first paragraph of the proof).

By Theorem 8.11 the short exact sequence $0\rightarrow \pi _{\alpha }\rightarrow \pi (\overline {\rho }_2)\rightarrow \tau _1\rightarrow 0$ induces an exact sequence

(8.20)\begin{align} 0\rightarrow \mathcal{S}^1(\pi_{\alpha})\rightarrow \mathcal{S}^1(\pi(\overline{\rho}_2))\rightarrow \mathcal{S}^1(\tau_1)\rightarrow0. \end{align}

Lemma 8.22 Both $\mathcal {S}^1(\pi _{\alpha })$ and $\mathcal {S}^1(\kappa )$ are $\omega ^{-1}$-typic (when restricted to $G_{\mathbb {Q}_p}$).

Proof. We claim that $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\mathbf {1}_{G_{\mathbb {Q}_p}},\mathcal {S}^1(\pi _{\alpha }))=\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\mathbf {1}_{G_{\mathbb {Q}_p}},\mathcal {S}^1(\kappa ))=0$. Combining (8.20) with Proposition 8.20, we obtain an embedding

\[ \mathcal{S}^1(\pi_{\alpha})\hookrightarrow \overline{\rho}_2(-1)\otimes \mathrm{JL}(\overline{\rho}_2). \]

As $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\mathbf {1}_{G_{\mathbb {Q}_p}},\overline {\rho }_2(-1))=0$, we deduce that $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\mathbf {1}_{G_{\mathbb {Q}_p}},\mathcal {S}^1(\pi _{\alpha }))=0$, as claimed. Using Proposition 8.15(i) and Corollary 8.18, the sequence $0\rightarrow \mathbf {1}_G\rightarrow \kappa \rightarrow \pi _{\alpha }\rightarrow 0$ induces an exact sequence

(8.21)\begin{equation} 0\rightarrow \mathcal{S}^1(\kappa)\rightarrow \mathcal{S}^1(\pi_{\alpha})\rightarrow \omega^{-1}\otimes\mathbf{1}_{D^{\times}}\rightarrow0, \end{equation}

which implies the claim for $\mathcal {S}^1(\kappa )$.

The claim implies that the surjection $\overline {\rho }_1(-1)\otimes \mathrm {JL}(\overline {\rho }_1)\twoheadrightarrow \mathcal {S}^1(\kappa )$ in (8.13) must factor as

\[ \overline{\rho}_1(-1)\otimes\mathrm{JL}(\overline{\rho}_1)\twoheadrightarrow\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1)\twoheadrightarrow \mathcal{S}^1(\kappa), \]

where the first quotient map is induced by the natural projection $\overline {\rho }_1(-1)\sim \big (\begin{smallmatrix} {\mathbf {1}} & {*}\\ {0} & {\omega ^{-1}}\end{smallmatrix}\big )\twoheadrightarrow \omega ^{-1}$. In particular, $\mathcal {S}^1(\kappa )$ is $\omega ^{-1}$-typic. Note that, being a subrepresentation of $\overline {\rho }_2(-1)\otimes \mathrm {JL}(\overline {\rho }_2)$, $\mathcal {S}^1(\pi _{\alpha })$ does not admit any $G_{\mathbb {Q}_p}$-subquotient isomorphic to a nontrivial self-extension of $\omega ^{-1}$, so $\mathcal {S}^1(\pi _{\alpha })$ is also $\omega ^{-1}$-typic by (8.21).

As a consequence of (8.16), there exists a $D^{\times }$-equivariant surjection

(8.22)\begin{equation} \mathrm{JL}(\overline{\rho}_2)\twoheadrightarrow (\mathbf{1}_{D^{\times}})^{\oplus2}. \end{equation}

We denote its kernel by $V_2$. Then $\overline {\rho }_2(-1)\otimes \mathrm {JL}(\overline {\rho }_2)$ can be filtered by subrepresentations such that the graded pieces are isomorphic to

\[ \omega^{-1}\otimes V_2,\quad (\omega^{-1}\otimes(\mathbf{1}_{D^{\times}})^{\oplus 2}) \oplus (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V_2),\quad \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes (\mathbf{1}_{D^{\times}})^{\oplus 2}. \]

Using (8.16) again, we obtain the following short exact sequences:

(8.23)\begin{gather} 0\rightarrow \omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_2)\rightarrow \mathcal{S}^1(\pi(\overline{\rho}_2))\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V_2\rightarrow0, \end{gather}

(8.24)\begin{gather}0\rightarrow \omega^{-1}\otimes V_2\rightarrow \mathcal{S}^1(\pi(\overline{\rho}_2))\rightarrow (\omega^{-1}\otimes \mathbf{1}_{D^{\times}})^{\oplus 2}\oplus(\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V_2)\rightarrow0. \end{gather}

Recall the definition of $V_{\rm fd}$ for an admissible smooth $D^{\times }$-representation $V$ from Definition 8.1, and that taking $(-)_{\rm fd}$ is right exact by Lemma 8.4.

Corollary 8.23 The following statements hold:

1. (i) $(\mathcal {S}^1(\kappa ))_{\rm fd}=0$ and $(\mathcal {S}^1(\pi _{\alpha }))_{\rm fd}\cong \omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$;

2. (ii) $(\mathcal {S}^1(E_{\phi }))_{\rm fd}$ is $\omega ^{-1}$-typic;

3. (iii) $(\mathcal {S}^1(\tau _1))_{\rm fd}$ is $\omega ^{-1}$-typic;

4. (iv) $(V_2)_{\rm fd}=0$.

Proof. (i) Since $\mathcal {S}^1(\kappa )$ is a quotient of $\overline {\rho }_1(-1)\otimes \mathrm {JL}(\overline {\rho }_1)$ by (8.13) and $(\mathrm {JL}(\overline {\rho }_1))^{\vee }$ is Cohen–Macaulay by Proposition 8.17, we have $\mathrm {JL}(\overline {\rho }_1)_{\rm fd}=0$ by Lemma 8.3, hence $(\mathcal {S}^1(\kappa ))_{\rm fd}=0$ as well by the right exactness of $(-)_{\rm fd}$. The second assertion follows from this, by applying Lemma 8.4 to (8.21).

(ii) Recall the exact sequence (8.14)

\[ 0\rightarrow \mathcal{S}^1(E_{\phi})\rightarrow \overline{\rho}_1(-1)\otimes \mathrm{JL}(\overline{\rho}_1)\rightarrow \mathcal{S}^1(\pi_{\alpha})\rightarrow \omega^{-1}\otimes\mathbf{1}_{D^{\times}}\rightarrow0. \]

Since $\mathcal {S}^1(\pi _{\alpha })$ is $\omega ^{-1}$-typic by Lemma 8.22, the morphism $\overline {\rho }_1(-1)\otimes \mathrm {JL}(\overline {\rho }_1)\rightarrow \mathcal {S}^1(\pi _{\alpha })$ factors through the quotient $\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho }_1)$. Let $W$ be the admissible $\mathbb {F}$-representation of $D^{\times }$ such that

\[ \omega^{-1}\otimes W=\operatorname{{\mathrm{Ker}}}(\omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1)\rightarrow \mathcal{S}^1(\pi_{\alpha})). \]

Then one checks that $\mathcal {S}^1(E_{\phi })$ fits in the following exact sequence:

(8.25)\begin{equation} 0\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes \mathrm{JL}(\overline{\rho}_1)\rightarrow \mathcal{S}^1(E_{\phi})\rightarrow \omega^{-1}\otimes W\rightarrow0. \end{equation}

Since $(\mathrm {JL}(\overline {\rho }_1))_{\rm fd}=0$ as seen in part (i), we deduce

\[ (\mathcal{S}^1(E_{\phi}))_{\rm fd}= (\omega^{-1}\otimes W)_{\rm fd}\cong\omega^{-1}\otimes W_{\rm fd}. \]

In particular, $(\mathcal {S}^1(E_{\phi }))_{\rm fd}$ is $\omega ^{-1}$-typic.

(iii) Note that there is a short exact sequence $0\rightarrow E_{\phi }\rightarrow \tau _1\rightarrow \mathbf {1}_{G}\rightarrow 0$ by the definition of $\tau _1$, see (8.11). By Proposition 8.15(i) it induces an exact sequence

(8.26)\begin{equation} 0\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathbf{1}_{D^{\times}}\rightarrow \mathcal{S}^1(E_{\phi})\rightarrow \mathcal{S}^1(\tau_1)\rightarrow0. \end{equation}

The assertion then follows from part (ii) using Lemma 8.4.

(iv) We view $\mathcal {S}^1(\pi _{\alpha })$ as a subrepresentation of $\mathcal {S}^1(\pi (\overline {\rho }_2))$ via (8.20). Since $\mathcal {S}^1(\pi _{\alpha })$ is $\omega ^{-1}$-typic by Lemma 8.22, it is contained in $\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho }_2)$, see (8.23). As a consequence, the snake lemma applied to (8.20) and (8.23) implies that $\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes V_2$ is a quotient of $\mathcal {S}^1(\tau _1)$, thus $(\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes V_2)_{\rm fd}$ is a quotient of $(\mathcal {S}^1(\tau _1))_{\rm fd}$. However, $(\mathcal {S}^1(\tau _1))_{\rm fd}$ is $\omega ^{-1}$-typic by part (iii), which forces $(\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes V_2)_{\rm fd}=0$ or equivalently $(V_2)_{\rm fd}=0$.

Corollary 8.24 We have isomorphisms $\mathcal {S}^1(\kappa )\cong \omega ^{-1}\otimes V_2$ and

\[ \mathcal{S}^1(\tau_1)\cong(\omega^{-1}\otimes \mathbf{1}_{D^{\times}})\oplus(\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V_2). \]

Proof. We may identify $\mathcal {S}^1(\kappa )$ with a subrepresentation of $\mathcal {S}^1(\pi (\overline {\rho }_2))$ via (8.20) and (8.21). As in the proof of Corollary 8.23(iv), $\mathcal {S}^1(\kappa )$ is contained in $\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho }_2)$. However, since $(\mathcal {S}^1(\kappa ))_{\rm fd}=0$ by Corollary 8.23(i), $\mathcal {S}^1(\kappa )$ is, in fact, contained in $\omega ^{-1}\otimes V_2$ by the definition of $V_2$, see (8.22). Denote by $\iota$ the inclusion

\[ \iota:\ \mathcal{S}^1(\kappa)\hookrightarrow \omega^{-1}\otimes V_2. \]

We need to prove that $\iota$ is an isomorphism or, equivalently, $\operatorname {{\mathrm {Coker}}}(\iota )=0$. Since $(V_2)_{\rm fd}=0$ by Corollary 8.23(iv), it suffices to prove that $\operatorname {{\mathrm {Coker}}}(\iota )$ is finite-dimensional.

Denote by $\widetilde {\iota }$ the embedding $\mathcal {S}^1(\kappa )\hookrightarrow \mathcal {S}^1(\pi (\overline {\rho }_2))$. Then (8.20) and (8.21) imply

(8.27)\begin{equation} 0\rightarrow \omega^{-1}\otimes\mathbf{1}_{D^{\times}}\rightarrow\operatorname{{\mathrm{Coker}}}(\widetilde{\iota})\rightarrow \mathcal{S}^1(\tau_1)\rightarrow0. \end{equation}

Since $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},\mathcal {S}^1(\tau _1))$ is finite-dimensional by Lemma 8.19, so is $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},\operatorname {{\mathrm {Coker}}}(\widetilde {\iota }))$. On the other hand, using (8.24) we have a commutative diagram

hence an exact sequence

(8.28)\begin{equation} 0\rightarrow \operatorname{{\mathrm{Coker}}}(\iota)\rightarrow\operatorname{{\mathrm{Coker}}}(\widetilde{\iota})\rightarrow (\omega^{-1}\otimes \mathbf{1}_{D^{\times}})^{\oplus 2}\oplus(\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V_2)\rightarrow0. \end{equation}

Consequently, $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\omega ^{-1},\operatorname {{\mathrm {Coker}}}(\iota ))$ is finite-dimensional. However, since $\operatorname {{\mathrm {Coker}}}(\iota )$ is $\omega ^{-1}$-typic (being a quotient of $\omega ^{-1}\otimes V_2$), this implies that $\operatorname {{\mathrm {Coker}}}(\iota )$ is itself finite-dimensional. As explained in last paragraph, we deduce that $\iota$ is an isomorphism and, consequently, by (8.28)

\[ \operatorname{{\mathrm{Coker}}}(\widetilde{\iota})\cong (\omega^{-1}\otimes \mathbf{1}_{D^{\times}})^{\oplus 2} \oplus(\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V_2). \]

Finally, the second isomorphism in the corollary follows from this by using (8.27).

We note the following consequence of the proof of Corollary 8.24.

Corollary 8.25

1. (i) There exists a short exact sequence of $G_{\mathbb {Q}_p}\times D^{\times }$-representations

   \[ 0\rightarrow \omega^{-1}\otimes V_2\rightarrow \mathcal{S}^1(\pi_{\alpha})\rightarrow \omega^{-1}\otimes \mathbf{1}_{D^{\times}}\rightarrow0. \]

2. (ii) There exists a $G_{\mathbb {Q}_p}\times D^{\times }$-equivariant surjection $\mathcal {S}^1(\operatorname {{\mathrm {Sp}}})\twoheadrightarrow \omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$ whose kernel admits only $\mathbf {1}_{G_{\mathbb {Q}_p}}$ as subquotients when restricted to $G_{\mathbb {Q}_p}$. In particular, $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\mathcal {S}^1(\operatorname {{\mathrm {Sp}}}),\omega ^{-1})$ is one-dimensional.

Proof. Part (i) follows from (8.21) and Corollary 8.24.

Part (ii) follows from (8.15) and Corollary 8.24.

Recall from Propositions 6.1 and 6.2 that we always have $\mathbf {1}_{\mathcal {O}_D^{\times }}\in W_D(\overline {\rho }_1)$ (no matter $\overline {\rho }_1$ is peu ramifié or très ramifié), so

\[ \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}(\mathbf{1}_{\mathcal{O}_D^{\times}},\mathrm{JL}(\overline{\rho}_1))\neq0. \]

Let $W_1$ be the $\mathbf {1}_{\mathcal {O}_D^{\times }}$-typic component of $\operatorname {{\mathrm {soc}}}_{\mathcal {O}_D^{\times }}\mathrm {JL}(\overline {\rho }_1)$. It is easy to see that $W_1$ is stable under $D^{\times }$. Define $V_1$ to be the quotient

(8.29)\begin{equation} V_1:=\mathrm{JL}(\overline{\rho}_1)/W_1. \end{equation}

The main result of this subsection is the following.

Theorem 8.26 There exists a $D^{\times }$-equivariant isomorphism $V_1\cong V_2$.

Proof. Recall the exact sequence (8.13)

\[ 0\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes \mathbf{1}_{D^{\times}}\rightarrow \mathcal{S}^1(\operatorname{{\mathrm{Sp}}})\rightarrow \overline{\rho}_1(-1)\otimes \mathrm{JL}(\overline{\rho}_1)\overset{j}{\rightarrow} \mathcal{S}^1(\kappa)\rightarrow0. \]

By Corollary 8.25(ii), $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\mathcal {S}^1(\operatorname {{\mathrm {Sp}}}),\omega ^{-1})$ is one-dimensional over $\mathbb {F}$, so the last sequence shows that $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\operatorname {{\mathrm {Ker}}}(j),\omega ^{-1})$ is also one-dimensional. Since $\mathcal {S}^1(\kappa )$ is $\omega ^{-1}$-typic by Lemma 8.22, the surjection $j$ factors as

\begin{align*} \overline{\rho}_1(-1)\otimes \mathrm{JL}(\overline{\rho}_1)\twoheadrightarrow \omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1)\overset{j'}{\twoheadrightarrow} \mathcal{S}^1(\kappa). \end{align*}

We clearly have a short exact sequence

\[ 0\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes \mathrm{JL}(\overline{\rho}_1)\rightarrow \operatorname{{\mathrm{Ker}}}(j)\rightarrow \operatorname{{\mathrm{Ker}}}(j')\rightarrow0, \]

which implies that $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\operatorname {{\mathrm {Ker}}}(j'),\omega ^{-1})$ is also one-dimensional. Moreover, by Corollary 8.25(ii) again, it is easy to see that the one-dimensional $\omega ^{-1}$-typic quotient of $\operatorname {{\mathrm {Ker}}}(j')$ is isomorphic to $\omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$. But $\operatorname {{\mathrm {Ker}}}(j')$ is itself $\omega ^{-1}$-typic (being a subrepresentation of $\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho }_1)$), so $\operatorname {{\mathrm {Ker}}}(j')$ is, in fact, isomorphic to $\omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$.

On the other hand, since $\mathcal {S}^1(\kappa )\cong V_2$ as representations of $D^{\times }$ by Corollary 8.24, we have

(8.30)\begin{equation} \operatorname{{\mathrm{soc}}}_{\mathcal{O}_D^{\times}}\mathcal{S}^1(\kappa)=\operatorname{{\mathrm{soc}}}_{\mathcal{O}_D^{\times}}V_2\subset \operatorname{{\mathrm{soc}}}_{\mathcal{O}_D^{\times}}\mathrm{JL}(\overline{\rho}_2)\cong (\alpha\oplus\alpha^{-1})^{\oplus m_2}, \end{equation}

for some integer $m_2\geq 1$, where the last isomorphism is given by Proposition 6.1, Proposition 6.2 and Corollary 6.8. Indeed, taking $r=p-3$ and $s=0$ in Proposition 6.1(ii)(c), we get $W_D(\overline {\rho }_2)=\{\xi ^{p-3}\alpha ^{-1}\zeta,\xi ^{p(p-3)}\alpha \zeta \}=\{\alpha,\alpha ^{-1}\}$. We deduce that the composition

\[ \omega^{-1} \otimes W_1\hookrightarrow \omega^{-1}\otimes \mathrm{JL}(\overline{\rho}_1)\overset{j'}{\twoheadrightarrow} \mathcal{S}^1(\kappa) \]

is zero, where the first morphism is induced from the natural inclusion $W_1\hookrightarrow \mathrm {JL}(\overline {\rho }_1)$, see (8.29). In other words, $\operatorname {{\mathrm {Ker}}}(j')$ contains $\omega ^{-1}\otimes W_1$. Combining with what has been proved in the last paragraph, this implies $\operatorname {{\mathrm {Ker}}}(j')=\omega ^{-1}\otimes W_1$. In particular, $W_1\cong \mathbf {1}_{D^{\times }}$ and

\[ \mathcal{S}^1(\kappa)\cong (\omega^{-1}\otimes\mathrm{JL}(\overline{\rho}_1))/(\omega^{-1}\otimes W_1) \overset{(8.29)}=\omega^{-1}\otimes V_1. \]

Taking into account Corollary 8.24, we obtain

\begin{align*} V_1\cong\operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\omega^{-1},\mathcal{S}^1(\kappa))\cong V_2 \end{align*}

as representations of $D^{\times }$.

Lemma 8.27 Let $\chi :\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ be a smooth character.

1. (i) If $\chi \notin W_D(\overline {\rho }_1)$, then $\operatorname {{\mathrm {Ext}}}^i_{\mathcal {O}_D^{\times }/Z_D^1}(\chi,\mathrm {JL}(\overline {\rho }_1))=0$ for $i\geq 0$.

2. (ii) If $\chi \notin W_D(\overline {\rho }_2)$, then $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\chi,\mathrm {JL}(\overline {\rho }_2))=\operatorname {{\mathrm {Ext}}}^1_{\mathcal {O}_D^{\times }/Z_D^1}(\chi,\mathrm {JL}(\overline {\rho }_2))=0$.

Proof. (i) The proof is as in [HW22, Proposition 10.10(i)]. The point is that $R_{\overline {\rho }_1}^{\psi \varepsilon }$ is formally smooth, so by Proposition 8.9 $\widetilde {H}^1 (U^{v} , \mathcal {O})_{\mathfrak {m}_{\overline {r}_1}}^{d}$ is flat over $\mathbb {T}(U^v)_{\mathfrak {m}_{\overline {r}_1}}$ with fiber isomorphic to $\mathrm {JL}(\overline {\rho }_1)^{\vee }$. Here we write $\overline {r}_1$ instead of $\overline {r}$ to indicate that we are considering the case where $\overline {r}_v(1)=\overline {\rho }_1$.

(ii) The difference to part (i) is that $R_{\overline {\rho }_2}^{\psi \varepsilon }$ is not formally smooth. It is clear that $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\chi,\mathrm {JL}(\overline {\rho }_2))=0$ for $\chi \notin W_D(\overline {\rho }_2)$. For the vanishing of $\operatorname {{\mathrm {Ext}}}^1_{\mathcal {O}_D^{\times }/Z_D^1}(\chi,\mathrm {JL}(\overline {\rho }_2))$, choose a set of generators $(f_1,\ldots,f_m)$ of $\mathfrak {m}_{\overline {r}_2}$, then they induce an exact sequence (recall assumption (H))

\[ 0\rightarrow \mathrm{JL}(\overline{\rho}_2)\rightarrow \Pi_2\rightarrow \prod_{i=1}^m\Pi_2, \]

where $\Pi _2:=\widetilde {H}^1 (U^{v} , \mathbb {F})_{\mathfrak {m}_{\overline {r}_2}}$. Since $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\chi,\Pi _2)=0$ and since $\Pi _2$ is an injective representation of $\mathcal {O}_D^{\times }/Z_D^1$ by Theorem 7.2(ii), the result easily follows.

Thanks to Theorem 8.26, we write $V$ for $V_1$ and $V_2$ from now on.

Corollary 8.28 There exists a short exact sequence

(8.31)\begin{equation} 0\rightarrow \mathbf{1}_{D^{\times}}\rightarrow \mathrm{JL}(\overline{\rho}_1)\rightarrow V\rightarrow0. \end{equation}

Proof. This is a direct consequence of the proof of Theorem 8.26.

Corollary 8.29 The following statements hold:

1. (i) $V_{\rm fd}=0$ and $\mathrm {soc}_{D^{\times }}(V)\cong \operatorname {{\mathrm {Ind}}}_{\mathcal {O}_D^{\times }Z_D}^{D^{\times }} \alpha$;

2. (ii) $\dim _{\mathbb {F}}\operatorname {{\mathrm {Ext}}}^1_{D^{\times }/Z_D}(\operatorname {{\mathrm {Ind}}}_{\mathcal {O}_D^{\times }Z_D}^{D^{\times }} \alpha,V)=1$;

3. (iii) $\dim _{\mathbb {F}}\operatorname {{\mathrm {Ext}}}^1_{D^{\times }/Z_D}(\mathbf {1}_{D^{\times }},V)=2$.

Proof. (i) The first assertion is just Corollary 8.23(iv). For the second assertion, by Frobenius reciprocity it suffices to show $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\alpha,V)$ has dimension $1$. We take $\overline {\rho }_1$ to be très ramifié. By Theorem 6.1, we have $\chi \in W_D(\overline {\rho }_1)$ if and only if $\chi = \mathbf {1}_{\mathcal {O}_D^{\times }}$, so that $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\alpha,\mathrm {JL}(\overline {\rho }_1))=0$. Hence, by applying $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\alpha,-)$ to (8.31) we obtain a long exact sequence

(8.32)\begin{align} & 0\rightarrow \operatorname{{\mathrm{Hom}}}(\alpha,V)\rightarrow \operatorname{{\mathrm{Ext}}}^1(\alpha,\mathbf{1}_{\mathcal{O}_D^{\times}})\rightarrow \operatorname{{\mathrm{Ext}}}^1(\alpha,\mathrm{JL}(\overline{\rho}_1))\rightarrow \operatorname{{\mathrm{Ext}}}^1(\alpha,V) \nonumber\\ &\quad {\buildrel {\partial} \over \longrightarrow} \operatorname{{\mathrm{Ext}}}^2(\alpha,\mathbf{1}_{\mathcal{O}_D^{\times}})\rightarrow \operatorname{{\mathrm{Ext}}}^2(\alpha,\mathrm{JL}(\overline{\rho}_1)), \end{align}

where $\operatorname {{\mathrm {Ext}}}^i$ means $\operatorname {{\mathrm {Ext}}}^i_{\mathcal {O}_D^{\times }/Z_D^1}$. Since $\alpha \notin W_D(\overline {\rho }_1)$, see Theorem 6.1(ii)(a), we have $\operatorname {{\mathrm {Ext}}}^1(\alpha,\mathrm {JL}(\overline {\rho }_1))=0$ by Lemma 8.27(i). The result follows as $\dim _{\mathbb {F}}\operatorname {{\mathrm {Ext}}}^1(\alpha,\mathbf {1}_{\mathcal {O}_D^{\times }})=1$ by Proposition 2.13.

(ii) By Frobenius reciprocity, it is equivalent to proving $\dim _{\mathbb {F}}\operatorname {{\mathrm {Ext}}}^1_{\mathcal {O}_D^{\times }/Z_D^1}(\alpha,V)=1$. Since $\alpha \notin W_D(\overline {\rho }_1)$ (again we take $\overline {\rho }_1$ to be très ramifié), Lemma 8.27(i) implies that the map $\partial$ in (8.32) is an isomorphism. On the other hand, using Propositions 2.13 and 2.14 we know that $\dim _{\mathbb {F}}\operatorname {{\mathrm {Ext}}}^2_{\mathcal {O}_D^{\times }/Z_D^1}(\alpha,\mathbf {1}_{\mathcal {O}_D^{\times }})=1$, from which the assertion follows.

(iii) Note that $\mathbf {1}_{\mathcal {O}_D^{\times }}\notin W_D(\overline {\rho }_2)$, see (8.30). Using Lemma 8.27(ii) this implies

\[ \operatorname{{\mathrm{Ext}}}^i_{\mathcal{O}_D^{\times}/Z_D^1}(\mathbf{1}_{\mathcal{O}_D^{\times}},\mathrm{JL}(\overline{\rho}_2))=0, \]

hence by Frobenius reciprocity $\operatorname {{\mathrm {Ext}}}^i_{D^{\times }/Z_D}(\operatorname {{\mathrm {Ind}}}_{\mathcal {O}_D^{\times }Z_D}^{D^{\times }}\mathbf {1},\mathrm {JL}(\overline {\rho }_2))=0$ for $i=0,1$. Since $\mathbf {1}_{D^{\times }}$ is a direct summand of $\operatorname {{\mathrm {Ind}}}_{\mathcal {O}_D^{\times }Z_D}^{D^{\times }}\mathbf {1}$ as $[D^{\times }:\mathcal {O}_D^{\times }Z_D]=2$ and $p>2$, we deduce

\[ \operatorname{{\mathrm{Hom}}}_{D^{\times}}(\mathbf{1}_{D^{\times}},\mathrm{JL}(\overline{\rho}_2))= \operatorname{{\mathrm{Ext}}}^1_{D^{\times}/Z_D}(\mathbf{1}_{D^{\times}},\mathrm{JL}(\overline{\rho}_2))=0. \]

Now, applying $\operatorname {{\mathrm {Hom}}}_{D^{\times }}(\mathbf {1}_{D^{\times }},-)$ to $0\rightarrow V\rightarrow \mathrm {JL}(\overline {\rho }_2)\rightarrow (\mathbf {1}_{D^{\times }})^{\oplus 2}\rightarrow 0$ gives the result.

Together with Theorem 6.1, we deduce the $D^{\times }$-socle of $\mathrm {JL}(\overline {\rho }_i)$.

Corollary 8.30

1. (i) If $\overline {\rho }_1$ is peu ramifié, then $\operatorname {{\mathrm {soc}}}_{D^{\times }}\mathrm {JL}(\overline {\rho }_1)\cong \mathbf {1}_{D^{\times }}\oplus \operatorname {{\mathrm {Ind}}}_{\mathcal {O}_D^{\times }Z_D}^{D^{\times }}\alpha$; if $\overline {\rho }_1$ is très ramifié, then $\operatorname {{\mathrm {soc}}}_{D^{\times }}\mathrm {JL}(\overline {\rho }_1)\cong \mathbf {1}_{D^{\times }}$.

2. (ii) We have $\operatorname {{\mathrm {soc}}}_{D^{\times }}\mathrm {JL}(\overline {\rho }_2)\cong \operatorname {{\mathrm {Ind}}}_{\mathcal {O}_D^{\times }Z_D}^{D^{\times }}\alpha$.

Remark 8.31 Unlike the generic case treated in § 8.2, we see that $\mathrm {JL}(\overline {\rho }_1)$ detects the extension type of $\overline {\rho }_1$.

We also deduce from Corollary 8.30 that $\operatorname {{\mathrm {soc}}}_{\mathcal {O}_D^{\times }}\mathrm {JL}(\overline {\rho }_1)$ and $\operatorname {{\mathrm {soc}}}_{\mathcal {O}_D^{\times }}\mathrm {JL}(\overline {\rho }_2)$ are multiplicity free. This corresponds to the fact that $\operatorname {{\mathrm {soc}}}_{K}\pi (\overline {\rho }_1)$ and $\operatorname {{\mathrm {soc}}}_{K}\pi (\overline {\rho }_2)$ are multiplicity free, and seems to be a nontrivial fact.

Remark 8.32 We can show that the kernel of $\mathcal {S}^1(\operatorname {{\mathrm {Sp}}})\twoheadrightarrow \omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$ in Corollary 8.25(ii), which we denote by $U$, is $\mathbf {1}_{G_{\mathbb {Q}_p}}$-typic, i.e. it does not admit self-extensions of $\mathbf {1}_{G_{\mathbb {Q}_p}}$ as subquotients when restricted to $G_{\mathbb {Q}_p}$. Indeed, take $\overline {\rho }_1$ to be très ramifié and $\overline {\rho }_1'$ to be peu ramifié, we obtain two embeddings

\[ i,i':\ \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes \mathbf{1}_{D^{\times}}\hookrightarrow\mathcal{S}^1(\operatorname{{\mathrm{Sp}}}) \]

from (8.13). One checks that

\begin{gather*} 0\rightarrow \mathrm{Im}(i)\rightarrow U\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathrm{JL}(\overline{\rho}_1)\rightarrow0,\\ 0\rightarrow \mathrm{Im}(i')\rightarrow U\rightarrow \mathbf{1}_{G_{\mathbb{Q}_p}}\otimes \mathrm{JL}(\overline{\rho}_1')\rightarrow0. \end{gather*}

As a consequence, $\mathrm {Im}(i)\neq \mathrm {Im}(i')$ because $\mathrm {JL}(\overline {\rho }_1)$ and $\mathrm {JL}(\overline {\rho }_1')$ are non-isomorphic by Corollary 8.30(i). It is then easy to deduce that $U$ is isomorphic to $\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes (\mathrm {JL}(\overline {\rho }_1)\times _{V}\mathrm {JL}(\overline {\rho }_1'))$, where the fibered product is taken with respect to (8.31).

8.3.1 Summary

We summarize the results proved above in the following theorem.

Theorem 8.33 We have the following:

1. (i) $\mathcal {S}^0(\mathbf {1}_G)=\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }}$, $\mathcal {S}^1(\mathbf {1}_G)=0$, $\mathcal {S}^2(\mathbf {1}_G)=\omega ^{-1}\otimes \mathbf {1}_{D^{\times }}$;

2. (ii) $\mathcal {S}^0(\operatorname {{\mathrm {Sp}}})=\mathcal {S}^2(\operatorname {{\mathrm {Sp}}})=0$, and there exists a short exact sequence

   \[ 0\rightarrow (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathbf{1}_{D^{\times}})^{\oplus 2}\rightarrow \mathcal{S}^1(\operatorname{{\mathrm{Sp}}}) \rightarrow (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes V)\oplus (\omega^{-1}\otimes\mathbf{1}_{D^{\times}})\rightarrow0; \]

3. (iii) $\mathcal {S}^0(\pi _{\alpha })=\mathcal {S}^2(\pi _{\alpha })=0$ and there exists a short exact sequence

   \[ 0\rightarrow \omega^{-1}\otimes V\rightarrow \mathcal{S}^1(\pi_{\alpha})\rightarrow \omega^{-1}\otimes \mathbf{1}_{D^{\times}}\rightarrow0; \]

4. (iv) there exist exact sequences

   \begin{align*} 0\rightarrow \mathbf{1}_{D^{\times}}\rightarrow \mathrm{JL}(\overline{\rho}_1)\rightarrow V\rightarrow0 \end{align*}
   and
   \begin{align*} 0\rightarrow V\rightarrow \mathrm{JL}(\overline{\rho}_2)\rightarrow (\mathbf{1}_{D^{\times}})^{\oplus 2}\rightarrow0; \end{align*}
   moreover, $\mathrm {JL}(\overline {\rho }_2)$ is isomorphic to the universal extension of $(\mathbf {1}_{D^{\times }})^{\oplus 2}$ by $V$.

8.4 The non-minimal case

We briefly explain how to modify the arguments in §§ 8.2 and 8.3 to handle the non-minimal case (i.e. $d\neq 1$ in Theorem 7.7).

Let $\overline {\rho }=\overline {r}_v(1)$ and assume $\operatorname {{\mathrm {End}}}_{G_{\mathbb {Q}_p}}(\overline {\rho })=\mathbb {F}$; in particular, $\overline {\rho }$ is allowed to be irreducible. We put

(8.33) \begin{equation} \mathrm{JL}(\overline{\rho}):=\left\{\begin{array}{@{}lll} \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi\omega^{-1},\mathcal{S}^1(\pi(\overline{\rho}))) & \mathrm{if}\ \overline{\rho}\sim \big(\begin{smallmatrix} {\chi} & {*}\\ {0} & {\chi\omega}\end{smallmatrix}\big), \\ \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\overline{\rho}(-1),\mathcal{S}^1(\pi(\overline{\rho}))) & \mathrm{otherwise}. \end{array}\right. \end{equation}

Proposition 8.34

1. (i) If $\overline {\rho }$ is not of the form $\big (\begin{smallmatrix} {\chi } & {*}\\ {0} & {\chi \omega }\end{smallmatrix}\big )$, then

   \[ \widetilde{H}^1 (U^{v},\mathbb{F})[\mathfrak{m}_{\overline{r}}]\cong (\overline{\rho}(-1)\otimes\mathrm{JL}(\overline{\rho}))^{\oplus d} \]
   and $\mathcal {S}^1(\pi (\overline {\rho }))\cong \overline {\rho }(-1)\otimes \mathrm {JL}(\overline {\rho })$.

2. (ii) If $\overline {\rho }\sim \big (\begin{smallmatrix} {1} & {*}\\ {0} & {\omega }\end{smallmatrix}\big )$, then

   \[ \widetilde{H}^1 (U^{v},\mathbb{F})[\mathfrak{m}_{\overline{r}}]\cong (\overline{\rho}(-1)\otimes\mathrm{JL}(\overline{\rho}))^{\oplus d}, \]
   and there exists a short exact sequence
   \[ 0\rightarrow \mathcal{S}^1(\pi(\overline{\rho})){\buildrel {f} \over \longrightarrow} \overline{\rho}(-1)\otimes\mathrm{JL}(\overline{\rho})\rightarrow (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathbf{1}_{D^{\times}})^{\oplus 2}\rightarrow0. \]

Proof. (i) We claim that $\mathcal {S}^1(\widetilde {S}(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}])=\widetilde {H}^1 (U^{v},\mathbb {F})[\mathfrak {m}_{\overline {r}}]$. If $\overline {\rho }^{\rm ss}\nsim \chi \oplus \chi \omega$, it is proved in [Paš22, Lemma 6.1]. If $\overline {\rho }^{\rm ss}\sim \chi \oplus \chi \omega$, then the assumption on $\overline {\rho }$ implies that $R_{\overline {\rho }}^{\psi \varepsilon ^{-1}}$ is formally smooth, so we may apply Proposition 8.9 (using Corollary 7.9).

The claim implies that $\mathcal {S}^1(\widetilde {S}(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}])$ is $\overline {\rho }(-1)$-typic. Since $\widetilde {S} (U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}]\cong \pi (\overline {\rho })^{\oplus d}$ by Theorem 7.7, $\mathcal {S}^1(\pi (\overline {\rho }))$ is also $\overline {\rho }(-1)$-typic by Lemma 8.6(ii). The result easily follows.

(ii) First, the proof of Proposition 8.20 shows that

\[ 0\rightarrow \mathcal{S}^1(\widetilde{S}(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]){\buildrel {\phi} \over \longrightarrow} \widetilde{H}^1 (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]\rightarrow (\mathbf{1}_{G_{\mathbb{Q}_p}}\otimes\mathbf{1}_{D^{\times}})^{\oplus 2d}\rightarrow0, \]

which implies

\begin{align*} \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\omega^{-1},\mathcal{S}^1(\widetilde{S} (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}])) & \overset{\phi^*}{\xrightarrow{\sim}}\operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\omega^{-1}, \widetilde{H}^1 (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}])\\ &\simeq\operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\overline{\rho}(-1),\widetilde{H}^1 (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]), \end{align*}

where the second isomorphism holds because $\widetilde {H}^1(U^{v},\mathbb {F})[\mathfrak {m}_{\overline {r}}]$ is $\overline {\rho }(-1)$-typic. Choose an isomorphism $\iota : \pi (\overline {\rho })^{\oplus d}\xrightarrow {\sim } \widetilde {S}(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}]$; it induces an isomorphism

\[ \mathrm{JL}(\overline{\rho})^{\oplus d}\overset{\iota^*}{\xrightarrow{\sim}} \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\omega^{-1},\mathcal{S}^1(\widetilde{S} (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}])). \]

Thus, we get an isomorphism

(8.34)\begin{equation} \widetilde{H}^1 (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]\cong \overline{\rho}(-1)\otimes \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\overline{\rho}(-1),\widetilde{H}^1(U^{v},\mathbb{F})[\mathfrak{m}_{\overline{r}}]) \overset{(\phi^*\circ\iota^*)^{-1}}{\xrightarrow{\sim}} \overline{\rho}(-1)\otimes \mathrm{JL}(\overline{\rho})^{\oplus d} \end{equation}

as desired. Let $f'$ be the composite map

\[ \mathcal{S}^1(\pi(\overline{\rho}))^{\oplus d}\overset{\iota}{\xrightarrow{\sim}} \mathcal{S}^1(\widetilde{S} (U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}]) {\buildrel {\phi} \over \longrightarrow} \widetilde{H}^1(U^{v}, \mathbb{F})[\mathfrak{m}_{\overline{r}}] \overset{(8.34)}{\xrightarrow{\sim}} \overline{\rho}(-1)\otimes \mathrm{JL}(\overline{\rho})^{\oplus d}. \]

Since $\mathcal {S}^1(\pi (\overline {\rho }))$ is contained in $\widetilde {H}^1(U^{v}, \mathbb {F})[\mathfrak {m}_{\overline {r}}]$ which is $\overline {\rho }(-1)$-typic, we may apply Lemma 8.35 below to obtain an embedding

\[ 0\rightarrow \mathcal{S}^1(\pi(\overline{\rho})) {\buildrel {f} \over \longrightarrow} \overline{\rho}(-1)\otimes \mathrm{JL}(\overline{\rho}), \]

extending the natural embedding $\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho })\hookrightarrow \overline {\rho }(-1)\otimes \mathrm {JL}(\overline {\rho })$. Moreover, $f$ is $G_{\mathbb {Q}_{p}}\times D^{\times }$-equivariant by construction. We are left to show $\mathrm {Coker}(f)\cong (\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }})^{\oplus 2}$. It is clear that $f^{\oplus d}$ and $f'$ coincide when restricted to $\omega ^{-1}\otimes \mathrm {JL}(\overline {\rho })^{\oplus d}$, so $f'=f^{\oplus d}$ by the uniqueness part of Lemma 8.35. Since $\operatorname {{\mathrm {Coker}}}(f')\cong (\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }})^{\oplus 2d}$, we obtain $\mathrm {Coker}(f)\cong (\mathbf {1}_{G_{\mathbb {Q}_p}}\otimes \mathbf {1}_{D^{\times }})^{\oplus 2}$ as required.

Lemma 8.35 Let $\overline {\rho }\sim \big (\begin{smallmatrix} {\chi _1} & {*}\\ {0} & {\chi _2}\end{smallmatrix}\big )$ with $\operatorname {{\mathrm {End}}}_{G_{\mathbb {Q}_p}}(\overline {\rho })\cong \mathbb {F}$. If $M$ is a $\overline {\rho }$-typic $\mathbb {F}[G_{\mathbb {Q}_p}]$-module, then for any submodule $M'\subset M$ there exists a unique embedding

\[ 0\rightarrow M'\rightarrow \overline{\rho}\otimes \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_1,M') \]

extending the embedding $\chi _1\otimes \operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _1,M')\hookrightarrow \overline {\rho }\otimes \operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _1,M')$ induced from $\chi _1\hookrightarrow \overline {\rho }$.

Proof. Since $M$ is $\overline {\rho }$-typic, it is naturally isomorphic to $\overline {\rho }\otimes M_0$ where $M_0:=\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\overline {\rho },M)$. Actually, the assumption on $\overline {\rho }$ implies that $M_0= \operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _1,M)$. Writing $M'_0:=\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _1,M')$, we claim that $M'$ is contained in $\overline {\rho }\otimes M_0'$, both regarded as subspaces of $\overline {\rho }\otimes M_0$. Indeed, letting $\widetilde {M}':=M'+\overline {\rho }\otimes M_0'$, we need to prove $\widetilde {M}'=\overline {\rho }\otimes M_0'$. It is clear that $\chi _1\otimes M_0'$ is identified with $M'\cap (\chi _1\otimes M_0)$, thus $M'/(\chi _1\otimes M_0')$ embeds in $\chi _2\otimes M_0$ and is $\chi _2$-typic. Using the natural isomorphism $\widetilde {M}'/(\overline {\rho }\otimes M_0')\cong M'/(M'\cap (\overline {\rho }\otimes M_0'))$, we see that $\widetilde {M}'/(\overline {\rho }\otimes M_0')$ is a quotient of $M'/(\chi _1\otimes M_0')$, thus

\[ \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_1,\widetilde{M}'/(\overline{\rho}\otimes M_0'))=0. \]

On the other hand, if $\widetilde {M}'/(\overline {\rho }\otimes M_0')$ is nonzero, then it embeds in $\overline {\rho }\otimes (M_0/M_0')$ and we must have $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(\chi _1,\widetilde {M}'/(\overline {\rho }\otimes M_0'))\neq 0$, a contradiction.

The claim implies that the given inclusion $M'\subset M$ provides an embedding required in the lemma, so we are left to prove the uniqueness.

Consider the exact sequence

\[ 0\rightarrow \chi_1\otimes M_0'\rightarrow M'\rightarrow Q\rightarrow0 \]

with $Q$ being the quotient. As seen above, $Q$ is $\chi _2$-typic. Applying $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(-,\overline {\rho }\otimes M_0')$ to it, we obtain an exact sequence

(8.35)\begin{equation} 0\rightarrow \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(Q,\overline{\rho}\otimes M_0')\rightarrow \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(M',\overline{\rho}\otimes M_0') {\buildrel {\gamma} \over \longrightarrow} \operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_1\otimes M_0',\overline{\rho}\otimes M_0'). \end{equation}

The result follows because $\operatorname {{\mathrm {Hom}}}_{G_{\mathbb {Q}_p}}(Q,\overline {\rho }\otimes M_0')=0$ (as $Q$ is $\chi _2$-typic).

Using Proposition 8.34, the arguments in §§ 8.2 and 8.3 when $\overline {\rho }$ is reducible with $\operatorname {{\mathrm {End}}}_{G_{\mathbb {Q}_p}}(\overline {\rho })=\mathbb {F}$, taking into account multiplicities everywhere, go through and give similar results in the non-minimal case as in Theorems 8.12, 8.26 and 8.33.

Remark 8.36 If $\overline {\rho }_0=\chi _1\oplus \chi _2$ with $\chi _1\chi _2^{-1}\neq \mathbf {1},\omega ^{\pm 1}$, we put

\[ \mathrm{JL}(\overline{\rho}_0):=\operatorname{{\mathrm{Hom}}}_{G_{\mathbb{Q}_p}}(\chi_1\omega^{-1},\mathcal{S}^1(\pi(\overline{\rho}))). \]

Combining with Proposition 8.34, the proof of Theorem 8.12 shows $\mathcal {S}^1(\pi (\overline {\rho }_0))=\overline {\rho }_0(-1)\otimes \mathrm {JL}(\overline {\rho }_0)$.