# Serre weights and Breuil's lattice conjecture in dimension three

Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra

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Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra. Serre weights and Breuil's lattice conjecture in dimension three. Forum of Mathematics, Pi, 2020, 8, pp.e5. 10.1017/fmp.2020.1. hal-03840824

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Distributed under a Creative Commons Attribution 4.0 International License
SERRE WEIGHTS AND BREUIL'S LATTICE
CONJECTURE IN DIMENSION THREE

DANIEL LE¹, BAO V. LE HUNG²,
BRANDON LEVIN³ and STEFANO MORRA⁴

¹ Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada;
email: le@math.toronto.edu

² Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA;
email: lhvietbao@googlemail.com

³ Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, P.O. Box 210089, Tucson, Arizona 85721, USA;
email: bwlevin@math.arizona.edu

⁴ Université Paris 8, Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France;
email: morra@math.univ-paris13.fr

Received 31 August 2016; accepted 28 January 2020

Abstract

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a *U*(3)-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above *p*. This is a generalization to $GL_3$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights (2, 1, 0) as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame *n*-dimensional Galois representations’, *Duke Math. J.* **149**(1) (2009), 37–116] over an unramified field extending the results of Le *et al*. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, *Invent. Math.* **212**(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $GL_3(\mathbb{F}_q)$.

2010 Mathematics Subject Classification: 11F80 (primary); 11F33, 20C33 (secondary)
# 1. Introduction

One of the most important developments in the Langlands program in recent years has been the *p*-adic local Langlands correspondence for GL<sub>2</sub>(*Q*<sub>*p*</sub>). Unfortunately, extending this correspondence even to GL<sub>2</sub>(*K*) has proven to be exceedingly difficult and all evidence suggests that the desired correspondence will be much more complicated. On the other hand, there has been some progress on several avatars of the *p*-adic local Langlands correspondence, namely, (generalized) Serre weight conjectures, geometric Breuil–Mézard conjecture, and Breuil’s lattice conjecture. These conjectures inform our understanding of what the sought after *p*-adic correspondence should look like. In this paper, we prove versions of each of these three conjectures for GL<sub>3</sub>(*K*) when *K*/*Q*<sub>*p*</sub> is unramified.

**1.1. Breuil's lattice conjecture.** Motivated by Emerton's local-global compatibility for completed cohomology, [CEG+16] constructs a candidate for one direction of the *p*-adic local Langlands correspondence for GL<sub>*n*</sub>(*K*). Namely, they associate to any continuous *n*-dimensional *Q*<sub>*p*</sub>-representation *ρ* of Gal(<span style="text-decoration: overline">*K*</span>/*K*<sub>0</sub>) an admissible Banach space representation *V*(*ρ*) of GL<sub>*n*</sub>(*K*) by patching completed cohomology. However, the construction depends on a choice of global setup, and one expects it to be a deep and difficult problem to show that the correspondence *ρ* ↦ *V*(*ρ*) is purely local.

In [Bre14], Breuil formulates a conjecture on lattices in tame types cut out by completed cohomology of Shimura curves which is closely related to the local nature of $V(\rho)$. This conjecture was proven subsequently in the groundbreaking work of Emerton–Gee–Savitt [EGS15]. Our first main theorem is a generalization of Breuil’s conjecture to three-dimensional Galois representations and the completed cohomology of $U(3)$-arithmetic manifolds.

Specifically, let $p$ be a prime, $F/F^+$ a CM extension (i.e. $F$ a totally imaginary quadratic extension of a totally real field $F^+$) unramified everywhere, and $r : G_F \to \text{GL}_3(\overline{\mathbb{Q}}_p)$ a Galois representation. Let $\lambda$ be the Hecke eigensystem corresponding to $r$, which appears in the cohomology of a $U(3)$-arithmetic manifold. Choose a place $v|p$ of $F^+$ which splits in $F$, and let $\tilde{H}$ be the integral $p$-adically completed cohomology with infinite level at $v$. One expects completed cohomology to realize a global $p$-adic Langlands correspondence generalizing the case of $\text{GL}_2/\mathbb{Q}$. That is, by letting $\tilde{v}$ denote a place of $F$ above $v$ and by letting $G_{F_{\tilde{v}}}^-$ be the absolute Galois group of $F_{\tilde{v}}$, the $\text{GL}_3(F_{\tilde{v}})$-representation on the Hecke eigenspace $\tilde{H}[\lambda]$ corresponds to $r|_{G_{F_{\tilde{v}}}^-}$ via a hypothetical $p$-adic local Langlands correspondence (when the level outside $v$ is chosen minimally). In particular, the globally constructed object $\tilde{H}[\lambda]$ should depend only on $r|_{G_{F_{\tilde{v}}}^-}$.
Suppose that $r$ is tamely potentially crystalline with Hodge–Tate weights (2, 1, 0) at each place above $p$. For simplicity, in Section 1, we suppose that $r$ is unramified away from $p$, although our results hold if $r$ is minimally split ramified. Assume that each place $v|p$ in $F^+$ splits in $F$ and fix a place $\tilde{v}|v$ for all $v|p$ in $F^+$. Let $\sigma(\tau)$ be the tame type corresponding to the Weil–Deligne representations associated with $r|_{G_{F_{\tilde{v}}}}$ for all $v|p$ under the inertial local Langlands correspondence. Throughout Section 1, the tame type $\sigma(\tau)$ is assumed to be sufficiently generic. If $r$ is modular, then by classical local–global compatibility, $\tilde{H}[\lambda][1/p]$ contains $\sigma(\tau)$ with multiplicity one.

**THEOREM 1.1.1** (Breuil's conjecture, cf. Theorem 5.3.5). Assume that $p$ is unramified in $F^+$ and that $\bar{r}$ satisfies Taylor–Wiles hypotheses and is semisimple at places above $p$. Assume that the level of $\tilde{H}$ outside $p$ is minimal with respect to $r$. Then, the lattice

$$ \sigma(\tau)^0 := \sigma(\tau) \cap \tilde{H}[\lambda] \subset \sigma(\tau) $$

depends only on the collection $\{r|_{G_{F_{\tilde{v}}}}\}_{v|p}$.

Let $\tilde{H}$ be the mod $p$ reduction of $\tilde{H}$ so that $\bar{H}$ is the mod $p$ cohomology with infinite level at places above $p$ of a $U(3)$-arithmetic manifold. We prove the following 'mod $p$ multiplicity one' result (cf. Theorem 5.3.4).

**THEOREM 1.1.2** (Theorem 5.3.4). Keep the assumptions of Theorem 1.1.1. Let $\sigma(\tau)^\sigma$ be a lattice in $\sigma(\tau)$ such that its mod $p$ reduction $\bar{\sigma}(\tau)^\sigma$ has an irreducible upper alcove cosocle. Then $\text{Hom}_{K_p}(\bar{\sigma}(\tau)^\sigma, \tilde{H}[\lambda])$ is one-dimensional.

These theorems should be compared to [EGS15, Theorems 8.2.1 and 10.2.1] in dimension two. In the special case where $p$ is split in $F^+$ and $\bar{r}$ is irreducible above $p$, both theorems were proven by the first author in [Le18].

The main ingredients used in [EGS15] are the Taylor–Wiles patching method, the geometric Breuil–Mézard conjecture for potentially Barsotti–Tate Galois deformation rings (building on the work of [Bre14]), and a classification of lattices in tame types (extending [Bre14, BP13]). When we began this project, only the first of these tools was available in the case of $GL_3$. The analogue of potentially Barsotti–Tate Galois deformation rings are potentially crystalline deformation rings with Hodge–Tate weights (2, 1, 0). In [LLHLM18], we develop a technique for computing these Galois deformation rings when the descent data is tame and sufficiently generic. We discuss in Section 1.2 the geometric Breuil–Mézard conjecture for these rings. The representation-theoretic results are discussed in Section 1.3.
Several key differences which distinguish our situation from [EGS15] are worth mentioning. Breuil’s conjecture for GL<sub>2</sub> gave an explicit description of the lattice σ(τ)<sup>0</sup> in terms of the Dieudonné module of r|<sub>GF<sub>v</sub></sub>. We prove abstractly that σ(τ)<sup>0</sup> is ‘purely local’ (Theorems 5.2.3 and 5.3.5) but without giving any explicit description of the lattice. The lattice σ(τ)<sup>0</sup> is determined by the parameters of the Galois deformation ring but in a complicated way.

Let $\sigma(\tau)^{\sigma}$ be a lattice in $\sigma(\tau)$ whose reduction has irreducible cosocle $\sigma$.
To prove Theorem 1.1.2, we show that a certain (minimal) patched module
$M_{\infty}(\sigma(\tau)^{\sigma})$ is free of rank one over the local Galois deformation ring (with
patching variables) $R_{\infty}(\tau)$ (Theorem 5.1.1). In fact, this result is also a key step
in our proof of Theorem 1.1.1. In loc. cit., the analogue of this result is Theorem
10.1.1 where they show that the patched module of any lattice with irreducible
cosocle is free of rank one. In our situation, it is no longer true that all such
patched modules are cyclic. Rather, this is only true when the cosocle $\sigma$ is upper
alcove in every embedding. As a consequence of this, one can deduce that the
isomorphism class as an $R_{\infty}(\tau)$-module of $M_{\infty}(\sigma(\tau)^{\sigma})$ is purely local for any
lattice $\sigma(\tau)^{\sigma}$; however, it need not be free.

For the proof that $M_\infty(\sigma(\tau)^\sigma)$ is free of rank one when $\sigma$ is upper alcove, we induct on the complexity of the deformation ring. The simplest deformation rings resemble those for $\text{GL}_2$ and so we follow the strategy similar to [EGS15]. For the most complicated deformation rings, we build up $M_\infty(\sigma(\tau)^\sigma)$ from its subquotients relying on the description of the submodule structure of reduction $\bar{\sigma}(\tau)^\sigma$ discussed in Section 1.3 and crucially intersection theory results for components of mod $p$ fiber of the Galois deformation ring.

**1.2. Serre weight and Breuil–Mézard conjectures.** There is an analogous global context for a mod $p$ Galois representation $\bar{r} : G_F \to \text{GL}_3(\mathbb{F})$ whose corresponding Hecke eigensystem $m$ appears in the mod $p$ cohomology with infinite level at $v$ of a $U(3)$-arithmetic manifold. One expects $\bar{H}[m]$ to correspond to $\bar{r}|_{\text{GF}_v}$ via a hypothetical mod $p$ local Langlands correspondence. Furthermore, if we let $W_v(\bar{r})$ be the set of Jordan–Hölder factors in the $\text{GL}_3(\mathcal{O}_{\text{F}_v^+})$-socle of $\bar{H}[m]$ and $\bar{r}|_{\text{GF}_v}$ is tamely ramified, [Her09, GHS18] predict that $W_v(\bar{r}) = W^?(\bar{r}|_{\text{IF}_v})$, where $\text{IF}_v$ denotes the inertia subgroup of $\text{G}_{\text{F}_v}$ and $W^?(\bar{r}|_{\text{IF}_v})$ is a set which is explicitly defined in terms of $\bar{r}|_{\text{IF}_v}$ (Strictly speaking, the Hecke eigenspace $\bar{H}[m]$ is a smooth representation of $\text{G}(\text{F}_v^+)$, where $\text{G}_{\text{IF}_v^+}$ is a reductive group having a reductive model $\mathcal{G}_v$ over $\mathcal{O}_{\text{F}_v^+}$ together with an isomorphism $\iota_v : \mathcal{G}_v(\mathcal{O}_{\text{F}_v^+}) \to \text{GL}_3(\mathcal{O}_{\text{F}_v^+})$). We have the following version of the weight part of Serre’s conjecture (cf. Theorem 5.3.3).
**THEOREM 1.2.1** (The weight part of Serre's conjecture). Assume that $p$ is unramified in $F^+$ and that $\bar{r}$ satisfies Taylor-Wiles hypotheses, has split ramification, and is semisimple and sufficiently generic at places above $p$. Then $W_v(\bar{r}) = W^2(\bar{r}|_{I_{F^{\bar{v}}}})$.

In [LLHLM18], we prove this theorem with the additional assumption that $p$ is split in $F^+$. The strategy is to show the numerical Breuil-Mézard conjecture for the simplest deformation rings (where the shape has length of at least 2) using [LLHLM18, Section 6.2]. The key new tool is a more conceptual and robust combinatorial technique for computing the intersection between the predicted weights $W^j(\bar{r}|_{I_{F^{\bar{v}}}})$ and the Jordan-Hölder factors of a type and which is developed in Section 2. This allows us to inductively prove that all predicted weights are modular.

Using a patching functor which is constructed globally, we show that the generic fibers of tamely potentially crystalline deformation rings of Hodge-Tate weight (2, 1, 0) are connected for $\bar{\rho}$ generic and deduce the full numerical Breuil-Mézard conjecture for these Galois deformation rings. Using the numerical formulation, we prove the following geometric version of the Breuil-Mézard conjecture (cf. [EG14], Proposition 3.6.1).

**THEOREM 1.2.2** (Proposition 3.6.1). Assume that $\bar{r}|_{G_{F^{\bar{v}}}}$ is semisimple and sufficiently generic. There is a unique assignment $\sigma \mapsto \mathfrak{p}(\sigma)$ taking Serre weights $\sigma \in W^j(\bar{r}|_{G_{F^{\bar{v}}}})$ to prime ideals in the unrestricted framed deformation ring $R_{\bar{r}|_{G_{F^{\bar{v}}}}}^{\square}$ such that the special fiber $\text{Spec}(\bar{R}_{\bar{r}|_{G_{F^{\bar{v}}}}}^{\tau})$ of the potentially crystalline framed deformation ring $R_{\bar{r}|_{G_{F^{\bar{v}}}}}^{\tau}$ of Hodge-Tate weight (2, 1, 0) and tame type $\tau$ is the reduced underlying subscheme of

$$ \bigcup_{\sigma \in W^j(\bar{r}|_{I_{F^{\bar{v}}}}) \cap \text{JH}(\sigma(\tau))} \text{Spec} \left( R_{\bar{r}|_{G_{F^{\bar{v}}}}}^{\square} / \mathfrak{p}(\sigma) \right). $$

Moreover, this is compatible with any patching functor.

**1.3. Representation theory results.** In order to deduce Breuil's lattice conjecture from the Breuil-Mézard conjecture, we need (and prove) new results on integral structures in Deligne-Lusztig representations, which may be of independent interest. The main theorem (Theorem 4.1.9) is a classification of integral lattices with irreducible cosocle in tame types, by means of an extension graph, which plays a key role in the proofs of Theorems 1.1.1 and 1.1.2.

We now briefly describe the extension graph. In Section 2.1, we introduce a graph on the set of $p$-regular Serre weights (with fixed central character), with
vertices corresponding to *p*-regular Serre weights and adjacency between
vertices described in a combinatorially explicit way. We then show in
Lemma 4.2.6 that two vertices are adjacent if and only if the corresponding
Serre weights have a nontrivial $GL_3(\mathbb{F}_q)$-extension between them, justifying the
terminology. This gives a natural notion of *graph distance* $d_{\text{gph}}$ between two
*p*-regular Serre weights. Theorem 4.1.9 states the following.

**THEOREM 1.3.1** (Theorem 4.1.9). Assume that $R$ is a sufficiently generic Deligne–Lusztig representation of $GL_3(\mathbb{F}_q)$. (In particular, the Jordan–Hölder factors of $\overline{R}$ occur with multiplicity one.) If $\sigma$ is a Jordan–Hölder factor of $\overline{R}$, let $\overline{R}^{\sigma}$ be the unique lattice up to homothety with cosocle $\sigma$. If $\sigma, \sigma' \in \text{JH}(\overline{R})$ and that $d_{\text{gph}}(\sigma, \sigma') = d$, then:

(1) $\sigma'$ is a direct summand of the $d$th layer of the cosocle filtration of $\overline{R}^{\sigma}$;

(2) if $\sigma'' \in \text{JH}(\overline{R})$ is such that $d_{\text{gph}}(\sigma, \sigma'') = d+1$ and $d_{\text{gph}}(\sigma', \sigma'') = 1$, then $\overline{R}^{\sigma}$ has a subquotient which is isomorphic to the unique nonsplit extension of $\sigma'$ by $\sigma''$; and

(3) if $R^{\sigma'} \subseteq R^{\sigma}$ is a saturated inclusion of lattices, then $p^d R^{\sigma} \subseteq R^{\sigma'}$ is also a saturated inclusion of lattices.

The argument is involved, using a mixture of local and global techniques, but we can distinguish two main steps in its proof. In the first step (Sections 4.2.3 and 4.2.4), we prove the first two items of Theorem 1.3.1 in the case when $\sigma$ is a lower alcove weight of *defect zero* (cf. Definition 2.3.8 and Theorem 4.2.16). The proof uses methods from the modular representation theory of algebraic groups, embedding $\overline{R}^{\sigma}$ in a Weyl module with non-$p$-restricted highest weights. The key local argument is a careful study of the restriction of algebraic representations to rational points (Proposition 4.2.10), which lets us constrain the submodule structure of (part of) the $GL_3(\mathbb{F}_q)$-restriction of an algebraic Weyl module in terms of the extension graph. This method does not work for all weights $\sigma \in \text{JH}(\overline{R})$, as the corresponding lattices will not always have simple socle and thus cannot be embedded into a Weyl module.

In the second step (Section 4.3), we reduce the theorem for the remaining lattices to the case treated in the first step. We relate the first two items and the last item of Theorem 1.3.1. The last item, a statement in characteristic zero, is amenable to an inductive analysis. First, we show that for a fixed weight $\sigma \in \text{JH}(\overline{R})$, item (3) of Theorem 1.3.1 actually implies the other two items (cf. Proposition 4.3.16). This crucially uses Theorem 1.3.1(3) in the case $d=1$ (cf. Proposition 4.3.7), which is proved using the computation of deformation
rings in [LLHLM18], combinatorics of Section 3, and the Kisin–Taylor–Wiles patching method. This argument follows the suggestion in [EGS15, Section B.2] that tamely potentially crystalline deformation rings strongly reflect aspects of local representation theory through global patching constructions.

Next, we show that the first two items of Theorem 1.3.1 applied to $R$ and its dual in the case of lower alcove defect zero weights imply Theorem 1.3.1(3) in the case of lower alcove defect zero weights. From this starting point, an inductive argument proves Theorem 1.3.1(3), thus concluding the proof of Theorem 1.3.1.

**1.4. Notation.** If $F$ is any field, we write $G_F \stackrel{\text{def}}{=} \text{Gal}(\overline{F}/F)$ for the absolute Galois group, where $\overline{F}$ is a separable closure of $F$. If $F$ is a number field and $v$ is a prime of $F$, we write $G_{F_v}$ for the decomposition group at $v$ and $I_{F_v}$ for the inertia subgroup of $G_{F_v}$. If $F$ is a $p$-adic field, we write $I_F$ to denote the inertia subgroup of $G_F$.

We fix once and for all an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$. All number fields are considered as subfield of our fixed $\overline{\mathbb{Q}}$. Similarly, if $\ell \in \mathbb{Q}$ is a prime, we fix algebraic closures $\overline{\mathbb{Q}}_\ell$ as well as embeddings $\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_\ell$. All finite extensions of $\overline{\mathbb{Q}}_\ell$ will thus be considered as subfields in $\overline{\mathbb{Q}}_\ell$. Moreover, the residue field of $\overline{\mathbb{Q}}_\ell$ is denoted by $\overline{F}_\ell$.

Let $p > 3$ be a prime. For $f > 0$, we let $K$ be the unramified extension of $\mathbb{Q}_p$ of degree $f$. We write $k$ for its residue field (of cardinality $q = p^f$) and $\mathcal{O}_K = W(k)$ for its ring of integers. For $r \ge 1$, we set $e_r \stackrel{\text{def}}{=} p^{fr} - 1$ and fix a compatible system of roots $\pi_r \stackrel{\text{def}}{=} (-p)^{\frac{1}{\sigma}} \in \overline{K}$. We write $e$ for $e_1$ and $\pi$ for $\pi_1$. Define the extension $L = K(\pi)$ and set $\Delta_0 \stackrel{\text{def}}{=} \text{Gal}(L/K)$. The choice of the root $\pi$ lets us define a character

$$ \begin{gathered} \tilde{\omega}_{\pi} : \Delta_0 \to W(k)^{\times} \\ g \mapsto \frac{g(\pi)}{\pi} \end{gathered} $$

whose associated residual character is denoted by $\omega_\pi$. In particular, for $f = 1$, $\omega_\pi$ is the mod $p$ cyclotomic character, which will be simply denoted by $\omega$. If $F_w/\mathbb{Q}_p$ is a finite extension and $W_{F_w} \le G_{F_w}$ denotes the Weil group, we normalize Artin's reciprocity map $\text{Art}_{F_w} : F_w^{\times} \to W_{F_w}^{\text{ab}}$ in such a way that uniformizers are sent to geometric Frobenius elements.

Let $E \subset \mathbb{Q}_p$ be a finite extension of $\mathbb{Q}_p$, which will be our coefficient field. We write $\mathcal{O}$ for its ring of integers, fix an uniformizer $\omega \in \mathcal{O}$, and let $\mathfrak{m}_E = (\omega)$. We write $\mathbb{F} \stackrel{\text{def}}{=} \mathcal{O}/\mathfrak{m}_E$ for its residue field. We will always assume that $E$ is
sufficiently large. In particular, we will assume that any embedding $\sigma: K \hookrightarrow \overline{\mathbb{Q}}_p$ factors through $E \subset \overline{\mathbb{Q}}_p$.

We fix an embedding $\sigma_0: K \hookrightarrow E$. The embedding $\sigma_0$ induces maps $O_K \hookrightarrow O$ and $k \hookrightarrow \mathbb{F}$; we will abuse notation and denote all these by $\sigma_0$. We let $\varphi$ denote the $p$th power Frobenius on $k$ and set $\sigma_i \stackrel{\text{def}}{=} \sigma_0 \circ \varphi^{-i}$. The choice of $\sigma_0$ gives $\omega_f \stackrel{\text{def}}{=} \sigma_0 \circ \tilde{\omega}_\pi: I_K \to O^\times$, a fundamental character of niveau $f$. We fix once and for all a sequence $\underline{p} \stackrel{\text{def}}{=} (p_n)_{n \in \mathbb{N}}$ where $p_n \in \overline{\mathbb{Q}}_p$ satisfies $p_{n+1}^p = p_n$ and $p_0 = -p$. We let $K_\infty \stackrel{\text{def}}{=} \bigcup_{n \in \mathbb{N}} K(p_n)$ and $G_{K_\infty} \stackrel{\text{def}}{=} \text{Gal}(\overline{\mathbb{Q}}_p/K_\infty)$.

Let $\rho: G_K \to \text{GL}_n(E)$ be a $p$-adic, de Rham Galois representation. For $\sigma: K \hookrightarrow E \subset \overline{\mathbb{Q}}_p$, we define $\text{HT}_\sigma(\rho)$ to be the multiset of $\sigma$-labeled Hodge–Tate weights of $\rho$, that is, the set of integers $i$ such that $\dim_E (\rho \otimes_{\sigma,K} \mathbb{C}_p(-i))^{G_K} \neq 0$ (with the usual notation for Tate twists). In particular, the cyclotomic character has Hodge–Tate weights 1 for all embedding $\sigma \hookrightarrow E$. For $\mu = (\mu_j)_j \in X^*(\mathcal{T})$, we say that $\rho$ has Hodge–Tate weighs $\mu$ if

$$ \text{HT}_{\sigma_j}(\rho) = \{\mu_{1,j}, \mu_{2,j}, \dots, \mu_{n,j}\}. $$

The inertial type of $\rho$ is the isomorphism class of $\text{WD}(\rho)|_{I_K}$, where $\text{WD}(\rho)$ is the Weil–Deligne representation attached to $\rho$ as in [CDT99], Appendix B.1 (in particular, $\rho \mapsto \text{WD}(\rho)$ is covariant). An inertial type is a morphism $\tau: I_K \to \text{GL}_n(E)$ with open kernel and which extends to the Weil group $W_K$ of $G_K$. We say that $\rho$ has type $(\mu, \tau)$ if $\rho$ has Hodge–Tate weights $\mu$ and inertial type given by (the isomorphism class of) $\tau$.

Let $G \stackrel{\text{def}}{=} \text{GL}_{3/\mathbb{F}}$, denote by $T \subseteq G$ the torus of diagonal matrices, and write $W$ (respectively $W_a$, respectively $\tilde{W}$) for the Weyl group (respectively the affine Weyl group, respectively the extended affine Weyl group) of $G$. We let $X^*(T)$ denote the group of characters of $T$, which we identify with $\mathbb{Z}^3$ in the usual way. Let $R$ (respectively $R^\vee$) denote the set of roots (respectively coroots) of $G$ and $\Lambda_R \subseteq X^*(T)$ the root lattice. We then have

$$ W_a = \Lambda_R \rtimes W(G), \quad \tilde{W} = X^*(T) \rtimes W(G). \qquad (1.1) $$

Let $\varepsilon'_1$ and $\varepsilon'_2$ be $(1, 0, 0)$ and $(0, 0, -1)$, respectively. Let $G^{\text{der}}$ be $\text{SL}_{3/\mathbb{F}}$ with torus $T^{\text{der}}$. Let $\Lambda_W = X^*(T^{\text{der}})$ denote the weight lattice for $G^{\text{der}}$. Let $X^0(T)$ denote the kernel of the restriction map $X^*(T) \to \Lambda_W$. We write $\varepsilon_1$ and $\varepsilon_2$ for the images of $\varepsilon'_1$ and $\varepsilon'_2$ in $\Lambda_W$, respectively. We define in a similar fashion the various Weyl groups $W^{\text{der}}$, $W_a^{\text{der}}$, $\tilde{W}^{\text{der}}$ for $G^{\text{der}}$. Note that we have canonical isomorphisms $W \cong W^{\text{der}}$ and $W_a \cong W_a^{\text{der}}$.

Let $\mathcal{S}$ be a finite set. For each $\tilde{v} \in \mathcal{S}$, let $F_{\tilde{v}}$ be an unramified extension of $\mathbb{Q}_p$ of degree $f_{\tilde{v}}$, and let $k_{\tilde{v}}$ be the residue field of $F_{\tilde{v}}$. Let $\mathcal{J}$ be the set of ring
homomorphisms $\prod_{\tilde{v} \in S} k_{\tilde{v}} \to \mathbb{F}$. If we fix an embedding $\sigma_{\tilde{v},0} : k_{\tilde{v}} \hookrightarrow \mathbb{F}$ and set $\sigma_{\tilde{v},i}$ to be $\sigma_{\tilde{v},0} \circ \varphi^{-i}$, then $\mathcal{J}$ is naturally identified with the set of pairs $(\tilde{v}, i_{\tilde{v}})$ with $\tilde{v} \in S$ and $i_{\tilde{v}} \in \mathbb{Z}/f_{\tilde{v}}$. In applications, $S$ will be a finite set of places dividing $p$ of a number field $F$. Sometimes, $S$ will have cardinality one, in which case we might drop the subscripts from $f_{\tilde{v}}$ and $k_{\tilde{v}}$ and denote the single unramified extension $F_{\tilde{v}}$ of $\mathbb{Q}_p$ by $K$.

Let $\mathcal{G}_0$ be the algebraic group $\prod_{\tilde{v} \in S} \text{Res}_{k_{\tilde{v}}/\mathbb{F}_p} \text{GL}_3$ with $\mathcal{T}_0$ the diagonal torus and center $\mathcal{Z}_0$. Let $\mathcal{G}$ be the base change $\mathcal{G}_0 \times_{\mathbb{F}_p} \mathbb{F}$, and similarly define $\mathcal{T}$ and $\mathcal{Z}$. There is a natural isomorphism $\mathcal{G} \cong \prod_{i \in \mathcal{J}} \text{GL}_3/\mathbb{F}$. One has similar isomorphisms for $\mathcal{T}$, $\mathcal{Z}$, $X^*(\mathcal{T})$, $\mathcal{R}$, $\mathcal{R}^{\vee}$, where $\mathcal{R}$ (respectively $\mathcal{R}^{\vee}$) denotes the set of roots (respectively coroots) of $\mathcal{G}$. If $\mu \in X^*(\mathcal{T})$, then we correspondingly write $\mu = \sum_{i \in \mathcal{J}} \mu_i = \sum_{\tilde{v} \in S} \mu_{\tilde{v}}$. We use similar notation for similar decompositions. Again, we identify $X^*(\mathcal{T})$ with $(\mathbb{Z}^3)^{\mathcal{J}}$ in the usual way and let $\epsilon'_{1,i}$ and $\epsilon'_{2,i}$ be (1, 0, 0) and (0, 0, -1), respectively, in the $i$th coordinate and 0 otherwise. Let $\mathcal{R}^+ \subseteq \mathcal{R}$ (respectively $\mathcal{R}^{\vee,+} \subseteq \mathcal{R}^{\vee}$) be the subset of positive roots (respectively coroots) of $\mathcal{G}$ with respect to the upper triangular Borel in each embedding. We define dominant (co)characters with respect to these choices. Let $X_+^*(\mathcal{T})$ be the set of dominant weights. We denote by $X_1(\mathcal{T}) \subset X_+^*(\mathcal{T})$ the subset of weights $\lambda \in X_+^*(\mathcal{T})$ satisfying $0 \le \langle \lambda, \alpha^\vee \rangle \le p-1$ for all simple roots $\alpha \in \mathcal{R}^+$. We call $X_1(\mathcal{T})$ the set of $p$-restricted weights. Let $\eta'_i \in X^*(\mathcal{T})$ be (1, 0, -1) in the $i$th coordinate and 0 otherwise, and let $\eta'$ be $\sum_{i \in \mathcal{J}} \eta'_i \in X^*(\mathcal{T})$. Let $\eta_i \in X^*(\mathcal{T})$ be (2, 1, 0) in the $i$th coordinate and 0 otherwise, and let $\eta$ be $\sum_{i \in \mathcal{J}} \eta_i \in X^*(\mathcal{T})$. Then $\eta$ is a lift of the half-sum of the positive roots of $\mathcal{G}$.

Let $\mathcal{W}$ be the Weyl group of $\mathcal{G}$ with longest element $w_0$. Let $\mathcal{W}_a$ and $\tilde{\mathcal{W}}$ be the affine Weyl group and extended affine Weyl group, respectively, of $\mathcal{G}$. Let $\Delta_R \subset X^*(\mathcal{T})$ denote the root lattice of $\mathcal{G}$. As above, we have identifications $\mathcal{W} \cong \mathcal{W}^{\mathcal{J}}$, $\mathcal{W}_a \cong \mathcal{W}_a^{\mathcal{J}}$, $\tilde{\mathcal{W}} \cong \tilde{\mathcal{W}}^{\mathcal{J}}$ and isomorphisms analogous to (1.1).

The Weyl groups $\mathcal{W}$, $\tilde{\mathcal{W}}$, $\mathcal{W}_a$ act naturally on $X^*(\mathcal{T})$. The image of $\lambda \in X^*(\mathcal{T})$ via the injection $X^*(\mathcal{T}) \hookrightarrow \tilde{\mathcal{W}}$ is denoted by $t_\lambda$. Our convention is that the dot action is always a $p$-dot action, that is, $t_\lambda w \cdot \mu = t_{p_\lambda} w(\mu + \eta) - \eta$.

Recall that for $(\alpha, n) \in \mathcal{R}^+ \times \mathbb{Z}$, we have the root hyperplane $H_{\alpha,n} \stackrel{\text{def}}{=} \{\lambda : \langle \lambda + \eta, \alpha^\vee \rangle = np\}$. An alcove (or sometimes $p$-alcove) is a connected component of the complement $X^*(\mathcal{T}) \otimes_{\mathbb{Z}} \mathbb{R} \setminus (\bigcup_{(\alpha,n)} H_{\alpha,n})$. We say that an alcove $\mathcal{C}$ is $p$-restricted if $0 < \langle \lambda + \eta, \alpha^\vee \rangle < p$ for all simple roots $\alpha \in \mathcal{R}^+$ and $\lambda \in \mathcal{C}$. If $\mathcal{C}_0 \subset X^*(\mathcal{T}) \otimes_{\mathbb{Z}} \mathbb{R}$ denotes the dominant base alcove, we let

$$ \tilde{\mathcal{W}}^+ \stackrel{\text{def}}{=} \{\tilde{w} \in \tilde{\mathcal{W}} : \tilde{w} \cdot \mathcal{C}_0 \text{ is dominant}\} $$

and

$$ \tilde{\mathcal{W}}_1^+ \stackrel{\text{def}}{=} \{\tilde{w} \in \tilde{\mathcal{W}}^+ : \tilde{w} \cdot \mathcal{C}_0 \text{ is } p\text{-restricted}\}. $$
Let $\tilde{w}_h = (\tilde{w}_{h,i}) \in \tilde{W}_1^+$ be the element $w_0 t_{-\eta}$. The discussion in this paragraph also applies for $G$, from which we define the dominant base alcove $C_0$, $\tilde{W}^+$, and $\tilde{W}_1^+$ for $G$.

There is a Frobenius action on $X^*(\mathcal{T})$, denoted by $\pi$ and an induced action of $\pi$ on $\mathcal{W}$ defined by $\pi(w)(\pi(\lambda)) = \pi(w(\lambda))$. If $\lambda \in X^*(\mathcal{T})$, then $\pi(\lambda)_{\tilde{v},i\tilde{v}} = \lambda_{\tilde{v},i\tilde{v}-1}$ under the standard identification and similarly $\pi(s)_{\tilde{v},i\tilde{v}} = s_{\tilde{v},i\tilde{v}-1}$ for an element $s$ of one of the Weyl groups above.

Let $\mathcal{G}_0^{\text{der}}$ be $\prod_{\tilde{v} \in S} \text{Res}_{k_{\tilde{v}}/F_p} \text{SL}_3$ with torus $\mathcal{T}_0^{\text{der}}$. Let $\mathcal{G}^{\text{der}}$ be the base change $\mathcal{G}_0^{\text{der}} \times_{F_p} \mathbb{F}$, and similarly define $\mathcal{T}^{\text{der}}$. Let $\mathcal{A}_W = X^*(\mathcal{T}^{\text{der}})$ denote the weight lattice for $\mathcal{G}^{\text{der}}$. Let $X^0(\mathcal{T})$ denote the kernel of the restriction map $X^*(\mathcal{T}) \to \mathcal{A}_W$. We write $\varepsilon_{1,i}$ and $\varepsilon_{2,i}$ for the images of $\varepsilon'_{1,i}$ and $\varepsilon'_{2,i}$, respectively. We define in a similar fashion the various Weyl groups $\mathcal{W}^{\text{der}}$, $\mathcal{W}_a^{\text{der}}$, $\tilde{W}^{\text{der}}$, $\tilde{W}^{\text{der,+}}$, $\tilde{W}_1^{\text{der,+}}$ for $\mathcal{G}^{\text{der}}$ (and $\tilde{W}^{\text{der,+}}$, $\tilde{W}_1^{\text{der,+}}$ for $G$), with analogous product decompositions.

Let $\alpha, \beta, \gamma^+$ denote the generators for the affine Weyl group $W_a$ of $\text{GL}_3$ given by reflection over the walls of $C_0$ with $\alpha, \beta \in W$, and $\gamma^+$ is the affine reflection. Note that $\alpha, \beta$ satisfy $\alpha(\varepsilon'_2) = \varepsilon'_2$ and $\beta(\varepsilon'_1) = \varepsilon'_1$.

Let $S_3$ denote the symmetric group on $\{1, 2, 3\}$. We fix an injection $S_3 \hookrightarrow \text{GL}_3(\mathbb{Z})$ sending $s$ to the permutation matrix whose $(k, m)$-entry is $\delta_{k,s(m)}$ and $\delta_{k,s(m)} \in \{0, 1\}$ is the Kronecker $\delta$ specialized at $\{k, s(m)\}$. We will abuse notation and simply use $s$ to denote the corresponding permutation matrix. We consider the embedding $X^*(T) \hookrightarrow \text{GL}_3(\mathbb{F}(v))$ defined by $\lambda \mapsto v^\lambda$ where then $v^{(a,b,c)}$ is the diagonal matrix with entries $v^a, v^b, v^c$, respectively. In this way, we get an embedding $\tilde{W} \hookrightarrow \text{GL}_3(\mathbb{F}(v))$. Finally, for $m \ge 0$ and a collection $(B_j)_{j=0,...,m}$ of square matrices of the same size, we write $\prod_{j=0}^m B_j = B_0 \cdot B_1 \cdot \dots \cdot B_m$.

Let $V$ be a representation of a finite group $\Gamma$ over an $E$-vector space. We write $\text{JH}(\bar{V})$ to denote the set of Jordan–Hölder factors of the mod $\varpi$-reduction of an $O$-lattice in $V$. This set is independent of the choice of the lattice.

If $R$ is a ring, we let $\text{Irr}(\text{Spec}(R))$ denote the set of minimal primes of $R$.

## 2. Extension graph

In this section, we give a description of the extension graph of generic irreducible representations of the group $G \stackrel{\text{def}}{=} \mathcal{G}_0(F_p)$ and describe the constituents of the mod $p$ reduction of *generic Deligne–Lusztig representations* of $G$. This also gives a description of the set $W^?(\bar{\rho})$ of predicted Serre weights defined by Herzig [Her09]. In Section 3, we use these descriptions to prove the Serre weight conjectures. The distance in the extension graph also plays an important role in computing the cosocle filtration for the reductions of lattices in tame types in Section 4.3.
**2.1. Definition and properties of the extension graph.** The surjection $X^*(T) \to \Lambda_W$ identifies $p$-restricted alcoves of $\mathrm{GL}_3$ and $\mathrm{SL}_3$. Let $C_0$ and $C_1$ denote the $p$-restricted lower and upper alcoves, respectively, of the weight space of $\mathrm{GL}_3$ so that $\mathcal{A} \stackrel{\mathrm{def}}{=} \{C_0, C_1\}^{\mathcal{J}}$ is naturally identified with the set of $p$-restricted alcoves for $\mathcal{G}$ (or $\mathcal{G}^{\mathrm{der}}$ when convenient). Our alcoves are fundamental domains with respect to the dot action, so the base alcove will have a vertex at $-\eta$. For notational convenience, we sometimes write 0 and 1 instead of $C_0$ and $C_1$ so that $\mathcal{A}$ is identified with $\{0, 1\}^{\mathcal{J}}$. We recall some notions from Section 1.4. Let $X_1(\mathcal{T})$ denote the set of $p$-restricted weights of $\mathcal{G}$ and $C_0 \subseteq X(\mathcal{T}) \otimes \mathbb{R}$ denote the lowest alcove $(C_0)_{i \in \mathcal{J}}$. We consider $\tilde{W}^{\mathrm{der}} = \Lambda_W \rtimes \mathcal{W}$ acting via the $p$-dot action on $\Lambda_W \otimes \mathbb{R}$. Let $\tilde{W}_1^{\mathrm{der,+}} \subset \tilde{W}^{\mathrm{der}}$ denote the set of elements which take $C_0$ to an element of $\mathcal{A}$. There are six elements of $\tilde{W}_1^{\mathrm{der,+}}$, namely id, $(12)t_{-(\varepsilon_1-\varepsilon_2)}$, $(23)t_{-(\varepsilon_2-\varepsilon_1)}$, $(123)t_{-\varepsilon_2}$, $(132)t_{-\varepsilon_1}$, and $(13)t_{-(\varepsilon_1+\varepsilon_2)}$.

Observe that the inclusion $\Lambda_W \hookrightarrow \tilde{W}^{\mathrm{der}}$ (respectively $X^*(\mathcal{T}) \hookrightarrow \tilde{W}$) induces an isomorphism $\iota^{\mathrm{der}} : \Lambda_W/\Lambda_R \xrightarrow{\sim} \tilde{W}^{\mathrm{der}}/W_a^{\mathrm{der}}$ (respectively $\iota : X^*(\mathcal{T})/\Lambda_R \cong X^*(\mathcal{Z}) \xrightarrow{\sim} \tilde{W}/W_a$). Let $\mathcal{P}^{\mathrm{der}} \subset \Lambda_W \times \tilde{W}_1^{\mathrm{der,+}}$ be the subset of pairs $(\omega, \tilde{w})$ with $\iota^{\mathrm{der}}(-\pi^{-1}(\omega) + \Lambda_R) = \tilde{w}W_a^{\mathrm{der}}$. We similarly define $\mathcal{P} \subset X^*(\mathcal{T}) \times \tilde{W}_1^{+}$. Note that restriction gives a natural surjection $\mathcal{P} \dashrightarrow \mathcal{P}^{\mathrm{der}}$.

**LEMMA 2.1.1.** *The map*

$$ \beta : \mathcal{P}^{\mathrm{der}} \to \Lambda_W \times \mathcal{A} \\ (\omega, \tilde{w}) \mapsto (\omega, \pi(\tilde{w}) \cdot C_0) $$

is a bijection.

*Proof.* The map $\beta$ is a bijection because $\iota^{\mathrm{der}}$ is a bijection and $W_a^{\mathrm{der}}$ acts simply transitively on the set of alcoves for $\mathcal{G}^{\mathrm{der}}$. $\square$

Let $\mu$ be an element of $C_0/(p-\pi)X^0(\mathcal{T})$. We will often have some lift of $\mu$ in $X^*(\mathcal{T})$ in mind, but what we write will not depend on the choice of this lift. We define a map

$$ \mathcal{P}^{\mathrm{der}} \to X^*(\mathcal{T})/(p-\pi)X^0(\mathcal{T}) \qquad (2.1) \\ (\omega, \tilde{w}) \mapsto \tilde{w}' \cdot (\mu - \eta + \omega'), $$

where $(\omega', \tilde{w}') \in \mathcal{P}$ is a lift of $(\omega, \tilde{w})$. The map (2.1) does not depend on the choice of lift. Then we define

$$ \mathfrak{Tr}'_{\mu} : \Lambda_W \times \mathcal{A} \to X^*(\mathcal{T})/(p-\pi)X^0(\mathcal{T}) $$

to be the composition of $\beta^{-1}$ with (2.1).
**DEFINITION 2.1.2.** We say that a weight $\lambda \in X^*(\underline{T})$ is p-regular if $\langle \lambda + \eta, \alpha^\vee \rangle \notin p\mathbb{Z}$ for all positive coroots $\alpha^\vee \in \underline{\mathbb{R}}^{\vee,+}$.

Note that a p-regular element belongs to a unique alcove.

Define $\underline{\Delta}_W^\mu$ to be the set

$$ \{\omega \in \underline{\Delta}_W : \omega + \mu - \eta \in \underline{C}_0\} $$

(taking the image of $\mu - \eta$ in $\underline{\Delta}_W$). Let $\mathfrak{T}\mathfrak{r}_\mu$ be the restriction of $\mathfrak{T}\mathfrak{r}'_\mu$ to $\underline{\Delta}_W^\mu \times \mathcal{A}$. We establish some basic properties of $\mathfrak{T}\mathfrak{r}_\mu$.

**PROPOSITION 2.1.3.** (1) (Any lift of) $\mathfrak{T}\mathfrak{r}_\mu(\omega, a)$ in $X^*(\underline{T})$ is p-regular and is in alcove $\pi^{-1}(a)$.

(2) The map $\mathfrak{T}\mathfrak{r}_\mu$ is injective.

*Proof.* For (1), suppose that $(\omega', \tilde{w}') \in \mathcal{P}$ is a lift of $\beta^{-1}(\omega, a)$. Then $\mu - \eta + \omega'$ is p-regular in alcove $\underline{C}_0$ by definition of $\underline{\Delta}_W^\mu$ so that $\mathfrak{T}\mathfrak{r}_\mu(\omega, a) = \tilde{w}' \cdot (\mu - \eta + \omega')$ is p-regular in alcove $\pi^{-1}(a) = \tilde{w} \cdot \underline{C}_0$.

For (2), suppose that $\mathfrak{T}\mathfrak{r}_\mu(\omega, a) = \mathfrak{T}\mathfrak{r}_\mu(v, b)$. Let $(\omega', \tilde{w}')$ and $(v', \tilde{x}') \in \mathcal{P}$ be lifts of $\beta^{-1}(\omega, a)$ and $\beta^{-1}(v, b)$, respectively. Then $\tilde{w}' \cdot (\mu - \eta + \omega') \equiv \tilde{x}' \cdot (\mu - \eta + v')$ (mod $(p-\pi)X^0(\underline{T}))$, from which we conclude from the definition of $\mathcal{P}^{\text{der}}$ that $\omega'|_{\underline{Z}} \equiv v'|_{\underline{Z}}$ (mod $3X^*(\underline{Z})$). Combining this with the fact that $a = b$ from (1) and using $\iota$, we see that $\tilde{w}' \equiv \tilde{x}'$ (mod $X^0(\underline{T}))$. This implies that $\omega = v$. $\square$

**PROPOSITION 2.1.4.** Let $\mu \in X^*(\underline{T})$ be a character. We have

$$ \mathfrak{T}\mathfrak{r}_\mu(\underline{\Delta}_W^\mu \times \mathcal{A}) = \left\{ \lambda \in X_1(\underline{T})/(p-\pi)X^0(\underline{T}) : \begin{array}{l} \lambda \text{ is p-regular and} \\ (\lambda - \mu + \eta)|_{\underline{Z}} \in (p-\pi)X^*(\underline{Z}) \end{array} \right\} \quad (2.2) $$

*Proof.* The inclusion of the left-hand side of (2.2) in the right follows from Proposition 2.1.3(1) and the formula for $\mathfrak{T}\mathfrak{r}_\mu$. We show now the reverse inclusion. Let $\lambda$ be an element in the right-hand side of (2.2). Then we let

* $a \in \mathcal{A}$ denote the unique alcove that contains $\pi(\lambda)$;

* $\omega' \in X^*(\underline{T})$ be such that $(\lambda - \mu + \eta)|_{\underline{Z}} = (1 - p\pi^{-1})\omega'|_{\underline{Z}}$;

* $\omega \in \underline{\Delta}_W$ be the image of $\omega'$;

* $(\omega', \tilde{w}') \in \mathcal{P}$ be a lift of $\beta^{-1}(\omega, a)$; and
• $v$ be $\lambda - \tilde{w}' \cdot (\mu - \eta + \omega') \in \underline{\Delta}_R$.

Then $(w'^{-1}(v) + \omega', \tilde{w}') \in \mathcal{P}$ is a lift of $\beta^{-1}(w'^{-1}(v) + \omega, a)$ so that

$$ \mathfrak{Tr}_{\mu}(w'^{-1}(v) + \omega, a) = \lambda $$

in $X^*(\underline{T})/(p-\pi)X^0(\underline{T})$.

□

**PROPOSITION 2.1.5.** Let $\lambda - \eta$ be a lift of $\mathfrak{Tr}_{\mu}(\varepsilon, \underline{C}_0)$ and $\beta^{-1}(\varepsilon, \underline{C}_0) = (\varepsilon, \tilde{w})$. Then

$$ \mathfrak{Tr}_{\lambda}(v, a) = \mathfrak{Tr}_{\mu}(w^{-1}(v) + \varepsilon, a) $$

for $(v, a) \in \underline{\Delta}_W^{\lambda} \times \mathcal{A}$, where $w \in \underline{W}$ is the image of $\tilde{w}$.

**Proof.** Let $(v', \tilde{x}') \in \mathcal{P}$ be a lift of $\beta^{-1}(v, a)$. Similarly, let $(\varepsilon', \tilde{w}') \in \mathcal{P}$ be a lift of $(\varepsilon, \tilde{w})$. Then

$$ \begin{aligned} \mathfrak{Tr}_{\lambda}(v, a) &= \tilde{x}' \cdot (\lambda - \eta + v') \\ &= \tilde{x}' \cdot (\tilde{w}' \cdot (\mu - \eta + \varepsilon') + v') \\ &= \tilde{x}' \tilde{w}' \cdot (\mu - \eta + \varepsilon' + w^{-1}(v')) \\ &= \mathfrak{Tr}_{\mu}(w^{-1}(v) + \varepsilon, a). \end{aligned} \quad \square $$

□

Recall that a *Serre weight* is an irreducible $\overline{\mathbb{F}}_p$-representation of $G$. Each Serre weight is obtained by restriction to $G$ from an irreducible algebraic representation of $G$ of highest weight $\lambda \in X_1(\underline{T})$, and this process gives a bijection between $X_1(\underline{T})/(p-\pi)X^0(\underline{T})$ to the set of Serre weights of $G$ (as described in [Her09, Theorem 3.10]; cf. also the beginning of Section 4.2). If $\lambda \in X_1(\underline{T})$, we write $F(\lambda)$ for the Serre weight corresponding to $\lambda$. We say that a Serre weight $F$ is *p-regular* if $F \cong F(\lambda)$ where $\lambda \in X_1(\underline{T})$ is p-regular (cf. Definition 2.1.2). Given $\mu \in \underline{C}_0$ and $(\omega, a) \in \underline{\Delta}_W^{\mu} \times \mathcal{A}$, we get a corresponding p-regular Serre weight $F(\mathfrak{Tr}_{\mu}(\omega, a))$. Propositions 2.1.3(2) and 2.1.4 show that $F(\mathfrak{Tr}_{\mu}(-))$ induces a bijection between the set $\underline{\Delta}_W^{\mu} \times \mathcal{A}$ and the set of p-regular Serre weights of $G$ with the same central character as $F(\mu - \eta)$.

**DEFINITION 2.1.6.** We say that $(\omega, a)$ and $(v, b)$ in $\underline{\Delta}_W^{\mu} \times \mathcal{A}$ are *adjacent* if there exists $j \in \mathcal{J}$ such that both $a_i = b_i$ and $\omega_i = v_i$ for $i \in \mathcal{J}$ with $i \neq j$, $a_j \neq b_j$, and

$$ \omega_j - v_j \in \{0, \pm(\varepsilon_{1,j} - \varepsilon_{2,j}), \pm\varepsilon_{1,j}, \pm\varepsilon_{2,j}\}. \quad (2.3) $$

If $F$ and $F'$ are $p$-regular Serre weights, we say that they are *adjacent* if we can write $F = F(\mathfrak{Tr}_{\mu}(\omega, a))$ and $F' = F(\mathfrak{Tr}_{\mu}(v, b))$ for some $p$-regular $\mu \in \underline{C}_0/(p-\pi)X^0(\underline{T})$ and some $(\omega, a)$ and $(v, b)$ which are adjacent in $\underline{\Delta}_W^{\mu} \times \mathcal{A}$.
By Proposition 2.1.5, this definition does not depend on the choice of the lowest alcove weight $\mu \in \mathcal{C}_0$. Indeed, if $F = F(\mathfrak{F}\mathfrak{t}_{\mu}(\omega, a))$, $F' = F(\mathfrak{F}\mathfrak{t}_{\mu}(\nu, b))$ are adjacent $p$-regular Serre weights and $\lambda \in \mathcal{C}_0$ such that $F(\lambda - \eta)$ and $F(\mu - \eta)$ have the same central character, then $F(\mathfrak{F}\mathfrak{t}_{\lambda}(w(\omega - \varepsilon), a)) = F(\mathfrak{F}\mathfrak{t}_{\mu}(\omega, a))$ and $F(\mathfrak{F}\mathfrak{t}_{\lambda}(w(\nu - \varepsilon), b)) = F(\mathfrak{F}\mathfrak{t}_{\mu}(\nu, b))$ for some $w \in \mathcal{W}$ and $\varepsilon \in \mathcal{A}_{\mathcal{W}}$, and $(w(\omega - \varepsilon), a)$, $(w(\nu - \varepsilon), b)$ are again adjacent since the set in (2.3) is $\mathcal{W}$-invariant.

**REMARK 2.1.7.** Geometrically, $\mathcal{A}_{\mathcal{W}}^{\mu}$ is the intersection of $\mathcal{A}_{\mathcal{W}}$ with a translate of a $p$-alcove, and two pairs $(\omega, a)$ and $(\nu, b)$ are adjacent if and only if $\omega$ and $\nu$ are either equal or neighbors (that is, differ by a $\mathcal{W}$-conjugate of a fundamental weight) and either $\omega \neq \nu$ and $a$ and $b$ have different labels in the unique component where $\omega$ and $\nu$ differ or $\omega = \nu$ and $a$ and $b$ differ in exactly one component. Note that for any $(\omega, a) \in \mathcal{A}_{\mathcal{W}}^{\mu}$ with all its neighbors in $\mathcal{A}_{\mathcal{W}}^{\mu}$, there are $7^{\#\mathcal{J}}$ adjacent vertices.

**Definition 2.1.6** endows $\mathcal{A}_{\mathcal{W}}^{\mu} \times \mathcal{A}$ with a graph structure with edges given by adjacency. By the above description, this graph is connected and thus it is endowed with a metric. By Proposition 2.1.4, any $\mu \in \mathcal{C}_0$ thus endows the set of $p$-regular Serre weights with the same central character as $F(\mu - \eta)$ with a metric. By Proposition 2.1.5, the metric on a fixed set is independent of the choice of $\mu$.

**Definition 2.1.8.** Given $p$-regular Serre weights $F$, $F'$ with the same central character, we denote their distance by $d_{\text{gph}}(F, F')$. In particular, $F$ and $F'$ are adjacent if and only if $d_{\text{gph}}(F, F') = 1$.

We conclude this section by showing the relation between $p$-regular Serre weights and $G$-extensions. We start by recalling the following definition.

**Definition 2.1.9.** Let $\lambda \in X^*(T)$ be a weight. We say that $\lambda$ lies $n$-deep in its alcove if, for all $\alpha^{\vee} \in \mathbb{R}^{\vee,+}$, there exist integers $m_{\alpha} \in \mathbb{Z}$ such that $pm_{\alpha} + n < \langle \lambda + \eta, \alpha^{\vee} \rangle < p(m_{\alpha} + 1) - n$. We have the analogous definition for weights in $X^*(T)$.

For instance, a dominant weight $\lambda \in X^*(T)$ lies $n$-deep in alcove $\mathcal{C}_0$ if $n < \langle \lambda + \eta, \alpha^{\vee} \rangle < p - n$ for all $i = 0, \dots, f-1$ and all positive coroots $\alpha^{\vee} \in \mathbb{R}^{\vee,+}$.

A Serre weight $F$ is said to be $n$-deep if we can write $F \cong F(\mu)$ for some $\mu \in X_1(T)$ which is $n$-deep. We call the graph on the $p$-regular Serre weights defined by adjacency the *extension graph*. The terminology is justified by the following theorem.
**THEOREM 2.1.10.** Let $F, F'$ be Serre weights which are both 6-deep. Then $\mathrm{Ext}_G^1(F, F') \neq 0$ if and only if $F$ and $F'$ are adjacent.

*Proof.* This is Lemma 4.2.6, whose proof only uses modular representation theory and does not rely on any results from other sections. □

**REMARK 2.1.11.** We partition the set of vertices $(\omega, a)$ into two sets, according to the class $\Sigma_i a_i$ mod 2 (by interpreting $a_i \in \{0, 1\}$). This makes the extension graph into a bipartite graph.

**2.2. Types and Serre weights.** Now suppose that $S$ contains exactly one element $\tilde{\nu}$, and let $K$ be $F_{\tilde{\nu}}$. An inertial type is a representation $\tau : I_K \to \mathrm{GL}_3(E)$ with open kernel which extends to the Weil group of $K$. An inertial type is *tame* if it factors through the tame inertial quotient. All our tame inertial types are defined over $\mathcal{O}$ and we use $\bar{\tau} : I_K \to \mathrm{GL}_3(\mathbb{F})$ to denote the reduction to the residue field.

Tame inertial types have a combinatorial description which we will now recall (cf. [Her09, (6.15)] or [GHS18, Definition 8.2.2]). Let $(w, \mu) \in \mathcal{W} \times \mathrm{X}^*(\mathcal{T})$. As in [Her09, (4.1)] or the paragraph preceding [GHS18, Definition 10.1.12], for any $(\nu, \sigma) \in \mathrm{X}^*(\mathcal{T}) \times \mathcal{W}$, define

$$ {}^{(\nu, \sigma)}(w, \mu) = (\sigma w \pi(\sigma)^{-1}, \sigma(\mu) + p\nu - \sigma w \pi(\sigma)^{-1} \pi(\nu)) \quad (2.4) $$

and we write $(w, \mu) \sim (w', \mu')$ if there exists $(\nu, \sigma)$ such that ${}^{(\nu, \sigma)}(w, \mu) = (w', \mu')$. The following describes all isomorphism classes of tame inertial types for $K$.

**DEFINITION 2.2.1.** Define an inertial type $\tau(w, \mu) : I_K \to \mathrm{GL}_3(\mathcal{O})$ as follows: If $w = (s_0, \dots, s_{f-1})$, then set $s_\tau = s_0 s_{f-1} s_{f-2} \cdots s_1 \in \mathcal{W}$ and $\alpha \in \mathrm{X}^*(\mathcal{T})$ such that $\alpha_0 = \mu_0$ and $\alpha_j = s_1^{-1} s_2^{-1} \cdots s_j^{-1}(\mu_j)$ for $1 \le j \le f-1$. Let $r$ denote the order of $s_\tau$, and set $f' = fr$. Then,

$$ \tau(w, \mu) \stackrel{\text{def}}{=} \bigoplus_{1 \le i \le 3} \omega_{f'}^{\sum_{0 \le k \le r-1} \mathbf{a}_{s_\tau(i)}^{(0)} p^{f_k}} \quad (2.5) $$

where $\mathbf{a}^{(0)} := \sum_{j=0}^{f-1} \alpha_j p^j \in \mathbb{Z}^3$. Note that $(w, \mu) \sim ((s_\tau, 1, \dots, 1), \alpha)$ and $\tau(w, \mu) \cong \tau((s_\tau, 1, \dots, 1), \alpha)$ by construction.

**DEFINITION 2.2.2.** Let $\tau$ be a tame inertial type.

(1) We say that $\tau$ is *regular* if the characters appearing in the right-hand side of (2.5) are pairwise distinct.
(2) We say that $\tau$ is $n$-generic if there is an isomorphism $\tau \cong \tau(s, \lambda + \eta)$ for some $s \in \mathbb{W}$ and $\lambda \in X_+^*(\mathbb{T})$ which is $n$-deep in alcove $\mathcal{C}_0$.

(3) We say that $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ is $n$-generic if $\bar{\rho}^{ss}|_{I_K} \cong \bar{\tau}$ for a tame inertial type $\tau$ which is $n$-generic.

(4) A lowest alcove presentation of $\tau$ is a pair $(s, \mu) \in \mathbb{W} \times X^*(\mathbb{T})$ where $\mu \in \mathcal{C}_0$ such that $\tau \cong \tau(s, \mu + \eta)$ (which by definition exists exactly when $\tau$ is $0$-generic).

**REMARK 2.2.3.** The notion of genericity given in [LLHLM18, Definition 2.1] is slightly different than Definition 2.2.2(2). In particular, an inertial type which is $n$-generic in the sense of Definition 2.2.2(2) is a fortiori $n$-generic in the sense of [LLHLM18]. Furthermore, our notion of genericity differs from that of [GHS18, Definition 10.1.12] by a shift by $\eta$.

**REMARK 2.2.4.** If $\tau$ is $1$-generic, then $\tau$ is regular.

Now let $S$ have arbitrary (finite) cardinality rather than one. For $s \in \mathbb{W}$ and $\mu \in X^*(\mathbb{T})$, we can define $s_{\tilde{v}}$ and $\mu_{\tilde{v}}$ in the evident way. Then we set $\tau_S(s, \mu)$ to be the collection $(\tau_{\tilde{v}}(s_{\tilde{v}}, \mu_{\tilde{v}}))_{\tilde{v} \in S}$.

The notions of regular and $n$-generic are extended in the evident way to a collection $\tau_S(s, \mu)$ of tame inertial types. Similarly, there is an evident notion of $n$-generic for a collection $\bar{\rho}_S$ of Galois representations $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$ for $\tilde{v} \in S$.

We now introduce the Deligne–Lusztig representations which are relevant for this paper. See also [GHS18, Section 9.2] and [LLHL19, Section 2.2], though we note that our context is slightly more general than [LLHL19, Section 2.2] since $S$ may have size greater than one. Let $(s, \mu) \in \mathbb{W} \times X^*(\mathbb{T})$ be a good pair [LLHL19, Section 2.2]. Following [GHS18, Propositions 9.2.1 and 9.2.2], we can attach to $(s, \mu)$ a genuine Deligne–Lusztig representation $R_s(\mu)$ of $G \stackrel{\text{def}}{=} \mathcal{G}_0(\mathbb{F}_p)$ defined over $E$ (taking $E$ to be sufficiently large). Note that $R_s(\mu)$ is denoted as $R(s, \mu)$ in [GHS18].

**DEFINITION 2.2.5.** Let $(s, \mu) \in \mathbb{W} \times X^*(\mathbb{T})$ be a good pair and let $n \ge 0$. We say that $R_s(\mu)$ is $n$-generic if there exists an isomorphism $R_s(\mu) \cong R_{s'}(\mu')$ where $\mu' - \eta$ is $n$-deep in alcove $\mathcal{C}_0$ (note that the evident generalization of [LLHL19, Lemma 2.2.3] implies that $(s', \mu')$ is good). By [GHS18, Proposition 9.2.1] and the evident generalization of [LLHL19, Proposition 2.2.4], when $n \ge 0$, then $\tau_S(s, \mu)$ is $n$-generic (cf. Definition 2.2.2(2)) if and only if $R_s(\mu)$ is $n$-generic.
By inflation, we consider $R_s(\mu)$ as a smooth $\prod_{\tilde{v} \in S} \text{GL}_3(\mathcal{O}_{\text{F}_{\tilde{v}}})$-representation. We recall some basic results on $R_s(\mu)$. Let $\sigma(\tau_{\tilde{v}})$ be a smooth $\text{GL}_3(\mathcal{O}_{\text{F}_{\tilde{v}}})$-representation associated with $\tau_{\tilde{v}}$ as in [CEG+16, Theorem 3.7], and let $\sigma(\tau_S)$ be the $\prod_{\tilde{v} \in S} \text{GL}_3(\mathcal{O}_{\text{F}_{\tilde{v}}})$-representation $\otimes_{\tilde{v} \in S} \sigma(\tau_{\tilde{v}})$.

**PROPOSITION 2.2.6.** Let $\mu - \eta \in X^*(\mathcal{I})$ be 1-deep in alcove $\mathcal{C}_0$. Then:

(1) [DL76, Theorem 6.8] $R_s(\mu)$ is irreducible.

(2) [LLHL19, Corollary 2.3.5] $\sigma(\tau_S)$ can be taken to be $R_s(\mu)$.

We conclude this section with some background on Serre weights associated with semisimple Galois representations. Let $\bar{\tau}_S$ be a collection of tame representations $\bar{\tau}_{\tilde{v}} : I_{\text{F}_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$ which extend to $G_{\text{F}_{\tilde{v}}}$. By [GHS18, Proposition 9.2.1], one attaches to $\bar{\tau}_{\tilde{v}}$ an $E$-valued representation $V(\bar{\tau}_{\tilde{v}})$ of $\text{GL}_3(k_{\tilde{v}})$. Let $V(\bar{\tau}_S)$ be the $G$-representation $\otimes_{\tilde{v} \in S} V(\bar{\tau}_{\tilde{v}})$. When $\bar{\tau}_S$ is $\bar{\tau}_S(s, \mu)$ for a good pair $(s, \mu)$, $V(\bar{\tau}_S)$ is isomorphic to $R_s(\mu)$ [GHS18, Proposition 9.2.3]. We write $\bar{\rho}_S|_{I_{\text{F}_S}}$ for the restriction to inertia of the collection $\bar{\rho}_S$.

Let $\tilde{w}_h \stackrel{\text{def}}{=} w_0t_{-\eta} \in \tilde{W}$. Recall the self-bijection $\mathcal{R}$ on $p$-regular Serre weights defined in [GHS18, Section 9.2]:

$$ \mathcal{R}(F(\lambda)) \stackrel{\text{def}}{=} F(\tilde{w}_h \cdot \lambda). $$

**DEFINITION 2.2.7 [GHS18, Definition 9.2.5]**. Let $\bar{\rho}_S$ be a collection of 2-generic semisimple Galois representations $\bar{\rho}_{\tilde{v}} : G_{\text{F}_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$. The set of predicted weights for $\bar{\rho}_S$ is defined to be

$$ W^2(\bar{\rho}_S) \stackrel{\text{def}}{=} \{ \mathcal{R}(F) : F \in \text{JH}(\overline{V(\bar{\rho}_S|_{I_{\text{F}_S}})}) \}. $$

If $\tau_S$ is a collection of tame inertial types $\tau_{\tilde{v}} : I_{\text{F}_{\tilde{v}}} \to \text{GL}_3(\mathcal{O})$, we furthermore define

$$ W^2(\bar{\rho}_S, \tau_S) \stackrel{\text{def}}{=} W^2(\bar{\rho}_S) \cap \text{JH}(\overline{\sigma(\tau_S)}). $$

**REMARK 2.2.8.** The condition that $\bar{\rho}_S|_{I_{\text{F}_S}}$ is 2-generic is to ensure that the elements of $\text{JH}(\overline{V(\bar{\rho}_S|_{I_{\text{F}_S}})})$ are all 0-deep so that $\mathcal{R}$ is defined (cf. Lemma 4.2.13).

Recall from [GHS18, Definition 7.1.3] that there is a subset $W_{\text{obv}}(\bar{\rho}) \subseteq W^2(\bar{\rho})$ of obvious Serre weights of $\bar{\rho}$. The set $W_{\text{obv}}(\bar{\rho}_S)$ is defined in the evident way.
**2.3. Combinatorics of types and Serre weights.** Recall the notation $\bar{\rho}_S$ from the previous section. We will always assume that $\bar{\rho}_v$ is 2-generic and semisimple for all $\tilde{v} \in S$ in what follows. The following proposition describes $W_{\text{obv}}(\bar{\rho}_S)$ in terms of the extension graph.

**PROPOSITION 2.3.1.** Assume that $\bar{\rho}_S$ is a collection of semisimple Galois representations. Suppose that $\bar{\rho}_S|_{I_{F_S}} \cong \bar{\tau}_S(s, \lambda)$, where $\lambda - \eta$ is 3-deep in $\underline{C}_0$. Then $W_{\text{obv}}(\bar{\rho}_S)$ is the set

$$ \{F(\mathfrak{t}\mathfrak{r}_{\lambda}(s(\omega), \pi(\tilde{w}) \cdot \underline{C}_0)) : \tilde{w} = wt_{-\pi^{-1}\omega} \in \underline{W}_1^{+,\text{der}}\}. $$

*Proof.* This follows from [LLHL19, Corollary 2.2.13 and (2.6)]. □

**DEFINITION 2.3.2.** Define

$$ \Sigma_0^{\text{def}} = \left\{ \begin{array}{c} (\varepsilon_1 + \varepsilon_2, 0), (\varepsilon_1 - \varepsilon_2, 0), (\varepsilon_2 - \varepsilon_1, 0) \\ (0, 1), (\varepsilon_1, 1), (\varepsilon_2, 1) \\ (0, 0), (\varepsilon_1, 0), (\varepsilon_2, 0) \end{array} \right\}. $$

Define $\Sigma^{\text{def}} = \Sigma_0^{\mathcal{J}} \subset \underline{\Lambda}_W \times \mathcal{A}$.

The set of predicted weights $W^r(\bar{\rho}_S)$ and the Jordan–Hölder factors of a Deligne–Lusztig representation $R_s(\mu)$ will be described in terms of the ‘triangle’ $\Sigma_0$ inside $\Lambda_W \times \{C_0, C_1\}$ (see Figure 1).

**DEFINITION 2.3.3.** Define the following subsets of $\Sigma_0$:

$$ \begin{aligned} \Sigma_0^{\text{obv}}^{\text{def}} &= \left\{ \begin{array}{c} (\varepsilon_1 + \varepsilon_2, 0), (\varepsilon_1 - \varepsilon_2, 0), (\varepsilon_2 - \varepsilon_1, 0) \\ (0, 1), (\varepsilon_1, 1), (\varepsilon_2, 1) \end{array} \right\}, \\ \Sigma_0^{\text{inn}}^{\text{def}} &= \Sigma_0 \setminus \Sigma_0^{\text{obv}}. \end{aligned} $$

We say that $(\omega, a) \in \Sigma$ is *obvious* (respectively *inner*) in component $i$ if $(\omega_i, a_i) \in \Sigma_0^{\text{obv}}$ (respectively $(\omega_i, a_i) \in \Sigma_0^{\text{inn}}$).

We define an involution $r$ of $\underline{\Lambda}_W^\mu \times \mathcal{A}$ by $r(\omega, a) = (\omega, a')$ with $a_i \neq a'_i$ for all $i \in \mathcal{J}$.

**PROPOSITION 2.3.4.** Let $\mu \in X^*(\underline{T})$. Suppose that $V(\bar{\rho}_S|_{I_{F_S}}) = R_s(\mu + \nu)$ where $\nu \in \underline{\Lambda}_R$ and $\mu + \nu - \eta$ is 3-deep in alcove $\underline{C}_0$. Then $W^r(\bar{\rho}_S)$ is $F(\mathfrak{t}\mathfrak{r}_{\mu}(t_\nu sr(\Sigma)))$.
<figure><img src="image_2.png" /><figcaption>Figure 1. The set $\Sigma_0$.</figcaption></figure>

*Proof.* By Proposition 2.3.1, $W_{\text{obv}}(\bar{\rho}_S)$ is given by

$$ \{ F(\mathfrak{F}\mathfrak{r}_{\mu+\nu}(s(\omega), \pi(\tilde{w}) \cdot \underline{C}_0)) : \tilde{w} = wt_{-\pi^{-1}\omega} \in \mathcal{W}_1^{+,\text{der}} \}. $$

Since $\nu \in \underline{\Delta}_R$, we have $\mathfrak{F}\mathfrak{r}_{\mu+\nu}(s(\omega), \pi(\tilde{w}) \cdot \underline{C}_0) = \mathfrak{F}\mathfrak{r}_{\mu}(s(\omega) + \nu, \pi(\tilde{w}) \cdot \underline{C}_0)$. By the generalization of Jantzen's formula in [Her09, Appendix, Theorem 3.4], the reduction of $R_s(\mu + \nu)$ is given in terms of reductions of Weyl modules. The fact that $\mu + \nu - \eta$ is 3-deep implies that these Weyl modules decompose in a generic pattern. Hence, $W^?(\bar{\rho}_S)$ is given by [GHS18, Proposition 10.1.8], which is

$$ F(\{\mathfrak{F}\mathfrak{r}_{\mu}(t_{\nu}s(\omega), a) \mid \tilde{w} = wt_{-\pi^{-1}\omega} \in \mathcal{W}_1^{+,\text{der}}, \pi(\tilde{w}) \cdot \underline{C}_0 \uparrow a\}). $$

This set is precisely $F(\mathfrak{F}\mathfrak{r}_{\mu}(t_{\nu}sr(\Sigma)))$.

<figure><img src="image_3.png" /></figure>

**PROPOSITION 2.3.5.** If $\nu \in X^*(\underline{T})$ such that $\nu + \eta \in \underline{\Delta}_R$ and $\mu + \nu - \eta$ is 3-deep in alcove $\underline{C}_0$, then $\text{JH}(\overline{R_s}(\lambda + \underline{1})) = F(\mathfrak{F}\mathfrak{r}_{\mu}(t_{\nu}s(\Sigma)))$.

*Proof.* Since $t_{\perp} \circ \mathcal{R} \circ \mathfrak{F}\mathfrak{r}_{\mu} = \mathfrak{F}\mathfrak{r}_{\mu} \circ r$, $F(\mathfrak{F}\mathfrak{r}_{\mu}(\omega, a)) \in \text{JH}(\overline{R_s}(\lambda - \underline{1}))$ if and only if $F(\mathfrak{F}\mathfrak{r}_{\mu}(r(\omega, a))) \in \mathcal{R}(\text{JH}(\overline{R_s}(\lambda))) = W^?(\bar{\rho}_S)$ where $V(\bar{\rho}|_{I_{F_S}}) = R_s(\lambda)$. The result then follows from Proposition 2.3.4. $\square$

<figure><img src="image_4.png" /></figure>
**DEFINITION 2.3.6.** For any $\tilde{w} \in \tilde{W}$, we set $R_{\tilde{w}}(\mu) \stackrel{\text{def}}{=} R_w(\mu + \tilde{w}(0))$, where $w \in W$ is the projection of $\tilde{w}$.

There is an action of $\tilde{W}$ on $\underline{\Lambda}_W \times \mathcal{A}$ where it acts through $\tilde{W}^{\text{der}}$ on just the first factor (via the usual action, not the dot action). By Propositions 2.3.4 and 2.3.5, we have the following.

**PROPOSITION 2.3.7.** Let $\tilde{w} \in \underline{W}_a t_\eta$ and assume that $\mu + \tilde{w}^{-1}(0) - \eta$ is 3-deep in $C_0$. Then the set $\text{JH}(R_{\tilde{w}^{-1}(\mu)})$ is $F(\mathfrak{F}\tau_\mu(\tilde{w}^{-1}(\Sigma)))$. If $\tilde{w} \in \underline{W}_a$ and $V(\bar{\rho}_S|_{I_{FS}}) = R_{\tilde{w}}(\mu)$, then the set $W^\gamma(\bar{\rho}_S)$ is $F(\mathfrak{F}\tau_\mu(\tilde{w}(r(\Sigma))))$.

For the rest of this section, fix a character $\mu$ such that $\mu - \eta$ is 3-deep in $C_0$ and an element $s \in W$.

For $(s\omega, a) \in \underline{\Lambda}_W^\mu \times \mathcal{A}$, we let $\sigma_{(\omega,a)}^{(s,\mu)} = F(\mathfrak{F}\tau_\mu(s\omega, a))$. When the pair $(s, \mu)$ is clear from the context, we will simply write $\sigma_{(\omega,a)}$ instead of $\sigma_{(\omega,a)}^{(s,\mu)}$ to lighten notation. Let $\tilde{w} \in \underline{W}_a t_\eta$ such that $s\tilde{w}^{-1}(\Sigma) \subset \underline{\Lambda}_W^\mu \times \mathcal{A}$ and $\mu + s\tilde{w}^{-1}(0) - \eta$ is 3-deep. Then the map

$$ \tilde{w}^{-1}(\Sigma) \to \text{JH}(R_{s\tilde{w}^{-1}(\mu)}) \qquad (2.6) \\ \tilde{w}^{-1}(\omega, a) \to \sigma_{(\tilde{w}^{-1}(\omega),a)}^{(s,\mu)} $$

is a bijection by Proposition 2.3.7. Similarly, if $V(\bar{\rho}_S|_{I_{FS}}) = R_s(\mu)$, then the map

$$ r(\Sigma) \to W^\gamma(\bar{\rho}_S) \qquad (2.7) \\ r(\omega, a) \to \sigma_{r(\omega,a)}^{(s,\mu)} $$

is a bijection.

**DEFINITION 2.3.8.** We say that $\sigma \in \text{JH}(R_{s\tilde{w}^{-1}(\mu)})$ is an outer (respectively inner) weight for $R \stackrel{\text{def}}{=} R_{s\tilde{w}^{-1}(\mu)}$ in component $i \in \mathcal{J}$ if $\sigma = \sigma_{(\tilde{w}^{-1}(\omega),a)}$ with $(\omega_i, a_i) \in \Sigma_0^{\text{obv}}$ (respectively $(\omega_i, a_i) \in \Sigma_0^{\text{inn}}$). We define the defect of $\sigma$ with respect to $R$ as

$$ \text{Def}_R(\sigma) \stackrel{\text{def}}{=} \#\{i \in \mathcal{J} \text{ such that } \sigma \text{ is an inner weight of } R \text{ in component } i\}. \qquad (2.8) $$

Similarly, if $V(\bar{\rho}_S|_{I_{FS}}) = R_s(\mu)$, we say that $\sigma \in W^\gamma(\bar{\rho}_S)$ is an obvious (respectively shadow) weight for $\bar{\rho}_S$ in component $i \in \mathcal{J}$ if $\sigma = \sigma_{r(\omega,a)}$ with $(\omega_i, a_i) \in \Sigma_0^{\text{obv}}$ (respectively $(\omega_i, a_i) \in \Sigma_0^{\text{inn}}$). We define the defect of $\sigma$ with
<figure><img src="image_5.png" /><figcaption>Table 1. The graph $\Sigma_0$.</figcaption></figure>

respect to $\bar{\rho}_S$ as

$$ \mathrm{Def}_{\bar{\rho}_S}(\sigma) \stackrel{\mathrm{def}}{=} \#\{i \in \mathcal{J} \text{ such that } \sigma \text{ is a shadow weight of } \bar{\rho}_S \text{ in component } i\}. \qquad (2.9) $$

We give the set $\Sigma_0$ the structure of a graph in Table 1. We write $\Sigma_0^J$ for the corresponding product graph. The above bijections respect the notion of adjacency for $\mathrm{JH}(\bar{R}_{s\tilde{w}^{-1}(\mu)}, W^J(\bar{\rho}_S))$, and $\Sigma_0^J$.

We now describe the intersections of $\mathrm{JH}(\bar{R}_{s\tilde{w}^{-1}(\mu)})$ and $W^J(\bar{\rho}_S)$ for $\bar{\rho}_S|_{G_{\bar{\rho}_S}} \cong \bar{\tau}_S(s, \mu)$ in terms of the action of $\tilde{W}$ on $\Delta_W$ using Proposition 2.3.7. We first introduce a subset of $\tilde{W}$, the admissible set, which appears in Corollary 2.3.11 and again in Section 3.

Recall from Section 1.4 that $\tilde{W}$ denotes the extended affine Weyl group of $G \stackrel{\mathrm{def}}{=} \mathrm{GL}_{3/\mathbb{F}}$. The choice of the dominant base alcove endows $\tilde{W}$ with a Bruhat order which is denoted by $\leq$. We also have the natural generalization of the Bruhat order $\leq$ on $\tilde{W}$ associated with the choice of the dominant base alcove.

**DEFINITION 2.3.9.** Let $\lambda \in X^*(T)$. We define

$$ \mathrm{Adm}(\lambda) = \{\tilde{w} \in \tilde{W} \mid \tilde{w} \leq t_{s(\lambda)} \text{ for some } s \in W\}. $$

Similarly, for a weight $\mu = (\mu_j)_j \in X^*(T)$, define

$$ \mathrm{Adm}(\mu) = \prod_j \mathrm{Adm}(\mu_j) \subset \tilde{W}. $$

**REMARK 2.3.10.** For combinatorics of weights as in this section, we use the dominant base alcove and the corresponding affine reflection $\gamma^+ = t_{(1,0,-1)}(13)$
as a generator for $W_a^{\text{der}}$. When working on the Galois side, as in Section 3, the antidominant base alcove appears naturally, and the analogous set $\text{Adm}^{\vee}(\eta)$ for the antidominant base alcove plays an important role (see Definition 3.1.1 and the discussion before). The affine generator for the antidominant choice of base alcove is $\gamma = (13)t_{(1,0,-1)}$. Note that in [LLHLM18], everything is in terms of antidominant and so what we call $\text{Adm}(\eta)$ there, we denote by $\text{Adm}^{\vee}(\eta)$ here.

For $\tilde{w} \in \tilde{W}$, define $\Sigma_{\tilde{w}} \stackrel{\text{def}}{=} \Sigma_0 \cap \tilde{w}(r(\Sigma_0))$. Similarly, for $\tilde{w} \in \tilde{W}$, let $\Sigma_{\tilde{w}} \stackrel{\text{def}}{=} \Sigma \cap \tilde{w}(r(\Sigma))$. One can directly compute these sets using Figure 1. For example, using Figure 2, one can check that $\Sigma_{\tilde{w}}$ is empty if and only if $\tilde{w}$ is not in $\text{Adm}(\eta)X^0(T)$ (in essence, [LLHL19, Proposition 4.4.2] is a generalization of this fact). Otherwise, see Table 2 for a description of $\Sigma_{\tilde{w}}$.

**COROLLARY 2.3.11.** Let $\tilde{w} \in \tilde{W}_a t_\eta$. Assume that both $\mu - \eta$ and $\mu + s\tilde{w}^{-1}(\underline{0}) - \eta$ are 3-deep in $\underline{C}_0$. The set $\mathcal{R}(\text{JH}(\overline{R_s(\mu)})) \cap \text{JH}(\overline{R_{s\tilde{w}^{-1}(\mu)}})$ is the set of weights $\sigma_{r(\omega,a)}^{(s,\mu)}$ where $(\omega, a) \in \Sigma_{\tilde{w}^{-1}}$. Similarly, the set $\text{JH}(\overline{R_s(\mu - 1)}) \cap \mathcal{R}(\text{JH}(\overline{R_{s\tilde{w}(\mu - 1)}}))$ is the set of weights $\sigma_{(\omega,a)}^{(s,\mu)}$ where $(\omega, a) \in \Sigma_{\tilde{w}}$.

*Proof.* By Proposition 2.3.7, one can reduce the case where $s = \text{id}$. In that case, (2.6) and (2.7) imply that the set of Serre weights in the intersection $\mathcal{R}(\text{JH}(\overline{R_s(\mu)})) \cap \text{JH}(\overline{R_{s\tilde{w}^{-1}(\mu)}})$ is $\mathfrak{I}\mathfrak{t}_\mu(\tilde{w}^{-1}(\Sigma) \cap r(\Sigma))$. We conclude by noting that $\tilde{w}^{-1}(\Sigma) \cap r(\Sigma) = r(\Sigma_{\tilde{w}^{-1}})$. The statement for $\text{JH}(\overline{R_s(\mu - 1)}) \cap \mathcal{R}(\text{JH}(\overline{R_{s\tilde{w}(\mu - 1)}}))$ is proved similarly. $\square$

**LEMMA 2.3.12.** If $(\omega, a) \in \underline{\Delta}_W \times \mathcal{A} \setminus r(\Sigma)$, there exists $\tilde{w} \in \underline{W}_a t_\eta$ such that $(\omega, a) \in \tilde{w}^{-1}(\Sigma)$ and $\tilde{w} \notin \text{Adm}(\eta)$.

*Proof.* We can work one component at a time and, thus, reduce to the case where $\#\mathcal{J} = 1$. This can then be checked directly using Figure 2. $\square$

**COROLLARY 2.3.13.** Let $\lambda \in X_1(T)$ be 5-deep and $\mu - \eta \in \underline{C}_0$ be 3-deep. If $F(\lambda) \notin \mathcal{R}(\text{JH}(\overline{R_s(\mu)}))$, then there exists a 3-generic Deligne–Lusztig representation $\overline{R_{s'}}(\mu')$ such that $F(\lambda) \in \text{JH}(\overline{R_{s'}(\mu')})$ but

$$ \mathcal{R}(\text{JH}(\overline{R_s(\mu)})) \cap \text{JH}(\overline{R_{s'}(\mu')}) = \emptyset. $$

*Proof.* If the central characters do not match, that is, $(\lambda - \mu + \eta)|_{\underline{Z}} \notin (p - \pi)X^*(\underline{Z})$, then any Deligne–Lusztig representation which contains $F(\lambda)$ works. Otherwise, by Proposition 2.1.4, there exists $(s\omega, a) \in \underline{\Delta}_W^\mu \times \mathcal{A}$ such that
<table><caption>Table 2. Intersections.</caption><thead><tr><th>\tilde{w}_{t_{-1}}</th><th>Σ<sub>\tilde{w}</sub></th><th>\tilde{w}_{t_{-1}}</th><th>Σ<sub>\tilde{w}</sub></th></tr></thead><tbody><tr><td>γ<sup>+</sup>αβα = t<sub>(1,0,-1)</sub></td><td>{(ε<sub>1</sub> + ε<sub>2</sub>, 0)}</td><td>αβαγ<sup>+</sup> = t<sub>(-1,0,1)</sub></td><td>{(0, 1)}</td></tr><tr><td>βγ<sup>+</sup>βα = t<sub>(1,-1,0)</sub></td><td>{(ε<sub>1</sub>, 1)}</td><td>αβγ<sup>+</sup>β = t<sub>(-1,1,0)</sub></td><td>{(ε<sub>2</sub> - ε<sub>1</sub>, 0)}</td></tr><tr><td>αγ<sup>+</sup>αβ = t<sub>(0,1,-1)</sub></td><td>{(ε<sub>2</sub>, 1)}</td><td>βγ<sup>+</sup>αγ<sup>+</sup> = t<sub>(0,-1,1)</sub></td><td>{(ε<sub>1</sub> - ε<sub>2</sub>, 0)}</td></tr><tr><td>βαγ<sup>+</sup></td><td>{(ε<sub>1</sub> - ε<sub>2</sub>, 0), (0, 1)}</td><td>αβγ<sup>+</sup></td><td>{(ε<sub>2</sub> - ε<sub>1</sub>, 0), (0, 1)}</td></tr><tr><td>αβα</td><td>{(0, 1), (0, 0)}</td><td></td><td></td></tr><tr><td>βα</td><td>{ (0, 1), (0, 0) (ε<sub>1</sub> - ε<sub>2</sub>, 0), (ε<sub>1</sub>, 1) }</td><td>αβ</td><td>{ (0, 1), (0, 0) (ε<sub>2</sub> - ε<sub>1</sub>, 0), (ε<sub>2</sub>, 1) }</td></tr><tr><td>α</td><td>{ (0, 1), (0, 0), (ε<sub>2</sub>, 1) (ε<sub>2</sub>, 0), (ε<sub>2</sub> - ε<sub>1</sub>, 0), (ε<sub>1</sub>, 1) }</td><td>Id</td><td>{ (0, 1) (ε<sub>1</sub>, 1) (ε<sub>2</sub>, 1) (0, 0) (ε<sub>1</sub>, 0) (ε<sub>2</sub>, 0) }</td></tr></tbody></table>

Not all elements of $\mathrm{Adm}(\eta)t_{-1}$ appear in the table. There is an order-three symmetry of $\mathrm{Adm}(\eta)t_{-1}$ induced by the outer automorphism of $\tilde{W}^{\mathrm{der}}$, and we include at least one representative for each orbit.

$\sigma_{(\omega,a)}^{(s,\mu)} \cong F(\lambda)$. Moreover, since $F(\lambda) \notin \mathcal{R}(\mathrm{JH}(\overline{R_s(\mu)}))$, we have $(\omega, a) \notin r(\Sigma)$. By Lemma 2.3.12, there exists $\tilde{w} \in \overline{W_a} t_\eta \setminus \mathrm{Adm}(\eta)$ such that $(\omega, a) \in \tilde{w}^{-1}(\Sigma)$. We take $s' = sw^{-1}$ and $\mu' = \mu + s\tilde{w}^{-1}(0)$ so that $R_{s'}(\mu') = R_{s\tilde{w}^{-1}}(\mu)$. Since $(\omega, a) \in \tilde{w}^{-1}(\Sigma)$, $\omega - \tilde{w}^{-1}(0)$ is in $\overline{W(\Sigma)}$, where $\Sigma$ denotes the image of $\Sigma$ in $\overline{\Delta_W}$. A direct computation shows that since $\lambda$ is 5-deep, $\mu + s\tilde{w}^{-1}(0) - \eta$ is 3-deep in $\overline{C_0}$. By Proposition 2.3.7, we have that $F(\lambda) \in \overline{\mathrm{JH}(R_{s'}(\mu'))}$. By Corollary 2.3.11 and the remark before it, we have that $\mathcal{R}(\overline{\mathrm{JH}(R_s(\mu))}) \cap \overline{\mathrm{JH}(R_{s'}(\mu'))} = \emptyset$. $\square$

### 3. Serre weight conjectures

In Sections 3.1–3.3, we improve results of [LLHLM18] on Kisin modules and étale $\varphi$-modules with descent data. Most of these results are generalized
<figure><img src="image_6.png" /><figcaption>Figure 2. The standard apartment of SL<sub>3</sub>. Labeled alcoves are alcoves in Adm(η).</figcaption></figure>

to GL<sub>n</sub> in [LLHL19]. In Section 3.2, we prove a key structure result relating the Frobenius of a Kisin module with tame descent data to the Frobenius of the corresponding étale φ-module (Proposition 3.2.1). This plays an important role in computing the Galois representation associated with semisimple Kisin modules (Proposition 3.3.8) and later establishing an explicit geometric Breuil–Mézard conjecture (see Section 3.6). The remainder of Section 3.3 is devoted to bounding reductions of potentially crystalline representations of type (η, τ) for the purpose of weight elimination (Theorem 3.3.12).
In Section 3.4, we connect the results of Sections 3.1–3.3 with the combinatorics of Serre weights from Section 2.3. In particular, Proposition 3.4.2 describes the intersection of the set of predicted weights of a semisimple $\bar{\rho}$ with the set of Jordan–Hölder constituents of a Deligne–Lusztig representation, as a function of its shape. In Section 3.5, we prove the Serre weight conjectures for semisimple $\bar{\rho}$ by matching the number of predicted Serre weights to the number of irreducible components of the Galois deformation rings which were computed in [LLHLM18, Theorem 3.5.3]. In Section 3.6, we prove a geometric Breuil–Mézard conjecture which assigns to each (predicted) Serre weight a prime ideal of the universal framed deformation ring (Proposition 3.6.1). In Section 3.6.1, we make this assignment explicit for certain tamely potentially crystalline deformation rings. The explicit ideals play an important role in the proof of Breuil’s lattice conjecture in Section 5.

Throughout this section, we assume that $\mathcal{S} = \{\tilde{v}\}$ unless otherwise stated and write $K \stackrel{\text{def}}{=} F_{\tilde{v}}$. We drop the subscript $\tilde{v}$ from notation in this situation. The set $\mathcal{J}$ is identified with the set of embeddings $K \hookrightarrow E$ and also with $\mathbb{Z}/f\mathbb{Z}$ using our chosen embedding $\sigma_0$ (cf. Section 1.4). For $i \in \mathcal{J}$, we will use the word component $i$ and embedding $i$ interchangeably.

**3.1. Background.** In this subsection, we recall some basic facts on Kisin modules with tame descent data. We start with the following involution on $\tilde{W}$, which naturally appears when passing from Kisin modules to $G_{K_\infty}$-representations. We let $\tilde{W}^\vee$ (respectively $\tilde{W}^\wedge$) be the partially ordered group which is identified with $\tilde{W}$ (respectively $\tilde{W}$) as a group, and whose Bruhat order is defined by the *antidominant* base alcove (and still denoted as $\leq$).

**DEFINITION 3.1.1.** Define a bijection $\tilde{w} \mapsto \tilde{w}^*$ between $\tilde{W}^\vee$ and $\tilde{W}$ as follows:

(1) For $w = (w_j) \in \tilde{W}$, define $w^* = (w_j^*) \in \tilde{W}$ by $w_j^* = w_{j-1}^{-1}$.

(2) For $v = (v_j) \in X^*(\tilde{W})$, define $v^* = (v_j^*) \in X^*(\tilde{W})$ by $v_j^* = v_{j-1}^{-1}$.

(3) For $\tilde{w} = w t_v \in \tilde{W}^\vee$, define $\tilde{w}^* \in \tilde{W}$ by $\tilde{w}^* = t_{v^*} w^*$.

Note that $\tilde{w} \mapsto \tilde{w}^*$ is an *antihomomorphism* of groups. For $j \in \mathcal{J}$, we write $\tilde{w}_j^*$ for the $j$th component of $\tilde{w}^*$.

Let $\tau$ be a tame inertial type which we always assume to be 1-generic. Fix a lowest alcove presentation $(s, \mu)$ for $\tau$ (Definition 2.2.1).

If $s = (s_0, \dots, s_{f-1})$ and $\mu = (\mu_j)_{0 \le j \le f-1} \in X^*(\tilde{W})$, we take

$$s_{\tau} \stackrel{\text{def}}{=} s_0 s_{f-1} s_{f-2} \cdots s_1 \in W$$
and $\alpha_{(s,\mu)} \in X^*(\underline{T})$ such that $\alpha_{(s,\mu),j} = s_1^{-1}s_2^{-1}\dots s_j^{-1}(\mu_j+\eta_j)$ for $1 \le j \le f-1$ and $\alpha_{(s,\mu),0} = \mu_0 + \eta_0$. Let $r \in \{1, 2, 3\}$ be the order of $s_\tau$, and write $K'$ for the unramified extension of $\mathbb{Q}_p$ of degree $f' \stackrel{\text{def}}{=} f r$.

REMARK 3.1.2. In [LLHLM18], the notion of lowest alcove presentation does not appear. Everything is written for presentations of the form $\tau((s_\tau, 1, \dots, 1))$, $\alpha_{(s,\mu)}$ (see, for example, the beginning of [LLHLM18, Sections 2.1, 6.1]). In the notation of loc. cit., $\alpha_{(s,\mu),j} = (a_1, \dots, a_{j-1}, a_{j+1}, \dots)$.

$$s_{\text{or}} \stackrel{\text{def}}{=} (s_1^{-1}s_2^{-1}\dots s_{f-1}^{-1}, s_1^{-1}s_2^{-1}\dots s_{f-2}^{-1}, \dots, s_1^{-1}, 1) \in \underline{W}$$

has the property that $s_{\text{or}}^*(\alpha_{(s,\mu)}) = \mu + \eta$, and, hence, $(0, s_{\text{or}}^*)(((s_\tau, 1, \dots, 1)), \alpha_{(s,\mu)})) = (s, \mu + \eta)$. The element $s_{\text{or}}$ is called the orientation of $\alpha_{(s,\mu)}$ [LLHLM18, Definition 2.6 and equation (2.2)].

If $r=1$, we say that $\tau$ is a *principal series type*. Otherwise, we write $\tau'$ for the base change of $\tau$ to $K'/K$ (which is just $\tau$ considered as a principal series type for $G_{K'}$). We record the relevant data for $\tau'$. Define $\alpha'_{(s,\mu)} \in X^*(T)^{\text{Hom}(k',\mathbb{F})} \cong X^*(\underline{T})^r$ (using a choice of embedding $\sigma'_0 : k' \hookrightarrow \mathbb{F}$ extending $\sigma_0$) by

$$\alpha'_{(s,\mu),j+kf} \stackrel{\text{def}}{=} s_{\tau}^{-k}(\alpha_{(s,\mu),j}) \quad \text{for } 0 \le j \le f-1, 0 \le k \le r-1.$$

If $\tau_K(w', \mu')$ is the analogous construction of tame types over $K'$ for $(w', \mu') \in (\underline{W} \times X^*(\underline{T}))^r$ as in Definition 2.2.1, then $\tau' \cong \tau_{K'}(1, \alpha'_{(s,\mu)})$ by direct comparison using equation (2.5). The orientation $s'_{\text{or}} \in \underline{W}^r$ of $\alpha'_{(s,\mu)}$ in the sense of [LLHLM18, Definition 2.6] is given by

$$s'_{\text{or},j+kf} \stackrel{\text{def}}{=} s_{\tau}^{k+1} s_{\text{or},j} \quad \text{for } 0 \le j \le f-1, 0 \le k \le r-1 \quad (3.1)$$

(compare with [LLHLM18, Proposition 6.1]).

Let $L' = K'(\pi_r) = K'((-p)^{\frac{1}{p^r-1}})$ and $\Delta' \stackrel{\text{def}}{=} \text{Gal}(L'/K') \subseteq \Delta \stackrel{\text{def}}{=} \text{Gal}(L'/K)$. Note that $\tau$ defines a $\mathcal{O}$-valued representation of $\Delta'$. For any complete local Noetherian $\mathcal{O}$-algebra $R$ with residue field $\mathbb{F}'$ finite over $\mathbb{F}$, let $\mathcal{S}_{L',R} \stackrel{\text{def}}{=} (W(k') \otimes_{\mathbb{Z}_p} R)[[u']]$. We endow $\mathcal{S}_{L',R}$ with an action of $\Delta$ as follows: for any $g$ in $\Delta'$, $g(u') = \frac{g(\pi_r)}{\pi_r}u'$ and $g$ acts trivially on the coefficients; if $\sigma \in \text{Gal}(L'/\mathbb{Q}_p)$ is the lift of Frobenius on $W(k')$ which fixes $\pi_r$, then $\sigma^f$ generates $\text{Gal}(K'/K)$ acting in natural way on $W(k')$ and trivially on both $u'$ and $R$. Set $v = (u')^{p^f-1}$, and note that

$$(\mathcal{S}_{L',R})^{\Delta'=1} = (W(k) \otimes_{\mathbb{Z}_p} R)[[v]].$$

As usual, $\varphi : \mathcal{S}_{L',R} \to \mathcal{S}_{L',R}$ acts as $\sigma$ on $W(k')$, trivially on $R$, and sends $u'$ to $(u')^p$.
If $\tau$ is a principal series types, let $Y^{[0,h],\tau}(R)$ be the category of Kisin modules over $L'$ with tame descent of type $\tau$ and height in $[0, h]$ as defined in [CL18, Section 3]. More generally, we have the following.

**DEFINITION 3.1.3.** An element $(\mathfrak{M}, \phi_{\mathfrak{M}}, \{\hat{g}\}) \in Y^{[0,h],\tau}(R)$ is a Kisin module $(\mathfrak{M}, \phi_{\mathfrak{M}})$ over $\mathfrak{S}_{L',R}$ [LLHLM18, Definition 2.3] with height less than $h$ together with a semilinear action of $\Delta$ which commutes with $\phi_{\mathfrak{M}}$ such that for each $0 \le j \le f' - 1$

$$ \mathfrak{M}^{(j)} \mod u' \cong \tau^{\vee} \otimes_{\mathcal{O}} R $$

as $\Delta'$-representations. In particular, the semilinear action induces an isomorphism $\iota_{\mathfrak{M}} : (\sigma^f)^*(\mathfrak{M}) \cong \mathfrak{M}$ (see [LLHLM18, Section 6.1]) as elements of $Y^{[0,h],\tau'}(R)$.

**REMARK 3.1.4.** As explained in [LLHLM18, Section 6.1], the data of an extension of the action of $\Delta'$ to an action of $\Delta$ is equivalent to the choice of an isomorphism $\iota_{\mathfrak{M}} : (\sigma^f)^*(\mathfrak{M}) \cong \mathfrak{M}$ satisfying an appropriate cocycle condition. We will use both points of view interchangeably.

**REMARK 3.1.5.** The appearance of $\tau^{\vee}$ in the definition is due to the fact that we are using the contravariant functors to Galois representations to be consistent with [LLHLM18] as opposed to the covariants versions which appear in [CL18, EGH13]. In [LLHLM18], we did not use the notation $\tau^{\vee}$. Instead, we included it in our description of descent data by having a minus sign in the equation before Definition 2.1 of loc. cit. The notion of Kisin module with tame descent data of type $\tau$ here is consistent with what appears in loc. cit.

**DEFINITION 3.1.6.** An *eigenbasis* of $\mathfrak{M} \in Y^{[0,h],\tau}(R)$ is an eigenbasis $\beta' = (\beta'^{(j')})_{j'}$ for $\mathfrak{M}$ considered as a Kisin module with descent data of type $\tau'$ in the sense of [LLHLM18, Definition 2.8], which is compatible with the isomorphism $\iota_{\mathfrak{M}}$: by letting $\beta'^{(j')} = (f_1'^{(j)}, f_2'^{(j)}, f_3'^{(j)})$ then

$$ \mathfrak{M}^{(j')} \leftarrow\sim \left( (\sigma^f)^*(\mathfrak{M}) \right)^{(j'+f)} \xrightarrow{\iota_{\mathfrak{M}}} \mathfrak{M}^{(j'+f)} $$

$$ \{f_1'^{(j)}, f_2'^{(j)}, f_3'^{(j)}\} \leftarrow\sim \{f_1'^{(j'+f)}, f_2'^{(j'+f)}, f_3'^{(j'+f)}\} $$

for all $0 \le j' \le f' - 1$, where the isomorphism on the left-hand side is obtained from [LLHLM18, Lemma 6.2]. For short, the compatibility above will be written as $\iota_{\mathfrak{M}}(\beta'^{(j')}) = \beta'^{(j'+f)}$. (See also [LLHL19, Definition 3.2.8].)
**3.2. Étale $\varphi$-modules with descent data.** The main result of this subsection is Proposition 3.2.1, describing the Frobenius action on étale $\varphi$-modules with tame descent data.

Let $\mathcal{O}_{\mathcal{E},K'}$ (respectively $\mathcal{O}_{\mathcal{E},L'}$) be the $p$-adic completion of $(W(k')[[v]])[1/v]$ (respectively of $(W(k')[[u']')][1/u'])$. Recall from [LLHLM18, Section 2.3] that for a complete local Noetherian $\mathcal{O}$-algebra $R$, we have the category $\Phi$-Mod$^{\acute{e}t}_{K'}(R)$ (respectively $\Phi$-Mod$^{\acute{e}t}_{dd,L'}(R)$) of étale $(\varphi, \mathcal{O}_{\mathcal{E},K'}\widehat{\otimes}_{\mathbb{Z}_p}R)$-modules (respectively étale $(\varphi, \mathcal{O}_{\mathcal{E},L'}\widehat{\otimes}_{\mathbb{Z}_p}R)$-modules with descent data from $L'$ to $K'$). There is an analogous definition of $\Phi$-Mod$^{\acute{e}t}_K(R)$ and $\Phi$-Mod$^{\acute{e}t}_{dd,L'}(R)$. Given $(\mathfrak{M}, \iota_{\mathfrak{M}}) \in Y^{[0,h],\tau}(R)$, the element $\mathfrak{M} \otimes \mathcal{O}_{\mathcal{E},L'}$ is naturally an object $\Phi$-Mod$^{\acute{e}t}_{dd,L'}(R)$.

We define an étale $\varphi$-module $\mathcal{M} \in \Phi$-Mod$^{\acute{e}t}_K(R)$ by

$$ \mathcal{M} \stackrel{\text{def}}{=} (\mathfrak{M} \otimes \mathcal{O}_{\mathcal{E},L'})^{\Delta=1} $$

with the induced Frobenius. This defines a functor from $Y^{[0,h],\tau}(R)$ to $\Phi$-Mod$^{\acute{e}t}_K(R)$. Finally, recall the functor $\varepsilon_0 : \Phi$-Mod$^{\acute{e}t}_K(R) \to \Phi^f$-Mod$^{\acute{e}t}_{W(k)}(R)$ from [LLHLM18, Section 2.3]. It is obtained by considering the $f$-fold composite of the partial Frobenius which act on $\mathcal{M}^{(0)}$.

Let $R$ be a local, Artinian $\mathcal{O}$-algebra with finite residue field $\mathbb{F}$. We have the usual functor $\mathbb{V}_K^*$ from $\Phi$-Mod$^{\acute{e}t}_K(R)$ to representations of $G_{K_\infty}$ over $R$. Recall from [LLHLM18, Section 6.1] the functor

$$ T_{dd'}^* : Y^{[0,h],\tau}(R) \to \mathrm{Rep}_R(G_{K_\infty}) $$

which is defined as $(\mathfrak{M}, \iota_{\mathfrak{M}}) \mapsto \mathbb{V}_K^*(\mathcal{M})$. From now onward, we write $T_{dd}^*$ for the functor which was written as $T_{dd'}^*$ in [LLHLM18, Section 6.1].

We can now state the main result of this section, which is a sharpening of [LLHLM18, Proposition 2.26].

**PROPOSITION 3.2.1.** Let $\tau$ be a 1-generic type with lowest alcove presentation $(s, \mu)$. Let $\mathfrak{M} \in Y^{[0,h],\tau}(R)$ and $\beta$ be an eigenbasis of $\mathfrak{M}$. Let $s_{\mathrm{or}} = (s_{\mathrm{or},j})_j$ be the orientation for $\tau$ and write $A^{(j)} = \mathrm{Mat}_{\beta}(\phi_{\mathfrak{M},s_{\mathrm{or},j+1}(3)}^{(j)})$ for $j \in \{0, \dots, f-1\}$. Let $\mathcal{M} = (\mathfrak{M} \otimes \mathcal{O}_{\mathcal{E},L'})^{\Delta=1} \in \Phi$-Mod$^{\acute{e}t}_K(R)$.

Then there is a basis $f = (f^{(j)})_j$ for $\mathcal{M}$ such that

$$ \mathrm{Mat}_f(\phi_{\mathcal{M}}^{(j)}) = A^{(j)} s_j^* v^{\mu_j^* + n_j^*}. $$

*Proof.* The proof is a direct computation using [LLHLM18, Propositions 2.26 and 6.1].
Let $\alpha'_{(s,\mu)}$ and $(s'_{\text{or},j'})$ be as in the discussion after Remark 3.1.2. Let $\beta'^{(j')} = (f_1'^{(j')}, f_2'^{(j')}, f_3'^{(j')})$ be an eigenbasis for $\mathfrak{M}$ and write $A^{(j')} \stackrel{\text{def}}{=} \text{Mat}_{\beta'}(\phi_{\mathfrak{M}',s'_{\text{or},j'+1}}^{(j')})$ for $0 \le j' < f' - 1$.

For any $0 \le j' \le f' - 1$, set

$$ \mathbf{a}'_{(s,\mu)} = \sum_{i=0}^{f'-1} \alpha'_{(s,\mu),-j'+i} p^i \in \mathbb{Z}^3, $$

where $-j' + i$ is taken modulo $f'$ (compare with [LLHLM18, Section 2.1]). Then

$$ f'^{(j')} \stackrel{\text{def}}{=} \{(u')^{\mathbf{a}'_{(s,\mu),1}} f_1'^{(j')}, (u')^{\mathbf{a}'_{(s,\mu),2}} f_2'^{(j')}, (u')^{\mathbf{a}'_{(s,\mu),3}} f_3'^{(j')}\} $$

is a basis for $\mathcal{M}^{(j')} \stackrel{\text{def}}{=} (\mathfrak{M}^{(j')}[1/u'])^{\Delta'=1}$ for $0 \le j' < f' - 1$.

For any $0 \le j' \le f' - 1$, a direct computation (using [LLHLM18, Proposition 2.13]) shows that the matrix for $\phi_{\mathcal{M}'}^{(j')} : \mathcal{M}^{(j')} \to \mathcal{M}^{(j'+1)}$ with respect to the bases above is

$$ s'_{\text{or},j'+1} A^{(j')}(s'_{\text{or},j'+1})^{-1} (u')^{p \mathbf{a}'_{(s,\mu)} - \mathbf{a}'_{(s,\mu)}^{(j'+1)}}. $$

Since $p \mathbf{a}'_{(s,\mu)} - \mathbf{a}'_{(s,\mu)}^{(j'+1)} = (p^{f'} - 1) \alpha'_{(s,\mu),f'-1-j'}$, this is same as

$$ s'_{\text{or},j'+1} A^{(j')}(s'_{\text{or},j'+1})^{-1} v^{\alpha'_{(s,\mu),f'-1-j'}}. \quad (3.2) $$

Define $\tilde{\beta}$ by $\tilde{\beta}^{(j')} = f'^{(j')} s'_{\text{or},j'}$ (reordering the basis vectors). Let $j' = j + if$ for $0 \le j \le f - 1$. Then the matrix for $\phi_{\mathcal{M}'}^{(j')}$ with respect to $\tilde{\beta}$ is given by

$$ A^{(j')}(s'_{\text{or},j'+1})^{-1} s'_{\text{or},j'} v^{(s'_{\text{or},j'})^{-1} (\alpha'_{(s,\mu),f'-1-j'})} = A^{(j)} s_j^* v^{\mu_j^* + \eta_j^*} \quad (3.3) $$

using equation (3.1), Remark 3.1.2, and the fact that $A^{(j')}$ only depends on $j'$ mod $f$ (cf. [LLHLM18, Proposition 6.9]).

Finally, recall that the eigenbasis $\beta'$ is required to be compatible with $\iota_{\mathfrak{M}} : (\sigma^f)^*(\mathfrak{M}) \cong \mathfrak{M}$ as in Definition 3.1.6, which gives $\iota_{\mathfrak{M}}^{(j')}(\beta'^{(j')}) = \beta'^{(j'+f)} s_{\tau}^{-1}$. Hence, $\tilde{\beta}$ descends to a collection of ordered bases $(f^{(j)})_{0 \le j \le f-1}$ of the $\mathcal{M}^{(j)} = ((\mathfrak{M}[1/u'])^{\Delta'=1})^{(j)}$ such that the matrix for the partial Frobenius map $\mathcal{M}^{(j)} \to \mathcal{M}^{(j+1)}$ is given by equation (3.3). $\square$

**PROPOSITION 3.2.2.** Let $\mathfrak{M} \in Y^{[0,h],\tau}(R)$ and let $\mathcal{M} = (\mathfrak{M} \otimes \mathcal{O}_{\mathcal{E},L})^{\Delta'=1} \in \Phi$-Mod$^{\text{etl}}_K(R)$ be the associated étale $\varphi$-module over $K$.
If $\mathfrak{f}$ is a basis of $\mathcal{M}$ and $B^{(j)} \stackrel{\text{def}}{=} \text{Mat}_f(\phi_{\mathcal{M}}^{(j)})$ are the matrices of the partial Frobenii, then the étale $\varphi^f$-module $\varepsilon_0(\mathcal{M})$ is described by

$$ \text{Mat}_{f^{(0)}}(\phi_{\mathcal{M}^{(0)}}^{f}) = \prod_{j=0}^{f-1} \varphi^{j}(B^{(f-j-1)}). $$

LEMMA 3.2.3. Let $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ be semisimple and 0-generic. Let $\mathcal{M} \in \Phi^f$-Mod$^{\dot{\text{et}}}_{W(k)}(\mathbb{F})$ such that $\mathbb{V}_K^*(\mathcal{M}) \cong \bar{\rho}|_{G_{K\infty}}$. Assume that there exists $s_0 \in W$, $\lambda = (\lambda_j) \in X_1^*(\mathbb{T})$, and a basis for $\mathcal{M}$ such that the $\phi^f$-action on $\mathcal{M}$ is given by $Ds_0^{-1}v^\mu$ where $D \in T(\mathbb{F})$ and $\mu \stackrel{\text{def}}{=} \sum_{j=0}^{f-1} p^j \lambda_j \in X^*(T)$. Then by letting $s = (s_0, 1, \dots, 1) \in \mathbb{W}$,

$$ V(\bar{\rho}|_{I_K}) = R_s(\lambda). $$

*Proof.* Let $k$ be the order of $s_0$ and write $\mu = (\mu_1, \mu_2, \mu_3) \in X^*(T)$. Then

$$ \begin{aligned} \phi^{fk}(e_i) &= \left( \prod_{m=0}^{k-1} D_{s_0^{-m}(i)} \right) v^{\sum_{m=0}^{k-1} p^{f(k-1-m)} \mu_{s_0^{-m}(i)} e_i} \\ &= \left( \prod_{m=0}^{k-1} D_{s_0^{m+1}(i)} \right) v^{\sum_{m=0}^{k-1} p^{fm} \mu_{s_0^{m+1}(i)} e_i. \end{aligned} $$

Hence, $\bar{\rho}|_{I_K}$ is isomorphic to $\bigoplus_{i=1}^3 \omega_{kf}^{\sum_{m=0}^{k-1} p^{fm} \mu_{s_0^{m+1}(i)}}$. The lemma now follows from (2.5). $\square$

**3.3. Semisimple Kisin modules.** The statement of [LLHLM18, Theorem 2.21] gives a classification of Kisin modules with descent datum which can arise as reductions of potentially crystalline representations with Hodge-Tate weights $(2, 1, 0)^f$. In this subsection, we identify Kisin modules which correspond to semisimple Galois representations in an explicit way (Theorem 3.3.12).

For $\mathbb{F}'/\mathbb{F}$ finite, let $\mathcal{I}(\mathbb{F}') \subset \text{GL}_3(\mathbb{F}'[\mathbb{V}])$ be the Iwahori subgroup of elements which are upper triangular modulo $\mathbb{V}$. Recall from Section 3.1 the partially ordered group $\tilde{\mathbb{W}}^{\vee}$ (respectively $\tilde{\mathbb{W}}^{\vee}$) which is identified with $\tilde{\mathbb{W}}$ (respectively $\tilde{\mathbb{W}}$) as a group but whose Bruhat order is defined by the *antidominant* base alcove (and still denoted as $\leq$). For any character $\mu \in X^*(\mathbb{T})$, we define $\text{Adm}^{\vee}(\mu)$ as in Definition 2.3.9 but for antidominant base alcove.

Let $h$ be a positive integer. As before, let $\tau$ be a 1-generic tame type with fixed lowest alcove presentation $(s, \mu)$. Recall from [LLHLM18, Definition 2.17] the notion of *shape*.
**DEFINITION 3.3.1.** Let $\tau$ be a principal series type, and let $\tilde{w} = (\tilde{w}_0, \tilde{w}_1, \dots, \tilde{w}_{f-1}) \in \tilde{W}^\vee$. A Kisin module $\mathfrak{M} \in Y^{[0,h],\tau}(\mathbb{F}')$ has shape $\tilde{w}$ if for any eigenbasis $\beta$, the matrices $(A^{(j)})_j = (\mathrm{Mat}_\beta (\phi_{\mathfrak{M},s_{\mathrm{or},j+1}(3)}))_j$ have the property that $A^{(j)} \in \mathcal{I}(\mathbb{F}')\tilde{w}_j \mathcal{I}(\mathbb{F}')$.

For a nonprincipal series type, we define the shape of a Kisin module $\mathfrak{M} \in Y^{[0,h],\tau}(\mathbb{F}')$ in terms of its base change $\mathrm{BC}(\mathfrak{M})$; see [LLHL19, Definition 3.2.11] or [LLHLM18, Definition 6.10] for details.

For any $\lambda \in X^*(\mathcal{T})$ effective (that is, $\lambda_j = (a_{i,j})_i$ with $a_{i,j} \ge 0$ for all $i, j$) and $h \gg 0$, let $Y^{\lambda, \tau} \subset Y^{[0,h], \tau}$ denote the closed substack defined in [CL18, Section 5] (cf. also [LLHLM18, Section 2.2]). Then for any finite extension $\mathbb{F}'/\mathbb{F}$, $Y^{\lambda, \tau}(\mathbb{F}') \subset Y^{[0,h], \tau}(\mathbb{F}')$ is the subgroupoid consisting of Kisin modules with shape in $\mathrm{Adm}^\vee(\lambda)$.

**DEFINITION 3.3.2.** Let $\tau$ be a 3-generic type with lowest alcove presentation $(s, \mu)$ and let $\bar{\rho} : G_K \to \mathrm{GL}_3(\mathbb{F})$. Assume that there exists $\overline{\mathfrak{M}}_{\bar{\rho}} \in Y^{\eta, \tau}(\mathbb{F})$ such that $T_{dd}^*(\overline{\mathfrak{M}}_{\bar{\rho}}) \cong \bar{\rho}|_{G_{K\infty}}$. We define $\tilde{w}(\bar{\rho}, \tau) \in \mathrm{Adm}^\vee(\eta)$ to be the shape of $\overline{\mathfrak{M}}_{\bar{\rho}}$.

This is well defined because there exists at most one Kisin module $\overline{\mathfrak{M}}_{\bar{\rho}}$ with height in $[0, 2]$ such that $T_{dd}^*(\overline{\mathfrak{M}}_{\bar{\rho}}) \cong \bar{\rho}|_{G_{K\infty}}$ (see [LLHLM18, Theorem 3.2] for a principal series type and Section 6.2 and the discussion before Lemma 6.13 in loc. cit. for the general case).

**REMARK 3.3.3.** More generally, if $\#S > 1$, for $\tilde{w} = (\tilde{w}_{\tilde{v}})_{\tilde{v} \in S}$, we define $\tilde{w}^*$ as the collection $(\tilde{w}_{\tilde{v}}^*)_{\tilde{v} \in S}$, where each $\tilde{w}_{\tilde{v}}^* \in \tilde{W}_{\tilde{v}}$ is as in Definition 3.1.1. Similarly, if $\tau_S$ is a collection of 3-generic tame inertial type and $\bar{\rho}_S$ is a collection of Galois representations, we let $\tilde{w}(\bar{\rho}_S, \tau_S) \stackrel{\mathrm{def}}{=} (\tilde{w}(\bar{\rho}_{\tilde{v}}, \tau_{\tilde{v}}))_{\tilde{v} \in S}$.

We now introduce the notion of semisimple Kisin module of shape $\tilde{w}$.

**DEFINITION 3.3.4.** Let $\overline{\mathfrak{M}} \in Y^{[0,h],\tau}(\mathbb{F}')$, where $\mathbb{F}'/\mathbb{F}$ is a finite extension. We say that $\overline{\mathfrak{M}}$ is semisimple of shape $\tilde{w} = (\tilde{w}_j)$ if there exists an eigenbasis $\beta$ of $\overline{\mathfrak{M}}$ such that

$$A^{(j)} \in T(\mathbb{F}'[\mathbb{V}])\tilde{w}_j$$

for $0 \le j \le f' - 1$.

**PROPOSITION 3.3.5.** If $\overline{\mathfrak{M}}$ is semisimple of shape $\tilde{w}$, then $T_{dd}^*(\overline{\mathfrak{M}})$ is semisimple.

*Proof.* If $\overline{\mathfrak{M}}$ is semisimple, we can choose a basis such that $A^{(j)} \in T(\mathbb{F}'[\mathbb{V}])\tilde{w}_j$ where $\mathbb{F}'/\mathbb{F}$ is a finite extension. Using this basis, we see by Propositions 3.2.1
and **3.2.2** that the matrix for the étale $\varphi^f$-module $\varepsilon_0(\overline{\mathfrak{M}}[1/u']^{\Delta=1})$ lies in $T(\mathbb{F}'((v)))$ for some $s \in W$. If $s$ has order $d$, then $T_{dd}^*(\overline{\mathfrak{M}})$ restricted to the unramified extension of $K_\infty$ of degree $d$ is a direct sum of characters (as $p \nmid d$) and so $T_{dd}^*(\overline{\mathfrak{M}})$ is semisimple. $\square$

**PROPOSITION 3.3.6.** If $\overline{\mathfrak{M}}$ is semisimple of shape $\tilde{w} = (\tilde{w}_j)$, then there exists an eigenbasis $\beta$ such that

$$A^{(j)} = \tilde{w}_j \quad \text{for } 0 \le j \le f-2, \quad A^{(f-1)} \in T(\mathbb{F}')\tilde{w}_{f-1}.$$

*Proof.* See [LLHL19, Proposition 3.2.16]. $\square$

**DEFINITION 3.3.7.** For any $\tilde{w} = (w_j t_{\lambda_j}) \in \tilde{W}^\vee$, define $\mathcal{M}(\tilde{w}) \in \Phi$-Mod$^{\text{ét}}_K(\mathbb{F})$ to be the free étale $\varphi$-module over $\mathcal{O}_{\varepsilon,K} \otimes \mathbb{F}$ of rank three such that the matrix of

$$\phi_{\mathcal{M}}^{(j)} : \mathcal{M}(\tilde{w})^{(j)} \to \mathcal{M}(\tilde{w})^{(j+1)}$$

is given by $w_j v^{\lambda_j}$ (with respect to the standard basis).

**PROPOSITION 3.3.8.** Let $(s, \mu)$ be a lowest alcove presentation of $\tau$. If $\overline{\mathfrak{M}} \in Y^{[0,h],\tau}(\mathbb{F}')$ is semisimple of shape $\tilde{w} \in \tilde{W}^\vee$, then

$$T_{dd}^*(\overline{\mathfrak{M}})|_{I_K} \cong \mathbb{V}_K^*(\mathcal{M}(\tilde{w}s^*t_{\mu^*+\eta^*}))|_{I_K} \cong \bar{\tau}(w, v+\eta)$$

where $\tilde{w}s^*t_{\mu^*} = w^*t_{v^*}$. (Note that since both $T_{dd}^*(\overline{\mathfrak{M}})$ and $\mathbb{V}_K^*(\mathcal{M}(\tilde{w}s^*t_{\mu^*+\eta^*}))$ are tame, they canonically extend to $G_K$.)

*Proof.* This is the special case when $n=3$ of [LLHL19, Corollary 3.2.17]. For the sake of completeness, we include an argument here.

The first isomorphism follows from Proposition **3.3.6** combined with Proposition **3.2.1**.

Let $w, v$ satisfy $\tilde{w}s^*t_{\mu^*} = w^*t_{v^*}$. One easily checks that the $\phi^f$-action on $\varepsilon_0(\mathcal{M}(w^*t_{v^*+\eta^*})) \in \Phi^f$-Mod$^{\text{ét}}_{W(k)}(\mathbb{F})$ is given by $\prod_{j=0}^{f-1} w_j^{-1} v^{p^j(v_j+\eta_j)}$ using Proposition **3.2.2**.

Let $s \in \mathbb{W}$. The $\phi^f$-action of $\varepsilon_0(\mathcal{M}((st_{v+\eta}w\pi(s)^{-1})^*)) \in \Phi^f$-Mod$^{\text{ét}}_{W(k)}(\mathbb{F})$ is given by

$$\prod_{j=0}^{f-1} s_{j-1} w_j^{-1} s_j^{-1} v^{p^j s_j(v_j+\eta_j)} = s_{f-1} \left( \prod_{j=0}^{f-1} w_j^{-1} v^{p^j(v_j+\eta_j)} \right) s_{f-1}^{-1}.$$

We conclude that $\varepsilon_0(\mathcal{M}(w^*t_{v^*+\eta^*})) \cong \varepsilon_0(\mathcal{M}((st_{v+\eta}w\pi(s)^{-1})^*))$. Therefore, $\mathbb{V}_K^*(\mathcal{M}(w^*t_{v^*+\eta^*})) \cong \mathbb{V}_K^*(\mathcal{M}((st_{v+\eta}w\pi(s)^{-1})^*))$.
As $(w, v + \eta) \sim (sw\pi(s)^{-1}, s(v + \eta))$, by [LLHL19, Proposition 2.2.4] and [Her09, Lemma 4.2], we reduce to the case where $w_i = 1$ for $i \neq f - 1$. The second isomorphism then follows from Lemma 3.2.3. $\square$

**PROPOSITION 3.3.9.** Let $\bar{\rho}, \bar{\rho}'$ be semisimple representations of $G_K$. Assume that there exists $\overline{\mathfrak{M}}$ semisimple of shape $\tilde{w}$ such that $T_{dd}^*(\overline{\mathfrak{M}}) \cong \bar{\rho}|_{G_{K\infty}}$. Then $\bar{\rho}'|_{I_K} \cong \bar{\rho}|_{I_K}$ if and only if there exists a semisimple $\overline{\mathfrak{M}}$ of shape $\tilde{w}$ such that $T_{dd}^*(\overline{\mathfrak{M}}) \cong \bar{\rho}|_{G_{K\infty}}$.

*Proof.* This follows from counting unramified twists using Proposition 3.3.6. $\square$

<figure><img src="image_7.png" /></figure>

The remainder of the subsection is devoted to results used for weight elimination.

**LEMMA 3.3.10.** If $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ admits a potentially crystalline lift of type $(\lambda, \tau)$, then so does $\bar{\rho}^{ss}$ (after possibly replacing $E$ with a ramified extension).

*Proof.* Fix a characteristic 0 lift $\rho$ of $\bar{\rho}$ which is potentially crystalline of type $(\lambda, \tau)$. Then by enlarging the coefficient ring of $\rho$, we can always find a lattice whose reduction is semisimple by [Enn, Lemma 5(2)]. $\square$

<figure><img src="image_8.png" /></figure>

The following two theorems are key inputs in weight elimination by giving an upper bound on the semisimple representations which are reductions of potentially crystalline representations of type $(\lambda, \tau)$.

**THEOREM 3.3.11.** Let $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$. Assume that $\bar{\rho}$ has a potentially crystalline lift of type $(\lambda, \tau)$ where $\lambda \in X^*(\Gamma)$ is effective. Assume that either (1) $\tau$ is a regular principal series type or (2) $\lambda = \eta$ and $\tau$ is 3-generic. Then, for a sufficiently large $h$, there is a Kisin module $\overline{\mathfrak{M}} \in Y^{[0,h]\cdot\tau}(\mathbb{F})$ of shape $\tilde{w} = (\tilde{w}_0, \tilde{w}_1, \dots, \tilde{w}_{f-1}) \in \text{Adm}^{\vee}(\lambda)$ such that $T_{dd}^*(\overline{\mathfrak{M}}) \cong \bar{\rho}|_{G_{K\infty}}$. In particular, $\overline{\mathfrak{M}} \in Y^{\lambda, \tau}(\mathbb{F})$.

*Proof.* See [LLHL19, Theorem 3.2.20]. $\square$

<figure><img src="image_9.png" /></figure>

**THEOREM 3.3.12.** Let $\bar{\rho}$ be a semisimple representation of $G_K$ and assume $\tau$ is a 1-generic tame type. Assume that either (1) $\bar{\rho}$ is a direct sum of characters or (2) $\lambda = \eta$ and $\tau$ is 3-generic.

If there exists a Kisin module $\overline{\mathfrak{M}} \in Y^{\lambda, \tau}(\mathbb{F})$ such that $T_{dd}^*(\overline{\mathfrak{M}}) \cong \bar{\rho}|_{G_{K\infty}}$, then there exists a finite extension $\mathbb{F}'/\mathbb{F}$ and a semisimple Kisin module $\overline{\mathfrak{M}}' \in Y^{\lambda, \tau}(\mathbb{F}')$
such that (after extending scalars) $T_{dd}^*(\overline{\mathfrak{M}}') \cong \overline{\rho}|_{G_{K_\infty}}$. Furthermore, we can take $\mathbb{F}' = \mathbb{F}$ in case (2).

**REMARK 3.3.13.** Theorems 3.3.11 and 3.3.12 together say that for a fixed 3-generic type $\tau$ the set of semisimple $\overline{\rho}|_{I_K}$ which arise as reductions of potentially crystalline representations of type $(\eta, \tau)$ are in bijection with a subset of $\mathrm{Adm}^\vee((2, 1, 0))^f$. In fact, it turns out that this admissibility condition exactly captures those semisimple $\overline{\rho}$ which are reductions of crystalline representations of type $(\eta, \tau)$. Checking this is equivalent to checking that the potentially crystalline deformation ring of type $(\eta, \tau)$ of the semisimple $\overline{\rho}$ corresponding to $\tilde{w} \in \mathrm{Adm}^\vee((2, 1, 0))^f$ is nonzero. When $\tilde{w}$ is such that $\ell(\tilde{w}_i) > 1$ for all $i$, this nontriviality follows from [LLHLM18, Table 7]. When $K = \mathbb{Q}_p$, [LLHLM18, Section 8] shows the nontriviality for all admissible $\tilde{w}$. For general unramified $K$ and admissible $\tilde{w}$, the nontriviality of the deformation ring will follow from Theorem 3.5.3.

**Proof of Theorem 3.3.12.** We first treat the case where $\overline{\rho}$ is a direct sum of characters. Let $M_{dd} = \overline{\mathfrak{M}}[1/u']$ be an étale $\varphi$-module with descent datum for $L'/K$ (that is, a semilinear action of $\Delta$). Since there is an equivalence of categories between étale $\varphi$-modules over $L'$ with descent datum to $K$ and $G_{K_\infty}$-representations (cf. [LLHLM18, page 24] for principal series case and Section 6.1 in general), if $\overline{\rho}$ is a direct sum of characters, then $M_{dd} = M_1 \oplus M_2 \oplus M_3$, where each $M_i$ is stable under $\phi_{M_{dd}}$ and the descent datum.

Let $Y_{M_{dd}}^{\lambda, \tau}$ be the Kisin variety parametrizing lattices in $M_{dd}$ which lie in $Y^{\lambda, \tau}$. [LLHLM18, Definition 3.1] defines $Y_{M_{dd}}^{\lambda, \tau}$ in the principal series case. In general, $Y_{M_{dd}}^{\lambda, \tau}$ is the closed subscheme of fixed points of $Y_{M_{dd}}^{\lambda, \tau'}$ under the natural action of $\sigma^f$. The torus $T = \mathbb{G}_m^3$ acts on $M_{dd}$ by scaling individually in each factor. As a consequence, we get an algebraic action of $T$ on the projective variety $Y_{M_{dd}}^{\lambda, \tau'}$.

Any such action has a fixed point over some finite extension $\mathbb{F}'/\mathbb{F}$. Let $\overline{\mathfrak{M}}' \subset Y_{M_{dd}}^{\lambda, \tau}(\mathbb{F}')$ be a $T$-fixed point. Let $\chi_i : T \to \mathbb{G}_m$ denote the projection onto each coordinate and set $\overline{\mathfrak{M}}'_i \stackrel{\mathrm{def}}{=} (\overline{\mathfrak{M}}')^{\chi_i}$. Then

$$ \overline{\mathfrak{M}}' = \overline{\mathfrak{M}}'_1 \oplus \overline{\mathfrak{M}}'_2 \oplus \overline{\mathfrak{M}}'_3. \qquad (3.4) $$

Since the $T$-action commutes with $\phi_{M_{dd}}$ and $\Delta$, each $\overline{\mathfrak{M}}'_i$ is stable under both; hence, $\overline{\mathfrak{M}}'_i$ is a rank-one Kisin module with descent datum. Any choice of eigenbasis which respects this decomposition shows that $\overline{\mathfrak{M}}'$ is semisimple. Because $\overline{\mathfrak{M}}'$ is in $Y^{\lambda, \tau}(\mathbb{F}')$, it is semisimple with an admissible shape $\tilde{w} \in \mathrm{Adm}^\vee(\lambda)$.
Now suppose that $\tau$ is 3-generic, but $\bar{\rho}$ is not necessarily a direct sum of characters. In this case, the Kisin module $\overline{\mathfrak{M}}$ of type $(\eta, \tau)$ is unique by [LLHLM18, Theorem 3.2]. We make a base change to the unramified extension $\tilde{K}/K$ of degree 6. Let $\tilde{\tau}$ be the base change of $\tau$ to $\tilde{K}$, and let $\overline{\mathfrak{M}} = \text{BC}(\overline{\mathfrak{M}})$ be the base change of the Kisin module $\overline{\mathfrak{M}}$. Since $\overline{\mathfrak{M}}$ is the unique Kisin module of type $(\eta, \tilde{\tau})$, by the above argument, it must be semisimple.

Recall the notion of gauge basis [LLHLM18, Definition 2.22]. Fix a gauge basis $\beta$ of $\overline{\mathfrak{M}}$ and let $\tilde{\beta}$ be the induced gauge basis on $\overline{\mathfrak{M}}$. The eigenbasis $\tilde{\beta}_2$ for $\overline{\mathfrak{M}}$ which puts the partial Frobenii in the form as in Proposition 3.3.6 is also a gauge basis. By [LLHLM18, Theorem 4.16], $\tilde{\beta}$ and $\tilde{\beta}_2$ differ by embedding-wise scaling by torus elements. We conclude that the matrices for the partial Frobenii with respect to $\tilde{\beta}$ and hence $\beta$ are monomial. Since the only monomial matrices $\mathcal{I}(\mathbb{F})\tilde{w}\mathcal{I}(\mathbb{F})$ are $T(\mathbb{F}[\langle v \rangle])\tilde{w}$, we see that $\overline{\mathfrak{M}}$ is semisimple. $\square$

**3.4. Shapes and Serre weights.** We continue to assume $S = \{\tilde{v}\}$. We compute $V(\bar{\rho}|_{I_K})$ in terms of shape for a semisimple $\bar{\rho}$. This will effectively allow us to determine $W^\zeta(\bar{\rho}) \cap \text{JH}(\sigma(\tau))$ for a 3-generic tame type $\tau$ (Proposition 3.4.2) via the combinatorics of Section 2 (especially Corollary 2.3.11).

**PROPOSITION 3.4.1.** Let $(s, \mu)$ be a lowest alcove presentation of $\tau$. Let $\overline{\mathfrak{M}} \in Y^{[0,h]_\tau}(\mathbb{F}')$ be semisimple of shape $\tilde{w} = (\tilde{w}_j) \in \overline{\mathbb{W}}^\vee$. Then

$$V(T_{dd}^*(\overline{\mathfrak{M}})|_{I_K}) = R_s\tilde{w}^*(\mu + \eta).$$

*Proof.* By Proposition 3.3.8, $T_{dd}^*(\overline{\mathfrak{M}})|_{I_K} \cong \bar{\tau}(w, v + \eta)$ where $\tilde{w}s^*t_{\mu^*} = w^*t_{v^*}$. By [GHS18, Proposition 9.2.3],

$$V(T_{dd}^*(\overline{\mathfrak{M}})|_{I_K}) = R_w(v + \eta).$$

Finally, an easy calculation shows that $R_s\tilde{w}^*(\mu + \eta) = R_w(v + \eta)$ using Definition 2.3.6. $\square$

**PROPOSITION 3.4.2.** Let $\tau$ be a 3-generic type with lowest alcove presentation $(s, \mu)$ and let $\overline{\mathfrak{M}} \in Y^{[0,h]_\tau}(\mathbb{F}')$ be semisimple of shape $\tilde{w} = (\tilde{w}_j) \in \overline{\mathbb{W}}^\vee$. Let $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ be semisimple and 3-generic and assume that $\bar{\rho}|_{G_{K_\infty}} \cong T_{dd}^*(\overline{\mathfrak{M}})$. Then

$$W^\zeta(\bar{\rho}, \tau) \stackrel{\text{def}}{=} W^\zeta(\bar{\rho}) \cap \text{JH}(\sigma(\tau)) = \{\sigma_{(\omega,a)}^{(s,\mu+\eta)} : (\omega, a) \in \Sigma_{\tilde{w}^*}\} \subset \text{JH}(\overline{R_s}(\mu + \eta)).$$

*Proof.* This follows from Corollary 2.3.11 and Proposition 3.4.1. $\square$
$F$, and $F_{\tilde{v}}$ will be the completion at $\tilde{v}$ of $F$. Let $\bar{\rho}_S$ be a collection $(\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S}$ where $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$ is a continuous Galois representation. Let $R_{\tilde{v}}^{\square} = R_{\bar{\rho}_{\tilde{v}}}^{\square}$ denote the unrestricted universal framed deformation ring of $\bar{\rho}_{\tilde{v}}$. Fix a natural number $h$ and let

$$R_{\infty} = \left(\bigotimes_{\tilde{v} \in S} R_{\tilde{v}}^{\square}\right) [[x_1, x_2, \dots, x_h]] \quad \text{and} \quad X_{\infty} = \text{Spf} R_{\infty}.$$

If $\tau_{\tilde{v}}$ is an inertial type for $G_{F_{\tilde{v}}}$, then let $R_{\tilde{v}}^{\tau_{\tilde{v}}} = R_{\bar{\rho}_{\tilde{v}}}^{\tau_{\tilde{v}}}$ be the universal framed deformation ring of $\bar{\rho}_{\tilde{v}}$ of inertial type $\tau_{\tilde{v}}$ and (parallel) Hodge–Tate weights (2, 1, 0). If $\tau_S = (\tau_{\tilde{v}})_{\tilde{v} \in S}$, then let

$$R_{\infty}(\tau_S) = \widehat{\bigotimes}_{\tilde{v} \in S} R_{\tilde{v}}^{\tau_{\tilde{v}}} \otimes \widehat{\bigotimes}_{\tilde{v} \in S} R_{\tilde{v}}^{\square} R_{\infty} \quad \text{and} \quad X_{\infty}(\tau_S) = \text{Spf} R_{\infty}(\tau_S).$$

Let $d+1$ be the dimension of $X_{\infty}(\tau_S)$ (the dimension is independent of $\tau_S$ by [Kis08, Theorem 3.3.4]). We denote by $\bar{R}_{\tilde{v}}^{\square}$, $\bar{R}_{\infty}$, and so on, the reduction of these objects modulo $\mathfrak{w}$.

Let $K_{\tilde{v}}$ be the group $\text{GL}_3(\mathcal{O}_{F_{\tilde{v}}})$ and $K$ be the product $\prod_{\tilde{v} \in S} K_{\tilde{v}}$. Results toward the inertial local Langlands correspondence (cf. Proposition 2.2.6(2)) associate a $\mathbb{Q}_p$-valued $K_{\tilde{v}}$-representation $\sigma(\tau_{\tilde{v}})$ to a 1-generic tame inertial type $\tau_{\tilde{v}}$ (and $\sigma(\tau_{\tilde{v}})$ can be realized over $\mathcal{O}$).

**DEFINITION 3.5.1.** Let $\text{Rep}_K(\mathcal{O})$ denote the category of continuous $K$-representations over finitely generated $\mathcal{O}$-modules and $\text{Mod}(X_{\infty})$ the category of coherent sheaves over $X_{\infty}$.

A weak minimal patching functor for $\bar{\rho}_S = (\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S}$ is defined to be a nonzero covariant exact functor $M_{\infty}: \text{Rep}_K(\mathcal{O}) \to \text{Mod}(X_{\infty})$ satisfying the following axioms:

(1) For each $\tilde{v} \in S$, let $\tau_{\tilde{v}}$ be an inertial type for $G_{F_{\tilde{v}}}$. Let $\sigma(\tau_S)$ be the $K$-representation $\bigotimes_{\tilde{v} \in S} \sigma(\tau_{\tilde{v}})$. If $\sigma(\tau_S)^{\circ}$ is an $\mathcal{O}$-lattice in $\sigma(\tau_S)$, then $M_{\infty}(\sigma(\tau_S)^{\circ})$ is $p$-torsion-free and is maximally Cohen–Macaulay over $R_{\infty}(\tau_S)$.

(2) If $M_{\infty}(R_1(\mu))$ is nonzero, then $\bar{\rho}_{\tilde{v}}$ has a semistable lift of type $\tau_{\tilde{v}}(1, \mu_{\tilde{v}})$ for all $\tilde{v} \in S$.

(3) If $\sigma$ is an irreducible $\prod_{\tilde{v} \in S} \text{GL}_3(k_{\tilde{v}})$-representation over $\mathbb{F}$, the module $M_{\infty}(\sigma)$ is either 0 or maximal Cohen–Macaulay over its support, which is equidimensional of dimension $d$.

(4) The locally free sheaf $M_{\infty}(\sigma(\tau_S)^{\circ})[1/p]$ (being maximal Cohen–Macaulay over the regular ring $R_{\infty}(\tau_S)[1/p]$) has rank at most one on each connected component.
Assume that a weak minimal patching functor $M_\infty$ for $\bar{\rho}_S$ exists. Following [GHS18, Definition 3.2.6], let $W^{\text{BM}}(\bar{\rho}_S)$ be the set of irreducible G-representations (recall that $G \stackrel{\text{def}}{=} \prod_{\tilde{v} \in S} \text{GL}_3(k_{\tilde{v}})$) $\sigma$ over $\mathbb{F}$ such that $M_\infty(\sigma)$ is nonzero (note that a priori this set depends on the choice of $M_\infty$). For a finitely generated $R_\infty$-module $M$ with scheme-theoretic support Spec $A$ of dimension at most $d$, define $e(M)$ to be $d!$ times the coefficient of the degree $d$-term of the Hilbert polynomial of $M$ (considered as an $A$-module). In particular, $e(M)$ is the Hilbert–Samuel multiplicity of $M$ as an $A$ module when $\dim A = d$ and is 0 otherwise.

We now assume that for all $\tilde{v} \in S$, $F_{\tilde{v}}$ is an unramified extension of $\mathbb{Q}_p$. Recall Definition 2.2.7 of $W^?(\bar{\rho}_S)$.

**THEOREM 3.5.2.** Suppose that $\bar{\rho}_S$ is semisimple and 10-generic and that $M_\infty$ is a weak patching functor for $\bar{\rho}_S$. Then for all Serre weights $\sigma$, $e(M_\infty(\sigma)) = 1$ if $\sigma \in W^?(\bar{\rho}_S)$ and $e(M_\infty(\sigma)) = 0$ otherwise. In particular, $W^{\text{BM}}(\bar{\rho}_S) = W^?(\bar{\rho}_S)$.

**THEOREM 3.5.3.** Let $K/\mathbb{Q}_p$ be a finite unramified extension of degree $f$. Let $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ be a continuous, 10-generic, and semisimple Galois representation, and let $\tau$ be a tame inertial type. If $\tau$ is not 1-generic, $R_{\bar{\rho}}^\tau$ is 0. If $\tau$ is 1-generic, then the number of irreducible components of $\bar{R}_{\bar{\rho}}^\tau \stackrel{\text{def}}{=} R_{\bar{\rho}}^\tau/\omega$ is equal to the number of elements in $W^?(\bar{\rho}, \tau)$. The ring $R_{\bar{\rho}}^\tau$ is a normal domain and is Cohen–Macaulay. Moreover, $\bar{R}_{\bar{\rho}}^\tau$ is reduced and its components are formally smooth of the same dimension.

The proofs of Theorems 3.5.2 and 3.5.3 appear in Section 3.5.3.

**3.5.2. Types, weights, and the Breuil–Mézard philosophy.** In this subsection, we describe the basic inductive argument toward the proofs of Theorems 3.5.2 and 3.5.3.

Recall that we have a length function $\ell : \tilde{W}^\vee \to \mathbb{Z}$.

**LEMMA 3.5.4.** Let $K/\mathbb{Q}_p$ be a finite unramified extension of degree $f$. Let $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ be a semisimple Galois representation. Let $\tau$ be 5-generic with lowest alcove presentation $(s, \mu)$. Assume that $V(\bar{\rho}|_{I_K}) = R_s\tilde{w}^*$ $(\mu + \eta)$ where $\tilde{w} = \tilde{w}(\bar{\rho}, \tau) = (\tilde{w}_0, \dots, \tilde{w}_{f-1}) \in \text{Adm}^\vee(\eta)$ with $\ell(\tilde{w}_j) > 1$ for every $j$. Then $\bar{R}_{\bar{\rho}}^\tau \neq 0$ and the number of irreducible components of $\bar{R}_{\bar{\rho}}^\tau$ is equal to the number of elements in $W^?(\bar{\rho}, \tau)$. The ring $R_{\bar{\rho}}^\tau$ is a normal domain and is Cohen–Macaulay. Moreover, $\bar{R}_{\bar{\rho}}^\tau$ is reduced and its components are formally smooth of the same dimension.
*Proof.* By Proposition 3.4.1, our hypotheses imply that there exists a Kisin module $\mathfrak{M}_{\bar{\rho}} \in Y^{\eta, \tau}(\mathbb{F})$ such that $T_{dd}^*(\mathfrak{M}_{\bar{\rho}}) \cong \bar{\rho}|_{G_{K_{\infty}}} \text{ and that } \tilde{w} = \tilde{w}(\bar{\rho}, \tau)$. In [LLHLM18, Sections 5.3, 6], we gave an explicit presentation for $R_{\bar{\rho}}^{\tau}$ whenever the length of $\tilde{w}_j$ is at least 2 and the type $\tau$ is 5-generic. For each shape, there were at most two cases depending on $\mathfrak{M}_{\bar{\rho}}$. By Propositions 3.3.9 and 3.3.12, whenever $\bar{\rho}$ is semisimple, $\mathfrak{M}_{\bar{\rho}}$ is semisimple in the sense of Definition 3.3.1, and so $\mathfrak{M}_{\bar{\rho}}$ lies in the special locus for the given shape.

We claim that

$$ \#\mathrm{Irr}(\mathrm{Spec}(\bar{R}_{\bar{\rho}}^{\tau})) = \prod_{j} 2^{4-\ell(\tilde{w}_j)} $$

and that $\bar{R}_{\bar{\rho}}^{\tau}$ is reduced and Cohen–Macaulay. Indeed, from Table 7 in loc. cit., we see that $\bar{R}_{\bar{\rho}}^{\tau}$ is the completion of a tensor product over $\mathbb{F}$ of rings of the form $\mathbb{F}[X]$, $\mathbb{F}[X, Y]/XY$ at the maximal ideal generated by the variables $X, Y$. Since such a tensor product is Cohen–Macaulay, so is its completion by [Sta19, Tag 07NX]. Since such a tensor product is reduced and excellent (being a finite type $\mathbb{F}$-algebra), its completion is reduced by [Sta19, Tag 07NZ]. Finally, we also note that since each irreducible components of such a tensor product is smooth, the completion of the irreducible components stay irreducible and are formally smooth.

This implies the remaining properties (see the proof of [LLHLM18, Corollary 8.9]). Note that the reducedness of $\bar{R}_{\bar{\rho}}^{\tau}$ implies that $R_{\bar{\rho}}^{\tau}$ is a normal domain).

Finally, we compare this with the size of $W^{\tau}(\bar{\rho}, \tau)$, which is $\prod_{j \in \mathcal{J}} \#\Sigma_{(\tilde{w}_j^*)}$ by Proposition 3.4.2 using Table 2. $\square$

**COROLLARY 3.5.5.** Suppose that for each $\tilde{v} \in S$, $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to GL_3(\mathbb{F})$ is a semisimple Galois representation and $\tau_{\tilde{v}}$ is a 5-generic tame inertial type satisfying the hypotheses of Lemma 3.5.4. Let $\tau_S$ be $(\tau_{\tilde{v}})_{\tilde{v} \in S}$. Then $e(\bar{R}_{\infty}(\tau_S))$ is equal to $\#W^{\tau}(\bar{\rho}_S, \tau_S)$.

*Proof.* This follows immediately from Lemma 3.5.4 and [EG14, Lemma 2.2.14] and properties of tensor products of representations. $\square$

**PROPOSITION 3.5.6 (Weight elimination).** Suppose that for each $\tilde{v}$, $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to GL_3(\mathbb{F})$ is a 10-generic Galois representation. Then

$$ W^{\mathrm{BM}}(\bar{\rho}_S) \subset W^{\tau}(\bar{\rho}_S^{\mathrm{ss}}). $$

*Proof.* Suppose that $\sigma$ is a Serre weight such that $\sigma \in W^{\mathrm{BM}}(\bar{\rho}_S) \setminus W^{\tau}(\bar{\rho}_S^{\mathrm{ss}})$. Note that $W^{\mathrm{BM}}(\bar{\rho}_S)$ satisfies the (evident generalization of) [Enn, Axiom (WE)]
by 3.5.1(2). By [Enn, Theorem 8] and [LLHL19, Remark 2.2.8], $\bar{\rho}_S$ is 10-generic implies that $\sigma$ is 3-generic. Then $\sigma \in \text{JH}(\overline{\sigma(\tau_S)})$ for some collection $\tau_S \stackrel{\text{def}}{=} (\tau_{\tilde{v}})_{\tilde{v} \in S}$ of 1-generic tame types (for example, taking a tame principal series type containing $\sigma$). Since $\bar{\rho}_S$ is 10-generic, we see by Proposition 3.4.5 and Definition 3.5.1(1) that $M_\infty(\sigma(\tau_S)) \neq 0$ implies that $\tau_S$ is 7-generic. By Lemma 4.2.13 (whose proof only uses modular representation theory and is independent of the results in Section 3), $\sigma$ is 5-deep.

By Corollary 2.3.13, $\tau_S$ above can be taken so that $W^?(\bar{\rho}_S^{ss}, \tau_S) = \emptyset$ and $\sigma \in \text{JH}(\overline{\sigma(\tau_S)})$. By Proposition 3.4.4, $\bar{\rho}_{\tilde{v}}^{ss}$ is not the reduction of a potentially crystalline representation of type $(\eta_{\tilde{v}}, \tau_{\tilde{v}})$ for some $\tilde{v} \in S$. By Proposition 3.3.10, $\bar{\rho}_{\tilde{v}}$ is also not the reduction of a potentially crystalline representation of type $(\eta_{\tilde{v}}, \tau_{\tilde{v}})$ for some $\tilde{v} \in S$. Then $X_\infty(\tau_S) = \emptyset$ and $M_\infty(\sigma(\tau_S)^\circ) = 0$ for any $\mathcal{O}$-lattice $\sigma(\tau_S)^\circ$ in $\sigma(\tau_S)$. By exactness of $M_\infty$, $M_\infty(\sigma) = 0$ which is a contradiction. $\square$

**REMARK 3.5.7.** Instead of appealing to [Enn], one can show that provided that $\bar{\rho}_S$ is generic enough, every element of $W^{\text{BM}}(\bar{\rho}_S)$ appears as a Jordan–Hölder factor of the reduction of some 1-generic tame type as follows. Even though not all Serre weights come from the reduction $\sigma(\tau_S)$ for a collection $\tau_S$ of 1-generic tame types, they do occur in the reduction of $V_\lambda \otimes \sigma(\tau_S)$ where $\lambda$ is sufficiently large (independent of $p$). The results of Section 3.3 hold also for potentially crystalline representations of type $(\lambda_{\tilde{v}} + \eta_{\tilde{v}}, \tau_{\tilde{v}})_{\tilde{v} \in S}$ (possibly with a stronger genericity hypothesis), and then the same argument as in the proof of Proposition 3.5.6 gives the result.

Given a collection $\tau_S$ of tame inertial types and a $\mathcal{O}$-lattice $\sigma(\tau_S)^\circ$ in $\sigma(\tau_S)$, we write $\bar{\sigma}(\tau_S)^\circ$ to denote the reduction of $\sigma(\tau_S)^\circ$ modulo $\bar{\omega}$.

**LEMMA 3.5.8.** *Suppose that for each $\tilde{v} \in S$, $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$ is a semisimple Galois representation and $\tau_{\tilde{v}}$ is a 5-generic tame inertial type satisfying the hypotheses of Lemma 3.5.4. Let $\tau_S = (\tau_{\tilde{v}})_{\tilde{v} \in S}$. Let $\sigma(\tau_S)^\circ$ be an $\mathcal{O}$-lattice in $\sigma(\tau_S)$. Then either $M_\infty(\bar{\sigma}(\tau_S)^\circ) = 0$ or $e(M_\infty(\bar{\sigma}(\tau_S)^\circ))$ is equal to $e(\bar{R}_\infty(\tau_S))$, and the alternative does not depend on the lattice.*

*Proof.* The generic fiber of $X_\infty(\tau_S)$ is irreducible since its special fiber is reduced (see Lemma 3.5.4). Then since $M_\infty(\sigma(\tau_S)^\circ)$ is maximal Cohen–Macaulay over $X_\infty(\tau_S)$, either $M_\infty(\sigma(\tau_S)^\circ)$ has full support on $X_\infty(\tau_S)$ or $M_\infty(\sigma(\tau_S)^\circ)$ is 0. The lemma now follows from the proof of [LLHLM18, Proposition 7.14]. $\square$
LEMMA 3.5.9. Suppose that $\bar{\rho}_{\bar{v}}: G_{F_{\bar{v}}} \to GL_3(\mathbb{F})$ is semisimple and 7-generic for all $\bar{v} \in S$. If $\sigma \in W^\gamma(\bar{\rho}_S)$ has defect $\delta$, then there are tame inertial types $\tau_{\bar{v}}$ and $\tau'_{\bar{v}}$ satisfying the hypotheses of Corollary 3.5.5 such that if $\tau_S = (\tau_{\bar{v}})_{\bar{v} \in S}$ and $\tau'_S = (\tau'_{\bar{v}})_{\bar{v} \in S}$, we have

$$ (1) \quad \sigma \in W^\gamma(\bar{\rho}_S, \tau_S) \subset W^\gamma(\bar{\rho}_S, \tau'_S); $$

$$ (2) \#W^\gamma(\bar{\rho}_S, \tau_S) = 2^\delta \text{ and } \#W^\gamma(\bar{\rho}_S, \tau'_S) = 2^{\delta+1}; $$

$$ (3) \text{ all Serre weights in } W^\gamma(\bar{\rho}_S, \tau_S) \text{ and } W^\gamma(\bar{\rho}_S, \tau'_S) \text{ have defect at most } \delta \text{ and if } \delta > 0, \sigma \text{ is the unique Serre weight in } W^\gamma(\bar{\rho}_S, \tau'_S) \text{ with defect } \delta; $$

$$ (4) \text{ if } \delta = 0, \text{ for any } \sigma' \in W^\gamma(\bar{\rho}_S) \text{ of defect } 0 \text{ which is adjacent to } \sigma, \text{ one can choose the } \tau'_S \text{ above so that } W^\gamma(\bar{\rho}_S, \tau'_S) = \{\sigma, \sigma'\}. $$

*Proof.* This is essentially a consequence of results in Section 2.3. Suppose that $V(\bar{\rho}_S|_{I_{F_S}}) = R_S(\mu)$ with $\mu - \eta$ 7-deep in alcove $C_0$, and label $W^\gamma(\bar{\rho}_S)$ by $r(\Sigma)$ as in Proposition 2.3.4. Let $\tilde{w} = (\tilde{w}_{\bar{v}})_{\bar{v} \in S} = (\tilde{w}_i)_{i \in \mathcal{J}} \in \text{Adm}(\eta)$ and $\tau_S$ be the tame inertial type with lowest alcove presentation $(sw^{-1}, \mu + s\tilde{w}^{-1}(0) - \eta)$, where $w \in W$ is the image of $\tilde{w}$. Then $\mu + s\tilde{w}^{-1}(0) - \eta$ is 5-deep in alcove $C_0$ so that $\tau_S$ is 5-generic. Corollary 2.3.11 says that

$$ W^\gamma(\bar{\rho}_S, \tau_S) = \left\{ \sigma_{r(\omega, a)}^{(s, \mu)} : r(\omega, a) \in \prod_{i \in \mathcal{J}} r(\Sigma_{\tilde{w}_i^{-1}}) \right\}, $$

where $\Sigma_{\tilde{w}_i^{-1}} = \Sigma_0 \cap \tilde{w}_i^{-1}(r(\Sigma_0))$. If $\ell(\tilde{w}_i) > 1$ for all $i$, then $\tau_S$ satisfies the hypotheses of Corollary 3.5.5 (noting that $\ell(\tilde{w}) = \ell(\tilde{w}^*)$ by the proof of [LLHL19, Lemma 2.1.3], where the lengths are as elements of $\tilde{W}^\vee$ and $\tilde{W}$, respectively).

Let $(\omega, a) = ((\omega_i, a_i))_i$ be such that $\sigma = \sigma_{r(\omega, a)}^{(s, \mu)}$. We will construct the required types by appropriately choosing $(\tilde{w}_i)_i$.

If $(\omega_i, a_i) \in \Sigma_0^{\text{obv}}$, we can find an element $\tilde{w}_i \in \text{Adm}((2, 1, 0))$ such that $\ell(\tilde{w}_i) = 4$ and $\Sigma_{\tilde{w}_i^{-1}} = \{(\omega_i, a_i)\}$. If $(\omega_i, a_i) \in \Sigma_0^{\text{inn}}$, we can find an element $\tilde{w}_i t_{-1} \in \{\alpha\beta\alpha, \beta\gamma\beta, \alpha\gamma\alpha\}$ such that $\Sigma_{\tilde{w}_i^{-1}}$ contains exactly $\{(\omega_i, a_i), r(\omega_i, a_i)\}$. This choice of $(\tilde{w}_i)_{i \in \mathcal{J}}$ gives a type $\tau_S = (\tau_{\bar{v}})_{\bar{v} \in S}$ such that $\sigma \in W^\gamma(\bar{\rho}_S, \tau_S)$, $\#W^\gamma(\bar{\rho}_S, \tau_S) = 2^\delta$, and all weights in $W^\gamma(\bar{\rho}_S, \tau_S)$ have defect at most $\delta$.

To construct $\tau_S$ satisfying (2) and (3), we proceed similarly. We first deal with the case $\delta > 0$. In this case, we can find a $j_0 \in \mathcal{J}$ such that $\ell(\tilde{w}_{j_0}) = 3$, and consider any $\tilde{w}'_{j_0} \le \tilde{w}_{j_0}$ of length 2. Then

$$ \Sigma_{\tilde{w}_{j_0}^{-1}} \subset \Sigma_{\tilde{w}'_{j_0}^{-1}} $$
and has size 4 (see Table 2). Furthermore, $\Sigma_{\tilde{w}'_j}^{-1} \setminus \Sigma_{\tilde{w}'_j}^{-1} \subset \Sigma_0^{\text{obv}}$. Let $\tilde{w}'$ be such that $\tilde{w}'_i = \tilde{w}_i$ if $i \neq j_0$ and $\tilde{w}'_{j_0}$ is the element chosen above. Then the type $\tau'_S$ such that $\sigma(\tau'_S) = R_{s(\tilde{w}'^{-1})}(\mu)$ satisfies items (2) and (3).

Finally, assume that $\delta = 0$. Let $\sigma' \in W^r(\bar{\rho}_S)$ be a defect 0 weight adjacent to $\sigma$, and write $\sigma' = \sigma_{r(\omega', a')}^{(s, \mu)}$. Then there is a unique $j_0 \in \mathcal{J}$ such that $(\omega'_{j_0}, a'_{j_0}) \neq (\omega_{j_0}, a_{j_0})$. There are six possible pairs $\{(\omega_j, a_j), (\omega'_j, a'_j)\} \in \Sigma_0^{\text{obv}}$ which are adjacent in Table 1, each of which is $\Sigma_{\tilde{w}'_j}^{-1}$ for some length 3 nonshadow element $\tilde{w}'_j$ (see [LLHLM18, Table 1]). We let $\tau'_S$ be the inertial type corresponding to $\tilde{w}'$ such that $\tilde{w}'_i = \tilde{w}_i$ for $i \neq j_0$ and $\tilde{w}'_{j_0}$ chosen as in the previous sentence. This gives the type satisfying items (2), (3), and (4). $\square$

REMARK 3.5.10. From Table 2, we see that the type $\tau_S$ constructed in Lemma 3.5.9 is uniquely characterized by requiring that $\sigma \in W^r(\bar{\rho}_S, \tau_S)$ and $\#W^r(\bar{\rho}_S, \tau_S) = 2^\delta$. We call it the minimal type of $\sigma$ with respect to $\bar{\rho}$.

In what follows, our $\bar{\rho}_S = (\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S}$ will be assumed to be 10-generic so that Proposition 3.5.6 applies. We observe that if $\tau_S$ is a 3-generic tame type, then $\bar{\sigma}(\tau_S)^\circ$ is multiplicity free for any choice of lattice $\sigma(\tau_S)^\circ$. Then

$$e(M_\infty(\bar{\sigma}(\tau_S)^\circ)) = \sum_{\sigma \in \mathcal{J}\mathcal{H}(\bar{\sigma}(\tau_S))} e(M_\infty(\sigma))$$

by Definition 3.5.1, and, in fact,

$$e(M_\infty(\bar{\sigma}(\tau_S)^\circ)) = \sum_{\sigma \in W^r(\bar{\rho}_S^{\text{sg}}, \tau_S)} e(M_\infty(\sigma)).$$

Finally, observe that if $\bar{\rho}_S$ is 10-generic and $W^r(\bar{\rho}_S, \tau_S)$ is nonempty, then $\tau_S$ is 7-generic by Proposition 3.4.5.

LEMMA 3.5.11. Suppose that $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$ is semisimple and 10-generic for all $\tilde{v} \in S$. If there exists $\sigma \in W^{\text{BM}}(\bar{\rho}_S)$ with defect 0, then for all $\sigma' \in W^r(\bar{\rho}_S)$, $e(M_\infty(\sigma')) = 1$. In particular, $W^{\text{BM}}(\bar{\rho}_S) = W^r(\bar{\rho}_S)$.

*Proof.* We first prove the lemma assuming $\sigma'$ has defect 0 by induction on $d \stackrel{\text{def}}{=} d_{\text{gph}}(\sigma, \sigma')$. By Lemma 3.5.9, one can choose a 1-generic tame type $\tau_S$ such that $W^r(\bar{\rho}_S, \tau_S) = \{\sigma\}$. Note that $\tau_S$ is then 7-generic under our assumptions. Then $e(M_\infty(\sigma)) = 1$ by Corollary 3.5.5 and Lemma 3.5.8. This establishes the case $d = 0$.

Suppose that $\sigma' \in W^r(\bar{\rho}_S)$ has defect 0 and that $d > 0$. Then there is a $\sigma'' \in W^r(\bar{\rho}_S)$ adjacent to $\sigma'$, with defect 0, and such that $d_{\text{gph}}(\sigma, \sigma'') = d - 1$.
We choose a type $\tau'_S$ as in Lemma 3.5.9(4) for the adjacent weights $\sigma'$ and $\sigma''$ and an $O$-lattice $\sigma(\tau'_S)^\circ$ in $\sigma(\tau'_S)$. Then by inductive hypothesis, $M_\infty(\sigma'')$, and hence $M_\infty(\bar{\sigma}(\tau'_S)^\circ)$, is nonzero. By Corollary 3.5.5 and Lemma 3.5.8, $e(M_\infty(\bar{\sigma}(\tau'_S)^\circ)) = 2$. We deduce from the inductive hypothesis that $e(M_\infty(\sigma')) = 1$.

We now prove the general case of the lemma by induction on the defect. Suppose that $\sigma' \in W'(\bar{\rho}_S)$ has defect $\delta > 0$. We choose $\tau_S$ as in Lemma 3.5.9 (and an $O$-lattice $\sigma(\tau_S)^0$ in $\sigma(\tau_S)$) taking $\sigma$ to be $\sigma'$. By Lemma 3.5.9(2) and (3), $W'(\bar{\rho}_S, \tau_S)$ contains a weight of lower defect. The inductive hypothesis implies that $M_\infty(\bar{\sigma}(\tau_S)^\circ)$ is nonzero. By Lemma 3.5.8 and Corollary 3.5.5, $e(M_\infty(\bar{\sigma}(\tau_S)^\circ)) = 2^\delta$. By Lemma 3.5.9(3) and induction, $e(M_\infty(\bar{\sigma}(\tau_S)^\circ)) - e(M_\infty(\sigma'))$ is the number of weights in $W'(\bar{\rho}_S, \tau_S) \setminus \{\sigma'\}$, which is $2^\delta - 1$. We conclude that $e(M_\infty(\sigma')) = 1$. $\square$

LEMMA 3.5.12. Suppose that $\bar{\rho}_{\tilde{v}} : G_{F_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$ is semisimple and 10-generic for all $\tilde{v} \in S$. If $\sigma \in W^{\text{BM}}(\bar{\rho}_S)$ has defect $\delta > 0$, then there exists $\sigma' \in W^{\text{BM}}(\bar{\rho}_S)$ with defect less than $\delta$.

*Proof.* Choose $\tau_S$ and $\tau'_S$ as in Lemma 3.5.9 and fix lattices $\sigma(\tau_S)^\circ$ and $\sigma(\tau'_S)^\circ$. Then $M_\infty(\sigma(\tau_S)^\circ)$ and $M_\infty(\sigma(\tau'_S)^\circ)$ are nonzero. Hence, by Lemmas 3.5.8 and 3.5.9(2) and Corollary 3.5.5, $e(M_\infty(\bar{\sigma}(\tau'_S)^\circ)) - e(M_\infty(\bar{\sigma}(\tau_S)^\circ)) = 2^\delta$. By Proposition 3.5.9(1), $e(M_\infty(\bar{\sigma}(\tau'_S)^\circ)) - e(M_\infty(\bar{\sigma}(\tau_S)^\circ))$ is the sum of $e(M_\infty(\sigma'))$ as $\sigma'$ runs over the Serre weights in $W'(\bar{\rho}_S, \tau'_S) \setminus W'(\bar{\rho}_S, \tau_S)$. By Lemma 3.5.9(3), the Serre weights in $W'(\bar{\rho}_S, \tau'_S) \setminus W'(\bar{\rho}_S, \tau_S)$ have defect less than $\delta$. We conclude that there must be a Serre weight $\sigma' \in W^{\text{BM}}(\bar{\rho}_S)$ of defect less than $\delta$. $\square$

### 3.5.3. Proofs

*Proof of Theorem 3.5.2.* Since $M_\infty$ is nonzero, there is a Serre weight $\sigma \in W^{\text{BM}}(\bar{\rho}_S) \subset W'(\bar{\rho}_S)$ by Proposition 3.5.6. By induction on the defect using Lemma 3.5.12, we can assume, without loss of generality, that the defect of $\sigma$ is 0. The theorem now follows from Lemma 3.5.11. $\square$

REMARK 3.5.13. Our axioms for $M_\infty$ imply that if $\sigma \in \text{JH}(\overline{\sigma(\tau_S)})$, then the support of $M_\infty(\sigma)$ is a (possibly empty) union of irreducible components of $\text{Spec } \bar{R}_\infty(\tau_S)$. As $\text{Spec } R_\infty(\tau_S)$ is the preimage of

$$ \text{Spec} \bigwedge_{\tilde{v} \in S} R_{\tilde{v}}^{\tau_{\tilde{v}}} \subset \text{Spec} \bigwedge_{\tilde{v} \in S} R_{\tilde{v}}^{\square} $$
inside Spec $R_\infty$, we have shown that if $\sigma \in W^{\text{BM}}(\bar{\rho}_S)$, then the scheme-theoretic support of $M_\infty(\sigma)$ is irreducible with generic point given by the preimage in $R_\infty$ of a prime ideal in $\bigotimes_{\tilde{v} \in S} R_{\tilde{v}}^{\square}$.

We now give two examples of weak minimal patching functors using the setup from [LLHLM18, Section 7.1]. Recall the definitions from loc. cit. of $F/F^+$, $\Sigma_p^+$, $G_{F^+}$, and $\iota_w$ (see also Section 5.3). Suppose that $\bar{r} : G_F \to \text{GL}_3(\mathbb{F})$ is

* automorphic (of some weight) in the sense of [LLHLM18, Definition 7.1];

* satisfies the Taylor–Wiles hypotheses in the sense of [LLHLM18, Definition 7.3]; and

* if $\bar{r}$ is ramified at a finite place $w \notin \Sigma_p$ of $F$, then $w|_{F^+}$ splits in $F$ (we say that $\bar{r}$ has split ramification outside of $p$).

Then [LLHLM18, Proposition 7.15] constructs a weak minimal patching functor for $\bar{r}$ in the sense of [LLHLM18, Definition 7.11], which we will denote by $\tilde{M}_\infty$. Let $h$ be the corresponding integer.

For each $v \in \Sigma_p^+$, choose a place $\tilde{v}$ of $F$ lying above $v$. Let $S_p$ be the set $\{\tilde{v} : v \in \Sigma_p^+\}$. Let $\bar{\rho}_{S_p}$ be $(\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S_p}$ where we define $\bar{\rho}_{\tilde{v}} \stackrel{\text{def}}{=} \bar{r}|_{G_{F_{\tilde{v}}}} : G_{F_{\tilde{v}}} \to \text{GL}_3(\mathbb{F})$. Define $R_\infty$, $R_\infty(\tau_{S_p})$, and so on, as before with respect to $h$ above. Let $K$ be $\prod_{S_p} K_{\tilde{v}}$ as before. From the proof of [LLHL19, Proposition 4.2.6] we have the following.

**PROPOSITION 3.5.14.** Let $M_\infty : \text{Rep}_K(\mathcal{O}) \to \text{Mod}(X_\infty)$ be the functor $\tilde{M}_\infty \circ \prod_{\tilde{v} \in S_p} \iota_{\tilde{v}}$. Then $M_\infty$ is a weak minimal patching functor for $\bar{\rho}_{S_p}$.

*Proof.* Definition 3.5.1(2) follows from the proof of [LLHL19, Proposition 4.2.6]. The remaining properties follow easily from definitions and [LLHLM18, Proposition 7.15]. □

Now suppose that $p > 3$ and $K/\mathbb{Q}_p$ is a finite extension. Let $\bar{\rho} : G_K \to \text{GL}_3(\mathbb{F})$ be a continuous Galois representation with a potentially diagonalizable lift of type $(\eta, \tau)$ such that $R_{\bar{\rho}}^\tau$ is formally smooth. For example, if $\bar{\rho}$ is 6-generic and semisimple, we can take $\tau$ so that if $\tilde{w}(\bar{\rho}, \tau) = (\tilde{w}_j)_j$, then $\ell(\tilde{w}_j) = 4$ for all $j$. We now construct a weak minimal patching functor for $\bar{\rho}$ (here $\#S = 1$ in the notation of Definition 3.5.1).

[EG14, Corollary A.7] constructs a CM extension $F/F^+$, a choice of places $S_p$ above $\Sigma_p^+$ as before, and an automorphic Galois representation $\bar{r} : G_F \to \text{GL}_3(\mathbb{F})$ satisfying the above itemized properties such that there is an isomorphism $K \cong F_{\tilde{v}}$ and $\bar{r}|_{G_{F_{\tilde{v}}}} \cong \bar{\rho}$ for all $\tilde{v} \in S_p$. Fix a place $\tilde{v} \in S_p$
and let $R_\infty$ be $R_\bar{\rho} \otimes \widehat{\otimes}_{v' \in \Sigma_\rho^+, v' \neq v} R_{\bar{\rho}_{v'}}^\tau$ $[[x_1, \dots, x_h]]$. Note that $R_\infty$ is as defined in Section 3.5.1 (by increasing $h$, and the assumption of formal smoothness of $R_\bar{\rho}^\tau$) and we identify $R_\infty$ with $R_{\bar{r}|_{G_{F_\bar{v}}}}^\Box \otimes \widehat{\otimes}_{v' \in \Sigma_\rho^+, v' \neq v} R_{\bar{r}|_{G_{F_\bar{v}}}}^\tau$ $[[x_1, \dots, x_h]]$. Let $X_\infty = \text{Spf} R_\infty$ as usual. We abusively let $K$ be the group $\text{GL}_3(\mathcal{O}_K)$ (the meaning of each $K$ will be clear from the context). Fix a lattice $\sigma(\tau)^\circ$ in the $K$-module $\sigma(\tau)$.

Let $\tilde{M}_\infty$ be the weak minimal patching functor for $\bar{r}$ constructed above. The following proposition follows from the construction of $\bar{r}$ and Proposition 3.5.14.

**PROPOSITION 3.5.15.** Assume the above setup. Let $M_\infty : \text{Rep}_K(\mathcal{O}) \to \text{Mod}(X_\infty)$ be the functor

$$ \tilde{M}_{\infty} \circ \prod_{v \in \Sigma_{\rho}^{+}} l_{\tilde{v}} \circ \left( - \otimes \widehat{\otimes}_{v \in \Sigma_{\rho}^{+}, v' \neq v} \sigma(\tau)^{\circ} \right). $$

Then $M_\infty$ is a patching functor for $\bar{\rho}$.

**COROLLARY 3.5.16.** Suppose that $\bar{\rho}_S$ is a collection of continuous Galois representations satisfying the properties of $\bar{\rho}$ in Proposition 3.5.15. Then there exists a patching functor for $\bar{\rho}_S$.

*Proof.* We can take the completed tensor product of the patching functors constructed in Proposition 3.5.15 for each $\bar{\rho}_{\tilde{v}}$. $\square$

*Proof of Theorem 3.5.3.* Suppose that $\tau$ is not 1-generic. We claim that $R_\bar{\rho}^\tau = 0$. It suffices to show that after restriction to $G_K'$ for any unramified extension $K'/K$, $\bar{\rho}$ does not have a potentially crystalline lift of type $\tau$ and Hodge–Tate weights (2, 1, 0). Moreover, after such a restriction, $\tau$ is still not 1-generic and $\bar{\rho}$ is 10-generic. We can then assume, without loss of generality, that $\tau$ is a principal series. By [LLHL19, Remark 2.2.8], $\tau$ is not 2-generic and $\bar{\rho}$ is 10-generic in the sense of [Enn, Definition 2.2]. Then $\bar{\rho}$ does not have a potentially crystalline lift of type $\tau$ and Hodge–Tate weights (2, 1, 0) by [Enn, Proposition 7].

If $\tau$ is 1-generic and $R_\bar{\rho}^\tau \neq 0$, then Proposition 3.4.5 implies that $\tau$ is 7-generic. Proposition 3.4.2 implies that $W^\tau(\bar{\rho}, \tau) \neq \emptyset$. Suppose that $\tau$ is 1-generic and $W^\tau(\bar{\rho}, \tau) \neq \emptyset$. By Proposition 3.5.15, a patching functor $M_\infty$ for $\bar{\rho}$ exists. For any $\mathcal{O}_E$-lattice $\sigma(\tau)^\circ \subset \sigma(\tau)$, $M_\infty(\sigma(\tau)^\circ) \neq 0$ by Theorem 3.5.2. This implies that $R_\bar{\rho}^\tau$ is nonzero.

We now show that if $\tau$ is 1-generic and $R_\bar{\rho}^\tau \neq 0$, then

$$ e(\bar{R}_{\bar{\rho}}^{\tau}) = e(M_{\infty}(\bar{\sigma}(\tau)^{\circ})). \qquad (3.5) $$
As $R_{\bar{\rho}}^{\tau} \neq 0$, we deduce from Proposition 3.4.5 that $\tau$ is 7-generic. The proof of (3.5) is now obtained by a direct generalization of the arguments of [LLHLM18, Section 8]; so we only explain the key details. There is a ring $\tilde{R}_{\bar{\rho}}^{\tau}$ of the same dimension as $\bar{R}_{\bar{\rho}}^{\tau}[[X_1, \dots, X_{3f}]]$ which is a power series ring over the completed tensor product over $\mathbb{F}$ of rings $\tilde{R}$ in [LLHLM18, Section 8] and $\bar{R}_{\mathfrak{M}, \tilde{w}}^{\text{expl}, \nabla}$ for some $\mathfrak{M}$ and $\tilde{w}$ from [LLHLM18, Table 7], and which admits a surjection to $\bar{R}_{\bar{\rho}}^{\tau}[[X_1, \dots, X_{3f}]]$. Then we have

$$e(\tilde{R}_{\bar{\rho}}^{\tau}) = e(M_{\infty}(\bar{\sigma}(\tau)^{\circ}))$$

for any $\mathcal{O}$-lattice $\sigma(\tau)^{\circ} \subset \sigma(\tau)$ by computing both sides using [LLHLM18, Table 7, Propositions 8.5 and 8.12] and Theorem 3.5.2 (using that $\bar{\rho}$ is 10-generic). The proof of [LLHLM18, Proposition 7.14] implies

$$e(M_{\infty}(\bar{\sigma}(\tau)^{\circ})) \leq e(\bar{R}_{\bar{\rho}}^{\tau}).$$

Then we must have that $e(\tilde{R}_{\bar{\rho}}^{\tau}) = e(\bar{R}_{\bar{\rho}}^{\tau}[[X_1, \dots, X_{3f}]]) = e(\bar{R}_{\bar{\rho}}^{\tau})$ by the above surjection. Furthermore, $\tilde{R}_{\bar{\rho}}^{\tau}$ is reduced and Cohen-Macaulay by the same argument as in the proof of Lemma 3.5.4. Then $\bar{R}_{\bar{\rho}}^{\tau}[[X_1, \dots, X_{3f}]]$ is isomorphic to $\tilde{R}_{\bar{\rho}}^{\tau}$ by [LLHLM18, Lemma 8.8] and the desired properties of $\bar{R}_{\bar{\rho}}^{\tau}$ hold because they do for $\tilde{R}_{\bar{\rho}}^{\tau}$. The desired ring-theoretic properties of $\bar{R}_{\bar{\rho}}^{\tau}$ follow from the proof of [LLHLM18, Corollary 8.9]. $\square$

**REMARK 3.5.17.** Since the number of irreducible components of $\bar{R}_{\bar{\rho}}^{\tau}$ is equal to $\#W^{?}(\bar{\rho}, \tau)$, $\bar{R}_{\bar{\rho}}^{\tau}$ is reduced, and $M_{\infty}(\bar{\sigma}(\tau)^{\circ})$ has full support over $\bar{R}_{\infty}(\tau)$, the proof of Theorem 3.5.3 shows that the irreducible support of $M_{\infty}(\sigma)$ must be different for each $\sigma \in W^{?}(\bar{\rho}, \tau)$.

**3.6. The geometric Breuil-Mézard conjecture.** We now show that weak minimal patching functors can be used to assign components in deformation rings to Serre weights.

**PROPOSITION 3.6.1.** (1) Let $\bar{\rho}$ be as in Theorem 3.5.3. There is a unique assignment $\sigma \mapsto \mathfrak{p}(\sigma)$ for $\sigma \in W^{?}(\bar{\rho})$ such that $\mathfrak{p}(\sigma) \subset \bar{R}_{\bar{\rho}}^{\square}$ is a prime ideal and

$$\text{Spec}(\bar{R}_{\bar{\rho}}^{\tau}) = \bigcup_{\sigma \in W^{?}(\bar{\rho}, \tau)} \text{Spec}(\bar{R}_{\bar{\rho}}^{\square}/\mathfrak{p}(\sigma)) \quad (3.6)$$

for any tame type $\tau$ where the right-hand side is given the reduced scheme structure. Moreover, the image of $\mathfrak{p}(\sigma)$ in $\bar{R}_{\bar{\rho}}^{\tau}$ is a minimal prime ideal and (3.6) is the decomposition of $\bar{R}_{\bar{\rho}}^{\tau}$ into irreducible components.
(2) If $M_\infty$ is a weak minimal patching functor for $\bar{\rho}_S = (\bar{\rho}_{\tilde{v}})_{\tilde{v} \in S}$, then the scheme-theoretic support of $M_\infty(\otimes_{\tilde{v} \in S} \sigma_{\tilde{v}})$ is $\text{Spec}(R_\infty / \sum_{\tilde{v} \in S} \mathfrak{p}(\sigma_{\tilde{v}}) R_\infty)$.

*Proof.* We first prove uniqueness. Suppose there is such an assignment. This closely follows the procedure of induction on the defect with respect to $W^2(\bar{\rho})$ in the proof of Theorem 3.5.2. If the defect of $\sigma \in W^2(\bar{\rho})$ is 0, then by letting $\tau$ be the minimal type of $\sigma$ with respect to $\bar{\rho}$, we have $\#W^2(\bar{\rho}, \tau) = 1$ by Lemma 3.5.9. Then we must have $\mathfrak{p}(\sigma) = \text{Ann}_{R_\infty} \bar{R}_\rho^\tau$. If the defect of $\sigma$ is $\delta > 0$, then choose $\tau$ to be the minimal type of $\sigma$ with respect to $\bar{\rho}$ as in Lemma 3.5.9. Then by induction and Lemma 3.5.9(3), there is a unique component of $\text{Spec}(\bar{R}_\rho^\tau)$ whose defining ideal is not $\mathfrak{p}(\sigma')$ for some $\sigma' \in W^2(\bar{\rho}, \tau)$ of lower defect. Then $\mathfrak{p}(\sigma)$ must be this defining ideal.

We now show existence of an assignment. By Proposition 3.5.15, there is a weak minimal patching functor $M_\infty$ for $\bar{\rho}$, which we fix. By Remark 3.5.13, the generic point of the scheme-theoretic support of $M_\infty(\sigma)$ is of the form $\mathfrak{p}'(\sigma)R_\infty$ for some prime ideal $\mathfrak{p}'(\sigma) \subset R_\rho^{\square}$. We claim that $\sigma \mapsto \mathfrak{p}'(\sigma)$ is an assignment satisfying (3.6). Indeed, since the generic fiber of $R_\rho^\tau$ is connected by 3.5.3, $\text{Spec } \bar{R}_\infty(\tau)$ is the scheme-theoretic support of $M_\infty(\bar{\sigma}(\tau)^\circ)$. On the other hand, $M_\infty(\bar{\sigma}(\tau)^\circ)$ is filtered by $M_\infty(\sigma)$ for $\sigma \in W^2(\bar{\rho}, \tau)$ so that the support of $M_\infty(\bar{\sigma}(\tau)^\circ)$ is

$$ \bigcup_{\sigma \in W^2(\bar{\rho}, \tau)} \text{Spec} \left( \frac{R_\infty}{\mathfrak{p}'(\sigma)} R_\infty \right). \qquad (3.7) $$

Equation (3.7) is a decomposition of $\bar{R}_\infty(\tau)$ into irreducible components by Remark 3.5.17. Finally, we observe that this statement descends to $\bar{R}_\rho^\tau$.

We now show part (2). Suppose that $M_\infty$ is a weak minimal patching functor for $\bar{\rho}_S$. Let $\sigma = \otimes_{\tilde{v} \in S} \sigma_{\tilde{v}} \in W^2(\bar{\rho}_S)$. Again by the proof of Theorem 3.5.2, the scheme-theoretic support of $M_\infty(\sigma)$ is $\text{Spec } R_\infty / (\sum_{\tilde{v} \in S} \mathfrak{p}(\sigma)_{\tilde{v}} R_\infty)$ for some prime ideals $\mathfrak{p}(\sigma)_{\tilde{v}} \subset R_\rho^{\square}$. We will show that $\mathfrak{p}(\sigma)_{\tilde{v}} = \mathfrak{p}(\sigma_{\tilde{v}})$, where $\mathfrak{p}$ is the assignment in part (1). We induct on $\delta = \text{Def}_{\bar{\rho}_S}(\sigma)$. If $\delta = 0$, then one can choose $\tau_S$ as in Lemma 3.5.9 so that $\bar{R}_\infty(\tau_S) = R_\infty / \mathfrak{p}(\sigma)$. We conclude that $\mathfrak{p}(\sigma)_{\tilde{v}} = \mathfrak{p}(\sigma_{\tilde{v}})$ for all $\tilde{v} \in S$ in this case. Suppose that $\delta > 0$. Again choose $\tau_S$ as in Lemma 3.5.9. Then by the inductive hypothesis, for any weight $\sigma' = \otimes_{\tilde{v} \in S} \sigma'_{\tilde{v}} \in W^2(\bar{\rho}_S, \tau_S)$ with $\sigma' \not\cong \sigma$, the scheme-theoretic support of $M_\infty(\sigma')$ is $R_\infty / (\sum_{\tilde{v} \in S} \mathfrak{p}(\sigma'_{\tilde{v}}) R_\infty)$. Since the generic fiber of $\text{Spec}(\otimes_{\tilde{v} \in S} \mathfrak{R}_{\bar{\rho}_{\tilde{v}}}^{\tau_{\tilde{v}}})$ is connected by 3.5.3, by item (1), the scheme-theoretic support of $M_\infty(\bar{\sigma}(\tau_S)^\circ)$ is

$$ \bigcup_{\sigma' \in W^2(\bar{\rho}_S, \tau_S)} \text{Spec} \left( \frac{R_\infty}{\left( \sum_{\tilde{v} \in S} \mathfrak{p}(\sigma'_{\tilde{v}}) R_\infty \right)} \right). \qquad (3.8) $$
From this, we see that since $e(M_\infty(\sigma)) = 1$, the scheme-theoretic support of $M_\infty(\sigma)$ is forced to be $\text{Spec } R_\infty/\left(\sum_{\tilde{v} \in S} \mathfrak{p}(\sigma_{\tilde{v}})R_\infty\right)$. $\square$

We have the following refinement of Proposition 3.6.1(2).

LEMMA 3.6.2. Assume that $\bar{\rho}$ is 10-generic. Let $\tau_S = (\tau_{\tilde{v}})_{\tilde{v} \in S}$ be a collection of tame inertial types and let $V$ be a subquotient of $\bar{\sigma}(\tau_S)^\circ$ for some $\mathcal{O}$-lattice $\sigma(\tau_S)^\circ$ in $\sigma(\tau_S)$. Define the closed subscheme $\bar{X}_\infty(V) \hookrightarrow \text{Spec } \bar{R}_\infty$ to be the reduced subscheme underlying $\bigcup_{\sigma \in \text{JH}(V)} \text{Supp}(M_\infty(\sigma))$. Then the scheme-theoretic support of $M_\infty(V)$ is $\bar{X}_\infty(V)$. In particular, if $M_\infty(V)$ is a cyclic $R_\infty$-module, then $M_\infty(V) \cong R_\infty/I(V)$, where

$$I(V) \stackrel{\text{def}}{=} \bigcap_{\sigma \in \text{JH}(V)} \text{Ann}_{R_\infty}(M_\infty(\sigma)).$$

*Proof.* Since $\bar{\rho}$ is 10-generic, there is nothing to prove unless $\tau_S$ is 7-generic. The proof now follows exactly as in the second paragraph of the proof of [EGS15, Proposition 8.1.1]. We recall the argument. The support of $M_\infty(V)$ is (the topological space) $\bar{X}_\infty(V)$ since $M_\infty(V)$ is filtered by $M_\infty(\sigma)$ for $\sigma \in \text{JH}(V)$. It suffices to show that the scheme-theoretic support of $M_\infty(V)$ is reduced. The scheme-theoretic support of $M_\infty(V)$ is generically reduced (since the same is true for $M_\infty(\bar{\sigma}(\tau_S)^\circ)$). Now since each $M_\infty(\sigma)$ is maximal Cohen–Macaulay over $\bar{R}_\infty(\tau_S)$ (by Definition 3.5.1(3) and the fact that $\dim \bar{R}_\infty(\tau_S) = d$) and the maximal Cohen–Macaulay property is preserved under extension (by the characterization of depth in terms of Ext groups), $M_\infty(V)$ is maximal Cohen–Macaulay over $\bar{R}_\infty(\tau_S)$. This guarantees that the scheme-theoretic support of $M_\infty(V)$ has no embedded associated primes and, hence, is reduced. $\square$

3.6.1. *Matching components.* Recall that Proposition 3.6.1 gives a canonical parametrization of the irreducible components of the special fiber of the potentially crystalline deformation ring $R_{\bar{\rho}}^\tau$ in terms of $W^\gamma(\bar{\rho}, \tau)$. Given a lowest alcove presentation $(s, \mu - \eta)$ of $\tau = \tau(s, \mu)$, we will define in item (4) below explicit rings $\bar{R}_{\bar{\sigma}, \tilde{w}}^{\text{expl}, \vee}$, building on [LLHLM18, Sections 5.3.1, 8]. The rings $\bar{R}_{\bar{\sigma}, \tilde{w}}^{\text{expl}, \vee}$ will be formally smooth modifications of $\bar{R}_{\bar{\rho}}^\tau$. Thus, Proposition 3.6.1 gives a bijection between minimal primes of $\bar{R}_{\bar{\sigma}, \tilde{w}}^{\text{expl}, \vee}$ and $W^\gamma(\bar{\rho}, \tau)$. On the other hand, by Corollary 2.3.11, the data $(s, \mu)$ gives a description of $W^\gamma(\bar{\rho}, \tau)$ as $\sigma_{(\omega, a)}^{(s, \mu)}$, for $(\omega, a) \in \Sigma_{\tilde{w}}$. In this subsection, we will make the bijection between minimal primes of $\bar{R}_{\bar{\sigma}, \tilde{w}}^{\text{expl}, \vee}$ and $\Sigma_{\tilde{w}}$ explicit. This will be needed in Section 5, where we need to check relations between ideals corresponding to various subquotients of $\bar{\sigma}(\tau)^0$ for certain lattices $\sigma(\tau)^0$.
We begin by recalling the relationship between $\overline{R}_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$ and $\overline{R}_{\tilde{\rho}}^{\tau}$. For the rest of this section, we assume that $\overline{\rho}$ is 10-generic. Recall that we have chosen a lowest alcove presentation $(s, \mu - \eta)$ of $\tau = \tau(s, \mu)$. We assume that $R_{\tilde{\rho}}^{\tau} \neq 0$, and, thus, $\tau$ is 7-generic and there exists a unique $\overline{\mathfrak{M}} \in Y^{\eta, \tau}(\mathbb{F})$ such that $T_{dd}^*(\overline{\mathfrak{M}}) \cong \overline{\rho}|_{G_{K_\infty}}$. Let $\tilde{w} = \tilde{w}(\overline{\rho}, \tau)$ be the shape of $\overline{\mathfrak{M}}$ and let $\overline{\mathcal{M}} \stackrel{\text{def}}{=} (\overline{\mathfrak{M}}[1/u'])^{\Delta=1}$. We also recall the notion of gauge basis [LLHLM18, Definitions 4.15, 6.11], and we fix a gauge basis $\tilde{\beta}$ for $\overline{\mathfrak{M}}$. We also fix a framing (that is, a basis) for $\tilde{\rho}$. Recall from [LLHLM18, Definition 2.11 and Section 6] (see also the discussion after [LLHL19, Definition 3.2.8], which is more aligned with the notation of this paper) the notation $A^{(i)} \stackrel{\text{def}}{=} \text{Mat}_{\beta}(\phi_{\overline{\mathfrak{M}}, s_{\text{or},i+1}(3)}^{(i)})$, for an eigenbasis $\beta$ of a Kisin module $\overline{\mathfrak{M}}$ with descent data of type $\tau$ and $s_{\text{or}} \in \mathcal{W}$ the orientation of $\tau = \tau(s, \mu)$. We say that $A^{(i)}$ is the matrix of the $i$th partial Frobenius $\phi_{\overline{\mathfrak{M}}, s_{\text{or},i+1}(3)}^{(i)}$ with respect to the eigenbasis $\beta$.

We have the following canonical diagram (cf. [LLHLM18, Diagram (5.9)]):

$$
\begin{tikzcd}
& & \mathrm{Spf}(\bar{R}_{\overline{\mathfrak{M}}, \bar{\rho}}^{\tau, \bar{\beta}, \Box}) \arrow[r, "f.s."'] & \mathrm{Spf}(\bar{R}_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}) \arrow[r, "f.s."'] & \bar{D}_{\overline{\mathfrak{M}}}^{\tau, \bar{\beta}} \arrow[d, "f.s."'] \\
& & \mathrm{Spf}(\bar{R}_{\bar{\rho}}^{\tau}) = \mathrm{Spf}(\bar{R}_{\overline{\mathfrak{M}}, \bar{\rho}}^{\tau, \Box}) \arrow[r, "f.s."'] & \left[ \mathrm{Spf}(\bar{R}_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}) / \hat{\mathbf{G}}_m^{3f} \right] \arrow[r, "f.s."'] & \bar{Y}_{\overline{\mathfrak{M}}}^{\eta, \tau} \\
& & \Phi\text{-Mod}_{\bar{\mathcal{M}}}^{\acute{\text{et}}, \Box} \arrow[r, "f.s."'] & \Phi\text{-Mod}_{\bar{\mathcal{M}}}^{\acute{\text{et}}} \arrow[r, "t_{\tau}"] \arrow[d, "l'_{\tau}", "t_{\tau}"] & \Phi\text{-Mod}_{\bar{\mathcal{M}}}^{\acute{\text{et}}} \arrow[d, "l'_{\tau}", "t_{\tau}"] \\
& & & &
\end{tikzcd}
\quad (3.9)
$$

where $f.s.$ stands for a formally smooth morphism. We explain the diagram as follows:

(1) $\bar{R}_{\overline{\mathfrak{M}}, \bar{\rho}}^{\tau, \bar{\beta}, \Box}$, $\bar{R}_{\overline{\mathfrak{M}}, \bar{\rho}}^{\tau, \Box}$, are defined in [LLHLM18, Sections 4.3, 5.2, 6.2]. They parametrize various deformation problems of $\bar{\rho}$ and $\overline{\mathfrak{M}}$ with extra data such as framings on Galois representations and gauge bases on $\overline{\mathfrak{M}}$. We note that in loc. cit., we used the symbol $\mu$ for what we call $\eta$ in this paper.

(2) $\Phi$-Mod$_{\overline{\mathcal{M}}}^{\acute{\text{et}}}$ (respectively $\Phi$-Mod$_{\overline{\mathcal{M}}}^{\acute{\text{et}}, \Box}$) denotes the groupoid of étale $\varphi$-modules deforming $\overline{\mathcal{M}}$ (respectively deformations with a basis on the associated $G_{K_\infty}$-representation).

(3) $\bar{Y}_{\overline{\mathfrak{M}}}^{\eta, \tau}$ is the groupoid whose values on a local Artinian $\mathbb{F}$-algebra $A$ is given by the groupoid of pairs $(\mathfrak{M}_A, J_A)$ where $\mathfrak{M}_A \in Y^{\eta, \tau}(A)$ and $J_A : \mathfrak{M}_A \otimes_A \mathbb{F} \to \overline{\mathfrak{M}}$ is an isomorphism in $Y^{\eta, \tau}(\mathbb{F})$. The groupoid $\bar{D}_{\overline{\mathfrak{M}}}^{\tau, \bar{\beta}}$ parametrizes
the same data plus the data of a gauge basis lifting $\bar{\beta}$. By [LLHLM18, Theorem 6.12], there is an action of $\hat{G}_m^{3f}$ on $\overline{D_{\mathfrak{M}}^{\tau, \bar{\beta}}}$ by scaling the gauge basis, and one has $[\overline{D_{\mathfrak{M}}^{\tau, \bar{\beta}}}/\hat{G}_m^{3f}] \cong \overline{Y}^{\eta, \tau}$. By [LLHLM18, Theorem 4.17 and Section 6.2], $\overline{D_{\mathfrak{M}}^{\tau, \bar{\beta}}}$ is representable by $\overline{R_{\mathfrak{M}}^{\tau, \bar{\beta}}} = \hat{\otimes}_i (R_{\tilde{w}_i}^{\text{expl}, \text{flat, red}}/\varpi)$. Over $\overline{R_{\mathfrak{M}}^{\tau, \bar{\beta}}}$, we have a universal pair $(\mathfrak{M}^{\text{univ}}, \beta^{\text{univ}})$ and, hence, the universal matrices of partial Frobenii $A^{(i),\text{univ}}$. By construction, $(R_{\tilde{w}_i}^{\text{expl}, \text{flat, red}}/\varpi)$ is a quotient of the power series ring over $\mathbb{F}$ generated by (suitable modifications of) the coefficients of the entries of $A^{(i),\text{univ}}$ subject to certain 'finite height' equations. The map $\iota'_\tau$ is the map sending $\mathfrak{M}^{\text{univ}}$ to $(\mathfrak{M}^{\text{univ}}[\frac{1}{\tilde{w}'_i}])^{\Delta=1}$.

(4) The ring $\overline{R_{\mathfrak{M}, \tilde{w}}^{\text{expl}, \nabla}} = \hat{\otimes}_i \overline{R_{\mathfrak{M}, \tilde{w}_i}^{\text{expl}, \nabla}}$. We recall the description of each component ring from [LLHLM18] (see also Table 3). We let $(a, b, c) \in \mathbb{F}_p^3$ be the mod $p$ reduction of $s_{f-1-i}^{-1}(\mu_{f-1-i}) \in X^*(T) \cong \mathbb{Z}^3$.

(1) When $\ell(\tilde{w}_i) > 1$: $\overline{R_{\mathfrak{M}, \tilde{w}_i}^{\text{expl}, \nabla}}$ is the quotient of the power series ring over $\mathbb{F}$ generated by (suitable modifications of) the coefficients of the entries of $A^{(i),\text{univ}}$ by an explicit list of relations given by [LLHLM18, Section 5.3 and Table 7]. In this case, we even have the rings $R_{\mathfrak{M}, \tilde{w}_i}^{\text{expl}, \nabla}$ and diagram (3.9) can be lifted to a diagram over $\mathcal{O}$ with the same properties; cf. [LLHLM18, Diagram (5.9)].

Note that, strictly speaking, [LLHLM18, Table 7] only has entries for $\tilde{w}_i$ belonging to a certain set of representatives under the action of the outer automorphisms of $\tilde{W}_a$.

(2) When $\ell(\tilde{w}_i) = 1$: By symmetry, we may assume $\tilde{w}_i = \alpha t_\perp$. The matrix $A^{(i),\text{univ}}$ has the form

$$A^{(i), \text{univ}} = \begin{pmatrix} c_{11} & c_{12} + v c_{12}^* & c_{13} \\ v c_{21}^* & c_{22} + v d_{22} & c_{23} \\ v c_{31} & v c_{32} & c_{33} + v c_{33}^* \end{pmatrix}.$$

Set $\tilde{c}_{32} \stackrel{\text{def}}{=} \frac{c_{32}c_{21} - d_{22}c_{31}}{c_{21}^*}$. We define $\overline{R_{\mathfrak{M}, \tilde{w}_i}^{\text{expl}, \nabla}}$ to be the quotient of the power series ring

$$\mathbb{F}[\llbracket c_{11}, c_{12}, c_{13}, c_{22}, c_{23}, c_{31}, \tilde{c}_{32}, c_{33}, d_{22}, c_{12}^* - [\tilde{c}_{12}^*], c_{21}^* - [\tilde{c}_{21}^*], c_{33}^* - [\tilde{c}_{33}^*] \rrbracket]$$

by the following relations:

$$c_{11}c_{23} = 0,$$

$$c_{33}^*c_{11}\tilde{c}_{32} = c_{13}c_{31}\tilde{c}_{32},$$
<table><caption>Table 3. Universal matrix, presentation, and component labeling for the ring $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$.</caption><thead><tr><th></th><th>A<sup>(f-1-i)</sup></th><th>(ω<sub>i</sub>, a<sub>i</sub>) ∈ Σ<sub>20</sub></th><th>c<sub>ω<sub>i</sub>,a<sub>i</sub></sub></th><th>ζ<sub>i</sub></th></tr></thead><tbody><tr><td rowspan="2">βγ</td><td rowspan="2">(e<sub>1</sub> + ε<sub>2</sub>, 0)</td><td>(e<sub>11</sub>)</td><td>βγ'β</td></tr><tr><td>(ε<sub>2</sub>, 1)</td><td>α</td></tr><tr><td rowspan="2">aβγ</td><td rowspan="2">(e<sub>1</sub> + ε<sub>2</sub>, 0)</td><td>(e<sub>22</sub>)</td><td>αγ'α</td></tr><tr><td>(e<sub>1</sub>, 1)</td><td>β</td></tr><tr><td rowspan="2">aβ</td><td rowspan="2">(0, 0)</td><td>(e<sub>11</sub>)</td><td>γ*</td></tr><tr><td>(0, 1)</td><td>((a-b)c<sub>23</sub>c<sub>23</sub> - (a-c)c<sub>31</sub>c<sub>31</sub>)</td><td>id</td></tr><tr><td rowspan="4">aβ</td><td rowspan="4">(e<sub>1</sub>, 1)</td><td>(e<sub>12</sub>, c<sub>31</sub>)</td><td>γ'β</td></tr><tr><td>(e<sub>1</sub> - ε<sub>2</sub>, 0)</td><td>(e<sub>11</sub>, d<sub>33</sub>)</td><td>βγ'αβ</td></tr><tr><td>(0, 0)</td><td>(e<sub>12</sub>, c<sub>22</sub>)</td><td>αγ*</td></tr><tr><td>(0, 1)</td><td>(c<sub>22</sub>, (b-c)d<sub>31</sub>c<sub>31</sub> + (a-b)c<sub>31</sub>d<sub>33</sub>)</td><td>α</td></tr><tr><td rowspan="4">βα</td><td rowspan="4">(e<sub>2</sub>, 1)</td><td>(d<sub>22</sub>, c<sub>11</sub>)</td><td>γ'α</td></tr><tr><td>(e<sub>2</sub> - ε<sub>1</sub>, 0)</td><td>(d<sub>22</sub>, c<sub>32</sub>)</td><td>αγ'βα</td></tr><tr><td>(0, 0)</td><td>(e<sub>11</sub>, c<sub>31</sub>)</td><td>βγ*</td></tr><tr><td>(0, 1)</td><td>((a-b)c<sub>33</sub>c<sub>33</sub> - (a-c)d<sub>22</sub>d<sub>33</sub>, c<sub>31</sub>c<sub>31</sub> - c<sub>11</sub>d<sub>33</sub>)</td><td>β</td></tr><tr><td rowspan="6">a</td><td rowspan="6">(e<sub>1</sub>, 0)</td><td>(e<sub>11</sub>, c<sub>31</sub>, c<sub>31</sub>)</td><td>βγ'β</td></tr><tr><td>(e<sub>2</sub>, 0)</td><td>(c<sub>11</sub>, c<sub>31</sub>, c<sub>31</sub>)</td><td>γ'αβ</td></tr><tr><td>(e<sub>2</sub>, 1)</td><td>(c<sub>11</sub>, c<sub>23</sub>c<sub>23</sub> - d<sub>22</sub>c<sub>31</sub>, (a-b)c<sub>23</sub>d<sub>22</sub> + (-1 + a)O(c<sub>23</sub>c<sub>23</sub>)</td><td>αbγ'βα</td></tr><tr><td>(e<sub>2</sub> - ε<sub>1</sub>, 0)</td><td>(c<sub>23</sub>, d<sub>22</sub>, c<sub>11</sub> - c<sub>23</sub>c<sub>23</sub>)</td><td>βαγ'</td></tr><tr><td>(0, 0)</td><td>(e<sub>11</sub>c<sub>31</sub> = c<sub>11</sub>c<sub>31</sub>, c<sub>21</sub>, (a-b)c<sub>23</sub>d<sub>22</sub> + (c-b)(c<sub>23</sub>c<sub>23</sub> - d<sub>22</sub>c<sub>33</sub>))</td><td>βα</td></tr><tr><td>(0, 1)</td><td>(c<sub>11</sub>c<sub>31</sub> = c<sub>11</sub>c<sub>31</sub>, c<sub>21</sub>, (a-b)c<sub>23</sub>d<sub>22</sub> + (c-b)(c<sub>23</sub>c<sub>23</sub> - d<sub>22</sub>c<sub>33</sub>))</td><td>βα</td></tr><tr><td rowspan="6">id</td><td rowspan="6">(e<sub>1</sub>, 0)</td><td>(e<sub>11</sub>, c<sub>31</sub>, c<sub>31</sub>)</td><td>βγ'β</td></tr><tr><td>(e<sub>1</sub>, 1)</td><td>(c<sub>11</sub>, c<sub>33</sub>, c<sub>11</sub>, (-1 + a) + c(c<sub>33</sub>c<sub>33</sub> - (-1 + a) + c(c<sub>33</sub>c<sub>33</sub> - c<sub>21</sub>c<sub>21</sub>)</td><td>βγ'β</td></tr><tr><td>(e<sub>2</sub>, 0)</td><td>(c<sub>11</sub>, c<sub>31</sub>, c<sub>31</sub>, c<sub>21</sub>c<sub>21</sub> + c<sub>23</sub>c<sub>23</sub>)</td><td>αγ'αβ</td></tr><tr><td>(e<sub>2</sub>, 1)</td><td>(c<sub>12</sub>, d<sub>22</sub>, c<sub>11</sub>, (a-b)c<sub>23</sub>d<sub>22</sub> + (-1 + b) + O(c<sub>23</sub>c<sub>23</sub> - c<sub>31</sub>c<sub>31</sub>)</td><td>αγ'α</td></tr><tr><td>(0, 0)</td><td>(c<sub>23</sub>, c<sub>31</sub>, c<sub>31</sub>, c<sub>23</sub>c<sub>23</sub> + (a-b)c<sub>23</sub>c<sub>23</sub> - (a-c)d<sub>22</sub>d<sub>33</sub>)</td><td>αbγ'</td></tr><tr><td>(0, 1)</td><td>(c<sub>23</sub>, c<sub>31</sub>, c<sub>31</sub>, (a-b)c<sub>23</sub>c<sub>23</sub> - (a-c)c<sub>21</sub>c<sub>21</sub>)</td><td>αbα</td></tr></tbody></table>

Data relevant to Theorem 3.6.4. The first column records the components of the shape $\tilde{w} = \tilde{w}(\bar{\rho}, \tau)$. The second column records the form of the matrix of partial Frobenius $A^{(f-1-i)}$ and the presentation of the ring $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ in terms of the entries of $A^{(f-1-i)}$. The fourth column records the prime ideal $c_{(ω_i, a_i)}$ in the statement of Theorem 3.6.4. The last column records the element $\tilde{z}^* \in W_a$ that occurs in Proposition 3.6.9, which controls the minimal type $\tau'$ of the weight given by the third column with respect to $\bar{\rho}$. Thus, $\tau' = \tau(sz^*, \mu + s\tilde{z}^*(0))$. Note that $\tilde{w}(\bar{\rho}, \tau') = \tilde{w}(\bar{\rho}, \tau)\tilde{z}^{-1}$. Finally, the structure constants that feature in the presentation of $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ are given by $(a, b, c) \in \mathbb{F}_p^2$ with $(a, b, c) \equiv s_i^{-1}(\mu_i) \mod p$.
<table><caption>Table 3. (continued)</caption><thead><tr><th></th><th>A(f-1-i)</th><th>(ω<sub>i</sub>, a<sub>i</sub>) ∈ Σ<sup>af</sup><sub>n</sub></th><th>w<sub>ω<sub>i</sub></sub> = c<sub>(ω<sub>i</sub>,0)</sub> ∩ c<sub>(ω<sub>i</sub>,1)</sub></th><th>z̃<sub>i</sub></th></tr></thead><tbody><tr><td>αβ</td><td><table><tr><td>c<sub>13</sub>c<sub>12</sub>(c<sub>32</sub><sup>*</sup>)<sup>-1</sup></td><td>c<sub>12</sub></td><td>c<sub>13</sub> + vc<sub>13</sub><sup>*</sup></td></tr><tr><td>vc<sub>21</sub><sup>*</sup></td><td>c<sub>22</sub></td><td>c<sub>23</sub> + vd<sub>23</sub></td></tr><tr><td>vc<sub>31</sub><sup>*</sup></td><td>vc<sub>32</sub><sup>*</sup></td><td>c<sub>31</sub>c<sub>23</sub>(c<sub>21</sub><sup>*</sup>)<sup>-1</sup> + vd<sub>33</sub></td></tr></table></td><td>(0, 1), (0, 0)</td><td>(c<sub>22</sub>)</td><td>α</td></tr><tr><td>βα</td><td><table><tr><td>c<sub>11</sub></td><td>(c<sub>31</sub><sup>*</sup>)<sup>-1</sup>c<sub>11</sub>c<sub>32</sub> + vc<sub>12</sub><sup>*</sup></td><td>c<sub>13</sub></td></tr><tr><td>0</td><td>vd<sub>22</sub></td><td>vc<sub>23</sub><sup>*</sup></td></tr><tr><td>vc<sub>31</sub><sup>*</sup></td><td>vc<sub>32</sub></td><td>c<sub>33</sub> + vd<sub>33</sub></td></tr></table></td><td>(0, 1), (0, 0)</td><td>(c<sub>13</sub>c<sub>31</sub><sup>*</sup> - c<sub>11</sub>d<sub>33</sub>)</td><td>β</td></tr><tr><td>α</td><td><table><tr><td>c<sub>11</sub></td><td>c<sub>12</sub> + vc<sub>12</sub><sup>*</sup></td><td>c<sub>13</sub></td></tr><tr><td>vc<sub>21</sub><sup>*</sup></td><td>c<sub>22</sub> + vd<sub>22</sub></td><td>c<sub>23</sub></td></tr><tr><td>vc<sub>31</sub><sup>*</sup></td><td>vc<sub>32</sub></td><td>(c<sub>21</sub><sup>*</sup>)<sup>-1</sup>c<sub>31</sub>c<sub>23</sub> + vc<sub>33</sub><sup>*</sup></td></tr></table></td><td>(ε<sub>2</sub>, 1), (ε<sub>2</sub>, 0)</td><td>(c<sub>32</sub>c<sub>21</sub><sup>*</sup> - d<sub>22</sub>c<sub>31</sub>, c<sub>11</sub>)</td><td>γ<sup>*</sup>α</td></tr><tr><td>id</td><td><table><tr><td>c<sub>11</sub> + vc<sub>11</sub><sup>*</sup></td><td>c<sub>12</sub></td><td>c<sub>13</sub></td></tr><tr><td>vc<sub>21</sub><sup>*</sup></td><td>c<sub>22</sub> + vc<sub>22</sub><sup>*</sup></td><td>c<sub>23</sub></td></tr><tr><td>vc<sub>31</sub><sup>*</sup></td><td>vc<sub>32</sub></td><td>c<sub>33</sub> + vc<sub>33</sub><sup>*</sup></td></tr></table></td><td>(ε<sub>1</sub>, 1), (ε<sub>1</sub>, 0)</td><td>(c<sub>33</sub>, c<sub>11</sub>, c<sub>31</sub>, c<sub>11</sub>c<sub>23</sub> - c<sub>21</sub>c<sub>13</sub>)</td><td>βγ<sup>*</sup>β</td></tr><tr><td></td><td>see §3.6.2 for the relations among coefficients</td><td>(0, 1), (0, 0)</td><td>(c<sub>11</sub>c<sub>33</sub><sup>*</sup> - c<sub>13</sub>c<sub>31</sub>, c<sub>23</sub>)</td><td>αγ<sup>*</sup>α</td></tr><tr><td></td><td>see §3.6.2 for the relations among coefficients</td><td>(0, 1), (0, 0)</td><td>(c<sub>22</sub>, c<sub>33</sub>, c<sub>23</sub>, c<sub>33</sub>c<sub>12</sub> - c<sub>13</sub>c<sub>32</sub>)</td><td>αβα</td></tr></tbody></table>

Further data relevant to Theorem 3.6.4. The description of the columns is the same as that in Table 3.

$$ 
\begin{aligned}
c_{11}d_{22}c_{33}^* &= \frac{b-c}{a-b}c_{21}^*c_{13}\tilde{c}_{32}, & c_{13}c_{23}\tilde{c}_{32} &= 0, \\
c_{23}c_{31}\tilde{c}_{32} &= 0, & c_{21}^*c_{33} &= c_{31}c_{23}, \\
c_{12}c_{33}^* &= \frac{a-c}{a-b}c_{13}\tilde{c}_{32}, & c_{22}c_{33}^* &= \frac{(-1-a+c)}{(-1-a+b)}c_{23}\tilde{c}_{32},
\end{aligned}
$$

and

$$ (a-b)c_{13}c_{31}d_{22} + (c-b)c_{13}\tilde{c}_{32}c_{21}^* + (-1-a+c)c_{23}c_{31}c_{12}^* = 0. $$

These equations come from [LLHLM18, Proposition 8.11] and its proof, by restoring the units $c_{12}^*$, $c_{21}^*$, and $c_{33}^*$ (which were set to be 1 in loc. cit.); the first six equations above are deduced from the equations
appearing in the statement of loc. cit. (where $d_{22}$ above is denoted by $c'_{22}$ in loc. cit.), the seventh and eighth equations above appear in the proof of loc. cit., and the last equation above is implicit in loc. cit. (where we solved $c_{33}$ using the $p$-saturation of the 2 by 2 minor condition). In particular, $\bar{R}^{\text{expl},\nabla}_{\bar{\mathfrak{m}}, \tilde{w}_i}$ is a formal power series ring over the ring $\tilde{R}$ appearing in [LLHLM18, Proposition 8.11].

(3) When $\ell(\tilde{w}_i) = 0$, the matrix $A^{(i),\text{univ}}$ has the form

$$A^{(i),\text{univ}} = \begin{pmatrix} c_{11} + vc_{11}^* & c_{12} & c_{13} \\ vc_{21} & c_{22} + vc_{22}^* & c_{23} \\ vc_{31} & vc_{32} & c_{33} + vc_{33}^* \end{pmatrix}.$$

We define $\bar{R}^{\text{expl},\nabla}_{\bar{\mathfrak{m}}, \tilde{w}_i}$ to be the quotient of $\mathbb{F}\llbracket c_{ij} \rrbracket$, $c_{kk}^* - [\bar{c}_{kk}^*]$, $1 \le i, j, k \le 3\rrbracket$ by the relations

$$ \begin{aligned} c_{ii}c_{jj} &= 0 \quad \text{for } i \neq j, & c_{11}c_{23} &= 0 & c_{31}c_{22} &= 0, \\ c_{33}c_{12} &= 0, & c_{12}c_{23} &= c_{22}c_{13}, & c_{11}c_{32} &= c_{12}c_{31}, \\ c_{21}c_{33} &= c_{31}c_{23}, & & & \end{aligned} $$

and

$$ \begin{aligned} &(-1 - a + c)c_{22}^*c_{33} + (-1 - a + b)c_{22}c_{33}^* \\ &\quad - (-1 - a + c)c_{23}c_{32} = 0, \\ &(a - b)c_{33}^*c_{11} + (-1 - b + c)c_{33}c_{11}^* - (a - b)c_{13}c_{31} = 0, \\ &(b - c)c_{11}^*c_{22} + (a - c)c_{11}c_{22}^* - (b - c)c_{12}c_{21} = 0, \\ &c_{11}c_{22}^*c_{33}^* + c_{22}c_{11}^*c_{33}^* + c_{33}c_{11}^*c_{22}^* - c_{11}^*c_{23}c_{32} - c_{22}^*c_{13}c_{31} - c_{33}c_{12}c_{21} \\ &\quad + c_{13}c_{32}c_{21} = 0. \end{aligned} $$

These equations come from [LLHLM18, Corollary 8.4] by restoring the units $c_{kk}^*$ (which were set to be 1 in loc. cit.) and we added the equation $c_{12}c_{33} = 0$, which was missing in loc. cit. and which is obtained by the 2 by 2 minor condition on $A^{(i),\text{univ}}$. In particular, $\bar{R}^{\text{expl},\nabla}_{\bar{\mathfrak{m}}, \tilde{w}_i}$ is a formal power series ring over the ring $\tilde{R}$ appearing in [LLHLM18, Corollary 8.4].

Note that each object with a superscript $\square$ receives a $\hat{\text{GL}}_3$-action, corresponding to changing the framing on the Galois representation.

We now justify the diagram:
• We claim that there is a canonical isomorphism $[\text{Spf}(\overline{R}^{\tau, \beta, \square}_{\overline{\mathfrak{M}}, \overline{\rho}})/\widehat{\text{GL}}_3] \cong \text{Spf}(\overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{M}}, \overline{w}})$. When $\ell(\tilde{w}_i) > 1$ for all $i$, this is [LLHLM18, Theorem 5.12, Theorem 6.14 and Table 7]. When $\ell(\tilde{w}_i) \le 1$ for some $i$, this follows from the same arguments as [LLHLM18, Section 8] together with Theorem 3.5.2, as explained in the proof of Theorem 3.5.3. This justifies the existence and the properties of the first row of Diagram (3.9). It is clear from the construction that the $\widehat{G}_m^{3f}$-action on $\overline{D}^{\tau, \beta}_{\overline{\mathfrak{M}}}$ preserves $\text{Spf}(\overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{M}}, \overline{w}})$. Let $(\mathfrak{M}_{\tilde{w}, \tau}, \beta_{\tilde{w}, \tau})$ be the restriction of $(\mathfrak{M}^{\text{univ}}, \beta^{\text{univ}})$ to $\text{Spf}(\overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{M}}, \overline{w}})$. Thus, the map $\iota_{\tau}$ sends $\mathfrak{M}_{\tilde{w}, \tau}$ to $\mathcal{M}_{\tilde{w}, \tau} \stackrel{\text{def}}{=} (\mathfrak{M}_{\tilde{w}, \tau}[\frac{1}{u'}])^{\Delta=1}$.

• The second row is obtained from the first row by quotienting by the $\widehat{G}_m^{3f}$-action coming from scaling the gauge basis and, hence, inherits all properties from the first row. The top squares are Cartesian.

• The second column is obtained from the first column by quotienting by the $\widehat{\text{GL}}_3$-action coming from changing the framing of the Galois representation; hence, the bottom square is Cartesian.

The following proposition finishes our justification of the diagram.

**PROPOSITION 3.6.3.** Assume that $\tau$ is 3-generic and that $\overline{\mathfrak{M}} \in \overline{Y}^{\eta, \tau}(\mathbb{F})$ is semisimple of shape $\tilde{w} = (\tilde{w}_j)$. Let $\beta$ be a gauge basis for $\overline{\mathfrak{M}}$. Then the map

$$i'_{\tau} : \overline{Y}^{\eta, \tau}_{\overline{\mathfrak{M}}} \to \Phi \text{-Mod}^{\text{et}}_{\overline{\mathcal{M}}}$$

is a monomorphism.

*Proof.* We need to prove that the map on the groupoids of $\mathbb{F}[\varepsilon]/(\varepsilon^2)$-points

$$\overline{Y}^{\eta, \tau}_{\overline{\mathfrak{M}}} (\mathbb{F}[\varepsilon]/(\varepsilon^2)) \to \Phi \text{-Mod}^{\text{et}}_{\overline{\mathcal{M}}} (\mathbb{F}[\varepsilon]/(\varepsilon^2))$$

induced by $i'_\tau$ is fully faithful. But this follows from [LLHL19, Proposition 3.2.18], noting that the right-hand side is equivalent to $\text{Rep}_{\overline{\mathbb{F}}[\varepsilon]/\varepsilon^2}(G_{K_\infty})_{\overline{\rho}}$ in loc. cit. □

Diagram (3.9) gives a bijection between the set of minimal primes $\text{Irr}(\text{Spec}(\overline{R}^{\tau}_{\overline{\rho}}))$ and $\text{Irr}(\text{Spec}(\overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{M}}, \overline{w}_i})) = \prod_i \text{Irr}(\text{Spec}(\overline{R}^{\text{expl}, \nabla}_{\overline{\mathfrak{M}}, \overline{w}_i}))$. By Proposition 3.6.1, this set is in bijection with the set $\mathfrak{p}(\sigma)$ for $\sigma \in W^?(\overline{\rho}, \tau)$. On the other hand, by Proposition 3.4.2, we have a bijection

$$\prod_i \Sigma_{\tilde{w}_i^*} \xrightarrow{\sim} W^?(\overline{\rho}, \tau)$$
$$ (\omega, a) \mapsto \sigma_{(\omega, a)}^{(s, \mu)}. $$

The following theorem is the main result of this subsection, which computes the above bijections in terms of the explicit rings.

**THEOREM 3.6.4.** Assume $\bar{\rho}$ is 10-generic. Via the above bijections, we have

$$ \begin{gathered} \prod_{i=0}^{f-1} \Sigma_{\tilde{w}_i^*} \xrightarrow{\sim} \prod_i \mathrm{Irr}(\mathrm{Spec}(\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}_i}^{\mathrm{expl}, \nabla})) \xrightarrow{\sim} \mathrm{Irr}(\mathrm{Spec}(\bar{R}_{\bar{\rho}}^{\tau})) \\ ((\omega_i, a_i))_i \mapsto ((c_{(\omega_{f-1-i}, a_{f-1-i})} \bar{R}_{\bar{\mathfrak{M}}, \tilde{w}_i}^{\mathrm{expl}, \nabla})_i \mapsto \mathfrak{p}(\sigma_{(\omega, a_i)}^{(s, \mu)}), \end{gathered} $$

where $c_{(\omega_i, a_i)}$ is the minimal prime of $\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}_{f-1-i}}^{\mathrm{expl}, \nabla}$ given by Table 3.

**REMARK 3.6.5.** Note that Table 3 only gives the ideals $c_{(\omega_i, a_i)} \subseteq \bar{R}_{\bar{\mathfrak{M}}, \tilde{w}_{f-1-i}}^{\mathrm{expl}, \nabla}$ for a set of representatives for $\tilde{w}_{f-1-i} \in \mathrm{Adm}^{\vee}((2, 1, 0))$ for the action of the outer automorphisms of $\tilde{W}^{\vee}$. This is sufficient because this action corresponds to changing the lowest alcove presentation of $\tau$.

We now describe the strategy of proof. The main idea is that by the proof of Proposition 3.6.1, the prime $\mathfrak{p}(\sigma)$ can be characterized by the relation (3.6) for a specific type $\tau'$ (inducting on defect of $\sigma$). Namely, one can use the minimal type $\tau'$ of $\sigma$ with respect to $\bar{\rho}$. For this type, each component of $\tilde{w}' = \tilde{w}(\bar{\rho}, \tau')$ has length $\ge 3$. Furthermore, the minimal type $\tau'$ has the property that $\mathrm{Spf}\bar{R}_{\bar{\rho}}^{\tau'} \subset \mathrm{Spf}\bar{R}_{\bar{\rho}}^{\tau}$ inside $\mathrm{Spf}\bar{R}_{\bar{\rho}}^{\Box}$. Thus, it suffices to determine the subset of $\mathrm{Irr}(\mathrm{Spec}(\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}}^{\mathrm{expl}, \nabla}))$ which occurs in $\bar{R}_{\bar{\rho}}^{\tau'}$. This is achieved by 'matching' the universal étale $\varphi$-module over a union of irreducible components of $\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}}^{\mathrm{expl}, \nabla}$ with the universal étale $\varphi$-module living over $\bar{R}_{\bar{\rho}}^{\tau'}$ (or rather $\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}'}^{\mathrm{expl}, \nabla}$).

The precise formulation of this matching mechanism is given by the following.

**LEMMA 3.6.6.** Let $\tau = \tau(s, \mu)$ and $\tau' = \tau(s', \mu')$. Assume that $R_{\bar{\rho}}^{\tau}$, $R_{\bar{\rho}}^{\tau'} \ne 0$, and the running hypothesis that $\bar{\rho}$ is 10-generic. Consider diagram (3.9) for $R_{\bar{\rho}}^{\tau}$, $R_{\bar{\rho}}^{\tau'}$, constructed using the above presentations. We decorate the objects that occur in the diagram for $\tau'$ with the same symbol as those in the diagram for $\tau$ but with a superscript added (so we have, for example, $\tilde{w}'$, $\mathfrak{M}_{\tilde{w}', \tau'}$, and so on). Assume that there are ideals $I$ (respectively $I'$) of $\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}}^{\mathrm{expl}, \nabla}$ (respectively $\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}'}^{\mathrm{expl}, \nabla}$) such that

* $I, I'$ are intersections of minimal primes.

* There is an isomorphism $\bar{R}_{\bar{\mathfrak{M}}, \tilde{w}}^{\mathrm{expl}, \nabla} / I \cong \bar{R}_{\bar{\mathfrak{M}}, \tilde{w}'}^{\mathrm{expl}, \nabla} / I'$.
* There exists an isomorphism of étale $\varphi$-modules between the base change of $\mathcal{M}_{\tilde{w}, \tau}$ to $\overline{\mathcal{R}}_{\tilde{\mathfrak{m}}, \tilde{w}}^{\text{expl}, \nabla} / I$ and the base change of $\mathcal{M}_{\tilde{w}', \tau'}$ to $\overline{\mathcal{R}}_{\tilde{\mathfrak{m}}', \tilde{w}'}^{\text{expl}, \nabla} / I'$ compatible with the above ring homomorphism.

Let $J$ (respectively $J'$) denote the intersection of minimal primes in $\overline{R}_{\overline{\rho}}^{\tau}$ (respectively $\overline{R}_{\overline{\rho}}^{\tau'}$) which correspond to $I$ (respectively $I'$). Then, $J$ and $J'$ induce the same ideals in $\overline{R}_{\overline{\rho}}^{\square}$.

*Proof.* Our hypotheses imply that there is an isomorphism

$$ \mathrm{Spf}(\overline{R}^{\mathrm{expl}, \nabla}_{\overline{\mathfrak{m}}, \overline{\tilde{w}}}/I) \xrightarrow{\sim} \mathrm{Spf}(\overline{R}^{\mathrm{expl}, \nabla}_{\overline{\mathfrak{m}'}', \overline{\tilde{w}'}'}/I') \quad (3.10) $$

$\Phi$-Mod$^{\mathrm{ét}, \square}_{\overline{M}} \to \Phi$-Mod$^{\mathrm{ét}, \square}_{M}$ gives a commutative diagram

$$ \begin{tikzcd} \mathrm{Spf}(\overline{R}^{\tau, \overline{\rho}, \square}_{\overline{\mathfrak{m}}, \overline{\tilde{w}}})/I^{\square} \arrow[r, "\sim"] & \mathrm{Spf}(\overline{R}^{\tau', \overline{\rho}', \square}_{\overline{\mathfrak{m}'}', \overline{\tilde{w}'}'})/I'^{\square} \arrow[d, "f.s."] \\ \mathrm{Spf}(\overline{R}^{\tau}_{\overline{\rho}}/J) \arrow[u, "f.s."] & \mathrm{Spf}(\overline{R}^{\tau'}_{\overline{\rho}'}/J') \arrow[u, "f.s."] \\ \mathrm{Spf}(\overline{R}^{\square}_{\overline{\rho}}) \arrow[u, "f.s."] & \mathrm{Spf}(\overline{R}^{\square}_{\overline{\rho}}) \arrow[u, "f.s."] \\ \multicolumn{4}{c}{\Phi\text{-Mod}^{\mathrm{ét}, \square}_{\overline{M}}} \end{tikzcd} \quad (3.11) $$

where we use [LLHLM18, Proposition 3.12] to see that the natural map $\mathrm{Spf}(\overline{R}_{\overline{\rho}}^{\square}) \to \Phi$-$\mathrm{Mod}_{\overline{M}}^{\mathrm{ét}, \square}$ is a monomorphism (as $\overline{\rho}$ 10-generic implies $\mathrm{ad}(\overline{\rho})$ is cyclotomic free). But this implies that $\mathrm{Spf}(\overline{R}_{\overline{\rho}}^{\tau}/J)$ and $\mathrm{Spf}(\overline{R}_{\overline{\rho}}^{\tau'}/J')$ define the same subfunctor of $\Phi$-$\mathrm{Mod}_{\overline{M}}^{\mathrm{ét}, \square}$ and, hence, also the same subfunctor of $\mathrm{Spf}(\overline{R}_{\overline{\rho}}^{\square})$. $\square$

In practice, we will apply the lemma by matching matrices of partial Frobenii.
COROLLARY 3.6.7. Keep the notations and setting of Lemma 3.6.6. Assume that there exists $\tilde{z} = (\tilde{z}_i)_i \in W_\alpha^\vee$ with corresponding $\tilde{z}^* = (\tilde{z}_i^*)_i \in W_\alpha$ and $z$ is the image of $\tilde{z}$ in $W^\vee$ such that $s' = sz^*$ and $\mu' = \mu + s\tilde{z}^*(0)$. Write $(I_i)_i$ for the collection of ideals in $\overline{R}_{\tilde{M}, \tilde{w}_i}^{\text{expl}, \vee}$ giving rise to $I$, write $A^{(i)}$ for the matrix of the $i$th partial Frobenius of $\mathfrak{M}_{\tilde{w}, \tau}$ with respect to $\beta_{\tilde{w}, \tau}$, and let $(I'_i)_i$ and $A^{(i)}$ be the analogous objects for $\tau'$. Assume the following:

* The isomorphism $\overline{R}_{\tilde{M}, \tilde{w}}^{\text{expl}, \vee} / I \cong \overline{R}_{\tilde{M}', \tilde{w}'}^{\text{expl}, \vee} / I'$ is induced by a collection of isomorphisms $\overline{R}_{\tilde{M}, \tilde{w}_i}^{\text{expl}, \vee} / I_i \cong \overline{R}_{\tilde{M}', \tilde{w}'_i}^{\text{expl}, \vee} / I'_i$.

* For each $i$,

$$A^{(f-1-i)} \mod I_{f-1-i} = A^{(f-1-i)}\tilde{z}_{f-1-i} \mod I'_{f-1-i}$$

via the above isomorphisms.

Then the same conclusion as Lemma 3.6.6 holds.

*Proof.* This follows from Lemma 3.6.6, Proposition 3.2.1, and the fact that $\tilde{z}s^*t_{\mu^*} = s'^*t_{\mu'^*}$. $\square$

REMARK 3.6.8. Note that the second condition in Corollary 3.6.7 implies in particular that

$$(\overline{A}^{(i)})_i = (\overline{A'}^{(i)}\tilde{z}_i)_i, \qquad (3.12)$$

where $\overline{A}^{(i)}$, $\overline{A'}^{(i)}$ denote the reductions modulo the maximal ideal. We now explain how, in the situations where we apply Corollary 3.6.7, we can always arrange this.

If we have $\overline{\rho}$ semisimple and $\tau, \tau', \tilde{z}$ as in Proposition 3.6.9, then

$$\tilde{w} = \tilde{w}'\tilde{z}. \qquad (3.13)$$

Furthermore, for any choices of gauge bases $\beta, \beta'$, for $\mathfrak{M}_{\tilde{w}, \tau}$ and $\mathfrak{M}_{\tilde{w}', \tau'}$, respectively, one has $\overline{A}^{(i)} \in T(\mathbb{F})\tilde{w}_i$ and $\overline{A'}^{(i)} \in T(\mathbb{F})\tilde{w}'_i$ by Proposition 3.3.6. Let $\mathfrak{M}_{\tilde{w}, \tau}$ denote the element of $Y^{\eta, \tau}(\mathbb{F})$ with eigenbasis $\beta_{\mathfrak{M}}$ such that the partial Frobenii is given by $(\overline{A}^{(i)}\tilde{z}_i)_i$. (Note that $\mathfrak{M}_{\tilde{w}, \tau} \in Y^{\eta, \tau}(\mathbb{F})$ because it has shape $\tilde{w} \in \text{Adm}(\eta)$.) Since $\tilde{z}s^*t_{\mu^*} = s'^*t_{\mu'^*}$, $(\mathfrak{M}_{\tilde{w}, \tau})^{\Delta=1} \cong (\mathfrak{M}_{\tilde{w}', \tau'})^{\Delta=1}$ by Proposition 3.2.1. Thus, $T_{dd}^*(\mathfrak{M}_{\tilde{w}, \tau}) \cong \overline{\rho}$. By the triviality of Kisin variety (see [LLHLM18, Theorem 3.2]), $\mathfrak{M}_{\tilde{w}, \tau} \cong \mathfrak{M}_{\tilde{w}', \tau}$ and $\beta_{\mathfrak{M}}$ defines a gauge basis of $\mathfrak{M}_{\tilde{w}, \tau}$. Thus, by replacing $\beta_{\mathfrak{M}}$ by $\beta_{\mathfrak{M}'}$ (this can be done by scaling by an element of $\overline{T}(\mathbb{F})$ by [LLH19, Proposition 3.2.22]), we can then ensure that condition (3.12) holds. In all the matching that we perform below, we will always assume that the gauge bases have been chosen so that condition (3.12) holds.
The following proposition describes all the types $\tau'$ for which we need to perform some matching with $\tau$.

**PROPOSITION 3.6.9.** Assume that $\bar{\rho}$ is 10-generic. Let $\tau = \tau(s, \mu)$ be a tame inertial type such that $R_{\bar{\rho}}^{\tau} \neq 0$, and let $\tilde{w} = \tilde{w}(\bar{\rho}, \tau) = (\tilde{w}_i)_i$. Let $(\omega, a) = ((\omega_i, a_i))_i \in \prod_i \Sigma_{\tilde{w}_i^*}$ and $\sigma = \sigma_{(\omega, a)}^{(s, \mu)} \in W^-(\bar{\rho}, \tau)$. Let $\tau'$ be the minimal type of $\sigma$ with respect to $\bar{\rho}$; cf. Remark 3.5.10. Then we have the following:

* For $\tilde{z}^* = (\tilde{z}_i^*)_i \in \tilde{W}_a$ as given by Table 3, we have $\tau' = \tau(s', \mu')$ where

$$s' = sz^*, \quad \mu' = \mu + s\tilde{z}^*(0)$$

* $\tilde{w}(\bar{\rho}, \tau')\tilde{z} = \tilde{w}$; and

* $W^-(\bar{\rho}, \tau') \subset W^-(\bar{\rho}, \tau)$.

*Proof.* This is immediate by computing the pairwise intersections among $\Sigma_0$, $\tilde{w}_i^*(r(\Sigma_0))$, and $\tilde{z}_i^*(\Sigma_0)$. $\square$

**Proof of Theorem 3.6.4.** By Proposition 3.6.9 and the proof of Proposition 3.6.1, for each $\sigma \in W^-(\bar{\rho}, \tau)$ with minimal type $\tau'$ with respect to $\bar{\rho}$, we need to show that the intersection of minimal primes

$$\bigcap_{(\omega, a) \in \Sigma \cap \tilde{z}^*(\Sigma)} \left( \sum_{i=0}^{f-1} c_{(\omega_i, a_i)} \overline{R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}} \right) = \sum_{i=0}^{f-1} \left( \bigcap_{(\omega_i, a_i) \in \Sigma_0 \cap \tilde{z}_i^*(\Sigma_0)} c_{(\omega_i, a_i)} \overline{R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}} \right)$$

of $\overline{R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}}$ corresponds to the intersection of primes $\bigcap_{\sigma' \in W^-(\bar{\rho}, \tau')} \mathfrak{p}(\sigma')$ of $\overline{R_{\bar{\rho}}^{\square}}$. But this follows from Corollary 3.6.7 (which allows us to work for each $i$ separately) and the explicit computations in Section 3.6.2. $\square$

We record the following lemma, which follows from Theorem 3.6.4, for future use. Let $\tilde{w} = \tilde{w}(\bar{\rho}, \tau)$. Suppose that $\ell(\tilde{w}_i) \ge 2$ for all $0 \le i \le f-1$. Let $N \stackrel{\text{def}}{=} \sum_i (4 - \ell(\tilde{w}_i))$, and $R_N \stackrel{\text{def}}{=} \bigotimes_{j=1}^N \mathcal{O}[[x_j, y_j]]/(x_j y_j - p)$. In this case, there exists a formally smooth map $R_N \to R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$, which we fix. We think of $R_N$ as a subalgebra of $R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$. For $\sigma \in W^-(\bar{\rho}, \tau)$, let $\mathfrak{p}^{\text{expl}}(\sigma) \subset R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$ be the prime ideal corresponding to the weight $\sigma$. The minimal prime ideals of $R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$ containing $\sigma$ are of the form $((z_j)_{j=1}^N + (\sigma))R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$ where $z_j \in \{x_j, y_j\}$ for all $1 \le j \le N$. Define $z_j(\sigma)$ so that $\mathfrak{p}^{\text{expl}}(\sigma) = ((z_j(\sigma))_{j=1}^N + (\sigma))R_{\overline{\mathfrak{M}}, \tilde{w}}^{\text{expl}, \nabla}$.
LEMMA 3.6.10. Assume that $\bar{\rho}$ is 10-generic and that $\ell(\tilde{w}_i) \ge 2$ for all $0 \le i \le f-1$. For $\sigma_1$ and $\sigma_2 \in W^\ell(\bar{\rho}, \tau)$,

$$ \#(\{z_j(\sigma_1)\}_{j=1}^N \Delta\{z_j(\sigma_2)\}_{j=1}^N) = 2d_{\text{gph}}(\sigma_1, \sigma_2), $$

where $\Delta$ denotes the symmetric difference.

*Proof.* From the $\alpha\beta\alpha$, $\alpha\beta$, and $\beta\alpha$ rows of Table 3, we check that if $\sigma_1, \sigma_2 \in W^\ell(\bar{\rho}, \tau)$, then $\mathfrak{p}^{\text{expl}}(\sigma_1) + \mathfrak{p}^{\text{expl}}(\sigma_2)$ is a prime ideal of $R_{\overline{\mathfrak{m}}, \overline{\tilde{w}}}$ of height

$$ d_{\text{gph}}(\sigma_1, \sigma_2) + 1. $$

On the other hand, the height of the intersection of $((z_j)_j + (\omega)) + ((z'_j)_j + (\omega))$ is easily seen to be

$$ \frac{1}{2} \#(\{z_j\}_{j=1}^N \Delta\{z'_j\}_{j=1}^N) + 1. \quad \square $$

For convenience, we list in Table 4 the length 4 shapes and their universal families over $R_{\overline{\mathfrak{m}}, \overline{\tilde{w}}_j}^{\text{expl}, \nabla}$.

### 3.6.2. Explicit computations.
In this section, we record the matching of matrices of partial Frobenii needed in the proof of Theorem 3.6.4. We are always in the setting of the theorem. In particular, we have two types $\tau, \tau'$ with chosen presentations related by the element $\tilde{z}^*$ together with matrices of partial Frobenii $A^{(f-1-i)}$, $A^{(f-1-i)}$. We will fix $i$ throughout.

We will frequently recall presentations of rings from [LLHLM18], and since we work in characteristic $p$, all occurrences of the symbol $e$ in loc. cit. will become $-1$ here. We let $(a, b, c)$ and $(a', b', c') \in \mathbb{F}_p^3$ be such that

$$ (a, b, c) = s_i^{-1}(\mu_i) \mod p, \quad (a', b', c') = (s'_i)^{-1}(\mu'_i) \mod p. $$

Note that $(a', b', c') = \tilde{z}_{f-1-i}(a, b, c)$. These are the structure constants that feature in the presentation of our explicit rings. We will also replace occurrences of the symbols $c'_{ij}$ in loc. cit. by $d_{ij}$ as we wish to decorate objects associated with $\tau'$ with a prime superscript.

Case $\tilde{w}_{f-1-i} = \alpha\beta t_1$. From [LLHLM18, Table 5], the matrix of the partial Frobenius $A^{(f-1-i)}$ has the form

$$ \begin{pmatrix} c_{31}c_{12}(c_{32}^*)^{-1} & c_{12} & c_{13} + v c_{13}^* \\ v c_{21}^* & c_{22} & c_{23} + v d_{23} \\ v c_{31} & v c_{32}^* & (c_{31}c_{23}(c_{21}^*)^{-1} + v d_{33}) \end{pmatrix} $$
<table><caption>Table 4. Universal matrix and presentation for the rings $\overline{R}_{\mathbb{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ for extremal and shadow shapes.</caption><thead><tr><th>$$\tilde{w}'_{f-1-i}t^{-1}$$</th><th>A'(f-1-i)</th><th>$$\tilde{w}'_{f-1-i}t^{-1}$$</th><th>A'(f-1-i)</th></tr></thead><tbody><tr><td>$$\alpha\beta\gamma$$</td><td>$$\begin{pmatrix} v^2c_{11}^* & 0 & 0 \\ v^2c_{21}^* & v_{c'22}^* & 0 \\ v^2d_{31}^* & v_{c'32}^* & c_{33}^* \end{pmatrix}$$</td><td>$$\beta\gamma\alpha\gamma$$</td><td>$$\begin{pmatrix} vc_{11}^* & vc_{12}^* & 0 \\ vc_{31}^* & v^2d_{22}^* & 0 \\ 0 & vc_{32}^* + v^2d_{32}^* & c_{33}^* \end{pmatrix}$$</td></tr><tr><td>$$\beta\gamma\beta\alpha$$</td><td>$$\begin{pmatrix} c'_{11}^* & vd_{12}^* & c'_{13} \\ 0 & v^2c_{22}^* & 0 \\ 0 & v^2c_{32}^* & vc_{33}^* \end{pmatrix}$$</td><td>$$\gamma\alpha\beta\alpha$$</td><td>$$\begin{pmatrix} c'_{11}^* & c'_{12} & c'_{13} + vd_{13} \\ 0 & vc_{22}^* & vc_{23}^* \\ 0 & 0 & v^2c_{33}^* \end{pmatrix}$$</td></tr><tr><td>$$\alpha\gamma\alpha\beta$$</td><td>$$\begin{pmatrix} vc_{11}^* & 0 & vc_{13}^* \\ vc_{12}^* & c_{22}^* & vd_{23}^* \\ 0 & 0 & v^2c_{33}^* \end{pmatrix}$$</td><td>$$\alpha\beta\gamma\beta$$</td><td>$$\begin{pmatrix} v^2c_{11}^* & 0 & 0 \\ v(c_{21} + vd_{21}^*) & c_{22}^* & c_{23}^* \\ v^2c_{31}^* & 0 & vc_{33}^* \end{pmatrix}$$</td></tr><tr><td>$$\alpha\beta\alpha$$</td><td>$$\begin{pmatrix} c'_{11} & c'_{11}c_{32}^{*(s)} - 1 & c'_{11}d_{33}^{*(c)} - 1 + vc_{13}^* \\ 0 & vc_{22}^* & vc_{23}^* \\ vc_{31}^* & vc_{32}^* & vd_{33}^* \end{pmatrix}$$</td><td>$$\beta\gamma\beta$$</td><td>$$\begin{pmatrix} vd_{11}^* & 0 & 0 \\ v(c_{22}d_{11}^{*(c_{12}^*)} + vc_{21}^*) & c_{12}^* & c_{13}^* \\ v^2c_{31}^* & 0 & vc_{33}^* \end{pmatrix}$$</td></tr><tr><td>$$\gamma\alpha\gamma$$</td><td>$$\begin{pmatrix} vc_{11}^* & vc_{12}^* & 0 \\ vc_{21}^* & vd_{22}^* & c_{23}^* \\ vc_{33}^*c_{21}^{*(c_{23}^*)} - 1 & v(c_{33}d_{22}^{*(c_{23}^*)} + vc_{32}^*) & c_{33}^* \end{pmatrix}$$</td><td></td><td>$$c_{22}^{*(b' - c')}c_{31}^*c_{13}^* - (1 - a' + b')c_{33}^*d_{11}^* = 0$$</td></tr></tbody></table>

The relevant data for the type $\tau'$ occurring in the proof of Theorem 3.6.4. The first column records the components of the shape $\tilde{w}' = \tilde{w}(\bar{\rho}, \tau')$. The second column records the form of the matrix of partial Frobenius $A'(f-1-i)$ and the presentation of the ring $\overline{R}_{\mathbb{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ in terms of the entries of $A'(f-1-i)$. The structure constants that feature in the presentation of $\overline{R}_{\mathbb{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ are given by $(a', b', c') \in \mathbb{F}_p^3$ where $(a', b', c') \equiv (s_i')^{-1}(\mu_i') \mod p$. Note that $(s_i')^{-1}(\mu_i') \equiv \tilde{z}_{f-1-i}(a, b, c) \mod p$.

The ring $\overline{R}_{\mathbb{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of $\mathbb{F}[c_{12}, c_{13}, c_{22}, c_{23}, d_{23}, c_{31}, d_{33}, c_{13}^* - [c_{13}^*], c_{21}^* - [c_{21}^*], c_{32}^* - [c_{32}^*]]$ by the relations

$$c_{12}c_{23} - c_{22}c_{13} = 0; \qquad (3.14)$$
$$c_{22}c_{31} = 0;$$
$$c_{32}^{*}c_{13} - d_{33}c_{12} = 0;$$

$$c_{12}((b-c)d_{33}c_{21}^{*} + (a-b)c_{31}d_{23}) = 0;$$

$$(-1 - a + c)c_{23}c_{32}^{*} = (-1 - a + b)c_{22}d_{33}.$$

Note that all equations except for the last one are mod $\omega$ reductions of equations in [LLHLM18, Section 5.3.3]. We explain how to justify the last equation from the computations in [LLHLM18, Section 5.3.3], which was implicit in [LLHLM18, Table 7]. For the remainder of this paragraph, we adopt the notation of loc. cit. We have $c_{12}((a-b)c_{31}c_{23}^{*} + (b-c)c_{21}^{*}c_{33}^{*}) = p z^{*}$. From the equation $c_{32}^{*}c_{13} - pc_{32}^{*}c_{13} - c_{33}^{*}c_{12} = 0$, replacing $p$ by $(z^{*})^{-1}(c_{12}((a-b)c_{31}c_{23}^{*} + (b-c)c_{21}^{*}c_{33}^{*}))$, we deduce that $c_{13}$ is a certain multiple of $c_{12}$. Finally, using the equation $c_{12}c_{23} = c_{22}c_{13}$ and canceling out $c_{12}$ (as it is a unit after inverting $p$), we get an equation solving $c_{23}$ in terms of the remaining variables. The mod $\omega$ reduction of this equation gives the last equation above.

The minimal primes of $\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ are

$$c_{(0,0)} = (c_{12}, c_{22}); \quad c_{(\varepsilon_1, 1)} = (c_{12}, c_{31}); \quad c_{(\varepsilon_1 - \varepsilon_2, 0)} = (c_{31}, d_{33})$$

and

$$c_{(0,1)} = (c_{22}, (b-c)d_{33}c_{21}^{*} + (a-b)c_{31}d_{23}).$$

(Note that in $\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$, there are no more relation than those listed in (3.14). Indeed, let $\tilde{R}$ be the quotient of $\mathbb{F}\llbracket c_{12}, c_{13}, c_{22}, c_{23}, d_{23}, c_{31}, d_{33}, c_{13}^{*} - [c_{13}^{*}], c_{21}^{*} - [c_{21}^{*}], c_{32}^{*} - [c_{32}^{*}]\rrbracket$ by the relations (3.14). Then the discussion above proves that there is a surjection $\tilde{R} \to \overline{R}_{\overline{\mathfrak{m}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$. A direct check on the relations (3.14) shows that $\tilde{R}$ is reduced, equidimensional of the same dimension as $\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$, and with the same Hilbert–Samuel multiplicity. Therefore, the surjection is an isomorphism; see Lemma 3.6.11.) We need to perform matching for the ideals $c_{(0,0)}$, $c_{(\varepsilon_1, 1)}$, $c_{(\varepsilon_1 - \varepsilon_2, 0)}$, $w_{(0)} \stackrel{\text{def}}{=} c_{(0,1)} \cap c_{(0,0)}$. We provide details for the matching of the ideal $c_{(0,0)}$ in Table 3. We have

$$\overline{R}_{\overline{\mathfrak{m}}, \tilde{w}_{f-1}}^{\text{expl}, \nabla} / c_{(0,0)} = \mathbb{F}\llbracket d_{23}, c_{31}, d_{33}, c_{13}^{*} - [c_{13}^{*}], c_{21}^{*} - [c_{21}^{*}], c_{32}^{*} - [c_{32}^{*}]\rrbracket$$

and

$$A^{(f-1-i)} \mod c_{(0,0)} = \begin{pmatrix} 0 & 0 & vc_{13}^{*} \\ vc_{21}^{*} & 0 & vd_{23} \\ vc_{31} & vc_{32}^{*} & vd_{33} \end{pmatrix}.$$
On the other hand, $\tilde{w}'_{f-1-i} = \alpha\beta\gamma t_1$, $\tilde{z}_i^* = \alpha\gamma^+$, $(a', b', c') = (c-1, a, b+1)$, and $A^{(f-1-i)}$ has the form

$$ \begin{pmatrix} v^2 c_{11}^* & 0 & 0 \\ v^2 c_{21}^* & v c_{22}^* & 0 \\ v^2 d_{31}^* & v c_{32}^* & c_{33}^{*\prime} \end{pmatrix}. $$

Now we note that $\tilde{z}_{f-1-i} = t_{(-1,0,1)}(123)$ and

$$ \begin{pmatrix} 0 & 0 & v c_{13}^* \\ v c_{21}^* & 0 & v d_{23} \\ v c_{31} & v c_{32}^* & v d_{33} \end{pmatrix} = \begin{pmatrix} v^2 c_{11}^* & 0 & 0 \\ v^2 c_{21}^* & v c_{22}^* & 0 \\ v^2 d_{31}^* & v c_{32}^* & c_{33}^{*\prime} \end{pmatrix} \begin{pmatrix} 0 & 0 & v^{-1} \\ 1 & 0 & 0 \\ 0 & v & 0 \end{pmatrix} $$

under the isomorphism

$$ \overline{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / \mathfrak{c}_{(0,0)} \cong \overline{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla} $$

given by the change of variable

$$ \begin{aligned} c_{21}^* &= c_{22}^{*\prime}, & c_{13}^* &= c_{11}^{*\prime}, & d_{23} &= c_{21}^{\prime}, \\ c_{31}^* &= c_{32}^{\prime}, & c_{32}^* &= c_{33}^{*\prime}, & d_{33} &= d_{31}^{\prime}. \end{aligned} $$

Such a change of variable is allowed, provided the units that are matched with each other agree modulo the maximal ideal because $\tilde{c}_{22}^{*\prime} = \tilde{c}_{21}^*, \tilde{c}_{11}^* = \tilde{c}_{13}^*$, and $\tilde{c}_{33}^* = \tilde{c}_{32}^*$ as elements of $\mathbb{F}$ (see Remark 3.6.8). Thus, 3.6.7 applies and we are done with this case. The matching for $\mathfrak{c}_{(\varepsilon_1-\varepsilon_2,0)}$ can be performed in a similar fashion.

Next we provide details for the matching of $\mathfrak{c}_{(\varepsilon_1,1)}$. We have

$$ \overline{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / \mathfrak{c}_{(\varepsilon_1, 1)} = \frac{\mathbb{F}[c_{22}, c_{23}, d_{23}, d_{33}, c_{13}^* - [\tilde{c}_{13}^*], c_{21}^* - [\tilde{c}_{21}^*], c_{32}^* - [\tilde{c}_{32}^*]]}{((-1-a+c)c_{23}c_{32}^* - (-1-a+b)c_{22}d_{33})}. $$

Moreover,

$$ A^{(f-1-i)} \mod \mathfrak{c}_{(\varepsilon_1, 1)} = \begin{pmatrix} 0 & 0 & v c_{13}^* \\ v c_{21}^* & c_{22} & c_{23} + v d_{23} \\ 0 & v c_{32}^* & v d_{33} \end{pmatrix}. $$

On the other hand, $\tilde{w}'_{f-1-i} = \alpha\beta\gamma t_1$, $\tilde{z}_i^* = \gamma^+\beta$, $(a', b', c') = (c-1, a, b+1)$ and $A^{(f-1-i)}$ has the form

$$ A^{(f-1-i)} = \begin{pmatrix} v^2 c_{11}^{*\prime} & 0 & 0 \\ v c_{21}^{*} + v^2 d_{21}^{*} & c_{22}^{*\prime} & c_{23}^{*} \\ v^2 c_{31}^{*} & 0 & v c_{33}^{*\prime} \end{pmatrix}. $$
The ring $\bar{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of $\mathbb{F}\llbracket c'_{21}, d'_{21}, c'_{23}, c'_{31}, c'^*_{11} - [\tilde{c}'^*_{11}], c'^*_{22} - [\tilde{c}'^*_{22}], c'^*_{33} - [\tilde{c}'^*_{33}]\rrbracket$ by the relation

$$(-1 - a' + b')c'_{21}c'^*_{33} - (b' - c')c'_{31}c'_{23} = 0.$$

Now we note that

$$\begin{pmatrix} 0 & 0 & vc_{13}^* \\ vc_{21}^* & c_{22} & c_{23} + vd_{23} \\ 0 & vc_{32}^* & vd_{33} \end{pmatrix} = \begin{pmatrix} v^2 c'_{11}^* & 0 & 0 \\ vc'_{21} + v^2 d'_{21} & c'_{22} & c'_{23} \\ v^2 c'_{31} & 0 & vc'^*_{33} \end{pmatrix} \begin{pmatrix} 0 & 0 & v^{-1} \\ v & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$$

under the isomorphism

$$\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / \mathfrak{c}_{(\epsilon_1, 1)} \cong \bar{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla}$$

given by the change of variable

$$\begin{align*} c_{22} &= c'_{23}, & c_{23} &= c'_{21}, & d_{23} &= d'_{21}, \\ d_{33} &= c'_{31}, & c'_{13} &= c'^*_{11}, & c'_{21} &= c'^*_{22}, & c'_{32} &= c'^*_{33}. \end{align*}$$

We finish the matching in this case.

Finally, we explain the matching of the ideal $w_0 \stackrel{\text{def}}{=} \mathfrak{c}_{(0,1)} \cap \mathfrak{c}_{(0,0)} = (c_{22})$. We have

$$\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / w_0 = \frac{\mathbb{F}\llbracket c_{12}, d_{23}, c_{31}, d_{33}, c^*_{13} - [\tilde{c}^*_{13}], c^*_{21} - [\tilde{c}^*_{21}], c^*_{32} - [\tilde{c}^*_{32}]\rrbracket}{(c_{12}((b-c)d_{33}c^*_{21} + (a-b)c_{31}d_{23}))}.$$

Moreover,

$$A^{(f-1-i)} \mod w_0 = \begin{pmatrix} c_{31}c_{12}(c_{32}^*)^{-1} & c_{12} & d_{33}c_{12}(c_{32}^*)^{-1} + vc_{13}^* \\ vc_{21}^* & 0 & vd_{23} \\ vc_{31} & vc_{32}^* & vd_{33} \end{pmatrix}.$$

On the other hand, $\tilde{w}'_{f-1-i} = \alpha\beta\alpha t_\perp, \tilde{z}_i^* = \alpha, (a', b', c') = (b, a, c)$ and $A^{(f-1-i)}$ has the form

$$A^{(f-1-i)} = \begin{pmatrix} c'_{11} & c'_{11}c'_{32}(c'^*_{31})^{-1} & d'_{33}c'_{11}(c'^*_{31})^{-1} + vc'^*_{13} \\ 0 & vc'^*_{22} & vc'_{23} \\ vc'^*_{31} & vc'_{32} & vd'_{33} \end{pmatrix}.$$

The ring $\bar{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of $\mathbb{F}\llbracket c'_{11}, c'_{23}, c'_{32}, d'_{33}, c^*_{31} - [\tilde{c}^*_{31}], c^*_{22} - [\tilde{c}^*_{22}], c^*_{13} - [\tilde{c}^*_{13}]\rrbracket$ by the relations

$$c'_{11}((a' - b')c'_{23}c'_{32} - (a' - c')c'^*_{33}d'_{33}) = 0.$$
(We remark that the form of the (3, 3) entry of $A'^{(f-1-i)}$ is as above because in the notation of [LLHLM18, Section 5.3.1], we have $c_{33} = -y'_{33}c_{13}c_{31}^* = 0$ mod $p$ since $c_{13}c_{31}^* = c_{11}c'_{33}$ mod $p$ and $y'_{33}c_{11} = 0$ mod $p$.)

Now we note that

$$ 
\begin{pmatrix}
c_{31}c_{12}(c_{32}^*)^{-1} & c_{12} & d_{33}c_{12}(c_{32}^*)^{-1} + vc_{13}^* \\
vc_{21}^* & 0 & vd_{23} \\
vc_{31} & vc_{32}^* & vd_{33}
\end{pmatrix}
=
\begin{pmatrix}
c'_{11} & c'_{11}c'_{32}(c'_{31})^{-1} & d'_{33}c'_{11}(c'_{31}^*)^{-1} + vc'_{13}^* \\
0 & vc'_{22} & vc'_{23} \\
vc'_{31}^* & vc'_{32} & vd'_{33}
\end{pmatrix}
\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
$$

under the isomorphism

$$ \overline{R}_{\overline{\mathfrak{M}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / \mathfrak{w}_0 \cong \overline{R}_{\overline{\mathfrak{M}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} $$

given by the change of variable

$$ 
\begin{array}{llll}
c_{12} = c'_{11}, & c_{31} = c'_{32}, & d_{23} = d'_{23}, & d_{33} = d'_{33}, \\
c_{32}^* = c'_{31}^*, & c_{21}^* = c'_{22}^*, & c_{13}^* = c'_{13}^*.
\end{array}
$$

Case $\tilde{w}_{f-1-i} = \alpha t_1$. From [LLHLM18, Table 5], the matrix of the partial Frobenius $A^{(f-1-i)}$ has the following form:

$$ \begin{pmatrix} c_{11} & c_{12} + vc_{12}^* & c_{13} \\ vc_{21}^* & c_{22} + vd_{22} & c_{23} \\ vc_{31} & vc_{32} & c_{33} + vc_{33}^* \end{pmatrix}. $$

Set $\tilde{c}_{32} \stackrel{\text{def}}{=} \frac{c_{32}c_{21}^* - d_{22}c_{31}^*}{c_{21}^*}$. Recall from item (2) above that $\overline{R}_{\overline{\mathfrak{M}}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of

$$ \mathbb{F}[\![c_{11}, c_{12}, c_{13}, c_{22}, c_{23}, c_{31}, \tilde{c}_{32}, c_{33}, d_{22}, c_{12}^* - [\tilde{c}_{12}^*], c_{21}^* - [\tilde{c}_{21}^*], c_{33}^* - [\tilde{c}_{33}^*]] \! $$

by the following relations:

$$ 
\begin{align*}
c_{11}c_{23} &= 0, & c_{33}^*c_{11}\tilde{c}_{32} &= c_{13}c_{31}\tilde{c}_{32}, \\
c_{11}d_{22}c_{33}^* &= \frac{b-c}{a-b}c_{21}^*c_{13}\tilde{c}_{32}, & c_{13}c_{23}\tilde{c}_{32} &= 0, \\
c_{23}c_{31}\tilde{c}_{32} &= 0, & c_{21}^*c_{33} &= c_{31}c_{23}, \\
c_{12}c_{33}^* &= \frac{a-c}{a-b}c_{13}\tilde{c}_{32}, & c_{22}c_{33}^* &= \frac{(-1-a+c)}{(-1-a+b)}c_{23}\tilde{c}_{32},
\end{align*}
$$
and

$$ (a - b)c_{13}c_{31}d_{22} + (c - b)c_{13}\tilde{c}_{32}c_{21}^{*} + (-1 - a + c)c_{23}c_{31}c_{12}^{*} = 0. $$

(Note that $\bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla}$ is formally smooth of relative dimension three over the ring $\tilde{R}$ defined in [LLHLM18, Proposition 8.11].)

We provide details for the matching of the ideal $c_{(\varepsilon_{1,1})}$. We have

$$ \overline{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / c_{(\varepsilon_{1,1})} = \frac{\mathbb{F}[c_{22}, c_{23}, c_{32}, d_{22}, c_{21}^* - [\tilde{c}_{21}^*], c_{12}^* - [\tilde{c}_{12}^*], c_{33}^* - [\tilde{c}_{33}^*]]}{((-1 - a + b)c_{22}c_{33}^* - (-1 - a + c)c_{23}c_{32})} $$

and

$$ A^{(f-1-i)} \mod c_{(\varepsilon_{1,1})} = \begin{pmatrix} 0 & vc_{12}^* & 0 \\ vc_{21}^* & c_{22} + vd_{22} & c_{23} \\ 0 & vc_{32} & vc_{33}^* \end{pmatrix}. $$

On the other hand, $\tilde{w}'_{f-1-i} = \alpha\beta\gamma\beta, \tilde{z}_i^* = \beta\gamma^+\beta, (a', b', c') = (b-1, a+1, c)$ and $A^{(f-1-i)}$ has the form

$$ \begin{pmatrix} v^2 c'_{11}^* & 0 & 0 \\ v(c'_{21} + vd'_{21}) & c'_{22}^* & c'_{23} \\ v^2 c'_{31} & 0 & vc'_{33}^* \end{pmatrix}. $$

The ring $\bar{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla}$ is the quotient of $\mathbb{F}[c'_{21}, d'_{21}, c'_{23}, c'_{31}, c'_{11}^* - [\tilde{c}'_{11}^*], c'_{22}^* - [\tilde{c}'_{22}^*], c'_{33}^* - [\tilde{c}'_{33}^*]]$ by the relation

$$ (-1 - a' + b')c'_{21}c'_{33}^* - (b' - c')c'_{31}c'_{23} = 0. $$

We now note that

$$ \begin{pmatrix} 0 & vc_{12}^* & 0 \\ vc_{21}^* & c_{22} + vd_{22} & c_{23} \\ 0 & vc_{32} & vc_{33}^* \end{pmatrix} = \begin{pmatrix} v^2 c'_{11}^* & 0 & 0 \\ v(c'_{21} + vd'_{21}) & c'_{22}^* & c'_{23} \\ v^2 c'_{31} & 0 & vc'_{33}^* \end{pmatrix} \begin{pmatrix} 0 & v^{-1} & 0 \\ v & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$

under the isomorphism

$$ \bar{R}_{\mathfrak{M}, \tilde{w}_{f-1-i}}^{\text{expl}, \nabla} / c_{(\varepsilon_{1,1})} \cong \bar{R}_{\mathfrak{M}', \tilde{w}'_{f-1-i}}^{\text{expl}, \nabla} $$

given by the change of variables

$$ \begin{align*} c_{12}^* &= c_{11}^{*'}, & c_{21}^* &= c_{22}^{*'}, & d_{22} &= d_{21} \\ c_{23} &= c_{23}^{'}, & c_{32} &= c_{31}^{'}, & c_{33}^* &= c_{33}^{*'} \end{align*} $$