Serre weights for three-dimensional wildly ramified Galois representations

We formulate and prove the weight part of Serre's conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption in \cite{arXiv:1512.06380}, \cite{arXiv:1608.06570} that the representation be tamely ramified at $p$. We also prove a version of Breuil's lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton--Gee stacks \cite{arXiv:2012.12719} using the local models introduced in \cite{arXiv:2007.05398}.

Breuil's lattice conjecture for these representations and the Breuil-Mézard conjecture for generic tamely potentially crystalline deformation rings of parallel weight (2, 1, 0). For a detailed discussion of these conjectures, see [LLHLM20] where we establish the tame case of these conjectures.
1.1.1. The weight part of Serre's conjecture. Let p be a prime, and let F/F + be a CM extension of a totally real field F + = Q. Assume that all places in F + above p split in F/F + . Let G be a definite unitary group over F + split over F which is isomorphic to U (3) at each infinite place and split at each place above p. A (global) Serre weight is an irreducible F p -representation V of G(O F + ,p ). These are all of the form ⊗ v|p V v with V v an irreducible F p -representation of G(k v ), where k v is the residue field of F + at v. For a mod p Galois representation r : G F → GL 3 (F p ), let W (r) denote the collection of modular Serre weights for r. That is, V ∈ W (r) if the Hecke eigensystem attached to r appears in a space of mod p automorphic forms on G of weight V for some prime to p level.
For each place v | p, fix a place v of F dividing v which identifies G(k v ) with GL 3 (k v ). Define ρ v := r| Gal(F v /F v ) . We can now state the main theorem: Theorem 1.1.1 (Theorem 5.4.2). Assume that p is unramified in F and that ρ v is 8-generic for all v | p. Assume that r is modular (i.e. W (r) is nonempty) and satisfies Taylor-Wiles hypotheses. Then is an explicit set of irreducible F p -representations of GL 3 (k v ) attached to ρ v (see Definition 1.2.1 below).
In particular, this affirms the expectation from local-global compatibility that W (r) depends only on the restrictions of r to places above p.
Remark 1.1.2. This is the first complete description of W (r) in dimension greater than two for representations r that are wildly ramified above p. Some lower bounds were previously obtained in [GG12,MP17,HLM17,LMP18,LLHLM18].
The first obstacle we overcome is the lack of a conjecture. One basic problem is that while tame representations (when restricted to inertia) depend only on discrete data, wildly ramified representations vary in nontrivial moduli. Buzzard-Diamond-Jarvis defined a recipe in terms of crystalline lifts in dimension two. However, after [LLHLM18], it was clear that the crystalline lifts perspective is insufficient in higher dimension. In higher dimensions, Herzig defined a combinatorial/representation theoretic recipe for a collection of weights W ? (ρ v ) when ρ v is tame. For possibly wildly ramified ρ, [GHS18] makes a conjectural conjecture: they define a conjectural set conditional on a version of the Breuil-Mézard conjecture. Our first step is to prove a version of the Breuil-Mézard conjecture (Theorem 1.2.2 and Remark 1.2.3) when n = 3.
Having established a version of the Breuil-Mézard conjecture when n = 3, the weight set from [GHS18] turns out (in generic cases) to have a simple geometric interpretation. Let X 3 be the moduli stack of (ϕ, Γ)-modules recently constructed by Emerton-Gee [EG]. The irreducible components of X 3 are labelled by irreducible mod p representations of GL 3 (k v ) and W g (ρ v ) is defined so that V v ∈ W g (ρ v ) if and only if ρ v lies on C Vv . However, this definition of W g (ρ v ) gives very little insight into its structure. We study W g (ρ v ) using the local models developed in [LLHLMa] combined with the explicit calculations of tamely potentially crystalline deformation rings in [LLHLM18,LLHLM20]. We ultimately arrive at an explicit description of all possible weight sets W g (ρ v ) which allows us to then employ the Taylor-Wiles patching method to prove Theorem 1.1.1.
1.1.2. Breuil's lattice conjecture and mod p multiplicity one. The weight part of Serre's conjecture can be viewed as a local-global compatibility result in the mod p Langlands program. In this section, we mention two further local-global compatibility results-one mod p and one p-adic. We direct the reader to the introduction of [LLHLM20] for further context for the following two results.
In the global setup above assume further that F/F + is unramified at all finite places and G is quasi-split at all finite places. Let r : G F → GL 3 (Q p ) be a modular Galois representation which is tamely potentially crystalline with Hodge-Tate weights (2, 1, 0) at each place above p and unramified outside p (though our results hold true when r is minimally split ramified, cf. §5.4). Write λ for the Hecke eigensystem corresponding to r. We fix places v|v|p of F and F + respectively. We let H be the integral p-adically completed cohomology with infinite level at v, hyperspecial level outside v, and constant coefficients. Set ρ def = r| G F v and let σ(τ ) be the tame type corresponding to the Weil-Deligne representation associated to ρ under the inertial local Langlands correspondence (so that H[λ][1/p] contains σ(τ ) with multiplicity one). Let r and ρ be the reductions of r and ρ, respectively.
Theorem 1.1.3 (Theorem 5.4.4). Assume that p is unramified in F + , r is unramified outside p, ρ is 11-generic, and r satisfies Taylor-Wiles hypotheses. Then, the lattice depends only on ρ.
We now let H be the mod p reduction of H. Thus, H is the mod p cohomology with infinite level at v (and hyperspecial level at places outside v with constant coefficients) of a U (3)-arithmetic manifold. (See §1.5, 2.1.6, for the notion of upper and lower alcove for Serre weights for GL 3 .) The statement of Theorem 1.1.4 is also true when the cosocle is not necessarily upper alcove if one imposes a condition on the shape of ρ with respect to τ ; see Theorem 5.4.3.
1.2.1. Local methods: Geometry of the Emerton-Gee stack and local models. We begin by recalling the set W g (ρ) that appears in Theorem 1.1.1. Let K be a finite unramified extension of Q p of degree f , with ring of integers O K and residue field k. Let X K,n be the Noetherian formal algebraic stack over Spf Z p defined in [EG, Definition 3.2.1]. It has the property that for any complete local Noetherian Z p -algebra R, the groupoid X K,n (R) is equivalent to the groupoid of rank n projective R-modules equipped with a continuous G K -action, see [EG,§3.6.1]. In particular, X K,n (F p ) is the groupoid of continuous Galois representations ρ : G K → GL n (F p ). As explained in [LLHLMa,§7.4], there is a bijection σ → C σ between irreducible F p -representations of GL n (k) and the irreducible components of the reduced special fiber of X K,n . (This is a relabeling of the bijection of [EG, Theorem 6.5.1].) Definition 1.2.1. Let ρ ∈ X K,n (F p ). Define the set of geometric weights of ρ to be W g (ρ) = {σ | ρ ∈ C σ (F p )}.
While Definition 1.2.1 is simple, it does not appear to be an easy task to determine the possible sets W g (ρ). The irreducible components of X K,n are described in terms of closures of substacks, but we expect the closure relations and component intersections in X K,n to be rather complicated.
We now specialize to the case n = 3. A key tool in the analysis of the sets W g (ρ) in this setting is the description of certain potentially crystalline substacks. For a tame inertial type τ , let X η,τ ⊂ X K,3 be the substack parametrizing potentially crystalline representations of type τ and parallel weight (2, 1, 0). Recall that σ(τ ) denotes the representation of GL 3 (O K ) obtained by applying the inertial local Langlands correspondence to τ (it is the inflation of a Deligne-Lusztig representation; see §2.1.3). The following is an application of the theory of local models of [LLHLMa]: Theorem 1.2.2 (Corollary 3.3.3). If τ is a 4-generic tame inertial type, then X η,τ is normal and Cohen-Macaulay and its special fiber X η,τ F is reduced. Moreover, X η,τ F is the scheme-theoretic union σ∈JH(σ(τ )) C σ .
Remark 1.2.3. This shows that the choice of cycles Z σ = C σ solves the Breuil-Mézard equations for the above X η,τ (cf. [ The equality of the underlying reduced X η,τ F,red and the scheme-theoretic union σ∈JH(σ(τ )) C σ is proved in Theorem 1.3.5 in [LLHLMa] though we reprove it here with a weaker genericity condition (see Remark 1.2.4). The key point is to prove that the special fiber of X η,τ is in fact reduced. (If we replace η by λ + η with λ dominant and nonzero or n = 3 by n > 3, the Breuil-Mézard conjecture predicts that the analogous stacks never have reduced special fiber.) The special fiber of X η,τ has an open cover with open sets labeled by f -tuples of (2, 1, 0)-admissible elements ( w j ) in the extended affine Weyl group of GL 3 . The complexity of the geometry of the open sets increases as the lengths of the w j decrease. When the length of w j is greater than 1 for all j, the reducedness immediately follows from the calculations in [LLHLM18,§5.3]. Otherwise, the calculations of [LLHLM18,§8] give an explicit upper bound on the special fiber which when combined with X η,τ F,red = σ∈JH(σ(τ )) C σ must be an equality, and the reducedness follows.
Remark 1.2.4. An inexplicit genericity condition appears in the main theorems of [LLHLMa] (see §1.2.1 of loc. cit.). While we use the models constructed in loc. cit., we reprove some of its main theorems in §3.2, 3.3 with the inexplicit condition replaced by the more typical genericity condition on the gaps between the exponents of the inertial characters in τ . This is possible because of the computations in [LLHLM18,LLHLM20].
Finally, we analyze W g (ρ) using local models. The special fibers of the local models embed inside the affine flag variety where irreducible components appear as subvarieties of translated affine Schubert varieties. In §4, we introduce a subset W obv (ρ) ⊂ W g (ρ) of obvious weights for (possibly) wildly ramified ρ, which has a simple interpretation in terms of the affine flag variety. Obvious weights generalize the notion of ordinary weights that appear in [GG12] and the additional weights appearing in the exceptional cases of [MP17,HLM17,LMP18]. The set W obv (ρ) gives upper and lower bounds for W g (ρ). We finally show that, in almost all cases, one can determine W g (ρ) from W obv (ρ) (Theorem 4.2.5). This last part uses a curious piece of numerology from the calculations of [LLHLM18]-points in the special fibers of the local models never lie on exactly three components.
1.2.2. Global methods: Patching. To prove Theorems 1.1.3 and 1.1.4 we combine the explicit description of the weight sets W (r), coming from Theorems 1.1.1 and 4.2.5, with the Kisin-Taylor-Wiles methods developed in [EGS15] and employed in [LLHLM20,§5]. A crucial ingredient is the analysis of certain intersections of cycles in the special fiber of deformation rings. The local models introduced in [LLHLMa] allow us to algebraize the computations made for the tame case in [LLHLM20,§3.6].
We now turn to Theorem 1.1.1. The key input into its proof beyond the Kisin-Taylor-Wiles method is the fact that the local Galois deformation rings of type (η, τ ) are domains when τ is 4-generic. This is guaranteed by the fact that the stacks X η,τ are normal (Theorem 1.2.2). Then the supports of the patched modules of type τ are either empty or the entire potentially crystalline deformation rings of type (η, τ ). The proof is then similar to the tame case in [LLHLM20]-one propagates modularity between obvious weights and then to shadow weights using carefully chosen types-except for one new wrinkle. From the axioms of a weak patching functor, one cannot deduce the modularity of an obvious weight to get started! Indeed one cannot rule out that ρ v lies on a unique component C σv and W (r) contains exactly one Serre weight σ = σ def = ⊗ v|p σ v with the property that for any tame inertial type τ , if JH(σ(τ )) contains σ , then it also contains σ. We use a patched version of the weight cycling technique introduced in [EGH13] to rule out this pathology. In fact, we axiomatize our setup to make clear the ingredients that our method requires.
§3 establishes the main results about the geometry of local deformation rings. We specialize the theory of local models in [LLHLMa] to dimension three. The main results are Theorem 3.3.2 and Corollary 3.3.3 which establish the geometric properties that we need, some of which are specific to dimension three.
In §4, we analyze possible sets of geometric weights using the affine flag variety. Theorem 4.2.5 gives a complete explicit description when ρ is sufficiently generic.
§5 contains our global applications. In §5.1, we introduce the axioms of patching functors following [LLHLMa,§6] and prove the weight part of Serre's conjecture assuming the modularity of at least one obvious weight (Proposition 5.1.11). The latter condition is then removed in §5.2 using modules with an arithmetic action (Theorem 5.2.6). In §5.3, we prove results on mod p multiplicity one and Breuil's lattice conjectures for patched modules (Theorems 5.3.1, 5.3.13), generalizing analogous results in [LLHLM20] to the wildly ramified case. Finally, §5.4 proves our main global theorems.
1.4. Acknowledgements. The main ideas of this article date back to the summer of 2016, but the formulation of the results were rather awkward due to the lack of the Emerton-Gee stack and its local model theory at that time. For this reason we decided to write up this paper only after the release of [LLHLMa]. We apologize for the long delay. Part of the work has been carried out during visits at the Institut Henri Poinaré (2016), the Institute for Advanced Study (2017), Mathematisches Forschungsinstitut Oberwolfach (2019), University of Arizona, and Northwestern University. We would like to heartily thank these institutions for the outstanding research conditions they provided, and for their support. 1.5. Notation. For any field K we fix once and for all a separable closure K and let G K def = Gal(K/K). If K is a nonarchimedean local field, we let I K ⊂ G K denote the inertial subgroup. We fix a prime p ∈ Z >0 . Let E ⊂ Q p be a subfield which is finite-dimensional over Q p . We write O to denote its ring of integers, fix an uniformizer ∈ O and let F denote the residue field of E. We will assume throughout that E is sufficiently large.
We consider the group G def = GL 3 (defined over Z). We write B for the subgroup of upper triangular matrices, T ⊂ B for the split torus of diagonal matrices and Z ⊂ T for the center of G. Let Φ + ⊂ Φ (resp. Φ ∨,+ ⊂ Φ ∨ ) denote the subset of positive roots (resp. positive coroots) in the set of roots (resp. coroots) for (G, B, T ). Let ∆ (resp. ∆ ∨ ) be the set of simple roots (resp. coroots). Let X * (T ) be the group of characters of T which we identify with Z 3 by letting the standard i-th basis element ε i = (0, . . . , 1, . . . , 0) (with the 1 in the i-th position) correspond to extracting the i-th diagonal entry of a diagonal matrix. In particular, we let ε 1 and ε 2 be (1, 0, 0) and (0, 0, −1) respectively.
We write W (resp. W a , resp. W ) for the Weyl group (resp. the affine Weyl group, resp. the extended affine Weyl group) of G. If Λ R ⊂ X * (T ) denotes the root lattice for G we then have W a = Λ R W, W = X * (T ) W and use the notation t ν ∈ W to denote the image of ν ∈ X * (T ). The Weyl groups W , W , and W a act naturally on X * (T ) and on X * (T ) ⊗ Z A for any ring A by extension of scalars.
Let , denote the duality pairing on X * (T ) × X * (T ), which extends to a pairing on (X * (T ) ⊗ Z A)×(X * (T )⊗ Z A) for any ring A. We say that a weight λ ∈ X * (T ) is dominant if 0 ≤ λ, α ∨ for all α ∈ ∆. Set X 0 (T ) to be the subgroup consisting of characters λ ∈ X * (T ) such that λ, α ∨ = 0 for all α ∈ ∆, and X 1 (T ) to be the subset consisting of characters λ ∈ X * (T ) such that 0 ≤ λ, α ∨ < p for all α ∈ ∆ We fix an element η ∈ X * (T ) such that η, α ∨ = 1 for all positive simple roots α. We define the p-dot action as t λ w · µ = t pλ w(µ + η) − η. By letting w 0 denote the longest element in W define Recall that for (α, n) ∈ Φ + × Z, we have the p-root hyperplane H α,n def = {λ : λ + η, α ∨ = np}. A p-alcove is a connected component of the complement X * (T ) ⊗ Z R \ (α,n) H α,n . We say that a p-alcove C is p-restricted (resp. dominant) if 0 < λ + η, α ∨ < p (resp. 0 < λ + η, α ∨ ) for all simple roots α ∈ ∆ and λ ∈ C. If C 0 ⊂ X * (T ) ⊗ Z R denotes the dominant base alcove (i.e. the alcove defined by the condition 0 < λ + η, α ∨ < p for all positive roots α ∈ Φ + , we let We sometimes refer to C 0 as the lower alcove and C 1 def = w h · C 0 as the upper alcove. should be clear as should the natural isomorphisms X * (T ) = X * (T ) J and the like. Given an element j ∈ J , we use a subscript notation to denote j-components obtained from the isomorphism G ∼ = G J /O (so that, for instance, given an element w ∈ W we write w j to denote its j-th component via the induced identification W ∼ = W J ). For sake of readability, we abuse notation and still write w 0 to denote the longest element in W , and fix a choice of an element η ∈ X * (T ) such that η, α ∨ = 1 for all α ∈ ∆. The meaning of w 0 , η and w h def = w 0 t −η should be clear from the context.
The absolute Frobenius automorphism on O p /p lifts canonically to an automorphism ϕ of O p . We define an automorphism π of the identified groups X * (T ) and X * (T ∨ ) by the formula π(λ) σ = λ σ•ϕ −1 for all λ ∈ X * (T ) and σ : O p → O. We assume that, in this case, the element η ∈ X * (T ) we fixed is π-invariant. We similarly define an automorphism π of W and W . Let where Gal(E/Q p ) acts on the set of homomorphisms F + p → E by post-composition. We now specialize to the case where S p = {v} is a singleton. Hence F + p = K is an unramified extension of degree f with ring of integers O K and residue field k. Let W (k) be ring of Witt vectors of k, which is also the ring of integers of K.
We denote the arithmetic Frobenius automorphism on W (k) by ϕ; it acts as raising to p-th power on the residue field.
Recall that we fixed a separable closure K of K. We choose π ∈ K such that π p f −1 = −p and let ω K : G K → O × K be the character defined by g(π) = ω K (g)π, which is independent of the choice of π. We fix an embedding σ 0 : K → E and define σ j = σ 0 • ϕ −j , which identifies J = Hom(k, F) = Hom Qp (K, E) with Z/f Z. We write ω f : Let ε denote the p-adic cyclotomic character. If W is a de Rham representation of G K over E, then for each κ ∈ Hom Qp (K, E), we write HT κ (W ) for the multiset of Hodge-Tate weights labelled by embedding κ normalized so that the p-adic cyclotomic character ε has Hodge-Tate weight {1} for every κ. For µ = (µ j ) j∈J ∈ X * (T ), we say that a 3-dimensional representation W has Hodge-Tate weights µ if Our convention is the opposite of that of [EG,CEG + 16], but agrees with that of [GHS18].
We say that a 3-dimensional potentially semistable representation ρ : G K → GL n (E) has type (µ, τ ) if ρ has Hodge-Tate weights µ and the restriction to I K of the Weil-Deligne representation attached to ρ (via the covariant functor ρ → WD(ρ)) is isomorphic to the inertial type τ . Note that this differs from the conventions of [GHS18] via a shift by η.
Let Γ be a group. If V is a finite length Γ-representation, we let JH(V ) be the (finite) set of Jordan-Hölder factors of V . If V • is a finite O-module with a Γ-action, we write V • for the If P is a statement, the symbol δ P ∈ {0, 1} takes value 1 if P is true, and 0 if P is false.
2. Background 2.1. Affine Weyl group, tame inertial types and Deligne-Lusztig representations. Throughout this section, we assume that S p = {v}. Thus O p = O K is the ring of integers of a finite unramified extension K of Q p and G 0 = Res O K /Zp G /O K . We drop subscripts v from notation and 2.1.1. Admissibility. We follow [LLHLMa, §2.1- §2.4], specializing to the case of n = 3. We denote by ≤ the Bruhat order on W ∼ = X * (T ) W associated to the choice of the dominant base alcove C 0 and set We will also consider the partially ordered group W ∨ which is identified with W as a group, but whose Bruhat order is defined by the antidominant base alcove (and still denoted as ≤). Then Adm ∨ (η) is defined as above, using now the antidominant order. We have an order reversing bijection w → w * between W and W 2.1.2. Tame inertial types and Deligne-Lusztig representations. An inertial type (for K) is the GL 3 (E)-conjugacy class of a homomorphism τ : with open kernel and which extends to the Weil group of G K . An inertial type is tame if it factors through the tame quotient of I K .
We will sometimes identify a tame inertial type with a fixed choice of a representative in its class. Given s = (s 0 , . . . , s f −1 ) ∈ W and µ ∈ X * (T ) ∩ C 0 , we have an associated integer r ∈ {1, 2, 3} (which is the order of the element s 0 s 1 . . . s f −1 ∈ W ), integers a (j ) ∈ Z 3 for 0 ≤ j ≤ f r − 1 and a tame inertial type τ (s, µ + η) defined as τ (s, µ + η) i (see [LLHLMa,Example 2.4.1, equations (5.2), (5.1)] for the details of this construction). We say that (s, µ) is the lowest alcove presentation for the tame inertial type τ (s, µ + η) and that τ (s, µ + η) is N-generic if µ is N -deep in alcove C 0 . We say that a tame inertial type τ has a lowest alcove presentation if there exists a pair (s, µ) as above such that τ ∼ = τ (s, µ + η) (in which case we will say that (s, µ) is a lowest alcove presentation for τ ), and that τ is N -generic if τ has a lowest alcove presentation (s, µ) such that µ is N -deep in alcove C 0 . We remark that different choices of pairs (s, µ) as above can give rise to isomorphic tame inertial types (see [LLHL19, Proposition 2.2.15]). If τ is a tame inertial type of the form τ = τ (s, µ + η), we write w(τ ) for the element t µ+η s ∈ W . (In particular, when writing w(τ ) we use an implicit lowest alcove presentation for τ ).
Repeating the above with E replaced by F, we obtain the notion of inertial F-types and lowest alcove presentations for tame inertial F-types. We use the notation τ to denote a tame inertial F-type τ : I K → GL 3 (F). We say that a tame inertial F-type is N -generic if it admits a lowest alcove presentation (s, µ) such that µ is N -deep in C 0 .
If µ is 1-deep in C 0 , then for each 0 ≤ j ≤ f r − 1 there is a unique element s or,j ∈ W such that (s or,j ) −1 (a (j ) ) is dominant. (In the terminology of [LLHLM18], cf. Definition 2.6 of loc. cit., the f r-tuple (s or,j ) 0≤j ≤f r−1 is the orientation of (a (j ) ) 0≤j ≤f r−1 .) To a pair (s, µ) ∈ W ×X * (T ), we can also associate a virtual G 0 (F p )-representation over E which we denote R s (µ) (cf. [GHS18, Definition 9.2.2], where R s (µ) is denoted as R(s, µ)). In particular, R 1 (µ) is a principal series representation. If µ − η is 1-deep in C 0 then R s (µ) is an irreducible representation. In analogy with the terminology for tame inertial type, if µ − η is N -deep in alcove C 0 for N ≥ 0, we call (s, µ − η) an N -generic lowest alcove presentation for R s (µ), and say that R s (µ) is N -generic.
2.1.3. Inertial local Langlands correspondence. Given a tame inertial type τ : I K → GL 3 (E), [CEG + 16, Theorem 3.7] gives an irreducible smooth E-valued representation σ(τ ) of G 0 (F p ) = GL 3 (k) over E satisfying results towards the inertial local Langlands correspondence (see loc. cit. for the properties satisfied by σ(τ )). (By inflation, we will consider σ(τ ) as a smooth representation of G 0 (Z p ) without further comment.) This representation need not be uniquely determined by τ and in what follows σ(τ ) will denote either a particular choice that we have made or any choice that satisfies the properties of [CEG + 16, Theorem 3.7] (see also [LLHLMa,Theorem 2.5.3] and the discussion following it).
2.1.4. Serre weights. We finally recall the notion of Serre weights for G 0 (F p ), and the notion of lowest alcove presentations for them, following [LLHLMa, §2.2]. A Serre weight for G 0 (F p ) is the isomorphism class of an (absolutely) irreducible representation of G 0 (F p ) over F. (We will sometimes refer to a representative for the isomorphism class as a Serre weight.) Given λ ∈ X 1 (T ), we write F (λ) for the Serre weight with highest weight λ; the assignment λ → F (λ) induces a bijection between X 1 (T )/(p−π)X 0 (T ) and the set of Serre weights (cf. [GHS18, Lemma 9.2.4]). We say that F (λ) is N -deep if λ is (this does not depend on the choice of λ).
Recall from [LLHLMa, §2.2] the equivalence relation on W ×X * (T ) defined by ( w, ω) ∼ (t ν w, ω − ν) for all ν ∈ X 0 (T ). For (an equivalence class of) a pair ( w 1 , ω − η) ∈ W + 1 × (X * (T ) ∩ C 0 )/ ∼ the Serre weight F ( w 1 ,ω) def = F (π −1 ( w 1 ) · (ω − η)) is well defined i.e. is independent of the representative of the equivalence class of ( w 1 , ω). The equivalence class of ( w 1 , ω) is called a lowest alcove presentation for the Serre weight F ( w 1 ,ω) . The Serre weight F ( w 1 ,ω) is N -deep if and only if ω − η is N -deep in alcove C 0 . As above, we sometimes implicitly choose a representative for a lowest alcove presentation to make a priori sense of an expression, though it is a posteriori independent of this choice.
2.1.5. Compatibility for lowest alcove presentations. Recall that we have a canonical isomorphism W /W a ∼ = X * (Z) where W a ∼ = Λ R W is the affine Weyl group of G. Given an algebraic character ζ ∈ X * (Z), we say tha t an element w ∈ W is ζ-compatible if it corresponds to ζ via the isomorphism W /W a ∼ = X * (Z). In particular, a lowest alcove presentation (s, µ) for a tame inertial type (resp. a lowest alcove presentation (s, µ − η) for a Deligne-Lusztig representation) is ζ-compatible if the element t µ+η s ∈ W (resp. t µ s ∈ W ) is ζ-compatible. Similarly, a lowest alcove presentation ( w 1 , ω) for Serre weight is ζ-compatible if the element t ω−η w 1 ∈ W is ζ-compatible.
2.1.6. A comparison to [LLHLM20]. In [LLHLM20], the parametrization of Serre weights is slightly different from the one in [LLHLMa]. Here, we give a dictionary between the two.
We let Λ W and W der be X * (T )/X 0 (T ) and W /X 0 (T ), respectively. Recall from [LLHLM20, §2.1] the set Letting A be the set of p-restricted alcoves in X * (T ) ⊗ Z R, the map is a bijection by [LLHLM20, Lemma 2.1.1].
2.1.9. L-parameters. We now assume that S p has arbitrary finite cardinality. An L-parameter (over E) is a G ∨ (E)-conjugacy class of a continuous homomorphism ρ : G Qp → L G(E) which is compatible with the projection to Gal(E/Q p ) (such homomorphism is called L-homomorphism). An inertial L-parameter is a G ∨ (E)-conjugacy class of a homomorphism τ : I Qp → G ∨ (E) with open kernel, and which admits an extension to an L-homomorphism. An inertial L-parameter is tame if some (equivalently, any) representative in its equivalence class factors through the tame quotient of I Qp .
we have a bijection between L-parameters (resp. tame inertial L-parameters) and collections of the form ( . We have similar notions when E is replaced by F. Again we will often abuse terminology, and identify an L-parameter (resp. a tame inertial L-parameter) with a fixed choice of a representative in its class. This shall cause no confusion, and nothing in what follows will depend on this choice.
The definitions and results of §2.1.2-2.1.8 generalize in the evident way for tame inertial Lparameters and L-homomorphism. (In the case of the inertial local Langlands correspondence of §2.1.3, given a tame inertial L-parameter τ corresponding to the collection of tame inertial types (τ v ) v∈Sp , we let σ(τ ) be the irreducible smooth E-valued representation of G 0 (Z p ) given by ⊗ v∈Sp σ(τ v ).) 2.2. Breuil-Kisin modules. We recall some background on Breuil-Kisin modules with tame descent data. We refer the reader to [LLHLM20, §3.1,3.2] and [LLHLMa, §5.1] for further detail, with the caveat that we are following the conventions of the latter on the labeling of embeddings for tame inertial types and Breuil-Kisin modules (see loc. cit. Remark 5.1.2).
Let τ = τ (s, µ + η) be a tame inertial type with lowest alcove presentation (s, µ) which we fix throughout this section (recall that µ is 1-deep in C 0 ). Recall that r ∈ {1, 2, 3} is the order of s 0 s 1 s 2 · · · s f −1 ∈ W . We write K /K for the unramified extension of degree r contained in K and is a fixed choice for an embedding extending σ 0 : K → E. In this way, restriction of embeddings corresponds to reduction modulo f in the above identifications.
Let π ∈ K be an e -th root of −p, let L def = K (π ) and ∆ π for g ∈ ∆ (this does not depend on the ]. The latter is endowed with an endomorphism ϕ : S L ,R → S L ,R acting as Frobenius on W (k ), trivially on R, and sending u to (u ) p . It is endowed moreover with an action of ∆ as follows: for any g in ∆ , g(u ) = g(π ) π u = ω K (g)u and g acts trivially on the coefficients; if σ f ∈ Gal(L /K) is the lift of p f -Frobenius on W (k ) which fixes π , then σ f is a generator for Gal(K /K), acting in natural way on W (k ) and trivially on both u and R. (1) a finitely generated projective S L ,R -module M which is locally free of rank 3; (2) an injective S L ,R -linear map φ M : ϕ * (M) → M whose cokernel is annihilated by E(v) 2 ; and (3) a semilinear action of ∆ on M which commutes with φ M , and such that, for each j ∈ Hom Qp (K , E), with a semilinear action of ∆ and the Frobenius φ M induces ∆ -equivariant morphisms φ In particular, by letting τ denote the tame inertial type for K obtained from τ via the identification I K = I K induced by the inclusion K ⊆ K, the semilinear action of ∆ induces an isomorphism ι M : with respect to the bases ϕ * (β (j −1) ) and β (j ) and set , is upper triangular modulo v and only depends on the restriction of j to K (see [LLHLMa, §5.1]).
Let I(F) denote the Iwahori subgroup of GL 3 (F((v))) relative to the Borel of upper triangular matrices. We define the shape of a mod p Breuil-Kisin module M ∈ Y [0,2],τ (F) to be the element z = ( z j ) ∈ W ∨ such that for any eigenbasis β and any j ∈ J , the matrix A

Local models in mixed characteristic and the Emerton-Gee stack
We assume throughout this section that 3.1. Local models in mixed characteristic. We now define the mixed characteristic local models which are relevant to our paper. We follow closely [LLHLMa, §4] and the notation therein.
For any Noetherian O-algebra R, define and are upper triangular modulo v .

The fpqc quotients
LG O (R) are representable by a projective scheme Gr For any z = ( z j ) j∈J ∈ W ∨ and any Noetherian O-algebra R, define z j (k)k ) = ν j,k and the coefficient of the leading term is a unit of R; where we have written z = zt ν and ν = (ν j,1 , ν j,2 , ν j,3 ) j∈J where we have written z = zt ν and ν = (ν j,1 , ν j,2 , ν j,3 ) j∈J (and recall from §1.5 the notation for the Kronecker deltas δ i>k , δ i<z j (k) ). We now compare the objects above with groupoids of Breuil-Kisin modules with tame descent. Let (s, µ) ∈ W × X * (T ) be a lowest alcove presentation for the tame inertial type τ def = τ (s, µ + η). We have the twisted shifted conjugation action of By [LLHLMa,Theorem 5.3.1], whenever µ is 3-deep in C 0 , we have a morphism of p-adic formal algebraic stacks over O: where the left map is a T ∨ O -torsor for the twisted shifted conjugation action on the source and the second map is an open immersion.

The fiber Gr
. The computations of [LLHLM18,§4] give the following: Table 1. Moreover, let Then we have a morphism of p-adic formal algebraic stacks over O where the left map is a T ∨ O -torsor for the twisted shifted conjugation action on the source and the second map is an open immersion.
Recall the element η ∈ X * (T ) we fixed in §1.5. Let τ def = τ (s, µ + η) be a tame inertial type with lowest alcove presentation (s, µ), where µ is 1-deep in alcove C 0 . By [LLHLMa, Proposition 7.1.6] the datum of a p-adically complete, topologically finite type flat O-algebra R, and a morphism It has the property that for any complete local Noetherian O-algebra R with finite residue field, the groupoid X K,3 (R) is equivalent to the groupoid of rank 3 projective R-modules equipped with a continuous G K -action, see [EG, §3.6.1]. (In particular, we will consider closed points of X K,3 (F) as continuous Galois representations ρ : G K → GL 3 (F), and conversely.) Moreover, by [EG,Theorem 4.8.12], there is a unique O-flat closed formal substack X η,τ of X K,3 which parametrizes, over finite flat O-algebras, those G K -representations which after inverting p are potentially crystalline with Hodge-Tate weight η and inertial type τ . We define X ≤η,τ in the same way, except that the condition on Hodge-Tate weights becomes ≤η. In particular, X ≤η,τ is the scheme theoretic union of the substacks X λ,τ for λ dominant and λ ≤ η.
Using Proposition 3.1.1, we can finally relate the objects introduced so far: Theorem 3.2.2. Let z ∈ Adm ∨ (η) and assume that the character µ (appearing in the lowest alcove presentation (s, µ) of τ ) is 4-deep. We have a commutative diagram of p-adic formal algebraic stacks over Spf O: where: • all the stacks appearing in the left column and in the central column are defined so that all the squares in the diagram are cartesian; • the hooked horizontal arrows are open immersion; • the left horizontal arrows are T ∨ O -torsor for the twisted shifted conjugation action on the source (induced by the twisted shifted conjugation action on U ( z, ≤η) ∧p ); • the vertical hooked arrows are closed immersions and the vertical arrows decorated with " ∼ =" are isomorphisms.
In particular, U ( z, ≤η, ∇ τ,∞ ) is an affine p-adic formal scheme over Spf O, topologically of finite type. Furthermore, if ( z j ) ≥ 2 for all j, then U ( z, ≤η, ∇ τ,∞ ) is the p-adically completed tensor product over O of the rings in Table 2.
Proof. This is [LLHLMa, Proposition 7.2.3]. The last assertion follows from the computations in [LLHLM18, §5.3] noting that the whole discussion there applies to the p-adic completion (as opposed to completions at closed points), and that the computations of loc. cit. can be performed with less stringent genericity assumptions (see the proof of Theorem 3.3.2 below for the precise genericity).
Remark 3.2.3. Note that X η,τ ⊆ X ≤η,τ can be characterized as the union of the (1+3f )-dimensional irreducible components (which is the maximal possible dimension). In particular, by letting 3.3. Special fibers. Let Fl denote the affine flag variety over F for GL 3/F (with respect to the Iwahori relative to the upper triangular Borel), identified with the special fiber of Gr G,O . As in [LLHLMa, (4.7)], we define the closed sub-ind scheme Fl ∇ 0 → Fl. [LLHLMa,Proposition 4.3.3]). It does not depend on the equivalence class of the pair ( w 1 , ω) defined in §2.1.4.

T ∨
F -torsors. Replacing the Iwahori with the pro-v Iwahori in the construction of Fl yields a T ∨ F -torsor Fl J → Fl J . We use · to denote the pullback via this T ∨ F -torsor of objects introduced so far (e.g.
⊂ Fl be the pullback of the special fibre of Gr between lowest alcove presentations ( w 1 , ω) of 0-deep Serre weights and 0-deep Serre weights σ with the choice of an algebraic central character ζ ∈ X * (Z) lifting the central character of σ. If ( w 1 , ω) maps to (σ, ζ) under this bijection, then we set C ζ σ def = C ( w 1 ,ω) . If the algebraic central character ζ ∈ X * (Z) is understood, we will simply write C σ .
3.3.4. The local model diagram in characteristic p. As explained in [LLHLMa,§7.4] there is a bijection σ → C σ between Serre weights and the top dimensional (namely, 3f -dimensional) irreducible components of X K,3 . (This is a relabeling of the bijection of [EG, Theorem 6.5.1].) The main result of this section describes sufficiently generic C σ in terms of the coordinate charts of Theorem 3.2.2.
We have a commutative diagram • All the squares are cartesian (this defines the previously undefined objects C σ ( z), C σ ( z) and P σ, z ). • All the hooked arrows decorated with a circle are open immersions; all the hooked undecorated arrows are monomorphisms and, except ι 0 , are moreover closed immersions; all the arrow decorated with T ∨,J • The map ι 0 is defined, fpqc locally, by sending the class of a tuple (A (j) ) j∈J to the free rank nétale ϕ-module with Frobenius given by (A (j) ) j∈J in the standard basis. Table 3 (or the unit ideal if P (ε j ,a j ), z j , up to symmetry, does not appear in Table 3 for Proof. Theorem 3.2.2 (together with Remark 3.2.3) and [LLHLMa, Proposition 5.4.7, Theorem 7.4.2], imply the existence of the portion of diagram (3.2) which excludes the leftmost vertical column, the top triangle, and the identification of U ( z, η, ∇ τ,∞ ) F with entries of Table 3. (In the notation of [LLHLMa,Proposition 5.4.7] the monomorphism ι 0 would be denoted as ι s * t µ * +η * , the morphism ι (s,µ) would be the diagonal arrow.) Furthermore, all stated properties of this portion of the diagram are already known to hold, except possibly for the last item. We now explain how to fill in the missing parts with all the desired properties except for the last item.
(1) We first deal with the case ( z j ) ≥ 2 for all 0 ≤ j ≤ f −1. In this situation, the computations identifies with the scheme given by Table 3. Indeed, we note that: The computations are performed with an unnecessary strong genericity condition: indeed, by using the "(1,3)-entry" of the leading term in the monodromy condition, one recovers the last displayed equation at page 59 of loc. cit. with n − 3 replaced by n − 1. Choosing the closed subscheme P σ, z according to Table 3, Table 3, the image of P σ, z along the top horizontal arrows is, in the notation of loc. cit., a T ∨,J F -torsor over S ∇ 0 F ( w 1 , w 2 , s) for suitable choices of w 1 , w 2 , and for s taken to be t µ+η s, and this is exactly an open subscheme of C ζ σ . For item (b), we use the just established item (a), and then use the same argument as in the third paragraph in the proof of [LLHLMa,Theorem 7.4.2] to recognize that C κ is actually C σ .
(2) We deal with the general case. The computations in [ the last displayed equations at page 60 and 61 of [LLHLM18]) obtaining the "monodromy equations" claimed in loc. cit. as soon as µ is 4-deep in alcove C 0 . of U ( z, η, ∇ τ,∞ ) F → U ( z, ≤ η) F factors through U ( z, η, ∇ τ,∞ ) table,F , where we temporarily write U ( z, η, ∇ τ,∞ ) table,F for the scheme defined in the second column of Table 3.
We have to prove that this closed immersion is actually an isomorphism. Let n z be # W ? (τ (sz * , µ + s(ν * ) + η)) ∩ JH(R s (µ + η)) . Note that the arguments of [LLHLM18,§8] show that U ( z, η, ∇ τ,∞ ) table,F is reduced, and that its number of irreducible component is n z . We will show that there are at least n z irreducible components of X η,τ F which intersect the open substack X η,τ ( z) F . Now, from the previously established cases of diagram (3.2), we see that X η,τ must contain all the C σ which occurs in diagram 3.2 for z such that ( z j ) ≥ 3 for all 0 ≤ j ≤ f − 1. In particular, X η,τ F contains all C σ such that σ ∈ JH(R s (µ + η)). Note that by definition, We are thus reduced to showing that there are at least n z choices of σ as above such that C σ ∩ U ( z, ≤ η) F s * t µ * +η * /T ∨,J F -sh.cnj = ∅. But this last condition is equivalent to C ζ σ ∩ U ( z, ≤ η) F s * t µ * +η * = ∅, and in turn equivalent to zs * t µ * +η * ∈ C ζ σ . To summarize, we need to show the combinatorial statement that the number of C ζ σ which contain zs * t µ * +η * is exactly n z . But this is the same combinatorial statement as [LLHLMa,Theorem 4.7.6], and we observe that the conclusion of that Theorem holds in our current setup: this follows from the invariance property [LLHLMa, Proposition 4.3.5] of C ζ σ , as well as the fact that z s * t µ * +η * ∈ C ζ σ whenever C ζ σ occurs in diagram (3.2) for z such that ( z j ) ≥ 3 for 0 ≤ j ≤ f − 1. At this point, we have shown that U ( z, η, ∇ τ,∞ ) F identifies with U ( z, η, ∇ τ,∞ ) table,F , and thus we establish the top horizontal arrow of diagram (3.2). The rest of the proof now is exactly the same as in the previous case. Finally, it remains to check the last item in the theorem. But the reducedness follow from the reducedness of each U ( z, η, ∇ τ,∞ ) F , while the identification of the irreducible components was already established in the arguments above.
Remark 3.3.4. It can be showed that both O U ( z, η, ∇ τ,∞ ) and R η,τ ρ are Cohen-Macaulay. This can be done by either explicit inspection of the schemes occurring in Table 3 as in [LLHLM18,§8], or by using the cyclicity of patched modules proven in Theorem 5.3.1 below.
Proof. The fact that O U ( z, η, ∇ τ,∞ ) is a normal domain follows from the fact that its special fiber is reduced, as in the proof of [LLHLM18, Proposition 8.5]. The statement for R η,τ ρ follows in the same way, noting that R η,τ ρ,F , being (equisingular to) a completion of the excellent reduced ring O U ( z, η, ∇ τ,∞ ) F , is reduced.
We can finally introduce the notion of Serre weights attached to a continuous Galois representation ρ : G K → GL 3 (F).
Finally, we say that a Serre weight is generic if it is a Jordan-Hölder constituent of a 4-generic Deligne-Lusztig representation. (By equation (2.3) and §2.1.7, a Serre weight is generic if and only if it admits a lowest alcove presentation ( w 1 , ω) such that (ω, w 1 · C 0 ) ∈ t µ+η s(Σ) for some µ ∈ C 0 ∩ X * (T ) which is 4-deep.) A generic Serre weight is necessarily 2-deep by [LLHLMa, Proposition 2.3.7]. We let W g gen (ρ) and W ? gen (ρ) denote the subsets of generic Serre weights of W g (ρ) and W ? (ρ), respectively.  Proof. Let σ be a generic Serre weight so that σ ∈ JH R s (µ + η) for some 4-deep µ ∈ C 0 and s ∈ W . By Theorem 3.3.2, if ρ / ∈ X η,τ F then σ / ∈ W g (ρ). Else ρ, being semisimple, corresponds to a point in T ∨,J F z ∈ U ( z, ≤ η) F in the diagram (3.2). The proof of Theorem 3.3.2 (precisely, the end of the third paragraph in the proof of item (2) of loc. cit.) shows that ρ ∈ C σ if and only if σ ∈ W ? (ρ).
Remark 3.3.10. The notions and the results of this section hold true, mutatis mutandis, when the set S p has arbitrary finite cardinality, and τ , ρ are a tame inertial L-parameter and a continuous F-valued L-homomorphism respectively. In this case U ( z), U [0,2] ( z), etc. are fibered products, over Spec O and over the elements v ∈ S p , of objects of the form E). Analogously, the algebraic stacks Y [0,2],τ , X K,3 , X η,τ etc. are fibered products, over Spf O and over the elements v ∈ S p , of Y [0,2],τv , X F + v ,3 , X ηv,τv etc. The results of this section hold true in this more general setting.

Geometric Serre weights
The irreducible components of X K,n from [EG, Definition 3.2.1] give rise to a partition of X K,n with locally closed parts indexed by sets W + of Serre weights. It is of interest to determine the geometric properties of these pieces e.g. when they are nonempty. In principle, one can directly study for a set W + of generic Serre weights using the relationship between X K,n and Fl ∇ 0 , but this seems to be complicated even when n = 4. In this section, we determine when (4.1) is nonempty in generic cases when n = 3 using a notion of obvious weights for wildly ramified representations.

4.1.
Intersections of generic irreducible components in Fl ∇ 0 . We first study the geometry of Fl ∇ 0 . The set J will be a singleton, and so we will omit it from the notation. For n ∈ N, let C n-deep be the set of ω ∈ X * (T ) such that ω − η is n-deep in C 0 . Recall from §3.3.1 that given ( w, ω) ∈ W 1 × C 2-deep , we have the irreducible subvariety C ( w,ω) of Fl ∇ 0 . We define Fl ∇ 0 2-deep as the union of the C ( w,ω) with ( w, ω) ∈ W 1 × C 2-deep (in particular, these C ( w,ω) are its irreducible components). The action of T ∨ F (resp. G m ) on Fl ∇ 0 induced by right multiplication (resp. loop rotation t · v = t −1 v) preserves Fl ∇ 0 2-deep and its irreducible components. We let T ∨ F be the extended torus Recall that the equivalence relation is given by ( w, ω) ∼ (t ν w, ω − ν) for any ν ∈ X 0 (T ).
The main result of this section classifies the sets W g 2-deep (x * ) for x * ∈ Fl ∇ 0 2-deep (F). Combining the proof of Theorem 3.3.2 (namely, the combinatorial statement in the proof of item (2) of loc. cit.) and Corollary 3.3.7 (see also [LLHLMa, Prop. 2.6.2]), we obtain the following description of It is defined as follows (cf. [LLHLMb]).
We call elements in S(x * ) specializations of x * . The set SP (x * ) is the set of specialization pairs consisting of a specialization and an obvious weight. [LLHLMa,(4
To determine W g 2-deep (x * ), we first determine SP (x * ). The idea is that W obv (x * ) gives a lower bound for W g 2-deep (x * ) (Remark 4.1.5(2)) while S(x * ) gives an upper bound by Lemma 4.1.7(2) and Theorem 4.1.2, and SP (x * ) combines these invariants into a more uniformly behaved set (see Corollary 4.1.10).
The following results are key to our analysis of SP (x * ). For x * ∈ Fl ∇ 0 2-deep , let θ x * : SP (x * ) → W be the map that takes (y, ( w, ω)) to the image of y w −1 in W .
Proof. This is [LLHLMb, Prop. 3.6.4]. (It can also be proven by direct computation in the case of GL 3 .) Let I def = L + G F be the Iwahori group scheme. For w τ ∈ W and x * ∈ Fl, let w(x * , w τ ) ∈ W be the unique element such that x * ∈ I\I w(x * , w τ ) * I w * τ .
Proof. That y ∈ S(x * ) implies that x * ∈ T ∨ F \ U (y * ) or equivalently that y * is in the T ∨ F -orbit closure of x * . For (1), y * is in the ( T ∨ F -orbit) closure of I\I w(x * , w τ ) * I w * τ which implies the desired inequality. For (2), if x * ∈ C ( w,ω) , then y * ∈ C ( w,ω) since C ( w,ω) is T ∨ F -stable and closed.
The following result provides a method to start with an element of SP (x * ) and produce another using a simple reflection in W .
Let M (≤ η) F ⊂ Fl be the reduced closure of ∪ w∈W I\It w −1 (η) I (this is compatible with the notation in §3.1).
hence w(x * , w τ ) ≤ t w −1 (η) . Combining this with the last paragraph, we have w −1 w −1 h w 0 s w ≤ w(x * , w τ ) ≤ t w −1 (η) . Since ( w −1 w −1 h w 0 s w) = 3 = (t w −1 (η) ) − 1 (this is a consequence of a more general result in [LLHLMb], but can be checked directly using [LLHLM18, Table 1]), we see that w(x * , w τ ) = t w −1 (η) or w −1 w −1 h w 0 s w. If w(x * , w τ ) = t w −1 (η) , then (y w −1 s w, ( w, y w −1 (0))) ∈ SP (x * ) (this is represented by the red and blue parts in Figure 1). We claim that if w(x * , w τ ) = w −1 w −1 h w 0 s w, then x * ∈ C ( sw,y sw −1 (0)) (y * ) (this is represented by the arrows in Figure 1). It suffices to show that Using (1)    We illustrate the dichotomy given by the last paragraph of the proof of Proposition 4.1.8. We represent the data (y, ( w, ω)) ∈ SP (x * ) by the alcove labeled by y and the dot (resp. circle) at ω ∈ X * (T )/X 0 (T ) when w·C 0 is lower (resp. upper) alcove (thus the left picture is the case where w · C 0 is upper alcove, and the right picture the case where w · C 0 is lower alcove). The starting pair (y, ( w, y w −1 (0))) ∈ SP (x * ) is given by the red triangle (with vertexes labeled by the set W obv (y * )) and the source of the arrow (labeled by the obvious weight ( w, y w −1 (0)) ∈ W obv (x * )). The dotted triangle represents the possible new specialization, while the tip of the arrow represents the new obvious weight.
Suppose now that w · C 0 is the upper p-restricted alcove. For z ≤ w * w * 0 , either z * ∈ W w * , or z * ∈ W w , where w is the unique element in W 1 such that w < w. In particular w · C 0 = C 0 . There are two cases: • If x * ∈ C ( w,ω) ( zt ω * ) for some z * ∈ W w * , we get (t ω z * , ( w, ω)) ∈ SP (x * ) as above.
. Repeating our arguments with ( w, ω) replaced by ( w , ω), we are also done in this case.
Remark 4.1.12. The sets in the second part of Lemma 4.1.11 are the minimal sets containing the corresponding sets in the first part closed under changing a 0 in the second argument to a 1. Since the set in the second part are obtained by taking intersections ∩ y∈S(x * ) W g 2-deep (y * ) which are closed under this operation, these sets are a natural upper bound for W g 2-deep (x * ). Proof. We will illustrate the proof with various figures, all of which follow the same graphic conventions as in Figure 1.
Recall that we have canonical isomorphisms W /W a ∼ → X * (Z) and π 0 (Fl) ∼ → X * (Z). In this proof we will choose various λ ∈ X * (T ) with the property that the image of t λ in X * (Z) is the same as the image of x * , and use (2.2) to identify ( W 1 × (X * (T ) ∩ C 0 + η) λ−η| Z )/ ∼} and Λ λ W × A, since the latter set is more convenient to work with here.
By Corollary 4.1.10, one obtains the elements of W obv (x * ) by repeatedly applying the process described in Proposition 4.1.8 which we call a simple walk. We use the following two basic facts repeatedly.
• Next, if s 1 and s 2 ∈ W denote the simple reflections and x * is generic in I\Is 1 s 2 Is 2 s 1 (t λ w) * ∩ I\Is 2 s 1 Is 1 s 2 (t λ w) * , then t λ w, t λ ww 0 ∈ S(x * ) so that 7. This rules out (6). One can furthermore check that the image of W obv (x * ) under (2.2) contains (4) so that out of the six possibilities it must equal (4).
• If x * is generic in I\Iw 0 I(t λ w) * , then W g 2-deep (x * ) only has one element, namely (1). • If x * is generic in an upper alcove component, then W g 2-deep (x * ) only has one element corresponding to (2).
Proof. We explain how the bounds on W obv (x * ) and Table 3 can be used to determine W g 2-deep (x * ). Let x * ∈ Fl ∇ 0 2-deep be as in the statement of the theorem, and let λ and w be as in Lemma 4.1.11. Define Σ g (x * ) to be the image of W g 2-deep (x * ) under (2.2). Table 3 (with s j in the notation of loc. cit. taken to be w in this proof) implies that the number of irreducible components of the completion of U ( z, η, ∇ w −1 (µ+η) ) F at an F-point is never three (and that each irreducible component is smooth). Theorem 3.3.2 then implies that #Σ g (x * ) ∩ t ν s(Σ 0 ) = 3 for all t ν s ∈ W a . (The relevant type τ in Theorem 3.3.2 is 4-generic since w(y * 0 , w(τ )) ≤ w(x * , w(τ )) ∈ Adm(η) by Lemma 4.1.7(1) and y 0 (0) ∈ C 6-deep .) This is the key fact that we will use in our analysis of Σ g (x * ).
Finally, we show that every possibility arises. Since Lemma 4.1.11 showed that every one of the six possibilities arises, we only need to show that the two possibilities in case (1 ) arise. The case w{(0, 0)} arises when x * is generic on a lower alcove component so that W g 2-deep (x * ) only has one element corresponding to this component. The case w{(0, 0), (0, 1)} arises when x * is generic in the intersection of the two components corresponding to w{(0, 0), (0, 1)}. Indeed, this intersection is two-dimensional so as long as x * is not in cases (5 ) or (6 ) which are of dimensions one and zero, respectively, the case w{(0, 0), (0, 1)} must apply.
Remark 4.1.14. The notions in this section extend to the case of products: if x * = (x * i ) i∈J ∈ (Fl ∇ 0 2-deep ) J , we let W g 2-deep (x * ) and W obv (x * ) be the subsets i∈J W g 2-deep (x * i ) and i∈J W obv (x * i ) of ( W 1 × (X * (T ) ∩ C 0 + η))/ ∼, respectively. The natural analogues of Theorems 4.1.2 and 4.1.13 generalize to this setting.
Definition 4.2.2. We say that a Galois representation ρ : G K → GL 3 (F) is m-generic if the tame inertial F-type ρ ss | I K is m-generic and ρ has an m-generic specialization.
(2) It is shown (in greater generality under a suitable genericity assumption) in [LLHLMb] that ρ ss | I K ∈ S(ρ) so that the requirement that ρ has an m-generic specialization in Definition 4.2.2 is superfluous when m is sufficiently large.
We now recall the setting of Theorem 3.3.2. In particular, we have a pair (σ, ζ) which corresponds to a lowest alcove presentation ( w, ω) of a Serre weight σ. Given the auxiliary choice of an appropriate tame inertial type τ (and letting (s, µ) be the compatible lowest alcove presentation), we have the diagram: In particular the composite of the middle column gives a map C ζ σ → Φ-Modé t,n K,F which does not depend on τ and which factors through X K,3 . Note also that C ζ σ is a subvariety of ( Fl ∇ 0 ) J , and that the rightmost vertical arrow factors through (Fl ∇ 0 ) J . The following Proposition relates the Galois theoretic notions in §3.3 with the geometric notions in §4.1 (or rather, its product version as in Remark 4.1.14): Proposition 4.2.4. Let ρ ∈ X K,3 (F) be 6-generic and ( w, ω) be a lowest alcove presentation of an element of W g gen (ρ) compatible with a 6-generic lowest alcove presentation of ρ ss . If Proof. This follows from Theorem 3.3.2, applied to suitably chosen auxiliary 4-generic types τ containing F ( w,ω) .
Remark 4.2.6. By the proof of Theorem 4.2.5 (and Lemma 4.1.11), if ρ is 6-generic and has an m-generic specialization, then every specialization is (m − 4)-generic.
If ρ ∈ C σ for a generic weight σ, then the same argument above shows that ρ ∈ C σ .
Proof. This follows from Theorem 4.2.7 since there is a contracting T ∨ F -cocharacter for each translated Schubert cell (see [LLHLMa,Lemma 3.4.7]).

Results for patching functors
We start in §5.1 by recalling the formalism of weak (minimal) patching functors and we prove abstract versions of Serre weight conjectures assuming the modularity of an obvious weight (see Propositions 5. 1.10, 5.1.11 below). This assumption is removed in Section 5.2 if the weak patching functor comes from an arithmetic module. In §5.3, we prove results on cyclicity of patching functors arising from arithmetic modules and we finally give global applications of the above results in 5.4. 5.1. Patching functors and Serre weights. We recall the setup and the basic definitions for weak minimal patching functors. Recall from §1.5 that we write the finiteétale Z p -algebra O p as the product v∈Sp O v , where S p is a finite set and for each v ∈ S p , O v is the ring of integers in a finite unramified extension F + v of Q p , and that L G denotes the Langlands dual group of G 0 def = Res Op/Zp (GL 3/O p ). Following §2.1.9, an F-valued L-homomorphism ρ : G Qp → L G(F) (resp. a tame inertial L-parameter τ : and R p is a (nonzero) complete local Noetherian equidimensional flat O-algebra with residue field F such that each irreducible component of Spec R p and of Spec R p is geometrically irreducible (we remind the reader that M denotes M ⊗ O F for any O-module M ). We suppress the dependence on R p below. For a Weil-Deligne inertial L-parameter τ , let Let X ∞ , X ∞ (τ ), and X ∞ (τ ) be Spec R ∞ , Spec R ∞ (τ ), and Spec R ∞ (τ ) respectively. Let Mod(X ∞ ) be the category of coherent sheaves over X (1) M ∞ (σ • (τ )) is either zero or a maximal Cohen-Macaulay sheaf on X ∞ (τ ); and (2) for all σ ∈ JH(σ • (τ )), M ∞ (σ) is a maximal Cohen-Macaulay sheaf on X ∞ (τ ) (or is 0).
(3) Suppose σ • is an O-lattice in a principal series representation R 1 (µ). Then M ∞ (σ • ) is supported on the potentially semistable locus of type (η, τ (1, µ)) in X ∞ . We say that a weak patching functor M ∞ is minimal if R p is formally smooth over O and whenever τ is an inertial L-parameter, M ∞ (σ • (τ ))[p −1 ], which is locally free over (the regular scheme) Spec R ∞ (τ )[p −1 ], has rank at most one on each connected component.
Remark 5.1.2. The above definition of weak patching functor is slightly weaker than that in [LLHLMa, Definition 6.2.1] and closer in spirit to that of [LLHL19, Definition 4.2.1]: the purpose of the third item is to eliminate non-regular Serre weights.
Let d be the (common) dimension of X ∞ (τ ) for any inertial L-parameter τ . If M is an R ∞module whose action factors through R ∞ (τ ) for some inertial L-parameter τ , let Z(M ) be the associated d-dimensional cycle. Note that Z(M ∞ (−)) is additive in exact sequences.
We now fix an L-homomorphism ρ : W Qp → L G(F) and a weak patching functor M ∞ . Let W (ρ) be the set of Serre weights σ such that M ∞ (σ) = 0.
For a Serre weight σ, let p(σ) be the prime ideal or unit ideal in R ρ corresponding to the pullback of the stack C σ to Spec R ρ . For an inertial L-parameter τ , let I(τ ) be the kernel of the surjection R ρ R η,τ ρ . Observe that if I(τ ) ⊂ p(σ) = 1, then p(σ) induces a minimal prime of R η,τ ρ , and all minimal primes arise this way.
Lemma 5.1.5. Suppose that τ is an inertial L-parameter corresponding to a collection of 4-generic tame inertial types (τ v ) v∈Sp . Then any minimal prime ideal of R ∞ (τ ) is of the form I(τ )R ∞ + pR ∞ for some minimal prime ideal p ⊂ R p .
Proof. Since R η,τ ρ is geometrically irreducible (its special fiber is reduced after arbitrary finite extension of F and hence is normal; see the proof of [LLHLM20, Lemma 3.5.4]), the first part follows from [BGHT11, Lemma 3.3(5)]. Similarly, any minimal prime of R ∞ (τ ) is of the form p(σ)R ∞ + pR ∞ , where p(σ) corresponds to a minimal prime of R η,τ ρ , and p is a minimal prime of R p .
If M is a nonzero finitely generated maximal Cohen-Macaulay R ∞ (τ )-module, then Z(M ) is at least the reduction of the cycle in Spec R ∞ (τ )[1/p] corresponding to a minimal prime of R ∞ (τ ). In particular, for any prime p(σ) of R ρ inducing a minimal prime of R η,τ ρ , Ann R∞(τ ) (M ) is contained in a prime induced by p(σ)R ∞ + pR ∞ for some minimal prime p of R p . Since R ∞ /(p(σ)R ∞ + pR ∞ ) ∼ = R ρ /p(σ) ⊗R p /p, (p(σ)R ∞ + pR ∞ ) ∩ R ρ = p(σ) by Lemma 5.1.6. We conclude that The reverse inclusion is clear.
Lemma 5.1.6. Let F be a field. If R and S are complete Noetherian local F-algebras with residue field F, then the natural map R → R ⊗ F S, r → r ⊗1 is an injection.
Proof. Let m S ⊂ S be the maximal ideal. The composition R → R ⊗ F S → R ⊗ F (S/m S ) ∼ = R ⊗ F F is the isomorphism given by r → r ⊗ 1. The result follows.
For the rest of the section, we assume that ρ is 8-generic. In particular, every element of S(ρ) is 4-generic by Remark 4.2.6.
In fact, τ is unique. We say that τ is minimal with respect to ρ and σ.
Proposition 5.1.11. Let ρ : W Qp → L G(F) be an 8-generic L-homomorphism and let M ∞ be a weak minimal patching functor. Assume that W obv (ρ) ∩ W (ρ) is nonempty. Then Z(M ∞ (σ)) is the irreducible or zero cycle corresponding to the prime or unit ideal p(σ)R ∞ . In particular, W (ρ) = W g (ρ).
Proof. Let τ be a 4-generic tame inertial type. Let C σ (ρ) be the irreducible or zero cycle corresponding to the ideal p(σ)R ∞ . Then where the first inequality follows from the fact that M ∞ is minimal (see [LLHLM18,Proposition 7.14]) and the second inequality follows from Proposition 5.1.10. However the first and last expression are equal by Theorem 3.3.2, which forces the inequalities to be equalities. We conclude that the result holds for all σ ∈ JH(σ(τ )) for a 4-generic tame inertial type τ . In particular, the result holds for all generic σ.

Arithmetic patched modules.
Let R ∞ be as in §5.1 and set F p is a non-zero O-module M ∞ with commuting actions of R ∞ and GL 3 (F p ) satisfying the following axioms: , is a weak patching functor; (3) the action of H where the map η ∞ is the map denoted by η in [CEG + 16, Theorem 4.1] except with r p normalized so that r p (π) = rec p (π ⊗ | det | (n−1)/2 ).
We say that an arithmetic Let I be the preimage of B 0 (F p ) under the reduction map G 0 (Z p ) → G 0 (F p ). Let I 1 be the (unique) pro-p Sylow subgroup of I. Let χ : I/I 1 → O × be a character. Let θ(χ) be ind If χ is regular i.e. χ = χ s implies s = 1 for s ∈ W (GL Sp 3 ), then θ(χ)[1/p] is absolutely irreducible.
Proof. Up to a unit, for each v ∈ S p , the image of η ∞ (U τ v,1 ,τv ) (mod ) ∈ F in (the completion of) the second column of Table 1 is a nonempty product of diagonal elements modulo v by [DL21, Corollary 3.7]. One can check that each of these diagonal elements modulo v is contained in each of the ideals in the final column corresponding to (0, 0), (ε 1 , 0), or (ε 2 , 0).
If M ∞ is minimal, then the last part now follows from Proposition 5.1.11. 5.3. Cyclicity for patching functors. In this section, we show that certain patched modules for tame types are locally free of rank one over the corresponding local deformation space. The argument follows closely that of [LLHLM20, §5.2]. Recall from §2.1.3 the irreducible smooth E-representation σ(τ ) attached to a tame inertial Lparameter τ . Given σ ∈ JH(σ(τ )) we write σ(τ ) σ for an O-lattice, unique up to homothety, in σ(τ ) with cosocle σ. For an L-parameter ρ : G Qp → L G(F), we write W g (ρ, τ ) for the intersection W g (ρ) ∩ JH(σ(τ )). Throughout this section, we fix an L-parameter ρ and a weak minimal patching functor M ∞ for ρ which comes from an arithmetic R ∞ [GL 3 (F p )]-module. The main result of this section is the following: Theorem 5.3.1. Suppose that ρ : G Qp → L G(F) is a 11-generic L-parameter arising from an F-point of X η,τ for tame inertial L-parameter τ (in particular, τ is 9-generic) and let z def = w * (ρ, τ ). Let F (λ) ∈ W g (ρ, τ ) be a Serre weight such that for all j ∈ J Then M ∞ (σ(τ ) F (λ) ) is a free R ∞ (τ )-module of rank 1.
The proof is similar to the case when ρ is semisimple ([LLHLM20, Theorem 5.1.1] with slightly weaker genericity assumptions), and we will indicate the necessary modifications. First, [LLHLM20, Theorem 5.1.1] relies on a structure theorem for lattices in generic Deligne-Lusztig representations of G 0 (F p ) (Theorem 4.1.9 in loc. cit.). The following proposition improves the genericity hypothesis of that result. We refer the reader to loc. cit. for unexplained notation or terminology.
Then the radical filtration of R σ is predicted by the extension graph with respect to σ, and the graph distance, the radical distance and the saturation distance from σ all coincide on Γ(R σ ).
Proof. First, the scheme-theoretic support of M ∞ (σ) is (nonempty and) generically reduced by Theorem 5.2.6 and hence reduced e.g. by the proof of [LLHLM20, Lemma 3.6.2]. It is then formally smooth by Table 3, and so M ∞ (σ) is free over its scheme-theoretic support by the Auslander-Buchsbaum-Serre theorem and the Auslander-Buchsbaum formula.
Lemma 5.3.5. Suppose that ( z j ) > 1 or Proof. This follows from the proof of [LLHLM20, Lemma 5.1.5] with the usual modifications. (In the reduction step in the first paragraph of the proof, one possibly changes τ and so possibly changes ρ sp . This only affects this proof.) References to [LLHLM20, Theorem 4.1.9] are replaced by Proposition 5.3.2 above, and references to [LLHLM20, Lemma 5.1.4] are replaced by references to Lemma 5.3.4.
Finally, we can and do choose V 2 in the final paragraph of the proof of [LLHLM20, Lemma 5.1.5] so that if (ε , 0) ∈ V 2 i in the notation of loc. cit., then (ε , 0) ∈ Σ g i . Indeed, ( z i ) ≤ 1 and [LLHLM18, §8] ensure that U ( z i , η i , ∇ s −1 i (µ i +η i ) ) F has at least 5 components, where τ = τ (s, µ + η), so that #Σ g i ≥ 6 and contains two of w −1 i ((0, 0), (ε 1 , 0), (ε 2 , 0)) by Theorem 4.2.5. Then by Theorem 5.2.6, we can apply [EGS15, Lemma 10.1.13] as described in loc. cit. Remark 5.3.6. There was a gap in the proof of [LLHLM20, Lemma 5.1.5]: in the proof of loc. cit. one needs to possibly change the type τ to an auxiliary type, which may cause a loss of 2 in the genericity. Since we need to apply [LLHLM20, Theorem 4.1.9] to this auxiliary type, one needs to increase the genericity assumption by 2 in [LLHLM20, Theorem 5.1.1].
Proof. This follows from the proof of [LLHLM20, Lemma 5.1.6] with the usual modifications and using Lemma 5.3.5 and Lemmas 5.3.11 and 5.3.12 below (completed at x). In the lemmas below we refer the reader to §6 for unexplained notation. These lemmas are algebraizations of [LLHLM20, Lemmas 3.6.12, 3.6.14, 3.6.16(3.19), 3.6.16 (3.17) and (3.18)]. Their proof follows verbatim in our setting by replacing R expl,∇ M, z and the ideals c (ω,a) of loc. cit. with U ( z, η j , ∇ s −1 j (µ j +η j ) ) F (with µ j ∈ X * (T ) 4-deep) and the ideals P (ω,a),αt 1 respectively. (The second displayed equation in the statement of the Lemma 5.3.12 is not covered by [LLHLM20, Lemma 3.6.16(3.17)], but the proof is analogous.) Alternatively, one observes that all the ideal equalities we need to verify can be checked after projecting U ( z j , η j , ∇ s −1 j (µ j +η j ) ) F to Fl, where there is a contracting T ∨ F -action with unique fixed point z j . Since all the ideals involved are T ∨ F -equivariant one only needs to check the equalities after completion at z j , which are exactly the results of [LLHLM20, §3.6.3].
The scheme X ∞ (τ ) is normal and Cohen-Macaulay by Theorem 3.3.3 and Remark 3.3.4. We let Z ⊂ X ∞ (τ ) be the locus of points lying on two irreducible components of the special fiber of X ∞ (τ ) (in particular Z ⊂ X ∞ (τ ) has codimension at least two) and be the natural open immersion.
where p(θ) is the minimal prime ideal of (R τ ρ ) F corresponding to θ via Theorem 3.3.2 and localization at ρ ∈ X η,τ F . Proof. The proof follows verbatim the argument of [LLHLM20, Lemmas 5.2.1 and 5.2.2, Theorem 5.2.3] after replacing occurrences of W ? (ρ S ) in loc. cit. with W g (ρ).

Global applications.
In this section, we deduce global applications of Theorems 5.2.6, 5.3.1, and 5.3.13 generalizing results of [LLHLM20, §5.3] in the tamely ramified setting. Let F + be a totally real field, S p the set of places of F + dividing p, and F/F + a CM extension. We assume that all places of S p are unramified over Q and split in F . We start with the following modularity lifting result.
Theorem 5.4.1. Let F/F + be a CM extension, and let r : G F → GL 3 (E) be a continuous representation such that • r is unramified at all but finitely many places; • r is potentially crystalline at places dividing p of type (η, τ ) where τ is a tame inertial type that admits a lowest alcove presentation (s, µ) with µ 4-deep in alcove C 0 ; • r c ∼ = r ∨ ε −2 ; • ζ p / ∈ F ker adr and r(G F (ζp) ) ⊂ GL 3 (F) is an adequate subgroup; and • r ∼ = r ι (π) for some π a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation of GL 3 (A F ) of weight 0 so that σ(τ ) is a K-type for π at places dividing p.
Proof. This follows from standard base change and Taylor We now suppose that F + = Q. Let O F + ,p def = O F + ⊗ Z Z p be the finiteétale Z p -algebra denoted O p in §5.1, 5.2. We fix an outer form G /F + of GL n which splits over F and is definite at all archimedean places of F + . There exists N ∈ N, a reductive model G of G defined over O F + [1/N ], and an isomorphism  [LLHLMa,(9.2)]. If U is unramified at places in P, then S(U, W ) has a natural T P -action. Let T P (U, W ) be the quotient of T P acting faithfully on S(U, W ).
Let G 3 be the group scheme over Z defined in [CHT08, §2.1]. We consider a continuous Galois representation r : G F + → G 3 (F) which is automorphic in the sense of [LLHLMa], i.e. for which there exists a maximal ideal m ⊆ T P (U, W ), for some level U and coefficients W satisfying det (1 − r(Frob w )X) = 2 j=0 (−1) j (N F/Q (w)) ( j 2 ) T (j) w X j mod m for all w ∈ P. Note that the collection (r| G F + v ) v∈Sp defines an F-valued L-parameter, which will be denoted as r p in what follows. For such r, we define as in [LLHLMa, Definition 9.1.1] the set W (r) of modular Serre weights for r.
Theorem 5.4.2. Let r : G F + → G 3 (F) be an automorphic Galois representation. Assume further that • r| G F (G F (ζp) ) is adequate; and • r p is 8-generic.
Then W (r) = W g (r p ).
Proof. The proof of [LLHLM20, Theorem 5.3.3] applies verbatim after replacing Theorem 3.5.2 of loc. cit. with Theorem 5.2.6 above.
5.4.1. Mod p multiplicity one. We continue using the setup from §5.4. We assume further that F/F + is unramified at all finite places. We now let S 0 denote the set of finite places of F + away from p where r ramifies and assume that every place of S 0 splits in F . For each v ∈ S 0 , with fixed lift v in F , we let τ v be the minimally ramified type in the sense of [CHT08, Definition 2.4.14] corresponding to r| G F v : G F v → GL 3 (F) and σ(τ v ) def = σ(τ v ) • ι v be the G(O F + v )-representation attached to it (where ι v is the localization at v of the isomorphism (5.3); σ(τ v ) is independent of the choice of v|v, cf. [LLHLM20, §5.3]). Fix an O-lattice W S 0 in ⊗ v∈S 0 σ(τ v ). We have the following mod p multiplicity one result.
Theorem 5.4.3. Let r : G F + → G 3 (F) be a continuous Galois representation such that r p is 11generic. Let τ and F (λ) ∈ W g (r p , τ ) be as in the statement of Theorem 5.3.1. Assume moreover that: • r : G F + → G 3 (F) is automorphic; • r| G F (G F (ζp) ) is adequate; and • the places at which r ramifies split in F .
is one-dimensional over F, where m is the maximal ideal in the Hecke algebra T P corresponding to r.
Proof. The proof of [LLHLM20, Theorem 5.3.4] applies verbatim after replacing the reference to [LLHLM20, Theorem 5.2.1] by a reference to Theorem 5.3.1 above.

5.4.2.
Breuil's lattice conjecture. We now consider an automorphic Galois representation r : G F → GL 3 (E) as in Theorem 5.4.1 which is minimally ramified, i.e. for any place v of F lying above some v ∈ S 0 , the Galois representation r| G F v is minimally ramified in the sense of [CHT08, Definition 2.4.14]. Let λ be the kernel of the system of Hecke eigenvalues α : T P → O associated to r, i.e. α satisfies det 1 − r ∨ (Frob w )X = 3 j=0 (−1) j (N F/Q (w)) ( j 2 ) α(T (j) w )X j for all w ∈ P. For U p ≤ G(A ∞,p F + ) and a finite O-module W with a continuous U p -action, let Theorem 5.4.4. Let r : G F → GL 3 (E) and τ be as in Theorem 5.4.1. Assume furthermore that r is minimally ramified and that the places at which r ramifies are split in F . Finally, assume that r p is 11-generic. Then the lattice σ(τ ) ∩ S(U p , W S 0 )[λ] ⊂ σ(τ ) ∩ S(U p , W S 0 )[λ] ⊗ O E depends only on r p .
Proof. The proof of [LLHLM20, Theorem 5.3.5] applies verbatim after replacing occurrences of W ? (r S ) in loc. cit. with W g (r p ) and the reference to [LLHLM20, Theorem 5.2.3] with a reference to Theorem 5.3.13.

Appendix: tables
In the following tables we write α, β and γ for the elements of W corresponding to (12), (23) and w 0 t (1,0,−1) respectively. Moreover, the image of 1 def = (1, 1, 1) ∈ X * (T ) in W is denoted as t 1 . We identify the elements above with matrices in GL 3 (Z((v))) via the embedding W → GL 3 (Z((v))) defined by α →  (v + p)c12 0 0 (v + p) 2 c *   Table 3 z j t −1 U ( z j , η j , ∇ s −1 j (µ j +η j ) ) F (ε j , a j ) ∈ Σ 0 P (ε j ,a j ), z j The table records data relevant to Theorem 3.3.2. The first column records the components of z. The second column records the coordinates of ( U ( z, η, ∇ τ,∞ ) F ) in terms of the universal matrix A (j) and the relations between its coefficients. Recall that in the statement of Theorem 3.3.2 the Serre weight σ is parametrized by (µ + η − λ + s(ε), a) ∈ Λ λ R × A. The ideal corresponding to the closed immersion P σ, z → U ( z, η, ∇ τ,∞ ) F is of the form f −1 j=0 P (ε j ,a j ), z j , where each P (ε j ,a j ), z j is a minimal prime ideal of O( U ( z j , η j , ∇ s −1 j (µ j +η j ) ) F ). The elements (ε j , a j ) ∈ Σ 0 are specified in the third column and the ideal P (ε j ,a j ), z j specified in the fourth column records. The structure constants that feature in the presentation are given by (a, b, c) ∈ F 3 p with (a, b, c) ≡ s −1 j (µ j + η j ) mod p.