On Mod P Local-Global Compatibility for Unramified GL by John Enns A

On mod p local-global compatibility for unramified GL3

John Enns

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Mathematics University of Toronto

On mod p local-global compatibility for unramiﬁed GL3

John Enns Doctor of Philosophy Department of Mathematics University of Toronto 2018

Let F/F + be a CM number ﬁeld and w|p a place of F lying over a place of F + that splits in F .

Ifr ¯ : GF → GLn(Fp) is a global Galois representation automorphic with respect to a deﬁnite + 0 unitary group over F then there is a naturally associated GLn(Fw)-representation H (¯r) over

0 Fp constructed using mod p automorphic forms, and it is hoped that H (¯r) will correspond to r¯| under a conjectural mod p local Langlands correspondence. The goal of the thesis is to GFw study how to recover the data ofr ¯| from H0(¯r) in certain cases. GFw

Assume that Fw is unramified and n = 3. Under certain technical hypotheses onr ¯, we give an explicit recipe to find all the data ofr ¯| inside the GL (F )-action on H0(¯r) when GFw 3 w r¯| is maximally nonsplit, Fontaine-Laffaille, generic, and has a unique Serre weight. This GFw generalizes work of Herzig, Le and Morra [HLM17] who found analogous results when Fw = Qp as well as work of Breuil and Diamond [BD14] in the case of unramified GL2.

ii For Cathleen

iii Acknowledgements

I want to thank my advisor Florian for suggesting this problem and for his advice since the beginning stages of this work. I am grateful for having had the opportunity to learn about some deep and diﬃcult areas of math under his supervision. I also thank him for helpful comments on an earlier version of this thesis. It would not have been possible for me to complete graduate school without the support of my family. Thank you Mom, Dad, William, and Noah. It’s been hard being away from home. Finally, to Cathleen: I can’t begin to describe the positive impact you’ve had on my life. Thank you for your unconditional love and support. This thesis is dedicated to you.

iv Contents

Acknowledgements ...... iv Table of contents ...... v

0 Introduction 1 0.1 Notation and conventions ...... 9

1 Integral p-adic Hodge theory 11 1.1 The local mod p Galois representation ...... 11 1.2 Fontaine-Laﬀaille modules and invariants ...... 12 1.2.1 Fontaine-Laﬀaille invariants ...... 14 1.3 Summary of categories used ...... 17

1.4 Lattices inside potentially semistable GK0 -representations ...... 18 1.4.1 Filtered isocrystals with monodromy and descent data ...... 18 1.4.2 Strongly divisible modules ...... 19 1.4.3 Breuil modules ...... 23 1.5 Kisin modules and ϕ-modules ...... 25 1.5.1 Etale´ ϕ-modules ...... 25 1.5.2 Kisin modules with descent data ...... 28 1.5.3 Kisin modules of Hodge type µ ...... 31 1.6 Shape computations ...... 33 1.7 Main results ...... 41

2 Modular representation theory of GL3 48 2.1 Representations of reductive groups ...... 49 2.2 Restriction of algebraic representations to ﬁnite subgroups ...... 50 2.3 Summary of main results ...... 55 2.4 Weyl modules and the distribution algebra ...... 59 2.5 Proof of Theorem 2.3.2 ...... 64 Proof of Theorem 2.3.2(1) ...... 64 Proof of Theorem 2.3.2(2) and (3), part I ...... 80 Proof of Theorem 2.3.2(3), part 2 ...... 85 2.6 Characteristic 0 principal series ...... 89

v 3 Mod p local-global compatibility 96 3.1 Deﬁnite unitary groups ...... 96 3.2 p-adic, mod p and classical automorphic forms ...... 97 3.3 Galois representations and local-global compatibility ...... 99 3.4 Serre weights and modularity ...... 101 3.5 Main result ...... 102 3.6 Freeness over the Hecke algebra ...... 105

Bibliography 109

vi Chapter 0

Introduction

This thesis is about relationships between automorphic forms and Galois representations. Specif- ically, it is motivated by the search for a mod p local Langlands correspondence, and how such a correspondence might be realized inside the cohomology of arithmetic manifolds attached to reductive groups over number ﬁelds. The words “local-global compatibility” in the title refer to this possibility. We begin by illustrating an example of local-global compatibility in the classical Langlands program, following [Bre12]. Let ` be a prime and K a ﬁnite extension of Q` with absolute Galois group GK and Weil group WK . The classical local Langlands correspondence for GLn posits the existence of a unique family of bijections

∼ recn,p :{irreducible admissible smooth representations of GLn(K) over Qp}/ = ∼ → {rank n Frobenius-semisimple Weil-Deligne representations of K over Qp}/ = (0.0.1) for each n characterized by the list of conditions found in the introduction to [HT01], in particular compatibility with local class ﬁeld theory. When ` 6= p, Grothendieck’s `-adic monodromy theorem gives an equivalence of categories between continuous Galois representations

GK → GLn(Qp) and bounded Weil-Deligne representations of K over Qp. Via recn,p this allows us to attach an irreducible admissible Qp-representation of GLn(K) to any continuous p-adic representation of GK after Frobenius semisimplifying. We say that a local-global compatibility occurs in any situation where a purely local correspondence, such as (0.0.1), arises in connection with a correspondence between global objects, typically the cohomology of a Shimura variety or more generally an arithmetic quotient of a locally symmetric space attached to a reductive group over a number ﬁeld. For example, the cohomology of modular curves, which are Shimura varieties attached to the reductive group

GL2/Q, exhibits local-global compatibility. We use this example to explain the idea. Let A denote the ring of ad`elesof Q. Associated with any suﬃciently small open compact ∞ subgroup U ≤ GL2(A ) there is an algebraic curve Y (U) (the classical modular curve of level U)

1 Chapter 0. Introduction 2

∞ defined over Q whose space of complex points is isomorphic to GL2(Q)\(C − R) × GL2(A )/U. We may consider the p-adic étalecohomology1 H1(Y (U) , ), which is a finite-dimensional Q Qp Qp-vector space with a continuous action of GQ. As U varies, there are corresponding maps between the curves Y (U) and it makes sense to consider all levels U at once by defining 1 1 ∞ H = lim H (Y (U)¯ , ). This space has a natural smooth action of GL ( ) commut- −→U Q Qp 2 A ing with its GQ-action. One possible strong statement of classical local-global compatibility is the following. Suppose that r : GQ → GL2(Qp) is a continuous global Galois representation such that Hom (r, H1) 6= 0 (this assumption means that r is the p-adic representation asso- GQ ciated with a classical weight 2 modular newform by the construction of Deligne). Then as a ∞ 2 GL2(A )-representation, we have O Hom (r, H1) =∼ π ⊗ 0 rec (r| ) (0.0.2) GQ p 2,p G` `6=p

Here G` is a decomposition group for the prime ` inside GQ and πp denotes a certain smooth representation of GL2(Qp) over Qp that we will return to in a moment. The key point here is that we observe each local Langlands correspondence occurring inside the cohomology of the tower of Shimura varieties (Y (U))U for each prime not equal to p. But as one sees in (0.0.2), something different happens when ` = p, that is, when the field of coefficients and the local Galois representation are both p-adic. In fact there is a recipe for constructing πp from r|Gp , but the procedure loses a great deal of information. In other words, it is not possible to reconstruct r|Gp from πp. This is because there can exist no analogue to (0.0.1) that attaches smooth representations of GLn(K) to continuous p-adic representations of

GK when ` = p in any way that doesn’t lose information. The issue is that since the topologies of GK and Qp are compatible, there is a much richer supply of local Galois representations, but on the other hand smooth representations ignore the topology of the coefficient field by definition. The problem of how to recover the local Galois representation at p in (0.0.2) is one possible starting point for the p-adic (and mod p) Langlands correspondences. In order to get a p-adic local Langlands correspondence the idea is to replace smooth representations of GLn(K) with something that sees the p-adic topology, namely Banach space representations of GLn(K) over (a finite subextension of) Qp. The attempt to prove results in this direction has been a point of heavy research activity for some time, and the only cases that have been settled are n = 1 (which is as usual a consequence of local class field theory) and n = 2,K = Qp. See [Bre10] for an overview. Indeed, it is now known that there is a func- torial link between categories of unitary Banach representations of GL2(Qp) and continuous p-adic representations of GQp inducing a bijection between irreducible objects on both sides (see [CDPs14] for a precise statement). Analogues of these statements for n > 2 or K 6= Qp are highly sought after.

1Technically one should have parabolic cohomology in (0.0.2). See [Eme06] for more precise statements. 2Again, we actually need to use the “Tate normalization” of (0.0.1) here, and we refer to [Eme06] for the precise version. Chapter 0. Introduction 3

In [Eme11], Emerton proved the p-adic analogue of (0.0.2). Of course, one cannot use simply the p-adic ´etalecohomology exactly as above since we just saw this doesn’t work. Emerton has instead introduced a theory of p-adically completed cohomology where H1 is replaced by a kind 1 of completion Hb that is not a smooth representation of GL2(Qp), but a Banach representation. Then the analogue of (0.0.2) holds. See the introduction to [Eme11] for the precise statement, or below for the mod p version. Since Banach representations admit a canonical notion of reduction mod p (by reducing their unit ball), it is hoped that in addition to a p-adic correspondence there will also be a mod p local Langlands correspondence compatible with reducing lattices mod p on the Galois side. Precisely, the original hope for the mod p local Langlands correspondence was the existence of a map

∼ {continuous representations GK → GLn(Fp)}/ = ∼ ,→ {admissible representations of GLn(K) over Fp}/ = . (0.0.3) which we denoteρ ¯ 7→ Πp(¯ρ) compatible with the (conjectural) p-adic local Langlands correspondence and hopefully with the mod p cohomology of Shimura varieties. In fact, the evidence we currently have suggests that a 1-1 correspondence of this form may be too much to ask in general, and that we should instead look for a family of representations Πp(¯ρ). As with the p-adic case, the existence of a 1-1 correspondence is known in the case of

GL2(Qp) (it is actually easier than the p-adic result and was known before), and the analogue of (0.0.2) holds as a corollary to Emerton’s p-adic result. We will actually state a weaker version, since it is analogous to the situation considered in this thesis. Fix an open compact subgroup p ∞,p U ≤ GL2(A ) (a tame level). Consider the mod p étalecohomology of the modular curve 1 p 1 p with infinite level at p: H¯ (U ) := lim H (Y (UpU ) , p) where Up runs over compact open −→Up Q F subgroups of GL2(Qp). By its construction this object has a smooth action of GL2(Qp) commuting with its GQ-action. Suppose thatr ¯ : GQ → GL2(Fp) is a continuous irreducible modular representation satisfying some technical assumptions. Then as a smooth Fp-representation of GL2(Qp) we have p Hom (¯r, H¯ 1(U p)) =∼ Π (¯r| )d(U ) (0.0.4) GQ p Gp p where d(U ) ∈ N is some multiplicity and Πp is the previously established mod p local Langlands correspondence for GL2(Qp) [Eme11]. The reason that no mod p (or p-adic) local Langlands correspondence has yet been established for groups other than GL2(Qp) is that the mod p representation theory of any other group appears to be infinitely more difficult than for GL2(Qp). For example, there are infinitely many irreducible admissible Fp-representations of GL2(K) for any field K 6= Qp with a fixed central character (whereas for GL2(Qp) there are only finitely many). Breuil and Paˇsk¯unas in [BP12] made a detailed study of smooth Fp-representations of GL2(Qpf ). Their results made Chapter 0. Introduction 4

it clear that one could not expect a 1-1 correspondence in (0.0.3) for GL2(Qpf ) if f > 1, and in light of this the same is expected for any group other than GL2(Qp). The main problem in the classiﬁcation of irreducible smooth Fp-representations of GLn(K) is in understanding the supersingular representations (which are analogous to the supercuspidals in the characteristic 0 case). Despite recent advances in classifying the irreducible admissible representations of

GLn(K) (and even more general reductive groups) in terms of supersingular representations ([AHHV17]), the classification of the supersingular representations themselves does not appear to be forthcoming. Since none of these difficulties appear on the Galois side of (0.0.3) (it is easy to classify representations GK → GLn(Fp) for any p-adic field K), these discoveries have caused some uncertainty about the nature of the conjectural mod p local Langlands correspondence. Nev- ertheless, we expect some kind of local-global compatibility and it has been the viewpoint of a number of papers (and is the viewpoint of this thesis) that the shape of the hypothetical Πp(¯ρ) for groups other than GL2(Qp) might be studied via analogues of the globally defined space Hom (¯r, H¯ 1(U p)). In particular, one would like to determine whether (an analogue of) this GQ space depends only on the local Galois representationr ¯|Gp . Unfortunately this seems to be too much to ask right now, given how poorly smooth mod p representations are understood.

This work

This thesis studies local-global compatibility for the group GL3(Qpf ). It is customary in the burgeoning literature of the area to use the global set-up of a definite unitary group G defined over a totally imaginary CM field F/F +. This is because the associated arithmetic manifolds are 0-dimensional and hence admit cohomology only in degree 0, which allows for numerous simplifications. On the other hand, one knows how to attach Galois representations to automorphic representations of G satisfying (classical) local-global compatibility and the Taylor-Wiles patching machinery is well-developed in this context. So these groups form the simplest possible global set-up in which to test local conjectures. We now describe our results. Let F be a number field which is a quadratic extension of a totally real field F + such that p is unramified in F + and each place of F + above p splits in + F . Fix a definite unitary group G defined over F that becomes isomorphic to GL3 over F and choose places w|v of F/F + unramified of degree f over p. We begin with an irreducible automorphic global Galois representationr ¯ : GF → GL3(Fp). Here automorphic means thatr ¯ is the reduction mod p of a p-adic Galois representation attached to an automorphic representation of G. We also assume thatr ¯ satisfies some technical hypotheses needed for the Taylor-Wiles v ∞,v method. Let U ≤ G(AF + ) be a fixed well-chosen compact open subgroup which is in particular unramified at all places of F + inert in F , and consider

H0(U v, ) := lim H0(U U v, ), Fp −→ v Fp Uv Chapter 0. Introduction 5

+ 0 where Uv runs over compact open subgroups of G(Fv ) and H (U, Fp) denotes the 0th coho- 3 + ∞ 0 v mology of the (ﬁnite) arithmetic quotient G(F )\G(AF + )/U. By its construction H (U , Fp) + ∼ has a smooth admissible action of G(Fv ) = GL3(Fw) = GL3(Qpf ). There is also a commuting + action of a Hecke algebra T consisting of Hecke operators at all places of F not dividing p that split in F . Let mr¯ denote the maximal ideal of T corresponding tor ¯ via the Satake isomorphism. Since there is no Galois action on the cohomology group we have chosen, the natural analogue of ther ¯-isotypic subspace that was considered in (0.0.4) above is the mr¯-torsion sub- 0 v space H (U , Fp)[mr¯]. This is a smooth admissible Fp-representation of GL3(Fw), nonzero by the assumption of automorphy. By analogy with the local-global compatibility for GL2(Qp) in (0.0.4) we expect this representation to be a number of copies of Π (¯r| ), where the number p GFw of copies depends only on U v. 0 v In particular, one might optimistically hope that the smooth GL3(Fw)-representation H (U , Fp)[mr¯] depends only on the local Galois representationr ¯| . Unfortunately, this is out of reach at GFw the moment. We are instead concerned with whether the passage fromr ¯| to H0(U v, )[m ] GFw Fp r¯ loses information. Our main result (Theorem 3.5.6 + Theorem 3.6.1) is that it does not. A somewhat vague version of this result is Theorem 0.0.5 below; see Chapter 3 for the precise version. Unfamiliar terminology will be described in the following paragraphs.

Theorem 0.0.5. In the above setting, let r¯| be upper-triangular and maximally nonsplit. GFw Assume that it is Fontaine-Laﬀaille and suﬃciently generic, and has a single modular Serre weight. Then it is possible to recover the data of the local Galois representation r¯| : G → GFw Qpf 0 v GL3(Fp) from the GL3(Qpf )-action on H (U , Fp)[mr¯] via an explicit recipe.

In order to explain the meaning of the hypotheses in the theorem we need some additional background. Let us begin with the concept of a Serre weight of a representationr ¯. If K is any p-adic ﬁeld, a Serre weight of K is deﬁned to be an (isomorphism class of an) irreducible

Fp-representation of GLn(OK ). All Serre weights factor through GLn(k) where k is the residue field of K, so there are finitely many of them. Going back to the setting of (0.0.3), we see that if the correspondence Πp exists, then associated withρ ¯ there should be the set of Serre weights given by the set of irreducible constituents of the GLn(OK )-socle of Πp(¯ρ) (if Πp(¯ρ) is a family, this socle should be constant throughout). In the global setting described above, the Serre weights ofr ¯| should therefore arise as GFw 0 v the irreducible constituents of the GL3(OFw )-socle of H (U , Fp)[mr¯], so we define the set of modular Serre weights ofr ¯ to be this set and call it W (¯r| ) even though a priori it depends GFw on all ofr ¯ (technically, one should take into account all places above p when defining this set but we ignore this point in the Introduction; see Definition 3.4.2). Conjectures about the modular Serre weights of an automorphic Galois representation have played an important role in the development of the mod p Langlands program, since the GLn(OK )-socle is the most accessible piece of Π (¯ρ). In particular, whenr ¯| is semisimple, [Her09] gave a conjectural p GFw

3Technically we take coefficients in a certain local system V 0, but this is irrelevant for the Introduction. Chapter 0. Introduction 6 set W ?(¯r| ) of regular weights, this time truly depending only onr ¯| , and in fact only GFw GFw on the restriction to inertia, that should coincide with the set of regular weights in W (¯r| ). GFw Here regularity is a genericity condition. Ifr ¯| is sufficiently generic then it is known that GFw all weights occurring in W (¯r| ) are regular ([Enn17]) so this point is not so important for us. GFw We also refer to [GHS] for an overview of various sets of Serre weights that can be attached to a local mod p Galois representation. There has been a great deal of progress on comparing the sets W (¯r| ) and W ?(¯r| ) GFw GFw in the setting of definite unitary groups ([EGH13, LLHL17]) whenr ¯| is semisimple. On GFw the other hand, whenr ¯| is not semisimple, the conjecture of [Her09] does not apply and GFw there isn’t yet a concrete way to predict the set of modular weights. Nevertheless, one expects ? ss W (¯r|G ) ⊆ W (¯r| ) in all cases and that there will be a geometric interpretation of how the Fw GFw weight set behaves ([GHS], Section 6). Moreover, forthcoming work of Le, Le Hung, Levin and Morra [LLHLM] should give the complete picture on the modular Serre weights in the setting considered in this paper. Going back to the statement of Theorem 0.0.5, it is expected that whenr ¯| is semisimple GFw the data ofr ¯| may be read off from its (still conjectural) set of modular Serre weights, that IFw 0 v is, off from the GL3(OFw )-socle of H (U , Fp)[mr¯], and it should be possible to extend this to r¯| by considering the natural action of Hecke operators on this socle. But whenr ¯| is GFw GFw not semisimple, the GL (O )-socle will not contain all of the data ofr ¯| . For example, we 3 Fw GFw assume thatr ¯| is upper-triangular and maximally nonsplit, which is to say GFw

  ψ2 ∗ ∗ r¯| ∼  ψ ∗  (0.0.6) GFw  1  ψ0

× 1 for some characters ψi : GFw → Fp where the two off-diagonal Ext classes are nonzero. Then one expects to see the data of the diagonal characters ψ2, ψ1, ψ0 in the GL3(OFw )-socle of 0 v H (U , Fp)[mr¯] with its Hecke action, but this socle cannot contain the data of the extension classes. So the question to answer in Theorem 0.0.5 is how to recover these Ext1 classes from 0 v the GL3(Fw)-action on H (U , Fp)[mr¯]. To do so is then to investigate the structure of the cohomology representation H0(U v, )[m ] (hence Π(¯r| )) beyond its GL (O )-socle. Fp r¯ GFw 3 Fw The remaining hypotheses in Theorem 0.0.5 are there to simplify the question. The strongest of them is the assumption thatr ¯| is Fontaine-Laffaille. This condition is crucial for our GFw method since we use Fontaine-Laffaille theory to normalize the extension classes; see the strategy of proof below. The hypothesis of genericity is a condition on the inertial weights ofr ¯| GFw (Definition 1.1.2) and is critical in the sense that were it absent nearly everything in the proof would go wrong, but not so important in the sense that for a fixed degree of genericity the “proportion” of local mod p Galois representations that are generic approaches 1 as p → ∞. But also note that the assumption of genericity does imply a lower bound on p. Finally, it was first observed (almost, up to a shadow weight, and for f = 1) in [HLM17] that Chapter 0. Introduction 7 most upper-triangular maximally nonsplit Galois representations had a unique modular Serre weight (see Remark 3.5.5). This is confirmed for all f, based on discussions with the authors of the forthcoming [LLHLM], so the final assumption in Theorem 0.0.5 again represents the most generic case. We should add that technically we make a stronger hypothesis in specifying the precise Serre weight that occurs, but in some cases we can prove that this stronger hypothesis is actually equivalent. See Remark 3.5.4.

Strategy of proof

We now briefly explain the explicit recipe in Theorem 0.0.5. The case f = 1 was done by Herzig, Le and Morra in [HLM17], and our strategy follows theirs. The first step is local - using Fontaine- × Laffaille theory we define parameters FLτ ∈ Fp which, along with the diagonal characters in (0.0.6) determine the isomorphism class ofr ¯| (Definition 1.2.14). These Fontaine-Laffaille GFw parameters are indexed by a collection of inertial types τ (principal series types). 0 For each type τ, we define explicit group algebra operators Sτ ,Sτ ∈ Fp[GL3(kw)] and our main result (Theorem 3.5.6) shows (roughly) that there is an equality of injective operators

0 Sτ Π = κτ FLτ Sτ (0.0.7)

0 v × between explicit Iwahori eigenspaces of H (U , Fp)[mr¯] where κτ is an explicit constant in Fp , and Π ∈ GL3(Fw) is a normalizer of the Iwahori subgroup. 0 v The main idea of the proof is to relate the mod p cohomology H (U , Fp)[mr¯] directly to spaces of classical automorphic forms on G in characteristic 0, which is easy since cohomology occurs only in degree 0. It is known how to attach Galois representations to automorphic representations of G (Theorem 3.3.1) satisfying classical local-global compatibility (0.0.1) linking 0 the local Galois and automorphic representations. We study how lifts of the operators Sτ ,Sτ behave in characteristic 0, where we can relate the local Galois and automorphic sides, and then study how both sides reduce mod p. The automorphic representations which contribute to the Iwahori eigenspaces in question are principal series at w by design, and their Galois representations at w are potentially crystalline lifts ofr ¯| of parallel Hodge type µ = (2, 1, 0) and inertial type τ. Accordingly, the first GFw chapter of the thesis studies potentially crystalline lifts ofr ¯| using p-adic Hodge theory. GFw The main result (Theorem 1.7.1) relates the Fontaine-Laffaille invariants ofr ¯| with certain GFw Frobenius eigenvalues on the weakly admissible modules associated with the lifts. We make use of the recent work of [LLHLM18a] giving explicit descriptions of potentially crystalline deformation rings of Hodge type µ. The subject of Chapter 2 is the local automorphic side. The results in characteristic 0 are easier compared to the Galois side (Section 2.6), but the most technical part of this thesis is to show the injectivity of the operators in (0.0.7). This boils down to a statement about 0 nonvanishing of Sτ and Sτ on certain quotients of principal series representations for the finite Chapter 0. Introduction 8

group GL3(kw). The proof of this result (Theorem 2.3.2) is inspired by [LLHLM18b] - by embedding the principal series into an algebraic representation of Qf−1 GL , we may use j=0 3/Fp Lie theory and known results about the structure of Weyl modules to do explicit computations that show nonvanishing. We remark that one technical improvement in this thesis to previous work consists of using nonsplit algebraic groups, which makes the aforementioned computations feasible. We also spend some time in this chapter generalizing a result of Pillen about restricting algebraic representations to finite subgroups to the nonsplit case (Section 2.2), although we don’t end up needing this. The proof of (0.0.7) in Chapter 3 uses a certain assumption about freeness over a Hecke algebra. We conclude the thesis by showing that this hypothesis is satisfied when U v is well- chosen andr ¯ satisfies some technical conditions using Taylor-Wiles patching and the Diamond- Fujiwara method (Theorem 3.6.1). The proof of this result is a very mild generalization of the analogous result in [HLM17]; in fact it is easier in our case because of the assumption about a single Serre weight. We remark that on both the Galois and automorphic sides, the particular choice of types τ is very important and it is not yet clear why these types work whereas others don’t.

Relation to previous work

As mentioned above, the f = 1 case is due to [HLM17]. They work in a similar setting to us, but without the assumption of a single Serre weight. It would be possible to study the Serre weights ofr ¯| when f > 1 using generalizations of their methods, and in fact we use our GFw group operators to prove a mild result on the Serre weights ofr ¯| in our main result Theorem GFw 3.5.6, but we avoided a more serious investigation because the Serre weights ofr ¯ are the subject of the forthcoming [LLHLM]. Our method follows that of [HLM17], except we use the new technology of [LLHLM18a] on 0 the Galois side, and our technique for dealing with the operators Sτ ,Sτ on the local automorphic side is new. When f = 1, our operators contain those considered in [HLM17] (see Remark 2.3.9). Similar operators also appeared in work of Breuil and Diamond [BD14] for the group

GL2(Qpf ). It may be interesting to note that some of our operators (the ones used to recover the Fontaine-Laffaille invariants corresponding to the 2-dimensional subquotients ofr ¯| ) have GFw a strong affinity with theirs, though the context is different and there is no obvious relation between the two settings. The case of GL ( ) whenr ¯| is wildly ramified with a 2-dimensional irreducible subquo- 3 Qp GFw tient was studied by similar means in [LMP16]. Finally, [HLM17] has recently been generalized in a different direction to GLn(Qp) by Park and Qian in [PQ17], who again use a similar method. Chapter 0. Introduction 9

0.1 Notation and conventions

If L is any field, GL denotes the absolute Galois group Gal(L/L) with respect to a fixed choice of algebraic closure L. If L is a p-adic field then WL ≤ GL denotes the Weil group of L and ∞ IL ≤ WL the inertia group. We write AL for the ring of adèlesof L and AL for the ring of finite adèles.

Throughout the thesis, K0 denotes an unramified p-adic field of residue degree f ≥ 1, and k its residue field. We write q = pf . The symbol E denotes a p-adic field serving as the coefficient

field. It is always assumed to be finite over Qp but large enough to contain the image of all embeddings HomQp (K0, Qp). The residue field of E is called F. Let K = K0(π) be the Galois totally ramified extension of K0 of degree e := q −1 generated e by (any) eth root π of −p. Write E(u) = u + p ∈ K0[u] for the minimal polynomial of π. With × this choice of π there is associated a fundamental character ωK0,π : GK0 → k (of niveau f) sending g to g(π)/π mod p. Its restriction to IK0 does not depend on the choice of π or even of the uniformizer −p, so we denote it by simply ω . It is equal to Art−1| mod p, where K0 K0 IK0 Art : K× → Gab denotes the reciprocity map of local class field theory normalized in such a 0 K0 way that it takes uniformizers to geometric Frobenius automorphisms. 1/p We make a fixed choice of field embedding τ0 : k ,→ F, and define inductively τj+1 = τj . × We then define the character ωj : GK0 → F to be τj ◦ωK0,π. The notation is cyclic, so ωf = ω0. We will generally prefer to use ωf instead of ω0. Note that with this choice of ordering for our field embeddings, we have f−1 Pf−1 j Y aj j=0 af−j p ωj = ωf j=0

Pf−1 j for any integers aj ∈ Z. We will often consider p-adic expansions of the form j=0 af−jp ; the coeﬃcients are always considered to be indexed modulo f. In order to simplify notation, all sums without any explicit limits written go from 0 to f − 1.

We often consider modules M over rings like k⊗Fp F. Recall that since F is suﬃciently large, ∼ Qf−1 k ⊗ = as an -algebra. Letting j ∈ k ⊗ denote the idempotent corresponding to Fp F j=0 F F Fp F Qf−1 j j the jth factor, we have M = j=0 M where M = jM is the submodule {m ∈ M | (α⊗β)m = (1⊗τj(α)β ∀α ∈ k, β ∈ F}. It will often be the case that M is equipped with a φ⊗1-semilinear map ϕM : M → M, where φ denotes the absolute Frobenius of k (we also use φ for the induced j j+1 Frobenius of W (k)). We may therefore view ϕM as a collection of F-linear maps M → M . We often do this without comment, including in variants of this situation with rings other than k ⊗Fp F. ¯ p Fix a choice of elements (pn)n≥0 in K obeying pn+1 = pn and p0 = −p. Given this and ¯ e the choice of π already made, there exist unique elements (πn)n≥0 in K such that πn = pn, p πn+1 = πn, and π0 = π. We write (K0)∞ = K0((pn)n≥0) and K∞ = K((πn)n≥0). Observe that restriction deﬁnes a canonical isomorphism Gal(K∞/(K0)∞) = Gal(K/K0) =: ∆, via which we × consider ωK0,π as a character Gal(K∞/(K0)∞) → k . Chapter 0. Introduction 10

An inertial type is a representation τ : IK0 → GLn(E) with open kernel that extends to WK0 . A tamely ramified inertial type factors as a sum of fundamental characters ω : I → O×; enf K0 E we say that τ is a principal series type if it is tamely ramified of niveau f, which is to say Ln−1 Ai τ = i=0 ωef for Ai ∈ Z. Equivalently, τ factors through ∆. × The convention on Hodge-Tate weights is that the cyclotomic character : GQp → Qp has Hodge-Tate weight {1}. We write ω for the cyclotomic character mod p. Since K0 is unramified, ∼ we may canonically identify Hom(K0,E) = Hom(k, F). Let HTj denote the set of Hodge-Tate weights of a representation of GK0 with respect to embedding τj. Then our convention means × that a character ψ : GK0 → E is Hodge-Tate with HTj(ψ) = {aj} iff ψ agrees with the aj character Qf−1 τ ◦ Art−1 : I → E× on an open neighbourhood of the identity in I , j=0 j K0 K0 K0 f−1 for aj ∈ Z. We will be interested in representations having Hodge type µ := (2, 1, 0)j=0 . × The notation nrλ stands for the unramified character GK0 → F taking any geometric × Frobenius to λ ∈ F . The symbol δ∗ is the Kronecker delta ∈ {0, 1}. When dealing with the algebraic group GL3 we always identify its Weyl group with S3 the group of permutations of {0, 1, 2} and we embed S3 inside GL3 by identifying ς ∈ S3 with the matrix whose (i, j)-th entry is δi=ς(j). Then we have

−1 ςDiag(d0, d1, d2)ς = Diag(dς−1(0), dς−1(1), dς−1(2)).

We write T3 and B3 for the standard diagonal torus and upper-triangular Borel in GL3, re- ∗ 3 spectively. We identify the character group X (T3) with Z in the usual way and define η = (1, 0, −1). If v is a weight vector in some algebraic representation, we write wt(v) for its weight. If M is a module then soc(M) and rad(M) denote the socle and radical of M, respectively. The socle series of M is defined by setting soci(M) = soc M/ soci−1(M) inductively, where soc1(M) = soc(M). The cosocle of M is cosoc(M) = M/ rad(M). We often draw lattices to indicate the submodule structure of a module. It is understood that these are Alperin diagrams [Alp80] with the socle at the bottom and the cosocle at the top. If J ⊆ {0, . . . , f − 1} we define ∂J “the boundary of J” to be

∂J = {j | (j ∈ J and j + 1 ∈/ J) or (j∈ / J and j + 1 ∈ J)}.

Note that |∂J| is always even. We write J c for the complement {0, . . . , f − 1}\ J. All other notation will be introduced in the body of the thesis. Chapter 1

Integral p-adic Hodge theory

In this chapter we carry out the calculations on the local Galois side. We begin by specifying our assumptions about the local mod p Galois representation, which throughout the chapter is calledρ ¯ rather thanr ¯ , and defining the Fontaine-Laffaille invariants mentioned in the GFw Introduction. This comprises Sections 1.1 and 1.2. The main result of this chapter, Theorem 1.7.1, shows how to recover these invariants as the reduction mod p of Frobenius eigenvalues on the weakly admissible filtered ϕ-module associated to certain well-chosen potentially crystalline lifts ofρ ¯. In order to establish this result we make use of various categories of integral p-adic Hodge theoretic data that describe lattices inside potentially crystalline representations, so as to make the link between characteristic 0 and p. Our argument blends the approach of [HLM17] using strongly divisible modules with the theory developed in [LLHLM18a], using Kisin modules and the notion of shape. Thus we spend some time explaining the relationships between the categories used in the two approaches in Sections 1.3-1.5, starting with an overview of the argument in Section 1.3. The main calculations and results are then concentrated in Sections 1.6 and 1.7.

1.1 The local mod p Galois representation

This section establishes our basic assumptions and notation concerning the local mod p Galois representation.

Letρ ¯ : GK0 → GL3(F) be continuous. We assume that it is maximally nonsplit, meaning that it is uniserial and the graded pieces of its socle ﬁltration are 1-dimensional. Then it may be expressed as

 Pf−1 2 j  j=0 (af−j +1)p ωf nrµ2 ∗21 ∗  Pf−1(a1 +1)pj  ρ¯ ∼  j=0 f−j  (1.1.1)  ωf nrµ1 ∗10   Pf−1 0 j  j=0 (af−j +1)p ωf nrµ0

11 Chapter 1. Integral p-adic Hodge theory 12

i × where aj ∈ Z, µi ∈ F , and the extension classes ∗21 and ∗10 are nonzero. An essential assumption that will be made throughout this thesis is thatρ ¯ is Fontaine-Laﬀaille and δ-generic:

Deﬁnition 1.1.2. Let δ ≥ 0 be an integer. Letρ ¯ : GK0 → GL3(F) be a continuous maximally nonsplit representation as in (1.1.1). We say thatρ ¯ satisﬁes (FLδ), or alternativelyρ ¯ ∈ (FLδ), i if we can choose the aj so that

2 1 1 0 2 0 aj − aj , aj − aj , aj − aj ∈ [δ, p − 1 − δ] (1.1.3) for each 0 ≤ j ≤ f − 1.

Definition 1.1.4. Letρ ¯ ∈ (FLδ) where δ ≥ 1. For a presentation ofρ ¯ as in Definition 1.1.2, Pf−1 0 j 0 −A00 we define A00 = j=0 (af−j − 1)p and setρ ¯ :=ρ ¯ ⊗ ωf . This depends only onρ ¯.

0 Remark 1.1.5. The notation is chosen becauseρ ¯ ∈ (FLδ) guarantees thatρ ¯ is in the essential image of the Fontaine-Laﬀaille functor if δ ≥ 3. See Proposition 1.2.7 below.

1.2 Fontaine-Laﬀaille modules and invariants

In this section we first recall relevant facts about certain categories of Fontaine-Laffaille modules and their relations with Galois representations. Then we define the Fontaine-Laffaille invariants of a Galois representationρ ¯ ∈ (FL3).

[0,p−2] Deﬁnition 1.2.1. The category FLfr (OE) consists of ﬁnite free W (k) ⊗Zp OE-modules M having the following structures:

i • a decreasing ﬁltration Fil (M) by W (k)⊗Zp OE-submodules, each a W (k)-direct summand of M, such that Fil0(M) = M and Filp−1(M) = 0;

i • φ-semilinear and OE-linear Frobenius maps ϕi : Fil (M) → M obeying ϕi|Fili+1(M) = Pp−2 i pϕi+1 and i=0 ϕi(Fil (M)) = M. [0,p−2] Morphisms in FLfr (OE) are defined to be W (k) ⊗Zp OE-linear maps compatible with the filtrations and Frobenii. [0,p−2] The category FL (F) consists of finite free k ⊗Fp F-modules M having the analogous structures: a filtration Fili(M) obeying Fil0(M) = M and Filp−1(M) = 0 and Frobenius maps i Pp−2 i ϕi : Fil (M) → M obeying ϕi|Fili+1(M) = 0 and M = i=0 ϕi(Fil (M)). [0,p−2] [0,p−2] There is an obvious functor − ⊗ F : FLfr (OE) → FL (F). There are functors from these categories of Fontaine-Laffaille modules to categories of Galois representations making use of a period ring Acris, which is a W (k¯)-algebra admitting a decreas- 0 1 ing filtration by ideals Acris = Fil (Acris) ⊇ Fil (Acris) ⊇ · · · , a φW (k¯)-semilinear ring endomor- i phism ϕ, and a continuous GK0 -action preserving Fil (Acris) and commuting with ϕ. For 0 ≤ i i i i i ≤ p − 2 we have ϕ(Fil (Acris)) ⊆ p Acris so we may define ϕi = ϕ/p : Fil (Acris) → Acris. See Chapter 1. Integral p-adic Hodge theory 13

∗ [0,p−2] [Hat18] for more details. We deﬁne contravariant functors T : FL (OE) → Rep (OE), cris fr GK0 where the latter denotes the category of ﬁnite free OE-modules with continuous GK0 -action, by

∗ • Tcris(M) = HomW (k),ϕ•,Fil (M,Acris) and T ∗ : FL[0,p−2]( ) → Rep ( ) by cris F GK0 F

∗ • Tcris(M) = HomW (k),ϕ•,Fil (M,Acris/p).

[0,p−2] [0,p−2] Qf−1 j An object M in FLfr (OE) or FL (F) decomposes as a product M = j=0 M , where j j M = εjM as explained in the Notation section. Each M inherits a filtration by OE- submodules or F-submodules and we define the set of Fontaine-Laffaille weights of M with i j respect to j to be the multiset of integers FLj(M) = {i : gr (M ) 6= 0} where i is counted with i j i j multiplicity equal to rkOE gr (M ) or dimF gr (M ), respectively. We remark that in the case i j of OE-coefficients, it is not hard to see from the definitions that gr M is finite free over OE, so this definition makes sense. Moreover, clearly FLj(M) = FLj(M ⊗ F). We say that M is of regular weight if FLj(M) is multiplicity-free for each j.

[0,p−2] [0,p−2] Proposition 1.2.2. 1. The categories FLfr (OE) and FL (F) are abelian. The func- ∗ [0,p−2] ∗ tors Tcris are exact and fully faithful and for M ∈ FLfr (OE) we have Tcris(M ⊗ F) = T ∗ (M)⊗ . Moreover, the functors T ∗ preserve rank in the sense that rk M = cris F cris W (k)⊗Zp OE rk T ∗ (M) for M ∈ FL[0,p−2](O ) and rk M = dim T ∗ (M) for M ∈ FL[0,p−2]( ). OE cris fr E k⊗Fp F F cris F

∗ [0,p−2] 2. Tcris deﬁnes an anti-equivalence of categories between FLfr (OE) and the category

of GK0 -stable OE-lattices inside crystalline E-representations of GK0 whose Hodge-Tate weights are all in [0, p − 2]. If M corresponds to a lattice in the crystalline representation

ρ, then for τ : K0 → E,

HTτ (ρ) = FLτ (M).

3. The essential image of T ∗ : FL[0,p−2]( ) → Rep ( ) is closed under taking subquo- cris F GK0 F tients.

[0,p−2] a j 4. If each aj ∈ [0, p − 2] let M ∈ FL (F) denote a rank 1 object such that gr j (M ) 6= 0. P a pj Then T ∗ (M)| = ω j f−j . cris IK0 f

Proof. These facts are well-known but aren’t easily available in the form stated above in the literature. For a source that treats OE- and F-coefficients as we do in slightly more generality, see Section 2.2 of [GL14] and the references therein. The first three claims above may be deduced by following the proof of Theorem 2.2.1 of loc. cit.. We remark that the concurrence of Hodge-Tate weights with Fontaine-Laffaille weights uses our chosen normalization HT() = +1; otherwise they would differ by a minus sign. For the fourth claim, see for example [FL82], Section 5. Chapter 1. Integral p-adic Hodge theory 14

We make some general deﬁnitions concerning the representation of ϕ• via matrices.

[0,p−2] Deﬁnition 1.2.3. Let M be an object of FL (F) of rank n. A basis e of M is equivalent j j j j to a choice of F-basis e = (e0, . . . , en−1) = εje for each M . We say that e is adapted to the ﬁltration if for each 0 ≤ j ≤ f − 1 we have

k j X j Fil (M ) = Fei i≥ik for some integers 0 ≤ i0 ≤ · · · ≤ ip−2 ≤ n − 1.

Lemma 1.2.4. There always exists a basis adapted to the ﬁltration. If M is of regular weight then an adapted basis e is unique up to a transformation of the form ej 7→ ejLj where Lj ∈

Bn(F) is lower-triangular. Proof. This is clear.

Definition 1.2.5. Let e be a basis adapted to the filtration and assume M is of regular weight. We define [ϕ ]j to be the matrix such that ϕ (ej ) = ej+1 · [ϕ ]j for k such that grk(M j) 6= 0. M e k ik M e j j Then [ϕM ]e ∈ GLn(F) and for L ∈ Bn(F) we have

j j+1−1 j j [ϕM ]eL = L · [ϕM ]e · π(L ) (1.2.6)

where π : Bn(F) → Tn(F) is the natural projection.

[0,p−2] If M is of regular weight then the isomorphism class of M ∈ FL (F) is completely j determined by the sets FLj(M) and matrices [ϕM ]e for 0 ≤ j ≤ f − 1.

1.2.1 Fontaine-Laﬀaille invariants

We now use the results of the previous section to deﬁne invariants forρ ¯ ∈ (FL3).

j Proposition 1.2.7. Assume that ρ¯ ∈ (FL3) and define ai as in (1.1.1). Up to isomorphism [0,p−2] ∗ ∼ 0 0 0 0 0 there exists a unique object M of FL (F) such that Tcris(M) = ρ¯ . Let 0 =ρ ¯3 ⊂ ρ¯2 ⊂ ρ¯1 ⊂ ρ¯ denote the socle filtration of ρ¯0. Then M has a unique filtration by subobjects

0 ⊂ M0 ⊂ M1 ⊂ M2 = M

∗ ∼ 0 0 1 such that Tcris(Mi) = ρ¯ /ρ¯i+1 for 0 ≤ i ≤ 2 and the corresponding elements of Ext (M1/M0,M0) 1 and Ext (M2/M1,M1,M0) are nonzero.

0 0 Proof. Becauseρ ¯ ∈ (FL3) the hypotheses of [GG12] Lemma 3.1.5 are satisﬁed forρ ¯ . Henceρ ¯ 2 0 1 0 has an ordinary crystalline lift with Hodge-Tate weights HTj = {aj − aj + 2, aj − aj + 2, 2}. 2 0 As aj − aj + 2 ≤ p − 2 it follows from Proposition 1.2.2(2) that this lift is in the essential image ∗ 0 of Tcris and by (1) thatρ ¯ is too. The other statements now follow from full faithfulness and ∗ exactness of Tcris and the fact thatρ ¯ is nonsplit. Chapter 1. Integral p-adic Hodge theory 15

j Corollary 1.2.8. There exists a basis e adapted to the ﬁltration on M such that each [ϕM ]e is upper-triangular. This basis is unique up to a transformation of the form ej 7→ ejDj where j D ∈ T3(F). We have   αj xj yj j   [ϕM ]e =  βj zj  (1.2.9) γj

× where αj, βj, γj ∈ F and xj, yj, zj ∈ F, and there exist j0 and j1 such that xj0 6= 0 and zj1 6= 0.

Proof. Continue with notation as in Proposition 1.2.7. We in particular have FLj(Mi) = Pf−1 ai pj i 0 ∗ ∼ j=0 f−j {2, . . . , aj − aj + 2} and Tcris(Mi/Mi−1) = ωf nrµi for 0 ≤ i ≤ 2. Define e by letting i 0 j j j j j aj −aj +2 j e = (e0, e1, e2) where ei is a basis of the 1-dimensional space Fil (Mi ). Then e is a basis k k j j j of M adapted to the filtration since Fil (M)∩Mi = Fil (Mi), and we have Mi = Fe0 +···+Fei j so clearly [ϕM ]e is upper-triangular. That it has the form in (1.2.9) follows from Proposition 1.2.2(4) and nonsplitness. Conversely any basis f of M adapted to the filtration whose Frobenius matrices are upper triangular defines a filtration by subobjects 0 ⊂ N0 ⊂ N1 ⊂ N2 = M. There is a unique such j j j j filtration, so f = e B for B ∈ B3(F) . On the other hand, f and e are both adapted to the j j j j j j filtration so by Lemma 1.2.4 f = e L for L ∈ B3(F). Hence B = L is diagonal.

Remark 1.2.10. The reason we have normalizedρ ¯0 so that its Fontaine-Laﬀaille weights start at 2 rather than 0 is for technical reasons which become clear in (1.6.9) below.

Deﬁnition 1.2.11. We refer to a basis of M as in Corollary 1.2.8 as a framed basis.

We use framed bases to define our Fontaine-Laffaille invariants forρ ¯. However, for our × invariants to be well-defined elements of F , we require some extra hypotheses.

Definition 1.2.12. Suppose thatρ ¯ satisfies (FLδ) with δ ≥ 3, and pick a framed basis e of M ∗ so (1.2.9) holds. We say thatρ ¯ satisfies (FLδ) if

xj, zj 6= 0

∗∗ for each 0 ≤ j ≤ f − 1. We say thatρ ¯ satisﬁes (FLδ ) if

xj, zj, yj, yjβj − xjzj 6= 0 for each 0 ≤ j ≤ f − 1.

Remark 1.2.13. 1. It is easy to see using 1.2.6 that these conditions do not depend on the choice of the framed basis e.

∗∗ 2. It is clear that most representationsρ ¯ satisfying (FLδ) will satisfy (FLδ ), so making this assumption is not very restrictive. Chapter 1. Integral p-adic Hodge theory 16

∗∗ 3. In our global argument in Chapter 3, we will show that whenρ ¯ ∈ (FLδ) with δ ≥ 15, (FLδ ) is a consequence of the more basic Assumption (SW) that the global Galois representation has a unique Serre weight. See Deﬁnition 3.5.3, Theorem 3.5.6 and Remark 3.5.5 for the precise statements and discussion.

∗∗ Deﬁnition 1.2.14. Assume thatρ ¯ ∈ (FLδ ) and pick a framed basis of M as in (1.2.9). Let S ⊆ {0, . . . , f − 1} be any subset. We deﬁne quantities

Y Y Y −1 Y XS = αj · βj · xj · xj j∈S j+1∈/S j∈S,j+1∈/S j∈ /S,j+1∈S Y Y Y −1 Y ZS = βj · γj · zj · zj j∈S j+1∈/S j∈S,j+1∈/S j∈ /S,j+1∈S Y Y yjβj − xjzj YS = βj · yj j∈ /S j∈S

× lying in F . It is not hard to show using the uniqueness in Corollary 1.2.8 that XS,ZS,YS do not depend on the choice of framed basis e for any S. We call them the Fontaine-Laﬀaille invariants of ρ¯.

j Together with the set of inertial exponents ai ofρ ¯, these invariants (over)determine the isomorphism class ofρ ¯. However, we will only be able to recover some of them. The next lemma shows that they are still enough to determineρ ¯.

Lemma 1.2.15. The isomorphism class of ρ¯ is determined uniquely by the inertial exponents j 0≤j≤f−1 (ai )0≤i≤2 together with the collection of invariants XT ,ZT for any T ⊆ {0, . . . , f − 1} and YS where S is of the form

S = {j | (j ∈ ∂J and j + 1 ∈/ K) or (j∈ / ∂J and j + 1 ∈ K)} for some J, K ⊆ {0, . . . , f − 1} obeying J ⊆ K with K 6= ∅.

Proof. By making a diagonal change of basis we may assume that αj, βj, γj = 1 for 0 ≤ j ≤ f −2 and x0 = z0 = 1. Then X{0,...,f−1} = αf−1, Z{0,...,f−1} = βf−1 and Z∅ = γf−1. For 1 ≤ k ≤ f −2 −1 −1 we have X{1,...,k} = βf−1xk and X{1,...,f−1} = αf−1βf−1xf−1. From this and a similar argument using Z{1,...,k} we see that we may recover all αj, βj, γj, xj, zj from the invariants XT and ZT . It follows that knowing the Y is equivalent to knowing all products Q yj βj −xj zj . If we take S j∈S yj J = K = {j0} then S = {j0} (consider f = 1 and f ≥ 2 separately). So we can recover each individual yj βj −xj zj , and hence each y . Finally, we have already remarked earlier that the aj yj j i as well as the αj, βj, γj, xj, yj, zj constitute all the data of M and hence ofρ ¯. Chapter 1. Integral p-adic Hodge theory 17

1.3 Summary of categories used

Our calculations make use of a number of categories of p-adic Hodge theoretic data. Figure 1.3.1 below depicts these categories and their interrelations.

µ,τ −1 −⊗F µ,τ −1 Y (OE) / Y ( ) _ _ F incl incl [0,r],dd Θh [0,r],dd −⊗F [0,r],dd S.D.Mod (OE)0 o ∼ Y (OE) / Y (F) V ^

´et −⊗F ´et ϕModK0 (OE) / ϕModK0 (F) ` M V ∗ o V ∗ o

(−)0 −⊗F RepG (OE) / RepG ( ) (K0)∞ (K0)∞ F F O O res res

K-sst −⊗E K-sst −⊗F [0,p−2] Rep (E)o Rep (OE) / RepG ( ) o FL ( ) GK0 GK0 K0 F T ∗ F O O O cris ∗ ∗ ∗ Tst Tst Tst

dd S0 [0,r],dd S [0,r],dd Mod(ϕ, N) (E) o S.D.Mod (OE) / BrMod (F) Figure 1.3.1: Categories of p-adic Hodge theoretic data and functors between them. Hooked arrows denotes fully faithful functors and tildes denote (anti)equivalences of categories. The diagram is organized so that the ring of coeﬃcients changes from E to OE to F as one moves from left to right, and so the bottom two rows relate to representations of GK0 and the top four rows relate to representations of G(K0)∞ . All descent data (dd) is from K to K0.

We explain a few of these categories now: RepG ( ) (resp. RepG ( )) means contin- K0 F (K0)∞ F K-sst uous representations of GK0 (resp. G(K ) ) on finite-dimensional -vector spaces, Rep (E) 0 ∞ F GK0 denotes continuous representations of GK0 on finite dimensional E-vector spaces becoming K-sst semistable when restricted to GK , and Rep (OE) is the category of GK0 -stable OE lattices GK0 inside them. Finally RepG (OE) is the category of continuous representations of G(K ) (K0)∞ 0 ∞ on finite free OE-modules. The remaining categories and functors will be defined in Sections 1.4 and 1.5. We give a brief overview of the argument that establishes the main result Theorem 1.7.1. The goal is to relate data (Frobenius eigenvalues) in the weakly admissible module in Mod(ϕ, N)dd(E) associated with a potentially crystalline lift ρ ofρ ¯ becoming crystalline over K and having Hodge f−1 type µ = (2, 1, 0)j=0 and (a certain) inertial type τ. This lift corresponds to a strongly divisible module M ∈ S.D.Mod[0,2],dd. It follows from Theorem 1.5.13 that the corresponding object M [0,2],dd µ,τ −1 in Y (OE) lies in the subcategory Y (OE). Equivalence classes of Kisin modules in µ,τ −1 Y (F) are classified in [LLHLM18a] by their shape. Accordingly we need to compute the shape of M ⊗ F. This is achieved in Section 1.6 using the Breuil module S(M) and the functor Chapter 1. Integral p-adic Hodge theory 18

M. Once the shape of M ⊗ F is known, there is also a canonical form of M determined in [LLHLM18a]. Using this, the desired result is obtained by tracing through the diagram back dd [0,p−2] to Mod(ϕ, N) (E) as well as in the other direction to FL (F) using the functor F. We remark that in many cases we actually use variants of the functors in Figure 1.3.1 having the opposite covariance, in order to most easily use existing results.

1.4 Lattices inside potentially semistable GK0 -representations

In this section we recall three categories that describe potentially semistable representations of GK0 , the integral lattices in these representations, and their reductions mod p respectively. These are the categories of weakly admissible ﬁltered isocrystals with monodromy, of strongly divisible modules, and of Breuil modules, all with descent data and coeﬃcients. Much of this material is taken from [EGH13], to which we refer the reader for more details.

1.4.1 Filtered isocrystals with monodromy and descent data

0 Let L be a finite extension of Qp with maximal unramified subextension L0. Let L be a 0 subextension of L/Qp such that L/L is Galois. Definition 1.4.1. A filtered isocrystal with monodromy, descent data from L to L0, and E- 0 coefficients, alternatively referred to as a filtered (ϕ, N, L/L ,E)-module, is a finite free L0 ⊗Qp E-module D together with

• a φ ⊗ 1-semilinear bijection ϕD : D → D;

• a nilpotent L0 ⊗Qp E-linear endomorphism N obeying Nϕ = pϕN;

i • an exhausted and separated decreasing ﬁltration (Fil DL)i∈Z on DL by L⊗Qp E-submodules;

0 • a L0-semilinear and E-linear action of Gal(L/L ) commuting with ϕ and N and preserving

the ﬁltration on DL. We denote this category by Mod(ϕ, N, L/L0,E). An object D is called weakly admissible if the underlying (ϕ, N, L/L, Qp)-module is weakly admissible. The corresponding abelian category of weakly admissible objects is denoted Modw.a.(ϕ, N, L/L0,E).

L-sst Let Rep (E) denote the category of continuous E-representations of G 0 that become GL0 L semistable upon restriction to GL. We refer the reader to Section 3.1 of [EGH13] for details about the antiequivalence of abelian tensor categories RepL−st(E) −→∼ Modw.a.(ϕ, N, L/L0,E) GL0 deﬁned by 0 D∗,L (ρ) = Hom (ρ, B ) st GL0 st ∗,L0 and its inverse functor Tst . For an integer r we also deﬁne a covariant version of this equivalence by L0,r ∗,L0 ∨ r ∼ ∗,L0 ∨ Dst (ρ) = Dst (ρ ⊗ ) = Dst (ρ) (r) (1.4.2) Chapter 1. Integral p-adic Hodge theory 19

Here we are referring to the duality D 7→ D∨ and Tate twist D 7→ D(r) on the category 0 ∨ Mod(ϕ, N, L/L ,E), which are defined as follows. We set D = HomL0 (D,L0) with Frobenius −1 ∨ ϕD (f) = FrobL0 ◦f ◦ ϕD , the natural induced monodromy and filtration, and Galois action g · f = g ◦ f ◦ g−1 for g ∈ Gal(L/L0), f ∈ D∨. The Tate twist D(r) is just D with the same r i i−r monodromy operator and Galois action but ϕD(r) = p ϕD and Fil (D(r)L) = Fil (DL). For all this and the isomorphism in (1.4.2), see for example Section 8.3 of [BC]1. Remark 1.4.3. Here and throughout this section, we use the notation of [HLM17] rather than ∗,L0 L0 that of [EGH13]. For example, the functor Dst above is denoted Dst in [EGH13]. The same ∗ is true of the functors Tst used below. 0 0 We will only be interested in the niveau 1 tamely ramified case L = K, L = K0, Gal(L/L ) = ∆. In this case, since ϕj : Dj → Dj+1 is an E[∆]-linear isomorphism, there exists a principal Ln−1 Ai j j j j j series type τ = i=0 ωef such that D has an E-basis e = (e0, . . . , en−1) with ∆ acting on ei Ai by ωef . Definition 1.4.4. In the above situation we say that D is of type τ and that e is a framed basis. Given a choice of framed basis e (and its concomitant implicit ordering of τ) we define j j j+1 j Mate(ϕ) ∈ GLn(E) to be the matrix such that ϕ(e ) = e · Mate(ϕ) .

Lemma 1.4.5. If D has a framed basis e of type τ then D∨(r) has a framed basis e∨ of type τ −1 such that j r t −1 Mate∨ (ϕD∨ ) = p · Mate(ϕD) .

Proof. This follows easily from the description of the dual and Tate twist above.

1.4.2 Strongly divisible modules

Let L and L0 be as in the previous subsection and assume from now on that L/L0 is tamely ramified with ramification index e(L/L0). Let e denote the absolute ramification index of L and write ` for the residue field of L. Choose a uniformizer π of L such that πe(L/L0) ∈ L0 (which can be done since the ramification is tame) and let E(u) ∈ L0[u] denote its minimal polynomial. 0 For g ∈ Gal(L/L ) write g(π) = hgπ where hg ∈ W (`). We define the Breuil ring S as in ie Section 3.1 of [EGH13]. It is the p-adic completion of W (`)[u, u /i!]i≥1, or more concretely the

W (`)-subalgebra of L0[[u]] given by

( ∞ ie ) X u S = wi wi → 0 in W (`) . i! i=0

It is W (`)-ﬂat and we can give it the following structures: the unique continuous φ-semilinear p ie ipe map ϕS taking u to u and u /i! to u /i!, a W (`)-linear monodromy map NS which for us will i be irrelevant, and a decreasing ﬁltration by ideals (depending on the choice of π) {Fil (S)}i≥0

1Note the definition of the Tate twist given in Definition 8.3.1 is backwards if we want it to commute with contravariant functors as stated there. Chapter 1. Integral p-adic Hodge theory 20 where Fili(S) is defined to be the closure of the ideal generated by E(u)j/j! for all j ≥ i (it is easy to check that E(u)j/j! does in fact lie in S). If i ≤ p − 1 then ϕ(Fili(S)) ⊆ piS and we i i i × write ϕi for ϕ/p : Fil (S) → S. For 0 ≤ i ≤ p − 2 we define ci = ϕr(E(u) ) ∈ S . It is easy to 0 check that ci ≡ 1 mod p. There is an action of Gal(L/L ) by continuous ring automorphisms ui i ui gˆ defined byg ˆ(w bi/ec! ) = g(w)hg bi/ec! for any i ≥ 0 and w ∈ W (`). It is helpful to observe that 1 p S is not noetherian but it is local, with maximal ideal mS = (p, Fil (S)) = (p, u, Fil (S)) and residue field `. We will also use that S/ Filp(S) is p-torsion free below. i 0 We write SOE for S ⊗Zp OE. The data of ϕ, ϕi,N, Fil and the action of Gal(L/L ) all extend OE-linearly to this ring.

[0,r],dd Definition 1.4.6. Fix an integer 0 ≤ h ≤ p − 2. The category S.D.Mod (OE) of strongly 0 divisible modules of height ≤ r with OE-coefficients and descent data from L to L is defined on page 13 of [EGH13]. In particular an object M is a finite free SOE -module together with a submodule Mr, a SOE -semilinear Frobenius map ϕ : M → M, an SOE -linear monodromy map N : M → M, and a semilinear action of Gal(L/L0) that commutes with ϕ and N and preserves

Mr. Moreover, M must obey

r • Mr ⊇ (Fil (S))M;

• Mr ∩ jM = jMr for each ideal j ≤ OE;

r • ϕ(Mr) is contained in p M and generates it over SOE ;

• axioms involving N, which for us are irrelevant.

[0,r],dd The category S.D.Mod (OE)0 of quasi-strongly divisible modules of height ≤ r with OE- coefficients and descent data from L to L0 is defined as above but we forget all data involving the monodromy operator N (thus the axioms above constitute its complete definition). We write M 7→ M0 for the natural forgetful functor. r If M is a (quasi-)strongly divisible module, we write ϕr for ϕ/p : Mr → M.

There is a covariant functor T ∗ :S.D.Mod[0,r],dd(O ) → Rep (O ) deﬁned as T ∗ (M) = st E GL0 E st ˆ ˆ HomS,(−)r,ϕ,N (M, Ast) for a certain period ring Ast deﬁned in [EGH13] and [Bre99]. This [0,r],dd functor induces an equivalence of categories between S.D.Mod (OE) and the category of

GL0 -stable OE-lattices inside potentially semistable E-representations of GL0 which become semistable when restricted to GL and have Hodge-Tate weights in [0, r] (Proposition 3.1.4, L0,r ∗,L0 ∨ r [EGH13]). We will use the contravariant version Tst (M) := Tst (M) ⊗ , which clearly has the same property. 0 We now specialize to our tamely ramiﬁed niveau 1 case L = K, L = K0, π = −p. There is Q j j ie ∧ j an O -algebra decomposition S = S where S = εjS = O [u, u /i!] , so ϕ : S → E OE j E i≥1 SOE j+1 uie uipe j S is the OE-linear map taking bi/ec! to bi/ec! . Each S is a local OE-algebra, with maximal 1 j ideal mSj = ($E, Fil (S )) and residue ﬁeld F. Accordingly given a (quasi-)strongly divisible Q j Q j j j+1 module M we write M = j M , Mr = j Mr and ϕr : Mr → Mr . Chapter 1. Integral p-adic Hodge theory 21

Ln−1 Ai Deﬁnition 1.4.7. Let τ = i=0 ωef be a principal series type. We say that a (quasi-)strongly divisible module M is of type τ if it is of rank n and there exist Sj-bases ej of Mj such that ∆ j Ai acts on ei via ωef for 0 ≤ i ≤ n − 1 and 0 ≤ j ≤ f − 1. We refer to any basis with this property as a framed basis of M.

Ln−1 Ai Lemma 1.4.8. Let τ = i=0 ωef be a principal series type and let M be a (quasi-)strongly divisible module. The following are equivalent:

1. M has type τ;

j ∼ 2. M /mSj = τ¯ as an F[∆]-module for each 0 ≤ j ≤ f − 1;

j j j j 3. for each 0 ≤ j ≤ f − 1 there exists a tuple f = (f0 , . . . , fn−1) of elements of Mr which j p j j j j Ai generate Mr/(Fil (S ))M over S such that ∆ acts on fi by ωef .

j j p j j Proof. It is obvious that (1) implies (2), so assume that (2) holds. Set N = Mr/(Fil (S ))M . j p j This is a submodule of M / Fil (S ), hence is free over OE. By the second axiom in Def- ep j j p j inition 1.4.6, we have an inclusion of F[u]/(u )-modules N /p ,→ M /(p, Fil (S )). Since uerMj/(p, Filp(Sj)) ⊆ Nj/p, Lemme 2.2.1.1 (which is stated for e = 1 but generalizes easily) of [Bre98] applies and we obtain

j j j+1 dimF N /(p, u) = dimF M /mSj = dimF M /mSj+1 .

But ϕr induces an F[∆]-linear surjection

j j+1 N /(p, u) M /mSj+1 , (1.4.9)

j which must therefore be an isomorphism. Now N /u is a free OE[∆]-module whose reduction mod p isτ ¯ for each 0 ≤ j ≤ f − 1. It follows immediately that Nj/u = τ. Moreover, as uNj is j an OE[∆]-submodule of N such that the corresponding quotient is free over OE, and p - |∆|, it j j follows that uN has an OE[∆]-complement in N . Finally by Nakayama’s lemma we see that there exists a generating set of Nj of the desired form. Applying the complement argument again, we deduce the existence of a set f j as in (3). Finally, assume (3). We run the argument above backwards: from the isomorphism (1.4.9) j j p j we get M /mSj =τ ¯ for each 0 ≤ j ≤ f − 1, hence M /(u, Fil (S )) = τ. Now apply the same argument about a complement and Nakayama’s lemma to see that Mj has a generating set ej j Ai j j such that ∆ acts on ei by ωef . As M is free it follows that e is a basis as in (1).

The next corollary to the proof will be used later.

j j j Lemma 1.4.10. Let M be a strongly divisible module of rank n. A tuple f = (f0 , . . . , fn−1) ⊂ j j p j j j j+1 j+1 Mr generates Mr/(Fil (S ))M iﬀ ϕr(f ) is an S -basis of M . Chapter 1. Integral p-adic Hodge theory 22

Proof. We use the notation of the proof of the previous lemma. By Nakayama’s lemma the j j+1 j+1 j j isomorphism (1.4.9) shows that ϕr(f ) is an S -basis of M iﬀ f is an F-basis of N /(p, u). But by Nakayama’s lemma again this is equivalent to being a generating set of Nj.

[0,h],dd Deﬁnition 1.4.11. Let M be an object of S.D.Mod (OE)0 of type τ. Let e be a framed K/K0 basis. Then there exists a tuple f as in Lemma 1.4.8(3). We refer to any such tuple as a framed j p j j system of generators of Mr/(Fil (S ))M . We also refer to the pair (e, f) as simply a framed j j system of generators of M. We deﬁne Mate,f (Mr) to be the matrix in Matn(S ) such that

j j j f = e · Mate,f (Mr)

j j+1 and Mate,f (ϕr) to be the matrix in GLn(S ) such that

j j+1 j ϕr(f ) = e · Mate,f (ϕr) .

j j Remark 1.4.12. In fact, we always have Mate,f (Mr) ∈ GLn(S [1/p]). To see this, note that by deﬁnition we have

j+1 j j j e · Mate,f (ϕr) = ϕr(e · Mate,f (Mr) ) j r j = ϕ(e ) · p ϕ Mat (Mr) SOE e,f

j j+1 j j j+1 j Writing ϕ(e ) = e B for some B ∈ Matn(S ) we deduce that ϕ Mat (Mr) ∈ SOE e,f j+1 j GLn(S ). The claim now follows from the fact that S is local. We now explain the relationship between strongly divisible modules and ﬁltered isocrystals.

Let s0 : S[1/p] → K0 denote the Galois and Frobenius equivariant map taking u 7→ 0. Following [EGH13] we deﬁne a functor

[0,r],dd w.a. S0 :S.D.Mod (OE) → Mod (ϕ, N, K/K0,E)

by setting S0(M) = D where D = M[1/p] ⊗S[1/p],s0 K0 with ϕD and the Gal(K/K0)-action defined as the semilinear extensions of the corresponding structures on M, ND defined as the linear extension of NM, and the filtration on DK defined via the recipe in the proof of Proposition 3.1.4 of [EGH13]. See loc. cit. for more details. It is also shown in the proof in ∗ ∗ question that Tst(M) is naturally a GK0 -stable OE-lattice in Tst(D). Hence we have a natural isomorphism of functors ∗ ∼ ∗ Tst(S0(M)) = Tst(M) ⊗OE E. (1.4.13)

Lemma 1.4.14. Suppose that ρ is a GK0 -stable OE-lattice inside a potentially semistable representation of GK0 having Hodge-Tate weights inside [0, r] that becomes semistable when restricted −1 ∗ to GK . Let M be a strongly divisible module of type τ such that Tst(M) = ρ and let (e, f) be K0,r 0 a framed system of generators of M. Then Dst (ρ) is of type τ and has a framed basis e such Chapter 1. Integral p-adic Hodge theory 23 that −1 j j j Mate0 (ϕ) = s0 Mate,f (ϕr) · s0 Mate,f (Mr) (1.4.15)

Proof. Let D = S0(M) and let e0 = e ⊗ 1 ∈ D. Then e0 is clearly a basis of D, which we see is −1 j j of type τ by the deﬁnition of the Galois action on D. The equation ϕr e Mate,f (Mr) ) = j+1 j j j j+1 r j e · Mate,f (ϕr) becomes ϕD(e1) · s0 Mate,f (Mr) = e1 p Mate,f (ϕr) in D. By Remark 1.4.12, this implies that

−1 j r t j t M j Mate1 (ϕ) = p · s0 Mate,f (ϕr) · s0 Mate,f ( r) .

K0,r ∼ ∨ Note that Dst (ρ) = D (r) by (1.4.13). The result now follows from Lemma 1.4.5 by taking 0 ∨ e to be the basis of D (r) obtained from e1 there.

1.4.3 Breuil modules

Let L, π and L0 be as at the beginning of Section 1.4.2. In this section we recall aspects of the theory of Breuil modules, which are (in some sense) the reductions mod p of the strongly divisible modules in the previous section, and are used to describe the mod p reductions of stable lattices inside potentially semistable Galois representations. The relevant mod p version of the Breuil ring is S := S/(p, Filp(S)) = `[u]/(uep), having 0 the Frobenius ϕS, monodromy, ﬁltration and Gal(L/L )-action induced from S. We also write

SF = S ⊗Fp F, with all structures extended F-linearly.

[0,r],dd Definition 1.4.16. Let 0 ≤ r ≤ p−2. The category BrMod (F) of Breuil modules of height 0 ≤ r with F-coefficients and descent data from L to L is defined on pages 17-18 of [EGH13]. In particular an object is a finite free S -module M with a submodule Mr, a ϕ -semilinear F SF map ϕr : Mr → M, a SF-linear monodromy map N : M → M, and a SF-semilinear action of 0 Gal(L/L ) that commutes with ϕr and N and preserves Mr. Moreover, the following axioms must hold:

er • Mr ⊇ u M;

• the image of ϕr generates M over SF;

• other conditions on N, which we don’t need.

[0,r],dd p If M is in S.D.Mod (OE) then M := M/($E, Fil (SOE )) has a natural structure of an [0,r],dd object of BrMod (F) by defining Mr to be the image of Mr and giving it the naturally induced monodromy and Galois action. By the second axiom in Definition 1.4.6, ϕr : Mr → M [0,r],dd induces a well-defined map (ϕr)M : Mr → M. This defines a functor S :S.D.Mod (OE) → [0,r],dd BrMod (F). ∗ [0,r],dd 0 In [EGH13], Section 3.2 there is defined a functor Tst : BrMod (F) → RepF(GL ) by ∗ ˆ Tst(M) = HomBrMod(M, A) for the period ring Ab = Abst ⊗S S. Lemma 3.2.2 op. cit. shows that Chapter 1. Integral p-adic Hodge theory 24

Tst is faithful, rank-preserving and compatible with strongly divisible modules in the sense that [0,r],dd for any M in S.D.Mod (OE) we have a natural isomorphism

∗ ∼ ∗ Tst(S(M)) = Tst(M) ⊗OE F. (1.4.17)

r ∗ ∨ r In fact we will use the variant of this functor defined by Tst(M) = Tst(M) ⊗ ω , for the sole reason that certain results in [HLM17] we wish to quote are stated using this version. We recall [0,r],dd that there is a duality theory for the category BrMod (F) which is to say there is a functor ∗ ∗∗ ∼ r ∼ ∗ ∗ M 7→ M obeying M = M and Tst(M) = Tst(M ). This result is due to Caruso [Car08]; see [EGH13] Definition 3.2.8 for an explicit definition. 0 We once again restrict to the case L = K, L = K0 and π = −p, and make some brief remarks and definitions analogous to those made in the previous subsection. We have natural Q j Q j decompositions SF = j S and M = j M as before.

Ln−1 Ai Deﬁnition 1.4.18. Let τ = i=0 ωef be a principal series type. We say that a Breuil module [0,r],dd j j j M in BrModL/L0 (F) is of type τ if it is of rank n and there exist S bases e of M such that j Ai ∆ acts on ei by ωf for 0 ≤ i ≤ n − 1 and 0 ≤ j ≤ f − 1. We refer to any basis with this property as a framed basis of M.

Ln−1 Ai [0,r],dd Lemma 1.4.19. Let τ = i=0 ωef be a principal series type and M an object of BrModL/L0 (F). The following are equivalent:

1. M has type τ;

j ∼ 2. M /u = τ¯ as an F[∆]-module for each 0 ≤ j ≤ f − 1;

j j j j j 3. for each 0 ≤ j ≤ f − 1 there exists a set of S -generators f = (f0 , . . . , fn−1) of Mr such j Ai that ∆ acts on fi by ωf .

Proof. This is entirely analogous to Lemma 1.4.8, but easier.

[0,r],dd Definition 1.4.20. Let M be an object of BrMod (F) of type τ. Let e be a framed basis. Then there exists a tuple f as in Lemma 1.4.19(3). We refer to any such tuple as a framed j j j j+1 system of generators of Mr. We define Mate,f (Mr) ∈ Matn(S ) and Mate,f (ϕr) ∈ GLn(S ) exactly as in Definition 1.4.11.

We close this section with a lemma about change of basis in Breuil modules. It is a slight generalization of Lemma 2.2.8 of [HLM17].

Lemma 1.4.21. Let M be an object in BrMod[0,r],dd( ) of type τ, and let e, f be a framed K/K0 F j j basis and system of generators respectively. Suppose that there are matrices V ∈ Matn(S ) j j and B ∈ GLn(S ) such that

j−1 j j j (r+1)e Mate,f (ϕr) V ≡ Mate,f (Mr) B mod u . Chapter 1. Integral p-adic Hodge theory 25 for each 0 ≤ j ≤ f − 1. Then there exists a framed basis e0j and framed system of generators 0j j j j f of type τ such that Mate0,f 0 (Mr) = V and Mate0,f 0 (ϕr) = ϕ (B ). SF

j e(r+1) j j Proof. The proof is the same as Lemma 2.2.8 of [HLM17]. We have Mate,f (Mr) + u M B = Mat (ϕ )j−1V j for some M j ∈ Mat (S¯j) because Bj is invertible. Then f j := ej(Mat (M )j+ e,f r n 1 e,f r j j ue(r+1)M j) is a framed system of generators of M . As f j = f j + ue(something in M ) we see r 1 r j j 0j j j−1 0j 0j j j j that Mate,f (ϕr) = Mate,f (ϕr) . Now set e = e Mate,f (ϕr) and f = e V = f B . 1 1 j 0j j Since B is invertible, f is also a framed system of generators for Mr. We ﬁnally compute

0j j j j+1 j j 0j+1 j ϕ (f ) = ϕ (f )ϕ ¯(B ) = e Mat (ϕ ) ϕ ¯(B ) = e ϕ ¯(B ) r r 1 S e,f r S S which proves the claim.

1.5 Kisin modules and ϕ-modules

The theory in the preceding section dealt with potentially semistable representations having Hodge-Tate weights in the range [0, r]. The lifts ofρ ¯ arising from automorphic representations in Chapter 3 are more specialized: they are potentially crystalline with parallel Hodge-Tate weights (2, 1, 0). In order to capitalize on this fact it is convenient to use the results in [LLHLM18a] which deals with this exact situation. Since the work in question uses the theory of Kisin modules, we begin with a review of the relevant deﬁnitions and the closely related ϕ-modules.

Unlike the categories of the previous section, these categories describe representations of G(K0)∞ .

1.5.1 Etale´ ϕ-modules

Let L/Qp be an arbitrary extension with residue ﬁeld `, and $L a uniformizer. Make a choice p of elements ($n)n≥0 in L such that $n+1 = $n and $0 = $L, and set L∞ = L(($n)n≥0). In this situation (and more general situations) the theory of norm ﬁelds associates with L∞ an

“imperfect norm field” that is (noncanonically) isomorphic to `((uL)) together with a Cohen ring OEL ⊂ W (Frac(R))[1/p] that can be used to describe representations of GL∞ , constructed as follows. Let R = lim O /p where the transition maps are the Frobenius. Write $ = ←− L ($n)n≥0 ∈ R and let [$] ∈ W (R) denote the Teichmüllerlift. Define SL = W (`)[[uL]] where uL is a variable and embed S ,→ W (R) by identifying uL with [$]. Let OE,L denote the p-adic completion of S[1/uL], so OE,L is a discrete valuation ring with residue field `((uL)). Concretely, ( ∞ ) X i OE,L = wiuL wi ∈ W (`), lim wi = 0 . i→−∞ i=−∞

Let EL denote its ﬁeld of fractions. Then the inclusion SL ,→ W (R) extends to an inclusion un EL ,→ W (Frac(R))[1/p]. Let EL ⊂ W (Frac(R))[1/p] denote the maximal unramiﬁed extension of EL contained in W (Frac(R))[1/p] and let OEun,L ⊂ W (Frac(R)) denote its ring of integers. Chapter 1. Integral p-adic Hodge theory 26

As Frac(R) is algebraically closed the residue ﬁeld OEun,L/pOEun,L is a separable closure of `((u )). Write O for the p-adic completion of O and Eun for its ﬁeld of fractions. L Edun,L Edun,L dL un Finally, write SL = OEun,L ∩W (R) ⊂ W (Frac(R)). The ring W (Frac(R)) has a natural action of G . The induced action of G preserves O . L L∞ Edun,L p We endow SL with the φ-semilinear Frobenius map taking uL to uL. This extends uniquely all the way to O un . Let R be a complete local noetherian OE-algebra (for example, R = OE EdL or R = ). Then we extend this Frobenius map R-linearly to O un ⊗ R. F EdL b Zp

Definition 1.5.1. The category of ϕ-modules with coefficients in R is denoted ϕ ModL(R) and consists of finite free OE,L⊗b Zp R-modules M equipped with a φ⊗1-semilinear map ϕM : M → M. We say that M is étale if the image of ϕ generates M (equivalently, the linearization of ϕ is an et isomorphism) and write ϕ ModL (R) for the full subcategory of étaleobjects. It is abelian. The point of these constructions is that we obtain an exact antiequivalence of abelian tensor categories [Fon90]

ϕ Modet(R) → Rep (R) L GL∞ ∗ M 7→ V (M) := HomO ,ϕ(M, O un ) L E,L EdL where Rep (R) denote the category of p-adically continuous representations of G on finite GL∞ L∞ free R-modules. This construction is also compatible with base change OE → F. In certain cases of interest to us it is possible to add descent data to the above construction. 0 0 m Following [CDM14], let L be a subfield of L such that L = L ($L) where $L0 = $L is a uniformizer of L0, for some integer m prime to p. We also assume that L0 contains a primitive 0 0 m mth root of 1 so that L/L is Galois. Choose ($n)n≥0 as before and set $n = $n . Then we 0 may perform all the constructions above with uL0 = [$ ], identifying SL0 and OE,L0 with the m W (`)-subalgebras of SL and OE,L generated by uL0 = uL . Define a W (`)-semilinear action 0 0 of Gal(L∞/L∞) = Gal(L/L ) on OE,L such that g(uL) = ω$L (g)uL, so the fixed subring is 0 × 0 OE,L (here ω$L is the character Gal(L/L ) → W (`) such that g($L) = ω$L (g)$L which is well defined because we assumed L0 contains the mth roots of 1). This leads to the following definition. Definition 1.5.2. A ϕ-module with descent data from L to L0 and coefficients in R is an 0 object M of ϕ ModL(R) with an OE,L-semilinear action of Gal(L/L ) that commutes with the

Frobenius ϕM . It is ´etale if the underlying object of ϕ ModL(R) is. We denote the resulting et,dd category by ϕ ModL/L0 (R). There is an exact antiequivalence of abelian tensor categories

et,dd ϕ Mod 0 (R) → RepG 0 (R) L/L L∞ M 7→ V ∗ (M) := Hom (M, O ) dd OE,L,ϕ Edun,L0 Chapter 1. Integral p-adic Hodge theory 27

−1 0 where we deﬁne the action of GL∞ on the Hom group by g(f) = g ◦ f ◦ g¯ , where bar denotes 0 projection to Gal(L∞/L∞). Moreover the diagram

∗ ét,dd Vdd ϕ Mod 0 (R) / RepG 0 (R) L/L L∞ O 7 V ∗ L0 ét ϕ ModL0 (R) where the vertical arrow is the base change M 7→ M ⊗ O is a commutative diagram of OE,L0 E,L equivalences of categories. One sees that the vertical arrow is an equivalence with inverse given 0 0 ∗ by taking Gal(L/L )-invariants using Galois descent (valid since p - | Gal(L/L )|) and that Vdd is an equivalence by arguing as in Lemma A.2.6 of [HLM17]. See also Section 2.1.3 of [CDM14]. 0 Now that we are done with generalities we specialize to the case L = K, L = K0, $L0 = −p e and $L = π, so m = e. We use the notation u = uK and v = uK0 = u . As before, given M ∈ ϕ Mod (R) (resp. ϕ Moddd (R)) of rank n we decompose M = Qf−1 M j and write K0 K/K0 j=0 j j+1 j j+1 j Mate(ϕM ) for the matrix in GLn(R ) such that ϕ(e ) = e · Mate(ϕM ) , where e is a basis (resp. framed basis) of M. Here the notion of framed basis with respect to a principal series type has the obvious definition. [0,p−2] et We now explain the functor F : FL (F) → ϕ ModK0 (F) in Figure 1.3.1. We define F exactly as it appears in Appendix A.1 of [HLM17]. The next lemma describes its key properties.

Lemma 1.5.3. 1. F is fully faithful.

[0,p−2] 2. Let M ∈ FL (F) be of rank n, having regular Fontaine-Laﬀaille weights FLj(M) = j j j {m0 < m1 < ··· < mn−1}. If e0 is any basis of M adapted to the ﬁltration then F(M) has a basis e such that

j+1 j+1 j j+1 m0 mn−1 j ϕF(M)(e ) = e · Diag(v , . . . , v ) Mate0 (ϕM )

3. The diagram et RepG o ∗ ϕ ModK ( ) (K0)∞ V 0 F O O res F

Rep ( ) FL[0,p−2]( ) GK F o ∗ F 0 Tcris commutes up to equivalence.

Proof. The fact that F is fully faithful is proved during its construction in Appendix A.1 of [HLM17]. The second part is a mild generalization of Lemma 2.2.7 op. cit. and is proved in exactly the same way. The third statement is shown in the proof of Lemma 2.2.8 op. cit.. Chapter 1. Integral p-adic Hodge theory 28

1.5.2 Kisin modules with descent data

The theory of Kisin modules is a refinement of that of ϕ-modules used to study integral structures in G(K0)∞ -representations. We will use the variant involving descent data developed in 0 [LLHLM18a, CL18]. In this section we specialize to the case L = K, L = K0. Let R be a Ln−1 Ai complete local noetherian OE-algebra. We also fix a principal series type τ = i=0 ωef . Definition 1.5.4. A Kisin module with height in [0, r] with coefficients in R and descent data from K to K0 is a finite free S⊗b Zp R-module M together with a ϕS ⊗ 1-semilinear Frobenius map ϕ : M → M such that the cokernel of the linearization ϕ∗M → M is killed by E(u)r, and also a semilinear action of ∆ that commutes with ϕ. We refer to any such basis as a framed basis of M. These objects form a category Y [0,r],dd(R). j j j Ai If for each 0 ≤ j ≤ f − 1 there exists a basis e of M such that ∆ acts on ei by ωef ⊗ R then we say that M is of type τ. The full subcategory of objects of type τ is denoted Y [0,r],τ (R).

Exactly as in Lemma 1.4.8, saying that M is of type τ is equivalent to asking that M j/u =∼

τ ⊗OE R as R[∆]-modules for each 0 ≤ j ≤ f − 1. j Deﬁnition 1.5.5. If e is a framed basis of M then Mate(ϕ) is the matrix in Matn(R[[u]]) such j j+1 j that ϕ(e ) = e · Mate(ϕ) for 0 ≤ j ≤ f − 1.

[0,r],dd ét,dd ∗ There is a functor Y (R) → ϕ Mod (R) taking M to M ⊗S O . We write T for K/K0 Eb,K0 dd [0,r],dd ∗ ∆=1 the resulting functor Y (R) → RepG (R) obtained by composing with V ((−) ). (K0)∞ We now explain the connection between Kisin modules and quasi-strongly divisible modules with descent data. In [Car08] it is shown that there is an equivalence of categories Θr : [0,r] ∼ [0,r] Y (Zp) −→ S.D.Mod (Zp)0 where the notation means the categories analogous to the ones we have defined but with Zp-coefficients and no descent data. We now update this result to include this extra data. Following [Car08] we define a functor

[0,r],dd [0,r],dd Θr : Y (OE) →S.D.Mod (OE)0

M 7→Θr(M) := S ⊗ϕS ,S M

ϕ where the notation means we take the tensor product along the map S ,→ S −−→S S. The extra structures on M = Θr(M) are given by setting

r Mr = {x ∈ M : (1 ⊗ ϕM )(x) ∈ Fil (S) ⊗S M ⊆ S ⊗S M} where we note that 1 ⊗ ϕM deﬁnes an S-linear map M → S ⊗S M. We let (ϕM)r : Mr → M be the composition 1⊗ϕM r (ϕS )r⊗1 Mr −−−−→ Fil (S) ⊗S M −−−−−→ M.

1 r We deﬁne ϕM(x) = r (ϕM)r(E(u) x). Finally the ∆-action on M is the obvious S-semilinear c1 [0,r],dd one. It is easy to check that M is an object of S.D.Mod (OE)0 (see [Car08]). Chapter 1. Integral p-adic Hodge theory 29

Proposition 1.5.6. Let 0 ≤ r ≤ p − 2. The functor Θr deﬁned above is an equivalence of categories.

Proof. In the case OE = Zp and trivial descent data this is Theorem 2.2.1 of [Car08]. We simply verify that it continues to hold more generally. Given M with OE-coefficients, by the result of [0,r] ∼ Caruso we know there exists a Kisin module M ∈ Y (Zp) such that Θr(M) = M, and since Θr induces a bijection between the Zp-endomorphism rings of M and M, it follows that M has an OE-action compatible with its other structures. The only non-formal thing is to verify that ˆ M is free over S⊗Zp OE. To do this we use a trick from Lemma 1.2.5 of [Kis09]: note that if M has rank d over S ⊗ OE then since W (k) ⊗S M = W (k) ⊗φ,S M and φ ⊗ 1 is an automorphism of W (k) ⊗ OE it follows that W (k) ⊗S M = M/uM is free of rank d over W (k) ⊗ OE (where we view W (k) as an S- and S-module via u 7→ 0). By Nakayama’s lemma we can find a surjection d (S ⊗ OE) M. The left hand side is free of rank dk over S, where k = rankZp OE. But we also have rankS M = rankS M = dk. Hence the map is an isomorphism. This shows that Θr is essentially surjective between the categories with OE-coefficients. The proof of full faithfulness is formal. [0,r] We now add descent data. Note, for example, that if M ∈ Y (OE) then for g ∈ ∆ we can (g) [0,r] define M = S ⊗g,S M with Frobenius ϕS ⊗ ϕM , and this is again an object of Y (OE). To say that M has a descent datum is equivalent to asking that there exist morphismsg ˆ : (g) [0,r] ˆ(g) M → M in Y (OE) for each g ∈ ∆ obeying the cocycle relations ghc =g ˆ ◦ h (note M (gh) = (M (h))(g)). Similar remarks are true on the strongly divisible module side, and one (g) ∼ (g) checks that Θr(M ) = Θr(M) . It is now formal to show that Θr induces an equivalence of categories when descent data is taken into account.

[0,r],τ Corollary 1.5.7. Let M be an object of Y (OE) and let M = Θr(M). If ˆe is a framed j basis of M then M has a framed system of generators (e, f) of type τ such that Mate,f (Mr) j j j−1 j r is the unique matrix V in Matn(OE[[u]]) ⊆ Matn(S ) obeying Matˆe(ϕ) V = E(u) · id, and j Mate(ϕr) = cr · id.

j j r Proof. Let us abbreviate Matê(ϕM ) = A . Since E(u) M ⊆ him(ϕM )i it follows that there j j j j j r j j−1 exists a matrix V ∈ Matn(OE[[u]]) such that A V = V A = E(u) · id. Set e = 1 ⊗ ê ∈ j j−1 j j j j j j−1 S ⊗ϕS ,S M = M and define f = e V . Then a calculation shows that (1 ⊗ ϕM )(f ) = r j+1 j j j r j+1 j+1 E(u) e , so f lies in Mr. Moreover it follows that ϕr(f ) = (ϕS)r(E(u) )e = cre . From Lemma 1.4.10 we see that (e, f) is a system of generators, and it is easy to check from the definitions that it is framed. This proves the claim. Chapter 1. Integral p-adic Hodge theory 30

Lemma 1.5.8. The diagram of functors

[0,r],dd [0,r],dd S.D.Mod (OE)0 o Y (OE) O Θr ∆=1 (−)0 −⊗O Eb,K0 [0,r],dd et S.D.Mod (OE) ϕ ModK0 (OE)

T ∗ V ∗ st K0 res RepG (OE) / RepG (OE) K0 (K0)∞ commutes up to equivalence.

[0,r],dd [0,r],dd Proof. Let M be an object of S.D.Mod (OE) and M in Y (OE). The result can be K/K0 extracted from the literature as follows: by Lemma 3.3.4 of [CL09] that there is a natural bijection un HomS,ϕ(M, S ) = HomS,ϕ(Θr(M),Acris) and from (3.1.5) in the same paper that

∗ HomS,ϕ(M0,Acris) = Tst(M).

un The result now follows from Corollary 2.1.4 of [Kis06] which says that HomS,ϕ(M, S ) = ∗ Tdd(M) and the fact that Θr is an equivalence of categories. Note that the references above do not deal with descent data or coeﬃcients exactly as we do, but it doesn’t matter because the functors are the same.

[0,r],dd [0,r],dd We now deﬁne the functor M : BrMod (F) → Y (F) appearing in Figure 1.3.1.

A closely related functor is deﬁned in Appendix A.1 [HLM17], where it is called M k((u)) and et,dd [0,r],dd takes values in the category ϕ Mod (F). Since M k((u)) factors through Y (F) by its [0,r],dd deﬁnition, we let M be M k((u)) but having target ϕ (F).

∗ Lemma 1.5.9. 1. If M is of type τ with framed system of generators (e, f) then M(M ) has a basis e0 of type τ −1 such that

−1 0j 0j+1 t j+1 j ϕ(e ) = e · Mate,f (Mr) Mate,f (ϕr) .

2. The diagram

RepG ( ) / RepG ( ) K0 F (K0)∞ F O O ∗ ∗ Tst Tdd

[0,r],dd M [0,r],dd BrMod (F) / Y (F) commutes up to equivalence. Chapter 1. Integral p-adic Hodge theory 31

Proof. The proof of the ﬁrst claim is a mild generalization of the proof of Lemma 2.2.6, [HLM17]. The second follows from Lemma A.2.7 op. cit..

1.5.3 Kisin modules of Hodge type µ

Let τ be a principal series type and µ ∈ ([0, r])Hom(K0,E) a p-adic Hodge type. In [CL18], inspired by the theory of local Shimura varieties, the authors construct subcategories Y µ,τ (R) ⊆ Y [0,r],τ (R) consisting of Kisin modules of height ≤ r and of type τ which morally speaking are supposed to correspond to Galois representations of Hodge type µ. Unfortunately the construction of these categories is somewhat abstruse and to recall details would take us astray. So in this section we will only explain why these categories are useful for us and recall the results we need. On the other hand, when R is OE-flat and reduced one can show a posteriori that conditions for membership in Y µ,τ (R) are fairly concrete. See for example [LLHLM18a] Proposition 4.18. f−1 We will specialize to the case n = 3 and µ = (2, 1, 0)j=0 . In [LLHLM18a], under the assumption that τ is 3-generic (Definition 1.5.10 below), the authors manage to give a classification µ,τ µ,τ of points of Y (F) in terms of an associated datum called the shape of an object of Y (F), which we recall below. They also compute the universal framed deformation of a Kisin module of a given shape and give a necessary and sufficient monodromy criterion for the associated

G(K0)∞ -representation to come from a potentially crystalline GK0 -representation. The utility of this for us is that these constructions are all explicit and we can make use of the “canonical forms” in the tables of [LLHLM18a]. In order to go further we now recall some basic deﬁnitions and notation from [LLHLM18a].

L2 Ai Pf−1 i j Deﬁnition 1.5.10. Let τ = i=0 ωef be a principal series type where Ai = − j=0 bf−jp i for some bj ∈ [0, p − 1] (note the minus sign). We say that τ is δ-generic if

i i0 bj − bj ∈ [δ, p − 1 − δ] mod p − 1

f−1 for each 0 ≤ i 6= i0 ≤ 2 and 0 ≤ j ≤ f − 1. Write bi = P bi pj so that bi ≡ pj0 bi (j0) j=0 f−j+j0 (j0) (0) i mod e and Ai = −b(0) by deﬁnition. i If τ is 1-generic (which implies that the bj are unique) we deﬁne the orientation of τ to be f−1 f the element (sj)j=0 ∈ S3 such that

sj (0) sj (1) sj (2) bj+1 > bj+1 > bj+1 .

sj (0) sj (1) sj (2) This is equivalent to asking that b(j) > b(j) > b(j) . Deﬁnition 1.5.11. Let M be an object of Y [0,2],τ (R) of type τ with eigenbasis e. We deﬁne j e associated matrices Ae ∈ GL3(R[[v]]) (where v = u ) for 0 ≤ j ≤ f − 1 by the formula

sj+1(i) sj+1(i) j −b(j+1) −1 j b(j+1) Ae = Diagi(u )sj+1 Mate(ϕ) sj+1Diagi(u ) Chapter 1. Integral p-adic Hodge theory 32

Remark 1.5.12. It is easy to check that this matrix does actually lie in GL3(R[[v]]) by the deﬁni- tion of the orientation. The purpose of this deﬁnition is simply that the tables of [LLHLM18a] j are given in terms of the matrices Ae.

We now explain the notion of the shape of a Kisin module. Let I(F) ≤ GL3(F[[v]]) denote the subgroup of matrices that are upper-triangular mod v, and let Wf = NGL3 (T3)(F((v)))/T3(F[[v]]) a b c denote the extended aﬃne Weyl group of GL3 which is equal to the group of matrices Diag(v , v , v )ς for ς ∈ S3. Bruhat-Tits theory gives a double coset decomposition

G GL3(F((v))) = I(F)weI(F). we∈Wf

[0,2],τ f−1 f j Following [LLHLM18a], we say that M ∈ Y (F) has shape (wej)j=0 ∈ Wf if the matrix Ae of Deﬁnition 1.5.11 lies in I(F)wejI(F) for 0 ≤ j ≤ f − 1 for any choice of framed basis e.

If τ is 3-generic andρ ¯ : GK0 → GL3(F) is continuous then the proof of Theorem 3.2 in [0,2],τ ∗ ∼ [LLHLM18a] shows that if there exists M ∈ Y ( ) such that T (M) = ρ¯|G then M is F dd (K0)∞ f−1 unique up to isomorphism. This allows us to deﬁne the shape of the pair w(¯ρ, τ) = (wj(¯ρ, τ))j=0 to be the shape of M. The next theorem collects the main results from [LLHLM18a] that we need.

Theorem 1.5.13. Suppose that ρ is a GK0 -stable OE-lattice inside an n-dimensional potentially f−1 crystalline E-representation having inertial type τ and Hodge type µ = (2, 1, 0)j=0 . If M is a [0,2],dd ∗ ∼ strongly divisible module in S.D.Mod (OE) such that Tst(M) = ρ then there exists M in µ,τ −1 ∼ Y (OE) of rank n such that Θr(M) = M0. [0,2],τ −1 ∗ ∼ In particular there exists M ∈ Y ( ) such that T (M) = ρ¯|G so if τ is 3-generic F dd (K0)∞ the shape w(¯ρ, τ −1) is defined. j Moreover M has a framed basis e such that Ae is given by the matrix in the final column and −1 row corresponding to shape wj(¯ρ, τ ) ∈ Wf of Table 5 of [LLHLM18a], subject to the conditions j j there. In particular Ae = Ae mod $E must be the matrix given in the first column of the table. Finally, the monodromy condition in the first column of Table 7 loc. cit. holds on the entries j of Ae.

Proof. The ﬁrst paragraph follows from the uniqueness in the proof of Proposition 5.17 of

[CL18], the fact that Θ2 is an equivalence of categories, and Lemma 1.5.8. The second paragraph j follows immediately (see also [CL18] Corollary 5.18). The statement about Ae follows from µ,τ −1 the computation of the universal framed deformation of M to Y (R) for R flat over OE in Theorem 4.17 of [LLHLM18a]. The final statement regarding monodromy follows from Theorem 5.1 and Theorem 5.6 of [LLHLM18a] (see also the definition of the “monodromy condition” in Definition 5.7. Table 6 presents one of the equations coming from this monodromy condition; the monodromy condition in Table 7 is a consequence of the vanishing of the expression in Table 6). Chapter 1. Integral p-adic Hodge theory 33

Remark 1.5.14. Table 7 in [LLHLM18a] only contains results for some of the possible shapes. However, all the shapes we compute in Section 1.6 belong to this table.

1.6 Shape computations

i Letρ ¯ : GK0 → GL3(F) be a Galois representation satisfying (FLδ) and choose aj as in Deﬁnition 1.1.2. For the remainder of the thesis we will interested in the principal series types τ of Deﬁnition 1.6.1 whose choice is explained later in Remark 2.3.7. The goal in this section is to compute the shapes w(¯ρ, τ −1). The strategy is to use Breuil modules following a method simlar to that of [HLM17].

f−1 f Deﬁnition 1.6.1. Let σ = (σj)j=0 ∈ S3 and K ⊆ {0, . . . , f − 1}. We deﬁne a principal series L2 Ai type τ = τ(σ, K) by setting τ = i=0 ωef where

f−1 −1 X σf−k(i) −1 k Ai := af−k + δf−k+1∈/K (1 − σf−k+1(i)) p . k=0

The main result of this section is the computation of w(¯ρ, τ −1) in the next proposition.

Proposition 1.6.2. Let ρ¯ : GK0 → GL3(F) be a Galois representation satisfying (FLδ) with

δ ≥ 5. Assume that ρ¯ is the reduction mod p of a GK0 -stable OE-lattice inside a potentially f−1 crystalline E-representation of inertial type τ = τ(σ, K) and parallel Hodge type (2, 1, 0)j=0 (so the shape w(¯ρ, τ −1) is defined according to Theorem 1.5.13). Let M be a Fontaine-Laffaille [0,p−2] ∗ module in FL (F) such that Tcris(M) =ρ ¯ and define xj, yj, zj ∈ F as in Corollary 1.2.8.

∗∗ 1. If ρ¯ satisﬁes (FLδ ) then  −1 w0v if j + 2 ∈ K wj(¯ρ, τ ) = Diag(v2, v, 1) if j + 2 ∈/ K.

 id j ∈ J 2. Assume that K = ∅ and σj = for some J ⊆ {0, . . . , f − 1}. Then (01) j∈ / J

 2 −1 sβDiag(v , v, 1) if j + 1 ∈ ∂J and xj+1 = 0 wj(¯ρ, τ ) = Diag(v2, v, 1) else. Chapter 1. Integral p-adic Hodge theory 34

 id j ∈ J 3. Assume that K = ∅ and σj = for some J ⊆ {0, . . . , f − 1}. Then (12) j∈ / J

 2 −1 sαDiag(v , v, 1) if j + 1 ∈ ∂J and zj+1 = 0 wj(¯ρ, τ ) = Diag(v2, v, 1) else.

 ∗ id j ∈ J 4. Assume that ρ¯ satisﬁes (FLδ), that σj = for some J ⊆ {0, . . . , f − 1} and (02) j∈ / J that ∂J ⊆ K − 1. Then  w0v if j + 1 ∈ ∂J   −1 sαsβv if j + 1 ∈/ ∂J, j + 2 ∈ K and yj+1 = 0 wj(¯ρ, τ ) = s s v if j + 1 ∈/ ∂J, j + 2 ∈ K and y β − x z = 0  β α j+1 j+1 j+1 j+1  Diag(v2, v, 1) if j + 2 ∈/ K.

Proof. Write ρ for the stable OE-lattice in the statement of the proposition and let M ∈ [0,2],dd K0,2 ∼ S.D.Mod (OE) be a strongly divisible module of type τ such that Tst (M) = ρ. Then K0,2 ∼ setting M = S(M) we have Tst (M) = ρ¯ by (1.4.17). We refer to [HLM17] Section 2.3 for definitions concerning the notion of Breuil submodule. The key fact we want to use is Proposi- tion 2.3.5 loc. cit., which implies that sinceρ ¯ is upper-triangular, we may find a filtration 0 ⊂ P (ai +1)pj M ⊂ M ⊂ M = M by Breuil submodules such that T K0,2(M /M )| = ω j f−j 2 1 0 st i i+1 IK0 f for 0 ≤ i ≤ 2 (the notion of a quotient of Breuil submodules is explained in the reference cited).

[0,2],dd Since Mi/Mi+1 is in particular a rank 1 object of BrMod (F), the classiﬁcation of rank j j 1 Breuil modules in Lemma 3.3.2 of [EGH13] together with the inclusion Mi /u ,→ M /u and j Lemma 1.4.19 imply that there exist for each 0 ≤ j ≤ f − 1 and 0 ≤ i ≤ 2 integers ri ∈ [0, 2e] j j rj j j j j j and ψj(i) ∈ {0, 1, 2} such that (Mi /Mi+1)2 = u i (Mi /Mi+1), and Mi /Mi+1 has a S -basis A ψj (i) upon which ∆ acts by ωf , and we have

j j ri ≡ p Aψj+1(i) − Aψj (i) mod e (1.6.3) and X 1 X (ai + 1)pj ≡ A + rjpf−j mod e. (1.6.4) f−j ψ0(i) e i j j

Observe that despite the notation, ψj is not a priori a permutation. We now exploit these two Chapter 1. Integral p-adic Hodge theory 35 congruences. Observe that by (1.6.3) we may write

−1 −1 j j X σf−k+j (ψj+1(i)) σf−k+j (ψj (i)) ri = δi e + af−k+j − af−k+j k −1 −1 k + δf−k+j+1∈/K (σf−k+j+1(ψj(i)) − σf−k+j+1(ψj+1(i))) p (1.6.5)

j where δi ∈ [0, 2]. A rather unpleasant computation now shows that

−1 1 X X σ (ψf−j (i)) A + rjpf−j = a f−j + δ (1 − σ−1 (ψ (i))) pj ψ0(i) e i f−j f−j+1∈/K f−j+1 f−j j j X j f−j + δi p . j

Hence by (1.6.4) we obtain

−1 X i j X σf−j (ψf−j (i)) f−j −1 j (af−j + 1)p ≡ af−j + δi + δf−j+1∈/K (1 − σf−j+1(ψf−j(i)) p mod e. j j

By 3-genericity we deduce that ψj = σj for each 0 ≤ j ≤ f − 1, and also  1 if j + 1 ∈ K δj = i −1 σj+1σj(i) if j + 1 ∈/ K.

At this point we pause to introduce some new notation. For 0 ≤ i, i0 ≤ 2 deﬁne

−1 −1 0 j X σf−k+j (σj+1(i)) σf−k+j (σj (i )) k −1 0 −1 k dii0 = af−k+j −af−k+j )p +δf−k+j+1∈/K (σf−k+j+1(σj(i ))−σf−k+j+1(σj+1(i)) p k and

−1 −1 0 j X σf−k+j (σj (i)) σf−k+j (σj (i )) k −1 0 −1 0 k bii0 = af−k+j −af−k+j )p +δf−k+j+1∈/K (σf−k+j+1(σj(i ))−σf−k+j+1(σj(i )) p . k

h j i j j h j i j j 0 0 Then one checks that p (Aσj+1(i) − Aσj (i )) = uii0 e+dii0 and p (Aσj (i) − Aσj (i )) = vii0 e+bii0 , where we deﬁne  −1 0 j 0 if σj+1σj(i ) ≤ i u 0 = ii −1 0 1 if σj+1σj(i ) > i and  −1 0 −1 j 0 if σj+1σj(i ) ≤ σj+1σj(i) v 0 = ii −1 0 −1 1 if σj+1σj(i ) > σj+1σj(i). Chapter 1. Integral p-adic Hodge theory 36

Note that in this new notation (1.6.5) becomes

j j j ri = δi e + dii.

A A A j j j j j σj (0) σj (1) σj (2) j We deduce that M has a basis e = (e0, e1, e2) of type ωf ⊕ ωf ⊕ ωf and M2 A A A j j j j σj+1(0) σj+1(1) σj+1(2) has a system of generators f = (f0 , f1 , f2 ) of type ωf ⊕ ωf ⊕ ωf such that

 δj e+dj  u 0 00 j j j j j j Mat (M ) =  eu01+d01 δ1e+d11  e,f 2 x01u u  j euj +dj j euj +dj δj e+dj x02u 02 02 x12u 12 12 u 2 22 and  j+1  α00 j j+1 evj+1+bj+1 j+1 Mat (ϕ ) =  01 01  e,f 2 α01 u α11  j+1 evj+1+bj+1 j+1 evj+1+bj+1 j+1 α02 u 02 02 α12 u 12 12 α22

j j+1 e ep j+1 e ep × j j+1 where xii0 , αii0 ∈ F[u ]/(u ) and αii ∈ (F[u ]/(u )) (because ϕ2(M2) generates M ). This concludes the preliminary analysis of the structure of M. We next do a base change to reﬁne these matrices. j j For the next claim, note that δi0 − uii0 ∈ [0, 1] if i 6= 2. j e(δj −uj ) i0 ii0 j Claim 1.6.6. We may assume that xii0 ∈ F + Fu and that Mate,f (ϕ2) is diagonal with j+1 × entries αii ∈ F for each 0 ≤ j ≤ f − 1.

Proof. Set  δj e+dj  u 0 00 j 0j j j j j V =  eu01+d01 δ1e+d11  x01u u  0j euj +dj 0j euj +dj δj e+dj x02u 02 02 x12u 12 12 u 2 22 and  j  α00 Bj =  j j  . β01 α11  j j j β02 β12 α22

0j j e ep j−1 j j j where xii0 , βii0 ∈ F[u ]/(u ) are unknowns. The equation Mate,f (ϕ2) V = Mate,f (M2) B reduces to the equations

j δj e+dj euj +dj j e(δj +vj −uj ) j 0j j j 1. β01u 1 11 = u 01 01 [α01u 0 01 01 + α11x01 − α00x01]

j δj e+dj euj +dj j e(δj +vj −uj ) j 0j j j 2. β12u 2 22 = u 12 12 [α12u 1 12 12 + α22x12 − α11x12]

j δj +dj euj +dj j e(δj +vj −uj ) j 0j e(vj +uj −uj ) j 0j j j j j euj +dj 3. β02u 2 22 = u 02 02 [α02u 0 02 02 +α12x01u 12 01 02 +α22x02−α00x02]−β01x12u 12 12 .

j j j 0 j j j using that bii0 + dki − dki0 = 0 for any 0 ≤ i, i , k ≤ 2. One checks easily that δi + vii0 − uii0 ≥ 0 Chapter 1. Integral p-adic Hodge theory 37

0 j j j 0 for i 6= i and v12 + u01 − u02 ≥ 0 so these equations make sense. For i < i deﬁne

j j cii0 = di0i0 − dii0 −1 0 −1 −1 X σf−k+j (σj+1(i )) σf−k+j (σj+1(i)) = af−k+j − af−k+j k −1 −1 0 k + δf−k+j+1∈/K (σf−k+j+1σj+1(i) − σf−k+j+1σj+1(i )) p .

This lies in (0, e). j 0j e(δj −uj ) Since α11 is a unit we can pick x01 ∈ F+Fu 1 01 so that the expression in square brackets j j e(1+δ1−u ) j e−c e ep in the ﬁrst equation is divisible by u 01 . Then we can choose β01 ∈ u 01 F[u ]/(u ) 0j e(δj −uj ) j e−cj e ep satisfying the equation. Similarly we can pick x12 ∈ F + Fu 2 12 and β12 ∈ u 12 F[u ]/(u ) j e−c01 ˜j j j j satisfying the second equation. Write β01 = u β01. As d12 −c01 −d02 = 0 the third equation becomes

j j j j j j j j j j j δ2+d22 eu02+d02 j e(δ0+v02−u02) j 0j e(v12+u01−u02) β02u = u [α02u +α12x01u j j j 0j j j ˜j j e(1+u12−u02) + α22x02 − α00x02 − β01x12u ].

j 0j e(δj −uj ) Since α22 is a unit we can pick x02 ∈ F + Fu 2 02 so that the expression in square brackets e(1+δj −uj ) j e−cj e ep is divisible by u 2 02 . Then we can pick β02 ∈ u 02 F[u ]/(u ) satisfying the equation. Now note that for i < i0 we have

j i0 i −1 0 −1 p(e − cii0 ) = e(p − 1 − aj+1+aj+1 + δj+2∈/K (σj+2σj+1(i ) − σj+2σj+1(i))) −1 0 −1 X σf−k+j+1σj+1(i ) σf−k+j+1σj+1(i) + p − 1 − af−k+j+1 + af−k+j+1 k −1 0 −1 k + δf−k+j+2∈/K (σf−k+j+2σj+1(i ) − σf−k+j+2σj+1(i)) p .

j × 3e This is ≥ 3e by 5-genericity. Thus ϕ j (B ) is diagonal with entries in modulo u . Thus S F performing the change of basis in Lemma 1.4.21 allows us to reset all notation and assume j (δj −uj )e j+1 j+1 j+1 i0 ii0 j 3e from the beginning that xii0 ∈ F + Fu and A ≡ Diag(α00 , α11 , α22 ) mod u j+1 × where αii ∈ F . We now perform a second change of basis. This time let

 δj e+dj  u 0 00  αj j j j j j  j 00 eu01+d01 δ1e+d11 V =  j x01u u   α11   αj j j j αj j j j j j  00 eu02+d02 11 eu12+d12 δ2e+d22 j x02u j x12u u α22 α22 and j j j j B = Diag(α00, α11, α22).

j−1 j j j 3e Then clearly we have Mate,f (ϕ2) V ≡ Mate,f (M2) B mod u and making the change of basis in Lemma 1.4.21 proves the claim. Chapter 1. Integral p-adic Hodge theory 38

j j e(δj −uj ) j i0 ii0 Claim 1.6.7. Building on Claim 1.6.6 we can in fact take xii0 = yii0 u with yii0 ∈ F obeying  j × y01 ∈ F xj   j × y ∈ F yj 02 (1.6.8) j × j y12 ∈ F z   j j j × y02 − y01y12 ∈ F (yjβj − xjzj)

∗ ∗ ∼ 0 Proof. Let M denote the twist of M(M ) such that V (M) = ρ¯ |G . It follows from (K0)∞ j A00−A0 A00−A1 A00−A2 j Lemma 1.5.9 that M, has a basis e of type ωf ⊕ ωf ⊕ ωf such that ϕM (e ) = 0 ∆=1 j j+1 t j+1 t j −1 −1 0j j [p (Ai−A00)] e σj+2( V )( A ) σj+1. Hence M := M has a basis e = e Diagi(u ) such that

0j 0j+1 −[pj+1(A −A )] t j+1 t j −1 −1 p[pj (A −A )] 0 i 00 i 00 ϕM (e ) = e · Diagi(u )σj+2( V )( A ) σj+1Diagi(u ) −[pj+1(A −A )] 0j+1 σj+2(i) 00 = e · σj+2Diagi(u )  j+1 −1 δj+1e+dj+1 j+1 −1 j+1 euj+1+dj+1 j+1 −1 j+1 euj+1+dj+1  (α00 ) u 0 00 (α11 ) x01 u 01 01 (α22 ) x02 u 02 02 j+1 j+1 j+1 j+1 j+1 j+1 j+1 ·  −1 δ1 e+d11 −1 eu12 +d12   (α11 ) u (α22 ) x12 u  j+1 −1 δj+1e+dj+1 (α22 ) u 2 22 p[pj (A −A )] σj+1(i) 00 −1 · Diagi(u )σj+1

Now using the fact that

j j+1 j+1 p[p (Aσ (i0) − A00)]+d 0 − [p (Aσ (i) − A00)] j+1 ii j+2 (1.6.9) i0 0 −1 0 = e(aj+1 − aj+1 + 1 + δj+2∈/K (1 − σj+2σj+1(i )))

j+1 −1 as well as the fact that δi + 1 + δj+2∈/K (1 − σj+2σj+1(i)) = 2 for each 0 ≤ i ≤ 2 the equation above becomes

 j+1 uj+1−δj+1 j+1 uj+1−δj+1  1 x01 v 01 1 x02 v 02 2 0j 0j+1 j+1 j+1 j+1 j+1 −1 ai −a0 +2 −1 ϕ 0 (e ) = e ·σ  u12 −δ2  Diag (α ) v j+1 j+1 σ . M j+2  1 x12 v  i ii j+1 1

Making the change of basis

 j uj −δj j uj −δj  1 x01v 01 1 x02v 02 2 00j 0j j j j e = e σ  u12−δ2  j+1  1 x12v  1 Chapter 1. Integral p-adic Hodge theory 39 we get

 j uj −δj j uj −δj  1 φ(x01v 01 1 ) φ(x02v 02 2 ) 00j 00j+1 j+1 −1 ai −a0 +2 j j j ϕ 0 (e ) = e · Diag (α ) v j+1 j+1  u12−δ2  . M i ii  1 φ(x12v ) 1

j j uj −δj ii0 i0 −p Note by Claim 1.6.6 gii0 := φ(xii0 v ) lies in Fv + F. On the other hand, we know from j Lemma 1.5.3 that there must exist matrices X ∈ GL3(F((v))) for 0 ≤ j ≤ f − 1 such that

 j+1 −1 2 j+1 −1 j 2 j+1 −1 j 2  (α00 ) v (α00 ) g01v (α00 ) g02v j+1 1 0 j+1 j 1 0  −1 aj+1−aj+1+2 −1 aj+1−aj+1+2 j (α11 ) v (α11 ) g12v φ(X )  2 0  j+1 −1 aj+1−aj+1+2 (α22 ) v (1.6.10) αjv2 xjv2 yjv2  1 0 1 0 j+1  j aj+1−aj+1+2 j aj+1−aj+1+2 = X  β v z v  . a2 −a0 +2 γjv j+1 j+1

j j We now show that (1.6.10) implies X ∈ T3(F) and gii0 ∈ F. This is enough to prove Claim j j j j 1.6.7. Write X = (fii0 )0≤i,i0≤2 for fii0 ∈ F((v)). First one shows that X is upper-triangular. For example, from the bottom left entry of (1.6.10) we obtain the equations

−1 2 0 j+1 j+1 aj+1−aj+1 j f20 = α22 αj v φ(f20)

j which imply each f20 = 0 by looking at the ﬁrst nonzero term of positive degree and the term of lowest degree. Once it has been shown that Xj is upper-triangular, it is easy to see that × its diagonal coeﬃcients lie in F by looking at the diagonal entries of (1.6.10). Finally the claim will then follow from arguments like that in Example 1.6.11 by considering the remaining entries of (1.6.10).

Example 1.6.11. Let us show that if fj ∈ F((v)) for 0 ≤ j ≤ f − 1 obey the equations −d −p × fj+1 = v j [cjφ(fj) + ζjv + θj] for some dj ∈ [2, p − 2], cj ∈ F and ζj, θj ∈ F then each fj, ζj, θj = 0.

If f0 has terms of positive degree and the minimal such degree is n0 > 0, then the minimal positive degree term of fj is nj deﬁned recursively by nj+1 = pnj − dj. But then the equation n0 = nf is impossible since each dj ≤ p − 2. This shows that each fj contains terms of nonpositive degree only. If the smallest term of f0 is of degree m0 ≤ −2, then the smallest term of fj will be of degree mj where mj+1 = pmj − dj. But again the equation m0 = mf is −1 impossible. This implies that each fj ∈ Fv + F. But since dj ≥ 2 it is now easy to see that this implies ζj, θj, fj = 0.

From Claim 1.6.7 we conclude that in the original Breuil module M with e, f as before we Chapter 1. Integral p-adic Hodge theory 40 actually have

 δj e+dj  u 0 00 j j j j j j j j+1 j+1 j+1 Mat (M ) =  δ1e+d01 δ1e+d11  and Mat (ϕ ) = Diag(α , α , α ) e,f 2 y01u u  e,f 2 00 11 22 j δj e+dj j δj e+dj δj e+dj y02u 2 02 y12u 2 12 u 2 22

j ∗ [0,2],τ −1 where yii0 ∈ F satisfy (1.6.8). Deﬁne M = M(M ) ∈ Y . It follows from Lemma 1.5.9 that M has a framed basis e of type τ −1 such that

 j+1 −1 δj+1e+dj+1 j+1 −1 j+1 δj+1e+dj+1 j+1 −1 j+1 δj+1e+dj+1  (α00 ) u 0 00 (α11 ) y01 u 1 01 (α22 ) y02 u 2 02 j j+1 j+1 j+1 j+1 j+1 j+1 j+1 j+1 −1 ϕ(e ) = e ·σ  −1 δ1 e+d11 −1 δ2 e+d12  σ . j+2  (α11 ) u (α22 ) y12 u  j+1 j+1 −1 δj+1e+dj+1 (α22 ) u 2 22

j i −1 In the notation of Definition 1.5.10 we have bi = aj + δj+1∈/K (1 − σj+1(i)) and it is easy to j see that the orientation is sj = σj+1w0. (Technically, we defined bi ∈ [0, p − 1] following i [LLHLM18a] but this doesn’t matter because we can shift each aj by the same multiple of p − 1 to achieve this.) Applying Definition 1.5.11 we compute

σj+2(i) σj+1(i) j −b(j+1) j b(j+1) −1 Ae = w0Diagi u Mate(ϕM ) Diagi u σj+1σj+2w0

0 j+1 σj+2(i) σj+1(i ) and using the deﬁnition dii0 = b(j+1) − b(j+1) this becomes

j+1 j+1  −1   δ0  (α22 ) v j j+1 −1 j+1 j+1 −1 j+1 −1 A =   w  δ1  σ σ w . e (α22 ) y12 (α11 )  0  v  j+1 j+2 0 j+1 −1 j+1 j+1 −1 j+1 j+1 −1 δj+1 (α22 ) y02 (α11 ) y01 (α00 ) v 2

If j + 2 ∈ K then the diagonal matrix is constant = v. If j + 2 ∈/ K then this is equal to

 j+1 −1   2  (α22 ) v  j+1 −1 j+1 j+1 −1  w σ−1 σ w   . (α22 ) y12 (α11 )  0 j+1 j+2 0  v  j+1 −1 j+1 j+1 −1 j+1 j+1 −1 (α22 ) y02 (α11 ) y01 (α00 ) 1

∗ ∼ ∗ ∗ ∼ K0,2 ∼ Since Tdd(M) = Tst(M ) = Tst (M) = ρ¯ using Lemma 1.5.9(2), the four statements of Proposition 1.6.2 now follow from (1.6.8) and the cases in Lemma 1.6.12. This completes the proof of Proposition 1.6.2.

The next lemma was used above. Its proof is an elementary computation.

a    Lemma 1.6.12. Let A = u b  w0ςw0 ∈ GL3(F) where ς ∈ S3. Write [·] for the image t w c in the double coset space B3(F)\ GL3(F)/B3(F). Chapter 1. Integral p-adic Hodge theory 41

1. If uw − bt, u, w, t 6= 0 then [A] = [w0] for any ς. If ς = w0 then the same is always true.

2. Assume that ς = id and u, w 6= 0. If t = 0 then [A] = [sαsβ]. If uw − bt = 0 then

[A] = [sβsα].

a    2 Now let B = u b  w0ςw0Diag(v , v, 1) ∈ GL3(F((v))) where ς ∈ S3 and a, b, c, u, w, t ∈ t w c F. Write [·] for the image in the double coset space I(F)\ GL3(F((v)))/I(F).

1. We have [B] = [Diag(v2, v, 1)] in any of the following cases:

• ς = id,

• ς = sα and w 6= 0,

• ς = sβ and u 6= 0, • ς is arbitrary and u, w, t, uw − bt 6= 0.

2 2. If ς = sα and w = 0 then [B] = [sβDiag(v , v, 1)]. If ς = sβ and u = 0 then [B] = 2 sα[Diag(v , v, 1)].

1.7 Main results

Letρ ¯ : GK0 → GL3(F) be a continuous representation satisfying (FLδ) and let τ = τ(σ, K) denote an inertial types from Deﬁnition 1.6.1. In this section we show our main results relating K0,2 the Fontaine-Laﬀaille invariants of Section 1.2 to Frobenius eigenvalues of Dst (ρ) for potentially crystalline lifts ρ of Hodge type µ and inertial type τ. Given the shape computations of the previous section, the proofs of these results are simply a matter of combining the tables of [LLHLM18a] with the calculations of Sections 1.4-1.5.

∗∗ Theorem 1.7.1. Assume ρ¯ ∈ (FLδ ) with δ ≥ 5 and let ρ be a potentially crystalline of ρ¯ of Hodge type µ and inertial type τ = τ(σ, K). In particular the invariants of Deﬁnition 1.2.14 K0,2 f make sense. Write D = Dst (ρ) and let λi ∈ E denote the eigenvalue of ϕD on the isotypic A ∆=ω i component D ef .

1. Assume that  id j ∈ J σj = (1.7.2) (02) j∈ / J

for some J ⊆ {0, . . . , f − 1}. Then

f −1 |S| p λ1 = (−1) YS Chapter 1. Integral p-adic Hodge theory 42

where S = {j | (j ∈ ∂J and j + 1 ∈/ K) or (j∈ / ∂J and j + 1 ∈ K)} . (1.7.3)

2. Assume that  id j ∈ J σj = (1.7.4) (01) j∈ / J

for some J ⊆ {0, . . . , f − 1} and K = ∅. Then

1 2f−|J| −1 f−|J|+ |∂J| p λ1 = (−1) 2 XJc .

3. Assume that  id j ∈ J σj = (1.7.5) (12) j∈ / J

for some J ⊆ {0, . . . , f − 1} and K = ∅. Then

1 f−|J| −1 |J|+ |∂J| p λ2 = (−1) 2 ZJc .

Notation 1.7.6. For any of the types τ appearing in this theorem, we deﬁne mτ ∈ Z, iτ ∈ × {0, 1, 2} and FLτ ∈ F so that the conclusion of the theorem in each case is that

−1 pmτ λ = FL . iτ τ

Proof. We will do the calculation at ﬁrst without any assumptions on σ or K. Let M be a −1 ∗ ∼ strongly divisible module of type τ such that Tst(M) = ρ. By Theorem 1.5.13 there exists µ,τ −1 ∼ M ∈ Y (OE) such that Θ2(M) = M0. By Proposition 1.6.2, we know that M has a framed j basis e such that Ae is given by the ﬁnal column of Table 5 in [LLHLM18a] in row w0v if j + 2 ∈ K and Diag(v2, v, 1) if j + 2 ∈/ K. That is

• if j + 2 ∈ K we have

 j j j ∗j −1 j e ∗j   e ∗j  c00 c00c21(c20) c02 + (u + p)c02 u c¯02 Aj =  e ∗j e j  ; A¯j =  e ∗j e j  e  0 (u + p)c11 (u + p)c12  e  u c¯11 u c¯12 e ∗j e j j e 0j e ∗j e j e 0j u c20 u c21 c22 + (u + p)c22 u c¯20 u c¯21 u c¯22 (1.7.7) j ∗j × for cii0 ∈ OE, cii0 ∈ OE subject to all condition coming from the requirement that the j j j j j ∗j reduction mod $E of Ae be the matrix Ae given, as well as the equations c00c22 = −pc02c20 j 0j ∗j ∗j ∗j ∗j j j and c00c22 − c02c20 = −pc02c20. In particular ordp(c00), ordp(c22) > 0. Chapter 1. Integral p-adic Hodge theory 43

• if j + 2 ∈/ K we have

 e 2 ∗j   2e¯∗j  (u + p) b00 0 0 u b00 0 0 Aj =  e e j e ∗j  ; A¯j =  2e¯j e¯∗j  (1.7.8) e  u (u + p)b10 (u + p)b11 0  e u b21 u b11 0  e j e 0j e j ∗j 2e¯0j e¯j ¯∗j u (b20 + (u + p)b20) u b21 b22 u b20 u b21 b22

j ∗,j × for bii0 ∈ OE, bii0 ∈ OE subject to all equations coming from the requirement that the j j reduction mod $E of Ae is the matrix Ae given. Note that we have applied the monodromy ¯j equation b20 = 0 in accordance with Theorem 1.5.13.

According to Deﬁnition 1.5.11 and Corollary 1.5.7, M has a framed system of generators (e, f) j such that Mate,f (ϕ2) = c2 · id and

sj (i) E(u)2 sj (i) j −b(j) j−1 adj b(j) −1 Mate,f (M2) = sjDiagi u j−1 (A ) Diagi u sj det(Ae )

−1 i σj (i) −1 where sj = σj+1w0 and bj = aj + δj+1∈/K (1 − σj+1(i)) is the notation of Section 1.5, and adj denotes the adjugate matrix. By Lemma 1.4.14, D has a framed basis e0 of type τ such that

j −1 j Mate0 (ϕD) = s0(c2) · s0 Mate,f (M2) .

[Note the conﬂict between two uses of s0 here.] One easily computes that the right hand side of this formula is  −cj−1 2  −1 −1 00 p p s0(c2) · σ Diag ∗(j−1) ∗(j−1) , ∗(j−1) , j−1 σj+1 j + 1 ∈ K  j+1 c c c c 02 20 11 00 (1.7.9) 2 s (c )−1 · σ−1 Diag −p , p , −1 σ j + 1 ∈/ K.  0 2 j+1 ∗(j−1) ∗(j−1) ∗(j−1) j+1 b22 b11 b00

In particular we now easily calculate that

−f f Q ∗(j−1) −1 Q ∗(j−1) −1 • if (1.7.2) holds then λ1 = s0(c2) p j+1∈K (c11 ) j+1∈/K (b11 )

−f 2f−|J| f−|J| Q ∗j −1 Q ∗j −1 • if (1.7.4) holds and K = ∅ then λ1 = s0(c2) p (−1) j+1∈J (b11) j+1∈/J (b22)

−f f−|J| |J| Q ∗j −1 Q ∗j −1 • if (1.7.5) holds and K = ∅ then λ2 = s0(c2) p (−1) j+1∈J (b00) j+1∈/J (b11) .

The next step is to compute the Fontaine-Laﬀaille module ofρ ¯0 in terms of these matrix j j j j entries. Write Ae = F D where F ∈ GL3(F) and  ue j + 2 ∈ K Dj := Diag(u2e, ue, 1) j + 2 ∈/ K.

∆=1 b0 b1 b2 Letting e also denote the framed basis of M, a basis of M is given by e0j := ejDiag(u (j) , u (j) , u (j) ) Chapter 1. Integral p-adic Hodge theory 44 with respect to which we calculate

0j 0j+1 j −1 e(ai +1) −1 0 j+1 ϕM (e ) = e · sj+1F w0σj+2σj+1Diag(u )σj+1 in a manner similar to the computations surrounding (1.6.9). With respect to the basis e00j = 0j j−1 −1 e sjF sj sj−1w0 we have

j e(1+ai ) j−1 −1 00 0 j+1 Mate (ϕM ) = Diagi(u )w0F w0σj+1σj. (1.7.10)

∗ ∼ Let M0 denote the Fontaine-Laffaille module such that Tcris(M0) = ρ¯. By the full faithfulness of F, Lemma 1.5.3(2) and (1.7.10) we deduce that M0 has a basis e0 adapted to the filtration j j−1 −1 such that Mate0 (ϕ) = w0F w0σj+1σj. By Lemma 1.2.4 and (1.2.5) there exists a unique j j j j−1 −1 lower-unipotent matrix L with coefficients in F such that Fe := L w0F w0σj+1σj is the upper-triangular matrix of Corollary 1.2.9. Table 1.7.11 computes Fej+1.

Table 1.7.11: The Fontaine-Laﬀaille matrix of Corollary 1.2.8 forρ ¯. −1 j+1 j+1 σj+2σj+1 Fe if j + 2 ∈ K Fe if j + 2 ∈/ K  0j j ∗j  c¯22 c¯21 c¯20  ∗j j 0j  j j j ∗j ¯b ¯b ¯b ∗j c¯ c¯ −c¯ c¯ 22 21 20  c¯ − 12 21 12 20  ∗j j id  11 c¯0j c¯j   ¯b ¯b   22 22   11 10 ∗j ∗j ∗j ∗j  c¯02c¯11c¯20  ¯b ∗j 0j j j 00 c¯11c¯22−c¯21c¯12  0j ∗j j   j ∗j 0j  c¯22 c¯20 c¯21 ¯b ¯b ¯b ∗j j j j 21 22 20 −c¯ c¯ c¯ c¯ ∗j ∗j 0j ∗j  20 12 ∗j 12 21   −¯b ¯b j ¯b ¯b  (01)  0j c¯11 − 0j  22 11 ¯b − 20 11  c¯22 c¯22   ¯bj 10 ¯bj   −c¯∗j c¯∗j   21 21  02 11 ¯∗j j b00 c¯12  j 0j ∗j  c¯ c¯ c¯ ¯∗j ¯0j ¯j  21 22 20 b22 b20 b21 c¯∗j c¯0j −c¯∗j c¯∗j  c¯j − 11 22 11 20   ¯bj ¯b∗j  (12)  12 c¯j c¯j   10 11   21 21  ¯∗j¯∗j  −c¯∗j c¯∗j c¯∗j   −b11b00  02 11 20 ¯j ∗j 0j j j b10 c¯11c¯22−c¯21c¯12  0j ∗j j   ∗j 0j j  ¯b ¯b ¯b c¯20 c¯22 c¯21 20 22 21 −¯bj ¯b∗j ¯bj ¯bj  j ∗j   10 22 ¯∗j 10 21  (012) c¯12 c¯11  ¯0j b11 − ¯0j   ∗j ∗j   b20 b20   −c¯02c¯11   −¯b∗j¯b∗j  j 00 11 c¯12 ¯j b10  j ∗j 0j  ¯j ¯0j ¯∗j  c¯ c¯ c¯ b21 b20 b22 21 20 22 ∗j 0j ∗j ∗j ∗j ∗j 0j ∗j ¯b ¯b −¯b ¯b  −c¯ c¯ j c¯ c¯   ¯j 11 20 11 22  (021) 11 20 c¯ − 22 11  b10 − ¯j ¯j   c¯j 12 c¯j   b21 b21   21 21  ¯∗j¯∗j¯∗j j  −b00b11b22  c¯02 ¯ ¯0j ¯j ¯j b11b20−b10b21 ¯j ¯j ¯∗j   ∗j j 0j  b20 b21 b22 c¯ c¯ c¯ j j j ∗j 20 21 22 ∗j ¯b ¯b −¯b ¯b ∗j j  ¯b − 10 21 10 22  (02)  c¯ c¯   11 ¯b0j ¯b0j   11 12  20 20  ∗j ¯∗j¯∗j¯∗j c¯  b00b11b22  02 ¯j ¯j ¯0j ¯∗j b21b10−b20b11

From the matrices in this table it is possible to compute the Fontaine-Laﬀaille invariants ofρ ¯ Chapter 1. Integral p-adic Hodge theory 45 directly. We obtain

f−|S| Q ∗j Q ¯∗j • if (1.7.2) holds then deﬁning S as in (1.7.3) we have YS = (−1) j+2∈K c¯11 j+2∈/K b11.

1 |∂J| Q ¯∗j Q ¯∗j • if (1.7.4) holds and K = ∅ then XJc = (−1) 2 j+2∈J b11 j+2∈/J b22.

1 |∂J| Q ¯∗j Q ¯∗j • if (1.7.5) holds and K = ∅ then ZJc = (−1) 2 j+2∈J b00 j+2∈/J b11.

By comparing these items with the Frobenius eigenvalues computed just after (1.7.9) we complete the proof of Theorem 1.7.1.

Proposition 1.7.12. Assume ρ¯ ∈ (FLδ) with δ ≥ 5 and let ρ be a potentially crystalline lift of K0,2 ρ¯ of Hodge type µ and inertial type τ = τ(σ, K). Write D = Dst (ρ) and let λi ∈ E denote Ai f ∆=ωef the eigenvalue of ϕD on the isotypic component D .

1. Assume that (1.7.4) holds and K = ∅. Then

X X X ordp(λ1) = f + 1 + 1 + j j∈ /J j∈J j∈∂J j+1∈/J j+1∈/J xj =0 xj 6=0

where j ∈ (0, 1).

2. Assume that (1.7.5) holds and K = ∅. Then

X X X ordp(λ2) = 1 + 1 + j j∈ /J j∈J j∈∂J j+1∈/J j+1∈/J zj =0 zj 6=0

where j ∈ (0, 1).

∗ 3. Assume that ρ¯ ∈ (FLδ) and (1.7.2) holds and ∂J ⊆ K − 1. Then

X X 0 ordp(λ1) = f + j − j j∈ /∂J j∈ /∂J j+1∈K j+1∈K yj =0 yj βj −xj zj =0

0 where j, j ∈ (0, 1).

Proof. The method of proof is the same as the previous theorem, the diﬀerence being that the vanishing of the various Fontaine-Laﬀaille parameters implies that w(¯ρ, τ) will take different values according to Proposition 1.6.2. We will show part 3 only, since the others are similar. Choose M and M as in the proof of Theorem 1.7.1. By Proposition 1.6.2(4) and the j corresponding column of Table 5 in [LLHLM18a] we can choose e and Ae such that

j • If j + 1 ∈ ∂J then Ae is exactly as in (1.7.7). Chapter 1. Integral p-adic Hodge theory 46

j • If j + 2 ∈/ K then Ae is exactly as in (1.7.8).

• If j + 1 ∈/ ∂J and yj+1 = 0 then

 j j ∗j −1 j j e ∗j   e ∗j  c20c01(c21) c01 c02 + (u + p)c02 0 0 u c¯02 Aj =  e ∗j j j e 0j  ; A¯j =  e ∗j e 0j  e  u c10 c11 c12 + (u + p)c12  e u c¯10 0 u c¯12 e j e ∗j j j ∗j −1 e 0j e ∗j e 0j u c20 u c21 c20c12(c10) + (u + p)c22 0 u c¯21 u c¯22

j j ∗j ∗j j j j j ∗j j 0j j ∗j ∗j satisfying the equations c11c20 = −pc10c21, c01c12 = c11c02, and c21c02 − c22c01 = pc21c02. j In particular ordp(c20) ∈ (0, 1).

• If j + 1 ∈/ ∂J and yj+1βj+1 − xj+1zj+1 = 0 then

 j ∗j −1 j j e ∗j j   e ∗j  c00 (c20) c00c21 + (u + p)c01 c02 0 u c¯01 0 Aj =  e 0j e ∗j  ; A¯j =  e ∗j   0 (u + p)c11 (u + p)c12   0 0 u c¯12 e ∗j e j j e 0j e ∗j e j e 0j u c20 u c21 c22 + (u + p)c22 u c¯20 u c¯21 u c¯22

j j ∗j j 0j j 0j 0j j ∗j ∗j ∗j ∗j satisfying the equations c00c22 = −pc20c02 and c11c00c22 − c11c02c20 = pc12c01c20. In 0j particular ordp(c11) ∈ (0, 1).

Exactly as in (1.7.9) we compute the Frobenius on D as

j s0(c2) · Mate(ϕD) =

 ( −c j−1) 2  t −1 00 p p t · σ Diag ∗(j−1) ∗(j−1) , ∗(j−1) , ( σj+1 j ∈ ∂J  j+1 c c c c j−1)  02 20 11 00  2 ·tσ−1 Diag −p , p , −1 tσ j + 1 ∈/ K  j+1 ∗(j−1) ∗(j−1) ∗(j−1) j+1 b22 b11 b00 ∗(j−1) −pc cj−1 −pcj−1 pcj−1  t −1 10 02 20 12 t · σ Diag ∗(j−1) , ∗(j−1) ∗(j−1) , ∗(j−1) σj+1 j∈ / ∂J, j + 1 ∈ K and yj = 0  j+1 cj−1cj−1c c c cj−1c  12 20 02 10 21 02 10  0(j−1)  −pcj−1c pcj−1 ·tσ−1Diag 02 11 , p , 22 tσ j∈ / ∂J, j + 1 ∈ K and y β − x z = 0.  j j−1 ∗(j−1) ∗(j−1) 0(j−1) j−1 ∗(j−1) j+1 j j j j c22 c01 c12 c11 c02 c20

This proves part 3 of the proposition.

Corollary 1.7.13. Let ρ¯ and ρ be as in the previous proposition. If there exists j0 such that xj0 = 0 then it is possible to choose J as in (1.7.4) and K = ∅ so that ordp(λ1) > 2f − |J|. If there exists j0 such that zj0 = 0 then it is possible to choose J as in (1.7.5) and K = ∅ so that ∗ ordp(λ2) > f −|J|. If ρ¯ ∈ (FLδ) and there exists j0 such that either yj0 = 0 or yj0 βj0 −xj0 zj0 = 0 then it is possible to choose J as in (1.7.2) and J ⊆ K so that ordp(λ1) 6= f.

Proof. If there exists j0 such that xj0 = 0, then sinceρ ¯ is nonsplit there must exist j1 such that xj1 6= 0. Choose J so that ∂J = {j0, j1} and j1 ∈ J. Then from Proposition 1.7.12(1) c we obtain ordp(λ1) = f + |J | + j0 > 2f − |J| and the claim follows. The proof of the second statement is similar using Proposition 1.7.12(2). Chapter 1. Integral p-adic Hodge theory 47

∗ Ifρ ¯ ∈ (FLδ) then pick J = ∅ so ∂J = ∅. If there exists j0 such that either yj0 = 0 or yj0 βj0 − xj0 zj0 = 0, deﬁne K = {j0 + 1}. Then we deduce the claim from Proposition 1.7.12(3). Chapter 2

Modular representation theory of GL3

In this chapter we are mainly interested in modular representations of the finite group GLn(Fq) when n = 3. Representations of these groups behave differently depending on the characteristic ` of the coefficient field. When ` 6= p (referred to as the cross-characteristic case), they are closely related to the characteristic 0 theory and studied by way of Deligne-Lusztig characters. This case is better understood than the alternative ` = p, which is called the defining characteristic case and which is the topic of this chapter. In this case the theory is closely tied to the representation theory of the algebraic group GLn, since we can view GLn(Fq) as the set of fixed points under the Frobenius morphism of the algebraic group. For example, one of the possible classifications of irreducible Fp-representations of GLn(Fq) (Proposition 2.2.1 below) makes use of this connection. The main result of this chapter is a statement about the nonvanishing of certain group algebra operators on mod p principal series representations of GL3(Fq) (Theorem 2.3.2). The proof makes heavy use of representations of algebraic groups. Accordingly we begin in Section 2.1 with an overview of some background on the representation theory of reductive algebraic groups (fairly general ones, despite the title of this chapter). Then in Section 2.2 we generalize a result first due to Pillen [Pil97] about restricting algebraic representations of reductive groups to finite subgroups. Although this result turned out to not be necessary for the remainder of the chapter (see Remark 2.5.8), it is closely related to the method of proof of our main result. For this reason, and because it may be of independent interest, we decided to leave it in. Section 2.3 we state the main result Theorem 2.3.2 of this chapter. Since the proof is long and technical, we also include in this section an overview of the strategy. In Section 2.4, we review the necessary background material and then proceed with the proof in Section 2.5. The final section is in a somewhat different vein than the others. In it we study how lifts of the group algebra operators to characteristic 0 behave on smooth principal series representations of GL(Qpf ).

48 Chapter 2. Modular representation theory of GL3 49

Throughout this section let Fp be an algebraic closure of the prime field Fp, and fix a choice of embedding ωf : Fq ,→ Fp via which we consider Fq as a subfield of Fp. When we speak of a representations V of an algebraic group G defined over Fp, we in particular envision V as an Fp-module with an action of G(Fp). A choice of Fq-structure via ωf on G then induces an action of G(Fq) on V .

2.1 Representations of reductive groups

In this section we recall some important facts about representations of split reductive groups. All references in this section are to [Jan03] unless otherwise noted. Let G be a reductive group defined and split over a field k, which for the moment is arbitrary. In fact all such groups arise via base change from Z. Let T ⊆ G be a k-split maximal torus. This choice determines a root datum (X(T ),R,Y (T ),R∨, h , i) where R ⊂ X(T ) is the set of roots together with a Weyl group W acting on X(T ). Choose a positive system R+ ⊂ R with associated simple roots S. This determines a Borel subgroup B = TU (whose roots are R+, note this is opposite the convention of [Jan03]). The irreducible G-modules may be classified in terms of T and R+: let X(T )+ = {λ ∈ X(T ): hλ, α∨i ≥ 0 ∀α ∈ R+} be the set of dominant weights with respect to R+. The isomorphism classes of irreducible G-modules are in bijection with X(T )+. We write L(λ) for the module corresponding to λ ∈ X(T )+. It is uniquely determined among the irreducible modules by the fact that it has a (unique up to scalar) weight λ vector killed by U (II.2.7). We call a vector with this property (in any G-module) a highest weight vector of weight λ. For any λ ∈ X(T )+ there exists a G-module V (λ) universal for the property that it is generated by a highest weight vector of weight λ (II.2.13). Hence L(λ) = cosoc(V (λ)). It is called a Weyl module of highest weight λ. Its character (as a T -module) is given by the well-known Weyl character formula involving ρ ∈ X(T )⊗Z Q which is defined to be half the sum of the positive roots (II.5.10). We define the dot action of W on X(T ) by w · λ = w(λ + ρ) − ρ. If char(k) = 0 then V (λ) = L(λ) is irreducible. Now suppose that k is a perfect field of characteristic p. Since G arises via base change from Z, it also arises via base change from Fp and hence has a Frobenius morphism F : G → G. n This gives rise to the Frobenius kernels Gn := ker(F ) ≤ G. We get a restriction functor

Rep(G) → Rep(Gn); G-modules are often studied this way. We also get a notion of Frobenius twist: if V is a G-module then V [n] denotes the twist of V by F n. An important result is + Steinberg’s factorization theorem (II.3.16), which says that if λ ∈ Xn(T ) and µ ∈ X(T ) then n ∼ [n] + L(λ + p µ) = L(λ) ⊗ L(µ) . Here, for any positive integer n we deﬁne Xn(T ) = {λ ∈ X(T ) : hλ, α∨i < pn ∀α ∈ S}. In particular by induction we have

 n  n X j ∼ O [j] L  λjp  = L(λj) j=0 j=0 Chapter 2. Modular representation theory of GL3 50

for λ1, . . . , λn ∈ X1(T ).

The irreducible GnT -modules are in bijection with X(T ) (II.9.6). For λ ∈ X(T ) we write ∼ Lbn(λ) for the associated irreducible. If λ ∈ Xn(T ) then L(λ)|GnT = Lbn(λ)(loc. cit.). We n remark that if Xn(T ) is a system of representatives for X(T )/p X(T ) (which is the case below) then for µ ∈ X(T )+ and λ ∈ X(T )

∼ n L(µ)|GnT = Lbn(λ) ⇒ µ − λ ∈ p X(T ). (2.1.1)

n + [n] To see this, write µ = µ0 + p µ1 with µ0 ∈ Xn(T ), µ1 ∈ X(T ) . Note that L(µ1) |GnT = L pnν, so by Steinberg’s factorization theorem and II.9.6(f) we have ν∈L(µ1)

∼ M n L(µ)|GnT = Lbn(µ0 + p ν)

ν∈L(µ1) and (2.1.1) follows.

We write Qbn(λ) for the injective hull of Lbn(λ) in the category of GnT -representations. If p ≥ 2h − 2, where h is the Coxeter number of G (defined below) and λ ∈ Xn(T ) then Qb(λ) has a unique G-module structure, which in fact makes it the injective hull of L(λ) in the full subcategory of pn-bounded G-modules (this term is also defined below). In particular socG(Qb(λ)) = L(λ). See II.11.11 for these results. Many results in the representation theory of G make use of the geometry of the system of facets and alcoves in X(T ) under the dot action of the p-affine Weyl group Wp := pR o W . An alcove is a set of the form

∨ + {λ ∈ X(T ):(nα − 1)p < hλ + ρ, α i < nαp ∀α ∈ R } for some collection of integers (nα)α∈R+ . The closure of any alcove (i.e. replace the < by ≤ in the deﬁnition) is a fundamental domain for the action of Wp on X(T ) and Wp permutes the alcoves simply transitively (II.6.1(5)). Any two weights in the same Wp-orbit are said to 1 be linked. The linkage principle (II.6.17) says that if ExtG(L(µ),L(λ)) 6= 0 then λ and µ are linked. Since it is also true that irreducible modules exhibit no self-extensions (II.2.12(1)), it 1 follows that ExtG(L(µ),L(λ)) = 0 for any µ, λ lying in the same alcove.

2.2 Restriction of algebraic representations to ﬁnite subgroups

The goal of this section is to prove a theorem about restricting algebraic representations of a reductive group G to a ﬁnite subgroup Γ (Proposition 2.2.2), generalizing work of Pillen /Fp ([Pil97], Lemma 3.1) who showed it in the split case. Since the main result of this section won’t be used in the sequel, the reader won’t lose anything by skipping its proof. However, we do use some of the preparatory material in this section later on. The reason for our interest in this problem is best illustrated by referring to the summary of proof of the main result in Chapter 2. Modular representation theory of GL3 51

Section 2.3. There we make use of the device of embedding a representation of Γ = GL3(Fq) into a representation of the algebraic group G = Qf−1 GL . Now it is clear that Γ can be j=0 3/Fp thought of as the set of fixed points of the Frobenius morphism Frobf on GL coming from 3/Fp the standard Fq-structure on the latter. In this case, when the finite subgroup is the fixed points of a Frobenius morphism coming from an Fq-structure on G with respect to which G is split, Pillen’s theorem applies. But for technical reasons, we really do want to embed Γ into the product group G which involves thinking of Γ as the set of fixed points for the Frobenius coming from the nonsplit Fp-structure on G given by ResFq/Fp (GL3). In fact the ideas in Pillen’s proof work in this generality and the differences are largely technical, but there are some small modifications to his argument. Moreover Pillen only considers the case of G semisimple and simply connected, so at the same time we extend the result to the case when G is reductive with simply connected derived subgroup. We remark that [LLHLM18b] also extends Pillen’s original result to the case of split GL3/Fq via a different method. Again in this section all references are to [Jan03] unless otherwise noted. For the remainder of the section let G be a connected reductive algebraic group defined 0 + and split over Fp such that the derived group G is simply connected. Let T,R,R , S, W be 0 0 as in Section 2.1. Then T1 := T ∩ G is a split maximal torus of G . The restriction map 0 X(T ) → X(T1) is surjective and identifies the roots R and the Weyl groups W of G and G . It will be denoted by a bar: µ 7→ µ¯. Its kernel is X0(T ) = {µ ∈ X(T ): hµ, α∨i = 0 ∀α ∈ W 0 0 R} = X(T ) . Since G is simply connected, for any simple root α ∈ S there exists ωα ∈ X(T ) 0 ∨ satisfying hωα, β i = δα,β for all β ∈ S. This element is any choice of lift of the fundamental 0 0 0 P 0 weight ωα for G , and is therefore unique up to an element of X (T ). Define ρ = α∈S ωα. We define the dot action of the extended p-affine Weyl group Wfp := pX(T ) o W on X(T ) by 0 0 0 w · µ = w(µ + ρ ) − ρ . Note that this is independent of the choice of lifts ωα. Ff−1 ∨ Decompose R = j=0 Rj into irreducible components, and let α0,j denote the longest ∨ ∨ positive coroot of Rj. Then λ ≤ µ in X(T ) implies hλ, α0,ji ≤ hµ, α0,ji for each j and for + ∨ ∨ + λ ∈ X(T ) := {λ ∈ X(T ) | hλ, α i ≥ 0 ∀α ∈ S} we have hλ, α0,ji ≥ 0. Moreover if λ ∈ X(T ) ∨ ∨ + Pf−1 ∨ 0 then hλ, α i ≤ max0≤j≤f−1hλ, α0,ji for each α ∈ R , and j=0 hλ, α0,ji = 0 iff λ ∈ X (T ). Let 0 ∨ + h = max{hρ , α i + 1 | α ∈ R } be the maximum among the Coxeter numbers of the Rj. For Qf−1 example, in the case G = j=0 GLn, h = n. We say that a weight λ ∈ X(T ) is δ-deep in its alcove, or just is δ-deep, if hλ + ρ0, α∨i ∈ [δ, p − δ] mod p for all α ∈ R+.A G-module is said to be δ-deep if the highest weights of all of its irreducible factors are δ-deep. We call a G-module pn-bounded if all the highest weights ∨ n of its irreducible factors µ satisfy hµ, α0,ji < 2p (h − 1) for each 0 ≤ j ≤ f − 1. This implies that G is pn-bounded in the sense of [Jan03] p.333. 0 + n Since G is semisimple and simply connected, each λ ∈ X(T ) may be written as µ0 + p µ1 + where µ0 ∈ Xn(T ) and µ1 ∈ X(T ) (II.3.15, Remark (2)). In particular (2.1.1) holds. Let π be a finite-order automorphism of the based root datum of G given by the choice of + ∨ R . Since it preserves h , i and the set of (simple) roots by definition, it permutes the α0,j and Chapter 2. Modular representation theory of GL3 52

0 0 0 0 preserves X (T ). It moreover permutes the ωα and preserves ρ , both up to elements of X (T ), and hence preserves the dot action. It permutes W , and preserves ≤. If we fix a pinning (xα)α∈R of G as in II.1.2, then by Theorem II.1.15 it follows that π may be lifted to an automorphism −1 of (G, B, T, xα) over Fp such that, using π for both maps, we have π(λ) = λ ◦ π |T . It necessarily commutes with the Frobenius endomorphism of G, since Frobn ◦π and π ◦ Frobn induce the same map Uα → Uπ(α) and by II.1.3(10). Identifying G with its Fp-points, we let n n F F = Frob ◦π = π◦Frob and set Γn = G . This is a finite group, which is the set of Fpn -points of a form of G defined over Fpn (see Proposition 1.4.27 in [GM]). In fact the map taking π to this form induces a bijection between conjugacy classes of finite-order automorphisms of the based root datum and forms of G defined over Fpn . This is seen using the Galois cohomology classification of forms of G. Any G-module becomes a Γn-module by restriction. The next result is Proposition 1.3 of the Appendix in [Her09].

Proposition 2.2.1. If λ ∈ Xn(T ) then L(λ) restricts to an irreducible representation of Γn ∼ 0 0 n 0 and L(λ)|Γn = L(λ )|Γn iﬀ λ − λ ∈ (p − π)X (T ). All irreducible Fp-representations of Γn arise this way.

We write F (λ) = L(λ)|Γn and let Un(λ) be the injective hull of F (λ) in the category of Fp[Γn]- modules. The main result of this section is the next proposition.

Proposition 2.2.2. Assume that h ≥ 2 and let δ ∈ (0, p − h) be an integer. Let V be a G- ∨ n module such that each irreducible constituent L(µ) of V obeys max0≤j≤f−1hµ, α0,ji < δp . If V is (δ + h − 1)-deep, soci (V ) = soci (V ) for all i ≥ 1. G Γn The proof of this proposition closely follows the proof of Lemma 3.1 of [Pil97], which is the case where G is semisimple and π = id. However, since the proof therein references multiple sources and because there are some small but nontrivial modifications to be made as well as a small gap to be filled, we have opted to present the complete argument. The first part of the next result is often known as the “translation principle” or the “tensor product theorem”.

Proposition 2.2.3. Suppose that λ ∈ X(T )+ lies in the interior of a p-alcove and that ν ∈ X(T )+ is small enough so that λ + ν0 lies in the same alcove as λ for all weights ν0 of L(ν). Then M 0 L(λ) ⊗ L(ν) = dim(L(ν)ν0 ) · L(λ + ν ). ν0∈L(ν)

Moreover if λ ∈ Xn(T ) then

M 0 Qbn(λ) ⊗ L(ν) = dim(L(ν)ν0 ) · Qbn(λ + ν ) ν0∈L(ν) as GnT -modules, and if p ≥ 2h − 2 then also as G-modules. Chapter 2. Modular representation theory of GL3 53

Proof. The ﬁrst isomorphism is well-known in this generality: see [Hum06] Proposition 6.4.

The second isomorphism of GnT -modules follows exactly as in [Pil93], Lemma 5.1. In order to upgrade it to an isomorphism of G-modules, we proceed as follows: the right hand side is injective in the full subcategory of pn-bounded G-modules. The G-socle of the right-hand side is L 0 ν0∈L(v) dim(L(ν)ν0 )·L(λ+ν ) and this injects into the left hand side by the ﬁrst isomorphism. n Since Qbn(λ) ⊗ L(ν) is p -bounded (II.11.11), injectivity gives an inclusion of G-modules from the right-hand side to the left; by comparing dimensions it must be an isomorphism.

We start with the easier inclusion, whose proof is essentially the same as Lemma 2.3 of [Pil97].

Lemma 2.2.4. Let V be a G-module and δ ≥ 1 an integer such that each irreducible constituent L(µ) of V obeys hµ, α∨ i < δpn for each 0 ≤ j ≤ f −1. If V is δ-deep then soci (V ) ⊆ soci (V ) 0,j G Γn for all i ≥ 1.

Proof. Since the hypotheses on V are preserved under taking quotients, it suﬃces to show that socG(V ) is completely reducible as a Γn-module. The G-socle of V is a direct sum of modules n ∼ [n] + L(µ0 + p µ1) = L(µ0) ⊗ L(µ1) where µ0 ∈ Xn(T ) and µ1 ∈ X(T ) . As a Γn-module this is isomorphic to L(µ0) ⊗ L(π(µ1)). Since V is δ-deep, µ0 has distance ≥ δ from any alcove wall ∨ + and by assumption we must have hµ1, α i < δ for all α ∈ R . Since π preserves the positive ∨ + roots, it follows that hπ(µ1), α i < δ for all α ∈ R . Now Proposition 2.2.3 implies that

L(µ0) ⊗ L(π(µ1)) is a direct sum of irreducible algebraic representations with highest weights in the same alcove as that of µ0. In particular they are in Xn(T ). Therefore L(µ0) ⊗ L(π(µ1)) is a semisimple Γn-representation by Proposition 2.2.1.

Proposition 2.2.5. Let λ ∈ Xn(T ). Let p be an odd prime. Assume that p ≥ 2h − 2 so that Qbn(λ) has a unique G-module structure. If λ is δ-deep in its alcove, where 1 ≤ δ := 0 ∨ max0≤j≤f−1hρ , α0,ji = h − 1 then Qbn(λ)|Γn = Un(λ).

Proof. By [Jan81], 2.10, Un(λ) occurs with multiplicity 1 inside Qbn(λ) and it is enough to n 0 show that if µ ∈ Xn(T ) is such that µ − λ∈ / (p − π)X (T ) then we have [L(µ) ⊗ L(π(ν)) : n + n L(p ν + λ)]G = 0 for all ν ∈ X(T ) . If this was nonzero we would have p ν + λ ≤ µ + π(ν), n ∨ hence (p − π)ν ≤ µ − λ. Choose j such that hν, α0,ji is maximal. Then

n ∨ n ∨ ∨ n 0 ∨ (p − 1)hν, α0,ji ≤ h(p − π)ν, α0,ji ≤ hµ − λ, α0,ji < (p − 1)hρ , α0,ji

∨ where the last inequality is because µ, λ ∈ Xn(T ) and λ is 1-deep in its alcove. Thus maxjhν, α0,ji < δ. It follows that hπν, α∨i ∈ [0, δ) for each α ∈ R+. n n Now if [L(µ) ⊗ L(π(ν)) : L(p ν + λ)]G 6= 0 then [L(µ) ⊗ L(π(ν)) : p ν ⊗ L(λ)]GnT 6= 0. Since Qbn(λ) is the projective envelope of L(λ) in the category of GnT -modules (II.11.5(3)), we get a n nonzero map Qbn(λ) → L(µ − p ν) ⊗ L(πν). Therefore

∨ n HomGnT (Qbn(λ) ⊗ L(πν) ,L(µ − p ν)) 6= 0. Chapter 2. Modular representation theory of GL3 54

Since λ is δ-deep, we get by the translation principle

0 n HomGnT (Qbn(λ + ν ),L(µ − p ν)) 6= 0 for some weight ν0 ∈ L(πν)∨. By (2.1.1) this implies that µ and λ + ν0 diﬀer by an element n 0 n 0 of p X(T ). But since µ, λ + ν ∈ Xn(T ), they actually diﬀer by an element of p X (T ), say µ = λ+ν0 +pnz. It follows that pnν +λ ≤ µ+πν = λ+ν0 +πν +pnz, hence (pn −π)ν ≤ ν0 +pnz. Picking j as before we have

n ∨ 0 n ∨ 0 ∨ ∨ ∨ (p − 1)hν, α0,ji ≤ hν + p z, α0,ji = hν , α0,ji ≤ h−w0πν, α0,ji ≤ hν, α0,ji.

∨ 0 Since p 6= 2 it follows that hν, α0,ji = 0 which implies by the choice of j that ν ∈ X (T ). But then L(µ) ⊗ L(π(ν)) = L(µ + πν) so we obtain λ + pnν = µ + πν, hence µ − λ ∈ (pn − π)X0(T ), which is a contradiction.

Now we prove Proposition 2.2.2. Note since V is (h − 1)-deep, we automatically have p ≥ 2h − 2.

Proof. Since the hypotheses on V are preserved under taking quotients, we just need to show L n that the socles coincide. Assume that socG(V ) = L(λ0 + p λ1) where each λ0 ∈ Xn(T ) + n n n and λ1 ∈ X(T ) . Note that L(λ0 + p λ1) = L(λ0) ⊗ L(p λ1) ,→ Qbn(λ0) ⊗ L(p λ1). The L n ﬁrst step is to show that the embedding socG(V ) ,→ N := Qbn(λ0) ⊗ L(p λ1) extends to all n of V . It is enough to show that for each composition factor L(µ) = L(µ0 + p µ1) of V that 1 n ExtG(L(µ), Qbn(λ0) ⊗ L(p λ1)) = 0 because of the exact sequence 1 V · · · → HomG(V,N) → HomG(socG V,N) → ExtG ,N = 0. socG V

n Since the Qbn(·) are injective Gn-modules, so are the Qbn(λ0) ⊗ L(p λ1) by I.3.10(c) and one n has for µ = µ0 + p µ1 that

1 n 1 [n] [n] ExtG(L(µ), Qbn(λ0) ⊗ L(p λ1)) = ExtG(L(µ1) , Hom(L(µ0), Qbn(λ0) ⊗ L(λ1) ))

[n] [n] since this is a general property of Ext. Since Gn acts trivially on L(µ1) we have HomG(L(µ1) , −) = Gn HomG/Gn (L(µ1), (−) ) so passing this equality to the derived functors we obtain that the above is equal to

Ext1 (L(µ )[n], Hom (L(µ ), Q (λ ) ⊗ L(λ )[n])) G/Gn 1 Gn 0 bn 0 1 = Ext1 (L(µ )[n], Hom (L(µ ), Q (λ )) ⊗ L(λ )[n]). G/Gn 1 Gn 0 bn 0 1

Now since socGn (Qbn(λ0)) = L(λ0)|Gn , if HomGn (L(µ0), Qbn(λ0)) is nonzero we must have n n 0 µ0 −λ0 = p ν ∈ p X (T ). Then the Hom space is 1-dimensional by II.3.10(3). Since it contains Chapter 2. Modular representation theory of GL3 55

HomG(L(µ0), Qbn(λ0)), which is nonzero, the two must be equal and hence HomGn (L(µ0), Qbn(λ0)) is the trivial G-module. So in this case the Ext space reduces to Ext1 (L(µ )[n],L(λ )[n]) = G/Gn 1 1 1 ∨ ∨ ExtG(L(µ1),L(λ1)). The boundedness condition on V implies that hλ1, α i, hµ1, α i ∈ [0, δ) + for all α ∈ R . Since δ < p − h both λ1 and µ1 lie in the lowest alcove and hence this space of extensions is 0 as desired. L We know from Proposition 2.2.4 that the Γn-socle contains the G-socle of V . Since L(λ0)⊗ n L n L(p λ1) = socG V ⊆ socG N ⊆ socΓn N, if we can show that L(λ0) ⊗ L(p λ1) is equal to the

Γn-socle of N then we will be done because socΓn V sits between socG V and socΓn N. Therefore n n we just need to show that the Γn-socle of Qbn(λ0) ⊗ L(p λ1) is equal to L(λ0) ⊗ L(p λ1). n ∨ As a Γn-module, Qbn(λ0) ⊗ L(p λ1) is equal to Qbn(λ0) ⊗ L(π(λ1)). We have hπ(λ1), α i ∈ + [0, δ) for all α ∈ R , and the deepness on V implies that λ0 is (δ + h − 1)-deep. It follows by the translation principle that as a G-module we have

M 0 Qbn(λ0) ⊗ L(π(λ1)) = dim(L(πλ1)ν0 ) · Qbn(λ0 + ν ). 0 ν ∈L(πλ1)

0 All the weights λ0 + ν are still (h − 1)-deep. By Proposition 2.2.5, restricting this to Γn we get

M 0 dim(L(πλ1)ν0 ) · Un(λ0 + ν ) 0 ν ∈L(πλ1)

L 0 which has Γ -socle equal to 0 dim(L(πλ ) 0 ) · L(λ + ν ). By the translation principle, n ν ∈L(πλ1) 1 ν 0 this is the restriction to Γn of L(λ0) ⊗ L(πλ1) which is the same as the restriction to Γn of n L(λ0) ⊗ L(p λ1). This proves the claim.

2.3 Summary of main results

Qf−1 From now on we specialize the notation of the previous section to the case where G = j=0 GL3 and T and B are the standard diagonal torus and upper-triangular Borel respectively. We 3 f f−1 3 identify X(T ) with (Z ) and L(λ) with j=0 L(λj), adopting the convention that when λj ∈ Z , L(λj) refers to the irreducible representation of the jth factor GL3 of highest weight λj. Let f−1 f−1 π : G → G be the left shift automorphism (gj)j=0 7→ (gj+1)j=0 . The map

f f−1 g 7→ (Frob (ωf (g)), Frob (ωf (g)),..., Frob(ωf (g))) (2.3.1)

∼ is an isomorphism GL3(Fq) −→ Γ1 and we let F (λ) denote the irreducible representation of GL3(Fq) coming from Proposition 2.2.1 via this isomorphism. Thus F (λ) is characterized by the P λi pj N2 j f−j fact that it has a B3(Fq)-invariant vector on which T3(Fq) acts by the character i=0 ωf . 2 1 0 f−1 3 f Let (aj , aj , aj )j=0 ∈ (Z+) be a tuple of integers satisfying (1.1.3). Throughout this chapter Chapter 2. Modular representation theory of GL3 56 and the next, we are interested in a particular Serre weight

0 1 2 f−1 F0 := F ((−aj − 1, −aj , −aj + 1)j=0 ).

The next theorem is the main result of this chapter. It is easy to see that for any distinct × characters χi : Fq → Fp the space of B3(Fq)-invariant vectors upon which T3(Fq) acts by GL3(Fq) χ0 ⊗χ1 ⊗χ2 in the principal series representation π = Ind (χ0 ⊗χ1 ⊗χ2) is 1-dimensional. B3(Fq) We refer to any nonzero element of this subspace (somewhat abusively) as a highest weight vector of π.

f Theorem 2.3.2. For σ ∈ S3 and K ⊆ {0, . . . , f − 1} define a principal series type τ = L2 Ai i τ(σ, K) = i=0 ωef as in Definition 1.6.1. Assume that the integers aj satisfy (1.1.3) with 0 δ ≥ 15. In each case below, we define operators Sτ ,Sτ ∈ Fp[GL3(Fq)].

1. Assume that  id if j ∈ J σj = (2.3.3) (02) if j∈ / J

for some J ⊆ {0, . . . , f − 1} and that J ⊆ K and K 6= ∅. Deﬁne

σ−1 (2) P f−j 0 j X (p−a +a −δf−j+1∈/K )p Sτ = ωf (x) j f−j f−j x,y,z∈Fq 1 x y  1 P (p−a2 +a1 )pj j f−j f−j     · ωf (z)  1 z  1  1 1

and

P 1 0 j 0 X j (p−af−j +af−j )p Sτ = ωf (x) x,y,z∈Fq     σ−1 (0) 1 x y 1 P (p−a2 +a f−j −δ )pj j f−j f−j f−j+1∈/K     · ωf (z)  1 z  1  . 1 1

Let π = IndGL3(Fq) ω−A1 ⊗ ω−A0 ⊗ ω−A2 and π0 = IndGL3(Fq) ω−A0 ⊗ ω−A2 ⊗ ω−A1 B3(Fq) f f f B3(Fq) f f f and write v, v0 for their highest weight vectors respectively.

0 0 0 Then Sτ v and Sτ v are nonzero in the unique quotients of π and π with socle F0, respectively.

2. Assume that  id if j ∈ J σj = (2.3.4) (01) if j∈ / J Chapter 2. Modular representation theory of GL3 57

and that K = ∅. Deﬁne

1 x y  1 X P (p−a1 +a0 )pj f−j∈J f−j f−j     Sτ = ωf (x)  1  1  x,y∈Fq 1 1

and 1 x   1  X P (p−a1 +a0 )pj 0 f−j∈ /J f−j f−j     Sτ = ωf (x)  1  1  . x∈Fq 1 1 Write π = IndGL3(Fq) ω−A0 ⊗ ω−A2 ⊗ ω−A1 and π0 = IndGL3(Fq) ω−A1 ⊗ ω−A0 ⊗ ω−A2 . B3(Fq) f f f B3(Fq) f f f 0 0 0 Then the images of Sτ v and Sτ v are nonzero in the unique quotient of π and π with socle F0 respectively.

3. Assume that  id if j ∈ J σj = (2.3.5) (12) if j∈ / J

and that K = ∅. Deﬁne

1 y  1  X P (p−a2 +a1 )pj f−j∈ /J f−j fj     Sτ = ωf (z)  1 z  1 . y,z∈Fq 1 1

and 1  1  X P (p−a2 +a1 )pj 0 f−j∈J f−j f−j     Sτ = ωf (z)  1 z  1 . z∈Fq 1 1 Let π = IndGL3(Fq) ω−A2 ⊗ ω−A0 ⊗ ω−A1 and π0 = IndGL3(Fq) ω−A0 ⊗ ω−A1 ⊗ ω−A2 B3(Fq) f f f B3(Fq) f f f 0 0 0 with highest weight vectors v, v , respectively. Then Sτ v and Sτ v are nonzero in the 0 unique quotients of π and π with socle F0, respectively.

Notation 2.3.6. In each of the cases considered in this theorem, we deﬁne ς = ς(τ) ∈ S3 to −A −A −A be the permutation such that π = IndGL3(Fq) ω ς(0) ⊗ ω ς(1) ⊗ ω ς(2) . B3(Fq) f f f

Remark 2.3.7. It is not obvious that F0 is actually an irreducible constituent of any of these principal series, but it may be deduced from either [GHS], Proposition 10.1.7, or the results of

[LLHLM18b] on the submodule structure of principal series for GL3(Fq). However, we don’t need these results because it also follows from the proof of the theorem. In fact, the choice of types τ in Deﬁnition 1.6.1 was motivated by the fact that the princi- GL3(Fq) A0 A1 A2 pal series representations Ind ω ⊗ ω ⊗ ω are the only ones containing F0 as a B3(Fq) f f f Jordan-H¨olderfactor, as can be checked using [GHS] Proposition 10.1.7. Chapter 2. Modular representation theory of GL3 58

j Remark 2.3.8. If the integers ai satisfy (1.1.3) when δ ≥ 15, then Theorem 1.3.1 of [LLHLM18b] implies that for any of the principal series representations π, π0 in Theorem 2.3.2 all Jordan-

Hölderconstituents occur with multiplicity 1. In particular, given that F0 is constituent, there exists a unique quotient with irreducible socle F0. 0 Remark 2.3.9. Taking f = 1, τ as in (1.7.2) and K = {0}, the operators Sτ and Sτ recover the operators denoted S and S0 in [HLM17]. Remark 2.3.10. While we have been working with algebraically closed coefficients, the irreducible representations F (λ) may all be defined over Fp so the Theorem is equally valid over a finite subfield F ⊆ Fp containing the image of ωf . The method of proof of this theorem is to embed π and π0 into well-chosen Weyl modules f−1 0 V = j=0 V (λj) for G, and then interpret the action of the operators Sτ and Sτ by way of the distribution algebra of G. Since the structure of Weyl modules for GL3 is known in the cases we need, this allows us to do explicit computations to prove nonvanishing of the group algebra operators. This technique was inspired by [Pil97] and [LLHLM18b], which both embed principal series representations into algebraic representations, the latter deducing the entire submodule structure of (sufficiently generic) principal series representations of GL3(Fq).

However, [LLHLM18b] embed principal series into Weyl modules for split GL3/Fq . Using the product group G with the nonsplit Fp-structure above allows for simpliﬁcations to their set- up and in particular makes the kind of computations we want to do feasible. Our method of computation was also inspired by [Irv86], which determines the submodule structure of certain

Weyl modules for SL3. We need the following general lemmas.

Lemma 2.3.11. Let G1 and G2 be split reductive groups over a ﬁeld k. If Li is a simple module for Gi (i = 1, 2), then L1 L2 is a simple module for G1 × G2. Moreover, every simple module for G1 × G2 is of this form for unique L1,L2.

Proof. If N ⊆ L1 L2 is a G1 × G2-submodule, then picking a k-basis of L2, we have N ⊆ ∼ n L1 L2 = L1 as a G1-module. The image of N under this isomorphism must be a sub-direct n sum of L1 by [Jan03] II.2.8, so we must have N = L1 W for some k-subspace W ⊆ L2. If φ⊗1 φ : L1 → k is any nonzero linear functional, then the image of N ⊆ L1 L2 −−→ L2 is W . On the other hand, this map is G2-equivariant. It follows that W is a G2-submodule of L2 and hence W = L2. This proves that L1 L2 is simple. If Ti ⊂ Gi are split maximal tori then T1 ×T2 is a split maximal torus of G1 ×G2. Identifying + + + X(T1 ×T2) = X(T1) ×X(T2) for some choice of positive roots, it is clear that L(λ1)L(λ2) has highest weight (λ1, λ2), so the other statements follow.

Corollary 2.3.12. In the situation of the previous Lemma let Mi be a finite dimensional Gi- module, and for simplicity assume that the Jordan-Hölderconstituents of Mi all occur with multiplicity 1 (i = 1, 2). Then the Jordan-Hölderconstituents of M1 M2 are exactly the Chapter 2. Modular representation theory of GL3 59

L1 L2 where Li is a constituent of Mi each occurring with multiplicity 1, and the submodule lattice of M1 M2 is the product lattice of the submodule lattices of M1 and M2.

Proof. The first claim about the Jordan-Hölderconstituents is easy to prove by induction on the length of M1 and M2, using the Lemma (which is the length 1 case). For the second, it now suffices to show that for each constituent L1 L2, if Qi denotes the unique quotient of Mi having socle Li, then Q1 Q2 is the unique quotient of M1 M2 having socle L1 L2. It is clear that Q1 Q2 is a quotient of M1 M2 having L1 L2 in its socle, so it suffices to 0 0 0 prove that the socle is irreducible. But if L1 L2 ,→ Q1 Q2 for some irreducibles Li, then 0n m considering the G1 module structure on both sides we obtain L1 ,→ Q1 for some m, n ≥ 1, 0 0 which implies L1 = L1. Similarly L2 = L2 and we are done.

2.4 Weyl modules and the distribution algebra

In this section we recall facts about Weyl modules for the algebraic group GL and their 3/Fp 0 Z-forms. Let G0 = GL3/Z, and let G0 = SL3/Z be the derived subgroup. We consider the standard diagonal torus T3 and upper-triangular Borel B3 in G0, with root system R. Identify 3 X(T3) with Z , and let α = (1, −1, 0) and β = (0, 1, −1) denote the simple positive roots and + γ = (1, 0, −1) the nonsimple positive root. For δ ∈ R write sδ for the corresponding reflection 0 in the Weyl group and let w0 = sαsβsα denote the longest element. For each δ ∈ R let Uδ ≤ G0 ∼ denote the standard root subgroup; we fix a pinning (xδ : Ga/Z −→ Uδ)δ∈R as in [Jan03] II.1.2 such that xδ(1) is a matrix having all entries equal to 0 or 1. This choice gives rise to raising and lowering operators X±α,X±β,X±γ and weight operators Hα,Hβ,Hγ = Hα + Hβ inside the 0 distribution algebra U := Dist(G0). We refer to [Jan03] for the definition of the distribution algebra. By [Jan03] II.1.12, U is the Z-subalgebra of the universal enveloping C-algebra of 0 (n) n Lie(G0) generated by elements X±δ := X±δ/n! and Hα,m,Hβ,m with n, m ∈ N, δ ∈ {α, β, γ}, Hδ where Hδ,m := m . It is free over Z with basis given by

(nα) (nγ ) (nβ ) (n−α) (n−γ ) (n−β ) Xα Xγ Xβ Hα,mα Hβ,mβ X−α X−γ X−β

+ (nδ) + for n±δ, mδ ≥ 0. We let U denote the subalgebra of U generated by Xδ with δ ∈ R , nδ ≥ 1 − + 0 and similarly for U with δ ∈ −R . Moreover we have Dist((G ) ) = U ⊗ p. 0 Fp Z F + By deﬁnition we have [Xδ,X−δ] = Hδ for δ ∈ R . With our choice of pinning, we also have for example

[Xβ,X−γ] = X−α and [X−β,X−α] = X−γ. (2.4.1)

The following elementary lemma applies to U and therefore also to U by noting that X Fp δ1 and Xδ2 commute whenever δ1 + δ2 is not a root and nonzero. We will use it heavily in the calculations of Section 2.5.

Lemma 2.4.2. Let A be a ring of characteristic 0. For x ∈ A, write x(n) = xn/n!. If a, b ∈ A Chapter 2. Modular representation theory of GL3 60 and [a, b] = γ, where = ±1 and γ commutes with a and b, then for all m, n ≥ 1 we have

min(m,n) X a(m)b(n) = `b(n−`)a(m−`)γ(`). `=0

Proof. By induction.

Any G0-module (resp. (G0) -module) has the structure of a U-module (resp. U -module) Fp Fp + by restriction. If V is a G0-module and λ a weight, then for δ ∈ R , Hδ,m acts by the scalar hλ,δ∨i (n) m on Vλ. Moreover, X±δ maps Vλ → Vλ±nδ. The same is true over Fp. This implies the next lemma.

Lemma 2.4.3. If V is a G0- or (G0) -module and v ∈ V is a weight vector, then Fp

(n) (n) ∨ (n−1) XδX−δ v = X−δ Xδv + [hwt(v), δ i − (n − 1)]X−δ v for any δ ∈ R+.

Proof. By induction using the above remarks.

The formula in the next lemma may be thought of as a Taylor expansion.

Lemma 2.4.4. If V is a (G0) -module and v ∈ V then for δ ∈ R and u ∈ p we have Fp F

X i (i) xδ(u)v = u Xδ v. i≥0

Proof. This is II.1.19(6) in [Jan03].

Remark 2.4.5. In the formula above, 00 is interpreted as 1. This plays a role below. 3 Next we recall some facts about Weyl modules. If λ ∈ Z is a dominant weight let V (λ)Z denote the integral Weyl module for G0 of highest weight λ, deﬁned in [Jan03] II.8.2, and set

V (λ) := V (λ)Z ⊗ Fp. Briefly, V (λ)Z is defined to be the finitely generated Z-module generated by Dist(G0)v where v is a highest weight vector inside the Weyl module of GL3/Q of highest weight λ. This is G0-stable and it is well known (loc. cit.) that V (λ) is then the Weyl module of highest weight λ for (G0) defined by the same symbol in Section 2.1. We note that V (λ)| 0 Fp G0 0 is equal to the corresponding Weyl module for G0. This is important because some of the references below only work with semisimple groups and this fact allows us to extend results from SL3 to GL3. For example there is a similar construction of the weight λ integral Weyl 0 module for G0 in Section 27 of [Hum72]. The submodule structure of V (λ) is completely known for small λ (at least when λ lies in the interior of a p-alcove, which is the only case we need). In order to express it, we use Humphreys’ labelling A-J of the 10 lowest dominant p-alcoves

[Hum06] of (G0) . In order to be explicit, we list the interiors of these alcoves in the table below. Fp Chapter 2. Modular representation theory of GL3 61

λ ∈ A B C D E F G H I J hλ + η, α∨i ∈ (0, p) (0, p) (0, p) (p, 2p) (0, p) (p, 2p) (p, 2p) (0, p) (2p, 3p) (p, 2p) hλ + η, β∨i ∈ (0, p) (0, p) (p, 2p) (0, p) (p, 2p) (0, p) (p, 2p) (2p, 3p) (0, p) (p, 2p) hλ + η, γ∨i ∈ (0, p) (p, 2p) (p, 2p) (p, 2p) (2p, 3p) (2p, 3p) (2p, 3p) (2p, 3p) (2p, 3p) (3p, 4p)

Proposition 2.4.6. Suppose that p ≥ 5. Let λ be a weight in the interior of one the alcoves A-J. The submodule structure of V (λ) is given by the diagram: GJ

E F H C G D I

C A D E F

H F

E C A D D B

ABBA For example, if λ ∈ G then the submodule structure is given by the top left diagram, where each alcove label corresponds to the unique irreducible representation L(µ) with µ lying in that alcove and linked to λ. The cases I,E,C not pictured are symmetric to H,F,D, respectively. If λ ∈ A then V (λ) = L(λ) is already irreducible.

Proof. This proposition is taken directly from [BDM15], Section 4. Note in that paper they

work with SL3. Alternatively, it is possible to re-derive these structures using direct calculations as in Lemma 2.4.13 below.

Remark 2.4.7. In the previous proposition, the alcove J is far enough from the boundary of the

dominant region to exhibit Jantzen’s generic decomposition pattern [Jan77b] for GL3. We now make some further remarks on the structure of V (λ) as a U -module, with special Fp attention to the case when λ lies in alcove G or J, since these are the cases that ﬁgure in the proofs of Section 2.5. The next deﬁnition is taken from [Jan77a].

Deﬁnition 2.4.8. A primitive element in a (G0) -module is a nonzero weight vector killed Fp (n) + by Xδ for all δ ∈ R and n ≥ 1.

For example, a highest weight vector inside Weyl module is a primitive element. Chapter 2. Modular representation theory of GL3 62

Lemma 2.4.9. A primitive element in a (G0) -module generates a quotient of V (λ). Fp

(n) + Proof. This follows from the universal property of Weyl modules, since Xδ v = 0 for all δ ∈ R and n ≥ 1 implies v is U-invariant by Lemma 2.4.4.

Lemma 2.4.10. Let λ be a dominant weight and let J denote the left ideal of (U) generated λ Fp (n) hλ,α∨i hλ,β∨i + (mα) (mβ ) by Xδ , Hα,n − n ·1 and Hβ,n − n ·1 for δ ∈ R and n ≥ 1, as well as X−α ,X−β ∨ ∨ ∼ for all mα ≥ hλ, α i + 1 and m ≥ hλ, β i + 1. Then V (λ) = (U) /J as a (U) -module. β Fp λ Fp Proof. This follows from Satz 2 of [Jan77a], which is the analogous result for the integral Weyl module V (λ)Z.

Lemma 2.4.11. Let v be a highest weight vector of V (λ) for some dominant weight λ. If ∨ (c) (b) ∨ (a) (c) (b) b + c ≥ hλ, γ i + 1 then X−γX−βv = 0. Similarly if a + c ≥ hλ, γ i + 1 then X−αX−γX−βv = 0 for any b ≥ 0.

Proof. This follows from the well-known fact that all weights of V (λ) satisfy w0λ ≤ µ ≤ λ; one checks easily that the stated inequalities imply that the respective vectors are of weights outside this allowed range.

Lemma 2.4.12. Let λ be a dominant weight and let ξ = λ−rα−sβ where 0 ≤ r ≤ hλ, α∨i and (r−i) (i) (s−i) ∨ 0 ≤ s. A basis of the V (λ)ξ is given by the elements X−α X−γX−β v for max(0, s−hλ, β i) ≤ i ≤ min(r, s), where v is a highest weight vector in V (λ). In particular these elements are nonzero. [Note: if r < s − hλ, β∨i then there are no integers i satisfying this inequality and the conclusion in this case is that V (λ)ξ = 0.]

(r−i) (i) (s−i) Proof. It is clear that V (λ)ξ is spanned by the vectors X−α X−γX−β v for 0 ≤ i ≤ min(r, s). But this vector is 0 by Lemma 2.4.10 if s − i ≥ hλ, β∨i + 1. Then one checks by cases using ∨ the Kostant multiplicity formula that dim(V (λ)ξ) = min(r, s) − max(0, s − hλ, β i) + 1, and the result follows.

Lemma 2.4.13. Let v denote a highest weight vector in V (λ).

(hλ,β∨i−p+1) (hλ,α∨i−p+1) (hλ,γ∨i−2p+2) (hλ,β∨i−p+1) 1. If λ ∈ G then vF = X−β v, vE = X−α v, vC = X−α X−β v (hλ,γ∨i−2p+2) (hλ,α∨i−p+1) and vD = X−β X−α v are primitive and generate the unique submodule of V (λ) having irreducible cosocle with highest weight in alcove F,E,C,D, respectively.

(hλ,β∨i−p+1) (hλ,α∨i−p+1) 2. If λ ∈ J then vI = X−β v and vH = X−α v are primitive and generate the unique submodule of V (λ) having irreducible cosocle with highest weight in alcove I and H respectively.

Proof. That these elements are primitive is shown via direct calculation. Once this is known, by Lemma 2.4.9 we know they must generate a quotient of the Weyl module of the appropriate weight and the other claims follow by examining the structures in Proposition 2.4.6. We include Chapter 2. Modular representation theory of GL3 63

two calculations as an example. Suppose λ ∈ G. First note vF 6= 0 by Lemma 2.4.12. Since (n) Xα commutes with X−β, is it immediate that Xα vF = 0 for all n ≥ 1. And using Lemma (hλ,β∨i−p+1) (hλ,β∨i−p) (n) 2.4.3 we compute XβvF = X−β Xβv + pX−β v = 0. This proves Xβ vF = 0 for all n < p; for n ≥ p it follows from weight considerations. Since Xγ = [Xα,Xβ] this is enough to prove vF primitive and hence the submodule generated by vF is a quotient of a Weyl module in alcove F. Using this, together with Lemma 2.4.12 again, we deduce that vC 6= 0, and then similar calculations prove that vC is primitive.

Corollary 2.4.14. If λ ∈ G then the submodules of V (λ) having irreducible cosocle C, D, E, and F respectively are themselves Weyl modules. If λ ∈ J then the same is true of A, H, I.

Proof. This follows from Lemma 2.4.9, Lemma 2.4.13 and Proposition 2.4.6.

Finally in the remainder of this section we collect some important results allowing us to completely describe the U -module structure of the irreducible algebraic representations L(λ), Fp 3 where λ ∈ Z is a dominant weight. In any case, L(λ) = cosoc V (λ) so Lemmas 2.4.10 and 2.4.11 apply. But we can say more. As we know from Section 2.1, L(λ) decomposes via the Steinberg tensor product theorem.

It is easy to check directly from the deﬁnitions that the Frobenius endomorphism of (G0) Fp induces an p-algebra endomorphism Frob∗ of U taking F Fp

 (n/p) (n) Xδ p|n Xδ 7→ (2.4.15) 0 p - n for δ ∈ R. Moreover, if L0 and L1 are U-modules then by deﬁnition of the Hopf algebra structure on U, U acts on L0 ⊗ L1 via the formula

n (n) X (n−`) (`) Xδ (v0 ⊗ v1) = (Xδ v0) ⊗ (Xδ v1) (2.4.16) `=0 for vi ∈ Li. Given these statements we are reduced to understanding the U -module structure of L(λ) Fp for λ ∈ X1(T3). In this case there is a result due to Nanhua Xi [Xi96] giving a complete description of L(λ). In order to state it, we ﬁrst deﬁne a certain monomial in U− . Fp

− Deﬁnition 2.4.17. Given λ ∈ X1(T3) we deﬁne its Xi monomial to be the element of U given Fp by ∨ ∨ ∨ (hθ,β i) (hsβ θ,α i) (hsαsβ θ,β i) Ξ(λ) = X−β X−α X−β where θ = (p − 1)η − λ. An equivalent expression for it is:

∨ ∨ ∨ (hθ,α i) (hsαθ,β i) (hsβ sαθ,α i) Ξ(λ) = X−α X−β X−α . Chapter 2. Modular representation theory of GL3 64

Unless otherwise stated, we use the ﬁrst expression in our calculations.

− (n) + Let (U )1 denote the subalgebra of U generated by X for 0 ≤ n ≤ p − 1 and δ ∈ R . In Fp Fp −δ fact Ξ(λ) lies in (U−) ([Xi96] Theorem 6.5(i)) but we don’t need this. F 1

Proposition 2.4.18. Let λ ∈ X1(T3) and let v ∈ L(λ) be a highest weight vector. For u ∈ − (U )1 we have uv = 0 in L(λ) iﬀ u · Ξ(λ) = 0 in U . Fp Fp Proof. This follows from Theorem 6.7(i) in [Xi96], if we observe that the intersection with − 0 (U )1 of the ideal there denoted Iλ is precisely Fp

− n = {u ∈ (U )1 | u · Ξ(λ) = 0}. Fp

− To see this, consider the map (U )1 → L(λ) sending u to uv, which is surjective because L(λ) Fp − 0 is an irreducible representation of the Frobenius kernel of (G0) . Its kernel is (U )1 ∩ Iλ, Fp Fp which contains n. On the other hand, Theorem 6.7(iii) of loc. cit. says that dim(L(λ)) = − dim(U )1 − dim(n), which proves the claim. Fp

2.5 Proof of Theorem 2.3.2

We now present the proof of Theorem 2.3.2 following the approach explained in Section 2.3. Throughout the computations in this section we often use without comment in addition to the lemmas of the previous section the following elementary observations about U : Fp

(n) (m) n+m (n+m) n m n+m • X∗ X∗ = n X∗ is 0 if b p c + b p c < b p c;

(ap+n) (ap) (n) • as a corollary to the previous point, if 0 ≤ n < p then X∗ = X∗ X∗ for any a ≥ 0 ap+n since ap ≡ 1 mod p;

p−1 n p−2 n • similarly if 0 ≤ n ≤ p − 3 then we have n ≡ (−1) , n ≡ (−1) (n + 1), and p−3 n (n+1)(n+2) n ≡ (−1) 2 mod p.

The reader should keep these in mind in order to follow the calculations.

Proof of Theorem 2.3.2(1)

By applying the automorphism

 1  1   t −1   g 7→  −1  g  −1  1 1 Chapter 2. Modular representation theory of GL3 65

2 1 0 0 1 2 of GL3(Fq) and replacing (aj , aj , aj ) with (−aj , −aj , −aj ) we deduce the second part of Theorem 0 2.3.2(1) (the statement about Sτ ) from the first. For the remainder of the proof, let π, σ,J,K,ς be as in the first part of the theorem. f−1 The first step is to embed π into a Weyl module V = j=0 V (λj) for the algebraic group Qf−1 GL . To do this, for 0 ≤ i ≤ 2 write −A ≡ P ξi pj mod e where (ξ0, ξ1, ξ2) ∈ j=0 3/Fp ς(i) j f−j j j j X1(T3) for each 0 ≤ j ≤ f − 1. We may take

−1 i σj (ς(i)) i i −1 ξj = −aj + pj − j+1 − δj+1∈/K (1 − σj+1ς(i)) where  (1, 0, 0) j ∈ J j = (0, 0, −1) j∈ / J.

Deﬁne λj = ξj + (p − 1)η. Then it is easy to check that

ξj ∈ B and λj ∈ J ⇐⇒ j ∈ J

ξj ∈ A and λj ∈ G ⇐⇒ j∈ / J.

Make a choice of highest weight vector v = ⊗jvj in V . It is clear that under the embedding P ξi pj N2 j f−j in (2.3.1), T3(Fq) acts on v via the character i=0 ωf . Hence by Frobenius reciprocity, we obtain a nonzero map of GL3(Fq)-representations

π → V sending the highest weight vector of π to v. In fact, this map is injective as one sees by comparing socles, using Proposition 2.2.2 with n = 1 and the fact that π is multiplicity-free. We won’t need this, however. It will be helpful for the reader to consult the diagrams of the submodule structures of the

V (λj) in Figures 2.5.2 and 2.5.3.

Notation 2.5.1. We make the following conventions: if X is the name of an alcove, then Lj(X) denotes the (unique) Jordan-H¨olderconstituent of V (λj) having highest weight in X. We let

Ξj(X) denote the Xi monomial of the tensor factor of Lj(X) under the Steinberg decomposition having p-restricted highest weight. We deﬁne Qj(X) to be the unique (algebraic) quotient of

V (λj) having socle Lj(X). Moreover, vX,j denotes the primitive element of V (λj) corresponding to alcove X in Lemma 2.4.13. Finally, since they correspond under the embedding π → V , we use v to denote both our ﬁxed highest weight vector of V and the highest weight vector of π when appropriate.

There is a particular quotient of V in which we are interested in because it contains F0 in Chapter 2. Modular representation theory of GL3 66

0 1 2 L(ξj −1,ξj ,ξj +1) ⊗L(1, 0, −1)[1] J

1 0 2 0 2 1 2 1 0 1 0 2 0 2 1 L(ξj −1,ξj −p,ξj +1) L(ξj −1,ξj +p,ξj +1) L(ξj +p−1,ξj ,ξj −p+1) L(ξj −1,ξj −p,ξj +1) L(ξj −1,ξj +p,ξj +1) ⊗L(1, 1, −1)[1] ⊗L(0, 0, −1)[1] ⊗L(1, 0, −1)[1] ⊗L(1, 0, 0)[1] ⊗L(1, −1, −1)[1] H C G D I

1 2 0 2 0 1 L(ξj +p−1,ξj +p,ξj −p+1) L(ξj +p−1,ξj −p,ξj −p+1) ⊗L(0, 0, −1)[1] ⊗L(1, 0, 0)[1] E F

2 1 0 L(ξj + p − 1, ξj , ξj − p + 1) A

Figure 2.5.2: The submodule structure of V (λj) when λj ∈ J.

0 1 2 [1] L(ξj − 1, ξj , ξj + 1) ⊗ L(1, 0, −1) G

1 0 2 [1] 0 2 1 [1] L(ξj + p − 1, ξj , ξj + 1) ⊗ L(0, 0, −1) L(ξj − 1, ξj , ξj − p + 1) ⊗ L(1, 0, 0) E F

2 0 1 [1] 0 1 2 1 2 0 [1] L(ξj + p − 1, ξj , ξj + 1) ⊗ L(0, 0, −1) L(ξj − 1, ξj , ξj + 1) L(ξj − 1, ξj , ξj − p + 1) ⊗ L(1, 0, 0) C A D

1 1 0 L(ξj + p − 1, ξj , ξj − p + 1) B

Figure 2.5.3: The submodule structure of V (λj) when λj ∈ G. Chapter 2. Modular representation theory of GL3 67

f−1 its socle. This is Q := j=0 Qj(Xj) where  D j ∈ J Xj = . C j∈ / J

We also write Qj = Qj(Xj). By Corollary 2.3.12 and Figures 2.5.2 and 2.5.3, the (algebraic) socle of Q is

f−1 0 1 2 [1] soc(Q) = j=0 L (−aj − 1, −aj , −aj + 1) − ζj,j+1 ⊗ L(j) where ζj,j+1 is given in the following table.

ζj,j+1 = j + 1 ∈ K j + 1 ∈/ K j, j + 1 ∈ J (0, 1, 0) j ∈ J, j + 1 ∈/ J (0, 0, −1) (−1, 0, 0) j∈ / J, j + 1 ∈ J (0, 1, 0) j, j + 1 ∈/ J (−1, 0, 0) (0, 0, −1)

Note the unﬁlled entries don’t occur because we assumed J ⊆ K. Now we have

O 0 1 2 [f−j] [f−j+1] soc(Q)|GL ( ) = L((−aj − 1, −aj , −aj + 1)j − ζj,j+1) ⊗ L(j) 3 Fq GL ( ) GL ( ) j 3 Fq 3 Fq O 0 1 2 [f−j] = L((−aj − 1, −aj , −aj + 1)j − ζj,j+1) ⊗ L(j+1) (2.5.4) GL ( ) j 3 Fq f−1  M Y 0 1 2 f−1 =  dim L(j)vj  F ((−aj − 1, −aj , −aj + 1) − ζj,j+1 + vj+1)j=0 f−1 j=0 {(vj )j=0 | vj ∈L(j )} where the last step holds by the translation principle, and we write vj ∈ L(j) to mean vj is a weight of nonzero multiplicity in L(j). Since ζj,j+1 is a permutation of j+1 it is clear that this sum contains F0 with multiplicity 1. In fact, note that each weight of L(j) occurs with multiplicity 1 so each dim L(j)vj = 1 in the last line of (2.5.4). Claim 2.5.5. If we can show that

1. the image of Sτ v in Q lies in soc(Q), and (2.5.6)

2. it is nonzero under the projection soc(Q) F0 (2.5.7) then it follows that the image of Sτ v in the unique quotient of π having socle F0 is nonzero, and we will have proved Theorem 2.3.2(1).

Proof. Let πb denote the (nonzero) image of π under the GL3(Fq)-equivariant morphism π → V Q. By (2.5.6), the image of Sv in Q lies in π ∩ soc (Q) = soc(π) (observing b GL3(Fq) b Chapter 2. Modular representation theory of GL3 68

that soc(Q)|GL3(Fq) is visibly semisimple due to the computation in (2.5.4), hence lies inside soc (Q)), and by (2.5.7), there is a map soc(π) → F under which the image of Sv is GL3(Fq) b 0 nonzero (that is, the map soc(π) ,→ soc (Q) soc(Q)| F , where the second b GL3(Fq) GL3(Fq) 0 map is any projection using semisimplicity). In particular F0 is a constituent of soc(πb) and the claim follows using Remark 2.3.8.

Remark 2.5.8. The argument above could have been made slightly simpler if we had applied Proposition 2.2.2 but there was no need. We did use the easy direction of Proposition 2.2.2, but did not have to quote the result since its conclusion was obvious due to (2.5.4).

Our goal is now to prove (2.5.6) and (2.5.7). To begin, we use Lemma 2.4.4 together with the fact that 1 x y 1 0 0 1 0 y 1 x 0          1 z =  1 z  1 0  1 0 . 1 1 1 1 To ease notation, let us temporarily write

1 x y X B C   Sτ = ωf (x) ωf (z)  1 z w0. x,y,z∈Fq 1

We compute:

  f−1 1 ωj(x) ωj(y) X B C O   Sτ v = ωf (x) ωf (z)  1 ωj(z) w0vj x,y,z∈Fq j=0 1   P j P j P j X X B+ if−j p `f−j p C+ kf−j p =  ωf (x) j ωf (y) j ωf (z) j  (2.5.9) (ij ,`j ,kj )≥0 x,y,z∈Fq f−1 O (kj ) (`j ) (ij ) · Xβ Xγ Xα w0vj j=0 f−1 X O (kj ) (`j ) (ij ) = −w0 X−α X−γ X−β vj f−1 j=0 (ij ,`j ,kj )j=0 ∈D where in the last step we used [Jan03] I.7.18(1) as well as the fact that  X −1 if q − 1 | n and n 6= 0 an =

a∈Fq 0 else Chapter 2. Modular representation theory of GL3 69

f−1 (recall Remark 2.4.5). Here D is deﬁned to be the set of tuples (ij, `j, kj)j=0 obeying

−1 X σf−j (2) 0 j p − af−j + af−j − δf−j+1∈/K + if−j p ≡ 0 mod e j X j `f−jp ≡ 0 mod e and 6= 0 (2.5.10) j X 2 1 j p − af−j + af−j + kf−j p ≡ 0 mod e j as well as

ij, `j, kj ≥ 0. (2.5.11)

f−1 P j Any f-tuple of integers (nj)j=0 obeying j nf−jp ≡ 0 mod e may be written as nj = mjp − f−1 mj+1 for some unique mj ∈ Z. It follows that for any tuple (ij, `j, kj)j=0 ∈ D we may write

−1 σj (2) 0 ij =(rj − 1)p + aj − aj + δj+1∈/K − rj+1

`j =tjp − tj+1 (2.5.12) 2 1 kj =(uj − 1)p + aj − aj − uj+1 for some unique rj, tj, uj ∈ Z. It is easy to see that (2.5.11) plus the nonzero condition in (2.5.10) imply rj, tj, uj ≥ 1 for all 0 ≤ j ≤ f − 1. For example, in the case of tj, we have m tj+1 ≤ tjp for all j, so if tj0 ≤ −1 for some j0 then by induction tj0+m ≤ −p which is impossible. And if tj0 = 0 for some j0 then by the same argument all tj = 0 which contradicts the nonzeroness in (2.5.10). The proofs for rj, uj are similar but easier.

Given rj ≥ 1 for all 0 ≤ j ≤ f − 1, (2.5.11) implies that if j∈ / J, j + 1 ∈ K then rj ≥ 2, and if j∈ / J, j + 1 ∈/ K and rj = 1 then we must have rj+1 = 1. These two statements give the implication

j∈ / J and rj = 1 ⇒ rj+1 = 1 and j + 1 ∈/ K.

Since J ⊆ K this means if there is a single j0 such that j0 ∈/ J and rj0 = 1 then all j∈ / K, that is K = ∅. Since we assumed this was not the case, we must have

rj ≥ 2 if j∈ / J.

f−1 Via (2.5.12) we may consider D as a subset of the set of all tuples (rj, tj, uj)j=0 of integers satisfying the conditions just found, namely rj, uj, tj ≥ 1 and rj ≥ 2 if j∈ / J. Since it suﬃces to prove (2.5.6) and (2.5.7) for −w0Sτ v, our strategy is to now consider each of the terms Chapter 2. Modular representation theory of GL3 70 appearing in

f−1 2 1 X O ((uj −1)p+aj −aj −uj+1) (tj p−tj+1) −w0Sτ v = X−α X−γ f−1 j=0 (rj ,tj ,uj )j=0 ∈D (2.5.13)

σ−1(2) j 0 ((rj −1)p+aj −aj +δj+1∈/K −rj+1) · X−β vj individually and analyze their images in the quotient Q. We write

σ−1(2) 2 1 j 0 ((uj −1)p+aj −aj −uj+1) (tj p−tj+1) ((rj −1)p+aj −aj +δj+1∈/K −rj+1) Θj(u, t, r) = X−α X−γ X−β .

Lemma 2.5.14. Only terms of (2.5.13) such that rj, tj ≤ 2, uj ≤ 3 for each j may be nonzero in V .

Proof. By Lemma 2.4.10 we must have  2 0 ∨ p + aj − aj j ∈ J ij ≤ hλj, β i ≤ 2 0 2p − aj + aj j∈ / J.

In particular the term is zero unless ij ≤ 2p − 4 for 0 ≤ j ≤ f − 1. We claim this implies each rj ≤ 2. Indeed, we immediately obtain rj+1 ≥ (rj − 3)p + 4 for each j. If some rj0 ≥ 3, then rj0+1 ≥ 4 and rj0+2 ≥ p + 4. But (resetting notation) as soon as some rj0 ≥ p, by induction rj0+m ≥ bmp where bm is the sequence deﬁned recursively by b0 = 1 and bm+1 = bmp − 3. Since bm → ∞ this is impossible. This proves that rj ≤ 2 for 0 ≤ j ≤ f − 1. Similarly, by Lemma 2.4.11 we have  2 1 ∨ 3p + aj − aj − 2 j ∈ J ij + `j ≤ hλj, γ i ≤ 1 0 3p − aj + aj − 2 j∈ / J.

−1 σj (2) 0 As we now know ij ≥ aj − aj + δj+1∈/K − 2, this implies `j ≤ 3p − 4 for 0 ≤ j ≤ f − 1. An argument similar to that above now shows tj ≤ 2 in all cases. ∨ Finally, since `j + kj ≤ hλj, γ i by Lemma 2.4.11 again, the bound above together with the fact that we now know `j ≥ p − 2 for each j implies kj ≤ 3p − 5 for 0 ≤ j ≤ f − 1. Hence uj+1 ≥ (uj − 4)p + 5 and a similar argument as above shows we must have uj ≤ 3 for each j as claimed.

The purpose of this lemma is that since rj, tj, uj are now known to be constrained in a small interval, we may essentially forget about them in many arguments by genericity. From now on we often do this without comment.

So far in our computation we have not used anything about the σj, but starting soon we must divide into cases because the behaviour of the terms in (2.5.13) genuinely varies depending Chapter 2. Modular representation theory of GL3 71 on both j and j + 1.

Lemma 2.5.15. Only terms of (2.5.13) obeying  1 j ∈ J rj = 2 j∈ / J and tj = 1 if j∈ / J may be nonzero in Q.

0 [1] Proof. First assume j ∈ J and write L(λj) = L(λj) ⊗ L(1, 0, −1) . Then  2 1 0 ∨ p + aj − aj − 3 j + 1 ∈ K hλj, γ i = 2 1 p + aj − aj − 2 j + 1 ∈/ K.

From Lemma 2.4.11 as well as (2.4.15) and (2.4.16) we deduce that

1 0 2 0 (p−aj +aj +rj+1−2) (aj −aj +δj+1∈/K −rj+1) X−γ X−β vj ∈ rad(V (λj))

(the point here is that since the exponents are in the (0, p) range, these operators act only on the first tensor factor of cosoc(V (λj))). In particular the image of this element in Qj is in (p) soc(Qj) as Qj has length 2. But X−β kills soc(Qj) by weight considerations, so in order for Θj(u, t, r)vj to have nonzero image we must have rj = 1. Since we already know rj = 2 if j∈ / J by Lemma 2.5.14, this proves the first claim. 2 0 0 (p−aj +aj +j ) In order to prove the second, assume j∈ / J. Observe that vF,j = X−β vj where we temporarily define  0 j + 1 ∈ J  0  j = −1 j + 1 ∈ K \ J  1 j + 1 ∈/ K.

−1 (a2−a0+δ −r −0 ) (p−a2+a0+0 ) (ij ) ij j j j+1∈/K j+1 j j j j Since ij < p in all cases here, we can split X−β = 2 0 0 X−β X−β p−aj +aj +j and hence Θj(u, r, t)vj is a nonzero multiple of

2 1 2 0 0 ((uj −1)p+aj −aj −uj+1) (tj p−tj+1) (aj −aj +δj+1∈/K −rj+1−j ) X−α X−γ X−β vF,j.

∨ 2 1 Since hwt(vF,j), γ i ≤ 2p + aj − aj − 2 it now follows from Corollary 2.4.14 and Lemma 2.4.11 that tj = 1 as desired.

0 Lemma 2.5.16. Consider the alcove J Weyl module V (λj) of Figure 2.5.2. Let vj denote a choice of highest weight vector of soc(Qj) in Qj. Then

0 1 1 2 0 2 (ξj −ξj ) (ξj −ξj −1) (ξj −ξj −p−1) 0 1. the image of X−γ X−β vj in Qj is a nonzero multiple of X−β vj; Chapter 2. Modular representation theory of GL3 72

0 1 1 2 (ξj −ξj +1) (ξj −ξj −2) 2. the image of X−γ X−β vj in Qj is a nonzero multiple of

0 2 0 2 (ξj −ξj −p−2) 0 1 (ξj −ξj −p−1) 0 X−γX−β − (ξj − ξj )X−αX−β vj.

Proof. We will prove the second statement since it is similar to the first but harder. The goal 0 0 1 is to find an explicit element Q ∈ V (λj) whose image in Qj is vj. Let µ = λj − (ξj − ξj )α − pβ be the highest weight of soc(Qj) and let M = hvH,j, vI,jiGL3 . By Lemma 2.4.12 and Lemma 2.4.13 we find the following bases:

0 1 (ξj −ξj −`) (`) (p−`) 0 1 V (λj)µ = Span X X X vj 0 ≤ ` ≤ ξ − ξ −α −γ −β j j 0 1 (ξj −ξj −`) (`) (p−`) 1 2 hvI,jiµ = Span X X X vj 1 ≤ ` ≤ p − ξ + ξ −α −γ −β j j  0 1  ξj −ξj 0 1  X (ξj −ξj −`) (`) (p−`)  hvH,jiµ = Span X−α X−γX−β vj .  `=0 

0 1 (ξj −ξj ) (p) Consider θ := X−α X−βvj ∈ V (λj)µ. From the bases just computed we see that θ∈ / M. 0 1 (ξj −ξj ) (p) Moreover, by (2.4.15) and (2.4.16) X−α X−β kills the highest weight vector of cosoc V (λj), so the image of θ in V (λj)/M is a nonzero element of rad(V (λj)/M) = soc(V (λj)/M). It is easy to check that soc(V (λj)/M))µ is 1-dimensional and this immediately implies that the image of

θ in Qj is a highest weight vector for soc(Qj). Since we are free to modify θ by elements of Mµ without changing this statement, we therefore take

0 1 ξj −ξj 0 1 X (ξj −ξj −`) (`) (p−`) Q := X−α X−γX−β vj. 1 2 `=p−ξj +ξj +1

We now prove the lemma. We compute directly from Lemma 2.4.2

0 1 0 1 ξj −ξj ξj −ξj −` 0 2 0 1 0 2 (ξj −ξj −p−1) X X (ξj −ξj −`−j) (j) (`) (ξj −ξj −p−1−j) (p−`) X−αX−β Q = X−αX−α X−γX−γX−β X−β vj. 1 2 j=0 `=p−ξj +ξj +1

0 1 In this equation, terms such that ` + j ≥ ξj − ξj are either outright zero or lie in hvI,jiGL3 ⊂ M. 0 1 The only remaining terms obey ` + j = ξj − ξj and we conclude that

 0 1  ξj −ξj (ξ0−ξ2−p−1) X ξ0 − ξ1ξ1 − ξ2 − 1 (ξ0−ξ1) (ξ1−ξ2−1) X X j j Q ≡  j j j j  X X j j X j j v mod M −α −β  i p − i  −α −γ −β j 1 2 i=p−ξj +ξj +1 0 1 1 2 (ξj −ξj ) (ξj −ξj −1) = X−αX−γ X−β vj mod M Chapter 2. Modular representation theory of GL3 73 where the second step follows from the fact that for any m ≤ n in (0, p) we have1

n X np − m ≡ 1 mod p. i p − i i=m

Similarly from Lemma 2.4.2 we directly compute

0 1 0 1 0 2 ξj −ξj min(ξj −ξj −`,ξj −ξj −p−2) 0 2 0 1 (ξj −ξj −p−2) X X (ξj −ξj −`−j) X−γX−β Q = X−α 1 2 j=0 `=p−ξj +ξj +1 0 2 (j) (`) (ξj −ξj −p−2−j) (p−`) · X−γX−γX−γX−β X−β vj.

0 1 0 1 Like before, only terms such that j = ξj − ξj − 1 − ` or ξj − ξj − ` survive modulo M. Splitting up the sum into these two cases we obtain

0 1 ξj −ξj −1 0 2 0 1 1 2 0 1 1 2 (ξ −ξ −p−2) X (ξj − ξj )! ξ − ξ − 1 (ξ −ξ ) (ξ −ξ −1) X X j j Q ≡ j j X X j j X j j v −γ −β `!(ξ0 − ξ1 − 1 − `)! p − ` −α −γ −β j 1 2 j j `=p−ξj +ξj +1 0 1 ξj −ξj 0 1 1 2 0 1 1 2 X (ξj − ξj + 1)! ξ − ξ − 2 (ξ −ξ +1) (ξ −ξ −2) + j j X j j X j j v mod M `!(ξ0 − ξ1 − `)! p − ` −γ −β j 1 2 j j `=p−ξj +ξj +2 0 1 1 2 0 1 (ξj −ξj ) (ξj −ξj −1) = (ξj − ξj )X−αX−γ X−β vj 0 1 1 2 0 1 (ξj −ξj +1) (ξj −ξj −2) + (ξj − ξj + 1)X−γ X−β vj mod M.

The lemma follows by putting together these two calculations.

0 Lemma 2.5.17. Consider the alcove G Weyl module V (λj) of Figure 2.5.3. Let vj denote a choice of highest weight vector of soc(Qj) in Qj. Then

0 2 0 1 (ξj −ξj ) (p−1) (p+ξj −ξj −1) 0 1. The image of X−γ X−β vj in Qj is a nonzero multiple of X−β vj.

0 2 (ξj −ξj +1) (p−2) 2. The image of X−γ X−β vj in Qj is a nonzero multiple of

0 1 0 1 (p+ξj −ξj −2) 0 2 (p+ξj −ξj −1) 0 X−γX−β − (ξj − ξj )X−αX−β vj.

Proof. We prove the second statement, since the proof of the ﬁrst is similar but easier. Observe 0 2 1 2 (ξj −ξj +1) (p−ξj +ξj −2) that the given element of V (λj) is clearly a nonzero multiple of X−γ X−β vF,j. 0 2 (ξj −ξj ) 0 Moreover the image of vC,j = X−α vF,j in Qj is a nonzero multiple of vj by Lemma 2.4.13.

1 p−m −m This may be proved as follows: writing n = m + k and noting that p−m−i ≡ i mod p, the sum reduces Pk m+k−m m+k−m to j=0 k−j j mod p which is = k = 1 by the Vandermonde-Chu identity. Chapter 2. Modular representation theory of GL3 74

We may therefore directly compute

0 2 ξj −ξj 0 1 0 2 0 1 (p+ξj −ξj −2) X (ξj −ξj −`) (`+1) (p+ξj −ξj −2−`) X−γX−β vC,j = (` + 1)X−α X−γ X−β vF,j `=0 0 2 1 2 0 2 1 2 0 2 (ξj −ξj ) (p−ξj +ξj −1) 0 2 (ξj +ξj +1) (p−ξj +ξj −2) = (ξj − ξj )X−αX−γ X−β vF,j + (ξj − ξj + 1)X−γ X−β vF,j where the second equality follows from Lemma 2.4.10. Similarly we compute

0 1 0 2 1 2 (p+ξj −ξj −1) (ξj −ξj ) (p−ξj +ξj −1) X−αX−β vC,j = X−αX−γ X−β vF,j.

The lemma follows.

With these two lemmas we now study the image of Θj(u, t, r)vj in Qj on a case-by-case 0 basis. From now on we fix for each 0 ≤ j ≤ f − 1 a highest weight vector vj ∈ soc(Qj). 0 0 1 Decomposing soc(Qj) via the Steinberg tensor product theorem we in fact write vj = vj ⊗ vj 0 0 1 2 1 where vj is a highest weight vector for L((−aj − 1, −aj , −aj + 1) − ζj,j+1) and vj is a highest [1] 0 weight vector for L(j) . In fact we fix the choice of vj such that the final sentence in each of Cases I-VI below is true (a priori it is true up to a nonzero multiple not depending on u, t, r).

Case I. If j, j +1 ∈ J and j +1 ∈ K then by Lemma 2.5.16 we see that the image of Θj(u, t, r)vj in Qj is equal to

−1 2 1 1 0 2 1 p − tj+1 (aj −aj −uj+1) (aj −aj +1−tj+1) (aj −aj −2) 0 (uj −1) (tj −1) 1 1 0 X−α X−γ X−β vj ⊗ X−α X−γ vj . aj − aj + 1 − tj+1

Case II. If j ∈ J, j +1 ∈/ J, j +1 ∈ K then by Lemma 2.5.16 we see that the image of Θj(u, t, r)vj in Qj is equal to

2 1 1 0 2 1 (aj −aj −uj+1) (aj −aj −1) (aj −aj −2) 0 (uj −1) (tj −1) 1 X−α X−γ X−β vj ⊗ X−α X−γ vj .

Case III. If j ∈ J, j + 1 ∈/ J, j + 1 ∈/ K then by Lemma 2.5.16 we see that the image of

Θj(u, t, r)vj in Qj is equal to

2 1 1 0 2 1 h 1 0 (aj −aj −uj+1) (aj −aj ) (aj −aj −2) (aj − aj )X−α X−γ X−β 2 1 1 0 2 1 1 0 2 1 (aj −aj +1−uj+1) (aj −aj −1) (aj −aj −1)i 0 + (aj − aj + 1)(aj − aj + 1 − uj+1)X−α X−γ X−β vj

(uj −1) (tj −1) 1 ⊗ X−α X−γ vj .

Case IV. If j∈ / J, j+1 ∈ J, j+1 ∈ K then by Lemma 2.5.17 we see that the image of Θj(u, t, r)vj Chapter 2. Modular representation theory of GL3 75

in Qj is equal to

−1 2 1 1 0 2 1 p − tj+1 (aj −aj −uj+1) (aj −aj +1−tj+1) (aj −aj −2) 0 (uj −1) 1 1 0 X−α X−γ X−β vj ⊗ X−α X−βvj . aj − aj + 1 − tj+1

Case V. If j, j + 1 ∈/ J, j + 1 ∈ K then by Lemma 2.5.17 we see that the image of Θj(u, t, r)vj in Qj is equal to

(uj −1) 1 ⊗ X−α X−βvj .

Case VI. If j, j + 1 ∈/ J, j + 1 ∈/ K then by Lemma 2.5.17 we see that the image of Θj(u, t, r)vj in Qj is equal to

2 1 1 0 2 1 (aj −aj −uj+1) (aj −aj −1) (aj −aj −2) 0 (uj −1) 1 X−α X−γ X−β vj ⊗ X−α X−βvj .

Observe that the ﬁrst tensor factors are the same in Cases II and VI, I and IV, and III and V.

This is signiﬁcant because ζj,j+1 is also the same in these three pairs of cases, which allows us to cut down the amount of work needed. From now on, we write the expressions appearing in Cases I - VI above as

0 1 image of Θj(u, t, r)vj in Qj = θj(uj+1, tj+1)vj ⊗ ηj(uj, tj)vj .

Incidentally, these calculations have already proven (2.5.6).

Lemma 2.5.18. The image of Θj(u, t, r)vj is only possibly nonzero in Qj if  {(1, 2), (2, 1)} j ∈ J (uj, tj) ∈ {(1, 1), (2, 1)} j∈ / J

1 Proof. If j ∈ J then vj is a highest weight vector of L(1, 0, 0) and from cases I - III above we 1 (uj −1) (tj −1) 1 1 have ηj(uj, tj)vj = X−α X−γ vj which is 0 if uj + tj > 3. If j∈ / J then vj is a highest 1 (uj −1) 1 weight vector of L(0, 0, −1) and we have ηj(uj, tj)vj = X−α X−βvj , which is 0 if uj > 2. From these two cases we deduce that uj ≤ 2 in all cases, and (uj, tj) 6= (2, 2). In cases I and IV (which are the same), we easily compute that

0 θj(uj+1, tj+1)Ξ(wt(vj )) = 0 if uj+1 = tj+1 = 1. Together with the previous paragraph, this proves that j ∈ J ⇒ (uj, tj) ∈ Chapter 2. Modular representation theory of GL3 76

{(1, 2), (2, 1)} and the lemma follows.

This lemma allows us to more simply write

0 1 0 1 θj(uj+1, tj+1)vj ⊗ ηj(uj, tj)vj = θj(uj+1)vj ⊗ ηj(uj)vj  3 − uj+1 j ∈ J by taking tj+1 = . From these calculations and (2.5.13), we deduce that the 1 j∈ / J image of −w0Sτ v inside Q, under the identiﬁcation

O 0 1 2 [f−j] soc(Q) = L((−a − 1, −a , −a + 1)j − ζj,j+1) ⊗ L(j+1) GL3(Fq) j j j GL3(Fq) j from (2.5.4) is equal to

f−1 X O 0 1 [θj(uj+1)vj ⊗ ηj+1(uj+1)vj+1]. (2.5.19) f−1 f j=0 (uj )j=0 ∈{1,2}

0 1 We therefore need to compute the image of θj(uj+1)vj ⊗ ηj+1(uj+1)vj+1 under the projection

0 1 2 0 1 2 prj : L((−aj − 1, −aj , −aj + 1) − ζj,j+1) ⊗ L(j+1) L(−aj − 1, −aj , −aj + 1).

This is accomplished in Lemma 2.5.20 below, which shows that

0 1 0 1 prj(θj(1)vj ⊗ ηj+1(1)vj+1) = µj prj(θj(2)vj ⊗ ηj+1(2)vj ) for some constant µj such that 1 + µj 6= 0. To be precise, we have  0 in cases II and IV by Lemma 2.5.20(1)   1 µj = 1 0 in cases I and IV by Lemma 2.5.20(2) aj −aj  −(a2−a0)  j j  2 1 in cases III and V by Lemma 2.5.20(3).  aj −aj −1

Given this result, we deduce from (2.5.19) that the image of −w0Sτ v under the projection soc(Q) F0 is equal to X Y O 0 1 µj prj(θj(2)vj ⊗ ηj+1(2)vj ) f−1 f {j:uj+1=1} j (uj )j=0 ∈{1,2}   Y O 0 1 =  (1 + µj) prj(θj(2)vj ⊗ ηj+1(2)vj ). j j

0 1 Since Lemma 2.5.20 also shows that prj(θj(2)vj ⊗ ηj+1(2)vj ) 6= 0, this ﬁnishes the proof of Chapter 2. Modular representation theory of GL3 77

Theorem 2.3.2(1).

3 Lemma 2.5.20. Let (a, b, c) ∈ Z be a 5-deep weight inside the lowest alcove for GL3. Let u = 1, 2. In each item below v0 and v1 denote highest weight vectors of the corresponding tensor factors.

(a−b−u) (b−c−1) (a−b−2) (u−1) 1. Let θ(u) = X−α X−γ X−β v0 ⊗ X−α X−βv1 inside L(−c − 1, −b, −a + 2) ⊗ L(0, 0, −1). If pr denotes the projection to L(−c − 1, −b, −a + 1) then pr(θ(2)) 6= 0 and pr(θ(1)) = 0.

0 p−3+u −1 (a−b−u) (b−c−2+u) (a−b−2) (u−1) (2−u) 2. Let θ (u) = b−c−2+u X−α X−γ X−β v0 ⊗ X−α X−γ v1 inside L(−c − 1, −b − 1, −a + 1) ⊗ L(1, 0, 0). If pr denotes the projection to L(−c − 1, −b, −a + 1) then 0 1 0 pr(θ (1)) = b−c pr(θ (2)) 6= 0

00 (a−b−u) (b−c) (a−b−2) (a−b+1−u) (b−c−1) (a−b−1) 3. Let θ (u) = [(b−c)X−α X−γ X−β +(b−c+1)(a−b+1−u)X−α X−γ X−β ]v0⊗ (u−1) X−α X−βv1 inside L(−c, −b, −a + 1) ⊗ L(0, 0, −1). If pr denotes the projection to 00 −(a−c) 00 L(−c − 1, −b, −a + 1) then pr(θ (1)) = a−b−1 pr(θ (2)) 6= 0.

0 00 Proof. We ﬁrst remark that θ(u), θ (u), θ (u) are all of weight w0λ0 := (−a + 1, −b, −c + 1). The statements are given in increasing order of diﬃculty of calculation. We will give the most details for the third statement and only summarize the proofs of the others. From the tensor product theorem the w0λ0 weight space in L(−c, −b, −a + 1) ⊗ L(0, 0, −1) is 4-dimensional, equal to

L(−c, −b, −a)w0λ0 ⊕ L(−c, −b − 1, −a + 1)w0λ0 ⊕ L(−c − 1, −b, −a + 1)w0λ0 , (2.5.21) where the ﬁrst summand has dimension 2. A basis of this weight space is given by

   (a−b−2) (b−c) (a−b−2)  e0 X−α X−γ X−β v0 ⊗ X−γv1    (a−b−1) (b−c−1) (a−b−1)  e1 X−α X−γ X−β v0 ⊗ X−γv1   =   . (2.5.22) e   X(a−b)X(b−c+1)X(a−b−1)v ⊗ X v   2  −α −γ −β 0 −β 1  (a−b−1) (b−c) (a−b−1) e3 X−α X−γ X−β v0 ⊗ v1

To see this one uses the Xi monomial to check that the given vectors are all nonzero and the first two are linearly independent. Some of these calculation are carried out in Example 2.5.25(1). For the remainder of the proof, all vectors will be expressed with respect to this basis. 0 (a−b−1) (b−c) (a−b−1) Using the Xi monomial, one sees that the vector v := X−α X−γ X−β v0 ⊗ X−γv1 is nonzero and therefore must be a lowest weight vector for the summand L(−c, −b, −a) ⊆ L(−c, −b, −a + 1) ⊗ L(0, 0, −1). It follows that a basis of the first direct summand in 2.5.21 0 0 is {XβXαv ,XαXβv }. After a great deal of computation (see Example 2.5.25(2) where one of Chapter 2. Modular representation theory of GL3 78 these is carried out), we find that

    1 0     0 a − b − 1 0  1  XβXαv =   and XαXβv =   . (2.5.23)  −1  −1     −1 0

In order to ﬁnd a basis vector for the second direct summand in (2.5.21), we observe ﬁrst that 00 v = (a − b − 1)v0 ⊗ X−βv1 − X−βv0 ⊗ v1 is killed by Xα and Xβ and is therefore a highest weight vector for the summand L(−c, −b − 1, −a + 1) ⊆ L(−c, −b, −a + 1) ⊗ L(0, 0, −1). Thus (a−b−2) (b−c+1) (a−b−2) 00 a lowest weight vector for this factor is given by X−α X−γ X−β v and a basis of (a−b−2) (b−c+1) (a−b−2) 00 the second summand in (2.5.21) by XβX−α X−γ X−β v . Calculating in the same manner as before, this ends up being equal to   −1   −(a − b − 1)   . −(a − b − 1)   −(a − b − 1)

Now ﬁnally one checks that     0 −(b − c)      0  −(b − c + 1)(a − b − 1) θ(1) =   θ(2) =   a − c  0      0 0 and the unique constant µ such that       1 0 −1 *     + a − b − 1  1  −(a − b − 1) θ(1) − µθ(2) ∈ span   ,   ,    −1  −1 −(a − b − 1)       −1 0 −(a − b − 1)

−(a−c) is µ = a−b−1 which proves the claim.

Now we prove the second statement. In this case the w0λ0 weight space is 2-dimensional, equal to

L(−c, −b − 1, −a + 1)w0λ0 ⊕ L(−a − 1, −b, −a + 1)w0λ0 . (2.5.24)

A highest weight vector for L(−c, −b − 1, −a + 1) ⊆ L(−c − 1, −b − 1, −a + 1) ⊗ L(1, 0, 0) is given by v0 ⊗ v1. Using the Xi monomial (to check nonvanishing) one sees that a lowest weight (a−b−2) (b−c) (a−b−2) vector for L(−c, −b − 1, −a + 1) is given by X−α X−γ X−β v0 ⊗ X−γv1. Then a basis Chapter 2. Modular representation theory of GL3 79

for the ﬁrst direct summand in 2.5.24 is given by Xβ times this, which works out to be

(a−b−1) (b−c−1) (a−b−2) (a−b−2) (b−c) (a−b−2) X−α X−γ X−β v0 ⊗ X−γv1 + X−α X−γ X−β v0 ⊗ X−αv1.

This is (−1)b−c[θ0(1) − (b − c)θ0(2)] and it follows that pr(θ0(1)) = (b − c) pr(θ0(2)) as desired. Finally, the ﬁrst statement is easy because one checks directly using the Xi monomial that

θ(1) = 0 and θ(2) 6= 0. Since the w0λ0 weight space of L(−c − 1, −b, −a + 2) ⊗ L(0, 0, −1) is 1-dimensional by the tensor product theorem the result follows.

Example 2.5.25. We do two example computations with the Xi monomial that were used in the above proof.

1. We show that the ﬁrst two vectors in (2.5.22) are nonzero and linearly independent. (a−b−2) (b−c) (a−b−2) (a−b−1) (b−c−1) (a−b−1) It suﬃces to show that X−α X−γ X−β v0 and X−α X−γ X−β v0 are linearly independent inside L(−c, −b, −a + 1). The Xi monomial of this representation is (p−a+b) (2p−a+c−1) (p−b+c−1) Ξ = Ξ(−c, −b, −a + 1) = X−β X−α X−β . We compute

p − 2 X(a−b−2)X(b−c)X(a−b−2)Ξ = X(a−b−2)X(b−c)X(p−2)X(2p−a+b−1)X(p−b+c−1) −α −γ −β a − b − 2 −α −γ −β −α −β a−b (a−b−2) (b−c) = (−1) (a − b − 1)X−α X−γ p−2 X (2p−a+c−1−`) (`) (p−2−`) (p−b+c−1) X−α X−γX−β X−β . `=0

The `th term here is zero if either b − c + ` ≥ p or p − 2 − ` + p − b + c − 1 ≥ p. So only two terms ` = p − b + c − 2, p − b + c − 1 survive and we are left with

b−c (p−1) (p−2) (p−1) (p−1) (p−1) (p−2) = (−1) (a−b−1)(b−c+1)X−α X−γ X−β +(a−b−1)(b−c)X−α X−γ X−β .

On the other hand we similarly calculate

(a−b−1) (b−c−1) (a−b−1) (p−1) (p−2) (p−1) (p−2) (p−1) (p−2) X−α X−γ X−β Ξ = (b−c)X−α X−γ X−β +(a−b)(b−c)X−α X−γ X−β .

Since these two elements of U1 are linearly independent the result follows from Proposi- Fp tion 2.4.18.

0 2. We calculate the element XαXβv in (2.5.23):

0 (a−b−1) (b−c) (a−b−1) Xβv =X−α XβX−γ X−β v0 ⊗ X−γv1 (a−b−1) (b−c) (a−b−2) (a−b) (b−c−1) (a−b−1) = X−α X−γ X−β v0 ⊗ X−γv1 + (a − b)X−α X−γ X−β v0 ⊗ X−γv1. Chapter 2. Modular representation theory of GL3 80

Applying Xα we obtain

(a−b−1) (b−c−1) (a−b−1) (a−b−1) (b−c) (a−b−2) X−α X−γ X−β v0 ⊗ X−γv1 − X−α X−γ X−β v0 ⊗ X−βv1 (a−b) (b−c−1) (a−b−1) − (a − b)X−α X−γ X−β v0 ⊗ X−βv1.

(a−b−1) (b−c) (a−b−2) (a−b) (b−c−1) (a−b−1) Finally, it is easy to check that X−α X−γ X−β −(a−b−1)X−α X−γ X−γ kills Ξ, so we get (2.5.23).

Proof of Theorem 2.3.2(2) and (3), part I

We now move on to the proof of the other parts of Theorem 2.3.2. Unfortunately, the method of proof is the same - embed the principal series inside a Weyl module and compute directly, and the specifics of this case differ enough to necessitate full details. On the other hand, the computations turn out to be simpler. t −1 First, by applying the automorphism g 7→ w0 g w0 of GL3(Fq) we easily deduce Theorem 2.3.2(2) from part (3), so we may focus on proving the two halves of part (3). Thus, from now 0 0 on let σ, τ, J, K = ∅, π, π ,Sτ ,Sτ be as in the statement of part (3). The statement about Sτ 0 0 and π is proved in this subsection, and the statement about Sτ and π in the next. As before, we first embed π into a Weyl module V = jV (λj). We set

−1 i σj ς(i) i −1 ξj = −aj + (p − 1)ε − 1 + σj+1ς(i)

0 1 2 where ε = (1, 0, 0) so that (ξj , ξj , ξj ) ∈ X1(T3) for each 0 ≤ j ≤ f −1, and set λj = ξj +(p−1)η. Then we have

ξj ∈ A and λj ∈ G ⇐⇒ j ∈ J

ξj ∈ B and λj ∈ J ⇐⇒ j∈ / J.

Though the notation has changed, the submodule structure of the V (λj) may still be read oﬀ of Figures 2.5.2 and 2.5.3.

In the same way as before, after picking a highest weight vector v = ⊗jvj ∈ V we obtain a nonzero GL3(Fq)-equivariant map π → V by Frobenius reciprocity. This time we deﬁne f−1 f−1 Q = j=0 = j=0 Qj(Xj) where  D j ∈ J Xj = H j∈ / J.

Then we have

f−1 0 1 2 [1] soc(Q) = j=0 L (−aj − 1, −aj , −aj + 1) − ζj,j+1 ⊗ L(j) Chapter 2. Modular representation theory of GL3 81 where  (1, 0, 0) j ∈ J j = (1, −1, −1) j∈ / J and ζj,j+1 is determined by the following table.

ζj,j+1 = j, j + 1 ∈ J (1, 0, 0) j ∈ J, j + 1 ∈/ J (1, −1, 1) j∈ / J, j + 1 ∈ J (1, 0, 0) j, j + 1 ∈/ J (1, 1, −1)

The equation (2.5.4) holds verbatim with the new notation (and note that all weights in

L(j) still occur with multiplicity 1, so the multiplier in the ﬁnal line is again equal to 1). And since ζj,j+1 is a permutation of j+1 we see that soc(Q)GL3(Fq) contains F0 with multiplicity 1. As in Claim 2.5.5, our goal is to show that the image of Sτ v in Q lies in soc(Q) and is nonzero under the projection soc(Q) F0. Computing as in (2.5.9) we deduce that in V we have

X O (`j ) (kj ) Sτ v = −sαsβ X−α X−γ vj (2.5.26) f−1 j (`j ,kj )j=0 ∈D

f−1 where D denotes the set of tuples (`j, kj)j=0 such that

`j, kj ≥ 0 (2.5.27) and

X j `f−jp ≡ 0 mod e and 6= 0 (2.5.28) j X 2 1 j (kf−j + δf−j∈ /J (p − af−j + af−j))p ≡ 0 mod e and 6= 0. j

In particular we may write

`j = tjp − tj+1  ujp − uj+1 j ∈ J kj = 2 1 (uj − 1)p + aj − aj − uj+1 j∈ / J for some integers tj, uj ∈ Z. By (2.5.27) and (2.5.28) we must have tj, uj ≥ 1 for each 0 ≤ j ≤ f − 1.

Lemma 2.5.29. The only terms in (2.5.26) that are possibly nonzero in V obey tj = 1 for Chapter 2. Modular representation theory of GL3 82

0 ≤ j ≤ f − 1 and  1 j ∈ J uj = 1, 2, 3 j∈ / J.

∨ 1 0 Proof. By Lemma 2.4.10 we must have `j ≤ hλj, α i ≤ 2p − aj + aj in all cases, which implies tj = 1 for each j by an argument similar to that in the proof of Lemma 2.5.14. Lemma 2.4.11 implies  2 1 ∨ 3p − aj + aj − 2 j ∈ J `j + kj ≤ hλj, γ i ≤ 2 1 3p + aj − aj − 2 j∈ / J.

As we now know `j = p − 1 for all j, we get uj ≤ 3 by another similar argument.

Like before, this lemma shows that we can consider uj to be small enough to ignore by generic- (`j ) (kj ) (p−1) (kj ) ity in the remainder of our arguments. Let us now write Θj(u) = X−α X−γ vj = X−α X−γ . 0 1 The next step is to analyze the image of Θj(u)vj in Qj on a case-by-case basis. Let vj ⊗ vj ∈ 0 soc(Qj) denote a ﬁxed choice of highest weight vector, where vj is a highest weight vector of 0 1 2 1 [1] L (−aj − 1, −aj , −aj + 1) − ζj,j+1 and vj is a highest weight vector of L(j) . In each case 0 1 below, we make the choice of vj and vj such that the ﬁnal equation is true.

(p−1) (p−1) Case I. Assume that j, j + 1 ∈ J. Then clearly Θj(u)vj = X−α X−γ vj is a nonzero mul- 2 0 2 0 2 1 (aj −aj −2) (p−1) (aj −aj −2) (p−aj +aj ) tiple of X−α X−γ vE,j. One checks that X−α X−γ Ξj(E) = 0, so Xj := 2 0 2 1 (aj −aj −2) (p−aj +aj ) X−α X−γ vE,j lies in the radical of the sub-Weyl module generated by vE,j. More- over, Xj is nonzero in V by Lemma 2.4.12, and one can easily check that its weight is not a possible weight of the irreducible constituent of V with highest weight lying in A or C. Hence 1 0 it lies in the sub-Weyl module of alcove D. As wt(Xj) = wt(vD,j) − (p + aj − aj − 2)α, we 1 0 (p+aj −aj −2) deduce from Lemma 2.4.12 that Xj is a nonzero multiple of X−α vD,j. Thus we see that the image of Θj(u)vj in Qj is (after rescaling)

1 0 2 1 (aj −aj −2) (aj −aj −1) 0 1 X−α X−γ vj ⊗ X−αvj .

Case II. Assume that j ∈ J, j + 1 ∈/ J. Then arguing exactly as in the previous case, we see (p−1) (p−uj+1) that (after rescaling) the image of Θj(u)vj = X−α X−γ in Qj is

−1 1 0 2 1 p − uj+1 (aj −aj −3) (aj −aj +2−uj+1) 0 1 2 1 X−α X−γ vj ⊗ X−αvj . aj − aj + 2 − uj+1

1 0 (p−aj +aj +1) Case III. Assume that j∈ / J, j + 1 ∈ J. Since vH,j = X−α in this case we obviously Chapter 2. Modular representation theory of GL3 83 have

2 1 ((uj −1)p+aj −aj −1) (p−1) Θj(u)vj = X−γ X−α vj 1 0 ((u −1)p) (a2−a1−1) (a1−a0−2) aj −aj j j j j j = (−1) X−γ X−γ X−α vH,j and thus the image of Θj(u)vj in Qj is (after rescaling)

1 0 2 1 (aj −aj −2) (aj −aj −1) 0 (uj −1) 1 X−α X−γ vj ⊗ X−γ vj .

Case IV. Assume that j, j + 1 ∈/ J. One checks in a manner very similar to the previous case that the image of Θj(u)vj in Qj is

1 0 2 1 (aj −aj −1) (aj −aj −uj+1) 0 (uj −1) 1 X−α X−γ vj ⊗ X−γ vj .

The conclusion in each of the four cases just discussed was that the image of Θj(u)vj in Qj is equal to an element of the form

0 1 θj(uj+1)vj ⊗ ηj(uj)vj .

In particular, we have already shown that the image of Sτ v in Q lies in soc(Q). Now in (2.5.19), we deduce that the image of −sβsαSτ v inside Q under the identiﬁcation

f−1 O 0 1 2 [f−j] soc(Q) = L((−a − 1, −a , −a + 1)j − ζj,j+1) ⊗ L(j+1) GL3(Fq) j j j GL3(Fq) j=0 is equal to f−1 X O 0 1 [θj(uj+1)vj ⊗ ηj+1(uj+1)vj+1]. (2.5.30) f−1 f j=0 (uj )j=0 ∈{1,2,3}

(In this sum by deﬁnition let us take only terms with uj+1 = 1 whenever j + 1 ∈ J to be nonzero). Let

0 1 2 0 1 2 prj : L((−aj − 1, −aj , −aj + 1) − ζj,j+1) ⊗ L(j+1) L(−aj − 1, −aj , −aj + 1) denote the projection. Lemma 2.5.31 below shows that if j + 1 ∈/ J then

0 1 2 0 1 prj(θj(2)vj ⊗ ηj+1(2)vj+1) = µj prj(θj(3)vj ⊗ ηj+1(3)vj+1) 0 1 1 0 1 prj(θj(1)vj ⊗ ηj+1(1)vj+1) = µj prj(θj(3)vj ⊗ ηj+1(3)vj+1) Chapter 2. Modular representation theory of GL3 84

1 2 1 2 where µj , µj are constants obeying 1 + µj + µj 6= 0. Precisely, we have

2 1 2 1 2 1 2 1  −(aj −aj −1)(aj −aj )(aj −aj +1)  2(aj −aj )  2 1 j ∈ J  2 1 j ∈ J 1 2(aj −aj +3) 2 (aj −aj +3) µj = and µj = 0 j∈ / J 0 j∈ / J.

 1 j ∈ J If we write simply θ for the term of (2.5.30) such that uj = then we have shown 3 j∈ / J that the image of −sαsβSτ v under the projection pr : soc(Q) F0 is equal to       X Y 1 Y 2 Y 1 2  µj   µj  pr(θ) =  (1 + µj + µj ) pr(θ). f (uj )∈{1,2,3} j+1∈/Juj+1=1 j+1∈/J,uj+1=2 j+1∈/J

Since Lemma 2.5.31 also shows that pr(θ) 6= 0, we have completed the proof of the ﬁrst half of Theorem 2.3.2(3).

3 Lemma 2.5.31. Let (a, b, c) ∈ Z be a 5-deep weight in the lowest alcove for GL3. Let u = 1, 2, 3. In each item below v0 and v1 denote highest weight vectors of the corresponding tensor factors.

(b−c−1) (a−b−u) (u−1) 1. The image of X−α X−γ v0 ⊗ X−γ v1 under the projection L(−c − 2, −b − 1, −a + 2) ⊗ L(1, 1, −1) L(−c − 1, −b, −a + 1) is 0 if u = 1, 2 and nonzero if u = 3.

(b−c−2) (a−b−1) 2. The image of X−α X−γ v0 ⊗ X−αv1 under the projection L(−c − 2, −b, −a + 1) ⊗ L(1, 0, 0) L(−c − 1, −b, −a + 1) is nonzero.

p−u −1 (b−c−3) (a−b+2−u) (u−1) 3. Let θ(u) = a−b+2−u X−α X−γ v0 ⊗ X−γ v1 inside L(−c − 2, −b + 1, −a) ⊗ L(1, 1, −1). If pr denotes the projection onto L(−c − 1, −b, −a + 1) then we have

−(a − b − 1)(a − b)(a − b + 1) pr(θ(1)) = pr(θ(3)), 2(a − b + 3) 2(a − b) pr(θ(2)) = pr(θ(3)) a − b + 3

and pr(θ(3)) 6= 0.

Proof. The proof is similar to that of Lemma 2.5.20 so we give the main steps only. For the (b−c−1) (a−b−u) ﬁrst statement it is easy to check that X−α X−γ Ξ(−c−2, −b−1, −a+2) = 0 if u = 1, 2 (b−c−1) (a−b−3) (2) and is nonzero if u = 3. By considering the weight we see that X−α X−γ v0 ⊗ X−γ v1 is a nonzero multiple of sαsβv0 ⊗ w0v1 = sαsβ(v0 ⊗ v1) and the result follows. The second statement is very similar. We now prove the third part of the lemma. Note wt(θ(u)) = (−a + 1, −c − 1, −b) and this Chapter 2. Modular representation theory of GL3 85 weight space in L(−c − 2, −b + 1, −a) ⊗ L(1, 1, −1) has dimension 3 equal to

L(−c − 1, −b + 2, −a − 1)wt(θ(u)) ⊕ L(−c − 1, −b, −a + 1)wt(θ(u)) (2.5.32) ⊕ L(−c − 1, −b + 1, −a)wt(θ(u)) by the translation principle. We ﬁnd all constants µi such that the projection of θ(i) − µiθ(3) to the second summand is zero, for i = 1, 2. Let

   (b−c−2) (a−b)  e0 X−α X−γ X−βv0 ⊗ v1   =  (b−c−3) (a−b)  . e1  X−α X−γ v0 ⊗ X−γv1  (b−c−2) (a−b−2) (2) e2 X−α X−γ X−βv0 ⊗ X−γ v1

This is a basis of the wt(θ(u)) weight space with respect to which we express elements from now on. That these elements are nonzero can be checked using the Xi monomial in a manner similar to Example 2.5.25. Starting from the fact that v0 ⊗ v1 is a highest weight vector for the summand L(−c − 1, −b + 2, −a − 1) one computes that bases of the ﬁrst and third summand in (2.5.32) are  −1    a−b+1 1  1  and  −(a−b−1)     2  −1 a−b−1 1

(−1)a−b (−1)a−b (−1)a−b2 respectively. Since θ(1) = a−b+1 e0, θ(2) = a−b+1 e1 and θ(3) = (a−b−1)(a−b)(a−b+1) e2 we now 2(a−b) −(a−b−1)(a−b)(a−b+1) easily calculate that there are unique solutions µ2 = a−b+3 and µ1 = 2(a−b+3) which proves the lemma.

Proof of Theorem 2.3.2(3), part 2

We now ﬁnish the proof of Theorem 2.3.2 by addressing the second part of statement (3) 0 0 regarding Sτ and π . The proof uses the same method, but fortunately the calculations in this case are easiest of all. This time, deﬁne −1 i σj (i) 0i 0i −1 ξj = −aj + pj − j+1 − 1 + σj+1(i) where  0 (0, 0, 0) j ∈ J j := (0, 0, −1) j∈ / J.

0 f−1 In the same way as before, we obtain a nonzero map π → V := j=0 V (λj) where λj = ξj + (p − 1)η via the action of T3(Fq) on a choice of highest weight vector v ∈ V , and in fact Chapter 2. Modular representation theory of GL3 86 this map is an embedding, though we don’t need this result. Then it is clear that

ξj ∈ A and λj ∈ G ⇐⇒ j ∈ J

ξj ∈ B and λj ∈ J ⇐⇒ j∈ / J.

As before, the structure of V (λj) may be seen in Figures 2.5.2 and 2.5.3. We deﬁne the quotient f−1 f−1 of V by Q := j=0 Qj = j=0 Qj(Xj) where  G j ∈ J Xj = I j∈ / J.

Then we observe that

f−1 0 1 2 [1] soc(Q) = j=0 L (−aj − 1, −aj , −aj + 1) − ζj,j+1 ⊗ L(j) where  (1, 0, −1) j ∈ J j = (1, −1, −1) j∈ / J and ζj,j+1 is determined by the following table.

ζj,j+1 = j, j + 1 ∈ J (1, 0, −1) j ∈ J, j + 1 ∈/ J (1, −1, −1) j∈ / J, j + 1 ∈ J (1, −1, 0) j, j + 1 ∈/ J (1, −1, −1)

Note that L(1, 0, −1) is 8-dimensional, having as weights the 6 permutations of (1, 0, −1) occurring with multiplicity 1 and (0, 0, 0) occurring with multiplicity 2 (this is easy to see, since

(0, 0, 0) is the only other allowed weight). The equation (2.5.4) holds verbatim and since ζj,j+1 always occurs in L(j+1) with multiplicity 1 we observe that F0 is a constituent of soc(Q)|GL3(Fq) 0 occurring with multiplicity 1. As in Claim 2.5.5, if we can show that the image of Sτ v in Q lies in soc(Q) and is nonzero under the projection soc(Q) F0 then we will have proved Theorem 2.3.2(3), part 2. Much as in (2.5.9) we compute

0 X O (kj ) Sτ v = −sβ X−β vj (2.5.33) f−1 j (kj )j=0 ∈D

f−1 where D is deﬁned to be the set of tuples (kj)j=0 of integers such that

kj ≥ 0 (2.5.34) Chapter 2. Modular representation theory of GL3 87 and X 2 1 j kf−j + δf−j∈J (p − af−j + af−j) p ≡ 0 mod e and 6= 0 j From this last equation it is clear that we can write  2 1 (uj − 1)p + aj − aj − uj+1 j ∈ J kj = (2.5.35) ujp − uj+1 j∈ / J

f−1 for some choice of integers (uj)j=0 . Then by (2.5.34) and genericity we clearly must have uj ≥ 1 if j ∈ J and uj ≥ 0 if j∈ / J. It then follows that if uj = 0 we must have j∈ / J and uj+1 = 0, so by the requirement that the sum in (2.5.35) be nonzero we deduce that uj ≥ 1 for all 0 ≤ j ≤ f − 1.

Lemma 2.5.36. Only terms in (2.5.33) obeying uj ∈ [1, 2] for each 0 ≤ j ≤ f − 1 may be nonzero in V .

Proof. We have already shown that uj ≥ 1. And it is clear that  2 1 ∨ p + aj − aj − 1 j ∈ J hλj, β i ≤ 2 2 2p − aj + aj − 1 j∈ / J so the result follows from Lemma 2.4.10.

(kj ) Let us write Θj(u) = X−β . We now analyze the image of Θj(u)vj inside Qj on a case-by- 0 1 case basis. In each case below, we make the choice of highest weight vector vj ⊗ vj ∈ soc(Qj) so that the ﬁnal equation is true.

2 1 ((uj −1)p) (aj −aj −uj+1) Case I. Assume that j, j + 1 ∈ J. Then Θj(u)vj = X−β X−β vj. If uj+1 = 1 then this clearly lies in hvFi and hence has image 0 inside Qj. If uj+1 = 2 then the image in Qj is 2 1 (aj −aj −2) 0 (uj −1) 1 X−β vj ⊗ X−β vj .

Case II. Assume that j ∈ J, j + 1 ∈/ J. The image of Θj(u)vj in Qj is just

2 1 (aj −aj −uj+1) 0 (uj −1) 1 X−β vj ⊗ X−β vj .

−1 ((u −1)p+a2−a1+1−u ) p−uj+1 j j j j+1 Case III. Assume that j∈ / J, j+1 ∈ J. Then Θj(u)vj = 2 1 X−β vI aj −aj +1−uj+1 so its image inside Qj is

−1 2 1 p − uj+1 (aj −aj +1−uj+1) 0 (uj −1) 1 2 1 X−β vj ⊗ X−β vj . aj − aj + 1 − uj+1 Chapter 2. Modular representation theory of GL3 88

Note that if uj = 2 then this is zero.

Case IV. Assume that j, j + 1 ∈/ J. Exactly the same as the previous case, the image of

Θj(u)vj in Qj is

−1 2 1 p − uj+1 (aj −aj +1−uj+1) 0 (uj −1) 1 2 1 X−β vj ⊗ X−β vj . aj − aj + 1 − uj+1

Again this is zero if uj = 2.

From Cases III and IV we see that j∈ / J ⇒ uj = 1. Following the notation of earlier compu- 0 1 tations, we write the image found in each of the four cases above as θj(uj+1)vj ⊗ ηj(uj)vj . In 0 particular, we have already shown that the image of Sτ v in Q lies in soc(Q). Now from (2.5.33) 0 the image of −sβSτ v inside Q, under the identiﬁcation

O 0 1 2 [f−j] soc(Q) = L((−a − 1, −a , −a + 1)j − ζj,j+1) ⊗ L(j+1) GL3(Fq) j j j GL3(Fq) j is equal to X O 0 1 [θj(uj+1)vj ⊗ ηj+1(uj+1)vj+1] (2.5.37) f−1 f (uj )j=0 ∈{1,2} where all terms with j + 1 ∈/ J and uj+1 = 2 are deﬁned to be zero. Letting

0 1 2 0 1 2 prj : L((−aj − 1, −aj , −aj + 1) − ζj,j+1) ⊗ L(j+1) L(−aj − 1, −aj , −aj + 1) denote the projection map, Lemma 2.5.38 below shows that if j + 1 ∈ J then

0 1 0 1 prj(θj(1)vj ⊗ ηj+1(1)vj ) = µj(prj(θj(2)vj ⊗ ηj+1(2)vj ) for some constant µj obeying 1 + µj 6= 0. To be precise, we have  0 j ∈ J µj = . 2 1 aj − aj j∈ / J

The same lemma shows moreover that the term of (2.5.37) corresponding to the tuple uj =  2 j ∈ J , which we call θ, obeys pr(θ) 6= 0, where pr : soc(Q) F0 is the projection. We 1 j∈ / J 0 conclude that the image of −sβSτ v under the projection soc(Q) F0 is     X Y Y  µj pr(θ) =  (1 + µj) pr(θ) 6= 0 f−1 f j+1∈J,uj+1=1 j+1∈J (uj )j=0 ∈{1,2} Chapter 2. Modular representation theory of GL3 89

This completes the proof of Theorem 2.3.2.

3 Lemma 2.5.38. Let (a, b, c) ∈ Z be a 5-deep weight in the lowest alcove for GL3. In each case below v0 and v1 denote highest weight vectors for the corresponding tensor factors.

(a−b−2) 1. The image of X−β v0⊗X−βv1 ∈ L(−c−2, −b, −a+2)⊗L(1, 0, −1) under the projection to L(−c − 1, −b, −a + 1) is nonzero.

(a−b−1) 2. The image of X−β v0 ⊗ v1 ∈ L(−c − 2, −b + 1, −a + 2) ⊗ L(1, −1, −1) under the projection to L(−c − 1, −b, −a + 1) is nonzero.

p−u −1 (a−b+1−u) (u−1) 3. For u = 1, 2 let θ(u) = a−b+1−u X−β v0 ⊗ X−β v1 inside L(−c − 2, −b + 1, −a + 1) ⊗ L(1, 0, −1). If pr denotes the projection to L(1, −1, −1) then pr(θ(1)) = (a − b) pr(θ(2)) 6= 0.

Proof. In the ﬁrst statement, the given vector is a nonzero multiple of sβv0 ⊗sβv1 so the result is immediate. In the second statement, again the given vector is a nonzero multiple of sα(v0 ⊗ v1) so the result follows. In the third statement, wt(θ(u)) = (−c−1, −a+1, −b) and this weight space is 2-dimensional, equal to

L(−c − 1, −b + 1, −a)wt(θ(u)) ⊕ L(−c − 1, −b, −a + 1)(wt(θ(u)).

Using the Xi monomial, one checks that a basis for the ﬁrst direct summand above is

(a−b) (a−b) (a−b−1) X−β [v0 ⊗ v1] = X−β v0 ⊗ v1 + X−β v0 ⊗ X−βv1 = (−1)a−bθ(1) + (−1)a−b−1(a − b)θ(2) so the claim follows.

2.6 Characteristic 0 principal series

In this section we prove some results in a somewhat different direction, working in characteristic 0. We continue to make use of inertial types τ = τ(σ, K) as in Definition 1.6.1. We are interested in the action of natural lifts of the operators defined in Section 2.3 on certain smooth principal series representations. Let Fw denote an unramified extension of Qp of degree f and identify its residue field with Fq (the subscript w is irrelevant in this section; we are simply anticipating the case that Fw arises as the completion of a number field F at the place w that occurs later). Let Qp denote an algebraic closure of Qp whose residue field is identified with Fp. Let the Teichmüllerlift Fp → Qp be denoted by a tilde x 7→ xe. It induces a function Fp[GL3(Fq)] → Qp[GL3(Fq)]. If S∗ is any of the operators defined in Section 2.3, we let Se∗ denote its image under this function.

Let K = GL3(OFw ) denote the standard maximal compact subgroup of GL3(Fw) (not to be confused with K ⊆ {0, . . . , f − 1} or the ﬁeld K of Chapter 1) and Iw the standard Iwahori Chapter 2. Modular representation theory of GL3 90

subgroup of GL3(Fw), which is the preimage of the upper-triangular matrices under the map 3 K GL3(Fq). We also write K(1) for the kernel of this map. If (a, b, c) ∈ Z is a tuple of integers and πw is a smooth Qp-representation of GL3(Fw) then we deﬁne the Iwahori eigenspace Iw=(a,b,c) πw to be the subspace of πw on which Iw acts via the character

a b c ωef ⊗ωef ⊗ωef × Iw / B3(Fq) / T3(Fq) / Qp .

× The normalizer of the Iwahori subgroup in GL3(Fw) is generated by Iw,Fw , and the matrix

 1    Π =  1 . p

× Iw=(a,b,c) Iw=(b,c,a) Observe that Π has order 3 modulo Fw and that it maps πw 7→ πw . We write Iw1 for the maximal pro-p subgroup of Iw, which is the preimage of the upper-triangular unipotent matrices in GL3(Fq). Iw=(a,b,c) The operators Se∗ have a well defined action on πw since the latter is contained in K(1) πw , and it is easy to check that they translate between different eigenspaces of T3(Fq) within K(1) πw . From now on, we prefer to work with a non-algebraically closed coefficient field. In the statement of the proposition, E ⊆ Qp is any subfield finite over Qp large enough to contain the Teichmüllerlifts of all elements in the image of ωf .

L2 Ai Proposition 2.6.1. Write τ(σ, K) = i=0 ωef as in Deﬁnition 1.6.1 and suppose that ψi : × −Ai Fw → Qp is a smooth character obeying ψi|O× = ωf . We also set Fw e   X 0 j X 1 j X 2 j (−B0, −B1, −B2) =  (−aj − 1)p , −aj p , (−aj + 1)p  . j j j

GL3(Fw) 1. Assume that (2.3.3) holds and that J ⊆ K. Let πw = Ind (ψ1 ⊗ ψ0 ⊗ ψ2). Then B3(Fw) we have an equality of operators

0 f SeY Π = p κY ψ1(p)SeY

Iw=(−A1,−A0,−A2) Iw=(−B0,−B1,−B2) between the 1-dimensional spaces πw → πw , where κY ∈ × OE obeys

Q 0 0 mj!nj!m !n ! A0−A2+f−1 j j j κY ≡ (−1) Q 0 0 mod $E (2.6.2) j(p − 1 − mj − nj)!(mj + nj)! Chapter 2. Modular representation theory of GL3 91

where

−1 1 σj (2) mj =aj − aj + tjp − tj+1 − δj+1∈/K 1 0 nj =p − aj + aj −1 0 1 σj (0) mj =aj − aj + sjp − sj+1 + δj+1∈/K −1 0 2 σj (0) nj =p − aj + aj − sjp + sj+1 − δj+1∈/K

with  1 j ∈ J tj = and sj = 1 − tj. 0 j∈ / J

GL3(Fw) 2. Assume that (2.3.4) holds and that K = ∅. Let πw = Ind (ψ0 ⊗ ψ2 ⊗ ψ1). Then B3(Fw) we have an equality of operators

0 −1 |J| −1 SeX Π = p κX ψ1(p) SeX

Iw=(−A0,−A2,−A1) Iw=(−B0,−B1,−B2) between the 1-dimensional spaces πw → πw , where κX ∈ × OE obeys Y mj!nj! κ ≡ (−1)A1+f−|J|+1 mod $ X (p − a1 + a0 − n )! E j j j j where  1 0 p − aj + aj j, j + 1 ∈ J    1 0 p − aj + aj − 1 j ∈ J, j + 1 ∈/ J 0 j ∈ J mj = and nj = 1 0 1 0 aj − aj j∈ / J, j + 1 ∈ J p − aj + aj j∈ / J.   1 0 aj − aj − 1 j, j + 1 ∈/ J

GL3(Fw) 3. Assume that (2.3.5) holds and that K = ∅. Let πw = Ind (ψ2 ⊗ ψ0 ⊗ ψ1). Then B3(Fw) we have an equality of operators

0 f−|J| SeZ Π = p κZ ψ2(p)SeZ

Iw=(−A2,−A0,−A1) Iw=(−B0,−B1,−B2) between the 1-dimensional spaces πw → πw , where κZ ∈ × OE obeys Y mj!nj! κ ≡ (−1)A2+|J|−1 mod $ Z (p − a2 − a1 − n )! E j j j j Chapter 2. Modular representation theory of GL3 92

where  2 1 aj − aj − 1 j, j1 ∈ J    2 1 2 1 aj − aj j ∈ J, j + 1 ∈/ J p − aj + aj j ∈ J mj = and nj = 2 1 p − aj + aj − 1 j∈ / J, j + 1 ∈ J 0 j∈ / J.   2 1 p − aj + aj j, j + 1 ∈/ J

Remark 2.6.3. To see that the Iwahori eigenspaces in this proposition are actually 1-dimensional GL3(Fw) i as claimed, use the fact that if πw = IndB (F ) (ψa ⊗ ψb ⊗ ψc) where ψi|O× = ωf then we 3 w Fw e 0 0 0 0 0 0 B3(Fq)=(a ,b ,c ) K(1) GL3(Fq) a b c Iw=(a ,b ,c ) K(1) have πw = Ind (ω ⊗ ω ⊗ ω ). As πw = πw , with obvious B3(Fq) f f f notation. The claim now follows from [Her11], Lemma 2.3 and the fact that the principal series for GL3(Fq) are multiplicity-free. It also follows easily using this observation that a basis for Iw=(a,b,c) × πw is given by the unique function v : GL3(K0) → Qp with support B3(K0) · Iw such that v(1) = 1.

Notation 2.6.4. For any of the types τ considered in the theorem, we deﬁne nτ ∈ Z, τ ∈ {±1} × and κτ ∈ OE so that the conclusion of the theorem in each case is

0 τ nτ τ Seτ Π = p κτ ψiτ (p) Seτ .

GL3(Fw) Also note that with the deﬁnition of ς in Notation 2.3.6 we have πw = Ind (ψ ⊗ψ ⊗ B3(Fw) ς(0) ς(1) ψς(2)) in each case.

For the proof of the proposition we will need Theorem 2.5.1 of [BD14] (Stickelberger’s theorem), which is reproduced here for convenience in slightly diﬀerent notation. It gives the dominant p-adic term of certain Jacobi sums.

Theorem 2.6.5. 1. Let m, n ∈ (0, e] such that e - m + n. Write

f−1 X j m = mf−jp j=0 f−1 X j n = nf−jp j=0 f−1 X j m + n = (m + n)f−jp j=0

where mj, nj, (m + n)j ∈ [0, p − 1]. Then in OE we have

X m n u ωef (s) ωef (1 − s) = κp + C s∈Fq Chapter 2. Modular representation theory of GL3 93

1 Pf−1 ≥0 where u = p−1 j=0 (p − 1 − (mj + nj − (m + n)j)) ∈ Z ,

Qf−1 mj!nj! κ = (−1)f−1−u j=0 ∈ × Qf−1 Zp j=0 (m + n)j!

and the p-valuation of C is > u.

2. If a ∈ (0, e) then P ω (s)aω (1 − s)e−a = (−1)a+1. s∈Fq ef ef

Proof. The ﬁrst statement is Theorem 2.5.1 of [BD14]. The second statement is easy; just note that s 7→ s/(1 − s) deﬁnes a bijection Fq \{0, 1} → Fq \{0, −1}.

Proof. The proof strategy is the same as [HLM17], Proposition 3.2.2. For part 1 of the theorem, the proof is in fact essentially the same. And since the proof of part 3 is similar to that of part 2 we will concentrate on proving part 2 only. Iw=(−A0,−A2,−A1) Let v denote the basis vector of πw mentioned in Remark 2.6.3. We ﬁrst remark that λ µ 1 −1 −1 X   Π v = ψ2(p) 1  v. (2.6.6) λ,µ∈Fq 1

λ µ 1 P Iw=(a,b,c) Iw=(c,a,b) To see this, one checks easily that θ := 1  sends πw to πw , so λ,µ∈Fq   1 Πθv must be a multiple of v. Then compute

1  1  X     −1 (Πθv)(1) = v  1   1  = ψ2(p) λ,µ∈Fq p λ µ 1 as the matrix in the centre lies in the support of v iﬀ λ = µ = 0. Now from (2.6.6) we ﬁnd

P 1 0 j 0 −1 −1 X f−j∈ /J (p−a +a )p Se Π v = ψ2(p) ωef (x) f−j f−j Aλ,µ,x (2.6.7) x,λ,µ∈Fq where 1 + xλ xµ x   Aλ,µ,x =  λ µ 1 . 1 Note that 1  1 xµ x     A0,µ,x =  µ 1  1  . 1 1 Chapter 2. Modular representation theory of GL3 94

As v is Iw1-invariant, the sum over terms with µ = 0 in (2.6.7) is 0 since we can pull out the coeﬃcient of x (this happens even if J c 6= ∅). If λ 6= 0, then

1 x + λ−1 −µλ−1  1 λ µ 1        Aλ,µ,x =  1 0  1   1 0  . 1 1 −λ−1

Hence (2.6.7) is equal to

P 1 0 j A1 −1 X X A1−A0 f−j∈ /J (p−a +a )p (−1) ψ2(p) ωef (λ) ωef (x) f−j f−j × x,µ∈Fq λ∈Fq 1 x + λ−1 −µλ−1  1     ·  1 0  1  v. 1 1

Making the change of variables (λ0, µ0, x0) = (λ−1, −µλ−1, x + λ−1) we get

P 1 0 j 0 −1 A1 −1 X X 0 A0−A1 0 0 f−j∈ /J (p−a +a )p Se Π v = (−1) ψ2(p) ωef (λ ) ωef (x − λ ) f−j f−j 0 0 0 × x ,µ ∈Fq λ ∈Fq 1 x0 µ0  1 (2.6.8)     ·  1 0  1  v. 1 1

P 1 0 j P 1 0 j Observe that A0 − A1 + f−j∈ /J (p − af−j + af−j)p ≡ f−j∈J (p − af−j + af−j)p mod e. If J 6= ∅, this implies that the contribution of the terms with x0 = 0 above is zero. Let us make the assumptions J, J c 6= ∅ for now; we will deal with the remaining cases after. 0 × The sum (2.6.8) running over x ∈ Fq , make the change of variables

(λ, µ, x) = (λ0x0−1, µ0, x0)

(these are new variables) to get

P 1 0 j A1 −1 X X A0−A1 f−j∈ /J (p−a +a )p (−1) ψ2(p) ωef (λ) ωef (1 − λ) f−j f−j × µ∈Fq x,λ∈Fq 1 x µ  1 P (p−a1 +a0 )pj f−j∈J f−j f−j     ωef (x) ·  1 0 1  v. 1 1 which is clearly equal to   P 1 0 j A1 −1 X A0−A1 f−j∈ /J (p−a +a )p (−1) ψ2(p)  ωef (λ) ωef (1 − λ) f−j f−j  Sv.e λ∈Fq Chapter 2. Modular representation theory of GL3 95

It remains to evaluate the Jacobi sum in square brackets; using the notation of Theorem 2.6.5(1) we may take  1 0 p − aj + aj j, j + 1 ∈ J    1 0 p − aj + aj − 1 j ∈ J, j + 1 ∈/ J 0 j ∈ J mj = and nj = 1 0 1 0 aj − aj j∈ / J, j + 1 ∈ J p − aj + aj j∈ / J   1 0 aj − aj − 1 j, j + 1 ∈/ J

1 0 and (m + n)j = p − aj + aj − nj. We easily calculate that u = |J| and κ is as claimed in the proposition. Now we consider the case J = ∅. Beginning from (2.6.8), we may do the exact same change of variables on the terms of the sum with x0 6= 0, this time using Theorem 2.6.5(2) to simplify the Jacobi sum. We ﬁnd that the contribution of these terms is

1 x µ  1 A +1+P (p−a1 +a0 )pj X X 1 j f−j f−j −1     (−1) ψ2(p)  1 0 1  v. × µ∈Fq x∈Fq 1 1

On the other hand, the contribution of the terms with x0 = 0 is directly calculated to be (renaming µ0 = µ)

1 0 µ  1 X X P (p−a1 +a0 )pj A1 −1 0 0 j f−j f−j     (−1) ψ2(p) ωef (λ ) (−1)  1  1  0 × µ∈Fq λ ∈Fq 1 1 1 0 µ  1 A +1+P (p−a1 +a0 )pj X 1 j f−j f−j −1     = (−1) ψ2(p)  1  1  µ∈Fq 1 1

Adding these together, we get exactly what is claimed in the theorem statement. The case J c = ∅ follows similarly. Chapter 3

Mod p local-global compatibility

In this chapter we put the results on the Galois side from Chapter 1 and the results on the local automorphic side from Chapter 2 together to get our main local-global compatibility theorem (Theorem 3.5.6) in the cohomology of deﬁnite unitary groups.

3.1 Deﬁnite unitary groups

We first review the global set-up of definite unitary groups of rank 3 considered in the papers [EGH13, HLM17, LLHL17]. Let F be a totally imaginary CM number field with maximal totally 1 + + real subfield F 6= Q. Let c be the non-identity element of Gal(F/F ) and for simplicity assume that all finite primes of F + are unramified in F . Moreover assume that all primes of + + + F dividing p split in F . We write Σp (resp. Σp) for the places of F (resp. F ) lying above + c p. For each place v ∈ Σp , choose a place w ∈ Σp lying above it, so that v splits as ww in F . + Write Fp = F ⊗Q Qp and OF +,p = OF + ⊗Z Zp. Let G be a definite unitary group over F + quasi-split at all finite places and let G be a model + ∼ of G defined over OF + . That is, G(Fv ) = U3(R) is compact for each v|∞ and G is an outer form of GL3 that splits over F . Section 2.1 of [Ger10] explains how to concretely construct such a group and its model by considering an involution of the second kind on Matn(F ) and an order in Matn(F ) preserved by this involution. The details of the construction are not important and it would be possible to simply specify a list of axioms G needs to satisfy (for example, see [EGH13], Section 7.1). We will review all relevant properties of G below. + c + ∼ If v is a place of F splitting as ww in F , then the isomorphism Fv = Fw induces an + ∼ ∼ isomorphism ι : G(F ) −→ GL (F ) which restricts to an isomorphism ι : G(O + ) −→ w v 3 w w Fv

GL3(OFw ). This isomorphism depends on the choice of w, and [Ger10] shows how we can t c ensure that ιwc = (ιw) .

1This is required for technical reasons in [Lab11]. Without it, Theorem 3.3.1 might not hold.

96 Chapter 3. Mod p local-global compatibility 97

3.2 p-adic, mod p and classical automorphic forms

For general reductive groups G there is currently no notion of “p-adic” or “mod p” automorphic form to play a role in the hypothetical p-adic and mod p Langlands correspondences. Neverthe- less a good substitute seems to be the (completed) cohomology of arithmetic quotients of the locally symmetric space associated to G having p-adic or mod p coefficients (see [CE12, Eme14]). When G is compact at infinity (which is the case here) there is also a related notion of algebraic automorphic form ([Gro99]). In this section we define the relevant spaces for the group G and explain their connections with classical automorphic forms on G.

Definition 3.2.1. If W is a finitely generated OE-module with an action of G(OF +,p) and U ≤ ∞,p G(AF + ) × G(OF +,p) is a compact open subgroup we define the space of algebraic automorphic forms on G with level U and coefficients in W to be the OE-module

0 + ∞ −1 ∞ H (U, W ) := {f : G(F )\G(AF + ) → W | f(gu) = up · f(g) ∀g ∈ G(AF + ), u ∈ U}

0 where u 7→ up denotes the projection U → G(OF +,p). The reason for the notation H (from a more general viewpoint than we really need) is that this module may be interpreted as the 0th cohomology of the local system defined by W on the arithmetic quotient of level U associated to G; see [Eme14] 2.1.3. This arithmetic quotient is finite, since G is compact at infinity and so the degree 0 cohomology is the only nonzero one. 0 When W is the Weyl module VOE (λ) defined below, H (U, W ) is an example of a space of algebraic modular forms in the sense of [Gro99].

∞ F + For any compact open subgroup U as above we may write G(AF + ) = i∈Λ G(F )tiU for ∞ ﬁnitely many ti ∈ G(AF + ). It is easy to check that there is an isomorphism of OE-modules

0 ∼ M U∩t−1G(F +)t H (U, W ) −→ W i i i∈Λ

0 given by f 7→ (f(ti))i∈Λ. In particular H (U, W ) is finitely generated over OE. Note that 0 ∞,p lim H (U, W ) where U runs over all compact open subgroups of G( ) × G(O + ) has a −→U AF + F ,p ∞ natural G(AF + )-action given by (g · f)(x) = gp · f(xg). The result just stated then shows that ∞ this is an admissible G(AF + )-module over OE. We say that U is sufficiently small if for some finite place v of F + the projection of U to + −1 + G(Fv ) contains no element of exact order p. The groups U ∩ ti G(F )ti are finite, and if U is sufficiently small then they contain no elements of exact order p so if W is free over OE and A = E or F there is an isomorphism

0 ∼ 0 H (U, W ) ⊗OE A −→ H (U, W ⊗OE A).

There is a Hecke action on the space H0(U, W ). Let us say that the level U is unramiﬁed + v v ∞,v at a place v of F if U = G(O + ) × U for some compact open U ≤ G( + ). Let P denote Fv AF U Chapter 3. Mod p local-global compatibility 98

the set of finite places w of F such that v := w|F + splits in F , v - p, and U is unramified at v. If P ⊆ PU is a subset of finite complement then we define the abstract Hecke algebra P (i) (i) 0 T = OE[Tw : w ∈ P, 0 ≤ i ≤ 3] where the Hecke operator Tw acts on H (U, W ) via the double coset operator " ! # −1 $widi ιw GL3(OFw ) GL3(OFw ) , id3−i where $w is a uniformizer of Fw (the operator is independent of $w). Next we make the connection with spaces of classical automorphic forms on G. Let A denote the space of automorphic forms on G(AF + ). In the compact case this means the space of smooth + + functions G(F )\G(AF + ) → C that are G(F∞)-finite and right invariant under some compact ∞ open subgroup of G(AF + ). It is a pre-unitary representation of G(AF + ) by right translation, which preserves the natural inner product. Let Irr∞ denote the set of isomorphism classes + of continuous complex representations of G(F∞). Compactness implies that all W ∈ Irr∞ + are finite-dimensional and the action of G(F∞) is completely reducible. Hence the G(AF + )- representation A decomposes into a direct sum of G(AF + )-modules as M A = W ⊗ Hom + (W, A). (3.2.2) C G(F∞) W ∈Irr∞

∞ ∨ For W ∈ Irr∞ write SW for the space of functions f : G(AF + ) → W right invariant under ∞ ∞ some compact open subgroup of G(AF + ) obeying f(γg) = γ∞f(g) for all g ∈ G(AF + ) and + ∞ γ ∈ G(F ). This space is a G(AF + )-module by right translation. Then the map

−1 ∞ f 7→ (w 7→ (g 7→ (g∞ f(g ))(w)))

∞ ∼ is an isomorphism of G( + )-modules S −→ Hom + (W, A). AF W G(F∞) ∼ Now fix an algebraic closure Qp ⊃ E and an isomorphism of fields ι : Qp −→ C. Let Hom(F, ) 3,+ Qp 3,+ Hom(F,Qp) (Z )c denote the set of elements λ = (λτ )τ in (Z ) such that λτ,i = −λτ c,2−i for each τ : F,→ and 0 ≤ i ≤ 2. Identifying Hom(F, ) = F Hom(F , ), we write Qp Qp w∈Σp w Qp 3,+ Hom(Fw,Qp) λw, for the projection of λ to (Z ) . Given λw, we define a representation VE(λw) + of G(Fv ) over E, where v = w|F + , as follows: for τ : Fw ,→ Qp let VFw (λw,τ ) denote the Weyl + module of highest weight λw,τ for GL3/Fw defined in Section 2.1 and give it an action of G(Fv ) via ιw. Then set O VE(λw) = VFw (λw,τ ) ⊗Fw,τ E.

τ:Fw,→Qp We also deﬁne V (λ ) similarly, by taking V (λ ) to be the integral Weyl module of OE w OFw w,τ highest weight λ for GL ([Jan03], II.8.2) and giving it an action of G(O + ) via ι . So w,τ 3/OFw Fv w V (λ ) is an O -representation of G(O + ) and we have V (λ ) ⊗ E =∼ V (λ ). OE w E Fv OE w OE E w 3,+ Hom(F,Qp) Since λ ∈ ( )c , the representations of G(O + ) given by V (λ ) and V (λ c ) Z Fv OE w OE w Chapter 3. Mod p local-global compatibility 99

+ are isomorphic (and similarly with VE(λw) and VE(λwc ) as G(Fv )-representations). To see this, note that twisting by the inverse transpose automorphism of GL3 changes the Weyl module

V (µ) to V (−w0µ), by the universal property for example. We denote them by VOE (λv) and + VE(λv) respectively. Finally, we deﬁne a representation of G(Fp ) over E by O VE(λ) := VE(λv) + v∈Σp

N and an integral model over O having an action of G(O + ) by V (λ) = + V (λ ). E F ,p OE v∈Σp OE v Fix a set of embeddings {τ˜ : F,→ Qp} such that Hom(F, Qp) = {τe} ∪ {τe ◦ c}. Since the Weyl modules in question are all deﬁned over F (or even Z, as we used in Chapter 2), the space ∼ N + VO (λ) ⊗O ,ι = VE(λ) ⊗E,ι = VF (λτ ) ⊗F,τ has a natural action of G(F ) where E E C C τe e e C ∞ + + ∼ for v|∞, G(Fv ) acts via G(Fv ) → G(C) = GL3(C), both of these maps being induced by the (unique) embedding ι ◦ τ deﬁning v. For γ ∈ G(F +), v ∈ V (λ) ⊗ this new action obeys e OE OE ,ι C ∞ γ∞ · v = γp · v. Finally, one checks that there is an isomorphism of G(AF + )-modules

0 ∼ lim H (U, VO (λ)) ⊗O ,ι −→S(V (λ)⊗ )∨ (3.2.3) −→ E E C OE OE ,ιC U

f 7→(g 7→ gp · f(g)).

In particular, since we have already seen the left hand side is admissible we deduce the well- known result that A is an admissible G(AF + )-module. Together with the fact that A is pre- unitary, this implies that A decomposes as a direct sum

M A = m(π)π π where π runs over isomorphism classes of automorphic representations of G(AF + ) and m(π) ∈ N is a ﬁnite multiplicity.

3.3 Galois representations and local-global compatibility

It is known how to attach p-adic Galois representations to automorphic representations of G, and that these representations satisfy local-global compatibility at all places. In this section we ∼ recall these facts. We continue to make use of our ﬁxed isomorphism ι : Qp −→ C. Let ` be a prime, possibly equal to p. Recall that the local Langlands correspondence for

GLn over an `-adic ﬁeld L is a family of bijections

F−ss recL,C : IrrC(GLn(L)) → WDn (L) between isomorphism classes of irreducible admissible C-representations of GLn(L) and n- dimensional Frobenius-semisimple Weil-Deligne representations of WL over C, where WL de- Chapter 3. Mod p local-global compatibility 100 notes the Weil group of L. The correspondence is natural in the sense that it is the unique map satisfying the list of natural properties in the Introduction of [HT01]. The only speciﬁc thing GL3(L) we need to know about the correspondence is that rec sends Ind (ψ0 ⊗ ψ1 ⊗ ψ2) to the L,C B3(L) −1 2 −1 −1 Weil-Deligne representation ψ0 | · | ⊕ ψ1 | · | ⊕ ψ2 having trivial monodromy for any smooth × × −1 ±1 characters ψi : L → C such that ψiψi+1 6= | · | (note this is an unnormalized principal series representation).

We define the local Langlands correspondence over Qp, denoted recL, by ι◦recL = recL,C ◦ι. × × −1 In the theorem below, |·| : F → Qp denotes ι ◦|·|. We recall that if (σ, N) is a Weil-Deligne representation then (σ, N)F−ss means (σss,N). If ` 6= p, recall that Grothendieck’s `-adic monodromy theorem gives an equivalence of categories between continuous representations ρ : GL → GLn(Qp) and bounded Weil-Deligne representations of L over Qp, which we denote ρ 7→ WD(ρ) (see [Gee], Proposition 2.17). On the other hand, if ` = p then there is a construction allowing us to associate a Weil- Deligne representation WD(ρ) over E with any potentially semistable (equivalently, de Rham) representation GL → GLn(E), for E ⊂ Qp sufficiently large and finite over Qp. The construction is defined for example in Appendix B of [CDT99]; we briefly review the details since it will be used below. Suppose that ρ becomes semistable when restricted to GK for some finite Galois ∗ ∨ extension K ⊇ L. Then we have the associated (ϕ, N, Fil, K/L, E)-module D = Dst(ρ ) of Section 1.4. We define an action of g ∈ WL on D by making g act as the product of the −n commuting operators given by the image of g in Gal(K/L) and ϕD , where the image of g on the residue field of WL is the nth power of absolute Frobenius. This makes D into a finite free L ⊗ E-module with a continuous action of W . Then we define WD(ρ) = D ⊗ 0 Qp L L0⊗Qp E,τ0 Qp for any choice of embedding τ0 : L0 ,→ E. This is independent of τ0. These constructions are used in the statement of local-global compatibility for Galois representations associated with cuspidal automorphic representations of G.

Theorem 3.3.1. If π is an automorphic representation of G contributing to lim H0(U, V (λ)) −→U OE 3,+ Hom(F,Qp) ⊗OE ,ι C in (3.2.3) for some λ ∈ (Z )c , then there exists a unique continuous semisimple representation

rπ : GF → GL3(Qp) such that

c ∼ −2 1. rπ = rπ ⊗ ,

2. rπ is de Rham at all places dividing p and crystalline if π has level prime to p. If τ : F,→

Qp then HTτ (rπ) = −w0λτ + (2, 1, 0).

3. if v is a place of F + which splits as v = wwc in F (including possibly v|p) then

WD(r | )F−ss ∼ rec ((π ◦ ι−1) ⊗ | · |−1). π GFw = Fw v w Chapter 3. Mod p local-global compatibility 101

Proof. This is Theorem 7.2.1 of [EGH13]; see the list of references in the proof there. The only thing to note in addition is that item (3) holds without semisimpliﬁcation at places dividing p by [Car14].

Remark 3.3.2. Note that a Galois representation of GF is uniquely determined by its local + factors at Fw for places w of F lying above a place of F that is split in F . This is by the Chebotarev density theorem because the set of such places has Dirichlet density 1 in F . Remark 3.3.3. It is not actually the most natural that we should have our Galois representations attached to π taking values in GL3 from the point of view of general philosophy concerning the association of Galois representations to automorphic representations. Rather, they should take values in the C-group of G, which is described in Section 8.3 of [BG14]. And indeed it is known that the Galois representations above extend to the C-group (loc. cit.). But since we are only interested in studying local-global compatibility for GL3 anyway, we don’t need this extra structure.

3.4 Serre weights and modularity

Deﬁnition 3.4.1. A Serre weight for G is an isomorphism class of an (absolutely) irreducible

F-representation of G(OF +,p). The Serre weights are parametrized by the set Q ( 3 )Hom(kw,Fp) of p-restricted tuples w∈Σp Z+ res (λw) obeying λτ,i = −λτ c,2−i for any τ : kw ,→ Fp. Namely, let F (λw) be the representation of + GL3(kw) so denoted in Section 2.2. Then for v ∈ Σp , F (λv) = F (λw) ◦ ιw is independent of the N choice of w dividing v and we deﬁne the Serre weight F (λ) = + F (λ ). v∈Σp v If λw lies in the lowest alcove for each w ∈ Σp then using the fact that Hom(Fw, Qp) and Hom(kw, Fp) are in natural bijection (since each Fw is unramiﬁed), it makes sense to write ∼ VOE (λ), and we have VOE (λ) ⊗OE F = F (λ) by Proposition 2.4.6.

Definition 3.4.2. We now consider a continuous irreducible representationr ¯ : GF → GL3(F). ∞,p Let U be a compact open subgroup of G(AF + ) × G(OF +,p) and P ⊆ PU a subset as in Section 3.2, and assume thatr ¯ is unramified at all places in P. Thenr ¯ gives rise to a maximal ideal (i) mr¯ ≤ TP by defining the reduction mod mr¯ of Tw so the equation

3 ∨ X j (j) (j) j det(1 − r¯ (Frobw)X) = (−1) p 2 (Tw mod mr¯)X j=0 in F[X] is satisfied for each w ∈ P. Let V be a Serre weight for G. We say thatr ¯ is automorphic of weight V , or that V is a modular weight ofr ¯, if there exists a compact open U unramified at p, and a set P as 0 above such thatr ¯ is unramified at all places in P, such that H (U, V )mr¯ 6= 0 (equivalently 0 H (U, V )[mr¯] 6= 0 by commutative algebra). The set of weights ofr ¯ is denoted W (¯r). We say thatr ¯ itself is automorphic if W (¯r) 6= ∅. Chapter 3. Mod p local-global compatibility 102

Remark 3.4.3. As one might expect from the terminology,r ¯ being automorphic is equivalent tor ¯ being the reduction mod p of one of the p-adic representations rπ of Theorem 3.3.1. This follows from an argument like in (3.5.8) below. f From now on, ﬁx a choice of preferred place w|p in F and let v = w|F + and q = p = |kw|. We write kw = Fq so the notation in what follows agrees with that of Chapter 2. In order to be able to most easily focus on the single place v in what follows, we assume thatr ¯ is a

Galois representation automorphic of weight V = F (λ) where λv0 lies in the lowest alcove for 0 0 0 N 0 N 0 all v 6= v. Write V = v0|p,v06=v F (λv ) and Ve = v0|p,v06=v VOE (λv ). Then by the remarks 0 0 above we have Ve ⊗OE F = V . ∞,p We fix a choice of a compact open subgroup U ≤ G(AF + ) × G(OF +,p) that is sufficiently small and unramified at all places dividing p, and fix a subset P ⊆ PU as above such that 0 H (U, V )mr¯ 6= 0. We define

W (¯r| ) = {Serre weights F of GL ( ) | H0(U, (F ◦ ι ) ⊗ V 0)[m ] 6= 0}. GFw n Fq w r¯

(Despite the notation, it is not yet known that this set depends only onr ¯| , or that it is GFw independent of V 0.)

3.5 Main result

In this section we prove our main result about recovering the Galois representation from cohomology. Having ﬁxed U and V 0 at the end of the previous section, we deﬁne H0(U v,V 0) = 0 v 0 lim H (U Uv,V ) where the limit runs over compact open subgroups Uv ≤ G(O + ). This −→Uv Fv + space has a smooth action of G(Fv ), and therefore of GL3(Fw) = GL3(Qpf ) via ιw. By the analogue of mod p local-global compatibility for GL2(Qp) described in the introduction, we expect H0(U v,V 0)[m ] to correspond tor ¯| under a mod p local Langlands correspondence. r¯ GFw Lemma 3.5.1. W (¯r| ) is the set of Serre weights F for GL ( ) such that GFw 3 Fq

∨ 0 v 0 Hom F ,H (U ,V )[mr¯] 6= 0. GL3(Zpf )

Proof. This follows from the proof of Lemma 7.4.3 of [EGH13], which in fact shows that for + any ﬁnite-dimensional F-module Vv with a smooth Uv-action, where Uv ≤ G(Fv ) is compact open, we have ∨ 0 v 0 ∼ 0 v 0 HomUv (Vv ,H (U ,V )) = H (U Uv,Vv ⊗ V ) (3.5.2)

P as T -modules.

For our main result we will make the following assumption. Deﬁnition 3.5.3. Letr, ¯ U, V 0 be as above. We say thatr ¯ satisﬁes Assumption (SW) ifr ¯| ∈ GFw (FL ) with δ ≥ 3, so we can pick a presentation ofr ¯| as in (1.1.1) and W (¯r| ) = {F ∨} δ GFw GFw 0 in the notation of Section 2.3. Chapter 3. Mod p local-global compatibility 103

Remark 3.5.4. Note that F ∨ is the unique ordinary weight ofr ¯| in the terminology of 0 GFw [BH15]. With certain technical assumptions on U andr ¯ and ifr ¯|G is Fontaine-Laffaille for Fw0 each place w0|p it follows from Theorem 4.4.1 of [BLGGT14] and Lemma 3.1.5 of [GG12] that (ifr ¯ is automorphic) then F ∨ always lies in W (¯r| ) (for some choice of V 0). So in these cases 0 GFw the above assumption is equivalent to asking that W (¯r| ) be a singleton. This should be GFw true more generally. Remark 3.5.5. Based on forthcoming work [LLHLM] that determines W (¯r) in cases we are considering, it should be true that “most”r ¯ such thatr ¯| ∈ (FL ) (for sufficiently large GFw δ δ) obey Assumption (SW), so this assumption again represents the most generic case. We show below that Assumption (SW) implies thatr ¯| ∈ (FL∗∗) in certain cases. The converse GFw δ statement is not true, but there should be a slightly more complicated condition on the Fontaine- Laffaille invariants ofr ¯| which is necessary and sufficient for W (¯r| ) to be a singleton. GFw GFw

Theorem 3.5.6. Let r¯ : GF → GLn(Fp) be a continuous representation satisfying Assumption (SW) for a choice of U, V 0 as above. Assume that (1.1.3) holds with δ ≥ 15. Let τ be any of the Aς(0) Aς(1) Aς(2) principal series types of Theorem 2.3.2 and write ηe = ωf ⊗ωf ⊗ωf where ς = ς(τ) ∈ S3 P 0 v 0 Iw=η−1 is deﬁned in Notation 2.6.4. Let T denote the image of T inside EndOE (H (U ,V )[mr¯] . 0 v 0 Iw=η−1 Assume that the OE-dual of H (U ,V )[mr¯] is free over T for each τ. Then 1. r¯| ∈ (FL∗∗), and GFw δ 2. For each τ there is an equality

0 τ Sτ Π =κ ¯τ FLτ Sτ (3.5.7)

0 v 0 Iw=η−1 0 v 0 Iw=η−1 × of maps H (U ,V )[mr¯] → H (U ,V )[mr¯] 0 , where κτ ∈ F ,τ ∈ {±1} are × defined in Notation 2.6.4 and FLτ ∈ F is defined in Notation 1.7.6. Moreover, these maps are injective with nonzero domain. Proof. We prove both parts at the same time. Let τ and η be as in the statement of the theorem. In addition to ς, κτ , FLτ we make use of nτ , mτ ∈ Z and iτ ∈ {0, 1, 2} defined in Notations 1.7.6 and 2.6.4. Let

−1 M := S(U v, V 0)Iw=ηe =∼ S(U v · Iw, η ⊗ V 0) . e mr¯ e e mr¯

∨ (the second isomorphism by (3.5.2). Then M 6= 0 because F0 is a Jordan-Hölderfactor of GL3(OFw ) GL3(kw) Ind (η) ⊗O = Ind (η) and F0 is a modular weight ofr ¯|G by Assumption Iw e E F B3(kw) Fw ∼ (SW). Moreover M is a finite free OE-module. Using the fixed isomorphism ι : Qp −→ C we obtain from (3.2.3) and (3.2.2) a decomposition

M −1 v M =∼ m(π) · πIw=ηe ⊗ (π∞,v)U (3.5.8) Qp v π

v where π runs over the irreducible Qp-representations π∞ ⊗ πv ⊗ π of G(AF + ) such that π ⊗ι C is a cuspidal automorphic representation of G of multiplicity m(π) contributing to Sλ where λ Chapter 3. Mod p local-global compatibility 104 is determined by the highest weight of Ve 0 via the recipe in Section 3.2, and moreover whose ∨ associated Galois representation rπ (which is absolutely irreducible) liftsr ¯ (by local-global compatibility in Theorem 3.3.1). For any π contributing to (3.5.8) we must have

∼ GL3(Fw) × × • πv = Ind ψ ⊗ ψ ⊗ ψ where ψi : F → is a smooth character (de- B3(Fw) ς(0) ς(1) ς(2) w Qp −1 pending on π) such that ψi|O× = ηi , and Fw e

∨ • rπ| is a potentially crystalline lift ofr ¯|G having parallel Hodge-Tate weights (2, 1, 0) GFw Fw and ∨ F−ss ∼ −1 2 −1 −1 WD(rπ| ) = ψ | · | ⊕ ψ | · | ⊕ ψ . GFw ς(0) ς(1) ς(2)

The ﬁrst point follows from Frobenius reciprocity and the fact that the principal series type τ is the inertial type associated to πv via the inertial local Langlands correspondence (see [EGH13], Theorem 2.4.1). The second point follows from local global compatibility in Theorem 3.3.1 at places dividing p as well as the special case of the local Langlands correspondence mentioned near the beginning of Section 3.3. From the second point and the description of WD(r | ) π GFw in Section 3.3 we immediately see that the ϕf -eigenvalue on

Fw,2 ∨ IF =ηi D (rπ| ) w e st GFw is equal to ς−1(i)f −1 p ψi(p) . (3.5.9) L We have ME = p ME[pE] where p runs over the minimal primes of TE. Extending coeﬃcients, ME[pE] is a direct summand of (3.5.8). By the Chebotarev density theorem (see

Remark 3.3.2), the πv contributing to ME[pE] are all identical. Then by Proposition 2.6.1 the equation

0 τ nτ τ Seτ Π = p κτ ψiτ (p) Seτ holds on M[p] ⊆ ME[pE]. Hence the equation

0 τ nτ τ Sτ Π = p κτ ψiτ (p) Sτ (3.5.10)

−1 0 v 0 Iw=ηe holds on M[p]F ⊆ MF[m] = H (U ,V )[mr¯] . The assumption of freeness over the Hecke algebra implies that this last inclusion is an equality. Indeed, observe that

d d • M[p] = HomOE (M /p, OE) since M is OE-free, hence M[p]F = HomOE (M /p, OE)F,

• the natural inclusion

d d HomOE (M /p, OE)F ,→ HomF((M /p)F, F) Chapter 3. Mod p local-global compatibility 105

d is an isomorphism if M /p is free over OE, and

d d • M /p is free over OE since M is free over T and T/p = OE.

0 v 0 Iw=η−1 We deduce that (3.5.10) holds on H (U ,V )[mr¯] e . By Assumption (SW) and Frobenius 0 v 0 Iw=η−1 reciprocity, the GL3(OF )-representation generated by any vector v ∈ H (U ,V )[mr¯] e is w GL3(kw) N2 −Aς(i) the unique quotient of Ind ω with socle F0. From Theorem 2.3.2, we know B3(kw) i=0 f 0 τ that both Sτ Π v and Sτ v are nonzero in the equation

0 τ nτ τ Sτ Π v = p κτ ψiτ (p) Sτ v.

In particular we must have ordp(ψiτ (p)) = −τ nτ or equivalently by (3.5.9)

−1 ordp(λiτ ) = ς (iτ )f + τ nτ . (3.5.11)

But observe that m = ς−1(i )f + n in all cases. By Corollary 1.7.13, ifr ¯| ∈/ (FL∗∗) τ τ τ τ GFw δ we can therefore choose τ so that (3.5.11) is false. This provesr ¯| ∈ (FL∗∗). Now Theorem GFw δ 1.7.1 implies that (3.5.7) holds. This proves Theorem 3.5.6.

3.6 Freeness over the Hecke algebra

The assumption of freeness over the Hecke algebra is important in the proof of Theorem 3.5.6. In this section we illustrate that ifr ¯ satisﬁes certain technical assumptions it is possible to choose U, V 0 so that Hecke freeness holds under Assumption (SW), using the Taylor-Wiles patching method of [CEG+16]. The method used here follows very closely the techniques used to prove the analogous result in Section 5.3 of [HLM17], so we only give a summary. In fact, our argument is simpler because of (SW): we don’t need to consider any extra operators like the Up-operators of op. cit. in order to deal with the possible presence of extra Serre weights. Let F/F + be a CM ﬁeld such that

(i) F/F + is unramiﬁed at all ﬁnite places and all places of F + above p split in F .

We ﬁx a place w|p of F and let v = w|F + . Letr ¯ : GF → GL3(F) be a continuous Galois representation such that

(ii)r ¯ is unramiﬁed outside p

(iii)r ¯ is Fontaine-Laﬀaille with regular weights at all places dividing p

(iv)r ¯ has image containing GL3(k) for some k ⊆ F with |k| > 9 ker(ad(¯r)) (v) F does not contain F (ζp).

+ As in [HLM17] the assumption (v) allows us to choose a ﬁnite place v1 of F prime to p satisfying Chapter 3. Mod p local-global compatibility 106

c • v1 splits in F as v1 = w1w1;

• v1 does not split completely in F (ζp);

±1 • r¯(Frobw1 ) has distinct F-rational eigenvalues, no two of which have ratio (Nv1) .

+ v We choose a unitary group G/F and a model G/OF + as in Section 3.1. Let U = Q ∞,v v06=v Uv0 ≤ G(AF + ) be a compact open subgroup such that

0 + (vi) Uv0 = G(OF + ) for all places v of F that split in F , except for v1 and the places dividing v0 p;

(vii) Uv1 is the preimage of the upper-triangular matrices under the isomorphism

∼ G(O + ) → G(kv1 ) = GL3(kw1 ) Fv1

induced by ιw1 ;

+ 0 (viii) Uv0 is a hyperspecial maximal compact subgroup of G(Fv0 ) if v is inert in F .

v The choice of Uv1 implies that U Uv is sufficiently small in the sense of Section 3.2 for any + compact open Uv ≤ G(Fv ). Assume there exists a choice of λ defining V 0 and Ve 0 as in Section 3.4, so in particular 0 ∼ 0 0 Ve ⊗OE F = V . For w ∈ Sp let ψw denote −w0λw. Let P denote the set of finite places w of F + P lying over a place of F that splits in F , and which do not divide p or v1, and define mr¯ ≤ T as in Definition 3.4.2. We make the crucial automorphy assumption that

v 0 (ix) S(U ,V )mr¯ 6= 0.

v 0 By the choice of U and v1 we remark that this implies (iii), and by the choice of V this also implies (iv). We ﬁnally assume that

(x)r ¯ obeys Assumption (SW).

v 0 Iw=˜η−1 As in the proof of Theorem 3.5.6, this implies that S(U ,V )mr¯ 6= 0 for any of the characters ηe used in the statement of that theorem. Let η be any of the characters deﬁned in Theorem 3.5.6. Let T denote the OE subalgebra e −1 P 0 v 0 Iw=ηe generated by T inside EndOE (H (U , Ve )mr¯ .

v Theorem 3.6.1. Let r¯ : GF → GL3(F) be a continuous Galois representation and U ≤ ∞,v 0 G(AF + ), Ve as above satisfy (i)-(x). Then

−1 0 v 0 Iw=ηe HomOE (H (U , Ve )mr¯ , OE) is free over T. Chapter 3. Mod p local-global compatibility 107

Proof. The proof uses a modiﬁcation due to [HLM17] of the Taylor-Wiles patching method + developed in [CEG 16] in order to construct a patching functor M∞, along with the Diamond- Fujiwara trick to deduce freeness over the Hecke algebra. Since the argument we use is essentially the same as the one in [HLM17] (but simpler), we give a summary of the method and refer the reader to [HLM17] for more details.

By Lemma 5.1.2 of [HLM17], we may assume that E is large enough so that T[1/p] is reduced. + 0 Let S be the set of places of F dividing p together with v1. For each v ∈ S ﬁx a choice 0 ˜ of placev ˜ of F lying over it, and takev ˜ = w. Write S for the set of these places. Let δF/F + denote the quadratic character of F/F +. We consider the deformation problem

+ ∨ −2 ,ψv˜ ˜ + 0 0 S = (F/F , S, S, OE, r¯ , δF/F , {Rv˜1 } ∪ {Rv˜0 }v |p,v 6=v ∪ {Rv˜ })

0 in the terminology of [CHT08]. Here for a place w of F we write Rw0 for the universal OE-lifting ∨ ,ψw0 ring ofr ¯ | , and R for the universal crystalline lifting ring of Hodge type ψ 0 + (2, 1, 0) GFw w0 w if w0 lies over p. Set loc Oˆ ,ψv˜ R = R ⊗ˆ R ⊗ˆ R 0 . v˜1 v˜ v0|p,v06=v v˜ + Let q ≥ 3[F : Q] be an integer, and set

loc R∞ = R [[x1, . . . , xq−3[F +:Q]]].

◦ GL3(OFw ) If ηe denotes any of the characters used in the proof of Theorem 3.5.6 let τ denote IndIw (ηe) denote the corresponding choice of OE-lattice inside the principal series type over E. The construction of [CEG+16], slightly modiﬁed, provides us with a local map

S∞ := OE[[z1, . . . , z9#S, y1, . . . , yq]] → R∞

and an R∞[[GL3(OFw )]]-module M∞ having a compatible GL3(Fw)-action with the properties (i)-(iv) listed in Theorem 5.2.1 of [HLM17], except that in (iii) and (iv) we have a surjection R∞ T with kernel containing aR∞ and equivariant identiﬁcations

◦ d ∼ 0 v ˜ 0 Iw=˜η−1 (M∞(τ )/a) = H (U , V )mr¯ and

◦ d ∼ 0 v 0 Iw=η−1 (M∞(τ )/p) =H (U , Ve )[p] e ◦ ∨ ∼ 0 v 0 Iw=η−1 (M∞(τ )/mr¯) =H (U ,V )[mr¯] respectively where p is any OE-point of Spec(T) (the point being that this holds for each η Chapter 3. Mod p local-global compatibility 108 of interest). The only difference to the proof of Theorem 5.2.1 of [HLM17] in our case is that we are allowing places above p to be unramified, and this has no effect on the constructions of [CEG+16]. For the remainder of the proof we freely use the notation of [HLM17] and [CEG+16]. ◦ ◦ Note that R(τ ) by definition is the image of R∞ inside EndOE (M∞(τ ). By (iii) the action ◦ ◦ ◦ of R∞/a on M∞(τ )/a factors through T. Hence the action of R∞(τ )/a on M∞(τ ) factors ◦ ◦ through T, which gives rise to a surjection R∞(τ )/a T. As R∞(τ ) acts faithfully on ◦ ◦ ◦ M∞(τ ), the R∞(τ )/a-module M∞(τ )/a has full support by Lemma 2.2(1) of [Tay08]. Since ◦ ∼ T is reduced, we get an induced isomorphism (R∞(τ ))red −→ T. ◦ ◦ ◦ It suffices to prove that M∞(τ ) is free over R∞(τ ), for then M∞(τ )/a is free over ◦ ◦ R∞(τ )/a and the map R∞(τ )/a T must be an isomorphism. In order to show this we now follow the proof of Theorem 5.2.3 in [HLM17]. Let n = ◦ ◦ ∼ dimF(M∞(F0)/mr¯). We have M∞(τ ) M∞(τ ) = M∞(F0), where the isomorphism is by ◦ Assumption (SW) and Proposition 5.1.1 of [Le]. Hence by Nakayama’s lemma, M∞(τ ) is ◦ ◦ n ◦ generated over R∞(τ ) by n elements. We may therefore fix a surjection R∞(τ ) M∞(τ ) with kernel P , and get an exact sequence

◦ n ◦ 0 → P [1/p] → R∞(τ )[1/p] → M∞(τ )[1/p] → 0.

◦ ◦ As M∞(τ )[1/p] is projective over R∞(τ )[1/p] by (ii), it now suffices to show that it is projective of constant rank n, since then P [1/p] is projective of rank 0, hence equal to 0 and this implies ◦ P = 0 because R∞(τ ) is p-torsion free. But the argument in the third paragraph of the proof of Theorem 5.2.3 in [HLM17] shows ◦ that each irreducible component of Spec(R∞(τ )[1/p] has an E-point in the closed subscheme Spec(T). Therefore it suffices to observe that for any OE-point of Spec(T) with corresponding ◦ prime p, M∞(τ )/p is OE-torsion free by (iv) and has rank n modulo $E by definition, hence has generic rank = n. This proves the claim.

Remark 3.6.2. For completeness, we should also make some remarks to the eﬀect that the assumptions (i)-(x) in Theorem 3.6.1 are not overly restrictive in the sense that there exist globalizationsr ¯ to which Theorem 3.6.1 and hence Theorem 3.5.6 apply. If one had conditions on the local representationr ¯| guaranteeing Assumption (SW) then it would not be hard to GFw deduce a result of this form from the globalization argument of [HLM17] Theorem 5.3.7. But since this local condition won’t be known until [LLHLM] is available, we have not pursued this. Bibliography

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