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

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On Mod P Local-Global Compatibility for Unramified GL by John Enns A On mod p local-global compatibility for unramified GL3 by John Enns A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Mathematics University of Toronto c Copyright 2018 by John Enns Abstract 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 wjp 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¯j under a conjectural mod p local Langlands correspondence. The goal of the thesis is to GFw study how to recover the data ofr ¯j 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 ¯j inside the GL (F )-action on H0(¯r) when GFw 3 w r¯j 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 '-modules . 25 1.5.1 Etale´ '-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 :firreducible admissible smooth representations of GLn(K) over Qpg= = ∼ ! frank n Frobenius-semisimple Weil-Deligne representations of K over Qpg= = (0.0.1) for each n characterized by the list of conditions found in the introduction to [HT01], in par- ticular compatibility with local class field theory. When ` 6= p, Grothendieck's `-adic mon- odromy 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 corre- spondence, 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 1 subgroup U ≤ GL2(A ) there is an algebraic curve Y (U) (the classical modular curve of level U) 1 Chapter 0. Introduction 2 1 defined over Q whose space of complex points is isomorphic to GL2(Q)n(C − R) × GL2(A )=U. We may consider the p-adic ´etalecohomology1 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 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 1 2 GL2(A )-representation, we have O Hom (r; H1) =∼ π ⊗ 0 rec (rj ) (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 rjGp , but the procedure loses a great deal of information. In other words, it is not possible to reconstruct rjGp 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 rep- resentations 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.
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