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Explicit Serre Weights for Two-Dimensional Galois Representations

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Explicit Serre Weights for Two-Dimensional Galois Representations Explicit Serre weights for two-dimensional Galois representations Misja Frederik Alban Steinmetz Supervisor: Prof. Fred Irvin Diamond Second Supervisor: Prof. Kevin Mark Buzzard Submitted for the degree of Doctor of Philosophy Department of Mathematics King's College London August 2019 1 2 Abstract Let F be a totally real field and p a prime number. Given a Galois representation ρ : GF Ñ GL2pFpq, we have precise conjectures (see [BLGG13]) in terms of non-explicit p-adic Hodge theory giving the sets of weights of Hilbert modular forms such that the reduction of the associated Galois representation is isomorphic to ρ. Under the assumption that p is unramified in F an alternative explicit formulation of these sets of weights was proposed in the paper [DDR16] replacing the p-adic Hodge theory by local class field theory. Subsequently, the equivalence of the reformulated conjecture to the original conjecture was proved in [CEGM17]. In this thesis we generalise the conjecture of [DDR16] and the proof of equivalence of the two conjectures of [CEGM17] to hold for any totally real field F . Thereby, we give an equivalent explicit version of the conjectures on the modularity of two-dimensional Galois representations over totally real fields. The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without proper acknowledgement. Contents Chapter 1. Introduction 5 1.1. Notation 8 1.2. Conventions in p-adic Hodge theory 9 1.3. Acknowledgements 10 Chapter 2. The weight in Serre's conjecture 13 2.1. Historical notes on Serre's conjecture 13 2.2. Cohomological reformulation of the weight 18 2.3. Serre weights and crystalline lifts 26 Chapter 3. Serre weights for totally real fields 29 3.1. Hilbert modular forms 29 3.2. The weight recipe for totally real fields 32 3.3. The distinguished subspaces LV 37 Chapter 4. The peu ramifi´esubspace 41 4.1. Introduction 41 4.2. The trivial Serre weight case 44 4.3. The general Serre weight case 47 4.4. Redefining the peu ramifi´espace 51 Chapter 5. Explicit bases of spaces of extensions 53 1 5.1. The filtration on H pGK ; Fppχqq 53 5.2. Constructing the basis elements 61 1 5.3. An explicit basis of H pGK ; Fppχqq 66 Chapter 6. Explicit Serre weights over ramified bases 69 6.1. Reductions of pseudo-Barsotti Tate representations 69 6.2. Restricting to GM8 73 6.3. An explicit basis of LV 78 Chapter 7. Explicit Formulae 85 7.1. Choice-independent subspaces 86 7.2. The unramified case (d'apr`esDemb´el´e{Diamond{Roberts) 98 7.3. The totally ramified case 107 7.4. Explicit formulae in generic cases 113 7.5. Explicit formulae for e “ 2 118 Index 131 Bibliography 133 3 CHAPTER 1 Introduction For a totally real field F , a prime p and a mod p Galois representation ρ : GF Ñ GL2pFpq, the question whether this representation can arise as the reduction of the Galois representation associated to a Hilbert modular form was first rigorously studied in [BDJ10] under the hypothesis that p is unramfied in F . This being a generalisation of Serre's famous conjecture [Ser87] for F “ Q, it was long believed that a similar conjecture could be formulated for totally real fields. However, in the conjecture of [BDJ10], the authors were for the first time able to predict the set of weights of Hilbert modular forms leading to the given representation precisely. The unramifiedness hypothesis was later removed in Schein [Sch08], [Gee11] and [BLGG13]. Building on previous work of Gee and co-authors it was proved in [GLS15] that the predicted set of weights of [BDJ10] and its extensions is correct in the sense that if ρ comes from a Hilbert modular form then it must come from a form of weight predicted by the conjectures and every predicted weight occurs. One unifying aspect of all these conjectures is that the sets of weights are predicted using abstract p-adic Hodge theory making it complicated to do explicit computations in many cases. This problem was addressed in the paper [DDR16] under the assumption that p is unramified in F . The authors conjecture an alternative explicit formulation of the predicted sets of weights in terms of local class field theory and the Artin{Hasse expo- nential. Subsequently, it was proved in [CEGM17] that their reformulation was equivalent to the original conjectures. The main result of this thesis is the generalisation of the conjecture of [DDR16] as well as a a generalisation of the proof of the equivalence of the two conjectures of [CEGM17] to the case of an arbitrary totally real field F . As an immediate consequence, we can give an alternative, explicit and equivalent formulation of the sets of weights appearing in the conjectures on the modularity of two-dimensional mod p Galois representations over an arbitrary totally real field. Let us now give a more careful chapter-by-chapter overview of the con- tents of this thesis. In Chapter 2 and 3 we give some history and back- ground in order to put the results of this thesis in a broader context. In 5 6 1. INTRODUCTION Chapter 2 we describe Serre's modularity conjecture on Galois representa- tion ρ: GalpQ{Qq Ñ GL2pFpq through some of the original letters between Serre, Tate and Grothendieck. We move on to give a cohomological re- formulation of the weight part of Serre's conjecture in which we use the Eichler{Shimura isomorphism to redefine ρ being modular in terms of the non-vanishing to localisations of certain group cohomology groups. Lastly, we will explain how this cohomological formulation is related to crystalline lifts of ρ| of certain prescribed Hodge{Tate weights. These reformula- GQp tions are important for the generalisations of Serre's conjecture treated in the following chapter. In Chapter 3 we introduce generalisations of Serre's modularity conjec- ture to Galois representations ρ: GalpF {F q Ñ GL2pFpq for a totally real field F . We start with a very brief introduction to the theory of Hilbert modular forms. Next we sketch how to define modularity of Galois repre- sentations in this context following [BDJ10, x2]. To define the associated sets of weights W pρq we restrict to a decomposition group at a prime p of F dividing p. Writing ρp for the resulting local representation at this prime, we define W pρpq explicitly if ρp is irreducible. The reducible case requires more attention. Suppose χ1 ˚ ρp „ ˜ 0 χ2¸ ˆ for continuous characters χ1; χ2 : GFp Ñ Fp . Then ρp defines a cocy- 1 cle in the space of extensions H pGK ; Fppχqq of Fppχ2q by Fppχ1q, where ´1 χ :“ χ1χ2 : We include a weight V in W pρpq if and only if this associated 1 cocycle lies in a mysterious subspace LV Ď H pGK ; Fppχqq depending on the weight V . At the end of the chapter we give a definition of the sub- spaces LV in terms of the existence of crystalline lifts of certain prescribed Hodge{Tate weights. All this results a precise statement of the generalisa- tion of Serre's conjecture to two dimensional mod p Galois representations of totally real fields. In Chapter 4 we prove a result that seems to be well-known to ex- perts, but for which we have been unable to find an appropriate refer- ence. If χ is the cyclotomic character, then we can use Kummer theory 1 ˆ to identify H pGK ; Fppχqq – K b Fp: The peu ramifi´e subspace is the subspace of cocycles corresponding to elements of the codimension-one sub- ˆ space OK b Fp: The result we prove in this chapter is that the subspaces 1 LV Ď H pGK ; Fppχqq are contained in the peu ramifi´esubspace (except in one exceptional case). We prove this by first proving the result in the trivial 1. INTRODUCTION 7 weight case using the theory of finite flat group schemes. Then we prove that we have inclusions LV Ď LVtriv using the classification of [GLS15]. In the hope of making this chapter useful without the need to read the rest of the thesis, we have tried to keep it somewhat self-contained. 1 In Chapter 5 we define an explicit basis of H pGK ; Fppχqq by making use of local class field theory and the Artin{Hasse exponential. That is, we generalise [DDR16, x3,x5] to the case of an arbitrary local field K. We 1 start by defining a filtration on H pGK ; Fppχqq via restrictions to higher ramification groups and we find the jumps of this filtration as well as their dimension. Then we study the dual filtration which is defined using local class field theory and given in terms of the higher unit groups of a tamely ramified extension of K. Using the Artin{Hasse exponential we define basis elements tui;ju in these higher unit groups and we obtain a basis tci;ju of 1 H pGK ; Fppχqq dually. In Chapter 6 we study the relationship between the explicitly defined 1 basis elements of H pGK ; Fppχqq from the previous chapter and the spaces LV defined using crystalline lifts. This is a generalisation of [CEGM17, x3] to the case of an arbitrary local field K. We start by recalling some p-adic Hodge theory as well as the classification theorem of [GLS15] which classifies the space LV in terms of the ´etale '-modules that it corresponds to. Via Artin{Schreier theory and the theory of the field of norms, we use an explicit reciprocity law to write down a precise criterion for when AH the pairing of (the image of) LV with ui;j does not vanish.
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