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General Serre Weight Conjectures GENERAL SERRE WEIGHT CONJECTURES TOBY GEE, FLORIAN HERZIG, AND DAVID SAVITT Abstract. We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GLn over an arbitrary number field, motivated by the formalism of the Breuil{M´ezardconjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We gen- eralise one of these conjectures to the case of connected reductive groups which are unramified over Qp, and we also generalise the second author's previous conjecture for GLn=Q to this setting, and show that the two conjectures are generically in agreement. Contents 1. Introduction 1 2. A global setting 12 3. The Breuil{M´ezardformalism for GLn and Serre weights 17 4. Patching functors and the Breuil{M´ezard formalism 21 5. Crystalline lifts and Serre weights 24 6. The picture 28 7. Explicit weight conjectures in the semisimple case 30 8. Existing conjectures in the literature 42 9. Unramified groups 50 10. Comparison with [Her09] 61 Appendix A. Wang's result on the "-ordering of alcoves 71 Appendix B. Wobv(ρ) is non-empty 75 References 79 1. Introduction The goal of this paper is to formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GLn over an arbitrary number field. Since this is a problem with a long and involved history and since we work in significant generality in this paper, we begin with an extended introduction, in which we try to summarize this history (Sections 1.1 to 1.5) and give a detailed overview of the approach that we have taken (Sections 1.6 and 1.7). The first author was partially supported by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie Career Integration Grant 303605, and by ERC Starting Grant 306326. The second author was partially supported by NSF grants DMS-0902044 and DMS-0932078, a Sloan Fellowship, and an NSERC grant. The third author was partially supported by NSF grant DMS-0901049 and NSF CAREER grant DMS-1054032. 1 2 TOBY GEE, FLORIAN HERZIG, AND DAVID SAVITT 1.1. Serre's conjecture for GL2 over Q. Let p be a prime. Serre's conjecture, as originally formulated in 1973 (see [SD73, p. 9] and [Ser75, x3]), predicted that every odd irreducible continuous representationr ¯ : GQ ! GL2(Fp) which is unramified outside p has a twist by a power of the mod p cyclotomic character which arises from a cuspidal modular Hecke eigenform of level 1 and weight at most p+1. The theory of the θ-operator then implies thatr ¯ itself is modular of weight at most p2 −1. This was a bold conjecture, for at the time there was little evidence outside of the cases p = 2; 3. In those cases, since there are no cusp forms of level 1 and weight less than 12 the conjecture simply predicts that there are no such representations. This can be established via discriminant bounds, as in [Tat94] and [Ser86, p. 710]. Serre later formulated ([Ser87]) a version of the conjecture with no restriction on the ramification ofr ¯, which included a precise recipe for both the weight and the level of a modular eigenform giving rise tor ¯. In this way the conjecture became computationally verifiable, and was tested in a number of cases in whichr ¯ has small image. At least as far back as [Ser75], it had been known that in the theory of mod p modular forms, one can trade off the weight and level (and Nebentypus) at p. For this reason Serre restricted his attention to modular forms of level prime to p. He conjectured that the minimal possible level of the candidate eigenform giving rise tor ¯ could be taken to be the prime-to-p Artin conductor ofr ¯, while his conjectural recipe for the minimal possible weight of the eigenform (in prime-to-p-level) was more intricate, and depended on the ramification behaviour ofr ¯ at p. The part of Serre's conjecture which predicts that every odd irreducible con- tinuous representationr ¯ : GQ ! GL2(Fp) arises from some modular eigenform is often referred to as \the weak form of Serre's conjecture", while the form of the conjecture that includes the precise recipes for the minimal weight and level is called \the strong form of Serre's conjecture". Much of the early work concerning Serre's conjecture was focussed on proving that the weak form implies the strong form, and it is natural to expect that work on generalisations of Serre's conjec- ture will follow the same pattern. (Indeed, the eventual proof of Serre's conjecture [KW09a, KW09b, Kis09b] relied on the work that had been done to prove the equivalence of the weak and strong forms.) Serre's conjectural recipe for the minimal weight of an eigenform of prime-to-p level giving rise tor ¯ was more subtle than the recipe for the level, but essentially amounted to providing the minimal weight k that was consistent with the known properties of the restriction to a decomposition group at p of the Galois repre- sentations associated to eigenforms. To make this precise one nowadays uses the language of p-adic Hodge theory. Given a modular eigenform f of weight k ≥ 2 and prime-to-p level, the associated p-adic Galois representation rf : GQ ! GL2(Qp) has the property that the restriction rf jG to a decomposition group at p is crys- Qp talline with Hodge{Tate weights k−1 and 0. Therefore any results on the reduction mod p of crystalline representations of GQp with Hodge{Tate weights k − 1 and 0, such as the early results of Deligne and Fontaine{Serre when k ≤ p, give information (purely in terms ofr ¯jG ) about the possible weights k of the modular eigenforms Qp giving rise tor ¯. GENERAL SERRE WEIGHT CONJECTURES 3 To give a concrete example, let " denote the cyclotomic character, and " its reduction mod p. Suppose that k−1 ∼ " ∗ (1.1.1)r ¯jI = Qp 0 1 where IQp is the inertia group at p, and 2 < k < p + 1. Then the minimal weight predicted by Serre's recipe is k. Indeed, it is known that any crystalline represen- tation ρ : GQp ! GL2(Qp) with Hodge{Tate weights k − 1 and 0 (with k in the given range) and whose reduction mod p is reducible must be an extension of an unramified character by an unramified twist of "k−1, and therefore the shape of ρjI must be as on the right-hand side of (1.1.1). Qp We make one further remark about the above example. Suppose that the ex- tension class ∗ vanishes, and assume for simplicity that k < p − 1. Serre observed that p−k 1−k ∼ " 0 (1.1.2) (¯r ⊗ " )jI = Qp 0 1 and therefore has minimal weight p + 1 − k. Thus, although Serre's conjecture predicts that anyr ¯ has a twist which is modular with weight at most p + 1, in this split case there are actually two such twists. This is the so-called \companion forms" phenomenon. 1.2. Serre weights. We now explain a representation-theoretic reformulation of the weight k in Serre's conjecture. This optic first appears in the work of Ash{ Stevens [AS86], and both simplifies the original weight recipe for GL2 over Q and has proved crucial for formulating the weight part of Serre's conjecture for other groups and over other fields. The Eichler{Shimura isomorphism allows one to reinterpret Serre's conjecture in terms of the cohomology of arithmetic groups. If V is an Fp-representation of GL2(Fp) and N is prime to p, then we have a natural action of the Hecke algebra of 1 Γ1(N) on H (Γ1(N);V ), and so it makes sense to speak of a continuous representa- tionr ¯ : GQ ! GL2(Fp) being associated to an eigenclass in that cohomology group. Ifr ¯ is odd and irreducible, then the Eichler{Shimura isomorphism implies thatr ¯ is modular of weight k and prime-to-p level N if and only ifr ¯ is associated to an 1 k−2 2 k−2 2 eigenclass in H (Γ1(N); Sym Fp), where Sym Fp is the (k − 2)th symmetric 2 power of the standard representation of GL2(Fp) on Fp. By d´evissageone deduces thatr ¯ is modular of weight k and prime-to-p level N if and only ifr ¯ is associated 1 k−2 2 to an eigenclass in H (Γ1(N);V ) for some Jordan–H¨olderfactor V of Sym Fp. k−2 2 (Recall that the representation Sym Fp is reducible as soon as k > p + 1.) It is then natural to associate tor ¯ the set W (¯r) of irreducible Fp-representations 1 V of GL2(Fp) such thatr ¯ is associated to an eigenclass in H (Γ1(N);V ) for some prime-to-p level N. Thanks to the argument in the previous paragraph, the (finite) set W (¯r) determines all weights in whichr ¯ occurs in prime-to-p level, and not just the minimal such weight. For this reason such representations of GL2(Fp) are now often referred to as Serre weights, or even simply weights. To illustrate, suppose once again thatr ¯ is as in (1.1.1), with 2 < k < p − 1. If k−2 2 the extension class ∗ is non-split, then we have W (¯r) = fSym Fpg. However, in the companion forms case where the extension class ∗ is split, we have k−2 2 k−1 p−1−k 2 W (¯r) = fSym Fp; det ⊗ Sym Fpg 4 TOBY GEE, FLORIAN HERZIG, AND DAVID SAVITT instead.
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