The Local Langlands Conjecture for GL(2)

Bushnell, Colin J.; Henniart, Guy

doi:10.1007/3-540-31511-x

Cited by 263 publications

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“…We deduce this (in somewhat circular fashion) from the irreducibility of Ind Now suppose that r is the restriction to I F of an irreducible representation of G F . Then σ(τ ) = Ind K J λ for an irreducible representation λ of J extending an irreducible representation η of a pro-p normal subgroup J 1 of J (see [BH06], sections 15.5, 15.6 and 15.7 -note that our J is the maximal compact subgroup of their J α , but our J 1 agrees with their J 1 α ). We have J/J 1 = k × , where k is the residue field of a quadratic extension of F , and so J has a normal subgroupJ of pro-order coprime to l such that J/J is an l-group.…”

Section: Reduction Of Types -Proofsmentioning

confidence: 97%

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Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p

Shotton

2016

Algebra Number Theory

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Abstract. We compute the deformation rings of two dimensional mod l representations of Gal(F /F ) with fixed inertial type, for l an odd prime, p a prime distinct from l, and F/Qp a finite extension. We show that in this setting an analogue of the Breuil-Mézard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL 2 (O F ). IntroductionLet p be a prime, and let F be a finite extension of Q p with absolute Galois group G F . We study the (framed) deformation rings for two-dimensional mod l representations of G F , where l is an odd prime distinct from p. More specifically, let E be a finite extension of Q l , with ring of integers O, uniformiser λ, and residue field F. Let ρ : G F → GL 2 (F) be a continuous representation. Then there is a universal lifting (or framed deformation) ring R (ρ) parametrising lifts of ρ. Our main result relates congruences between irreducible components of Spec R (ρ) to congruences between certain representations of GL 2 (O F ), where O F is the ring of integers of F . Our method is to give explicit equations for the components of Spec R (ρ), which may be of independent use.If τ : I F → GL 2 (E) is a continuous representation that extends to a representation of G F (an inertial type), then we say that a representation ρ : G F → GL 2 (E) has type τ if its restriction to I F is isomorphic to τ . Say that an irreducible component of Spec R (ρ) has type τ if a Zariski dense subset of its E-points correspond to representations of type τ . We define (definition 4.1) a formal sum C(ρ, τ ) of irreducible components of the special fibre Spec R (ρ) ⊗ O F. For semisimple τ , this is obtained as the intersection with the special fibre of those components of Spec R (ρ) having type τ ; for non-semisimple τ this must be slightly modified.To an inertial type τ we also associate an irreducible E-representation σ(τ ) of GL 2 (O F ), by a slight variant on the definition of [Hen02] (see section 3.3). For an irreducible F-representation θ of GL 2 (O F ), define m(θ, σ(τ )) to be the multiplicity of θ as a Jordan-Hölder factor of the mod λ reduction of σ(τ ). Then we can state our main theorem (theorem 4.2):Theorem. Let ρ : G F → GL 2 (F) be a continuous representation. For each irreducible F-representation θ of GL 2 (O F ), there is a formal sum C(ρ, θ) of irreducible components of Spec R (ρ) ⊗ F such that, for each inertial type τ , we have the In fact the C(ρ, θ) are uniquely determined (at least for those θ which actually occur in some σ(τ )).This theorem is an analogue for mod l representations of G F of the BreuilMézard conjecture [BM02], which pertains to mod p representations of G Qp . Our statement is not in the language of Hilbert-Samuel multiplicities used in [BM02], but rather in the geometric language of [EG14]. The original conjecture of BreuilMézard was proved in most cases by Kisin [Kis09a]; further cases were proved by Paskunas [Paš15] by local methods, and the full conjecture was proved when p > 3 in ...

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Section: Reduction Of Types -Proofsmentioning

confidence: 97%

“…• If τ = (rec(π)| IF , 0) for a cuspidal representation π of GL 2 (F ), then by [BH06], 15.5 Theorem, there is a certain subgroup J ⊂ G, containing the center of G and compact modulo center, and a representation Λ of J such that π = c-Ind…”

Section: Deformation Rings With Fixed Typementioning

confidence: 99%

Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p

Shotton

2016

Algebra Number Theory

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“…This depends only on ı −1 ( √ p), and if we define r p (π) := rec p (π ⊗ | det | (1−n)/2 ), then r p is independent of the choice of ı. Furthermore, if V is a Frobenius semisimple Weil-Deligne representation of W F over E, then r −1 p (V ) is also defined over E by [Clo90,Prop 3.2] and the fact that r p commutes with automorphisms of C. (The claims about the dependence of rec p and r p on the choice of ı follow from the main theorem of [Hen93], together with a study of the behaviour of ε-factors under automorphisms of C. In the case n = 2, this is explained in [BH06,§35], and the same argument goes through in general, with the required input on ε-factors being provided by [BH00, Thm. …”

Section: Notationmentioning

confidence: 99%

Patching and the $p$-adic local Langlands correspondence

Caraiani

Emerton

Gee

et al. 2016

Cambridge Journal of Mathematics

166

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“…We refer to Section 7 of [BH06] for the vocabulary and assertions concerning the WeilDeligne representations of the Weil group of F and their local constants. Let W F be the Weil group of F .…”

Section: Local Langlands Correspondence and Local Factorsmentioning

confidence: 99%