The Breuil–mézard Conjecture for Potentially
               Barsotti–tate Representations

Gee, Toby; Kisin, Mark

doi:10.1017/fmp.2014.1

Cited by 91 publications

(238 citation statements)

References 36 publications

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“…The map cyc also appears in the l = p situation when working with potentially semistable (rather than potentially crystalline) deformation rings. Our proof of the main theorem is 'global', making use of the methods of [GK14] and [EG14]. We use the Taylor-Wiles-Kisin patching method to produce an exact functor θ −→ H ∞ (θ) from the category of finitely generated O-modules with a smooth GL n (O F )-action to the category of finitely generated R (ρ)-modules, such that the support of H ∞ (θ) -counted with multiplicity -is cyc(θ).…”

Section: Introductionmentioning

confidence: 99%

The Breuil–Mézard conjecture when l≠p

Shotton¹

2018

Duke Math. J.

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be a continuous representation. Let R (ρ) be the universal framed deformation ring for ρ. If l = p, then the Breuil-Mézard conjecture (as formulated in [EG14]) relates the mod l reduction of certain cycles in R (ρ) to the mod l reduction of certain representations of GLn(O F ). We state an analogue of the Breuil-Mézard conjecture when l = p, and prove it whenever l > 2 using automorphy lifting theorems. We give a local proof when l is "quasi-banal" for F and ρ is tamely ramified. We also analyse the reduction modulo l of the types σ(τ ) defined by Schneider and Zink [SZ99]. IntroductionWhen F is a p-adic field and ρ is an n-dimensional mod p representation of its absolute Galois group G F , the Breuil-Mézard conjecture relates singularities in the deformation ring of ρ to the mod p representation theory of GL n (O F ). It was first formulated, for F = Q p and n = 2, in [BM02], and (mostly) proved in this case in [Kis09a]. In full generality, the conjecture is formulated in [EG14] but is not known in any cases with n > 2. In this article we prove an analogue of the Breuil-Mézard conjecture for mod l representations of G F and GL n (O F ), with F a p-adic field and l an odd prime distinct from p.We give a precise statement, after setting up a little notation. Let F be a finite extension of Q p with ring of integers O F , residue field k F of order q, and absolute Galois group G F , and let l be a prime distinct from p. 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 n (F) 1 2 JACK SHOTTON be a continuous representation. Then there is a universal framed deformation ring R (ρ) parameterizing lifts of ρ. Our main result, stated below, relates congruences between irreducible components of Spec R (ρ) to congruences between representations of GL n (O F ).It is known that Spec R (ρ) is flat and equidimensional of relative dimension n 2 over Spec O -see Theorem 2.5. Let Z(R (ρ)) be the free abelian group on the irreducible components 1 of Spec R (ρ); similarly we have the group Z(R (ρ)) where R (ρ) = R (ρ) ⊗ O F. There is a natural homomorphism red : Z(R (ρ)) −→ Z(R (ρ)) taking an irreducible component of Spec(R (ρ)) to its intersection with the special fibre (counted with multiplicities).An inertial type is an isomorphism class of continuous representation τ : I F → GL n (E) that may be extended to G F . If τ is an inertial type, then there is a quotient R (ρ, τ ) of R (ρ) that (roughly speaking) parameterizes representations of type τ ; that is, whose restriction to I F is isomorphic to τ . Then Spec R (ρ, τ ) is a union of irreducible components of Spec R (ρ).Let R E (GL n (O F ) (resp. R F (GL n (O F ))) be the Grothendieck group of finite dimensional smooth representations of GL n (O F ) over E (resp. F), and letbe the surjective map given by reducing a representation modulo l. In section 4 we define a homomorphism cyc : R E (GL n (O F )) → Z(R (ρ)) by the formula cyc(θ) = τ m(θ ∨ , τ )Z(R (ρ, τ )) where the sum is over all inertial types, Z(R (ρ, τ ...

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Section: Introductionmentioning

confidence: 99%

The Breuil–Mézard conjecture when l≠p

Shotton¹

2018

Duke Math. J.

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“…Cela se démontre en utilisant les techniques de multiplicité un issues de la méthode de Taylor-Wiles comme inauguré par Fujiwara ( [23]) et l'un d'entre nous ( [16] n'a qu'un seul de ses constituants qui apparaît dans ce GL 2 (O Fv )-socle :à savoir le poids de Serre ⊗ σ (Sym rv,σ F p 2 ) σ ci-dessus. Cela se déduit par exemple directement de [25] et d'un calcul facile (mais peut aussi se démontrer de manière plusélémentaire sans utiliser [25]). Une fois ces deux ingrédients disponibles, l'existence de x(J) se ramène essentiellementà de la théorie des représentations (cf.…”

Section: Introductionunclassified

“…Les auteurs remercient Toby Gee pour leur avoir signalé les résultats récents de [2] et [25] On désigne par L une extension finie de Q p non ramifiée de degré…”

Section: Introductionunclassified