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 ...

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 ...

Let F be a finite extension of Q p . We prove that the category of finitely presented smooth Z-finite representations of GL 2 (F ) over a finite extension of F p is an abelian subcategory of the category of all smooth representations. The proof uses amalgamated products of completed group rings.

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We determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$ -adic field, when $l \neq p$ , relating them to the space of $q$ -power-stable semisimple conjugacy classes in the dual group. As a consequence, we give a local proof of the $l \neq p$ Breuil–Mézard conjecture of the author, in the tame case.