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

Abstract

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 ...

On crystabelline deformation rings of $$\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)$$ Gal ( Q ¯ p / Q p ) (with an appendix by Jack Shotton)

“…Global application. Our result combined with the results of Jack Shotton proved in [47], when p > 2, and in §B, when p = 2, imply that certain global potentially semi-stable deformation rings are O-torsion free. This was one of our motivations to prove the Cohen-Macaulayness of local deformation rings.…”

“…If v ∤ p we let R ,ψ v be the maximal reduced and p-torsion free quotient of R ,ψ v all of whose Lpoints give rise to representations ρ of G Fv , such that the semisimplification of the restriction of ρ to I Fv is isomorphic to τ v . Jack Shotton has proved in [47] and in the appendix below that these rings are Cohen-Macaulay.…”

The Breuil–Mézard conjecture when l≠p

“…This is a modification, due to Choi [Cho09], of [CHT08] Corollary 2.4.13 to take into account the framings. See [Sho16] Lemma 2.3.…”

“…If multiple components of Spec R (ρ) have the same type, then H will be a strict subgroup of Z(R (ρ, τ )); this happens, for instance, if n = 2, ρ = ½ ⊕ χ where χ is the cyclotomic character, and q ≡ −1 mod l (see [Sho16] Proposition 5.6).…”

“…We can also give local proofs of (variants of) the theorem in some special cases. In [Sho16], we studied the case n = 2 and l > 2; explicitly calculated the rings R (ρ, τ ) in this case and gave a local proof of the theorem.…”