2010 Mathematics Subject Classification. 11F80, 11F33.
Key words and phrases. Fontaine–Mazur conjecture.
Introduction
The mixed case
Let F F F p p p ρ : G F → G L 2 ( Q ‾ p ) \rho:G_F \to\mathrm{GL}_2(\overline{\mathbb{Q}}_p) ρ : G F → GL 2 ( Q p ) ρ \rho ρ ρ ∣ G F v \rho|_{G_{F_v}} ρ ∣ G F v v ∣ p v \mid p v ∣ p ρ \rho ρ modular , i.e. it arises from a regular algebraic cuspidal automorphic representation of G L 2 ( A F ) \mathrm{GL}_2(\mathbb{A}_F) GL 2 ( A F )
When the residual representation ρ ˉ \bar{\rho} ρ ˉ [29 ] , under the assumption that ρ \rho ρ p p p [22 ] introduced a method based on the completed cohomology of definite quaternion algebras and Paškūnas’ theory of blocks [23 ] , which treats the case where F F F Q \mathbb{Q} Q p p p F F F ρ ∣ G F v \rho|_{G_{F_v}} ρ ∣ G F v v ∣ p v \mid p v ∣ p [32 ] removed the remaining exceptional residual configuration at p = 3 p=3 p = 3 [24 ] . The mixed case, in which ρ ∣ G F v \rho|_{G_{F_v}} ρ ∣ G F v v ∣ p v \mid p v ∣ p
Setting
Throughout, p p p F F F Q \mathbb{Q} Q p p p F v = Q p F_v=\mathbb{Q}_p F v = Q p v ∣ p v \mid p v ∣ p E / Q p E/\mathbb{Q}_p E / Q p O \mathcal{O} O F \mathbb{F} F
ρ : G F ⟶ G L 2 ( E ) \rho:G_F \longrightarrow\mathrm{GL}_2(E) ρ : G F ⟶ GL 2 ( E ) be a continuous, absolutely irreducible, odd representation, unramified outside finitely many places. We write ρ v = ρ ∣ G F v \rho_v=\rho|_{G_{F_v}} ρ v = ρ ∣ G F v Σ p \Sigma_p Σ p F F F p p p Σ p = Σ n ⊔ Σ o \Sigma_p=\Sigma_n\sqcup\Sigma_o Σ p = Σ n ⊔ Σ o
(N) For v ∈ Σ n v\in\Sigma_n v ∈ Σ n ρ v \rho_v ρ v
(O) For v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ v \rho_v ρ v
(R) ρ ˉ s s = χ ˉ 1 ⊕ χ ˉ 2 \bar{\rho}^{\mathrm{ss}}=\bar{\chi}_1\oplus\bar{\chi}_2 ρ ˉ ss = χ ˉ 1 ⊕ χ ˉ 2 χ ˉ i : G F → F × \bar{\chi}_i:G_F\to\mathbb{F}^{\times} χ ˉ i : G F → F × χ ˉ = χ ˉ 1 χ ˉ 2 − 1 \bar{\chi}=\bar{\chi}_1\bar{\chi}_2^{-1} χ ˉ = χ ˉ 1 χ ˉ 2 − 1 G Q G_{\mathbb{Q}} G Q χ ˉ ∣ G F v ≠ 1 \bar{\chi}|_{G_{F_v}}\ne1 χ ˉ ∣ G F v = 1 v ∈ Σ p v\in\Sigma_p v ∈ Σ p
Here and below we freely extend scalars: a local representation ρ v \rho_v ρ v E E E E E E ρ v \rho_v ρ v irreducible means absolutely irreducible. Condition (R) does not depend on the order of χ ˉ 1 , χ ˉ 2 \bar{\chi}_1,\bar{\chi}_2 χ ˉ 1 , χ ˉ 2 ρ v \rho_v ρ v H T ( ε ) = − 1 \mathrm{HT}(\varepsilon)=-1 HT ( ε ) = − 1 ρ v \rho_v ρ v ρ v \rho_v ρ v classical character of ρ v \rho_v ρ v
The non-split case
Our first result treats the case where ρ \rho ρ
Theorem A. Assume (N), (O) and (R), and that ρ v \rho_v ρ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ \rho ρ modular .
When Σ o = ∅ \Sigma_o=\varnothing Σ o = ∅ Σ n = ∅ \Sigma_n=\varnothing Σ n = ∅ [22 Theorem 5.1.2] ; the new content is the mixed case. No condition is imposed on the Hodge–Tate weights beyond (N) and (O); in particular they need not be parallel. The hypothesis that ρ v \rho_v ρ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ v \rho_v ρ v one of its two subcharacters occurs in the relevant Hida family, and the non-classical one gives no classical vectors.
We should be clear about what is new in the proof of Theorem A. The pro-modularity input is [32 Theorem 5.5.1] (for p ≥ 5 p\geq5 p ≥ 5 [22 ] ), applied verbatim: its proof does not use any local condition at p p p [14 , 15 ] , the local–global compatibility of Pan and Zhang for Paškūnas blocks [22 , 32 ] , the results of Breuil–Emerton on locally algebraic vectors in ordinary principal series [6 ] , and those of Colmez–Dospinescu–Paškūnas at the non-ordinary places [7 ] . The only new point is to combine them, taking ordinary parts only at the places of Σ o \Sigma_o Σ o [22 Remark 8.0.3] , and we do not claim any originality for it. Moreover, the non-split hypothesis avoids precisely the genuine difficulty of the mixed case, namely showing that the classical character occurs. The main interest of the paper lies instead in the applications of Theorem A given in § 5 and § 6 , all the more so since they will be used to prove new modularity results that do not depend on any extension, not yet available, of Pan’s theorem to the mixed case (see Remark 1.2).
Remark 1.1. Under the hypothesis of potential modularity, which is available in the applications we have in mind (Remark 1.2), Greenberg’s question is exactly what is needed to deduce from Theorem A the modularity of ρ \rho ρ ρ v \rho_v ρ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ \rho ρ θ \theta θ G K G_K G K K / F K/F K / F ρ \rho ρ ρ ∣ G W \rho|_{G_W} ρ ∣ G W W / F W/F W / F ρ ∣ G W \rho|_{G_W} ρ ∣ G W [35 ] ) predicts that the Galois representation attached to such a Hilbert eigenform is non-split at every place above p p p ρ ∣ G W \rho|_{G_W} ρ ∣ G W w w w v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ v \rho_v ρ v ρ \rho ρ ρ v \rho_v ρ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o § 5 and § 6 ), and the more cases of Greenberg’s question are settled, the more general this result becomes.
Three applications of Theorem A
Theorem A also applies whenever the local types at the places above p p p ρ v \rho_v ρ v ρ v \rho_v ρ v ρ v \rho_v ρ v ρ v \rho_v ρ v ρ v \rho_v ρ v Lemma 5.1 ). Hence ρ \rho ρ v ∈ Σ p v\in\Sigma_p v ∈ Σ p ρ v \rho_v ρ v Corollary 5.3 ); over a real quadratic field it suffices that the Hodge–Tate numbers differ by 1 at both places above p p p ρ \rho ρ ( 2 , k ) (2,k) ( 2 , k ) p p p ρ ′ \rho' ρ ′ ρ \rho ρ Theorem 5.5 reduces the modularity of ρ \rho ρ [22 Theorem 5.1.2] , or to a residually irreducible case covered by the modularity lifting theorems of Kisin, Hu–Tan and Tung.
Abelian varieties of GL2 _2 2
Theorem A applies to abelian varieties. Let A / F A/F A / F 2 _2 2 K ⊂ End F ( A ) ⊗ Q K \subset\operatorname{End}_F(A) \otimes\mathbb{Q} K ⊂ End F ( A ) ⊗ Q [ K : Q ] = dim A [K:\mathbb{Q}] = \dim A [ K : Q ] = dim A λ \lambda λ K K K p p p λ \lambda λ A A A
ρ A , λ : G F ⟶ GL 2 ( K λ ) . \rho_{A,\lambda}:G_F \longrightarrow\operatorname{GL}_2(K_\lambda). ρ A , λ : G F ⟶ GL 2 ( K λ ) . Theorem B. Assume that ρ A , λ \rho_{A,\lambda} ρ A , λ ρ A , λ \rho_{A,\lambda} ρ A , λ F F F
No irreducibility assumption is needed: since F F F ρ A , λ \rho_{A,\lambda} ρ A , λ Lemma 6.2 ). No assumption is made on the reduction of A A A p p p v ∣ p v \mid p v ∣ p ρ A , λ \rho_{A,\lambda} ρ A , λ Σ o \Sigma_{\mathrm{o}} Σ o Σ n \Sigma_{\mathrm{n}} Σ n A A A Q \mathbb{Q} Q [42 Theorem 5.2] , which extends a theorem of Hida [36 ] , shows that ρ A , λ ∣ G F v \rho_{A,\lambda}|_{G_{F_v}} ρ A , λ ∣ G F v v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o
As a consequence, we obtain in § 6.4 the modularity of potentially modular representations of parallel weight 2 in the mixed case (Theorem B′ ' ′ [ F : Q ] [F:\mathbb{Q}] [ F : Q ] ρ \rho ρ p p p [10 ] and of the proof of [2 Theorem 5.5.1] , and under this condition a result of Snowden [44 ] , applied to a member of the system with large residual image, shows that it comes from an abelian variety of GL2 _2 2 F F F
Remark 1.2. Although the results of the present paper do not settle the mixed case of Pan’s theorem in general, they are sufficient for significant applications to new modularity results. In particular, two of the results of this paper, Theorem B′ ' ′ Corollary 5.3 , are already used in [9 ] . In such applications, and more generally in the propagation of modularity along safe chains of congruences, the hypothesis of potential modularity in Theorems B′ ' ′ Q \mathbb{Q} Q [19 , 20 ] and by the author [11 ] . In inductive arguments of this kind, the residual hypotheses of Theorem 5.5 (c) on the entangled representation ρ ′ \rho' ρ ′ ρ ′ \rho' ρ ′ ρ \rho ρ
Strategy
After a solvable base change F 1 / F F_1/F F 1 / F [32 ] shows that ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 p \mathfrak{p} p S S S p p p
Ordinary parts and classicality. We apply Emerton’s functor of ordinary parts [14 , 15 ] to S S S F 1 F_1 F 1 Σ o \Sigma_{\mathrm{o}} Σ o S o S^{\mathrm{o}} S o Σ n \Sigma_{\mathrm{n}} Σ n Σ o \Sigma_{\mathrm{o}} Σ o T o \mathbb{T}^{\mathrm{o}} T o ψ v , 1 \psi_{v,1} ψ v , 1 T o \mathbb{T}^{\mathrm{o}} T o ψ v , 1 \psi_{v,1} ψ v , 1 Lemma 3.4 ); since the central character need not be locally constant at p p p p p p S S S ρ ∣ G F 1 , v \rho|_{G_{F_{1,v}}} ρ ∣ G F 1 , v Proposition 4.1 ). If it is the classical character, the corresponding principal series and the non-ordinary local factors have dense locally algebraic vectors, and the adjunction map produces classical forms (Theorem 4.2 ). If ρ v \rho_v ρ v
Abelian varieties. For Theorem B , the main additional input is the local indecomposability theorem of Hida and Zhao, which is a statement about abelian varieties and does not use modularity; the CM case is classical. Theorem B ′ follows from Theorem B , since by a result of Snowden, applied to a suitable member of the compatible system containing ρ \rho ρ G L 2 \mathrm{GL}_2 GL 2
Organisation
Section 2 contains local results on reducible representations of G Q p G_{\mathbb{Q}_p} G Q p p p p Corollary 5.3 ), to representations of parallel weight 2 over real quadratic fields with Steinberg type at one place above p p p ( 2 , k ) (2,k) ( 2 , k ) Theorem 5.5 ). Section 6 is devoted to abelian varieties: it contains the proof of Theorem B and of its consequence Theorem B ′ for potentially modular representations of parallel weight 2.
Acknowledgments
The author acknowledges the use of Claude (Anthropic), an AI assistant, during the preparation of this work: it was used to check the relevant parts of the literature, to work out and verify technical details of the arguments and to help draft the manuscript. The author has checked all the mathematical content and takes full responsibility for it.
Notation
For a field k k k G k G_k G k K K K S S S G K , S G_{K,S} G K , S S S S S S S G v = G K v G_v=G_{K_v} G v = G K v v v v H T ( ε ) = − 1 \mathrm{HT}(\varepsilon)=-1 HT ( ε ) = − 1 [22 , 32 ] . Characters of G Q p G_{\mathbb{Q}_p} G Q p Q p × \mathbb{Q}_p^\times Q p × Art \operatorname{Art} Art [22 , 32 ] , so that ε ∘ Art ( x ) = x ∣ x ∣ p \varepsilon\circ\operatorname{Art}(x)=x|x|_p ε ∘ Art ( x ) = x ∣ x ∣ p ε ∘ Art ( u ) = u \varepsilon\circ\operatorname{Art}(u)=u ε ∘ Art ( u ) = u u ∈ Z p × u\in\mathbb{Z}_p^\times u ∈ Z p × B B B B ‾ \overline{B} B G L 2 \mathrm{GL}_2 GL 2 N N N B B B T T T V V V ad 0 V \operatorname{ad}^{0}V ad 0 V V ( 1 ) = V ⊗ ε V(1)=V\otimes\varepsilon V ( 1 ) = V ⊗ ε K K K π \pi π G L 2 ( A K ) \mathrm{GL}_2(\mathbb{A}_K) GL 2 ( A K ) ℓ \ell ℓ ι : Q ‾ ℓ ≅ C \iota:\overline{\mathbb{Q}}_\ell\cong\mathbb{C} ι : Q ℓ ≅ C r π , ι : G K → G L 2 ( Q ‾ ℓ ) r_{\pi,\iota}:G_K\to\mathrm{GL}_2(\overline{\mathbb{Q}}_\ell) r π , ι : G K → GL 2 ( Q ℓ ) π \pi π ρ \rho ρ G K G_K G K ρ ⊗ Q ‾ ℓ ≅ r π , ι \rho\otimes\overline{\mathbb{Q}}_\ell\cong r_{\pi,\iota} ρ ⊗ Q ℓ ≅ r π , ι π \pi π ι \iota ι E E E
Local preliminaries
In this section G = G Q p G=G_{\mathbb{Q}_p} G = G Q p
Reducible de Rham representations
Lemma 2.1. Let ψ 1 , ψ 2 : G → E × \psi_1,\psi_2:G\to E^\times ψ 1 , ψ 2 : G → E × H T ( ψ 1 ) ≠ H T ( ψ 2 ) \mathrm{HT}(\psi_1)\ne\mathrm{HT}(\psi_2) HT ( ψ 1 ) = HT ( ψ 2 ) H T ( ψ 1 ) < H T ( ψ 2 ) \mathrm{HT}(\psi_1)<\mathrm{HT}(\psi_2) HT ( ψ 1 ) < HT ( ψ 2 ) ψ 2 \psi_2 ψ 2 ψ 1 \psi_1 ψ 1 H T ( ψ 1 ) > H T ( ψ 2 ) \mathrm{HT}(\psi_1)>\mathrm{HT}(\psi_2) HT ( ψ 1 ) > HT ( ψ 2 ) ψ 2 \psi_2 ψ 2 ψ 1 \psi_1 ψ 1
Proof. Let η = ψ 1 ψ 2 − 1 \eta= \psi_{1}\psi_{2}^{-1} η = ψ 1 ψ 2 − 1 H 1 ( G , E ( η ) ) H^{1}(G,E(\eta)) H 1 ( G , E ( η )) H g 1 H^{1}_{g} H g 1 [3 Corollary 3.8.4] and local duality, H g 1 = H 1 H^{1}_{g} = H^{1} H g 1 = H 1 HT ( η ) < 0 \operatorname{HT}(\eta) < 0 HT ( η ) < 0 H g 1 = 0 H^{1}_{g} = 0 H g 1 = 0 HT ( η ) > 0 \operatorname{HT}(\eta) > 0 HT ( η ) > 0
Let ρ v : G → GL 2 ( E ) \rho_{v}: G \to\operatorname{GL}_{2}(E) ρ v : G → GL 2 ( E ) ψ c l , ψ n c \psi^{\mathrm{cl}}, \psi^{\mathrm{nc}} ψ cl , ψ nc HT ( ψ c l ) < HT ( ψ n c ) \operatorname{HT}(\psi^{\mathrm{cl}}) < \operatorname{HT}(\psi^{\mathrm{nc}}) HT ( ψ cl ) < HT ( ψ nc ) Lemma 2.1 , ψ c l \psi^{\mathrm{cl}} ψ cl ρ v \rho_{v} ρ v ρ v \rho_{v} ρ v classical character . It is the subcharacter occurring for ordinary modular forms.
Lemma 2.2. Let μ = μ 1 ⊗ μ 2 \mu= \mu_{1} \otimes\mu_{2} μ = μ 1 ⊗ μ 2 unitary character of T ( Q p ) T(\mathbb{Q}_{p}) T ( Q p ) μ i ∣ Z p × = x a i θ i \mu_{i}|_{\mathbb{Z}_{p}^{\times}} = x^{a_{i}}\theta_{i} μ i ∣ Z p × = x a i θ i a i ∈ Z a_{i} \in\mathbb{Z} a i ∈ Z θ i \theta_{i} θ i ( Ind B ‾ GL 2 ( Q p ) μ ) c o n t (\operatorname{Ind}_{\overline{B}}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\mu)_{\mathrm{cont}} ( Ind B GL 2 ( Q p ) μ ) cont nonzero locally algebraic vectors if and only if a 1 ≥ a 2 a_{1} \geq a_{2} a 1 ≥ a 2 ψ c l , ψ n c \psi^{\mathrm{cl}}, \psi^{\mathrm{nc}} ψ cl , ψ nc μ c l = ψ c l ⊗ ψ n c ε \mu^{\mathrm{cl}} = \psi^{\mathrm{cl}} \otimes\psi^{\mathrm{nc}}\varepsilon μ cl = ψ cl ⊗ ψ nc ε ( Ind B ‾ G L 2 ( Q p ) μ c l ) c o n t (\operatorname{Ind}_{\overline{B}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\mu^{\mathrm{cl}})_{\mathrm{cont}} ( Ind B GL 2 ( Q p ) μ cl ) cont
Proof. Algebraic functions f f f GL 2 \operatorname{GL}_{2} GL 2 f ( b ‾ g ) = μ a l g ( b ‾ ) f ( g ) f(\overline{b}g) = \mu_{\mathrm{alg}}(\overline{b})f(g) f ( b g ) = μ alg ( b ) f ( g ) det a 2 \det^{a_{2}} det a 2 a 1 − a 2 a_{1}-a_{2} a 1 − a 2 a 1 < a 2 a_{1} < a_{2} a 1 < a 2 a 1 ≥ a 2 a_{1} \geq a_{2} a 1 ≥ a 2 W ⊗ ( Ind B ‾ GL 2 ( Q p ) μ s m ) s m W \otimes(\operatorname{Ind}_{\overline{B}}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\mu_{\mathrm{sm}})_{\mathrm{sm}} W ⊗ ( Ind B GL 2 ( Q p ) μ sm ) sm W = Sym a 1 − a 2 ⊗ det a 2 W = \operatorname{Sym}^{a_{1}-a_{2}} \otimes\det^{a_{2}} W = Sym a 1 − a 2 ⊗ det a 2 μ s m = μ μ a l g − 1 \mu_{\mathrm{sm}} = \mu\mu_{\mathrm{alg}}^{-1} μ sm = μ μ alg − 1 ( Ind B ‾ GL 2 ( Q p ) μ ) c o n t (\operatorname{Ind}_{\overline{B}}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\mu)_{\mathrm{cont}} ( Ind B GL 2 ( Q p ) μ ) cont B ‾ \ GL 2 ( Q p ) = P 1 ( Q p ) \overline{B}\backslash\operatorname{GL}_{2}(\mathbb{Q}_{p}) = \mathbb{P}^{1}(\mathbb{Q}_{p}) B \ GL 2 ( Q p ) = P 1 ( Q p ) P 1 ( Q p ) \mathbb{P}^{1}(\mathbb{Q}_{p}) P 1 ( Q p ) V i V_{i} V i s i s_{i} s i W ⊗ ( Ind B ‾ GL 2 ( Q p ) μ s m ) s m W \otimes(\operatorname{Ind}_{\overline{B}}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\mu_{\mathrm{sm}})_{\mathrm{sm}} W ⊗ ( Ind B GL 2 ( Q p ) μ sm ) sm ∑ i f i s i \sum_{i}f_{i}s_{i} ∑ i f i s i f i f_{i} f i V i V_{i} V i f i f_{i} f i [6 Proposition 2.2.1] for a 1 > a 2 a_{1} > a_{2} a 1 > a 2 a 1 = − HT ( ψ c l ) a_{1} = -\operatorname{HT}(\psi^{\mathrm{cl}}) a 1 = − HT ( ψ cl ) a 2 = 1 − HT ( ψ n c ) a_{2} = 1 - \operatorname{HT}(\psi^{\mathrm{nc}}) a 2 = 1 − HT ( ψ nc ) a 1 − a 2 = HT ( ψ n c ) − HT ( ψ c l ) − 1 ≥ 0 a_{1} - a_{2} = \operatorname{HT}(\psi^{\mathrm{nc}}) - \operatorname{HT}(\psi^{\mathrm{cl}}) - 1 \geq0 a 1 − a 2 = HT ( ψ nc ) − HT ( ψ cl ) − 1 ≥ 0
Ordinary lines
The following lemma concerns the notion of ordinary pseudo-representation of [22 Definition 5.3.2] . Let K K K v v v K K K T : G K → R T: G_{K} \to R T : G K → R T ∣ G v = ψ 1 + ψ 2 T|_{G_{v}} = \psi_{1} + \psi_{2} T ∣ G v = ψ 1 + ψ 2 T T T ψ 1 \psi_{1} ψ 1
T ( σ τ η ) − ψ 1 ( σ ) T ( τ η ) − ψ 2 ( τ ) T ( σ η ) + ψ 1 ( σ ) ψ 2 ( τ ) T ( η ) = 0 ( σ , τ ∈ G v , η ∈ G K ) . T(\sigma\tau\eta) - \psi_{1}(\sigma)T(\tau\eta) - \psi_{2}(\tau)T(\sigma\eta) + \psi_{1}(\sigma)\psi_{2}(\tau)T(\eta) = 0 \qquad(\sigma,\tau\in G_{v},\ \eta\in G_{K}). T ( σ τ η ) − ψ 1 ( σ ) T ( τ η ) − ψ 2 ( τ ) T ( σ η ) + ψ 1 ( σ ) ψ 2 ( τ ) T ( η ) = 0 ( σ , τ ∈ G v , η ∈ G K ) . If T T T ρ : G K → GL 2 ( R ) \rho: G_{K} \to\operatorname{GL}_{2}(R) ρ : G K → GL 2 ( R ) tr [ ( ρ ( σ ) − ψ 1 ( σ ) ) ( ρ ( τ ) − ψ 2 ( τ ) ) ρ ( η ) ] = 0 \operatorname{tr}[(\rho(\sigma) - \psi_{1}(\sigma))(\rho(\tau) - \psi_{2}(\tau))\rho(\eta)] = 0 tr [( ρ ( σ ) − ψ 1 ( σ )) ( ρ ( τ ) − ψ 2 ( τ )) ρ ( η )] = 0
Lemma 2.3. Let A A A O \mathcal{O} O k k k ρ A : G K → GL 2 ( A ) \rho_{A}: G_{K} \to\operatorname{GL}_{2}(A) ρ A : G K → GL 2 ( A ) ρ k \rho_{k} ρ k tr ρ A ∣ G v = ψ 1 + ψ 2 \operatorname{tr}\rho_{A}|_{G_{v}} = \psi_{1} + \psi_{2} tr ρ A ∣ G v = ψ 1 + ψ 2 ψ 1 ( σ 0 ) − ψ 2 ( σ 0 ) ∈ A × \psi_{1}(\sigma_{0}) - \psi_{2}(\sigma_{0}) \in A^{\times} ψ 1 ( σ 0 ) − ψ 2 ( σ 0 ) ∈ A × σ 0 ∈ G v \sigma_{0} \in G_{v} σ 0 ∈ G v
(1) tr ρ A \operatorname{tr}\rho_{A} tr ρ A ψ 1 \psi_{1} ψ 1 ρ A ∣ G v \rho_{A}|_{G_{v}} ρ A ∣ G v G v G_{v} G v G v G_{v} G v ψ 1 \psi_{1} ψ 1
(2) If tr ρ A \operatorname{tr}\rho_{A} tr ρ A ψ 1 \psi_{1} ψ 1 ψ 2 \psi_{2} ψ 2 ρ A ∣ G v ≃ ψ 1 ⊕ ψ 2 \rho_{A}|_{G_{v}} \simeq\psi_{1} \oplus\psi_{2} ρ A ∣ G v ≃ ψ 1 ⊕ ψ 2
Proof. Since ρ k \rho_{k} ρ k k k k ρ k ( G K ) \rho_{k}(G_{K}) ρ k ( G K ) M 2 ( k ) M_{2}(k) M 2 ( k ) A A A ρ A ( G K ) \rho_{A}(G_{K}) ρ A ( G K ) M 2 ( A ) M_{2}(A) M 2 ( A ) M 2 ( A ) M_{2}(A) M 2 ( A ) ψ 1 \psi_{1} ψ 1
( ρ A ( σ ) − ψ 1 ( σ ) ) ( ρ A ( τ ) − ψ 2 ( τ ) ) = 0 ( σ , τ ∈ G v ) . (\rho_{A}(\sigma) - \psi_{1}(\sigma))(\rho_{A}(\tau) - \psi_{2}(\tau)) = 0 \qquad(\sigma,\tau\in G_{v}). ( ρ A ( σ ) − ψ 1 ( σ )) ( ρ A ( τ ) − ψ 2 ( τ )) = 0 ( σ , τ ∈ G v ) . The characteristic polynomial of ρ A ( σ 0 ) \rho_{A}(\sigma_{0}) ρ A ( σ 0 ) ( X − ψ 1 ( σ 0 ) ) ( X − ψ 2 ( σ 0 ) ) (X - \psi_{1}(\sigma_{0}))(X - \psi_{2}(\sigma_{0})) ( X − ψ 1 ( σ 0 )) ( X − ψ 2 ( σ 0 )) A 2 = L 1 ⊕ L 2 A^{2} = L_{1} \oplus L_{2} A 2 = L 1 ⊕ L 2 L i L_{i} L i ρ A ( σ 0 ) = ψ i ( σ 0 ) \rho_{A}(\sigma_{0}) = \psi_{i}(\sigma_{0}) ρ A ( σ 0 ) = ψ i ( σ 0 ) L i L_{i} L i ρ A ( σ 0 ) − ψ 2 ( σ 0 ) \rho_{A}(\sigma_{0}) - \psi_{2}(\sigma_{0}) ρ A ( σ 0 ) − ψ 2 ( σ 0 ) L 1 L_{1} L 1
(1) If tr ρ A \operatorname{tr}\rho_{A} tr ρ A ψ 1 \psi_{1} ψ 1 τ = σ 0 \tau= \sigma_{0} τ = σ 0 ρ A ( σ ) − ψ 1 ( σ ) \rho_{A}(\sigma) - \psi_{1}(\sigma) ρ A ( σ ) − ψ 1 ( σ ) L 1 L_{1} L 1 σ ∈ G v \sigma\in G_{v} σ ∈ G v L 1 L_{1} L 1 L L L L L L ( ρ A ( σ ) − ψ 1 ( σ ) ) ( ρ A ( τ ) − ψ 2 ( τ ) ) (\rho_{A}(\sigma) - \psi_{1}(\sigma))(\rho_{A}(\tau) - \psi_{2}(\tau)) ( ρ A ( σ ) − ψ 1 ( σ )) ( ρ A ( τ ) − ψ 2 ( τ )) ψ 1 \psi_{1} ψ 1 ψ 1 ( σ 0 ) \psi_{1}(\sigma_{0}) ψ 1 ( σ 0 ) ρ A ( σ 0 ) \rho_{A}(\sigma_{0}) ρ A ( σ 0 ) L 1 L_{1} L 1 ψ 1 \psi_{1} ψ 1 ψ 2 \psi_{2} ψ 2 L 1 L_{1} L 1 L 2 L_{2} L 2 G v G_{v} G v ψ 1 \psi_{1} ψ 1 ψ 2 \psi_{2} ψ 2
Completed cohomology and ordinary parts
In this section F 1 F_{1} F 1 Q \mathbb{Q} Q d 1 d_{1} d 1 p p p Σ p \Sigma_{p} Σ p p p p Σ p = Σ o ⊔ Σ n \Sigma_{p}=\Sigma_{o}\sqcup\Sigma_{n} Σ p = Σ o ⊔ Σ n [22 §3] and [32 §§3–4] .
Completed cohomology
3.1. Completed cohomology. Let D D D F 1 F_{1} F 1 ( D ⊗ F 1 A F 1 ∞ ) × ≃ G L 2 ( A F 1 ∞ ) (D\otimes_{F_{1}}\mathbb{A}_{F_{1}}^{\infty})^{\times}\simeq\mathrm{GL}_{2}(\mathbb{A}_{F_{1}}^{\infty}) ( D ⊗ F 1 A F 1 ∞ ) × ≃ GL 2 ( A F 1 ∞ ) D p × = ∏ v ∈ Σ p G L 2 ( Q p ) D_{p}^{\times}=\prod_{v\in\Sigma_{p}}\mathrm{GL}_{2}(\mathbb{Q}_{p}) D p × = ∏ v ∈ Σ p GL 2 ( Q p ) U p U^{p} U p ψ \psi ψ ( A F 1 ∞ ) × / F 1 , > 0 × (\mathbb{A}_{F_{1}}^{\infty})^{\times}/F_{1,>0}^{\times} ( A F 1 ∞ ) × / F 1 , > 0 × ξ \xi ξ Σ ⊇ Σ p \Sigma\supseteq\Sigma_{p} Σ ⊇ Σ p [32 §4.1] . Completed cohomology
S ψ , ξ ( U p ) = l i m ← n l i m → U p S ψ , ξ ( U p U p , O / ϖ n ) S_{\psi,\xi}(U^{p})=\varprojlim_{n}\varinjlim_{U_{p}}S_{\psi,\xi}(U^{p}U_{p},\mathcal{O}/\varpi^{n}) S ψ , ξ ( U p ) = n lim U p lim S ψ , ξ ( U p U p , O / ϖ n ) is an admissible unitary O \mathcal{O} O D p × D_{p}^{\times} D p × T = T ψ , ξ ( U p ) \mathbb{T}=\mathbb{T}_{\psi,\xi}(U^{p}) T = T ψ , ξ ( U p ) T w T_{w} T w w ∉ Σ w\notin\Sigma w ∈ / Σ m \mathfrak{m} m T m \mathbb{T}_{\mathfrak{m}} T m T m : G F 1 , Σ → T m T_{\mathfrak{m}}:G_{F_{1},\Sigma}\to\mathbb{T}_{\mathfrak{m}} T m : G F 1 , Σ → T m T m ( Frob w ) = T w T_{\mathfrak{m}}(\operatorname{Frob}_{w})=T_{w} T m ( Frob w ) = T w ψ ε − 1 \psi\varepsilon^{-1} ψ ε − 1 1 + χ ˉ 1+\bar{\chi} 1 + χ ˉ χ ˉ \bar{\chi} χ ˉ χ ˉ ∣ G v ≠ 1 \bar{\chi}|_{G_{v}}\ne1 χ ˉ ∣ G v = 1 v ∈ Σ p v\in\Sigma_{p} v ∈ Σ p S = S ψ , ξ ( U p ) m S=S_{\psi,\xi}(U^{p})_{\mathfrak{m}} S = S ψ , ξ ( U p ) m M = Hom O ( S ⊗ O E / O , E / O ) M=\operatorname{Hom}_{\mathcal{O}}(S\otimes_{\mathcal{O}}E/\mathcal{O},E/\mathcal{O}) M = Hom O ( S ⊗ O E / O , E / O ) p \mathfrak{p} p T m \mathbb{T}_{\mathfrak{m}} T m ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) T m m o d p T_{\mathfrak{m}}\bmod\mathfrak{p} T m mod p
Paškūnas blocks
3.2. Paškūnas blocks. For v ∈ Σ p v\in\Sigma_{p} v ∈ Σ p B v B_{v} B v v v v 1 1 1 χ ˉ ∣ G v \bar{\chi}|_{G_{v}} χ ˉ ∣ G v p = 3 p=3 p = 3 χ ˉ ∣ G v = ω \bar{\chi}|_{G_{v}}=\omega χ ˉ ∣ G v = ω [24 ] . Let P B v P_{B_{v}} P B v E B v = End ( P B v ) E_{B_{v}}=\operatorname{End}(P_{B_{v}}) E B v = End ( P B v ) R v p s R_{v}^{\mathrm{ps}} R v ps v v v [23 ] for p ≥ 5 p\geq5 p ≥ 5 [24 Theorems 1.3 and 6.13] in general (including the exceptional block), there is a natural map from R v p s R_{v}^{\mathrm{ps}} R v ps Z B v Z_{B_{v}} Z B v E B v E_{B_{v}} E B v Z B v Z_{B_{v}} Z B v O \mathcal{O} O R v p s R_{v}^{\mathrm{ps}} R v ps R v p s [ 1 / p ] ≃ Z B v [ 1 / p ] R_{v}^{\mathrm{ps}}[1/p]\simeq Z_{B_{v}}[1/p] R v ps [ 1/ p ] ≃ Z B v [ 1/ p ] E B v E_{B_{v}} E B v R v p s R_{v}^{\mathrm{ps}} R v ps R v p s R_{v}^{\mathrm{ps}} R v ps [22 ] , extended by Zhang to the exceptional block Theorem 3.3.3 [32 ] , states the following. Let B = ∏ v B v B=\prod_{v}B_{v} B = ∏ v B v m = Hom ( P B , M ) \mathfrak{m}=\operatorname{Hom}(P_{B},M) m = Hom ( P B , M ) R v p s R_{v}^{\mathrm{ps}} R v ps m \mathfrak{m} m E B v E_{B_{v}} E B v T m T_{\mathfrak{m}} T m G v G_{v} G v m \mathfrak{m} m T m \mathbb{T}_{\mathfrak{m}} T m [32 ] .
For a point x x x R v p s [ 1 / p ] R_{v}^{\mathrm{ps}}[1/p] R v ps [ 1/ p ] Irr ( x ) \operatorname{Irr}(x) Irr ( x ) x x x
Lemma 3.1. Let x x x T x T_{x} T x
(1) If T x T_{x} T x ρ x \rho_{x} ρ x Irr ( x ) = { Π x } \operatorname{Irr}(x)=\{\Pi_{x}\} Irr ( x ) = { Π x } Π x \Pi_{x} Π x ρ x \rho_{x} ρ x p p p [32 §3.3] , so that Colmez’s functor gives V ( Π x ) ≃ ρ x ⊗ ε \mathbf{V}(\Pi_{x})\simeq\rho_{x}\otimes\varepsilon V ( Π x ) ≃ ρ x ⊗ ε ρ x \rho_{x} ρ x Π x \Pi_{x} Π x
(2) If T x = ψ 1 + ψ 2 T_{x}=\psi_{1}+\psi_{2} T x = ψ 1 + ψ 2 Irr ( x ) \operatorname{Irr}(x) Irr ( x ) η ∘ det \eta\circ\det η ∘ det ( Ind B ˉ G L 2 ( Q p ) μ ) c o n t (\operatorname{Ind}_{\bar{B}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\mu)_{\mathrm{cont}} ( Ind B ˉ GL 2 ( Q p ) μ ) cont μ \mu μ Q p × \mathbb{Q}_{p}^{\times} Q p ×
(3) The absolutely irreducible admissible unitary Banach representations in B = ∏ v B v B=\prod_{v}B_{v} B = ∏ v B v x = ( x v ) v x=(x_{v})_{v} x = ( x v ) v ⨂ ^ v Π v \widehat{\bigotimes}_{v}\Pi_{v} ⨂ v Π v Π v ∈ Irr ( x v ) \Pi_{v}\in\operatorname{Irr}(x_{v}) Π v ∈ Irr ( x v ) B B B
Proof. (1) The first assertion is [32 Proposition A.2.5] , that is, [24 Proposition 6.11] . Assume ρ x \rho_x ρ x π \pi π ρ x ⊗ ε \rho_x \otimes\varepsilon ρ x ⊗ ε W W W [7 Theorem 1.3(ii)] to the full admissible filtered ( φ , N , G Q p ) (\varphi,N,G_{\mathbb{Q}_p}) ( φ , N , G Q p ) D p s t ( ρ x ⊗ ε ) D_{\mathrm{pst}}(\rho_x \otimes\varepsilon) D pst ( ρ x ⊗ ε ) F ( Δ , W ) \mathcal{F}(\Delta,W) F ( Δ , W ) [7 §1.1] , we get an absolutely irreducible non-ordinary unitary completion Π \Pi Π π ⊗ W \pi\otimes W π ⊗ W V ( Π ) ≃ ρ x ⊗ ε V(\Pi) \simeq\rho_x \otimes\varepsilon V ( Π ) ≃ ρ x ⊗ ε Π ≃ Π x \Pi\simeq\Pi_x Π ≃ Π x [7 Theorem 1.1] . Being a completion of π ⊗ W \pi\otimes W π ⊗ W Π x \Pi_x Π x π ⊗ W \pi\otimes W π ⊗ W [7 ] hold for every prime p p p p = 3 p=3 p = 3 [32 Proposition A.2.4] . (3) follows from [32 Theorem A.1.6 and Lemma 3.3.1] . □
Lemma 3.2. Let p \mathfrak{p} p T m [ 1 / p ] \mathbb{T}_{\mathfrak{m}}[1/p] T m [ 1/ p ] ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) ( S ⊗ E ) [ p ] (S \otimes E)[\mathfrak{p}] ( S ⊗ E ) [ p ] η ∘ det \eta\circ\det η ∘ det v ∈ Σ p v \in\Sigma_p v ∈ Σ p
Proof. See [22 proof of Proposition 4.4.6] : otherwise, by strong approximation for the derived group of D × D^\times D × ρ ( p ) \rho(\mathfrak{p}) ρ ( p )
Ordinary parts at the places of Σ o \Sigma_o Σ o
Write G p = D p × = G o × G n G_p=D_p^\times=G_o\times G_n G p = D p × = G o × G n G o = ∏ v ∈ Σ o G L 2 ( Q p ) G_o=\prod_{v\in\Sigma_o}\mathrm{GL}_2(\mathbb{Q}_p) G o = ∏ v ∈ Σ o GL 2 ( Q p ) G n = ∏ v ∈ Σ n G L 2 ( Q p ) G_n=\prod_{v\in\Sigma_n}\mathrm{GL}_2(\mathbb{Q}_p) G n = ∏ v ∈ Σ n GL 2 ( Q p )
P = B o × G n , L P = T o × G n , P=B_o\times G_n,\qquad L_P=T_o\times G_n, P = B o × G n , L P = T o × G n , where B o = ∏ v ∈ Σ o B ( Q p ) B_o=\prod_{v\in\Sigma_o}B(\mathbb{Q}_p) B o = ∏ v ∈ Σ o B ( Q p ) T o = ∏ v ∈ Σ o T ( Q p ) T_o=\prod_{v\in\Sigma_o}T(\mathbb{Q}_p) T o = ∏ v ∈ Σ o T ( Q p ) N 0 = ∏ v ∈ Σ o N ( Z p ) N_0=\prod_{v\in\Sigma_o}N(\mathbb{Z}_p) N 0 = ∏ v ∈ Σ o N ( Z p ) Ord P \operatorname{Ord}_P Ord P P P P L P L_P L P [14 Definition 3.1.9] , and extends it to ϖ \varpi ϖ [14 Definition 3.4.1] . We use the following properties.
(O1) Ord P \operatorname{Ord}_P Ord P [14 Proposition 3.2.4] .
(O2) Ord P \operatorname{Ord}_P Ord P [14 Theorems 3.3.3 and 3.4.8] .
(O3) For V V V Ord P ( V ) M j = e V I j , j \operatorname{Ord}_P(V)^{M_j}=eV^{I_{j,j}} Ord P ( V ) M j = e V I j , j V I j , j V^{I_{j,j}} V I j , j U v = ∑ a ∈ Z p / p ( p a 0 1 ) U_v=\sum_{a\in\mathbb{Z}_p/p}\begin{pmatrix}p&a\\0&1\end{pmatrix} U v = ∑ a ∈ Z p / p ( p 0 a 1 ) v ∈ Σ o v\in\Sigma_o v ∈ Σ o M j M_j M j T o ( Z p ) × ∏ v ∈ Σ n G L 2 ( Z p ) T_o(\mathbb{Z}_p)\times\prod_{v\in\Sigma_n}\mathrm{GL}_2(\mathbb{Z}_p) T o ( Z p ) × ∏ v ∈ Σ n GL 2 ( Z p ) 1 1 1 p j p^j p j I j , j = N ‾ j M j N 0 I_{j,j}=\overline{N}_jM_jN_0 I j , j = N j M j N 0 [14 proof of Theorem 3.3.3] ; writing Z M Z_M Z M L P L_P L P Z M + = { z ∈ Z M : z N 0 z − 1 ⊂ N 0 } Z_M^+=\{z\in Z_M:zN_0z^{-1}\subset N_0\} Z M + = { z ∈ Z M : z N 0 z − 1 ⊂ N 0 } A = O / ϖ n A=\mathcal{O}/\varpi^n A = O / ϖ n ϖ n V = 0 \varpi^nV=0 ϖ n V = 0 V I j , j V^{I_{j,j}} V I j , j Z M + Z_M^+ Z M + [14 Lemma 3.3.2] , and Hom A [ Z M + ] ( A [ Z M ] , V I j , j ) \operatorname{Hom}_{A[Z_M^+]}(A[Z_M],V^{I_{j,j}}) Hom A [ Z M + ] ( A [ Z M ] , V I j , j ) U v U_v U v [14 Lemma 3.1.5] and its proof).
(O4) (Adjunction [14 Theorem 4.4.6] ) For U U U V V V ϖ \varpi ϖ
Hom G p ( Ind P ‾ G p U , V ) ≃ Hom L P ( U , Ord P V ) . \operatorname{Hom}_{G_p}(\operatorname{Ind}_{\overline{P}}^{G_p}U,V)\simeq\operatorname{Hom}_{L_P}(U,\operatorname{Ord}_PV). Hom G p ( Ind P G p U , V ) ≃ Hom L P ( U , Ord P V ) . By considering unit balls, the same holds for admissible unitary Banach representations.
Put S ∘ = Ord P ( S ) S^\circ=\operatorname{Ord}_P(S) S ∘ = Ord P ( S ) c ≥ 1 c\geq1 c ≥ 1 U ( c ) = U p ⋅ ∏ v ∈ Σ n K v ( c ) ⋅ ∏ v ∈ Σ o I w 1 ( v c ) U(c)=U^p\cdot\prod_{v\in\Sigma_n}K_v(c)\cdot\prod_{v\in\Sigma_o}\mathrm{Iw}_1(v^c) U ( c ) = U p ⋅ ∏ v ∈ Σ n K v ( c ) ⋅ ∏ v ∈ Σ o Iw 1 ( v c ) K v ( c ) K_v(c) K v ( c ) p c p^c p c I w 1 ( v c ) \mathrm{Iw}_1(v^c) Iw 1 ( v c ) [22 5.2.1] .
Central characters. The character ψ \psi ψ p p p ψ = ε 2 \psi=\varepsilon^2 ψ = ε 2 ψ ( z ) ≠ 1 \psi(z)\ne1 ψ ( z ) = 1 z ∈ U z\in U z ∈ U f ( g ) = f ( g z ) = ψ ( z ) f ( g ) f(g)=f(gz)=\psi(z)f(g) f ( g ) = f ( g z ) = ψ ( z ) f ( g ) S ψ , ξ ( U , O ) S_{\psi,\xi}(U,\mathcal{O}) S ψ , ξ ( U , O ) [22 §§3.2.7, 3.3.3] . Let ψ 0 \psi_0 ψ 0 ψ ‾ \overline{\psi} ψ ψ 0 ψ − 1 \psi_0\psi^{-1} ψ 0 ψ − 1 1 + ϖ O 1+\varpi\mathcal{O} 1 + ϖ O p p p θ \theta θ 1 + ϖ O 1+\varpi\mathcal{O} 1 + ϖ O θ ∘ Nrd \theta\circ\operatorname{Nrd} θ ∘ Nrd S ψ , ξ ( U p ) S_{\psi,\xi}(U^p) S ψ , ξ ( U p ) S ψ 0 , ξ 0 ( U p ) S_{\psi_0,\xi_0}(U^p) S ψ 0 , ξ 0 ( U p ) Σ \Sigma Σ G p G_p G p θ ∘ det \theta\circ\det θ ∘ det ϖ n \varpi^n ϖ n T w T_w T w θ ( Frob w ) T w \theta(\operatorname{Frob}_w)T_w θ ( Frob w ) T w T T T θ T \theta T θ T Ord P \operatorname{Ord}_P Ord P ⟨ γ ⟩ v \langle\gamma\rangle_v ⟨ γ ⟩ v θ v ( Art ( γ ) ) \theta_v(\operatorname{Art}(\gamma)) θ v ( Art ( γ )) θ \theta θ
Lemma 3.3. S ∘ / ϖ n ≃ Ord P ( S / ϖ n ) ≃ lim ⟵ c e S ψ , ξ ( U ( c ) , O / ϖ n ) m S^\circ/\varpi^n \simeq\operatorname{Ord}_P(S/\varpi^n) \simeq\lim_{\longleftarrow_c} eS_{\psi,\xi}(U(c),\mathcal{O}/\varpi^n)_\mathfrak{m} S ∘ / ϖ n ≃ Ord P ( S / ϖ n ) ≃ lim ⟵ c e S ψ , ξ ( U ( c ) , O / ϖ n ) m e = ∏ v ∈ Σ o lim m U v m e=\prod_{v\in\Sigma_o}\lim_m U_v^m e = ∏ v ∈ Σ o lim m U v m
Proof. The second isomorphism is (O3) together with (O1). Since S S S O \mathcal{O} O Ord P \operatorname{Ord}_P Ord P Ord P ( S ) / ϖ n \operatorname{Ord}_P(S)/\varpi^n Ord P ( S ) / ϖ n Ord P ( S / ϖ n ) \operatorname{Ord}_P(S/\varpi^n) Ord P ( S / ϖ n ) ψ \psi ψ ψ 0 \psi_0 ψ 0 c c c ψ 0 \psi_0 ψ 0 U ( c ) U(c) U ( c ) [22 Corollary 3.1.3] applies and S ψ 0 , ξ 0 ( U ( c ) , O ) → S ψ 0 , ξ 0 ( U ( c ) , O / ϖ n ) S_{\psi_0,\xi_0}(U(c),\mathcal{O})\to S_{\psi_0,\xi_0}(U(c),\mathcal{O}/\varpi^n) S ψ 0 , ξ 0 ( U ( c ) , O ) → S ψ 0 , ξ 0 ( U ( c ) , O / ϖ n ) e e e
Let T ∘ \mathbb{T}^\circ T ∘ O \mathcal{O} O S ∘ S^\circ S ∘ T m \mathbb{T}_m T m Ord P \operatorname{Ord}_P Ord P ⟨ γ ⟩ v \langle\gamma\rangle_v ⟨ γ ⟩ v γ ∈ Q p × \gamma\in\mathbb{Q}_p^\times γ ∈ Q p × v ∈ Σ o v\in\Sigma_o v ∈ Σ o diag ( γ , 1 ) ∈ T ( Q p ) \operatorname{diag}(\gamma,1)\in T(\mathbb{Q}_p) diag ( γ , 1 ) ∈ T ( Q p ) v v v ⟨ p ⟩ v = U v \langle p\rangle_v=U_v ⟨ p ⟩ v = U v v ∈ Σ o v\in\Sigma_o v ∈ Σ o ψ v , 1 : G v → ( T ∘ ) × \psi_{v,1}:G_v\to(\mathbb{T}^{\circ})^\times ψ v , 1 : G v → ( T ∘ ) × ψ v , 1 ∘ Art ( γ ) = ⟨ γ ⟩ v \psi_{v,1}\circ\operatorname{Art}(\gamma)=\langle\gamma\rangle_v ψ v , 1 ∘ Art ( γ ) = ⟨ γ ⟩ v
Lemma 3.4. There is a 2-dimensional pseudo-representation T ∘ : G F 1 , Σ → T ∘ T^\circ:G_{F_1,\Sigma}\to\mathbb{T}^\circ T ∘ : G F 1 , Σ → T ∘ with determinant ψ ε − 1 \psi\varepsilon^{-1} ψ ε − 1 and T ∘ ( Frob w ) = T w T^\circ(\operatorname{Frob}_w)=T_w T ∘ ( Frob w ) = T w for w ∉ Σ w\notin\Sigma w ∈ / Σ For every v ∈ Σ o v\in\Sigma_o v ∈ Σ o we have T ∘ ∣ G v = ψ v , 1 + ψ ε − 1 ψ v , 1 − 1 T^\circ|_{G_v}=\psi_{v,1}+\psi\varepsilon^{-1}\psi_{v,1}^{-1} T ∘ ∣ G v = ψ v , 1 + ψ ε − 1 ψ v , 1 − 1 and T ∘ T^\circ T ∘ is ψ v , 1 \psi_{v,1} ψ v , 1 ordinary .
Proof. The pseudo-representation is the push-forward of T m T_m T m θ \theta θ ψ 0 \psi_0 ψ 0 A c A_c A c O \mathcal{O} O e S ψ 0 , ξ 0 ( U ( c ) , O ) m eS_{\psi_0,\xi_0}(U(c),\mathcal{O})_\mathfrak{m} e S ψ 0 , ξ 0 ( U ( c ) , O ) m T w T_w T w ⟨ γ ⟩ v \langle\gamma\rangle_v ⟨ γ ⟩ v O \mathcal{O} O e S ψ 0 , ξ 0 ( U ( c ) , O / ϖ n ) m eS_{\psi_0,\xi_0}(U(c),\mathcal{O}/\varpi^n)_\mathfrak{m} e S ψ 0 , ξ 0 ( U ( c ) , O / ϖ n ) m A c / ϖ n A_c/\varpi^n A c / ϖ n S ∘ / ϖ n S^\circ/\varpi^n S ∘ / ϖ n A c A_c A c A c A_c A c A c [ 1 / p ] A_c[1/p] A c [ 1/ p ]
We have e S ψ 0 , ξ 0 ( U ( c ) , E ) = ⨁ π e ( π U ( c ) ) eS_{\psi_0,\xi_0}(U(c),E)=\bigoplus_\pi e(\pi^{U(c)}) e S ψ 0 , ξ 0 ( U ( c ) , E ) = ⨁ π e ( π U ( c ) ) π \pi π D × D^\times D × π \pi π U v U_v U v p p p T w T_w T w v ∈ Σ o v\in\Sigma_o v ∈ Σ o T ( Z p ) T(\mathbb{Z}_p) T ( Z p ) π Iw 1 ( v ) \pi^{\operatorname{Iw}_1(v)} π Iw 1 ( v ) [22 §5.2.1] . Hence A c [ 1 / p ] A_c[1/p] A c [ 1/ p ] π \pi π v ∈ Σ o v\in\Sigma_o v ∈ Σ o [31 Theorem 2] , whose statement is place by place (see also [25 , 26 ] ), ρ Π , v \rho_{\Pi,v} ρ Π , v ψ v , 1 ( π ) \psi_{v,1}(\pi) ψ v , 1 ( π ) ψ v , 1 ( π ) ∘ Art \psi_{v,1}(\pi)\circ\operatorname{Art} ψ v , 1 ( π ) ∘ Art ⟨ ⋅ ⟩ v \langle\cdot\rangle_v ⟨ ⋅ ⟩ v tr ρ π \operatorname{tr}\rho_\pi tr ρ π ρ k \rho_k ρ k
Lemma 3.5. For v ∈ Σ n v\in\Sigma_n v ∈ Σ n and every closed G p G_p G p stable subspace Π ⊂ S ⊗ E \Pi\subset S\otimes E Π ⊂ S ⊗ E stable under T m \mathbb{T}_m T m the action of R v p s R_v^{\mathrm{ps}} R v ps on Ord P ( Π ) \operatorname{Ord}_P(\Pi) Ord P ( Π ) through the centre of E B v E_{B_v} E B v coincides with its action through T ∘ ∣ G v T^\circ|_{G_v} T ∘ ∣ G v
Proof. An element z z z E B v E_{B_v} E B v S S S t ( z ) ∈ T m t(z)\in\mathbb{T}_m t ( z ) ∈ T m GL 2 ( Q p ) \operatorname{GL}_2(\mathbb{Q}_p) GL 2 ( Q p ) z z z GL 2 ( Q p ) \operatorname{GL}_2(\mathbb{Q}_p) GL 2 ( Q p ) S S S G p G_p G p G p G_p G p Ord P ( z ) = t ( z ) \operatorname{Ord}_P(z)=t(z) Ord P ( z ) = t ( z ) S ∘ S^\circ S ∘ Ord P ( S / ϖ n ) ↪ S / ϖ n \operatorname{Ord}_P(S/\varpi^n)\hookrightarrow S/\varpi^n Ord P ( S / ϖ n ) ↪ S / ϖ n [14 Theorem 3.3.3] is equivariant for the factors at v ∈ Σ n v\in\Sigma_n v ∈ Σ n N 0 N_0 N 0 M + M^+ M + [14 §3.1] , and evaluation at 1 is M + M^+ M + [14 Lemma 3.1.7 and Definition 3.1.10] . So the action of z z z Ord P ( S ) \operatorname{Ord}_P(S) Ord P ( S ) S S S t ( z ) t(z) t ( z ) Ord P ( Π ) ⊂ S ∘ ⊗ E \operatorname{Ord}_P(\Pi)\subset S^\circ\otimes E Ord P ( Π ) ⊂ S ∘ ⊗ E
Classicality and the proof of Theorem A
We keep the notation of the previous section.
Local–global compatibility at ordinary points
Proposition 4.1. Let p \mathfrak{p} p T m [ 1 / p ] \mathbb{T}_{\mathfrak{m}}[1/p] T m [ 1/ p ] ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v v ∈ Σ o v \in\Sigma_o v ∈ Σ o Π = ( S ⊗ E ) [ p ] \Pi= (\mathcal{S} \otimes E)[\mathfrak{p}] Π = ( S ⊗ E ) [ p ] E E E μ = ⨂ v ∈ Σ o μ v \mu= \bigotimes_{v \in\Sigma_o} \mu_v μ = ⨂ v ∈ Σ o μ v T o T_o T o μ v = μ v , 1 ⊗ μ v , 2 \mu_v = \mu_{v,1} \otimes\mu_{v,2} μ v = μ v , 1 ⊗ μ v , 2 Ord P ( Π ) [ T o = μ ] ≠ 0 \operatorname{Ord}_{P}(\Pi)[T_o = \mu] \ne0 Ord P ( Π ) [ T o = μ ] = 0 v ∈ Σ o v \in\Sigma_o v ∈ Σ o μ v , 1 ∘ Art − 1 \mu_{v,1} \circ\operatorname{Art}^{-1} μ v , 1 ∘ Art − 1 ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v
Proof. Since m \mathfrak{m} m T m \mathbb{T}_{\mathfrak{m}} T m Π ≠ 0 \Pi\ne0 Π = 0 Lemma 3.1 (3) it has finite length. Let Π ′ ⊂ Π \Pi' \subset\Pi Π ′ ⊂ Π Π ′ ≃ ⨂ v ∈ Σ n Π v ′ \Pi' \simeq\bigotimes_{v \in\Sigma_n} \Pi_v' Π ′ ≃ ⨂ v ∈ Σ n Π v ′ Lemmas 3.2 and 3.1 (2), for each v ∈ Σ o v \in\Sigma_o v ∈ Σ o μ v \mu_v μ v T ( Q p ) T(\mathbb{Q}_p) T ( Q p ) f v : ( Ind B ‾ G μ v ) c o n t ↠ Π v ′ f_v : (\operatorname{Ind}_{\overline{B}}^{G}\mu_v)_{\mathrm{cont}} \twoheadrightarrow\Pi_v' f v : ( Ind B G μ v ) cont ↠ Π v ′ U 0 = μ ⊗ ⨂ v ∈ Σ n Π v ′ U_0 = \mu\otimes\bigotimes_{v \in\Sigma_n} \Pi_v' U 0 = μ ⊗ ⨂ v ∈ Σ n Π v ′ L P L_P L P ⨂ f v ⊗ id \bigotimes f_v \otimes\operatorname{id} ⨂ f v ⊗ id Ind P ‾ G p U 0 → Π ′ ⊂ Π \operatorname{Ind}_{\overline{P}}^{G_p} U_0 \to\Pi' \subset\Pi Ind P G p U 0 → Π ′ ⊂ Π L P L_P L P ϕ : U 0 → Ord P ( Π ) \phi: U_0 \to\operatorname{Ord}_{P}(\Pi) ϕ : U 0 → Ord P ( Π ) Ord P ( Π ) [ T o = μ ] \operatorname{Ord}_{P}(\Pi)[T_o = \mu] Ord P ( Π ) [ T o = μ ] Ord P ( Π ) ⊂ S o ⊗ E \operatorname{Ord}_{P}(\Pi) \subset\mathcal{S}^o \otimes E Ord P ( Π ) ⊂ S o ⊗ E ϕ \phi ϕ T m \mathbb{T}_{\mathfrak{m}} T m p \mathfrak{p} p T o T_o T o μ \mu μ T o T^o T o P \mathfrak{P} P ψ v , 1 ( P ) = μ v , 1 ∘ Art − 1 \psi_{v,1}(\mathfrak{P}) = \mu_{v,1} \circ\operatorname{Art}^{-1} ψ v , 1 ( P ) = μ v , 1 ∘ Art − 1 Lemma 3.4 modulo P \mathfrak{P} P tr ρ ( p ) \operatorname{tr}\rho(\mathfrak{p}) tr ρ ( p ) ψ v , 1 ( P ) \psi_{v,1}(\mathfrak{P}) ψ v , 1 ( P ) ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) [22 Lemma 5.3.3] (or Lemma 2.3 over the residue field of p \mathfrak{p} p ψ v , 1 ( P ) \psi_{v,1}(\mathfrak{P}) ψ v , 1 ( P ) ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v
Classicality
Theorem 4.2. Let p \mathfrak{p} p T m [ 1 / p ] \mathbb{T}_{\mathfrak{m}}[1/p] T m [ 1/ p ] Π = ( S ⊗ E ) [ p ] \Pi= (\mathcal{S} \otimes E)[\mathfrak{p}] Π = ( S ⊗ E ) [ p ] μ = ⨂ v ∈ Σ o μ v \mu= \bigotimes_{v \in\Sigma_o} \mu_v μ = ⨂ v ∈ Σ o μ v T o T_o T o Ord P ( Π ) [ T o = μ ] ≠ 0 \operatorname{Ord}_{P}(\Pi)[T_o = \mu] \ne0 Ord P ( Π ) [ T o = μ ] = 0
ρ ( p ) \rho(\mathfrak{p}) ρ ( p )
for v ∈ Σ n v \in\Sigma_n v ∈ Σ n ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v
for v ∈ Σ o v \in\Sigma_o v ∈ Σ o ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v μ v , 1 ∘ Art − 1 \mu_{v,1} \circ\operatorname{Art}^{-1} μ v , 1 ∘ Art − 1
Then ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) GL 2 ( A F 1 ) \operatorname{GL}_2(\mathbb{A}_{F_1}) GL 2 ( A F 1 )
Proof. The centre of GL 2 ( Q p ) \operatorname{GL}_2(\mathbb{Q}_p) GL 2 ( Q p ) v v v Π \Pi Π μ v , 2 \mu_{v,2} μ v , 2 μ v = μ v c l \mu_v = \mu_v^{\mathrm{cl}} μ v = μ v cl U 1 = Ord P ( Π ) [ T o = μ ] ≠ 0 U_1 = \operatorname{Ord}_{P}(\Pi)[T_o = \mu] \ne0 U 1 = Ord P ( Π ) [ T o = μ ] = 0 G n G_n G n L P L_P L P T o T_o T o μ ≡ 1 ( m o d ϖ ) \mu\equiv1 \pmod{\varpi} μ ≡ 1 ( mod ϖ ) Lemma 3.5 , the centre of B n = ∏ v ∈ Σ n B v B_n = \prod_{v \in\Sigma_n} B_v B n = ∏ v ∈ Σ n B v U 1 U_1 U 1 x v x_v x v ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v v ∈ Σ n v \in\Sigma_n v ∈ Σ n Lemma 3.1 (1),(3) it contains a closed subrepresentation isomorphic to Π n = ⨂ v ∈ Σ n Π x v \Pi_n = \bigotimes_{v \in\Sigma_n} \Pi_{x_v} Π n = ⨂ v ∈ Σ n Π x v Σ n = ∅ \Sigma_n = \varnothing Σ n = ∅
Φ : ⨂ v ∈ Σ o ( Ind B ‾ G μ v c l ) c o n t ⊗ ^ Π n ⟶ Π . \Phi: \bigotimes_{v \in\Sigma_o}(\operatorname{Ind}_{\overline{B}}^{G}\mu_v^{\mathrm{cl}})_{\mathrm{cont}} \widehat{\otimes} \Pi_n \longrightarrow\Pi. Φ : v ∈ Σ o ⨂ ( Ind B G μ v cl ) cont ⊗ Π n ⟶ Π. By Lemmas 2.2 and 3.1 (1), each factor of the source has dense locally algebraic vectors, hence so does the source, and Φ \Phi Φ Π \Pi Π D × D^\times D × [22 3.2.10] with Hecke eigenvalues given by p \mathfrak{p} p ρ ( p ) \rho(\mathfrak{p}) ρ ( p )
Pro-modularity
In this subsection ρ \rho ρ § 1.2 and satisfies (N), (O) and (R); we do not assume that ρ v \rho_v ρ v v ∈ Σ o v \in\Sigma_o v ∈ Σ o θ \theta θ χ ˉ − 1 \bar{\chi}^{-1} χ ˉ − 1 ρ \rho ρ ρ ⊗ θ \rho\otimes\theta ρ ⊗ θ ρ v \rho_v ρ v ρ \rho ρ ρ ˉ s s = 1 ⊕ χ ˉ \bar{\rho}^{ss}=1 \oplus\bar{\chi} ρ ˉ ss = 1 ⊕ χ ˉ χ = det ρ \chi=\det\rho χ = det ρ χ ˉ \bar{\chi} χ ˉ ψ = χ ε \psi=\chi\varepsilon ψ = χ ε ρ \rho ρ χ ˉ = det ρ ˉ \bar{\chi}=\det\bar{\rho} χ ˉ = det ρ ˉ Σ \Sigma Σ F F F p p p ρ \rho ρ
For a totally real field K ⊃ F K \supset F K ⊃ F Σ K \Sigma_K Σ K K K K Σ \Sigma Σ R K p s R_K^{\mathrm{ps}} R K ps ( 1 + χ ˉ ) ∣ G K (1+\bar{\chi})|_{G_K} ( 1 + χ ˉ ) ∣ G K G K , Σ K G_{K,\Sigma_K} G K , Σ K χ ∣ G K \chi|_{G_K} χ ∣ G K
Lemma 4.3. Let x x x Spec R K p s [ 1 / p ] \operatorname{Spec} R_K^{\mathrm{ps}}[1/p] Spec R K ps [ 1/ p ] ρ x \rho_x ρ x Spec R K p s \operatorname{Spec} R_K^{\mathrm{ps}} Spec R K ps x x x 1 + 2 [ K : Q ] 1+2[K:\mathbb{Q}] 1 + 2 [ K : Q ]
Proof. The completed local ring of R K p s R_K^{\mathrm{ps}} R K ps x x x T x T_x T x k ( x ) k(x) k ( x ) [22 Proposition 2.2.2] ; the proof there only uses that G K , Σ K G_{K,\Sigma_K} G K , Σ K k ( x ) k(x) k ( x ) E E E ρ x \rho_x ρ x T x T_x T x k ( x ) k(x) k ( x ) ρ x \rho_x ρ x [21 , 28 ] , so this functor coincides with the deformation functor of ρ x \rho_x ρ x [22 Corollary 2.2.3] . Note that this uses irreducibility of ρ x \rho_x ρ x k ( x ) k(x) k ( x ) ρ ˉ \bar{\rho} ρ ˉ k ( x ) [ [ X 1 , … , X h 1 ] ] / ( f 1 , … , f h 2 ) k(x)[[X_1,\ldots,X_{h^1}]]/(f_1,\ldots,f_{h^2}) k ( x ) [[ X 1 , … , X h 1 ]] / ( f 1 , … , f h 2 ) h i = dim H i ( G K , Σ K , ad 0 ρ x ) h^i=\dim H^i(G_{K,\Sigma_K},\operatorname{ad}^0\rho_x) h i = dim H i ( G K , Σ K , ad 0 ρ x ) h 1 − h 2 = 3 [ K : Q ] − [ K : Q ] = 2 [ K : Q ] h^1-h^2=3[K:\mathbb{Q}]-[K:\mathbb{Q}]=2[K:\mathbb{Q}] h 1 − h 2 = 3 [ K : Q ] − [ K : Q ] = 2 [ K : Q ] H 0 ( G K , Σ K , ad 0 ρ x ) = 0 H^0(G_{K,\Sigma_K},\operatorname{ad}^0\rho_x)=0 H 0 ( G K , Σ K , ad 0 ρ x ) = 0 ρ x \rho_x ρ x 2 [ K : Q ] 2[K:\mathbb{Q}] 2 [ K : Q ] R K p s R_K^{\mathrm{ps}} R K ps x x x x x x 2 [ K : Q ] + 1 2[K:\mathbb{Q}]+1 2 [ K : Q ] + 1
Lemma 4.4. Let F ′ / F F'/F F ′ / F ρ ∣ G F ′ \rho|_{G_{F'}} ρ ∣ G F ′
Proof. Suppose not. Since G F ′ G_{F'} G F ′ G F G_F G F ρ ∣ G F ′ ⊗ E ‾ \rho|_{G_{F'}}\otimes\overline{E} ρ ∣ G F ′ ⊗ E χ 1 , χ 2 \chi_1,\chi_2 χ 1 , χ 2 G F G_F G F χ 1 = χ 2 \chi_1=\chi_2 χ 1 = χ 2 ρ ( G F ′ ) \rho(G_{F'}) ρ ( G F ′ ) ρ \rho ρ v ∣ p v\mid p v ∣ p ρ v \rho_v ρ v χ 1 ≠ χ 2 \chi_1\ne\chi_2 χ 1 = χ 2 χ 1 \chi_1 χ 1 G M G_M G M M / F M/F M / F F ′ F' F ′ ρ ⊗ E ‾ ≃ Ind G M G F χ 1 ′ \rho\otimes\overline{E}\simeq\operatorname{Ind}_{G_M}^{G_F}\chi'_1 ρ ⊗ E ≃ Ind G M G F χ 1 ′ χ 1 ′ \chi'_1 χ 1 ′ G M G_M G M χ 1 \chi_1 χ 1 χ 1 ′ \chi'_1 χ 1 ′ p p p M M M [39 Chapter III] . Since M M M χ 1 ′ \chi'_1 χ 1 ′ ε n \varepsilon^n ε n ρ v ∣ G M w \rho_{v}|_{G_{M_w}} ρ v ∣ G M w w ∣ v ∣ p w\mid v\mid p w ∣ v ∣ p ε n ⋅ \varepsilon^n\cdot ε n ⋅ ρ v \rho_v ρ v
Proposition 4.5. There is a totally real field F 1 ⊃ F F_1\supset F F 1 ⊃ F Q \mathbb{Q} Q p p p F 1 F_1 F 1 Σ 1 \Sigma_1 Σ 1 F 1 F_1 F 1 Σ \Sigma Σ C C C Spec R F 1 p s \operatorname{Spec}R_{F_1}^{\mathrm{ps}} Spec R F 1 ps Σ F 1 = Σ 1 \Sigma_{F_1}=\Sigma_1 Σ F 1 = Σ 1 ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 [32 Definition 4.1.3] : there are a tame level U p U^p U p Σ ′ ⊃ Σ 1 \Sigma'\supset\Sigma_1 Σ ′ ⊃ Σ 1 ξ \xi ξ [32 §4.1] such that C C C Spec T ψ , ξ ( U p ) m → Spec R F 1 , Σ ′ p s \operatorname{Spec}\mathbb{T}_{\psi,\xi}(U^p)_{\mathfrak m}\to\operatorname{Spec}R_{F_1,\Sigma'}^{\mathrm{ps}} Spec T ψ , ξ ( U p ) m → Spec R F 1 , Σ ′ ps R F 1 , Σ ′ p s R_{F_1,\Sigma'}^{\mathrm{ps}} R F 1 , Σ ′ ps G F 1 , Σ ′ G_{F_1,\Sigma'} G F 1 , Σ ′ R F 1 p s R_{F_1}^{\mathrm{ps}} R F 1 ps
Proof. Take F 1 F_1 F 1 [32 Theorem 5.5.1] , applied with the set Σ \Sigma Σ χ ˉ \bar{\chi} χ ˉ G Q G_{\mathbb{Q}} G Q χ ˉ ∣ G v ≠ 1 \bar{\chi}|_{G_v}\ne1 χ ˉ ∣ G v = 1 v ∣ p v\mid p v ∣ p ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 C C C 1 + 2 [ F 1 : Q ] 1+2[F_1:\mathbb{Q}] 1 + 2 [ F 1 : Q ] [32 Theorem 5.5.1] , and by [32 Remark 4.1.2] a single pair ( U p , ξ ) (U^p,\xi) ( U p , ξ ) C C C p p p C C C
From now on we also write Σ o \Sigma_o Σ o Σ n \Sigma_n Σ n F 1 F_1 F 1 Σ o \Sigma_o Σ o Σ n \Sigma_n Σ n p p p F 1 F_1 F 1 v v v F 1 F_1 F 1 u ∈ Σ p u \in\Sigma_p u ∈ Σ p G v = G F u G_v = G_{F_u} G v = G F u ρ ∣ G v = ρ u \rho|_{G_v} = \rho_u ρ ∣ G v = ρ u
Proof of Theorem A
Let F 1 F_1 F 1 C C C ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 ( U p , ξ ) (U^p,\xi) ( U p , ξ ) ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 p \mathfrak{p} p T m [ 1 / p ] \mathbb{T}_{\mathfrak{m}}[1/p] T m [ 1/ p ] ρ ( p ) = ρ ∣ G F 1 \rho(\mathfrak{p})=\rho|_{G_{F_1}} ρ ( p ) = ρ ∣ G F 1 μ \mu μ Ord P ( ( S ⊗ E ) [ p ] ) [ T o = μ ] ≠ 0 \operatorname{Ord}_{P}((S\otimes E)[\mathfrak{p}])[T_o=\mu]\ne0 Ord P (( S ⊗ E ) [ p ]) [ T o = μ ] = 0 μ v , 1 ∘ Art − 1 \mu_{v,1}\circ\operatorname{Art}^{-1} μ v , 1 ∘ Art − 1 ρ ∣ G v \rho|_{G_v} ρ ∣ G v v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ ∣ G v \rho|_{G_v} ρ ∣ G v ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 F 1 / F F_1/F F 1 / F ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 ρ \rho ρ [2 Lemma 2.2.2] , [1 ] . □
Three applications of Theorem A
We say that a representation r : G Q p → GL 2 ( E ) r:G_{\mathbb{Q}_p}\to\operatorname{GL}_2(E) r : G Q p → GL 2 ( E ) Steinberg type if r ⊗ η r\otimes\eta r ⊗ η η \eta η G Q p G_{\mathbb{Q}_p} G Q p r r r r r r supercuspidal type if it is de Rham and its Weil–Deligne representation (attached to D p s t ( r ) D_{\mathrm{pst}}(r) D pst ( r ) discrete series type if it is of Steinberg or of supercuspidal type, that is, if its Frobenius-semisimple Weil–Deligne representation ( r W D , N ) (r_{\mathrm{WD}},N) ( r WD , N ) N ≠ 0 N\ne0 N = 0
Lemma 5.1. Let r : G Q p → GL 2 ( E ) r:G_{\mathbb{Q}_p}\to\operatorname{GL}_2(E) r : G Q p → GL 2 ( E )
If r r r r r r r r r r r r
If r r r r r r
Proof. (1) Twisting, we may assume that r r r D = D s t ( r ) D=D_{\mathrm{st}}(r) D = D st ( r ) ( φ , N ) (\varphi,N) ( φ , N ) E ⊗ Q p = E E\otimes\mathbb{Q}_p=E E ⊗ Q p = E [8 ] , with N ≠ 0 N\ne0 N = 0 e 1 , e 2 e_1,e_2 e 1 , e 2 N e 2 = e 1 Ne_2=e_1 N e 2 = e 1 N e 1 = 0 Ne_1=0 N e 1 = 0 N φ = p φ N N\varphi=p\varphi N N φ = pφN E e 1 = ker N Ee_1=\ker N E e 1 = ker N φ \varphi φ φ ( e 1 ) = α e 1 \varphi(e_1)=\alpha e_1 φ ( e 1 ) = α e 1 φ ( e 2 ) ∈ p α e 2 + E e 1 \varphi(e_2)\in p\alpha e_2+Ee_1 φ ( e 2 ) ∈ p α e 2 + E e 1 t N ( D ) = 2 v p ( α ) + 1 t_N(D)=2v_p(\alpha)+1 t N ( D ) = 2 v p ( α ) + 1 h 1 ≤ h 2 h_1\le h_2 h 1 ≤ h 2 t H ( D ) = h 1 + h 2 t_H(D)=h_1+h_2 t H ( D ) = h 1 + h 2 v p ( α ) = ( h 1 + h 2 − 1 ) / 2 v_p(\alpha)=(h_1+h_2-1)/2 v p ( α ) = ( h 1 + h 2 − 1 ) /2 h 1 = h 2 h_1=h_2 h 1 = h 2 t H ( E e 1 ) = h 1 > v p ( α ) t_H(Ee_1)=h_1>v_p(\alpha) t H ( E e 1 ) = h 1 > v p ( α ) h 1 < h 2 h_1<h_2 h 1 < h 2 ( φ , N ) (\varphi,N) ( φ , N ) ker N \ker N ker N E e 1 Ee_1 E e 1 r r r E e 1 Ee_1 E e 1 t H ( E e 1 ) = v p ( α ) t_H(Ee_1)=v_p(\alpha) t H ( E e 1 ) = v p ( α ) t H ( E e 1 ) ∈ { h 1 , h 2 } t_H(Ee_1)\in\{h_1,h_2\} t H ( E e 1 ) ∈ { h 1 , h 2 } v p ( α ) = ( h 1 + h 2 − 1 ) / 2 v_p(\alpha)=(h_1+h_2-1)/2 v p ( α ) = ( h 1 + h 2 − 1 ) /2 h 2 = h 1 + 1 h_2=h_1+1 h 2 = h 1 + 1 h 2 = h 1 + 1 h_2=h_1+1 h 2 = h 1 + 1 v p ( α ) = h 1 v_p(\alpha)=h_1 v p ( α ) = h 1 D D D t H ( E e 1 ) ≤ h 1 t_H(Ee_1)\le h_1 t H ( E e 1 ) ≤ h 1 t H ( E e 1 ) = h 1 = v p ( α ) t_H(Ee_1)=h_1=v_p(\alpha) t H ( E e 1 ) = h 1 = v p ( α ) r r r N = 0 N=0 N = 0
(2) If r r r ψ 2 \psi_2 ψ 2 ψ 1 \psi_1 ψ 1 D p s t D_{\mathrm{pst}} D pst ψ 1 \psi_1 ψ 1
Remark 5.2. The argument in (2) is the same as in [17 Theorem 3.2] , where it is used to show that the restriction to a decomposition group at p p p p p p f f f π p ( f ) \pi_p(f) π p ( f ) G Q p G_{\mathbb{Q}_p} G Q p [4 ] , [5 ] .
Corollary 5.3. Let p p p F F F ρ \rho ρ § 1.2 , and assume that ρ \rho ρ v ∈ Σ p v\in\Sigma_p v ∈ Σ p ρ v \rho_v ρ v ρ \rho ρ
Proof. Let Σ o \Sigma_o Σ o v ∈ Σ p v \in\Sigma_p v ∈ Σ p ρ v \rho_v ρ v Σ n = Σ p ∖ Σ o \Sigma_n = \Sigma_p \setminus\Sigma_o Σ n = Σ p ∖ Σ o ρ v \rho_v ρ v v ∈ Σ n v \in\Sigma_n v ∈ Σ n v ∈ Σ o v \in\Sigma_o v ∈ Σ o
For the remaining applications, let F F F p p p F F F Σ p = { v 1 , v 2 } \Sigma_p = \{v_1,v_2\} Σ p = { v 1 , v 2 }
Corollary 5.4. Let ρ : G F → GL 2 ( E ) \rho:G_F \to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E ) v ∈ Σ p v \in\Sigma_p v ∈ Σ p ρ v \rho_v ρ v ρ v s \rho_{v_s} ρ v s v s ∈ Σ p v_s \in\Sigma_p v s ∈ Σ p ρ \rho ρ
Proof. By Lemma 5.1, ρ v s \rho_{v_s} ρ v s v v v p p p ρ v \rho_v ρ v Σ o = { v s } \Sigma_o = \{v_s\} Σ o = { v s } ρ v \rho_v ρ v ρ \rho ρ p p p ρ w \rho_w ρ w w ∈ Σ p w \in\Sigma_p w ∈ Σ p ρ \rho ρ [22 Theorem 5.1.2] , which allows ρ v \rho_v ρ v ρ \rho ρ
Theorem 5.5. Let ρ : G F → GL 2 ( E ) \rho:G_F \to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E )
(1) for v ∈ Σ p v \in\Sigma_p v ∈ Σ p ρ v \rho_v ρ v v c v_c v c k − 1 ≥ 2 k-1 \ge2 k − 1 ≥ 2 v k v_k v k
(2) ρ \rho ρ W / F W/F W / F π \pi π GL 2 ( A W ) \operatorname{GL}_2(\mathbb{A}_W) GL 2 ( A W ) ι : Q p ≃ C \iota:\mathbb{Q}_p \simeq\mathbb{C} ι : Q p ≃ C ρ ∣ G W ≃ r π , ι \rho|_{G_W} \simeq r_{\pi,\iota} ρ ∣ G W ≃ r π , ι
(3) ρ v s \rho_{v_s} ρ v s v s ∈ Σ p v_s \in\Sigma_p v s ∈ Σ p
Then:
(a) There is a continuous representation ρ ′ : G F → GL 2 ( E ′ ) \rho':G_F \to\operatorname{GL}_2(E') ρ ′ : G F → GL 2 ( E ′ ) E ′ / Q p E'/\mathbb{Q}_p E ′ / Q p ρ \rho ρ ρ ′ \rho' ρ ′ λ 0 , λ ′ \lambda_0,\lambda' λ 0 , λ ′ M M M w w w F F F D p s t D_{\mathrm{pst}} D pst w ∣ p w \mid p w ∣ p M M M M → Q ‾ p M \to\overline{\mathbb{Q}}_p M → Q p λ 0 \lambda_0 λ 0 λ ′ \lambda' λ ′ ρ w \rho_w ρ w ρ w ′ \rho'_w ρ w ′ ρ ′ \rho' ρ ′ v ∈ Σ p v \in\Sigma_p v ∈ Σ p ρ v 1 \rho_{v_1} ρ v 1 ρ v 2 ′ \rho'_{v_2} ρ v 2 ′ ρ v 2 \rho_{v_2} ρ v 2 ρ v 1 \rho_{v_1} ρ v 1 ρ \rho ρ ρ ′ \rho' ρ ′
(b) If v s = v c v_s = v_c v s = v c ρ \rho ρ
(c) If v s = v k v_s = v_k v s = v k ρ ′ \rho' ρ ′ ρ ′ \rho' ρ ′ ρ ′ \rho' ρ ′ p = 3 p=3 p = 3 ρ ′ ∣ G F ( ζ 3 ) \rho'|_{G_{F(\zeta_3)}} ρ ′ ∣ G F ( ζ 3 ) ρ \rho ρ
Note that k k k det ρ \det\rho det ρ F F F ε \varepsilon ε [39 Chapter III] , and the sums of the two Hodge–Tate numbers at v c v_c v c v k v_k v k k − 1 k-1 k − 1
Strictly speaking, the isomorphisms in (a) are not isomorphisms of Q ‾ p \overline{\mathbb{Q}}_p Q p ρ \rho ρ ρ ′ \rho' ρ ′ ρ \rho ρ ρ ′ \rho' ρ ′
Proof. (a) We use the descent construction of [10 ] and of the proof of [2 Theorem 5.5.1] , starting from the potential modularity given by (2); we do not use the residual hypotheses in the statement of [2 Theorem 5.5.1] . Solvable descent gives, for the intermediate fields W ⊃ W i ⊃ F W \supset W_i \supset F W ⊃ W i ⊃ F W / W i W/W_i W / W i p p p ρ ∣ G W i \rho|_{G_{W_i}} ρ ∣ G W i ( ρ λ ) λ (\rho_\lambda)_\lambda ( ρ λ ) λ G F G_F G F M M M λ \lambda λ M M M ρ ⊗ Q ‾ p = ρ λ 0 ⊗ Q ‾ p \rho\otimes\overline{\mathbb{Q}}_p = \rho_{\lambda_0} \otimes\overline{\mathbb{Q}}_p ρ ⊗ Q p = ρ λ 0 ⊗ Q p λ 0 ∣ p \lambda_0 \mid p λ 0 ∣ p p p p [25 ] , [26 Theorem 1] . As in [2 §5.1] , the Hodge–Tate numbers of the members of the system are given by multisets H τ H_\tau H τ τ : F → M ‾ \tau:F \to\overline{M} τ : F → M Gal ( M ‾ / M ) \operatorname{Gal}(\overline{M}/M) Gal ( M / M ) λ ∣ ℓ \lambda\mid\ell λ ∣ ℓ v ∣ ℓ v \mid\ell v ∣ ℓ ρ λ ∣ G F v \rho_\lambda|_{G_{F_v}} ρ λ ∣ G F v H τ H_\tau H τ τ \tau τ F → F v → M ‾ λ F \to F_v \to\overline{M}_\lambda F → F v → M λ M ‾ \overline{M} M H τ 1 ≠ H τ 2 H_{\tau_1} \ne H_{\tau_2} H τ 1 = H τ 2 τ 1 , τ 2 \tau_1,\tau_2 τ 1 , τ 2 F F F Gal ( M ‾ / M ) \operatorname{Gal}(\overline{M}/M) Gal ( M / M ) M M M τ 1 ( F ) \tau_1(F) τ 1 ( F ) F F F τ 1 ( F ) ⊂ M \tau_1(F) \subset M τ 1 ( F ) ⊂ M λ ∣ p \lambda\mid p λ ∣ p M M M u ( λ ) u(\lambda) u ( λ ) F F F λ \lambda λ ρ λ ∣ G F u ( λ ) \rho_\lambda|_{G_{F_{u(\lambda)}}} ρ λ ∣ G F u ( λ ) H τ 1 H_{\tau_1} H τ 1 ρ λ \rho_\lambda ρ λ H τ 2 H_{\tau_2} H τ 2 p p p p p p F F F λ ′ ∣ p \lambda' \mid p λ ′ ∣ p M M M u ( λ ′ ) ≠ u ( λ 0 ) u(\lambda') \ne u(\lambda_0) u ( λ ′ ) = u ( λ 0 ) ρ ′ = ρ λ ′ \rho'=\rho_{\lambda'} ρ ′ = ρ λ ′ p p p ρ \rho ρ v 1 v_1 v 1 v 2 v_2 v 2 ρ \rho ρ ρ ′ ∣ G W ≃ r π ′ , ι ′ \rho'|_{G_W} \simeq r_{\pi',\iota'} ρ ′ ∣ G W ≃ r π ′ , ι ′ ι ′ : Q ‾ p ≃ C \iota':\overline{\mathbb{Q}}_p \simeq\mathbb{C} ι ′ : Q p ≃ C ρ ′ \rho' ρ ′ W W W ρ ′ ⊗ Q ‾ p ≃ r π ′ , ι ′ ′ \rho' \otimes\overline{\mathbb{Q}}_p \simeq r_{\pi',\iota''} ρ ′ ⊗ Q p ≃ r π ′ , ι ′′ π ′ \pi' π ′ GL 2 ( A F ) \operatorname{GL}_2(\mathbb{A}_F) GL 2 ( A F ) ι ′ ′ : Q ‾ p ≃ C \iota'':\overline{\mathbb{Q}}_p \simeq\mathbb{C} ι ′′ : Q p ≃ C ι : Q ‾ p ≃ C \iota:\overline{\mathbb{Q}}_p \simeq\mathbb{C} ι : Q p ≃ C ι ∘ j 0 = ι ′ ′ ∘ j ′ \iota\circ j_0=\iota''\circ j' ι ∘ j 0 = ι ′′ ∘ j ′ M M M j 0 , j ′ : M → Q ‾ p j_0,j':M \to\overline{\mathbb{Q}}_p j 0 , j ′ : M → Q p λ 0 \lambda_0 λ 0 λ ′ \lambda' λ ′ ρ \rho ρ r π ′ , ι r_{\pi',\iota} r π ′ , ι ρ ⊗ Q ‾ p ≃ r π ′ , ι \rho\otimes\overline{\mathbb{Q}}_p \simeq r_{\pi',\iota} ρ ⊗ Q p ≃ r π ′ , ι ρ \rho ρ
(b) The proof of Corollary 5.4 applies verbatim with v s = v c v_s=v_c v s = v c ρ v s \rho_{v_s} ρ v s
(c) By Lemma 5.1 , ρ v k \rho_{v_k} ρ v k ρ ′ \rho' ρ ′ ρ ′ \rho' ρ ′ ρ v k \rho_{v_k} ρ v k ρ v c \rho_{v_c} ρ v c Lemma 5.1 , ρ ‾ v k \overline{\rho}_{v_k} ρ v k ρ v c ′ \rho'_{v_c} ρ v c ′ ρ ‾ ′ \overline{\rho}' ρ ′ ρ ′ \rho' ρ ′ ρ ′ \rho' ρ ′ Σ o = { v k } \Sigma_o=\{v_k\} Σ o = { v k } ρ v c ′ \rho'_{v_c} ρ v c ′ [22 Theorem 5.1.2] if ρ v c ′ \rho'_{v_c} ρ v c ′ ρ ‾ ′ \overline{\rho}' ρ ′ ρ ‾ ′ ∣ G F ( ζ p ) \overline{\rho}'|_{G_{F(\zeta_p)}} ρ ′ ∣ G F ( ζ p ) p = 3 p=3 p = 3 p > 3 p>3 p > 3 p p p F F F Gal ( F ( ζ p ) / F ) ≃ ( Z / p Z ) × \operatorname{Gal}(F(\zeta_p)/F) \simeq(\mathbb{Z}/p\mathbb{Z})^\times Gal ( F ( ζ p ) / F ) ≃ ( Z / p Z ) × K = F ( p ∗ ) K=F(\sqrt{p^*}) K = F ( p ∗ ) p ∗ = ( − 1 ) ( p − 1 ) / 2 p p^*=(-1)^{(p-1)/2}p p ∗ = ( − 1 ) ( p − 1 ) /2 p [12 Lemma 1.13] , written over Q \mathbb{Q} Q F F F ρ ‾ ′ ∣ G F ( ζ p ) \overline{\rho}'|_{G_{F(\zeta_p)}} ρ ′ ∣ G F ( ζ p ) ρ ‾ ′ \overline{\rho}' ρ ′ G K G_K G K K / F ′ K/F' K / F ′ v k v_k v k F v k = Q p F_{v_k}=\mathbb{Q}_p F v k = Q p ρ ‾ ′ ∣ G F v k \overline{\rho}'|_{G_{F_{v_k}}} ρ ′ ∣ G F v k G Q p ( p ∗ ) G_{\mathbb{Q}_p(\sqrt{p^*})} G Q p ( p ∗ ) [12 Lemma 1.14] , which follows [27 Proposition 2.2] , only uses this local information at p p p ρ ‾ ′ ∣ G F v k \overline{\rho}'|_{G_{F_{v_k}}} ρ ′ ∣ G F v k k 0 k_0 k 0 p = 2 k 0 − 1 p=2k_0-1 p = 2 k 0 − 1 p = 2 k 0 − 3 p=2k_0-3 p = 2 k 0 − 3 ρ ′ \rho' ρ ′ ρ v k ′ \rho'_{v_k} ρ v k ′ ρ v k ′ \rho'_{v_k} ρ v k ′ k 0 ∈ { 2 , p + 1 } k_0 \in\{2,p+1\} k 0 ∈ { 2 , p + 1 } p p p v k v_k v k k 0 ≠ p + 1 k_0 \ne p+1 k 0 = p + 1 k 0 = 2 k_0=2 k 0 = 2 p = 3 p=3 p = 3 ρ ′ \rho' ρ ′ [18 ] , which hold over totally real fields in which p p p p p p [16 ] and Tung [30 ] to all odd primes. □
Remark 5.6. Theorem 5.5 may look somewhat technical, but it applies to representations of weight ( 2 , k ) (2,k) ( 2 , k ) p p p ρ ′ \rho' ρ ′ ρ \rho ρ ρ \rho ρ ρ \rho ρ p p p ρ \rho ρ ( ρ , ρ ′ ) (\rho,\rho') ( ρ , ρ ′ ) entangled pair . The two members of the pair are members of the same compatible system, the only difference being the place of the coefficient field above p p p p p p p > 3 p > 3 p > 3 ρ \rho ρ ρ \rho ρ [22 Theorem 5.1.2] , or to a residually irreducible case, where modularity lifting theorems covering all cases, including the mixed ones, are available [18 , 16 , 30 ] . Theorem 5.5 is therefore a modularity lifting theorem of Wiles type for a pair of entangled representations. Finally, this type of result is meant to be applied in inductive arguments, where modularity is proved by induction on the size of invariants such as the weight and the level, and is propagated through safe chains of congruences. In such arguments potential modularity is available, and the entangled representation ρ ′ \rho' ρ ′ ρ \rho ρ ρ \rho ρ p p p ρ ˉ ′ \bar{\rho}' ρ ˉ ′
Abelian varieties of GL2-type
In this section p p p F F F A / F A/F A / F g g g K ⊂ End F ( A ) ⊗ Q K \subset\operatorname{End}_F(A) \otimes\mathbb{Q} K ⊂ End F ( A ) ⊗ Q [ K : Q ] = g [K:\mathbb{Q}]=g [ K : Q ] = g A A A F F F O K ⊂ End F ( A ) \mathcal{O}_K \subset\operatorname{End}_F(A) O K ⊂ End F ( A ) λ \lambda λ K K K p p p T λ A T_\lambda A T λ A λ \lambda λ V λ A = T λ A ⊗ O K λ K λ V_\lambda A=T_\lambda A\otimes_{\mathcal{O}_{K_\lambda}}K_\lambda V λ A = T λ A ⊗ O K λ K λ
Lemma 6.1. (1) V λ A V_\lambda A V λ A K λ K_\lambda K λ ρ A , λ : G F → G L 2 ( K λ ) \rho_{A,\lambda}:G_F\to\mathrm{GL}_2(K_\lambda) ρ A , λ : G F → GL 2 ( K λ )
(2) ρ A , λ \rho_{A,\lambda} ρ A , λ p p p v ∣ p v\mid p v ∣ p − 1 -1 − 1
(3) ρ A , λ \rho_{A,\lambda} ρ A , λ
Proof. (1) is due to Ribet [37 , 43 ] . In (2), unramifiedness is the criterion of Néron–Ogg–Shafarevich [40 ] , semisimplicity is due to Faltings [33 ] , and the de Rham property, with the Hodge–Tate decomposition of V p A V_pA V p A [34 ] . For the Hodge–Tate numbers of each coefficient factor, decompose H 1 ( A Q ‾ ( C ) , C ) = ⨁ τ H τ H_1(A_{\overline{\mathbb{Q}}}(\mathbb{C}),\mathbb{C})=\bigoplus_\tau H_\tau H 1 ( A Q ( C ) , C ) = ⨁ τ H τ τ : K → C \tau:K\to\mathbb{C} τ : K → C H τ H_\tau H τ K K K H − 1 , 0 H^{-1,0} H − 1 , 0 H 0 , − 1 H^{0,-1} H 0 , − 1 H τ H_\tau H τ c c c σ \sigma σ F F F H 1 ( A σ ( C ) , Q ) H_1(A_{\sigma}(\mathbb{C}),\mathbb{Q}) H 1 ( A σ ( C ) , Q ) K K K K K K ± 1 \pm1 ± 1 Q \mathbb{Q} Q g g g c c c H − 1 , 0 H^{-1,0} H − 1 , 0 H 0 , − 1 H^{0,-1} H 0 , − 1 K K K det ρ A , λ ( c ) = − 1 \det\rho_{A,\lambda}(c)=-1 det ρ A , λ ( c ) = − 1 H 1 ( A σ ( C ) , Q ) ⊗ K K λ ≃ V λ A H_1(A_{\sigma}(\mathbb{C}),\mathbb{Q})\otimes_K K_\lambda\simeq V_\lambda A H 1 ( A σ ( C ) , Q ) ⊗ K K λ ≃ V λ A
:::
Lemma 6.2. ρ A , λ \rho_{A,\lambda} ρ A , λ
Proof. This is well known (see [37 ] ); we recall the argument. Suppose that ρ A , λ ⊗ K λ ‾ \rho_{A,\lambda}\otimes\overline{K_\lambda} ρ A , λ ⊗ K λ G F G_F G F η \eta η η \eta η p p p F F F [39 Chapter III] . Since F F F η = ε n η 0 \eta= \varepsilon^n\eta_0 η = ε n η 0 n ∈ Z n \in\mathbb{Z} n ∈ Z η 0 \eta_0 η 0 w ∤ p w \nmid p w ∤ p η ( Frob w ) \eta(\operatorname{Frob}_w) η ( Frob w ) N ( w ) n N(w)^n N ( w ) n Frob w \operatorname{Frob}_w Frob w V λ A V_\lambda A V λ A N ( w ) 1 / 2 N(w)^{1/2} N ( w ) 1/2
The CM case
Lemma 6.3. Assume that A Q ‾ A_{\overline{\mathbb{Q}}} A Q ρ A , λ \rho_{A,\lambda} ρ A , λ
Proof. This is classical. By the theory of complex multiplication [41 ] , ρ A , λ \rho_{A,\lambda} ρ A , λ F F F ρ A , λ \rho_{A,\lambda} ρ A , λ G F G_F G F ρ A , λ ⊗ K ‾ λ ≃ Ind G M G F θ \rho_{A,\lambda}\otimes\overline{K}_\lambda\simeq\operatorname{Ind}_{G_M}^{G_F}\theta ρ A , λ ⊗ K λ ≃ Ind G M G F θ M / F M/F M / F θ ≠ θ c \theta\ne\theta^c θ = θ c G M G_M G M θ \theta θ M M M [39 Chapter III] , which is pure of weight 1 by Weil’s bounds for A A A M M M a a a 2 a = 1 2a = 1 2 a = 1 M M M θ \theta θ θ ≠ θ c \theta\ne\theta^c θ = θ c F F F ρ A , λ \rho_{A,\lambda} ρ A , λ [45 , 1 ] .
Local indecomposability
Lemma 6.4. Assume that A Q ‾ A_{\overline{\mathbb{Q}}} A Q v ∣ p v \mid p v ∣ p ρ v = ρ A , λ ∣ G F v \rho_v = \rho_{A,\lambda}|_{G_{F_v}} ρ v = ρ A , λ ∣ G F v ρ v \rho_v ρ v
Proof. It suffices to show that ρ A , λ ∣ G k w \rho_{A,\lambda}|_{G_{k_w}} ρ A , λ ∣ G k w k w k_w k w F v F_v F v [38 Exp. IX] , there is a finite extension k / F k/F k / F w ∣ v w \mid v w ∣ v k k k A k A_k A k w w w k = F ( A [ N ] ) k = F(A[N]) k = F ( A [ N ]) N ≥ 3 N \ge3 N ≥ 3 p p p t t t A k A_k A k w w w Z \mathbb{Z} Z t t t O K \mathcal{O}_K O K t t t [ K : Q ] = g [K : \mathbb{Q}] = g [ K : Q ] = g t ∈ { 0 , g } t \in\{0,g\} t ∈ { 0 , g }
If t = g t = g t = g [38 Exp. IX] , ρ A , λ ∣ G k w \rho_{A,\lambda}|_{G_{k_w}} ρ A , λ ∣ G k w ρ A , λ ∣ G k w \rho_{A,\lambda}|_{G_{k_w}} ρ A , λ ∣ G k w
If t = 0 t = 0 t = 0 A k A_k A k w w w ρ A , λ ∣ G k w \rho_{A,\lambda}|_{G_{k_w}} ρ A , λ ∣ G k w ρ v \rho_v ρ v 0 0 0 − 1 -1 − 1 K λ K_\lambda K λ K λ K_\lambda K λ G F v = G Q p G_{F_v} = G_{\mathbb{Q}_p} G F v = G Q p k w k_w k w ε \varepsilon ε V λ A V_\lambda A V λ A V p A V_p A V p A λ \lambda λ 0 0 0 1 1 1
We now pass to real multiplication, keeping track of this factor. Since A Q ‾ A_{\overline{\mathbb{Q}}} A Q A A A F F F [42 Propositions 1.1 and 1.4] A Q ‾ A_{\overline{\mathbb{Q}}} A Q B e B^e B e B B B End 0 ( B ) \operatorname{End}^0(B) End 0 ( B ) L L L dim B \dim B dim B k k k B B B L L L k k k O L \mathcal{O}_L O L B B B [42 Theorem 5.6] , comparing the actions of K K K L L L End 0 ( A k ) \operatorname{End}^0(A_k) End 0 ( A k ) E ′ E' E ′ V λ A ⊗ K λ E ′ ≃ V μ B ⊗ L μ E ′ V_\lambda A \otimes_{K_\lambda} E' \simeq V_\mu B \otimes_{L_\mu} E' V λ A ⊗ K λ E ′ ≃ V μ B ⊗ L μ E ′ G k w G_{k_w} G k w μ ∣ p \mu\mid p μ ∣ p μ \mu μ B B B w w w [42 Theorem 5.2] shows that V μ B V_\mu B V μ B w w w ρ A , λ ∣ G k w \rho_{A,\lambda}|_{G_{k_w}} ρ A , λ ∣ G k w ρ v \rho_v ρ v
Remark 6.5. For elliptic curves Lemma 6.4 is classical: if E E E ρ E , p ∣ G k w \rho_{E,p}|_{G_{k_w}} ρ E , p ∣ G k w E E E [36 ] when p p p k k k K K K [42 ] in general. Zhao’s theorem is a statement about abelian varieties and does not assume modularity.
Proof of Theorem B
Write ρ = ρ A , λ \rho=\rho_{A,\lambda} ρ = ρ A , λ E = K λ E=K_\lambda E = K λ A Q ‾ A_{\overline{\mathbb{Q}}} A Q ρ \rho ρ Lemma 6.3 . Assume it does not. By Lemmas 6.1 and 6.2 , ρ \rho ρ v ∣ p v\mid p v ∣ p § 1.2 , let Σ o \Sigma_o Σ o v ∣ p v\mid p v ∣ p ρ v \rho_v ρ v Σ n \Sigma_n Σ n ρ v \rho_v ρ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o Lemma 6.4 . By Theorem A, ρ \rho ρ − 1 -1 − 1 v ∣ p v\mid p v ∣ p
Potentially modular representations of parallel weight 2
Theorem B′. Let ρ : G F → GL 2 ( E ) \rho:G_F\to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E )
for every v ∣ p v\mid p v ∣ p ρ v \rho_v ρ v − 1 -1 − 1
ρ \rho ρ
ρ \rho ρ W / F W/F W / F π \pi π GL 2 ( A W ) \operatorname{GL}_2(\mathbb{A}_W) GL 2 ( A W ) ι : Q ‾ p ≃ C \iota:\overline{\mathbb{Q}}_p\simeq\mathbb{C} ι : Q p ≃ C ρ ∣ G W ≃ r π , ι \rho|_{G_W}\simeq r_{\pi,\iota} ρ ∣ G W ≃ r π , ι
either [ F : Q ] [F:\mathbb{Q}] [ F : Q ] v 0 v_0 v 0 F F F ρ ∣ G F v 0 \rho|_{G_{F_{v_0}}} ρ ∣ G F v 0
Then ρ \rho ρ
In (4), if v 0 ∣ p v_0\mid p v 0 ∣ p D p s t ( ρ v 0 ) D_{\mathrm{pst}}(\rho_{v_0}) D pst ( ρ v 0 ) ρ ∣ G F v \rho|_{G_{F_v}} ρ ∣ G F v v v v F F F
Proof. By Theorem B , it suffices to show that ρ ≃ ρ A , λ \rho\simeq\rho_{A,\lambda} ρ ≃ ρ A , λ A / F A/F A / F GL 2 \operatorname{GL}_2 GL 2 λ ∣ p \lambda\mid p λ ∣ p W W W ρ ∣ G W ≃ r π , ι \rho|_{G_W}\simeq r_{\pi,\iota} ρ ∣ G W ≃ r π , ι ρ \rho ρ π \pi π
Assume first that π \pi π ρ ∣ G W \rho|_{G_W} ρ ∣ G W M W M_W M W W W W ρ ( G F ) \rho(G_F) ρ ( G F ) ρ \rho ρ G F G_F G F ρ ⊗ E ‾ ≃ Ind G M G F θ \rho\otimes\overline{E}\simeq\operatorname{Ind}_{G_M}^{G_F}\theta ρ ⊗ E ≃ Ind G M G F θ M / F M/F M / F θ ≠ θ c \theta\ne\theta^c θ = θ c G M G_M G M G W G_W G W M W = M M_W=M M W = M M M M M W M_W M W W W W M M M θ \theta θ M M M [39 Chapter III] , and ρ \rho ρ Lemma 6.3 .
Assume now that π \pi π Theorem 5.5 (a), which does not require F F F [44 §9.2] ), shows that ρ \rho ρ ( ρ ℓ ) ℓ (\rho_\ell)_\ell ( ρ ℓ ) ℓ ρ ℓ : G F → GL 2 ( Q ‾ ℓ ) \rho_\ell:G_F\to\operatorname{GL}_2(\overline{\mathbb{Q}}_\ell) ρ ℓ : G F → GL 2 ( Q ℓ ) ρ p = ρ ⊗ Q ‾ p \rho_p=\rho\otimes\overline{\mathbb{Q}}_p ρ p = ρ ⊗ Q p ρ ℓ ∣ G W ≃ r π , ℓ \rho_\ell|_{G_W}\simeq r_{\pi,\ell} ρ ℓ ∣ G W ≃ r π , ℓ ι ℓ : Q ‾ ℓ ≃ C \iota_\ell:\overline{\mathbb{Q}}_\ell\simeq\mathbb{C} ι ℓ : Q ℓ ≃ C v v v F F F ρ ℓ ∣ G F v \rho_\ell|_{G_{F_v}} ρ ℓ ∣ G F v D p s t D_{\mathrm{pst}} D pst v ∣ ℓ v\mid\ell v ∣ ℓ ℓ \ell ℓ ρ ℓ \rho_{\ell} ρ ℓ [44 §1.2] , since its restriction to G W G_{W} G W r π , ι ℓ r_{\pi,\iota_{\ell}} r π , ι ℓ π \pi π π \pi π ℓ \ell ℓ r ‾ π , ι ℓ \overline{r}_{\pi,\iota_{\ell}} r π , ι ℓ SL 2 ( F ℓ ) \operatorname{SL}_{2}(\mathbb{F}_{\ell}) SL 2 ( F ℓ ) [13 ] . For ℓ ≥ 5 \ell\ge5 ℓ ≥ 5 SL 2 ( F ℓ ) \operatorname{SL}_{2}(\mathbb{F}_{\ell}) SL 2 ( F ℓ ) G W ( ζ ℓ ) G_{W(\zeta_{\ell})} G W ( ζ ℓ ) W ( ζ ℓ ) / W W(\zeta_{\ell})/W W ( ζ ℓ ) / W ρ ‾ ℓ ∣ G W ( ζ ℓ ) \overline{\rho}_{\ell}|_{G_{W(\zeta_{\ell})}} ρ ℓ ∣ G W ( ζ ℓ ) ρ ‾ ℓ ∣ G F ( ζ ℓ ) \overline{\rho}_{\ell}|_{G_{F(\zeta_{\ell})}} ρ ℓ ∣ G F ( ζ ℓ )
Choose such a prime ℓ ∉ { 2 , 5 , p } \ell\notin\{2,5,p\} ℓ ∈ / { 2 , 5 , p } v 0 v_{0} v 0 ρ ℓ \rho_{\ell} ρ ℓ [44 Corollary 1.1.2(4)] : either [ F : Q ] [F:\mathbb{Q}] [ F : Q ] v 0 ∤ ℓ v_{0} \nmid\ell v 0 ∤ ℓ ρ ℓ ∣ G F v 0 \rho_{\ell}|_{G_{F_{v_{0}}}} ρ ℓ ∣ G F v 0 ρ ∣ G F v 0 \rho|_{G_{F_{v_{0}}}} ρ ∣ G F v 0 ρ ℓ \rho_{\ell} ρ ℓ λ ℓ \lambda_{\ell} λ ℓ A / F A/F A / F GL 2 \operatorname{GL}_{2} GL 2 λ ℓ ∣ ℓ \lambda_{\ell} \mid\ell λ ℓ ∣ ℓ K K K λ \lambda λ A A A λ \lambda λ K K K ρ \rho ρ λ ℓ \lambda_{\ell} λ ℓ p p p ρ \rho ρ λ ∣ p \lambda\mid p λ ∣ p K K K ρ \rho ρ ρ A , λ \rho_{A,\lambda} ρ A , λ ρ \rho ρ ρ ⊗ Q ‾ p ≃ ρ A , λ ⊗ Q ‾ p \rho\otimes\overline{\mathbb{Q}}_{p} \simeq\rho_{A,\lambda} \otimes\overline{\mathbb{Q}}_{p} ρ ⊗ Q p ≃ ρ A , λ ⊗ Q p
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585. 08007 Barcelona
Email address: 1dieulefait@ub.edu
