Introduction
Setting
1.1. Setting. Let F F F Q \mathbb{Q} Q 2 2 2 F v = Q 2 F_v=\mathbb{Q}_2 F v = Q 2 v ∣ 2 v\mid2 v ∣ 2 E / Q 2 E/\mathbb{Q}_2 E / Q 2 O \mathcal{O} O ϖ \varpi ϖ F \mathbb{F} F ρ : G F → G L 2 ( E ) \rho:G_F\to\mathrm{GL}_2(E) ρ : G F → GL 2 ( E ) Σ 2 \Sigma_2 Σ 2 2 2 2 Σ 2 = Σ n ⊔ Σ o \Sigma_2=\Sigma_n\sqcup\Sigma_o Σ 2 = Σ n ⊔ Σ o
(N) for v ∈ Σ n v\in\Sigma_n v ∈ Σ n ρ v = ρ ∣ G F v \rho_v=\rho|_{G_{F_v}} ρ v = ρ ∣ G F 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 = 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 χ ~ \widetilde{\chi} χ 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 ∣ 2 v\mid2 v ∣ 2
Since 2 2 2 χ ˉ ∣ G F v = χ ~ ∣ G Q 2 \bar{\chi}|_{G_{F_v}}=\widetilde{\chi}|_{G_{\mathbb{Q}_2}} χ ˉ ∣ G F v = χ ∣ G Q 2 v ∣ 2 v\mid2 v ∣ 2 χ ~ \widetilde{\chi} χ χ ~ ( Frob 2 ) ≠ 1 \widetilde{\chi}(\operatorname{Frob}_2)\ne1 χ ( Frob 2 ) = 1 2 2 2 χ ˉ ∣ G F v ≠ 1 \bar{\chi}|_{G_{F_v}}\ne1 χ ˉ ∣ G F v = 1 [15 ] . Under (O), the Jordan–Hölder character of ρ v \rho_v ρ v ρ v \rho_v ρ v classical character ψ v c l \psi_v^{\mathrm{cl}} ψ v cl
For odd p p p Σ o = ∅ \Sigma_o=\varnothing Σ o = ∅ [26 ] and Zhang [38 ] , the case Σ n = ∅ \Sigma_n=\varnothing Σ n = ∅ [26 Theorem 5.1.2] , and the mixed case is treated in [Die26a, Die26b]. Over Q \mathbb{Q} Q 2 2 2 2 2 2 p = 2 p=2 p = 2 F F F
The pro-modularity input.
Pseudo-representations are understood as determinants in the sense of Chenevier [11 ] . Let Σ \Sigma Σ F F F 2 2 2 ρ \rho ρ χ = det ρ \chi= \det\rho χ = det ρ K ⊃ F K \supset F K ⊃ F R K p s R^{\mathrm{ps}}_K R K ps ρ ˉ s s ∣ G K \bar{\rho}^{\mathrm{ss}}|_{G_K} ρ ˉ ss ∣ G K G K , Σ K G_{K,\Sigma_K} G K , Σ K χ ∣ G K \chi|_{G_K} χ ∣ G K Σ K \Sigma_K Σ K Σ \Sigma Σ K K K T K ( U p ) \mathbb{T}_K(U^p) T K ( U p ) 2 2 2 K K K U p U^p U p χ ε \chi\varepsilon χ ε ρ ˉ s s \bar{\rho}^{\mathrm{ss}} ρ ˉ ss R K p s R^{\mathrm{ps}}_K R K ps
Hypothesis 1.1 (PM). There are totally real fields F ⊂ K ⊂ F 1 F \subset K \subset F_1 F ⊂ K ⊂ F 1 2 2 2 K / F K/F K / F F 1 / Q F_1/\mathbb{Q} F 1 / Q C C C Spec R K p s \operatorname{Spec} R^{\mathrm{ps}}_K Spec R K ps ρ ∣ G K \rho|_{G_K} ρ ∣ G K C C C Spec R F 1 p s \operatorname{Spec} R^{\mathrm{ps}}_{F_1} Spec R F 1 ps G F 1 G_{F_1} G F 1 Spec T F 1 ( U p ) \operatorname{Spec} \mathbb{T}_{F_1}(U^p) Spec T F 1 ( U p ) U p U^p U p
For odd p p p [38 Theorem 5.5.1] gives the stronger statement with K = F 1 K = F_1 K = F 1 p ≥ 5 p \ge5 p ≥ 5 p p p p = 2 p = 2 p = 2 G Q G_{\mathbb{Q}} G Q § 8 we prove:
Theorem 1.2 (Theorem 8.4). Assume (R), that ρ \rho ρ 2 2 2 det ρ = ε w χ 0 \det\rho= \varepsilon^w\chi_0 det ρ = ε w χ 0 χ 0 \chi_0 χ 0
The last condition holds under (N) and (O), by §2.2; it is needed because the propagation theorem and the Hecke algebras of [OAI] use a determinant of this form.
The proof follows [OAI, §§6–7]. The base field Q \mathbb{Q} Q F F F χ ˉ ∣ G Q 2 ≠ 1 \bar{\chi}|_{G_{\mathbb{Q}_2}} \ne1 χ ˉ ∣ G Q 2 = 1 F F F Q \mathbb{Q} Q 2 2 2 2 2 2
Main results.
Theorem A. Assume (N), (O) and (R), and that ρ v \rho_v ρ v v ∈ Σ 0 v \in\Sigma_0 v ∈ Σ 0 ρ \rho ρ
This includes the case Σ 0 = ∅ \Sigma_0 = \varnothing Σ 0 = ∅ 2 2 2 Σ n = ∅ \Sigma_n = \varnothing Σ n = ∅ ρ \rho ρ
(St) there is a finite place v 0 ∤ 2 v_0 \nmid2 v 0 ∤ 2 F F F ρ \rho ρ ρ ∣ G F v 0 \rho|_{G_{F_{v_0}}} ρ ∣ G F v 0
For odd p p p [15 ] uses Pan’s finiteness theorem for fully ordinary pseudo-deformation rings [26 Theorem 5.1.2(1)] . For p = 2 p = 2 p = 2 § 8 and the structure of the generic blocks of GL 2 ( Q 2 ) \operatorname{GL}_2(\mathbb{Q}_2) GL 2 ( Q 2 ) K K K 2 2 2 R K r e d R^{\mathrm{red}}_K R K red R K p s R^{\mathrm{ps}}_K R K ps 2 2 2 Λ K = O [ [ ∏ v ∣ 2 ( 1 + 4 Z 2 ) ] ] \Lambda_K = \mathcal{O}[[\prod_{v|2}(1 + 4\mathbb{Z}_2)]] Λ K = O [[ ∏ v ∣2 ( 1 + 4 Z 2 )]] ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) R K r e d R^{\mathrm{red}}_K R K red Λ K \Lambda_K Λ K K K K
Theorem B. Assume (N), (O), (R) and (St). Then ρ \rho ρ
No condition is imposed on the splitness of ρ v \rho_v ρ v v ∈ Σ o v \in\Sigma_o v ∈ Σ o
Over real quadratic fields we can remove the condition χ ˉ ∣ G v ≠ 1 \bar{\chi}|_{G_v} \ne1 χ ˉ ∣ G v = 1 q ≡ 1 ( m o d 8 ) q \equiv1 \pmod{8} q ≡ 1 ( mod 8 ) F = Q ( q ) F = \mathbb{Q}(\sqrt{q}) F = Q ( q ) w q w_q w q F F F q q q w h + w^{h^+} w h + w ∣ 2 w \mid2 w ∣ 2 h + h^+ h + q = 17 q = 17 q = 17
Theorem C (Theorem 9.4). Let F = Q ( q ) F = \mathbb{Q}(\sqrt{q}) F = Q ( q ) 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 = 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 ρ v \rho_v ρ v v ∈ Σ o v \in\Sigma_o v ∈ Σ o χ ˉ ∣ G v ≠ 1 \bar{\chi}|_{G_v} \ne1 χ ˉ ∣ G v = 1 v ∣ 2 v \mid2 v ∣ 2 χ ˉ = 1 \bar{\chi} = 1 χ ˉ = 1 F F F χ ˉ ≠ 1 \bar{\chi} \ne1 χ ˉ = 1 ρ \rho ρ w q w_q w q χ ˉ ( Frob w q ) ≠ 1 \bar{\chi}(\operatorname{Frob}_{w_q}) \ne1 χ ˉ ( Frob w q ) = 1 ρ \rho ρ
For F = Q ( 17 ) F = \mathbb{Q}(\sqrt{17}) F = Q ( 17 ) χ ˉ ≠ 1 \bar{\chi} \ne1 χ ˉ = 1 ρ \rho ρ w 17 w_{17} w 17 χ ˉ ( Frob w 17 ) = 1 \bar{\chi}(\operatorname{Frob}_{w_{17}}) = 1 χ ˉ ( Frob w 17 ) = 1 ρ ˉ s s \bar{\rho}^{\mathrm{ss}} ρ ˉ ss ρ \rho ρ χ ˉ \bar{\chi} χ ˉ G Q G_{\mathbb{Q}} G Q F F F § 11 ).
Finally, when ρ ˉ \bar{\rho} ρ ˉ
Theorem D (Theorem 10.6). Let F F F ρ ˉ \bar{\rho} ρ ˉ ρ v \rho_v ρ v v ∈ Σ o v \in\Sigma_o v ∈ Σ o ρ \rho ρ
1.4. The method. Theorem A is proved as [Die26a, Theorem A]: (PM) places ρ \rho ρ Σ o \Sigma_o Σ o ρ v \rho_v ρ v ρ v \rho_v ρ v ({xref}Section 3`).
For Theorem B we follow [15 ] . Let C C C x x x ρ \rho ρ Spec R ∘ \operatorname{Spec} R^{\circ} Spec R ∘ Σ o \Sigma_o Σ o C C C Σ n \Sigma_n Σ n D ⊂ C D \subset C D ⊂ C 1 + d 1+d 1 + d y ∈ D y \in D y ∈ D T ∘ \mathbb{T}^{\circ} T ∘ C C C Spec T ∘ \operatorname{Spec}\mathbb{T}^{\circ} Spec T ∘ x x x p p p D D D p = 2 p = 2 p = 2
By Theorem 8.9, D D D Λ \Lambda Λ G F 1 G_{F_1} G F 1 F 1 F_1 F 1 some ordering of the characters at each place above 2.
By (St), the ratio of the Frobenius eigenvalues at v 0 v_0 v 0 N ( v 0 ) ± 1 N(v_0)^{\pm1} N ( v 0 ) ± 1 C C C D D D [35 Theorem B] , the corresponding Hilbert modular forms are of Steinberg type at v 0 v_0 v 0
Their regularity follows from the theorem of Newton and Thorne [22 Theorem B] , which holds for p = 2 p = 2 p = 2 v 0 v_0 v 0
Condition (St) is also what excludes CM points from D D D p p p p = 2 p = 2 p = 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.
Notation. We use the conventions of [Die26a]: HT ( ε ) = − 1 \operatorname{HT}(\varepsilon)=-1 HT ( ε ) = − 1 G Q 2 G_{\mathbb{Q}_2} G Q 2 Q 2 × \mathbb{Q}_2^{\times} Q 2 × G v = G F v G_v=G_{F_v} G v = G F v I v I_v I v ϕ v \phi_v ϕ v q v = N ( v ) q_v=N(v) q v = N ( v ) D D D R R R r \mathfrak{r} r R R R ρ ( r ) \rho(\mathfrak{r}) ρ ( r ) k ( r ) k(\mathfrak{r}) k ( r ) D m o d r D\bmod\mathfrak{r} D mod r [11 Theorem 2.12] ; for a point z z z ρ z \rho_z ρ z T T T irreducible if ρ z \rho_z ρ z h i ( V ) = dim H i ( V ) h^i(V)=\dim H^i(V) h i ( V ) = dim H i ( V ) d = [ F : Q ] d=[F:\mathbb{Q}] d = [ F : Q ] § 2.2 .
Preliminaries
Determinants and ordinary conditions
For p = 2 p=2 p = 2 [11 ] throughout. Let A A A k k k D D D G G G A A A det ( ψ ‾ 1 ⊕ ψ ‾ 2 ) \det(\overline{\psi}_1\oplus\overline{\psi}_2) det ( ψ 1 ⊕ ψ 2 ) ψ ‾ 1 ≠ ψ ‾ 2 \overline{\psi}_1\ne\overline{\psi}_2 ψ 1 = ψ 2 A [ G ] A[G] A [ G ] ( A B C A ) \begin{pmatrix} A & B \\ C & A \end{pmatrix} ( A C B A ) A / B C A/BC A / B C B C = 0 BC=0 B C = 0 ψ 1 , ψ 2 : G → A × \psi_1,\psi_2:G\to A^\times ψ 1 , ψ 2 : G → A × ψ ‾ 1 , ψ ‾ 2 \overline{\psi}_1,\overline{\psi}_2 ψ 1 , ψ 2 D = det ( ψ 1 ⊕ ψ 2 ) D=\det(\psi_1\oplus\psi_2) D = det ( ψ 1 ⊕ ψ 2 ) [26 Definition 5.3.2] , if D ∣ H = det ( ψ 1 ⊕ ψ 2 ) D|_H=\det(\psi_1\oplus\psi_2) D ∣ H = det ( ψ 1 ⊕ ψ 2 ) H ⊂ G H\subset G H ⊂ G D D D ψ 1 \psi_1 ψ 1 H H H
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 ∈ H , η ∈ G ) . 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 H,\ \eta\in G). 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 ∈ H , η ∈ G ) . The proof of [Die26a, Lemma 2.3] is characteristic free and gives: if ρ A : G → GL 2 ( A ) \rho_A:G\to\operatorname{GL}_2(A) ρ A : G → GL 2 ( A ) 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 ∈ H \sigma_0\in H σ 0 ∈ H det ρ A \det\rho_A det ρ A ψ 1 \psi_1 ψ 1 H H H ρ A ∣ H \rho_A|_H ρ A ∣ H ψ 1 \psi_1 ψ 1 det ρ A \det\rho_A det ρ A ψ 1 \psi_1 ψ 1 ψ 2 \psi_2 ψ 2 ρ A ∣ H ≃ ψ 1 ⊕ ψ 2 \rho_A|_H\simeq\psi_1\oplus\psi_2 ρ A ∣ H ≃ ψ 1 ⊕ ψ 2
Lemma 2.1. Let ψ 1 , ψ 2 : G Q 2 → E × \psi_1,\psi_2:G_{\mathbb{Q}_2}\to E^\times ψ 1 , ψ 2 : G Q 2 → E × HT ( ψ 1 ) ≠ HT ( ψ 2 ) \operatorname{HT}(\psi_1)\ne\operatorname{HT}(\psi_2) HT ( ψ 1 ) = HT ( ψ 2 ) HT ( ψ 1 ) < HT ( ψ 2 ) \operatorname{HT}(\psi_1)<\operatorname{HT}(\psi_2) HT ( ψ 1 ) < HT ( ψ 2 ) ψ 2 \psi_2 ψ 2 ψ 1 \psi_1 ψ 1 HT ( ψ 1 ) > HT ( ψ 2 ) \operatorname{HT}(\psi_1)>\operatorname{HT}(\psi_2) HT ( ψ 1 ) > HT ( ψ 2 ) ψ 2 \psi_2 ψ 2 ψ 1 \psi_1 ψ 1
Proof. This is [Die26a, Lemma 2.1], whose proof uses the Bloch–Kato dimension formulas [7 Corollary 3.8.4] , which hold for p = 2 p=2 p = 2 □ \square □
Lemma 2.2. Let v ∣ 2 v\mid2 v ∣ 2 χ ‾ ∣ G v ≠ 1 \overline{\chi}|_{G_v}\ne1 χ ∣ G v = 1 R v p s R_v^{\mathrm{ps}} R v ps ( χ ‾ 1 ⊕ χ ‾ 2 ) ∣ G v (\overline{\chi}_1\oplus\overline{\chi}_2)|_{G_v} ( χ 1 ⊕ χ 2 ) ∣ G v χ ∣ G v \chi|_{G_v} χ ∣ G v f v ∈ R v p s f_v\in R_v^{\mathrm{ps}} f v ∈ R v ps f v f_v f v
Proof. Since χ ‾ ∣ G v ≠ 1 \overline{\chi}|_{G_v}\ne1 χ ∣ G v = 1 G v G_v G v B B B C C C B C BC B C B B B C C C h 1 ( G Q 2 , χ ‾ ) h^1(G_{\mathbb{Q}_2},\overline{\chi}) h 1 ( G Q 2 , χ ) h 1 ( G Q 2 , χ ‾ − 1 ) ) h^1(G_{\mathbb{Q}_2},\overline{\chi}^{-1})) h 1 ( G Q 2 , χ − 1 )) h 1 ( G Q 2 , χ ‾ ± 1 ) = 1 + h 0 ( χ ‾ ± 1 ) + h 0 ( χ ‾ ∓ 1 ) = 1 h^1(G_{\mathbb{Q}_2},\overline{\chi}^{\pm1})=1+h^0(\overline{\chi}^{\pm1})+h^0(\overline{\chi}^{\mp1})=1 h 1 ( G Q 2 , χ ± 1 ) = 1 + h 0 ( χ ± 1 ) + h 0 ( χ ∓ 1 ) = 1 B C BC B C f v = b c f_v=bc f v = b c □ \square □
Lemma 2.3. Let G G G k k k α ≠ β \alpha\ne\beta α = β η ≠ 1 \eta\ne1 η = 1 G → k × G \to k^{\times} G → k × r = ( α 0 β β ) r = \begin{pmatrix} \alpha& 0 \\ \beta& \beta\end{pmatrix} r = ( α β 0 β ) H 0 ( G , ad 0 r ⊗ η ) ≠ 0 H^{0}(G,\operatorname{ad}^{0}r \otimes\eta) \ne0 H 0 ( G , ad 0 r ⊗ η ) = 0 α β − 1 = η − 1 \alpha\beta^{-1} = \eta^{-1} α β − 1 = η − 1
Proof. This is [15 Lemma 3.2] , whose proof is characteristic free. □
Lemma 2.4. Let K K K r : G K → GL 2 ( Q ‾ 2 ) r : G_{K} \to\operatorname{GL}_{2}(\overline{\mathbb{Q}}_{2}) r : G K → GL 2 ( Q 2 ) 2 2 2 K ′ / K K'/K K ′ / K r ∣ G K ′ r|_{G_{K'}} r ∣ G K ′
Proof. Let H H H r ( G K ) r(G_{K}) r ( G K ) H ∘ H^{\circ} H ∘ H ∘ H^{\circ} H ∘ H H H r r r H H H r r r H ∘ ⊃ SL 2 H^{\circ} \supset\operatorname{SL}_{2} H ∘ ⊃ SL 2 r ∣ G K ′ r|_{G_{K'}} r ∣ G K ′ H ∘ H^{\circ} H ∘
The base field and the reducible locus
If ρ \rho ρ θ \theta θ G K G_{K} G K K / F K/F K / F θ \theta θ K K K ρ \rho ρ ρ \rho ρ K ⊂ F 1 K \subset F_{1} K ⊂ F 1 ρ \rho ρ G K G_{K} G K G F 1 G_{F_{1}} G F 1 F 1 , w = F v = Q 2 F_{1,w} = F_{v} = \mathbb{Q}_{2} F 1 , w = F v = Q 2 w ∣ v ∣ 2 w \mid v \mid2 w ∣ v ∣ 2 ρ v \rho_{v} ρ v v ∈ Σ o v \in\Sigma_{o} v ∈ Σ o F 1 / F F_{1}/F F 1 / F K / F K/F K / F ρ \rho ρ ρ ∣ G F 1 \rho|_{G_{F_{1}}} ρ ∣ G F 1 ρ ∣ G K \rho|_{G_{K}} ρ ∣ G K [8 Lemma 2.2.2] , [1 ] . We replace F F F K K K F 1 F_{1} F 1 Σ n \Sigma_{n} Σ n Σ o \Sigma_{o} Σ o Σ \Sigma Σ v 0 v_{0} v 0 v 0 v_{0} v 0 d = [ F : Q ] d = [F : \mathbb{Q}] d = [ F : Q ]
Twisting ρ \rho ρ χ ‾ − 1 \overline{\chi}^{-1} χ − 1 ρ ‾ s s = 1 ⊕ χ ‾ \overline{\rho}^{\mathrm{ss}} = 1 \oplus\overline{\chi} ρ ss = 1 ⊕ χ δ F \delta_{F} δ F F F F 2 2 2 [36 ] , the 2 2 2 F F F δ F ≤ ( d − 1 ) / 2 \delta_{F} \le(d - 1)/2 δ F ≤ ( d − 1 ) /2 2 2 2 G F a b G_{F}^{\mathrm{ab}} G F ab Z 2 \mathbb{Z}_{2} Z 2 1 + δ F ≤ ( d + 1 ) / 2 1 + \delta_{F} \le(d + 1)/2 1 + δ F ≤ ( d + 1 ) /2 det ( ψ ⊕ χ ψ − 1 ) \det(\psi\oplus\chi\psi^{-1}) det ( ψ ⊕ χ ψ − 1 ) Spec R F p s \operatorname{Spec} R^{\mathrm{ps}}_{F} Spec R F ps ( d + 3 ) / 2 < 1 + d (d + 3)/2 < 1 + d ( d + 3 ) /2 < 1 + d R p s = R F p s R^{\mathrm{ps}} = R^{\mathrm{ps}}_{F} R ps = R F ps G F G_{F} G F ε k χ 0 \varepsilon^{k}\chi_{0} ε k χ 0 χ 0 \chi_{0} χ 0
Completed cohomology at 2 and ordinary parts
Completed cohomology
Let B B B F F F ψ = χ ε \psi= \chi\varepsilon ψ = χ ε U p = ∏ v ∤ p U v U^{p} = \prod_{v\nmid p} U_{v} U p = ∏ v ∤ p U v U v = GL 2 ( O F v ) U_{v} = \operatorname{GL}_{2}(\mathcal{O}_{F_{v}}) U v = GL 2 ( O F v ) v ∉ Σ v \notin\Sigma v ∈ / Σ S = S ψ ( U p ) m S = S_{\psi}(U^{p})_{m} S = S ψ ( U p ) m m m m 1 + χ ‾ 1 + \overline{\chi} 1 + χ T = T ( U p ) m \mathbb{T} = \mathbb{T}(U^{p})_{m} T = T ( U p ) m T w T_{w} T w w ∉ Σ w \notin\Sigma w ∈ / Σ D m D_{m} D m G F G_{F} G F T ( Frob w ) = T w T(\operatorname{Frob}_{w}) = T_{w} T ( Frob w ) = T w χ \chi χ T \mathbb{T} T R p s R^{\mathrm{ps}} R ps U p U^{p} U p Spec T \operatorname{Spec}\mathbb{T} Spec T Spec R p s \operatorname{Spec}R^{\mathrm{ps}} Spec R ps F F F F 1 F_{1} F 1 T F 1 \mathbb{T}_{F_{1}} T F 1 T F 1 ∘ \mathbb{T}^{\circ}_{F_{1}} T F 1 ∘ 2 2 2
(C1) [OAI, Proposition 4.1]. At sufficiently small levels the spaces of forms are free over the group algebras of the relevant finite 2 2 2 S S S B × B^{\times} B ×
(C2) [OAI, Proposition 4.2]. Local–global compatibility. Put S t o r = S ⊗ O E O S_{\mathrm{tor}}=S\otimes_{\mathcal{O}_E} \mathcal{O} S tor = S ⊗ O E O M = S t o r ∨ M=S_{\mathrm{tor}}^{\vee} M = S tor ∨ P P P 2 2 2 m = Hom ( P , M ) \mathbf{m}=\operatorname{Hom}(P,M) m = Hom ( P , M ) T \mathbb{T} T ⨂ ^ v ∣ 2 R v p s \widehat{\bigotimes}_{v\mid2}R_v^{\mathrm{ps}} ⨂ v ∣ 2 R v ps ψ = χ ε \psi=\chi\varepsilon ψ = χ ε v v v ( ρ x ∣ G v ⊗ ε ) s s (\rho_x|_{G_v}\otimes\varepsilon)^{\mathrm{ss}} ( ρ x ∣ G v ⊗ ε ) ss χ v ε 2 \chi_v\varepsilon^2 χ v ε 2 R v p s R_v^{\mathrm{ps}} R v ps χ \chi χ ε \varepsilon ε x x x Spec T [ 1 / 2 ] \operatorname{Spec}\mathbb{T}[1/2] Spec T [ 1/2 ] E E E ( S ⊗ E ) [ x ] (S\otimes E)[x] ( S ⊗ E ) [ x ] ⨂ ^ v ∣ 2 Π v \widehat{\bigotimes}_{v\mid2}\Pi_v ⨂ v ∣ 2 Π v Π v \Pi_v Π v ρ x ∣ G v \rho_x|_{G_v} ρ x ∣ G v Π v \Pi_v Π v p p p [10 ] , which has dense locally algebraic vectors if ρ x ∣ G v \rho_x|_{G_v} ρ x ∣ G v [10 Theorem 1.3] .
(C3) [OAI, Proposition 4.4]. Every irreducible component of Spec T \operatorname{Spec}\mathbb{T} Spec T 1 + 2 d 1+2d 1 + 2 d
We also use the classification of the absolutely irreducible objects of the blocks of GL 2 ( Q 2 ) \operatorname{GL}_2(\mathbb{Q}_2) GL 2 ( Q 2 ) [27 Corollary 6.10 and Proposition 6.11] : if the local parameter at x v x_v x v η ∘ det \eta\circ\det η ∘ det ( Ind B ‾ GL 2 ( Q 2 ) μ ) c o n t (\operatorname{Ind}_{\overline{B}}^{\operatorname{GL}_2(\mathbb{Q}_2)}\mu)_{\mathrm{cont}} ( Ind B GL 2 ( Q 2 ) μ ) cont ( S ⊗ E ) [ x ] (S\otimes E)[x] ( S ⊗ E ) [ x ]
Ordinary parts
Let P = B o × G n ⊂ ∏ v ∣ 2 GL 2 ( Q 2 ) P=B_o\times G_n\subset\prod_{v\mid2}\operatorname{GL}_2(\mathbb{Q}_2) P = B o × G n ⊂ ∏ v ∣ 2 GL 2 ( Q 2 ) Σ o \Sigma_o Σ o Σ n \Sigma_n Σ n S o = Ord P ( S ) S^o=\operatorname{Ord}_P(S) S o = Ord P ( S ) [17 ] . Properties (O1)–(O4) of [Die26a, §3.3] hold for any p p p S o / ϖ n ≃ l i m ← e e S ψ ( U p U ( c ) , O / ϖ n ) m S^o/\varpi^n\simeq\varprojlim_e eS_{\psi}(U^pU(c),\mathcal{O}/\varpi^n)_{\mathfrak{m}} S o / ϖ n ≃ lim e e S ψ ( U p U ( c ) , O / ϖ n ) m e e e Σ o \Sigma_o Σ o T o \mathbb{T}^o T o S o S^o S o T w T_w T w ⟨ γ ⟩ v \langle\gamma\rangle_v ⟨ γ ⟩ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o ψ v , 1 : G v → ( T o ) × \psi_{v,1}:G_v\to(\mathbb{T}^o)^\times ψ v , 1 : G v → ( T o ) × 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 Σ o = Σ 2 \Sigma_o=\Sigma_2 Σ o = Σ 2 T o r d \mathbb{T}^{\mathrm{ord}} T ord T o \mathbb{T}^o T o
Lemma 3.1. There is a determinant of G F , Σ G_{F,\Sigma} G F , Σ T o \mathbb{T}^o T o T ( Frob w ) = T w T(\operatorname{Frob}_w)=T_w T ( Frob w ) = T w χ \chi χ v ∈ Σ o v\in\Sigma_o v ∈ Σ o det ( ψ v , 1 ⊕ χ ψ v , 1 − 1 ) \det(\psi_{v,1}\oplus\chi\psi_{v,1}^{-1}) det ( ψ v , 1 ⊕ χ ψ v , 1 − 1 ) G v G_v G v ψ v , 1 \psi_{v,1} ψ v , 1 G v G_v G v
Proof. Write χ = ε k χ 0 \chi=\varepsilon^k\chi_0 χ = ε k χ 0 χ 0 \chi_0 χ 0 ψ = χ ε \psi=\chi\varepsilon ψ = χ ε 2 2 2 k k k ε = η 4 ⟨ ε ⟩ \varepsilon=\eta^4\langle\varepsilon\rangle ε = η 4 ⟨ ε ⟩ ν \nu ν 2 2 2 ≡ 1 ( m o d ϖ ) \equiv1\pmod{\varpi} ≡ 1 ( mod ϖ ) ν 2 = ⟨ ε ⟩ \nu^2=\langle\varepsilon\rangle ν 2 = ⟨ ε ⟩ λ = ν − ( k + 1 ) \lambda=\nu^{-(k+1)} λ = ν − ( k + 1 ) χ λ 2 = ε − 1 χ 0 η 4 k + 1 \chi\lambda^2=\varepsilon^{-1}\chi_0\eta_4^{k+1} χ λ 2 = ε − 1 χ 0 η 4 k + 1 ( χ λ 2 ) ε = χ 0 η 4 k + 1 (\chi\lambda^2)\varepsilon=\chi_0\eta_4^{k+1} ( χ λ 2 ) ε = χ 0 η 4 k + 1 λ ∘ nrd \lambda\circ\operatorname{nrd} λ ∘ nrd ψ v , 1 \psi_{v,1} ψ v , 1 λ v ψ v , 1 \lambda_v\psi_{v,1} λ v ψ v , 1 λ v ( Art v ( 2 ) ) = 1 \lambda_v(\operatorname{Art}_v(2))=1 λ v ( Art v ( 2 )) = 1
For the normalized determinant, finite-level forms are scalar-valued of parallel weight 2 2 2 [37 Theorem 2] replaced by local–global compatibility at 2 2 2 [29 , 31 ] : if π \pi π v v v D p s t ( ρ π ∣ G v ) D_{\mathrm{pst}}(\rho_\pi|_{G_v}) D pst ( ρ π ∣ G v ) ρ π ∣ G v \rho_\pi|_{G_v} ρ π ∣ G v A c [ 1 / 2 ] A_c[1/2] A c [ 1/2 ] [11 Theorem 2.12] , hence over their torsion-free integral models and in the completed limit. Undoing the twist gives the assertions for χ \chi χ ψ v , 1 \psi_{v,1} ψ v , 1 λ \lambda λ
Lemma 3.2. For v ∈ Σ n v \in\Sigma_n v ∈ Σ n R v p s R_v^{\mathrm{ps}} R v ps M o M^o M o S o S^o S o v v v G v G_v G v m o = Hom ( P n , M o ) \mathfrak{m}^o = \operatorname{Hom}(P_n, M^o) m o = Hom ( P n , M o ) P n P_n P n Σ n \Sigma_n Σ n T o \mathbb{T}^o T o ( S o ⊗ E ) [ P ] ≠ 0 (S^o \otimes E)[\mathfrak{P}] \ne0 ( S o ⊗ E ) [ P ] = 0 P \mathfrak{P} P T o [ 1 / 2 ] \mathbb{T}^o[1/2] T o [ 1/2 ]
Proof. This is the proof of [Die26a, Lemma 3.5], with the local–global compatibility of Pan and Zhang replaced by (C2).
Proposition 3.3. Let p \mathfrak{p} p T [ 1 / 2 ] \mathbb{T}[1/2] T [ 1/2 ] 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 P \mathfrak{P} P T o [ 1 / 2 ] \mathbb{T}^o[1/2] T o [ 1/2 ] 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 v ∈ Σ o v \in\Sigma_o v ∈ Σ o
Proof. The proof of [Die26a, Proposition 4.1] applies: by (C2) the eigenspace Π = ( S ⊗ E ) [ p ] \Pi= (S \otimes E)[\mathfrak{p}] Π = ( S ⊗ E ) [ p ] Π ′ = ⨂ ^ v ∈ Σ o Π v ′ \Pi' = \widehat{\bigotimes}_{v \in\Sigma_o}\Pi'_v Π ′ = ⨂ v ∈ Σ o Π v ′ Π v ′ \Pi'_v Π v ′ v ∈ Σ o v \in\Sigma_o v ∈ Σ o Π v ′ \Pi'_v Π v ′ ( Ind B ‾ G v μ ‾ v ) c o n t (\operatorname{Ind}_{\overline{B}}^{G_v}\overline{\mu}_v)_{\mathrm{cont}} ( Ind B G v μ v ) cont μ ⊗ ⨂ ^ v ∈ Σ n Π v ′ → Ord P ( Π ) = ( S o ⊗ E ) [ p ] \mu\otimes\widehat{\bigotimes}_{v \in\Sigma_n}\Pi'_v \to\operatorname{Ord}_{P}(\Pi) = (S^o \otimes E)[\mathfrak{p}] μ ⊗ ⨂ v ∈ Σ n Π v ′ → Ord P ( Π ) = ( S o ⊗ E ) [ p ] P \mathfrak{P} P ψ v , 1 ( P ) \psi_{v,1}(\mathfrak{P}) ψ v , 1 ( P )
Theorem 3.4. Let p \mathfrak{p} p T [ 1 / 2 ] \mathbb{T}[1/2] T [ 1/2 ] P \mathfrak{P} P T o [ 1 / 2 ] \mathbb{T}^o[1/2] T o [ 1/2 ] ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v v ∈ Σ n v \in\Sigma_n v ∈ Σ n v ∈ Σ o v \in\Sigma_o v ∈ Σ o ψ v , 1 ( P ) \psi_{v,1}(\mathfrak{P}) ψ v , 1 ( P ) ρ ( p ) \rho(\mathfrak{p}) ρ ( p )
Proof. This is the proof of [Die26a, Theorem 4.2], using Lemma 3.2, (C1), (C2) and the following density statement: if μ = μ 1 ⊗ μ 2 \mu= \mu_1 \otimes\mu_2 μ = μ 1 ⊗ μ 2 T ( Q 2 ) T(\mathbb{Q}_2) T ( Q 2 ) μ i ∣ Z × = x a i θ i \mu_i|_{Z^\times} = x^{a_i}\theta_i μ i ∣ Z × = x a i θ i θ i \theta_i θ i a 1 ≥ a 2 a_1 \ge a_2 a 1 ≥ a 2 ( Ind B ‾ G μ ‾ ) c o n t (\operatorname{Ind}_{\overline{B}}^{G}\overline{\mu})_{\mathrm{cont}} ( Ind B G μ ) cont a 1 = a 2 a_1 = a_2 a 1 = a 2 a 1 > a 2 a_1 > a_2 a 1 > a 2 [6 Proposition 2.2.1] ; see Remark 11.5.
Proposition 3.5. Every minimal prime of T m ′ o \mathbb{T}^{o}_{\mathfrak{m}'} T m ′ o m ′ \mathfrak{m}' m ′ T o \mathbb{T}^o T o 1 + 2 d − ∣ Σ o ∣ 1 + 2d - |\Sigma_o| 1 + 2 d − ∣ Σ o ∣
Proof. For Σ o = ∅ \Sigma_o = \varnothing Σ o = ∅ p = 2 p = 2 p = 2 [26 Theorem 3.6.1] . In general, the proof of (C3) applies to S o S^o S o [15 Proposition 6.1] : the freeness of M o / ϖ M^o/\varpi M o / ϖ 3 ∣ Σ n ∣ + ∣ Σ o ∣ 3|\Sigma_n| + |\Sigma_o| 3∣ Σ n ∣ + ∣ Σ o ∣ S o S^o S o m o \mathfrak{m}^o m o dim ( N ⊗ ^ P n ) ≤ dim A N + ∣ Σ n ∣ \dim(N \widehat{\otimes} P_n) \le\dim_A N + |\Sigma_n| dim ( N ⊗ P n ) ≤ dim A N + ∣ Σ n ∣ A = ⨂ ^ v ∈ Σ n R v p s / ϖ ⊗ ^ Λ T / ϖ A = \widehat{\bigotimes}_{v \in\Sigma_n} R_v^{\mathrm{ps}}/\varpi\widehat{\otimes} \Lambda_T/\varpi A = ⨂ v ∈ Σ n R v ps / ϖ ⊗ Λ T / ϖ ∣ Σ n ∣ |\Sigma_n| ∣ Σ n ∣ ∏ v ∈ Σ n G L 2 ( Q 2 ) \prod_{v \in\Sigma_n}\mathrm{GL}_2(\mathbb{Q}_2) ∏ v ∈ Σ n GL 2 ( Q 2 )
Proof of Theorem A
Let x x x Spec R p s \operatorname{Spec} R^{\mathrm{ps}} Spec R ps ρ \rho ρ x x x ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 p \mathfrak{p} p T F 1 [ 1 / 2 ] \mathbb{T}_{F_1}[1/2] T F 1 [ 1/2 ] ρ ( p ) ≃ ρ ∣ G F 1 \rho(\mathfrak{p}) \simeq\rho|_{G_{F_1}} ρ ( p ) ≃ ρ ∣ G F 1 F 1 F_1 F 1 ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 v ∣ 2 v \mid2 v ∣ 2 ρ v \rho_v ρ v F 1 F_1 F 1 P \mathfrak{P} P p \mathfrak{p} p ψ w , 1 ( P ) \psi_{w,1}(\mathfrak{P}) ψ w , 1 ( P ) ρ v \rho_v ρ v w ∣ v ∈ Σ o w \mid v \in\Sigma_o w ∣ v ∈ Σ o ρ v \rho_v ρ v F 1 F_1 F 1 ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 ρ \rho ρ
Setup for Theorem B
Deformation rings
For v ∣ 2 v \mid2 v ∣ 2 G v G_v G v det ( ψ ⊕ χ ψ − 1 ) \det(\psi\oplus\chi\psi^{-1}) det ( ψ ⊕ χ ψ − 1 ) ψ \psi ψ χ ‾ ∣ G v \overline{\chi}|_{G_v} χ ∣ G v χ ‾ ∣ G v ≠ 1 \overline{\chi}|_{G_v} \ne1 χ ∣ G v = 1 v ∈ Σ o v \in\Sigma_o v ∈ Σ o ψ ‾ v ∈ { 1 , χ ‾ ∣ G v } \overline{\psi}_v \in\{1,\overline{\chi}|_{G_v}\} ψ v ∈ { 1 , χ ∣ G v } ψ v c l \psi_v^{\mathrm{cl}} ψ v cl
Definition 4.1. (1) R o R^{\mathrm{o}} R o R p s R^{\mathrm{ps}} R ps v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o det ( ψ v ⊕ χ ψ v − 1 ) \det(\psi_v \oplus\chi\psi_v^{-1}) det ( ψ v ⊕ χ ψ v − 1 ) G v G_v G v ψ v \psi_v ψ v ψ ˉ v \bar{\psi}_v ψ ˉ v ψ v \psi_v ψ v G v G_v G v
(2) R r e d R^{\mathrm{red}} R red R p s R^{\mathrm{ps}} R ps G v G_v G v v ∣ 2 v \mid2 v ∣ 2 Spec R r e d \operatorname{Spec}R^{\mathrm{red}} Spec R red det ( ψ v ( 1 ) ⊕ χ ψ v ( 1 ) , − 1 ) \det(\psi_v^{(1)} \oplus\chi\psi_v^{(1),-1}) det ( ψ v ( 1 ) ⊕ χ ψ v ( 1 ) , − 1 ) G v G_v G v ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) 1 1 1
Let Λ = O [ [ ∏ v ∣ 2 ( 1 + 4 Z 2 ) ] ] \Lambda= \mathcal{O}[[\prod_{v \mid2}(1 + 4\mathbb{Z}_2)]] Λ = O [[ ∏ v ∣ 2 ( 1 + 4 Z 2 )]] O \mathcal{O} O d d d ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) Art ( 1 + 4 Z 2 ) ⊂ I v \operatorname{Art}(1 + 4\mathbb{Z}_2) \subset I_v Art ( 1 + 4 Z 2 ) ⊂ I v R r e d R^{\mathrm{red}} R red Λ \Lambda Λ χ ( 1 + 4 Z 2 ) ⊂ 1 + ϖ O \chi(1 + 4\mathbb{Z}_2) \subset1 + \varpi\mathcal{O} χ ( 1 + 4 Z 2 ) ⊂ 1 + ϖ O χ ˉ \bar{\chi} χ ˉ [ u ] v ↦ χ ( u ) [ u ] v − 1 [u]_v \mapsto\chi(u)[u]_v^{-1} [ u ] v ↦ χ ( u ) [ u ] v − 1 Λ \Lambda Λ ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) χ ψ v ( 1 ) , − 1 \chi\psi_v^{(1),-1} χ ψ v ( 1 ) , − 1 Λ \Lambda Λ
The point x x x ρ \rho ρ Spec R o [ 1 / 2 ] \operatorname{Spec}R^{\mathrm{o}}[1/2] Spec R o [ 1/2 ] ψ v ( x ) = ψ v c l \psi_v(x) = \psi_v^{\mathrm{cl}} ψ v ( x ) = ψ v cl v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o
By (PM), the restriction to G F 1 G_{F_1} G F 1 Spec R p s \operatorname{Spec}R^{\mathrm{ps}} Spec R ps x x x Spec T F 1 p ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{p}}(U_1^{\mathrm{p}}) Spec T F 1 p ( U 1 p ) U 1 p U_1^{\mathrm{p}} U 1 p F 1 F_1 F 1 F F F U p U^{\mathrm{p}} U p m ′ \mathfrak{m}' m ′ T o = T o ( U p ) \mathbb{T}^{\mathrm{o}} = \mathbb{T}^{\mathrm{o}}(U^{\mathrm{p}}) T o = T o ( U p ) ψ v , 1 \psi_{v,1} ψ v , 1 ψ ˉ v \bar{\psi}_v ψ ˉ v v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o R o ↠ T m ′ o R^{\mathrm{o}} \twoheadrightarrow\mathbb{T}^{\mathrm{o}}_{\mathfrak{m}'} R o ↠ T m ′ o ψ v ↦ ψ v , 1 \psi_v \mapsto\psi_{v,1} ψ v ↦ ψ v , 1 R o R^{\mathrm{o}} R o Spec T m ′ o \operatorname{Spec}\mathbb{T}^{\mathrm{o}}_{\mathfrak{m}'} Spec T m ′ o m ′ \mathfrak{m}' m ′
A lower bound for the component of ρ \rho ρ
Proposition 4.2. There is an irreducible component C C C Spec R o \operatorname{Spec}R^{\mathrm{o}} Spec R o x x x dim C ≥ 1 + 2 d − ∣ Σ o ∣ \dim C \ge1 + 2d - |\Sigma_{\mathrm{o}}| dim C ≥ 1 + 2 d − ∣ Σ o ∣ O \mathcal{O} O C [ 1 / 2 ] C[1/2] C [ 1/2 ] Spec R F 1 p s \operatorname{Spec}R_{F_1}^{\mathrm{ps}} Spec R F 1 ps Spec T F 1 p ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{p}}(U_1^{\mathrm{p}}) Spec T F 1 p ( U 1 p )
Proof. Let p x \mathfrak{p}_x p x x x x R ^ x \widehat{R}_x R x R o [ 1 / 2 ] R^{\mathrm{o}}[1/2] R o [ 1/2 ] p x \mathfrak{p}_x p x [26 Proposition 2.2.2] and [20 §2.3] , whose arguments only use Mazur’s finiteness condition and apply to determinants [11 Theorem 2.22] , R ^ x \widehat{R}_x R x ρ x \rho_x ρ x E E E χ \chi χ v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o G v G_v G v ψ v c l \psi_v^{\mathrm{cl}} ψ v cl V = ad 0 ρ x V = \operatorname{ad}^{0}\rho_x V = ad 0 ρ x p = 2 p = 2 p = 2 d d d g g g r r r g − r ≥ 3 − 6 d − h 0 ( G F , V ( 1 ) ) = 3 − 6 d g-r \ge3 - 6d - h^{0}(G_F,V(1)) = 3 - 6d g − r ≥ 3 − 6 d − h 0 ( G F , V ( 1 )) = 3 − 6 d H 0 ( G F , V ( 1 ) ) = 0 H^{0}(G_F,V(1)) = 0 H 0 ( G F , V ( 1 )) = 0 ρ x ≄ ρ x ( 1 ) \rho_x \not\simeq\rho_x(1) ρ x ≃ ρ x ( 1 ) Σ o \Sigma_{\mathrm{o}} Σ o v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o v ∈ Σ n v \in\Sigma_{\mathrm{n}} v ∈ Σ n ρ x \rho_x ρ x
dim R ^ x ≥ 6 ∣ Σ n ∣ + 5 ∣ Σ o ∣ + 2 d + ( 3 − 6 d ) − 3 = 2 d − ∣ Σ o ∣ . \dim\widehat{R}_x \ge6|\Sigma_{\mathrm{n}}| + 5|\Sigma_{\mathrm{o}}| + 2d + (3 - 6d) - 3 = 2d - |\Sigma_{\mathrm{o}}|. dim R x ≥ 6∣ Σ n ∣ + 5∣ Σ o ∣ + 2 d + ( 3 − 6 d ) − 3 = 2 d − ∣ Σ o ∣. Choose a minimal prime q ⊂ p x \mathfrak{q} \subset\mathfrak{p}_x q ⊂ p x R o R^{\mathrm{o}} R o dim ( R o / q ) p x = dim R p x o = dim R ^ x \dim(R^{\mathrm{o}}/\mathfrak{q})_{\mathfrak{p}_x} = \dim R^{\mathrm{o}}_{\mathfrak{p}_x} = \dim\widehat{R}_x dim ( R o / q ) p x = dim R p x o = dim R x R o / q R^{\mathrm{o}}/\mathfrak{q} R o / q p x \mathfrak{p}_x p x C = V ( q ) C = V(\mathfrak{q}) C = V ( q ) 1 + 2 d − ∣ Σ o ∣ 1 + 2d - |\Sigma_{\mathrm{o}}| 1 + 2 d − ∣ Σ o ∣ O \mathcal{O} O Spec R p s \operatorname{Spec}R^{\mathrm{ps}} Spec R ps x x x x x x G F 1 G_{F_1} G F 1 Spec T F 1 p ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{p}}(U_1^{\mathrm{p}}) Spec T F 1 p ( U 1 p ) □ \square □
We fix such a component C C C A C = R o / q A_C = R^{\mathrm{o}}/\mathfrak{q} A C = R o / q
The Steinberg place
Let v = v 0 v = v_0 v = v 0 q = q v 0 \mathfrak{q} = \mathfrak{q}_{v_0} q = q v 0 ϕ = ϕ v 0 \phi= \phi_{v_0} ϕ = ϕ v 0 ρ ∣ I v 0 ≃ μ ⊗ u \rho|_{I_{v_0}} \simeq\mu\otimes u ρ ∣ I v 0 ≃ μ ⊗ u μ \mu μ I v 0 I_{v_0} I v 0 u u u
P v 0 = ( 1 + q ) 2 χ ( ϕ ) − q T ( ϕ ) 2 ∈ R p s . P_{v_0} = (1 + q)^2\chi(\phi) - qT(\phi)^2 \in R^{\mathrm{ps}}. P v 0 = ( 1 + q ) 2 χ ( ϕ ) − q T ( ϕ ) 2 ∈ R ps . If a , b a,b a , b ρ z ( ϕ ) \rho_z(\phi) ρ z ( ϕ ) P v 0 ( z ) = ( a − q b ) ( b − q a ) P_{v_0}(z) = (a - qb)(b - qa) P v 0 ( z ) = ( a − q b ) ( b − q a )
Definition 4.1. (1) R o R^{\mathrm{o}} R o R p s R^{\mathrm{ps}} R ps v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o det ( ψ v ⊕ χ ψ v − 1 ) \det(\psi_v \oplus\chi\psi_v^{-1}) det ( ψ v ⊕ χ ψ v − 1 ) G v G_v G v ψ v \psi_v ψ v ψ ˉ v \bar{\psi}_v ψ ˉ v ψ v \psi_v ψ v G v G_v G v
(2) R r e d R^{\mathrm{red}} R red R p s R^{\mathrm{ps}} R ps G v G_v G v v ∣ 2 v \mid2 v ∣ 2 Spec R r e d \operatorname{Spec}R^{\mathrm{red}} Spec R red det ( ψ v ( 1 ) ⊕ χ ψ v ( 1 ) , − 1 ) \det(\psi_v^{(1)} \oplus\chi\psi_v^{(1),-1}) det ( ψ v ( 1 ) ⊕ χ ψ v ( 1 ) , − 1 ) G v G_v G v ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) 1 1 1
Let Λ = O [ [ ∏ v ∣ 2 ( 1 + 4 Z 2 ) ] ] \Lambda= \mathcal{O}[[\prod_{v \mid2}(1 + 4\mathbb{Z}_2)]] Λ = O [[ ∏ v ∣ 2 ( 1 + 4 Z 2 )]] O \mathcal{O} O d d d ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) Art ( 1 + 4 Z 2 ) ⊂ I v \operatorname{Art}(1 + 4\mathbb{Z}_2) \subset I_v Art ( 1 + 4 Z 2 ) ⊂ I v R r e d R^{\mathrm{red}} R red Λ \Lambda Λ χ ( 1 + 4 Z 2 ) ⊂ 1 + ϖ O \chi(1 + 4\mathbb{Z}_2) \subset1 + \varpi\mathcal{O} χ ( 1 + 4 Z 2 ) ⊂ 1 + ϖ O χ ˉ \bar{\chi} χ ˉ [ u ] v ↦ χ ( u ) [ u ] v − 1 [u]_v \mapsto\chi(u)[u]_v^{-1} [ u ] v ↦ χ ( u ) [ u ] v − 1 Λ \Lambda Λ ψ v ( 1 ) \psi_v^{(1)} ψ v ( 1 ) χ ψ v ( 1 ) , − 1 \chi\psi_v^{(1),-1} χ ψ v ( 1 ) , − 1 Λ \Lambda Λ
The point x x x ρ \rho ρ Spec R o [ 1 / 2 ] \operatorname{Spec}R^{\mathrm{o}}[1/2] Spec R o [ 1/2 ] ψ v ( x ) = ψ v c l \psi_v(x) = \psi_v^{\mathrm{cl}} ψ v ( x ) = ψ v cl v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o
By (PM), the restriction to G F 1 G_{F_1} G F 1 Spec R p s \operatorname{Spec}R^{\mathrm{ps}} Spec R ps x x x Spec T F 1 p ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{p}}(U_1^{\mathrm{p}}) Spec T F 1 p ( U 1 p ) U 1 p U_1^{\mathrm{p}} U 1 p F 1 F_1 F 1 F F F U p U^{\mathrm{p}} U p m ′ \mathfrak{m}' m ′ T o = T o ( U p ) \mathbb{T}^{\mathrm{o}} = \mathbb{T}^{\mathrm{o}}(U^{\mathrm{p}}) T o = T o ( U p ) ψ v , 1 \psi_{v,1} ψ v , 1 ψ ˉ v \bar{\psi}_v ψ ˉ v v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o R o ↠ T m ′ o R^{\mathrm{o}} \twoheadrightarrow\mathbb{T}^{\mathrm{o}}_{\mathfrak{m}'} R o ↠ T m ′ o ψ v ↦ ψ v , 1 \psi_v \mapsto\psi_{v,1} ψ v ↦ ψ v , 1 R o R^{\mathrm{o}} R o Spec T m ′ o \operatorname{Spec}\mathbb{T}^{\mathrm{o}}_{\mathfrak{m}'} Spec T m ′ o m ′ \mathfrak{m}' m ′
A lower bound for the component of ρ \rho ρ
Proposition 4.2. There is an irreducible component C C C Spec R o \operatorname{Spec}R^{\mathrm{o}} Spec R o x x x dim C ≥ 1 + 2 d − ∣ Σ o ∣ \dim C \ge1 + 2d - |\Sigma_{\mathrm{o}}| dim C ≥ 1 + 2 d − ∣ Σ o ∣ O \mathcal{O} O C [ 1 / 2 ] C[1/2] C [ 1/2 ] Spec R F 1 p s \operatorname{Spec}R_{F_1}^{\mathrm{ps}} Spec R F 1 ps Spec T F 1 p ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{p}}(U_1^{\mathrm{p}}) Spec T F 1 p ( U 1 p )
Proof. Let p x \mathfrak{p}_x p x x x x R ^ x \widehat{R}_x R x R o [ 1 / 2 ] R^{\mathrm{o}}[1/2] R o [ 1/2 ] p x \mathfrak{p}_x p x [26 Proposition 2.2.2] and [20 §2.3] , whose arguments only use Mazur’s finiteness condition and apply to determinants [11 Theorem 2.22] , R ^ x \widehat{R}_x R x ρ x \rho_x ρ x E E E χ \chi χ v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o G v G_v G v ψ v c l \psi_v^{\mathrm{cl}} ψ v cl V = ad 0 ρ x V = \operatorname{ad}^{0}\rho_x V = ad 0 ρ x p = 2 p = 2 p = 2 d d d g g g r r r g − r ≥ 3 − 6 d − h 0 ( G F , V ( 1 ) ) = 3 − 6 d g-r \ge3 - 6d - h^{0}(G_F,V(1)) = 3 - 6d g − r ≥ 3 − 6 d − h 0 ( G F , V ( 1 )) = 3 − 6 d H 0 ( G F , V ( 1 ) ) = 0 H^{0}(G_F,V(1)) = 0 H 0 ( G F , V ( 1 )) = 0 ρ x ≄ ρ x ( 1 ) \rho_x \not\simeq\rho_x(1) ρ x ≃ ρ x ( 1 ) Σ o \Sigma_{\mathrm{o}} Σ o v ∈ Σ o v \in\Sigma_{\mathrm{o}} v ∈ Σ o v ∈ Σ n v \in\Sigma_{\mathrm{n}} v ∈ Σ n ρ x \rho_x ρ x
dim R ^ x ≥ 6 ∣ Σ n ∣ + 5 ∣ Σ o ∣ + 2 d + ( 3 − 6 d ) − 3 = 2 d − ∣ Σ o ∣ . \dim\widehat{R}_x \ge6|\Sigma_{\mathrm{n}}| + 5|\Sigma_{\mathrm{o}}| + 2d + (3 - 6d) - 3 = 2d - |\Sigma_{\mathrm{o}}|. dim R x ≥ 6∣ Σ n ∣ + 5∣ Σ o ∣ + 2 d + ( 3 − 6 d ) − 3 = 2 d − ∣ Σ o ∣. Choose a minimal prime q ⊂ p x \mathfrak{q} \subset\mathfrak{p}_x q ⊂ p x R o R^{\mathrm{o}} R o dim ( R o / q ) p x = dim R p x o = dim R ^ x \dim(R^{\mathrm{o}}/\mathfrak{q})_{\mathfrak{p}_x} = \dim R^{\mathrm{o}}_{\mathfrak{p}_x} = \dim\widehat{R}_x dim ( R o / q ) p x = dim R p x o = dim R x R o / q R^{\mathrm{o}}/\mathfrak{q} R o / q p x \mathfrak{p}_x p x C = V ( q ) C = V(\mathfrak{q}) C = V ( q ) 1 + 2 d − ∣ Σ o ∣ 1 + 2d - |\Sigma_{\mathrm{o}}| 1 + 2 d − ∣ Σ o ∣ O \mathcal{O} O Spec R p s \operatorname{Spec}R^{\mathrm{ps}} Spec R ps x x x x x x G F 1 G_{F_1} G F 1 Spec T F 1 p ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{p}}(U_1^{\mathrm{p}}) Spec T F 1 p ( U 1 p ) □ \square □
We fix such a component C C C A C = R o / q A_C = R^{\mathrm{o}}/\mathfrak{q} A C = R o / q
The Steinberg place
Let v = v 0 v = v_0 v = v 0 q = q v 0 \mathfrak{q} = \mathfrak{q}_{v_0} q = q v 0 ϕ = ϕ v 0 \phi= \phi_{v_0} ϕ = ϕ v 0 ρ ∣ I v 0 ≃ μ ⊗ u \rho|_{I_{v_0}} \simeq\mu\otimes u ρ ∣ I v 0 ≃ μ ⊗ u μ \mu μ I v 0 I_{v_0} I v 0 u u u
P v 0 = ( 1 + q ) 2 χ ( ϕ ) − q T ( ϕ ) 2 ∈ R p s . P_{v_0} = (1 + q)^2\chi(\phi) - qT(\phi)^2 \in R^{\mathrm{ps}}. P v 0 = ( 1 + q ) 2 χ ( ϕ ) − q T ( ϕ ) 2 ∈ R ps . If a , b a,b a , b ρ z ( ϕ ) \rho_z(\phi) ρ z ( ϕ ) P v 0 ( z ) = ( a − q b ) ( b − q a ) P_{v_0}(z) = (a - qb)(b - qa) P v 0 ( z ) = ( a − q b ) ( b − q a )
Lemma 4.3. (1) For every closed point z z z C [ 1 / 2 ] C[1/2] C [ 1/2 ] ρ z ∣ I v 0 ≃ μ ⊗ u z \rho_z|_{I_{v_0}} \simeq\mu\otimes u_z ρ z ∣ I v 0 ≃ μ ⊗ u z u z u_z u z P v 0 P_{v_0} P v 0 C C C
(2) Let K / F K/F K / F w w w K K K v 0 v_0 v 0 z z z C [ 1 / 2 ] C[1/2] C [ 1/2 ] ρ z ∣ G K \rho_z|_{G_K} ρ z ∣ G K Π \Pi Π G L 2 ( A K ) \mathrm{GL}_2(\mathbb{A}_K) GL 2 ( A K ) Π w \Pi_w Π w ρ z ∣ G K w \rho_z|_{G_{K_w}} ρ z ∣ G K w
Proof. (1) For σ ∈ I v 0 \sigma\in I_{v_0} σ ∈ I v 0 ρ z ( σ ) \rho_z(\sigma) ρ z ( σ ) σ \sigma σ ρ z \rho_z ρ z σ \sigma σ q q q T ( σ ) ∈ A C [ 1 / 2 ] T(\sigma) \in A_{C[1/2]} T ( σ ) ∈ A C [ 1/2 ] A C [ 1 / 2 ] A_{C[1/2]} A C [ 1/2 ] T ( σ ) T(\sigma) T ( σ ) C [ 1 / 2 ] C[1/2] C [ 1/2 ] T x ( σ ) T_x(\sigma) T x ( σ ) det ρ z ( σ ) = χ ( σ ) \det\rho_z(\sigma)=\chi(\sigma) det ρ z ( σ ) = χ ( σ ) T ( σ σ ′ ) T(\sigma\sigma') T ( σ σ ′ ) σ , σ ′ ∈ I v 0 \sigma,\sigma' \in I_{v_0} σ , σ ′ ∈ I v 0 ρ z ∣ I v 0 \rho_z|_{I_{v_0}} ρ z ∣ I v 0 ρ x ∣ I v 0 \rho_x|_{I_{v_0}} ρ x ∣ I v 0 ρ z ∣ I v 0 ≃ μ ⊗ u z \rho_z|_{I_{v_0}} \simeq\mu\otimes u_z ρ z ∣ I v 0 ≃ μ ⊗ u z u z u_z u z σ 0 ∈ I v 0 \sigma_0 \in I_{v_0} σ 0 ∈ I v 0 2 2 2 μ ( σ 0 ) = 1 \mu(\sigma_0)=1 μ ( σ 0 ) = 1 σ 0 \sigma_0 σ 0 Z N ⊂ C [ 1 / 2 ] Z_N \subset C[1/2] Z N ⊂ C [ 1/2 ] T ( σ 0 g ) = T ( g ) T(\sigma_0g)=T(g) T ( σ 0 g ) = T ( g ) g g g z z z z ∈ Z N z \in Z_N z ∈ Z N ρ z ( σ 0 ) = 1 \rho_z(\sigma_0)=1 ρ z ( σ 0 ) = 1 ρ z ∣ G v 0 \rho_z|_{G_{v_0}} ρ z ∣ G v 0 x ∉ Z N x \notin Z_N x ∈ / Z N Z N Z_N Z N C [ 1 / 2 ] C[1/2] C [ 1/2 ] ϕ N ϕ − 1 = q N \phi N\phi^{-1}=qN ϕN ϕ − 1 = q N N ≠ 0 N\ne0 N = 0 ϕ \phi ϕ ker N \ker N ker N q ± 1 q^{\pm1} q ± 1 P v 0 ( z ) = 0 P_{v_0}(z)=0 P v 0 ( z ) = 0 C [ 1 / 2 ] C[1/2] C [ 1/2 ] C C C O \mathcal{O} O P v 0 P_{v_0} P v 0 C C C
(2) By (1), the inertia group I w I_w I w μ ⊗ u z \mu\otimes u_z μ ⊗ u z ϕ w \phi_w ϕ w ϕ f \phi^f ϕ f f f f ρ z ∣ G v 0 \rho_z|_{G_{v_0}} ρ z ∣ G v 0 ρ z ( ϕ w ) \rho_z(\phi_w) ρ z ( ϕ w ) q w ± 1 q_w^{\pm1} q w ± 1 [9 , 33 , 19 ] , the semisimplified Langlands parameter of Π w \Pi_w Π w ν ⊕ ν ∣ ⋅ ∣ ± 1 \nu\oplus\nu|\cdot|^{\pm1} ν ⊕ ν ∣ ⋅ ∣ ± 1 ν \nu ν Π w \Pi_w Π w Π \Pi Π 2 2 2 [9 Théorème (A)] , and ρ z ∣ G K w \rho_z|_{G_{K_w}} ρ z ∣ G K w
Lemma 4.4. Let z z z C [ 1 / 2 ] C[1/2] C [ 1/2 ] ρ z \rho_z ρ z 2 2 2 ρ z \rho_z ρ z
Proof. Suppose ρ z ≃ Ind G K G F θ \rho_z \simeq\operatorname{Ind}_{G_K}^{G_F}\theta ρ z ≃ Ind G K G F θ K / F K/F K / F θ \theta θ 2 2 2 K K K [30 Chapter III] . If K K K θ \theta θ θ = ε n θ 0 \theta=\varepsilon^n\theta_0 θ = ε n θ 0 θ 0 \theta_0 θ 0 ρ z \rho_z ρ z 2 2 2 K K K θ ≠ θ c \theta\ne\theta^c θ = θ c ρ z \rho_z ρ z Π \Pi Π θ \theta θ 2 2 2 ρ z \rho_z ρ z K = F K=F K = F Π v 0 \Pi_{v_0} Π v 0
Parity
Write χ = ε k χ 0 \chi=\varepsilon^k\chi_0 χ = ε k χ 0 χ 0 \chi_0 χ 0 { h , h + 1 } \{h,h+1\} { h , h + 1 } 2 2 2 k k k k k k η 4 : G F → { ± 1 } \eta_4:G_F\to\{\pm1\} η 4 : G F → { ± 1 } ε η 4 − 1 \varepsilon\eta_4^{-1} ε η 4 − 1 1 + 4 Z 2 1+4\mathbb{Z}_2 1 + 4 Z 2 ⟨ ε ⟩ = ε η 4 − 1 \langle\varepsilon\rangle=\varepsilon\eta_4^{-1} ⟨ ε ⟩ = ε η 4 − 1 E E E c c c 5 5 5 c ≡ 1 ( m o d ϖ ) c\equiv1\pmod{\varpi} c ≡ 1 ( mod ϖ ) ν \nu ν ν = c log 5 ⟨ ε ⟩ \nu=c^{\log_5\langle\varepsilon\rangle} ν = c l o g 5 ⟨ ε ⟩ ν 2 = ⟨ ε ⟩ \nu^2=\langle\varepsilon\rangle ν 2 = ⟨ ε ⟩ 2 2 2 ν ≡ 1 ( m o d ϖ ) \nu\equiv1\pmod{\varpi} ν ≡ 1 ( mod ϖ ) ν ( c ∞ ) = 1 \nu(c_\infty)=1 ν ( c ∞ ) = 1 c ∞ c_\infty c ∞ ν \nu ν R 0 \mathcal{R}^0 R 0 R r e d 0 \mathcal{R}^0_{\mathrm{red}} R red 0 R \mathcal{R} R R r e d \mathcal{R}^{\mathrm{red}} R red χ \chi χ χ ′ = χ ⟨ ε ⟩ \chi'=\chi\langle\varepsilon\rangle χ ′ = χ ⟨ ε ⟩ k k k k + 1 k+1 k + 1 ψ ν \psi_\nu ψ ν ψ ν ν \psi_{\nu\nu} ψ ν ν P v 0 P_{v_0} P v 0 ν ( ϕ ) 2 P v 0 \nu(\phi)^2P_{v_0} ν ( ϕ ) 2 P v 0 C C C x x x C ′ C' C ′ x ′ x' x ′ x ′ x' x ′ x x x k k k h = ( − k − 1 ) / 2 h=(-k-1)/2 h = ( − k − 1 ) /2
The fully ordinary subfamily
For v ∈ Σ n v \in\Sigma_n v ∈ Σ n f v ∈ A C f_v \in A_C f v ∈ A C R v p s → R p s → A C R_v^{\mathrm{ps}} \to R^{\mathrm{ps}} \to A_C R v ps → R ps → A C G v G_v G v z z z C C C V ( f v ) V(f_v) V ( f v ) ρ z ∣ G v \rho_z|_{G_v} ρ z ∣ G v f v f_v f v A C A_C A C D \mathfrak{D} D ( f v ) v ∈ Σ n (f_v)_{v\in\Sigma_n} ( f v ) v ∈ Σ n dim A C / D ≥ 1 + 2 d − ∣ Σ o ∣ − ∣ Σ n ∣ = 1 + d \dim A_C/\mathfrak{D} \geq1+2d-|\Sigma_o|-|\Sigma_n|=1+d dim A C / D ≥ 1 + 2 d − ∣ Σ o ∣ − ∣ Σ n ∣ = 1 + d D = Spec A C / D D=\operatorname{Spec}A_C/\mathfrak{D} D = Spec A C / D D ⊂ Spec R r e d D\subset\operatorname{Spec}R^{\mathrm{red}} D ⊂ Spec R red
Proposition 5.1. D D D Λ \Lambda Λ dim D = 1 + d \dim D=1+d dim D = 1 + d D → Spec Λ D\to\operatorname{Spec}\Lambda D → Spec Λ D D D O \mathcal{O} O
Proof. D D D Spec R r e d \operatorname{Spec}R^{\mathrm{red}} Spec R red Λ \Lambda Λ dim D ≥ 1 + d = dim Λ \dim D\geq1+d=\dim\Lambda dim D ≥ 1 + d = dim Λ Λ \Lambda Λ dim D = 1 + d \dim D=1+d dim D = 1 + d D → Spec Λ D\to\operatorname{Spec}\Lambda D → Spec Λ Spec Λ \operatorname{Spec}\Lambda Spec Λ d d d D D D
For odd p p p [26 Theorem 5.1.2(1)] in [15 ] .
Lemma 5.2. For every v ∣ 2 v\mid2 v ∣ 2 ψ v , s ∈ { ψ v , χ ψ v ( 1 ) , ψ v ( 1 ) , − 1 } \psi_{v,s}\in\{\psi_v,\chi\psi_v^{(1)},\psi_v^{(1),-1}\} ψ v , s ∈ { ψ v , χ ψ v ( 1 ) , ψ v ( 1 ) , − 1 } ψ v , s \psi_{v,s} ψ v , s G v G_v G v D D D v ∈ Σ o v\in\Sigma_o v ∈ Σ o ψ v , s = ψ v \psi_{v,s}=\psi_v ψ v , s = ψ v
Proof. As in [15 Lemma 5.1] : at irreducible points one of the two characters is a subcharacter, the two conditions are closed, and the irreducible locus of D D D
Definition 5.3. A closed point y y y D [ 1 / 2 ] D[1/2] D [ 1/2 ] ρ y \rho_y ρ y v ∣ 2 v\mid2 v ∣ 2 ψ v , s ( y ) ∣ I v = ε − h θ v \psi_{v,s}(y)|_{I_v}=\varepsilon^{-h}\theta_v ψ v , s ( y ) ∣ I v = ε − h θ v χ ψ v , s ( y ) − 1 ∣ I v = ε − h − 1 θ v ′ \chi\psi_{v,s}(y)^{-1}|_{I_v}=\varepsilon^{-h-1}\theta'_v χ ψ v , s ( y ) − 1 ∣ I v = ε − h − 1 θ v ′ θ v , θ v ′ \theta_v,\theta'_v θ v , θ v ′ θ v ≠ θ v ′ \theta_v\ne\theta'_v θ v = θ v ′
At a good point, the Jordan–Hölder characters of ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v h < h + 1 h<h+1 h < h + 1 ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v χ ψ v , s ( y ) − 1 \chi\psi_{v,s}(y)^{-1} χ ψ v , s ( y ) − 1 ψ v , s ( y ) \psi_{v,s}(y) ψ v , s ( y ) ψ v , s ( y ) \psi_{v,s}(y) ψ v , s ( y ) ρ y \rho_y ρ y
Lemma 5.4. Good points are Zariski dense in D D D
Proof. The restrictions of the ψ v , s \psi_{v,s} ψ v , s Art ( 1 + 4 Z 2 ) \operatorname{Art}(1+4\mathbb{Z}_2) Art ( 1 + 4 Z 2 ) Λ \Lambda Λ D → Spec Λ D\to\operatorname{Spec}\Lambda D → Spec Λ D [ 1 / 2 ] D[1/2] D [ 1/2 ] Spec Λ \operatorname{Spec}\Lambda Spec Λ v v v ψ v , s = ε − h ζ v \psi_{v,s}=\varepsilon^{-h}\zeta_v ψ v , s = ε − h ζ v Art ( 1 + 4 Z 2 ) \operatorname{Art}(1+4\mathbb{Z}_2) Art ( 1 + 4 Z 2 ) ζ v \zeta_v ζ v ζ v \zeta_v ζ v v v v ψ v , s ∣ I v \psi_{v,s}|_{I_v} ψ v , s ∣ I v ε − h \varepsilon^{-h} ε − h χ ψ v , s − 1 ∣ I v \chi\psi_{v,s}^{-1}|_{I_v} χ ψ v , s − 1 ∣ I v ε − 2 \varepsilon^{-2} ε − 2 Art ( 1 + 4 Z 2 ) \operatorname{Art}(1+4\mathbb{Z}_2) Art ( 1 + 4 Z 2 ) ζ v − 2 \zeta_v^{-2} ζ v − 2 ζ v \zeta_v ζ v Art ( 5 ) \operatorname{Art}(5) Art ( 5 ) v v v ε − h ( Art ( 5 ) ) ζ v ( Art ( 5 ) ) \varepsilon^{-h}(\operatorname{Art}(5))\zeta_v(\operatorname{Art}(5)) ε − h ( Art ( 5 )) ζ v ( Art ( 5 )) ζ v ( Art ( 5 ) ) ∈ μ 2 ∞ \zeta_v(\operatorname{Art}(5))\in\mu_{2^\infty} ζ v ( Art ( 5 )) ∈ μ 2 ∞ d d d d d d Spec Λ \operatorname{Spec}\Lambda Spec Λ D D D D → Spec Λ D\to\operatorname{Spec}\Lambda D → Spec Λ D D D D D D
Good points
Proposition 6.1. Let y y y
ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v v ∣ 2 v \mid2 v ∣ 2
there is a maximal ideal P y \mathfrak{P}_y P y T F 1 ( U p ) [ 1 / 2 ] \mathbb{T}_{F_1}(U^p)[1/2] T F 1 ( U p ) [ 1/2 ] ρ y ∣ G F 1 \rho_y|_{G_{F_1}} ρ y ∣ G F 1 ψ w , 1 ( P y ) = ψ v , s ( y ) \psi_{w,1}(\mathfrak{P}_y)=\psi_{v,s}(y) ψ w , 1 ( P y ) = ψ v , s ( y ) w w w F 1 F_1 F 1 v ∣ 2 v \mid2 v ∣ 2
ρ y \rho_y ρ y π \pi π G L 2 ( A F ) \mathrm{GL}_2(\mathbb{A}_F) GL 2 ( A F ) v ∣ 2 v \mid2 v ∣ 2 π v 0 \pi_{v_0} π v 0
y y y o o o ψ v , 1 = ψ v ( y ) \psi_{v,1}=\psi_v(y) ψ v , 1 = ψ v ( y ) v ∈ Σ o v \in\Sigma_o v ∈ Σ o U p U^p U p F F F
Proof. (1) Put r = ρ y ⊗ ε h r=\rho_y\otimes\varepsilon^h r = ρ y ⊗ ε h v ∣ 2 v\mid2 v ∣ 2 r ∣ G v r|_{G_v} r ∣ G v ε − 1 θ v ′ \varepsilon^{-1}\theta'_v ε − 1 θ v ′ θ v \theta_v θ v { 0 , 1 } \{0,1\} { 0 , 1 } θ v ≠ θ v ′ \theta_v\ne\theta'_v θ v = θ v ′ r ∣ G v r|_{G_v} r ∣ G v [35 §1.1] , which uses the convention H T ( ε ) = − 1 \mathrm{HT}(\varepsilon)=-1 HT ( ε ) = − 1 0 0 0 1 1 1 [35 Theorem B] , there are a finite totally real extension F ′ ′ / F F''/F F ′′ / F Π \Pi Π G L 2 ( A F ′ ′ ) \mathrm{GL}_2(\mathbb{A}_{F''}) GL 2 ( A F ′′ ) r ∣ G F ′ ′ ≃ r Π , ι r|_{G_{F''}}\simeq r_{\Pi,\iota} r ∣ G F ′′ ≃ r Π , ι Π \Pi Π ρ y \rho_y ρ y r ∣ G F ′ ′ r|_{G_{F''}} r ∣ G F ′′ Π \Pi Π w ∣ 2 w\mid2 w ∣ 2 F ′ ′ F'' F ′′ r ∣ G F w ′ ′ r|_{G_{F''_w}} r ∣ G F w ′′ 0 0 0 [29 , 31 ] , Π \Pi Π w w w Π \Pi Π v 0 v_0 v 0 Π \Pi Π r ∣ G F w ′ ′ r|_{G_{F''_w}} r ∣ G F w ′′ w ∣ 2 w\mid2 w ∣ 2 ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v v ∣ 2 v\mid2 v ∣ 2
(2) Since y ∈ C y\in C y ∈ C ρ y \rho_y ρ y G F 1 G_{F_1} G F 1 p y \mathfrak{p}_y p y T F 1 ( U p ) [ 1 / 2 ] \mathbb{T}_{F_1}(U^p)[1/2] T F 1 ( U p ) [ 1/2 ] F 1 F_1 F 1 P y \mathfrak{P}_y P y p y \mathfrak{p}_y p y ψ w , 1 ( P y ) \psi_{w,1}(\mathfrak{P}_y) ψ w , 1 ( P y ) ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v w ∣ v ∣ 2 w\mid v\mid2 w ∣ v ∣ 2 ψ v , s ( y ) \psi_{v,s}(y) ψ v , s ( y )
(3) By (2), ψ w , 1 ( P y ) \psi_{w,1}(\mathfrak{P}_y) ψ w , 1 ( P y ) ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v w ∣ v ∣ 2 w\mid v\mid2 w ∣ v ∣ 2 F 1 F_1 F 1 a 1 = a 2 a_1=a_2 a 1 = a 2 ρ y ∣ G F 1 \rho_y|_{G_{F_1}} ρ y ∣ G F 1 ρ y \rho_y ρ y π \pi π ρ y \rho_y ρ y [29 , 31 ] , and π v 0 \pi_{v_0} π v 0
(4) This is the argument of [15 Proposition 5.4(3)] , with [37 Theorem 2] replaced by [29 , 31 ] . For v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v π v \pi_v π v ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v ψ v ( y ) \psi_v(y) ψ v ( y ) d d d B × B^\times B × U p U^p U p Σ o \Sigma_o Σ o S o S^o S o S o ⊗ E S^o\otimes E S o ⊗ E tr ρ y ( Frob w ) \operatorname{tr}\rho_y(\operatorname{Frob}_w) tr ρ y ( Frob w ) ψ v ( y ) \psi_v(y) ψ v ( y ) v ∈ Σ o v\in\Sigma_o v ∈ Σ o □ \square □
Proposition 6.2. Let y y y p y \mathfrak{p}_y p y R o R^o R o R p y o R^o_{\mathfrak{p}_y} R p y o 2 d − ∣ Σ o ∣ 2d-|\Sigma_o| 2 d − ∣ Σ o ∣ C C C Spec R o \operatorname{Spec}R^o Spec R o y y y dim C = 1 + 2 d − ∣ Σ o ∣ \dim C=1+2d-|\Sigma_o| dim C = 1 + 2 d − ∣ Σ o ∣
Proof. We follow [15 Proposition 5.5] . Let V = ad 0 ρ y V=\operatorname{ad}^0\rho_y V = ad 0 ρ y dim R p y o ≥ 2 d − ∣ Σ o ∣ \dim R^o_{\mathfrak{p}_y}\ge2d-|\Sigma_o| dim R p y o ≥ 2 d − ∣ Σ o ∣ t y t_y t y H L 1 ( F , V ) H^1_{\mathcal{L}}(F,V) H L 1 ( F , V ) L v \mathcal{L}_v L v H 1 ( G v , b v 0 ) H^1(G_v,\mathfrak{b}^0_v) H 1 ( G v , b v 0 ) v ∈ Σ o v \in\Sigma_o v ∈ Σ o L v = H 1 ( G v , V ) \mathcal{L}_v = H^1(G_v,V) L v = H 1 ( G v , V ) Σ \Sigma Σ t y t_y t y ρ y \rho_y ρ y [11 Theorem 2.22] and §2.
Local terms at 2. Let v ∣ 2 v \mid2 v ∣ 2 ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v ψ ′ \psi' ψ ′ ψ \psi ψ ψ ′ ψ − 1 = ε − 1 μ v \psi'\psi^{-1} = \varepsilon^{-1}\mu_v ψ ′ ψ − 1 = ε − 1 μ v μ v = θ v 0 θ v − 1 \mu_v = \theta_v^0\theta_v^{-1} μ v = θ v 0 θ v − 1 I v I_v I v V ( 1 ) V(1) V ( 1 ) ε \varepsilon ε ε 2 μ v − 1 \varepsilon^2\mu_v^{-1} ε 2 μ v − 1 μ v \mu_v μ v H 0 ( G v , V ( 1 ) ) = 0 H^0(G_v,V(1)) = 0 H 0 ( G v , V ( 1 )) = 0 m = 1 m = 1 m = 1 V / b v ≅ ε − 1 μ v V/\mathfrak{b}_v \cong\varepsilon^{-1}\mu_v V / b v ≅ ε − 1 μ v ε \varepsilon ε μ v \mu_v μ v ( b v ∨ ) ( 1 ) (\mathfrak{b}_v^\vee)(1) ( b v ∨ ) ( 1 ) dim L v − h 0 ( G v , V ) = 2 \dim L_v - h^0(G_v,V) = 2 dim L v − h 0 ( G v , V ) = 2 v ∈ Σ o v \in\Sigma_o v ∈ Σ o h 1 ( G v , V ) − h 0 ( G v , V ) = 3 h^1(G_v,V)-h^0(G_v,V) = 3 h 1 ( G v , V ) − h 0 ( G v , V ) = 3 v ∈ Σ n v \in\Sigma_n v ∈ Σ n H f 1 ( G v , V ) ⊂ L v H^1_f(G_v,V) \subset L_v H f 1 ( G v , V ) ⊂ L v H g 1 ( G v , ε − 1 μ v ) = 0 H^1_g(G_v,\varepsilon^{-1}\mu_v) = 0 H g 1 ( G v , ε − 1 μ v ) = 0 [7 Corollary 3.8.4] .
Local terms away from 2. Let v ∈ Σ v \in\Sigma v ∈ Σ v ∤ 2 v \nmid2 v ∤ 2 Proposition 6.1 (3) and Carayol’s theorem [9 Théorème (A)] , the Weil–Deligne representation of ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v π v \pi_v π v π v \pi_v π v ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v ρ y ∣ G v ≇ ρ y ∣ G v ( 1 ) \rho_y|_{G_v} \not\cong\rho_y|_{G_v}(1) ρ y ∣ G v ≅ ρ y ∣ G v ( 1 ) ε ∣ G v \varepsilon|_{G_v} ε ∣ G v π v \pi_v π v ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v ε ± 1 \varepsilon^{\pm1} ε ± 1 π v \pi_v π v ρ y ∣ G v \rho_y|_{G_v} ρ y ∣ G v ν \nu ν ν ε \nu\varepsilon ν ε Lemma 2.3 applies. In all cases H 0 ( G v , V ( 1 ) ) = 0 H^0(G_v,V(1)) = 0 H 0 ( G v , V ( 1 )) = 0 h 1 ( G v , V ) − h 0 ( G v , V ) = h 2 ( G v , V ) = 0 h^1(G_v,V)-h^0(G_v,V) = h^2(G_v,V) = 0 h 1 ( G v , V ) − h 0 ( G v , V ) = h 2 ( G v , V ) = 0
Global estimate. As in loc. cit., the Greenberg–Wiles formula [12 Theorem 2.19] , valid for p = 2 p = 2 p = 2 0 0 0 h 0 ( G v , V ) = 1 h^0(G_v,V) = 1 h 0 ( G v , V ) = 1 d d d h L 1 ( F , V ) − h L ⊥ 1 ( F , V ( 1 ) ) = 2 d − ∣ Σ o ∣ h^1_{\mathcal{L}}(F,V)-h^1_{\mathcal{L}^{\perp}}(F,V(1)) = 2d - |\Sigma_o| h L 1 ( F , V ) − h L ⊥ 1 ( F , V ( 1 )) = 2 d − ∣ Σ o ∣ h f 1 ( F , V ) = h f 1 ( F , V ( 1 ) ) h^1_f(F,V) = h^1_f(F,V(1)) h f 1 ( F , V ) = h f 1 ( F , V ( 1 )) H L ⊥ 1 ( F , V ( 1 ) ) ⊂ H f 1 ( F , V ( 1 ) ) H^1_{\mathcal{L}^{\perp}}(F,V(1)) \subset H^1_f(F,V(1)) H L ⊥ 1 ( F , V ( 1 )) ⊂ H f 1 ( F , V ( 1 )) [22 Theorem B] , which holds for every prime, including p = 2 p = 2 p = 2 H f 1 ( F , ad ρ y ) = 0 H^1_f(F,\operatorname{ad}\rho_y) = 0 H f 1 ( F , ad ρ y ) = 0 π \pi π dim t y ≤ 2 d − ∣ Σ o ∣ \dim t_y \le2d - |\Sigma_o| dim t y ≤ 2 d − ∣ Σ o ∣
Proof of Theorem B
We work with the data of §4.4, twisted if necessary. By Lemma 5.4 there is a good point y ∈ D ⊂ C y \in D \subset C y ∈ D ⊂ C Proposition 6.1 (4) it is the image of a prime of T m ′ ∘ \mathbb{T}^{\circ}_{\mathfrak{m}'} T m ′ ∘ W W W Spec T m ′ ∘ \operatorname{Spec}\mathbb{T}^{\circ}_{\mathfrak{m}'} Spec T m ′ ∘ Spec R ∘ \operatorname{Spec}R^{\circ} Spec R ∘ [15 §7] , W = V ( p W ) W = V(\mathfrak{p}_W) W = V ( p W ) p W \mathfrak{p}_W p W T m ′ ∘ \mathbb{T}^{\circ}_{\mathfrak{m}'} T m ′ ∘ q ′ ⊂ p W \mathfrak{q}' \subset\mathfrak{p}_W q ′ ⊂ p W R ∘ R^{\circ} R ∘ q ′ ⊂ p y \mathfrak{q}' \subset\mathfrak{p}_y q ′ ⊂ p y Proposition 6.2 , q ′ = q \mathfrak{q}' = \mathfrak{q} q ′ = q W ⊂ C W \subset C W ⊂ C Propositions 3.5 and 6.2 , dim W ≥ 1 + 2 d − ∣ Σ o ∣ = dim C \dim W \ge1+2d-|\Sigma_o| = \dim C dim W ≥ 1 + 2 d − ∣ Σ o ∣ = dim C W = C W = C W = C
In particular x ∈ W x \in W x ∈ W ν \nu ν k k k P \mathfrak{P} P T m ′ ∘ [ 1 / 2 ] \mathbb{T}^{\circ}_{\mathfrak{m}'}[1/2] T m ′ ∘ [ 1/2 ] x x x ψ v , 1 ( P ) = ψ v ( x ) = ψ v c l \psi_{v,1}(\mathfrak{P}) = \psi_v(x) = \psi_v^{\mathrm{cl}} ψ v , 1 ( P ) = ψ v ( x ) = ψ v cl v ∈ Σ o v \in\Sigma_o v ∈ Σ o p \mathfrak{p} p T [ 1 / 2 ] \mathbb{T}[1/2] T [ 1/2 ] ρ ( p ) ≅ ρ \rho(\mathfrak{p}) \cong\rho ρ ( p ) ≅ ρ Theorem 3.4 applies by (N) and (O). Hence ρ \rho ρ
Pro-modularity at 2
In this section F F F Q \mathbb{Q} Q 2 2 2 α = χ ˉ \alpha= \bar{\chi} α = χ ˉ G F G_F G F α ˉ \bar{\alpha} α ˉ G Q G_{\mathbb{Q}} G Q ρ ˉ s s = 1 ⊕ α \bar{\rho}^{\mathrm{ss}} = 1 \oplus\alpha ρ ˉ ss = 1 ⊕ α α ˉ \bar{\alpha} α ˉ 2 2 2 α ∣ G v = α ˉ ∣ G Q 2 ≠ 1 \alpha|_{G_v} = \bar{\alpha}|_{G_{Q_2}} \ne1 α ∣ G v = α ˉ ∣ G Q 2 = 1 v ∣ 2 v \mid2 v ∣ 2 k 0 = F 2 ( α ) k_0 = F_2(\alpha) k 0 = F 2 ( α ) q = ∣ k 0 ∣ ≥ 4 q = |k_0| \ge4 q = ∣ k 0 ∣ ≥ 4 K α K_\alpha K α α \alpha α F F F f f f 4 4 4 Theorem 8.4 , and enlarge Σ \Sigma Σ Spec R k 0 p s \operatorname{Spec}R^{\mathrm{ps}}_{k_0} Spec R k 0 ps ρ ∣ G K \rho|_{G_K} ρ ∣ G K e = [ F : Q ] e = [F:\mathbb{Q}] e = [ F : Q ] Σ 0 \Sigma_0 Σ 0 Σ \Sigma Σ 2 2 2 E E E β ∈ { α , α − 1 } \beta\in\{\alpha,\alpha^{-1}\} β ∈ { α , α − 1 } β \beta β k 0 ( β ) k_0(\beta) k 0 ( β ) k 0 ( β − 1 ) k_0(\beta^{-1}) k 0 ( β − 1 )
An auxiliary set of places
By Chebotarev’s theorem in the compositum of the Hilbert class field of F F F K α K_{\alpha} K α F F F P 0 P_{0} P 0 F F F Σ 0 \Sigma_{0} Σ 0 α ( Frob v ) ≠ 1 \alpha(\operatorname{Frob}_{v}) \ne1 α ( Frob v ) = 1 v ∈ P 0 v \in P_{0} v ∈ P 0 S ′ = Σ 2 ∪ P 0 S^{\prime}=\Sigma_{2}\cup P_{0} S ′ = Σ 2 ∪ P 0 O S ′ × \mathcal{O}_{S^{\prime}}^{\times} O S ′ × S ′ S^{\prime} S ′ S ′ S^{\prime} S ′ F F F v ∉ S ′ v \notin S^{\prime} v ∈ / S ′ x v ∈ F × x_{v}\in F^{\times} x v ∈ F × ( x v ) = v h v a v (x_{v})=v^{h_{v}}a_{v} ( x v ) = v h v a v h v h_{v} h v a v a_{v} a v S ′ S^{\prime} S ′
Lemma 8.1. Let n ≥ 1 n\ge1 n ≥ 1 w ∉ S ′ ∪ Σ 0 w\notin S^{\prime}\cup\Sigma_{0} w ∈ / S ′ ∪ Σ 0 F F F F ( μ 2 n + 1 , ( O S ′ × ) 1 / 2 n ) F(\mu_{2^{n+1}},(\mathcal{O}_{S^{\prime}}^{\times})^{1/2^{n}}) F ( μ 2 n + 1 , ( O S ′ × ) 1/ 2 n ) L w / F L_{w}/F L w / F 2 n 2^{n} 2 n w w w S ′ S^{\prime} S ′
For a place v ∉ S ′ ∪ { w } v\notin S^{\prime}\cup\{w\} v ∈ / S ′ ∪ { w } Frob v \operatorname{Frob}_{v} Frob v Gal ( L w / F ) \operatorname{Gal}(L_{w}/F) Gal ( L w / F ) x v x_{v} x v w w w Gal ( L w / F ) 2 \operatorname{Gal}(L_{w}/F)^{2} Gal ( L w / F ) 2 x v x_{v} x v w w w
Proof. Let Cl w , S ′ \operatorname{Cl}_{w,S^{\prime}} Cl w , S ′ w w w S ′ S^{\prime} S ′
( O F / w ) × / im ( O S ′ × ) ⟶ Cl w , S ′ ⟶ Cl S ′ ⟶ 1. (\mathcal{O}_{F}/w)^{\times}/\operatorname{im}(\mathcal{O}_{S^{\prime}}^{\times})\longrightarrow\operatorname{Cl}_{w,S^{\prime}}\longrightarrow\operatorname{Cl}_{S^{\prime}}\longrightarrow1. ( O F / w ) × / im ( O S ′ × ) ⟶ Cl w , S ′ ⟶ Cl S ′ ⟶ 1. As Cl S ′ \operatorname{Cl}_{S^{\prime}} Cl S ′ Cl w , S ′ \operatorname{Cl}_{w,S^{\prime}} Cl w , S ′ A w = ( O F / w ) × / im ( O S ′ × ) A_{w}=(\mathcal{O}_{F}/w)^{\times}/\operatorname{im}(\mathcal{O}_{S^{\prime}}^{\times}) A w = ( O F / w ) × / im ( O S ′ × ) ( O F / w ) × (\mathcal{O}_{F}/w)^{\times} ( O F / w ) × N ( w ) − 1 N(w)-1 N ( w ) − 1 2 n + 1 2^{n+1} 2 n + 1 O S ′ × \mathcal{O}_{S^{\prime}}^{\times} O S ′ × 2 n 2^{n} 2 n A w A_{w} A w 2 n 2^{n} 2 n 2 n 2^{n} 2 n L w L_{w} L w w w w ( O F / w ) × (\mathcal{O}_{F}/w)^{\times} ( O F / w ) × S ′ S^{\prime} S ′ v ∉ S ′ ∪ { w } v\notin S^{\prime}\cup\{w\} v ∈ / S ′ ∪ { w } v h v v^{h_{v}} v h v x v x_{v} x v A w A_{w} A w a v a_{v} a v Cl w , S ′ \operatorname{Cl}_{w,S^{\prime}} Cl w , S ′ h v h_{v} h v A w A_{w} A w Frob v \operatorname{Frob}_{v} Frob v Gal ( L w / F ) / Gal ( L w / F ) 2 \operatorname{Gal}(L_{w}/F)/\operatorname{Gal}(L_{w}/F)^{2} Gal ( L w / F ) / Gal ( L w / F ) 2 x v x_{v} x v ( O F / w ) × (\mathcal{O}_{F}/w)^{\times} ( O F / w ) ×
We write γ w : Gal ( L w / F ) → ∼ Z / 2 n Z \gamma_{w}:\operatorname{Gal}(L_{w}/F)\xrightarrow{\sim}\mathbb{Z}/2^{n}\mathbb{Z} γ w : Gal ( L w / F ) ∼ Z / 2 n Z κ w = γ w m o d 2 \kappa_{w}=\gamma_{w}\bmod2 κ w = γ w mod 2 δ w \delta_{w} δ w G F w G_{F_{w}} G F w κ w \kappa_{w} κ w
Lemma 8.2. Fix finitely many classes in H 1 ( G F , k 0 ( β ) ) H^{1}(G_{F},k_{0}(\beta)) H 1 ( G F , k 0 ( β )) β ∈ { α , α − 1 } \beta\in\{\alpha,\alpha^{-1}\} β ∈ { α , α − 1 } E ′ / F E^{\prime}/F E ′ / F w w w F F F K α E ′ K_{\alpha}E^{\prime} K α E ′ B B B F F F β \beta β k 0 k_{0} k 0 w w w
Proof. Put N = G K α E ′ N=G_{K_{\alpha}E^{\prime}} N = G K α E ′ K α / F K_{\alpha}/F K α / F E ′ / F E^{\prime}/F E ′ / F Gal ( K α E ′ / F ) = C ⋊ P \operatorname{Gal}(K_{\alpha}E^{\prime}/F)=C\rtimes P Gal ( K α E ′ / F ) = C ⋊ P C C C k 0 ( β ) k_{0}(\beta) k 0 ( β ) P P P N N N N N N β ( C ) \beta(C) β ( C ) k 0 k_{0} k 0
The fields
Proposition 8.3. There is a constant U U U F F F α \alpha α Σ 0 \Sigma_{0} Σ 0 P 0 P_{0} P 0 n > a ≥ 1 n>a\ge1 n > a ≥ 1 F ⊂ F a ⊂ F n F\subset F_{a}\subset F_{n} F ⊂ F a ⊂ F n 2 a 2^{a} 2 a 2 n 2^{n} 2 n W W W t ≤ U t\le U t ≤ U F n F_{n} F n Σ 0 \Sigma_{0} Σ 0 S n S_{n} S n F n F_{n} F n ∞ \infty ∞ Σ 0 \Sigma_{0} Σ 0 W W W u n ≤ ∣ Σ 0 ∣ + t u_{n}\le|\Sigma_{0}|+t u n ≤ ∣ Σ 0 ∣ + t β ∈ { α , α − 1 } \beta\in\{\alpha,\alpha^{-1}\} β ∈ { α , α − 1 }
{ c ∈ H 1 ( G F n , S n , k 0 ( β ) ) : loc v ( c ) = 0 for all v ∈ S n } = 0 ; (1) \{c \in H^{1}(G_{F_n,S_n}, k_0(\beta)) : \operatorname{loc}_v(c)=0 \text{ for all } v \in S_n\}=0; \tag*{(1)} { c ∈ H 1 ( G F n , S n , k 0 ( β )) : loc v ( c ) = 0 for all v ∈ S n } = 0 ; (1) dim H 1 ( G F n , S n , k 0 ( β ) ) ≤ [ F n : Q ] + u n ; (2) \dim H^{1}(G_{F_n,S_n}, k_0(\beta)) \le[F_n:\mathbb{Q}]+u_n; \tag*{(2)} dim H 1 ( G F n , S n , k 0 ( β )) ≤ [ F n : Q ] + u n ; (2) restriction H 1 ( G F a , S a , k 0 ( β ) ) → H 1 ( G F n , S n , k 0 ( β ) ) is injective, and α ∣ G F n ≠ 1 , α ∣ G F n , w ≠ 1 for w ∣ 2. (3) \text{restriction }H^{1}(G_{F_a,S_a}, k_0(\beta))\to H^{1}(G_{F_n,S_n}, k_0(\beta))\text{ is injective, and }\alpha|_{G_{F_n}}\ne1,\ \alpha|_{G_{F_{n,w}}}\ne1\text{ for }w\mid2. \tag*{(3)} restriction H 1 ( G F a , S a , k 0 ( β )) → H 1 ( G F n , S n , k 0 ( β )) is injective, and α ∣ G F n = 1 , α ∣ G F n , w = 1 for w ∣ 2. (3) Proof. We follow the case B = Q B=\mathbb{Q} B = Q w 1 , … , w t w_1,\ldots,w_t w 1 , … , w t Lemma 8.1 , let F n F_n F n γ = ∑ j c j γ w j \gamma=\sum_j c_j\gamma_{w_j} γ = ∑ j c j γ w j c j c_j c j F a F_a F a 2 a 2^a 2 a β ∈ { α , α − 1 } \beta\in\{\alpha,\alpha^{-1}\} β ∈ { α , α − 1 } X β X^\beta X β F F F k 0 ( β ) k_0(\beta) k 0 ( β ) k 0 δ w j k_0\delta_{w_j} k 0 δ w j w j w_j w j Y β Y^\beta Y β k 0 ( β − 1 ) k_0(\beta^{-1}) k 0 ( β − 1 )
dim X β − dim Y β = − e , \dim X^\beta-\dim Y^\beta=-e, dim X β − dim Y β = − e , since a real zero condition contributes − 1 -1 − 1 α ∣ G v ≠ 1 \alpha|_{G_v}\ne1 α ∣ G v = 1 k 0 δ w j k_0\delta_{w_j} k 0 δ w j 1 − 1 = 0 1-1=0 1 − 1 = 0 Y β ≠ 0 Y^\beta\ne0 Y β = 0
We choose the w j w_j w j w j w_j w j S ′ ∪ Σ 0 ∪ { w 1 , … , w j − 1 } S'\cup\Sigma_0\cup\{w_1,\ldots,w_{j-1}\} S ′ ∪ Σ 0 ∪ { w 1 , … , w j − 1 } N ( w j ) ≡ 1 ( m o d 4 ) N(w_j)\equiv1\pmod{4} N ( w j ) ≡ 1 ( mod 4 ) K α K_\alpha K α F ( μ 2 n + 1 , ( O S × ) 1 / 2 n ) F(\mu_{2^{n+1}},(\mathcal{O}_S^\times)^{1/2^n}) F ( μ 2 n + 1 , ( O S × ) 1/ 2 n )
(i) if j = 1 j=1 j = 1 x v x_v x v w 1 w_1 w 1 v ∈ Σ 0 v\in\Sigma_0 v ∈ Σ 0 j ≥ 2 j\ge2 j ≥ 2 x v x_v x v w j w_j w j v ∈ Σ 0 v\in\Sigma_0 v ∈ Σ 0
(ii) for i < j i<j i < j x w i x_{w_i} x w i w j w_j w j w j w_j w j L w i L_{w_i} L w i
(iii) if j ≥ 2 j\ge2 j ≥ 2 w j w_j w j X β X^\beta X β Y β Y^\beta Y β w 1 , … , w j − 1 w_1,\ldots,w_{j-1} w 1 , … , w j − 1
As for the first conductor in [OAI, §6.3], no evaluation condition is imposed at w 1 w_1 w 1 M n = F ( μ 2 n + 1 , ( O S × ) 1 / 2 n ) M_n=F(\mu_{2^{n+1}},(\mathcal{O}_S^\times)^{1/2^n}) M n = F ( μ 2 n + 1 , ( O S × ) 1/ 2 n ) F F F F F F j = 1 j=1 j = 1 K α M n K_\alpha M_n K α M n F ( x v ) F(\sqrt{x_v}) F ( x v ) v ∈ Σ 0 v\in\Sigma_0 v ∈ Σ 0 v ∉ S ′ v\notin S' v ∈ / S ′ M n M_n M n v v v F ( x v ′ ) F(\sqrt{x_{v'}}) F ( x v ′ ) v ′ ∈ Σ 0 ∖ { v } v'\in\Sigma_0\setminus\{v\} v ′ ∈ Σ 0 ∖ { v } x v ′ x_{v'} x v ′ v v v F ( x v ) F(\sqrt{x_v}) F ( x v ) v v v x v x_v x v
Gal ( M n ∏ v ∈ Σ 0 F ( x v ) / F ) = Gal ( M n / F ) × ( Z / 2 ) Σ 0 , \operatorname{Gal}\left(M_n\prod_{v\in\Sigma_0}F(\sqrt{x_v})/F\right)=\operatorname{Gal}(M_n/F)\times(\mathbb{Z}/2)^{\Sigma_0}, Gal ( M n v ∈ Σ 0 ∏ F ( x v ) / F ) = Gal ( M n / F ) × ( Z /2 ) Σ 0 , and, K α K_\alpha K α w 1 w_1 w 1 j ≥ 2 j\ge2 j ≥ 2 w j w_j w j K α E j ′ K_\alpha E'_j K α E j ′ E j ′ E'_j E j ′ M n M_n M n F ( x v ) F(\sqrt{x_v}) F ( x v ) v ∈ Σ 0 v\in\Sigma_0 v ∈ Σ 0 F ( x w i ) F(\sqrt{x_{w_i}}) F ( x w i ) L w i L_{w_i} L w i i < j i<j i < j E j ′ / F E'_j/F E j ′ / F K α K_\alpha K α Lemma 8.2 with E ′ = E j ′ E'=E'_j E ′ = E j ′ X β X^\beta X β Y β − 1 Y^{\beta^{-1}} Y β − 1 1 / q 1/q 1/ q 2 / q < 1 2/q<1 2/ q < 1 w j w_j w j
Adding w j w_j w j w j w_j w j k 0 δ w j k_0\delta_{w_j} k 0 δ w j γ w j \gamma_{w_j} γ w j w i w_i w i Lemma 8.1 ). For j ≥ 2 j\ge2 j ≥ 2 w j w_j w j K α K_\alpha K α N ( w j ) ≡ 1 ( m o d 4 ) N(w_j)\equiv1\pmod{4} N ( w j ) ≡ 1 ( mod 4 ) X β X^\beta X β C 0 = ∑ β dim H 1 ( G F , Σ 0 ∪ Σ 2 ∪ ∞ , k 0 ( β ) ) C_0=\sum_\beta\dim H^1(G_{F,\Sigma_0\cup\Sigma_2\cup\infty},k_0(\beta)) C 0 = ∑ β dim H 1 ( G F , Σ 0 ∪ Σ 2 ∪ ∞ , k 0 ( β )) X β X^\beta X β dim H 1 ( G F , Σ 0 ∪ Σ 2 ∪ ∞ , k 0 ( β ) ) + 1 \dim H^1(G_{F,\Sigma_0\cup\Sigma_2\cup\infty},k_0(\beta))+1 dim H 1 ( G F , Σ 0 ∪ Σ 2 ∪ ∞ , k 0 ( β )) + 1 w 1 w_1 w 1 dim k 0 δ w 1 = 1 \dim k_0\delta_{w_1}=1 dim k 0 δ w 1 = 1 ∑ β dim X β ≤ C 0 + 2 \sum_\beta\dim X^\beta\le C_0+2 ∑ β dim X β ≤ C 0 + 2 j ≥ 2 j\ge2 j ≥ 2 t ≤ C 0 + 3 t\le C_0+3 t ≤ C 0 + 3 X β X^\beta X β n n n a a a
We check the properties. The places of S ′ S' S ′ 2 2 2 L w j L_{w_j} L w j F n F_n F n v ∈ Σ 0 v \in\Sigma_0 v ∈ Σ 0 γ ( Frob v ) = c 1 γ w 1 ( Frob v ) + ∑ j ≥ 2 c j γ w j ( Frob v ) \gamma(\operatorname{Frob}_v) = c_1\gamma_{w_1}(\operatorname{Frob}_v) + \sum_{j \ge2} c_j\gamma_{w_j}(\operatorname{Frob}_v) γ ( Frob v ) = c 1 γ w 1 ( Frob v ) + ∑ j ≥ 2 c j γ w j ( Frob v ) v v v F n F_n F n w j w_j w j γ \gamma γ G F w j G_{F_{w_j}} G F w j c j γ w j c_j\gamma_{w_j} c j γ w j 2 n 2^n 2 n w j w_j w j F n F_n F n δ w j \delta_{w_j} δ w j u n ≤ ∣ Σ 0 ∣ + t u_n \le|\Sigma_0| + t u n ≤ ∣ Σ 0 ∣ + t
For (1), let Z Z Z 2 2 2 Gal ( F n / F ) \operatorname{Gal}(F_n/F) Gal ( F n / F ) F F F k 0 ( β ) G F n = 0 k_0(\beta)^{G_{F_n}} = 0 k 0 ( β ) G F n = 0 X β X^\beta X β 2 2 2 v ∈ Σ 0 v \in\Sigma_0 v ∈ Σ 0 S n S_n S n F n / F F_n/F F n / F w j w_j w j k 0 δ w j k_0\delta_{w_j} k 0 δ w j X β = 0 X^\beta= 0 X β = 0 Z = 0 Z = 0 Z = 0 S n S_n S n β − 1 \beta^{-1} β − 1 0 0 0 1 1 1 α ∣ G v ≠ 1 \alpha|_{G_v} \ne1 α ∣ G v = 1 S n S_n S n h 2 ≤ 1 h^2 \le1 h 2 ≤ 1 α \alpha α F n / F F_n/F F n / F 2 2 2 α \alpha α G F n G_{F_n} G F n G F n , w = G F v G_{F_{n,w}} = G_{F_v} G F n , w = G F v w ∣ v ∣ 2 w \mid v \mid2 w ∣ v ∣ 2 U = ∣ Σ 0 ∣ + 3 + C 0 U = |\Sigma_0| + 3 + C_0 U = ∣ Σ 0 ∣ + 3 + C 0
Proof of (PM)
Theorem 8.4. Assume (R), that ρ \rho ρ 2 2 2 det ρ = ε w χ 0 \det\rho= \varepsilon^w\chi_0 det ρ = ε w χ 0 χ 0 \chi_0 χ 0
Proof. By Lemma 2.4, ρ \rho ρ α ∣ G Q 2 ≠ 1 \alpha|_{G_{\mathbb{Q}_2}} \ne1 α ∣ G Q 2 = 1 Q ⊂ A ⊂ F \mathbb{Q} \subset A \subset F Q ⊂ A ⊂ F F ⊂ F a ⊂ F n F \subset F_a \subset F_n F ⊂ F a ⊂ F n
Leopoldt’s conjecture. In [OAI, Lemma 3.8] and in the proof of [OAI, Lemma 7.3], abelian Leopoldt is used to bound by 1 1 1 F a F_a F a F n F_n F n Q \mathbb{Q} Q K K K d K d_K d K 1 + δ K ≤ ( d K + 1 ) / 2 1 + \delta_K \le(d_K + 1)/2 1 + δ K ≤ ( d K + 1 ) /2 ( d K + 1 ) / 2 (d_K+1)/2 ( d K + 1 ) /2 F n F_n F n d n + u n + 3 + ( d n − 1 ) / 2 d_n + u_n + 3 + (d_n - 1)/2 d n + u n + 3 + ( d n − 1 ) /2 d n = [ F n : Q ] = 2 n e d_n = [F_n : \mathbb{Q}] = 2^n e d n = [ F n : Q ] = 2 n e 3 + δ F n 3 + \delta_{F_n} 3 + δ F n d K d_K d K
Numerical conditions. In place of [OAI, (39)] we choose a a a n n n d n > 2 u n + 10 d_n > 2u_n+10 d n > 2 u n + 10 d a > 2 ( U + C ) + 10 d_a > 2(U+C)+10 d a > 2 ( U + C ) + 10 C C C F n F_n F n 2 d n − 1 2d_n - 1 2 d n − 1 d n > 2 u n + 7 d_n > 2u_n + 7 d n > 2 u n + 7 2 d n − 4 − u n > d n 2d_n - 4 - u_n > d_n 2 d n − 4 − u n > d n F a F_a F a d a d_a d a d K ≥ ( d K + 1 ) / 2 d_K \ge(d_K + 1)/2 d K ≥ ( d K + 1 ) /2
Fields and Selmer bounds. Proposition 8.3 replaces [OAI, Proposition 6.1]. Property (2) is the bound [OAI, (26)] in the unmarked case, and property (3) gives the residual data used in [OAI, Lemma 7.5 and §7.4]. There are no marks, so every point of the residual
fibre is transverse, and in [OAI, §7.4] any stable lattice of the seed and target curves gives a point of the residual scheme over F n F_n F n
(4) The seed. By [OAI, Lemma 7.2], applied over Q \mathbb{Q} Q 1 ⊕ α ~ 1 \oplus\widetilde{\alpha} 1 ⊕ α f f f 1 ⊕ α ~ 1 \oplus\widetilde{\alpha} 1 ⊕ α Σ \Sigma Σ Σ \Sigma Σ F a F_a F a Q ⊂ F ⊂ F a \mathbb{Q} \subset F \subset F_a Q ⊂ F ⊂ F a [1 ] , is cuspidal by regularity, as in [OAI, §7.4], and has residual representation ρ ‾ s s ∣ G F a \overline{\rho}^{\mathrm{ss}}|_{G_{F_a}} ρ ss ∣ G F a A A A
(5) The target. The target locus over F a F_a F a ρ ∣ G F a \rho|_{G_{F_a}} ρ ∣ G F a
The remaining arguments of [OAI, §7], together with the propagation theorem [OAI, Theorem 5.1], apply verbatim: they are formulated over totally real solvable fields of even degree in which 2 2 2 F a F_a F a F n F_n F n Q \mathbb{Q} Q 2 2 2 F F F Q \mathbb{Q} Q F F F 2 2 2 F n F_n F n F n / F a F_n/F_a F n / F a Spec R F a p s \operatorname{Spec}R^{\mathrm{ps}}_{F_a} Spec R F a ps ρ ∣ G F a \rho|_{G_{F_a}} ρ ∣ G F a 2 2 2 Q \mathbb{Q} Q K = F a K=F_a K = F a F F F F 1 F_1 F 1 K K K Q \mathbb{Q} Q K ′ K' K ′ ρ ∣ G K ′ \rho|_{G_{K'}} ρ ∣ G K ′ K K K K / F K/F K / F
Fully reducible loci
The argument of Theorem 8.4 applies to many more loci than the components through the point of ρ \rho ρ
Theorem 8.5. Let K = F a K=F_a K = F a F 1 ⊃ K F_1 \supset K F 1 ⊃ K Q \mathbb{Q} Q 2 2 2 U 1 p U^p_1 U 1 p Spec R K p s \operatorname{Spec}R^{\mathrm{ps}}_K Spec R K ps Spec T F 1 ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}(U^p_1) Spec T F 1 ( U 1 p )
Proof. Let d = [ K : Q ] d=[K:\mathbb{Q}] d = [ K : Q ] J \mathcal{J} J Spec R K p s \operatorname{Spec}R^{\mathrm{ps}}_K Spec R K ps 2 d − 3 2d-3 2 d − 3 C ^ ∈ J \widehat{C}\in\mathcal{J} C ∈ J a a a F 1 F_1 F 1 U 1 p U^p_1 U 1 p C ^ ∈ J \widehat{C}\in\mathcal{J} C ∈ J
Let Z Z Z Spec R K p s \operatorname{Spec}R^{\mathrm{ps}}_K Spec R K ps Z Z Z C ^ ∈ J \widehat{C}\in\mathcal{J} C ∈ J C ⊂ Z C\subset Z C ⊂ Z ρ C \rho_C ρ C Z Z Z C C C ρ ‾ b \overline{\rho}_b ρ b [32 Lemma 2.13] ; this defines a point of the framed representation scheme with fixed determinant, in the completion of this scheme at the closed point y y y ρ ‾ b \overline{\rho}_b ρ b 2 d 2d 2 d 2 d + 3 − h 0 ( ( ad 0 ρ ‾ ) ∨ ( 1 ) ) ≥ 2 d 2d+3-h^{0}((\operatorname{ad}^{0}\overline{\rho})\vee(1))\geq2d 2 d + 3 − h 0 (( ad 0 ρ ) ∨ ( 1 )) ≥ 2 d Y i Y_i Y i C C C P i P_i P i Spec R K p s \operatorname{Spec}R^{\mathrm{ps}}_{K} Spec R K ps 2 d − 3 2d-3 2 d − 3 C ⊂ P i C\subset P_i C ⊂ P i P i P_i P i Spec R K p s \operatorname{Spec}R^{\mathrm{ps}}_{K} Spec R K ps J \mathcal{J} J Z Z Z J \mathcal{J} J Z Z Z Z Z Z
In particular, Theorem 8.5 contains (PM) for the components through the point of ρ \rho ρ
Proposition 8.6. Let K K K F 1 F_1 F 1 Z Z Z Spec R K r e d \operatorname{Spec}R^{\mathrm{red}}_{K} Spec R K red O \mathcal{O} O Z Z Z Λ K \Lambda_K Λ K
Proof. Let Z 1 Z_1 Z 1 Z Z Z Spec R F 1 r e d \operatorname{Spec}R^{\mathrm{red}}_{F_1} Spec R F 1 red Z Z Z Z 1 Z_1 Z 1 Z 1 Z_1 Z 1 O \mathcal{O} O Z 1 Z_1 Z 1 Λ F 1 \Lambda_{F_1} Λ F 1 Λ F 1 → Λ K \Lambda_{F_1}\to\Lambda_K Λ F 1 → Λ K w w w F 1 F_1 F 1 K K K G F 1 , w = G K v G_{F_1,w}=G_{K_v} G F 1 , w = G K v Z 1 Z_1 Z 1 Λ F 1 \Lambda_{F_1} Λ F 1
If the generic point of Z 1 Z_1 Z 1 G F 1 G_{F_1} G F 1 Z 1 Z_1 Z 1 det ( ψ ⊕ χ ψ − 1 ) \det(\psi\oplus\chi\psi^{-1}) det ( ψ ⊕ χ ψ − 1 ) O [ [ G F 1 , Σ a b ( 2 ) ] ] \mathcal{O}[[G^{\mathrm{ab}}_{F_1,\Sigma}(2)]] O [[ G F 1 , Σ ab ( 2 )]] Λ F 1 \Lambda_{F_1} Λ F 1 ψ \psi ψ G F 1 , Σ a b ( 2 ) G^{\mathrm{ab}}_{F_1,\Sigma}(2) G F 1 , Σ ab ( 2 ) 1 + 4 Z 2 1+4\mathbb{Z}_2 1 + 4 Z 2 Σ \Sigma Σ Λ F 1 \Lambda_{F_1} Λ F 1
Otherwise, the generic point of Z Z Z Z 1 Z_1 Z 1 Spec T F 1 ( U p ) \operatorname{Spec}\mathbb{T}_{F_1}(U^{\mathrm{p}}) Spec T F 1 ( U p ) z z z Z 1 [ 1 / 2 ] Z_1[1/2] Z 1 [ 1/2 ] ρ z \rho_z ρ z Z 1 Z_1 Z 1 Z 1 Z_1 Z 1 O \mathcal{O} O z z z T F 1 [ 1 / 2 ] \mathbb{T}_{F_1}[1/2] T F 1 [ 1/2 ] F 1 F_1 F 1 z z z Spec T F 1 , m ′ ′ o r d \operatorname{Spec}\mathbb{T}^{\mathrm{ord}}_{F_1,\mathfrak{m}^{\prime\prime}} Spec T F 1 , m ′′ ord m ′ ′ \mathfrak{m}^{\prime\prime} m ′′ T F 1 o r d \mathbb{T}^{\mathrm{ord}}_{F_1} T F 1 ord Y m ′ ′ Y_{\mathfrak{m}^{\prime\prime}} Y m ′′ Spec R F 1 r e d \operatorname{Spec}R^{\mathrm{red}}_{F_1} Spec R F 1 red R F 1 r e d → T F 1 , m ′ ′ o r d R^{\mathrm{red}}_{F_1}\to\mathbb{T}^{\mathrm{ord}}_{F_1,\mathfrak{m}^{\prime\prime}} R F 1 red → T F 1 , m ′′ ord ψ w ( 1 ) \psi_w^{(1)} ψ w ( 1 ) χ ψ w ( 1 ) , − 1 \chi\psi_w^{(1),-1} χ ψ w ( 1 ) , − 1 ψ w , 1 \psi_{w,1} ψ w , 1 ψ w , 1 \psi_{w,1} ψ w , 1 m ′ ′ \mathfrak{m}^{\prime\prime} m ′′ Z 1 ⊂ Y m ′ ′ Z_1\subset Y_{\mathfrak{m}^{\prime\prime}} Z 1 ⊂ Y m ′′ T F 1 , m ′ ′ o r d \mathbb{T}^{\mathrm{ord}}_{F_1,\mathfrak{m}^{\prime\prime}} T F 1 , m ′′ ord Λ F 1 \Lambda_{F_1} Λ F 1 Ord B ( S F 1 ) \operatorname{Ord}_{B}(S_{F_1}) Ord B ( S F 1 ) Λ F 1 \Lambda_{F_1} Λ F 1 Λ F 1 \Lambda_{F_1} Λ F 1 Y m ′ ′ Y_{\mathfrak{m}^{\prime\prime}} Y m ′′ ψ w ( 1 ) \psi_w^{(1)} ψ w ( 1 ) ψ w , 1 \psi_{w,1} ψ w , 1 Λ F 1 \Lambda_{F_1} Λ F 1 Z 1 Z_1 Z 1 Λ F 1 \Lambda_{F_1} Λ F 1
Ordinary parts at arbitrary primes
We now remove the flatness assumption in Proposition 8.6, by an analogue of Proposition 3.3 for arbitrary primes of the Hecke algebra, including primes of characteristic 2, which are reducible at every place above 2. It uses the structure of the generic blocks of GL 2 ( Q 2 ) \operatorname{GL}_2(\mathbb{Q}_2) GL 2 ( Q 2 )
Fix a place v ∣ 2 v\mid2 v ∣ 2 F ′ F' F ′ B v \mathcal{B}_v B v 1 ⊕ α ∣ G v 1\oplus\alpha|_{G_v} 1 ⊕ α ∣ G v α ∣ G v ≠ 1 \alpha|_{G_v}\ne1 α ∣ G v = 1 B v = { π 1 , π 2 } \mathcal{B}_v=\{\pi_1,\pi_2\} B v = { π 1 , π 2 } π i = ( Ind B ‾ G χ i ) s m \pi_i=(\operatorname{Ind}_{\overline{B}}^{G}\chi_i)^{\mathrm{sm}} π i = ( Ind B G χ i ) sm χ 1 , χ 2 \chi_1,\chi_2 χ 1 , χ 2 T ( Q 2 ) T(\mathbb{Q}_2) T ( Q 2 ) [27 §4.1, type (ii)] . Let J v J_v J v π v \pi_v π v J v = J 1 ⊕ J 2 J_v=J_1\oplus J_2 J v = J 1 ⊕ J 2 P v = J v ∨ P_v=J_v^\vee P v = J v ∨ E v = End ( P v ) E_v=\operatorname{End}(P_v) E v = End ( P v ) e 1 , e 2 ∈ E v e_1,e_2\in E_v e 1 , e 2 ∈ E v Z v Z_v Z v E v E_v E v
We distinguish the block and the global normalizations. The block deformation ring R v b l k R^{\mathrm{blk}}_v R v blk ψ ε = χ ε 2 \psi\varepsilon=\chi\varepsilon^2 ψ ε = χ ε 2 R v p s R^{\mathrm{ps}}_v R v ps χ \chi χ t v b l k = ε t v t_v^{\mathrm{blk}}=\varepsilon t_v t v blk = ε t v f v f_v f v R v r e d = R v p s / ( f v ) R^{\mathrm{red}}_v=R^{\mathrm{ps}}_v/(f_v) R v red = R v ps / ( f v )
We use the following inputs.
(P1) Z v ≃ R v b l k Z_v \simeq R_v^{\mathrm{blk}} Z v ≃ R v blk Z v ≃ R v p s Z_v \simeq R_v^{\mathrm{ps}} Z v ≃ R v ps [25 Theorems 1.3 and 2.23] ; the result is stated there for all p p p p = 2 p=2 p = 2 χ 1 χ 2 − 1 ≠ 1 , ω ± 1 \chi_1\chi_2^{-1} \ne1, \omega^{\pm1} χ 1 χ 2 − 1 = 1 , ω ± 1
(P2) E v E_v E v Z v Z_v Z v 4 4 4 E v = Z v e 1 ⊕ Z v Φ ~ 12 ⊕ Z v Φ ~ 21 ⊕ Z v e 2 E_v = Z_v e_1 \oplus Z_v\widetilde{\Phi}_{12} \oplus Z_v\widetilde{\Phi}_{21} \oplus Z_v e_2 E v = Z v e 1 ⊕ Z v Φ 12 ⊕ Z v Φ 21 ⊕ Z v e 2 e i E v e i = Z v e i e_iE_ve_i = Z_ve_i e i E v e i = Z v e i Φ ~ 12 Φ ~ 21 = c e 1 \widetilde{\Phi}_{12}\widetilde{\Phi}_{21} = ce_1 Φ 12 Φ 21 = c e 1 Φ ~ 21 Φ ~ 12 = c e 2 \widetilde{\Phi}_{21}\widetilde{\Phi}_{12} = ce_2 Φ 21 Φ 12 = c e 2 ( c ) = ( f v ) (c) = (f_v) ( c ) = ( f v ) [25 Theorem 2.23] . The unnormalized Colmez module satisfies g 2 − t v b l k ( g ) g + χ ( g ) ε ( g ) 2 = 0 g^2-t_v^{\mathrm{blk}}(g)g+\chi(g)\varepsilon(g)^2=0 g 2 − t v blk ( g ) g + χ ( g ) ε ( g ) 2 = 0 V ˇ χ ( P v ) = V ˇ ( P v ) ⊗ ε − 1 \check{V}_{\chi}(P_v)=\check{V}(P_v)\otimes\varepsilon^{-1} V ˇ χ ( P v ) = V ˇ ( P v ) ⊗ ε − 1 g 2 − t v ( g ) g + χ ( g ) = 0 g^2-t_v(g)g+\chi(g)=0 g 2 − t v ( g ) g + χ ( g ) = 0 [25 Theorem 2.23 and proof of Corollary 2.25] . For principal series we likewise multiply the Colmez character by ε − 1 \varepsilon^{-1} ε − 1
(P3) Ord B ( J i ) \operatorname{Ord}_B(J_i) Ord B ( J i ) J χ i J_{\chi_i} J χ i χ i \chi_i χ i T ( Q 2 ) T(\mathbb{Q}_2) T ( Q 2 ) [27 Lemma 4.4] . Its endomorphism ring Λ i \Lambda_i Λ i χ i \chi_i χ i O [ [ Q 2 × ( 2 ) ] ] \mathcal{O}[[\mathbb{Q}_2^\times(2)]] O [[ Q 2 × ( 2 )]] Hom ( J χ 1 , J χ 2 ) = 0 \operatorname{Hom}(J_{\chi_1},J_{\chi_2})=0 Hom ( J χ 1 , J χ 2 ) = 0 χ 1 ≠ χ 2 \chi_1 \ne\chi_2 χ 1 = χ 2
Lemma 8.7. The ring E v E_v E v Ord B ( J v ) = J χ 1 ⊕ J χ 2 \operatorname{Ord}_B(J_v)=J_{\chi_1}\oplus J_{\chi_2} Ord B ( J v ) = J χ 1 ⊕ J χ 2 E v → Λ 1 × Λ 2 E_v\to\Lambda_1\times\Lambda_2 E v → Λ 1 × Λ 2 I v I_v I v e 1 E v e 2 e_1E_ve_2 e 1 E v e 2 e 2 E v e 1 e_2E_ve_1 e 2 E v e 1 f v f_v f v I v ∩ e i E v e i = f v Z v e i I_v\cap e_iE_ve_i=f_vZ_ve_i I v ∩ e i E v e i = f v Z v e i I v 2 ⊂ f v E v I_v^2\subset f_vE_v I v 2 ⊂ f v E v N N N E v E_v E v f v f_v f v N = I v N N=I_vN N = I v N N = N I v N=NI_v N = N I v N = 0 N=0 N = 0
Proof. By functoriality E v E_v E v Ord B ( J v ) \operatorname{Ord}_B(J_v) Ord B ( J v ) 0 0 0 K i K_i K i Z v → Λ i Z_v\to\Lambda_i Z v → Λ i ϑ \vartheta ϑ T ( Q 2 ) T(\mathbb{Q}_2) T ( Q 2 ) χ i \chi_i χ i γ \gamma γ ( Ind B ‾ T ϑ ) c o n t (\operatorname{Ind}_{\overline{B}}^T\vartheta)_{\mathrm{cont}} ( Ind B T ϑ ) cont γ ( g ) ≠ 0 \gamma(g)\ne0 γ ( g ) = 0 t v ( g ) ↦ γ ( g ) + χ ( g ) γ ( g ) − 1 t_v(g)\mapsto\gamma(g)+\chi(g)\gamma(g)^{-1} t v ( g ) ↦ γ ( g ) + χ ( g ) γ ( g ) − 1 Λ i \Lambda_i Λ i Spec Λ i \operatorname{Spec}\Lambda_i Spec Λ i O \mathcal{O} O Λ i \Lambda_i Λ i t v ↦ γ i u n i v + χ ( γ i u n i v ) − 1 t_v\mapsto\gamma_i^{\mathrm{univ}}+\chi(\gamma_i^{\mathrm{univ}})^{-1} t v ↦ γ i univ + χ ( γ i univ ) − 1 γ i u n i v \gamma_i^{\mathrm{univ}} γ i univ R v r e d → Λ i R_v^{\mathrm{red}}\to\Lambda_i R v red → Λ i χ \chi χ K i = ( f v ) K_i=(f_v) K i = ( f v ) E v → Λ 1 × Λ 2 E_v\to\Lambda_1\times\Lambda_2 E v → Λ 1 × Λ 2 f v f_v f v I v I_v I v I v 2 ⊂ f v E v I_v^2\subset f_vE_v I v 2 ⊂ f v E v N = I v N N=I_vN N = I v N N = N I v N=NI_v N = N I v N = 0 N=0 N = 0
Proposition 8.8. Let F ′ F' F ′ 2 2 2 T = T F ′ ( U p ) m \mathbb{T}=\mathbb{T}_{F'}(U^p)_m T = T F ′ ( U p ) m T o r d \mathbb{T}^{\mathrm{ord}} T ord q \mathfrak{q} q T \mathbb{T} T q \mathfrak{q} q 2 2 2 q \mathfrak{q} q Spec T o r d → Spec T \operatorname{Spec}\mathbb{T}^{\mathrm{ord}}\to\operatorname{Spec}\mathbb{T} Spec T ord → Spec T
Proof. Put S t o r = S ⊗ ^ O E / O S_{\mathrm{tor}}=S\widehat{\otimes}_{\mathcal{O}}\mathcal{E}/\mathcal{O} S tor = S ⊗ O E / O M = S t o r ∨ M=S_{\mathrm{tor}}^\vee M = S tor ∨ E = ⨂ v ∣ 2 E v \mathcal{E}=\bigotimes_{v\mid2}E_v E = ⨂ v ∣ 2 E v P = ⨂ v ∣ 2 P v P=\bigotimes_{v\mid2}P_v P = ⨂ v ∣ 2 P v J = P ∨ J=P^\vee J = P ∨ m = Hom ( P , M ) \mathfrak{m}=\operatorname{Hom}(P,M) m = Hom ( P , M ) E \mathcal{E} E m \mathfrak{m} m T \mathbb{T} T E \mathcal{E} E m \mathfrak{m} m E \mathcal{E} E M ≃ m ⊗ ^ E P M\simeq\mathfrak{m}\widehat{\otimes}_{\mathcal{E}}P M ≃ m ⊗ E P S t o r ≃ Hom E c t s ( m , J ) S_{\mathrm{tor}}\simeq\operatorname{Hom}_{\mathcal{E}}^{\mathrm{cts}}(\mathfrak{m},J) S tor ≃ Hom E cts ( m , J ) Ord B \operatorname{Ord}_B Ord B m \mathfrak{m} m Ord B ( S t o r ) ≃ Hom E c t s ( m , Ord B J ) \operatorname{Ord}_B(S_{\mathrm{tor}})\simeq\operatorname{Hom}_{\mathcal{E}}^{\mathrm{cts}}(\mathfrak{m},\operatorname{Ord}_B J) Ord B ( S tor ) ≃ Hom E cts ( m , Ord B J )
Put Q = ( Ord B J ) ∨ Q=(\operatorname{Ord}_B J)^\vee Q = ( Ord B J ) ∨ J J J Ord B J \operatorname{Ord}_B J Ord B J [27 Lemma 4.4] , applied to the product category, gives
Q ≃ ⨁ ( i v ) ∈ { 1 , 2 } { v ∣ 2 } ⨂ ^ v ∣ 2 Λ v , i v ≃ E / I , Q \simeq\bigoplus_{(i_v)\in\{1,2\}^{\{v\mid2\}}}\widehat{\bigotimes}_{v\mid2}\Lambda_{v,i_v}\simeq\mathcal{E}/I, Q ≃ ( i v ) ∈ { 1 , 2 } { v ∣ 2 } ⨁ ⨂ v ∣ 2 Λ v , i v ≃ E / I , where I I I I v I_v I v E v E_v E v
X : = ( Ord B S T ) ∨ ≃ m ⊗ E Q ≃ m / m I X := (\operatorname{Ord}_B S_T)^\vee\simeq\mathbf{m}\otimes_{\mathcal{E}}\mathbb{Q} \simeq\mathbf{m}/\mathbf{m}I X := ( Ord B S T ) ∨ ≃ m ⊗ E Q ≃ m / m I and by the finite-level description of §3 and the commutation of ordinary parts with filtered torsion limits, X X X T o r d \mathbb{T}^{\mathrm{ord}} T ord
The module X X X T \mathbb{T} T T o r d \mathbb{T}^{\mathrm{ord}} T ord T \mathbb{T} T T o r d ⊂ End T ( X ) \mathbb{T}^{\mathrm{ord}}\subset\operatorname{End}_{\mathbb{T}}(X) T ord ⊂ End T ( X ) T \mathbb{T} T Spec T o r d \operatorname{Spec}\mathbb{T}^{\mathrm{ord}} Spec T ord Spec T \operatorname{Spec}\mathbb{T} Spec T Supp T X \operatorname{Supp}_{\mathbb{T}}X Supp T X N = m ⊗ T k ( q ) N=\mathbf{m}\otimes_{\mathbb{T}}k(\mathfrak{q}) N = m ⊗ T k ( q ) E \mathcal{E} E m \mathbf{m} m T \mathbb{T} T f v f_v f v f v f_v f v T \mathbb{T} T q \mathfrak{q} q I v 2 I_v^2 I v 2 N N N N I s + 1 = 0 NI^{s+1}=0 N I s + 1 = 0 s s s N = N I N=NI N = N I N = 0 N=0 N = 0 X ⊗ T k ( q ) ≃ N / N I ≠ 0 X\otimes_{\mathbb{T}}k(\mathfrak{q})\simeq N/NI\ne0 X ⊗ T k ( q ) ≃ N / N I = 0 q ∈ Supp T X \mathfrak{q}\in\operatorname{Supp}_{\mathbb{T}}X q ∈ Supp T X
Theorem 8.9. Let K K K F 1 F_1 F 1 R K r e d R_K^{\mathrm{red}} R K red Λ K \Lambda_K Λ K
Proof. It suffices to show that every irreducible component Z Z Z Spec R K r e d \operatorname{Spec}R_K^{\mathrm{red}} Spec R K red Λ K \Lambda_K Λ K G K G_K G K K K K Z 1 Z_1 Z 1 Z Z Z Spec R r e d \operatorname{Spec}R^{\mathrm{red}} Spec R red Spec T F 1 ( U 1 p ) \operatorname{Spec}\mathbb{T}_{F_1}(U_1^p) Spec T F 1 ( U 1 p ) q \mathfrak{q} q T F 1 ( U 1 p ) \mathbb{T}_{F_1}(U_1^p) T F 1 ( U 1 p ) q \mathfrak{q} q Spec T F 1 o r d \operatorname{Spec}\mathbb{T}_{F_1}^{\mathrm{ord}} Spec T F 1 ord Z 1 ⊂ Y m ′ ′ Z_1\subset Y_{m''} Z 1 ⊂ Y m ′′ m ′ ′ \mathfrak{m}'' m ′′ T F 1 o r d \mathbb{T}_{F_1}^{\mathrm{ord}} T F 1 ord Z 1 Z_1 Z 1 Z Z Z Λ K \Lambda_K Λ K
In particular condition (OF) holds for the field K K K
Removing the dyadic condition over real quadratic fields
In Theorem A, the condition χ ‾ ∣ G v ≠ 1 \overline{\chi}|_{G_v}\ne1 χ ∣ G v = 1 Q \mathbb{Q} Q χ ‾ ∣ G Q 2 = 1 \overline{\chi}|_{G_{\mathbb{Q}_2}}=1 χ ∣ G Q 2 = 1 B B B F F F B B B G F G_F G F
Throughout this section, q q q q ≡ 1 ( m o d 8 ) q\equiv1\pmod{8} q ≡ 1 ( mod 8 ) F = Q ( q ) F=\mathbb{Q}(\sqrt{q}) F = Q ( q ) w 1 , w 2 w_1,w_2 w 1 , w 2 F F F w q w_q w q q q q F F F Q \mathbb{Q} Q F F F F F F
The field.
Lemma 9.1. (1) The narrow class number h + h^+ h + F F F F F F
(2) There is a totally positive π ∈ F \pi\in F π ∈ F ( π ) = w 1 h + (\pi)=w_1^{h^+} ( π ) = w 1 h + F w 2 × / ( F w 2 × ) 2 F_{w_2}^{\times}/(F_{w_2}^{\times})^2 F w 2 × / ( F w 2 × ) 2 π \pi π
(S) π \pi π F w 2 = Q 2 F_{w_2}=\mathbb{Q}_2 F w 2 = Q 2
does not depend on the numbering of w 1 , w 2 w_1,w_2 w 1 , w 2
(3) For q = 17 q=17 q = 17
Proof. (1) The quadratic extension F / Q F/\mathbb{Q} F / Q q q q Q \mathbb{Q} Q h + h^+ h + F F F
(2) The ideal w 1 h + w_1^{h^+} w 1 h + π \pi π σ \sigma σ F F F σ ( π ) \sigma(\pi) σ ( π ) w 2 h + w_2^{h^+} w 2 h + σ \sigma σ F w 2 → F w 1 F_{w_2} \to F_{w_1} F w 2 → F w 1 π \pi π σ ( π ) \sigma(\pi) σ ( π ) π \pi π F w 2 F_{w_2} F w 2 σ ( π ) \sigma(\pi) σ ( π ) F w 1 F_{w_1} F w 1
(3) By [OAI, §6.1], for q = 17 q=17 q = 17 h + = 1 h^+=1 h + = 1 π = ( 5 + 17 ) / 2 \pi=(5+\sqrt{17})/2 π = ( 5 + 17 ) /2 w 1 w_1 w 1 X 2 − 5 X + 2 X^2-5X+2 X 2 − 5 X + 2 F w 2 = Q 2 F_{w_2}=\mathbb{Q}_2 F w 2 = Q 2 Z 2 \mathbb{Z}_2 Z 2
Remark 9.2. Suppose that w 1 = ( π 0 ) w_1=(\pi_0) w 1 = ( π 0 ) π 0 \pi_0 π 0 π = π 0 h + \pi=\pi_0^{h^+} π = π 0 h + π 0 \pi_0 π 0 π 0 \pi_0 π 0 X 2 − t X + 2 X^2-tX+2 X 2 − tX + 2 t = Tr ( π 0 ) t=\operatorname{Tr}(\pi_0) t = Tr ( π 0 ) u u u F w 2 = Q 2 F_{w_2}=\mathbb{Q}_2 F w 2 = Q 2 u + 2 u − 1 = t u+2u^{-1}=t u + 2 u − 1 = t u ≡ 1 , 3 , 5 , 7 u\equiv1,3,5,7 u ≡ 1 , 3 , 5 , 7 t ≡ 3 , 1 , 7 , 5 t\equiv3,1,7,5 t ≡ 3 , 1 , 7 , 5 t ≢ 3 ( m o d 8 ) t\not\equiv3\pmod8 t ≡ 3 ( mod 8 ) π 0 = ( 5 + 17 ) / 2 , ( 7 + 41 ) / 2 , ( 9 + 73 ) / 2 \pi_0=(5+\sqrt{17})/2,(7+\sqrt{41})/2,(9+\sqrt{73})/2 π 0 = ( 5 + 17 ) /2 , ( 7 + 41 ) /2 , ( 9 + 73 ) /2 q = 17 , 41 , 73 q=17,41,73 q = 17 , 41 , 73 π 0 = ( 11 + 113 ) / 2 \pi_0=(11+\sqrt{113})/2 π 0 = ( 11 + 113 ) /2 q = 113 q=113 q = 113
Lemma 9.3. Let χ ˉ : G F → F × \bar{\chi}:G_F\to\mathbb{F}^{\times} χ ˉ : G F → F × G Q G_{\mathbb{Q}} G Q χ ~ \widetilde{\chi} χ χ ˉ ∣ G w i = χ ~ ∣ G Q 2 \bar{\chi}|_{G_{w_i}}=\widetilde{\chi}|_{G_{\mathbb{Q}_2}} χ ˉ ∣ G w i = χ ∣ G Q 2 i = 1 , 2 i=1,2 i = 1 , 2 χ ˉ \bar{\chi} χ ˉ w q w_q w q χ ~ \widetilde{\chi} χ q q q χ ~ ( Frob q ) = χ ˉ ( Frob w q ) \widetilde{\chi}(\operatorname{Frob}_q)=\bar{\chi}(\operatorname{Frob}_{w_q}) χ ( Frob q ) = χ ˉ ( Frob w q )
Proof. The group F × \mathbb{F}^{\times} F × χ ˉ \bar{\chi} χ ˉ Gal ( F / Q ) \operatorname{Gal}(F/\mathbb{Q}) Gal ( F / Q ) F × \mathbb{F}^{\times} F × χ ~ \widetilde{\chi} χ G Q 2 G_{\mathbb{Q}_2} G Q 2 Z 2 × \mathbb{Z}_2^{\times} Z 2 × F w i = Q 2 F_{w_i}=\mathbb{Q}_2 F w i = Q 2 χ ˉ \bar{\chi} χ ˉ w q w_q w q χ ~ ( I q ) \widetilde{\chi}(I_q) χ ( I q ) χ ˉ ( I w q ) = 1 \bar{\chi}(I_{w_q})=1 χ ˉ ( I w q ) = 1 χ ~ ( I q ) = 1 \widetilde{\chi}(I_q)=1 χ ( I q ) = 1 F w q / Q q F_{w_q}/\mathbb{Q}_q F w q / Q q G F w q G_{F_{w_q}} G F w q G Q q / I q G_{\mathbb{Q}_q}/I_q G Q q / I q χ ~ ( Frob q ) = χ ˉ ( Frob w q ) \widetilde{\chi}(\operatorname{Frob}_q)=\bar{\chi}(\operatorname{Frob}_{w_q}) χ ( Frob q ) = χ ˉ ( Frob w q )
Statement
Theorem 9.4. Let ρ : G F → GL 2 ( E ) \rho:G_F\to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E ) § 1.1 and assume (N) and (O). Assume that ρ ˉ s s = χ ˉ 1 ⊕ χ ˉ 2 \bar{\rho}^{ss}=\bar{\chi}_1\oplus\bar{\chi}_2 ρ ˉ ss = χ ˉ 1 ⊕ χ ˉ 2 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 χ ~ \widetilde{\chi} χ G Q G_{\mathbb{Q}} G Q ρ v \rho_v ρ v v ∈ Σ o v\in\Sigma_o v ∈ Σ o
(a) χ ˉ ∣ G w 1 ≠ 1 \bar{\chi}|_{G_{w_1}}\ne1 χ ˉ ∣ G w 1 = 1
(b) χ ˉ = 1 \bar{\chi}=1 χ ˉ = 1 F F F
(c) χ ˉ ≠ 1 \bar{\chi}\ne1 χ ˉ = 1 ρ \rho ρ w q w_q w q χ ˉ ( Frob w q ) ≠ 1 \bar{\chi}(\operatorname{Frob}_{w_q})\ne1 χ ˉ ( Frob w q ) = 1
Then ρ \rho ρ
By Lemma 9.3 , χ ˉ ∣ G w 1 = χ ˉ ∣ G w 2 = χ ~ ∣ G Q 2 \bar{\chi}|_{G_{w_1}}=\bar{\chi}|_{G_{w_2}}=\widetilde{\chi}|_{G_{\mathbb{Q}_2}} χ ˉ ∣ G w 1 = χ ˉ ∣ G w 2 = χ ∣ G Q 2 χ ˉ = 1 \bar{\chi}=1 χ ˉ = 1 F F F χ ˉ ≠ 1 \bar{\chi}\ne1 χ ˉ = 1 χ ~ ∣ G Q 2 = 1 \widetilde{\chi}|_{G_{\mathbb{Q}_2}}=1 χ ∣ G Q 2 = 1 ρ \rho ρ w q w_q w q χ ˉ ( Frob w q ) = 1 \bar{\chi}(\operatorname{Frob}_{w_q})=1 χ ˉ ( Frob w q ) = 1 F = Q ( 17 ) F=\mathbb{Q}(\sqrt{17}) F = Q ( 17 ) ρ ˉ s s \bar{\rho}^{ss} ρ ˉ ss
Corollary 9.5. Let ρ \rho ρ § 1.1 , with ρ ˉ s s = χ ˉ 1 ⊕ χ ˉ 2 \bar{\rho}^{ss}=\bar{\chi}_1\oplus\bar{\chi}_2 ρ ˉ ss = χ ˉ 1 ⊕ χ ˉ 2 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 χ ~ \widetilde{\chi} χ G Q G_{\mathbb{Q}} G Q F F F ρ \rho ρ χ ˉ = 1 \bar{\chi}=1 χ ˉ = 1 ρ ˉ s s \bar{\rho}^{ss} ρ ˉ ss det ρ ˉ = 1 \det\bar{\rho}=1 det ρ ˉ = 1 ρ ˉ s s \bar{\rho}^{ss} ρ ˉ ss F F F F = Q ( 17 ) F=\mathbb{Q}(\sqrt{17}) F = Q ( 17 ) ρ \rho ρ v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ \rho ρ
Proof. Let ℓ \ell ℓ ℓ ≠ q \ell\ne q ℓ = q F / Q F/\mathbb{Q} F / Q ℓ \ell ℓ χ ~ \widetilde{\chi} χ ℓ \ell ℓ χ ˉ \bar{\chi} χ ˉ ℓ \ell ℓ ℓ = q \ell=q ℓ = q χ ~ \widetilde{\chi} χ q q q Lemma 9.3 . Hence χ ~ \widetilde{\chi} χ Gal ( Q ( μ 2 ∞ ) / Q ) ≃ Z 2 × \operatorname{Gal}(\mathbb{Q}(\mu_{2^\infty})/\mathbb{Q})\simeq\mathbb{Z}_2^{\times} Gal ( Q ( μ 2 ∞ ) / Q ) ≃ Z 2 × F × \mathbb{F}^{\times} F × χ ~ = 1 \widetilde{\chi}=1 χ = 1 χ ˉ = 1 \bar{\chi}=1 χ ˉ = 1 det ρ ˉ = χ ˉ 1 χ ˉ 2 = 1 \det\bar{\rho}=\bar{\chi}_1\bar{\chi}_2=1 det ρ ˉ = χ ˉ 1 χ ˉ 2 = 1 χ ˉ = χ ˉ 1 2 = 1 \bar{\chi}=\bar{\chi}_1^2=1 χ ˉ = χ ˉ 1 2 = 1 χ ˉ 1 = 1 \bar{\chi}_1=1 χ ˉ 1 = 1 F × \mathbb{F}^{\times} F × χ ˉ 2 = 1 \bar{\chi}_2=1 χ ˉ 2 = 1 Theorem 9.4 . □
The first part of the argument, that χ ˉ = 1 \bar{\chi}=1 χ ˉ = 1 F F F F F F χ ˉ \bar{\chi} χ ˉ G Q G_{\mathbb{Q}} G Q
The arguments of Section 3 without the dyadic condition.
Lemma 9.6. Let Π \Pi Π G L 2 ( Q 2 ) \mathrm{GL}_2(\mathbb{Q}_2) GL 2 ( Q 2 ) ψ 1 + ψ 2 \psi_1+\psi_2 ψ 1 + ψ 2 Π \Pi Π η ∘ det \eta\circ\det η ∘ det ( Ind B ˉ G L 2 ( Q 2 ) μ ) c o n t (\operatorname{Ind}_{\bar{B}}^{\mathrm{GL}_2(\mathbb{Q}_2)}\mu)_{\mathrm{cont}} ( Ind B ˉ GL 2 ( Q 2 ) μ ) cont
Proof. Following [10 ] , Π \Pi Π ordinary if it is a subquotient of a unitary parabolic induction. If Π \Pi Π [10 Theorem 1.1] , which holds for every prime p p p [27 Theorem 1.4] , after inverting 2 the centre of the block is identified with the fixed-determinant pseudo-deformation ring, and it acts on Π \Pi Π p = 2 p=2 p = 2
Hence Π \Pi Π I = ( Ind B ˉ G L 2 ( Q 2 ) μ ) c o n t I=(\operatorname{Ind}_{\bar{B}}^{\mathrm{GL}_2(\mathbb{Q}_2)}\mu)_{\mathrm{cont}} I = ( Ind B ˉ GL 2 ( Q 2 ) μ ) cont μ \mu μ T ( Q 2 ) T(\mathbb{Q}_2) T ( Q 2 ) ϖ \varpi ϖ Ind B ˉ G L 2 ( Q 2 ) μ ˉ \operatorname{Ind}_{\bar{B}}^{\mathrm{GL}_2(\mathbb{Q}_2)}\bar{\mu} Ind B ˉ GL 2 ( Q 2 ) μ ˉ [4 ] , this has length at most 2, and when it is reducible its constituents are a character and a twist of the Steinberg representation. A closed G L 2 ( Q 2 ) \mathrm{GL}_2(\mathbb{Q}_2) GL 2 ( Q 2 ) I 1 ⊂ I I_1\subset I I 1 ⊂ I I 1 ≠ 0 , I I_1\ne0,I I 1 = 0 , I I I I
If I I I Π = I \Pi=I Π = I 0 → I 1 → I → I 2 → 0 0\to I_1\to I\to I_2\to0 0 → I 1 → I → I 2 → 0 I 1 , I 2 I_1,I_2 I 1 , I 2 Π ∈ { I 1 , I 2 } \Pi\in\{I_1,I_2\} Π ∈ { I 1 , I 2 } [17 Theorem 4.4.6] , in its form for admissible unitary Banach representations ((O4) of [Die26a, §3.3]), Hom ( I , η ∘ det ) = Hom T ( Q 2 ) ( μ , Ord B ( η ∘ det ) ) \operatorname{Hom}(I,\eta\circ\det)=\operatorname{Hom}_{T(\mathbb{Q}_2)}(\mu,\operatorname{Ord}_{B}(\eta\circ\det)) Hom ( I , η ∘ det ) = Hom T ( Q 2 ) ( μ , Ord B ( η ∘ det )) Ord B ( η ∘ det ) = 0 \operatorname{Ord}_{B}(\eta\circ\det)=0 Ord B ( η ∘ det ) = 0 O / ϖ n \mathcal{O}/\varpi^n O / ϖ n η ∘ det \eta\circ\det η ∘ det t = diag ( 2 , 1 ) t=\operatorname{diag}(2,1) t = diag ( 2 , 1 ) ∑ n ∈ N 0 / t N 0 t − 1 n t \sum_{n\in\mathbb{N}_0/t\mathbb{N}_0t^{-1}}nt ∑ n ∈ N 0 / t N 0 t − 1 n t 2 η ( 2 ) 2\eta(2) 2 η ( 2 ) I 2 I_2 I 2 G L 2 ( Q 2 ) \mathrm{GL}_2(\mathbb{Q}_2) GL 2 ( Q 2 ) det \det det I 2 I_2 I 2 I 1 I_1 I 1 I 1 I_1 I 1 Π \Pi Π I I I
Proposition 9.7. Let F ′ F' F ′ S S S T \mathbb{T} T S ∘ S^\circ S ∘ T ∘ \mathbb{T}^\circ T ∘ F ′ F' F ′ m \mathfrak{m} m r ˉ ′ \bar{r}' r ˉ ′ G F ′ G_{F'} G F ′ 1 ⊕ χ ˉ ′ 1\oplus\bar{\chi}' 1 ⊕ χ ˉ ′ χ ˉ ′ \bar{\chi}' χ ˉ ′
(1) (C1), (C2), (C3), the description of the absolutely irreducible objects with reducible parameter in §3.1, Lemmas 3.1 and 3.2, and Theorem 3.4 hold;
(2) Proposition 3.3 holds for every maximal ideal p \mathfrak{p} p T [ 1 / 2 ] \mathbb{T}[1/2] T [ 1/2 ] ρ ( p ) \rho(\mathfrak{p}) ρ ( p ) v ∈ Σ ∞ v\in\Sigma_\infty v ∈ Σ ∞ ρ ( p ) ∣ G v \rho(\mathfrak{p})|_{G_v} ρ ( p ) ∣ G v
Proof. (1) Propositions 4.1, 4.2 and 4.4 of [OAI] are proved for every residual factor of the Hecke algebra; their block inputs, from [27 ] , apply to every block, including the scalar residual block at 2 [OAI, proof of Proposition 4.2]. The description of the absolutely irreducible objects with reducible parameter is Lemma 9.6. The proof of Lemma 3.1 takes place at finite level and in characteristic 0, and uses no condition on r ˉ ′ \bar{r}' r ˉ ′ [10 ] at the irreducible local parameters, the product structure of the blocks [OAI, proof of Proposition 4.2], and the density statement in its proof, none of which involves r ˉ ′ \bar{r}' r ˉ ′ a 1 > a 2 a_1>a_2 a 1 > a 2 [6 Proposition 2.2.1] with k = a 1 − a 2 + 2 ≥ 3 k=a_1-a_2+2\ge3 k = a 1 − a 2 + 2 ≥ 3 k = 2 k=2 k = 2 a 1 = a 2 a_1=a_2 a 1 = a 2
:::
(2) In the proof of Proposition 3.3, (C2) gives Π ′ = ⨂ ^ Π v ′ \Pi'=\widehat{\bigotimes}\Pi'_v Π ′ = ⨂ Π v ′ ( S ⊗ E ) [ p ] (S\otimes E)[\mathfrak p] ( S ⊗ E ) [ p ] Π v ′ \Pi'_v Π v ′ v ∈ Σ o v\in\Sigma_o v ∈ Σ o Π v ′ \Pi'_v Π v ′ P \mathfrak P P P \mathfrak P P ψ v , 1 ( P ) \psi_{v,1}(\mathfrak P) ψ v , 1 ( P ) G v G_v G v det ( ψ v , 1 ( P ) ⊕ χ ψ v , 1 ( P ) − 1 ) \det(\psi_{v,1}(\mathfrak P)\oplus\chi\psi_{v,1}(\mathfrak P)^{-1}) det ( ψ v , 1 ( P ) ⊕ χ ψ v , 1 ( P ) − 1 ) ρ ( p ) ∣ G v \rho(\mathfrak p)|_{G_v} ρ ( p ) ∣ G v σ 0 ∈ G v \sigma_0\in G_v σ 0 ∈ G v § 2 , applied over the field k ( p ) k(\mathfrak p) k ( 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
Pro-modularity
Proposition 9.8. Let ρ : G F → GL 2 ( E ) \rho:G_F\to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E ) § 1.1 , de Rham with distinct Hodge–Tate weights at some place above 2, not induced, with det ρ = ε w χ 0 \det\rho=\varepsilon^w\chi_0 det ρ = ε w χ 0 χ 0 \chi_0 χ 0 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 = 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 Theorem 9.4 . Then (PM) holds. Moreover, if χ ˉ ∣ G w 1 = 1 \bar{\chi}|_{G_{w_1}}=1 χ ˉ ∣ G w 1 = 1 K K K Q \mathbb Q Q
Proof. If χ ˉ ∣ G w 1 ≠ 1 \bar{\chi}|_{G_{w_1}}\ne1 χ ˉ ∣ G w 1 = 1 K K K χ ˉ ∣ G w 1 = 1 \bar{\chi}|_{G_{w_1}}=1 χ ˉ ∣ G w 1 = 1 χ ~ ∣ G Q 2 = 1 \widetilde{\chi}|_{G_{\mathbb Q_2}}=1 χ ∣ G Q 2 = 1
Step 0: normalization. Twisting by the Teichmüller lift of χ ˉ 2 − 1 \bar{\chi}_2^{-1} χ ˉ 2 − 1 ρ ˉ s s = 1 ⊕ χ ˉ \bar{\rho}^{\mathrm{ss}}=1\oplus\bar{\chi} ρ ˉ ss = 1 ⊕ χ ˉ ρ \rho ρ w q w_q w q χ ˉ 2 \bar{\chi}_2 χ ˉ 2 ρ \rho ρ w q w_q w q α = χ ~ \alpha=\widetilde{\chi} α = χ r ˉ = 1 ⊕ α \bar{r}=1\oplus\alpha r ˉ = 1 ⊕ α G Q G_{\mathbb Q} G Q r ˉ ∣ G F = ρ ˉ s s \bar{r}|_{G_F}=\bar{\rho}^{\mathrm{ss}} r ˉ ∣ G F = ρ ˉ ss α \alpha α α = 1 \alpha=1 α = 1 α ≠ 1 \alpha\ne1 α = 1 α ∣ G Q 2 = 1 \alpha|_{G_{\mathbb Q_2}}=1 α ∣ G Q 2 = 1 α \alpha α q q q α ( Frob q ) ≠ 1 \alpha(\operatorname{Frob}_q)\ne1 α ( Frob q ) = 1 ρ ∣ G F ′ \rho|_{G_{F'}} ρ ∣ G F ′ F ′ / F F'/F F ′ / F χ = det ρ \chi=\det\rho χ = det ρ
Step 1: the seed and the allowed primes. By [OAI, Lemma 7.2] there is a cuspidal eigenform f f f N ℓ N\ell N ℓ r ˉ \bar{r} r ˉ N N N ≥ 5 \ge5 ≥ 5 α \alpha α ℓ \ell ℓ N 2 a N2^a N 2 a a a a α \alpha α q q q N N N q q q ℓ ≠ q \ell\ne q ℓ = q f f f q q q Σ Q \Sigma_{\mathbb Q} Σ Q N ℓ N\ell N ℓ F F F ρ \rho ρ r ˉ \bar{r} r ˉ Σ Q ∪ { 2 } \Sigma_{\mathbb Q}\cup\{2\} Σ Q ∪ { 2 } q ∉ Σ Q q\notin\Sigma_{\mathbb Q} q ∈ / Σ Q w q w_q w q F F F q q q Σ \Sigma Σ F F F Σ Q ∪ { 2 , ∞ } \Sigma_{\mathbb Q}\cup\{2,\infty\} Σ Q ∪ { 2 , ∞ }
Step 2: the fields. We apply [OAI, Proposition 6.1] to r ˉ \bar{r} r ˉ Σ Q \Sigma_{\mathbb Q} Σ Q Q ⊂ A ⊂ A ′ \mathbb Q\subset A\subset A' Q ⊂ A ⊂ A ′ α ∣ G Q 2 = 1 \alpha|_{G_{\mathbb Q_2}}=1 α ∣ G Q 2 = 1 B B B A = B L a A=BL_a A = B L a A ′ = B L A'=BL A ′ = B L L a ⊂ L L_a\subset L L a ⊂ L B = F B=F B = F F ⊂ A ⊂ A ′ F\subset A\subset A' F ⊂ A ⊂ A ′ A A A A ′ A' A ′ Q \mathbb Q Q
Case (b). In [OAI, §6.1], B = Q ( 17 ) B=\mathbb Q(\sqrt{17}) B = Q ( 17 ) B B B
2 splits in B B B
B B B
there is a totally positive π ∈ B \pi\in B π ∈ B w 1 h w_1^h w 1 h h h h
the condition that a conductor prime r r r B B B Σ Q \Sigma_{\mathbb Q} Σ Q
For B = F B=F B = F h = h + h=h^+ h = h + r r r F F F q q q r r r E n = Q ( ζ 2 n + 1 , 2 1 / 2 n ) E_n=\mathbb Q(\zeta_{2^{n+1}},2^{1/2^n}) E n = Q ( ζ 2 n + 1 , 2 1/ 2 n ) q ∈ Σ Q q\in\Sigma_{\mathbb Q} q ∈ Σ Q F L / F FL/F F L / F r r r A ′ A' A ′ β = 1 \beta= 1 β = 1 π \pi π ( 5 + 17 ) / 2 (5 + \sqrt{17})/2 ( 5 + 17 ) /2 π \pi π w 2 w_2 w 2 F w 2 = Q 2 F_{w_2} = \mathbb{Q}_2 F w 2 = Q 2
Every ℓ ∈ Σ Q \ell\in\Sigma_{\mathbb{Q}} ℓ ∈ Σ Q r r r Gal ( L / Q ) ≃ Gal ( F L / F ) \operatorname{Gal}(L/\mathbb{Q}) \simeq\operatorname{Gal}(FL/F) Gal ( L / Q ) ≃ Gal ( F L / F ) F ∩ L = Q F \cap L = \mathbb{Q} F ∩ L = Q F F F ℓ \ell ℓ A ′ A' A ′ u A , u A ′ ≤ 2 ∣ Σ Q ∣ u_A, u_{A'} \leq2|\Sigma_{\mathbb{Q}}| u A , u A ′ ≤ 2∣ Σ Q ∣
Case (c). In [OAI, §§6.3–6.5], B = Q ( q 0 ) B = \mathbb{Q}(\sqrt{q_0}) B = Q ( q 0 ) q 0 ∉ Σ Q ∪ { 2 } q_0 \notin\Sigma_{\mathbb{Q}} \cup\{2\} q 0 ∈ / Σ Q ∪ { 2 } q 0 ≡ 1 ( m o d 8 ) q_0 \equiv1 \pmod{8} q 0 ≡ 1 ( mod 8 ) α ( Frob q 0 ) ≠ 1 \alpha(\operatorname{Frob}_{q_0}) \ne1 α ( Frob q 0 ) = 1 q 0 q_0 q 0 q 0 = q q_0 = q q 0 = q q 0 q_0 q 0 α \alpha α q 0 q_0 q 0 α ( Frob q 0 ) ≠ 1 \alpha(\operatorname{Frob}_{q_0}) \ne1 α ( Frob q 0 ) = 1 B ∩ E 2 = Q B \cap E_2 = \mathbb{Q} B ∩ E 2 = Q q 0 ∉ Σ Q q_0 \notin\Sigma_{\mathbb{Q}} q 0 ∈ / Σ Q q 0 = q q_0 = q q 0 = q B = F B = F B = F
In both cases, [OAI, Proposition 6.1] holds for A ⊂ A ′ A \subset A' A ⊂ A ′ M M M α ∣ G Q 2 = 1 \alpha|_{G_{\mathbb{Q}_2}} = 1 α ∣ G Q 2 = 1 U U U u A , u A ′ u_A, u_{A'} u A , u A ′
Step 3: the target loci. The point of ρ ∣ G A \rho|_{G_A} ρ ∣ G A Spec R A p s \operatorname{Spec} R_A^{\mathrm{ps}} Spec R A ps r ˉ ∣ G A \bar{r}|_{G_A} r ˉ ∣ G A χ ∣ G A \chi|_{G_A} χ ∣ G A G A , S A G_{A,S_A} G A , S A S A S_A S A Σ Q ∪ { 2 , ∞ } \Sigma_{\mathbb{Q}} \cup\{2,\infty\} Σ Q ∪ { 2 , ∞ } C C C Spec R A p s \operatorname{Spec} R_A^{\mathrm{ps}} Spec R A ps ρ ∣ G A \rho|_{G_A} ρ ∣ G A C C C ρ ∣ G A \rho|_{G_A} ρ ∣ G A dim C ≥ 2 [ A : Q ] − 2 \dim C \geq2[A:\mathbb{Q}] - 2 dim C ≥ 2 [ A : Q ] − 2
Step 4: the argument of [OAI, §7]. We follow [OAI, §§7.2–7.4] with the fields A ⊂ A ′ A \subset A' A ⊂ A ′ f f f χ ∣ G A \chi|_{G_A} χ ∣ G A S A S_A S A C C C Q \mathbb{Q} Q
[OAI, §7.2]: the allowed set contains the bad primes of the target and of the seed, by Step 1. The degree conditions [OAI, (39)] only involve U U U
[OAI, Lemmas 7.3–7.5] concern the fixed-determinant framed representation scheme over A ′ A' A ′ r ˉ ∣ G A ′ \bar{r}|_{G_{A'}} r ˉ ∣ G A ′
the relative representation over the real local rings, which needs only that χ \chi χ
the bounds (25) and (26);
Leopoldt’s conjecture for A ′ A' A ′ A ′ A' A ′ Q \mathbb{Q} Q
the propagation theorem [OAI, Theorem 5.1], which applies to the determinant χ ∣ G A ′ = ε ω χ 0 ∣ G A ′ \chi|_{G_{A'}} = \varepsilon^{\omega}\chi_0|_{G_{A'}} χ ∣ G A ′ = ε ω χ 0 ∣ G A ′ G Q G_{\mathbb{Q}} G Q
[OAI, §7.4], seed locus: the base change of f f f A A A r ˉ ∣ G A \bar{r}|_{G_A} r ˉ ∣ G A χ ε \chi\varepsilon χ ε ϖ \varpi ϖ α ∣ G A \alpha|_{G_A} α ∣ G A C s e e d C_{\mathrm{seed}} C seed 1 + 2 [ A : Q ] 1 + 2[A:\mathbb{Q}] 1 + 2 [ A : Q ]
[OAI, §7.4], target locus: we use C C C r ˉ ∣ G A \bar{r}|_{G_A} r ˉ ∣ G A v 0 v_0 v 0 1 + u A 1+u_A 1 + u A dim C ≥ 2 [ A : Q ] − 2 \dim C \geq2[A:\mathbb{Q}] - 2 dim C ≥ 2 [ A : Q ] − 2 [ A : Q ] [A:\mathbb{Q}] [ A : Q ] [ A : Q ] [A:\mathbb{Q}] [ A : Q ]
and 3.9] over the abelian field A A A
(v) [OAI, §7.4], entering the transverse fibre over A ′ A' A ′ v 0 v_0 v 0 C C C
(vi) [OAI, §7.4], conclusion: Lemma 7.5 of loc. cit. makes the target curve potentially pro-modular, and [OAI, Theorem 5.1] over A A A C C C
Hence there are a totally real field F C F_C F C Q \mathbb{Q} Q C C C G F C G_{F_C} G F C
Step 5: conclusion. There are finitely many components C C C F 1 F_1 F 1 F C F_C F C Q \mathbb{Q} Q F 1 F_1 F 1 K = A K=A K = A Q \mathbb{Q} Q F F F F 1 F_1 F 1
Proof of Theorem 9.4
In case (a), condition (R) holds and the result is Theorem A. Assume (b) or (c). If ρ \rho ρ det ρ = ε w χ 0 \det\rho=\varepsilon^w\chi_0 det ρ = ε w χ 0 χ 0 \chi_0 χ 0 F ⊂ K ⊂ F 1 F\subset K\subset F_1 F ⊂ K ⊂ F 1
By (PM), the restriction ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 p \mathfrak{p} p T F 1 [ 1 / 2 ] \mathbb{T}_{F_1}[1/2] T F 1 [ 1/2 ] ρ ( p ) ≃ ρ ∣ G F 1 \rho(\mathfrak{p})\simeq\rho|_{G_{F_1}} ρ ( p ) ≃ ρ ∣ G F 1 w w w F 1 F_1 F 1 v ∈ Σ o v\in\Sigma_o v ∈ Σ o ρ ( p ) ∣ G w = ρ v \rho(\mathfrak{p})|_{G_w}=\rho_v ρ ( p ) ∣ G w = ρ v P \mathfrak{P} P p \mathfrak{p} p ψ w , 1 ( P ) \psi_{w,1}(\mathfrak{P}) ψ w , 1 ( P ) ρ v \rho_v ρ v w w w ρ v \rho_v ρ v F 1 F_1 F 1 ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 F 1 / F F_1/F F 1 / F ρ \rho ρ [8 Lemma 2.2.2] , [1 ] . □
Remark 9.9. Over a general field F F F F F F B B B Q \mathbb{Q} Q F = Q ( q ) F=\mathbb{Q}(\sqrt{q}) F = Q ( q ) χ ˉ ≠ 1 \bar{\chi}\ne1 χ ˉ = 1 ρ \rho ρ w q w_q w q χ ˉ ( Frob w q ) = 1 \bar{\chi}(\operatorname{Frob}_{w_q})=1 χ ˉ ( Frob w q ) = 1 q 0 = q q_0=q q 0 = q χ ˉ ( Frob w q ) = 1 \bar{\chi}(\operatorname{Frob}_{w_q})=1 χ ˉ ( Frob w q ) = 1 q q q ρ \rho ρ w q w_q w q q ∈ Σ Q q\in\Sigma_{\mathbb{Q}} q ∈ Σ Q q \sqrt{q} q E 2 E_2 E 2
Residually irreducible representations with solvable image
In this section F F F ρ : G F → GL 2 ( E ) \rho:G_F\to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E ) ρ ˉ \bar{\rho} ρ ˉ ρ ˉ \bar{\rho} ρ ˉ [34 ] . We treat the case of solvable image. Then residual modularity is automatic, by an odd Artin lift, and the method of Sections 3 and 8 applies with the simplifications of the residually irreducible case.
Residual representations
Lemma 10.1. Let ρ ˉ : G F → GL 2 ( F ) \bar{\rho}:G_F\to\operatorname{GL}_2(\mathbb{F}) ρ ˉ : G F → GL 2 ( F ) 2 n 2n 2 n n ≥ 3 n\ge3 n ≥ 3 F \mathbb{F} F L / F L/F L / F θ ˉ : G L → F × \bar{\theta}:G_L\to\mathbb{F}^{\times} θ ˉ : G L → F × θ ˉ ≠ θ ˉ c \bar{\theta}\ne\bar{\theta}^{c} θ ˉ = θ ˉ c ρ ˉ ≃ Ind G L G F θ ˉ \bar{\rho}\simeq\operatorname{Ind}_{G_L}^{G_F}\bar{\theta} ρ ˉ ≃ Ind G L G F θ ˉ c c c Gal ( L / F ) \operatorname{Gal}(L/F) Gal ( L / F ) θ ˉ / θ ˉ c \bar{\theta}/\bar{\theta}^{c} θ ˉ / θ ˉ c n n n
Proof. Let k k k F \mathbb{F} F k k k SL 2 ( k ) → PGL 2 ( k ) \operatorname{SL}_{2}(k) \to\operatorname{PGL}_{2}(k) SL 2 ( k ) → PGL 2 ( k ) G G G ρ ˉ \bar{\rho} ρ ˉ SL 2 ( k ) \operatorname{SL}_{2}(k) SL 2 ( k ) SL 2 ( k ) \operatorname{SL}_{2}(k) SL 2 ( k ) 2 m 2^{m} 2 m ( x − 1 ) 2 m = x 2 m − 1 = 0 (x-1)^{2^{m}}=x^{2^{m}}-1=0 ( x − 1 ) 2 m = x 2 m − 1 = 0 SL 2 ( k ) \operatorname{SL}_{2}(k) SL 2 ( k ) O 2 ( G ) O_{2}(G) O 2 ( G ) G G G ρ ˉ ⊗ k \bar{\rho}\otimes k ρ ˉ ⊗ k O 2 ( G ) = 1 O_{2}(G)=1 O 2 ( G ) = 1 N N N G G G G G G
Let H H H p p p N N N p p p H ≠ 1 H\ne1 H = 1 z ≠ 1 z\ne1 z = 1 z z z SL 2 ( k ) \operatorname{SL}_{2}(k) SL 2 ( k ) T ≃ k × T\simeq k^{\times} T ≃ k × H H H N N N z z z N ⊂ T N\subset T N ⊂ T N N N N N N G G G G G G G ⊂ N SL 2 ( k ) ( T ) = T ⋊ ⟨ s ⟩ G\subset N_{\operatorname{SL}_{2}(k)}(T)=T\rtimes\langle s\rangle G ⊂ N SL 2 ( k ) ( T ) = T ⋊ ⟨ s ⟩ s s s T T T G ⊂ T G\subset T G ⊂ T ρ ˉ ⊗ k \bar{\rho}\otimes k ρ ˉ ⊗ k C = G ∩ T C=G\cap T C = G ∩ T G G G n n n k × k^{\times} k × G G G 2 n 2n 2 n n = 1 n=1 n = 1 G G G n ≥ 3 n\ge3 n ≥ 3
Let L L L F F F G → G / C G\to G/C G → G / C ρ ˉ ( G L ) \bar{\rho}(G_L) ρ ˉ ( G L ) C ⊂ T C\subset T C ⊂ T T T T ρ ˉ ∣ G L ⊗ k = θ ˉ ⊕ θ ˉ c \bar{\rho}|_{G_L}\otimes k=\bar{\theta}\oplus\bar{\theta}^{c} ρ ˉ ∣ G L ⊗ k = θ ˉ ⊕ θ ˉ c ρ ˉ \bar{\rho} ρ ˉ θ ˉ ′ = θ ˉ c ≠ θ ˉ \bar{\theta}'=\bar{\theta}^{c}\ne\bar{\theta} θ ˉ ′ = θ ˉ c = θ ˉ ρ ˉ ⊗ k ≃ Ind G L G F θ ˉ \bar{\rho}\otimes k\simeq\operatorname{Ind}_{G_L}^{G_F}\bar{\theta} ρ ˉ ⊗ k ≃ Ind G L G F θ ˉ θ ˉ / θ ˉ c \bar{\theta}/\bar{\theta}^{c} θ ˉ / θ ˉ c G L G_L G L C C C ρ ˉ ( G L ) \bar{\rho}(G_L) ρ ˉ ( G L ) n n n L L L n n n C C C G G G
Lemma 10.2. Let ρ ˉ = Ind G L G F θ ˉ \bar{\rho}=\operatorname{Ind}_{G_L}^{G_F}\bar{\theta} ρ ˉ = Ind G L G F θ ˉ θ : G L → O × \theta:G_L\to\mathcal{O}^{\times} θ : G L → O × θ ˉ \bar{\theta} θ ˉ r = Ind G L G F θ r=\operatorname{Ind}_{G_L}^{G_F}\theta r = Ind G L G F θ ϖ \varpi ϖ ρ ˉ \bar{\rho} ρ ˉ
Proof. Let θ 0 \theta_{0} θ 0 θ ˉ \bar{\theta} θ ˉ θ 0 ( c w ) = 1 \theta_{0}(c_{w})=1 θ 0 ( c w ) = 1 c w ∈ G L c_{w}\in G_L c w ∈ G L v v v F F F v v v L L L c v c_{v} c v v v v G L G_L G L Ind θ ( c v ) \operatorname{Ind}\theta(c_{v}) Ind θ ( c v ) c v 2 = 1 c_{v}^{2}=1 c v 2 = 1 ± 1 \pm1 ± 1 − 1 -1 − 1 θ \theta θ v v v w , w ′ w,w' w , w ′ L L L Ind θ ( c w ) \operatorname{Ind}\theta(c_{w}) Ind θ ( c w ) diag ( θ ( c w ) , θ ( c w ′ ) ) \operatorname{diag}(\theta(c_{w}),\theta(c_{w'})) diag ( θ ( c w ) , θ ( c w ′ ))
By weak approximation there is x ∈ L × x\in L^{\times} x ∈ L × w w w v v v L L L η \eta η L ( x ) / L L(\sqrt{x})/L L ( x ) / L θ = θ 0 η \theta=\theta_{0}\eta θ = θ 0 η det r ( c v ) = η ( c w ) η ( c w ′ ) = − 1 \det r(c_{v})=\eta(c_{w})\eta(c_{w'})=-1 det r ( c v ) = η ( c w ) η ( c w ′ ) = − 1 r r r η \eta η ± 1 ≡ 1 ( m o d ϖ ) \pm1\equiv1\pmod{\varpi} ± 1 ≡ 1 ( mod ϖ ) θ \theta θ θ ˉ \bar{\theta} θ ˉ r m o d ϖ ≃ Ind G L G F θ ˉ = ρ ˉ r\bmod\varpi\simeq\operatorname{Ind}_{G_L}^{G_F}\bar{\theta}=\bar{\rho} r mod ϖ ≃ Ind G L G F θ ˉ = ρ ˉ θ ˉ ≠ θ ˉ c \bar{\theta}\ne\bar{\theta}^{c} θ ˉ = θ ˉ c θ ≠ θ c \theta\ne\theta^{c} θ = θ c r r r
:::
In characteristic 2 the field L L L η \eta η
Proposition 10.3. Let ρ ˉ \bar{\rho} ρ ˉ g g g F F F k ≥ 2 k\ge2 k ≥ 2 ρ g \rho_{g} ρ g ρ ˉ g ≃ ρ ˉ \bar{\rho}_{g}\simeq\bar{\rho} ρ ˉ g ≃ ρ ˉ
Proof. Let r = Ind G L G F θ r=\operatorname{Ind}_{G_L}^{G_F}\theta r = Ind G L G F θ π \pi π θ \theta θ L L L [LL79], {[citep]}[AC89, Chapter 3] is cuspidal, since θ ≠ θ c \theta\ne\theta^{c} θ = θ c v v v F F F r r r W F v W_{F_v} W F v Gal ( C / R ) \operatorname{Gal}(\mathbb{C}/\mathbb{R}) Gal ( C / R ) 1 ⊕ sgn 1\oplus\operatorname{sgn} 1 ⊕ sgn r ( c v ) r(c_{v}) r ( c v ) − 1 -1 − 1 r r r π v \pi_{v} π v π \pi π f f f r r r [RT83]. Its Hecke eigenvalues c ( v , f ) = tr r ( Frob v ) c(v,f)=\operatorname{tr}r(\operatorname{Frob}_{v}) c ( v , f ) = tr r ( Frob v ) v ∤ N v\nmid N v ∤ N N N N O \mathcal{O} O
We now raise the weight, as in {[citep]}[Wil88]; see {[citep]}[All14, §5.1] for its use at p = 2 p=2 p = 2 F F F Q \mathbb{Q} Q F F F [Bru67], and the 2-adic zeta function of F F F s = 1 s=1 s = 1 C ( m , ⋅ ) C(\mathfrak{m},\cdot) C ( m , ⋅ ) h + h^{+} h + t λ t_{\lambda} t λ 2 2 2 k k k N ( t λ ) − k / 2 N(t_{\lambda})^{-k/2} N ( t λ ) − k /2 k k k l = 2 m l=2^{m} l = 2 m m ≥ 2 m\geq2 m ≥ 2 G l G_l G l l l l C ( m , G l ) = ∑ a ∣ m N ( a ) l − 1 C(\mathfrak{m},G_l)=\sum_{\mathfrak{a}\mid\mathfrak{m}}N(\mathfrak{a})^{l-1} C ( m , G l ) = ∑ a ∣ m N ( a ) l − 1 2 − d ζ F ( 1 − l ) 2^{-d}\zeta_F(1-l) 2 − d ζ F ( 1 − l ) 2 2 2 2 2 2 F F F l l l ζ F , 2 ( 1 − l ) = ( 1 − 2 l − 1 ) d ζ F ( 1 − l ) \zeta_{F,2}(1-l)=(1-2^{l-1})^{d}\zeta_F(1-l) ζ F , 2 ( 1 − l ) = ( 1 − 2 l − 1 ) d ζ F ( 1 − l ) ζ F , 2 = ∏ χ L 2 ( s , χ ) \zeta_{F,2}=\prod_{\chi}L_2(s,\chi) ζ F , 2 = ∏ χ L 2 ( s , χ ) χ \chi χ F F F ζ 2 \zeta_2 ζ 2 s = 1 s=1 s = 1 L 2 ( 1 , χ ) ≠ 0 L_2(1,\chi)\ne0 L 2 ( 1 , χ ) = 0 χ ≠ 1 \chi\ne1 χ = 1 [5 ] , v 2 ( ζ F ( 1 − l ) ) → − ∞ v_2(\zeta_F(1-l))\to-\infty v 2 ( ζ F ( 1 − l )) → − ∞ m → ∞ m\to\infty m → ∞ E = 2 d G l / ζ F ( 1 − l ) E=2^dG_l/\zeta_F(1-l) E = 2 d G l / ζ F ( 1 − l ) m m m 2 2 2 2 2 2 1 1 1 ϖ \varpi ϖ [37 ] . Then f E fE f E l + 1 l+1 l + 1 N N N f f f C ( m , f E ) C(\mathfrak{m},fE) C ( m , f E ) f f f E E E C ( m , f E ) ≡ C ( m , f ) C(\mathfrak{m},fE)\equiv C(\mathfrak{m},f) C ( m , f E ) ≡ C ( m , f ) ϖ \varpi ϖ m \mathfrak{m} m M M M l + 1 l+1 l + 1 N N N 2 2 2 O \mathcal{O} O T v T_v T v v ∤ 2 N v\nmid2N v ∤ 2 N T v T_v T v 1 1 1 l + 1 l+1 l + 1 N ( v ) j N(v)^j N ( v ) j j ≥ 0 j\geq0 j ≥ 0 ≡ 1 \equiv1 ≡ 1 ϖ \varpi ϖ f E fE f E M / ϖ M M/\varpi M M / ϖ M T v T_v T v c ( v , f ) c(v,f) c ( v , f ) ϖ \varpi ϖ [16 Lemme 6.11] , after enlarging O \mathcal{O} O g ∈ M g\in M g ∈ M c ( v , g ) ≡ c ( v , f ) c(v,g)\equiv c(v,f) c ( v , g ) ≡ c ( v , f ) ϖ \varpi ϖ v ∤ 2 N v\nmid2N v ∤ 2 N f f f g g g N ( v ) l ≡ 1 N(v)^l\equiv1 N ( v ) l ≡ 1 ϖ \varpi ϖ ρ g ( Frob v ) \rho_g(\operatorname{Frob}_v) ρ g ( Frob v ) r ( Frob v ) r(\operatorname{Frob}_v) r ( Frob v ) ϖ \varpi ϖ v ∤ 2 N v\nmid2N v ∤ 2 N ρ ‾ g s s ≃ ρ ‾ \overline{\rho}_g^{\mathrm{ss}}\simeq\overline{\rho} ρ g ss ≃ ρ ρ ‾ g ≃ ρ ‾ \overline{\rho}_g\simeq\overline{\rho} ρ g ≃ ρ ρ ‾ \overline{\rho} ρ
Fields
Let Σ \Sigma Σ § 1.1 , enlarged by the places above the primes dividing the level of the form g g g Σ 0 \Sigma_0 Σ 0 Σ \Sigma Σ 2 2 2 P 0 P_0 P 0 F F F 2 2 2 Σ 0 \Sigma_0 Σ 0 2 2 2 S ′ S' S ′ x v x_v x v L w L_w L w χ ‾ \overline{\chi} χ F ( ρ ‾ ) F(\overline{\rho}) F ( ρ ) ρ ‾ \overline{\rho} ρ
Lemma 10.4. For all integers n > a ≥ 1 n>a\geq1 n > a ≥ 1 F ⊂ F a ⊂ F n F\subset F_a\subset F_n F ⊂ F a ⊂ F n 2 a 2^a 2 a 2 n 2^n 2 n 2 2 2 F n F_n F n Σ 0 \Sigma_0 Σ 0 F n F_n F n F n ∩ F ( ρ ‾ ) = F F_n\cap F(\overline{\rho})=F F n ∩ F ( ρ ) = F ρ ‾ ∣ G F n \overline{\rho}|_{G_{F_n}} ρ ∣ G F n F n F_n F n Σ 0 ∪ { w } \Sigma_0\cup\{w\} Σ 0 ∪ { w } ∣ Σ 0 ∣ + 1 |\Sigma_0|+1 ∣ Σ 0 ∣ + 1
Proof. Choose a place w w w F F F S ′ ∪ Σ 0 S'\cup\Sigma_0 S ′ ∪ Σ 0 F ( ρ ‾ ) F(\overline{\rho}) F ( ρ ) M n = F ( μ 2 n + 1 , ( O S ′ ) 1 / 2 n ) M_n=F(\mu_{2^{n+1}},(\mathcal{O}_{S'})^{1/2^n}) M n = F ( μ 2 n + 1 , ( O S ′ ) 1/ 2 n ) F ( x v ) F(\sqrt{x_v}) F ( x v ) v ∈ Σ 0 v\in\Sigma_0 v ∈ Σ 0 w w w Gal ( M n ∏ v ∈ Σ 0 F ( x v ) / F ) \operatorname{Gal}(M_n\prod_{v\in\Sigma_0}F(\sqrt{x_v})/F) Gal ( M n ∏ v ∈ Σ 0 F ( x v ) / F ) ≃ Gal ( M n / F ) × ( Z / 2 ) Σ 0 \simeq\operatorname{Gal}(M_n/F)\times(\mathbb{Z}/2)^{\Sigma_0} ≃ Gal ( M n / F ) × ( Z /2 ) Σ 0 F ( ρ ‾ ) F(\overline{\rho}) F ( ρ ) F n = L w F_n=L_w F n = L w F a F_a F a 2 a 2^a 2 a
By Lemma 8.1, F n F_n F n 2 2 2 w w w v ∈ Σ 0 v\in\Sigma_0 v ∈ Σ 0 x v x_v x v w w w Frob v \operatorname{Frob}_v Frob v Gal ( F n / F ) \operatorname{Gal}(F_n/F) Gal ( F n / F ) v v v F n / F F_n/F F n / F w w w F ( ρ ‾ ) / F F(\overline{\rho})/F F ( ρ ) / F w w w F n ∩ F ( ρ ‾ ) = F F_n\cap F(\overline{\rho})=F F n ∩ F ( ρ ) = F ρ ‾ ( G F n ) = ρ ‾ ( G F ) \overline{\rho}(G_{F_n})=\overline{\rho}(G_F) ρ ( G F n ) = ρ ( G F )
Pro-modularity and modularity.
Proposition 10.5. Let ρ \rho ρ § 1.1 , with ρ ˉ \bar{\rho} ρ ˉ det ρ = ε w χ 0 \det\rho= \varepsilon^{w}\chi_{0} det ρ = ε w χ 0 χ 0 \chi_{0} χ 0 ρ ˉ ∣ G K \bar{\rho}|_{G_K} ρ ˉ ∣ G K K / F K/F K / F
Proof. By Lemma 2.4 , ρ ∣ G F ′ \rho|_{G_{F'}} ρ ∣ G F ′ F ′ / F F'/F F ′ / F Q ⊂ A ⊂ F \mathbb{Q} \subset A \subset F Q ⊂ A ⊂ F F ⊂ F a ⊂ F n F \subset F_a \subset F_n F ⊂ F a ⊂ F n Theorem 8.4 , with the following simplifications and changes.
No reducible loci. Since ρ ˉ ∣ G F n \bar{\rho}|_{G_{F_n}} ρ ˉ ∣ G F n ρ ˉ ∣ G F a \bar{\rho}|_{G_{F_a}} ρ ˉ ∣ G F a F n F_n F n F a F_a F a d K d_K d K
Numerical conditions. With u n = ∣ Σ 0 ∣ + 1 u_n = |\Sigma_0|+1 u n = ∣ Σ 0 ∣ + 1 n n n a a a 2 d n − 4 − u n > d n 2d_n-4-u_n > d_n 2 d n − 4 − u n > d n d a > u a + C + 4 d_a > u_a + C + 4 d a > u a + C + 4 d n = [ F n : Q ] d_n = [F_n:\mathbb{Q}] d n = [ F n : Q ] d a = [ F a : Q ] d_a = [F_a:\mathbb{Q}] d a = [ F a : Q ] C C C
The seed. The base change of g g g F a F_a F a F a / F F_a/F F a / F [1 ] , is cuspidal by regularity, as in [OAI, §7.4]. Its residual representation is ρ ˉ ∣ G F a \bar{\rho}|_{G_{F_a}} ρ ˉ ∣ G F a Σ \Sigma Σ χ ε \chi\varepsilon χ ε ϖ \varpi ϖ det ρ ˉ g = det ρ ˉ \det\bar{\rho}_g = \det\bar{\rho} det ρ ˉ g = det ρ ˉ 1 + 2 d a 1+2d_a 1 + 2 d a
The target. The target loci are the irreducible components of Spec R F a p s \operatorname{Spec} R^{\mathrm{ps}}_{F_a} Spec R F a ps ρ ∣ G F a \rho|_{G_{F_a}} ρ ∣ G F a Proposition 9.8 , each is the pseudo-image of a component of the framed representation scheme and has dimension at least 2 d a − 2 2d_a-2 2 d a − 2
The remaining arguments of [OAI, §7] and the propagation theorem [OAI, Theorem 5.1] apply as in the proof of Theorem 8.4 . In [OAI, §7.4], any stable lattice of the seed and target curves gives a point of the residual scheme over F n F_n F n F n F_n F n K = F a K=F_a K = F a F F F
Theorem 10.6. Let F F F § 1.1 and ρ : G F → GL 2 ( E ) \rho:G_F \to\operatorname{GL}_2(E) ρ : G F → GL 2 ( E ) § 1.1 , satisfying (N) and (O), with ρ ˉ \bar{\rho} ρ ˉ ρ v \rho_v ρ v v ∈ Σ 0 v \in\Sigma_0 v ∈ Σ 0 ρ \rho ρ
Proof. If ρ \rho ρ § 2.2 . Otherwise, by (N), (O) and § 2.2 , det ρ = ε w χ 0 \det\rho= \varepsilon^w\chi_0 det ρ = ε w χ 0 χ 0 \chi_0 χ 0 Proposition 10.5 gives (PM) for fields F ⊂ K ⊂ F 1 F \subset K \subset F_1 F ⊂ K ⊂ F 1 ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 p \mathfrak{p} p ( ρ ˉ ∣ G F 1 ) s s (\bar{\rho}|_{G_{F_1}})^{\mathrm{ss}} ( ρ ˉ ∣ G F 1 ) ss Lemma 2.4 . The solvable extensions in Proposition 10.5 need not preserve residual irreducibility: if ( ρ ˉ ∣ G F 1 ) s s (\bar{\rho}|_{G_{F_1}})^{\mathrm{ss}} ( ρ ˉ ∣ G F 1 ) ss 1 ⊕ χ ˉ ′ 1 \oplus\bar{\chi}' 1 ⊕ χ ˉ ′ χ ˉ ′ = 1 \bar{\chi}'=1 χ ˉ ′ = 1 w ∣ v ∈ Σ 0 w \mid v \in\Sigma_0 w ∣ v ∈ Σ 0 ρ ∣ G F 1 , w = ρ v \rho|_{G_{F_{1,w}}}=\rho_v ρ ∣ G F 1 , w = ρ v P \mathfrak{P} P p \mathfrak{p} p ψ w , 1 ( P ) \psi_{w,1}(\mathfrak{P}) ψ w , 1 ( P ) ρ v \rho_v ρ v Lemma 2.1 . By Theorem 3.4 over F 1 F_1 F 1 ρ ∣ G F 1 \rho|_{G_{F_1}} ρ ∣ G F 1 F 1 / F F_1/F F 1 / F [1 ] , ρ \rho ρ
case: since F ( ζ 2 ) = F F(\zeta_2)=F F ( ζ 2 ) = F G F ( ζ p ) G_{F(\zeta_p)} G F ( ζ p ) [34 ] , is handled by the method of [OAI], whose patching takes place at a characteristic 2 2 2 [3 ] , with no condition on the splitness and with an additional condition at 2 2 2
Theorem 10.6 treats every absolutely irreducible residual representation with solvable image, under its stated hypotheses. For reducible residual semisimplifications, Theorem A requires the ratio to extend to G Q G_{\mathbb{Q}} G Q 2 2 2 F = Q ( q ) F=\mathbb{Q}(\sqrt{q}) F = Q ( q ) q ≡ 1 ( m o d 8 ) q\equiv1\pmod{8} q ≡ 1 ( mod 8 ) Theorem 9.4 adds precisely the cases (b) and (c), still assuming that the ratio extends. The uncovered cases are therefore: non-extendible ratios; scalar ratios when (S) fails; and the non-trivial ratios which are trivial at 2 2 2 Theorem 9.4 .
Remarks
Remark 11.1. Theorem 8.4 gives the potential form of pro-modularity stated in (PM): components over K K K F 1 F_1 F 1 K K K F 1 F_1 F 1 0 0 0 2 2 2 Proposition 8.8 and Theorem 8.9; it does not follow from points of characteristic 0 0 0 χ ‾ ∣ G Q 2 ≠ 1 \overline{\chi}|_{G_{\mathbb{Q}_2}}\ne1 χ ∣ G Q 2 = 1 χ ‾ ∣ G Q 2 = 1 \overline{\chi}|_{G_{\mathbb{Q}_2}}=1 χ ∣ G Q 2 = 1 Q \mathbb{Q} Q F = Q ( q ) F=\mathbb{Q}(\sqrt{q}) F = Q ( q ) § 11 under its explicit hypotheses, with F F F F F F F F F
Remark 11.2. For odd p p p Λ \Lambda Λ [26 Theorem 5.1.2(1)] , by an R = T R=T R = T p = 2 p=2 p = 2 2 2 2 Proposition 8.8 ). The second step cannot be obtained from Emerton’s adjunction at points alone, since primes of characteristic 2 2 2 0 0 0 2 2 2 Ord B \operatorname{Ord}_{B} Ord B 2 2 2 2 d 2d 2 d 2 2 2 Q 2 × \mathbb{Q}_2^{\times} Q 2 × 2 2 2 ad 0 \operatorname{ad}^{0} ad 0
Remark 11.3. Condition (St) is used in three places: to exclude CM good points (Lemma 4.4 ), which for odd p p p 2 2 2 C C C Lemma 4.3 preserves the scalar inertial semisimplification and the equation P v 0 = 0 P_{v_0}=0 P v 0 = 0 C C C 2 2 2 C C C v ∣ 2 v\mid2 v ∣ 2 v v v
Remark 11.4. The present proof of Theorem B uses χ ˉ ∣ G v ≠ 1 \bar{\chi}|_{G_v} \ne1 χ ˉ ∣ G v = 1 Σ n \Sigma_n Σ n Lemma 2.2 ), and in the generic-block argument of §8. This is a requirement of the proof; we do not claim that the conclusion fails without it. Since quadratic characters are trivial modulo 2, (R) excludes all cases in which ρ ˉ \bar{\rho} ρ ˉ G L 2 ( F 2 ) \mathrm{GL}_2(\mathbb{F}_2) GL 2 ( F 2 )
Remark 11.5. We record the verification of the results used for p = 2 p = 2 p = 2
[6 Proposition 2.2.1] is stated for an arbitrary prime p p p k ≥ 2 k \ge2 k ≥ 2 k = 2 k = 2 k = 2 a 1 > a 2 a_1 > a_2 a 1 > a 2 k ≥ 3 k \ge3 k ≥ 3
Saito’s local–global compatibility at p p p [29 Theorem 1] does not exclude p = 2 p = 2 p = 2 [ F : Q ] [F : \mathbb{Q}] [ F : Q ] π \pi π [31 ] , whose argument covers p = 2 p = 2 p = 2 Proposition 6.1 .
The theorem of Hida and Zhao [Zhao14, Theorem 1] is stated for an arbitrary prime p p p p p p [ F : Q ] [F : \mathbb{Q}] [ F : Q ] p p p [18 ] , apart from unramifiedness conditions in the latter, which Zhao removes.
In [35 §1.1] , with HT ( ε ) = − 1 \operatorname{HT}(\varepsilon) = -1 HT ( ε ) = − 1 { 0 , 1 } \{0, 1\} { 0 , 1 }
( ψ 1 ∗ 0 ε − 1 ψ 2 ) \begin{pmatrix} \psi_1 & * \\ 0 & \varepsilon^{-1}\psi_2 \end{pmatrix} ( ψ 1 0 ∗ ε − 1 ψ 2 ) with ψ 1 , ψ 2 \psi_1,\psi_2 ψ 1 , ψ 2 r ∣ G v r|_{G_v} r ∣ G v Proposition 6.1 (1), and [35 Theorem B] only assumes that the determinant is totally odd.
The presentation used in Proposition 4.2 is [OAI, Proposition 3.6 and Corollary 3.7], proved for p = 2 p = 2 p = 2
The theorem of Newton and Thorne [22 Theorem B] is stated for every prime p p p p = 2 p = 2 p = 2
Inputs (P1) and (P2) of Lemma 8.7 are [25 Theorems 1.3 and 2.23] , which treats generic blocks for p = 2 p = 2 p = 2 p > 2 p > 2 p > 2 [24 §B.1] ); the action of the centre on Ord B ( J i ) \operatorname{Ord}_B(J_i) Ord B ( J i ) V ˇ ( P ) \check{V}(P) V ˇ ( P ) [25 Corollary 2.25] , and (P3) is [27 Lemma 4.4] , see also [25 (2)] for its residual form. In the proof of Theorem 8.4 , the modifications of [OAI, §7] were checked against loc. cit.: Leopoldt’s conjecture is used there only in [OAI, Lemmas 3.8 and 7.3]; the numerical conditions of [OAI, (39)] are replaced by those of item (2); the seed must be fixed before the fields, as in [OAI, §7.2]; and K = F a K = F_a K = F a F F F Q \mathbb{Q} Q K / F K/F K / F Lemma 8.2 ; this costs one more conductor in the bound for t t t
Remark 11.6. In Theorem A, the condition χ ˉ ∣ G v ≠ 1 \bar{\chi}|_{G_v} \ne1 χ ˉ ∣ G v = 1 v v v Theorem 8.4 , whose proof uses χ ˉ ∣ G Q 2 ≠ 1 \bar{\chi}|_{G_{\mathbb{Q}_2}} \ne1 χ ˉ ∣ G Q 2 = 1 F = Q ( q ) F = \mathbb{Q}(\sqrt{q}) F = Q ( q ) Theorem 9.4 removes this condition outside one subcase.
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585. 08007 Barcelona
Email address: ldieulefait@ub.edu
