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86e7cefd687eba8df6e0586d0f263e6eeff8e0c6 | subsection | 163 | 399 | Deformations of rank | \!\!\!\!\!\!\!\!\!\!\!\!\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D,D\big )@> \cup >> E.
\end{}The top squares of (REF ) are induced from the bottom squares of (REF ). Recall \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}\big (D,D\big )=\kappa ^{-1}\big (\operatorname{\mathrm {E}xt}^1_{(\v... | {
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f0aef33720b710ee09a643145a36d4babc9d8de3 | subsection | 164 | 399 | Deformations of | We study certain Ext{}^1 groups in the category of locally analytic representations of \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p).For an integral weight \mu of \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p), we denote by \delta _{\mu } the algebraic character of the diagonal torus T(\mathbb {Q}_p) of weight \mu . We fi... | {
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412a953970bad3bde8acfd20b11b5bfa1e15aea6 | subsection | 165 | 399 | Deformations of | Then I(\lambda ) has the form I(\lambda )\cong L(\lambda )- \operatorname{\mathrm {S}t}_2^{\infty }(\lambda ) -I(s\cdot \lambda ) (recall - denotes a nonsplit extension), \operatorname{\mathrm {S}t}_2^{\infty }(\lambda ):=\operatorname{\mathrm {S}t}_2\otimes _E L(\lambda ) and where the subrepresentation L(\lambda )-\o... | {
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2fe9fe85d11f7b116d624fdcaa1fba98f43ab971 | subsection | 166 | 399 | Deformations of | If V is a locally analytic representation of \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p), we define the locally analytic homology groups H_i(\overline{N}(\mathbb {Q}_p),V) as in where \overline{N}(\mathbb {Q}_p) is the unipotent radical of \overline{B}(\mathbb {Q}_p).Lemma 3.13 We have the following isomorphisms:H_i(\... | {
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309fdd74feba89a37938cd8e8646aa2170cc1ae5 | subsection | 167 | 399 | Deformations of | The isomorphisms () and () follow from and .The following statement is not new, we include a proof for the reader's convenience.Theorem 3.14
We have natural isomorphisms:\small
\operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E){\sim } \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}... | {
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6db94cee601518a8b7669c33a6e184e70bc29dc9 | subsection | 168 | 399 | Deformations of | By we have \operatorname{\mathrm {E}xt}^i_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\widetilde{I}(s\cdot \lambda ), \operatorname{\mathrm {S}t}_2^{\infty }(\lambda ))=0 for i=1,2. By and (), we have \operatorname{\mathrm {E}xt}^i_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\widetilde{I}(s\cdot \lambda ), I(s\cd... | {
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02f9ebe0f24047c601caad3766a206101e268ec3 | subsection | 169 | 399 | Deformations of | \end{}Consider the short exact sequence:\footnotesize 0 \longrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(L(\lambda ), L(\lambda )) \longrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(L(\lambda ), I(\lambda )) \longrightarrow \operato... | {
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3a598dbb0f2967926645c8c1a03454abceb005fc | subsection | 170 | 399 | Deformations of | Let \psi \in \operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E) and choose \psi _i\in \operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E) for i=1,2 such that \psi _1-\psi _2=\psi . Let \sigma (\psi _1,\psi _2) be the following two dimensional representation of T(\mathbb {Q}_p):\sigma (\psi _1,\psi _2)\begin{... | {
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72107a629f7fe6c9d5cce91b3dae2090a3d28bcf | subsection | 171 | 399 | Deformations of | By and (REF ), (), we have \operatorname{\mathrm {H}om}(T(\mathbb {Q}_p),E)\sim \over \longrightarrow \operatorname{\mathrm {E}xt}^1_{T(\mathbb {Q}_p)}(\delta _{\lambda }, \delta _{\lambda }) {\sim } \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(i(\lambda ), I(\lambda )) and the composit... | {
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9b004bbd49cf081c2fba23171ecd07fb19bd3bad | subsection | 172 | 399 | Deformations of | Putting the above maps together, we obtain:\operatorname{\mathrm {H}om}(T(\mathbb {Q}_p),E){\sim } \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(i(\lambda ), I(\lambda )/L(\lambda )), \ (\psi _1, \psi _2)\longmapsto \pi (\lambda ,\psi _1, \psi _2)^-.Let 0\ne \psi \in \operatorname{\mathr... | {
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432488f2e22b062c5e76f1ee2f80b5e79ff5c99f | subsection | 173 | 399 | Deformations of | Let \chi _{\lambda }:=\delta _{\lambda }|_{Z(\mathbb {Q}_p)}, which is the central character of \pi (\lambda , \psi ).Lemma 3.15
For any \widetilde{\pi }\in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\pi (\lambda , \psi ), \pi (\lambda , \psi )), there exists a unique lifting \wideti... | {
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73ca83959f7cb19edb2fd43bb386ec3994f8a14a | subsection | 174 | 399 | Deformations of | We fix a v which is not in \pi (\lambda , \psi )=\epsilon \widetilde{\pi } and define:\pi (\widetilde{\chi }_{\lambda }):=\big \lbrace w\in \widetilde{\pi },\ (z-\widetilde{\chi }_{\lambda }(z))w=0 \ \forall \ z\in Z(\mathbb {Q}_p)\big \rbracewhich is a \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)-subrepresentation of ... | {
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014090406e3b139354bb26e84ab26e7508a37143 | subsection | 175 | 399 | Deformations of | Hence \pi (\widetilde{\chi }_{\lambda })=\widetilde{\pi }.Lemma 3.16
We have a short exact sequence:0 \longrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p),Z}(\pi (\lambda , \psi ), \pi (\lambda , \psi )) \longrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}... | {
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local-global compatibility | [
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ef8df1f73fc49232c67bb739761af371c5bce4b1 | subsection | 176 | 399 | Deformations of | The lemma follows.Lemma 3.17
(1) We have \operatorname{\mathrm {E}xt}^i_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(I(s\cdot \lambda ), \pi (\lambda , \psi ))=\operatorname{\mathrm {E}xt}^i_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\widetilde{I}(s\cdot \lambda ), \pi (\lambda , \psi ))=0 for all i\in \mathbb {... | {
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c0cb17d9a49b94bb6d2e3c30b98a49b744c796a2 | subsection | 177 | 399 | Deformations of | \end{array}(3) We have exact sequences:0 \longrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p),Z}(\pi (\lambda , \psi )/\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \pi (\lambda , \psi )) \longrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {... | {
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bd70ef2dc1d85fb03c05c514e5c983b43ebb3a14 | subsection | 178 | 399 | Deformations of | \operatorname{\mathrm {E}xt}^{i}_{Z}) for \operatorname{\mathrm {E}xt}^{i}_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)} (resp. \operatorname{\mathrm {E}xt}^{i}_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p),Z}).(1) We prove the case of I(s\cdot \lambda ), the proof for \widetilde{I}(s\cdot \lambda ) being parallel. By... | {
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72f120909d7cfb9f579ebec863a73232df1ddab0 | subsection | 179 | 399 | Deformations of | Together with (REF ) this implies (again by dévissage) \operatorname{\mathrm {E}xt}^i(I(s\cdot \lambda ), \pi (\lambda , \psi ))=0 for all i\ge 0. This concludes the proof of (1).(2) By , we have \operatorname{\mathrm {E}xt}^i_{Z}(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_2^{\infty ... | {
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... | |
eeeb5f8bad152d7008d9033c0a03daa591aa6fb5 | subsection | 180 | 399 | Deformations of | So we have:\widetilde{\pi }&\cong &\big (\widetilde{\pi }\otimes _{E[\epsilon ]/\epsilon ^2} (1-(\psi ^{\prime }/2)\epsilon )\circ \operatorname{\mathrm {d}et}\big )\otimes _{E[\epsilon ]/\epsilon ^2} (1+(\psi ^{\prime }/2)\epsilon )\circ \operatorname{\mathrm {d}et}\\
&\cong & \big (\operatorname{\mathrm {S}t}_2^{\inf... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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] | 2,018 | en | Mathematics | [
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94085e09ce6440d2b1c01b3857e52fbb6caebb2e | subsection | 181 | 399 | Deformations of | By and (), we have:\operatorname{\mathrm {E}xt}^1_Z(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \widetilde{I}(\lambda ))&\cong &\operatorname{\mathrm {E}xt}^1_{T(\mathbb {Q}_p),Z}(\delta _\lambda (|\cdot |^{-1}\otimes |\cdot |), \delta _\lambda (|\cdot |^{-1}\otimes |\cdot |))\\
\operatorname{\mathrm {E}xt}^i_Z(... | {
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local-global compatibility | [
"Christophe Breuil",
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3ecefc66dface406e571ee0360dfa04333aaf6f0 | subsection | 182 | 399 | Deformations of | Likewise we have:0=\operatorname{\mathrm {E}xt}^1_Z(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_2^{\infty }(\lambda )) \longrightarrow \operatorname{\mathrm {E}xt}^1_Z(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \pi (\lambda , \psi )) \\ \longrightarrow \operatorname{\mathrm {E... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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206b54d1568a371000bca1dbc9b04af1b6ea2848 | subsection | 183 | 399 | Deformations of | By and (REF ), we have \operatorname{\mathrm {E}xt}^i_Z(L(\lambda ), \widetilde{I}(s\cdot \lambda ))=0 for i\ge 0 and by , we have \operatorname{\mathrm {E}xt}^1_Z(L(\lambda ), L(\lambda ))=0. By dévissage, we deduce then \operatorname{\mathrm {E}xt}^1_Z(L(\lambda ), \widetilde{I}(\lambda )/\operatorname{\mathrm {S}t}_... | {
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local-global compatibility | [
"Christophe Breuil",
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38212ff8f1c269602c002fea5c894a67c024b641 | subsection | 184 | 399 | Deformations of | The second one follows easily from \operatorname{\mathrm {E}xt}^2_Z(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ),\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ))=0 (see the proof of (2) above). By and (), we have \dim _E \operatorname{\mathrm {E}xt}^1_Z(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \widetil... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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28dfc2d1f5a35469f4c31154ea76712ba6097df8 | subsection | 185 | 399 | Deformations of | We have seen \dim _E \operatorname{\mathrm {E}xt}^1(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_2^{\infty }(\lambda ))=2 in the proof of (2), the rest of (4) follows from lemma REF .Remark 3.18
It follows from (REF ) and (2) that, if \psi ^{\prime }\notin E\psi \subset \operatorname{... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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88d4b16478974c00cc538ea61cffe1ef6c65a7ef | subsection | 186 | 399 | Deformations of | It is not difficult then to deduce:\widetilde{\pi }^{\operatorname{\mathrm {l}alg}}\cong {\left\lbrace \begin{array}{ll} \operatorname{\mathrm {S}t}_2^{\infty }(\lambda )^{\oplus 2} & \psi \text{ not smooth}\\
\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ) \oplus \widetilde{i}(\lambda ) & \psi \text{ smooth}
\end{ar... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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1a21aa6347f53d466bbd25905759875ff2ab3aec | subsection | 187 | 399 | Deformations of | For a locally analytic character \delta : T(\mathbb {Q}_p) \rightarrow E^{\times }, denote by \operatorname{\mathrm {E}xt}^1_{T(\mathbb {Q}_p)}(\delta , \delta )_\psi \subseteq \operatorname{\mathrm {E}xt}^1_{T(\mathbb {Q}_p)}(\delta , \delta ) the E-vector subspace corresponding to \operatorname{\mathrm {H}om}(T(\math... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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9bca4b9c310f63cc3e4029e24b2b6eb63431e97a | subsection | 188 | 399 | Deformations of | Since \pi (\lambda , \psi ) is very strongly admissible, it is not difficult to prove they are equal using together with the left exactness of J_B and .Lemma 3.20
(1) Let V\in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \pi (\lambda ,... | {
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... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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6262dfce42694b3837f120f4bd5ed8dcfdde0272 | subsection | 189 | 399 | Deformations of | By , the map j_1 then induces a \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)-equivariant map:j_2: I_{\overline{B}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\widetilde{\delta }_{\lambda } \longrightarrow Vsuch that the morphism j_1 can be recovered from j_2 by applying the functor J_B(\cdot ) and whe... | {
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"raw": "Emerton, M. Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction. J. Inst. Math. Jussieu. to appear.",
"source_ref_id": "c860... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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6335486e1819c7b25b7f6d797181f500d088cc01 | subsection | 190 | 399 | Deformations of | By the exact sequence (REF ) together with the fact that \widetilde{I}(s\cdot \lambda ) is not an irreducible constituent of I_{\overline{B}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\widetilde{\delta }_{\lambda } (since it is not an irreducible constituent of (\operatorname{\mathrm {I}nd}_{\overlin... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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a8a6baa4c96da2e2f6a58309f3104ae111be76eb | subsection | 191 | 399 | Deformations of | In particular, taking J_B induces a bijection by (REF ):\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_2^{\infty }(\lambda )) {\sim } \operatorname{\mathrm {H}om}(Z(\mathbb {Q}_p), E) \big (\longrightarrow \ope... | {
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"raw": "Emerton, M. Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction. J. Inst. Math. Jussieu. to appear.",
"source_ref_id": "c860... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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bb449652ce6c5aecfa7db680507021a8080835cb | subsection | 192 | 399 | Deformations of | Moreover \pi (\lambda , \psi _1, \psi _2)^-\hookrightarrow W(\Psi )/L(\lambda ) and neither J_B(L(\lambda )) nor J_B((W(\Psi )/L(\lambda ))/\pi (\lambda , \psi _1, \psi _2)^-)=J_B(I(s\cdot \lambda )) contains \chi as a subquotient (the latter by ). By left exactness of J_B we deduce:\chi \otimes _E (1+\Psi \epsilon )\l... | {
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"raw": "Emerton, M. Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction. J. Inst. Math. Jussieu. to appear.",
"source_ref_id": "c8607... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
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9cdb4eefa33bae5595a4b5303a73c29dfeba3f44 | subsection | 193 | 399 | Deformations of | Indeed, let \lbrace \psi _1^{\prime }\circ \operatorname{\mathrm {d}et}, \psi _2^{\prime } \circ \operatorname{\mathrm {d}et}, \Psi _3:=\Psi \rbrace be a basis of the 3-dimensional space \operatorname{\mathrm {H}om}(T(\mathbb {Q}_p),E)_{\psi }\cong \operatorname{\mathrm {E}xt}^1_{T(\mathbb {Q}_p)}(\chi , \chi )_\psi wh... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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0.00610... | |
8f06a0eabf10bec32ad6809054b99e9b51a8e3c8 | subsection | 194 | 399 | Deformations of | This finishes the proof.Remark 3.21
From the proof of Lemma REF , we can explicitly describe the inverse of (REF ) as follows. Let \Psi \in \operatorname{\mathrm {H}om}(T(\mathbb {Q}_p), E)_{\psi }, define:\widetilde{\chi }=\delta _{\lambda }(|\cdot |\otimes |\cdot |^{-1})(1+\Psi \epsilon )\in \operatorname{\mathrm {E... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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5f88c20743eba2c7fd5bbd9808f91e8a423b7854 | subsection | 195 | 399 | Deformations of | If \Psi \notin \operatorname{\mathrm {H}om}(Z(\mathbb {Q}_p),E), the inverse image of \Psi is then isomorphic to \operatorname{\mathrm {p}r}^{-1}(i(\lambda ))/L(\lambda ).We now denote by \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(\pi (\lambda , \psi ), \pi (\lambda , \psi )) the kernel of the compos... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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0.0320... | |
a903aa93c0c203871c267d632cac86b595b54871 | subsection | 196 | 399 | Deformations of | In particular, by (REF ) we have \operatorname{\mathrm {I}m}(\iota _0)\subset \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(\pi (\lambda , \psi ), \pi (\lambda , \psi )).Lemma 3.22
(1) We have \dim _E \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(\pi (\lambda , \psi ), \pi (\lambda , \p... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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1d7ab0ad0f6eba4aa48da77e34311c4ae91a73cf | subsection | 197 | 399 | Deformations of | Together with (the proof of) Lemma REF (1), left exactness of J_B and (REF ), we easily deduce (2) and (3), where the third map of (REF ) is given by:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(\pi (\lambda , \psi ), \pi (\lambda , \psi )) \twoheadrightarrow \operatorname{\mathrm {I}m}(\iota _1) [\sim... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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5b10a1c28bc34fd4043ce416c844609db603b441 | subsection | 198 | 399 | Deformations of | From the exact sequence:0 \longrightarrow E \psi \longrightarrow \operatorname{\mathrm {H}om}(T(\mathbb {Q}_p), E)_{\psi } {\operatorname{\mathrm {p}r}_2} \operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E) \longrightarrow 0(where the injection is \psi \mapsto \Psi =(\psi ,0)), we obtain with (REF ) an exact seque... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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] | 2,018 | en | Mathematics | [
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269ec8e6d8276ed9beda83dc069eb63490405711 | subsection | 199 | 399 | Deformations of | \kappa _1([\widetilde{\pi }])\in E^{\times } \iota _1([\pi (\lambda , \psi ,0)^-]), then \widetilde{\pi } does not have central character \chi _{\lambda } (which is the central character of \pi (\lambda , \psi )), and it is enough to show that \pi (\lambda , \psi ,0)^- does not have central character \chi _{\lambda }. | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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79a4e5c83fbd5536fc42413e55a153b07621f106 | subsection | 200 | 399 | Deformations of | By the construction following Theorem REF and by Lemma REF (applied first to the extension W of V_1=\pi (\lambda , \psi ,0)^- by V_2=L(\lambda ) inside V:=(\operatorname{\mathrm {I}nd}_{\overline{B}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)} \delta _{\lambda }\otimes _E \sigma (\psi , 0))^{\operator... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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] | 2,018 | en | Mathematics | [
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e5ef9c95dfef27d74dfa137aec23078d19997460 | subsection | 201 | 399 | Deformations of | In particular:\dim _E \operatorname{\mathrm {E}xt}^1_g(\pi (\lambda , \psi ),\pi (\lambda , \psi ))={\left\lbrace \begin{array}{ll}
2 & \psi \text{ non smooth}\\
3 & \psi \text{ smooth}.
\end{array}\right.}(1) It is easy to see \operatorname{\mathrm {I}m}(\iota _0)\subseteq \operatorname{\mathrm {E}xt}^1_g(\pi (\lambda... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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] | 2,018 | en | Mathematics | [
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53a731dbdbc7313d5e948d603344eeaaec9832ef | subsection | 202 | 399 | Deformations of | By (1) and Remark REF , we know that there exists \Psi \in \operatorname{\mathrm {H}om}(T(\mathbb {Q}_p),E)_{\psi } such that\delta _{\lambda }(|\cdot |\otimes |\cdot |^{-1})\otimes _E (1+\Psi \epsilon )\longrightarrow J_B(\widetilde{\pi }).Moreover, the natural surjection \widetilde{\pi }\twoheadrightarrow \pi (\lambd... | {
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"raw": "Emerton, M. Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction. J. Inst. Math. Jussieu. to appear.",
"source_ref_id": "c860... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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2db43a026dfbadedacc4c0a96e7a987a2fbe1f27 | subsection | 203 | 399 | Deformations of | However, by the construction in Remark REF , if \Psi in Remark REF is smooth, then we see that the inverse image \widetilde{\pi }_1 of \Psi in (REF ) has extra locally algebraic vectors than (\pi (\lambda , \psi )^-)^{\operatorname{\mathrm {l}alg}}=\pi (\lambda , \psi )^{\operatorname{\mathrm {l}alg}}. Let \widetilde{\... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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86eab478b6ed12413d2f714129b73e24fc0dedbe | subsection | 204 | 399 | Notation and preliminaries | We define some useful locally analytic representations of \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p).We now switch to \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p) and we let B(\mathbb {Q}_p) (resp. \overline{B}(\mathbb {Q}_p)) be the Borel subgroup of upper (resp. lower) triangular matrices, T(\mathbb {Q}_p) the diag... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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cf91b4f4a00715ba66f2795a8b3fef58d3187555 | subsection | 205 | 399 | Notation and preliminaries | To lighten notation we set:I_{\overline{B}}^{\operatorname{\mathrm {G}L}_3}(\lambda )&:=&\big (\operatorname{\mathrm {I}nd}_{\overline{B}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)} \delta _{\lambda }\big )^{\operatorname{\mathrm {a}n}} \\
I_{\overline{P}_i}^{\operatorname{\mathrm {G}L}_3}(\lambda )&... | {
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"raw": "Schraen, B. Représentations localement analytiques de \\mathrm {GL}_3(\\mathbb {Q}_p). Ann. Sci. Éc. Norm. Sup. 44 (2011), 43–145.",
"source_ref_id": "70643133c0d312160f140ebf370cf9c4e10c77d6... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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f14d3c0ea8d4c1a8cc67b65acb767ad880cbec9c | subsection | 206 | 399 | Notation and preliminaries | Note that we have \overline{L}(-\lambda )\cong L(\lambda )^{\prime }. We use without comment the theory of , see e.g. for a summary. We often write \operatorname{\mathrm {G}L}_3, \overline{P}_i, Z (= the center of \operatorname{\mathrm {G}L}_3) instead of \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p), \overline{P}_i(\ma... | {
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... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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b9eeb124886ec9c1d163daa649a0249a5215b69b | subsection | 207 | 399 | Notation and preliminaries | We let \lambda _{1,2}:=(k_1,k_2) (which is thus a dominant weight for \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p) as in § REF ), it is easy to see that we have a commutative diagram (where we write \operatorname{\mathrm {G}L}_3 for \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p) etc.):\begin{}
\big (\operatorname{\mathrm... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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6515e2d061331af26e5a10835b89a71bf5991b53 | subsection | 208 | 399 | Notation and preliminaries | Together with (REF ), we deduce an exact sequence:0 \longrightarrow v_{\overline{P}_2}^{\operatorname{\mathrm {a}n}}(\lambda ) \longrightarrow \big (\operatorname{\mathrm {I}nd}_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3} \operatorname{\mathrm {S}t}_2^{\operatorname{\mathrm {a}n}}(\lambda _{1,2}) \otimes x^{k_3}\b... | {
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... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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f295e30070dad2d990366884d2c573bd44d09b9e | subsection | 209 | 399 | Notation and preliminaries | Indeed, as in the proof of , we have \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(C_{2,1}, v_{\overline{P}_2}^{\infty }(\lambda ))=0. Together with the fact that (REF ) is nonsplit, the claim follows by a straightforward dévissage. By replacing P_1 by P_2 and s_2 by s_1, we define in ... | {
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{
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"raw": "Breuil, C. \\mathrm {Ext}^1 localement analytique et compatibilité local-global. Amer. J. of Math.. to appear.",
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"start": 0
... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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a3f4f95f82f66c5332c244724a6d605d68c4168d | subsection | 210 | 399 | Notation and preliminaries | We let I_0 \supset I_1\supset I_2\supset \cdots \supset I_i \supset I_{i+1} \supset \cdots be a cofinal family of compact open subgroups of G(L) such that:I_i is normal in I_0
I_i admits an Iwahori decomposition.For i\in \mathbb {Z}_{\ge 0}, we put N_i:=N_P(L)\cap I_i, L_i:=L_P(L)\cap I_i and \overline{N}_i:=N_{\overl... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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64f6f6527c293ff5745ddd7000809fc9173eac6b | subsection | 211 | 399 | Notation and preliminaries | Let I_i:=\lbrace g\in \operatorname{\mathrm {G}L}_n(\mathcal {O}_L),\ g\equiv 1 \pmod {\varpi _L^i}\rbrace , we have:Z_{L_P}^+=\lbrace (a_1,\cdots , a_k)\in Z_{L_P}(L),\ \operatorname{\mathrm {v}al}_p(a_1)\ge \cdots \ge \operatorname{\mathrm {v}al}_p(a_k)\rbracewhere a_j\in L^{\times } is seen in (the center of) \opera... | {
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"sour... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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f94a1caaa5e4a4099aa32ec9bdacfe718589f406 | subsection | 212 | 399 | Notation and preliminaries | By , the functor \pi \mapsto \pi ^\vee induces an anti-equivalence of categories:\operatorname{\mathrm {M}od}_{G}^{\operatorname{\mathrm {s}m}}(A) {\sim } \operatorname{\mathrm {M}od}_{G}^{\operatorname{\mathrm {p}ro}\ \!\!\!\operatorname{\mathrm {a}ug}}(A).As in , we denote by \operatorname{\mathrm {M}od}_G^{\fg \ \!\... | {
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"sourc... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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d5c309a44ddfb5b682e17a02c65daa9b026c8824 | subsection | 213 | 399 | Notation and preliminaries | Recall that Colmez defined a covariant exact functor (called Colmez's functor, see ):\textbf {V}: \operatorname{\mathrm {M}od}^{\operatorname{\mathrm {f}in}}_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\mathcal {O}_E) \longrightarrow \operatorname{\mathrm {R}ep}^{\operatorname{\mathrm {f}in}}_{\operatorname{\mathrm ... | {
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... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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e77b51a7d580bdb25625e524ddbd9906b0716346 | subsection | 214 | 399 | Simple | We recall some facts on simple \mathcal {L}-invariants.We keep all the previous notation.Lemma 3.34
Let i,j\in \lbrace 1,2\rbrace , i\ne j.(1) We have \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_i}^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_{3}^{\operatorname{\... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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282294a9d715bd1ecd4cc84dd620fcb81d1151d7 | subsection | 215 | 399 | Simple | By the theory , we know that C is of the form \mathcal {F}_{\overline{P}_w}^{\operatorname{\mathrm {G}L}_3}(\overline{L}(-w\cdot \lambda ), \pi ^{\infty }) were w is a nontrivial element of the Weyl group distinct from s_i (since we mod out by S_{j,0}), P_w\subset \operatorname{\mathrm {G}L}_3 is the maximal parabolic ... | {
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... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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48566950ebd550087554c59c00450e41eff7d354 | subsection | 216 | 399 | Simple | If w=s_j, then we have \pi ^\infty =\operatorname{\mathrm {S}t}_{2}^{\infty }\otimes 1 if i=1 or \pi ^\infty =1\otimes \operatorname{\mathrm {S}t}_{2}^{\infty } if i=2, and \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_i}^{\infty }(\lambda ), C)=0 (via Lemma REF ) is th... | {
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local-global compatibility | [
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947a0983d85ae7553cb56ef982a5da880ee3da0a | subsection | 217 | 399 | Simple | If \Psi is smooth (i.e. all \psi _j are smooth, j\in \lbrace 1,2,3\rbrace ), by considering the following exact sequence (which is then “contained” in (REF )):0 \longrightarrow i_{\overline{B}}^{\operatorname{\mathrm {G}L}_3}(\lambda ) \longrightarrow \big (\operatorname{\mathrm {I}nd}_{\overline{B}(\mathbb {Q}_p)}^{\o... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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e07b66854f7f3f510eb399c69a8f8a72fd191675 | subsection | 218 | 399 | Simple | Moreover, we have a commutative diagram:\begin{}
\operatorname{\mathrm {H}om}_{\infty }(\mathbb {Q}_p^{\times }, E) @> \sim >>\operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_i}^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_3^{\infty }(\lambda ))\\
@VVV @VVV \\
\operato... | {
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local-global compatibility | [
"Christophe Breuil",
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3b6bd8f455eb647a2c4f0d8d1eba9a765075b911 | subsection | 219 | 399 | Simple | Moreover, the one dimensional subspace \operatorname{\mathrm {E}xt}^1_e(\mathcal {R}_E(\delta _{i+1}), \mathcal {R}_E(\delta _i)) of crystalline extensions is exactly annihilated by the subspace \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_i}^{\infty }(\lambda ), \oper... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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005622d2a017d834e295e2e94b5e4cca4a9f6253 | subsection | 220 | 399 | Simple | By , we have:\operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_n(\mathbb {Q}_p)}\big ((\operatorname{\mathrm {I}nd}_{\overline{B}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_n(\mathbb {Q}_p)} 1)^{\mathcal {C}^0}\otimes \chi _1\circ \operatorname{\mathrm {d}et}, \widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }... | {
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"raw": "Emerton, M. Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties. Astérisque 331 (2010), 355–402.",
"sourc... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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15fb009db5a31bbdea11c8ba6ceb1ac5558cafe4 | subsection | 221 | 399 | Simple | Together with (REF ), (REF ), (REF ) and Corollary REF , we deduce then m_B(x)\le m(\rho ). The lemma follows.Lemma 7.44
The \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{B-\operatorname{\mathrm {o}rd}}[1/p]-module M_B(U^{\wp })[1/p] is locally free at the point x.(a) Let X:= \operatorname{\mathrm {S}pec}A := ... | {
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... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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c0439ac486c13f0a056f8eb9a8f45b949d9dde3c | subsection | 222 | 399 | Simple | From (REF ), (REF ) together with m(\pi )=1 (which follows from and ), we then deduce m(x^{\prime })=m(\rho ) for all x^{\prime }\in Z, and hence m_B(x^{\prime })=m(\rho ) by (a) for all x^{\prime }\in Z.(c) Denote by \mathcal {M} the coherent sheaf on X attached to the A-module M_B(U^{\wp })[1/p]/\mathfrak {p}. For an... | {
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"source_ref_id": "97d8a7ba90c26c8972588cb1e7... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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e49122d807fb785ef3fac289cb9327d78b071dc2 | subsection | 223 | 399 | Simple | Recall that we have a natural morphism (see (REF )):\omega =(\omega _i)_{i=1, \cdots , n}: (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })^{B-\operatorname{\mathrm {o}rd}}_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}} \longrightarrow \prod _{i=1}^n (\operatorname{\mathrm {S}pf}R_{\overline{\rho }_i}... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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795bdc23663564c44707a898878f196f55a3a697 | subsection | 224 | 399 | Simple | Let 0\ne v\in V_x such that \overline{d}\omega _x(v)=0 and denote by \mathcal {I}_v:=\operatorname{\mathrm {K}er}(\widetilde{\mathbb {T}}(U^{\wp })^{B-\operatorname{\mathrm {o}rd}}_{\overline{\rho }}\rightarrow E[\epsilon ]/\epsilon ^2) the ideal attached to v (so \widetilde{\mathbb {T}}(U^{\wp })^{B-\operatorname{\mat... | {
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local-global compatibility | [
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3f2b006f0a6394e875ef0094658489a650a35e2b | subsection | 225 | 399 | Simple | Let {\chi }:=\otimes _{i=1}^n ({\chi }_i\otimes \varepsilon ^{s_i}) and \widetilde{\chi }:=\otimes _{i=1}^n (\widetilde{\chi }_i\otimes \varepsilon ^{s_i}) where the tensor product \otimes _{i=1}^n on the latter is over E[\epsilon ]/\epsilon ^2. By (REF ) together with applied with M=\widetilde{\mathbb {T}}(U^{\wp })_{... | {
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"start... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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ce9295148b5dad0b6732418a043dc66f923f3adc | subsection | 226 | 399 | Simple | The lemma follows.Recall that for i=1,\cdots , n-1 the (\varphi ,\Gamma )-module D_i^{i+1} was defined at the end of § REF , and that \mathcal {L}_{\operatorname{\mathrm {F}M}}(D_i^{i+1}: \mathcal {R}_E(\chi _i)) is the line in \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\chi _i), \mathcal {R}_... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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7013dd01c4a07bcf9704dd45985176f733ca4b08 | subsection | 227 | 399 | Simple | It follows from Proposition REF (2) that v can be seen as an E[\epsilon ]/\epsilon ^2-valued point of \operatorname{\mathrm {S}pec}R_{{\rho }_{\widetilde{\wp }},\lbrace \chi _i\rbrace }^{B-\operatorname{\mathrm {o}rd}}, hence that \widetilde{\rho } is isomorphic to a successive extension of the \widetilde{\chi }_i as \... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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9500667e64b2eb350f7a60af84d970120c27dfc4 | subsection | 228 | 399 | Simple | We have isomorphisms of smooth representations of L_{P_r}(\mathbb {Q}_p) over E:\small
\operatorname{\mathrm {O}rd}_{P_r}(\operatorname{\mathrm {S}t}_n^{\infty } \otimes \chi _1 \circ \operatorname{\mathrm {d}et})\cong J_{P_r}(\operatorname{\mathrm {S}t}_n^{\infty }\otimes \chi _1\circ \operatorname{\mathrm {d}et})(\d... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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298cb41a77c8acb196ab375148ced52e037639bb | subsection | 229 | 399 | Simple | In particular, we have (see (REF ) for m(\rho )):\dim _E \operatorname{\mathrm {H}om}_{L_{P_r}(\mathbb {Q}_p)}\Big (\big (\bigotimes _{{i=1, \dots , n-1 \\ i \ne r}} \chi _1\big ) \otimes ( \widehat{\pi }(\rho _r^{r+1})\otimes \varepsilon ^{r-1} \circ \operatorname{\mathrm {d}et}), \operatorname{\mathrm {O}rd}_{P_r}(\w... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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f379c365cfeafc16d9f12379c1e89d952616db6b | subsection | 230 | 399 | Simple | Let 0\ne \psi \in \mathcal {L}_{\operatorname{\mathrm {F}M}}(D_r^{r+1}:\mathcal {R}_E(\chi _r)), then we have the following restriction maps:[5] {\operatorname{\mathrm {H}om}_{L_{P_r}(\mathbb {Q}_p)}\Big (\big (\bigotimes _{{i=1, \dots , n-1 \\ i \ne r}} \chi _1\big ) \otimes ( \widehat{\pi }(\rho _r^{r+1})\otimes \var... | {
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local-global compatibility | [
"Christophe Breuil",
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4b8abac099f591cce20b2e247c55e3424aa0c160 | subsection | 231 | 399 | Simple | Using , (REF ) and Lemma REF , we deduce that the second and third morphisms are injective by the same type of argument as in the proof of Proposition REF . By the same arguments as in using Lemma REF (2) and Lemma REF , one can prove that the second morphism is moreover surjective (see also the end of the proof of Pro... | {
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local-global compatibility | [
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262fb28ff650c05c9bf2f98706ee28ddb780335a | subsection | 232 | 399 | Parabolic inductions | We study the locally analytic representation (\operatorname{\mathrm {I}nd}_{\overline{P}_1(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)}\pi (\lambda _{1,2}, \psi )\otimes x^{k_3})^{\operatorname{\mathrm {a}n}} (cf. § REF ) and some of its subquotients.We keep the previous notation and fix 0\ne \psi \in... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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6054c63f84a876c4394a68d487b8de8b2d56253c | subsection | 233 | 399 | Parabolic inductions | Exactness of parabolic induction gives the isomorphism (recalling that s is the unique nontrivial element in the Weyl group of \operatorname{\mathrm {G}L}_2):I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(I(s\cdot \lambda _{1,2}),k_3)\cong I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\operatorname{\mathrm {S}... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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fcc63c13c7db82cb0e17b8460d803c0b7d7ca501 | subsection | 234 | 399 | Parabolic inductions | From the theory of , one moreover easily deduces that the irreducible constituents of S_{1,1}/C_{1,1} are:\Big \lbrace \mathcal {F}_{\overline{P}_2}^{\operatorname{\mathrm {G}L}_3}(\overline{L}(-s_1s_2\cdot \lambda ), 1), \ \mathcal {F}_{\overline{P}_2}^{\operatorname{\mathrm {G}L}_3}(\overline{L}(-s_1s_2\cdot \lambda ... | {
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local-global compatibility | [
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cfd94e7f17fd7d11932a4abda0a79d497c550e5c | subsection | 235 | 399 | Parabolic inductions | Denote by:S_{1,2}:=v_{\overline{P}_1}^{\operatorname{\mathrm {a}n}}(\lambda ), \ \ C_{1,2}:=v_{\overline{P}_1}^{\infty }(\lambda )\cong \operatorname{\mathrm {s}oc}_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)} S_{1,2}.Since I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\operatorname{\mathrm {S}t}_2^{\operatorn... | {
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local-global compatibility | [
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1b366c3a8ac91e32241260bfa9c60b95b90990f1 | subsection | 236 | 399 | Parabolic inductions | \\
0 @>>> \operatorname{\mathrm {S}t}_3^{\operatorname{\mathrm {a}n}}(\lambda ) @>>> \widetilde{\Pi }^1(\lambda , \psi )^- @>>> v_{\overline{P}_1}^{\operatorname{\mathrm {a}n}}(\lambda ) @>>> 0
\end{}where \Pi ^1(\lambda , \psi )_0 denotes the image of \psi via the bottom isomorphism of (REF ).Let \Psi _1:=(\psi _1, \p... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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69cd276267aa7f2e09dec6dc556a78b8a5f8819a | subsection | 237 | 399 | Parabolic inductions | We have (by the transitivity of parabolic inductions):I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\pi (\lambda _{1,2}, \psi )^-,k_3) \longrightarrow I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}\big ((\operatorname{\mathrm {I}nd}_{\overline{B}_2(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}... | {
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local-global compatibility | [
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c216f68d151f0ee75e9f342abdec7fda668c8013 | subsection | 238 | 399 | Parabolic inductions | Using the formula in and (), it is not difficult to show:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)}\big (\widetilde{C}_{1,2}, \mathcal {F}_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\overline{M}(-s_2\cdot \lambda ),\operatorname{\mathrm {S}t}_2^{\infty }\otimes 1)\big )=0,and he... | {
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local-global compatibility | [
"Christophe Breuil",
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c860b3c6300a199bb09561a84e266417fae52682 | subsection | 239 | 399 | Parabolic inductions | Denote by \Pi ^1(\lambda , \psi )^- the push-forward of \Pi ^1(\lambda , \psi )_0 via S_{2,0}\hookrightarrow \operatorname{\mathrm {S}t}_3^{\infty }(\lambda )-C_{1,1}-\widetilde{C}_{1,2}, which, by Lemma REF , is a subrepresentation of \widetilde{\Pi }^1(\lambda , \psi )^-.Remark 3.38 If \psi is not smooth then \Pi ^1(... | {
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local-global compatibility | [
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7af522766d2ef48af1759711050286ee4d37bc28 | subsection | 240 | 399 | Parabolic inductions | The irreducible constituents of S_{1,3}/C_{1,3} are (from ):\Big \lbrace \mathcal {F}_{\overline{P}_2}^{\operatorname{\mathrm {G}L}_3}\big (\overline{L}(-s_1s_2\cdot \lambda ), |\cdot |^{-1}\otimes (\operatorname{\mathrm {I}nd}_{\overline{B}_2(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)} |\cdot |\otim... | {
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local-global compatibility | [
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2aaecff8d851e0b78ce906d0405582bfa45e967c | subsection | 241 | 399 | Parabolic inductions | By Step 3 of , it is left to show \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)}(C_{1,3},C_{2,1})=0. However, using and (), one can show:\operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(C_{1,3}, \mathcal {F}_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\... | {
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local-global compatibility | [
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535c50dd5d884b03a757591d7070c0229736e514 | subsection | 242 | 399 | Parabolic inductions | The lemma follows.Now consider the exact sequence (see (REF )):0 \longrightarrow I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\pi (\lambda _{1,2}, \psi )^-, x^{k_3}) \longrightarrow I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\pi (\lambda _{1,2}, \psi ), x^{k_3}) {\operatorname{\mathrm {p}r}} S_{1,3} \long... | {
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local-global compatibility | [
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af9ed9a95609ac1f92f183575ce2ef3ac5dd7668 | subsection | 243 | 399 | Parabolic inductions | But we don't need this fact in the paper.Denote by \widetilde{\Pi }^1(\lambda , \psi ) the push-forward of (REF ) along I_{\overline{P}_1}^{\operatorname{\mathrm {G}L}_3}(\pi (\lambda _{1,2}, \psi )^-, k_3) \twoheadrightarrow \widetilde{\Pi }^1(\lambda , \psi )^-, which thus has the following form by Lemma REF :\wideti... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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6bf9d8815f07df2267ed5ba68a36c9741df3dc29 | subsection | 244 | 399 | Ordinary part functor | In this section we give several properties of the ordinary part functor of and review the ordinary part of a locally algebraic representation that has an invariant lattice (). | {
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d9cef5477922bc495a975e8a89382767cc26220a | subsection | 245 | 399 | The functor | We review and/or prove useful results on the functor \operatorname{\mathrm {O}rd}_P of , .Let A\in \operatorname{\mathrm {C}omp}(\mathcal {O}_E) with \mathfrak {m}_A the maximal ideal of A and let V be a smooth representation of G(L) over A in the sense of . Recall we have in particular V\cong \varinjlim _n V[\mathfrak... | {
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local-global compatibility | [
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a92aed9c784dae21e86548cf573edc1404a3a8dc | subsection | 246 | 399 | The functor | We put:\operatorname{\mathrm {N}Ord}_P(V):=\big \lbrace v\in V^{N_0}\text{ such that there exists $z\in Z_{L_P}^+$ with $z\cdot v=0$}\big \rbracewhich is an A-submodule of V^{N_0} stable by L_P^+.Theorem 4.4
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82acfff5b8eb8888af2f6c2e42c848163c9c36e5 | subsection | 247 | 399 | The functor | Since B_i is artinian we have a natural decomposition:B_i \cong \prod _{\mathfrak {m}\text{\ ordinary}} \!\!\!(B_i)_{\mathfrak {m}} \ \ \times \!\!\prod _{\mathfrak {m}\text{\ non\ ordinary}} \!\!\!\!\!(B_i)_{\mathfrak {m}}\ \ =:\ \ B_{i,\operatorname{\mathrm {o}rd}} \times B_{i, \operatorname{\mathrm {n}ord}}and anoth... | {
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local-global compatibility | [
"Christophe Breuil",
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e554fd70d33a2707100feea7cbcf2677ee0123ba | subsection | 248 | 399 | The functor | From (V_i)_{\operatorname{\mathrm {n}ord}}=\operatorname{\mathrm {N}Ord}_P(V)\cap V_i in (a), we also see \operatorname{\mathrm {N}Ord}_P(V)\cong \varinjlim _i (V_i)_{\operatorname{\mathrm {n}ord}}.(c) By , we have \operatorname{\mathrm {O}rd}_P(V)=\varinjlim _i \operatorname{\mathrm {O}rd}_P(V)^{L_i} and \iota _{\oper... | {
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local-global compatibility | [
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9fad183cf9774ad408ab9211d319a28a77a952d3 | subsection | 249 | 399 | The functor | Since V^{I_{i,i}} is a finitely generated A-module, any element in \operatorname{\mathrm {H}om}_{A[Z_{L_P}^+]}(A[Z_{L_P}(L)], V^{I_{i,i}}) is locally Z_{L_P}(L)-finite, hence we have an inclusion:\operatorname{\mathrm {H}om}_{A[Z_{L_P}^+]}\big (A[Z_{L_P}(L)], V^{I_{i,i}}\big )\subseteq \operatorname{\mathrm {H}om}_{A[Z... | {
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b90c62628d8cfccb8d5414bbe4c8e6afe0c5e45f | subsection | 250 | 399 | The functor | Then \operatorname{\mathrm {O}rd}_P(V) is an injective object in the category of smooth representations of L_0 over A.By the same argument as in the proof of , there exists r>0 such that V is a direct factor of \mathcal {C}(I_0, A)^{\oplus r} as a representation of I_0 where \mathcal {C}(I_0, A) (= the A-module of cont... | {
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a5888404d29b0e3d958ddd99afe9b80ecce04af0 | subsection | 251 | 399 | The functor | For any compact group K we endow \mathcal {C}(K, \mathcal {O}_E) and \mathcal {C}(K, E) with the left action of K by right translation on functions.Corollary 4.6
Assume moreover that V^0|_{I_0} is isomorphic to a direct factor of \mathcal {C}(I_0, \mathcal {O}_E)^{\oplus r} for some integer r>0. Then \operatorname{\ma... | {
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local-global compatibility | [
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294bb4399894cc7fded2ecea714de86b4d9ce583 | subsection | 252 | 399 | Ordinary parts of locally algebraic representations | We review and generalize the ordinary part of a locally algebraic representation of G(L) that admits an invariant lattice (see ).We keep the notation of §§ REF & REF and now assume that G is split. We fix a split torus T over L and a Borel subgroup containing T such that B\subseteq P (where P is the parabolic subgroup... | {
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"start... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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bd48ba6c544f478f901157589553ca2d9f1b33ff | subsection | 253 | 399 | Ordinary parts of locally algebraic representations | Similarly to what we did in the proof of Theorem REF , a maximal ideal \mathfrak {m} of B_i is called of finite slope if {\rm Image}(Z_{L_P}^+)\cap \mathfrak {m}=\emptyset (inside \operatorname{\mathrm {E}nd}_E(V_i)). Let \mathfrak {m} be such a maximal ideal of finite slope and consider:Z_{L_P}^+ \longrightarrow B_i \... | {
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local-global compatibility | [
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701f930fb7c347113bd61e968d7617e447810b28 | subsection | 254 | 399 | Ordinary parts of locally algebraic representations | For *\in \lbrace \operatorname{\mathrm {f}s}, 0, \operatorname{\mathrm {n}ull}, >0\rbrace , we set:(V^{N_0})_{*}:=\varinjlim _i (V_i)_*which is an E-vector subspace of V^{N_0} stable by L_P^+ (indeed, each (V_i)_* is a generalized eigenspace of some sort for the action of Z_{L_P}^+ on V_i, and the action of L_P^+ on V^... | {
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573c147afb11eb55386472adb4c912dfdc7ede43 | subsection | 255 | 399 | Ordinary parts of locally algebraic representations | If W is a E[Z_{L_P}(L)]-module such that the Z_{L_P}(L)-orbit of any element of W is of finite dimension, by the very same construction as above we have a decomposition W=W_0\oplus W_{>0} analogous to (REF ).Lemma 4.7
Let W be an E-vector space equipped with a Z_{L_P}^+-action and let f: W \rightarrow V^{N_0} be an E-... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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aeba5673152e70db5b208c3aeebaba31026233e3 | subsection | 256 | 399 | Ordinary parts of locally algebraic representations | If W^0 is a Z_{L_P}(L)-invariant \mathcal {O}_E-lattice of W, then W^0\cap W_\alpha is a Z_{L_P}(L)-invariant \mathcal {O}_E-lattice in W_\alpha which easily implies (W_\alpha )_0= W_\alpha and (2) follows.Remark 4.8
It easily follows from the first statement in Lemma REF (2) and the fact the L_P(L)-representations (V... | {
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local-global compatibility | [
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b650903d5f926a328b38e007ba94d269d91145d1 | subsection | 257 | 399 | Ordinary parts of locally algebraic representations | As in the proof of Theorem REF , a maximal ideal \mathfrak {n} of A_i is called ordinary if {\rm Image}(Z_{L_P}^+) \cap \mathfrak {n}=\emptyset and we put:(V_i^0)_{\operatorname{\mathrm {o}rd}}:=\oplus _{\mathfrak {n}\text{ ordinary}} (V_i^0)_{\mathfrak {n}} \ \ \ \ (V_i^0)_{\operatorname{\mathrm {n}ord}}:=\oplus _{\ma... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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fbde291406f2f8e7ee5b819728903d85e4fa968b | subsection | 258 | 399 | Ordinary parts of locally algebraic representations | The lemma follows.The action of Z_{L_P}^+ on (V_i^0)_{\operatorname{\mathrm {o}rd}} being invertible, it (uniquely) extends to an action of Z_{L_P}(L) and the isomorphism (V_i^0)_{\operatorname{\mathrm {o}rd}}\otimes _{\mathcal {O}_E} E \cong (V_i)_0 of Lemma REF is equivariant under the action of Z_{L_P}(L). We set (u... | {
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local-global compatibility | [
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39fc6bd06ffc3bccc86c67bcf9d1e31e06fde3bf | subsection | 259 | 399 | Ordinary parts of locally algebraic representations | We have N_0\cong N_0^{\prime }\rtimes N_0^{\prime \prime } and thus an isomorphism V^{N_0}\cong (V^{N_0^{\prime }})^{N_0^{\prime \prime }}. By Lemma REF and (the first statement in) Lemma REF (2), we see that the embedding (((V^{N_0^{\prime }})_{0})^{N_0^{\prime \prime }})_{0} \hookrightarrow V^{N_0} factors through (V... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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5026c1468f01c76ced6cc4d84c1f0efac76ed869 | subsection | 260 | 399 | Ordinary parts of locally algebraic representations | We have a natural bijection between the maximal ideals \mathfrak {m} of A_i and the maximal ideals \overline{\mathfrak {m}} of A_i/\varpi _E^n (since any maximal ideal of A_i contains \varpi _E) and it is easy to see that \mathfrak {m}\subset A_i is ordinary if and only if \overline{\mathfrak {m}} is ordinary (see the ... | {
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local-global compatibility | [
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"Yiwen Ding"
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3c3378989e4b3c18cda47278c7c74a7985e1dd26 | subsection | 261 | 399 | Ordinary parts of locally algebraic representations | The first part of the lemma follows. By unwinding the maps, the second part also easily follows.Remark 4.15
(1) The embedding \varinjlim _i V_i^0/\varpi _E^n \cong (V^0)^{N_0}/\varpi _E^n \longrightarrow (V^0/\varpi _E^n)^{N_0} is not surjective in general. Consequently (e.g. by the proof of Lemma REF ), (REF ) might ... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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a4b62ebef533fa799888ce1d72d67296790128a2 | subsection | 262 | 399 | Ordinary parts of locally algebraic representations | We also have \cap _n \varpi _E^n \operatorname{\mathrm {O}rd}_P(V^0)=0 since the same holds for V^0, and thus we obtain an injection:\operatorname{\mathrm {O}rd}_P(V^0) \longrightarrow \varprojlim _n \big (\operatorname{\mathrm {O}rd}_P(V^0)/\varpi _E^n\big ) \cong \varprojlim _n \big ( \varinjlim _i (V_i^0/\varpi _E^n... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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