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8013e997f738564796a7e688f3edef73fbf0f0ae | subsection | 263 | 399 | Ordinary parts of locally algebraic representations | At last, if f is injective and V^0=W^0\cap V, we have \varpi _E^n V^0=(\varpi _E^n W^0)\cap V hence V^0/\varpi _E^n\hookrightarrow W^0/\varpi _E^n, and by (REF ) and the left exactness of \operatorname{\mathrm {O}rd}_P(\cdot ) () the morphisms in (REF ) are injective. | {
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ffa7e781c5cb673163495b1e8a1c70be364a4eb4 | subsection | 264 | 399 | An adjunction property | We study some adjunction property of the functor \operatorname{\mathrm {O}rd}_P(\cdot ) of § REF on locally algebraic representations.We keep the notation of §§ REF , REF & REF . If U is any E-vector space, denote by \mathcal {C}_c^{\infty }(N_P(L), U) the E-vector space of U-valued locally constant functions with com... | {
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4c291f62df2b19a7b42148a8b19456529fa6cf9b | subsection | 265 | 399 | An adjunction property | Then the locally algebraic representation (\operatorname{\mathrm {I}nd}_{\overline{P}(L)}^{G(L)} U_{\infty })^{\infty } \otimes _E L(\lambda ) is unitary as representation of G(L) and there is a natural L_P(L)-equivariant injection:U=U_{\infty }\otimes _E L_P(\lambda ) \longrightarrow \operatorname{\mathrm {O}rd}_P\big... | {
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1e431609528636130aea3f99c37b49de9948c7dd | subsection | 266 | 399 | An adjunction property | We have G(L)-equivariant embeddings:V \longrightarrow \big (\operatorname{\mathrm {I}nd}_{\overline{P}(L)}^{G(L)}U\big )^{\operatorname{\mathrm {a}n}}\hookrightarrow \big (\operatorname{\mathrm {I}nd}_{\overline{P}(L)}^{G(L)}\widehat{U^0}\otimes _{\mathcal {O}_E}E\big )^{\mathcal {C}^0}.Since the right hand side of (RE... | {
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local-global compatibility | [
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9508266116bd9162b91eb235646bd992f94deaed | subsection | 267 | 399 | An adjunction property | We have a natural commutative diagram:\begin{}
U @>>> \widehat{U} \\
@V (\ref {equ: ord-alg5}) VV @V \wr VV \\
\operatorname{\mathrm {O}rd}_P\big ((\operatorname{\mathrm {I}nd}_{\overline{P}(L)}^{G(L)} U_{\infty })^{\infty }\otimes _E L(\lambda )\big ) @>>> \operatorname{\mathrm {O}rd}_P\big ((\operatorname{\mathrm {I}... | {
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local-global compatibility | [
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16b9d0b796515548d13de3212d2d731f80c011da | subsection | 268 | 399 | An adjunction property | Then f induces G(L)-equivariant morphisms:(\operatorname{\mathrm {I}nd}_{\overline{P}(L)}^{G(L)} U_{\infty })^{\infty } \otimes _E L(\lambda ) \longrightarrow \big (\operatorname{\mathrm {I}nd}_{\overline{P}(L)}^{G(L)}\widehat{U}\big )^{\mathcal {C}^0} \longrightarrow Vfrom which f can be recovered as the following com... | {
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3616bb21cb2db591017768b0fff7bf749f682511 | subsection | 269 | 399 | Classical local Langlands correspondence | We give a sufficient condition in terms of the (usual) local Langlands correspondence for a p-adic Galois representation to be P-ordinary. The results of this section will be used in §§ REF & REF .Let \rho : \operatorname{\mathrm {G}al}_L \rightarrow \operatorname{\mathrm {G}L}_n(E) be a potentially semi-stable repres... | {
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9ffe986dcab88d864f83fa0d2fcf7722768a6a58 | subsection | 270 | 399 | Classical local Langlands correspondence | Each D_{L^{\prime },\sigma } is stable by the N-action. Moreover, for w\in \operatorname{\mathrm {W}}_L (the Weil group of L), we have that r(w):=\varphi ^{-\alpha (w)} \circ \overline{w} acts L_0^{\prime }\otimes _{\mathbb {Q}_p} E-linearly on D_{L^{\prime }} where \alpha (w)\in [L_0:\mathbb {Q}_p]\mathbb {Z} is such ... | {
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9eeda3a2ef147cf39407d68a85c5ea7acd333b4d | subsection | 271 | 399 | Classical local Langlands correspondence | In fact, we only make use of \operatorname{\mathrm {W}}(\rho ) in the sequel.Let \pi be the smooth irreducible (hence admissible) representation of \operatorname{\mathrm {G}L}_n(L) over E associated to \operatorname{\mathrm {W}}(\rho )^{\operatorname{\mathrm {s}s}} via the classical local Langlands correspondence norma... | {
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59ea79f65c798f224531ab1ebc39308e08268461 | subsection | 272 | 399 | Classical local Langlands correspondence | If there is an embedding \otimes _{i=1}^k \pi _i\hookrightarrow (\pi ^{N_0})_{0} of smooth representations of L_P(L)=\prod _{i=1}^k\operatorname{\mathrm {G}L}_{n_i}(L), then there exist \rho _i:\operatorname{\mathrm {G}al}_L\rightarrow \operatorname{\mathrm {G}L}_{n_i}(E) for i=1, \cdots , k such that:\rho is isomorphi... | {
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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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dea4a38650b5c4a3b21c5274fa726ca3832961bc | subsection | 273 | 399 | Classical local Langlands correspondence | Thus if \otimes _{i=1}^k\pi _i\hookrightarrow (\pi ^{N_0})_{0}, there exists a smooth irreducible representation \pi ^{\prime }\otimes \pi _k of L_{P^{\prime }}(L) over E such that \pi ^{\prime }\otimes \pi _k\hookrightarrow (\pi ^{N_0^{\prime }})_{0} and \otimes _{i=1}^k \pi _i \hookrightarrow ((\pi ^{\prime }\otimes ... | {
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local-global compatibility | [
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02e392bd72852a30d50062d93b8bcd417fffa30a | subsection | 274 | 399 | Classical local Langlands correspondence | Enlarging E if needed, there exists a \varphi -submodule D_1 of D_{L^{\prime }}=(B_{\operatorname{\mathrm {s}t}}\otimes _{\mathbb {Q}_p} \rho )^{\operatorname{\mathrm {G}al}_{L^{\prime }}} such that the \varphi ^{[L_0^{\prime }:\mathbb {Q}_p]}-semi-simplification of D_1 is isomorphic to D_1 as \varphi -modules over L_0... | {
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local-global compatibility | [
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3286f5d1d60655a06986f27edcdda2f0f3ae3c41 | subsection | 275 | 399 | Classical local Langlands correspondence | We equip D_1\otimes _{L^{\prime }_0}L^{\prime } with the Hodge filtration \operatorname{\mathrm {F}il}^\bullet (D_1\otimes _{L^{\prime }_0}L^{\prime }) induced by D_{L^{\prime }}\otimes _{L^{\prime }_0}L^{\prime }. Since \operatorname{\mathrm {H}T}_{\sigma }(\rho )=\lbrace 1-n, 2-n, \cdots , 0\rbrace for all \sigma \in... | {
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local-global compatibility | [
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52c541dd8342c65913ccd424f18c522f9907ce37 | subsection | 276 | 399 | Classical local Langlands correspondence | If D_{\sigma }^{\prime }\ne D_{1,\sigma } then there exists a (\varphi ^{f^{\prime }}, N)-submodule D_{\sigma }^{\prime \prime } of D_{\sigma }^{\prime } such that:\dim _E D_{\sigma }^{\prime \prime }=\dim _E D_{1,\sigma }=n_1\ \ {\rm and}\ \ t_N(D^{\prime \prime }) < t_N(D_1)where D^{\prime \prime } is the (\varphi , ... | {
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local-global compatibility | [
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e213649f596bdf61ed88a1cea8e2bc016823743a | subsection | 277 | 399 | Classical local Langlands correspondence | Since D_{\sigma }^{\prime }\ne D_{1,\sigma }, we have \operatorname{\mathrm {K}er}(N^{s-1})\nsubseteq D_{1,\sigma } or D_{1,\sigma }\nsubseteq \operatorname{\mathrm {K}er}(N^s) (indeed, otherwise we have \operatorname{\mathrm {K}er}(N^{s-1})\subseteq D_{1,\sigma }\subseteq \operatorname{\mathrm {K}er}(N^s) which implie... | {
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local-global compatibility | [
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47fde5c7bb4a519e0894fd8b8fe8a7e341761349 | subsection | 278 | 399 | Classical local Langlands correspondence | The claim then follows with D_{\sigma }^{\prime \prime }:=\oplus _{j\in J} D_{\sigma ,j}^{\prime \prime }.Assume now D^{\prime }_{\sigma }\ne D_{1,\sigma } and let D^{\prime \prime } be as in the claim. The same argument as in (c) with the induced Hodge filtration gives then t_H(D^{\prime \prime }\otimes _{L^{\prime }_... | {
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local-global compatibility | [
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0f36f2633e1c0ae36946b5253eaa14981b6be9e6 | subsection | 279 | 399 | Classical local Langlands correspondence | Assume not and let n_1^{\prime }<n_1 such that \dim _E (D_1)_{\le (n_1-1)f^{\prime }}=n_1^{\prime }f^{\prime } (note that \dim _E D_1=n_1f^{\prime } and that (D_1)_{\le (n_1-1)f^{\prime }} is free over L_0^{\prime }\otimes _{\mathbb {Q}_p} E (as is easily checked)). Then we deduce:t_N\big ((D_1)_{\le (n_1-1)f^{\prime }... | {
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local-global compatibility | [
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58f85eeb27354ff0e63062a8b091b54e89d4b80d | subsection | 280 | 399 | Classical local Langlands correspondence | Let \operatorname{\mathrm {D}F}_1^{\prime }:=(D_1^{\operatorname{\mathrm {s}s}}, \varphi , N=0, \operatorname{\mathrm {G}al}(L^{\prime }/L)) be the Deligne-Fontaine module associated to (\operatorname{\mathrm {W}}(\rho _1)^{\operatorname{\mathrm {s}s}}, N=0) () where D_1^{\operatorname{\mathrm {s}s}} denotes the semi-s... | {
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local-global compatibility | [
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a0cf36336875aa0a340bcb2fb07ae39a5193c7d8 | subsection | 281 | 399 | Classical local Langlands correspondence | However, by (e) we have D_1^{\operatorname{\mathrm {s}s}}=(D_{L^{\prime }}^{\operatorname{\mathrm {s}s}})_{\le (n_1-1)f^{\prime }}, thus we also have D_1=(D_{L^{\prime }}^{\operatorname{\mathrm {s}s}})_{\le (n_1-1)f^{\prime }} since (D_{L^{\prime }}^{\operatorname{\mathrm {s}s}})_{\le (n_1-1)f^{\prime }} is only define... | {
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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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881cf8d212ff28277d6e20ed660a6b346373b2da | subsection | 282 | 399 | Automorphic and | In this section we start the global theory: we give the global setup, state our local-global compatibility conjecture for \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p), and prove several useful results on the P-ordinary part of (localized) Banach spaces of p-adic automorphic forms on definite unitary groups. | {
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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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296275e9925af6c39289b2b2824996965dd63541 | subsection | 283 | 399 | Global setup and main conjecture | We introduce the global setup and state our main local-global compatibility conjecture for \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p).We fix field embeddings \iota _{\infty }: \overline{\mathbb {Q}} \hookrightarrow \mathbb {C}, \iota _p: \overline{\mathbb {Q}} \hookrightarrow \overline{\mathbb {Q}_p}. We also fix F^+... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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1cf43ee3d8c4b8514ada1fe9de494de645098c52 | subsection | 284 | 399 | Global setup and main conjecture | The lattice \widehat{S}(U^p,\mathcal {O}_E) is obviously stable by this action, so the Banach representation \widehat{S}(U^p,E) of G(F^+\otimes _{\mathbb {Q}} \mathbb {Q}_p) is unitary. Moreover, we know (see e.g. the proof of Lemma REF ) that \widehat{S}(U^p,E) is admissible. Let D(U^p) be the set of primes v of F^+ s... | {
"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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84b539394469b0292110949da88bfba22fc6c512 | subsection | 285 | 399 | Global setup and main conjecture | An automorphic representation \pi is isomorphic to \pi _{\infty }\otimes _{\mathbb {C}} \pi ^{\infty } where \pi _{\infty }=W_{\infty } is an irreducible algebraic representation of (\operatorname{\mathrm {R}es}_{F^+/\mathbb {Q}}G)(\mathbb {R})=G(F^+\otimes _{\mathbb {Q}} \mathbb {R}) over \mathbb {C} and \pi ^{\infty ... | {
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"source_ref_id": "dd425b4ffeec17... | 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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0c68a1750560734baa2088dc30f404294e582ce8 | subsection | 286 | 399 | Global setup and main conjecture | Denote by \widehat{S}(U^p,E)^{\operatorname{\mathrm {l}alg}} the subspace of \widehat{S}(U^p,E) of locally algebraic vectors for the (\operatorname{\mathrm {R}es}_{F^+/\mathbb {Q}}G)(\mathbb {Q}_p)=G(F^+\otimes _{\mathbb {Q}} \mathbb {Q}_p)-action, which is stable by \mathbb {T}(U^p). We have an isomorphism which is eq... | {
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"raw": "Breuil, C. Vers le socle localement analytique pour \\mathrm {GL}_n, II. Math. Annalen 361 (2015), 741–785.",
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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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c076928928200c756f40149d4f4feb29f337e120 | subsection | 287 | 399 | Global setup and main conjecture | We put U^\wp :=U^pU_p^\wp and:\widehat{S}(U^\wp ,W^\wp ):=\big (\widehat{S}(U^p,E)\otimes _E W^\wp \big )^{U_p^\wp }which is an admissible unitary Banach representation of G(F^+_\wp ) over E (see the discussion below) which is equipped with a natural action of \mathbb {T}(U^p) commuting with the action of G(F^+_\wp )\c... | {
"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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35f088a39a2e5051012b565930062eca543ce0f4 | subsection | 288 | 399 | Global setup and main conjecture | We also define for *\in \lbrace W^{\wp }, \mathbb {W}^{\wp }, \mathbb {W}^{\wp }/\varpi _E^s\rbrace :S(U^{\wp },*):=\Big \lbrace f: G(F^+) \setminus G(\mathbb {A}_{F^+}^{\infty })/U^p \longrightarrow *,\ \ f \text{ is locally constant and} \\ f(gg_p^{\wp })=(g_p^{\wp })^{-1}(f(g))\text{ for all }g\in G(\mathbb {A}_{F^+... | {
"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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8348169a0c32907da87721b6aadef4b43cc27f5f | subsection | 289 | 399 | Global setup and main conjecture | We have moreover \operatorname{\mathrm {G}L}_n(L)\times \mathbb {T}(U^p)-equivariant isomorphisms:S(U^{\wp }, \mathbb {W}^{\wp }/\varpi _E^s) &\cong &\widehat{S}(U^{\wp }, \mathbb {W}^{\wp }/\varpi _E^s) \ \ \ \ \cong \ \ S(U^{\wp }, \mathbb {W}^{\wp })/\varpi _E^s \\
\widehat{S}(U^{\wp }, \mathbb {W}^{\wp }) & \cong &... | {
"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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78e89eb2f7546ced7240fee1ef92715473636d19 | subsection | 290 | 399 | Global setup and main conjecture | We can then easily deduce from (REF ) a \operatorname{\mathrm {G}L}_n(L)\times \mathbb {T}(U^p)-equivariant isomorphism:\widehat{S}(U^{\wp },W^{\wp })^{\operatorname{\mathrm {l}alg}}\otimes _{E} \overline{\mathbb {Q}_p} \cong \bigoplus _{\pi } \big ( (\pi ^{\infty ,p})^{U^p} \otimes _{\overline{\mathbb {Q}}} (\otimes _... | {
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"raw": "Clozel, L., Harris, M., and Taylor, R. Automorphy for some \\ell -adic lifts of automorphic mod \\ell Galois representations. Pub. Math. I.H.É.S. 108 (2008), 1–181.",
"source_ref_id": "936fc5... | 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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a193371e8b9aa4520c2b0b36e511bfccdb0e93ee | subsection | 291 | 399 | Global setup and main conjecture | Since U^p is sufficiently small, we have U^pU_p\cap g G(F^+) g^{-1}=\lbrace 1\rbrace for all g\in G(\mathbb {A}_{F^+}^{\infty }) (the left hand side is a finite group as G(F^+) is discrete in G(\mathbb {A}_{F^+}^{\infty }), then U^p being sufficiently small implies it has to be \lbrace 1\rbrace ). From which we deduce ... | {
"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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193b18b4f0e0972e25581bbe275290bef4e73f36 | subsection | 292 | 399 | Global setup and main conjecture | Then from (a) we deduce using U_p=U_{\wp }U^{\wp }_p:\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })|_{U_{\wp }} \cong \mathcal {C}(U_{\wp }, \mathcal {O}_E)\widehat{\otimes }_{\mathcal {O}_E} [\mathcal {C}(U_p^{\wp }, \mathcal {O}_E)^{r^{\prime }} \otimes _{\mathcal {O}_E} \mathbb {W}^{\wp }]^{U^{\wp }_p}.Since [\mathcal {C... | {
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"raw": "Breuil, C., and Herzig, F. Ordinary representations of G(\\mathbb {Q}_p) and fundamental algebraic representations. Duke Math. J. 164 (2015), 1271–1352.",
"source_ref_id": "82002626a821a3509b... | 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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6bec96d655893848ace5d27d53a6fa48dc2cbabd | subsection | 293 | 399 | Global setup and main conjecture | We end this section by our main local-global compatibility conjecture when n=3 and L=\mathbb {Q}_p. If \rho _p:\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}\longrightarrow \operatorname{\mathrm {G}L}_3(E) is a semi-stable representation such that N^2\ne 0 on D_{\operatorname{\mathrm {s}t}}(\rho _p), there exists a uniqu... | {
"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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e3bf8410eed4ab70105dc29d3c59c036d99d62ff | subsection | 294 | 399 | Global setup and main conjecture | Let \rho : \operatorname{\mathrm {G}al}_F\rightarrow \operatorname{\mathrm {G}L}_3(E) be a continuous absolutely irreducible representation which is unramified at the places of D(U^p) and such that:\widehat{S}(U^{\wp }, W^{\wp })[\mathfrak {m}_{\rho }]^{\operatorname{\mathrm {l}alg}}\ne 0
\rho _{\widetilde{\wp }}:={\r... | {
"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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80f1656d2350a78b9b4536409050fe18d6bbbf4c | subsection | 295 | 399 | Hecke operators | We give (or recall) the definition of some useful pro-p-Hecke algebras and of their localisations.We keep the notation of § REF . For s\in \mathbb {Z}_{>0} and a compact open subgroup U_{\wp } of \operatorname{\mathrm {G}L}_n(L), we let \mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s) (resp. \mathbb {T}(U^... | {
"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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4492d0e7eaff926392fe6713f7963aee0b53cb1d | subsection | 296 | 399 | Hecke operators | From (REF ), it is also easy to see:\mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }){\sim } \varprojlim _s \mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s).For U_{\wp ,2}\subseteq U_{\wp ,1} an inclusion of compact open subgroups of \operatorname{\mathrm {G}L}_n(L), the natural injections:S(U^{\wp }U_{\wp... | {
"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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30849ae627bfb43ade0e5ffba4bfabbedf199aee | subsection | 297 | 399 | Hecke operators | From (REF ) we deduce isomorphisms:\small
\widetilde{\mathbb {T}}(U^{\wp }):=\varprojlim _s \varprojlim _{U_{\wp }} \mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s)
\cong \varprojlim _{U_{\wp }}\varprojlim _s \mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s) \cong \varprojlim _{U_{\wp }} \math... | {
"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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... | |
e7e7e6f9cf9692ae434047dc0eb166fcc8815b2e | subsection | 298 | 399 | Hecke operators | Since the operators in \mathbb {T}(U^p) acting on S(U^{\wp }, W^{\wp }) are semi-simple (which easily follows from () and (REF )), we deduce \widetilde{\mathbb {T}}(U^{\wp }) is reduced.To a continuous representation \overline{\rho }: \operatorname{\mathrm {G}al}_{F}\rightarrow \operatorname{\mathrm {G}L}_n(k_E) which ... | {
"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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a0ce6f2cd02a8ffe0336094012998107e4443b07 | subsection | 299 | 399 | Hecke operators | Let (U_p^{\wp })^{\prime }\subseteq U_p^{\wp } such that (U_p^{\wp })^{\prime } acts trivially on \mathbb {W}^{\wp }/\varpi _E, and let (U^{\wp })^{\prime }:=U^p(U_p^{\wp })^{\prime }\subseteq U^{\wp }. If \mathfrak {m} is (U^{\wp }, \mathbb {W}^{\wp })-automorphic, there exists a compact open subgroup U_{\wp ,2}\subse... | {
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local-global compatibility | [
"Christophe Breuil",
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df78f36710bd52ea73e149fdfa0f0c90a173379e | subsection | 300 | 399 | Hecke operators | Using the trace map \operatorname{\mathrm {t}r}_{U_{\wp ,1}/U_{\wp ,2}} (which is \mathbb {T}(U^p)-equivariant) and the proof of , we then deduce that if S((U^{\wp })^{\prime }U_{\wp ,2}, k_E)_{\mathfrak {m}^{\prime }}\ne 0 then S((U^{\wp })^{\prime }U_{\wp ,1}, k_E)_{\mathfrak {m}^{\prime }}\ne 0 (where we identify \m... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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79bc7359ef7d3d59bb1fd3821f68066fb8c425a5 | subsection | 301 | 399 | Hecke operators | It then easily follows from Lemma REF and its proof, (REF ) and from \mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s)\cong \prod _{\mathfrak {m}}\mathbb {T}(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s)_{\mathfrak {m}} (where the product is over the (U^{\wp }, \mathbb {W}^{\wp })-automorphic maximal id... | {
"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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8d8821f2b494929400e4b3301056a0e0f1733842 | subsection | 302 | 399 | Hecke operators | It is also easy to see that:\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })_{\mathfrak {m}}\cong \varprojlim _s \varinjlim _{U_{\wp }} S(U^{\wp }U_{\wp }, \mathbb {W}^{\wp }/\varpi _E^s)_{\mathfrak {m}}is a direct summand of \widehat{S}(U^{\wp }, \mathbb {W}^{\wp }) (where the localisation \widehat{S}(U^{\wp }, \mathbb {W}^{... | {
"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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ecef5a7e97426e8474209f8c0975e0213af19d9c | subsection | 303 | 399 | Dominant algebraic vectors | In this section, which is purely local, we prove density results of subspaces of algebraic functions.We fix H a connected reductive algebraic group over \mathbb {Z}_p and denote by A the finitely generated \mathbb {Z}_p-algebra which represents H. For any f\in A, the natural map \operatorname{\mathrm {H}om}_{\mathbb {Z... | {
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"s... | 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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25c9a98a1a6c57cfb5f95af1463a6d794e4c28fe | subsection | 304 | 399 | Dominant algebraic vectors | By we have an isomorphism:\mathcal {C}^{\operatorname{\mathrm {a}lg}}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p), E)\cong \bigoplus _{\sigma } \operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p)}(\sigma , \mathcal {C}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p), E)) \otimes _E \sigmawhere \sig... | {
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"s... | 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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269cbac9677b6eaa749b531c1dea568f6079f859 | subsection | 305 | 399 | Dominant algebraic vectors | For a\in \mathbb {Z}, we put:\mathcal {C}^{\operatorname{\mathrm {a}lg}}_{\le a}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p),E):=\bigoplus _{{\lambda =(\lambda _1,\cdots , \lambda _n)\\ \lambda _1\le a}} \operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p)}\big (L(\lambda ), \mathcal {C}(\operato... | {
"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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227136cc96f4efb9746e36ef3b9cb7aff89edc25 | subsection | 306 | 399 | Dominant algebraic vectors | This map is a homeomorphism and thus induces an isomorphism:h: \mathcal {C}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p), E) {\sim } \mathcal {C}(\operatorname{\mathrm {S}L}_r(\mathbb {Z}_p)\times \mathbb {Z}_p^{\times }, E) \cong \mathcal {C}(\operatorname{\mathrm {S}L}_r(\mathbb {Z}_p),E)\widehat{\otimes }_E \mathcal... | {
"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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c5ab5d4271b7cc15dcfa50e1214a2e57d4a34311 | subsection | 307 | 399 | Dominant algebraic vectors | We claim that h\vert _{\mathcal {C}^{\operatorname{\mathrm {a}lg}}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p),E)} induces an isomorphism via (REF ):\operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p)}\big (L(\lambda ), \mathcal {C}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p), E)\big )\otimes _... | {
"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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8985408e22f793fb02bc07e1a5e3e2f483d310c0 | subsection | 308 | 399 | Dominant algebraic vectors | Since we have from the proof of :\dim _E \operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p)}\big (L(\lambda ), \mathcal {C}(\operatorname{\mathrm {G}L}_r(\mathbb {Z}_p), E)\big )\\
=\dim _E \operatorname{\mathrm {H}om}_{\operatorname{\mathrm {S}L}_r(\mathbb {Z}_p)}\big (L(\lambda )_0, \mathcal {... | {
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"s... | 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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4b281de319546aac7b86c56714b9bc64394c413d | subsection | 309 | 399 | Dominant algebraic vectors | We have in particular:\mathcal {C}(L_P(\mathbb {Z}_p), E)\cong \widehat{\bigotimes }_{i=1,\cdots , k}\mathcal {C}(\operatorname{\mathrm {G}L}_{n_i}(\mathbb {Z}_p), E)\ {\rm and}\ \mathcal {C}^{\operatorname{\mathrm {a}lg}}(L_P(\mathbb {Z}_p), E) \cong \bigotimes _{i=1, \cdots , k} \mathcal {C}^{\operatorname{\mathrm {a... | {
"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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d1eec3e8c2a963b12d4175003044244a9dd7da9b | subsection | 310 | 399 | Dominant algebraic vectors | We call vectors in \mathcal {C}^{\operatorname{\mathrm {a}lg}}_+(L_P(\mathbb {Z}_p), E) dominant L_P(\mathbb {Z}_p)-algebraic vectors.Proposition 7.2
The vector spaces \mathcal {C}^{\operatorname{\mathrm {a}lg}}_{++}(L_P(\mathbb {Z}_p), E) and \mathcal {C}^{\operatorname{\mathrm {a}lg}}_+(L_P(\mathbb {Z}_p),E) are den... | {
"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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aa8ccddcd40abaa637f8c122215d74e083c1dbb7 | subsection | 311 | 399 | Dominant algebraic vectors | We deduce that the closure of \mathcal {C}^{\operatorname{\mathrm {a}lg}}_+(L_P(\mathbb {Z}_p), E) in \mathcal {C}(L_P(\mathbb {Z}_p), E) contains \oplus _{\underline{\lambda }_1}F_{\underline{\lambda }_1}\cong \mathcal {C}^{\operatorname{\mathrm {a}lg}}(\operatorname{\mathrm {G}L}_{n_1}(\mathbb {Z}_p), E)\otimes _E\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",
"math.RT"
] | 2,018 | en | Mathematics | [
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beef2134586ada58826b5c4539196dda610877f2 | subsection | 312 | 399 | Dominant algebraic vectors | If W is a closed subrepresentation of V, one easily checks that W^{L_P(\mathbb {Z}_p)-\operatorname{\mathrm {a}lg}}_*\cong W\cap V^{L_P(\mathbb {Z}_p)-\operatorname{\mathrm {a}lg}}_* with *\in \lbrace \emptyset , +, ++\rbrace .Corollary 7.3
Assume that V|_{L_P(\mathbb {Z}_p)} is isomorphic to a direct summand of \math... | {
"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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b3ad21dd5f768025c47dfdd4f6f780b6d9ea89a3 | subsection | 313 | 399 | Benign points | We define benign points of \operatorname{\mathrm {S}pec}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}}[1/p] and prove several results on them.We keep the previous notation. We also keep all the notation and assumption of § REF with L=\mathbb {Q}_p (in particular U^p is sufficientl... | {
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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.",
"sour... | 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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733000cdf87b672fd5b45c160b1e5817e9786d8b | subsection | 314 | 399 | Benign points | We denote by:\rho _x: \operatorname{\mathrm {G}al}_F \longrightarrow \operatorname{\mathrm {G}L}_n(R_{\overline{\rho }, S(U^p)})\longrightarrow \operatorname{\mathrm {G}L}_n(\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}})\longrightarrow \operatorname{\mathrm {G}L}_n(k(x))the conti... | {
"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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a9b4a3b4f80b4bb4606ea2a3699d9de39edc9976 | subsection | 315 | 399 | Benign points | If x classical, then it follows that there is an integral dominant \lambda =(\lambda _1, \cdots , \lambda _n) as in § REF such that :\big (\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }})[\mathfrak {m}_x]\otimes _E L_P(\lambda )^\vee \big )^{\operatorname{\mathrm {s}m}}\otimes _EL(\lambda )\hookrightarrow \widehat{S... | {
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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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209901ee7b777f988ae8dbec9510ab38b5d80ef5 | subsection | 316 | 399 | Benign points | The admissibility of the L_P(\mathbb {Q}_p)-continuous representation \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_x] together with and imply that there exist a smooth admissible representation \pi _x^{\infty } of L_P(\mathbb {Q}_p) over k(x) with (\pi _x^{\infty })^{L... | {
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"raw": "Schneider, P., and Teitelbaum, J. Algebras of p-adic distributions and admissible representations. Inv. Math. 153 (2003), 145–196.",
"s... | 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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a6bd4d14877623c0ccc881077d7bad0c870cdd0a | subsection | 317 | 399 | Benign points | By Lemma REF (2) and Corollary REF , \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }})^{L_P(\mathbb {Z}_p)-\operatorname{\mathrm {a}lg}}_{+} is dense in \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}). Since by Lemma REF (1) the action of \widetilde{\m... | {
"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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61e7bff2f7da96746a452a63146860768f6a95a2 | subsection | 318 | 399 | Benign points | Let V_{\infty } be the smooth L_P(\mathbb {Q}_p)-subrepresentation of (\operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }})\otimes _E L_P(\lambda )^\vee )^{\operatorname{\mathrm {s}m}} generated by v_{\infty } and consider the L_P(\mathbb {Q}_p)-equivariant injection (see ):V_{\infty }\oti... | {
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"doi": "10.48550/arxiv.math/0405137",
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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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7c13e2be2aceed7e2eccd821a6d18be733174bca | subsection | 319 | 399 | Benign points | Proposition REF ), we see (REF ) factors through:\operatorname{\mathrm {O}rd}_P\Big (\bigoplus _{x \text{ classical}} \widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_x]\Big )\cong \bigoplus _{x \text{ classical}} \operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\ma... | {
"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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a6638c6e3e3bd277b6ad5f5a6197307b0b71b600 | subsection | 320 | 399 | Benign points | For i=1,\cdots , k we denote by \widehat{\pi }(\rho _{x_i}) the continuous finite length representation of \operatorname{\mathrm {G}L}_{n_i}(\mathbb {Q}_p) over k(x) associated to \rho _{x_i} via the p-adic local Langlands correspondence for \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p) () normalized as in when n_i=2, v... | {
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"doi": "",
"end": 453,
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"raw": "Colmez, P. Représentations de \\mathrm {GL}_2(\\mathbb {Q}_p) et (\\varphi ,\\Gamma )-modules. Astérisque 330 (2010), 281–509.",
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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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] | 2,018 | en | Mathematics | [
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790a072b85695f39c1e617b991492d8b295c0847 | subsection | 321 | 399 | Benign points | Then it is easy to check that we have:\pi ^{\infty }\cong \bigotimes _{i=1,\cdots , k} \pi _i^{\infty }where if n_i=1, \psi _{s_i+1}:=\pi _i^{\infty } is an unramified character of \mathbb {Q}_p^{\times } and if n_i=2, either there exist unramified characters \psi _{s_i+1}, \psi _{s_i+2} of \mathbb {Q}_p^{\times } such... | {
"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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7320a4800d5dd2175a9cd061b56c7ebec2cdf695 | subsection | 322 | 399 | Benign points | As in (REF ), we have an L_P(\mathbb {Q}_p)-equivariant embedding:\Big (\bigotimes _{i=1,\cdots , k} \pi _i^{\infty }\Big )\otimes _{E} L_P(\lambda ) \longrightarrow \operatorname{\mathrm {O}rd}\big (\widehat{S}(U^{\wp },W^{\wp })_{\overline{\rho }}[\mathfrak {m}_x]\big )which, by Proposition REF , induces a nonzero mo... | {
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"start": 60... | 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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23f7994d10c5f2ad73f6420b5914656c46bd1a83 | subsection | 323 | 399 | Benign points | This implies that \pi _i^{\infty } is infinite dimensional when n_i=2 since otherwise it is easy to check that (\operatorname{\mathrm {I}nd}_{\overline{P}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_n(\mathbb {Q}_p)}\otimes _{i=1,\cdots , k} \pi _i^{\infty })^{\infty } has no generic irreducible constituent.(2) The fa... | {
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"start": 651... | 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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9de95834c6f98096e4740e5cce7005cec9944cdd | subsection | 324 | 399 | Benign points | Let D^{k-1} be a \varphi -submodule of D_{\operatorname{\mathrm {s}t}}(\rho _{x, \widetilde{\wp }}) such that the \varphi -semi-simplification (D^{k-1})^{\operatorname{\mathrm {s}s}} is isomorphic to \oplus _{j=1}^{n-n_k} \operatorname{\mathrm {u}nr}(\alpha _j), we thus have t_H(D^{k-1})\ge [k(x):\mathbb {Q}_p](\sum _{... | {
"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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581deced0f0f0620256c4f2220cc66f45c67878e | subsection | 325 | 399 | Benign points | Let \rho _{x_k}^{\prime }:=\rho _{x, \widetilde{\wp }}/\rho ^{k-1}, we have D_{\operatorname{\mathrm {s}t}}(\rho _{x_k}^{\prime })\cong D/D^{k-1} and in particular (note s_k+n_k=n):\lbrace -\mu _{s_k+1}, -\mu _{s_k+n_k}\rbrace =\operatorname{\mathrm {H}T}(\rho ^{\prime }_{x_k})
\lbrace \alpha _{s_k+1}, \alpha _{s_k+n_... | {
"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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720a4c45db02b2be63ea59cc87f5a25f7c901fb3 | subsection | 326 | 399 | Benign points | \lambda _j>\lambda _{j+1} for all j, we deduce \alpha _j \alpha _{j^{\prime }}^{-1}\notin \lbrace 1,p,p^{-1}\rbrace if j, j^{\prime } do not lie in \lbrace s_i+1, s_i+{n_i}\rbrace for any i\in \lbrace 1,\cdots ,k\rbrace .With the notation in the proof of Proposition REF , there exists m(x)\in \mathbb {Z}_{\ge 1} such t... | {
"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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edf64bb308be6c6cf37e4ab76c58fcbfaae919cc | subsection | 327 | 399 | Benign points | By (REF ) and the fact that each \pi _i^{\infty } for i=1,\cdots ,k, and thus \otimes _{i=1,\cdots , k} \pi _i^{\infty }, has an irreducible socle (see the proof of Proposition REF (1)), we deduce an L_P(\mathbb {Q}_p)-equivariant injection:\big (\bigotimes _{i=1,\cdots , k} \pi _i^{\infty }\big )\otimes _{E} L_P(\lamb... | {
"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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b70f89f2047cc9e1f295c0e133789d4e2e741bac | subsection | 328 | 399 | Benign points | On the other hand, we deduce from Remark REF :J_{B\cap L_P}\big ( \operatorname{\mathrm {O}rd}_P(\pi _{\wp }\otimes _{E}L(\lambda ))\big )\longrightarrow J_B(\pi _{\wp }\otimes _{E}L(\lambda ))(\delta _P^{-1}).Comparing with (REF ) (recall \pi _{\wp } is a constituent of (\operatorname{\mathrm {I}nd}_{\overline{P}(\mat... | {
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"raw": "Prasad, D., and Raghuram, A. Representation theory of \\mathrm {GL}(n) over non-Archimedean local fields. preprint.",
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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"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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2d0046c113036262cf24eb5257f4c1c655a2c4b8 | subsection | 329 | 399 | Benign points | If \chi ^{\prime } injects into J_{B\cap L_P}\big (\pi _P^{\infty }\otimes _{E}L_P(\lambda )\big ) (which is equivalent to \chi ^{\prime }\delta _{\lambda }^{-1}\hookrightarrow J_{B\cap L_P}(\pi _P^{\infty })), by we deduce a nonzero morphism :\big (\operatorname{\mathrm {I}nd}_{\overline{B}(\mathbb {Q}_p)\cap L_P(\mat... | {
"cite_spans": [
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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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23b31f96fd0d82c1f80107cc0e56aa07d6722a45 | subsection | 330 | 399 | Benign points | Since J_P(\pi _{\wp }) does not have cuspidal constituents and J_{B\cap L_P} is an exact functor, we deduce that the injection (REF ) must be bijective.Proposition 7.9
With the notation of Proposition REF and its proof, we have an L_P(\mathbb {Q}_p)-equivariant isomorphism:\operatorname{\mathrm {O}rd}_P\big (\widehat{... | {
"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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15c305bf0dbd872626e42f54c0a41d44e6ce3d23 | subsection | 331 | 399 | Benign points | By Lemma REF and the fact that \operatorname{\mathrm {O}rd}_P(\pi _{\wp }\otimes _{E} L(\lambda )) is a direct summand of J_P\big (\pi _{\wp }\otimes _{E} L(\lambda )\big )(\delta _P^{-1}) (which follows from (REF )), we deduce an L_P(\mathbb {Q}_p)-equivariant surjection J_P(\pi _{\wp }) \twoheadrightarrow (\otimes _{... | {
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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"
] | [
"math.NT",
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a9442580fb8dc2306da9868a91e492a9d5ff1c51 | subsection | 332 | 399 | Benign points | Using \varepsilon =z\operatorname{\mathrm {u}nr}(p^{-1}), we easily deduce:\bigotimes _{i=1,\cdots , k} \big (\widehat{\pi }(\rho _{x_i})^{\operatorname{\mathrm {l}alg}}\otimes \varepsilon ^{s_i}\circ \operatorname{\mathrm {d}et}\big ) \cong \Big (\bigotimes _{i=1,\cdots , k} \pi _i^{\infty }\Big )\otimes _{E} L_P(\lam... | {
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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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73f0cfcb9f620058241251e379413052c394ecdf | subsection | 333 | 399 | Local-global compatibility | We prove local-global compatibility results for the L_P(\mathbb {Q}_p)-representation \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}) by generalizing Emerton's method ().We keep the notation and assumptions of §§ REF , REF , REF , REF (in particular we assume Hypothesis REF ... | {
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60ddf068e0271e05da1b0a040dbb6154a1b342d8 | subsection | 334 | 399 | Local-global compatibility | We set:\pi ^{\otimes }_P(U^{\wp }):=\widetilde{\bigotimes }_{i=1, \cdots , k} \big (\pi _i(U^{\wp }) \otimes \varepsilon ^{s_i} \circ \operatorname{\mathrm {d}et}\big )(the \mathfrak {m}_{\overline{\rho }}-completed tensor product being over \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm ... | {
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d3e07e775da16dbb55d85ed64a66e69df6fcf9bd | subsection | 335 | 399 | Local-global compatibility | Note that X_P(U^{\wp }) is equipped with a natural action of \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}}.We fix a point x of (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}})^{\operatorname{\mathrm {r}ig}} and le... | {
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local-global compatibility | [
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f9e45606bb9d16b5418bca667f65276eb65706e0 | subsection | 336 | 399 | Local-global compatibility | We can deduce then (note that the \mathfrak {m}_{\overline{\rho }}-adic topology on \widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }}/\mathfrak {p}_x coincides with the p-adic topology):\pi ^{\otimes }_P(U^{\wp }) \otimes _{\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {... | {
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bf08a7589cf3cdc3c15762626999c4abbd9d8189 | subsection | 337 | 399 | Local-global compatibility | We have an injection of k_E-vector spaces:X_P(U^{\wp })/\varpi _E\longrightarrow \operatorname{\mathrm {H}om}_{\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}}[L_P(\mathbb {Q}_p)]}\Big (\bigotimes _{i=1,\cdots , k}(\overline{\pi }_i\otimes \overline{\varepsilon }^{s_i}\circ \operato... | {
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313a7df8a121b3819563962fe005a48dff3e1c25 | subsection | 338 | 399 | Local-global compatibility | The lemma follows.Theorem 7.32
(1) The \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}}-module X_P(U^{\wp }) is faithful.(2) For any point x\in (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}})^{\operatorname{\mathrm... | {
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8939f0a25a560de4ae65765790dee269b699633f | subsection | 339 | 399 | Local-global compatibility | The theorem then follows by the same argument as in the proof of (see also ).Corollary 7.33
Let x\in (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}})^{\operatorname{\mathrm {r}ig}}, there exists a nonzero morphism of admissible Banach representations o... | {
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2c5cb77e88482b48a4847952cacbb84c27c99ea6 | subsection | 340 | 399 | Local-global compatibility | Moreover, since \rho _{x_i} is irreducible, we know that \widehat{\pi }(\rho _{x_i}) is also irreducible as a continuous representation of \operatorname{\mathrm {G}L}_{n_i}(\mathbb {Q}_p). It then follows that \widehat{\otimes }_{i=1,\cdots , k} (\widehat{\pi }(\rho _{x_i}) \otimes \varepsilon ^{s_i}\circ \det ) is als... | {
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934ebf8fa6dc38a5ad43221dc1d1cb47fb0bc425 | subsection | 341 | 399 | Local-global compatibility | It follows that the morphism (REF ) is injective and thus restricts to an injective L_P(\mathbb {Q}_p)-equivariant morphism:\Big (\bigotimes _{i=1,\cdots , k} \pi _i^{\infty }\Big ) \otimes _{k(x)} L_P(\lambda )\\ \cong \bigotimes _{i=1,\cdots , k} \big (\widehat{\pi }(\rho _{x_i})^{\operatorname{\mathrm {l}alg}} \otim... | {
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4815cc6c0a42cb118f5ecc1c2f60409a43f45155 | subsection | 342 | 399 | Local-global compatibility | Moreover by , for any x\in (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}})^{\operatorname{\mathrm {r}ig}}, the \mathcal {O}_{k(x)}-modules M_P(U^{\wp }) /\mathfrak {p}_x and X_P(U^{\wp })[\mathfrak {p}_x] are finitely generated free of the same rank, t... | {
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920ec975f229b7bd1696925e3079cfc05ea3ef42 | subsection | 343 | 399 | Local-global compatibility | The first map is injective since \widehat{\otimes }_{i=1, \cdots , k} \pi _i^{\operatorname{\mathrm {a}n}} is dense in \widehat{\otimes }_{i=1, \cdots , k} \widehat{\pi }(\rho _{x_i}) (see the proof of Corollary REF ). By Corollary REF , the composition is surjective. By the proof of Proposition REF , the second map is... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
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666825eed4a7f9924d8908d00fc4c59745a5c5c5 | subsection | 344 | 399 | Local-global compatibility | From Proposition REF , we deduce an isomorphism:\operatorname{\mathrm {H}om}_{L_P(\mathbb {Q}_p)}\Big (\bigotimes _{i=1, \cdots , k} \big (\widehat{\pi }(\rho _{x_i})^{\operatorname{\mathrm {l}alg}}\otimes \varepsilon ^{s_i}\circ \operatorname{\mathrm {d}et}\big ), \operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp... | {
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c685c8ec67c3a608a4f7088eab4e175d91696760 | subsection | 345 | 399 | Local-global compatibility | Finally let S(\overline{\rho }_i) be the set of Serre weights attached to \overline{\rho }_i, that is the set of irreducible summands in \operatorname{\mathrm {s}oc}(\overline{\pi }_i\vert _{\operatorname{\mathrm {G}L}_{n_i}(\mathbb {Z}_p)}), and let S^{P-\operatorname{\mathrm {o}rd}} be the set of (isomorphism classes... | {
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16609def4b96f578f7727c4fb67cea9eca93881f | subsection | 346 | 399 | Local-global compatibility | Lemma REF (2)), we have an isomorphism (e.g. by ):\operatorname{\mathrm {H}om}_{L_P(\mathbb {Z}_p)}\big (\Theta , \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }})\big )/\varpi _E
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
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abca59c95e150ca7d85e1814b212ae70c336a7d8 | subsection | 347 | 399 | Local-global compatibility | Let x\in C and consider (recall \Theta is an \mathcal {O}_E-lattice in L_P(\lambda ) stable by L_P(\mathbb {Z}_p)):\pi _x^{\infty }:=\big (\operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_x]\big ) \otimes _{\mathcal {O}_E} L_P(\lambda )^{\vee }\big )^{\operatorname{\m... | {
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04a4a4c9246b00dc67b4aad43ab1cda54f48f317 | subsection | 348 | 399 | Local-global compatibility | But the latter isomorphism together with (\pi _i^{\infty })^{\operatorname{\mathrm {G}L}_{n_i}(\mathbb {Z}_p)}\ne 0 easily imply, using that \Theta _i is up to scaling the only \mathcal {O}_E-lattice in L_i(\underline{\lambda }_i) which is stable by \operatorname{\mathrm {G}L}_{n_i}(\mathbb {Z}_p):\sigma _i=\overline{\... | {
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e46aa0025c563129382c67121323163832610fcd | subsection | 349 | 399 | Local-global compatibility | By (REF ), it is enough to prove that the evaluation map:\operatorname{\mathrm {H}om}_{k_E[L_P(\mathbb {Q}_p)]}\Big (\bigotimes _{i=1,\cdots , k}(\overline{\pi }_i\otimes _{k_E} \overline{\varepsilon }^{s_i}\circ \operatorname{\mathrm {d}et}), \big (\operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, \mathbb {W}^{\wp ... | {
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ae22d4000b1ce489e5d4bdfa2063e7eb121f5294 | subsection | 350 | 399 | Local-global compatibility | By Lemma REF , Corollary REF and the same argument as in the proof of Proposition REF (2), it is enough to prove that for any benign point x, we have:\operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_x]\big )^{L_P(\mathbb {Z}_p)-\operatorname{\mathrm {a}lg}}_+\subset \... | {
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e64e5030e99082fa06625ec9aa053c6db2a87608 | subsection | 351 | 399 | Local-global compatibility | The corollary follows since (X_P(U^{\wp })/\varpi _E)[\mathfrak {m}_{\overline{\rho }}] is a finite dimensional k_E-vector space.Corollary 7.41
Keep the assumption of Theorem REF . Let x\in (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}})^{\operatornam... | {
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local-global compatibility | [
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72ecb0cfb011092a1a0fd7c9bb8c234a7446426a | subsection | 352 | 399 | Preliminaries | We start with easy preliminaries.Throughout § REF we keep the notation and assumptions of § REF and of all the subsections of § REF , in particular we assume Hypothesis REF and that the open compact subgroup U^{\wp } is such that U^p is sufficiently small, U_v is maximal for v|p, v\ne \wp , and U_v is maximal hyperspec... | {
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"start": 13... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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b106d7e16fc8f0907219e95f608c059d18e62965 | subsection | 353 | 399 | Preliminaries | We also easily deduce that \rho _{\widetilde{\wp }} is strictly P-ordinary for any parabolic subgroup P of \operatorname{\mathrm {G}L}_n containing B.Using , we see that there exists m(\rho ) such that:\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_{\rho }]^{\operatorname{\mathrm {l}alg}}\cong (\opera... | {
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33462349f02b7146d881bcbdfb5f36e32d6d1bb3 | subsection | 354 | 399 | Preliminaries | We easily check that:\operatorname{\mathrm {O}rd}_B(\operatorname{\mathrm {S}t}_n^{\infty } \otimes \chi _1\circ \operatorname{\mathrm {d}et})\cong J_B(\operatorname{\mathrm {S}t}_n^{\infty }\otimes \chi _1\circ \operatorname{\mathrm {d}et})(\delta _B^{-1})\cong \chi _1\circ \operatorname{\mathrm {d}et}.Lemma 7.42
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92e41fa7c5e9d93be3adeb679d237dccac7502ef | subsection | 355 | 399 | Preliminaries | By (REF ) and Lemma REF , we obtain that x\in (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}} for all P\supseteq B.For 1\le i <i^{\prime }\le n, we denote by \rho _i^{i^{\prime }} the (unique) subquotient of \rho _{\widet... | {
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local-global compatibility | [
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caae321c971e400c35a43103e05f426c9b3432c7 | subsection | 356 | 399 | Local-global compatibility for | In dimension 3, we finally use most of the previous material to prove our main local-global compatibility result (Corollary REF ).We keep all the notation of §§ REF , REF , REF and now assume n=3 (and thus p>3). For r=1, 2, we let \mathcal {L}_r\in E such that:\psi _{\mathcal {L}_r}:=\log _p -\mathcal {L}_r \operatorna... | {
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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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62b77dd661e34be411d8d11051dfd2f1208e401f | subsection | 357 | 399 | Local-global compatibility for | The assumptions on \rho _{\widetilde{\wp }} imply in particular that D is sufficiently generic in the sense of (the end of) § REF , and we can then define E(\widetilde{\Pi }^1(\alpha ,\lambda , \psi _{\mathcal {L}_1}), v_{\overline{P}_2}^{\infty }(\alpha ,\lambda )^{\oplus 2}, \mathcal {L}_{\operatorname{\mathrm {a}ut}... | {
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local-global compatibility | [
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f0dfb166ec441c199e935ad528ddeda4bf939f26 | subsection | 358 | 399 | Local-global compatibility for | One can check by using the functor J_B(\cdot ) that the composition in (REF ) gives a section of the restriction morphism:\operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)}\big (\widetilde{\Pi }^1(\alpha ,\lambda , \psi _{\mathcal {L}_1}), \widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mat... | {
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local-global compatibility | [
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a50c3be48bcdb65a8375103a6b77815b5809e852 | subsection | 359 | 399 | Local-global compatibility for | Consequently, the fourth injection in (REF ) is also bijective.(c) By (a) and (b), it is enough to prove that, for any line Ew\subseteq \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D: D_1^2), setting \Pi :=E(\widetilde{\Pi }^1(\alpha ,\lambda , \psi _{\mathcal {L}_1}), v_{\overline{P}_2}^{\infty }(\alpha ,\lambda ), Ew)... | {
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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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721c99494d253762db00886641035ffce470f6cf | subsection | 360 | 399 | Local-global compatibility for | (REF )), by Proposition REF (2) there exists v\in V_x such that d \omega _{1, x}^+(v)\mapsto w \in \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(D_1^2, \mathcal {R}_E(\chi _2)). Denote by \mathcal {I}_v the ideal of \tilde{\mathbb {T}}(U^{\wp })^{P_1-\operatorname{\mathrm {o}rd}}_{\overline{\rho }} attached to v ... | {
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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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f58f0fcfd7c1bb17e991d426d6da9dc23873af81 | subsection | 361 | 399 | Local-global compatibility for | \chi _1) over E[\epsilon ]/\epsilon ^2 attached to d\omega _{1,x}(v) (resp. d\omega _{2,x}(v)), we have a commutative diagram:\begin{}
\pi : = \widehat{\pi }(\rho _1^2)\boxtimes _E \chi _1 @> f>> \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }})[\mathfrak {m}_{\rho }] \\
@V\iota VV @VVV... | {
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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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8a4273b7bdf6ab2a44ea7ff58e4ff54125ead15a | subsection | 362 | 399 | Local-global compatibility for | From the definition of d \omega _{1, x}^+ we have:\widetilde{\pi } \cong ((\chi _1^{-1}\widetilde{\chi }_1)\circ \operatorname{\mathrm {d}et}_{\operatorname{\mathrm {G}L}_2} \otimes _{E[\epsilon ]/\epsilon ^2} \widetilde{\pi }_{w_0}) \boxtimes _{E[\epsilon ]/\epsilon ^2} \widetilde{\chi }_1\cong (\chi _1^{-1}\widetilde... | {
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local-global compatibility | [
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