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689f9e419de8ce28b81abf3a22a722f53404a7e7 | subsection | 63 | 399 | Body | From the constructions of \Pi ^1(\lambda , \psi ) and \widetilde{\Pi }^1(\lambda , \psi ) (and from Lemma REF ), it is not difficult to see that one has an injection:\Pi ^1(\lambda , \psi )^+:=\Pi ^1(\lambda , \psi )\oplus _{\operatorname{\mathrm {S}t}_3^{\infty }(\lambda )} S_{1,0} \ \hookrightarrow \ \widetilde{\Pi }... | {
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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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3f6762928d90a6325206c0a163f78f4341cfa38c | subsection | 64 | 399 | Body | It is sufficient to show \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_2}^{\infty }(\lambda ), \widetilde{\Pi }^1(\lambda , \psi )/\Pi ^1(\lambda , \psi )^+)=0. From and , we easily deduce that for any irreducible representation W in the union (REF ) \cup (REF ) we have... | {
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8a8f9c490b89763a13db10c899cbb0005286a20b | subsection | 65 | 399 | Body | By loc.cit. we also have:\dim _E \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p),Z}(v_{\overline{P}_2}^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_3^{\infty }(\lambda ))=\dim _E \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_2}^{\infty... | {
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4e86df8a44eccf3cfd0b8c152a89bd5e3837eb84 | subsection | 66 | 399 | Body | However, any irreducible constituent of \pi (\lambda _{1,2}, \psi )\otimes x^{k_3} is either locally algebraic or isomorphic to a locally analytic principal series, and hence satisfies the condition (FIN) (see the discussion in the beginning of for the locally algebraic case, and the discussion before Step 1 in the pro... | {
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a7cf10f1044b2938f3636736c3ed758faee50d08 | subsection | 67 | 399 | Body | L_1$\!-dominant}}} L_1(w\cdot \lambda )\Big ) \otimes \big ((\operatorname{\mathrm {S}t}_2^{\infty }\otimes 1) \oplus (|\cdot |^{-1}\circ \operatorname{\mathrm {d}et}\otimes |\cdot |^{2})\big ).For all w with w\cdot \lambda dominant with respect to B(\mathbb {Q}_p)\cap L_1(\mathbb {Q}_p) we have by considering the acti... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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bff24c3c70db4e6f4c2bb10f2edc941accd6c3ea | subsection | 68 | 399 | Body | By , we have \operatorname{\mathrm {E}xt}^i_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p), Z}(v_{\overline{P}_2}^{\infty }(\lambda ), I_{\overline{P}_2}^{\operatorname{\mathrm {G}L}_3}(\lambda ))=0 for all i\ge 0 and:\operatorname{\mathrm {E}xt}^i_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p), Z}\big (v_{\overline{... | {
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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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98a2859dd98735ca83e7ab901145df3dc2593e0c | subsection | 69 | 399 | Body | From the former equality we obtain an exact sequence:0 \longrightarrow \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p),Z}\big (v_{\overline{P}_2}^{\infty }(\lambda ), W\big ) \longrightarrow \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p),Z}\big (v_{\overlin... | {
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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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ac926ee00cac5896a769f7aba28499bf100835cb | subsection | 70 | 399 | Body | For any \widetilde{\pi }\in \operatorname{\mathrm {E}xt}^1_{L_1(\mathbb {Q}_p)}(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda _{1,2}) \otimes x^{k_3}, \pi (\lambda _{1,2}, \psi )\otimes x^{k_3}), the parabolic induction (\operatorname{\mathrm {I}nd}_{\overline{P}_1(\mathbb {Q}_p)}^{{\operatorname{\mathrm {G}L}_3}(\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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66fb3a4fa8c7ca388cb2dd0c78a095de65cb2768 | subsection | 71 | 399 | Body | In particular the composition sends \widetilde{\pi } to \operatorname{\mathrm {p}r}^{-1}(v_{\overline{P}_2}^{\infty }(\lambda ))/W.Consider the following composition:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\operatorname{\mathrm {S}t}_2^{\infty }(\lambda _{1,2}), \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"
] | [
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1367c46bbb4bc335c00cee0c0258416d70e551cd | subsection | 72 | 399 | Body | Consider \widetilde{\pi }^{\prime }:=\widetilde{\pi }\otimes _{E[\epsilon ]/\epsilon ^2} (\widetilde{\chi }^{-1}\circ \operatorname{\mathrm {d}et}), on which Z_2 acts thus by x^{k_3}. So there exists \widetilde{\pi }^{\prime }_0\in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\oper... | {
"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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92081424a3d301c3d0ea7a76217d7d747dabbade | subsection | 73 | 399 | Body | (REF )), \pi (\lambda , \psi ,0)^- is a subquotient of (\operatorname{\mathrm {I}nd}_{\overline{B}_2(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)} \delta _{\lambda _{1,2}}(1+\Psi _{1,2}\epsilon ))^{\operatorname{\mathrm {a}n}}, and thus (\operatorname{\mathrm {I}nd}_{\overline{P}_1(\mathbb {Q}_p)}^{{\o... | {
"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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348ebd7e9a0c828a3aac6fb612a8ec783d7916a3 | subsection | 74 | 399 | Body | The cup product (REF ) together with the isomorphisms:\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D_1^2, D_1^2\big )\cong \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D(p,\lambda _{1,2}^{\sharp }, \psi ),D(p,\lambda _{1,2}^{\sharp }, \psi )\big )\\
[\sim ]{(\ref {equ: hL-pLL0})} \operatorname{\... | {
"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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fda113dd13f34c4af50182bfbb31ca32f4f04c7d | subsection | 75 | 399 | Body | We have morphisms (see (REF ) for \kappa _1):\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\pi (\lambda _{1,2}, \psi ), \pi (\lambda _{1,2}, \psi )\big )
{\kappa _1} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\operatorname{\mathrm {S}t}_2^{\in... | {
"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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59b692be0544186874d26977db89854a7d00c915 | subsection | 76 | 399 | Body | Now consider:\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D_1^2, D_1^2\big ) {\sim } \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D(p,\lambda _{1,2}^{\sharp }, \psi ),D(p,\lambda _{1,2}^{\sharp }, \psi )\big ) \\ [\sim ]{(\ref {equ: hL-pLL0})} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathr... | {
"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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489eb5d3f39d99632238033ce10f87a19a7021fc | subsection | 77 | 399 | Body | \!\times @. \!\!\!\!\!\!\!\!\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (\mathcal {R}_E(\delta _3), D_1^2\big ) @> \cup >> E\\
\end{}where the left vertical map is the natural injection, the middle vertical map is the natural surjection, the bottom (perfect) pairing is the one in Theorem REF and the top (pe... | {
"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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c83bff8557904ee6bddba21cb9bd120e57d3e896 | subsection | 78 | 399 | Body | The same holds with (\operatorname{\mathrm {S}t}_3^{\operatorname{\mathrm {a}n}}(\lambda ),\widetilde{\Pi }^1(\lambda , \psi )) replaced by (S_{1,0},\Pi ^1(\lambda , \psi )^+).(a) We first show that the composition:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}\big (\pi (\lambda _{1,2},\psi ), \pi (\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",
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66f7d8b59a17615ad8789473c46f531ccc478dd2 | subsection | 79 | 399 | Body | It is thus sufficient to show that the composition:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\operatorname{\mathrm {S}t}_2^{\infty }(\lambda _{1,2}), \pi (\lambda _{1,2}, \psi )^-\big ) {\iota _1} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big ... | {
"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"
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69838191909c8a37db77d460db2bd7a9c57d3bc0 | subsection | 80 | 399 | Body | By the proof of Lemma REF (1), one has:\operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}\big (v_{\overline{P}_2}^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_3^{\operatorname{\mathrm {a}n}}(\lambda )\big ) {\sim } \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb ... | {
"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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0... | |
08ebb3f7e5954c4f7f72876f7b33b7b453ce77e9 | subsection | 81 | 399 | Body | From Lemma REF (1) and Proposition REF we deduce \dim _E\operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_2}^{\infty }(\lambda ), \operatorname{\mathrm {S}t}_3^{\operatorname{\mathrm {a}n}}(\lambda ))=2. Together with Lemma REF (1) and a dimension count, we obtain that the ... | {
"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 | [
-0.0632816031575203,
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0.0067927036434412,
-0.01693408377468586,
-0.... | |
6540b0b01e9dbfdd0add9f8bb5246cfefe12e373 | subsection | 82 | 399 | Body | In particular, for \widetilde{\pi }\in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\operatorname{\mathrm {S}t}_2^{\infty }(\lambda _{1,2}), \pi (\lambda _{1,2}, \psi )^-) , its image in \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_2}^{... | {
"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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0.005599601659923792,
0.04900795966386795,
0.0... | |
2fffec9b0c4a1a46cdcae629796baebbe2709d0e | subsection | 83 | 399 | Body | We deduce that the following diagram commutes (see (REF ) for \kappa ^{\operatorname{\mathrm {a}ut}}=\kappa and recall (REF ) comes from (REF ) by the proof of (a)):\begin{} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}\big (\pi (\lambda _{1,2}, \psi ), \pi (\lambda _{1,2}, \psi )\big )@>(\ref {equ: hL-... | {
"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 | [
-0.01928395964205265,
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0.03246540203690529,
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0.... | |
28750dabbc5ffaf51caa5fe30e7b4845126ef1c8 | subsection | 84 | 399 | Body | \!\times \!@. \ \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (\mathcal {R}_E(\delta _3), \mathcal {R}_E(\delta _2)\big ) @> \cup >> E\\
@V \kappa ^{\operatorname{\mathrm {a}ut}} VV @. @| @| \\
\ \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E(\delta _2),\mathcal {R}_E(\delta _2)) \!@. \ \ ... | {
"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 | [
-0.03696653991937637,
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-0.0026767165400087833,
0.008524282835423946... | |
c76a7d2324eecd2ae9b7e74f887b763969fda660 | subsection | 85 | 399 | Body | This concludes the proof.We fix a nonsplit extension D\in \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (\mathcal {R}_E(\delta _3), D_1^2\big ) and we let (assuming Hypothesis REF for D_1^2):\mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)\subseteq \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G... | {
"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 | [
-0.06867815554141998,
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-0.0053721582517027855,
0.009805714711546898,
... | |
5c68545572e5b6947aeef9f76cf5ba785fa8a557 | subsection | 86 | 399 | Body | Notation REF ):\widetilde{\Pi }^1(D)^-&:=&E\big (\widetilde{\Pi }^1(\lambda , \psi ), v_{\overline{P}_2}^{\infty }(\lambda )^{\oplus 2}, \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)\big ) \\
\Pi ^1(D)^-&:=& E\big (\Pi ^1(\lambda , \psi )^+, v_{\overline{P}_2}^{\infty }(\lambda )^{\oplus 2}, \mathcal {L}_{\opera... | {
"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 | [
-0.06707792729139328,
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0.03872697427868843,
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-0.005287193227559328,
0.01690681092441082,
0.0... | |
51c7404d343139b5ed9fd3d40e1f490d488a0c0f | subsection | 87 | 399 | Body | Notation REF ):\widetilde{\Pi }^1(D)^-_2&:=&E\big (\operatorname{\mathrm {S}t}_3^{\operatorname{\mathrm {a}n}}(\lambda ), v_{\overline{P}_2}^{\infty }(\lambda ), \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)_0\big )\\
\Pi ^1(D)^-_2&:=&E\big (S_{2,0}, v_{\overline{P}_2}^{\infty }(\lambda ), \mathcal {L}_{\operato... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1302,
"openalex_id": "",
"raw": "Berger, L. Équations différentielles p-adiques et modules filtrés. Astérisque 319 (2008), 13–38.",
"source_ref_id": "f0f49659872a231b8127b7a3c638d3b5ac221bea",
"start": 1101
}
]
} | 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 | [
-0.04509863257408142,
0.030330203473567963,
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0.043450914323329926,
0.00539... | |
cc3ee3e17a8ac58e292f788c7030adda60009cac | subsection | 88 | 399 | Body | Then there exists a unique subrepresentation:\Pi ^1(D)^{-}_1\in \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}\big (v_{\overline{P}_2}^{\infty }(\lambda ), \Pi ^1(\lambda , \psi )\big )\setminus \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}\big (v_{\ove... | {
"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 | [
-0.07085917890071869,
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0.03053170070052147,
0... | |
ea02f191097cfa2c0964a2fd3952b4ff86d85037 | subsection | 89 | 399 | Body | In particular, \Pi ^1(D)^- has the following form:{}[32]
(0,10)[a]{\operatorname{\mathrm {S}t}_3^{\infty }(\lambda )}
(12,2)[b]{C_{2,1}}
(25,2)[c]{v_{\overline{P}_2}^{\infty }(\lambda ).}
(12,18)[d]{C_{1,1}}
(23,10)[e]{C_{1,2}}
(34,18)[f]{C_{1,3}}
(47,10)[g]{v_{\overline{P}_2}^{\infty }(\lambda )}
(23, 26)[h]{\widetild... | {
"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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-0.029060333967208862,
-0.00... | |
e92141ccd8d7a345de8dd4772d4172f6d6f49e60 | subsection | 90 | 399 | Body | We first show:\operatorname{\mathrm {K}er}(\operatorname{\mathrm {p}r})\cap \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)=\operatorname{\mathrm {K}er}(\operatorname{\mathrm {p}r})\cap \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)_0=0.The first equality is clear since by definition and Lemma REF (1) we hav... | {
"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 | [
-0.06780408322811127,
0.04516203701496124,
-0.01569990999996662,
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0.0219554640352726,
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-0.0027387114241719246,
-0.034481827169656754,
... | |
0e02c8154a73446586cbb1268a268aa06fc6c929 | subsection | 91 | 399 | Body | Let \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)_1:=\operatorname{\mathrm {K}er}(\operatorname{\mathrm {p}r}_2)\cap \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_1^2)\subseteq \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}\big (v_{\overline{P}_2}^{\infty }(\lambda ), \Pi ^1(\l... | {
"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 | [
-0.036507025361061096,
0.0420013926923275,
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0.018635066226124763,
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0.014720328152179718,
-0.009363318793475628,
0.... | |
ac8745efa97dab6ef6f780c26a2476a8507acb18 | subsection | 92 | 399 | Body | The isomorphism (REF ) and (REF ) induce a perfect pairing:\operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}\big (v_{\overline{P}_1}^{\infty }(\lambda ),W\big ) \times \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D_2^3, \mathcal {R}_E(\delta _1)\big ) {\cup } Esuch that the fol... | {
"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 | [
-0.06529119610786438,
0.01896190457046032,
-0.010419132187962532,
-0.026406321674585342,
-0.034415170550346375,
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0.03710004314780235,
0.006334618665277958,
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0.0054650865495204926,
0.023614665493369102,
0.... | |
e5daeb57f0a668f2f0a6d355e0829c6852eed0f6 | subsection | 93 | 399 | Body | \!\!\!\!\!\!\!\!\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D_2^3, \mathcal {R}_E(\delta _1)\big ) @> \cup >> E\\
\end{}with (\Pi ^-,\Pi )=(S_{2,0},\Pi ^2(\lambda , \psi )^+) or (\operatorname{\mathrm {S}t}^{\operatorname{\mathrm {a}n}}_3(\lambda ),\widetilde{\Pi }^2(\lambda , \psi )) and where the top per... | {
"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 | [
-0.07050454616546631,
0.026217924430966377,
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0.028674902394413948,
-0.0012075048871338367,
... | |
05b79880a1c43ee632c39093a156c47d5196f13a | subsection | 94 | 399 | Body | We also define:\Pi ^2(D)^-&:=& E\big (\Pi ^2(\lambda , \psi )^+, v_{\overline{P}_1}^{\infty }(\lambda )^{\oplus 2}, \mathcal {L}_{\operatorname{\mathrm {a}ut}}(D:D_2^3)\big )\\
\widetilde{\Pi }^2(D)^- &:=&E\big (\widetilde{\Pi }^2(\lambda , \psi ), v_{\overline{P}_1}^{\infty }(\lambda )^{\oplus 2}, \mathcal {L}_{\opera... | {
"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 | [
-0.04570504650473595,
0.02828342840075493,
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0.0032265076879411936,
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0.04512534290552139,
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0.014637514017522335,
0.00408843532204628,
-0.0102... | |
e486e49f2b21637c0aa2476593a8df376080749c | subsection | 95 | 399 | Body | Similarly as in Proposition REF , assuming Hypothesis REF there exists a unique representation if N^2\ne 0:\Pi ^2(D)^-_2\in \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}\big (v_{\overline{P}_1}^{\infty }(\lambda ), \Pi ^2(\lambda , \psi )\big )\setminus \operatorname{\mathrm {E}xt}^1_{... | {
"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 | [
-0.04184238240122795,
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-0.0019332264782860875,
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0.04257485270500183,
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0.002216486493125558,
0.015793897211551666,
... | |
9c1df40be3f920f6942a1da83c65dda5f1206596 | subsection | 96 | 399 | Body | We can then associate to D the above representations:\Pi ^1(D)^-&\cong &\Pi ^1(D)^-_1\oplus _{\operatorname{\mathrm {S}t}_3^{\infty }(\lambda )} \Pi ^1(D)^-_2\ \hookrightarrow \ \widetilde{\Pi }^1(D)^-\\
\Pi ^2(D)^-&\cong &\Pi ^2(D)^-_1\oplus _{\operatorname{\mathrm {S}t}_3^{\infty }(\lambda )} \Pi ^2(D)^-_2\ \hookrigh... | {
"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 | [
-0.06769111007452011,
0.02264508791267872,
-0.007698414381593466,
-0.014046363532543182,
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0.042299315333366394,
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0.002460593357682228,
0.03589032590389252,
0.038... | |
19305721d3bc75b070b4f64bfb29a5986eae1f9c | subsection | 97 | 399 | Body | Denote by \Pi ^0(D)^- the following subrepresentation of \Pi ^1(D)^- and \Pi ^2(D)^-:{}[32]
(0,10)[a]{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Pi ^0(D)^-\ \ \cong \ \ \operatorname{\mathrm {S}t}_3^{\infty }(\lambda )}
(12,2)[b]{C_{2,1}}
(23,2)[c]{C_{2,2}}
(12,18)[d]{C_{1,1}}
(23,18)[e]{C_{1,2}}
... | {
"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 | [
-0.059852033853530884,
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-0.007409016601741314,
0.023119794204831123,
... | |
94ef12aee276a595514311cafdc2deada69ec47d | subsection | 98 | 399 | Body | \end{gathered}It follows from the previous results that the (\varphi ,\Gamma )-module D and the \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)-representation \Pi (D)^- determine each other. From the results of (see in particular ), there is a unique locally analytic representation \Pi (D) containing \Pi (D)^- of the form... | {
"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 | [
-0.05093850567936897,
0.008347323164343834,
-0.03589196130633354,
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0.02771250158548355,
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0.025728676468133926,
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0.020006107166409492,
0.010407447814941406,
0... | |
4d34cc8e7ad628d026eff8e8261a6063efb0135a | subsection | 99 | 399 | Body | \end{gathered}where the irreducible constituents C_{1,5}, C_{2,5}, \widetilde{C}_{1,4}, \widetilde{C}_{2,4} are defined in .For \chi : \mathbb {Q}_p^{\times } \rightarrow E^{\times } and D^{\prime }:=D\otimes _{\mathcal {R}_E} \mathcal {R}_E(\chi ), we finally set \Pi (D^{\prime })^-:=\Pi (D)^-\otimes \chi \circ \opera... | {
"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 | [
-0.0066843582317233086,
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0.03128523752093315,
0.02934... | |
dea0f85a6abb3c2dce9d1277f14feecb93249b0b | subsection | 100 | 399 | Body | We denote by L a finite extension of \mathbb {Q}_p.We define P-ordinary Galois deformations and recall some standard useful statements.We fix P a parabolic subgroup of \operatorname{\mathrm {G}L}_n containing the Borel subgroup of upper triangular matrices and with a Levi subgroup L_P given by (where \sum _{i=1}^k n_i=... | {
"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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50bc33e5f16c87f78404895e5c03172e1fa95003 | subsection | 101 | 399 | Body | We assume the following hypothesis on \overline{\rho } and the \overline{\rho }_i.Hypothesis 5.2
We have \operatorname{\mathrm {E}nd}_{\operatorname{\mathrm {G}al}_L}(\overline{\rho }) \cong k_E, \operatorname{\mathrm {E}nd}_{\operatorname{\mathrm {G}al}_L}(\overline{\rho }_i)\cong k_E for i=1, \cdots , k and \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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46be04ea43fae7fda97b3ff54b0e1d9005209955 | subsection | 102 | 399 | Body | By choosing basis, the functor \operatorname{\mathrm {D}ef}_{\overline{\rho }}(A) can also be described as the set:\lbrace \rho _A:\operatorname{\mathrm {G}al}_L\rightarrow \operatorname{\mathrm {G}L}_n(A)\ {\rm such\ that\ the\ composition\ with}\ \operatorname{\mathrm {G}L}_n(A)\twoheadrightarrow \operatorname{\mathr... | {
"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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3126859da201613e32d30b8c0baadcae09ffeb9b | subsection | 103 | 399 | Body | We define the functor \operatorname{\mathrm {D}ef}_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace }^{P-\operatorname{\mathrm {o}rd}}: \operatorname{\mathrm {A}rt}(\mathcal {O}_E) \rightarrow \lbrace \text{Sets}\rbrace by sending A\in \operatorname{\mathrm {A}rt}(\mathcal {O}_E) to the set:\left\lbrace \big ((\rho... | {
"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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cef215aec448f230cc511d8ad5be2bd78d3eedea | subsection | 104 | 399 | Body | The following two propositions are standard, we provide short proofs for the convenience of the reader.Lemma 5.3
The functor \operatorname{\mathrm {D}ef}_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace }^{P-\operatorname{\mathrm {o}rd}} is a subfunctor of \operatorname{\mathrm {D}ef}_{\overline{\rho }}.Let A\in \... | {
"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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944a5b75759fffd854cf97f43e568268028cc2ff | subsection | 105 | 399 | Body | The same argument replacing T_A/T^{(2)}_{A,1} by T^{\prime }_A/T^{\prime }_{A,1} shows that any equivariant isomorphism T_A\sim \over \rightarrow T^{\prime }_A must send T_{A,i} to T^{\prime }_{A,i}.Proposition 5.4
The functor \operatorname{\mathrm {D}ef}_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace }^{P-\oper... | {
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{
"arxiv_id": "",
"doi": "10.1090/s0002-9947-1968-0217093-3",
"end": 1370,
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"raw": "Schlessinger, M. Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208–222.",
"source_ref_id": "37ecdc68e5373237bab... | 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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d8242476f74348ca3acb347670a002a0754b3421 | subsection | 106 | 399 | Body | Moreover, for i\in \lbrace 1,\cdots , k\rbrace , we have a natural transformation of functors \operatorname{\mathrm {D}ef}^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace } \rightarrow \operatorname{\mathrm {D}ef}_{\overline{\rho }_i} sending ((\rho _A,T_A),T_{A,\bullet },i_A) to \... | {
"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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cf3392def6941e0c9b989324c46ebd53b727136c | subsection | 107 | 399 | Body | \operatorname{\mathrm {D}ef}_{\overline{\rho }_i}) but replacing \operatorname{\mathrm {A}rt}(E) by \operatorname{\mathrm {A}rt}(\mathcal {O}_E) and \overline{\rho } (resp. \overline{\rho }_i) by \rho (resp. \rho _i). Then from Hypothesis REF the functor \operatorname{\mathrm {D}ef}_{\rho } (resp. \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",
"math.RT"
] | 2,018 | en | Mathematics | [
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2788422fd34998803b504183b10e2dc3e90b136d | subsection | 108 | 399 | Body | \operatorname{\mathrm {D}ef}^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace } \rightarrow \operatorname{\mathrm {D}ef}_{\overline{\rho }_i}). In particular, \rho _{\xi }^0 is a representation of \operatorname{\mathrm {G}al}_L over a free \mathcal {O}_E-module T_{\mathcal {O}_E} en... | {
"cite_spans": [
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"arxiv_id": "",
"doi": "10.4007/annals.2009.170.1085",
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"raw": "Kisin, M. Moduli of finite flat group schemes, and modularity. Annals of Math. 170 (2009), 1085–1180.",
"source_ref_id": "73323a53b... | 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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2ca836dd8b5126b7e5d63562154ecb2ced33dd2e | subsection | 109 | 399 | Body | By , the generic fiber \operatorname{\mathrm {D}ef}_{\overline{\rho }, (\xi )} is isomorphic to \operatorname{\mathrm {D}ef}_{\rho _{\xi }}. Moreover, by the argument in the proof of loc.cit. (together with Lemma REF and Proposition REF ), the isomorphism \operatorname{\mathrm {D}ef}_{\overline{\rho }, (\xi )} \xrighta... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4007/annals.2009.170.1085",
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"raw": "Kisin, M. Moduli of finite flat group schemes, and modularity. Annals of Math. 170 (2009), 1085–1180.",
"source_ref_id": "73323a53bd... | 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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91ca6373fef493a9ec4f52c989eab098a420d483 | subsection | 110 | 399 | Body | Hence ((\rho _A,T_A),T_{A,\bullet },i_A)\in \operatorname{\mathrm {D}ef}_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace , (\xi )}^{P-\operatorname{\mathrm {o}rd}} (A) (see ) which implies (REF ) is also surjective, and thus an isomorphism.Definition 5.8
Let \overline{\rho } (resp. \rho ) be a P-ordinary represen... | {
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{
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"doi": "10.4007/annals.2009.170.1085",
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"source_ref_id": "73323a53bd... | 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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1576b46a2fc4146a68e672c94c3d3d89d86e9aeb | subsection | 111 | 399 | Body | Assume that \rho _{\xi }^0, and thus \rho _{\xi }:=\rho _{\xi }^0\otimes _{\mathcal {O}_E} E, are P-ordinary.(1) The morphism \xi factors through the quotient R_{\overline{\rho }, \lbrace \overline{\rho }_i\rbrace }^{P-\operatorname{\mathrm {o}rd}} of R_{\overline{\rho }}.(2) The representation \rho _{\xi } is strictly... | {
"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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b1142fae6a20b8ff927239aa92b49523a702d21b | subsection | 112 | 399 | Body | For any i\ge 0, (S(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}^{I_{i,i}})_{\operatorname{\mathrm {o}rd}}=(S(U^{\wp }I_{i,i}, \mathbb {W}^{\wp })_{\overline{\rho }})_{\operatorname{\mathrm {o}rd}} is stable by \mathbb {T}(U^{\wp }) (since the action of \mathbb {T}(U^{\wp }) on S(U^{\wp }, \mathbb {W}^{\wp })^{I_{i,... | {
"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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eea73942e0db2a079cb95f7a9fdae41bb9311a22 | subsection | 113 | 399 | Body | Moreover, as in the proof of Lemma REF , the operators in \mathbb {T}(U^p) acting on (S(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}^{I_{i,i}})_{\operatorname{\mathrm {o}rd}}\otimes _{\mathcal {O}_E}E are semi-simple (since they are so on S(U^{\wp }, W^{\wp })), and we have as in loc.cit. the following consequence.... | {
"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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a40bc81e8673a8c1bfaeaa21a3a4a90055edc8e3 | subsection | 114 | 399 | Body | Consequently, the action of \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\operatorname{\mathrm {o}rd}} on \operatorname{\mathrm {O}rd}_P(S(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}) extends to a faithful action on \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho... | {
"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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339dc551bcb168132c45bb4c13a73d6872955129 | subsection | 115 | 399 | Body | Thus the natural injection from (REF ):\small
\operatorname{\mathrm {O}rd}_P (S(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }})/\varpi _E^s \cong \big (\varinjlim _{i} \big (S(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}^{I_{i,i}}\big )_{\operatorname{\mathrm {o}rd}}\big )/\varpi _E^s \longrightarrow \operatorna... | {
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{
"arxiv_id": "",
"doi": "",
"end": 924,
"openalex_id": "https://openalex.org/W1563952581",
"raw": "Rogawski, J. Automorphic representations of unitary groups in three variable. Annals of Math. Studies 123 (1990).",
"source_ref_id": "97d8a7ba90c26c8972588cb1e7... | 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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898ef408ac7bc86c37910864f3966b5c2509d157 | subsection | 116 | 399 | Body | The functor A\mapsto \rho _A of (isomorphism classes of) deformations of \overline{\rho } on the category of local artinian \mathcal {O}_E-algebras A of residue field k_E satisfying that \rho _A is unramified outside S(U^p) and that \rho _A^{\vee }\circ c \cong \rho _A\otimes \varepsilon ^{n-1} is pro-representable by ... | {
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{
"arxiv_id": "",
"doi": "",
"end": 952,
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"raw": "Thorne, J. On the automorphy of \\ell -adic Galois representations with small residual image. J. Inst. Math. Jussieu 11 (2012), 855–920.",
"source_ref_id": "ebba5c78086b9d0ff45152be62b520ee54a... | 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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9bf4885560f1313a8381480f0d0de369fcc1c01d | subsection | 117 | 399 | Body | The restriction to \operatorname{\mathrm {G}al}_{F_{\widetilde{\wp }}} gives a natural morphism:R_{\overline{\rho }_{\widetilde{\wp }}} \longrightarrow R_{\overline{\rho }, S(U^p)}.We fix \rho : \operatorname{\mathrm {G}al}_F\rightarrow \operatorname{\mathrm {G}L}_n(E) a continuous representation such that \rho is unra... | {
"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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fa5b4628b261c9ec466c0dd127eb5bb4d4d1967b | subsection | 118 | 399 | Body | By () and (REF ), there is an automorphic representation \pi of G(\mathbb {A}_{F_+}) (with W_{\wp } trivial in (REF )) which contributes to:S(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_{\rho }]\cong S(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}[\mathfrak {p}_{\rho }] \otimes _{\mathcal {O}_E} E.By the lo... | {
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"raw": "Thorne, J. On the automorphy of \\ell -adic Galois representations with small residual image. J. Inst. Math. Jussieu 11 (2012), 855–920.",
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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b52838ec817091312fdb172681ff622736761bbb | subsection | 119 | 399 | Body | Denote by I^{P-\operatorname{\mathrm {o}rd}} the kernel of the natural surjection R_{\overline{\rho }_{\widetilde{\wp }}}\twoheadrightarrow R_{\overline{\rho }_{\widetilde{\wp }}}^{P-\operatorname{\mathrm {o}rd}}, which we also view as an ideal of \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }} via:R_{\overline{\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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ca17d7a710fb8852b2b81c051f7d0a3c8a4e6cc8 | subsection | 120 | 399 | Body | We deduce then:I^{P-\operatorname{\mathrm {o}rd}}\operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}\big )\subset \cap _{i\in \mathbb {Z}_{\ge 0}} \varpi _E^i \operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp }, \mathbb {W}^{\wp })_{\overline{\rho }}\big )=0and (1) foll... | {
"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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e7219081599fc2871124353924bafd55f92117d9 | subsection | 121 | 399 | Body | From the discussion above Proposition REF , we obtain that \rho _{\widetilde{\wp }} is P-ordinary.By Theorem REF (1) and the last part in Lemma REF (1), the surjection \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}\!\twoheadrightarrow \!\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P-\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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] | 2,018 | en | Mathematics | [
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30501b0e58a507ff637c7cf87b1573b8b4ffa5cd | subsection | 122 | 399 | Body | From Proposition REF we deduce that the image is in \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })^{\operatorname{\mathrm {l}alg}}), hence also in \operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })^{\operatorname{\mathrm {l}alg}})\lbrace \mathfrak {m}_x\rbrace . From (REF ) it is easy to che... | {
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"raw": "Emerton, M. Local-global compatibility in the p-adic Langlands programme for \\mathrm {GL}_2/\\mathbb {Q}. preprint.",
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"star... | 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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82f2d0c4145aebb25cf9e78fee5f9b1edbc4531c | subsection | 123 | 399 | Body | This is an essentially admissible representation of T(\mathbb {Q}_p) over E () which is equipped with an action of \widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }} commuting with T(\mathbb {Q}_p). Let \mathcal {T} be the rigid analytic space over E parametrizing the locally analytic... | {
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"doi": "10.48550/arxiv.math/0405137",
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"raw": "Emerton, M. Locally analytic vectors in representations of locally p-adic analytic groups. Memoirs Amer. Math. Soc. 248, 1175 (2017).",
... | 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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76ceb7e357eafb04fd5083b1ac95d76b46ab361b | subsection | 124 | 399 | Body | Then, following the continuous dual J_{B\cap L_P}(\operatorname{\mathrm {O}rd}_P(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }})^{\operatorname{\mathrm {a}n}})^\vee is the global sections of a coherent sheaf on the rigid analytic space (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }... | {
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"so... | 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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15d1f2982d6e5893bc517f3afcdaba665d9c82a2 | subsection | 125 | 399 | Body | If we have for all i=1, \cdots , k such that n_i=2:\operatorname{\mathrm {v}al}_p(\chi _{s_i+1}(p))<\lambda _{s_i+1}-\lambda _{s_i+2} \ \ \ (\text{equivalently }\operatorname{\mathrm {v}al}_p(\chi _{s_i+1}^{\infty }(p))<-\lambda _{s_i+2}),then y is P-ordinary classical.As in § REF we use without comment in this proof t... | {
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"doi": "10.1090/s0894-0347-2014-00803-1",
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"raw": "Orlik, S., and Strauch, M. On Jordan-Hölder series of some locally analytic representations. J. Amer. Math. Soc 28 (2015), 99–157.",
... | 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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d61d9efeb753e146704191e24939b519797a9424 | subsection | 126 | 399 | Body | If \operatorname{\mathrm {v}al}_p(p\chi ^{\infty }_{s_i+1}(p))<1-\lambda _{s_i+2}, by the representation:\mathcal {F}_{\overline{B}_2}^{\operatorname{\mathrm {G}L}_2}\big (\overline{L}_i(-s\cdot \underline{\lambda }_i), |\cdot |^{-1}\chi _{s_i+1}^{\infty }\otimes |\cdot |\chi _{s_i+2}^{\infty }\big )\\ \cong \big (\ope... | {
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"raw": "Breuil, C. Vers le socle localement analytique pour \\mathrm {GL}_n, I. Annales de l'Institut Fourier 66 (2016), 633–685.",
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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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cceef47fd3d60a381f7bf03eecde60f7ba61b5f5 | subsection | 127 | 399 | Body | The lemma follows (using the adjunction property of J_{B\cap L_P}(\cdot ) on locally algebraic representations).The proof of the following lemma is standard and we omit it (see e.g. the proof of ).Lemma 7.15
The set of points satisfying the conditions in Lemma REF is Zariski-dense in \mathcal {E}^{P-\operatorname{\mat... | {
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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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f249427852faeb5fdc6a1155aa582547f09072e2 | subsection | 128 | 399 | Body | By Lemma REF and J_{B\cap L_P}\circ \operatorname{\mathrm {O}rd}_P\hookrightarrow J_B(\delta _P^{-1}) (see § REF ), we see \iota ^{P-\operatorname{\mathrm {o}rd}}(y) is a classical point in \mathcal {E}. Together with Proposition REF , we deduce that \iota ^{P-\operatorname{\mathrm {o}rd}} induces a closed immersion of... | {
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"source_ref_id": "39edc5a035e574bed649... | 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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4c43bf6d9785f7636c330877d8e9fa537c7a43fa | subsection | 129 | 399 | Body | We call a point y=(x, \chi ) of \mathcal {E}^{P-\operatorname{\mathrm {o}rd}} noncritical if \iota ^{P-\operatorname{\mathrm {o}rd}}(y) is noncritical, or equivalently if \chi \delta _{B\cap L_P}^{-1}(1\otimes \varepsilon ^{-1}\otimes \cdots \otimes \varepsilon ^{1-n}) is a trianguline parameter of D_{\operatorname{\ma... | {
"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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56925c5fa11051c053949f423f87bbaf3ac155ed | subsection | 130 | 399 | Body | Together with the fact \rho _{x, \widetilde{\wp }} is isomorphic to a successive extension of the \rho _{x_i}, we deduce that \chi \delta _{B\cap L_P}^{-1}(1\otimes \varepsilon ^{-1}\otimes \cdots \otimes \varepsilon ^{1-n}) is a trianguline parameter of \rho _{x,\widetilde{\wp }}.We say that an r-dimensional crystalli... | {
"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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a09bef022ed22057ea1c3b16c684157c454fbb25 | subsection | 131 | 399 | Body | We deduce then from (REF ) that \rho _{x_i} is generic.Denote by \omega ^1 the following composition:\omega ^1: \mathcal {E}^{P-\operatorname{\mathrm {o}rd}} \longrightarrow (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}}... | {
"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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d4eede5e9a9c4a9189e47b8094c5c78b564f0ac6 | subsection | 132 | 399 | Body | \lambda _j>\lambda _{j+1} for all j.We let Z_1:=\omega ^1(Z_1^{\prime })\subseteq (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}}, which we can also view as a subset of (closed) points of the scheme \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"
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59976242ac2541e6b656d6e67ddddea0dd709796 | subsection | 133 | 399 | Body | The existence of y follows easily from Corollary REF (2) and its proof. By (1), Z_1^{\prime } accumulates at y, in particular y lies in the Zariski closure of (\omega ^1)^{-1}(Z_1) in \mathcal {E}^{P-\operatorname{\mathrm {o}rd}}, from which we easily deduce that \omega ^1(y)=x lies in the Zariski closure of Z_1 in the... | {
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"source_ref_id": "39edc5a035e574bed649... | 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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829ae610717865ad8e1eb3d6a2d0ff0d204bc3b7 | subsection | 134 | 399 | Body | Then any injection as in (REF ) extends to an injection of locally analytic representations of L_P(\mathbb {Q}_p) over k(x):\big (\operatorname{\mathrm {I}nd}_{\overline{B}\cap L_P(\mathbb {Q}_p)}^{L_P(\mathbb {Q}_p)} \chi \delta _{B\cap L_P}^{-1}\big )^{\operatorname{\mathrm {a}n}} \longrightarrow \operatorname{\mathr... | {
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"raw": "Orlik, S., and Strauch, M. On some properties of the functor {\\mathcal {F}}_P^G from Lie algebra to locally analytic representations. ArXiv preprint arXiv:1802.07514 (2018).",
"source_ref_id... | 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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3488366d435d7f1cb1b49cf7041e20ea41d52a36 | subsection | 135 | 399 | Body | Applying the functor J_{B\cap L_P}(\cdot ) to (REF ) gives a point y^{\prime }=(x, \chi ^{\prime })\in \mathcal {E}^{P-\operatorname{\mathrm {o}rd}} with (\chi ^{\prime })^{\infty }=\chi ^{\infty } (which is unramified) and \chi ^{\prime }_{s_i+1}=\chi _{s_i+1} x^{\lambda _{s_i+2}-\lambda _{s_i+1}-1}, \chi ^{\prime }_{... | {
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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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06eb2bd261b62bcae00c65b97d5bdc37eee0b248 | subsection | 136 | 399 | Body | The proposition then follows by the same arguments as in (or as in when k=1) using Lemma REF (2) as a replacement for and the above discussion as a replacement for .Corollary 7.24
Let x be a benign point, then any injection as in (REF ) extends to a closed injection of Banach representations of L_P(\mathbb {Q}_p) over... | {
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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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153d52beee5d527e4fb1ac42566fb954170dd890 | subsection | 137 | 399 | Body | From Proposition REF , we deduce a continuous L_P(\mathbb {Q}_p)-equivariant injection:\widehat{\bigotimes }_{i=1,\cdots , k} \pi _i^{\operatorname{\mathrm {a}n}} \cong \big (\operatorname{\mathrm {I}nd}_{\overline{B}\cap L_P(\mathbb {Q}_p)}^{L_P(\mathbb {Q}_p)} \chi \delta _{B\cap L_P}^{-1}\big )^{\operatorname{\mathr... | {
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"raw": "Berger, L., and Breuil, C. Sur quelques représentations potentiellement cristallines de \\mathrm {GL}_2 (\\mathbb {Q}_p). Astérisque 330 (2010), 155–211.",
"source_ref_id": "6d32f21fd6943258e... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
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8956375df19359140dbfa9a0331da39cd5882d7a | subsection | 138 | 399 | Body | We deduce that (REF ) induces a continuous L_P(\mathbb {Q}_p)-equivariant morphism:\widehat{\bigotimes }_{i=1,\cdots , k} \big (\widehat{\pi }(\rho _{x_i})\otimes _{k(x)} \varepsilon ^{s_i}\circ \operatorname{\mathrm {d}et}\big ) \longrightarrow \operatorname{\mathrm {O}rd}_P\big (\widehat{S}(U^{\wp } ,W^{\wp })_{\over... | {
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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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56bea1c86215303548a8426fc5cbefd2cfd11349 | subsection | 139 | 399 | Body | Recall we have assumed Hypothesis REF . We now assume one more condition till the end of the paper.Hypothesis 7.25
If n>3, we have U_v maximal hyperspecial at all inert places v.It then follows from and the smoothness of \mathcal {W} that the rigid variety \mathcal {E}_{\operatorname{\mathrm {r}ed}} is smooth at the p... | {
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"source_ref_id"... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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722f551169eadc94aeea7bbce3123c1a2c5307cf | subsection | 140 | 399 | Body | We fix i\in \lbrace 1, \cdots , k\rbrace and denote by \omega ^1_i the following composition:\omega ^1_i: \mathcal {E}_{\operatorname{\mathrm {r}ed}}^{P-\operatorname{\mathrm {o}rd}} {\omega ^1} (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }})^{\operato... | {
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"source_ref_id": "73323a53b... | 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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ced2129b2efb970027ad2ab9eb6b4410cc45546c | subsection | 141 | 399 | Body | Assume n_i=2, since \rho _{x_i} is crystalline, generic and nonsplit, we have \dim _{k(x)} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho _{x_i}, \rho _{x_i})=5 (e.g. by similar arguments as in Lemma REF ). For each refinement (\alpha _{s_i+w_i(1)}, \alpha _{s_i+w_i(2)}) on the Frobe... | {
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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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95caa16aeec881bfcb9c22583558cc1e061524e0 | subsection | 142 | 399 | Body | We denote by \operatorname{\mathrm {E}xt}^1_g( \rho _{x_i}, \rho _{x_i})\subseteq \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho _{x_i}, \rho _{x_i}) the {k(x)}-vector subspace of de Rham deformations, or equivalently of crystalline deformations (since \rho _{x_i} is crystalline gene... | {
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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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200c295fef3857b5e5fb67cb27f6840282ccb9ea | subsection | 143 | 399 | Body | We let \overline{d}\omega ^1_{i, y_J} be the composition:\overline{d}\omega ^1_{i, y_J}: V_J{d \omega ^1_{i,y_J}} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho _{x_i}, \rho _{x_i}) \twoheadrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho ... | {
"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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e57968c1473076785d48d2d2cdd562c611122c2d | subsection | 144 | 399 | Body | We set:\overline{d}\omega ^1_{y_J}:=(\overline{d}\omega ^1_{i, y_J})_{i=1,\cdots , k}:V_J \longrightarrow \bigoplus _{i=1, \cdots , k} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho _{x_i}, \rho _{x_i}) /\operatorname{\mathrm {E}xt}^1_g(\rho _{x_i}, \rho _{x_i}).Proposition 7.28
(1)... | {
"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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2f4dbedfd869ba81f9a36ae7f532cec2ac975e55 | subsection | 145 | 399 | Body | From the global triangulation theory (see for instance ) and Lemma REF , we derive:{\left\lbrace \begin{array}{ll} \mathcal {R}_{k(x)[\epsilon ]/\epsilon ^2}(\widetilde{\chi }_{J,s_i+1}\varepsilon ^{-s_i}) {\sim } D_{\operatorname{\mathrm {r}ig}}(\widetilde{\rho }_{x_i}) & n_i=1 \\
\mathcal {R}_{k(x)[\epsilon ]/\epsilo... | {
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local-global compatibility | [
"Christophe Breuil",
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9bdc1da35d096eb7b0bf310f99e30ccab5118277 | subsection | 146 | 399 | Body | It then follows from (REF ) and (REF ) that \widetilde{\rho }_{x_i}\notin \operatorname{\mathrm {E}xt}^1_g(\rho _{x_i}, \rho _{x_i}) if j\in \lbrace s_i+1, s_i+n_i\rbrace , whence \overline{d}\omega ^1_{y_J}(v)\ne 0.We denote by V_x the tangent space of the rigid variety (\operatorname{\mathrm {S}pf}\widetilde{\mathbb ... | {
"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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b3b64b3155bffb2ba9b60c1a1b3560fe2e413f7d | subsection | 147 | 399 | Body | Together with (REF ) it follows that the morphism V_x\rightarrow \oplus _{i=1, \cdots , k}\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho _{x_i}, \rho _{x_i}) /\operatorname{\mathrm {E}xt}^1_g(\rho _{x_i}, \rho _{x_i}) is in fact surjective. Since the right hand side has dimension n+\... | {
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"source_ref_id": "e3f2d... | 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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14354c4e9407e13268fe7fe15959a0ad2160d6e3 | subsection | 148 | 399 | Preliminaries on locally analytic representations | We recall some useful notation and statements on locally analytic representations.We fix the \mathbb {Q}_p-points G of a reductive algebraic group over \mathbb {Q}_p (we will only use its \mathbb {Q}_p-points).Lemma 3.1
Let V_1, V_2, V be locally \mathbb {Q}_p-analytic representations of G over E such that V is a stri... | {
"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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ce80f310fb94dcc0e4eb3b1d2118683a04791ee0 | subsection | 149 | 399 | Preliminaries on locally analytic representations | For v\in V, we have:g(f(v))=g(v\otimes e)=g(v)\otimes ((1+\psi \epsilon )\circ \operatorname{\mathrm {d}et}(g))e=g(v)\otimes e=f(g(v))where the last equality follows from the fact that g(v)\in V\hookrightarrow U is annihilated by \epsilon . Thus f|_{V} induces a G-equivariant automorphism of V if we still denote by V t... | {
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"doi": "10.1515/crelle.2011.013",
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"source_ref_id": "9cc4983e5... | 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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d46e16415a0ec1acd5adc3bb7f769318d12758b9 | subsection | 150 | 399 | Deformations of rank | We define and study certain subspaces of Ext{}^1 groups of (\varphi ,\Gamma )-module over \mathcal {R}_E and relate them to infinitesimal deformations of rank 2 special (\varphi ,\Gamma )-modules.We now assume L=\mathbb {Q}_p and let (D, (\delta _1,\delta _2)) be a special, noncritical and nonsplit (\varphi ,\Gamma )-m... | {
"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"
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f38aa121553b81e390ecc9e9b7dbf9101c777467 | subsection | 151 | 399 | Deformations of rank | By (REF ) we have \dim _E \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E(\delta _2),\mathcal {R}_E(\delta _2) )=2 and using Tate duality () we get \dim _E \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E(\delta _1),\mathcal {R}_E(\delta _2) )=1 and \operatorname{\mathrm {E}xt}^2_{(... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
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74ac7170f4fab2500dcb035df6703b01ebb018aa | subsection | 152 | 399 | Deformations of rank | We deduce a long exact sequence:0 \longrightarrow \operatorname{\mathrm {H}om}_{(\varphi ,\Gamma )}(\mathcal {R}_E(\delta _1), \mathcal {R}_E(\delta _1))\longrightarrow \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E(\delta _2), \mathcal {R}_E(\delta _1)) \\ {\iota _1} \operatorname{\mathrm {E}xt}^1_... | {
"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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... | |
033b65a2119515511ebab00fafab08c872e96430 | subsection | 153 | 399 | Deformations of rank | We denote by \kappa _0 the following composition:\kappa _0: \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(D,D) {\kappa } \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(D,\mathcal {R}_E(\delta _2)) {\kappa _2} \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E(\delta _1),\mathcal {R}_E(\delta ... | {
"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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f179960924e37e2150eb00f410ef4cb21943c5a4 | subsection | 154 | 399 | Deformations of rank | The lemma follows then from (REF ) and a dimension count using the first equality in Lemma REF .By (REF ), (REF ) and the proof of Lemma REF , we get a short exact sequence:0 \longrightarrow \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(D,\mathcal {R}_E(\delta _1)){\iota } \operatorname{\mathrm {E}xt}^1_{\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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8ed0796f9996634f56bb7c706d96378b10e61312 | subsection | 155 | 399 | Deformations of rank | The following formula (sometimes called a Colmez-Greenberg-Stevens formula) is a special case of Theorem REF (via the identification (REF )).Corollary 3.8
Let \widetilde{D}\in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(D,D) and (\widetilde{\delta }_1, \widetilde{\delta }_2) its above trianguline par... | {
"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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... | |
0aa293b6bf84061acf784a1918ebba32c2029e7a | subsection | 156 | 399 | Deformations of rank | From (REF ) and (the discussion after) (REF ), we deduce a short exact sequence:0 \longrightarrow \operatorname{\mathrm {I}m}(\iota _0) \longrightarrow \operatorname{\mathrm {K}er}(\kappa ) {\kappa _1} \mathcal {L}_{\operatorname{\mathrm {F}M}}(D: D_1) \longrightarrow 0.In particular, \operatorname{\mathrm {I}m}(\iota ... | {
"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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-0.0... | |
9dd3a7429552fd72fbbb2d88e59b4a00d99ad412 | subsection | 157 | 399 | Deformations of rank | Hence the latter is surjective and the lemma follows from the first equality in Lemma REF .Lemma 3.10
We have \dim _E \big (\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma ),Z}(D,D) \cap \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(D,D)\big )=2.From Lemma REF , Lemma REF and Lemma REF it is sufficien... | {
"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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e805661d90501c9bebac512ae5d47cf96a54d293 | subsection | 158 | 399 | Deformations of rank | If \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma ),Z}(D,D)\subseteq \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(D,D), this would imply \dim _E \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(D,D)\ge \dim _E \operatorname{\mathrm {I}m}(j)+\dim _E \operatorname{\mathrm {E}xt}^1_{(\varph... | {
"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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-... | |
9594820dc23ee7b57a61a2d0f6dbe9ecaa7fb2df | subsection | 159 | 399 | Deformations of rank | Since \operatorname{\mathrm {w}t}(\delta _2\delta _1^{-1})\in \mathbb {Z}_{<0} we have \operatorname{\mathrm {E}xt}^1_{g}(\mathcal {R}_E(\delta _1),\mathcal {R}_E(\delta _2))=H^1_{g}(\mathcal {R}_E(\delta _2\delta _1^{-1}))=0 and we deduce from (REF )) (since being de Rham is preserved by taking subquotients) \operator... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 666,
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"raw": "Ding, Y. \\mathcal {L}-invariants, partially de Rham families, and local-global compatibility. Annales de l'Institut Fourier 67 (2017), 1457–1519.",
"source_ref_id": "bdb8c59ae339878314705f21b... | 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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... | |
71063de77651710d6f695a3b8a87bc835212b40c | subsection | 160 | 399 | Deformations of rank | However, since \operatorname{\mathrm {w}t}(\delta _1\delta _2^{-1})\in \mathbb {Z}_{>0} and \mathcal {R}_{E[\epsilon ]/\epsilon ^2}(\widetilde{\delta }_1\widetilde{\delta }_2^{-1}) is de Rham, we know (e.g. by ) that any element in H^1_{(\varphi ,\Gamma )}\big (\mathcal {R}_{E[\epsilon ]/\epsilon ^2}(\widetilde{\delta ... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 368,
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"raw": "Ding, Y. \\mathcal {L}-invariants, partially de Rham families, and local-global compatibility. Annales de l'Institut Fourier 67 (2017), 1457–1519.",
"source_ref_id": "bdb8c59ae339878314705f21b... | 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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854b30b408cb47a08e29021e0700321d344eb02e | subsection | 161 | 399 | Deformations of rank | If \mathcal {L}_{\operatorname{\mathrm {F}M}}(D:D_1)=\operatorname{\mathrm {H}om}_{\infty }(\mathbb {Q}_p^{\times }, E), we have \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma ),Z}(D,D)\cap \operatorname{\mathrm {E}xt}^1_g(D,D)=\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma ),Z}(D,D)\cap \operatorname{\mathrm {E}xt... | {
"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.... | |
1183ea46bbab7bd70b3f53524211267fd8958be2 | subsection | 162 | 399 | Deformations of rank | The result then follows from (REF ).Now fix k\in \mathbb {Z}_{\ge 1}, set \delta _3:=\delta _2x^{-k} |\cdot |^{-1} and consider the special case of the pairing (REF ):\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _3), D) \times \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D, D) {\... | {
"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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0.01184... |
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