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f0907b9c7c10550666a04f6b906e60024ab9c5c1 | subsection | 22 | 59 | Weak Coupling – Two Loops | In fact, we carry out the two-loop calculation below without referring to the exponentiation formula (REF ), similar to the QCD calculation carried out in ref. . The diagrams for the two-loop calculation are collected in figure REF . Here, we have already accounted for the relations established by applying a reflection... | {
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Theory | [
"Hagen Münkler"
] | [
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306eabaf4320f04123dd3acfb8793c6ee12b2d33 | subsection | 23 | 59 | Relations Between Diagrams | Many of the diagrams depicted in figure REF can be related to each other or to products of one-loop diagrams. We work out these relations below, starting with the ladder-like diagrams. | {
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} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
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b6caa4647a815c9b92676d689e3f4a5ae39384fc | subsection | 24 | 59 | Ladder-like Diagrams. | \mbox{}We begin by considering the diagrams F^2 _1 and F^2 _2. Employing the abbreviations
\theta _{ij} = \theta (\tau _i - \tau _j) and D_{ij} = D(\tau _i v_1 - \tau _j v_2), we haveF^2 _1 &= \xi ^2 \! \int \limits _0 ^L {\mathrm {d}}^4 \tau \,
\theta _{31} \, \theta _{42} \,
D_{12} \,
D_{34} \, , &
F^2 _2 &= \xi ^2 \... | {
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} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
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de7a273c5fe7f7f581f6f94cf521fe358cab840c | subsection | 25 | 59 | Ladder-like Diagrams. | Here, the kinematic factor F^2_4 is the color-connected ladder-like diagram appearing in the calculation of the cusp anomalous dimension and F^2 _7 - F^2 _8 corresponds to the two-gluon exchange web.Using the reflection discussed for the one-loop diagrams, we find that F^2 _2 and F^2_4 are related in the following way:... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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96e48ad727faba66492a25dbe3a44c5509376f83 | subsection | 26 | 59 | Three-vertex and Self-energy Diagrams. | \mbox{}Next, we turn to the three-vertex and self-energy diagrams. A cancellation between these diagrams has been observed in the calculation of the cusp anomalous dimension in ref. . While the approach presented there has to be adapted slightly to the case of the cross anomalous dimension, the result remains the same.... | {
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"raw": "Yu. Makeenko, P. Olesen and G. W. Semenoff, “Cusped SYM Wilson loop at two loops and beyond”, Nucl. Phys. B748, 170 (2006), hep-th/0602... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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2e2377cdcd5c7b55efa531e34422f91e0e2c19cb | subsection | 27 | 59 | Three-vertex and Self-energy Diagrams. | They arise from Wick-contracting the terms& \left\langle \frac{i^3}{3!} \int {\mathrm {d}}\tau _1 \, {\mathrm {d}}\tau _2 \, {\mathrm {d}}\tau _3 \,
\mathcal {P} \left[ A( \tau _1) A( \tau _2) A( \tau _3 ) \right]
_{i i^\prime j j^\prime }
\left( \frac{1}{g^2 \mu ^{2 \epsilon }}
\int {\mathrm {d}}^\mathrm {D} y f^{a b ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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] | 2,018 | en | Physics | [
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0491043f07aa9319b3ebac11cf24c9b4003b05bb | subsection | 28 | 59 | Three-vertex and Self-energy Diagrams. | For any given diagram we must hence consider all combinations of (\tau _1, \tau _2, \tau _3) being on any of the edges. We see immediately that the integral vanishes for diagrams with \tau _1 and \tau _3 on the same line. In the case of \tau _2 and \tau _3 being on the same line, we have \dot{x}_2 = \dot{x}_3 and n_2 =... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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06cb07c5d882e9b694f110a0c413ae9ce3b848b6 | subsection | 29 | 59 | Three-vertex and Self-energy Diagrams. | For the kinematic factor F^2 _{14} we thus haveF^2 _{14} & = \frac{i \xi }{g^2 \mu ^{2 \epsilon }}
\int \limits _{-L} ^0 {\mathrm {d}}\tau _1 \, {\mathrm {d}}\tau _2 \,
\varepsilon ( \tau _2 - \tau _1 )
\int \limits _0 ^L {\mathrm {d}}\tau _3 \,
\frac{\partial }{\partial \tau _1} \, G(x_1 , x_2 , x_3) \\
& = \frac{ 2 i... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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] | 2,018 | en | Physics | [
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95d4e8e5fdb1882a096330d561262d87d38fabe7 | subsection | 30 | 59 | Three-vertex and Self-energy Diagrams. | The three-vertex contribution thus reduces to the case of one scalar being frozen to the intersection point, see also figure REF . It is described by the lowest-order coefficient of
F(\phi ) = F_0 (\phi ) + \mathrm {O}(\epsilon ) , which we compute in appendix to findF_0 (\phi ) = - \frac{2 i \phi \left( \pi ^2 - \phi ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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5a9c77e9639a7b0b89bdd64ce963099a88ea08c0 | subsection | 31 | 59 | Relation to the Cusp Anomalous Dimension | It is clear that some of the above kinematic factors appear also in the calculation of the cusp anomalous
dimension at the two-loop level. Specifically, these are the factors F^2_3, F^2_4, F^2_{13}, F^2_{14}
and F^2_{18} and hence the functions F_0 (\phi ) and K_0 (\phi ). Combining these with the appropriate color
fac... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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50251c10b86c4797b4a2ce04e14b1a7b3dfc6dd4 | subsection | 32 | 59 | The Two-loop Result | In terms of the color and kinematic factors, the correlator \mathcal {W}_a at the two-loop level is given by\mathcal {W}_a ^{\,(2)} = \sum \limits _m {\textstyle \frac{1}{N^2}} \, c_m \, F^2 _m \, C^2 _{a, m} \, .Here, the c_n denote combinatorial factors, for which we havec_5 = c_6 = 1 \, , \qquad c_n = 2 \quad \text{... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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58755836a6c48bb53152aff4acd43161a51b1df7 | subsection | 33 | 59 | The Two-loop Result | Due to the presence of the commutator term at the two-loop level, we also need the finite coefficients of the kinematic factors at the one-loop level,F^1 _1 &= \frac{g^2 \mu ^{2\epsilon }}{\epsilon } ( \cos \phi - \cos \rho )
\left( I_0 (\pi - \phi ) + \epsilon \, I_1 (\pi - \phi ) + \mathrm {O}( \epsilon ^2 ) \right) ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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15577ec1d9c5aa794cb854fa83dddc11387a766b | subsection | 34 | 59 | The Two-loop Result | Remarkably, also the contribution of the ladder diagramsThis behavior has also been observed for the cross anomalous dimension in QCD .
to the first line of the cross anomalous dimension vanishes, which can be seen by inserting the results
(REF ), (REF ) and (REF ) to find\Gamma _{A} ^{(2)} (\phi ,\rho ) = 0 \, .Using ... | {
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Theory | [
"Hagen Münkler"
] | [
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1c5ccccb58900fa25a31a6ba87a0f0dffbf82660 | subsection | 35 | 59 | The Two-loop Result | For the analytic continuation, note that the i 0-prescription for the position space propagator implies that
\gamma has a small negative imaginary part, such that the branch cuts of the polylogarithms appearing for terms
such as \mathrm {Li}_2 (e^\gamma ) are approached from below.The scaling (REF ) can be reduced to t... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
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cbced1378f43ef144bc125774071bf55e9e0582c | subsection | 36 | 59 | The Two-loop Result | Consider for example the first diagram for which we find the contribution\widetilde{F}^2 _1 &= (-\xi )^2 \, \frac{\Gamma (1-\epsilon )^2}{16 \pi ^{4-2\epsilon }} g^4 \mu ^{4 \epsilon } \,
\int \limits _{-L} ^0 {\mathrm {d}}\tau _2 \, {\mathrm {d}}\tau _4
\int \limits _0 ^L {\mathrm {d}}\tau _1 \, {\mathrm {d}}\tau _3 \... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
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7ce3b3d9720ca70bb547ef2df0f9ccad94b1c0fd | subsection | 37 | 59 | The Two-loop Result | We thus find\widetilde{F}^2 _{15} = - \frac{i \xi }{g^2 \mu ^{2 \epsilon } }
\int \limits _0 ^L {\mathrm {d}}^3 \tau \,
\partial _{\tau _1} \left[ G ( -v_2 \tau _1 , v_2 \tau _2 , v_1 \tau _3 )
+ G ( v_2 \tau _1 , - v_2 \tau _2 , v_1 \tau _3 ) \right]
= F ^2 _{15} \, .In summary, all kinematic factors can be related to... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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e08a88e0564024f1393e27b8bea414a97cffa495 | subsection | 38 | 59 | Conclusion and Outlook | In this paper, we have determined the cross anomalous dimension for Maldacena–Wilson loops in
\mathcal {N} \!=4 SYM theory at strong coupling and up to the two-loop level at weak coupling.
The strong-coupling discussion showed that the cross anomalous dimension displays Gross–Ooguri
phase transitions in certain regions... | {
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"Hagen Münkler"
] | [
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79e2e9ad8d88cfee041f2aa6dbcd4ee9fda847e0 | subsection | 39 | 59 | Color Factors | In this appendix, we provide the color factors arising from the contraction of the \mathrm {SU(N)}-generators T^a for the calculation of the cross anomalous dimension to two loops.
For the structure constants, we note the convention
\left[ T^a , T^b \right] = i \, f^{a b c} \, T^c. The generators are normalized in such... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
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] | [
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... |
a9d1c04c3745d1143cb798e34207a60bf2b59a94 | subsection | 40 | 59 | Color Factors | For the ladder-like diagrams we note\begin{}{4}
C_{1,1}^2 &= {\textstyle \frac{N^2 +1}{4 N^2}} \vert 1 \rangle - {\textstyle \frac{1}{2N}} \vert 2 \rangle \, , & \qquad C_{1,2}^2 &= {\textstyle \frac{N^2 -2}{4 N}} \vert 2 \rangle + {\textstyle \frac{1}{4N^2}} \vert 1 \rangle \, , & \qquad C_{1,3}^2 &= C_{1,2}^2 \, , & ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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8894f19a710113c43f2d932d7a31d48a6cd50a3b | subsection | 41 | 59 | Color Factors | \end{}For the three-vertex diagrams we have\begin{}{4}
C_{2,11}^2 &= {\textstyle \frac{i N}{4}} \vert 1 \rangle - {\textstyle \frac{i}{4}} \vert 2 \rangle \, , & \qquad \quad C_{2,12}^2 &= C_{2,11}^2 \, , & \qquad \quad C_{2,13}^2 &= {\textstyle \frac{i (N^2-1)}{4}} \vert 2 \rangle \, , & \\
C_{2,14}^2 &= C_{2,13}^2\, ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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ce1643da570b56dad1a13780bead7c2a5dadb251 | subsection | 42 | 59 | Color Factors | At the two-loop level, we find the following color factors
for the loop \widetilde{\mathcal {W}}_1:\widetilde{C}_{1,1}^2 &= {\textstyle \frac{(N^2-1)^2}{4 N^2}} \vert 1 \rangle \, , &
\widetilde{C}_{1,2}^2 &= - {\textstyle \frac{N^2-1}{4 N^2}} \vert 1 \rangle \, , &
\widetilde{C}_{1,5}^2 &= \widetilde{C}_{1,1}^2 \, , &... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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80f7e25c3f95bbfabad7fbc296c7bc5b0fa67a35 | subsection | 43 | 59 | Kinematic Factors | In this appendix, we calculate the relevant integrals for the cross anomalous dimension up to the two-loop level. While powerful methods for the evaluation of integrals of the kind encountered here exist , , , we shall take a pedestrian approach below, taking inspiration from ref. . The results of the integrals calcula... | {
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{
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"raw": "A. I. Davydychev, “Recursive algorithm of evaluating vertex type Feynman integrals”, J. Phys. A25, 5587 (1992).",
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"start": 11... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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1d8a8abcea103cbd53ecbed4f5e575af998f5bf1 | subsection | 44 | 59 | Cusp Anomalous Dimension | We begin by studying the integrals which appear also in the calculation of the cusp anomalous dimension. Explicit results for these were derived in ref. , partly based on the results of ref. .
We present an independent, elementary derivation below. | {
"cite_spans": [
{
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"doi": "10.1007/jhep06(2011)131",
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... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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cac5b4e8c115a2938e567d4800134d8c0e5694be | subsection | 45 | 59 | Three-vertex Diagrams. | \mbox{}We compute the function F_0 ( \phi ), which encodes the contribution of the three-vertex diagram to the cusp anomalous dimension. Starting from equation (REF ), we haveF ( \phi ) = - \frac{i \epsilon }{g^6 u^{6 \epsilon } } \,
\int \limits _{-L} ^0 {\mathrm {d}}\tau _2 \,
\int \limits _0 ^L {\mathrm {d}}\tau _3 ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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0b0ca863c9d66631f273a5516efc394b92ce5835 | subsection | 46 | 59 | Three-vertex Diagrams. | The integral can be further facilitated by substitutingr = \frac{\bar{z}}{z} \, , \qquad u = \frac{\beta }{\gamma } \, , \qquad v = \frac{\bar{\beta }}{\bar{\gamma }} \, ,to reach the formF_0 ( \phi ) & = - \frac{i }{( 4 \pi ) ^4} \,
\int \limits _0 ^{\infty } {\mathrm {d}}r
\int \limits _0 ^{1} {\mathrm {d}}u
\int \li... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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99b48e9f9654f4c78e3d55cc6be06af62e6e4504 | subsection | 47 | 59 | Three-vertex Diagrams. | Noting that\partial _\phi \left( \frac{1}{r + e^{i \phi }} - \frac{1}{r + e^{-i \phi }} \right) =
i \, \partial _r \left( \frac{e^{i\phi }}{r + e^{i \phi }} + \frac{e^{-i\phi }}{r + e^{-i \phi }} \right) ,and after integrating by parts, one finds (after some cancellations)f_0^\prime (\phi ) & = 4 \cos \phi \, \int \lim... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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148e3558483b99234ca8bc8069658dc4d94d480e | subsection | 48 | 59 | Crossed Propagators Diagram. | \mbox{}The ladder-like contribution to the cusp anomalous dimension is encoded in the diagram F^2_4, for which we note the expressionF^2_4 &= \frac{g^4}{16 \pi ^4} \left( \pi \mu ^2 L^2 \right)^{2 \epsilon }
\xi ^2
\int \limits _0 ^1
\frac{{\mathrm {d}}^4 \tau \,\theta ( \tau _4 - \tau _3 ) \, \theta ( \tau _1 - \tau _... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0419d803a6d00414d05a2931d0419a09a2bed5c7 | subsection | 49 | 59 | Crossed Propagators Diagram. | The result for \phi \ge \pi /2 can be deduced from analytic continuation. In the above expression, we substitutee ^{2 i \psi } = \frac{z + \bar{z}e^{i \phi }}{z + \bar{z}e^{-i \phi }}to reach the formK_{0} (\phi ) &= \frac{1}{2^6 \pi ^4 \sin ^2 \phi }
\int \limits _0 ^\phi {\mathrm {d}}\psi \,
\psi ( \phi - \psi ) \lef... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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ebac217af747a2f07338bafdd8cf5625177c6709 | subsection | 50 | 59 | Crossed Propagators Diagram. | By decomposing into partial fractions, one arrives atk_0^\prime ( \phi ) = \phi \, \mathrm {Re} \left[
\ln \big ( 1 + e^{i\phi } \big ) + \ln \big ( 1 - e^{i\phi } \big ) \right]
+ \mathrm {Im} \left[
\mathrm {Li}_2 \big ( e^{i \phi } \big ) +
\mathrm {Li}_2 \big ( - e^{i \phi } \big ) \right] .We may integrate the abo... | {
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... | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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527fe8872e33f6dbb9bd449a2e709da3705bc196 | subsection | 51 | 59 | Commutator Term. | \mbox{}The commutator term appearing in the cross anomalous dimension at the two-loop level involves the finite contributions to the diagrams F^1 _1 and F^1 _2. We recall the expressionsF^1 _2 &= -g^2 \frac{\Gamma (1-\epsilon )}{4 \pi ^2}
\left( \pi \mu ^2 L^2 \right) ^\epsilon \int \limits _0 ^1 {\mathrm {d}}\tau _1 {... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0222390a7712b930a34ee94a14cc3e4f53be31cc | subsection | 52 | 59 | Commutator Term. | \end{array}\right.}Then we haveI(\phi ) &= - \frac{(\pi L^2)^\epsilon \, \epsilon \, \Gamma ( 1 - \epsilon ) }{4 \pi ^2}
\int \limits _0 ^1 {\mathrm {d}}z
\frac{1}{\left( z^2 + \bar{z}^2 + 2 z \bar{z}\cos \phi \right)^{1- \epsilon } }
\int \limits _0 ^{f(z)} {\mathrm {d}}\kappa \, \kappa ^{-1+2 \epsilon } \\
&= - \frac... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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3286adbfb6741165a2c33ab1c4458194a11b3c47 | subsection | 53 | 59 | Commutator Term. | Recalling also our result
(REF ),I_0 (\phi ) = - \frac{\phi }{8 \pi ^2 \sin \phi } \, ,we obtainI_{1} (\phi ) &=
\frac{-1}{4 \pi ^2 \sin \phi } \left[
\phi \ln (2 \sin ( \phi /2 ) \sin \phi )
+ \mathrm {Im}\big (
2 \, \mathrm {Li}_2 \big ( e^{i \phi } \big )
+ 2 \, \mathrm {Li}_2 \big ( {\textstyle \frac{1}{2}}\big ( 1... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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2b9f9faff806729fa1b8d1b1b8f5c99ff9b9b07e | subsection | 54 | 59 | Two-gluon Exchange Web. | \mbox{}We calculate the difference between the kinematic factors F^2 _7 and F^2_8, which corresponds to the two-gluon exchange web. For F^2_7, we have the expressionF^2 _7 &= \frac{g^4 \, \Gamma (1-\epsilon )^2}{16 \pi ^4} \left( \pi \mu ^2 L^2\right)^{2 \epsilon } \xi ^2
\int \limits _0 ^1
\frac{ {\mathrm {d}}^4 \tau ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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2bd507b7c9f6350ef02d4831f693409fffd67b9f | subsection | 55 | 59 | Two-gluon Exchange Web. | In determining the upper boundary for \kappa , we need to discriminate between different cases. In the case \tau _4 > \tau _1 for example, we have x > y/\bar{y}, such that the pole term of the x-Integration disappears. In this case, the upper boundary for \kappa is irrelevant for the single pole term and we may set it ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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a2b79b25d49b305a06c460ac2d7c0c6547c5c54a | subsection | 56 | 59 | Two-gluon Exchange Web. | In the expressionF^2 _8
&= \frac{g^4 \,\Gamma (1-\epsilon )^2}{16 \pi ^4} \left( \pi \mu ^2 L^2\right)^{2 \epsilon } \xi ^2
\int \limits _0 ^1
\frac{ {\mathrm {d}}^4 \tau \, \theta ( \tau _3 - \tau _2 ) }{\left[ \left(\tau _1^2 + \tau _2^2 - 2 \tau _1 \tau _2 \cos \phi \right)
\left(\tau _3^2 + \tau _4^2 + 2 \tau _3 \t... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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75308d4c98a8f71432d68db5b893f26a2dcf86ee | subsection | 57 | 59 | Two-gluon Exchange Web. | After plugging in the boundary values (REF ), we are left withL_0 (\phi ) &= \frac{1}{32 \pi ^4}
\int \limits _0 ^{1/2} {\mathrm {d}}z
\int \limits _{z/ \bar{z}} ^1 {\mathrm {d}}y \, \ln \Big ( y \frac{\bar{z}}{z} \Big )
\left( \frac{ z \bar{z}}{R(\bar{y},z) }
- \frac{z \bar{z}}{R(y,z) } \right) .Now, we substituter = ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
"hep-th",
"hep-ph"
] | 2,018 | en | Physics | [
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0.0... |
965591f03c18e88cbf8c95ad4bca0c921d052281 | subsection | 58 | 59 | Two-gluon Exchange Web. | \int \limits _1 ^r
\frac{{\mathrm {d}}x}{(1+r)^4}
\left( \frac{\ln (x) }{R\big ( {\textstyle \frac{r-x}{r}} , {\textstyle \frac{1}{1+ r}} \big ) }
- \frac{ \ln (x) }{R \big ( {\textstyle \frac{x}{r}} , {\textstyle \frac{1}{1+ r}} \big ) } \right)
= \frac{1}{32 \pi ^4} \left( l_0 (\pi - \phi ) - l_0 (\phi ) \right) .Her... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)162 | 1805.06448 | The Cross Anomalous Dimension in Maximally Supersymmetric Yang--Mills
Theory | [
"Hagen Münkler"
] | [
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99352204747c0913051a5464db505b7da117d1a2 | abstract | 0 | 399 | Abstract | Let $\rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of
$\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ with Hodge-Tate weights
$(0,1,2)$ (up to shift) and such that $N^2\ne 0$ on $D_{\mathrm{st}}(\rho_p)$.
When $\rho_p$ comes from an automorphic representation $\pi$ of
$G(\mathbb{A}_{F^+})$ (for ... | {
"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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d8ee6bde421848f9c9f4e3e49c46f216585166ac | subsection | 1 | 399 | Introduction and notation | Let p be a prime number, n\ge 2 an integer, F^+ a totally real number field and F a totally imaginary quadratic extension of F^+ such that all places of F^+ dividing p split in F. We fix a unitary algebraic group G over F^+ which becomes \operatorname{\mathrm {G}L}_n over F and such that G(F^+\otimes _{\mathbb {Q}}\mat... | {
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local-global compatibility | [
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6cb7ab3150add17761858819bad5fd824c76103f | subsection | 2 | 399 | Introduction and notation | However, it is (quite reasonably) hoped that they determine the local Galois representation \rho _{\widetilde{\wp }}:=\rho \vert _{\operatorname{\mathrm {G}al}(\overline{F}_{\widetilde{\wp }}/F_{\widetilde{\wp }})} and (may-be less reasonably) hoped that they also only depend on \rho _{\widetilde{\wp }}. Note that the ... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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e31066313f9b6048d4038d0dd85dbd38c9a52b18 | subsection | 3 | 399 | Introduction and notation | Assume moreover that:\overline{\rho } is absolutely irreducible
\widehat{S}(U^{\wp }, W^{\wp })[\mathfrak {m}_{\rho }]^{\operatorname{\mathrm {l}alg}}\ne 0
\rho _{\widetilde{\wp }} is semi-stable with consecutive Hodge-Tate weights and N^2\ne 0 on D_{\operatorname{\mathrm {s}t}}(\rho _{\widetilde{\wp }})
any dimensi... | {
"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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c45d777d6ade4772d47d185dd5c02ff8631cb3e4 | subsection | 4 | 399 | Introduction and notation | \end{gathered}where the C_{i,j}, \widetilde{C}_{i,j} are certain explicit irreducible subquotients of locally analytic principal series of \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p) (see § REF or ), where \operatorname{\mathrm {S}t}_3^\infty = \operatorname{\mathrm {s}oc}_{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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ab27cd1edec162e9065a64bdc02e258e2fa6e49c | subsection | 5 | 399 | Introduction and notation | Hence \overline{\rho }_{\widetilde{\wp }} is up to twist isomorphic to \Big (\begin{} \overline{\varepsilon }^2 & * & * \\ 0 &\overline{\varepsilon }&*\\ 0 & 0&1\end{}\Big ), and the fourth assumption means that we require the two * above the diagonal in \overline{\rho }_{\widetilde{\wp }} to be nonzero, a kind of assu... | {
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"start... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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90e1f9f0ac78cc0858cde14ea8c5602a8450757e | subsection | 6 | 399 | Introduction and notation | For instance one could push a little bit further the methods of this paper to prove that \widehat{S}(U^{\wp }, W^{\wp })[\mathfrak {m}_{\rho }]^{\operatorname{\mathrm {a}n}} as in Theorem REF in fact contains (copies of) a representation of the form \widetilde{\Pi }\otimes \chi \!\circ \!\operatorname{\mathrm {d}et} wi... | {
"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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45015a3be470adec7cdda3d836ba015c10af608a | subsection | 7 | 399 | Introduction and notation | In (REF ), we denote by \operatorname{\mathrm {S}t}_3^{\operatorname{\mathrm {a}n}}, resp. v_{\overline{P}_i}^{\operatorname{\mathrm {a}n}}, the locally analytic Steinberg, resp. generalized Steinberg, and by (\operatorname{\mathrm {I}nd}_{\overline{B}(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)} \cdo... | {
"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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63ab19a88e29234811c1de74363e414caf4481b2 | subsection | 8 | 399 | Introduction and notation | In § REF , we show that one can associate to \rho _{\widetilde{\wp }}, assumed semi-stable with N^2\ne 0 on D_{\operatorname{\mathrm {s}t}}(\rho _{\widetilde{\wp }}) and sufficiently generic (we explain this below, any \rho _{\widetilde{\wp }} as in Theorem REF is sufficiently generic), a locally analytic representatio... | {
"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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f39b7dbcb2d009de556f4bd21d93916b708e2b5e | subsection | 9 | 399 | Introduction and notation | We conjecture the following statement.Conjecture 1.2 (Conjecture REF )
Assume n=3, F_{\widetilde{\wp }}=\mathbb {Q}_p and:\rho absolutely irreducible
\widehat{S}(U^{\wp }, W^{\wp })[\mathfrak {m}_{\rho }]^{\operatorname{\mathrm {l}alg}}\ne 0
\rho _{\widetilde{\wp }} semi-stable with N^2\ne 0 on D_{\operatorname{\mat... | {
"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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7e9235618d49cc4aaa4fcd586c65ac461ee7d2d3 | subsection | 10 | 399 | Introduction and notation | Twisting D_{\operatorname{\mathrm {r}ig}}(\rho _{\widetilde{\wp }}) if necessary (and twisting \Pi (\rho _{\widetilde{\wp }}) accordingly), we can assume \delta _1=x^{k_1}, \delta _2=x^{k_2}\varepsilon ^{-1} and \delta _3=x^{k_3}\varepsilon ^{-2} (note that D is not étale anymore if k_1\ne 0, but this won't be a proble... | {
"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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6eab9784b57f5aa12601cb24ae899dcb5b68b06b | subsection | 11 | 399 | Introduction and notation | Then the representations:{}[32]
(0,10)[a]{\operatorname{\mathrm {S}t}_3^{\infty }(\lambda )}
(14,10)[b]{C_{1,1}}
(26,4)[c]{v_{\overline{P}_1}^\infty (\lambda )}
(26,16)[d]{\widetilde{C}_{1,2}}
(38, 10)[e]{C_{1,3},}
{a}{b}{}[+1,]
{b}{c}{}[+1,]
{b}{d}{}[+1,]
{c}{e}{}[+1,]
{d}{e}{}[+1, \dashline ]
\ \ \ \ \ \ \ \ \ {}[32]... | {
"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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31653f13ee08b1823cb82e2e3a130405b61bfe2c | subsection | 12 | 399 | Introduction and notation | We consider the two following representations (see § REF where they are denoted \Pi ^1(\lambda ,\psi )^+ and \Pi ^2(\lambda ,\psi )^+):\ \ \ \ \ \ \ \ \ \ {}[32]
(0,10)[a]{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Pi ^{1}\ \ :=\ \ \operatorname{\mathrm {S}t}_3^{\infty }(\lambda )}
(12,2)[b]{C_{2,1}}
(12,18)[c]{C... | {
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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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ae9284e44c9b0ee22051f2ac6607660b9750c2ac | subsection | 13 | 399 | Introduction and notation | We prove in Lemma REF , Proposition REF and Proposition REF that such isomorphisms are true under mild genericity assumptions on the (\varphi ,\Gamma )-modules D_1^2 and D_2^3. Note that we couldn't find these isomorphisms in the literature (though we suspect they might be known), so we provided our own proofs, see e.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 | [
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63bb34c6c70353a966e579f4fd96c4e2e45b3786 | subsection | 14 | 399 | Introduction and notation | The (\varphi ,\Gamma )-module D gives an E-line in the left hand side of both (REF ), (), hence its orthogonal space gives a 2-dimensional subspace of \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_2}^{\infty }(\lambda ),\Pi ^1) and a 2-dimensional subspace of \operatorn... | {
"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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ecf0fbbf3d0e19e830003a4701f7a0ccffc0c1cf | subsection | 15 | 399 | Introduction and notation | Then results of show that there is a unique way to add constituents \widetilde{C}_{1,4}, \widetilde{C}_{2,4}, C_{1,5}, C_{2,5} on the right so that the resulting representation \Pi (D)=\Pi (\rho _{\widetilde{\wp }}) contains \Pi (D)^- and has the same form as (REF ) (see (REF )).We now assume k_1=k_2=k_3 and recall tha... | {
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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5aeb67ba2cf7d7c06d6ef631bd4fae245f890af6 | subsection | 16 | 399 | Introduction and notation | We show that \operatorname{\mathrm {O}rd}_{P_i}(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}) is a faithful module over a certain p-adic localized Hecke algebra \widetilde{\mathbb {T}}(U^{\wp })_{\overline{\rho }}^{P_i-\operatorname{\mathrm {o}rd}} (see Lemma REF ) and using the p-adic local Langlands corresponde... | {
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"raw": "Kisin, M. Deformations of G_{\\mathbb {Q}_p} and \\mathrm {GL}_2 (\\mathbb {Q}_p) representations. Astérisque 330 (2010), 511–528.",
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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7ad923cae39db09fb190b4ef80a519d7d40658c5 | subsection | 17 | 399 | Introduction and notation | Let w be a nonzero vector in the subspace D^\perp of \operatorname{\mathrm {E}xt}^1_{{\operatorname{\mathrm {G}L}_3}(\mathbb {Q}_p)}(v_{\overline{P}_{3-i}}^{\infty },\Pi ^i) orthogonal to D under the pairings (REF ), () and denote by \Pi ^w the corresponding extension \Pi ^i-v_{\overline{P}_{3-i}}^{\infty }. It is enou... | {
"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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7c25966d71f6ab02cfec2b31634c66fd914459ee | subsection | 18 | 399 | Introduction and notation | By a variation/generalization of the arguments in the \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)-case, we prove that the restriction induces an isomorphism (see Corollary REF ):\operatorname{\mathrm {H}om}_{L_{P_1}(\mathbb {Q}_p)}\big (\pi _{1,2}\boxtimes 1, (\operatorname{\mathrm {O}rd}_{P_1}(\widehat{S}(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",
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] | 2,018 | en | Mathematics | [
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96c29235c4026793cfff49442844fcd6c2cc82ee | subsection | 19 | 399 | Introduction and notation | The first one (see Theorem REF ) says that any extension D_1^2-D_1^2 which is contained as a (\varphi , \Gamma )-submodule in an extension D-D is sent (after a suitable twist) to an element of D^\perp via \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(D_1^{2},D_1^{2})\twoheadrightarrow \operatorname{\mathrm {E}xt}... | {
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"raw": "Greenberg, R., and Stevens, G. p-adic {L}-functions and p-adic periods of modular forms. Inv. Math. 111 (1993), 407–447.",
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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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] | 2,018 | en | Mathematics | [
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b6260b3cef0baa9c6c6858bb2120c92eeaf350fe | subsection | 20 | 399 | Introduction and notation | This implies that any L_{P_1}(\mathbb {Q}_p)-equivariant morphism \pi _{1,2}\boxtimes 1\rightarrow (\operatorname{\mathrm {O}rd}_{P_1}(\widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_{\rho }]))^{\operatorname{\mathrm {a}n}} extends to an E[\epsilon ]/\epsilon ^2-linear and L_{P_1}(\mathbb {Q}_p)-equiva... | {
"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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25504ba25470af97be9063d716b4b1ca1f186b22 | subsection | 21 | 399 | Introduction and notation | By the adjunction formula for \operatorname{\mathrm {O}rd}_{P_1}, we obtain a \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)-equivariant morphism:\big (\operatorname{\mathrm {I}nd}_{\overline{P}_1(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)}\widetilde{\pi }_{1,2}\boxtimes _{E[\epsilon ]/\epsilon ^2} \wi... | {
"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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9e24444425bc928898bfa456f2674c712ceeae77 | subsection | 22 | 399 | Introduction and notation | For instance one can ask for a more explicit (local) construction of the \operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)-representation \Pi (\rho _{\wp }), and in particular try to relate the two “branches” in (REF ) to the filtered (\varphi ,N)-module of \rho _{\wp } along the lines of . Though there is so far no constru... | {
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"doi": "",
"end": 286,
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"raw": "Breuil, C. \\mathrm {Ext}^1 localement analytique et compatibilité local-global. Amer. J. of Math.. to appear.",
"source_ref_id": "bfbc131d82cb2292dbd3029d0f1824eebfb6d241",
"start": 0
... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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37eae2b7b98cadaf855260e80f3e41eab663e85f | subsection | 23 | 399 | Introduction and notation | Given an E-bilinear map V \times W\xrightarrow{} E, for W^{\prime }\subseteq W we denote:(W^{\prime })^{\perp }:=\lbrace v\in V,\ v\cup w=0 \ \forall \ w\in W^{\prime }\rbrace .For L a finite extension of \mathbb {Q}_p, we let \Sigma _L be the set of embeddings of L into E (equivalently into \overline{\mathbb {Q}_p} by... | {
"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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39d3ecd3c9a99f6aa485d7fc5091e7b56e1cfec7 | subsection | 24 | 399 | Introduction and notation | We denote by \operatorname{\mathrm {E}xt}^i_{(\varphi ,\Gamma _L)}(\cdot ,\cdot ) the extensions groups in the category of (\varphi ,\Gamma _L)-modules over \mathcal {R}_{E,L} and by H^i_{(\varphi ,\Gamma _L)}(\cdot ):=\operatorname{\mathrm {E}xt}^i_{(\varphi ,\Gamma _L)}(\mathcal {R}_{E,L},\cdot ) (, , ). If \delta :L... | {
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"doi": "10.24033/ast.782",
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"raw": "Bellaïche, J., and Chenevier, G. Families of Galois representations and Selmer groups. Astérisque 324 (2009).",
"source_ref_id": "dd425b4ffeec179... | 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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23d62853f128b42576a5897a60663538e7ae5816 | subsection | 25 | 399 | Introduction and notation | Finally, if L=\mathbb {Q}_p, we denote by \operatorname{\mathrm {w}t}(\delta )\in E the Sen weight of \delta , for instance \operatorname{\mathrm {w}t}(x^k\operatorname{\mathrm {u}nr}(a))=k for k\in \mathbb {Z} and a\in E^\times .Let G be the L-points of a reductive algebraic group over \mathbb {Q}_p, we refer without ... | {
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"raw": "Schneider, P., and Teitelbaum, J. Locally analytic distributions and p-adic representation theory, with applications to \\mathrm {GL}_2. J. Amer. Math. Soc 15 (2002), 443–468.",
"source_ref_id... | 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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d8c3530a19370dc1f993f8a22a29b8b9b128ab01 | subsection | 26 | 399 | Introduction and notation | We denote by \delta _P the usual (smooth unramified) modulus character of P.If V, W are two locally \mathbb {Q}_p-analytic representations of G over E, we define the extension groups \operatorname{\mathrm {E}xt}^i_G(V,W) as in , that is, as the extension groups \operatorname{\mathrm {E}xt}^i_{D(G,E)}(W^\vee ,V^\vee ) o... | {
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"doi": "",
"end": 457,
"openalex_id": "",
"raw": "Schraen, B. Représentations localement analytiques de \\mathrm {GL}_3(\\mathbb {Q}_p). Ann. Sci. Éc. Norm. Sup. 44 (2011), 43–145.",
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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949bcebb108c3de8708212520eb3275f88120cce | subsection | 27 | 399 | Higher | In this section we define and study certain subspaces \mathcal {L}_{\operatorname{\mathrm {F}M}}(D: D_1^{n-1}) and \ell _{\operatorname{\mathrm {F}M}}(D: D_1^{n-1}) of some Ext{}^1 groups in the category of (\varphi ,\Gamma _L)-modules that will be used in the next sections.We fix a finite extension L of \mathbb {Q}_p ... | {
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"doi": "10.1007/s00222-016-0708-y",
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"source_ref_... | 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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6ed0364a356a5758a4de1c7bac48b8e108c30adb | subsection | 28 | 399 | Higher | We consider the following cup-product:\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _n), D_1^{n-1}) \times \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_1^{n-1}, D_1^{n-1}) {\cup } \operatorname{\mathrm {E}xt}^2_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _n), D_1^{n-1}).Lemma 2... | {
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"doi": "",
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"raw": "Nakamura, K. Classification of two-dimensional split trianguline representations of p-adic fields. Compositio Math. 145 (2009), 865–914.",
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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92e68704ef11fe64e73a69e47ad644f7404cf30a | subsection | 29 | 399 | Higher | \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_1^{n-1},D_1^{n-1}) @> \cup >> \operatorname{\mathrm {E}xt}^2_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _n), D_1^{n-1}) \\
@| @. @V \kappa VV @V \sim VV \\
\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _n), D_1^{n-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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15dfe138b14c4202a51631f8c218c290c4206bad | subsection | 30 | 399 | Higher | We have a natural isomorphism:\operatorname{\mathrm {E}xt}^2_{(\varphi ,\Gamma _L)}(D_1^{n-1}, \mathcal {R}_E(\delta _i))\cong H^2_{(\varphi ,\Gamma _L)}\big ((D_1^{n-1})^{\vee }\otimes _{\mathcal {R}_E} \mathcal {R}_E(\delta _i)\big ).Together with (see also ), we are thus reduced to show H^0_{(\varphi ,\Gamma _L)}\bi... | {
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"raw": "Liu, R. Cohomology and duality for (\\varphi ,\\Gamma )-modules over the Robba ring. Int. Math. Res. Not. 3 (2008).",
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local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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bf5ef7ed604836d4dd4efe42718559c6afb38243 | subsection | 31 | 399 | Higher | We have a commutative diagram:\footnotesize
\begin{} H^1_{(\varphi ,\Gamma _L)}((D_1^{n-1})^{\vee } \otimes _{\mathcal {R}_E} \mathcal {R}_E(\delta _{n-1}))@. \ \times @. H^1_{(\varphi ,\Gamma _L)}(D_1^{n-1}\otimes _{\mathcal {R}_E} \mathcal {R}_E(\delta _n^{-1}))@> \cup >> H^2_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\d... | {
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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": "bdb8c59ae339878314705f21... | 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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31fdd2dcfe59543ed6530238a5d90e3a990eee1d | subsection | 32 | 399 | Higher | By (REF ), we deduce:\operatorname{\mathrm {K}er}(\kappa )\subseteq \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _n), D_1^{n-1}) ^{\perp }.However, since the bottom cup-product of (REF ) is a perfect pairing and the bottom right map an isomorphism, we easily get \operatorname{\mathrm {E}x... | {
"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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4b61084c80a7ba524bb3471df1e6e313b0ef19b1 | subsection | 33 | 399 | Higher | In particular the E-vector subspace E[D] it generates is well defined and we define (with respect to the two bottom pairings in (REF )):\mathcal {L}_{\operatorname{\mathrm {F}M}}(D: D_1^{n-1})&:=&(E [D])^{\perp } \subseteq \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_1^{n-1},D_1^{n-1})\\
\ell _{\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",
"math.RT"
] | 2,018 | en | Mathematics | [
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63e4c25856bb32e620ecbb835063b9cab7ba3a16 | subsection | 34 | 399 | Higher | \ell _{\operatorname{\mathrm {F}M}}(D: D_1^{n-1})) is of codimension 1 in \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_1^{n-1},D_1^{n-1}) (resp. in \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_1^{n-1}, \mathcal {R}_E(\delta _{n-1}))).By functoriality we have a commutative diagram for i<n-1:\footno... | {
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"doi": "",
"end": 1482,
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"raw": "Liu, R. Cohomology and duality for (\\varphi ,\\Gamma )-modules over the Robba ring. Int. Math. Res. Not. 3 (2008).",
"source_ref_id": "17287f3b42ab85ed37656b42d38e9a77951046b8",
"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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] | 2,018 | en | Mathematics | [
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837556a8bdc2c9dc5ac2e685b5ef53a6abde1998 | subsection | 35 | 399 | Higher | By dévissage, we deduce that u_i is surjective, j_i is injective and \operatorname{\mathrm {K}er}(u_i)\cong \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _n), D_1^i). Also the two cup-products in (REF ) are perfect pairings by Proposition REF . In particular we obtain the following lemma.L... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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46b0ef5896ca77eebe4b1be74153328c7790df89 | subsection | 36 | 399 | Higher | By , the pairing (REF ) induces an equality of subspaces of \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _{n-1}), \mathcal {R}_E(\delta _{n-1})):\operatorname{\mathrm {E}xt}^1_{g}(\mathcal {R}_E(\delta _{n-1}), \mathcal {R}_E(\delta _{n-1})) \cong \operatorname{\mathrm {E}xt}^1_e(\mathcal... | {
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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"
] | [
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9b9b00c57e2fec6927e607f2f055f39733dcf8f1 | subsection | 37 | 399 | Higher | \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_2^{n},\mathcal {R}_E(\delta _1)) @> \cup >> \operatorname{\mathrm {E}xt}^2_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _2), \mathcal {R}_E(\delta _1))
\end{}where the right vertical map is an isomorphism of 1-dimensional E-vector spaces, the bottom cup-product is... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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] | 2,018 | en | Mathematics | [
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a22ec0f7e19888f956324990e7204f15793f778c | subsection | 38 | 399 | Higher | \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(D_2^{n}, \mathcal {R}_E(\delta _{1})) @> \cup >> \operatorname{\mathrm {E}xt}^2_{(\varphi ,\Gamma _L)}(\mathcal {R}_E(\delta _2), \mathcal {R}_E(\delta _{1})) \\
@A j_i AA @. @V u_i VV @| \\
\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma _L)}(\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"
] | [
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] | 2,018 | en | Mathematics | [
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a40c8abb75aadf528a427f3507c2f387a7447790 | subsection | 39 | 399 | Higher | \delta _1) over E[\epsilon ]/\epsilon ^2 such that \widetilde{D} sits in an exact sequence of (\varphi ,\Gamma _L)-modules over \mathcal {R}_{E[\epsilon ]/\epsilon ^2}:0 \longrightarrow \widetilde{D}_1^{n-1} \longrightarrow \widetilde{D} \longrightarrow \mathcal {R}_{E[\epsilon ]/\epsilon ^2}(\widetilde{\delta }_n) \lo... | {
"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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82c6fd59d549105e7ff27b727e7216157fe00a6b | subsection | 40 | 399 | Higher | Now consider the exact sequence 0 \longrightarrow D_1^{n-1}\longrightarrow \widetilde{D}_1^{n-1} \longrightarrow D_1^{n-1}\longrightarrow 0, taking cohomology, we get a long exact sequence:0\longrightarrow H^1_{(\varphi ,\Gamma _L)}(D_1^{n-1}) \longrightarrow H^1_{(\varphi ,\Gamma _L)}(\widetilde{D}_1^{n-1}) {\operator... | {
"cite_spans": [
{
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"doi": "",
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"raw": "Greenberg, R., and Stevens, G. p-adic {L}-functions and p-adic periods of modular forms. Inv. Math. 111 (1993), 407–447.",
"source_ref_id": "8d425eb7c0a4b7f674284390b7050107fb771487",
"... | 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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9c9940eb027af8a424f13bbbf6dd39670d7f2c6e | subsection | 41 | 399 | Higher | We have \operatorname{\mathrm {H}om}_{\infty }(L^{\times }, E) \subset \operatorname{\mathrm {H}om}_{\sigma }(L^{\times }, E) and \dim _E \operatorname{\mathrm {H}om}_{\sigma }(L^{\times }, E)=2. Let \log _p: L^{\times } \rightarrow L be the unique character which restricts to the p-adic logarithm on \mathcal {O}_L^{\t... | {
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{
"arxiv_id": "",
"doi": "",
"end": 1680,
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"raw": "Zhang, Y. \\mathcal {L}-invariants and logarithm derivatives of eigenvalues of Frobenius. Science China Math. 57 (2014), 1587–1604.",
"source_ref_id": "a7b82e6be47eeed7beb2e7014801d2185bce812... | 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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3d6da10c95c7ceaf20d52b97685123238e45ceee | subsection | 42 | 399 | Higher | A natural question in the p-adic Langlands program is to understand their counterpart on the automorphic side, e.g. in the setting of locally \mathbb {Q}_p-analytic representations of \operatorname{\mathrm {G}L}_n(L). The above results suggest that such invariants might be found in deformations of certain representatio... | {
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"doi": "",
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"raw": "Breuil, C. \\mathrm {Ext}^1 localement analytique et compatibilité local-global. Amer. J. of Math.. to appear.",
"source_ref_id": "bfbc131d82cb2292dbd3029d0f1824eebfb6d241",
"start": 391... | 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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6c4d2401e45546eae7871e74a1bb70746e851f83 | subsection | 43 | 399 | Higher | Together with Lemma REF , the lemma then follows by the same argument as in the proof of Lemma REF .We denote by V_x the tangent space of (\operatorname{\mathrm {S}pf}\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}} at x and by \overline{d}\omega _x 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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6fd840d6458427b0f3cf493386f64c69775741de | subsection | 44 | 399 | Higher | The following lemma is analogous to Lemma REF .Lemma 7.50
The morphism \overline{d}\omega _x is bijective.Since \dim _E V_x\ge n+1 by Proposition REF and the right hand side of (REF ) has dimension (n-2)+(5-2)=n+1 by Lemma REF and Lemma REF (3), it is enough to prove that \overline{d}\omega _x is injective.(a) Let v\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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1bfc526c092c7e4ad93fdc63094c9a3d03a066e4 | subsection | 45 | 399 | Higher | \rho _{x_i}\otimes \varepsilon ^{s_i}\Big )\otimes _E (\widehat{\pi }(\rho _{x_r})\otimes \varepsilon ^{r-1}\circ \operatorname{\mathrm {d}et})\widetilde{\pi }:=\big (\pi ^{\otimes }_P(U^{\wp }) \otimes _{\widetilde{\mathbb {T}}(U^{\wp })^{P-\operatorname{\mathrm {o}rd}}_{\overline{\rho }}} \widetilde{\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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d471b3259343c4a670eb6536b8eb168801c21b24 | subsection | 46 | 399 | Higher | It is enough to prove that \iota induces \pi \sim \over \rightarrow \widetilde{\pi }[\epsilon ] (since then we have a short exact sequence 0\rightarrow \pi \iota \over \rightarrow \widetilde{\pi } \rightarrow \epsilon \widetilde{\pi }\rightarrow 0 and we use that \widetilde{\pi } is an extension of \pi by \pi ). From w... | {
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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",
"math.RT"
] | 2,018 | en | Mathematics | [
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d97946ecc2633efbc10b1ea907c347b27b975032 | subsection | 47 | 399 | Higher | In particular we have a commutative diagram:\begin{}
0 @>>> \pi ^{\operatorname{\mathrm {l}alg}} @>\iota >> \widetilde{\pi }^{\operatorname{\mathrm {l}alg}} @>>> \pi ^{\operatorname{\mathrm {l}alg}} @>>> 0 \\
@. @VVV @VVV @VVV @. \\
0 @>>> \pi @>\iota >> \widetilde{\pi } @>>> \pi @>>> 0
\end{}where the vertical maps ar... | {
"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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3c80b4fe12141c9e5c878a39a01124bd70244362 | subsection | 48 | 399 | Higher | The lemma follows.We consider the E-linear injection \xi : \operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E) \hookrightarrow \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\rho _r^{r+1}, \rho _r^{r+1}), \ \psi \mapsto \rho _r^{r+1}\otimes _E (1+\psi \epsilon ) and set d\omega _{r,x... | {
"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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0f612c471404b4be34bb6744115c8d358ba6d2d9 | subsection | 49 | 399 | Higher | The following result is somewhat analogous to Proposition REF (see § for \mathcal {L}_{\operatorname{\mathrm {F}M}}(\cdot ) and \ell _{\operatorname{\mathrm {F}M}}(\cdot )).Proposition 7.51
(1) Inside \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(D_{r}^{r+1}\!, D_{r}^{r+1})\!\cong \!\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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... | |
f8cc99aafc6151ca4d3f7acaa862b65625bc9011 | subsection | 50 | 399 | Higher | \operatorname{\mathrm {I}m}(d\omega _{r,x}^-) \twoheadrightarrow \ell _{\operatorname{\mathrm {F}M}}(D_{r-1}^{r+1}: D_{r}^{r+1})\subseteq \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E(\chi _{r}),D_{r}^{r+1})).(1) From (REF ) we have \omega ^{\prime }: (\operatorname{\mathrm {S}pf}R_{\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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c25517a7f28bafded79d877bcdc6cf9b5b77a3f2 | subsection | 51 | 399 | Higher | By (1) and (REF ), we have an exact sequence (see (REF ) for the morphism \kappa ):0 \longrightarrow \operatorname{\mathrm {I}m}(d\omega _{r,x}^+)\cap \operatorname{\mathrm {K}er}(\kappa ) \longrightarrow \operatorname{\mathrm {I}m}(d\omega _{r,x}^+) \longrightarrow \ell _{\operatorname{\mathrm {F}M}}(D_{r}^{r+2}: D_{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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... | |
cdaf1b368bd299099d69734ebcbaa391ff50c14e | subsection | 52 | 399 | Higher | If \dim _E \operatorname{\mathrm {I}m}(d\omega _{r,x}^+)\cap \operatorname{\mathrm {K}er}(\kappa )=2, we have \operatorname{\mathrm {K}er}(\kappa )\subseteq \operatorname{\mathrm {I}m}(d\omega _{r,x}^+) since \dim _E \operatorname{\mathrm {K}er}(\kappa )=2, and thus \operatorname{\mathrm {I}m}(d\omega _{r,x}^+)\cap \op... | {
"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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366dde4e35f3ca26463a834d64f672b771f68c65 | subsection | 53 | 399 | Body | In this section we use the subspaces \mathcal {L}_{\operatorname{\mathrm {F}M}}(D: D_1^{n-1}) and \ell _{\operatorname{\mathrm {F}M}}(D: D_1^{n-1}) defined in § to associate to a given 3-dimensional semi-stable noncrystalline representation of \operatorname{\mathrm {G}al}_{\mathbb {Q}_p} with distinct Hodge-Tate weigh... | {
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"raw": "Breuil, C. \\mathrm {Ext}^1 localement analytique et compatibilité local-global. Amer. J. of Math.. to appear.",
"source_ref_id": "bfbc131d82cb2292dbd3029d0f1824eebfb6d241",
"start": 0
... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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] | 2,018 | en | Mathematics | [
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825aacfdcdc5435f393b6378cb1fc348b1168f7d | subsection | 54 | 399 | Body | For k\in \mathbb {Z}_{>0} and 0\ne \psi \in \operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E), we denote by D(k, \psi )\in \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}(\mathcal {R}_E, \mathcal {R}_E(|\cdot |x^k)) the unique (nonsplit) extension up to isomorphism such that:(ED(k, \psi ))^{\perp }=E \psi \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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94c5702d15fba5f2da4afa24c0347174860aa3f3 | subsection | 55 | 399 | Body | For \alpha \in E^{\times }, we set:D(\alpha , \lambda , \psi ):=D(\lambda , \psi )\otimes _{\mathcal {R}_E} \mathcal {R}_E(\operatorname{\mathrm {u}nr}(\alpha )).We also make the following hypotheses.Hypothesis 3.26
(1) There exists a natural isomorphism:\operatorname{\mathrm {p}LL}: \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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0.0... | |
221e049a5afc87b0659ec060683b3c7fd3ebc8cd | subsection | 56 | 399 | Body | The isomorphism (REF ) should also induce a bijection:\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma ), Z}(D(p,\lambda , \psi ), D(p,\lambda , \psi )) {\sim } \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p),Z}(\pi (\lambda ^{\flat }, \psi ), \pi (\lambda ^{\flat }, \psi )),but we won't ne... | {
"cite_spans": [
{
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"doi": "",
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"raw": "Berger, L. Équations différentielles p-adiques et modules filtrés. Astérisque 319 (2008), 13–38.",
"source_ref_id": "f0f49659872a231b8127b7a3c638d3b5ac221bea",
"start": 1489
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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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... | |
7e791e77ea8e116bf8c64010e071266f4c9d290c | subsection | 57 | 399 | Body | D(\alpha , \lambda , \psi )\cong D_{\operatorname{\mathrm {r}ig}}(\rho ) for a 2-dimensional continuous representation \rho of \operatorname{\mathrm {G}al}_{\mathbb {Q}_p} over E.If \alpha ^{\prime }\in E^{\times } is such that D(\alpha ^{\prime }, \lambda , \psi )\cong D_{\operatorname{\mathrm {r}ig}}(\rho ^{\prime })... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 868,
"openalex_id": "",
"raw": "Colmez, P. Représentations de \\mathrm {GL}_2(\\mathbb {Q}_p) et (\\varphi ,\\Gamma )-modules. Astérisque 330 (2010), 281–509.",
"source_ref_id": "a6e303f134b14f4b71e0382f5d5ffc38f4769f42",
... | 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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fbf32782e7f637a883244b86be9afb6eade5f0d8 | subsection | 58 | 399 | Body | By Corollary REF , Colmez's functor \textbf {V}_{\varepsilon ^{-1}} (see § REF ) induces a surjection:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\widehat{\pi }(\rho ), \widehat{\pi }(\rho )\big ) \twoheadrightarrow \operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D(\alpha... | {
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{
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"doi": "10.1007/s00222-002-0284-1",
"end": 1105,
"openalex_id": "https://openalex.org/W3098607373",
"raw": "Schneider, P., and Teitelbaum, J. Algebras of p-adic distributions and admissible representations. Inv. Math. 153 (2003), 145–196.",
"... | 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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90f86e23ce2517934ba4956b3598c63b43f3e2b1 | subsection | 59 | 399 | Body | Then, using that the universal unitary completion of \widehat{\pi }(\rho )^{\operatorname{\mathrm {a}n}} \cong \pi (p^{-1}\alpha , \lambda ^{\flat }, \psi ) is isomorphic to \widehat{\pi }(\rho ) (by ) together with the universal property of this universal completion and the exactness in , we easily deduce that the abo... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 618,
"openalex_id": "",
"raw": "Colmez, P., and Dospinescu, G. Complétés universels de représentations de \\mathrm {GL}_2(\\mathbb {Q}_p). Algebra Number Theory 8 (2014), 1447–1519.",
"source_ref_id": "b81bb87abca2c2778b5968... | 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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0.02853... | |
f2b4a0d1e072a6245276fd1ef5c7342c411159df | subsection | 60 | 399 | Body | The composition of (REF ) with the inverse of (REF ) gives an isomorphism:\operatorname{\mathrm {E}xt}^1_{(\varphi ,\Gamma )}\big (D(\alpha , \lambda , \psi ), D(\alpha , \lambda , \psi )\big ) {\sim } \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}\big (\widehat{\pi }(\rho )^{\operatornam... | {
"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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0.022326... | |
4662911b40bdf488cf41ca55b04cf3d11cbcd367 | subsection | 61 | 399 | Body | Twisting by \operatorname{\mathrm {u}nr}(p\alpha ^{-1})\circ \operatorname{\mathrm {d}et}, we deduce any element in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\pi (\lambda ^{\flat }, \psi ), \pi (\lambda ^{\flat }, \psi )) is also very strongly admissible, which is the second part of ... | {
"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... | |
1d3e52a74c785c210ee43728334480cce9f938b6 | subsection | 62 | 399 | Body | By Remark REF (see in particular (REF )), the existence of \widetilde{D} confirms the discussion in .We use the previous results for \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p) and the results of § to associate to a 3-dimensional semi-stable representation of \operatorname{\mathrm {G}al}_{\mathbb {Q}_p} with N^2\ne 0... | {
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"doi": "10.1112/s0010437x14007921",
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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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