-invariants, partially de Rham families, and local-global compatibility
Annales de l'Institut Fourier, Volume 67 (2017) no. 4, p. 1457-1519
Let F be a finite extension of p . By considering partially de Rham families, we establish a Colmez–Greenberg–Stevens formula (on Fontaine–Mazur -invariants) for (general) 2-dimensional semi-stable non-crystalline representations of the group Gal( p ¯/F ). As an application, we prove local-global compatibility results for completed cohomology of quaternion Shimura curves, and in particular the equality of Fontaine–Mazur -invariants and Breuil’s -invariants, in critical case.
Soit F une extension finie de p . En étudiant des familles de représentations galoisiennes partiellement de de Rham, on donne une formule de Colmez–Greenberg–Stevens (concernant les invariants de Fontaine–Mazur) pour les représentations semi-stables non cristallines de dimension 2 de Gal( p ¯/F ). Comme application, on montre dans le cas critique des résultats de compatibilité local-global pour le H 1 -complété d’une courbe de Shimura quaternionique, et en particulier l’égalité des invariants de Fontaine–Mazur et Breuil.
Received : 2016-03-18
Revised : 2016-10-04
Accepted : 2016-10-27
Published online : 2017-09-26
DOI : https://doi.org/10.5802/aif.3115
Classification:  11S37,  11S80,  22D12
Keywords: -invariants, partially de Rham families, locally analytic representations, local-global compatibility
@article{AIF_2017__67_4_1457_0,
     author = {Ding, Yiwen},
     title = {$\protect \mathcal{L}$-invariants, partially de Rham families, and local-global compatibility},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {4},
     year = {2017},
     pages = {1457-1519},
     doi = {10.5802/aif.3115},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_4_1457_0}
}
$\protect \mathcal{L}$-invariants, partially de Rham families, and local-global compatibility. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1457-1519. doi : 10.5802/aif.3115. https://aif.centre-mersenne.org/item/AIF_2017__67_4_1457_0/

[1] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent On overconvergent Hilbert modular cusp forms, p-adic arithmetic of Hilbert modular forms, Société Mathématique de France (Astérisque) Tome 382 (2016), pp. 163-193

[2] Bellaıche, Joël; Chenevier, Gaëtan Families of Galois representations and Selmer groups, Société Mathématique de France, Astérisque, Tome 324 (2009), xii+314 pages

[3] Bergdall, John On the variation of (ϕ,Γ)-modules over p-adic families of automorphic forms, Boston University, USA (2013) (Ph. D. Thesis)

[4] Bergdall, John Paraboline variation of p-adic families of (ϕ,Γ)-modules, Compos. Math., Tome 153 (2017), pp. 132-174 | Article

[5] Berger, Laurent Représentations p-adiques et équations différentielles, Invent. Math., Tome 148 (2002) no. 2, pp. 219-284 | Article

[6] Berger, Laurent Construction de (ϕ,Γ)-modules: représentations p-adiques et B-paires, Algebra Number Theory, Tome 2 (2008) no. 1, pp. 91-120 | Article

[7] Berger, Laurent Multivariable (ϕ,Γ)-modules and locally analytic vectors, Duke Math. J., Tome 165 (2016) no. 18, pp. 3567-3595 | Article

[8] Bijakowski, Stéphane Classicité de formes modulaires de Hilbert, p-adic arithmetic of Hilbert modular forms, Société Mathématique de France (Astérisque) Tome 382 (2016), pp. 49-71

[9] Breuil, Christophe Invariant et série spéciale p-adique, Ann. Sci. Éc. Norm. Supér., Tome 37 (2004) no. 4, pp. 559-610 | Article

[10] Breuil, Christophe Conjectures de classicité sur les formes de Hilbert surconvergentes de pente finie (2010) (unpublished note)

[11] Breuil, Christophe The emerging p-adic Langlands programme, Proceedings of the international congress of mathematicians, Vol. II, Hindustan Book Agency (2010), pp. 203-230

[12] Breuil, Christophe Remarks on some locally p -analytic representations of GL 2 (F) in the crystalline case, Non-abelian fundamental groups and Iwasawa theory, Cambridge University Press (London Mathematical Society Lecture Note Series) Tome 393 (2010), pp. 212-238

[13] Breuil, Christophe Série spéciale p-adique et cohomologie étale complétée, Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques, Société Mathématique de France (Astérisque) Tome 331 (2010), pp. 65-115

[14] Breuil, Christophe Vers le socle localement analytique pour GL n , II, Math. Ann., Tome 361 (2015) no. 3-4, pp. 741-785 | Article

[15] Breuil, Christophe Socle localement analytique I, Ann. Inst. Fourier, Tome 66 (2016) no. 2, pp. 633-685 | Article

[16] Breuil, Christophe; Hellmann, Eugen; Schraen, Benjamin Une interprétation modulaire de la variété trianguline (2015) (https://arxiv.org/abs/1411.7260v2, to appear in Math. Ann.)

[17] Buzzard, Kevin Eigenvarieties, L-functions and Galois representations, Cambridge University Press (London Mathematical Society Lecture Note Series) Tome 320 (2007), pp. 59-120

[18] Carayol, Henri Sur les représentations -adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér., Tome 19 (1986) no. 3, pp. 409-468 | Article

[19] Chenevier, Gaëtan Familles p-adiques de formes automorphes pour GL n , J. Reine Angew. Math, Tome 570 (2004), pp. 143-217

[20] Chenevier, Gaëtan Une correspondance de Jacquet-Langlands p-adique, Duke Math. J., Tome 126 (2005) no. 1, pp. 161-194 | Article

[21] Chenevier, Gaëtan On the infinite fern of Galois representations of unitary type, Ann. Sci. Éc. Norm. Supér., Tome 44 (2011) no. 6, pp. 963-1019 | Article

[22] Colmez, Pierre Invariants et dérivées de valeurs propres de Frobenius, Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques, Société Mathématique de France (Astérisque) Tome 331 (2010), pp. 13-28

[23] Ding, Yiwen Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global (2014) (https://sites.google.com/site/yiwendingmath/fmclg.pdf )

[24] Ding, Yiwen -invariants and local-global compatibility for GL 2 /F, Forum Math. Sigma, Tome 4 (2016) (Article ID e13, 49 p., electronic only) | Article

[25] Emerton, Matthew Locally analytic vectors in representations of locally p-adic analytic groups (2004) (https://arxiv.org/abs/math/0405137, to appear in Mem. Am. Math. Soc.)

[26] Emerton, Matthew Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties, Ann. Sci. Éc. Norm. Supér., Tome 39 (2006) no. 5, pp. 775-839 | Article

[27] Emerton, Matthew On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math., Tome 164 (2006) no. 1, pp. 1-84 | Article

[28] Emerton, Matthew Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction (2007) (to appear in J. Inst. Math. Jussieu)

[29] Greenberg, Ralph; Stevens, Glenn p-adic L-functions and p-adic periods of modular forms, Invent. Math., Tome 111 (1993) no. 2, pp. 407-447 | Article

[30] Kedlaya, Kiran S. Slope filtrations for relative Frobenius, Représentation p-adiques de groupes p-adiques I. Représentations galoisiennes et (φ,Γ)-modules, Société Mathématique de France (Astérisque) Tome 319 (2008), pp. 259-301

[31] Kedlaya, Kiran S.; Pottharst, Jonathan; Xiao, Liang Cohomology of arithmetic families of (ϕ,Γ)-modules, J. Am. Math. Soc., Tome 27 (2014) no. 4, pp. 1043-1115 | Article

[32] Liu, Ruochuan Triangulation of refined families, Comment. Math. Helv., Tome 90 (2015) no. 4, pp. 831-904 | Article

[33] Nakamura, Kentaro Classification of two-dimensional split trianguline representations of p-adic fields, Compos. Math., Tome 145 (2009) no. 4, pp. 865-914 | Article

[34] Nakamura, Kentaro Deformations of trianguline B-pairs and Zariski density of two dimensional crystalline representations, J. Math. Sci., Tokyo, Tome 20 (2013) no. 4, pp. 461-568

[35] Newton, James Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence, Math. Ann., Tome 355 (2013) no. 2, pp. 729-763 | Article

[36] Orlik, Sascha; Strauch, Matthias On Jordan–Hölder series of some locally analytic representations, J. Am. Math. Soc., Tome 28 (2015) no. 1, pp. 99-157 | Article

[37] Pottharst, Jonathan The -invariant, the dual -invariant, and families, Ann. Math. Qué., Tome 40 (2016) no. 1, pp. 159-165 | Article

[38] Saito, Takeshi Hilbert modular forms and p-adic Hodge theory, Compos. Math., Tome 145 (2009) no. 5, pp. 1081-1113 | Article

[39] Schneider, Peter; Teitelbaum, Jeremy Banach space representations and Iwasawa theory, Isr. J. Math., Tome 127 (2002) no. 1, pp. 359-380 | Article

[40] Schneider, Peter; Teitelbaum, Jeremy Algebras of p-adic distributions and admissible representations, Invent. Math., Tome 153 (2003) no. 1, pp. 145-196 | Article

[41] Schraen, Benjamin Représentations p-adiques de GL 2 (L) et catégories dérivées, Isr. J. Math., Tome 176 (2010) no. 1, pp. 307-361 | Article

[42] Shah, Shrenik Interpolating periods (2013) (https://arxiv.org/abs/1305.2872 )

[43] Tian, Yichao; Xiao, Liang p-adic cohomology and classicality of overconvergent Hilbert modular forms (to appear in Astérisque)

[44] Zhang, Yuancao -invariants and logarithm derivatives of eigenvalues of Frobenius, Sci. China, Math., Tome 57 (2014) no. 8, pp. 1587-1604 | Article