On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras

Matthew Emerton; Vytautas Paškūnas

doi:10.5802/jep.119

On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras
[Sur la densité des points supercuspidaux de poids régulier fixé dans des anneaux locaux de déformations et des algèbres de Hecke globales]

Matthew Emerton ¹ ; Vytautas Paškūnas ²

¹ University of Chicago, Department of Mathematics 5734 S. University Ave., Chicago, IL 60637, United States
² Universität Duisburg-Essen, Fakultät für Mathematik Thea-Leymann-Str. 9, 45127 Essen, Germany

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 337-371.

Résumé
Abstract

Nous étudions la clôture de Zariski, dans l’espace des déformations d’une représentation galoisienne locale, des points fermés correspondant à des représentations potentiellement semi-stables avec des propriétés de théorie de Hodge $p$ -adique prescrites. Nous montrons, dans les cas favorables, que la clôture contient une réunion de composantes irréductibles de l’espace des déformations. Nous étudions aussi une question analogue pour les algèbres de Hecke globales.

We study the Zariski closure in the deformation space of a local Galois representation of the closed points corresponding to potentially semi-stable representations with prescribed $p$ -adic Hodge-theoretic properties. We show in favourable cases that the closure contains a union of irreducible components of the deformation space. We also study an analogous question for global Hecke algebras.

Reçu le : 2018-11-09
Accepté le : 2020-01-16
Publié le : 2020-03-06

Zbl

DOI : 10.5802/jep.119

Classification : 11S37, 22E50
Keywords: $p$-adic Hodge theory, deformation theory of Galois representations, $p$-adic automorphic forms
Mot clés : Théorie de Hodge $p$-adique, théorie des déformations des représentations galoisiennes, formes automorphes $p$-adiques

Affiliations des auteurs :

Matthew Emerton ¹ ; Vytautas Paškūnas ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2020__7__337_0,
     author = {Matthew Emerton and Vytautas Pa\v{s}k\={u}nas},
     title = {On the density of supercuspidal points of fixed regular weight in local deformation rings and global {Hecke} algebras},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {337--371},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.119},
     zbl = {1198.22010},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.119/}
}

TY  - JOUR
AU  - Matthew Emerton
AU  - Vytautas Paškūnas
TI  - On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2020
SP  - 337
EP  - 371
VL  - 7
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.119/
DO  - 10.5802/jep.119
LA  - en
ID  - JEP_2020__7__337_0
ER  -

%0 Journal Article
%A Matthew Emerton
%A Vytautas Paškūnas
%T On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras
%J Journal de l’École polytechnique — Mathématiques
%D 2020
%P 337-371
%V 7
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.119/
%R 10.5802/jep.119
%G en
%F JEP_2020__7__337_0

Matthew Emerton; Vytautas Paškūnas. On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 337-371. doi : 10.5802/jep.119. https://jep.centre-mersenne.org/articles/10.5802/jep.119/

Bibliographie
Cité par

[1] K. Ardakov & K. A. Brown - “Ring-theoretic properties of Iwasawa algebras: a survey”, Doc. Math. (2006), p. 7-33, Extra Vol. | MR | Zbl

[2] T. Barnet-Lamb, T. Gee, D. Geraghty & R. Taylor - “Potential automorphy and change of weight”, Ann. of Math. (2) 179 (2014) no. 2, p. 501-609 | DOI | MR | Zbl

[3] C. Breuil, E. Hellmann & B. Schraen - “Une interprétation modulaire de la variété trianguline”, Math. Ann. 367 (2017) no. 3-4, p. 1587-1645 | DOI | Zbl

[4] C. Breuil & A. Mézard - “Multiplicités modulaires et représentations de ${GL}_{2} (Z_{p})$ et de $Gal ({\bar{Q}}_{p} / Q_{p})$ en $ℓ = p$ ”, Duke Math. J. 115 (2002) no. 2, p. 205-310, With an appendix by Guy Henniart | DOI

[5] C. J. Bushnell & A. Fröhlich - Gauss sums and $p$ -adic division algebras, Lect. Notes in Math., vol. 987, Springer-Verlag, Berlin-New York, 1983

[6] C. J. Bushnell & G. Henniart - “The essentially tame local Langlands correspondence. I”, J. Amer. Math. Soc. 18 (2005) no. 3, p. 685-710 | DOI | MR | Zbl

[7] C. J. Bushnell & P. C. Kutzko - The admissible dual of $GL (N)$ via compact open subgroups, Annals of Math. Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993 | DOI

[8] C. J. Bushnell & P. C. Kutzko - “Semisimple types in ${GL}_{n}$ ”, Compositio Math. 119 (1999) no. 1, p. 53-97 | DOI

[9] F. Calegari & M. Emerton - “Completed cohomology—a survey”, in Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser., vol. 393, Cambridge Univ. Press, Cambridge, 2012, p. 239-257 | MR | Zbl

[10] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas & S. W. Shin - “Patching and the $p$ -adic local Langlands correspondence”, Camb. J. Math. 4 (2016) no. 2, p. 197-287 | DOI | Zbl

[11] G. Chenevier - “Sur la densité des représentations cristallines de $Gal ({\bar{ℚ}}_{p} / ℚ_{p})$ ”, Math. Ann. 355 (2013) no. 4, p. 1469-1525 | DOI | MR

[12] P. Colmez, G. Dospinescu & V. Paškūnas - “The $p$ -adic local Langlands correspondence for ${GL}_{2} (ℚ_{p})$ ”, Camb. J. Math. 2 (2014) no. 1, p. 1-47 | DOI

[13] J. D. Dixon, M. P. F. du Sautoy, A. Mann & D. Segal - Analytic pro- $p$ groups, Cambridge Studies in Advanced Math., vol. 61, Cambridge Univ. Press, Cambridge, 1999 | DOI

[14] M. Emerton - “Jacquet modules of locally analytic representations of $p$ -adic reductive groups. I. Construction and first properties”, Ann. Sci. École Norm. Sup. (4) 39 (2006) no. 5, p. 775-839 | DOI | MR

[15] M. Emerton - “On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms”, Invent. Math. 164 (2006) no. 1, p. 1-84 | DOI | MR | Zbl

[16] M. Emerton - “Local-global compatibility in the $p$ -adic Langlands programme for ${GL}_{2} / ℚ$ ” (2011), Preprint, http://math.uchicago.edu/~emerton/preprints.html

[17] M. Emerton - “Completed cohomology and the $p$ -adic Langlands program”, in Proceedings of the International Congress of Mathematicians (Seoul, 2014). Vol. II, Kyung Moon Sa, Seoul, 2014, p. 319-342

[18] M. Emerton & T. Gee - “A geometric perspective on the Breuil-Mézard conjecture”, J. Inst. Math. Jussieu 13 (2014) no. 1, p. 183-223 | DOI | Zbl

[19] T. Gee & J. Newton - “Patching and the completed homology of locally symmetric spaces”, 2017 | arXiv

[20] B. H. Gross - “Algebraic modular forms”, Israel J. Math. 113 (1999), p. 61-93 | DOI | MR | Zbl

[21] M. Harris - “ $p$ -adic representations arising from descent on abelian varieties”, Compositio Math. 39 (1979) no. 2, p. 177-245

[22] M. Harris - “Correction to: “ $p$ -adic representations arising from descent on abelian varieties””, Compositio Math. 121 (2000) no. 1, p. 105-108 | DOI

[23] E. Hellmann, C. M. Margerin & B. Schraen - “Density of automorphic points”, 2018 | arXiv | Zbl

[24] E. Hellmann & B. Schraen - “Density of potentially crystalline representations of fixed weight”, Compositio Math. 152 (2016) no. 8, p. 1609-1647 | DOI | MR | Zbl

[25] P. Kutzko & D. Manderscheid - “On intertwining operators for ${GL}_{N} (F), F$ a non-Archimedean local field”, Duke Math. J. 57 (1988) no. 1, p. 275-293 | DOI

[26] H. Matsumura - Commutative ring theory, Cambridge Studies in Advanced Math., vol. 8, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[27] P. Monsky - “The Hilbert-Kunz function”, Math. Ann. 263 (1983) no. 1, p. 43-49 | DOI | MR | Zbl

[28] K. Nakamura - “Deformations of trianguline $B$ -pairs and Zariski density of two dimensional crystalline representations”, J. Math. Sci. Univ. Tokyo 20 (2013) no. 4, p. 461-568 | MR

[29] K. Nakamura - “Zariski density of crystalline representations for any $p$ -adic field”, J. Math. Sci. Univ. Tokyo 21 (2014) no. 1, p. 79-127 | MR

[30] J. Neukirch, A. Schmidt & K. Wingberg - Cohomology of number fields, Grundlehren Math. Wiss., vol. 323, Springer-Verlag, Berlin, 2013, corrected second printing | DOI | Zbl

[31] V. Paškūnas - “Unicity of types for supercuspidal representations of ${GL}_{N}$ ”, Proc. London Math. Soc. (3) 91 (2005) no. 3, p. 623-654 | DOI | MR

[32] V. Paškūnas - “The image of Colmez’s Montreal functor”, Publ. Math. Inst. Hautes Études Sci. 118 (2013), p. 1-191 | DOI | MR | Zbl

[33] V. Paškūnas - “On the Breuil-Mézard conjecture”, Duke Math. J. 164 (2015) no. 2, p. 297-359 | DOI | MR | Zbl

[34] V. Paškūnas - “On some consequences of a theorem of J. Ludwig”, 2018 | arXiv | Zbl

[35] P. Schneider & J. Teitelbaum - “Banach space representations and Iwasawa theory”, Israel J. Math. 127 (2002), p. 359-380 | DOI | MR | Zbl

[36] P. Scholze - “On the $p$ -adic cohomology of the Lubin-Tate tower”, Ann. Sci. École Norm. Sup. (4) 51 (2018) no. 4, p. 811-863, With an appendix by Michael Rapoport | DOI | MR

[37] S.-N. Tung - “On the automorphy of $2$ -dimensional potentially semi-stable deformation rings of $G_{ℚ_{p}}$ ”, 2018 | arXiv

[38] S.-N. Tung - “On the modularity of $2$ -adic potentially semi-stable deformation rings”, 2019 | arXiv

[39] O. Venjakob - “On the structure theory of the Iwasawa algebra of a $p$ -adic Lie group”, J. Eur. Math. Soc. (JEMS) 4 (2002) no. 3, p. 271-311 | DOI | MR

Cité par Sources :