Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence
-
Gabriel Dospinescu
and
Benjamin Schraen
Abstract
We bound the Gelfand–Kirillov dimension of unitary Banach space
representations of p-adic reductive groups, whose locally analytic vectors afford
an infinitesimal character. We use the bound to study Hecke eigenspaces in completed
cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over
Funding statement: Benjamin Schraen and Gabriel Dospinescu are members of the A.N.R. project CLap-CLap ANR-18-CE40-0026. Gabriel Dospinescu is a member of the A.N.R. project Coloss ANR-19-CE40-0015.
Acknowledgements
We would like to thank Yongquan Hu for pointing out a blunder in an earlier draft. We thank Toby Gee, Lue Pan and Peter Scholze for their comments on a preliminary version of the paper. We thank Konstantin Ardakov for simplifying the proof of theorem 4.1, and Olivier Taïbi for several enlightening discussions around the results in this paper.
References
[1]
K. Ardakov,
Equivariant
[2] K. Ardakov and K. A. Brown, Ring-theoretic properties of Iwasawa algebras: A survey, Doc. Math. 4 (2006), 7–33. 10.4171/dms/4/1Search in Google Scholar
[3] K. Ardakov and S. Wadsley, On irreducible representations of compact p-adic analytic groups, Ann. of Math. (2) 178 (2013), no. 2, 453–557. 10.4007/annals.2013.178.2.3Search in Google Scholar
[4] K. Ardakov and S. Wadsley, Verma modules for Iwasawa algebras are faithful, Münster J. Math. 7 (2014), no. 1, 5–26. Search in Google Scholar
[5]
K. Ardakov and S. J. Wadsley,
[6] D. J. Benson, Polynomial invariants of finite groups, London Math. Soc. Lecture Note Ser. 190, Cambridge University, Cambridge 1993. 10.1017/CBO9780511565809Search in Google Scholar
[7]
P. Berthelot,
[8] A. Beĭlinson and J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18. Search in Google Scholar
[9] J.-E. Björk, The Auslander condition on Noetherian rings, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris 1987/1988), Lecture Notes in Math. 1404, Springer, Berlin (1989), 137–173. 10.1007/BFb0084075Search in Google Scholar
[10]
C. Breuil,
Sur quelques représentations modulaires et p-adiques de
[11]
C. Breuil, F. Herzig, Y. Hu, S. Morra and B. Schraen,
Gelfand–Kirillov dimension and mod p cohomology for
[12] C. J. Bushnell and G. Henniart, Explicit functorial correspondences for level zero representations of p-adic linear groups, J. Number Theory 131 (2011), no. 2, 309–331. 10.1016/j.jnt.2010.09.003Search in Google Scholar
[13] F. Calegari and M. Emerton, Completed cohomology—a survey, Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser. 393, Cambridge University, Cambridge (2012), 239–257. 10.1017/CBO9780511984440.010Search in Google Scholar
[14] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas and S. W. Shin, Patching and the p-adic local Langlands correspondence, Camb. J. Math. 4 (2016), no. 2, 197–287. 10.4310/CJM.2016.v4.n2.a2Search in Google Scholar
[15] A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties, Ann. of Math. (2) 186 (2017), no. 3, 649–766. 10.4007/annals.2017.186.3.1Search in Google Scholar
[16]
P. Colmez and G. Dospinescu,
Complétés universels de représentations de
[17]
P. Colmez, G. Dospinescu and V. Paškūnas,
The p-adic local Langlands correspondence for
[18] N. Conze, Algèbres d’opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France 102 (1974), 379–415. 10.24033/bsmf.1786Search in Google Scholar
[19] G. Dospinescu, Actions infinitésimales dans la correspondance de Langlands locale p-adique, Math. Ann. 354 (2012), no. 2, 627–657. 10.1007/s00208-011-0736-2Search in Google Scholar
[20] G. Dospinescu, V. Paškūnas and B. Schraen, Infinitesimal characters in arithmetic families, preprint (2020), https://arxiv.org/abs/2012.01041. Search in Google Scholar
[21]
A. Dotto,
Restriction of p-adic representations of
[22]
A. Dotto and D. Le,
Diagrams in the
[23] M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques, Astérisque 331, Société Mathématique de France, Paris (2010), 355–402. Search in Google Scholar
[24] M. Emerton and V. Paškūnas, On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras, J. Éc. polytech. Math. 7 (2020), 337–371. 10.5802/jep.119Search in Google Scholar
[25] T. Gee and M. Kisin, The Breuil–Mézard conjecture for potentially Barsotti–Tate representations, Forum Math. Pi 2 (2014), Paper No. E1. 10.1017/fmp.2014.1Search in Google Scholar
[26] T. Gee and J. Newton, Patching and the completed homology of locally symmetric spaces, J. Inst. Math. Jussieu 21 (2022), no. 2, 395–458. 10.1017/S1474748020000158Search in Google Scholar
[27] T. J. Hodges and S. P. Smith, Differential operators on the flag variety and the conze embedding, preprint (1984), https://sites.math.washington.edu/~smith/Research/mypapers.html. Search in Google Scholar
[28]
Y. Hu and H. Wang,
On some mod p representations of quaternion algebra over
[29]
Y. Hu and H. Wang,
On the
[30]
Y. Hu and H. Wang,
On some p-adic and mod p representations of quaternion algebra over
[31] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. 10.1007/s00222-009-0206-6Search in Google Scholar
[32]
M. Kisin,
The Fontaine–Mazur conjecture for
[33] S. Lang, Algebra, 3rd ed., Grad. Texts in Math. 211, Springer, New York 2002. 10.1007/978-1-4613-0041-0_1Search in Google Scholar
[34] T. Levasseur, Some properties of noncommutative regular graded rings, Glasg. Math. J. 34 (1992), no. 3, 277–300. 10.1017/S0017089500008843Search in Google Scholar
[35] H. Li and F. van Oystaeyen, Zariskian filtrations, K-Monogr. Math. 2, Kluwer Academic, Dordrecht 1996. Search in Google Scholar
[36] K. Liu, Local-global compatibility of mod p Langlands program for certain Shimura varieties, preprint (2021), https://arxiv.org/abs/2106.10674. 10.1007/s00229-022-01410-1Search in Google Scholar
[37] J. Ludwig, A quotient of the Lubin–Tate tower, Forum Math. Sigma 5 (2017), Paper No. e17. 10.1017/fms.2017.15Search in Google Scholar
[38] J. Manning and J. Shotton, Ihara’s lemma for Shimura curves over totally real fields via patching, Math. Ann. 379 (2021), no. 1–2, 187–234. 10.1007/s00208-020-02048-8Search in Google Scholar
[39] L. Pan, The Fontaine–Mazur conjecture in the residually reducible case, J. Amer. Math. Soc. 35 (2022), no. 4, 1031–1169.10.1090/jams/991Search in Google Scholar
[40]
V. Paškūnas,
Extensions for supersingular representations of
[41] V. Paškūnas, The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci. 118 (2013), 1–191. 10.1007/s10240-013-0049-ySearch in Google Scholar
[42] V. Paškūnas, On the Breuil–Mézard conjecture, Duke Math. J. 164 (2015), no. 2, 297–359. 10.1215/00127094-2861604Search in Google Scholar
[43] V. Paškūnas, On 2-dimensional 2-adic Galois representations of local and global fields, Algebra Number Theory 10 (2016), no. 6, 1301–1358. 10.2140/ant.2016.10.1301Search in Google Scholar
[44] V. Paškūnas, On some consequences of a theorem of J. Ludwig, J. Inst. Math. Jussieu 21 (2022), no. 3, 1067–1106. 10.1017/S1474748020000547Search in Google Scholar
[45]
V. Paškūnas and S.-N. Tung,
Finiteness properties of the category of
[46] J. J. Rotman, An introduction to homological algebra, 2nd ed., Universitext, Springer, New York 2009. 10.1007/b98977Search in Google Scholar
[47] T. Schmidt, Auslander regularity of p-adic distribution algebras, Represent. Theory 12 (2008), 37–57. 10.1090/S1088-4165-08-00323-3Search in Google Scholar
[48] T. Schmidt, Analytic vectors in continuous p-adic representations, Compos. Math. 145 (2009), no. 1, 247–270. 10.1112/S0010437X08003825Search in Google Scholar
[49] T. Schmidt and M. Strauch, Dimensions of some locally analytic representations, Represent. Theory 20 (2016), 14–38. 10.1090/ert/475Search in Google Scholar
[50] P. Schneider and J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145–196. 10.1007/s00222-002-0284-1Search in Google Scholar
[51]
P. Schneider, J. Teitelbaum and D. Prasad,
[52]
P. Schneider and O. Venjakob,
[53] P. Scholze, On the p-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 4, 811–863. 10.24033/asens.2367Search in Google Scholar
[54] P. Scholze and J. Weinstein, Berkeley lectures on p-adic geometry, Ann. of Math. Stud. 207, Princeton University, Princeton 2020. 10.23943/princeton/9780691202082.001.0001Search in Google Scholar
[55] J.-P. Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in Math. 1500, Springer, Berlin 1992. 10.1007/978-3-540-70634-2_1Search in Google Scholar
[56] M. Strauch, Geometrically connected components of Lubin–Tate deformation spaces with level structures, Pure Appl. Math. Q. 4 (2008), no. 4, 1215–1232. 10.4310/PAMQ.2008.v4.n4.a10Search in Google Scholar
[57] R. Taylor, On the meromorphic continuation of degree two L-functions, Doc. Math. 2006 (2006), 729–779. 10.4171/dms/4/22Search in Google Scholar
[58] J. A. Thorne, A 2-adic automorphy lifting theorem for unitary groups over CM fields, Math. Z. 285 (2017), no. 1–2, 1–38. 10.1007/s00209-016-1681-2Search in Google Scholar
[59]
S.-N. Tung,
On the automorphy of 2-dimensional potentially semistable deformation rings of
[60] S.-N. Tung, On the modularity of 2-adic potentially semi-stable deformation rings, Math. Z. 298 (2021), no. 1–2, 107–159. 10.1007/s00209-020-02588-4Search in Google Scholar
[61] 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, 271–311. 10.1007/s100970100038Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Comparing the Kirwan and noncommutative resolutions of quotient varieties
- Supersingular elliptic curves over ℤ𝑝-extensions
- Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence
- Models of Jacobians of curves
- Hypercritical deformed Hermitian-Yang–Mills equation revisited
- Categorical action filtrations via localization and the growth as a symplectic invariant
- Boundary regularity of minimal graphs in the hyperbolic space
- Collapsing and noncollapsing in convex ancient mean curvature flow
Articles in the same Issue
- Frontmatter
- Comparing the Kirwan and noncommutative resolutions of quotient varieties
- Supersingular elliptic curves over ℤ𝑝-extensions
- Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence
- Models of Jacobians of curves
- Hypercritical deformed Hermitian-Yang–Mills equation revisited
- Categorical action filtrations via localization and the growth as a symplectic invariant
- Boundary regularity of minimal graphs in the hyperbolic space
- Collapsing and noncollapsing in convex ancient mean curvature flow
