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Gelfand–Kirillov dimension and mod \(p\) cohomology for \(\operatorname{GL}_{2}\)

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Let \(p\) be a prime number, \(F\) a totally real number field unramified at places above \(p\) and \(D\) a quaternion algebra of center \(F\) split at places above \(p\) and at no more than one infinite place. Let \(v\) be a fixed place of \(F\) above \(p\) and \(\overline {r}: {\mathrm{Gal}}(\overline{F}/F)\rightarrow \operatorname{GL}_{2}(\overline {\mathbb{F}}_{p})\) an irreducible modular continuous Galois representation which, at the place \(v\), is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of \(\operatorname{GL}_{2}(F_{v})\) over \(\overline {\mathbb{F}}_{p}\) associated to \(\overline {r}\) in the corresponding Hecke-eigenspaces of the mod \(p\) cohomology have Gelfand–Kirillov dimension \([F_{v}:\mathbb{Q}_{p}]\), as well as several related results.

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Notes

  1. Strictly speaking, this is not quite the Gelfand–Kirillov dimension of \(\pi \), see Remark 5.1.1 in the text, but this is the only dimension we will consider.

References

  1. Allen, P.B.: On automorphic points in polarized deformation rings. Am. J. Math. 141(1), 119–167 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Alperin, J.L.: Local Representation Theory. Modular representations as an introduction to the local representation theory of finite groups. Cambridge Studies in Advanced Mathematics, vol. 11. Cambridge University Press, Cambridge (1986)

    MATH  Google Scholar 

  3. Ardakov, K., Brown, K.A.: Ring-theoretic properties of Iwasawa algebras: a survey. Doc. Math. Extra Vol., 7–33 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R.: A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Bourbaki, N.: Éléments de Mathématique. Fasc. XXXVII. Groupes et Algèbres de Lie. Chapitre II: Algèbres de Lie Libres. Chapitre III: Groupes de Lie. Actualités Scientifiques et Industrielles, vol. 1349. Hermann, Paris (1972)

    MATH  Google Scholar 

  6. Breuil, C.: Sur quelques représentations modulaires et \(p\)-adiques de \(\mathrm{GL}_{2}(\mathbf{Q}_{p})\). I. Compos. Math. 138(2), 165–188 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Breuil, C.: Sur un problème de compatibilité local-global modulo \(p\) pour \(\mathrm{GL}_{2}\). J. Reine Angew. Math. 692, 1–76 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Breuil, C., Diamond, F.: Formes modulaires de Hilbert modulo \(p\) et valeurs d’extensions entre caractères galoisiens. Ann. Sci. Éc. Norm. Supér. (4) 47(5), 905–974 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Breuil, C., Mézard, A.: Multiplicités modulaires et représentations de \(\mathrm{GL}_{2}({ \mathbf {Z}}_{p})\) et de \({{\mathrm{Gal}}}(\overline{\mathbf {Q}}_{p}/{\mathbf {Q}}_{p})\) en \(l=p\). Duke Math. J. 115(2), 205–310 (2002). With an appendix by Guy Henniart

    MathSciNet  MATH  Google Scholar 

  10. Breuil, C., Paškūnas, V.: Towards a modulo \(p\) Langlands correspondence for \(\mathrm{GL}_{2}\). Mem. Am. Math. Soc. 216, no. 1016, vi+114 (2012)

    MATH  Google Scholar 

  11. Breuil, C., Herzig, F., Hu, Y., Morra, S., Schraen, B.: Conjectures and results on modular representations of \(\mathrm{GL}_{n}(k)\) for a \(p\)-adic field \(k\). Preprint (2021). https://arxiv.org/pdf/2102.06188.pdf

  12. Breuil, C., Herzig, F., Hu, Y., Morra, S., Schraen, B.: Multivariable \((\varphi ,\mathcal{O}_{K}^{\times})\)-modules and local-global compatibility. Preprint (2022). https://arxiv.org/pdf/2211.00438.pdf

  13. Buzzard, K., Diamond, F., Jarvis, F.: On Serre’s conjecture for mod \(\ell \) Galois representations over totally real fields. Duke Math. J. 155(1), 105–161 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Calegari, F.: Non-minimal modularity lifting in weight one. J. Reine Angew. Math. 740, 41–62 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Caraiani, A., Levin, B.: Kisin modules with descent data and parahoric local models. Ann. Sci. Éc. Norm. Supér. (4) 51(1), 181–213 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Caraiani, A., Emerton, M., Gee, T., Geraghty, D., Paškūnas, V., Shin, S.W.: Patching and the \(p\)-adic local Langlands correspondence. Camb. J. Math. 4(2), 197–287 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Caraiani, A., Emerton, M., Gee, T., Savitt, D.: Moduli stacks of two-dimensional Galois representations. Preprint (2019). https://arxiv.org/pdf/1908.07019

  18. Caraiani, A., Emerton, M., Gee, T., Savitt, D.: Components of moduli stacks of two-dimensional Galois representations. Preprint (2022). https://arxiv.org/pdf/2207.05237.pdf

  19. Clozel, L.: Globally analytic \(p\)-adic representations of the pro-\(p\)-Iwahori subgroup of \(GL(2)\) and base change, I: Iwasawa algebras and a base change map. Bull. Iranian Math. Soc. 43(4), 55–76 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Diamond, F.: A correspondence between representations of local Galois groups and Lie-type groups. In: \(L\)-Functions and Galois Representations. London Math. Soc. Lecture Note Ser., vol. 320, pp. 187–206. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  21. Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic Pro-\(p\) Groups, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 61. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  22. Dotto, A., Le, D.: Diagrams in the \(\operatorname{mod} p\) cohomology of Shimura curves. Compos. Math. 157(8), 1653–1723 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Elkik, R.: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. Éc. Norm. Supér. (4) 6, 553–603 (1973) (1974)

    MathSciNet  MATH  Google Scholar 

  24. Emerton, M.: Local-global compatibility in the \(p\)-adic Langlands program for \(\mathrm{GL}_{2/{\mathbb{Q}}}\). Preprint (2011). http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf

  25. Emerton, M.: Completed cohomology and the \(p\)-adic Langlands program. In: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. II: Invited Lectures, pp. 319–342. KM Kyung Moon Sa, Seoul (2014) (English)

    Google Scholar 

  26. Emerton, M., Paškūnas, V.: On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras. J. Éc. Polytech. Math. 7, 337–371 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Emerton, M., Gee, T., Herzig, F.: Weight cycling and Serre-type conjectures for unitary groups. Duke Math. J. 162(9), 1649–1722 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Emerton, M., Gee, T., Savitt, D.: Lattices in the cohomology of Shimura curves. Invent. Math. 200(1), 1–96 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Fontaine, J.-M.: Représentations \(p\)-Adiques des Corps Locaux. I, tHe Grothendieck Festschrift, Vol. II. Progr. Math., vol. 87, pp. 249–309. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  30. Gee, T., Kisin, M.: The Breuil-Mézard conjecture for potentially Barsotti-Tate representations. Forum Math. Pi 2, e1, 56 (2014)

    MATH  Google Scholar 

  31. Gee, T., Newton, J.: Patching and the completed homology of locally symmetric spaces. J. Inst. Math. Jussieu 21(2), 395–458 (2022)

    MathSciNet  MATH  Google Scholar 

  32. Gee, T., Liu, T., Savitt, D.: The Buzzard-Diamond-Jarvis conjecture for unitary groups. J. Am. Math. Soc. 27(2), 389–435 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Gee, T., Herzig, F., Liu, T., Savitt, D.: Potentially crystalline lifts of certain prescribed types. Doc. Math. 22, 397–422 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Gee, T., Herzig, F., Savitt, D.: General Serre weight conjectures. J. Eur. Math. Soc. 20(12), 2859–2949 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Publ. Math. Inst. Hautes Études Sci. 24, 231 (1965)

    MATH  Google Scholar 

  36. Hamann, E.: On power-invariance. Pac. J. Math. 61(1), 153–159 (1975)

    MathSciNet  MATH  Google Scholar 

  37. Herzig, F.: The weight in a Serre-type conjecture for tame \(n\)-dimensional Galois representations. Duke Math. J. 149(1), 37–116 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Hu, Y.: Sur quelques représentations supersingulières de \(\mathrm{GL}_{2}( \mathbb{Q}_{p^{f}})\). J. Algebra 324(7), 1577–1615 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Hu, Y., Wang, H.: Multiplicity one for the mod \(p\) cohomology of Shimura curves: the tame case. Math. Res. Lett. 25(3), 843–873 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Hu, Y., Wang, H.: On some mod \(p\) representations of quaternion algebra over \(\mathbb{Q}_{p}\). Preprint (2022). http://arxiv.org/abs/2201.01464

  41. Hu, Y., Wang, H.: On the mod \(p\) cohomology for \(\mathrm{GL}_{2}\): the non-semisimple case. Camb. J. Math. 10(2), 261–431 (2022)

    MathSciNet  MATH  Google Scholar 

  42. Humphreys, J.E.: Generic Cartan invariants for Frobenius kernels and Chevalley groups. J. Algebra 122(2), 345–352 (1989)

    MathSciNet  MATH  Google Scholar 

  43. Jantzen, J.C.: Representations of Algebraic Groups, 2nd edn. Mathematical Surveys and Monographs, vol. 107. Am. Math. Soc., Providence (2003)

    MATH  Google Scholar 

  44. Kisin, M.: Crystalline representations and \(F\)-crystals. In: Algebraic Geometry and Number Theory. Progr. Math., vol. 253, pp. 459–496. Birkhäuser, Boston (2006)

    Google Scholar 

  45. Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Kottwitz, R., Rapoport, M.: Minuscule alcoves for \(\mathrm{GL}_{n}\) and \(G\mathrm{Sp}_{2n}\). Manuscr. Math. 102(4), 403–428 (2000)

    MATH  Google Scholar 

  47. Lazard, M.: Groupes analytiques \(p\)-adiques. Publ. Math. Inst. Hautes Études Sci. 26, 389–603 (1965)

    MathSciNet  MATH  Google Scholar 

  48. Le, D.: Lattices in the cohomology of \(U(3)\) arithmetic manifolds. Math. Ann. 372(1–2), 55–89 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Le, D.: Multiplicity one for wildly ramified representations. Algebra Number Theory 13(8), 1807–1827 (2019)

    MathSciNet  MATH  Google Scholar 

  50. Le, D., Le Hung, B.V., Levin, B., Morra, S.: Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows. Invent. Math. 212(1), 1–107 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Le, D., Le Hung, B.V., Levin, B.: Weight elimination in Serre-type conjectures. Duke Math. J. 168(13), 2433–2506 (2019)

    MathSciNet  MATH  Google Scholar 

  52. Le, D., Le Hung, B.V., Levin, B., Morra, S.: Serre weights and Breuil’s lattice conjecture in dimension three. Forum Math. Pi 8, e5, 135 (2020)

    MathSciNet  MATH  Google Scholar 

  53. Le, D., Morra, S., Schraen, B.: Multiplicity one at full congruence level. J. Inst. Math. Jussieu 21(2), 637–658 (2022)

    MathSciNet  MATH  Google Scholar 

  54. Le, D., Le Hung, B.V., Levin, B., Morra, S.: Local models for Galois deformation rings and applications. Invent. Math. 231(3), 1277–1488 (2023)

    MathSciNet  MATH  Google Scholar 

  55. Li, H., van Oystaeyen, F.: Zariskian Filtrations. \(K\)-Monographs in Mathematics, vol. 2. Kluwer Academic, Dordrecht (1996)

    MATH  Google Scholar 

  56. Matsumura, H.: Commutative Ring Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. Reid

    MATH  Google Scholar 

  57. Morra, S.: Invariant elements for \(p\)-modular representations of \(\mathrm{GL}_{2}(\mathbb{Q}_{p})\). Trans. Am. Math. Soc. 365(12), 6625–6667 (2013)

    MATH  Google Scholar 

  58. Schneider, P., Teitelbaum, J.: Banach space representations and Iwasawa theory. Isr. J. Math. 127, 359–380 (2002)

    MathSciNet  MATH  Google Scholar 

  59. Scholze, P.: On the \(p\)-adic cohomology of the Lubin-Tate tower. Ann. Sci. Éc. Norm. Supér. (4) 51(4), 811–863 (2018). With an appendix by Michael Rapoport

    MathSciNet  MATH  Google Scholar 

  60. Serre, J.-P.: Sur la dimension cohomologique des groupes profinis. Topology 3, 413–420 (1965)

    MathSciNet  MATH  Google Scholar 

  61. Shotton, J.: Local deformation rings for \(\mathrm{GL}_{2}\) and a Breuil-Mézard conjecture when \(\ell \ne p\). Algebra Number Theory 10(7), 1437–1475 (2016)

    MathSciNet  MATH  Google Scholar 

  62. Venjakob, O.: On the structure theory of the Iwasawa algebra of a \(p\)-adic Lie group. J. Eur. Math. Soc. 4(3), 271–311 (2002)

    MathSciNet  MATH  Google Scholar 

  63. Wang, Y.: On the mod \(p\) cohomology for \(\mathrm{GL}_{2}\). Preprint (2023). https://arxiv.org/pdf/2209.09639.pdf

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Acknowledgements

The initial impetus for this work was a SQuaRE meeting at the American Institute of Mathematics at San Jose in August 2019 (though it was not quite clear at the time where we were really heading!). We heartily thank AIM for hosting and supporting us and for outstanding working conditions. We are also very grateful to Sug Woo Shin and Karol Kozioł for participating in this meeting and for sharing their thoughts with us. We are particularly grateful to Karol Kozioł for pointing out a mistake in an earlier version of this work. Finally, we heartily thank an anonymous referee for his or her report, especially for pointing out an embarrassing mistake in our previous use of multi-type deformation rings.

C.B. thanks X. Caruso for discussions in an early attempt to approach the Gelfand–Kirillov dimension via computational techniques, and Ahmed Abbes and all the organizers of the Séminaire de Géométrie Arithmétique Paris–Pékin–Tokyo for their invitation to give the very last talk of this seminar on this work in June 2020. Y.H. thanks Ahmed Abbes for inviting him to I.H.É.S. for the period of November–December 2019 and I.H.É.S. for its hospitality.

C.B., F.H., S.M., B.S. thank Y.H. and Haoran Wang for sharing a preliminary version of [41], which inspired us in the early stages of our project.

Funding

F.H. is partially supported by an NSERC grant. Y.H. is partially supported by National Key R&D Program of China 2020YFA0712600, National Natural Science Foundation of China Grants 12288201 and 11971028; National Center for Mathematics and Interdisciplinary Sciences and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences. S.M. and B.S. are partially supported by Institut Universitaire de France. C.B., S.M. and B.S. are members of the A.N.R. project CLap-CLap ANR-18-CE40-0026.

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Breuil, C., Herzig, F., Hu, Y. et al. Gelfand–Kirillov dimension and mod \(p\) cohomology for \(\operatorname{GL}_{2}\). Invent. math. 234, 1–128 (2023). https://doi.org/10.1007/s00222-023-01202-8

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