Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

Miaofen Chen; Sian Nie

doi:10.1515/crelle-2022-0011

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Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

Published/Copyright: February 25, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 785

Abstract

We consider the Kisin variety associated to an n-dimensional absolutely irreducible mod p Galois representation ρ¯ of a p-adic field K together with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds if K is totally ramified with n=3 or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to ρ¯ of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.

Funding source: National Key Research and Development Program of China

Award Identifier / Grant number: 2020YFA0712600

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11922119

Award Identifier / Grant number: 11671136

Award Identifier / Grant number: 12071135

Funding statement: The second author was partially supported by National Key R&D Program of China, No. 2020YFA0712600, and NSFC grant No. 11922119. The first author was partially supported by NSFC grant No. 11671136, No. 12071135 and STCSM grant No. 18dz2271000.

Acknowledgements

We would like to thank Brandon Levin who encouraged us to work on this topic. We thank Frank Calegari, Hui Gao, Mark Kisin for helpful discussions and comments. We also thank the referee for careful reading and useful comments.

References

[1] C. Breuil, Integral p-adic Hodge theory, Algebraic geometry 2000, Adv. Stud. Pure Math. 36, Mathematical Society of Japan, Tokyo (2002), 51–80. 10.2969/aspm/03610051Search in Google Scholar

[2] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41, (1972), 5–251. 10.1007/BF02715544Search in Google Scholar

[3] F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles method, Invent. Math. 211 (2018), no. 1, 297–433. 10.1007/s00222-017-0749-xSearch in Google Scholar

[4] A. Caraiani, M. Emerton, T. Gee and D. Savitt, Moduli stacks of two-dimensional Galois representations, preprint (2019), https://arxiv.org/abs/1908.07019. Search in Google Scholar

[5] X. Caruso, Sur la classification de quelques ϕ-modules simples, Mosc. Math. J. 9 (2009), no. 3, 562–568. 10.17323/1609-4514-2009-9-3-562-568Search in Google Scholar

[6] X. Caruso, Estimation des dimensions de certaines variétés de Kisin, J. reine angew. Math. 723 (2017), 1–77. 10.1515/crelle-2014-0066Search in Google Scholar

[7] X. Caruso, A. David and A. Mézard, Un calcul d’anneaux de déformations potentiellement Barsotti–Tate, Trans. Amer. Math. Soc. 370 (2018), no. 9, 6041–6096. 10.1090/tran/6973Search in Google Scholar

[8] X. Caruso, A. David and A. Mézard, Variétés de Kisin stratifiées et déformations potentiellement Barsotti–Tate, J. Inst. Math. Jussieu 17 (2018), no. 5, 1019–1064. 10.1017/S1474748016000232Search in Google Scholar

[9] L. Chen and S. Nie, Connected components of closed affine Deligne–Lusztig varieties, Math. Ann. 375 (2019), no. 3–4, 1355–1392. 10.1007/s00208-019-01804-9Search in Google Scholar

[10] L. Chen and S. Nie, Connected components of closed affine Deligne–Lusztig varieties for ResE/F⁢GLn, J. Algebra 546 (2020), 1–26. 10.1016/j.jalgebra.2019.10.031Search in Google Scholar

[11] M. Chen, M. Kisin and E. Viehmann, Connected components of affine Deligne–Lusztig varieties in mixed characteristic, Compos. Math. 151 (2015), no. 9, 1697–1762. 10.1112/S0010437X15007253Search in Google Scholar

[12] A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), no. 1, 209–241. 10.1090/S0894-0347-99-00322-7Search in Google Scholar

[13] M. Emerton and T. Gee, Moduli stacks of étale (ϕ,Γ)-modules and the existence of crystalline lifts, preprint (2019), https://arxiv.org/abs/1908.07185. Search in Google Scholar

[14] J.-M. Fontaine and Y. Ouyang, Theory of p-adic Galois representations, preprint (1999), http://staff.ustc.edu.cn/~yiouyang/research.html. Search in Google Scholar

[15] Q. R. Gashi, On a conjecture of Kottwitz and Rapoport, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 6, 1017–1038. 10.24033/asens.2138Search in Google Scholar

[16] T. Gee, A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), 805–811. 10.4310/MRL.2006.v13.n5.a10Search in Google Scholar

[17] U. Görtz, T. J. Haines, R. E. Kottwitz and D. C. Reuman, Dimensions of some affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), no. 3, 467–511. 10.1016/j.ansens.2005.12.004Search in Google Scholar

[18] U. Görtz, T. J. Haines, R. E. Kottwitz and D. C. Reuman, Affine Deligne–Lusztig varieties in affine flag varieties, Compos. Math. 146 (2010), no. 5, 1339–1382. 10.1112/S0010437X10004823Search in Google Scholar

[19] U. Görtz, X. He and S. Nie, 𝐏-alcoves and nonemptiness of affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 3, 647–665. 10.24033/asens.2254Search in Google Scholar

[20] P. Hamacher, The dimension of affine Deligne–Lusztig varieties in the affine Grassmannian, Int. Math. Res. Not. IMRN 2015 (2015), no. 23, 12804–12839. 10.1093/imrn/rnv081Search in Google Scholar

[21] P. Hamacher and E. Viehmann, Irreducible components of minuscule affine Deligne–Lusztig varieties, Algebra Number Theory 12 (2018), no. 7, 1611–1634. 10.2140/ant.2018.12.1611Search in Google Scholar

[22] X. He, Geometric and homological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), no. 1, 367–404. 10.4007/annals.2014.179.1.6Search in Google Scholar

[23] X. He and R. Zhou, On the connected components of affine Deligne–Lusztig varieties, Duke Math. J. 169 (2020), no. 14, 2697–2765. 10.1215/00127094-2020-0020Search in Google Scholar

[24] E. Hellmann, On the structure of some moduli spaces of finite flat group schemes, Mosc. Math. J. 9 (2009), no. 3, 531–561. 10.17323/1609-4514-2009-9-3-531-561Search in Google Scholar

[25] E. Hellmann, Connectedness of Kisin varieties for GL2, Adv. Math. 228 (2011), no. 1, 219–240. 10.1016/j.aim.2011.05.014Search in Google Scholar

[26] N. Imai, Finite flat models of constant group schemes of rank two, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3827–3833. 10.1090/S0002-9939-2010-10524-5Search in Google Scholar

[27] N. Imai, On the connected components of moduli spaces of finite flat models, Amer. J. Math. 132 (2010), no. 5, 1189–1204. 10.1353/ajm.2010.0006Search in Google Scholar

[28] N. Imai, Ramification and moduli spaces of finite flat models, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 5, 1943–1975. 10.5802/aif.2662Search in Google Scholar

[29] M. Kisin, Crystalline representations and F-crystals, Algebraic geometry and number theory, Progr. Math. 253, Birkhäuser, Boston (2006), 459–496. 10.1007/978-0-8176-4532-8_7Search in Google Scholar

[30] M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180. 10.4007/annals.2009.170.1085Search in Google Scholar

[31] R. Kottwitz and M. Rapoport, On the existence of F-crystals, Comment. Math. Helv. 78 (2003), no. 1, 153–184. 10.1007/s000140300007Search in Google Scholar

[32] D. Le, B. V. Le Hung and B. Levin, Weight elimination in Serre-type conjectures, Duke Math. J. 168 (2019), no. 13, 2433–2506. 10.1215/00127094-2019-0015Search in Google Scholar

[33] D. Le, B. V. Le Hung, B. Levin and S. Morra, Potentially crystalline deformation rings and Serre weight conjectures: Shapes and shadows, Invent. Math. 212 (2018), no. 1, 1–107. 10.1007/s00222-017-0762-0Search in Google Scholar

[34] D. Le, B. V. Le Hung, B. Levin and S. Morra, Local models for Galois deformation rings and applications, preprint (2020), https://arxiv.org/abs/2007.05398. 10.1007/s00222-022-01163-4Search in Google Scholar

[35] D. Le, B. V. Le Hung, B. Levin and S. Morra, Serre weights and Breuil’s lattice conjecture in dimension three, Forum Math. Pi 8 (2020), Paper No. e5. 10.1017/fmp.2020.1Search in Google Scholar

[36] B. Levin, G-valued crystalline representations with minuscule p-adic Hodge type, Algebra Number Theory 9 (2015), no. 8, 1741–1792. 10.2140/ant.2015.9.17Search in Google Scholar

[37] B. Levin and C. Wang-Erickson, A Harder–Narasimhan theory for Kisin modules, Algebr. Geom. 7 (2020), no. 6, 645–695. 10.14231/AG-2020-024Search in Google Scholar

[38] C. Lucarelli, A converse to Mazur’s inequality for split classical groups, J. Inst. Math. Jussieu 3 (2004), no. 2, 165–183. 10.1017/S1474748004000064Search in Google Scholar

[39] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95–143. 10.4007/annals.2007.166.95Search in Google Scholar

[40] A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), no. 1–3, 393–408. 10.1007/BF01231566Search in Google Scholar

[41] B. C. Ngô and P. Polo, Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Algebraic Geom. 10 (2001), no. 3, 515–547. Search in Google Scholar

[42] S. Nie, Connected components of closed affine Deligne–Lusztig varieties in affine Grassmannians, Amer. J. Math. 140 (2018), no. 5, 1357–1397. 10.1353/ajm.2018.0034Search in Google Scholar

[43] S. Nie, Irreducible components of affine Deligne–Lusztig varieties, preprint (2018), https://arxiv.org/abs/1809.03683. 10.4310/CJM.2022.v10.n2.a2Search in Google Scholar

[44] G. Pappas and M. Rapoport, Φ-modules and coefficient spaces, Mosc. Math. J. 9 (2009), no. 3, 625–663. 10.17323/1609-4514-2009-9-3-625-663Search in Google Scholar

[45] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compos. Math. 87 (1993), no. 3, 269–286. Search in Google Scholar

[46] M. Rapoport and M. Richartz, On the classification and specialization of F-isocrystals with additional structure, Compos. Math. 103 (1996), no. 2, 153–181. Search in Google Scholar

[47] M. Raynaud, Schémas en groupes de type (p,…,p), Bull. Soc. Math. France 102 (1974), 241–280. 10.24033/bsmf.1779Search in Google Scholar

[48] T. A. Springer, Linear algebraic groups, 2nd ed., Progr. Math. 9, Birkhäuser, Boston 1998. 10.1007/978-0-8176-4840-4Search in Google Scholar

[49] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. 10.2307/2118560Search in Google Scholar

[50] E. Viehmann, The dimension of some affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), no. 3, 513–526. 10.1016/j.ansens.2006.04.001Search in Google Scholar

[51] E. Viehmann, Connected components of closed affine Deligne–Lusztig varieties, Math. Ann. 340 (2008), no. 2, 315–333. 10.1007/s00208-007-0153-8Search in Google Scholar

[52] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. 10.1007/978-3-0348-9078-6_18Search in Google Scholar

[53] J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 59–89. 10.24033/asens.1440Search in Google Scholar

[54] L. Xiao and X. Zhu, Cycles on Shimura varieties via geometric Satake, preprint (2017), https://arxiv.org/abs/1707.05700. Search in Google Scholar

[55] R. Zhou and Y. Zhu, Twisted orbital integrals and irreducible components of affine Deligne–Lusztig varieties, Camb. J. Math. 8 (2020), no. 1, 149–241. 10.4310/CJM.2020.v8.n1.a3Search in Google Scholar

[56] X. Zhu, Affine Grassmannians and the geometric Satake in mixed characteristic, Ann. of Math. (2) 185 (2017), no. 2, 403–492. 10.4007/annals.2017.185.2.2Search in Google Scholar

Received: 2020-07-22

Revised: 2022-01-19

Published Online: 2022-02-25

Published in Print: 2022-04-01

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https://doi.org/10.1515/crelle-2022-0011