Irregular loci in the Emerton–Gee stack for GL2

Rebecca Bellovin; Neelima Borade; Anton Hilado; Kalyani Kansal; Heejong Lee; Brandon Levin; David Savitt; Hanneke Wiersema

doi:10.1515/crelle-2024-0045

Article

Irregular loci in the Emerton–Gee stack for GL₂

Rebecca Bellovin , Neelima Borade , Anton Hilado , Kalyani Kansal , Heejong Lee , Brandon Levin , David Savitt and Hanneke Wiersema

Published/Copyright: July 18, 2024

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 814

Abstract

Let K / Q p be unramified. Inside the Emerton–Gee stack X 2 , one can consider the locus of two-dimensional mod 𝑝 representations of Gal ⁢ ( K ̄ / K ) having a crystalline lift with specified Hodge–Tate weights. We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms. We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible. We also establish various inclusion relations between these loci.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1751281

Award Identifier / Grant number: DMS-1952556

Funding source: National Security Agency

Award Identifier / Grant number: H98230-21-1-0029

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: EXC-2047/1 – 390685813

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/W001683/1

Funding statement: This work began at the APAW Collaborative Research workshop supported by Ellen Eischen’s NSF CAREER grant DMS-1751281 and her NSA MSP conference grant H98230-21-1-0029. The work was completed at the trimester program “The Arithmetic of the Langlands Program” at the Hausdorff Institute for Mathematics, funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. B. Levin was supported by National Science Foundation grant DMS-1952556 and the Alfred P. Sloan Foundation. D. Savitt was supported by NSF grant DMS-1952566. H. Wiersema was supported by the Herchel Smith Postdoctoral Fellowship Fund, and the Engineering and Physical Sciences Research Council (EPSRC) grant EP/W001683/1.

Acknowledgements

This project began at the APAW Collaborative Research Workshop at the University of Oregon in August 2022 and was completed during various visits by six of us to the trimester program “The Arithmetic of the Langlands Program” at the Hausdorff Institute for Mathematics. We are grateful to Ellen Eischen, Maria Fox, Cathy Hsu, and Aaron Pollack for organizing the workshop, and we thank Frank Calegari, Ana Caraiani, Laurent Fargues, and Peter Scholze for their efforts organizing the trimester. We thank Matthew Emerton and Toby Gee for valuable conversations.

References

[1] T. Barnet-Lamb, T. Gee and D. Geraghty, Serre weights for rank two unitary groups, Math. Ann. 356 (2013), no. 4, 1551–1598. 10.1007/s00208-012-0893-ySearch in Google Scholar

[2] R. Bartlett, Inertial and Hodge–Tate weights of crystalline representations, Math. Ann. 376 (2020), no. 1–2, 645–681. 10.1007/s00208-019-01931-3Search in Google Scholar

[3] 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

[4] C. Breuil, Sur un problème de compatibilité local-global modulo 𝑝 pour GL 2 , J. reine angew. Math. 692 (2014), 1–76. 10.1515/crelle-2012-0083Search in Google Scholar

[5] C. Breuil and A. Mézard, Multiplicités modulaires et représentations de GL 2 ⁢ ( Z p ) et de Gal ⁢ ( Q ̄ p / Q p ) en l = p , Duke Math. J. 115 (2002), no. 2, 205–310. 10.1215/S0012-7094-02-11522-1Search in Google Scholar

[6] K. Buzzard, F. Diamond and F. Jarvis, On Serre’s conjecture for mod ℓ Galois representations over totally real fields, Duke Math. J. 155 (2010), no. 1, 105–161. 10.1215/00127094-2010-052Search in Google Scholar

[7] A. Caraiani, M. Emerton, T. Gee and D. Savitt, Local geometry of moduli stacks of two-dimensional Galois representations, Arithmetic Geometry 2020, TIFR Stud. Math. 24, Tata Institute of Fundamental Research Publications, Mumbai (2024), 127–202. Search in Google Scholar

[8] A. Caraiani, M. Emerton, T. Gee and D. Savitt, The geometric Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate Galois representions, preprint (2022), https://arxiv.org/abs/2207.05235; to appear in Algebra Number Theory. Search in Google Scholar

[9] A. Caraiani, M. Emerton, T. Gee and D. Savitt, Components of moduli stacks of two-dimensional Galois representations, Forum Math. Sigma 12 (2024), Paper No. e31. 10.1017/fms.2024.4Search in Google Scholar

[10] 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

[11] F. Diamond and S. Sasaki, A Serre weight conjecture for geometric Hilbert modular forms in characteristic 𝑝, J. Eur. Math. Soc. (JEMS) 25 (2023), no. 9, 3453–3536. 10.4171/jems/1265Search in Google Scholar

[12] M. Emerton, Formal algebraic stacks, http://www.math.uchicago.edu/~emerton/pdffiles/formal-stacks.pdf. Search in Google Scholar

[13] M. Emerton and T. Gee, ‘Scheme-theoretic images’ of morphisms of stacks, Algebr. Geom. 8 (2021), no. 1, 1–132. 10.14231/AG-2021-001Search in Google Scholar

[14] M. Emerton and T. Gee, Moduli sacks of étale ( φ , Γ )-modules and the existence of crystalline lifts, Ann. of Math. Stud. 215, Princeton University, Princeton 2023. 10.1515/9780691241364Search in Google Scholar

[15] M. Emerton, T. Gee and E. Hellmann, An introduction to the categorical 𝑝-adic Langlands program, preprint (2022), https://arxiv.org/abs/2210.01404. Search in Google Scholar

[16] 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

[17] T. Gee, T. Liu and D. Savitt, The Buzzard–Diamond–Jarvis conjecture for unitary groups, J. Amer. Math. Soc. 27 (2014), no. 2, 389–435. 10.1090/S0894-0347-2013-00775-4Search in Google Scholar

[18] T. Gee, T. Liu and D. Savitt, The weight part of Serre’s conjecture for GL ⁢ ( 2 ) , Forum Math. Pi 3 (2015), Paper No. e2. 10.1017/fmp.2015.1Search in Google Scholar

[19] 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

[20] 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

[21] D. Le, B. V. Le Hung, B. Levin and S. Morra, Local models for Galois deformation rings and applications, Invent. Math. 231 (2023), no. 3, 1277–1488. 10.1007/s00222-022-01163-4Search in Google Scholar

[22] 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

[23] H. Wiersema, Minimal weights of mod 𝑝 Galois representations, PhD thesis, King’s College London, 2021. Search in Google Scholar

[24] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. 10.2307/2118559Search in Google Scholar

[25] The Stacks project authors, The stacks project, http://stacks.math.columbia.edu,2018. Search in Google Scholar

Received: 2023-09-25

Revised: 2024-05-31

Published Online: 2024-07-18

Published in Print: 2024-09-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2024-0045