Modular representations of GL₂(𝔽_𝑞) using calculus

References

[1] J. L. Alperin, Local Representation Theory, Cambridge Stud. Adv. Math. 11, Cambridge University, Cambridge, 1986. Search in Google Scholar

[2] B. Arsovski, On the reductions of certain two-dimensional crystalline representations, Doc. Math. 26 (2021), 1929–1979. 10.4171/dm/861Search in Google Scholar

[3] N. Avni, U. Onn, A. Prasad and L. Vaserstein, Similarity classes of 3 × 3 matrices over a local principal ideal ring, Comm. Algebra 37 (2009), no. 8, 2601–2615. 10.1080/00927870902747266Search in Google Scholar

[4] J. Bergdall and B. Levin, Reductions of some two-dimensional crystalline representations via Kisin modules, Int. Math. Res. Not. IMRN 2022 (2022), no. 4, 3170–3197. 10.1093/imrn/rnaa240Search in Google Scholar

[5] J. Bergdall, B. Levin and T. Liu, Reductions of 2-dimensional semistable representations with large ℒ-invariant, J. Inst. Math. Jussieu 22 (2023), no. 6, 2619–2644. 10.1017/S1474748022000081Search in Google Scholar

[6] L. Berger, H. Li and H. J. Zhu, Construction of some families of 2-dimensional crystalline representations, Math. Ann. 329 (2004), no. 2, 365–377. 10.1007/s00208-004-0529-ySearch in Google Scholar

[7] S. Bhattacharya, Reduction of certain crystalline representations and local constancy in the weight space, J. Théor. Nombres Bordeaux 32 (2020), no. 1, 25–47. 10.5802/jtnb.1110Search in Google Scholar

[8] S. Bhattacharya and E. Ghate, Reductions of Galois representations for slopes in ( 1 , 2 ) , Doc. Math. 20 (2015), 943–987. 10.4171/dm/510Search in Google Scholar

[9] S. Bhattacharya, E. Ghate and S. Rozensztajn, Reductions of Galois representations of slope 1, J. Algebra 508 (2018), 98–156. 10.1016/j.jalgebra.2018.04.023Search in Google Scholar

[10] S. Bhattacharya, E. Ghate and R. Vangala, Reductions of crystalline representations of slope less than 𝑝, in preparation. Search in Google Scholar

[11] C. Breuil, Sur quelques représentations modulaires et 𝑝-adiques de GL 2 ⁢ ( Q p ) . I, Compos. Math. 138 (2003), no. 2, 165–188. 10.1017/S1474748003000021Search in Google Scholar

[12] C. Breuil, Representations of Galois and of GL 2 in characteristic 𝑝, Graduate course, Columbia University, 2007, https://www.imo.universite-paris-saclay.fr/{̃}christophe.breuil/publications/new-york.pdf. Search in Google Scholar

[13] C. Breuil and A. Mézard, Représentations semi-stables de GL 2 ⁢ ( Q p ) , demi-plan 𝑝-adique et réduction modulo 𝑝, Astérisque 331 (2010), 117–178. Search in Google Scholar

[14] C. Breuil and V. Paškūnas, Towards a modulo 𝑝 Langlands correspondence for GL 2 , Mem. Amer. Math. Soc. 216 (2012), no. 1016, 1–114. 10.1090/S0065-9266-2011-00623-4Search in Google Scholar

[15] K. Buzzard and T. Gee, Explicit reduction modulo 𝑝 of certain two-dimensional crystalline representations, Int. Math. Res. Not. IMRN 2009 (2009), no. 12, 2303–2317. 10.1093/imrn/rnp017Search in Google Scholar

[16] K. Buzzard and T. Gee, Explicit reduction modulo 𝑝 of certain 2-dimensional crystalline representations, II, Bull. Lond. Math. Soc. 45 (2013), no. 4, 779–788. 10.1112/blms/bds128Search in Google Scholar

[17] A. Chitrao, An Iwahori theoretic mod 𝑝 local Langlands correspondence, Canad. Math. Bull., to appear. Search in Google Scholar

[18] A. Chitrao and E. Ghate, Reductions of semi-stable representations using the Iwahori mod 𝑝 local Langlands correspondence, preprint (2024), https://arxiv.org/abs/2311.03740. Search in Google Scholar

[19] A. Chitrao, E. Ghate and S. Yasuda, Semi-stable representations as limits of crystalline representations, Algebra Number Theory, to appear. Search in Google Scholar

[20] F. Diamond, A correspondence between representations of local Galois groups and Lie-type groups, 𝐿-Functions and Galois Representations, London Math. Soc. Lecture Note Ser. 320, Cambridge University, Cambridge (2007), 187–206. 10.1017/CBO9780511721267.006Search in Google Scholar

[21] A. Ganguli and E. Ghate, Reductions of Galois representations via the mod ⁢ p local Langlands correspondence, J. Number Theory 147 (2015), 250–286. 10.1016/j.jnt.2014.07.011Search in Google Scholar

[22] A. Ganguli and S. Kumar, On the local constancy of certain mod ⁢ p Galois representations, Res. Number Theory 10 (2024), no. 2, Paper No. 52. 10.1007/s40993-024-00538-1Search in Google Scholar

[23] E. Ghate, A zig-zag conjecture and local constancy for Galois representations, RIMS Kôkyûroku Bessatsu B86 (2021), 249–268. Search in Google Scholar

[24] E. Ghate, Zig-zag for Galois representations, preprint (2024), https://arxiv.org/abs/2211.12114. Search in Google Scholar

[25] E. Ghate and V. Rai, Reductions of Galois representations of slope 3 2 , Kyoto J. Math., to appear. Search in Google Scholar

[26] E. Ghate and R. Vangala, The monomial lattice in modular symmetric power representations, Algebr. Represent. Theory 25 (2022), no. 1, 121–185. 10.1007/s10468-020-10013-xSearch in Google Scholar

[27] D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425–475. 10.1016/0021-8693(78)90116-3Search in Google Scholar

[28] J. Grünwald, Über duale Zahlen und ihre Anwendung in der Geometrie, Monatsh. Math. Phys. 17 (1906), no. 1, 81–136. 10.1007/BF01697639Search in Google Scholar

[29] L. Guerberoff and C. Park, Semistable deformation rings in even Hodge-Tate weights, Pacific J. Math. 298 (2019), no. 2, 299–374. 10.2140/pjm.2019.298.299Search in Google Scholar

[30] B. Haastert and J. C. Jantzen, Filtrations of the discrete series of SL 2 ⁢ ( q ) via crystalline cohomology, J. Algebra 132 (1990), no. 1, 77–103. 10.1016/0021-8693(90)90253-KSearch in Google Scholar

[31] Y. I. Hendel, On the universal mod ⁢ p supersingular quotients for GL 2 ⁢ ( F ) over F ̄ p for a general F / Q p , J. Algebra 519 (2019), 1–38. 10.1016/j.jalgebra.2018.10.021Search in Google Scholar

[32] W. Lee and C. Park, Semi-stable deformation rings in even Hodge–Tate weights: The residually reducible case, Int. J. Number Theory 18 (2022), no. 10, 2171–2209. 10.1142/S1793042122501111Search in Google Scholar

[33] E. Nagel and A. Pande, Reductions of modular Galois representations of slope ( 2 , 3 ) , preprint (2019), https://arxiv.org/abs/1801.08820. Search in Google Scholar

[34] D. Prasad, Notes on modular representations of 𝑝-adic groups, and the Langlands correspondence, preprint (2014), https://arxiv.org/abs/1404.6631. Search in Google Scholar

[35] D. A. Reduzzi, Reduction mod 𝑝 of cuspidal representations of GL 2 ⁢ ( F p n ) and symmetric powers, J. Algebra 324 (2010), no. 12, 3507–3531. 10.1016/j.jalgebra.2010.09.025Search in Google Scholar

[36] S. Rozensztajn, Asymptotic values of modular multiplicities for GL 2 , J. Théor. Nombres Bordeaux 26 (2014), no. 2, 465–482. 10.5802/jtnb.875Search in Google Scholar