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Topics on modular Galois representations modulo prime powers. (English) Zbl 1426.11046
Böckle, Gebhard (ed.) et al., Algorithmic and experimental methods in algebra, geometry, and number theory. Cham: Springer. 741-763 (2017).
Summary: This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.
For the entire collection see [Zbl 1394.14002].

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F80 Galois representations
##### Software:
ArtinAlgebras; Magma
Full Text:
##### References:
 [1] R. Adibhatla, J. Manoharmayum, Higher congruence companion forms. Acta Arith. 156(2), 159-175 (2012) · Zbl 1282.11041 [2] R. Adibhatla, P. Tsaknias, A characterisation of ordinary modular eigenforms with CM, in $$Arithmetic and Geometry$$. London Mathematical Society Lecture Note Series, vol. 420 (Cambridge University Press, Cambridge, 2015), pp. 24-35 · Zbl 1379.11052 [3] G. Böckle, On the density of modular points in universal deformation spaces. Am. J. Math. 123(5), 985-1007 (2001) · Zbl 0984.11025 [4] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24(3-4), 235-265 (1997). Computational algebra and number theory (London, 1993) · Zbl 0898.68039 [5] K. Buzzard, Questions about slopes of modular forms. Astérisque 298, 1-15 (2005). Automorphic forms. I · Zbl 1122.11025 [6] M. Camporino, A. Pacetti, Congruences between modular forms modulo prime powers (2013). arXiv:1312.4925 [7] I. Chen, I. Kiming, G. Wiese, On modular Galois representations modulo prime powers. Int. J. Number Theory 9(1), 91-113 (2013) · Zbl 1305.11047 [8] S.V. Deo, Structure of Hecke algebras of modular forms modulo $$p$$. Algebra Number Theory 11(1), 1-38 (2017) · Zbl 1361.11038 [9] F. Diamond, M. Flach, L. Guo, The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. Éc. Norm. Supér. (4) 37(5), 663-727 (2004) · Zbl 1121.11045 [10] N. Dummigan, Level-lowering for higher congruences of modular forms (2015). http://neil-dummigan.staff.shef.ac.uk/levell8.pdf [11] B. Edixhoven, The weight in Serre’s conjectures on modular forms. Invent. Math. 109(3), 563-594 (1992) · Zbl 0777.11013 [12] M. Emerton, Local-global compatibility in the $$p$$-adic Langlands programme for $$\(\text {GL}_{2/\mathbb {Q}}$$\) (2011). http://www.math.uchicago.edu/ emerton/pdffiles/lg.pdf [13] C. Khare, R. Ramakrishna, Lifting torsion Galois representations. Forum Math. Sigma 3, e14, 37 (2015) · Zbl 1400.11107 [14] C. Khare, J.P. Wintenberger, Serre’s modularity conjecture. I. Invent. Math. 178(3), 485-504 (2009) · Zbl 1304.11041 [15] I. Kiming, N. Rustom, G. Wiese, On certain finiteness questions in the arithmetic of modular forms. J. Lond. Math. Soc. 94(2), 479-502 (2016) · Zbl 1398.11082 [16] M. Kisin, The Fontaine-Mazur conjecture for GL_{2}. J. Am. Math. Soc. 22(3), 641-690 (2009) · Zbl 1251.11045 [17] K.A. Ribet, On modular representations of $$\(\text {Gal}(\overline {\mathbf {Q}}/{\mathbf {Q}})$$\) arising from modular forms. Invent. Math. 100(2), 431-476 (1990) · Zbl 0773.11039 [18] K.A. Ribet, Raising the levels of modular representations, in $$Séminaire de Théorie des Nombres, Paris 1987-88$$. Progress in Mathematics, vol. 81 (Birkhäuser Boston, Boston, MA, 1990), pp. 259-271 [19] J. Tilouine, Hecke algebras and the Gorenstein property, in $$Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995)$$ (Springer, New York, 1997), pp. 327-342 · Zbl 1155.11330 [20] P. Tsaknias, On higher congruences of modular Galois representations, Ph.D. thesis, University of Sheffield, 2009 [21] X.T.i. Ventosa, G. Wiese, Computing congruences of modular forms and Galois representations modulo prime powers, in $$Arithmetic, Geometry, Cryptography and Coding Theory 2009$$. Contemporary Mathematics, vol. 521 (American Mathematical Society, Providence, RI, 2010), pp. 145-166 · Zbl 1219.11069 [22] G. Wiese, Multiplicities of Galois representations of weight one. Algebra Number Theory 1(1), 67-85 (2007). With an appendix by Niko Naumann · Zbl 1171.11036 [23] G. Wiese, Magma package ArtinAlgebras (2008). http://math.uni.lu/ wiese/programs/ArtinAlgebras [24] G. Wiese, Magma package pAdicAlgebras (2014). http://math.uni.lu/ wiese/programs/pAdicAlgebras [25] G. Wiese, Magma package
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