Dans cet article, nous démontrons que la représentation de dans GL d’image dans PGL correspondant à l’exemple dans [B-K] est modulaire. Cette représentation est de conducteur et de déterminant . La modularité de cette représentation n’était pas encore prouvée ; en effet, elle ne vérifie pas les hypothèses des théorèmes de [B-D-SB-T] et [Tay2].
In this paper, we prove that the representation from in GL with image in PGL corresponding to the example in [B-K] is modular. This representation has conductor and determinant ; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].
@article{JTNB_2000__12_2_475_0, author = {Jehanne, Arnaud and M\"uller, Michael}, title = {Modularity of an odd icosahedral representation}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {475--482}, publisher = {Universit\'e Bordeaux I}, volume = {12}, number = {2}, year = {2000}, zbl = {01626657}, mrnumber = {1823197}, language = {en}, url = {http://www.numdam.org/item/JTNB_2000__12_2_475_0/} }
TY - JOUR AU - Jehanne, Arnaud AU - Müller, Michael TI - Modularity of an odd icosahedral representation JO - Journal de Théorie des Nombres de Bordeaux PY - 2000 DA - 2000/// SP - 475 EP - 482 VL - 12 IS - 2 PB - Université Bordeaux I UR - http://www.numdam.org/item/JTNB_2000__12_2_475_0/ UR - https://zbmath.org/?q=an%3A01626657 UR - https://www.ams.org/mathscinet-getitem?mr=1823197 LA - en ID - JTNB_2000__12_2_475_0 ER -
Jehanne, Arnaud; Müller, Michael. Modularity of an odd icosahedral representation. Journal de Théorie des Nombres de Bordeaux, Tome 12 (2000) no. 2, pp. 475-482. http://www.numdam.org/item/JTNB_2000__12_2_475_0/
[B-K] A table of A5-fields. In [Fre2], 37-46. | MR 1322317 | Zbl 0837.11066
, ,[Buh] Icosahedral Galois representations. Lecture note in math. 654, Springer-Verlag (1978). | MR 506171 | Zbl 0374.12002
,[B-D-SB-T] On icosahedral Artin representations. Preprint.
, , ,[B-S] A mod 5 approach to the modularity of icosahedral Galois representations. To appear in Pacific J. Math. | MR 1897901 | Zbl 1054.11028
,[B-T] Companion forms and weight one forms. Annals of Math 149 (1999), 905-919. | MR 1709306 | Zbl 0965.11019
,[Cre] Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge (1992). | MR 1201151 | Zbl 0758.14042
,[D-D-T] Fermat's Last Theorem. In: Current Development in Mathematics, International Press Boston (1995), 1-170. | Zbl 0877.11035
, ,[D-S] Formes modulaires de poids 1. Ann. scient. Éc. Norm. Sup. 4e série 7 (1974), 507-530. | Numdam | MR 379379 | Zbl 0321.10026
, ,[Edi] The weight in Serre's conjecture on modular forms. Invent. math. 109 (1992), 563-594. | MR 1176206 | Zbl 0777.11013
,[Fre1] Construction and arithmetical applications of modular forms of low weight. CRM Proceeding and Lecture Notes 4 (1994), 1-21. | MR 1260951 | Zbl 0814.11027
,[Fre2] On Artin's conjecture for odd 2-dimensional representations. Lecture note in math. 1585, Springer-Verlag (1994). | MR 1322315 | Zbl 0801.00004
,[Jeh] Realization over Q of the groups Ã5 and Â5. J. Number Th., to appear. | Zbl 0984.12003
,[Kim] On the experimental verification of the Artin conjecture for 2-dimensional odd Galois representations over Q. Lifting of 2-dimensional projective Galois representations over Q. In [Fre2], 1-36. | MR 1322316 | Zbl 0839.11056
,[K-W] Examples of 2-dimensional odd Galois repesentation of A5type over Q satisfying Artin conjecture. In [Fre2], 109-121. | MR 1322321 | Zbl 0841.11065
, ,[Lan] Base change for GL(2). Annals of Math. Studies 96, Princeton Univ. Press, Princeton (1980). | MR 574808 | Zbl 0444.22007
[Mer] Universal Fourier expansions of modular forms. In [Fre2], 59-94. | MR 1322319 | Zbl 0844.11033
,[PARI] PARI computation system.
, , , , ,[Rib] Report on mod representations of Gal(/Q). Proc. of Symp. in Pure Math. 55, Part 2 (1994), 179-230. | MR 1265566 | Zbl 0822.11034
,[Ser1] Modular forms of weight one and Galois representations. In Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham,1975), pp.193-268. Edited by A. Frohlich, Academic Press, London (1977). | MR 450201 | Zbl 0366.10022
,[Ser2] Corps locaux. (3ème edition), Hermann, Paris (1980). | MR 354618
,[Ser3] Sur les représentations modulaires de degré 2 de Gal(/Q). Duke math. Jour. 54 (1987), 179-230. | MR 885783 | Zbl 0641.10026
,[SB-T] Mod 2 and mod 5 icosahedral representations. J. AMS 10 (1997), 283-298. | MR 1415322 | Zbl 1015.11019
, ,[Shi] Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey (1971). | MR 314766 | Zbl 0872.11023
,[Tay1] Icosahedral Galois representations. Pacific Jour. of Math., Olga Taussky-Todd memorial issue (1997), 337-347. | MR 1610820 | Zbl 0942.11031
[Tay2] Icosahedral Galois representations II. Preprint.
[Tun] Artin's conjecture for representations of octahedral type. Bull. AMS 5 (1981), 173-175. | MR 621884 | Zbl 0475.12016