Modularity of an odd icosahedral representation

Jehanne, Arnaud; Müller, Michael

Jehanne, Arnaud ; Müller, Michael

Journal de Théorie des Nombres de Bordeaux, Tome 12 (2000) no. 2, pp. 475-482.

Résumé
Abstract

Dans cet article, nous démontrons que la représentation $ρ$ de $G_{ℚ}$ dans GL $_{2} (ℂ)$ d’image $A_{5}$ dans PGL $_{2} (A_{5})$ correspondant à l’exemple $16$ dans [B-K] est modulaire. Cette représentation est de conducteur $5203$ et de déterminant $χ_{- 43}$ . 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 $ρ$ from $G_{ℚ}$ in GL $_{2} (ℂ)$ with image $A_{5}$ in PGL $_{2} (A_{5})$ corresponding to the example $16$ in [B-K] is modular. This representation has conductor $5203$ and determinant $χ_{- 43}$ ; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].

MR 1823197 | Zbl 01626657

BibTeX
RIS

@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/

Bibliographie
Cité par

[B-K] J. Basmaji, I. Kiming, A table of A5-fields. In [Fre2], 37-46. | MR 1322317 | Zbl 0837.11066

[Buh] J. Buhler, Icosahedral Galois representations. Lecture note in math. 654, Springer-Verlag (1978). | MR 506171 | Zbl 0374.12002

[B-D-SB-T] K. Buzzard, M. Dickinson, N. Shepherd-Barron, R. Taylor. On icosahedral Artin representations. Preprint.

[B-S] K. Buzzard, W. Stein. A mod 5 approach to the modularity of icosahedral Galois representations. To appear in Pacific J. Math. | MR 1897901 | Zbl 1054.11028

[B-T] K. Buzzard, R. Taylor Companion forms and weight one forms. Annals of Math 149 (1999), 905-919. | MR 1709306 | Zbl 0965.11019

[Cre] J. Cremona, Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge (1992). | MR 1201151 | Zbl 0758.14042

[D-D-T] H. Darmon, F. Diamond, R. Taylor. Fermat's Last Theorem. In: Current Development in Mathematics, International Press Boston (1995), 1-170. | Zbl 0877.11035

[D-S] P. Deligne, J.-P. Serre, Formes modulaires de poids 1. Ann. scient. Éc. Norm. Sup. 4e série 7 (1974), 507-530. | Numdam | MR 379379 | Zbl 0321.10026

[Edi] S. Edixhoven, The weight in Serre's conjecture on modular forms. Invent. math. 109 (1992), 563-594. | MR 1176206 | Zbl 0777.11013

[Fre1] G. Frey, 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] G. Frey, On Artin's conjecture for odd 2-dimensional representations. Lecture note in math. 1585, Springer-Verlag (1994). | MR 1322315 | Zbl 0801.00004

[Jeh] A. Jehanne, Realization over Q of the groups Ã5 and Â5. J. Number Th., to appear. | Zbl 0984.12003

[Kim] I. Kiming, 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] I. Kiming, X. Wang, Examples of 2-dimensional odd Galois repesentation of A5type over Q satisfying Artin conjecture. In [Fre2], 109-121. | MR 1322321 | Zbl 0841.11065

[Lan] R.-P. Langlands Base change for GL(2). Annals of Math. Studies 96, Princeton Univ. Press, Princeton (1980). | MR 574808 | Zbl 0444.22007

[Mer] L. Merel, Universal Fourier expansions of modular forms. In [Fre2], 59-94. | MR 1322319 | Zbl 0844.11033

[PARI] C. Batut,K. Belabas,D. Bernardi,H. Cohen,M. Olivier, PARI computation system.

[Rib] K. Ribet, 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] J.-P. Serre, 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] J.-P. Serre, Corps locaux. (3ème edition), Hermann, Paris (1980). | MR 354618

[Ser3] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(/Q). Duke math. Jour. 54 (1987), 179-230. | MR 885783 | Zbl 0641.10026

[SB-T] N. Shepherd-Barron, R. Taylor, Mod 2 and mod 5 icosahedral representations. J. AMS 10 (1997), 283-298. | MR 1415322 | Zbl 1015.11019

[Shi] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey (1971). | MR 314766 | Zbl 0872.11023

[Tay1] R. Taylor Icosahedral Galois representations. Pacific Jour. of Math., Olga Taussky-Todd memorial issue (1997), 337-347. | MR 1610820 | Zbl 0942.11031

[Tay2] R. Taylor Icosahedral Galois representations II. Preprint.

[Tun] J. Tunnel Artin's conjecture for representations of octahedral type. Bull. AMS 5 (1981), 173-175. | MR 621884 | Zbl 0475.12016