La conjecture de modularité de Serre : le cas de conducteur 1  [ The conjecture of modularity of Serre: the case of conductor 1 ]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 956, p. 99-122

The conjecture says that an irreducible continuous odd representation of the Galois group of Q in a 2-dimensional vector space over a finite field F comes from a modular form. C. Khare just proved it in the case where the representation is unramified outside the characteristic of F.

La conjecture dit qu’une représentation continue irréductible impaire du groupe de Galois de Q dans un espace vectoriel de dimension 2 sur un corps fini F de caractéristique p provient d’une forme modulaire. C. Khare vient de la prouver pour les représentations qui sont non ramifiées hors de p.

Classification:  11F11,  11F80
Keywords: modular forms, Galois representations
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     author = {Wintenberger, Jean-Pierre},
     title = {La conjecture de modularit\'e de Serre : le cas de conducteur $1$},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:956},
     pages = {99-122},
     zbl = {1200.11036},
     mrnumber = {2359041},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2005-2006__48__99_0}
}
Wintenberger, Jean-Pierre. La conjecture de modularité de Serre : le cas de conducteur $1$, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 956, pp. 99-122. http://www.numdam.org/item/SB_2005-2006__48__99_0/

[1] V. A. Abrashkin - “Ramification in étale cohomology”, Invent. Math. 101 (1990), no. 3, p. 631-640. | MR 1062798 | Zbl 0761.14006

[2] J. Arthur & L. Clozel - Simple algebras, base change, and the advanced theory of the trace formula, Annals of Math. Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. | Article | MR 1007299 | Zbl 0682.10022

[3] L. Berger - “Limites de représentations cristallines”, Compos. Math. 140 (2004), no. 6, p. 1473-1498. | MR 2098398 | Zbl 1071.11067

[4] L. Berger, H. Li & H. J. Zhu - “Construction of some families of 2-dimensional crystalline representations”, Math. Ann. 329 (2004), no. 2, p. 365-377. | MR 2060368 | Zbl 1085.11028

[5] D. Blasius & J. D. Rogawski - “Motives for Hilbert modular forms”, Invent. Math. 114 (1993), no. 1, p. 55-87. | MR 1235020 | Zbl 0829.11028

[6] G. Böckle - “A local-to-global principle for deformations of Galois representations”, J. reine angew. Math. 509 (1999), p. 199-236. | MR 1679172 | Zbl 1040.11039

[7] -, “Presentations of universal deformation rings”, preprint, p. 1-27, 2005.

[8] C. Breuil - “Une remarque sur les représentations locales p-adiques et les congruences entre formes modulaires de Hilbert”, Bull. Soc. Math. France 127 (1999), no. 3, p. 459-472. | Numdam | MR 1724405 | Zbl 0933.11028

[9] C. Breuil & 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, p. 205-310, avec un appendice de Guy Henniart. | MR 1944572 | Zbl 1042.11030

[10] S. Brueggeman - “The nonexistence of certain Galois extensions unramified outside 5, J. Number Theory 75 (1999), no. 1, p. 47-52. | MR 1670870 | Zbl 0930.11036

[11] A. Brumer & K. Kramer - “Non-existence of certain semistable abelian varieties”, Manuscripta Math. 106 (2001), no. 3, p. 291-304. | MR 1869222 | Zbl 1073.14544

[12] K. Buzzard & R. Taylor - “Companion forms and weight one forms”, Ann. of Math. (2) 149 (1999), no. 3, p. 905-919. | MR 1709306 | Zbl 0965.11019

[13] H. Carayol - “Sur les représentations l-adiques associées aux formes modulaires de Hilbert”, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, p. 409-468. | Numdam | MR 870690 | Zbl 0616.10025

[14] -, “Sur les représentations galoisiennes modulo l attachées aux formes modulaires”, Duke Math. J. 59 (1989), no. 3, p. 785-801. | MR 1046750 | Zbl 0703.11027

[15] -, “Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet” 1991), Contemp. Math., vol. 165, Amer. Math. Soc., Providence, 1994, p. 213-237. | Zbl 0812.11036

[16] H. Darmon, F. Diamond & R. Taylor - “Fermat's last theorem”, in Current developments in mathematics, Internat. Press, Cambridge, MA, 1995, p. 1-154. | MR 1474977 | Zbl 0877.11035

[17] P. Deligne - “Formes modulaires et représentations -adiques”, in Séminaire Bourbaki, Lect. Notes in Math., vol. 179, Springer, Berlin, 1971, exp. no 355, p. 139-172. | Numdam | Zbl 0206.49901

[18] -, “Les constantes des équations fonctionnelles des fonctions L, in Modular functions of one variable, II (Antwerp 1972), Lect. Notes in Math., vol. 349, Springer, Berlin, 1973, p. 501-597. | MR 349635 | Zbl 0271.14011

[19] P. Deligne & J.-P. Serre - “Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), p. 507-530 (1975). | Numdam | MR 379379 | Zbl 0321.10026

[20] F. Diamond - “An extension of Wiles' results”, in Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, p. 475-489. | MR 1638490 | Zbl 0917.11021

[21] F. Diamond & R. Taylor - “Lifting modular mod l representations”, Duke Math. J. 74 (1994), no. 2, p. 253-269. | MR 1272977 | Zbl 0809.11025

[22] -, “Nonoptimal levels of mod l modular representations”, Invent. Math. 115 (1994), no. 3, p. 435-462. | MR 1262939 | Zbl 0847.11025

[23] L. V. Dieulefait - “Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture”, J. reine angew. Math. 577 (2004), p. 147-151. | MR 2108216 | Zbl 1065.11037

[24] B. Edixhoven - “The weight in Serre's conjectures on modular forms”, Invent. Math. 109 (1992), no. 3, p. 563-594. | MR 1176206 | Zbl 0777.11013

[25] -, “Serre's conjecture”, in Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, p. 209-242. | MR 1638480 | Zbl 0918.11023

[26] J. S. Ellenberg - “Serre’s conjecture over 𝔽 9 , Ann. of Math. (2) 161 (2005), no. 3, p. 1111-1142. | MR 2180399 | Zbl 1153.11312

[27] J.-M. Fontaine - “Représentations l-adiques potentiellement semi-stables”, in [29], p. 321-347. | MR 1293977 | Zbl 0873.14020

[28] -, “Il n’y a pas de variété abélienne sur 𝐙, Invent. Math. 81 (1985), no. 3, p. 515-538. | MR 807070 | Zbl 0612.14043

[29] -('ed.) - Périodes p-adiques (Bures-sur-Yvette 1988), Astérisque, vol. 223, Soc. Math. France, Paris, 1994. | Numdam | Zbl 0802.00019

[30] J.-M. Fontaine & G. Laffaille - “Construction de représentations p-adiques”, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, p. 547-608. | Numdam | MR 707328 | Zbl 0579.14037

[31] J.-M. Fontaine & B. Mazur - “Geometric Galois representations”, in Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong 1993), Ser. Number Theory, vol. I, Internat. Press, Cambridge, MA, 1995, p. 41-78. | MR 1363495 | Zbl 0839.14011

[32] B. H. Gross - “A tameness criterion for Galois representations associated to modular forms (mod p)”, Duke Math. J. 61 (1990), no. 2, p. 445-517. | MR 1074305 | Zbl 0743.11030

[33] H. Hida - “On p-adic Hecke algebras for GL 2 over totally real fields”, Ann. of Math. (2) 128 (1988), no. 2, p. 295-384. | MR 960949 | Zbl 0658.10034

[34] C. Khare - “Serre's modularity conjecture : the level one case”,Duke Math. J. 134 (2006), no. 3, p. 557-589. | MR 2254626 | Zbl 1105.11013

[35] -, “Serre's modularity conjecture : a survey of the level one case”. Preprint 2006, http://www.math.utah.edu/~shekhar/papers.html, à paraître dans les Actes de “L-functions and Galois representations”(Durham 2004).

[36] C. Khare & J.-P. Wintenberger - “On Serre’s reciprocity conjecture for 2-dimensional mod p representations of the Galois group G , arXiv : math.NT/0412076, 2004. | Zbl 1196.11076

[37] M. Kisin - “Modularity of some geometric Galois representations”. Preprint 2005, http://www.math.uchicago.edu/~kisin/preprints.html, à paraître dans les Actes de “L-functions and Galois representations”(Durham 2004). | MR 2392362 | Zbl 1171.11035

[38] J. Manoharmayum - “Serre's conjecture for mod 7 Galois representations”, in Modular curves and abelian varieties, Progr. Math., vol. 224, Birkhäuser, Basel, 2004, p. 141-149. | MR 2058648 | Zbl 1069.11020

[39] B. Mazur - “An introduction to the deformation theory of Galois representations”, in Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, p. 243-311. | MR 1638481 | Zbl 0901.11015

[40] L. Moret-Bailly - “Groupes de Picard et problèmes de Skolem. I, II”, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, p. 161-179, 181-194. | Numdam | MR 1005158 | Zbl 0704.14015

[41] R. Ramakrishna - “Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur”, Ann. of Math. (2) 156 (2002), no. 1, p. 115-154. | MR 1935843 | Zbl 1076.11035

[42] K. A. Ribet - “Galois representations attached to eigenforms with Nebentypus”, in Modular functions of one variable, V (Bonn 1976), Lect. Notes in Math., vol. 601, Springer, Berlin, 1977, p. 17-51. | MR 453647 | Zbl 0363.10015

[43] -, “Report on mod l representations of Gal (𝐐 ¯/𝐐), in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, 1994, p. 639-676. | MR 1265566

[44] K. A. Ribet & W. A. Stein - “Lectures on Serre's conjectures”, in Arithmetic algebraic geometry (Park City 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, 2001, p. 143-232. | MR 1860042 | Zbl 1160.11327

[45] T. Saito - “Hilbert modular forms and p-adic Hodge theory”. Preprint 2004, arXiv : math/0612077. | MR 2551990 | Zbl pre05625823

[46] -, “Modular forms and p-adic Hodge theory”, Invent. Math. 129 (1997), no. 3, p. 607-620. | MR 1465337 | Zbl 0877.11034

[47] D. Savitt - “On a conjecture of Conrad, Diamond, and Taylor”, Duke Math. J. 128 (2005), no. 1, p. 141-197. | MR 2137952 | Zbl 1101.11017

[48] R. Schoof - “Abelian varieties over with bad reduction in one prime only”, Compos. Math. 141 (2005), no. 4, p. 847-868. | MR 2148199 | Zbl 1173.11333

[49] J.-P. Serre - “Formes modulaires et fonctions zêta p-adiques”, in Modular functions of one variable, III (Antwerp 1972), Lect. Notes in Math., vol. 350, Springer, Berlin, 1973, p. 191-268. | MR 404145 | Zbl 0277.12014

[50] -, “Valeurs propres des opérateurs de Hecke modulo l, in Journées Arithmétiques (Bordeaux 1974), Astérisque, vol. 24-25, Soc. Math. France, Paris, 1975, p. 109-117. | Numdam | Zbl 0305.10021

[51] -, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986, 1972-1984. | Zbl 0849.01049

[52] -, “Sur les représentations modulaires de degré 2 de Gal (𝐐 ¯/𝐐), Duke Math. J. 54 (1987), no. 1, p. 179-230. | Zbl 0641.10026

[53] C. M. Skinner & A. J. Wiles - “Residually reducible representations and modular forms”, Publ. Math. Inst. Hautes Études Sci. 89 (2000), p. 5-126. | MR 1793414 | Zbl 1005.11030

[54] -, “Base change and a problem of Serre”, Duke Math. J. 107 (2001), no. 1, p. 15-25. | MR 1815248 | Zbl 1016.11017

[55] -, “Nearly ordinary deformations of irreducible residual representations”, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 1, p. 185-215. | Numdam | MR 1928993 | Zbl 1024.11036

[56] H. P. F. Swinnerton-Dyer - “On l-adic representations and congruences for coefficients of modular forms”, in Modular functions of one variable, III (Antwerp 1972), Lect. Notes in Math., vol. 350, Springer, Berlin, 1973, p. 1-55. | MR 406931 | Zbl 0267.10032

[57] J. Tate - “The non-existence of certain Galois extensions of 𝐐 unramified outside 2, in Arithmetic geometry (Tempe, AZ, 1993), Contemp. Math., vol. 174, Amer. Math. Soc., Providence, 1994, p. 153-156. | MR 1299740 | Zbl 0814.11057

[58] R. Taylor - “On Galois representations associated to Hilbert modular forms, I”, Invent. Math. 98 (1989), no. 2, p. 265-280. | MR 1016264 | Zbl 0705.11031

[59] -, “On Galois representations associated to Hilbert modular forms, II”, in Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong 1993), Ser. Number Theory, vol. I, Internat. Press, Cambridge, MA, 1995, p. 41-78. | MR 1363502 | Zbl 0836.11017

[60] -, “On the meromorphic continuation of degree two L-functions”, Doc. Math., extra volume : John H. Coates' Sixtieth Birthday (2006), p. 729-779. | MR 2290604 | Zbl 1138.11051

[61] -, “Remarks on a conjecture of Fontaine and Mazur”, J. Inst. Math. Jussieu 1 (2002), no. 1, p. 125-143. | MR 1954941 | Zbl 1047.11051

[62] -, “On icosahedral Artin representations. II”, Amer. J. Math. 125 (2003), no. 3, p. 549-566. | MR 1981033 | Zbl 1031.11031

[63] -, “Galois representations”, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, p. 73-119. | Numdam | MR 2060030 | Zbl 1074.11030

[64] R. Taylor & A. Wiles - “Ring-theoretic properties of certain Hecke algebras”, Ann. of Math. (2) 141 (1995), no. 3, p. 553-572. | MR 1333036 | Zbl 0823.11030

[65] T. Tsuji - Semi-stable conjecture of Fontaine-Jannsen : a survey, Astérisque, vol. 279, Soc. Math. France, Paris, 2002, Cohomologies p-adiques et applications arithmétiques, II. | Numdam | MR 1922833 | Zbl 1041.14003

[66] J. Tunnell - “Artin's conjecture for representations of octahedral type”, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, p. 173-175. | MR 621884 | Zbl 0475.12016

[67] A. Wiles - “Modular elliptic curves and Fermat's last theorem”, Ann. of Math. (2) 141 (1995), no. 3, p. 443-551. | MR 1333035 | Zbl 0823.11029

[68] J.-P. Wintenberger - “On p-Adic Representations of G Q , Doc. Math., extra volume : John H. Coates' Sixtieth Birthday (2006), p. 819-827. | MR 2290606 | Zbl 1137.11070