La conjecture de modularité de Serre : le cas de conducteur 1
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 956, pp. 99-122.

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.

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.

Classification : 11F11, 11F80
Mot clés : formes modulaires, représentations galoisiennes
Keywords: modular forms, Galois representations
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Wintenberger, Jean-Pierre. La conjecture de modularité de Serre : le cas de conducteur $1$, dans Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 956, pp. 99-122. http://archive.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. | EuDML | MR | Zbl

[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. | DOI | MR | Zbl

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

[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 | Zbl

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

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

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

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

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

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

[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. | EuDML | Numdam | MR | Zbl

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

[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

[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 | Zbl

[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. | EuDML | Numdam | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

[29] J.-M. Fontaine (éd.) - Périodes p-adiques (Bures-sur-Yvette 1988), Astérisque, vol. 223, Soc. Math. France, Paris, 1994. | Numdam | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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