Duke Mathematical Journal

Sur les représentations modulaires de degré $2$ de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$

Jean-Pierre Serre

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J. Volume 54, Number 1 (1987), 179-230.

Dates
First available: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077305511

Mathematical Reviews number (MathSciNet)
MR885783

Zentralblatt MATH identifier
0641.10026

Digital Object Identifier
doi:10.1215/S0012-7094-87-05413-5

Subjects
Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 14G15: Finite ground fields 14G25: Global ground fields 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

Citation

Serre, Jean-Pierre. Sur les représentations modulaires de degré 2 de Gal ( Q ¯ / Q ) . Duke Mathematical Journal 54 (1987), no. 1, 179--230. doi:10.1215/S0012-7094-87-05413-5. http://projecteuclid.org/euclid.dmj/1077305511.


Export citation

References

  • [1] E. Artin, Zur Theorie der $L$-Reihen mit allgemeinen Gruppencharakteren, Hamb. Abh. 8 (1930), 292–306, ( = coll. P., 165–179).
  • [2] A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Crelle Math. 365 (1986), 192–220.
  • [3] A. O. L. Atkin and W. Li, Twists of newforms and pseudo-eigenvalues of $W$-operators, Invent. Math. 48 (1978), no. 3, 221–243.
  • [4] B. J. Birch and W. Kuyk, Modular Forms of One Variable, IV, Lect. Notes in Math., vol. 476, Springer-Verlag, 1975.
  • [5] S. Bloch, Algebraic cycles and values of $L$-functions. II, Duke Math. J. 52 (1985), no. 2, 379–397.
  • [6] A. Brumer and K. Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715–743.
  • [7] J. P. Buhler, Icosahedral Galois Representations, Lecture Notes in Mathematics, vol. 654, Springer-Verlag, Berlin, 1978.
  • [8] H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468.
  • [9] P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 501–597. Lecture Notes in Math., Vol. 349.
  • [10] P. Deligne, Valeurs de fonctions $L$ et périodes d'intégrales, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 313–346.
  • [11] P. Deligne and J.-P. Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975).
  • [12] P. Dénes, Über die Diophantische Gleichung $x\sp l+y\sp l=cz\sp l$, Acta Math. 88 (1952), 241–251.
  • [13]1 G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366.
  • [13]2 G. Faltings, Erratum: “Finiteness theorems for abelian varieties over number fields”, Invent. Math. 75 (1984), no. 2, 381.
  • [14] G. Faltings, G. Wüstholz, et al., Rational points, Aspects of Mathematics, E6, Friedr. Vieweg & Sohn, Braunschweig, 1984.
  • [15] J.-M. Fontaine, Il n'y a pas de variété abélienne sur $\bf Z$, Invent. Math. 81 (1985), no. 3, 515–538.
  • [16] G. Frey, Rationale Punkte auf Fermatkurven und getwisteten Modulkurven, J. Crelle Math. 331 (1982), 185–191.
  • [17] G. Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1, iv+40.
  • [18] A. Grothendieck, Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972.
  • [19] Y. Hellegouarch, Courbes elliptiques et équation de Fermat, Thèse, Besançon, 1972.
  • [20] A. Hurwitz, Über endliche Gruppen linearer Substitutionen, welche in der Theorie der elliptischen Transzendenten aufreten, Math. Ann. 27 (1886), 183–233, = Math. W. XI.
  • [21] N. Jochnowitz, A study of the local components of the Hecke algebra mod $l$, Trans. Amer. Math. Soc. 270 (1982), no. 1, 253–267.
  • [22] N. Jochnowitz, Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math. Soc. 270 (1982), no. 1, 269–285.
  • [23] N. M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 69–190. Lecture Notes in Mathematics, Vol. 350.
  • [24] N. M. Katz, A result on modular forms in characteristic $p$, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, 53–61. Lecture Notes in Math., Vol. 601.
  • [25] S. LaMacchia, Polynomials with Galois group $\rm PSL(2,\,7)$, Comm. in Algebra 8 (1980), no. 10, 983–992.
  • [26] R. P. Langlands, Base change for $\rm GL(2)$, Annals of Mathematics Studies, vol. 96, Princeton Univ. Press, Princeton, N.J., 1980.
  • [27] W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.
  • [28] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).
  • [29] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.
  • [30] J.-F. Mestre, Courbes hyperelliptiques à multiplications réelles, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988), Univ. Bordeaux I, Talence, 1988, Exp. No. 34, 6.
  • [31] J.-F. Mestre, La méthode des graphes. Examples et applications, Taniguchi Symp., Kyoto, 1986, à paraître.
  • [32] I. Miyawaki, Elliptic curves of prime power conductor with $\bf Q$-rational points of finite order, Osaka J. Math. 10 (1973), 309–323.
  • [33] O. Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. II, Math. Nachr. 56 (1973), 269–280.
  • [34] J. Tate and F. Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1–21.
  • [35] M. Raynaud, Schémas en groupes de type $(p,\dots, p)$, Bull. Soc. Math. France 102 (1974), 241–280.
  • [36] K. A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804.
  • [37] C. Schoen, On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle, J. Crelle. Math. 364 (1986), 85–111.
  • [38] J.-P. Serre, Corps Locaux, 3ème édition ed., Hermann, Paris, 1968.
  • [39] J.-P. Serre, Représentations linéaires des groupes finis, 3ème édition ed., Hermann, Paris, 1978.
  • [40] J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), 1969/1970, Sém. Delange-Pisot-Poitou, exposé 19 (= Oe. 87).
  • [41] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.
  • [42] J.-P. Serre, Congruences et formes modulaires [d'après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Springer, Berlin, 1973, 319–338. Lecture Notes in Math., Vol. 317.
  • [43] J.-P. Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, 191–268. Lecture Notes in Math., Vol. 350.
  • [44] J.-P. Serre, Valeurs propres des opérateurs de Hecke modulo $l$, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Soc. Math. France, Paris, 1975, 109–117. Astérisque, Nos. 24-25.
  • [45] J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) ed. A. Fröhlich, Academic Press, London, 1977, pp. 193–268.
  • [46] J.-P. Serre, L'invariant de Witt de la forme $\rm Tr(x\sp 2)$, Comment. Math. Helv. 59 (1984), no. 4, 651–676.
  • [47] J.-P. Serre, Résumé des cours de 1984–1985, Annuaire du Collège de France (1985), 85–90.
  • [48] J.-P. Serre, Lettre à J.-F. Mestre, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) ed. K. Ribet, Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 263–268.
  • [49] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517.
  • [50] B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2) 10 (1975), 367–378.
  • [51] G. Shimura, Introduction to the arithmetic theory of automorphic functions, no. 11, Publ. Math. Soc. Japan, Princeton Univ. Press, 1971.
  • [52] G. Shimura, Class fields over real quadratic fields and Hecke operators, Ann. of Math. (2) 95 (1972), 130–190.
  • [53] J. Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173–175.
  • [54] J. Vélu, Courbes modulaires et courbes de Fermat, Séminaire de Théorie des Nombres, 1975-1976 (Univ. Bordeaux I, Talence), Exp. No. 16, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1976, p. 10.
  • [55] A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156.