Modular curves and the Eisenstein ideal
Publications Mathématiques de l'IHÉS, Tome 47 (1977), pp. 33-186. 
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     mrnumber = {488287},
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Mazur, Barry. Modular curves and the Eisenstein ideal. Publications Mathématiques de l'IHÉS, Tome 47 (1977), pp. 33-186. http://archive.numdam.org/item/PMIHES_1977__47__33_0/

[1] Artin (M.), Grothendieck topologies, Mimeographed notes, Harvard University, 1962. | Zbl

[2] Atkin (A. O. L.), Lehner (J.), Hecke operators on Γ0(m), Math. Ann., 185 (1970), 134-160. | MR | Zbl

[3] Bass (H.), On the ubiquity of Gorenstein rings, Math. Zeitschrift, 82 (1963), 8-28. | MR | Zbl

[4] Brumer (A.), Kramer (K.), On the rank of elliptic curves, I (in preparation). | Zbl

[5] Cassels (J. W. S.), Fröhlich (A.) (eds.), Algebraic Number Theory, London-New York, Academic Press, 1967. | Zbl

[6] Curtis (C. W.), Reiner (I.), Representation theory of finite groups and associative algebras, New York, Interscience, 1962. | MR | Zbl

[7] Deligne (P.), Formes modulaires et représentations l-adiques. Séminaire Bourbaki 68-69, no. 355, Lecture Notes in Mathematics, 179, Berlin-Heidelberg-New York, Springer, 1971, 136-172. | Numdam | Zbl

[8] Deligne (P.), Mumford (D.), The irreducibility of the space of curves of given genus, Publications Mathématiques I.H.E.S., 36 (1969), 75-109. | Numdam | MR | Zbl

[9] Deligne (P.), Rapoport (M.), Schémas de modules des courbes elliptiques. Vol. II of the Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Mathematics, 349, Berlin-Heidelberg-New York, Springer, 1973. | Zbl

[10] Deligne (P.), Serre (J.-P.), Formes modulaires de poids I, Ann. Scient. Éc. Norm. Sup., 4e série, t. 7 (1974), 507-530. | Numdam | MR | Zbl

[11] Demazure (M.), Gabriel (P.), Groupes algébriques, t. I, Amsterdam, North-Holland Publishing Co., 1970. | Zbl

[12] Demjanenko (V. A.), Torsion of elliptic curves [in Russian], Izv. Akad. Nauk. CCCP, 35 (1971), 280-307 [MR 44, 2755]. | MR | Zbl

[13] Drinfeld (G. I.), Elliptic modules [in Russian], Mat. Sbornik, 94 (136) (1974), No. 4. Engl. Trans.: A.M.S., vol. 23 (1976), 561-592. | Zbl

[14] Fontaine (J.-M.), Groupes finis commutatifs sur les vecteurs de Witt, C. R. Acad. Sc. Paris, t. 280 (1975), série A, 1423-1425. | MR | Zbl

[15] Grothendieck (A.), Le groupe de Brauer III: exemples et compléments (a continuation of Bourbaki exposés: 200, 297). Published in Dix exposés sur la cohomologie des schémas, Amsterdam, North-Holland Publ. Co., 1968, | Zbl

[16] Hadano (T.), On the conductor of an elliptic curve with a rational point of order 2, Nagoya Math. J., 53 (1974), 199-210. | MR | Zbl

[17] Hartshorne (R.), Residues and Duality, Lecture Notes in Mathematics, 20, Berlin-Heidelberg-New York. Springer, 1966. | MR | Zbl

[18] Hartshorne (R.), On the De Rham cohomology of algebraic varieties, Publications Mathématiques I.H.E.S., 45 (1975), 1-99. | Numdam | MR | Zbl

[19] Hecke (E.), Mathematische Werke, 2nd edition, Göttingen, Vandenhoeck & Ruprecht, 1970. | MR | Zbl

[20] Herbrand (J.), Sur les classes des corps circulaires, Journal de Math. pures et appliquées, 9e série, 11 (1932), 417-441. | JFM | Numdam

[21] Imai (H.), A remark on the rational points of abelian varieties with values in cyclotomic Zp-extensions, Proc. Japan Acad., 51 (1975), 12-16. | MR | Zbl

[22] Iwasawa (K.), Lectures on p-adic L-functions, Princeton, Princeton University Press and University of Tokyo Press, 1972. | MR | Zbl

[23] Iwasawa (K.), On p-adic L-functions, Ann. Math., 89 (1969), 198-205. | MR | Zbl

[24] Katz (N.), p-adic properties of modular schemes and modular forms, vol. III of The Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Mathematics, 350, Berlin-Heidelberg-New York, Springer, 1973, 69-190. | MR | Zbl

[25] Kiepert (L.), Ueber gewisse Vereinfachungen der Transformationsgleichungen in der Theorie der elliptischen Functionen, Math. Ann., 37 (1890), 368-398. | JFM

[26] Koike (M.), On the congruences between Eisenstein series and cusp forms (to appear).

[27] Kubert (D.), Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3), 33 (1976), 193-237. | MR | Zbl

[28] Kubert (D.), Lang (S.), Units in the modular function field, I, II, III, Math. Ann., 218 (1975), 67-96, 175-189, 273-285. | MR | Zbl

[29] Lang (S.), Elliptic Functions, Addison Wesley, Reading, 1974. | Zbl

[30] Ligozat (G.), Fonctions L des courbes modulaires, Séminaire Delange-Pisot-Poitou, Jan. 1970. Thesis : Courbes modulaires de genre I, Bull. Soc. math. France, mémoire 43, 1975. | Numdam | Zbl

[31] Manin (Y.), A uniform bound for p-torsion in elliptic curves [in Russian], Izv. Akad. Nauk. CCCP, 33 (1969), 459-465.

[32] Manin (Y.), Parabolic points and zeta functions of modular forms [in Russian], Izv. Akad. Nauk. CCCP, 36 (1972), 19-65. | Zbl

[33] Mazur (B.), Notes on étale cohomology of number fields, Ann. Scient. Éc. Norm. Sup., 4e série, t. 6 (1973), 521-556. | Numdam | MR | Zbl

[34] Mazur (B.), Rational points on abelian varieties with values in towers of number fields, Inventiones Math., 18 (1972), 183-266. | MR | Zbl

[35] Mazur (B.), Courbes elliptiques et symboles modulaires. Séminaire Bourbaki, No. 414, Lecture Notes in Mathematics, No. 317, Berlin-Heidelberg-New York, Springer, 1973. | Numdam | MR | Zbl

[36] Mazur (B.), p-adic analytic number theory of elliptic curves and abelian varieties over Q, Proc. of International Congress of Mathematicians at Vancouver, 1974, vol. I, 369-377, Canadian Math. Soc. (1975). | Zbl

[37] Mazur (B.), Messing (W.), Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, No. 370, Berlin-Heidelberg-New York, Springer, 1974. | MR | Zbl

[38] Mazur (B.), Serre (J.-P.), Points rationnels des courbes modulaires Xo(N). Séminaire Bourbaki, No. 469, Lecture Notes in Mathematics, No. 514, Berlin-Heidelberg-New York, Springer, 1976. | Numdam | Zbl

[39] Mazur (B.), Swinnerton-Dyer (P.), Arithmetic of Weil curves, Inventiones math., 25 (1974), 1-61. | MR | Zbl

[40] Mazur (B.), Tate (J.), Points of order 13 on elliptic curves, Inventiones math., 22 (1973), 41-49. | MR | Zbl

[41] Mazur (B.), Vélu (J.), Courbes de Weil de conducteur 26, C. R. Acad. Sc. Paris, t. 275 (1972), série A, 743-745. | MR | Zbl

[42] Miyawaka (I.), Elliptic curves of prime power conductor with Q-rational points of finite order, Osaka J Math., 10 (1973), 309-323. | MR | Zbl

[43] Mumford (D.), Geometric invariant theory, Ergebnisse der Math., 34, Berlin-Heidelberg-New York, Springer, 1965. | MR | Zbl

[44] Mumford (D.), Curves and their jacobians, Ann Arbor, The University of Michigan Press, 1975. | MR | Zbl

[45] Néron (A.), Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. I.H.E.S., 21 (1964), 361-483 [MR 31, 3424]. | Numdam | MR | Zbl

[46] Neumann (O.), Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten, I, II, Math. Nachr., 49 (1971), 107-123; Math. Nachr., 56 (1973), 269-280. | Zbl

[47] Oda (T.), The first De Rham cohomology group and Dieudonné modules, Ann. scient. Éc. Norm. Sup., 4e série, t. 2 (1969), 63-135. | Numdam | MR | Zbl

[48] Ogg (A.), Rational points on certain elliptic modular curves, Proc. Symp. Pure Math., 24 (1973), 221-231, A.M.S., Providence. | MR | Zbl

[49] Ogg (A.), Diophantine equations and modular forms, Bull. A.M.S., 81 (1975), 14-27. | MR | Zbl

[50] Ogg (A.), Hyperelliptic modular curves, Bull. Soc. Math. France, 102 (1974), 449-462. | Numdam | MR | Zbl

[51] Ogg (A.), Automorphismes des courbes modulaires, Séminaire Delange-Pisot-Poitou, déc. 1974 (mimeo. notes distributed by Secrétariat mathématique, 11, rue Pierre-et-Marie-Curie, 75231 Paris, Cedex 05). | Numdam | Zbl

[52] Ohta (M.), On reductions and zeta functions of varieties obtained from Γo(N) (to appear).

[53] Oort (F.), Commutative group schemes, Lecture Notes in Mathematics, No. 15, Berlin-Heidelberg-New York, Springer, 1966. | MR | Zbl

[54] Oort (F.), Tate (J.), Group schemes of prime order, Ann. Scient. Éc. Norm. Sup., série 4, 3 (1970), 1-21. | Numdam | MR | Zbl

[55] Raynaud (M.), Schémas en groupes de type (p, ...,p), Bull. Soc. Math. France, 102 (1974), 241-280. | Numdam | MR | Zbl

[56] Raynaud (M.), Spécialisation du foncteur de Picard, Publ. Math. I.H.E.S., 38 (1970), 27-76. | Numdam | MR | Zbl

[57] Raynaud (M.), Passage au quotient par une relation d'équivalence plate, Proc. of a Conference on Local Fields, NUFFIC Summer School held at Driebergen in 1966, 133-157, Berlin-Heidelberg-New York, Springer, 1967. | MR | Zbl

[58] Ribet (K.), Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101 (1975), 555-562. | MR | Zbl

[59] Sen (S.), Lie algebras of Galois groups arising from Hodge-Tate modules, Ann. of Math., 97 (1973), 160-170. | MR | Zbl

[60] Serre (J.-P.), Corps locaux, Paris, Hermann, 1962. | MR | Zbl

[61] Serre (J.-P.), Formes modulaires et fonctions zêta p-adiques, vol. III of the Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Mathematics, 350, 191-268, Berlin-Heidelberg-New York, Springer, 1973. | Zbl

[62] Serre (J.-P.), Algèbre locale. Multiplicités, Lecture Notes in Mathematics, No. 11 (3rd edition), Berlin-Heidelberg-New York, Springer, 1975. | MR | Zbl

[63] Serre (J.-P.), Abelian l-adic representations and elliptic curves, Lectures at McGill University, New York-Amsterdam, W. A. Benjamin Inc., 1968. | MR | Zbl

[64] Serre (J.-P.), Quelques propriétés des variétés abéliennes en caractéristique p, Amer. J. Math., 80 (1958), 715-739. | MR | Zbl

[65] Serre (J.-P.), p-torsion des courbes elliptiques (d'après Y. Manin). Séminaire Bourbaki, 69-70, No. 380, Lecture Notes in Mathematics, No. 180, Berlin-Heidelberg-New York, Springer, 1971. | Numdam | Zbl

[66] Serre (J.-P.), Congruences et formes modulaires (d'après H. P. F. Swinnerton-Dyer). Séminaire Bourbaki, 71-72, N° 416, Lecture Notes in Mathematics, No. 317, Berlin-Heidelberg-New York, Springer, 1973. | Numdam | Zbl

[67] Serre (J.-P.), Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Inventiones math., 15 (1972), 259-331. | MR | Zbl

[68] Setzer (B.), Elliptic curves of prime conductor, J. Lond. Math. Soc. (2), 10 (1975), 367-378. | MR | Zbl

[69] Shimura (G.), Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan, No. 11, Tokyo-Princeton, 1971. | Zbl

[70] Wada (H.), A table of Hecke operators, Proc. Japan Acad., 49 (1973), 380-384. | MR | Zbl

[71] Yamauchi (M.), On the fields generated by certain points of finite order on Shimura's elliptic curves, J. Math. Kyoto Univ., 14 (2) (1974), 243-255. | MR | Zbl

[72] Borevich (Z. I.), Shafarevich (I. R.), Number theory, London-New York, Academic Press, 1966. | Zbl

[73] Parry (W. R.), A determination of the points which are rational over Q of three modular curves (unpublished). | Zbl

[74] Serre (J.-P.), Tate (J.), Good reduction of abelian varieties, Ann. of Math., 88 (1968), 492-517. | MR | Zbl

[75] Tate (J.), Algorithm for determining the type of a singular fiber in an elliptic pencil, vol. IV of The Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Mathematics, 476, Berlin-Heidelberg-New York, Springer, 1975.

[SGA 3] Séminaire de Géométrie algébrique du Bois-Marie, 62-64. Directed by M. DEMAZURE and A. GROTHENDIECK, Lecture Notes in Mathematics, Nos. 151, 152, 153, Berlin-Heidelberg-New York, Springer, 1970.

[SGA 7] Séminaire de Géométrie algébrique du Bois-Marie, 67-69. P. Deligne and N. Katz, Lecture Notes in Mathematics, Nos. 288, 340, Berlin-Heidelberg-New York, Springer, 1972, 1973. | Zbl