×

Elliptic curves and automorphic representations. (English) Zbl 0336.14003


MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H45 Special algebraic curves and curves of low genus
11F12 Automorphic forms, one variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag)
[2] (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 4, Lecture Notes in Mathematics, No. 476 (1975), Springer-Verlag: Springer-Verlag New York/Berlin)
[3] Artin, A., Theorie der L-Riehen mit allgemeinen Gruppencharakteren, Abh. Math. Sem. Univ. Hamburg, 8, 292-306 (1930) · JFM 56.0173.02
[4] Atkin, A.; Lehner, J., Hecke operators on \(Γ_0\)(m), Math. Ann., 185, 134-160 (1970) · Zbl 0177.34901
[5] Birch, B. J.; Swinnerton-Dyer, H. P.F., Notes on elliptic curves, II, J. Reine Angew. Math., 218, 79-108 (1962) · Zbl 0147.02506
[6] Birch, B. J.; Swinnerton-Dyer, H. P.F., Elliptic curves and modular functions, (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 4, Lecture Notes in Mathematics, No. 476 (1975), Springer-Verlag: Springer-Verlag New York/Berlin), 2-32 · Zbl 1214.11081
[7] Borel, A., Formes automorphes et séries de Dirichlet, Sem. Bourbaki, No. 466 (1975), Juin
[8] Casselman, W., An assortment of results on representations of \(GL_2(k)\), (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag: Springer-Verlag New York/Berlin), 1-54 · Zbl 0301.22011
[9] Casselman, W., On representations of \(GL_2\) and the arithmetic of modular curves, (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag: Springer-Verlag New York/Berlin), 107-142
[10] Cassels, J. W.S., Diophantine equations with special reference to elliptic curves, J. London Math. Soc., 41, 193-291 (1966) · Zbl 0138.27002
[11] Deligne, P., La conjecture de Weil, I, Publ. I.H.E.S., 43, 273-307 (1974) · Zbl 0287.14001
[12] Deligne, P., Les constantes des équations fonctionnelles, Sem. Delange-Pisot-Poitou, No. 19 (1970), bis, Mai
[13] Deligne, P., Formes modulaires et représentations de \(GL_2\), (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag: Springer-Verlag New York/Berlin), 55-107
[14] Deligne, P., Formes modulaires et représentations \(l\)-adiques, (Lecture Notes in Mathematics (1971), Springer-Verlag: Springer-Verlag New York/Berlin), 139-186, No. 244
[15] Deligne, P., Les constantes des équations fonctionnelles des fonctions \(L\), (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag: Springer-Verlag New York/Berlin), 501-598
[16] Deligne, P.; Serre, J.-P., Formes modulaires de poids, 1, Ann. Sci. École Norm. Sup. \(4^e\) sér., 4, 507-530 (1974) · Zbl 0321.10026
[17] Dwork, B., On the rationality of the zeta-function of an algebraic variety, Amer. J. Math., 82, 631-648 (1960) · Zbl 0173.48501
[18] Gelbart, S., Automorphic Forms on Adele Groups, (Annals of Mathematical Studies (1975), Princeton Univ. Press: Princeton Univ. Press New York/Berlin), No. 83 · Zbl 0329.10018
[19] B. Gelbart and H. JacquetGLGLProc. Nat. Acad. Sci. U.S.A.; B. Gelbart and H. JacquetGLGLProc. Nat. Acad. Sci. U.S.A. · Zbl 0373.22008
[20] Goldstein, L., Analytic Number Theory (1973), Prentice-Hall: Prentice-Hall Princeton, N.J.
[21] Grothendieck, A., Formule de Lefschetz et rationalité des fonctions, L, Sem. Bourbaki, No. 279 (1964), December
[22] Howe, R., Review of Modular forms and \(l\)-adic representations, Zentralblatt für Math, 279, 94-96 (1974), Review 14007
[23] Igusa, J., Kroneckerian models of fields of elliptic modular functions, Amer. J. Math., 81, 561-577 (1959) · Zbl 0093.04502
[24] Jacquet, H.; Langlands, R. P., Automorphic Forms on GL(2), (Lecture Notes in Mathematics, No. 114 (1970), Springer-Verlag: Springer-Verlag Englewood Cliffs, N. J.) · Zbl 0236.12010
[25] \( \textsc{Kutzko}_2\)Amer. J. Math.; \( \textsc{Kutzko}_2\)Amer. J. Math.
[26] Lang, S., Elliptic Functions (1973), Addison-Wesley: Addison-Wesley New York/Berlin
[27] Langlands, R. P., Problems in the Theory of Automorphic Forms, (Lecture Notes in Mathematics, No. 170 (1970), Springer-Verlag: Springer-Verlag Reading, Mass.) · Zbl 0225.14023
[28] R. P. Langlands\(L\); R. P. Langlands\(L\)
[29] Langlands, R. P., Modular forms and \(l\)-adic representations, (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag: Springer-Verlag New York/Berlin), 361-500 · Zbl 0279.14007
[30] Langlands, R. P., Base change for GL(2): The theory of Saito-Shintani with applications (1975), Institute for Advanced Study: Institute for Advanced Study New York/Berlin, preprint
[31] Langlands, R. P., Automorphic forms on GL(2), Actes, Congrès Intern. Math., 2, 327-329 (1970) · Zbl 0236.12010
[32] Li, W., New forms and functional equations, Math. Ann., 212, 285-315 (1975) · Zbl 0278.10026
[33] Ogg, A. P., Elliptic curves with wild ramification, Amer. J. Math., 89, 1-21 (1967) · Zbl 0147.39803
[34] Ogg, A. P., Abelian curves of small conductor, J. Reine Angew. Math., 226, 204-215 (1967) · Zbl 0163.15404
[35] Pyatetskii-Shapiro, I. I., Zeta-functions of modular curves, (“Proceedings of the Antwerp Conference on Modular Functions of One Variable,” Vol. 2, Lecture Notes in Mathematics, No. 349 (1973), Springer-Verlag: Springer-Verlag Princeton, N. J.), 317-360
[36] Robert, A., Elliptic curves, (Lecture Notes in Mathematics (1973), Springer-Verlag: Springer-Verlag New York/Berlin), No. 326
[37] Sally, P. J.; Shalika, J., Characters of the discrete series of representations of SL(2) over a local field, (Proc. Nat. Acad. Sci. U.S.A., 63 (1969)), 1231-1237 · Zbl 0198.18203
[38] (Seminar on Complex Multiplication. Seminar on Complex Multiplication, Lecture Notes in Mathematics (1966), Springer-Verlag: Springer-Verlag New York/Berlin)
[39] Sen, S.; Tate, J., Ramification groups of local fields, J. Indian Math. Soc., 27, 197-202 (1963) · Zbl 0136.02702
[40] Serre, J.-P., Rationalité des Fonctions zêta des variétés algébriques (d’après Dwork), Sem. Bourbaki, No. 198 (1960) · Zbl 0119.36805
[41] Serre, J.-P., Course in Arithmetic (1973), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0256.12001
[42] Serre, J.-P., Facteurs locaux des fonctions zêta des variétés algébriques, Sem. Delange-Pisot-Poitou, No. 19 (1970), Mai · Zbl 0214.48403
[43] Serre, J.-P., Abelian \(l\)-adic Representations and Elliptic Curves (1968), Benjamin: Benjamin New York/Berlin
[44] Serre, J.-P., Une interprétation des congruences relative à la fonction τ de Ramanujan, Sem. Delange-Pisot-Poitou, No. 14 (1968), Février
[45] Serre, J.-P.; Tate, J., Good reduction of abelian varieties, Ann. Math., 88, 492-517 (1968) · Zbl 0172.46101
[46] Shafarevich, I. R., On Galois groups of \(p\)-adic fields, C. R. Dokl. Acad. Sci. USSR (N. S.), 53, 15-16 (1946) · Zbl 0063.06925
[47] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (1971), Iwanami Shoten: Iwanami Shoten New York, Princeton Univ. Press, Princeton, N. J. · Zbl 0221.10029
[48] Shimura, G., On elliptic curves with complex multiplication as factors of the Jacobians of modular functions fields, Nagoya Math. J., 43, 199-208 (1971) · Zbl 0225.14015
[49] Shimura, G., On the factors of the Jacobian variety of a modular function field, J. Math. Soc. Japan, 25, 523-544 (1973) · Zbl 0266.14017
[50] Swinnerton-Dyer, H. P.F., Analytic Theory of Abelian Varieties, (London Mathematical Society Lecture Note Series 14 (1974), Cambridge Univ. Press: Cambridge Univ. Press Tokyo) · Zbl 0299.14021
[51] Tate, J., The arithmetic of elliptic curves, Invent. Math., 23, 179-206 (1974) · Zbl 0296.14018
[52] Weil, A., Number of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55, 497-508 (1949) · Zbl 0032.39402
[53] Weil, A., Über die Bestimmung Dirichletscher Reiken durch Funktionalgleichungen, Math. Ann., 168, 149-156 (1967) · Zbl 0158.08601
[54] Weil, A., Dirichlet Series and Automorphic Forms, (Lecture Notes in Mathematics (1971), Springer-Verlag: Springer-Verlag London/New York), No. 189 · Zbl 0218.10046
[55] Weil, A., Sur la théorie du corps de classes, J. Math. Soc. Japan, 3, 1-35 (1951) · Zbl 0044.02901
[56] Weil, A., Number theory and algebraic geometry, (Proceedings of the International Congress of Mathematics 1950 (1952), American Mathematical Society: American Mathematical Society New York/Berlin), 90-100
[57] Weil, A., Exercices dyadiques, Invent. Math., 29 (1975) · Zbl 0307.12017
[58] Weil, A., Basic Number Theory (1973), Springer-Verlag: Springer-Verlag Provicence, R. I. · Zbl 0267.12001
[59] Yamamoto; Yamamoto
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.