Abstract
The aim of this Chapter is to give a brief survey of results, essentially without proofs, about elliptic curves, class groups and complex multiplication. No previous knowledge of these subjects is required. Some excellent references are listed in the bibliography. The basic general reference books are (Borevitch and Shafarevitch 1966), (Shimura 1971A), (Silverman 1986). In addition the algorithms and tables in (Birch and Kuyk 1975) are invaluable.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Birch, B. and Kuyk, W. eds. (1975) Modular Forms of One Variable IV. Lecture Notes in Math. 476, Springer (1975).
Birch, B. and Swinnerton-Dyer, H. P. F. (1963) Notes on elliptic curves (I) and (II). J. Reine Angew. Math. 212 (1963),7–25 and 218 (1965), 79-108.
Borevitch, Z. I. and Shafarevitch, I. R. (1966) Number Theory. Academic Press (1966).
Coates, J. and Wiles, A. (1977) On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977), 223–251.
Cohen, P. (1984) On the coefficients of the transformation polynomials for the elliptic modular function. Math. Proc. Camb. Phil. Soc. 95 (1984), 389–402.
Deligne, P. (1974) La conjecture de Weil. Publ. Math. IHES 43 (1974), 273–307.
Faltings, G. (1983) Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), 349–366.
Gross, B. and Zagier, D. (1983) Points de Heegner et dérivées de fonctions L. C. R. Acad. Sc. Paris 297 (1983), 85–87.
Gross, B. and Zagier, D. (1985) On singular moduli. J. Reine Angew. Math. 355 (1985), 191–219.
Gross, B. and Zagier, D. (1986) Heegner points and derivatives of L-functions. Invent. Math. 84 (1986), 225–320.
Hasse, H. (1933) Beweis des Analogons der Riemannschen Vermutung für die Artinschen u. F. K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen. Nachr. Gesell. Wissen. Göttingen I 42 (1933), 253–262.
Herrmann, O. (1975) Über die Berechnung der Fouriercoefficienten der Funktion j(τ). J. reine angew. Math. 274/275 (1975), 187–195.
Koblitz, N. Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Math. 97, Springer (1984).
Kolyvagin, V. A. (1988) Finiteness of E(â„š) and III (E/â„š) for a class of Weil curves. Izv. Akad. Nauk. SSSR 52 (1988).
Kolyvagin, V. A. (1989) Euler systems, in Progress in Math., Birkhäuser, to appear. Mazur, B. (1977) Modular curves and the Eisenstein ideal. Publ. Math. IHES 47 (1977), 33-186.
Mazur, B. (1978) Rational isogenies of prime degree. Invent Math. 44 (1978), 129–162.
Mestre, J.-F. (1981) Courbes elliptiques et formules explicites, in Sém. Th. Nombres Paris 1981–82. Progress in Math. 38, Birkhäuser, 179-188.
Mestre, J.-F. (1982) Construction of an elliptic curve of rank ≥ 12, C. R. Acad. Sc. Paris 295 (1982), 643–644.
Mordell, L. J. (1922) On the rational solutions of the indeterminate equations of the third and fourth degree. Proc. Camb. Philos. Soc. 21 (1922), 179–192.
Ogg, A. (1969) Modular Forms and Dirichlet Series. Benjamin (1969).
Ribet, K. (1990) Taniyama-Weil implies FLT.
Serre, J.-P. (1987) Sur les représentations modulaires de degré 2 de Gal (ℚ/ℚ). Duke Math. J. 54 (1987), 179–230.
Shimura, G. (1971A) Introduction to the arithmetic theory of automorphic functions. Princeton Univ. rress (1971).
Shimura, G. (1971B) On the zeta-function of an Abelian variety with complex multiplication. Ann. Math. 94 (1971), 504–533.
Shimura, G. (1971C) On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J. 43 (1971), 199–208.
Siegel, C.-L. (1936) Über die Classenzahl quadratischer Zahlkörper. Acta Arith. 1 (1936), 83–86.
Silverman, J. (1986) The Arithmetic of Elliptic Curves. Graduate Texts in Math. 106, Springer (1986).
Taniyama, Y. (1955), in the Problem session of the Tokyo-Nikko conference on number theory, problem 12 (1955).
Weil, A. (1930) Sur un théorème de Mordell. Bull. Sci. Math. 54 (1930), 182–191.
Weil, A. (1949) Number of solutions of equations in finite fields. Bull. AMS 55 (1949), 497–508.
Zagier, D. (1990) Introduction to modular forms, this volume.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cohen, H. (1992). Elliptic Curves. In: Waldschmidt, M., Moussa, P., Luck, JM., Itzykson, C. (eds) From Number Theory to Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02838-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-02838-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08097-5
Online ISBN: 978-3-662-02838-4
eBook Packages: Springer Book Archive