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On the conjecture of Birch and Swinnerton-Dyer. (English) Zbl 0359.14009

MSC:
14G10Zeta-functions and related questions
14H45Special curves and curves of low genus
14H25Arithmetic ground fields (curves)
14G25Global ground fields
11G15Complex multiplication and moduli of abelian varieties
11R42Zeta functions and L-functions of global number fields
References:
[1]Artin, E., Tate, J.: Class field theory. New York: Benjamin 1967
[2]Birch, B., Swinnerton-Dyer, P.: Notes on elliptic curves II. J. Reine Angew. Math.218, 79-108 (1965) · Zbl 0147.02506 · doi:10.1515/crll.1965.218.79
[3]Coates, J., Wiles, A.: Kummer’s criterion for Hurwitz numbers, to appear in Proceedings of the International Conference on Algebraic Number Theory held in Kyoto, Japan, 1976
[4]Damerell, R.:L-functions of elliptic curves with complex multiplication I. Acta Arith.17, 287-301 (1970)
[5]Deuring, D.: Die Zetafunktionen einer algebraischen Kurve vom Geschlechter Eins, I, II, III, IV. Nachr. Akad. Wiss. Göttingen, 85-94 (1953); 13-42 (1955); 37-76 (1956); 55-80 (1957)
[6]Fröhlich, A.: Formal Groups. Lecture Notes in Math.74. Berlin-Heidelberg-New York: Springer 1968
[7]Hasse, H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil II, 2nd ed. Würzburg-Wien: Physica 1965
[8]Iwasawa, K.: On some modules in the theory of cyclotomic fields. J. Math. Soc. Japan,16, 42-82 (1964) · Zbl 0125.29207 · doi:10.2969/jmsj/01610042
[9]Lubin, J.: One parameter formal Lie groups overp-adic integer rings. Ann. of Math.80, 464-484 (1964) · Zbl 0135.07003 · doi:10.2307/1970659
[10]Lubin, J., Tate, J.: Formal complex multiplication in local fields. Ann. Math.81, 380-387 (1965) · Zbl 0128.26501 · doi:10.2307/1970622
[11]Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Inventiones math.18, 183-266 (1972) · Zbl 0245.14015 · doi:10.1007/BF01389815
[12]Robert, G.: Unités elliptiques. Bull. Soc. Math. France, Mémoire36, 1973
[13]Robert, G.: Nombres de Hurwitz et Unités elliptiques. To appear
[14]Sah, H.: Automorphisms of finite groups. J. Algebra10, 47-68 (1968) · Zbl 0159.31001 · doi:10.1016/0021-8693(68)90104-X
[15]Serre, J.P., Tate, J.: Good reduction of abelian varieties. Ann. Math.88, 492-517 (1968) · Zbl 0172.46101 · doi:10.2307/1970722
[16]Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Pub. Math. Soc. Japan,11, 1971
[17]Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular functions of one variable IV. Lecture Notes in Math.476. Berlin-Heidelberg-New York: Springer 1975
[18]Wiles, A.: Higher explicit reciprocity laws. To appear