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Arithmetic on curves with complex multiplication by the Eisenstein integers

Published online by Cambridge University Press:  24 October 2008

A. R. Rajwade
A. R. Rajwade
Affiliation:
Panjab University, Chandigarh, India
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Extract

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication by i. In a previous paper (8), we had considered curves with complex multiplication by √ − 2. Here we shall look at the case when the ring of complex multiplications is isomorphic to the ring Z[ω], where ω3 = 1, ω ≠ 1.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 1 , January 1969 , pp. 59 - 73
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Birch, B. J. and Swinnerton-Dyer, H. P. F.Notes on elliptic curves. I. J. Reine Angew. Math. 212 (1963), 7–25 and II. J. Reine Angew. Math. 218 (1965), 79–108.CrossRefGoogle Scholar
(2)Cassels, J. W. S.Arithmetic on an elliptic curve. Proc. Intern. Congress Math., Stockholm (1962), 234–246.Google Scholar
(3)Cassels, J. W. S.Diophantine Equations with special reference to elliptic curves. J. London Math. Soc. 41 (1966), 193–291.CrossRefGoogle Scholar
(4)Cassels, J. W. S.A note on the division values of . Proc. Cambridge Philos. Soc. 45 (1948), 167–172.CrossRefGoogle Scholar
(5)Deuring, M.Die Typen der Multiplikatorenenringe elliptischer Funktionenkorper. Abh. Math. Sem. Univ. Hamburg. 14 (1941), 197–272.CrossRefGoogle Scholar
(6)Deuring, M.Die Zetafunktion einer algebraichen Kurve von Geschlechte Eins, I, II, III, IV. Nachr. Akad. Wiss Göttingen Math.-Phys. Kl. II (1953), 85–94, (1955) 13–42, (1956) 37–76, (1957) 55–80.Google Scholar
(7)Kubota, T.Reciprocities in Gauss' and Eisenstein's number fields. J. Reine Angew. Math. 208 (1961), 35–50.CrossRefGoogle Scholar
(8)Rajwade, A. R.Arithmetic on curves with complex multiplication by √ − 2. Proc. Cambridge Philos. Soc. 64 (1968), 659–672.CrossRefGoogle Scholar
(9)Weber, H.Algebra, vol. 3 (second edition) (Braunschweig, 1908).Google Scholar