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On the p-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of $\mathbbQ(\sqrt-3)$

Published online by Cambridge University Press:  17 October 2016

YUKAKO KEZUKA
YUKAKO KEZUKA*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WA. e-mail: [email protected]
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Abstract

We study infinite families of quadratic and cubic twists of the elliptic curve E = X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complex L-series at s=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the same L-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 67 - 98
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Cassels, J.W.S. and Fröhlich, A. Algebraic Number Theory. London Math. Soc., 2nd Edition (2010).Google Scholar
[2] Coates, J. and Wiles, A. On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977), pp. 223–251.Google Scholar
[3] Goldstein, C. and Schappacher, N. Séries d'Eisenstein et fonctions L de courbes elliptiques à multiplication complexe. J. Reine Angew. Math. 327 (1981), pp. 184–218.Google Scholar
[4] Rubin, K. The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991) pp. 25–68.CrossRefGoogle Scholar
[5] Silverman, J. H. The Arithmetic of Elliptic Curves, 2nd Edition. Graduate Texts in Math. vol. 106 (Springer–Verlag, 2006).Google Scholar
[6] Stephens, N. M. The diophantine equation X 3+Y 3=DZ 3 and the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 231 (1968), pp. 121–162.Google Scholar
[7] Tate, J. Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular Functions of One Variable IV. Lecture Notes in Math. vol. 476 (Springer, 1975), pp. 33–52.Google Scholar
[8] Zhao, C. A criterion for elliptic curves with second lowest 2-power in L(1). Math. Proc. Camb. Phil. Soc. 131 (2001), pp. 385–404.CrossRefGoogle Scholar
[9] Zhao, C. A criterion for elliptic curves with lowest 2-power in L(1) (II). Math. Proc. Camb. Phil. Soc. 134 (2003), pp. 407–420.Google Scholar