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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: Y.Kezuka@dpmms.cam.ac.uk
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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

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