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On the weak forms of the 2-part of Birch and Swinnerton-Dyer conjecture

Published online by Cambridge University Press:  05 September 2018

SHUAI ZHAI
SHUAI ZHAI*
Affiliation:
Institute for Advanced Research, Shandong University, Jinan, Shandong 250100, China. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road. Cambridge CB3 0WBU.K. e-mail: shuaizhai@gmail.com
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Abstract

In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 197 - 209
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

Supported by Shandong Province Natural Science Foundation (Grant No. ZR2016AP03)

References

REFERENCES

[1] Cai, L., Li, C. and Zhai, S.. On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves. arXiv:1712.01271 (2017).Google Scholar
[2] Česnavičius, K.. The Manin constant in the semistable case. arXiv:1703.02951 (2017).Google Scholar
[3] Coates, J.. Lectures on the Birch–Swinnerton-Dyer conjecture. Notices of the ICCM (2013).Google Scholar
[4] Coates, J., Kim, M., Liang, Z. and Zhao, C.. On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. Münster J. of Math. 7 (2014), 83–103.Google Scholar
[5] Coates, J., Li, Y., Tian, Y. and Zhai, S.. Quadratic twists of elliptic curves. Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 357–394.Google Scholar
[6] Coates, J.. The Conjecture of Birch and Swinnerton-Dyer. Open Problems in Math. (Springer, [Cham], 2016), 207–223.Google Scholar
[7] Cremona, J.. Algorithms for Modular Elliptic Curves. (Cambridge University Press, 1997).Google Scholar
[8] Cremona, J.. Computing the degree of the modular parametrisation of a modular elliptic curve. Math. Comp. 64 (1995), no. 211, 1235–1250.Google Scholar
[9] Kezuka, Y.. On the p-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of $\mathbb {Q}(\sqrt{-3})$. Math. Proc. Camb. Phil. Soc. 164 (2018), no. 1, 67–98.Google Scholar
[10] Manin, Ju. I.. Parabolic points and zeta-functions of modular curves. Math. USSR-Izv. 6 (1972), 19–64.Google Scholar
[11] Tian, Y.. Congruent numbers with many prime factors. Proc. Natl. Acad. Sci. USA 109 (2012), 21256–21258.Google Scholar
[12] Tian, Y.. Congruent numbers and Heegner points. Cambridge J. Math. 2 (2014), 117–161.Google Scholar
[13] Zhai, S.. Non-vanishing theorems for quadratic twists of elliptic curves. Asian J. Math. 20 (2016), no. 3, 475–502.Google Scholar
[14] Zhao, C.. A criterion for elliptic curves with lowest 2-power order in L(1). Proc. Camb. Phil. Soc. 121 (1997), 385–400.Google Scholar
[15] Zhao, C.. A criterion for elliptic curves with second lowest 2-power order in L(1). Proc. Camb. Phil. Soc. 131 (2001), 385–404.Google Scholar
[16] Zhao, C.. A criterion for elliptic curves with second lowest 2-power order in L(1) (II). Proc. Camb. Phil. Soc. 134 (2003), 407–420.Google Scholar
[17] Zhao, C.. A criterion for elliptic curves with second lowest 2-power order in L(1) (III). Acta Math. Sinica 21 (2005), 961–976.Google Scholar