Hostname: page-component-7b9c58cd5d-7g5wt Total loading time: 0 Render date: 2025-03-22T01:22:48.000Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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