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Non-vanishing theorems for central L-values of some elliptic curves with complex multiplication

John Coates Emmanuel College, Cambridge, England, UK Yongxiong Li Yau Mathematical Science Center, Tsinghua University, Beijing, China

Number Theory mathscidoc:2204.24005

Proceedings of the London Mathematical Society, 121, (6), 1531-1578, 2020.8
The paper uses Iwasawa theory at the prime 𝑝=2 to prove non-vanishing theorems for the value at 𝑠=1 of the complex 𝐿-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field 𝐾 = ℚ(√(-q)), where 𝑞 is any prime ≡ 7 mod 8. Our results establish some broad generalizations of the non-vanishing theorem first proven by Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell–Weil group and the Tate–Shafarevich group of the corresponding elliptic curves over the Hilbert class field of 𝐾. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to 𝐾 of the particular elliptic curve with complex multiplication introduced by Gross.
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@inproceedings{john2020non-vanishing,
  title={Non-vanishing theorems for central L-values of some elliptic curves with complex multiplication},
  author={John Coates, and Yongxiong Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422145035756472105},
  booktitle={Proceedings of the London Mathematical Society},
  volume={121},
  number={6},
  pages={1531-1578},
  year={2020},
}
John Coates, and Yongxiong Li. Non-vanishing theorems for central L-values of some elliptic curves with complex multiplication. 2020. Vol. 121. In Proceedings of the London Mathematical Society. pp.1531-1578. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422145035756472105.
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