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  • Cited by 8
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Xue, Hang 2016. Fourier–Jacobi periods and the central value of Rankin–Selberg L-functions. Israel Journal of Mathematics, Vol. 212, Issue. 2, p. 547.

    Yao, Cheng and Hung, Pin-Chi 2016. Non-vanishing of central L-values of canonical CM elliptic curves with quadratic twists. Manuscripta Mathematica, Vol. 150, Issue. 3-4, p. 293.

    Kim, Byoung Du Masri, Riad and Yang, Tong Hai 2011. Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture. Mathematische Annalen, Vol. 349, Issue. 2, p. 301.

    Masri, Riad and Yang, Tonghai 2011. Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties. Geometric and Functional Analysis, Vol. 21, Issue. 3, p. 648.

    Masri, R. 2010. Quantitative Nonvanishing of L-Series Associated to Canonical Hecke Characters. International Mathematics Research Notices,

    Masri, R. 2010. Asymptotics for Sums of Central Values of Canonical Hecke L-Series. International Mathematics Research Notices,

    Liu, Chunlei and Xu, Lanju 2004. The vanishing order of certain Hecke L-functions of imaginary quadratic fields. Journal of Number Theory, Vol. 108, Issue. 1, p. 76.

    Yang, Tonghai 2000. On the Central Derivative of Hecke L-Series. Journal of Number Theory, Vol. 85, Issue. 2, p. 130.


Nonvanishing of central Hecke L-values and rank of certain elliptic curves

  • DOI:
  • Published online: 01 July 1999

Let D≡ 7 mod 8 be a positive squarefree integer, and let h$_D$ be the ideal class number of E$_D$=Q($\sqrt{−D}$). Let d≡1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k[ges ]0 there is a constant M=M(k), independent of the pair (D,D), such that if (−1)$^k$=sign (d), (2k+1,h$_D$)=1, and$\sqrt{D}$>(12/π)d$^2$(log|d+M(k)), then the central L-value L(k+1, χ$^2k+1$$_D, d$>0. Furthermore, for k[les ]1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)$^d$ has Mordell–Weil rank 0 (over its definition field) when $\sqrt{D}$>(12/π)d$^2$ log d.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
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