Science China Mathematics

, Volume 60, Issue 4, pp 593–612 | Cite as

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

  • ZhangJie Wang


Given a large positive number x and a positive integer k, we denote by Q k(x) the set of congruent elliptic curves E (n): y 2 = z 3n 2 z with positive square-free integers nx congruent to one modulo eight, having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E (n)Q k(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to (ℤ/2ℤ)2. We also get a lower bound for the number of E (n)Q k(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to (ℤ/2ℤ)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers nx with k prime factors and residue symbols (quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.


Shafarevich-Tate group distribution congruent elliptic curve multiplicative number theory number field independence property residue symbol 


11G05 11N99 


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This work was supported by National Natural Science Foundation of China (Grant No. 11501541). The author is greatly indebted to his advisor Professor Ye Tian for many instructions and suggestions. The author thanks Lvhao Yan for carefully reading the manuscript and giving valuable comments.


  1. 1.
    Brown M, Calkin N, James K, et al. Trivial selmer groups and even partitions of a graph. Integers, 2006, 6: 1–17MathSciNetMATHGoogle Scholar
  2. 2.
    Cremona J, Odoni R. Some density results for negative Pell equations: An application of graph theory. J London Math Soc (2), 1989, 39: 16–28MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Fogels E. Über die Ausnahmenullstelle der Heckeschen L-Funktionen. Acta Arith, 1963, 8: 307–309MathSciNetMATHGoogle Scholar
  4. 4.
    Fogels E. On the zeros of L-functions. Acta Arith, 1965, 11: 67–96MathSciNetMATHGoogle Scholar
  5. 5.
    Fogels E. Corrigendum to the paper “On the zeros of L-functions” (Acta Arith, 1965, 11: 67–96). Acta Arith, 1968, 14: 435MathSciNetMATHGoogle Scholar
  6. 6.
    Hecke E. Lectures on the Theory of Algebraic Numbers. Berlin: Springer-Verlag, 1981CrossRefMATHGoogle Scholar
  7. 7.
    Hoffstein J, Ramakrishnan D. Siegel zeros and cusp forms. Int Math Res Not IMRN, 1995, 6: 279–308MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ireland K, Rosen M. A Classical Introduction to Modern Number Theory. Berlin: Springer-Verlag, 1990CrossRefMATHGoogle Scholar
  9. 9.
    Iwaniec H, Kowalski E. Analytic Number Theory. Providence: Amer Math Soc, 2004CrossRefMATHGoogle Scholar
  10. 10.
    Jung H, Yue Q. 8-ranks of class groups of imaginary quadratic number fields and their densities. J Korean Math Soc, 2011, 48: 1249–1268MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lang S. Algebraic Number Theory. Berlin: Springer-Verlag, 1994CrossRefMATHGoogle Scholar
  12. 12.
    Rhoades R. 2-Selmer groups and the Birch-Swinnerton-Dyer conjecture for the congruent number curves. J Number Theory, 2009, 129: 1379–1391MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Serre J. A Course in Arithmetic. Berlin: Springer-Verlag, 1973CrossRefMATHGoogle Scholar
  14. 14.
    Vatsal V. Rank-one twists of a certain elliptic curve. Math Ann, 1998, 311: 791–794MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wang Z J. Congruent elliptic curves with non-trivial Shafarevich-Tate groups. Sci China Math, 2016, 59: 2145–2166MathSciNetCrossRefGoogle Scholar

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© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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