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On several families of elliptic curves with arbitrary large Selmer groups

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Abstract

In this paper, we calculate the \( \phi (\hat \phi ) \)-Selmer groups S (φ)(E/ℚ) and \( S^{(\hat \phi )} (E'/\mathbb{Q}) \) of elliptic curves y 2=x(x+εpD)(x+εqD) via the descent method. In particular, we show that the Selmer groups of several families of such elliptic curves can be arbitrary large.

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References

  1. Dabrowski A, Wieczorek M. On the equation y 2 = x(x − 2m)(x + q − 2m). J Number Theory, 2007, 124: 364–379

    Article  MATH  MathSciNet  Google Scholar 

  2. Faulkner B, James K. A graphical approach to computing Selmer groups of congruent number curves. Ramanujan J, 2007, 14: 107–129

    Article  MATH  MathSciNet  Google Scholar 

  3. Feng K. Non-congruent numbers, odd graphs and the BSD conjecture. Acta Arith, 1996, 80: 71–83

    Google Scholar 

  4. Feng K, Xiong M. On elliptic curves y 2 = x 3 − n 2 x with rank zero. J Number Theory, 2004, 109: 1–24

    Article  MATH  MathSciNet  Google Scholar 

  5. Feng K. Non-Congruent Numbers and Elliptic Curves with Rank Zero (in Chinese). Hefei: Press of University of Science and Technology of China, 2008

    Google Scholar 

  6. Iwaniec H, Kowalski E. Analytic Number Theory. Providence, RI: American Mathematical Society, 2004

    MATH  Google Scholar 

  7. Lang S. Algebra. 3rd ed. New York: Springer-Verlag, 2002

    MATH  Google Scholar 

  8. Merriman J R, Siksek S, Smart N P. Explicit 4-descents on an elliptic curve. Acta Arith, 1996, lxxvii: 385–404

    MathSciNet  Google Scholar 

  9. Qiu D, Zhang X. Mordell-Weil groups and Selmer groups of two types of elliptic curves. Sci China Ser A, 2002, 45: 1372–1380

    Article  MATH  MathSciNet  Google Scholar 

  10. Rhoades R. 2-Selmer groups and Birch-Swinnerton-Dyer conjecture for the congruent number curves. J Number Theory, 2009, 129: 1379–1391

    Article  MATH  MathSciNet  Google Scholar 

  11. Schmitt S. Computation of the Selmer groups of certain parametrized elliptic curve. Acta Arith, 1997, lxxviii: 241–254

    Google Scholar 

  12. Silverman J H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1986

    MATH  Google Scholar 

  13. Strocker R J, Top J. On the equation y 2 = (x + p)(x 2 + p 2). Rocky Mountain J Math, 1994, 24: 1135–1161

    Article  MathSciNet  Google Scholar 

  14. Zhao C. A criterion for elliptic curves with lowest 2-power in L(1). Math Proc Camb Phil Soc, 1997, 121: 385–400

    Article  MATH  Google Scholar 

  15. Zhao C. A criterion for elliptic curves with second lowest 2-power in L(1). Math Proc Camb Phil Soc, 2001, 131: 385–404

    MATH  Google Scholar 

  16. Zhao C. A criterion for elliptic curves with lowest 2-power in L(1) (II). Math Proc Camb Phil Soc, 2003, 134: 407–420

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Institute of Mathematics and Interdisciplinary Science, Capital Normal University, Beijing, 100048, China

    Fei Li & DeRong Qiu

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  1. Fei Li
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  2. DeRong Qiu
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Correspondence to DeRong Qiu.

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Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

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Li, F., Qiu, D. On several families of elliptic curves with arbitrary large Selmer groups. Sci. China Math. 53, 2329–2340 (2010). https://doi.org/10.1007/s11425-010-4035-2

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  • DOI: https://doi.org/10.1007/s11425-010-4035-2

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