On Tate—Shafarevich groups of families of elliptic curves

Jerome T. Dimabayao and Fidel R. Nemenzo
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 18, 2012, Number 2, Pages 42—55
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Authors and affiliations

Jerome T. Dimabayao
Institute of Mathematics, University of the Philippines Diliman
Quezon City, Philippines

Fidel R. Nemenzo
Institute of Mathematics, University of the Philippines Diliman
Quezon City, Philippines

Abstract

We explicitly show that for some primes p ≡ 1 (mod 8), the elliptic curves y2 =x3 − p2x and y2 = x3 − 4p2x have Tate—Shafarevich groups with nontrivial elements. This involves obtaining Diophantine equations that violate the local-global principle.

Keywords

  • Elliptic curve
  • Congruent number
  • Rational point
  • Torsor, Mordell-Weil rank
  • Selmer group

AMS Classification

  • 11G05
  • 11D09

References

  1. Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein, Springer Verlag, Berlin,
    2000.
  2. Nemenzo, F.R., On the rank of the elliptic curve y2= x3 − 23792x, Proc. Japan Acad. Vol. 72, 1996, 206–207.
  3. Nemenzo, F.R., Congruent Numbers and the Tate—Shafarevich Group of the Elliptic Curve y2 = x3 − n2x. D.Sc. dissertation, Sophia University, 1997.
  4. Silverman, J.H. and Tate, J., Rational Points on Elliptic Curves, Springer Verlag, New York, 1992.
  5. Wada, H., On the rank of the elliptic curve y2 = x3 − 15132x, Proc. Japan Acad. Vol. 72, 1996, 34–35.

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Cite this paper

APA

Jerome T. Dimabayao and Fidel R. Nemenzo. (2012). On Tate—Shafarevich groups of families of elliptic curves, Notes on Number Theory and Discrete Mathematics, 18(2), 42-55.

Chicago

Jerome T. Dimabayao and Fidel R. Nemenzo. (2012). “On Tate—Shafarevich groups of families of elliptic curves ” Notes on Number Theory and Discrete Mathematics, 18, no. 2 (2012): 42-55.

MLA

Jerome T. Dimabayao and Fidel R. Nemenzo.(2012). “On Tate—Shafarevich groups of families of elliptic curves” Notes on Number Theory and Discrete Mathematics, 18.2 (2012): 42-55. Print.

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