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
Full paper (PDF, 228 Kb)

Details

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.