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

**Download 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 y^{2} =x^{3} − p^{2}x and y^{2} = x^{3} − 4p^{2}x 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

- Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein, Springer Verlag, Berlin,

2000. - Nemenzo, F.R., On the rank of the elliptic curve y
^{2}= x^{3}− 2379^{2}x, Proc. Japan Acad. Vol. 72, 1996, 206–207. - Nemenzo, F.R., Congruent Numbers and the Tate—Shafarevich Group of the Elliptic Curve y
^{2}= x^{3}− n^{2}x. D.Sc. dissertation, Sophia University, 1997. - Silverman, J.H. and Tate, J., Rational Points on Elliptic Curves, Springer Verlag, New York, 1992.
- Wada, H., On the rank of the elliptic curve y
^{2}= x^{3}− 1513^{2}x, Proc. Japan Acad. Vol. 72, 1996, 34–35.

## Related papers

## Cite this paper

APAJerome 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.

ChicagoJerome 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.

MLAJerome 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.