ON ELLIPTIC CURVES WHOSE 3-TORSION SUBGROUP SPLITS AS μ3 ⊕ℤ/3ℤ

Title & Authors
ON ELLIPTIC CURVES WHOSE 3-TORSION SUBGROUP SPLITS AS μ3 ⊕ℤ/3ℤ
Yasuda, Masaya;

Abstract
In this paper, we study elliptic curves E over $\small{\mathbb{Q}}$ such that the 3-torsion subgroup E[3] is split as $\small{{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}}$. For a non-zero intege $\small{m}$, let $\small{C_m}$ denote the curve $\small{x^3+y^3=m}$. We consider the relation between the set of integral points of $\small{C_m}$ and the elliptic curves E with $\small{E[3]{\simeq}{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}}$.
Keywords
elliptic curves;torsion points;V$\small{\acute{e}}$lu's formula;
Language
English
Cited by
References
1.
G. Frey, Some remarks concerning points of finite order on elliptic curves over global fields, Ark. Mat. 15 (1977), no. 1, 1-19.

2.
T. Hadano, Elliptic curves with a torsion point, Nagoya Math. J. 66 (1977), 99-108.

3.
I. Miyawaki, Elliptic curves of prime power conductor with ${\mathbb{Q}}$-rational points of finite order, Osaka J. Math. 10 (1973), 309-323.

4.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, Springer-Verlag, Berlin-Heidelberg New York, 1994.

5.
J. Velu, Isogenis entre courbs elliptiques, C. R. Acad. Sci. Paris Ser. A-B (1971), 238-241.