DOI QR코드

DOI QR Code

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

  • Received : 2011.07.20
  • Published : 2012.07.31

Abstract

In this paper, we study elliptic curves E over $\mathbb{Q}$ such that the 3-torsion subgroup E[3] is split as ${\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$. For a non-zero intege $m$, let $C_m$ denote the curve $x^3+y^3=m$. We consider the relation between the set of integral points of $C_m$ and the elliptic curves E with $E[3]{\simeq}{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$.

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. https://doi.org/10.1007/BF02386030
  2. T. Hadano, Elliptic curves with a torsion point, Nagoya Math. J. 66 (1977), 99-108. https://doi.org/10.1017/S0027763000017748
  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.