On the rank of the elliptic curves with a rational point of order 4

On the rank of the elliptic curves with a rational point of order 4

Shoichi Kihara

Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 80, Number 4 (2004), 26-27.

Abstract

We construct an elliptic curve of rank at least 4 over $Q(t)$ with a rational point of order 4. We also show an infinite family of elliptic curves of rank at least 5 over $Q$ with a rational point of order4, which is parametrized by the rational points of an elliptic curve of rank at least 1.

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Primary Subjects: 11G05

Keywords: Elliptic curve; rank

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pja/1116442212
Mathematical Reviews number (MathSciNet): MR2055072
Digital Object Identifier: doi:10.3792/pjaa.80.26

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References

Kihara, S.: On the rank of the elliptic curve $y^2=x^3+kx$. Proc. Japan Acad., 74A, 115--116 (1998).

Mathematical Reviews (MathSciNet): MR1658862

Mestre, J.-F.: Rang de courbes elliptiques d'invariant donné. C. R. Acad. Sci. Paris Sér. I Math., 314, 919--922 (1992).

Mathematical Reviews (MathSciNet): MR1168325

Nagao, K.: On the rank of elliptic curve $y^2=x^3-kx$. Kobe J. Math., 11, 205--210 (1994).

Mathematical Reviews (MathSciNet): MR1329432

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