Functiones et Approximatio Commentarii Mathematici

Elliptic curves with rank $0$ over number fields

Pallab Kanti Dey

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Abstract

Let $E: y^2 = x^3 + bx$ be an elliptic curve for some nonzero integer $b$. Also consider $K$ be a number field with $4 \nmid [K : \mathbb{Q}]$. Then in this paper, we obtain a necessary and sufficient condition for $E$ having rank $0$ over $K$.

Article information

Source
Funct. Approx. Comment. Math., Volume 56, Number 1 (2017), 25-37.

Dates
First available in Project Euclid: 19 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1484794815

Digital Object Identifier
doi:10.7169/facm/1585

Mathematical Reviews number (MathSciNet)
MR3629008

Zentralblatt MATH identifier
06864143

Subjects
Primary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]
Secondary: 11R04: Algebraic numbers; rings of algebraic integers

Keywords
elliptic curve number field Diophantine equation

Citation

Dey, Pallab Kanti. Elliptic curves with rank $0$ over number fields. Funct. Approx. Comment. Math. 56 (2017), no. 1, 25--37. doi:10.7169/facm/1585. https://projecteuclid.org/euclid.facm/1484794815


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References

  • R. Ayoub, An introduction to the analytic theory of numbers, American Mathematical Society, Providence, RI, (1963).
  • A. Bremner and J.W.S. Cassels, On the equation $Y^2 = X(X^2+p)$, Math. Comp. 42 (1984), 257–264.
  • A.J. Hollier, B.K. Spearman and Q. Yang, On the rank and integral points of ellipitc curves $y^2 = x^3 - px$, Int. J. of Algebra 3 (2009), 401–406.
  • A.J. Hollier, B.K. Spearman and Q. Yang, Elliptic Curves $y^2 = x^3 + pqx$ with maximal rank, Int. Math. Forum 5 (2010), 1105–1110.
  • T. Kudo and K. Motose, On group structures of some special elliptic curves, Math J. Okayam Univ. 47 (2005), 81–84.
  • L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Inventiones Mathematicae 124 (1996), 437–449.
  • J.H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, (1992).
  • J.H. Silverman and J.Tate, Rational points on elliptic curves, Springer-Verlag, New York, (1992).
  • B.K. Spearman, Elliptic curves $y^2 = x^3 - px$ of rank two, Math. J. Okayama Univ. 49 (2007), 183–184.
  • B.K. Spearman, On the group structure of elliptic curves $y^2 = x^3 - 2px$, Int. J. of Algebra 1 (2007), 247–250.
  • L.C. Washington, Elliptic curves number theory and cryptography, Chapman and Hall/CRC, Florida, (2003).