Missouri Journal of Mathematical Sciences

Real Preimages of Duplication on Elliptic Curves

John Cullinan

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Abstract

Let $E$ be an elliptic curve defined over the real numbers $R$ and let $P \in E(R)$. In this note we give an elementary proof of necessary and sufficient conditions for the preimages of $P$ under duplication to be real-valued.

Article information

Source
Missouri J. Math. Sci., Volume 29, Issue 1 (2017), 19-26.

Dates
First available in Project Euclid: 2 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1488423698

Digital Object Identifier
doi:10.35834/mjms/1488423698

Mathematical Reviews number (MathSciNet)
MR3619772

Zentralblatt MATH identifier
06759818

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
real elliptic curve Mordell-Weil Theorem torsion group quartic equation

Citation

Cullinan, John. Real Preimages of Duplication on Elliptic Curves. Missouri J. Math. Sci. 29 (2017), no. 1, 19--26. doi:10.35834/mjms/1488423698. https://projecteuclid.org/euclid.mjms/1488423698


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References

  • L. E. Dickson, Elementary Theory of Equations, Wiley, New York, 1914.
  • S. Lang, Elliptic Curves Diophantine Analysis, Grundlehren der Mathematischen Wissenschaften 231, Springer-Verlag, Berlin-New York, 1978.
  • J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106. Springer, Berlin-New York, 2009.
  • Y. Zarhin, Division by 2 on elliptic curves, preprint, posted 30 May, 2016, http://arxiv.org/pdf/1605.09279v1.pdf.