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A complete diophantine characterization of the rational torsion of an elliptic curve

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Abstract

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.

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References

  1. Cassels, J. W. S.: Lectures on Elliptic Curves, Cambridge University Press, Cambridge, 1991

    Book  MATH  Google Scholar 

  2. Silverman, J. H.: The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986

    Book  MATH  Google Scholar 

  3. Mordell, L. J.: On the rational solutions of the indeterminate equations of the third and fourth degrees. Math. Proc. Cambridge Philos. Soc., 21, 179–192 (1922)

    Google Scholar 

  4. Weil, A.: Sur un théor`eme de Mordell. Bull. Sci. Math., 54, 182–191 (1930)

    MATH  Google Scholar 

  5. Mazur, B.: Rational isogenies of prime degree. Invent. Math., 44 129–162 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math., 47, 33–186 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mordell, L. J.: Diophantine Equations, Academic Press, New York and London, 1969

    MATH  Google Scholar 

  8. Smart, N. P.: The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998

    Book  MATH  Google Scholar 

  9. García-Selfa, I., Olalla, M. A., Tornero, J. M.: Computing the rational torsion of an elliptic curve using Tate normal form. J. Number Theory, 96, 76–88 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nagell, T.: Solution de quelque problemes dans la théorie arithmétique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I, 1935

  11. Lutz, E.: Sur l’equation y 2 = x 3 − ax − b dans les corps p-adiques. J. Reine Angew. Math., 177, 237–247 (1937)

    Google Scholar 

  12. Ono, K.: Euler’s concordant forms. Acta Arith., 78(2), 101–123 (1996)

    MathSciNet  Google Scholar 

  13. Qiu, D., Zhang, X.: Explicit classification for torsion cyclic subgroups of rational points with even orders of elliptic curves. Chinese Sci. Bull., 44, 1951–1953 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Qiu, D., Zhang, X.: Explicit classification for torsion subgroups of rational points of elliptic curves. Acta Mathematica Sinica, English Series, 18, 539–548 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kwon, S.: Torsion subgroups of elliptic curves over quadratic extensions. J. Number Theory, 62, 144–162 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fujita, Y.: Torsion subgroups of elliptic curves with non-cyclic torsion over ℚ in elementary abelian 2-extensions of ℚ. Acta Arith., 115, 29–45 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujita, Y.: Torsion subgroups of elliptic curves in elementary abelian 2-extensions of ℚ. J. Number Theory, 114, 124–134 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. García-Selfa, I., Tornero, J. M.: Thue equations and torsion groups of elliptic curves. J. Number Theory, 129, 376–380 (2009)

    Article  Google Scholar 

  19. García-Selfa, I., González-Jiménez, E., Tornero, J. M.: Galois theory, discriminants and torsion subgroup of elliptic curves. Journal of Pure and Applied Algebra, 214, 1340–1346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Huelva, Facultad de Ciencias Experimentales, Campus de El Carmen, Avenida de las Fuerzas Armadas, s/n, 21071, Huelva, Spain

    Irene García-Selfa

  2. Departamento de Álgebra, Universidad de Sevilla, P. O. 1160, 41080, Sevilla, Spain

    José M. Tornero

Authors
  1. Irene García-Selfa
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  2. José M. Tornero
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Correspondence to Irene García-Selfa.

Additional information

Supported by MTM2007-66929 (Ministerio de Educación y Ciencia, Spain), FQM-218, P08-FQM-03894 (Junta de Andalucía, Spain), FSE (Fondo Social Europeo) and FEDER (Fondos Europeos de Desarrollo Regional)

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García-Selfa, I., Tornero, J.M. A complete diophantine characterization of the rational torsion of an elliptic curve. Acta. Math. Sin.-English Ser. 28, 83–96 (2012). https://doi.org/10.1007/s10114-012-9751-9

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  • DOI: https://doi.org/10.1007/s10114-012-9751-9

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