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Diophantine equations over ℂ(t) and complex multiplication

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Number Theory Carbondale 1979

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 751))

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Abstract

The diophantine equations tx3−(t−1)y2=1 (or x) can be completely solved in ℂ(t) by complex multiplication in the ring of sixth (or fourth) roots of unity. Relations to well-known diophantine equations in Φ are indicated.

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References

  1. Abel, N.H., Recherches sur les fonctions elliptiques, Journal für die reine und agnew. math., 3(1828) 160–190.

    Article  Google Scholar 

  2. Brumer, A. and Kramer, K., The rank of elliptic curves, Duke Math. Journal, 44(1977) 715–743.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cassels, J.W.S., Diophantine equations with special reference to elliptic curves, Journal London Math. Soc., 40(1966) 193–291.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cassels, J.W.S., A diophantine equation over a function field, Journal Australian Math. Soc., 25(1978) 385–422.

    Article  MathSciNet  MATH  Google Scholar 

  5. Davenport, H., On f3 (t)−g2(t), Norske. Vid. Selsk, Forh. 38(1965) 86–87.

    MathSciNet  Google Scholar 

  6. Königsberger, L., Die Transformation, die Multiplication und die Modulargleichungen der Elliptischen Functionen, Teubner, Leipzig, 1868.

    Google Scholar 

  7. Lang, S., Elliptic Curves, Diophantine Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1978.

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Mordell, L.J., Diophantine Equations, Academic, London, New York 1969.

    MATH  Google Scholar 

  10. Nagell, M.T., L'analyse indeterminée de degré supérieur, Mem. des Sci. Math., 1929, vol. 39.

    Google Scholar 

  11. Schmidt, W.M., Thues equation over function fields, Journal Australian Math. Soc., 25(1978) 385–422.

    Article  MathSciNet  Google Scholar 

  12. Weil, A., L'arithmétique sur les courbes algébriques, Acta Math. 52(1928) 281–315.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, City College of New York (CUNY), 10031, New York, New York

    Harvey Cohn

Authors
  1. Harvey Cohn
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Melvyn B. Nathanson

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© 1979 Springer-Verlag

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Cohn, H. (1979). Diophantine equations over ℂ(t) and complex multiplication. In: Nathanson, M.B. (eds) Number Theory Carbondale 1979. Lecture Notes in Mathematics, vol 751. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062702

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  • DOI: https://doi.org/10.1007/BFb0062702

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09559-0

  • Online ISBN: 978-3-540-34852-8

  • eBook Packages: Springer Book Archive

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