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Solutions of Specific Diophantine Equations and their Relationship to Complex Multiplication

Published online by Cambridge University Press:  20 November 2018

Clara Wajngurt
Clara Wajngurt*
Affiliation:
Queensborough Community College, Bayside, N.Y. 11364
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Abstract

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In this paper we establish a relationship between the rational solutions (x(t), y(t)), over C(t), of the diophantine equation:

and the solutions which parametrize the elliptic curve E, y2 = 4x3 — g2x — g3 admitting complex multiplication by λ. We first characterize the form of all rational solutions of diophantine equation (1). The rational solutions are derivable from the subsititutions

in which μ = 0,ω1,ω2,ω1 + ω2 = ω3. Using techniques established in elliptic function theory, we prove that the complex multiplier λ, associated with a unique rational solution (x(t), y(t)), must be of a certain form. Next we construct all rational solutions of diophantine equation (1) by using the addition theorems valid for the Weierstrass function, Specific examples are finally worked out for the cases

Keywords

11G05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 2 , 01 June 1989 , pp. 223 - 229
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Appell, P., Principles de la théorie des fonctions elliptiques, Gauther-Villars, Paris, 1922.Google Scholar
2. Borel, A., Seminar on Complex Multiplication, Lecture Notes in Mathematics 21, Springer-Verlag, N.Y., 1966.Google Scholar
3. Cohn, H., Diophantine Equations over C(t) and Complex Multiplication, Lecture Notes in Mathematics 751,Springer-Verlag, N.Y. (1979), pp. 70–81.Google Scholar
4. Du Val, Patrick, Elliptic Functions and Elliptic Curves, Cambridge Univ. Press, London Mathematical Society Lecture Note Series 9, 1973.Google Scholar
5. Hadano, Toshihiro, Conductor of Elliptic Curves with Complex Multiplication and Elliptic Curves of Prime Conductor, Proc. Japan Acad. 51 (1975), pp. 92–95.Google Scholar
6. Lang, Serge, Elliptic Functions, Addison-Wesley Publishing Co., Mass., 1973.Google Scholar
7. Mordell, L. J., Diophantine Equations, Academic, N.Y., 1969.Google Scholar
8. Olson, Loren D., Points of Finite Order on Elliptic Curves with Complex Multiplication, Manuscripta Math. 14 (1974), pp. 195-205.Google Scholar
9. Serre, J. P., Complex Multiplication, Algebraic Number Theory Proceedings, Academic, London, 1967, pp. 292–296.Google Scholar
10. Weber, Heinrich, Lehrbuch der Algebra III, Chelsea, New York, 1908.Google Scholar