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manuscripta mathematica

, Volume 14, Issue 2, pp 195–205 | Cite as

Points of finite order on elliptic curves with complex multiplication

  • Loren D. Olson
Article

Abstract

Let E be an elliptic curve defined overQ. The group ofQ- rational points of finite order on E is a finite group T(E). In this article T(E) is computed for all elliptic curves defined overQ admitting complex multiplication. The only possible values for the order t of T(E) are 1, 2, 3, 4, or 6 in these cases. A standard form for an affine equation describing an elliptic curve with a given j-invariant is obtained. This is used to show that if j ≠ 0, 26 33, then the number ofQ- rational points of order 2 on E depends only on j. The results are summarized in the accompanying table.

Keywords

Number Theory Standard Form Finite Group Algebraic Geometry Elliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

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    Fueter, R.: Uber kubische diophantische Gleichungen. Commentarii Math. Helv.2, 69–89 (1930).Google Scholar
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    Serre, J.P.: Complex multiplication in J.W.S. Cassels and A. Fröhlich, Algebraic Number Theory. Washington, D.C., Thompson Book Company 1967.Google Scholar
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    Serre, J.P.: Groupes de Lie l-adiques attachés aux courbes elliptiques. Coll. Internat. du C.N.R.S., No. 143 a Clermont-Ferrand, Editions du C.N.R.S.: Paris 1966.Google Scholar
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    Tate, J.: The arithmetic of elliptic curves. Inventiones mathematicae.23, Fasc. 3/4, 179–206 (1974).Google Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Loren D. Olson
    • 1
  1. 1.University of OsloNorway

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