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, 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.

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Bibliography

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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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