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Remarks on the Birch and Swinnerton-Dyer Conjecture

  • Dale Husemöller
Part of the Graduate Texts in Mathematics book series (GTM, volume 111)

Abstract

Let E be an elliptic curve over a number field K which satisfies the Taniyama-Weil conjecture for L E . The first assertion in the Birch and Swinnerton-Dyer conjecture is that L E(s) has a zero of order r = rk(E(K)). The rank of the Mordell-Weil group was the invariant of E that was completely inaccessible by elementary methods unlike, for example, the torsion subgroup of E(K). In the original papers where the conjecture was made, Birch and Swinnerton-Dyer checked the statement for a large family of curves of the form y 2 = x 3Dx and y 2 = x 3D, which, being curves with complex multiplication, have an L-function with analytic continuation.

Keywords

Analytic Continuation Elliptic Curve Complex Multiplication Elliptic Curf Number Field 
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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Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • Dale Husemöller
    • 1
  1. 1.Department of MathematicsHaverford CollegeHaverfordUSA

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