Remarks on the Birch and Swinnerton-Dyer Conjecture
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 3 — Dx and y 2 = x 3 — D, which, being curves with complex multiplication, have an L-function with analytic continuation.
KeywordsAnalytic Continuation Elliptic Curve Complex Multiplication Elliptic Curf Number Field
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