Remarks on the Birch and Swinnerton-Dyer Conjecture

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 ³ — Dx and y ² = x ³ — D, which, being curves with complex multiplication, have an L-function with analytic continuation.