The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An elliptic curve, say E, can be represented by points on a cubic equation as below with certain A, B ∈ Q:y2 = x3 + Ax +BA Theorem of Mordell says that that E(Q), the set of rational points of E, is a finitely generated abelian group, and thus,E(Q) = Zr ⊕ T,for some non-negative integer r and a finite group T. Here, r is called the algebraic rank of E.The Birch and Swinnerton-Dyer conjecture relates the algebraic rank of E to the value of the L-function, L(E, s), attached to E at s = 1.Further theoretical understanding, corroborated by computations lead to a stronger version of the BSD conjecture. This refined version of the BSD conjecture provides a very precise formula for the leading term of L(E, s) at s = 1, the coefficient of (s − 1)r, in terms of various arithmetical data atta...
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