Abstract
Let F be a finite extension of the rational field Q. If E is an elliptic curve defined over F, then the Mordell-Weil group E(F) of points on E with coordinates in F is a finitely generated abelian group. Let L(E/F, s) be the HasseWeil zeta function of E over F which, at least in the case where E has complex multiplication, is known to be an analytic function on the entire complex plane. Birch and Swinnerton-Dyer have conjectured that L(E/F,s) has a zero at s = l of order precisely equal to the rank of E(F) over Z. The strongest result known in support of this conjecture is the following theorem of Coates and Wiles [4].
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