On the Conjecture of Birch and Swinnerton-Dyer for Elliptic Curves with Complex Multiplication
Let E be an elliptic curve which is defined over a number field F. Let L(E,s) be its L-series over F, which is defined by an Euler product convergent in the half-plane Re(s) > 3/2 (2.4). Whenever L(E,s) has an analytic continuation to the entire complex plane, we can consider the first term in its Taylor expansion at s= 1; we write L(E,s) ~ c(E) (s − 1)r(E) as s → 1, where r(E) is a nonnegative integer and c(E) is a nonzero real number. Birch and Swinnerton-Dyer have conjectured that r(E) is equal to the rank of the finitely generated group E(F) of points over F; they have also given a conjectural formula for c(E) in terms of certain arithmetic invariants of the curve (2.10).
KeywordsTate ResO Fermat Exter Tame
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