CM elliptic curves and the Coates{Wiles Theorem

We describe one of the few cases of the Birch and Swinnerton-Dyer Conjecturethat has been already proved, the so called Coates{Wiles Theorem. Let K be animaginary quadratic eld with ring of integers O and class number 1 and let E be anelliptic curve dened over K with complex multiplication by O. The Coates{WilesTheorem states that if the L-series attached to E=K does not vanish at 1, thenthe set of K-rational points of E is nite. We explain a proof given by Karl Rubin,which uses the theory of Euler systems.Keywords: BSD Conjecture, Coates Wiles Theorem, L-series, elliptic curves with CM, Euler systems, ellip-tic units