The main conjecture for CM elliptic curves over totally real ?elds. Let M be an imaginary quadratic ?eld and let E be an elliptic curve over a totally real ?eld F with complex multiplication by the ring of integers O_M of M.Let p be an odd prime split in M. Let F_∞ be the cyclotomic Z_p-extension of F and let Λ_F:= Zp[[Gal(F_∞/F)]] be a one-variable Iwasawa algebra. We study thecyclotomicmainconjectureofIwasawatheoryforE which relates the size of Selmer groups to special values of the p-adic L-function attached to E. Recall that the Selmer group SelF∞(E) is de?ned by SelF_∞(E)=ker{H1(F_∞,E[p~∞])→∏vH~1(F_(∞,v),E)},where v runs over all places of F_∞.ItiswellknownthatSel_(F_∞)(E) is a co?nitely generated Λ_F-module. Denote by charΛFSelF∞(E) the characteristic power series of SelF_∞(E), which is an element in Λ_F unique up to a Λ_F-unit. Let W_p be the p-adic completion of the ring of integers of the maximal unrami?ed extension of Z_p. On the other hand, the specialization of a suitable twist of a Katz p-adic L-function to the cyclotomic line yields a p-adic L-function L_p(E/F_∞)∈Λ_F,W_p:= Λ_F?Z_pW_p, which roughly interpolates the algebraic part of central L-values L(E_(/F)? ν, 1) twisted by ?nite order characters ν:Gal(F_∞/F)→C×(see §8.6.2 for the precise de?nition). The main conjecture of Mazur and Swinnerton-Dyer for E predicts the following equality between ideals in Λ_F,W_p: Conjecture 1 (The main conjecture for CM elliptic curves). (char_(ΛF)Sel_(F∞) (E)) = (L_p(E/F_∞)).
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