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Let E be an elliptic curve over Q, let p be an ordinary prime for E, and let K be an imaginary quadratic field. Write K∞/K for the anticyclotomic Zp-extension of K and set G∞ = Gal(K∞/K). Following a construction of Section 2 of [BD1] which is recalled in Section 1, one attaches to the data (E,K, p) an anticyclotomic p-adic L-function Lp(E,K) which belongs to the Iwasawa algebra Λ := Zp[[G∞]]. This element, whose construction was inspired by a formula proved in [Gr1], is known, thanks to work of Zhang ([Zh, §1.4]), to interpolate special values of the complex L-function of E/K twisted by characters of G∞ | Annals of Mathematics Iwasawa s Main Conjecture for elliptic curves over anticyclotomic Zp-extensions By M. Bertolinin and H. Darmon Annals of Mathematics 162 2005 1 64 Iwasawa s Main Conjecture for elliptic curves over anticyclotomic Zp-extensions By M. Bertolini and H. Darmon Contents 1. p-adic L-functions 1.1. Modular forms on quaternion algebras 1.2. p-adic Rankin L-functions 2. Selmer groups 2.1. Galois representations and cohomology 2.2. Finite singular structures 2.3. Definition of the Selmer group 3. Some preliminaries 3.1. A-modules 3.2. Controlling the Selmer group 3.3. Rigid pairs 4. The Euler system argument 4.1. The Euler system 4.2. The argument 5. Shimura curves 5.1. The moduli definition 5.2. The Cerednik-Drinfeld theorem 5.3. Character groups 5.4. Hecke operators and the Jacquet-Langlands correspondence 5.5. Connected components 5.6. Raising the level and groups of connected components 6. The theory of complex multiplication 7. Construction of the Euler system 8. The first explicit reciprocity law 9. The second explicit reciprocity law References Introduction Let E be an elliptic curve over Q let p be an ordinary prime for E and let K be an imaginary quadratic field. Write K-ỵJK for the anticyclotomic Zp-extension of K and set G Gal K K . Partially supported by GNSAGA INdAM M.U.R.S.T. and the EC. Partially supported by CICMA and by an NSERC research grant. 2 M. BERTOLINI AND H. DARMON Following a construction of Section 2 of BD1 which is recalled in Section 1 one attaches to the data E K p an anticyclotomic p-adic L-function Lp E K which belongs to the Iwasawa algebra A Zp GixY This element whose construction was inspired by a formula proved in Gr1 is known thanks to work of Zhang Zh 1.4 to interpolate special values of the complex L-function of E K twisted by characters of G . Let Sel K Ep v be the Pontrjagin dual of the p-primary Selmer group attached to E over K equipped with its natural A-module structure as defined in Section 2. It is a .
