Iwasawa's Main Conjecture for elliptic curves over anticyclotomic $\Z_p$-extensions

This paper proves a version of Iwasawa's Main Conjecture for the anticyclotomic p-adic L-function attached to an elliptic curve E over the rationals and to an imaginary quadratic field K. More specifically, when p is an ordinary prime for E, it shows that the characteristic power series of the Pontrjagin-dual of the pro-p Selmer group of E over the anticyclotomic Z_p-extension of K divides the anticyclotomic p-adic L-function. The proof combines the theory of congruences between modular forms with the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves and the theory of complex multiplication of abelian varieties.