On the main conjecture of Iwasawa theory for certain elliptic curves with complex multiplication
Let E be a quadratic twist of the elliptic curve X_0(49), so that E has complex multiplication by the ring of integers of Q(sqrt(-7)). Using Iwasawa theory, Gonzalez-Aviles and Rubin proved that if L(E/Q,1) is nonzero, then the full Birch–Swinnerton-Dyer conjecture holds for E. We will consider a more general case: Take p to be any prime which is congruent to 7 modulo 8, and set K= Q(sqrt(-p)). We will discuss the main conjecture of Iwasawa theory for an infinite family of elliptic curves which are defined over the Hilbert class field of K with complex multiplication by the ring of integers of K.