The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $C$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result.

In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A=X0(27)$, so that they have complex multiplication by the ring of integers of $Q(-3)$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer.

In the remaining chapters, we let $K=Q(-q)$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/Q,1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=pp\ast$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H\infty =HK\infty$, where $K\infty$ is the unique $Zp$-extension of $K$ unramified outside $p$. We establish in this thesis the main conjecture for the extension $H\infty /H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H\infty /H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$.