Title:

On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication

The conjecture of Birch and SwinnertonDyer 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 $\mathbb{C}$, 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 SwinnertonDyer 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 $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{3})$. 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 SwinnertonDyer. In the remaining chapters, we let $K=\mathbb{Q}(\sqrt{q})$, 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 $q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and GonzalezAviles and Rubin proved, again using Iwasawa theory, that if $L(E/\mathbb{Q},1)$ is nonzero then the full BirchSwinnertonDyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\mathfrak{p}\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\infty=HK_\infty$, where $K_\infty$ is the unique $\mathbb{Z}_p$extension of $K$ unramified outside $\mathfrak{p}$. 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 BirchSwinnertonDyer conjecture for $E/H$.
