• elliptic curves
• Iwasawa theory
• number theory

In this thesis our main goal is to prove a theorem by Mazur called Control Theorem concerning the behaviour of certain groups called Selmer groups related to an elliptic curve with good ordinary reduction at all primes over $p$ in a $\mathbb{Z}_p$-extension. We will use Mazur's control theorem to find conditions to control the growth of $\rank_{\mathbb{Z}}E(L)$ where L varies over the subextensions of a $\mathbb{Z}_p$-extension. The idea is to study Selmer groups instead of studying directly E(K), and we do it using Galois cohomology. In fact, if we restrict to elliptic curves with good, ordinary reduction at a prime p, then the p-primary part of the Selmer group for E over a number field, or over certain infinite extension of $\mathbb{Q}$, can be easily described in terms of the Galois cohomology for the p-torsion part of E. We then use the explicit structure of $\Lambda = \Z_p[[T]]$-modules to derive results concerning the growth of the rank of an elliptic curve and the order of the Selmer groups.
We also give two examples: first we study an elliptic curve with positive $\mu$ invariant, then we briefly introduce L-functions and the Main Conjecture of Iwasawa theory for elliptic curves to show how Mazur's theorem and the Main Conjecture can be combined to provide partial results on the Birch and Swinnerton-Dyer conjecture.