Inspired by the paper of Klagsbrun, Mazur and Rubin [5], this thesis investigates the disparity of 2-Selmer ranks of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. In the first part, we calculate the density of quadratic twists of E with even 2-Selmer ranks under two different counting methods. First we count twists by elements inside a large convex body of the Euclidean space that contains the integer lattice of K. The second counting method is counting quadratic twists E^L by the norms of the finite part of conductors of quadratic extensions L/K. Under both counting methods we give an explicit formula for the densities, which are finite products of local factors. In the second part of the paper we give a method that uses Tate’s algorithm to calculate the size of the cokernel of the local norm maps of E at places over 2, assuming that E has good reduction. With this method we can extend Kramer’s early work on the cokernel of the local norm maps, and compute the local factors mentioned above in some additional cases.

Let $E_1$ and $E_2$ be elliptic curves defined over a number field $K$. Suppose that for all but finitely many primes $\ell$, and for all finite extension fields $L/K$,

$${\dim }_{{\mathbb{F}}_{\ell }}{\mathrm{Sel}}_{\ell }(L,E_1)={\dim }_{{\mathbb{F}}_{\ell }}{\mathrm{Sel}}_{\ell }(L,E_2).$$ We prove that $E_1$ and $E_2$ are isogenous over $K$.