Elliptic Curves over Quadratic Fields

Abstract: 

Given an elliptic curve C defined over a number field K, the set of K-rational points on C, denoted C(K) forms a finitely generated abelian group. Although they may not be easy to calculate in practice, C(K) is completely determined by its rank and torsion subgroup.

This talk focuses on the case where C is defined over the rational numbers, but where K is one of several quadratic fields. I give a brief overview of the possible torsion subgroups of C(K) and discuss what else one needs to know to determine the rank of C(K) when the rank of C(Q) is known, giving examples along the way.