Growth of Torsion of Elliptic Curves Over Imaginary Quadratic Fields with Class Number 1

In this talk, we will examine the effect that quadratic base change of an elliptic curve $E$ over a number field $K$ has on its torsion subgroup $E(K)_{\text{tor}}.$ If $K=\mathbb{Q},$ it has been extensively studied. As a step toward number fields greater than $\mathbb{Q}$, we will start with elliptic curves defined over a fixed quadratic field. The proofs require the use of several techniques ranging from the Galois action to the study of quadratic points on certain modular curves.