The Effect of Quadratic Base Change on Torsion of Elliptic Curves

Let $K$ be a quadratic number field and let $E$ be an elliptic curve defined over $K$ such that $E[2]\subseteq ̸E(K)\mathrm{.}$ In this paper, we study the effect of quadratic base change on $E(K{)}_{\text{tor}}\mathrm{.}$ Moreover, for a given elliptic curve $E\mathrm{/}K$ with prescribed torsion group over $K,$ (no restriction on its $2$-torsion part) we describe a fast algorithm to find all quadratic extensions $L\mathrm{/}K$ in which $E(K{)}_{\text{tor}}⊊E(L{)}_{\text{tor}}$ and describe $E(L{)}_{\text{tor}}$ in each such case. In particular, we determine the growth of $E(K{)}_{\text{tor}}$ upon quadratic base change when $K$ is any quadratic cyclotomic field, which completes the earlier work of the second author.