Injectivity of the specialization homomorphism of elliptic curves

Abstract

Let $E:y^2=(x-e_1)(x-e_2)(x-e_3)$ be a nonconstant elliptic curve over $\mathbb{Q}(t)$, where $e_j\in \mathbb{Z}[t]$. We describe a method for finding a specialization $t\mapsto t_0\in \mathbb{Q}$ such that the specialization homomorphism is injective. The method can be directly extended to elliptic curves with $e_j\in {\mathcal{R}}_K[t]$ where $K$ is a number field and ${\mathcal{R}}_K$ is some UFD such that ${\mathcal{O}}_K\subset {\mathcal{R}}_K\subset K$. Further, we make a simplification of the method for a special case of quadratic twists. The method is applied to obtain exactly the rank and prove that a set of points are free generators of several elliptic curves over $\mathbb{Q}(t)$ coming from \cite{Me}.