An Algorithm for Checking Injectivity of Specialization Maps from Elliptic Surfaces

Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. Given a subgroup $M$ of $E(\mathbb Q(t))$ with mild conditions and $t_0 \in \mathbb Q$ coming from a relatively large subset $S_M \subset \mathbb Q$, we provide an algorithm that can show that the specialization map $\sigma_{t_0} : E(\mathbb Q(t)) \to E_{t_0}(\mathbb Q)$ is injective when restricted to $M$. The set $S_M$ is effectively computable in certain cases, and we carry out this computation for some explicit examples where $E$ is given by a Weierstrass equation.