Injectivity of the specialization homomorphism of elliptic curves

TL;DR

This paper presents a method to find specializations of elliptic curves over rational function fields that preserve injectivity of the specialization homomorphism, aiding in rank determination and generator verification.

Contribution

The authors develop a new method for ensuring injectivity of the specialization homomorphism for elliptic curves over $Q(t)$ and its extensions, simplifying the process for quadratic twists.

Findings

Successfully applied to determine ranks of elliptic curves over $Q(t)$

Identified free generators for several elliptic curves

Extended the method to curves over number fields

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}.