Injectivity of the specialization homomorphism of elliptic curves

TL;DR

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

Contribution

It introduces a new technique for finding rational points where the specialization homomorphism is injective for elliptic curves over function fields, extendable to number fields.

Findings

Method successfully finds $t_0$ with injective specialization homomorphism.

Enables calculation of elliptic curve ranks over $Q(t)$.

Proves certain points are free generators.

Abstract

Let $E:y^2=x^3+Ax^2+Bx+C$ be a nonconstant elliptic curve over $\mathbb{Q}(t)$ with at least one nontrivial $\mathbb{Q}(t)$-rational $2$-torsion point. We describe a method for finding $t_0\in\mathbb Q$ for which the corresponding specialization homomorphism $t\mapsto t_0\in\mathbb{Q}$ is injective. The method can be directly extended to elliptic curves over $K(t)$ for a number field $K$ of class number $1$, and in principal for arbitrary number field $K$. One can use this method to calculate the rank of elliptic curves over $\mathbb Q(t)$ of the form as above, and to prove that given points are free generators. In this paper we illustrate it on some elliptic curves over $\mathbb Q(t)$ from an article by Mestre.