An Extension to the Gusi\'c-Tadi\'c Specialization Criterion

TL;DR

This paper extends the Gusić-Tadić specialization criterion for elliptic curves, allowing its application even when the curve lacks nontrivial rational 2-torsion points, with illustrative examples.

Contribution

It broadens the applicability of the Gusić-Tadić criterion to cases without rational 2-torsion points, providing new insights and examples.

Findings

Extended the criterion to cases without rational 2-torsion

Provided explicit examples demonstrating the extended criterion

Showed the criterion's applicability in broader contexts

Abstract

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. In 2015, Gusi\'c and Tadi\'c gave an easy-to-check criterion, based only on a Weierstrass equation for $E/\mathbb Q(t)$, that is sufficient to conclude that the specialization map at $t_0$ is injective. The criterion critically requires that $E$ has nontrivial $\mathbb Q(t)$-rational 2-torsion points. In this article, we explain how the criterion can be used in some cases where this requirement is not satisfied and provide some examples.