An Extension to the Gusić-Tadić Specialization Criterion

2021/09/21 by Tyler Raven Billingsley, Billingsley, Tyler Raven
Computer Science · Mathematics · #14H52 #Algebraic Geometry and Number Theory #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2109.10233

openalex publication_date 2021/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E/\mathbb Q(t) be an elliptic curve and let t₀ ∈ \mathbb Q be a rational number for which the specialization E_t0 is an elliptic curve. In 2015, Gusić and Tadić 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 t₀ 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.

Related

• Injectivity of the specialization homomorphism of elliptic curves
• Injectivity of the specialization homomorphism of elliptic curves
• Low rank specializations of elliptic surfaces
• On torsion sections of elliptic fibrations
• Counting rational points on elliptic curves with a rational 2-torsion point