Injectivity of the specialization homomorphism of elliptic curves

2012/11/16 by Ivica Gusic, Gusic, Ivica, Petra Tadic +1 · 1 citation
Mathematics · #11G05 #14H52 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G05 #msc:14H52

paper · pdf · doi:10.48550/arxiv.1211.3851

arxiv created 2012/11/16 · arxiv updated 2012/11/19

Abstract

Let E:y²=(x-e₁)(x-e₂)(x-e₃) be a nonconstant elliptic curve over ℚ(t), where e_j∈ ℤ[t]. We describe a method for finding a specialization t↦ t₀∈ℚ such that the specialization homomorphism is injective. The method can be directly extended to elliptic curves with e_j∈ R_K[t] where K is a number field and R_K is some UFD such that \mathcal O_K⊂\mathcal R_K⊂ 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 \citeMe.

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• Injectivity of the specialization homomorphism of elliptic curves

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