Injectivity of the specialization homomorphism of elliptic curves

Let E:y2=x3+Ax2+Bx+CE:y2=x3+Ax2+Bx+C be a nonconstant elliptic curve over Q(t)Q(t) with at least one nontrivial Q(t)Q(t)-rational 2-torsion point. We describe a method for finding t0∈Qt0∈Q for which the corresponding specialization homomorphism t↦t0∈Qt↦t0∈Q is injective. The method can be directly extended to elliptic curves over K(t)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 Q(t)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 Q(t)Q(t) from an article by Mestre.