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A remark on the injectivity of the specialization homomorphism. (English) Zbl 1300.11060

Consider a nonconstant elliptic curve \(E = E(T)\) defined over the polynomial ring. Silverman’s theorem guarantees that for all but finitely many \(t \in \mathbb Q\), the specialization homomorphism \(\theta: E(\mathbb Q(T)) \to E(t)(\mathbb Q)\) is injective, where \(E(t)\) is the curve defined (over \(\mathbb Q\)) by setting \(T = t\).
In this article, the elliptic curve over \(\mathbb Q[T]\) is assumed to have the form \(y^2 = (x-e_1)(x-e_2)(x-e_3)\) for \(e_1, e_2, e_3 \in \mathbb Z[T]\). The main theorem gives a sufficient condition for the injectivity of \(\theta\) based on the prime factors of the polynomial \((e_1-e_2)(e_2-e_3)(e_3-e_1)\) in \(\mathbb Z[T]\).
In the last section, the authors consider a family of elliptic curves defined by \(E_a: g_a(T)y^2 = x(x-1)(x+2a^2)\), where \(g_a(T)\) is an explicitly given polynomial of degree \(12\). Using their main theorem and Cremona’s program mwrank they show that \(E_a\) has rank \(3\) for all integers \(a\) with \(1 \leq a \leq 60\).

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves

Citations:

Zbl 1035.11025

Software:

mwrank
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