Injectivity of the specialization homomorphism of elliptic curves

Abstract

Let E : y 2 = x 3 +Ax 2 +Bx +C be a nonconstant elliptic curve over Q(t) with at least one nontrivial Q(t)-rational 2-torsion point. We describe a method for finding t 0 ∈ Q for which the corresponding specialization homomorphism t → t 0 ∈ Q is injective. The method can be directly extended to elliptic curves over 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) 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) from an article by Mestre.

The rank and generators of Kihara’s elliptic curve with torsion $\mathbf{Z}/4\mathbf{Z}$ over $\mathbf{Q}(t)$

“…The specialization for t 0 ¼ 6 satisfies the condition of [7,Theorem 1.3], and the specialized elliptic curve has rank 0, which proves our claim for that torsion group.…”

“…; W Ã 5 are free generators of the elliptic curve E À11=4 over Q. From the comments at the end of the introduction, we see that [7,Theorem 1.3] now implies that E has rank 5 over QðtÞ and that P 1 ; W 2 ; W 3 ; W 4 ; W 5 are its free generators. Since E has a point of fourth order and the torsion group of E À11=4 ðQÞ is Z=4Z, we conclude that the torsion group of EðQðtÞÞ is also Z=4Z.…”

“…In this paper, we will prove an analogous result for the curve with record rank over QðtÞ with torsion group Z=4Z found by Kihara [9] (see Theorem 2.1). Here we will use results from the recent paper [7], where the authors generalize and extend their method from [6]. In particular, by results of [7], now the method can be applied to curves with only one rational 2-torsion point.…”

Triples which are D(n)-sets for several n's

“…For the specialization t → 22 the conditions of the main theorem of [14] are satisfied, so the corresponding specialization homomorphism is injective. The specialized curve E 22 over Q is given by…”

High-rank elliptic curves with torsion induced by Diophantine triples

“…This improves previous records (with rank at least three) for curves with this torsion group, obtained by Lecacheux [15], Elkies [11] and Eroshkin (Personal communication, 2008). Moreover, since our curve has full 2-torsion, we can get more precise information by applying the algorithm by Gusić and Tadić [12, Theorem 3.1 and Corollary 3.2], see also [13]. Using this algorithm we can show that rank(E(Q(t))) = 4 and that the four points P 1 , P 2 , P 3 , P 4 are free generators of E(Q(t)).…”