Injectivity of the specialization homomorphism of elliptic curves

JOURNAL OF NUMBER THEORY

卷:148

Article

Gusic, Ivica¹  Tadic, Petra² 

[1] Univ Zagreb, Fac Chem Engn & Technol, Zagreb 10000, Croatia

[2] Juraj Dobrila Univ Pula, Dept Econ & Tourism, Pula 52100, Croatia

关键词: Elliptic curve;    Specialization homomorphism;    Number field;    Class number;    Quadratic field;    Cubic field;    Rank;    Pari;    Magma;   

DOI  :  10.1016/j.jnt.2014.09.023

来源: Elsevier

【 摘 要 】

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) is an element of Q for which the corresponding specialization homomorphism t bar right arrow t(0) is an element of 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. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free