Positive rank quadratic twists of four elliptic curves

Abstract

Let K be a number field and E-i/K an elliptic curve defined over K for i = 1, 2, 3, 4. We prove that there exists a number field L containing K such that there are infinitely many d(k) is an element of L-x/(L-x)(2) such that E-i(dk)(L) has positive rank, equivalently all four elliptic curves E-i have growth of the rank over each of quadratic extensions L-k := L(root d(k)), more strongly, for any, i(1), i(2), ... , i(m). rank (E-i(l(i1) .. L-m)) > rank(E-i(L-i1 ... Lim-1)) > ... >rank(E-i(L-i1)) > rank(E-i(L)). We also prove that if each elliptic curve E-i for i = 1, 2,3 can be written in Legendre form over a cubic extension K of a number field k, then there are infinitely many d is an element of k(x)/(k(x))(2) such that E-i(d)(K) for i = 1, 2, 3 is of positive rank. (C) 2012 Elsevier Inc. All rights reserved