Positive rank quadratic twists of four elliptic curves

Abstract: Let K be a number field and an elliptic curve defined over K for . We prove that there exists a number field L containing K such that there are infinitely many such that has positive rank, equivalently all four elliptic curves have growth of the rank over each of quadratic extensions , more strongly, for any , We also prove that if each elliptic curve for can be written in Legendre form over a cubic extension K of a number field k, then there are infinitely many such that for is of positive rank. [Copyright &y& Elsevier]