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 ∈ L × / ( L × ) 2 such that E i d k ( L ) has positive rank, equivalently all four elliptic curves E i have growth of the rank over each of quadratic extensions L k : = L ( d k ) , more strongly, for any i 1 , i 2 , … , i m , rank ( E i ( L i 1 ⋯ L i m ) ) > rank ( E i ( L i 1 ⋯ L i m − 1 ) ) > ⋯ > rank ( E i ( L i 1 ) ) > 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 ∈ k × / ( k × ) 2 such that E i d ( K ) for i = 1 , 2 , 3 is of positive rank.