Families of twists of tuples of hyperelliptic curves

Abstract Let f Qx be a square-free polynomial of degree at least 3, mᵢ, i=1, 2, 3, odd positive integers, and aᵢ, i=1, 2, 3, non-zero rational numbers. We show the existence of a rational function D Q (v₁, v₂, v₃, v₄) such that the Jacobian of the quadratic twist of y²=f (x) and the Jacobian of the mᵢ -twist, respectively, 2mᵢ -twist, of y²=x^mᵢ+aᵢ², i=1, 2, 3, by D are all of positive Mordell–Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell–Weil rank.