Families of twists of tuples of hyperelliptic curves

Let f ∈ Q[x] 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=xmᵢ+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.